Research Collection

Doctoral Thesis

Sequential Monte Carlo methods in air traffic management

Author(s): Lymperopoulos, Ioannis

Publication Date: 2010

Permanent Link: https://doi.org/10.3929/ethz-a-006159564

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ETH Library

Dissertation ETH Zürich No. 19004

Sequential Monte Carlo Methodsin Air Traffic Management

A dissertation submitted toETH Zürich

for the degree ofDoctor of Sciences

presented by

IOANNIS LYMPEROPOULOS

Dipl.-Ing, University of Patras, Greeceborn 11.04.1981 in Athens

citizen of Greece

accepted on the recommendation ofProf. Dr. John Lygeros, examiner

Prof. Dr. Fredrik Gustafsson, co-examiner

2010

Cover PhotoTheophilos Papadopoulos, "Air Traffic", 10 Jan. 2010, Online Image,Flickr, http://www.flickr.com/photos/theo_reth/4291205440

c© 2010 Ioannis LymperopoulosAll Rights Reserved

to my parents,Dimitri & Ioanna

����� ����� ���� ��� ��� ������

We must free ourselves of the hope that the sea will ever rest.We must learn to sail in high winds.

Aristoteles Onassis

Abstract

Demand for air transport services is constantly rising all over the world creating new routesof heavy traffic and exercising an additional pressure on the already existing ones. Thecurrent Air Traffic Management (ATM) system, rigidly built around human controllersand older concepts of operations, is put under severe strain. The expected future growthrequires drastic changes in infrastructure and technology to conserve and improve thecurrent levels of safety and efficiency. Novel automation and Decision Support Tools (DST)are required to alleviate the workload of air traffic controllers and provide new ways forhandling the increased traffic demand.

Accurate Trajectory Prediction (TP) plays a fundamental role in advanced air trafficcontrol and management operations. Uncertainty about the future position of the aircraftconstrains the growth of airspace capacity since large separation margins are requiredto ensure safety. Wind forecast errors are a core source for this uncertainty, becausecommercial aircraft fly level with constant airspeed, instead of constant ground-speed, forfuel efficiency reasons. We demonstrate how to decrease uncertainty about forecasts withthe use of ground radar measurements from multiple aircraft.

A hybrid stochastic model that captures the dynamics of multiple aircraft, equippedwith a 3D flight management system is developed. Aircraft follow a predefined flight planunder disturbances from varying wind and atmospheric conditions. Wind forecast errorsare modeled as a state space system that evolves with time and is spatially correlated.The correlation of the wind uncertainty allows for the creation of probabilistic maps thatrepresent the difference between the wind forecast and the real wind conditions. Enhancedknowledge about the current wind situation in the airspace of interest can be used forimproved TP.

Sequential Monte Carlo Methods (or particle filters) are used to filter the incomingground radar measurements and to provide estimates for the state of the aircraft and thewind. The situation is complicated by the fact that aircraft dynamics are nonlinear andwind is represented by a large number of states at each location where it is evaluated.Conventional particle filters can provide reasonable approximations to the filtering prob-lem, when very few aircraft are involved, at the expense of heavy computational burden.However, in order to sense the wind in different places and build a wind map that capturesuncertainties all over the airspace, measurements from multiple aircraft are required.

Conventional algorithms have great difficulties in assimilating such a large amount ofinformation under this formulation. A novel algorithm is developed, called SequentialConditional Particle Filter (SCPF) that aims to take advantage of the structure of thesituation. SCPF treats the linear and nonlinear parts of the system separately, filtersthe measurements sequentially and conditions the probabilistic wind map according tothe spatiotemporal correlation of the wind forecast errors. The effectiveness of the novelalgorithm is demonstrated on feasibility studies involving multiple aircraft, from one toseveral hundred.

Various air traffic management applications can take advantage of SCPF. TP lies at theheart of most Conflict Detection (CD) algorithms. By comparing the predicted trajectories

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of different aircraft against each other, we can detect real threats while avoiding falsealarms. We demonstrate here how SCPF can enhance CD performance over the simpleuse of meteorological forecasts. We show also how multiple aircraft can improve efficiencyin CD even further.

Finally, the particularly challenging problem of joint parameter and state estimation isaddressed as well. For air traffic management applications, considered here, we assume thatthe aircraft airspeeds are not available to the air traffic controllers. Radar measurementshave now to be used both for estimating the wind and aircraft states like before and forcalibrating the unknown parameters. We develop practical methods for augmenting SCPFin order to handle this problem. Simulation studies show how SCPF manages to improvetrajectory prediction while also identifying the unknown aircraft airspeeds.

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Zusammenfassung

Die Nachfrage nach Lufttransportdienstleistungen steigt stetig auf der ganzen Welt, im-mer mehr Luftstrassen kämpfen mit sehr dichtem Verkehr und bereits stark beflogeneRouten werden noch mehr belastet. Das momentane Air Traffic Management (ATM) Sys-tem, in welchem der menschliche Controller ein fester Bestandteil ist und das auf älterenBetriebskonzepten beruht, stösst an seine Belastungsgrenze. Das erwartete Wachstumerfordert erhebliche Veränderungen in Bezug auf die Infrastruktur und Technologie, umden momentanen Stand der Sicherheit und Effizienz zu wahren oder zu verbessern. NeueAutomatisierung und Decision Support Tools (DST) werden gebraucht, um die Last derFluglotsen zu erleichtern und um neue Wege zu ermöglichen, das erhöhte Verkehrsaufkom-men zu handhaben.

Die genaue Trajektorienplanung (TP) spielt eine fundamentale Rolle in der fortgeschrit-tenen Flugsicherung und dem fortgeschrittenen ATM. Das Wachstum der Flugraumkapa-zität wird durch die Unsicherheit über die zukünftige Position des Flugzeugs eingeschränkt,da generell grosse Distanzmargen notwendig sind um, Sicherheit zu gewährleisten. Vorher-sagefehler des Winds sind eine Hauptquelle für diese Unsicherheit, da Verkehrsflugzeuge ausEffizienzgründen mit konstanter relativer Geschwindigkeit zur Luftgeschwindigkeit anstattzur Bodengeschwindigkeit fliegen. Wir zeigen, wie die Unsicherheit der Vorhersagen durchden Einsatz von Bodenradarmessungen von mehreren Flugzeugen reduziert werden kann.

Ein hybrides stochastisches Modell wird entwickelt, welches die Dynamiken mehrererFlugzeuge, ausgestattet mit einem 3D Flug-Management-System, beschreibt. Die Flug-zeuge folgen einer vorgegebenen Trajektorie unter dem Einfluss von Störungen, verursachtdurch sich ändernde Wind- und atmosphärische Bedingungen. Fehler in der Vorhersageder Windbedingungen werden in einem sich zeitlich verändernden, räumlich korreliertenZustandsraum modelliert. Die Korrelation von Windunsicherheiten erlaubt die Erstel-lung von probabilistischen Karten, welche den Unterschied zwischen der Windvorhersageund den tatsächlichen Windbedingungen wiedergeben. Die umfangreichere Kenntnis dergegenwärtigen Windsituation im interessierenden Luftraum kann genutzt werden, um dieTrajektorienvorhersage zu verbessern.

Sequentielle Monte Carlo Methoden (oder Partikel-Filter) werden verwendet, um die ein-treffenden Bodenradarmessungen zu filtern und Zustandsschätzungen des Flugzeuges unddes Winds zu liefern. Dies wird verkompliziert durch die Nichtlinearität der Flugzeug-dynamik sowie dadurch, dass der Wind an jedem Ort, an dem er evaluiert wird, durcheine grosse Anzahl an Zuständen dargestellt wird. Konventionelle Partikel-Filter bietengute Approximationen des Filter-Problems, allerdings nur, wenn sehr wenige Flugzeugeinvolviert sind und unter hohem Rechenaufwand. Um jedoch den Wind an verschiedenenOrten zu erfassen und eine Windkarte zu erstellen, die die Unsicherheiten des gesamtenLuftraums darstellt, benötigt man Messungen von zahlreichen Flugzeugen.

Konventionelle Algorithmen haben grosse Schwierigkeiten, eine so grosse Menge an In-formationen wie in dieser Formulierung zu verarbeiten. Daher wird hier ein neuer Algo-rithmus entwickelt, der sogenannte Sequential Conditional Particle Filter (SCPF), der sichdie Struktur des Problems zu Nutze macht. SCPF behandelt die linearen und nichtlinearen

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Teile des Systems einzeln, filtert sie Messungen sequentiell und konditioniert die proba-bilitstische Windkarte je nach raumzeitlicher Korrelation der Windvorhersagefehler. DieEffektivität des neuen Algorithmus wird in Durchführbarkeitsstudien geprüft unter Ver-wendung von mehreren Flugzeugen, von einem bis zu mehreren hundert.

Zahlreiche ATM-Anwendungen können von SCPF profitieren. Trajektorienplanung liegtim Zentrum der meisten Conflict Detection (CD) Algorithmen. Durch Vergleich dervorhergesagten Trajektorien von unterschiedlichen Flugzeugen miteinander, können Bedro-hungen erkannt und Fehlalarme vermieden werden. Es wird hier gezeigt, wie SCPF dieCD Performance im Vergleich zur simplen Verwendung von Wetterprognosen verbessernkann. Weiterhin wird demonstriert, wie mehrere Flugzeuge die Effizienz von CD weitersteigern können.

Schliesslich wird das besonders schwierige Problem der gleichzeitigen Parameter- und Zu-standsschätzung adressiert. Für ATM-Anwendungen, wie hier betrachtet, wird angenom-men, dass die Flugzeugwindgeschwindigkeiten für die Fluglotsen nicht verfügbar sind.In dem Fall müssen Radarmessungen sowohl für die Zustandsschätzung des Winds unddes Flugzeugs genutzt werden als auch für die Kalibrierung der unbekannten Parameter.Simulationsstudien zeigen, wie SCPF die Trajektorienplanung verbessert, während es gle-ichzeitig die unbekannten Flugzeuggeschwindigkeiten identifiziert.

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Acknowledgements

I am indebted to my supervisor, Professor John Lygeros, for providing me with an extremelevel of support throughout the years of my PhD studies. I would like to gratefully thankhim for giving me the opportunity to take such a fantastic journey in the realms of scienceand academia. His guidance has been exemplar, always towards improvement, alwaystowards excelling, always towards new heights and achievements. ��������� ����� �

I need to specially thank all the current and former members of the Automatic ControlLaboratory in ETH Zürich. They have created an extraordinary and unique place. Mysincere thanks and respect to Prof. Manfred Morari. I am very glad that I have met Mrs.Martine Wassmer, Martha Mariani, Katerina Gnehm, Danielle Couson, Alice Vyskocil andDr. Franta Kraus. I sincerely thank them for their assistance and the beautiful times weshared together.

This place has always been packed. Packed with extraordinary people. I do not haveenough words to express my admiration for Thomas Besselman, Philipp Rostalski, Hel-fried Peyrl, Badis Djeridane, Marta Capiluppi, Colin Jones, Sébastien Mariéthoz, SasaV. Rakovic, Peter Al Hokayem, Stefan Almér, Daniel Axehill, Antonello Caruso, Deba-sish Chatterjee, Jack DiGiovanna, Riccardo Porreca, André Bertolace, Christian Conte,Michael Kearney, Henrik Manum, Alexander Domahidi, Kristian Nolde, Stephan Huck,Peyman Esfahani, Urban Mäder, Andreas Kuhn, Miroslav Baric, Robert Nguyen, StefanRichter, Joe Warrington, Silvestro Micera and Andrea Crema (or Hilfiger). I thank themfor the superb environment they have provided, for their kindness and for their maturity.It would not have been the same without you my friends.

During my studies I had the privilege to collaborate with exceptional people. I amthankful to Dr. Andrea Lecchini-Visintini, Dr. Thomas Schön, Prof. Jan Maciejowski,Dr. Eva Cruck, Ellie Siva and of course Dr. Federico Ramponi for their comments, insightsand contributions during our work together.

I would like to thank all the students that have been under my supervision for theirmaster and semester projects, Caterina Vitadello, Hours Jean-Hubert, Ioannis Tzanos,Stefan Scheidegger, Srdjan Curcic and Andreas Mehmann. My best wishes and may yourwildest dreams come true.

From the opposite part of my office I see a mural where blue seas, ancient lands, familiarfigures and places are merging together. Akin Sahin, merhaba Kardaş, we will alwaysmeet near the Aegean sea, under the hot summer sun and the clean Mediterranean skies.Eugenio Cinquemani, please flip once more, just for me. Sean Summers, thank you forbeing the daily note of unperturbed optimism. Aldo Zgraggen, keep up flying and coming

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back to earth neat and safe. Davide Raimondo, we still have to burn the city one daymy friend. Simone Gasparella, wir feiern die ganze nacht. Anastasia Stulova, I will steponce in the institute bearing your name. Melanie Zeilinger no coffee break will ever be thesame without you. Frauke Oldewurtel, I am deeply privileged to have met you my sister.Clara Verhelst, thank you for all the wonderful moments, may the power of Earth guardyou forever. Frank J. Christophersen, you have been a master and a pupil for me. I thankyou from the bottom of my heart for being there when I needed you and for accepting mysupport when you needed mine. Mark my words.

I am thankful to the Greek gang, Georgios Chaloulos, Konstantinos Koutroumpas, An-dreas Milias-Argeitis, Katerina Chalvatzi, Olga Moatsou, Ioannis Fotiou (and Stavroula),Kostas Margellos (and Maria), Alexandros Argyrakos (and Amaryllis) for being like afamily to me, for their friendship and support.

My family and my friends back in Greece deserve my deepest praise for their unshakenbelief in me throughout these years, for their unending love and their authentic smiles.

Finally, I would like to express my gratitude towards the people and the country ofHelvetia, die Schweizerische Eidgenossenschaft. They have been great hosts and a greatexample for me. I will always hold this place in my heart dearly, the high mountains, thegreat lakes, and their spirit of pride, democracy and independence...

Ioannis LymperopoulosZürich, Spring 2010

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Contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iZusammenfassung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vContents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiNotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiAcronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii

1. Introduction 11.1. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2. Trajectory Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3. Conflict Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4. Joint Parameter and State estimation . . . . . . . . . . . . . . . . . . . . . 81.5. Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2. Model Dynamics 112.1. Aircraft Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.1.1. Aerodynamic Forces . . . . . . . . . . . . . . . . . . . . . . . . . . 142.1.2. Fuel Consumption . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2. Discrete State Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3. Wind Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3.1. Wind Forecast . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.3.2. Wind Forecast Error Statistics . . . . . . . . . . . . . . . . . . . . . 182.3.3. Wind Forecast Error Generation . . . . . . . . . . . . . . . . . . . . 19

2.4. Flight Management System . . . . . . . . . . . . . . . . . . . . . . . . . . 222.4.1. Thrust control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.4.2. Flight Path Angle and ESF control . . . . . . . . . . . . . . . . . . 232.4.3. Bank Angle Control . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.5. Radar Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.6. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

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Contents

3. Non-Linear Filtering 273.1. Optimal Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.2. Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.3. Particle Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.3.1. Sequential Importance Sampling. . . . . . . . . . . . . . . . . . . . 323.3.2. Sequential Importance Resampling. . . . . . . . . . . . . . . . . . . 323.3.3. Marginalized Particle Filter . . . . . . . . . . . . . . . . . . . . . . 333.3.4. Kernel Smoothing and Auxiliary PF. . . . . . . . . . . . . . . . . . 35

3.4. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4. A novel Particle Filter for Trajectory Prediction 374.1. Trajectory Prediction and Filtering . . . . . . . . . . . . . . . . . . . . . . 384.2. Sequential Conditional Particle Filter . . . . . . . . . . . . . . . . . . . . . 404.3. Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.3.1. Benchmark Simulation Set-up . . . . . . . . . . . . . . . . . . . . . 434.3.2. Trajectory Prediction Results . . . . . . . . . . . . . . . . . . . . . 464.3.3. Algorithm Numerical Performance Analysis . . . . . . . . . . . . . 504.3.4. Effect of Number of Particles . . . . . . . . . . . . . . . . . . . . . 524.3.5. Maastricht airspace . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.4. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5. Comparison of Particle filters for Trajectory Prediction 575.1. Simulation Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585.2. Trajectory Prediction Comparison . . . . . . . . . . . . . . . . . . . . . . . 585.3. Filtering Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635.4. Algorithm Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 665.5. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

6. Conflict Detection using Sequential Conditional Particle Filter 696.1. Conflict Detection Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 706.2. Simulation Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 706.3. Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716.4. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

7. Joint Parameter and State Estimation 797.1. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 807.2. Methods for Parameter and State Estimation . . . . . . . . . . . . . . . . 807.3. Simulation Setup and Results . . . . . . . . . . . . . . . . . . . . . . . . . 847.4. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

8. Conclusions and Outlook 103

Appendix:

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Contents

A. Model Dynamics 115A.1. Wind Gradient Factors Derivation . . . . . . . . . . . . . . . . . . . . . . . 115A.2. Wind Forecast Error Derivation . . . . . . . . . . . . . . . . . . . . . . . . 117A.3. Thrust and ESF control: Flight Path Angle Control . . . . . . . . . . . . . 119

B. Curriculum Vitae 123

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x

xi

Notation

Aircraft Dynamics

X X-axis of the inertial reference frame (East)Y Y -axis of the inertial reference frame (North)Z Z-axis of the inertial reference frame (Up)P point in the inertial reference frame, P = (xI , yI , zI) ∈ R

3

xI aircraft position in the X-axisyI aircraft position in the Y -axiszI aircraft position in the Z-axisV aircraft true airspeed (TAS)ψ aircraft heading anglem aircraft massL aerodynamic liftD aerodynamic dragT engine thrustAOA angle of attackAOS sideslip angleγ flight path angleφ bank angleφnom nominal bank angle during turnsφ bank angle saturation limit (φ = 35◦)ψ heading angle saturation limit (ψ = 60◦)θ heading errorδ cross-track deviationk1, k2 bank angle controller gainsVmin minimum aircraft speedVnom nominal aircraft speedVmax maximum aircraft speedmmin empty aircraft massmmax maximum take-off massTmin minimum engine thrustTMaxClimb maximum available engine thrust during climbTmax maximum engine thrustγmin maximum allowed flight path angleγmax minimum allowed flight path angleφmin minimum allowed bank angleφmax maximum allowed bank anglefnom nominal fuel flowη fuel flow coefficientρ(zI) air density, at altitude zI

AP (zI) atmospheric pressure in mb, at altitude zI

g gravitational acceleration (g = 9.81ms−2)

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Aircraft parameters

CL aerodynamic lift coefficientCD aerodynamic drag coefficientCD0, CD2 drag coefficientsS total wing surface areaCTc1, CTc2, CTc3 engine thrust coefficientsCTdes thrust coefficient for descentCpowered reduced thrust CoefficientCf1, Cf2, Cf3, Cf4 fuel flow coefficients

Wind Dynamics

WI wind speed in the inertial reference frame at the aircraft positionwX wind speed component in the X-axiswY wind speed component in the Y -axiswZ wind speed component in the Z-axisWagf along-track wind gradient factorWcgf cross-track wind gradient factorWvgf vertical gradient factorw(t, P ) wind forecast error at point P ∈ R

3 and at time t ∈ R

R(t, P, t′, P ′) covariance matrix, of wind forecast error, between two points in space and timer(t, P, t′, P ′) covariance matrix, of wind forecast error, in the X (or Y) directionσ(zI) wind forecast error variance, at altitude zI

rt(·) time correlation of wind forecast errorrXY (·) horizontal correlation of wind forecast errorrZ(·) vertical correlation of wind forecast errorct, dt, Gt, et time correlation parameters of wind forecast errorcXY , GXY , bXY horizontal correlation parameters of wind forecast errorcZ , GZ vertical correlation parameters of wind forecast errorNX , NY , NZ grid points in the West-East, South-North and vertical directionδX , δY , δZ spacing of the NX , NY , NZ grid points (δX = δY = 60 km, δZ = 1000 ft)i, j, h indices of the grid points in the wind-field lattice, for the X, Y, Z axisWX ,WY forecast error wind-field in the X and Y axis, projected on gridR covariance matrix of WX and WY , R ∈ R

NXNY NZ×NXNY NZ

Nt number of time stepsδt time step width (δt = 30 s)k time step indexQ covariance matrix of the “a priori” wind forecast error distributionQ covariance matrix of the wind forecast error time evolutiona “state matrix” of the wind forecast error time evolution, a ∈ R

Γw angle of the wind vector WI with the horizontal planeΨw angle of the wind vector WI with the X− axis

xiii

Flight Plan

{O(i)}Λi=0 flight plan, a sequence of Λ + 1 way-points

O(i) way-point coordinates, O(i) ∈ R3

Ψ(i) angle of the ith segment of the flight plan with X−axisΓ(i) angle of the ith segment of the flight plan with the horizontal plane

Discrete statesFL Flight Level, FL ∈ {0, 500, 1000, 1500, 2000, 3000, 6000, 10000, 14000} ftWP Way point index, WP ∈ {0, 1, . . . ,Λ − 1}FP Flight Phase, FP ∈ {UD,LD,AP,LA}

(UD = upper descent, LD = lower descent, AP = approach, LA = landing)AM Acceleration Mode, AM ∈ (A = Acceleration, C = Cruise, D = Decceleration)CM Climb Mode, CM ∈ (C = em Climb, L = Level, D = Descent)TrM Troposphere Mode, TrM ∈ (ON,OFF )SHM Speed Hold Mode, SHM ∈ (ON,OFF )RPM Reduced Power Mode, RPM ∈ (ON,OFF )

Nonlinear Filtering

x(k) system state, at time step ky(k) system output (or measurement), at time step kf(x(k), v(k), k) system dynamicsh(x(k), n(k), k) measurement dynamicsv(k) process noise at time step kn(k) process noise at time step kp(x(0)) ‘a priori’ probability distribution of the system state at time 0,px(·|x(k − 1), k) transition density of the system statepy(·|x(k), k) probability distribution of the measurementsY(k) sequence of measurements up to time step k, Y(k) = {y(i)}i=0,...,k

X(k) sequence of states up to time step k, X(k) = {x(i)}i=0,...,k

p(X(k)|Y(k)) optimal bayesian estimate of the state

Particle FiltersN number of particlesp(X(k)|Y(k)) empirical estimate of the stateδXi(k) Dirac mass at particle X

i(k)qi(k) weight of particle i at time kgi(k) weight of particle i at time kqi(k) normalized weight of particle i at time kxi(k) estimate of the system state for particle ixn(k) nonlinear part of the system stateW (k) linear part of the system state

xiv

W (k) smoothed linear part of the system stateX

i(k) resampled particle path ixi(k) resampled particle state ifn(xn(k), k) system dynamics for the nonlinear part of the statehn(xn(k), k) measurement function of the nonlinear part of the stateAn(xn(k)) state evolution matrix for the nonlinear part of the stateAl(xn(k)) state evolution matrix for the linear part of the stateC(xn(k)) observation matrix for the linear part of the statevn(k) process noise of the linear part of the statevl(k) process noise of the nonlinear part of the stateQ(k) process noiseQn(k) process noise for the linear part of the stateQl(k) process noise for the nonlinear part of the stateQln(k) correlation between the process noise of the linear

and nonlinear part of the stateR(k) measurement noiseW (0) mean of the prior normal distribution of the linear part of the stateP (0) covariance of the prior normal distribution of the linear part of the stateV (k) monte carlo posterior variance matrixmi kernel locations after shrinkingμi prior point estimate of particle iα parameter for shrinkage of kernel locationsδ discount factorh smoothing parameter

Kalman Filter

C(k) observation matrixW (·|·) state estimate (mean)P (·|·) covariance matrix estimateK(k) kalman gainS(k) covariance matrix of the innovation term

Sequential Conditional Particle Filter

M(k) number of aircraft in the airspacel aircraft indexz vector containing all aircraft statesA interpolation matrix from the wind-field to wind at a certain locationμi

X , μiY mean of the wind forecast error distribution the X- and Y -axis

Σi covariance of the wind forecast error

xv

biX , biY wind estimate of particle i at the aircraft locationNeff effective sample size

Joint State and Parameter EstimationΩ parameter vectorμ0,Σ0 mean and covariance of the parameter “a priori” distributionσε standard deviation of the artificial noiseaf forgetting factor of the artificial dynamicsβ constant parameter in the artificial dynamicskf final filtering stepce1 shaping parameter of the exponential functioncs1, cs2 shaping parameters of the sigmoidal functionσrf weighting likelihood at time kf

cr shaping parameter for the weighting likelihood function

Mathematical Notation≈ approximately equal to∈ element of∑

summation∫integral

AT transpose of matrix AA−1 inverse of matrix AE [x] expected value of xR

n set of real numbers of dimension nN (μ,Σ) normal distribution with mean μ and covariance matrix Σ

xvi

Acronyms

ANSP Air Navigation Service ProviderAPF Auxiliary Particle FilterATC Air Traffic ControlATCC Air Traffic Control CenterATM Air Traffic ManagementBADA Base of Aircraft DataCAS Calibrated AirspeedCD Conflict DetectionCR Conflict ResolutionCD&R Conflict Detection and ResolutionCAS Calibrated AirspeedCFMU Central Flow Management UnitESF Energy Share FactorEKF Extended Kalman FilterFIR Flight Information RegionFMS Flight Management SystemHJS Hybrid Joint-Separable multi-target filterIATA International Air Transport AssociationIQR Interquartile RangeKDE Kernel Density EstimationKF Kalman FilterMCMC Markov Chain Monte CarloMPF Marginalized Particle FilterNEXTGEN Next Generation Air Transportation Systempdf Probability Density FunctionPF Particle FilterPMM Point Mass ModelRMS Root Mean Square ErrorROC Receiver Operating Characteristic CurveROCD Rate of Climb/DescentRTA Required Time of ArrivalRUC Rapid Update CycleSCPF Sequential Conditional Particle FilterSESAR Single European Sky ATM ResearchSIS Sequential Importance SamplingSIR Sequential Importance ResamplingSMC Sequential Monte CarloSOC System Operating Characteristic CurveTAS True AirspeedTEM Total Energy ModelTP Trajectory Prediction

xvii

xviii

Chapter 1Introduction

1.1. Overview

Air transportation has become an integral part of the modern world. Every location onthe planet is nowadays reachable in less than a day’s worth of travel. People and goodsare transported over vast distances, at speeds never experienced before. Novel business,leisure, cultural and scientific opportunities, are created. The existing ones are facilitatedand multiplied. All these possibilities have a profound effect on the global economy. Onlyin Europe aviation accounts, directly or indirectly, for 220 billion � of added value and 4million jobs [2, 103]. The respective figures for the whole world reach 722 billion � and13.5 million jobs [14]. These numbers include the combined output of airlines, airports,aerospace manufacturers and Air Navigation Service Providers (ANSP).

Air Traffic Management (ATM) constitutes a fundamental component of air transportthat allows for the safe and expeditious flow of traffic throughout the world. The primaryconcern of ATM is to ensure that accidents and risk-bearing incidents remain at low levelsdespite the increase of traffic [37, 98]. Moreover, ATM strives to increase the capacityof the airspace without compromising safety [10, 11, 77]. These objectives have to beprovided in a cost effective way that minimizes the expenses per flight in terms of fuel,delays and administration costs [4, 89, 110]. During the last years growing efforts forreducing the environmental impact of aviation, in terms of emissions and noise have alsobeen established [17, 22, 29, 35, 78, 82, 88, 94, 109, 112]. Finally, ATM requires that allthese services operate under certain degrees of predictability and flexibility, so that changes

1

1. Introduction

Figure 1.1: Air traffic controller in front of radar consoles.

and interventions can be possible [85, 90].Air transportation is built around a rigid structure of way-points and airways, an equiv-

alent of the crossroads and highways of the motorway system. Airways and way-pointsover Europe can be seen in Figure 1.2. Most of this architecture dates back to an era whenaircraft had to follow strict paths over radar beacons to enhance radar monitoring andseparation of air traffic. This system is not centralized but administered by different oper-ators. According to the Chicago convention [52], states are responsible for their nationalairspace. This has lead to the fragmentation of airspace and to monopolies, the ANSPs(usually government controlled), that provide services only inside the national boundaries.Recent research efforts focus on integrating the segregated airspace (following the SingleEuropean Sky ATM Research (SESAR) [104] initiative in Europe and Next GenerationAir Transportation System (NEXTGEN) [84] in the US).

Air Traffic Controll (ATC) and ATM services are provided inside controlled airspace.The largest regular division of airspace in use is the Flight Information Region (FIR). Anyportion of the atmosphere on the planet belongs to some specific FIR. Oceanic airspace isalso divided and part of the FIR bordering each region. The first subdivision of nationalairspace is the Air Traffic Control Center (ATCC) which some times extends outsidestate borders, see Figure 1.3. ATCC in its turn is composed of various sectors, see Fig-ure 1.4. This is the core component of the ATM system where air traffic controllers,operate (see Figure 1.1)1. Their task is to ensure safe operations of commercial and pri-

1Micah Maziar, "P1000171", 9 May 2005, Online Image, Flickr, http://www.flickr.com/photos/mzwp/292864558

2

1.1. Overview

50 NM

Figure 1.2: Way-points (green triangles) and airways (red lines) over central Europe.

vate aircraft. They must coordinate the movements of aircraft, keep them at safe distancesfrom each other, direct them during takeoff and landing from airports, direct them aroundbad weather and ensure that traffic flows smoothly with minimal delays. Early air trafficcontrollers had no direct radio link with aircraft and used telephones to stay in touch withairline dispatchers. The introduction of radars, modern computing systems and radio-communication with aircraft has revolutionized air traffic control and management.

More than 10 million flights take place inside the European airspace every year. Thecurrent record is 34476 flights over European skies in a single day. Developing economiesin Eastern Europe and the rise of the Asian economies have already a significant impactin the increase of air traffic [38]. Growth in air traffic is far from uniform, as is stronglyaffected, among others, by the economic cycle, fuel prices and ground transportation in-frastructure [40, 41]. However, long-term forecasts, predict that the traffic will double inmost of the world regions [28, 39] over the next 20 years.

To cope with the growing demand the ATM infrastructure and operations have to becontinuously updated. However, technologies and infrastructure are becoming old andsaturated. The rigid architecture of the current ATM system and the human-operated ATCservices [86], could potentially impose a constraint in the growth of air traffic. In support ofthis effort, a large variety of automation and decision support tools [5, 51, 111, 114, 115] arebeing developed to provide air traffic controllers with more accurate predictive informationabout aircraft trajectories, local and national traffic flow, weather and routing.

3

1. Introduction

40.0°

50.0°

-010

.0°

000.

0°

010 .

0 °

020.0°

030.0°

040.0°

100 NM

Figure 1.3: Air Traffic Control Center (ATCC) over Europe. Black circles represent major air-ports.

1.2. Trajectory Prediction

Aircraft Trajectory Prediction (TP) is central to many advanced ATM operational con-cepts. For example, TP is at the heart of most conflict detection and resolution algorithms.By comparing the predicted trajectories of different aircraft against each other, accurateTP can help detect real threats while avoiding false alarms. Large TP uncertainty forcesATC to use larger separations between aircraft to ensure safety, thus reducing the totalnumber of aircraft a given sector can accommodate. Moreover, changing a flight path forfear of a possible conflict, can decrease the fuel efficiency, or affect the time of arrival ofthe aircraft involved. Therefore more accurate TP may help ATC to accommodate forincreased traffic demand.

TP accuracy is naturally limited by the uncertainty inherent in every aircraft flight.Much of the TP uncertainty is due to meteorological forecast inaccuracies, especially thoseconcerning the wind. It has been shown that the deviation of wind forecasts from the realwind conditions is correlated in time and space [23], which implies that the effect of forecasterrors accumulates over time, leading to large uncertainties about the predicted position ofaircraft [20]. The design of the Flight Management System (FMS) also contributes to thepropagation of the error. Most aircraft do not correct along-track deviations over horizonsof tens of minutes, mainly for fuel efficiency reasons. Even though this has already startedto change with the introduction of 3.5D FMS systems that are able to track way-pointswith a Required Time of Arrival (RTA), for the current fleet and operating procedures

4

1.2. Trajectory Prediction

100 NM

Figure 1.4: Air traffic sectors between 40000 and 50000 ft. Red areas represent military restrictedairspace.

large along track uncertainty is the norm [34]. Chapter 2 describes a nonlinear hybridstochastic model for simulating aircraft dynamics, with a 3D FMS, under the effects ofwind forecast uncertainty.

Particle filters have been used extensively in air traffic and aircraft positioning appli-cations, especially for tracking [12, 13, 50, 65, 60, 87, 100, 108]. They pose effectivelyno restrictions on the process dynamics and measurements and are suitable for nonlinearsystems with non-Gaussian noise. Moreover, it can be shown that they converge to an op-timal solution as the computational resources tend to infinity. In Chapter 3 we present thebasics of nonlinear filtering and some of the most common particle filtering algorithms. Ithas been demonstrated that improved trajectory prediction accuracy can be achieved withparticle filters using radar measurements for a single aircraft [71, 72, 76], but the benefitsare expected to be much greater if one can fuse measurements from multiple aircraft atdifferent locations and time instants.

In Chapter 4, it is shown how the multi-aircraft sensor fusion problem can be formulatedas a high dimensional state estimation problem and a novel particle filtering algorithm,named Sequential Conditional Particle Filter (SCPF) is developed to solve it in realisticscale situations [73, 74]. By exploiting the structure of the problem, one can addressthe technical challenges that arise in the process: handling efficiently the information,dealing with the estimation of a very high dimensional state and dealing with the non-linear dynamics of aircraft motion and control. It is demonstrated how to reduce TPinaccuracies related to wind forecast errors. Roughly speaking, the aim is to build a

5

1. Introduction

probabilistic “wind map” to capture the difference between the wind forecast and the realwind conditions. This wind map is then added to the wind forecast and used to improvethe accuracy of TP. The key observation that enables this approach is that past aircrafttrajectories, available to the air traffic controllers through radar measurements or datalink,encapsulate information about the wind the aircraft has experienced. In other words, eachaircraft is treated as a moving, local sensor of the wind field. Then the spatiotemporalcorrelation of the error in the wind forecasts [23] is exploited, to fuse the informationfrom the multiple aircraft and to interpolate in space/time to points where/when aircraftmeasurements are not available. An implicit advantage of the proposed method is thatit provides more accurate wind estimates in regions with dense air traffic, where accurateTP is needed the most.

The SCPF aims to alleviate particle filter degeneracy by exploiting the problem struc-ture. The SCPF shares some of the insights of the Marginalized Particle Filter (MPF) [100],in the sense, that both treat the linear and the non-linear part of the state separately. Thetwo main novelties compared with the Marginalized Particle Filter (MPF) is the sequen-tial incorporation of information from different aircraft and the substitution of particlescarrying uncertainty realizations by particles carrying conditional distributions. A relatedidea was proposed in [21] for static instead of dynamic models. A closely related ideato SCPF, for multi-body visual tracking is the Hybrid Joint-Separable multi-target filter(HJS), which also handles targets in an independent way [64].

Related ideas for improving TP by reducing wind uncertainty are explored in [27, 49, 79].The impact of wind uncertainty on the TP along track error and the effect of initialconditions is investigated in [79]. Wind-field estimation is performed in [27], employing anExtended Kalman filter, by linearizing the model dynamics. The accuracy of the methodis demonstrated for a constant wind and requires a number of aircraft turns (with knownturn rate) to be utilized. Finally, [49] makes use of multiple aircraft wind measurementsand builds a Kalman filter to estimate the wind error, during the descend phase. Thekey innovation of the methods developed here is the generality of the Particle Filter (PF)approach, which allows us to model wind varying in time and space and treat the nonlineardynamics of the aircraft through which the wind information is obtained.

The performance of the proposed algorithm is documented through simulation basedfeasibility studies. The studies treat the case of multiple aircraft flying level, possibly atdifferent altitudes, with known airspeeds and aerodynamic equations (both depending onthe aircraft type). This is an ideal situation in an air traffic control context, since airspeedsand aerodynamic coefficients are only partially known in general. The aim is to determinewhether the proposed approach is viable, in particular whether aircraft trajectories containenough information to estimate wind forecast errors and whether the proposed filteringalgorithm can deal computationally with the problem. The results are very encouragingon both fronts. The proposed algorithm can deal with large numbers of aircraft (up tohundreds) while avoiding degeneracy. Moreover, in the presence of sufficient aircraft, thetrajectory prediction results using the improved wind map produced by the novel algo-rithm are close to the theoretical limit corresponding to perfect knowledge of the currentwind conditions in the entire region of interest. The studies reported here assume that

6

1.3. Conflict Detection

information is collected only through ground radar, but the methods can easily be adaptedif datalink information (for example, on-board measured wind speed) is also available.

In Chapter 5 we compare the performance of several different particle filtering algorithmsin aircraft TP [75]. The most generic methods such as Sequential Importance Sampling(SIS) and Sequential Importance Resampling (SIR) are usually applicable for single air-craft TP estimation, at the expense of computational effort. However, they have severedifficulties handling multi-aircraft estimation due to the detrimental increase in space di-mensions. Other methods, such as Kernel smoothing are employed as well, in order toseparately handle the slow evolving linear part of the state (in this case, the wind) andavoid sample attrition. Last, a technique to separate the linear substructure of the modelfrom the nonlinear, the MPF, is used. Unfortunately, our simulations demonstrate thatthese particle filtering algorithms cannot handle efficiently both the high dimensionalityof the wind state and the nonlinearities in multi-aircraft dynamics.

1.3. Conflict Detection

One important aspect for guaranteing safety in air travel is the separation assurance be-tween flight trajectories. Whenever a prescribed minimum separation between two aircraftis violated, a conflict occurs. Enhanced accuracy in aircraft conflict detection allows alsofor more efficient use of the airspace. For conflicts to be identified and prevented, anautomated mechanism for Conflict Detection (CD) is required. Once a possible futureconflict is detected, either a centralized [104] or a decentralized [26] Conflict Resolution(CR) scheme can be used to resolve it; for an overview see [63]. Trajectory prediction liesat the heart of most conflict detection algorithms. In Chapter 6 we show how trajectoryprediction tools that account for weather forecast errors can improve the performance ofa conflict detection scheme. The use of SCPF can considerably improve conflict detectionperformance in mid and short term horizon encounters.

CD is itself a challenging task and very often it is combined with the CR task. Mostcommon methods can be divided into three major categories, based on the predictionhorizon they consider. Roughly speaking Long term Conflict Detection and Resolution(CD&R) methods deal with horizons of more than 30 mins. Their main concern is typicallyflow management problems. Mid term CD&R, accounts for prediction horizons up to 30mins. Finally, short term CD&R, deals with horizons up to 10 mins.

For the conflict detection to be accurate, one should be able to compute a reliableprediction of the trajectory of an aircraft. Increasing levels of traffic require systems thatcan accurately predict conflicts earlier, in order to accommodate the extra traffic demand.An automated conflict detection mechanism can take advantage of data that might not bedirectly accessible, or possibly hard to interpret, by the air traffic controllers, such as theestimated state of the aircraft, weather information and weather uncertainty or differentaircraft performance models. This information combined with the data that an air trafficcontroller has access to, like the estimated position and aircraft flight plans, can lead to analgorithm that improves TP and assists the ATC in identifying early potential conflicting

7

1. Introduction

situations. The longer the horizon the aircraft trajectory is accurately predicted, the moreflexibility the ATC (or a conflict resolution algorithm) has to resolve a conflict, or toaccommodate more traffic.

We demonstrate how CD can be improved by reducing TP inaccuracies related to windforecast errors. To solve the problem we apply the Sequential Conditional Particle Filter(SCPF) that can deal with both the nonlinear and the high dimensional nature of theproblem [69]. The performance of the proposed algorithm is assessed with a series ofsimulated feasibility studies.

1.4. Joint Parameter and State estimation

Sequential Monte Carlo methods provide good approximations to the optimal filteringproblem for several applications despite non-linear or non-Gaussian formulation. However,a particularly challenging case arises when the state-space model depends on a set ofunknown fixed parameters. For air traffic management applications, considered here, weassume that the aircraft airspeeds are not available to the air traffic controller. Theunavailability of this information can have a detrimental effect on trajectory predictionaccuracy. Moreover, uncertainty about the airspeeds combined with additional uncertaintyabout the wind speeds can further deteriorate trajectory prediction results. In Chapter 7 wepropose different sequential monte carlo methods for jointly estimating aircraft airspeedsand wind speeds to decrease trajectory prediction errors.

Estimating model parameters can be carried in two different ways. If the data areavailable beforehand, the estimation of parameters can be done off-line (smoothing). Oth-erwise, if a particular application requires sequential estimation, the same task has tobe performed online. For off-line estimation several different techniques have been pro-posed. The most usual ones are Monte Carlo methods such as Markov Chain Monte Carlo(MCMC) [96]. Their main drawback is a heavy computational burden in order to con-verge to the optimal filtering solution. Various other methods have been proposed, such asGauss-Newton methods [68], stochastic gradient [32] or Expectation Maximization (EM)algorithms [16, 113].

For online methods a natural choice is to extend the state vector with the parameters asartificial states that have no dynamic evolution. Standard PF can be used to estimate thereformulated state of the system, that includes the static, unknown parameters. However,the absence of evolution for the additional artificial states, implies that the explorationof the parameter space depends only on the first extraction from the “a priori” densityfor the initial state of the system. A solution to this problem is the addition of smallrandom perturbations in the dynamics of parameters [66, 83]. For an overview, of differentPF methods in parameter estimation see [59, 15]. We demonstrate how the differentmethods can be combined with SCPF in order to improve TP accuracy and identify missingairspeeds. Feasibility studies show that SCPF can be successfully augmented to cope withthe joint parameter and state estimation problem.

8

1.5. Contributions

1.5. Contributions

The main goal of this thesis is the application of sequential monte carlo methods (orparticle filters) in air traffic related problems. Initially, a nonlinear stochastic hybridmodel was developed that describes the dynamics of an aircraft, equipped with a 3D flightmanagement system, under the influence of wind. Wind is treated as the sum of twocomponents, wind forecast and wind forecast uncertainty. A new linear gaussian modelfor describing the time and space varying wind uncertainty was created. The model wasdeveloped in a way that respects the experimental data concerning the spatiotemporalcorrelation of wind forecast uncertainty.

The primary objective was the use of the model for the improvement of trajectoryprediction (TP) accuracy in air traffic management. Particle filters have been success-fully implemented in single-aircraft cases and have provided substantial improvements inTP accuracy. For situations involving multiple aircraft and a time-varying wind-field anovel algorithm was developed called Sequential Conditional Particle Filter (SCPF). Itwas demonstrated how SCPF can increase TP accuracy even further, than PF for singleaircraft cases, and cope satisfactorily with problems of very high dimensions. The use of thealgorithm is not restricted only in air traffic applications but in any problem that a largenumber of nonlinear agents interacts with a time-varying spatially correlated environmentthat evolves linearly.

A further contribution was the successful demonstration that SCPF can be used inpractical air traffic control problems such as conflict detection (CD). A procedure for usingSCPF in CD was developed. It was shown that SCPF can improve significantly conflictdetection performance when compared with simply using the meteorological forecasts.SCPF can exploit information from aircraft that may precede the flights of interest andincrease performance even further.

Finally, SCPF has been used for solving the particularly demanding problem of on-line joint parameter and state estimation. Several different methods were developed foraugmenting SCPF to cope with this problem. The new algorithms were tested in tra-jectory prediction and airspeed identification providing again improved TP accuracy andsignificantly reducing airspeed uncertainty.

9

1. Introduction

10

Chapter 2Model Dynamics

In this chapter we present a model that simulates the dynamics of commercial air-craft from the point of view of an air traffic controller. Each aircraft has to follow anassigned flight plan despite disturbances from varying wind and atmospheric conditions.Variables of the atmosphere such as temperature, air density and pressure are calculatedusing standard models. Specialized models are developed to describe a time and spacevarying wind-field. Finally, a flight management system is developed that accounts forall these effects and tracks the desired flight plan. We outline the main parts of a pointmass model that describes the motion of commercial aircraft while maintaining a workabledegree of simplicity. The model is capable of capturing multiple instances of flights, eachwith a different flight plan, aircraft dynamics and flight management system, representingdifferent types of aircraft. The model is framed in the context of stochastic hybrid systems.The physical motion of the aircraft gives rise to the continuous part of the model. Theflight plan and the logic variables embedded in the FMS give rise to the discrete part. Theweather uncertainty (mainly the wind forecast error) is treated as a stochastic disturbanceto the model, that has its own dynamics. Figure 2.1 provides an overview of the differentcomponents of the model. In subsequent chapters this model is used as a basis for thedevelopment of filtering algorithms and their validation in simulations.

11

2. Model Dynamics

.

.

.

.

Forecast

Wind ErrorAircraft

Aircraft

Dynamics

Dynamics

FMS

FMS

Flight

Flight

Plan

Plan

Figure 2.1: Block diagram of the model components for multiple aircraft flying in the same windfield.

2.1. Aircraft Dynamics

The physical motion of an aircraft can be modeled using a Point Mass Model (PMM).From the point of view of ATM this is a reasonable approximation. A PMM does notreflect all the intricacies of aircraft flight, but is reasonably accurate and commonly usedin ATM research [36, 106, 53, 99, 91, 42]. To analyze the motion of the aircraft we useas an inertial reference frame a flat earth coordinate system centered at a location on thesurface of the earth. X-axis denotes the West to East direction, Y -axis denotes the Southto North direction and Z-axis denotes the up direction. The major variables of the PMMare the aircraft position in the inertial reference frame (xI , yI , zI), the True Airspeed(TAS)1 (V ), the aircraft mass (m), the flight path angle (γ), the heading angle (ψ) andbank angle (φ), summarized in Figure 2.2.

The forces applied to the aircraft are its weight (mg), the engine thrust (T ), and theaerodynamic forces of lift (L) and drag (D). The aerodynamic forces in practice dependon the angle of attack (AOA) and the side slip angle (AOS). From the point of viewof an ATC however, trimmed flight conditions can be assumed and thus one can setAOA = AOS = 0. Moreover by ignoring fast dynamics, T , γ and φ are treated as inputs.The movement of the aircraft is also affected by the wind which acts as a disturbance. Wewill model the wind through its speed WI = (wX , wY , wZ) ∈ R

3. To capture the effect of

1True Airspeed is the speed of the aircraft relative to the surrounding air.

12

2.1. Aircraft Dynamics

AOA

L T

Dmg

γX

Z

V

L

mg

φ

X

Z

AOS T

D

ψ

X

Y V

Figure 2.2: Forces acting on an aircraft

a moving air mass reference frame we employ wind gradient factors (Wagf - along trackand Wcgf - cross track) (see the Appendix A.1). The aircraft motion can be captured bya control system with six states, three inputs and three disturbances. Thus, the equationsof motion become⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

xI

yI

zI

V

ψ

m

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

V cos(ψ) cos(γ) + wX

V sin(ψ) cos(γ) + wY

V sin(γ) + wZ

[T −D −mg sin (γ)] /m−Wagf

L sin (φ)/m/V/ cos (γ) −Wcgf/V/ cos (γ)

−ηT

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦. (2.1)

The state and inputs are subject to constraints: zI > 0, V ∈ [Vmin, Vmax], m ∈[mmin, mmax], T ∈ [Tmin, Tmax], φ ∈ [φmin, φmax], γ ∈ [γmin, γmax]. Violation of thesebounds is seldom an issue with commercial aircraft which normally operate with largesafety margins (far from aerodynamic stall conditions, etc.) Values for the state and in-put bounds and the aerodynamic parameters, which depend on aircraft type, the phase offlight and aircraft configuration, can be obtained from the Base of Aircraft Data (BADA)

13

2. Model Dynamics

database [42]. Sections 2.1.1 and 2.1.2 outline the derivation of the the aerodynamic forces(L and D) and the fuel flow coefficent (η).

2.1.1. Aerodynamic Forces

Since we are interested primarily in the flight of airliners which normally operate aroundtrimmed flight conditions we have assumed that the angle of attack and side slip angle aresmall. In this case the lift and drag forces can be approximated by

L =CLSρ(zI)

2V 2 and D =

CDSρ(zI)

2V 2 , (2.2)

where S is the surface area of both wings, ρ(zI) is the air density (which depends onaltitude, zI) and CD, CL are aerodynamic lift and drag coefficients whose values dependon aircraft type and configuration (whether the flaps are extended, the landing gear down,etc.). CL is set to ensure that the vertical component of the lift exactly balances the weightof the aircraft, including a correction term for changes in the bank angle.

CL =2mg

ρ(zI)V 2S cos (φ). (2.3)

CL is used in the computation of the drag coefficient CD, which also depends on the phaseof the flight (whether the flaps are extended, the landing gear down, etc.). We set

CD = CD0(CM,FP) + CD2(CM,FP)C2L , (2.4)

where FP (Flight Phase) and CM (Climb Mode) are part of the discrete state of the FMSdiscussed in Section 2.2. The parameters CD0 and CD2 depend on the aircraft type [42].

2.1.2. Fuel Consumption

The η parameter represents the rate at which fuel is consumed and may not be constant.It depends on altitude (zI), the airspeed (V ), the aircraft engines and flight mode (climb,acceleration, etc.). The fuel flow, fnom (kg/min) can be calculated using the thrust andthe fuel flow coefficient (η)

fnom = ηT . (2.5)

Details for the calculation of the fuel flow for different flight phases can be found in [70].For example when the aircraft is climbing the fuel flow becomes

fnom = ηT =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

Cf1(1 +V

Cf2)T if Engine_Type=Jet

Cf1(1 − V

Cf2

)( V1000

)T if Engine_Type=Turboprop

Cf1 if Engine_Type=Piston

(2.6)

The coefficients Cf1, Cf2 depend on the aircraft type, obtained from [42].

14

2.2. Discrete State Dynamics

x4 ≤ Vnom

x4 ≥ Vnom + 1ms−1

x4 ≥ Vnom

x4 ≤ Vnom − 1ms−1 x4 ≥ Vnom + 1ms−1

x4 ≤ Vnom

|x4 − Vnom| ≤ 1ms−1

x4 ≤ Vnom x4 ≥ Vnom

AM = C

AM = DAM = A

Figure 2.3: Finite State Machine for AM.

2.2. Discrete State Dynamics

The discrete states of the model describe the different flight modes. The discrete dynamicsarise from the flight plan of the aircraft and the logic variables embedded in the FMS.The discrete state of the FMS is stored in 8 discrete variables: way-point index (WP),acceleration mode (AM), flight level (FL), climb mode (CM), speed hold mode (SHM),flight phase (FP), reduced power mode (RPM), and troposphere mode (TrM). Like alldiscrete quantities, the discrete state of the FMS evolves by “jumping” from one value toanother. The occurrence of the jumps depends on the values of the continuous state of theaircraft; the overall system is therefore hybrid. A different discrete state affects both thecontinuous dynamics (e.g. lift, drag, flow coefficients) and the outputs of the FMS (thrust,bank angle, flight path angle). For more details the reader is referred to [42], [46] and [70].

Each aircraft is assigned a flight plan which consists of a sequence of Λ + 1 way-points,{O(i)}Λ

i=0, in three dimensions, O(i) ∈ R3. The sequence of the way-points defines a

sequence of straight lines joining each way point to the next. The discrete variable repre-senting the way point index takes integer values reflecting the number of way points in theflight plan, WP ∈ {0, 1, . . . ,Λ − 1}. WP = i when the aircraft is on its way from the ithway-point at position O(i) ∈ R

3 to the (i + 1)st way point at position O(i+ 1) ∈ R3. To

determine when the aircraft switches from one way point to the next we adopt a “Fly-past”method2 for the way aircraft perform turns. This happens whenever an aircraft crosses avertical plane at the beginning of the turn, defined by at(i)

Tx = bt(i) in Figure 2.5. The

2The aircraft starts turning before it reaches the next way point (the method used for modern aircraft).

15

2. Model Dynamics

FP = LD

x4 ≤ V1 + 11kts∧x3 ≤ 3001ft

FP = AP

(x4 ≤ V2 + 11kts∧x3 ≤ 8001ft)

∧(x4 ≥ V1 + 11kts∨x3 ≥ 3001ft)

FP = DE

x3 ≤ hdes + 1ft∧(x4 ≥ V2 + 10kts∨x3 ≥ 8000ft)

FP = UD

x3 ≥ hdes

x4 ≥ V1 + 11kts∨x3 ≥ 3001ft

x4 ≤ V1 + 10kts∧x3 ≤ 3000ft

x4 ≥ V2 + 11kts∨x3 ≥ 8001ft

x4 ≤ V2 + 1kts∧x3 ≤ 8000ft

x3 ≥ hdes + 1

x3 ≤ hdes

Figure 2.4: Finite State Machine for FP.

location of the vertical plane (the values of at(i) and bt(i)) depends on the maximum bankangle and the locations of the way-points, see [70] for details.

The discrete variable for the acceleration mode reflects whether the aircraft is acceler-ating, decelerating, or cruising at constant speed, AM ∈ {A,D,C}. The value is resetwhenever the desired speed of the aircraft changes. This may be the case, for example,when the aircraft changes flight level, or the climb mode state changes. The state returnsto C when the aircraft reaches the desired speed (see Figure 2.3). A hysteresis of 1ms−1

is introduced to prevent chattering.The discrete variable representing the flight level takes on values representing the follow-

ing altitude discretization {0, 500, 1000, 1500, 2000, 3000, 6000, 10000, 14000} ft. It is usedprimarily to determine the nominal airspeed, Vnom. The climb mode reflects whether theaircraft is climbing, descending or flying level, CM ∈ {C,D, L}. The value is reset when-ever the aircraft starts a new segment of the reference path. CM affects the thrust settingsand the fuel flow coefficient (η). The speed hold mode represents whether the aircraft isholding constant Calibrated Airspeed (CAS)3 or holding constant Mach. The change takesplace at the transition altitude, which is the altitude where the desired TAS of the aircraftdetermined by CAS is equal to the desired TAS determined by the Mach number. Thisis to ensure that the maximum allowable Mach number imposes an upper bound on theaircraft speed. The reduced climb power has been introduced to allow climbs using lessthan the maximum thrust setting for climb. The troposphere mode represents whetherthe aircraft is above or below the tropopause, the boundary between the troposphere andthe stratosphere.

From the point of view of the FMS, the descent portion of flight is divided into 4 phases:upper descent (UD), lower descent (LD), approach (AP) and landing (LA). We use thediscrete variable FP ∈ {UD,LD,AP,LA} to store this information. The rules for changingthe values of FP are summarized in Figure 2.4. FP is used to set the thrust input (u1).As discussed in Section 2.1.1, it also has an effect on the aerodynamic forces acting onthe aircraft. For example during approach and landing the aircraft will have a high liftconfiguration which will also increase drag. When the aircraft is not descending, i.e. when

3 CAS is the speed shown by a conventional airspeed indicator after correction for instrument error andposition error. Under International Standard Atmosphere conditions CAS is the same as TAS

16

2.3. Wind Dynamics

at(i)Tx = bt(i)

O(i) O(i+ 1)

O(i+ 2)

Figure 2.5: Geometry of turn between way-points (assuming zero wind).

CM ∈ {C,L}, FP has no effect on the dynamics.

2.3. Wind Dynamics

We assume that aircraft fly inside a wind-field that varies in time and space. Each aircraftis affected by a different part of the wind-field through the effects of the wind speed vectorWI ∈ R

3 and the gradient factors (Wagf andWcgf). The wind speed vector and the gradientfactors are a function of the wind-field and depend on the position of the aircraft. The wind-field model is of fundamental importance in our studies since it is well accepted that a largepart of the uncertainty about the evolution of flights stems from the fact that meteorologicalforecasts are inherently inaccurate [8, 102, 9]. Moreover, among the different weatherphenomena affecting an aircraft, wind-speed is one of the most important [54, 80, 48]. Wemodel the wind-field as a sum of two components: a nominal component (representingweather forecasts) and a stochastic component (representing forecast errors).

2.3.1. Wind Forecast

The nominal part of the wind-field represents the meteorological predictions that are avail-able to the ATC. Those meteorological data are obtained mainly from the Rapid UpdateCycle (RUC), a numerical weather prediction model for the U.S.A. that incorporates air-craft measurements, balloon soundings and other sensor data [33, 7, 6]. The RUC modelis run every three hours and each run produces a set of three hourly forecasts. For eachrun the data are stored in a 3D grid with a horizontal resolution of 60 km and a verticalresolution of 50 mb.4 The wind is computed for time and space points not correspondingto grid nodes by linear interpolation between the neighboring grid nodes.

4With newer versions of RUC 40/20/13 km of horizontal resolution and hourly runs can be used.

17

2. Model Dynamics

2.3.2. Wind Forecast Error Statistics

The stochastic component of the wind is modeled as a random field:

w : R × R3 → R

2

where w(t, P ) represents the wind at point P ∈ R3 and at time t ∈ R. We assume here that

the wind in the vertical direction is zero. Restrict attention to the case where w(t, P ) ∈ R2

is Gaussian with zero mean and covariance matrix R(t, P, t′, P ′) ∈ R2×2. The zero mean

assumption reflects a hypothesis that weather forecasts contain all deterministic informa-tion about the wind. Furthermore, the wind field is assumed isotropic (invariant underrotations) and wind speeds in the south-north (Y-axis) and west-east (X-axis) directionsare uncorrelated. Under these assumptions the covariance matrix R simplifies to

R(t, P, t′, P ′) = E[w(t, P )wT (t′, P ′)] =

[r(t, P, t′, P ′) 0

0 r(t, P, t′, P ′)

], (2.7)

where E denotes expected value, and for P = (xI , yI , zI) ∈ R3, P ′ = (x′I , y

′I , z

′I) ∈ R

3,t, t′ ∈ R,

r(t, P, t′, P ′) = σ(zI)σ(z′I)rt(|t− t′|)rXY

(∥∥∥∥ xI − x′IyI − y′I

∥∥∥∥)rZ(|AP (zI) −AP (z′I)|). (2.8)

AP (zI) is the atmospheric pressure in mb and σ(zI) is the standard deviation of the winderror in m/s at altitude zI . AP (zI) is computed using the standard atmosphere model [42],and σ(zI) based on the data reported in [23]. The form and parameters of the functions rt,rXY and rZ were set using data from [23], which analyzes the statistics for the differencebetween the wind predicted by RUC and the wind measured by aircraft. According tothis, for s ≥ 0

rt(s) = ct + (1 − ct − dt)e−s/Gt + dtcos

(2πs− et

gt

)(2.9)

rXY (s) = cXY + (1 − cXY )e−s/GXY (2.10)

rZ(s) = cZ + (1 − cZ)e−s/GZ . (2.11)

Notice that the correlation decays exponentially as horizontal distance, altitude or timedifference increase. This difference between RUC and aircraft wind measurements are dueto the error between the wind predicted by RUC and the real wind, and the error betweenthe real wind and the wind measured by the aircraft. The parameters σ, ct, dt, Gt, gt, et,cXY , GXY , bXY , cZ , GZ are set to correct for this. The parameter values suggest a strongcorrelation between wind errors in the same horizontal plane, a very strong correlation intime and a weaker correlation across different altitudes.

18

2.3. Wind Dynamics

Figure 2.6: Wind forecast error projected on a horizontal grid. The intensity of the color fromblue to red indicates a high wind error magnitude. The left image displays the initial wind-field.while the right the evolution of the wind-field after 20 minutes.

2.3.3. Wind Forecast Error Generation

For simulation purposes and for the implementation of the filtering algorithms a methodfor generating wind realizations that respects the above correlation structure is needed.Different ways of doing this have been proposed [46, 70] aimed primarily at simulation. Wedescribe an approach, that is better suited for filtering since it generates wind realizationsefficiently and in a sequential manner, at the expense of a small approximation in thecorrelation structure.

To describe the wind-field the airspace is gridded into a lattice comprising:

• NX points in the West - East direction, spaced every δX (e.g. δX = 60km).

• NY points in the South - North direction, spaced every δY (e.g. δY = 60km).

• NZ points vertically, spaced every δZ (e.g. δZ = 1000ft).

• Nt+1 points in time, spaced every δt (e.g. δt = 30s).

For each point in the lattice two random numbers are generated, one for the south-northand one for the west-east direction of the wind error. Let i ∈ {1, . . . , NX}, j ∈ {1, . . . , NY }and h ∈ {1, . . . , NZ} denote the indices of a point in the lattice, k ∈ {0, . . . , Nt} denotethe current time step, and wX(i, j, h, k) ∈ R and wY (i, j, h, k) ∈ R denote the wind in the

19

2. Model Dynamics

two directions corresponding to the latice point; in the notation of the previous section

w

⎡⎣ kδt,

⎛⎝ iδX

jδYhδZ

⎞⎠

⎤⎦ =

[wX(i, j, h, k)wY (i, j, h, k)

]∈ R

2.

We order lexicographically in h, j, i, to form two vectors

WX(k) =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

wX(1, 1, 1, k)wX(2, 1, 1, k)

...wX(NX , 1, 1, k)wX(1, 2, 1, k)

...wX(NX , NY , NZ , k)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦∈ R

NXNY NZ , WY (k) =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

wY (1, 1, 1, k)wY (2, 1, 1, k)

...wY (NX , 1, 1, k)wY (1, 2, 1, k)

...wY (NX , NY , NZ , k)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦∈ R

NXNY NZ .

(2.12)To derive the dynamics of the vectors WX(k) and WY (k) the time correlation function is

approximated by rt(s) ≈ e−s/Gt . This is a very good approximation for the time horizonsof interest here (30 seconds to 1 hour). The quality of the approximation decreases for timedifferences of more than 3 hours, since the effect of the cosine term in Eq. (2.9) becomesstronger (Figure 2.7).

Let R ∈ RNXNY NZ×NXNY NZ denote the covariance matrix of WX(k) (by the isotropic

assumption, the matrix will be identical for WY (k)). Based on the discussion in theprevious section and the simplifying assumption on rt this covariance matrix is constant intime. Moreover, its elements (describing the correlation between two spatial grid points,say (i, j, h) and (i′, j′, h′), at the same point in time, k) can be calculated as

r

⎛⎝kδt,

⎡⎣ iδXjδYhδZ

⎤⎦ , kδt,

⎡⎣ i′δXj′δYh′δZ

⎤⎦⎞⎠

= σ(hδZ)σ(h′δZ)rXY (√

[(i− i′)2δ2X + (j − j′)2δ2

Y ])rZ(|AP (hδZ) − AP (h′δZ)|).

Samples that respect the simplified correlation structure can now be generated using thefollowing linear gaussian model

WX(0) =QvX(0), WX(k + 1) = aWX(k) +QvX(k) ,

WY (0) =QvY (0), WY (k + 1) = aWY (k) +QvY (k) ,(2.13)

where vX(k), vY (k) ∈ RNXNY NZ are standard (zero mean, identity covariance matrix) in-

dependent Gaussian random variables. Q and Q are derived by Cholesky Decompositionfrom the covariance matrix R according to

QQT = (1 − a2)R and QQT = R. (2.14)

20

2.3. Wind Dynamics

0 10 20 30 40 50 600

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

time (seconds)

corr

elat

ion

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

time (hours)

Cor

rela

tion

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 240

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

time (hours)

Cor

rela

tion

Figure 2.7: Comparison between the real (red solid line) and simplified (blue dashed line) timecorrelation functions.

Finally it is set

a = e−δt/Gt ∈ R. (2.15)

Clearly, a small time step implies a highly correlated wind-field between consequtive timesteps. It is easy to show that the resulting vectors indeed respect the approximate cor-relation structure; for completeness a proof of this fact is given in Appendix A.2. As forthe weather forecast, linear interpolation of the wind at the neighboring grid points of thewind-field is used to compute the wind at the aircraft position .

21

2. Model Dynamics

x

T

ESF

φ

FMS

Flight Plan

Thrust & ESFControl

Bank AngleControl

Figure 2.8: FMS controllers.

2.4. Flight Management System

The FMS can be thought of as a controller that measures the state of the aircraft and usesit together with the flight plan to determine the values of the inputs T , φ, γ, which aredetermined to some extent by conventional continuous controllers [46, 70]. The FMS isdivided into two components, one controlling along track and vertical motion through thethrust and flight path angle and the other controlling cross track motion through the bankangle. This arrangement is illustrated in Figure 2.8. The thrust and flight path angleare used to set the speed (V ) and the Rate of Climb/Descent (ROCD). Following currentpractice, the FMS tries to track a desired speed, Vnom, which is a function of altitudeand aircraft type and is determined by the airline. We assume that aircraft control theirhorizontal position using exclusively the bank angle (φ). This model reflects what is knownas a 3D FMS. Alternatives are the 4D FMS and, what is sometimes referred to as, the3.5D FMS (already in use). Aircraft equipped with 3.5 and 4D FMS correct their alongtrack position to meet the RTA at the way-points.

2.4.1. Thrust control

The settings for the thrust depend on the desired TAS of the aircraft, Vnom. Vnom is setbased on airline procedure tables. The setting of the thrust depends on the discrete statesAM, CM, SHM and on the engine type. If SHM = M , Vnom is set equal to a Mach numberwhich depends on aircraft type and whether the aircraft is climbing or descending (CM).If SHM = C, Vnom is set equal to a CAS which depends on aircraft type, FL and CM.First, engine type is used to set the value of a parameter TMaxClimb which depends on

22

2.4. Flight Management System

the aircraft altitude and speed and denotes the maximum thrust the engines can produceduring climbing.

TMaxClimb =

⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩

CTc1

(1 − zI

CTc2

+ CTc3z2I

)if Engine Type=Jet

CTc1

V

(1 − zI

CTc2

)+ CTc3 if Engine Type=Turboprop

CTc1

(1 − zI

CTc2

)+CTc3

Vif Engine Type=Piston

The values for the parameters CTc1, CTc2 and CTc3 depend on the aircraft type and canbe obtained from the BADA database. TMaxClimb is also used to set the available thrust,for the cruise and descent phases, based on the discrete states CM, FP, RPM and AM:

T =

⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

TMaxClimb (CM = C) ∧ (RPM = OFF )

TMaxClimbCpowred − CDSρ(zI)V2(1 − Cpowred)

2(CM = C) ∧ (RPM = ON)

CTdes(FP)TMaxClimb (CM = D) ∨ [(CM = L) ∧ (AM = D)]0.95TMaxClimb (CM = L) ∧ (AM = A)CDSρ(zI)

2V 2

nom (CM = L) ∧ (AM = C)

(2.16)The reduced climb power (RPM = ON) has been introduced to allow the simulation ofclimbs using less than the maximum climb setting,since in day-to-day operations, manyaircraft use a reduced setting during climb in order to extend engine life and save cost.The values for the parameters Cpowred and CTdes(FP) for the different values of FP dependon the aircraft type and are obtained from the BADA database.

2.4.2. Flight Path Angle and ESF control

In order to control the flight path angle and subsequently the ROCD we use what is knownas the Energy Share Factor (ESF). The ESF determines the ratio of the available powerwhich is allocated to climbing (increasing potential energy) versus accelerating (increasingkinetic energy) while following a selected speed profile during climb or descent. In ourcontext the ESF can be thought of as an alternative input u3 that replaces the flight pathangle (γ) in the equations. The two input parameterizations are related by(

T − CDSρ(zI)

2V 2

)u3 = mg sin(γ). (2.17)

When cruising at a constant altitude, the FMS sets ESF (and so the flight path angle)to zero, producing zero ROCD. For the most part the ESF is set to a constant value, asshown in Table 2.1. When AM = C and CM = C or D the ESF is set to ensure that theaircraft speed V tracks the nominal speed Vnom. Notice that the control in this case is

23

2. Model Dynamics

mostly open loop. No direct feedback is present from the continuous state of the aircraft,to the input γ (the same holds for the thrust settings). Feedback is present indirectly,however, since T and γ depend on the discrete state of the FMS, which, in turn, dependson the continuous state of the aircraft.

AM=A AM=C AM=DCM=C 0.3 Track Vnom 1.7CM=L 0 0 0CM=D 1.7 Track Vnom 0.3

Table 2.1: Settings for the energy share factor

This appears to be the procedure most commonly used in practice when aircraft aretaking off, cruising, or performing routine climbs and descents at altitude. The reason isthat, because speed is controlled directly, it allows for efficient climbs and descents. Insome cases, however, this procedure may be unacceptable, since it provides no control overthe ROCD. Climbing or descending at a specified ROCD may be desirable in some cases,especially in heavy traffic. In such cases, ATC may explicitly command an aircraft to usea given ROCD. In addition, controlling ROCD is essential for aircraft in final approach;using the above procedure in this case would typically result in aircraft over- or under-shooting the runway. To capture situations like these we have also developed specializedcontrollers for tracking a given ROCD. Details can be found in the Appendix A.3.

2.4.3. Bank Angle Control

To ensure that the aircraft do not stray too far off their reference path we have developeda model of a controller to correct such deviations. Our model assumes that the FMSsets the bank angle based on the heading error and the cross track deviation from thereference path. As reference path we define the line that connects the sequence of way-points {O(i)}Λ

i=0. The controller operates in continuous time and consists of a linearfeedback part, followed by non-linearities to ensure that the behavior is reasonable evenwith extreme inputs. Figure 2.9 summarizes the geometry of the situation. The headingerror at time t is defined as

θ(t) = Ψ(i) − ψ(t) , (2.18)

where Ψ is the angle of the reference path with X-axis. The cross-track deviation (d(t))at time t is defined as the distance of the aircraft position from the reference path. Thelinear part of the controller then is given by

φ1(t) = k1d(t) + k2θ(t) . (2.19)

The gains k1 and k2 are arbitrary at this stage; methods for setting their values to ensurestability and generate realistic cross track deviation statistics are discussed in [70]. The

24

2.4. Flight Management System

(X,Y )�

V

ψ

θ

d

O(i)

Ψ(i)

O(i+ 1)

O(i+ 2)Reference Path

Global Coordinate Frame

Figure 2.9: The aircraft FMS tracks the reference path between two subsequent way-points (O(i)and O(i+ 1)), in the presence of wind.

linear controller may command unrealistically large bank angles when faced with largedeviations from the reference path. To prevent this we introduce a saturation at someangle φ ≥ 0.

φ2(t) =

⎧⎨⎩

−φ if φ1(t) ≤ −φφ1(t) if − φ ≤ φ1(t) ≤ φ

φ if φ1(t) ≥ φ.

(2.20)

Even with the saturation, the aircraft may fail to converge to the reference path if itdeviates too far from it. Instead it may travel in arcs of circles. To prevent this weintroduce a further limit on the bank angle as a function of the heading error. This leadsto the final setting for the bank angle

φ(t) =

{min {φ2(t), 0} if π/2 ≥ ψ(t) ≥ ψ

max {φ2(t), 0} if − π/2 ≤ ψ(t) ≤ −ψ (2.21)

25

2. Model Dynamics

2.5. Radar Model

The position of all aircraft is measured using an ATC ground radar. This represents onlya partial observation of the aircraft state; for example mass and aircraft heading are notdirectly measured. Moreover, the radar measurements are assumed to be corrupted bynoise. For a more realistic model the polar description of the radar noise should be used.For simplicity, the noise is added in the X and Y position of the aircraft. In practice theaccuracy of the radar usually decreases as an aircraft moves away from the radar location.Here, the same measurement error statistics are used for all distances , and the varianceis selected high enough (σr = 80m) to err on the side of caution. Radar measurements areassumed to arrive sequentially at discrete time steps (δt).

Assume that at a discrete time instant kδt there are l = 1, ...,M(k) aircraft flying in aregion of the airspace. This number will change as aircraft enter and exit the region ofinterest. Define the position of aircraft l at time kδt as (X(kδt, l), Y (kδt, l)), Z(kδt, l)) ∈R

2. Assume a vector x(k) ∈ R6M(k) contains the full state for each of the aircraft in the

airspace. The measurement vector y(k) obtained by the radar is defined as follows

y(k) =

⎡⎢⎣ y(k, 1)

...y(k,M(k))

⎤⎥⎦ = h(x(k)) + n(k) ∈ R

2M(k). (2.22)

The function h is linear, a projection to the X and Y coordinates of the aircraft positions.The vector n(k) models the measurement noise and comprises independent Gaussian ran-dom variables with zero mean and standard deviation σr added to the radar measurementsof each aircraft. The altitude of each aircraft Z(kδt, l) is assumed constant and known.This assumption is reasonable since altitude is measured on-board the aircraft and trans-mitted to the ground through the secondary radar’s mode-s transponder.

2.6. Summary

In this chapter we provided an overview of a nonlinear hybrid stochastic model that cap-tures the dynamics of multiple aircraft equipped with a 3D flight management system(FMS). The aircraft use their FMS in an attempt to track a flight plan despite the effectsof the wind. A model that describes the evolution of the wind-forecast uncertainty wasalso developed. In subsequent chapters this formulation will be used both as the basis forthe development and the test-bed for evaluating the performance, of filtering, trajectoryprediction, conflict detection and airspeed identification methods.

26

Chapter 3Non-Linear Filtering

In this chapter we discuss the optimal estimation problem. We describe the formulationof the filtering problem for a discrete time stochastic process where the observations arecorrupted by noise. The optimal solution (with respect to the mean square error) for alinear and Gaussian system is the Kalman filter. For many cases however, where the stateor measurement dynamics include elements of nonlinearity or are affected by non-Gaussiannoise, analytical solutions to the filtering problem do not exist. Sequential Monte Carlomethods (or Particle Filters) are approximation techniques that solve the filtering problemfor these cases. We describe different particle filtering algorithms proposed in the literatureand discuss their strengths and weaknesses. In subsequent chapters the algorithms outlinedin this chapter form a basis of comparison for the novel filtering algorithms developed inour work.

27

3. Non-Linear Filtering

3.1. Optimal Estimation

Optimal bayesian estimation [44, 58, 24, 105] provides a rigorous approach for estimatingthe probability distribution of unknown variables by utilizing all of the available obser-vation data, and information about the system dynamics. In the Bayesian approach, aProbability Density Function (pdf) that embodies all available statistical information isconstructed. An optimal (with respect to any criterion) estimate of the state may be ob-tained from the pdf. Additionally, a measure of the accuracy of this estimate may also becalculated. The starting point is typically a discrete time model of the dynamics of theprocess and the measurements of the form

x(k + 1) = f(x(k), v(k), k)y(k) = h(x(k), n(k), k),

(3.1)

where x(k) ∈ Rn and y(k) ∈ R

p are the state and output of the system at time stepk ∈ N (we refer to k simply as time from now on), and f : R

n × Rn × N → R

n andh : R

n × Rp × N → R

p are (possibly nonlinear) functions. v(k) ∈ Rn and n(k) ∈ R

p areprocess and measurement noise, which are generally assumed to be independent, identicallydistributed stochastic processes, but not necessarily additive, or Gaussian. We also assumethat the initial state is independent of the noise processes and its distribution is giventhrough a pdf p(x(0)). If the pdf of the noise processes are known, the system can beequivalently represented by

x(k) ∼ px(·|x(k − 1), k)y(k) ∼ py(·|x(k), k) . (3.2)

Here px(·|x(k−1), k) is a conditional pdf that models the stochastic dynamics of the stateof the system, determined by f and the pdf of v(k). py(·|x(k), k) is a conditional pdf thatmodels the probability distribution of the measurements, determined by h and the pdf ofn(k).

Given k, k′ ∈ N let Y(k′) = {y(i)}i=0,...,k′ denote the sequence of measurements up totime k′ and X(k) = {x(i)}i=0,...,k denote the sequence of states up to time k. The aimis to estimate the pdf p(X(k)|Y(k′)). This density function embodies our best estimateof the state vector up to time k given all available information up to time k′. We makethe assumption that x(k) is Markovian. This means that its conditional probability giventhe past states, p(x(k)|X(k− 1)), depends only on x(k− 1) through the transition densitypx(·|x(k − 1), k). Depending on the relation of k to k′ we can formulate three differenttypes of estimation problem:

• Filtering (k = k′): estimate the state trajectory up to the current state.

• Prediction (k > k′): estimate the state trajectory up to a future state.

• Smoothing (k < k′): refine the estimate of an older part of the state trajectory.

28

3.1. Optimal Estimation

All three of these problems can be seen to be special cases of the problem of estimatingthe expected value

E[g(X(k), k)] =

∫R(k+1)×n

g(X(k), k)p(X(k)|Y(k′))dX(k) . (3.3)

of an arbitrary integrable function g : R(k+1)×n × N → R. (For example, think of taking

g(X(k), k) = X(k) for the filtering problem).The filtering, prediction and smoothing problems can be solved recursively by invoking

Bayes’ theorem. For example, in the case of filtering, consider the density

p(X(k)|Y(k)) =p(Y(k)|X(k))p(X(k))

p(Y(k))=p(y(k),Y(k − 1)|X(k))p(X(k))

p(y(k),Y(k − 1))

=p(y(k)|X(k),Y(k − 1))p(Y(k − 1)|X(k))p(X(k))

p(y(k)|Y(k − 1))p(Y(k − 1))

=p(y(k)|X(k))p(X(k)|Y(k − 1))p(Y(k − 1))p(X(k))

p(y(k)|Y(k − 1))p(Y(k − 1))p(X(k))

=p(y(k)|X(k))p(X(k)|Y(k − 1))

p(y(k)|Y(k − 1)),

where the Markov property is used in the last equation. Then p(y(k)|Y(k − 1)) andp(x(k + 1)|Y(k)) can be calculated using the Chapman-Kolmogorov equation [55]. Giventhat x(k) is assumed Markovian

p(y(k)|X(k),Y(k − 1)) = p(y(k)|X(k))

p(x(k + 1)|X(k),Y(k)) = p(x(k + 1)|X(k)) .

which simplifies the Chapman-Kolmogorov equation to

p(y(k)|Y(k − 1)) =

∫R(k+1)n

p(y(k)|X(k))p(X(k)|Y(k − 1))dX(k) , (3.4)

p(x(k + 1)|Y(k)) =

∫R(k+1)n

p(x(k + 1)|X(k))p(X(k)|Y(k))dX(k) . (3.5)

Now a recursive filter can be constructed, consisting of two steps, an update step (whena new measurement arrives the estimation of the state is updated, (3.4)) and a predictionstep (predict the state one step in the future, (3.5)).

This recursion represents the optimal bayesian solution. This conceptual solution isdeceptively simple, however, since the integrals involved are seldomly tractable. Twonotable exceptions are the Kalman filter [57] (see Section 3.2) for linear systems withadditive gaussian noise and the Grid-Based filter [3] for systems with a finite state space.In the general case the pdf of interest has to be approximated numerically.

29

3. Non-Linear Filtering

3.2. Kalman Filter

When the process and measurement models are both linear and the noise Gaussian and ad-ditive, the optimal solution to the filtering problem is given by the Kalman Filter (KalmanFilter (KF)) [57]. Consider the following model

W (k + 1) = Al(k)W (k) + v(k)y(k) = C(k)W (k) + n(k) , (3.6)

where W (k) ∈ Rl is the system state and y(k) ∈ R

p the system output. Also, matrixAl(k) ∈ R

l×l describes the dynamics and C(k) ∈ Rp×l the observation model. The quan-

tities v(k) ∈ Rn are process noise and n(k) ∈ R

p measurement noise respectively with,v(k) ∼ N (0, Q(k)) and n(k) ∼ N (0, R(k)). The prior distribution of the state is alsoGaussian with W (0) ∼ N (W (0), P (0)). Using the KF, the estimate of the state remainsGaussian and can be described using a mean and covariance matrix, which are computedanalytically. Define the mean (W (k|k′)) and covariance matrix (P (k|k′)) of the estimateat time k, given measurements up to to time k′ (with k ≥ k′) as

W (k|k′) = E[W (k) | Y(k′)]P (k|k′) = E[(W (k) − W (k|k′))(W (k) − W (k|k′)T ) | Y(k′)] .

(3.7)

The KF equations provide the following estimates for the state mean and covariance matrix

W (k|k) = W (k|k − 1) +K(k)(y(k) − C(k)W (k|k − 1))P (k|k) = P (k|k − 1) −K(k)C(k)P (k|k − 1)K(k) = P (k|k − 1)CT (k)S−1(k)S(k) = C(k)P (k|k − 1)CT (k) +R(k) ,

(3.8)

where K(k) is the Kalman gain and S(k) is the covariance matrix of the innovation term,y(k)−C(k)W (k|k − 1). The prediction estimates are also normally distributed accordingto

W (k + 1|k) = Al(k)W (k|k)P (k + 1|k) = Al(k)P (k|k)AT

l (k) +Q(k)(3.9)

KF can be used for non-linear systems, by approximating the model with a linear one,using Taylor series expansion and truncating the higher order terms. The filter obtainedthis way is called an Extended Kalman Filter (EKF) [56].

30

3.3. Particle Filters

−4−2

02

46

8 −4

−2

0

2

4

6

8

0

0.2

0.4

Y − Axis (km)

X − Axis (km)

−4−2

02

46

8 −4

−2

0

2

4

6

8

0

0.2

0.4

X − Axis (km)

Y − Axis (km)

Figure 3.1: Example of true continuous probability density function (left) and its particle approx-imation (right). The location of the particles reflect the value of the probability density in thatregion of the state space.

3.3. Particle Filters

Particle filters (or Sequential Monte Carlo (SMC) methods [30, 67, 31, 43, 95]) are estima-tion techniques that provide a numerical approximation to the optimal bayesian estimationusing simulation. SMC methods offer a number of significant advantages which arise pri-marily from the generality of the approach. They allow for inference of full posterior dis-tributions in general state-space models, which may be both nonlinear and non-Gaussianand might incorporate constraints in the system state or parameters. SMC methods allowfor the computation of different moments of the pdf, and not only first and second ordermoments (like the KF and the EKF, see Section 3.2).

The main idea is to approximate the continuous probability distribution of interest usinga discrete distribution comprising weighted samples (known as particles, Figure 3.1). Todo this N independent identically distributed particles, X

1(k), . . . ,XN(k) are extractedfrom p(X(k)|Y(k)), and an empirical estimate of the distribution is constructed

p(X(k)|Y(k)) =1

N

N∑i=1

δXi(k)(X(k)) , (3.10)

where δXi(k) denotes the Dirac mass at particle Xi(k). Then the expectation of any inte-

grable function, g, can be estimated by

E[g(X(k), k)] ≈∫g(X(k), k)p(dX(k)|Y(k)) =

1

N

N∑i=1

g(Xi(k), k) . (3.11)

It can be shown that this estimator is unbiased and (under weak assumptions) convergesto the true expectation as the number of particles N tends to infinity [25].

31

3. Non-Linear Filtering

3.3.1. Sequential Importance Sampling.

In cases where we are unable to extract from p(X(k)|Y(k)) directly, we can employ atechnique known as importance sampling [45]. An algorithm to perform these operationsrecursively is SIS, presented in Algorithm 1. SIS is a Monte Carlo (MC) method that formsthe basis for most sequential MC filters. Each particle is assigned a weight according tothe current measurement and this weight is updated each time new measurements arrive.

Algorithm 1 Sequential Importance Sampling

1: Initialization. Set k = 0 and extract N particles {Xi(0)}i=1,...,N = {xi(0)}i=1,...,N

from the initial distribution p(x(0)).

2: Measurement Update. Given the measurement y(k), evaluate the importanceweights qi(k) = py(y(k)|xi(k), k) and normalize them

qi(k) =qi(k)

ΣNj=1q

j(k)

3: Prediction. For i = 1, . . . , N , extract xi(k + 1) ∼ px(·|xi(k), k) and setX

i(k + 1) = (Xi(k), xi(k + 1)).

4: Iteration. Increment k and return to step 2.

While it can be shown that as N tends to infinity this algorithm leads to correct esti-mation, for a finite number of particles SIS easily leads to degeneracy. This is a situationwhere all the particles have negligible weight, except for a single one; indeed it can beshown that in some cases as k increases the SIS algorithm is guaranteed to fail in thisway [31].

3.3.2. Sequential Importance Resampling.

A standard method for avoiding this problem is to introduce an additional re-sampling step.The idea is to stop propagating particles that have low importance weights and multiplyparticles with high importance weights, thus focusing computational resources in areaswhere the pdf is higher. This leads to the Bayesian bootstrap [47, 18] or SIR algorithm,presented in Algorithm 2.

A potential problem of the SIR algorithm is loss of diversity as particles with largeweights are selected more and more often , eventually eliminating all other particles. Theloss of diversity diminishes the exploration capabilities of the algorithm. The algorithmusually works satisfactorily for low dimensional spaces, but might have difficulties converg-ing for higher dimensions.

32

3.3. Particle Filters

Algorithm 2 Sequential Importance Resampling

1: Initialization. Set k = 0 and extract N particles {Xi(0)}i=1,...,N = {xi(0)}i=1,...,N

from the initial distribution p(x(0)).

2: Measurement Update. Given the measurement y(k) evaluate the importanceweights qi(k) = py(y(k)|xi(k), k) and normalize them,

qi(k) =qi(k)

ΣNj=1q

j(k)

3: Resampling. Extract N particles {Xi(k)} from {X

i(k)}i=1,...,N with replacementaccording to importance weights qi(k).

4: Prediction. Extract xi(k + 1) ∼ px(·|xi(k), k) and set Xi(k + 1) = (Xi(k), xi(k)) for

i = 1, . . . , N .

5: Iteration. Increment k and return to step 2.

3.3.3. Marginalized Particle Filter

The MPF is a combination of a PF with the KF. It can be used when the process modelcontains a linear sub-structure, subject to Gaussian noise. The strategy of sampling someof the variables and marginalizing exactly the rest, with a KF, is also known as Rao-Blackwellisation [1], [19]. Consider the following model

xn(k + 1) = fn(xn(k), k) + An(xn(k))W (k) + vn(k)W (k + 1) = Al(xn(k))W (k) + vl(k)y(k) = hn(xn(k), k) + C(xn(k))W (k) + n(k) ,

(3.12)

where W (k) ∈ Rl and xn(k) ∈ R

n are the linear and nonlinear subparts of the staterespectively, and y(k) ∈ R

p the output of the system at time k ∈ N. f : Rn × N → R

n

and hn : Rn × N → R

p are (possibly nonlinear) functions. Matrices, An : Rn → R

n×l andAl : R

n → Rl×l determine the linear effects of the dynamics, and C : R

n → Rp×l, the

effect of the linear part on the measurement. The quantities vl(k) ∈ Rl and vn(k) ∈ R

n

are process noise for the linear and nonlinear dynamics while n(k) ∈ Rp is measurement

noise. The prior distribution of the linear state W (0) ∼ N (W (0), P (0)), the measurementnoise and the state noise are all assumed to be Gaussian,

n(k) ∼ N (0, R(k)) ,(vl(k)vn(k)

)∼ N (0, Q(k)) , where Q(k) =

(Ql(k) Qln(k)

(Qln)T (k) Qn(k)

).

The linear state variables can be marginalized out and be estimated using the KF, whichis the optimal filter for this case. Note that each particle carries a KF and makes its own

33

3. Non-Linear Filtering

estimate about the state. Moreover, the prediction of the nonlinear state is employed asan additional, artificial measurement, using again the KF. This improves even more theestimate of the linear state variables. The nonlinear state variables are estimated usingthe particle filter. MPF results in a particle representation of the nonlinear states, whereeach particle carries an associated Gaussian distribution for the remaining linear states.Algorithm 3 demonstrates this idea, for details see [62], [100], [101].

Algorithm 3 Marginalized Particle Filter

1: Initialization. Set k = 0 and extract i = 1, ..., N particles from the initial distribution

• xin(0) ∼ p(xn(0)) .

• {W i(0), P i(0)} = {W (0), P (0)} .

• Set {Xi(0)}i=1,...,N = {xi

n(0),W i(0)}i=1,...,N .

2: Measurement Update. For i = 1, ..., N , given the measurement y(k) evaluate theimportance weights qi(k) and normalize them,

qi(k) = py(y(k)|xi(k), k) , qi(k) =qi(k)

ΣNj=1q

j(k).

3: PF Resampling. Sample N particles with replacement according to importanceweights

Pr(xin(k) = xi

n(k)) = qi(k) .

4: Prediction and Kalman Filter.

i) KF measurement update. Filter the linear state using 3.8.

ii) Prediction. Extract xin(k + 1) ∼ px(·|xi

n(k), k)

iii) KF time update. Predict the mean and covariance of the linear state for thenext time step using 3.9.

5: Iteration. Set {Xi(k)}i=1,...,N = {xi

n(k),W i(k)}i=1,...,N . Increment k and return tostep 2.

34

3.4. Summary

3.3.4. Kernel Smoothing and Auxiliary PF.

Smoothing methods attempt to combat loss of diversity by adding small random perturba-tions, before updating to the next cluster of particles. Even though this artificial evolutionprovides an additional mechanism for exploring the state space the added noise impliesa loss of information, resulting in estimates that might be far too diffuse relative to thetheoretical posteriors. On the other hand, keeping the perturbations very small, to avoidthis drawback, might lead to degeneracy. The two risks can be balanced using an appro-priate kernel density to modify the evolution of the particles. The intuition behind thedevelopment of this algorithm comes from filtering methods for combining parameter andstate estimation [66]. The natural option in this case is treating the unknown parametersas part of the state with no dynamical evolution. This formulation is bound to fail as timeincreases, since the absence of evolution for the parameters suggests that the parameterspace is explored only in the initial step of the algorithm. Slowly evolving states can bealso treated as parameters under this formulation.

Assume again that a subpart of the system state evolves linearly while the rest of thestate evolves with nonlinear dynamics as in Eq. (3.12), and consider the following smoothkernel density

p(W (k)|Y(k)) ≈N∑

i=1

qi(k)N (W (k)|mi(k), h2V (k)) . (3.13)

N (·|m,S) is a multivariate Gaussian density distribution with meanm and variance matrixS. V (k) represents the Monte Carlo posterior variance matrix of p(W (k)|Y(k)) computedfrom the Monte Carlo sample. The mean of the kernel locations mi(k) are specified usinga shrinkage rule

mi(k) = αW i(k) + (1 − α)W (k) , (3.14)i.e. are displaced towards the Monte Carlo mean W (k). The resulting normal mixturepreserves the sample mean W (k) and variance [66], by setting α =

√1 − h2 and computing

α = (3δ−1)/2δ using a discount factor δ (typically around 0.95-0.99). To further alleviatesample degeneracy and make the filter robust to outliers, an auxiliary filtering step can beadded to the algorithm [92, 61]. Using the auxiliary step we can resample looking one stepahead, by propagating the mode or the mean of the particle distribution. The AuxilliaryParticle Filter (APF) is presented in Algorithm 4.

3.4. Summary

In this chapter we presented various sequential monte carlo methods for estimating andpredicting the state of non-linear non-Gaussian systems. The discussion serves as a back-ground for the development of a novel filtering algorithm for improving trajectory predic-tion in Chapter 4. The algorithms outlined in this chapter also form a basis for comparisonwith the new algorithm in Chapter 5.

35

3. Non-Linear Filtering

Algorithm 4 Auxiliary Particle Filter and Kernel Smoothing of Parameters

1: Initialization. At k = 0, extract N particles {Xi(0)}i=1,...,N = {(xi

n(0),W i(0))}i=1,...,N

from the initial distribution p(xn(0),W (0)).

2: Prior Point Estimates. Evaluate estimates of (xn(k),W (k)), as

μi(k + 1) = E(xin(k + 1)|xi

n(k),W i(k))mi(k) = αW i(k) + (1 − α)W (k) ,

where the expectation is computed by the state evolution density.3: Auxiliary Step. Generate an auxiliary integer j from {1, ..., N}, with probability

proportional to

gi(k + 1) ∝ qi(k)p(y(k + 1)|μi(k + 1), mi(k)) .

4: Resampling. Extract a new linear part W j(k) from the jth component of the kerneldensity,

W j(k) ∼ N (·|mj(k), h2V (k)) .

5: Prediction. Evolve to the next time step, according to system dynamics

xjn(k + 1),W j(k + 1)) ∼ px(·|xj

n(k), W j(k), k) .

6: Measurement Update. Given the measurement y(k + 1) evaluate the weight

qj(k + 1) ∝ p(y(k + 1)|xjn(k + 1),W j(k + 1))

p(y(k + 1)|μj(k + 1), mj(k)).

7: Iteration. Repeat steps 3 to 6 until a large number of samples (usually N) is gatheredto produce a final approximation (xj

n(k + 1),W j(k + 1)) and set

X(k + 1)i = {Xi(k), (xi

n(k + 1),W i(k + 1))} ,

for i = 1, . . . , N . Normalize weights qj(k + 1), increment k and return to step 2.

36

Chapter 4A novel Particle Filter for TrajectoryPrediction

In this chapter the multiple aircraft trajectory prediction problem is formulated as ahigh dimensional state estimation problem. A novel particle filtering algorithm is devel-oped to solve this problem in realistic scale situations. We show how, by exploiting thestructure of the problem, we can handle efficiently the very high dimensional state and thenonlinear dynamics of aircraft motion and control. The performance of the novel algorithmis demonstrated on feasibility studies involving multiple aircraft. The effect of the numberof aircraft on the trajectory prediction accuracy is also investigated. Finally, we presenta simulation study on the impact of the number of particles on the performance of thealgorithm.

37

4. A novel Particle Filter for Trajectory Prediction

4.1. Trajectory Prediction and Filtering

The trajectory prediction problem is reformulated so that its relation to nonlinear filteringbecomes more apparent. To keep the notation simple from now on the nominal (forecast)wind is assumed zero and only the stochastic error in the wind forecast is treated here;extension to non-zero wind forecast is straightforward.

Assume that at a discrete time instant kδt there areM(k) aircraft flying in a region of theairspace; this number can of course change over time as aircraft enter and exit the regionof interest. We restrict our attention here to level flights, flying with constant speed ataltitude zI . For this reason we truncate the aircraft states from 6 to 4, since the calculationsof airspeed acceleration (V ) and ROCD (zI) are not required any more. Likewise, the flightpath angle (γ) input is held constantly zero. Let z(t, l) = (xI(t, l), yI(t, l), ψ(t, l), m(t, l)) ∈R

4 denote the state of aircraft l = 1, . . . ,M(k) at time t ∈ [kδt, (k + 1)δt] and WX(k) andWY (k) denote the wind vectors. Aircraft might be flying at different flight levels zI(l). Let

x(k) =

⎡⎢⎢⎢⎢⎢⎣

z(kδt, 1)...

z(kδt,M(k))WX(k)WY (k)

⎤⎥⎥⎥⎥⎥⎦ ∈ R

4M(k)+2NXNY NZ

denote the overall state of the system (c.f. Eq. (3.1)).Using the information contained in x(k) one can compute the state of each aircraft

l = 1, . . . ,M(k) at the next discrete time instant (k + 1)δt by solving the differentialequation

z(t, l) =

⎡⎢⎢⎢⎢⎢⎣

V (l) cos(ψ(t, l)) + wX(t, l)

V (l) sin(ψ(t, l)) + wY (t, l)

CL(l)S(l)ρ(zI)V (l) sin(φ(t, l))

2m(t, l)

−η(l)T (t, l)

⎤⎥⎥⎥⎥⎥⎦ . (4.1)

for t ∈ [kδt, (k + 1)δt]. The values of the control inputs T (t, l) and φ(t, l) are computedusing the FMS model. Notice that the feedback controller encoded in the FMS is timevarying, since it depends on the flight plan; in other words, the same z(t, l) will lead todifferent T (t, l) and φ(t, l) depending on how far along the flight plan aircraft are. Thevalues of the wind wX(t, l) and wY (t, l) are computed by linear interpolation of WX(k)and WY (k) at the current position xI(t, l), xI(t, l), zI(l) of aircraft l. This implies a lineardependence of wX(t, l) and wY (t, l) on WX(k) and WY (k) of the form

wX(t, l) = A(xI(t, l), yI(t, l), zI(l))WX(k), wY (t, l) = A(xI(t, l), yI(t, l), zI(l))WY (k)(4.2)

38

4.1. Trajectory Prediction and Filtering

where the interpolating function A(xI(t, l), yI(t, l), zI(l)) can be computed explicitly. Tocapture the approximation introduced by the simulation, independent Gaussian randomvariables with zero mean and standard deviation σp = 5m are added to the resultingxI((k+ 1)δt, l) and yI((k+ 1)δt, l); let v(k, l) ∈ R

4 denote the vector comprising these tworandom variables followed by two 0’s.

The value of the state x(k + 1) is completed by updating WX(k) and WY (k) accordingto Eq. (2.13). The overall result is a new value

x(k + 1) = f(x(k), v(k), k)

where the process noise vector v(k) comprises both the aircraft state evolution errorsv(k, l) ∈ R

4, l = 1, . . . ,M(k) and the wind innovation vectors vX(k) ∈ RNXNY NZ and

vY (k) ∈ RNXNY NZ ; note again the dependence of f on the time k entering through the

flight plan. This defines a function f needed in Eq. (3.1), implicity (through simulation)for the aircraft states and explicitly (through Eq. (2.13)) for the wind states.

At time instant kδt the horizontal positions y(k, l) = (xI(kδt, l), yI(kδt, l)) ∈ R2 of all

aircraft, l = 1, . . . ,M(k) are measured through the radar; the altitude of each aircraftzI(l) is assumed constant and known, see Section 2.5 for a discussion. This gives rise to ameasurement vector

y(k) =

⎡⎢⎣ y(k, 1)

...y(k,M(k))

⎤⎥⎦ = h(x(k)) + n(k) ∈ R

2M(k).

The function h (c.f. Eq. (3.1)) is linear, a projection to the X and Y coordinates ofthe aircraft positions. The vector n(k) models the measurement noise and comprises theindependent Gaussian random variables with zero mean and standard deviation 80 madded to the radar measurements of each aircraft.

Trajectory prediction using this model is executed every time new radar measurementsare taken. Roughly speaking, the algorithm proceeds in two steps, a filtering step usingmeasurements to update the particles and a prediction step evolving particles using thedynamics, see Algorithm 5. The prediction step is straightforward both conceptually andcomputationally. The main difficulty lies in the filtering step, since the dimension of thestate and the nonlinear nature of the dynamics make it impossible to apply conventionalfiltering techniques. In the next section we show how the structure of the problem canbe exploited to develop specialized filtering algorithms to solve the problem at a realisticscale.

39

4. A novel Particle Filter for Trajectory Prediction

Algorithm 5 Trajectory Prediction

1: Initialization. Set k = 0 and extract i = 1, . . . , N particles for the entire state xi(0)

2: Filtering. Update the particles based on the measurement y(k).

3: Prediction. For each aircraft and for each particle simulate the continuous dynamicsof Eq.(4.1)-(4.2) over the prediction horizon, using the initial condition and windvectors contained in xi(k).

4: Iteration. Increment k, and return to step 2.

4.2. Sequential Conditional Particle Filter

The goal of SCPF is to exploit the structure of the TP problem. First, note that x(k)comprises two classes of state variables: the aircraft states z(kδt, l) and the wind statesWX(k) and WY (k). From the above discussion it is apparent that:

1. The aircraft states evolve according to non-linear dynamics, while the wind statesevolve according to linear dynamics.

2. The evolution of the wind states is independent of the evolution of the aircraft states.

3. The evolution of the states of different aircraft (z(kδt, l) and z(kδt, l′) for l = l′) are

only coupled to each other through the wind states.

The first two observations imply that the wind states should be easier to estimate, since,under Gaussianity assumptions, storing and manipulating them only requires keeping trackof their mean and covariance matrix. Moreover, given a probabilistic estimation of thewind at some points in the wind-field, the conditional distribution of the wind at all otherpoints can be explicitly derived. This distribution will also be Gaussian, hence easy tostore and manipulate. The first novelty of the proposed algorithm is that, instead of usingrealizations for the wind states carried by the particles (as in conventional particle filtering)the entire conditional probability distribution is stored and manipulated. The aircraft aretreated as flying sensors providing implicitly measurements for the wind they experience.

The latter two observations imply that, conditional on the wind states, the states ofdifferent aircraft are independent of each other. This is exploited by the second noveltyof the algorithm, which is the sequential incorporation of the information from differentaircraft. Every radar measurement contains information about the positions, y(k, l), of allaircraft in a region of the airspace, l = 1, . . . ,M(k). New measurements are processed one

40

4.2. Sequential Conditional Particle Filter

aircraft at the time: The measurement y(k, 1) of the position of aircraft 1 is used first to“filter” all the particles xi(k), updating the estimates of the states of aircraft 1, zi(k, 1),and the distributions of the wind states, WX(k) and WY (k). The updated zi(k, 1) is storeduntil the k+1 measurement while the distributions WX(k) and WY (k) are filtered using themeasurement y(k, 2) of the position of aircraft 2, then aircraft 3, etc. Experiments suggestthat the order in which the aircraft measurements are used is not particularly important.For consistency the order in which the aircraft entered the region of interest is used. Thecomplete algorithm is reported in Algorithm 6. A related idea, fo sequential filtering, wasproposed in [21] for static instead of dynamic models.

These two modifications provide a very substantial improvement in the performance ofthe filtering algorithm. Conventional particle filtering methods would normally carry real-izations of the wind states as part of the particles and would filter these (together with thestates of the aircraft) using the measurements from all aircraft simultaneously. In theorythis should not be a problem, provided the number of particles is high enough. In practice,due to computational limitations the number of particles is limited to a few thousands.Because of the large dimension of the wind states (which can be several hundreds, orthousands) it is unlikely that any one of the finite number of wind realizations carried byconventional particles would match well the wind in all the locations. Therefore, as theaircraft explore the wind field the importance weights tend to shift to just a few particles(the ones that match the wind field the least badly!) leading to degeneracy. This problemis especially acute in the presence of many aircraft. Experiments reported in Chapter 5suggest that none of the conventional particle filtering methods can handle more than 4aircraft: Even with 10, 000 particles degeneracy sets in only after 2 to 3 measurements forstandard particle filtering algorithms like the SIS and SIR. More sophisticated algorithmslike APF and MPF perform better but soon break down after the number of aircraft in-creases considerably. By contrast, the method proposed here appears to be stable withhundreds of aircraft and with 1, 000 (or even as few as 50) particles.

41

4. A novel Particle Filter for Trajectory Prediction

Algorithm 6 Sequential Conditional Particle Filter1: Initialization. Set k = 0. Generate i = 1, . . . , N particles, each comprising:

• An estimate, zi(kδ, l), of the state of each of the aircraft l = 1, . . . ,M(k) presentin the airspace. Set (X i(kδt, l), Y

i(kδt, l)) = y(k, l) (the measured position ofaircraft l at time kδt), ψi(kδt, l) the direction of the flight plan of aircraft l, andmi(kδt, l) the nominal mass of aircraft l.

• An estimate of the mean μiX and μi

Y and the covariance matrix Σi of the windstates. Set μi

X = 0, μiY = 0 and Σi = R.

2: Increment k.3: Update Aircraft List. Drop from the particles the state of all aircraft that have left

the region of interest between (k − 1)δt and kδt and add aircraft that have entered,initializing their states as above.

4: for aircraft l = 1, . . . ,M(k) do5: Extract wind realizations. For each particle i = 1, . . . , N extract Gaussian ran-

dom variablesW i

X ∼ N(μiX ,Σ

i) and W iY ∼ N(μi

Y ,Σi).

6: Evolution of Aircraft state. For each particle i = 1, . . . , N propagate zi((k −1)δt, l) to zi(kδt, l) using Eq.(4.1) and Eq.(4.2) with W i

X and W iY .

7: Measurement Update. Evaluate the weight of each particle i = 1, . . . , N accord-ing to the radar measurement qi = p(y(k, l)|zi(kδt, l)) and normalize qi = qi/ΣN

i=1qi.

8: Resampling. Draw N particles with replacement from among the existing particles,with probability of selecting particle j = 1, . . . , N equal to qj.

9: Conditioning. For ease of notation let Ai = A(zi(kδt, l)), biX = AiW iX and biY =

AiW iY . For each particle i = 1, . . . , N update the wind mean and covariance matrix

μiX = μi

X + Σi(Ai

)T(AiΣi

(Ai

)T)−1

(biX −AiμiX)

μiY = μi

Y + Σi(Ai

)T(AiΣi

(Ai

)T)−1

(biY − AiμiY )

Σi = Σi − Σi(Ai

)T(AiΣi

(Ai

)T)−1

AiΣi

to reflect the conditional distributions p(WX | AiW iX = biX) and p(WY | AiW i

Y = biY ).10: end for11: Evolution of Wind Dynamics. Propagate the wind distribution of each particle

i = 1, . . . , N to the next time step according to

μiX = aμi

X , μiY = aμi

Y , Σi = a2Σi + (1 − a2)R .

12: Iterate. Return to step 2.

42

4.3. Simulation Results

Figure 4.1: Flight plans for the 24-aircraft scenario overlayed over a map of Switzerland.

4.3. Simulation Results

4.3.1. Benchmark Simulation Set-up

To evaluate the algorithm, a series of artificial flight plans was devised, with a differentnumber of aircraft (1, 3, 6, 12 and 24) in a sector of interest. The flight plans for eachscenario can be seen in Figure 4.3; flight plan sets with more aircraft are supersets of thosewith fewer aircraft. To give an idea of the scale of the set-up, the 24 flight plans weresuperimposed over a map of Switzerland in Figure 4.1. The sector of interest is a squarewith dimensions 600×600 km. However, all flights are restricted in an area approximately180× 600 km. This is done in order to be able to assess the reduction in wind uncertaintyeven in areas where no aircraft have flown recently. All the aircraft fly level at 10000 m witha nominal airspeed of 215.6 m/s, and there are no turns included in the flight plans. Theparameters of the dynamical models for all aircraft represent a Boeing 737-200. The initialmass is considered known and is set to 46000 kg. Flights have a duration of approximately30 minutes and radar measurements arrive to the ATC every 30 seconds.

As discussed above, the weather forecast component of the wind is set to 0 throughoutthe region of interest for simplicity. The wind forecast error is evaluated in both thesouth-north and the west-east directions on a grid with 11 × 11 vertexes (i.e. the area ofinterest is split into cells of 60 × 60 km). Note that for the largest scenario (24 aircraft)the dimension of the non-linear state is 4 × 24 = 96 states, whereas the dimension of thelinear part of the state is 2 × 11 × 11 = 242 states.

To examine the uncertainty in mid-term TP the along-track error is used as a met-ric. Cross-track error was also evaluated, but was found to be considerably smaller, asone would expect due to the cross track corrections introduced by the 3D FMS model.

43

4. A novel Particle Filter for Trajectory Prediction

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20−1000

−500

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

5500

6000

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Alo

ng T

rack

Err

or (

m)

AC1 Bias

AC2 Bias

AC3 Bias

AC1 RMS

AC2 RMS

AC3 RMS

Figure 4.2: Comparison between the bias and RMS in TP for a 3 aircraft scenario.

The improvement in TP is therefore small in the cross-track direction as opposed to thealong-track directions; the cross-track results are omitted. For a thorough overview of thederivation of the along-track and cross-track error metrics the reader is referred to [81]and [97].

Unless otherwise stated, the results presented below are for TP based on filtering10 minutes of radar measurements (20 measurements total). The SCPF algorithm isrunning for this period, attempting to filter the aircraft and wind state. In all simula-tions 1000 particles are employed, independent of the number of aircraft. Having used theSCPF algorithm to filter 10 minutes of data the state estimate of all particles is extrap-olated for a further 20 minutes (40 time steps) into the future to get an estimate of thefuture trajectory for each aircraft. The extrapolation is done by simulating all the particlesfor the required time. The mean between these particles is considered as a TP estimate(β) for each flight. The real trajectory (β) is then compared with this estimate to computethe along track error metric.

To collect sufficient statistics, each scenario was simulated Ξ = 1000 times, keeping thesame flight plans, but different wind-fields, producing different trajectory estimates (βj)and real trajectories (βj), for j = 1, . . . ,Ξ. The estimator bias is then calculated as

bias =1

Ξ

Ξ∑j=1

(βj − βj) , (4.3)

44

4.3. Simulation Results

0 60000 120000 180000 240000 300000 360000 420000 480000 540000 6000000

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BestWorst

Figure 4.3: Flight Plans for 1, 3, 6, 12, 24 aircraft. Arrows display the direction of motion. Redand blue display the flight with the worst and best performance, in terms of uncertainty reduction,respectively.

45

4. A novel Particle Filter for Trajectory Prediction

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200

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Worst Performance BoundBest Performance BoundBest PerformanceLocal Known

Figure 4.4: Evolution of the along-track RMS, for 1 aircraft. Comparison of SCPF performancewith a perfect local filter (Local Known).

while the Root Mean Square Error (RMS) is calculated as

RMS =

√√√√ 1

Ξ

Ξ∑j=1

((βj − βj))2 . (4.4)

Experiments suggest that the bias is generally very small compared with the RMS (Fig-ure 4.2). Therefore report only the RMS statistics are reported below.

For comparison purposes in the figures two additional statistics are also reported. Thefirst one (referred to as “agnostic” below) describes the growth of the TP error when nofiltering is carried out. This is generally the case in current practice. In this case the aircraftfly in an uncertain wind environment (except for the forecast provided to the ATC) and noattempt is made to estimate the wind forecast errors from the radar measurements. Theagnostic statistic constitutes a worst case bound on the algorithm performance: at the veryleast the algorithm should improve on the current situation. The second statistic (referredto as “all-known” below) describes the growth of the TP error given perfect informationabout the wind-field all over the airspace at the current point in time. The TP uncertaintyin this case is only due to the time evolution of the wind-error, which is unpredictable giventhe current state. This is clearly an unrealistic, perfect filtering situation and constitutesan optimal, best case performance bound for the algorithm.

4.3.2. Trajectory Prediction Results

The first scenario that is examined, involves a single aircraft that is cruising at constantaltitude, with constant speed. In this case the algorithm is not fully exercised, since thereis no sequential incorporation of estimates between different aircraft. The results of thesimulations for the 1-aircraft scenario are presented in Figure 4.4. The agnostic and all-known performance bounds are also shown in the figure (approximately 5000 m and 1400 m

46

4.3. Simulation Results

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200

50010001500200025003000350040004500500055006000

time ahead (min)

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Worst Performance BoundBest Performance BoundBest performanceOther FlightsWorst Performance

(a) Flight Scenario with 3 aircraft

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200

50010001500200025003000350040004500500055006000

time ahead (min)

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(m

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Worst Performance BoundBest Performance BoundWorst PerformanceOther FlightsBest Performance

(b) Flight Scenario with 6 aircraft

Figure 4.5: Evolution of the RMS along-track error, for different flight scenarios, after performingfiltering for 10min of radar measurements. Blue represents the flight with the best performanceof SCPF and red represents the worst performance, while green any other flight.

respectively for a 20 min TP). The figure also includes a third statistic (referred to a “local-known” below) which also represents an ideal filter that perfectly estimates the position ofthe aircraft and uses ideal radar measurements (with no error) to estimate the wind. Thisis the best one can hope for with a single aircraft, since measurements are only availablein the aircraft location an nowhere else in the airspace. SCPF is closely tracking theideal local-known filter with an RMS approximately at 3780 m after 20 min of prediction.The deterioration in performance is only marginal. This clearly suggests that the filteringalgorithm has successfully managed to extract all the available information from the data.It is also interesting to note that the growth of the along-track error is almost linear intime, as expected, due to the accumulation of highly correlated wind errors.

The algorithm is exercised fully in the 3-aircraft scenario, as can be observed in Fig-ure 4.5(a). It is interesting to note, that for some aircraft performance is better than theideal local-known filter. This is expected since in this case measurements for the positionof 3 rather than 1 aircraft are acquired. Those measurements, even if they are inaccurate,provide an information advantage over the single aircraft case. Observing the flight plansof the 3 aircraft case (Figure 4.3) it is easy to note the advantage for the 2 trailing air-craft. The leading aircraft has no significant information for the wind-error it is going toencounter except through the spatiotemporal correlation of the wind in its own position.

47

4. A novel Particle Filter for Trajectory Prediction

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200

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time ahead (min)

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Worst Performance BoundBest Performance BoundWorst PerformanceOther FlightsBest Performance

(a) Flight Scenario with 12 aircraft

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200

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time ahead (min)

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Worst Performance BoundBest Performance BoundBest PerformanceOther flightsWorst Performance

(b) Flight Scenario with 24 aircraft

Figure 4.6: Evolution of the RMS along-track error, for different flight scenarios, after performingfiltering for 10min of radar measurements. Blue represents the flight with the best performanceof SCPF and red represents the worst performance, while green any other flight.

This is the reason that the along-track error for this aircraft is similar to that of the 1-aircraft scenario. The two trailing aircraft on the other hand take advantage of the radarmeasurements of the leading aircraft. The effect is stronger for the last aircraft which isfavored by measurements from both other aircraft.

The flight plans of the 6 aircraft scenario present even more favorable conditions for someof the aircraft involved. In this scenario another formation, consisting of 3 aircraft, moveswith almost an opposite direction to the first 3. Results are clearly better for the worstand mean case, since even the leading aircraft have now some informational advantagefrom each other. Along-track error growth for the 6 aircraft scenario is presented inFigure 4.5(b).

When more aircraft enter the airspace performance keeps improving, as can been seenin Figures 4.6(a) and 4.6(b). Both the maximum and minimum along-track error show asubstantial reduction. There even exist some flights that come close to the ideal all-knownperformance bound. Those aircraft have favorable flight plans that allow them to takeadvantage from the measurements of most of the other flights. Figures 4.7, 4.8, collectthe results from all different flight plan scenarios. The performance of the algorithm isexamined after the incorporation of 5 and 10 minutes of radar measurements. The figure

48

4.3. Simulation Results

0 1 3 6 12 240

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Worst Performance Bound (TP 20min)Best Performance Bound (TP 20min)Worst PerformanceAverage PerformanceBest Performance

Figure 4.7: Along-track error after 20 minutes of prediction for different numbers of aircraft inthe region of interest, after the incorporation of 5 minutes of radar measurements.

0 1 3 6 12 240

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Figure 4.8: Along-track error after 20 minutes of prediction for different number of aircraft in thesector, after the incorporation of 10 minutes of radar measurements.

depicts the RMS in the predicted along-track position of the aircraft 20 minutes into thefuture. The slight increase of the maximum along-track error in the 24 aircraft scenario(compared with the 12 aircraft scenario) is due to the unfavorable conditions for one of thenew flights. Even flights with unfavorable flight plans, however, have a significant reductionin uncertainty compared with the no filtering and 1 aircraft case. It is important to notethat after the 12 aircraft case there exists a saturation concerning the performance of thealgorithm when it comes to the best performing aircraft, even when doubling the numberof flights. This is not a surprise since the best performance is already close to the idealall-known case.

It is clear that there exists a strong improving trend when multiple aircraft are involved.For the 12 and 24 aircraft scenario, the mean TP along-track error for 20 minutes of predic-tion (after 5 minutes of filtering, Figure 4.7), drops to just 2629 m and 2506 m, compared

49

4. A novel Particle Filter for Trajectory Prediction

Figure 4.9: Diagonal elements of the Covariance Matrix of a particle after 20 radar measurementsfor 24 aircraft. Elements are positioned on their respective place in the horizontal plane. Thetransparent red surface displays the initial uncertainty, before any measurements are incorporated.

with 3784 m for the single aircraft case. When SCPF receives radar measurements foranother 5 min, Figure 4.8, the mean TP error drops to less than 2200 m. Considering,the minimum along-track error, results are as low as 1799 m for the 24 aircraft case and1792 m for the 12 aircraft case. This is a substantial reduction, since the ideal all-knownRMS bound for 20 minutes of prediction is approximately 1400 m. If multiple-aircraftfusion was not employed a significant number of such flights would experience as much asdouble the along-track TP uncertainty achieved here.

Figure 4.9 shows an estimate of a part of the covariance matrix of the wind-forecasterror distribution for the 24 aircraft scenario after the incorporation of 10 minutes ofmeasurements. It is apparent that where most aircraft fly uncertainty is very low. However,even in parts of the airspace where there are no flights, reduction in wind uncertainty canbe significant.

An important feature of the multiple-aircraft fusion is that even the worst cases performat least marginally (in some scenarios significantly) better than the single aircraft case. Ifsome flight, in a multi-aircraft scenario with SCPF, had an inferior performance comparedwith the single aircraft case (or, even worse, the no-filtering case) this would indicate asignificant drawback of the algorithm. It would actually suggest that the TP performanceof some flights deteriorates in accuracy in order for some others to gain an informationadvantage, an unfavorable trade-off. Happily, this does not appear to be the case with theproposed algorithm.

4.3.3. Algorithm Numerical Performance Analysis

In addition to the improvement in TP accuracy discussed above, the two novelties of theSCPF algorithm also provide a substantial improvement in numerical robustness. For allscenarios simulated and for all repetitions there was never a single case of particle collapsedue to insufficient number of particles and extreme degeneracy.

To quantify this observation a common indicator of the performance of a PF algorithm

50

4.3. Simulation Results

1 2 3 4 5 6 7 8 9 10 110

5

10

15

20

25

30

35

Time (min)

Mea

n E

ffect

ive

Sam

ple

Siz

e −

nef

f (%

)

Figure 4.10: Evolution of the mean Neff as new measurements arrive.

known as the effective sample size Neff is computed. Neff is a measure of the degeneracyof the algorithm and is usually computed by

Neff =1∑N

i=1(qi)2

(4.5)

(recall that qi is the normalized weight of particle i). Neff will be equal to the numberof particles, N , when all particles share the same weight and equal to 1, when all theparticles have zero weight except a single one. The former is the ideal case, the latterindicates extreme degeneracy.

In the case of SCPF for each measurement the normalized weights are computed repeat-edly, as the radar measurements of the different aircraft are sequentially conditioned. Theaverage of the Neff for the different aircraft for the 24 aircraft case scenario is displayedin Figure 4.10, as a percentage of the 1000 particles used in the simulation. Typicallyin PF algorithms Neff decays as time evolves after transients. SCPF, however, managesto keep Neff at a roughly constant level. This is an important advantage of the algorithmthat contributes to its robustness and reliability.

The computational resources devoted to the execution of the algorithm were 1000 par-ticles. The computational effort grows linearly with the number of aircraft involved ineach scenario (the number of particles remains the same but the dimension of the statebecomes higher). Even though generating the data reported in the figures required severaldays of computation time, one needs to keep in mind that each experiment was repeated1000 times to collect statistics. A single run of the algorithm for a scenario including 12different aircraft flying for 10 minutes (and filtering performed every 30 s) required onaverage 40 minutes of computational time. The same flights, projected 20 minutes intothe future require 30 minutes of computational time. It should be noted that the algo-rithm was implemented in Matlab and not a high performance language while it was alsonot optimized for speed (for example accelerating large matrix operations). Moreover, the

51

4. A novel Particle Filter for Trajectory Prediction

algorithm is easy to parallelize, since most of the computation for the evolution of theparticles can be done separately for each particle; only the computation of the normalizedweights and resampling requires any communication among particles.

4.3.4. Effect of Number of Particles

The number of particles can have a significant effect on the accuracy of the SCPF. Ana-lytical results for the effect of the number of particles in the performance of particle filters(for bounded functions) can be found in [25]. Here, this effect is investigated through theuse of extensive simulations. We simulate the 24-aircraft flight plan for 128 different wind-forecast error realizations. The SCPF was used to filter each of the realizations employinga different number of particles. The following discretization was used for the number ofparticles

N = {8, 12, 19, 29, 45, 69, 101, 159, 245, 375, 600, 900, 1400, 2100, 3250}Figure 4.11 and Figure 4.12 show the effect of the number of particles on the bias and

RMS error in the filtering of the xI aircraft position of all 24 aircraft (similar results areobtained for filtering the yI aircraft position), after 10 min of radar measurements. It isclear that an increase in the number of particles has a positive effect on the performanceof SCPF. This effect starts to saturate between 245 and 375 particles both for the bias andRMS of the error. This is expected since SCPF cannot outperform the optimal bayesianfiltering. Optimal filtering (under any criterion) does not imply that all uncertainty iseliminated but rather that all available information is fully exploited.

For 3250 particles the RMS error ranges from 46 m to 72 m (for different aircraft), whilefor 2100 particles RMS error ranges from 45 m to 71 m. Similarly, for the bias the range is−10 m to 14 m for 3250 particles and increases to just −10 m to 15 m for 2100 m. Becauseof this saturation every small gain in accuracy beyond these values imposes a substantialcomputational cost. On the other hand, it is apparent that very few particles (less than afew hundred) can greatly compromise the performance of the algorithm. For 45 particlesthe RMS error ranges from 75 m to 532 m, while for 159 particles it drops to a rangefrom 57 to 102 m. A similar behavior is observed for the bias in Figure 4.11. From theresults presented here it is apparent that the 1000 particles used for the artificial scenariosshould be adequate whereas the 50 particles used for the Maastricht scenarios (presentedin Section 4.3.5), are probably too few. This implies that the capabilities of SCPF werefully exploited in the first case while for the Maastricht case there is substantial place forimprovement without the need for a dramatic increase in computational effort.

52

4.3. Simulation Results

8 12 19 29 45 69 101 159 245 375 600 900 1400 2100 32500

200

400

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Err

or B

ias

Number of particles

Figure 4.11: Bias of the xI error after 10 minutes of filtering. Different colors represent differentaircraft. The figure presents the results using a logarithmic x-axis.

8 12 19 29 45 69 101 159 245 375 600 900 1400 2100 32500

200

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S e

rror

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Figure 4.12: RMS of the xI error after 10 minutes of filtering. Different colors represent differentaircraft. The figure presents the results using a logarithmic x-axis.

4.3.5. Maastricht airspace

After examining the performance of the algorithm with artificial flight plans, an initialassessment for a more realistic set-up was carried out. As a region of interest a 3D boxcentered in Brussels National (IATA: BRU) airport was used, with dimensions 600×600 kmin the horizontal plane and 33000 to 36000 feet in altitude. Real flight plans from CentralFlow Management Unit (CFMU) were used, more specifically all the flights that cross thissector throughout a single day and fly level for this part of their course. Figure 4.13 depictsthe flight plans from a top-down and side view. The total number of flights consideredwas 441, but aircraft were constantly entering and exiting the region of interest. The peakvalue of aircraft inside the 3D box was 330 and the algorithm was operating with at least150 aircraft for half of the simulation time. At the start and the end of the day, however,a single aircraft was left inside the sector.

This scenario put a considerable stress on SCPF since the dimension of the state in-creased dramatically. The non-linear part of the state (aircraft dynamics), topped at 1320

53

4. A novel Particle Filter for Trajectory Prediction

−300000 −240000 −180000 −120000 −60000 0 60000 120000 180000 240000 300000−300000

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Figure 4.13: Flight Plans centered at Brussels.

0

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Figure 4.14: Number of aircraft in “Brussels 3D sector”.

states, but was varying depending on the number of aircraft in the sector. The linear part(wind error dynamics), included the calculation of the wind error vector in the X and Yaxis, in 11 × 11 × 6 vertexes, a total of 1452 states. The flight plans included turns andwere positioned at different flight levels. The number of aircraft in the sector can be seenin Figure 4.14. Due to computational limitations just 50 particles were used throughoutthe simulation.

The performance of SCPF for TP can be seen in Figure 4.15 and bears many similaritieswith the 12 and 24 aircraft scenarios above. The average TP error over all aircraft is similar,but the best and worst aircraft performance are more extreme, since there exist so manydifferent flights and possibly complex situations. The flight with the worst performance isstill below the threshold of the agnostic and single aircraft case. The flight with the bestperformance is considerably closer to the ideal all-known case.

54

4.3. Simulation Results

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 200

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time ahead (min)

Alo

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Figure 4.15: Evolution of the RMS along-track error for flights in “Brussels 3D sector”.

Figure 4.16: Diagonal elements of the Covariance Matrix of a particle after 40 radar measure-ments for Brussels airspace. Elements are positioned on their respective place in the horizontalplane. The transparent surface displays the initial uncertainty, before any measurements areincorporated.

Figure 4.16 is the equivalent of Figure 4.9 for this case and shows an estimate of a partof the covariance matrix of the wind-forecast error distribution for one of the flight levelsof the Maastricht scenario. Like for the 24 aircraft case, in the region of the airspacewhere most aircraft fly uncertainty is very low. In this case specifically, where there existsignificantly more aircraft, wind uncertainty is dramatically reduced. This estimate is onlyafter 20 minutes of flight, so most of the aircraft have not yet entered the sector, or theaircraft that have, still have not traversed fully their flight paths. The incorporation ofdata from more flights is expected to reduce the uncertainty even more.

Despite the complexity of the simulation and the computational limitations, this initialassessment suggests that SCPF has a viable potential for both realistic and demandingsituations. Including flights at multiple flight levels case offers some advantage, since infor-mation about the wind can be passed from aircraft flying at different altitudes. However,

55

4. A novel Particle Filter for Trajectory Prediction

is has to be noted that wind forecast error correlation in the vertical direction decaysrather quickly. Moreover, the flight plans included multiple turns, which implicitly pro-vided advantageous information to the algorithm since the dynamic behavior of the aircraftexhibits a strong dependence on wind while turning [27]. Finally, despite the low numberof particles, the algorithm never collapsed due to degeneracy and the high-dimensionalityof the problem.

4.4. Summary

A novel Sequential Monte Carlo algorithm was developed, the Sequential Conditional Par-ticle Filter (SCPF) to improve Trajectory Prediction (TP) performance in a multi-aircraftenvironment from the perspective of a ground based Air Traffic Management (ATM) sys-tem. Simulation results suggest that this methodology is viable in realistic situations, sinceit is robust and can offer a substantial improvement in the accuracy of TP. The numberof aircraft in the simulation affect positively the performance of SCPF. Depending on theflight plan situation there exist aircraft cases where accuracy reaches the upper perfor-mance bound. Moreover, the algorithm is able to handle a very large number of aircraft,reaching hundreds, despite the dramatic increase of the state dimensions. The effectivesample size (Neff) of the particles remains always at a substantially high level. Finally,simulations show that the number of particles is crucial for the filtering accuracy, howeverthis effect appears to fade for a high enough number of particles.

56

Chapter 5Comparison of Particle filters for TrajectoryPrediction

In Chapter 4 we showed how a novel particle filter can provide improved trajectoryprediction accuracy by combining information from multiple aircraft at different locationsand time instants. In this chapter we demonstrate using case studies that more standardsequential Monte Carlo algorithms cannot provide the same improvement. In all studieswe assume that aircraft fly level (possibly at different altitudes) with known, constant,aircraft dependent airspeeds and estimate the wind forecast errors based only on groundradar measurements.

57

5. Comparison of Particle filters for Trajectory Prediction

5.1. Simulation Setup

In order to compare the different algorithms and evaluate the effectiveness of the proposedone we perform several simulation based feasibility studies. The setup includes multipleaircraft flying level, possibly at different altitudes, with known airspeeds and aerodynamicequations (both depending on the aircraft type). This represents an ideal situation inan air traffic control context, since both airspeeds and aerodynamic coefficients are notalways precisely known. Our goal is to investigate whether the proposed approach isviable, in particular whether aircraft trajectories contain enough information to estimatewind forecast errors. Moreover, we wish to determine how the proposed filtering algorithmsdeal computationally with the problem. The results indicate that generic particle filteringalgorithms are not able to cope with the high dimensionality of the problem. On the otherhand, SCPF manages to reduce TP uncertainty and keep the computational burden ata low level, even for a significantly large number of aircraft. Moreover, the presence ofsufficient aircraft, provides a TP estimate close to the theoretical limit for some of theflights involved.

We have designed a series of simulations to compare the performance of the differentalgorithms. We created a sequence of artificial flight plan scenarios, with a different numberof aircraft (1, 3, 6, 12, 24) in a sector of interest. The flight plans for each scenario aredisplayed in Figure 4.3. More details about the setup can be found in Section 4.3.1.

The results presented below are for TP based on filtering after 5 and 10 minutes ofradar measurements (10 and 20 measurements in total). All simulations employ 1000particles, independent of the number of aircraft. Having used the PF algorithms to filter5 or 10 minutes of data we extrapolate the state estimate of all particles for a further20 minutes into the future to get an estimate of the future trajectory for each aircraft. Theextrapolation is done by simulating all the particles for the required time. We consider asa TP estimate (β) for each flight the mean of these particles. The real trajectory (β) isthen compared with this estimate to compute the along track error metric, as discussed inSection 4.3.1.

To collect sufficient statistics, we simulated each scenario Ω = 256 times, keeping thesame flight plans, but different wind-fields, producing different trajectory estimates (βj)and real trajectories (βj), for j = 1, . . . ,Ω. The estimator bias and RMS are calculated asin Eq. (4.3) and Eq. (4.4). Our experiments suggest that the bias for all PF algorithms isgenerally very small compared with the RMS (Figure 4.2, for SCPF). We therefore reportonly the RMS statistics below. To make the comparison relevant we introduce the agnosticand unknown indicators as in Chapter 4.

5.2. Trajectory Prediction Comparison

A single aircraft is cruising at constant altitude, with constant speed, across the airspace forthe first scenario we consider. This represents the simplest case in terms of dimensionalityof the state space. Simulation results for this 1-aircraft scenario are presented in Figure 5.1

58

5.2. Trajectory Prediction Comparison

SIS SIR MPF APF SCPF0

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)Figure 5.1: Comparison of 20 minutes TP, for different PF algorithms, after 5 (left) and 10 (right)minutes of measurements. Single aircraft scenario.

using a box-and-whisker diagram1. The SIS algorithm does not improve the TP uncertaintycompared with the agnostic case, either for 5 or 10 minutes of filtering. The TP uncertaintyfor 20 min ahead remains close to 5000 m. The SIR filter, delivers a marginal improvement,for both cases. Both filters have difficulty handling the state-dimension, which despite thefew nonlinear states, still includes a large number of linear states concerning the windevolution.

In the applications considered here, the wind forecast error is handled as a parameter inthe case of APF, since it has slow varying dynamics, due to the strong temporal correlationof forecast errors (in contrast altitude correlation is weak and correlation with horizontaldistance is moderate) as is explained in Section 2.3.2.

The first significant accuracy improvement comes from MPF. The marginalized filter,can handle the high dimensionality of the problem, since it treats the linear state separatelyand in an optimal way. The nonlinear state is also manageable, since for this scenario itinvolves just 4 states (one aircraft). APF offers an additional improvement over MPF forthe 5 minutes filtering case. This advantage is not sustained for the 10 minutes case. Thischange in behavior will be more thoroughly explained in the analysis of the algorithmsin Section 5.4. Finally, SCPF clearly demonstrates the largest uncertainty reduction inboth cases. Note, that in this scenario the SCPF algorithm is not fully exercised. A singleaircraft does not allow for sequential incorporation of wind estimates. The nonlinearstate is treated as a whole and not in separate parts. Moreover, in MPF formulation

1A box-and-whisker diagram shows the median of a sample population (red line in our case) and the first(x.25) and third quartile (x.75) (the upper and lower bounds of the blue box). Any data observationwhich lies more than 1.5 · IQR lower than the first quartile or 1.5 · IQR higher than the third quartile(upper and lower black bounds, the whiskers) is considered an outlier (red star), where IQR is theinterquartile range (x.75 - x.25)

59

5. Comparison of Particle filters for Trajectory Prediction

SIS SIR MPF APF SCPF0

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)

Figure 5.2: Comparison of 20 minutes TP, for different PF algorithms, after 5 (left) and 10 (right)minutes of measurements. Three aircraft scenario.

linear dynamics must affect nonlinear dynamics and vice versa in an additive way, as itis described in Eq. (3.12). SCPF does not impose any such restriction and it is simplysufficient that a linear substructure exists in the dynamics. In the formulation of theTP problem presented here, due to the effect of the FMS the linear wind-forecast errordynamics have a non additive effect on the nonlinear state of the system. If we neglect thisand simply assume that linear dynamics have approximately an additive effect MPF willquickly diverge. To circumvent this problem and include MPF in the comparison studywe construct a matrix An(xn(k)) as in Eq. (3.12) that simulates the effect of the FMS ateach time step. This approximation improves results but should be taken into account fora fair comparison of MPF with the rest of the filters.

SCPF is exercised fully for the first time, in the 3-aircraft scenario. The simulation re-sults are presented in Figure 5.2. In this scenario we acquire measurements for the positionof 3 rather than 1 aircraft. Those measurements, despite their inaccuracies, provide aninformation advantage over the single aircraft case. By observing the flight plans of the 3aircraft case (Figure 4.3) we can see the advantage of the 2 trailing aircraft.

SIS, like before, cannot improve on the TP uncertainty, more than the agnostic case.For the first time, applying SIR and MPF increases TP uncertainty. This implies, thatthe filters diverge into a false solution. This deterioration is marginal for the MPF. SIRimproves the accuracy for one of the aircraft (lower box bound), does not affect the secondaircraft (median) and severely damages the TP performance of the third flight. Similarresults hold for both 5 and 10 minutes filtering cases. APF manages to increase the TPperformance for all aircraft. Moreover, the trailing flight gains a significant increase inaccuracy. Again, like in the single aircraft scenario the performance of APF deterioratesfor the 10 minutes case. SCPF maintains a superior performance, compared to all otheralgorithms. The only case, where an algorithm performs better, is for the trailing flight

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5.2. Trajectory Prediction Comparison

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Figure 5.4: Comparison of 20 minutes TP, for different PF algorithms, after 5 (left) and 10 (right)minutes of measurements. Twelve aircraft scenario.

performance, in the 5 min case, where APF is marginally better.Along-track error growth for the 6 aircraft scenario is presented in Figure 5.3. In this

scenario, we drop SIS from the comparisons, since it has not managed to perform satis-factorily in the simpler and less demanding configurations. SIR has again no effect in theTP accuracy, except marginally for the 10 minutes case. The performance of MPF nowdeteriorates significantly when compared to the agnostic case. This is a clear indicator thatthe filter has already diverged. The best performance for the 5 minutes case comes fromthe APF, while in the 10 minutes case, SCPF prevails. Both APF and SCPF demonstrate

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5. Comparison of Particle filters for Trajectory Prediction

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Figure 5.5: Comparison of 20 minutes TP, for different PF algorithms, after 5 (left) and 10 (right)minutes of measurements. 24 aircraft scenario.

a significant increase in performance compared with the 1 and the 3 aircraft configurations.When more aircraft enter the airspace, improvements in TP become even more substan-

tial as can been seen in Figure 5.4, for the 12-aircraft scenario. Both the maximum andminimum aircraft TP uncertainty show a reduction. There even exist flights that comeclose to the ideal all-known performance bound of approximately 1500 meters. Those air-craft have favorable flight plans that allow them to take advantage from the measurementsof the rest of the flights. MPF is not run extensively for this scenario since its performancedeteriorates even more than the 6 aircraft scenarios. SIR, like previously, exhibits the sameaccuracy as the agnostic case. APF is performing strongly in both the 5 and 10 minutescase. SCPF has a superior performance for the 10 minutes case. It is important to note,that for the APF, some of the simulations were numerically unstable (approximately 6 %of them). This means that the error was so high, and the likelihood so small, that it wasbelow the accuracy provided by the simulation software (10−3322). On the other hand,SCPF never experienced such a problem.

To further assess the performance of the two dominant algorithms, we created a lastscenario with 24 aircraft. For this configuration, APF failed numerically in 47 % of thesimulations, while SCPF carried them out without numerical problems. In Figure 5.5, thetwo algorithms seem to have a similar performance, APF even shows marginally betterresults. This is quite deceptive however, since almost half the repetitions were not includedin the calculation of the RMS error for APF. Taking into account that these numericallyfailed simulations had at least 3000 meters filtering error at the time of the numericalcollapse, for all their particles, (without adding the 20 min TP error), the reported resultssignificantly overstate the accuracy of the APF.

2This is the likelihood, for a radar error of approximately 3000 meters, with RMS 80m.

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5.3. Filtering Comparison

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5.3. Filtering Comparison

To demonstrate better the behavior of the different algorithms, under the various simu-lation settings, we compare their performance in estimating the state of the dynamicalsystem. We restrict our attention in the filtering of the aircraft positions, since this is themost compact statistic to present. Radar measurements have an RMS error of 80 m, andwith filtering an improvement over this is expected. The estimation bias is very small forthe PF we examined (usually a few meters), and we omit it. For this comparison we showresults both for the along-track and cross-track RMS error. Note, that cross-track error isgenerally smaller than the along-track due to the actions of the 3D-FMS.

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5. Comparison of Particle filters for Trajectory Prediction

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Figure 5.9: Evolution of the mean along-track (left) and cross-track (right) RMS filtering errorfor the twelve aircraft scenario.

The results for the single aircraft configuration are displayed in Figure 5.6. All thealgorithms, with the exception of SIS, improve on the accuracy of the radar. APF, SIRand SCPF have the best performance. Figure 5.7 shows the results for the three aircraftscenario. SCPF is the only algorithm that manages to improve the estimate of the aircraftstate beyond the measurement error, it reaches as low as 60 m for along-track and 50 mfor cross-track RMS. The rest of the filters diverge significantly as time increases. Anexplanation for the deterioration in the TP performance of APF for the 10 minutes case(see Figure 5.2) comes from these results. The filtering performance of APF starts toseverely deteriorate after 5 minutes of measurements, thus affecting the TP performanceas well.

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5.3. Filtering Comparison

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Figure 5.10: Effective sample size for 1, 3, 6, 12 aircraft, for all different algorithms.

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5. Comparison of Particle filters for Trajectory Prediction

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Figures 5.8 and 5.9, show the filtering results for the 6 and 12 aircraft case. For theseconfigurations the performance of all the algorithms, except for SCPF, is very poor. Thealong-track RMS error surpasses 500 meters for some cases, 6 times worse than the radarRMS error. On the other hand, SCPF consistently succeeds in filtering the aircraft state,providing superior performance than the radar. These results are indicative of the limita-tions of the different filters, since a good estimation of the current state greatly affects theoutcome of TP.

5.4. Algorithm Numerical Analysis

In addition to the improvement in TP and filtering accuracy discussed above, SCPF pro-vides a substantial improvement in numerical robustness. Indeed, this is most likely thereason for the improved TP performance. For SCPF, in all scenarios simulated and for allrepetitions there was never a numerical collapse due to degeneracy. In contrast, almost allof the other PF were numerically unstable in several simulations.

The proportion of failed simulations increased significantly with the number of flightsinvolved. For more than 12 aircraft, most of the algorithms were unable to finish thesimulation without a substantial number of failures and this is the reason their performancewas not reported. The exception was APF, which had the lowest failure rate. Still thefailure rate for APF was close to 50 % for the 24 aircraft case. To quantify this observationand compare the different algorithms we compute a common indicator of the performanceof PF known as the effective sample size Neff, see Section 4.3.3 for a definition.

Figure 5.10 compiles the results for all different flight scenarios and different algorithms.For SIS, Neff exhibits a sharp drop and quickly becomes degenerate. SIR has a more stableperformance, which diminishes with the number of aircraft involved in the setup. For MPFNeff drops very fast except for the single aircraft scenario, where it stops at 30%. The

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5.5. Summary

structure of the APF allows for a very high Neff. However, the effect of the number ofaircraft is crucial, as well that of time, for the multi-aircraft cases. Finally, SCPF exhibitsthe most robust performance, since Neff is always between 30 % and 40 % independent oftime and unaffected by the number of aircraft participating in the simulation. For SCPF,Neff is computed sequentially for all aircraft in the airspace. The Neff computed last (infiltering sequence) is displayed in Figure 5.10, as a percentage of the 1000 particles usedin the simulation. Figure 5.11 demonstrates the different Neff for various flights in the 6and 12 aircraft configurations.

5.5. Summary

A comparison study between different Sequential Monte Carlo methods for multi-aircrafttrajectory prediction was performed. Various algorithms were tested on different flightplan cases. The results indicate that standard particle filtering algorithms cannot handlesimultaneously a large number of aircraft. Simulations suggest that the most commonPF, such as SIS and SIR exhibit a very poor performance even for single aircraft cases.Specialized algorithms such as the MPF can handle the high dimensionality of the linearstates but not the nonlinear dynamics of multiple aircraft. The APF algorithm exhibits astronger performance for more aircraft but diverges when the number of aircraft becomesvery high. Simulations show that SCPF outperforms standard PF algorithms both for TPand filtering of the system state and can handle efficiently a very large number of aircraft.

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5. Comparison of Particle filters for Trajectory Prediction

68

Chapter 6Conflict Detection using SequentialConditional Particle Filter

In this chapter we investigate the effect of improved trajectory prediction accuracy onthe conflict detection (CD) performance. We apply SCPF to filter the wind forecast errorand the aircraft states using radar measurements from multiple aircraft. We then use thetrajectories predicted by the SCPF to estimate the conflict probability between any twoaircraft. Our results show that this can improve considerably conflict detection rates inmid and short term horizon encounters.

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6. Conflict Detection using Sequential Conditional Particle Filter

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Figure 6.1: Flight plans for two (left) and 6 aircraft (right).

6.1. Conflict Detection Algorithm

The conflict detection method described here is based on the use of the SCPF algorithm.Initially we filter the incoming radar measurements, concerning the multiple aircraft posi-tions, using SCPF. This way we get a number of particles that represent a current estimateof the state of the aircraft and the state of the wind-field uncertainty. We then project theparticles concerning wind uncertainty into the future. Moreover, a pair of aircraft for whichwe want to calculate the probability of conflict is selected. We can use now the aircraftdynamics to evolve the particles, concerning these two aircraft, into the future. For eachparticle the distance of the two aircraft is calculated for every second of their predictedtrajectories. The minimum separation between them is evaluated throughout the predictedtrajectories. Note here that for each of those aircraft we are using the same wind-field, inorder for their comparison to be relevant. Each particle will exhibit a different minimumseparation. So, there will be particles for which a conflict has occurred at some point intime and particles with no conflict for the same wind forecast error realizations. We usethe ratio of particles with conflict over the total number of particles as our estimate forthe probability of conflict.

6.2. Simulation Setup

We have devised a series of simulations to demonstrate the performance of the new algo-rithm in improving CD. The first scenario includes two aircraft approaching each otherwith a 45o angle of incidence. Nominally, without any wind forecast error, the two aircraftwill exhibit a minimum separation of 5 nmi, 25 minutes after the start of the simulation.In order to demonstrate how the algorithm can also exploit information from additionalaircraft that happen to be present in the airspace we have created an other scenario with6 aircraft in total. The 4 additional aircraft share similar flight plans with the first two,

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6.3. Simulation Results

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Figure 6.2: Time evolution of the separation between the two aircraft for different scenarios ofwind forecast errors. Blue lines represent conflicts and green lines represent scenarios where asafe separation distance was kept throughout the flight.

but precede the two flights of interest (where CD performance is evaluated) 50 nmi and100 nmi. The flight plans for the two cases can be seen in Figure 6.1.

Aircraft fly level at 10000 m altitude with a nominal airspeed of 419 knots. No turnsare included in the flight plans. The parameters of the dynamical models for all aircraftrepresent a Boeing 737-700. Flights have a duration of approximately 30 minutes and radarmeasurements arrive every 30 seconds. We simulate these flights under 1000 different windforecast error realizations. These represent different “real” scenario outcomes. The filteringalgorithm is evaluated on how accurately it can detect conflicts situations in each of thesereal outcomes. Figure 6.2 demonstrates the significance of the forecast error in aircraftseparation. For the different wind forecast errors, the minimum aircraft separation occurson average 25 minutes (1500 s) after the start of the two flights, as in the nominal case.However, the range of minimum separation time varies from 1440 s to 1570 s. Moreover,the minimum separation (nominally 5 nmi) ranges from 0.03 nmi to 14 nmi. In total, outof the 1000 real scenarios, 509 result in conflict (minimum separation less than 5 nmi).

To evaluate the efficiency of the algorithm we compare it against two benchmarks, theagnostic and all-known indicators as introduced in Chapter 4. The algorithm is run using1000 particles for the 2 and 6 aircraft scenarios for each of the 1000 “real” wind forecasterrors. We use SCPF to filter 5, 10 and 15 minutes of data. This represents equivalently20, 15 and 10 minutes before the nominal time of minimum separation.

6.3. Simulation Results

The results for the 509 realizations where a conflict actually happened are displayed inFigure 6.3. The results are displayed as distributions, using histograms. For example,

71

6. Conflict Detection using Sequential Conditional Particle Filter

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Figure 6.3: Evolution of the conflict probability after 5 (left column), 10 (middle column), 15(right column) minutes of filtering, for scenarios where a conflict occurred. Top row presents theunknown indicator where only meteorological forecasts were available. Bottom row presents theresults for SCPF with 2 aircraft. Red dots show the average minimum separation distance foreach percentage bin and black triangles the minimum among them. The left axis signifies thenumber of conflicts identified at each probability bin, while the right axis displays the minimumseparation between the aircraft in each bin.

all flights that were identified as conflicts with probability from 40 to 45% are placed inthe respective 40 − 45% bin. This way a distribution is constructed. Optimally we wouldprefer a CD scheme that places all the flights with conflict in the 95−100% bin, since thiswould imply that all conflicts were identified properly. Generally, a distribution skewedto the right implies that most of the conflicts (out of the 509) where identified with highprobability. The average minimum separation and the smallest minimum separation of allrealizations placed in each bin is also reported. These metrics can help us evaluate if theprobability of conflict, evaluated by SCPF, relates to the aircraft distance.

The agnostic predictions, Figure 6.3(a), for 20 and 15 minutes before nominal conflict

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6.3. Simulation Results

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Figure 6.4: Evolution of the conflict probability after 5 (left column), 10 (middle column), 15(right column) minutes of filtering, for scenarios where a conflict occurred. Top row presents theresults for SCPF with 6 aircraft. Bottom row presents the all-known results. Red dots show theaverage minimum separation distance for each percentage bin and black triangles the minimumamong them. The left axis signifies the number of conflicts identified at each probability bin,while the right axis displays the minimum separation between the aircraft in each bin.

(or otherwise 5 and 10 minutes of filtering), provide a rather flat distribution. This impliesthat most of the conflicts are not identified with high probability and that wind fore-casts alone, without filtering, are not particularly good predictors for this horizon. Theimprovement over the agnostic predictions is quite strong when SCPF is used for 2 air-craft (Figure 6.3(b)) and becomes even greater when we employ 6 aircraft (Figure 6.4(a)).SCPF with 6 aircraft, had better CD rates 20 minutes before conflict, than the agnosticcase, 10 minutes before conflict. However, 10 minutes before conflict (15 minutes of filter-ing) adding more aircraft does not increase significantly the CD rate. Note that 6 aircraftSCPF is not far from the upper bound of performance indicated by the perfect informationbenchmark, Figure 6.4(b).

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6. Conflict Detection using Sequential Conditional Particle Filter

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Figure 6.5: Evolution of the conflict probability after 5 (left column), 10 (middle column), 15(right column) minutes of filtering, for scenarios where a conflict did not occur. Top row presentsthe unknown indicator where only meteorological forecasts were available. Bottom row presentsthe results for SCPF with 2 aircraft. Black dots show the average minimum separation distance foreach percentage bin. The left axis signifies the number of no-conflicts identified at each probabilitybin, while the right axis displays the minimum separation between the aircraft in each bin.

Even 10 minutes before conflict there still exist conflicts that are not predicted well.SCPF, for some realizations, where a conflict actually occurred, gives a probability ofconflict as low as 10 − 20%. This is due to the conflict being on the bounds of whatis considered safe separation, between 4.5 and 5nmi, Figure 6.3 and 6.4. Accepting asconflicts flights that breach 5.5 or 6nmi, would increase margins, with an increase of falsealarms of course, but in many cases, this can be an acceptable trade-off.

An important metric that should be taken into consideration, is the false alarms rate.An overcautious, algorithm, that accepts all realizations as conflicts, would provide perfectresults in the conflict detection rate metric, since all realizations with conflicts would beperfectly identified. On the other hand, such a method would produce a great number of

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6.3. Simulation Results

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14Identified as Conflicts, after 15 minutes

Min

imum

Sep

arat

ion

(nm

i)

Probability of Conflict

(a) SCPF with 6 aircraft

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95100Probability of Conflict

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50

100

150

200

250

300

350

400

450

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Num

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cts

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95100Probability of Conflict

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1

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3

4

5

6

7

8

9

10

11

12

13

14

Min

imum

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(b) All known

Figure 6.6: Evolution of the conflict probability after 5 (left column), 10 (middle column), 15(right column) minutes of filtering, for scenarios where a conflict did not occur. Top row presentsthe results for SCPF with 6 aircraft. Bottom row presents the all-known results. Black dotsshow the average minimum separation distance for each percentage bin. The left axis signifiesthe number of no-conflicts identified at each probability bin, while the right axis displays theminimum separation between the aircraft in each bin.

false alarms. For this reason, we evaluate the performance of the algorithm when no conflictis present in realizations (Figures 6.5 and 6.6). This time we plot the 491 realizations whereno conflict occurred against the probability of conflict calculated by SCPF. An optimalCD scheme should be skewed to the left this time and place all non-conflict flights in the0 to 5% bin. SCPF with 6 aircraft, Figure 6.6(a) produces much less false alarms than theagnostic case, Figure 6.5(a), while improving over the SCPF with 2 aircraft, Figure 6.5(b).The all-known indicator is presented in Figure 6.6(b). This situation is almost symmetricto the conflict detection case. The results imply that SCPF is able to both increase theconflict detection rate while decreasing false alarms.

By choosing a probability threshold after which a scenario is considered a conflict we

75

6. Conflict Detection using Sequential Conditional Particle Filter

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Rate of alerts for No−Conflicts

Rat

e of

ale

rts

for

Con

flict

s

Agnostic

SCPF − 2AC

SCPF − 6AC

All−Known

random classifier

IncreasingSystem

Performance

Figure 6.7: System Operating Characteristic Curve (SOC), 15 minutes before conflict, for differentclassification thresholds.

Table 6.1: Conflict alerts matrix for 90 % threshold

Conflict No ConflictAgnostic 2-AC 6-AC All-known Agnostic 2-AC 6-AC All-known

Alert 51 % 75 % 80 % 94 % 51 % 27 % 22 % 7 %No Alert 49 % 25 % 20 % 6 % 49 % 73 % 78 % 93 %

can also estimate the false alarm and successful alert probabilities. Table 6.1 collects theresults 10 min before conflict for a threshold of 90%. We observe quite a significant increasein the successful alarms (80 % vs 51 %) and respectively a decrease in false alarms (22 % vs51 %) when the SCPF with 6 aircraft is employed, over the agnostic case. The SCPF withonly 2 aircraft has also a comparable performance.

For a different threshold different results might arise. In order to make a more com-plete evaluation of the performance of the algorithm we introduce the System OperatingCharacteristic Curve (SOC) (or Receiver Operating Characteristic Curve (ROC)). SOC is

76

6.3. Simulation Results

−32 −24 −16 −8 0 8 16 24 320

50

100

150

200

250

300

350

400

450

500Only Forecast, after 15 minutes

error in seconds

Num

ber

of s

cena

rios

−16−14−12−10−8 −6 −4 −2 0 2 4 6 8 10 12 14 16

SCPF with 6 aircraft

error in seconds−16−14−12−10−8 −6 −4 −2 0 2 4 6 8 10 12 14 16

Perfect Information available

error in seconds

Figure 6.8: Distribution of the time error for predicting the occurrence of minimum separationafter 15 minutes of filtering for different algorithms. Notice the different x-axis in the left his-togram.

−3 −2 −1 0 1 2 3

25

50

75

100

125

150Only Forecast

error in nmi

Num

ber

of s

cena

rios

−3 −2 −1 0 1 2 3

SCPF 6 aircraft

error in nmi−3 −2 −1 0 1 2 3

Perfect Information Available

error in nmi

Figure 6.9: Distribution of the error of minimum separation estimation after 15 minutes of filteringfor different algorithms

a graphical plot of the sensitivity, or true positives vs false positives, for a binary classifiersystem as its discrimination threshold is varied. In our case a true positive is a success-ful conflict alert while a false positive is an alert when there was no conflict. A perfectclassifier would give 100 % rate of true positives vs 0 % of false positives for any thresh-old choice. The worst classifier is not the one that gives opposite results to the perfect(since it could be simply reversed), but the one that provides 50 % vs 50 % rate. Sucha classifier calls an alert totally at random. The results for thresholds varying from 5 to95 % (with a 5 % discrete step), are presented in Figure 6.7. The curves correspond tothe classification results 15 min before conflict. Clearly, SCPF, especially when it employs6 aircraft provides a substantial improvement over the agnostic case, while it is very closeto the All-known indicator. It should be noted that in the agnostic case, in order to reacha successful alert close to 100 %, very large false positives are required. SCPF keeps this

77

6. Conflict Detection using Sequential Conditional Particle Filter

rate relatively small for all classification thresholds.Figure 6.8 shows how the algorithm improves the error in the prediction of the time

minimum separation occurs compared to the agnostic case. The error is reduced into a fewseconds and is not far away from the upper bound, produced by the perfect informationcase. Figure 6.9 shows the estimation of minimum separation distance, 10 minutes beforenominal conflict. The range of the error is between ±1 nmi, while for the agnostic case thiswas ±3 nmi. Both figures consider all wind realizations to derive these results. Standarddeviation for minimum separation error is 1.03 nmi and 11 s for time error in the agnosticcase, while this improves to 0.36 nmi and 2.7 s for the 6 aircraft SCPF case. The idealbound is 0.18 nmi and 1.9 s for the perfect information case.

6.4. Summary

A method for improving conflict detection (CD) using the Sequential Conditional ParticleFilter (SCPF) was presented. The performance of the algorithm was tested in flight plansincluding 2 and 6 aircraft against a large number of different wind forecast error realiza-tions. CD was improved considerably using SCPF compared with the agnostic case, whereonly meteorological forecasts were available and no inference about the forecast error wasmade. Moreover, the performance of the algorithm was close to the upper bound, for the6 aircraft case. The proposed method manages to both increase the successful alerts andreduce false alerts. Finally, simulations show how the error in the estimates of minimumseparation and time to minimum separation are reduced.

78

Chapter 7Joint Parameter and State Estimation

This chapter outlines different methodologies for overcoming the challenging problemof combined parameter and state estimation. For air traffic management applications,considered here, we assume that the aircraft airspeeds are not available to the ATC. Thisis a common case in practice. Nominal values of the airspeeds are known, given the aircrafttype and altitude. However, there is uncertainty about the airspeeds, the aircraft actuallyuse, since those are explicitly computed by airlines taking into account aircraft weight,settings, etc. We present different methods, used jointly with the Sequential ConditionalParticle Filter (SCPF), to improve trajectory prediction while also identifying the unknownaircraft airspeeds.

79

7. Joint Parameter and State Estimation

7.1. Overview

For a system that depends on unknown parameters, the problem of filtering the stateprocess X(k) = x(1), ..., x(k) given the observations Y(k) = y(1), ..., y(k) is particularlychallenging when this inference has to be done on-line. The situation is more compli-cated when the system has nonlinear dynamics with non-Gaussian noise. For the problemdiscussed in this chapter, the missing parameters will be the aircraft airspeeds for levelflights. Aircraft usually fly at constant airspeeds at each flight level mandated mostly bythe fuel efficiency. These might not be explicitly available to the ATC. For improving theperformance of TP, knowledge of the aircraft airspeeds is of crucial importance since it isone of the factors that greatly affect the aircraft future positions.

When the system dynamics depend on an unknown, constant parameter vector Ω ∈ RM

the state transition density, Eq. (3.2), becomes

x(k) ∼ px(·|x(k − 1),Ω, k) . (7.1)

For such cases, the output measurements of the system need to be used both for estimatingthe system state and also calibrating the parameters. Moreover, the statistics of themeasurements themselves can also be a function of the parameter vector

y(k) ∼ py(·|x(k),Ω, k) . (7.2)

Finally, we assume that some knowledge about the “a priori” distribution of the parametersis available. In the problem discussed here we choose Ω(0) ∼ N (μ0,Σ0). The index0 corresponds to an estimate of Ω at time step 0 and does not imply a change in theparameters with time.

Several different methodologies have been developed to address the combined parameterand state estimation problem. A natural choice is to simply augment the state vector andinclude the parameter vector as part of the state, with dynamics Ω(k + 1) = Ω(k). Anystandard PF algorithm can now be used to solve this problem. Unfortunately this methodresults in severe degeneracy. Because parameters do not have dynamics the parameterspace is explored only at the first step of the algorithm. At each resampling step less andless distinct values for the parameters remain. Various methodologies have been developedin order to cope with this problem on-line [66, 93, 107]. An overview of dedicated algorithmsfor parameter estimation using particle filters is provided in [59]. In the following sectionwe explore different practical methods for airspeed identification and TP in ATM.

7.2. Methods for Parameter and State Estimation

One common way to address the problem of joint parameter and state estimation is toadd artificial perturbations and dynamics to the evolution of the parameters. This way,

80

7.2. Methods for Parameter and State Estimation

at each filtering step, the parameters can still explore the parameter space, and movetowards a better match to the measurements. Choosing the magnitude and type of thisnoise is crucial. Adding very small perturbations will usually have no effect on degeneracy,while adding large perturbations will greatly diffuse the particles and diverge from theoptimal solution. For the TP problem concerned here, we augment the state of the aircraftto include airspeed as a state. We then add random gaussian noise to the parameterdynamics, and use SCPF to perform filtering. Each aircraft has its own airspeed Ω(l) (forl = 1, ...,M(k) aircraft at time step k). At each step of SCPF the part of the state thatcorresponds to airspeed is perturbed using

Ω(l, k + 1) = afΩ(l, k) + β + σε(k)εk . (7.3)

where εk ∼ N (0, 1) and σε(k) controls the magnitude of the artificial perturbations and canbe varying with time. Parameters af , β help us calibrate the above model. For example,by setting af = 1, β = 0 and σε(k) = 0 we get the model described before, where theparameters have no dynamics and remain unchanged with time. We call this, Method A.

We can devise a different scheme, called Method B, by adding some noise at each timestep. This is straightforward by setting σε(k) = σi, while letting af = 1, β = 0 likepreviously. If we neglect the effect of the filtering procedure, we can calculate the statisticsof the stationary distribution for the process described in Eq. (7.3). The distribution ofthis process at time step k is

Ω(l, k) ∼ N (μ0, σ20 + kσ2

i ) . (7.4)

The variance of the distribution becomes infinite as time goes to infinity. To avoid thiswe can choose a different setting by multiplying with a nonzero forgetting factor af < 1,letting β = 0 and σε(k) = σi like in Method B. Then the distribution of this process, calledMethod C, at time step k becomes

Ω(l, k) ∼ N (akfμ0, a

2kf σ

20 + σ2

i

n=k∑n=0

a2nf ) . (7.5)

Since af is smaller than 1, the series converges and the stationary distribution of theprocess becomes

limk→∞

Ω(l, k) ∼ N (0,σ2

i

1 − a2f

) . (7.6)

We can set the variance of the distribution with a proper choice of σε and af , but wehave no way of altering the mean, which will always converge to zero. In order to beable to affect the mean of the stationary distribution we can use a nonzero β = βi. Thedistribution of this process, called Method D, at time step k becomes

Ω(l, k) ∼ N (akfμ0 + βi

n=k∑n=0

anf , a

2kf σ

20 + σ2

i

n=k∑n=0

a2nf ) . (7.7)

81

7. Joint Parameter and State Estimation

Both series converge as time goes to infinity and the stationary distribution of the processbecomes

limk→∞

Ω(l, k) ∼ N (βi

1 − af

,σ2

i

1 − a2f

) . (7.8)

By making proper selections for af , β and σε(k) we can set the mean and variance of thestationary distribution. A natural selection would be to keep the same statistics with the“a priori” distribution. To do this we simply set β = (1− af )μ0 and σε(k) = σ0

√(1 − a2

f).We have investigated various ways for setting the above parameters. For example an

interesting choice is reducing the standard deviation of the perturbations, σε(k), with time.This idea is driven by the assumption that the algorithm converges towards the correctparameters as time passes so the artificial perturbations should become smaller and stayconcentrated around the particle selections. We have used three differen ways to reduceσε(k). The first one is linearly with time, the second exponentially with time and the lastone using a sigmoidal function

σε(k) = σi − k

kf

(σi − σf ) (7.9)

σε(k) = σie−ce1k (7.10)

σε(k) = σi

(1 − 1

1 + e−cs1(k−cs2)

), (7.11)

where σ2i is the initial variance and σ2

f is the final variance of the perturbation, at time stepkf . For k > kf we set the variance constant at σ2

f . Parameters ce1 and cs1, cs2 define theshape of the exponential and sigmoidal functions respectively. The three different methodsare named K, L and M respectively.

Moreover, driven from the same assumption mentioned above, we have extended thismethod by varying the likelihood variance as time passes. Initially, the weighting of thedifferent parameters is done more leniently and with time it becomes stricter since thealgorithm expects that it has already moved close to the real parameters. In the problemconsidered here the particles are weighted using the statistics of the radar measurementerror. We alter the standard deviation of this error with time, as follows

σr(k) = crσr − k

kf(crσr − σrf ) , (7.12)

for k = 1, ..., kf , where kf is the last step, σr is the actual standard deviation of the radar,σr(kf) = σrf is the final standard deviation and cr a constant that shapes the distribution.For k > kf we set the standard deviation constant at σrf . We call Method Q a procedurethat combines this idea with Method K (artificial dynamics and artificial noise decreasinglinearly with time).

82

7.2. Methods for Parameter and State Estimation

Algorithm SettingsMETHOD af β σε(k) σr(k) Filter Ω(l, k)

Method A 1 0 0 σr SCPF nonlinearMethod B 1 0 σi σr SCPF nonlinearMethod C < 1 0 σi σr SCPF nonlinearMethod D < 1 βi σi σr SCPF nonlinearMethod K < 1 βi Eq. (7.9) σr SCPF nonlinearMethod L < 1 βi Eq. (7.10) σr SCPF nonlinearMethod M < 1 βi Eq. (7.11) σr SCPF nonlinearMethod Q < 1 βi Eq. (7.9) Eq. (7.12) SCPF nonlinearMethod P n.a. n.a. n.a. σr SCPF + APF nonlinearMethod X n.a. n.a. n.a. σr SCPF linear

Table 7.1: Methods for joint on-line parameter and state estimation. The last column describeswhether the nonlinear or linear part of the state is augmented. The acronym n.a. stands for “notapplicable”.

These methods attempt to balance the risk of adding very small vs very large perturba-tions, mostly in a practical way. A more systematic method is the use of kernel smoothingwith auxiliary PF presented in Section 3.3.4. We have devised a mixed algorithm thatuses SCPF for filtering and APF for parameter estimation, called here Method P. Forevery aircraft state we resample looking one step ahead, by propagating the mean of theparticle distribution. To evolve the system dynamics we choose prior point estimates forthe parameters (airspeeds) as described in Eq.(3.14), that are displaced towards the MonteCarlo mean of the particles. We can then use a kernel density, Eq. 3.13, to extract newparameters. When new airspeeds are assigned to the particles describing the state of oneaircraft, we move to the next as dictated in SCPF.

Finally, for the problem discussed here, we investigated adding the parameters (air-speeds) to the linear part of the state (instead of the nonlinear aircraft dynamics), thatconcerns the evolution of the wind forecast error (called Method X). This action is justifiedsince there exists a strong coupling between the airspeed and the wind-speed. To applythis method, we extend the covariance and mean of the wind-forecast error to include thestatistics of the airspeed distribution. Initially, the two parts of the covariance matrix(one concerning the wind uncertainty covariance and the other concerning the airspeedsinitial variance) are uncorrelated. The part of the covariance of the augmented linear statethat concerns the airspeeds has zero non-diagonal elements, and σ0 variance. However, byconditioning at each time step, correlation terms start to appear. The underlying idea isthat uncertainty both about airspeeds and wind speeds will be reduced implicitly throughthe conditioning procedure of SCPF.

An overview of the different methods proposed here can be found in Table 7.1. Variousother approaches were also explored, however their performance was substantially inferior.

83

7. Joint Parameter and State Estimation

0 60000 120000 180000 240000 300000 360000 420000 480000 540000 6000000

60000

120000

180000

X − axis (m)

Y −

axi

s (m

)

Figure 7.1: Flight Plans for 24 aircraft. Arrows display the direction of motion.

7.3. Simulation Setup and Results

We compare the performance of the different parameter estimation algorithms using a 24aircraft scenario. The flight plans can be seen in Figure 7.1. All aircraft fly level at analtitude of 10000 m. The general setting is similar to the one presented in Section 4.3.1.To test the algorithms we create 400 different airspeed and wind forecast error realizationsand use each of the methods for identifying the unknown airspeeds and improving TPperformance by filtering ground radar measurements. The aircraft airspeeds are restrictedbetween the operational limits of a Boeing 737-200, which is used as a model in oursimulations. The lower limit is 187 m/s while the upper limit is 250 m/s. The “a priori”distribution for each of the airspeeds is Gaussian, with mean at 215 m/s, the nominalspeed at this altitude, and standard deviation 10 m/s, truncated at the operational limitsof the aircraft. Additionally a uniform distribution between these limits is used, to testthe robustness of SCPF when the “a priori” distribution is not known.

In order to create a performance benchmark for TP we run SCPF using the 400 airspeedand wind forecast error realizations, but assume that airspeeds are explicitly provided tothe ATC. This is similar to the results presented in Chapter 4 except that aircraft havedifferent airspeeds between them. Figure 7.2 presents the evolution of TP uncertaintyfor a varying number of radar measurements. Initially, without any filtering the along-track RMS error grows approximately to 5 km for 20 min TP. After 10 measurementsthe RMS error is already reduced below 4 km for all aircraft. Finally after incorporating40 measurements RMS error is further reduced to a range from 1.7 to 2.2 km. This isan upper performance bound for all the methods presented here, since it assumes thatairspeed identification was perfect. It should be noted that this benchmark does notrepresent optimal filtering, but an even more advantageous case in which parameters wereperfectly identified.

84

7.3. Simulation Setup and Results

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Figure 7.2: Trajectory Prediction 20 min ahead using SCPF with known airspeeds, after filtering0, 10, 20, 30 and 40 radar measurements.

85

7. Joint Parameter and State Estimation

1 2 3 4 5 6 7 8 9 101112131415161718192021222324−5

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Aircraft IndexA

irspe

ed s

.d. (

m/s

)

Figure 7.3: Airspeed estimation, bias (left figure) and RMS error (right figure) for Method A,after 20 min of filtering.

We start by examining Method A, where the aircraft states are augmented to includeairspeeds. Figure 7.3 compiles the results for the RMS and bias error in identifying theairspeeds of the 24 aircraft. For this algorithm, the initial uncertainty in the airspeeddistribution was represented by a Gaussian with mean 215 m/s and 10 m/s standarddeviation. The uncertainty about the airspeed drops below 4 m/s for most of the aircraftafter 20 min of filtering. The identification appears to be generally unbiased. The averagebias across the 24 airspeeds is −0.124 m/s. Moreover, the mean RMS error is 3.33 m/s, asignificant drop from the initial uncertainty of 10 m/s. The minimum RMS error achievedis 2.61 m/s for flight 12 while the maximum RMS error is 4.34 m/s for flight 4.

This algorithm was also tested against uncertainty about the airspeed distribution. Fig-ure 7.4 presents the airspeed identification results using Method A, when the initial un-certainty had a uniform distribution but the SCPF assumed it was Gaussian. Table 7.2collects all the different combinations between real uncertainty and the SCPF assump-tion. For uniform initial uncertainty and uniform assumption the results are similar tothe Gaussian-Gaussian (real and assumption respectively) case. However, when SCPFassumes a Gaussian distribution, whereas the real distribution is uniform, there is a quitesignificant increase in bias, that reaches 0.58 m/s. On the other hand, for the Gaussian-Uniform case, the bias is not dramatically affected but there is some deterioration in theRMS error that reaches 3.92 m/s. Despite these differences in performance it appears thatSCPF manages to reduce uncertainty even if the exact shape of the “a priori” airspeeddistribution is not precisely known.

Method A was also evaluated on how much it can improve TP uncertainty. Figure 7.5compiles the results for 20 min TP, after different number of radar measurements (0,10, 20, 30 and 40). It is apparent that uncertainty in airspeed has a dramatic effect inTP accuracy. The RMS error in TP, before any measurements are incorporated, ranges

86

7.3. Simulation Setup and Results

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peed

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/s)

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5

6

7

8

9

10

Aircraft Index

Airs

peed

s.d

. (m

/s)

Figure 7.4: Airspeed estimation, bias (left figure) and RMS error (right figure) for Method A,after 20 min of filtering. The initial uncertainty has a uniform distribution while SCPF assumesthat it is Gaussian.

from 12 to almost 14.5 km. This range is due to different airspeeds and flight plans fordifferent aircraft. This uncertainty represents an almost threefold increase, compared withthe results presented in Chapter 4, where aircraft airspeeds were assumed known. Afteronly 10 measurements SCPF manages to reduce the TP uncertainty in the range of 3.5to 4.2 km. Unfortunately, after 30 measurements the TP estimates deteriorate. After 40measurements, the along track RMS error for 20 min TP ranges from 3.6 to 6 km. Thisis due to sample degeneracy in the part of the particles concerning the parameters. Evenafter a small number of measurements, very few distinct airspeed values remain.

SCPF Uncertainty Assumptionmean Bias (m/s) mean RMS (m/s)

Real Uncertainty Uniform Gaussian Uniform GaussianUniform −0.11 0.58 3.47 3.46Gaussian −0.27 −0.12 3.92 3.33

Table 7.2: Bias and RMS error for different uncertainty assumptions.

To overcome the degeneracy problem we apply Method B, where small artificial pertur-bations are incorporated in the dynamics of the parameters. We used a standard deviationσε = 0.5 m/s for the added noise. This small extension improves significantly the airspeedidentification results as can be seen in Figure 7.6. The mean bias remains very small(−0.15 m/s) while the mean RMS error drops to 2.76 m/s. Also, the aircraft with the bestairspeed estimation (number 3) has only 1.33 m/s RMS error, while the worst estimate is4.30 m/s RMS error for aircraft number 1.

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7. Joint Parameter and State Estimation

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Figure 7.5: Trajectory Prediction 20 min ahead for Method A, after filtering 0, 10, 20, 30 and 40radar measurements.

88

7.3. Simulation Setup and Results

1 2 3 4 5 6 7 8 9 101112131415161718192021222324−5

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peed

s.d

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/s)

Figure 7.6: Airspeed estimation, bias (left figure) and RMS error (right figure) for Method B,after 20 min of filtering.

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7

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Aircraft Index

Airs

peed

s.d

. (m

/s)

Figure 7.7: Airspeed estimation, bias (left figure) and RMS error (right figure) for Method D,after 20 min of filtering.

Figure 7.8 shows the performance of Method B, for TP. There is a significant improve-ment when compared with method A. The added noise assists not only the parameteridentification, as reported before, but also the state estimation. The TP accuracy is im-proved significantly, since it now ranges from 3.6 to 4.6 km approximately. Moreover, theRMS error remains more or less stable for an increasing number of measurements and doesnot deteriorate like in Method A.

Method D provides a similar performance with Method B. The results are presented inFigure 7.7. The mean bias drops very low (−0.07 m/s) and the mean RMS error remainsalso quite small (2.82 m/s). The aircraft with the best identification of its airspeed is again

89

7. Joint Parameter and State Estimation

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7.3. Simulation Setup and Results

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number 3, with 1.61 m/s RMS error, and the aircraft with the worst identification is againnumber 1 with 4.11 m/s RMS error. TP performance, shown in Figure 7.11, is also verysimilar to that of Method B, with the addition that the minimum uncertainty was evenlower (3.4 km).

Figure 7.9 presents the airspeed identification results using the Method K, where theartificial noise in the parameters is decreasing linearly with time. The mean RMS errorwas as low as 2.54 m/s and the minimum and maximum RMS error ranged from 1.31to 4.08 m/s. Mean bias was also kept significantly low at 0.03 m/s, with a range from−0.33 m/s to 0.23 m/s. Method Q has a very similar performance. The mean bias

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7. Joint Parameter and State Estimation

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Figure 7.11: Trajectory Prediction for Method D after filtering 0, 10, 20, 30 and 40 radarmeasurements.

92

7.3. Simulation Setup and Results

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Figure 7.13: Airspeed estimation, bias (left figure) and RMS error (right figure) for Method X,after 20 min of filtering.

drops even more to −0.03 m/s and the mean RMS error is also smaller, 2.59 m/s. Theminimum and maximum RMS errors are 1.63 m/s and 3.86 m/s respectively. The resultsare presented in Figure 7.10. Both Methods K and Q had the best performance overall interms of airspeed identification.

The advantage of Method Q over Method K can be seen in the improvement of TPaccuracy. Figures 7.14 and 7.15 show the TP performance of the two methods. Method K,has a very similar performance with Methods B and D. However, for Method Q the worst20 min ahead TP uncertainty, among the different aircraft, drops to approximately 4 kmafter the incorporation of 10 radar measurements and remains in a similar level for 20,

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Figure 7.14: Trajectory Prediction for Method K after filtering 0, 10, 20, 30 and 40 radarmeasurements.

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7. Joint Parameter and State Estimation

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30 and 40 measurements. Previous methods were giving maximum TP uncertainty over4.5 km. The mean TP RMS error is also reduced to 3.3 km, significantly better than mostof the other methods. This method is a combination of all the ones presented up to here,combining random perturbations decreasing in amplitude with time, artificial dynamics,and varying measurement likelihood. Method Q has the best overall performance for bothairspeed identification and TP.

Although Methods P and X performed substantially worse than the rest of the inves-tigated methods, we present the results for the sake of completeness. Method P is thecombination of the APF with SCPF. At each filtering step a kernel for the estimationof the parameter vector is calculated. This algorithm did not provide any improvementsover the simpler methods as can been seen in Figure 7.12. The mean estimation bias wasvery small (0.03 m/s) while the mean RMS was higher even than the Method A reaching3.53 m/s. Maximum RMS error climbed to 4.83 m/s, while minimum RMS error was alsohigh, 3.03 m/s. Method X performed slightly better with a mean RMS error of 3.51 m/s,while the maximum and minimum RMS errors were 2.60 and 4.40 m/s respectively. Resultsare presented in Figure 7.13.

The performance of the two methods in TP is presented in Figures 7.16 and 7.17. MethodP had the worst accuracy in TP error among the methods investigated here, with RMSerror reaching more than 6 km. Method X was marginally better overall. What is apparentin both cases, is the large deviation of different flights from the mean TP accuracy. Sur-prisingly Method X had the best performance in terms of minimum TP RMS error withuncertainty being as low as 2.7 km, while Method X, the second best, had a minimumRMS error for 3.4 km. This was at the great expense of the maximum TP uncertaintywhich reached respectively 6.2 km.

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Figure 7.17: Trajectory Prediction for Method X after filtering 10, 20 and 30 radar measurements.

It is interesting to note a pattern in the airspeed identification, for aircraft index numbers1-2-3, 4-5-6, and so on. The increase in accuracy is due to the fact that 1 is a precedingflight to 2, and 2 a preceding flight to 3 (the same holds for all these triads). Intuitively,since the leading aircraft estimate both the wind and their airspeed, the trailing aircraftwill face a less uncertain wind environment and can “focus” their estimation effort in theunknown airspeed. This pattern appears, with different degree of magnitude, for all themethods reported here.

Figure 7.18, accumulates the results of all the methods presented here for airspeedidentification. In terms of bias it is clear that Methods D, K and Q are the ones mostconcentrated around zero. As far as it concerns the RMS error, Methods B and K havethe lowest minimum error but exhibit large maxima, while Method Q is the second best interms of average RMS error and best in terms of maximum RMS error. Clearly methodsA, P and X exhibit the worst performance.

Figure 7.19, gathers the TP performance results for all the methods, for different numberof radar measurements. Clearly, Method Q has the best accuracy on average. Moreover,different aircraft exhibit small differences between them in terms of accuracy. The max-imum uncertainty is also kept at the lowest level among methods. On the other hand

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7.3. Simulation Setup and Results

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Methods P and X show a very large degree of uncertainty between different aircraft. Fi-nally, it is easy to spot the negative effect, for Method A, of the number of measurementsdue to degeneracy.

Since Method Q exhibited the best performance overall we focus on a single run of thealgorithm for one of the airspeed and wind realizations. Figure 7.20 displays the resultsof Method Q, after filtering 40 radar measurements. The bars represent the real aircraftairspeeds and crosses are the respective particle estimates. Despite the large variety ofairspeeds, the mean of the particles is almost always very close to them. For some cases,where the mean deviates, there still exist particles close to the real airspeed. This helpsprevent the algorithm from becoming degenerate.

Figure 7.25 presents the process of filtering the airspeeds for some interesting aircraftcases. The blue straight line denotes the true, unknown airspeed of the aircraft, and thegreen crosses the individual particle estimates. When the initial mean of the particlesis very close to the true airspeed, like in case d), particles remain close to it, but donot steadily converge towards it. They constantly fluctuate around the real value of theairspeed. This happens also in cases where the mean of the particles identifies the trueairspeed at a later stage, like in case a). Moreover, we can observe that particles quicklymove close to the true airspeed even if this is far from the initial guess of the particleslike in cases b) and c). The bulk of the particles is located far from the the true airspeedwith very few outliers close to it. However, since parameters have artificial dynamics,they can still move towards the real airspeed. We can finally see that as time goes byand particles get concentrated close to the real values, the variance between the particlesbecomes smaller at each measurement.

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a) Aircraft number 13.

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Figure 7.25: Evolution of particles for the airspeed estimation of different aircraft, using MethodQ.

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7.4. Summary

7.4. Summary

A variety of different practical methods to address the joint parameter and state estimationproblem was presented. These methods were applied in the airspeed identification and TPproblem. Their performance was studied and evaluated through extensive simulations.Including the unknown parameters in the aircraft state dynamics provided encouragingresults. The deterioration in TP performance was corrected by the addition of artificialdynamics and noise. Further extensions with varying noise and measurement likelihoodimproved the performance even more. The use of more elaborate methods such as APFcombined with SCPF was not found sufficient to improve on the results of simpler methods.The simulation studies suggest that for this type of problem and setting, introducing noiseand artificial dynamics in the evolution of the parameters provides satisfactory results bothin parameter identification and trajectory prediction.

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Chapter 8Conclusions and Outlook

In this thesis, different sequential monte carlo methods were studied, implemented anddeveloped for Air Traffic Management (ATM) applications. The primary objective, toimprove ATM capabilities despite growing air traffic demand, revolves around two efforts.The first one is concerned with the creation of a realistic model that captures the systemdynamics and incorporates accurately all the related uncertainties. The second one isthe search for algorithmic methods that can use such a model efficiently to extract usefulinformation. This information can be used as the basis for decision support tools in airtraffic control and management.

We have created a model capturing the dynamics of multiple aircraft in a time-varyingwind-field that has been used both as a test-bed and as a basis for the development of airtraffic control algorithms. The validation of the proposed methods has been establishedthrough extensive simulations, under varying scenarios and settings. These feasibilitystudies have confirmed the potential of the existing and the proposed algorithms. However,in order to establish confidence in any method and to assess its true value, it is of paramountimportance that tests and experiments with real data in actual systems take place. Modelmismatch, corrupted data and unexpected events can have a significant impact on anymethod that is evaluated in a realistic setting. On the other hand, insights from theinteraction with the actual systems can reveal shortcomings but also opportunities forimprovement.

Future steps in this direction have to account for several inaccuracies not assessed inthis study. Wind forecast error, though important, is not the only uncertainty involvedin TP. Missing parameters, partially known intent, unknown airline settings contributesignificantly in TP errors. Moreover, the performance of the novel algorithms needs tobe tested in several different flight phases that include climb and descend. Due to theinclusion of the vertical direction, both the dimension of the nonlinear aircraft state andthe dimension of the linear wind field state are going to increase (reaching thousands ofstates) rendering filtering even more challenging. Furthermore, the wind forecast error isweakly correlated across different altitudes, so the information advantage might not besimilarly strong from one flight level to the other, as for flights in the same altitude.

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8. Conclusions and Outlook

Using the novel particle filtering algorithm developed here, the Sequential ConditionalParticle Filter (SCPF), we have attained significant results in various air traffic appli-cations. SCPF has managed to improve trajectory prediction accuracy, from single tomultiple aircraft situations. Moreover, the algorithm has been used successfully in con-flict detection (CD) methods. Simulations suggest that there is a substantial potential inimproving CD performance using SCPF over meteorological forecasts. Finally, for caseswhere aircraft airspeeds are unavailable, augmented SCPF was capable of performing on-line, both airspeed identification and wind and aircraft state estimation. This resultedin an increased trajectory prediction (TP) accuracy. Future work is needed to explorethe capabilities of the algorithm in providing combined conflict detection and resolution(CD&R).

The work presented here is based on a centralized scheme that utilizes ground-radarmeasurements from multiple aircraft in the airspace that can be afterwards used by SCPFor other filters. In the same direction new algorithms can be developed that incorporatedirect wind measurements from the aircraft, assuming that these can be data-linked tothe ground. Such methodologies have to account for potential delays in the transmissionof measurements and a different uncertainty structure, since wind measurements dependboth on the state of the aircraft and the magnitude of the wind measured. Moreover, workon decentralizing the methods proposed here could provide many practical advantages interms of computational efficiency and system deployment.

The methods developed here are not restricted only into air traffic applications but en-compass a larger class of problems. In general SCPF applies to problems where a largenumber of nonlinear agents interacts with a time-varying, spatially correlated environmentthat evolves linearly. The agents are affected by the environment but do not affect it back.The state of each agent does not directly depend on the state of the other agents but isindirectly coupled with them through the effects of the common environment. Measure-ments of the state of the agents can be used as indirect readings that reveal the state ofthe environment. Possible new applications include robot positioning, underwater vehiclenavigation and tracking and wind power forecasting.

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114

Appendix AModel Dynamics

A.1. Wind Gradient Factors Derivation

The time varying wind-field implies that the air-mass frame is moving relative to theinertial reference frame with some acceleration. This implies the creation of a fictitiousforce that acts on the aircraft dynamics. The effect of this force in the aircraft accelerationis included in the equations of motion with the use of wind gradient factors, Eq. (2.1).Wind gradient factors are derived as follows

⎡⎢⎢⎣wX

wY

wZ

⎤⎥⎥⎦ =

⎡⎢⎢⎢⎢⎢⎢⎣

∂wX

∂x

∂wX

∂y

∂wX

∂z∂wY

∂x

∂wY

∂y

∂wY

∂z∂wZ

∂x

∂wZ

∂y

∂wZ

∂z

⎤⎥⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎣xI

yI

zI

⎤⎥⎥⎦ +

⎡⎢⎢⎢⎢⎢⎢⎣

∂wX

∂t∂wY

∂t∂wZ

∂t

⎤⎥⎥⎥⎥⎥⎥⎦. (A.1)

The time derivatives of the aircraft position with respect to the inertial reference frame(xI ,yI ,zI) are available from Eq. (2.1)

⎡⎢⎢⎣wX

wY

wZ

⎤⎥⎥⎦ =

⎡⎢⎢⎢⎢⎢⎢⎣

∂wX

∂x

∂wX

∂y

∂wX

∂z∂wY

∂x

∂wY

∂y

∂wY

∂z∂wZ

∂x

∂wZ

∂y

∂wZ

∂z

⎤⎥⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎣V cos(ψ) cos(γ) + wX

V sin(ψ) sin(γ) + wY

V sin(γ) + wZ

⎤⎥⎥⎦ +

⎡⎢⎢⎢⎢⎢⎢⎣

∂wX

∂t∂wY

∂t∂wZ

∂t

⎤⎥⎥⎥⎥⎥⎥⎦. (A.2)

These equations calculate the derivative of the wind with respect to the inertial referenceframe. We have to transform them to the aircraft reference frame in order to include themin the equations of motion.

115

A. Model Dynamics

Define the inertial reference frame axes as

• X - East,

• Y - North,

• Z - Up.

and the aircraft reference frame axes as

• Xa - direction of motion relative to air mass,

• Ya - perpendicular to Xa in the horizontal plane (out of right wing),

• Za - perpendicular to Xa and Ya, approximately up.

Rotational translations between the two reference frames are defined as follows⎡⎢⎢⎣Xa

Ya

Za

⎤⎥⎥⎦ =

⎡⎢⎢⎣cos(γ) 0 −sin(γ)

0 1 0

sin(γ) 0 cos(γ)

⎤⎥⎥⎦⎡⎢⎢⎣

cos(ψ) sin(ψ) 0

−sin(ψ) cos(ψ) 0

0 0 1

⎤⎥⎥⎦⎡⎢⎢⎣X

Y

Z

⎤⎥⎥⎦ . (A.3)

which becomes⎡⎢⎢⎣Xa

Ya

Za

⎤⎥⎥⎦ =

⎡⎢⎢⎣cos(γ) sin(ψ) cos(γ)cos(ψ) −sin(γ)

cos(ψ) −sin(ψ) 0

sin(γ) sin(ψ) sin(γ)cos(ψ) cos(γ)

⎤⎥⎥⎦⎡⎢⎢⎣X

Y

Z

⎤⎥⎥⎦ . (A.4)

To calculate the wind gradient factors (Wagf, Wcgf, Wvgf) we transform the wind vectoraccelerations to the aircraft reference frame coordinates⎡

⎢⎢⎣Wagf

Wcgf

Wvgf

⎤⎥⎥⎦ =

⎡⎢⎢⎣cos(γ) sin(ψ) cos(γ)cos(ψ) −sin(γ)

cos(ψ) −sin(ψ) 0

sin(γ) sin(ψ) sin(γ)cos(ψ) cos(γ)

⎤⎥⎥⎦⎡⎢⎢⎣wX

wY

wZ

⎤⎥⎥⎦ . (A.5)

The effect of the vertical wind gradient factor (Wvgf) is not included in the equations ofmotion, since it normally affects the derivative of the flight path angle (γ). However, inour model this is explicitly derived by the FMS, as an input to the aircraft continuousdynamics.

116

A.2. Wind Forecast Error Derivation

A.2. Wind Forecast Error Derivation

The wind forecast error is generated using a linear gaussian model, with parameters a andQ.

WX(0) =QvX(0), WX(k + 1) = aWX(k) +QvX(k) ,

WY (0) =QvY (0), WY (k + 1) = aWY (k) +QvY (k) ,(A.6)

We prove that this model respects the correlation structure outlined in Section 2.3.2 andshow the derivation of a and Q. W below represents either WX or WY , since wind error isan isotropic field.

Proposition 1.Let

W (0) = Qn(0)

W (k + 1) = aW (k) +Qn(k)(A.7)

where n(k) are independent standard (zero mean, identity covariance matrix) Gaussianrandom variables. Then

E[W (k)] = 0

E[W (k)W T (k)] = R

E[W (k)W T (k′)] = e− |k−k′|δt

Gt R

(A.8)

Proof.By induction. At k = 0:

E[W (0)] = E[Qn(0)] = QE[n(0)] = 0 (A.9)

and

E[W (0)W T (0)] = E[Qn(0)nT (0)QT ] = QE[n(0)nT (0)]QT = QQT = R (A.10)

Assume (A.8) holds for some k ≥ 0, show that it also holds for k + 1. Clearly

E[W (k + 1)] = E[aW (k) +Qn(k)] = aE[W (k)] +QE[n(k)] = 0 (A.11)

Moreover

E[W (k + 1)W T (k + 1)] =E[(aW (k) +Qn(k))(aW T (k) + nT (k)QT )]

=a2E[W (k)W T (k)] + aQE[n(k)W T (k)]+

+ aE[W (k)nT (k)]QT +QE[n(k)nT (k)]QT

Since, W (k) depends only on n(0), n(1), ..., n(k − 1) and is independent of n(k),

117

A. Model Dynamics

E[W (k + 1)W T (k + 1)] = a2R +QQT = a2R+ (1 − a2)R = R (A.12)

Assume that for some k′ = 0, 1, 2, ..., k + 1

E[W (k + 1)W (k′)T ] = e− |k+1−k′|δt

Gt R (A.13)

Show that (backwards induction towards k′ = 0)

E[W (k + 1)W (k′ − 1)] = e(k+1−(k′−1))δt)

Gt R (A.14)

E[W (k + 1)W (k′ − 1)T ] = E[(aW (k) +Qn(k))W T (k − 1)]

= aE[W (k)W T (k′ − 1)] +QE[n(k)W T (k′ − 1)]

For k′ = 0, 1, 2, ..., k + 1, W (k′ − 1) depends only on n(0), n(1), ..., n(k − 2) and isindependent of n(k). So, E[n(k)W T (k′ − 1)] = 0. By the original induction hypothesis

E[W (k)W T (k′ − 1)] = e− (k−(k′−1))δt

Gt R (A.15)

Therefore,

E[W (k + 1)W T (k′ − 1)] = ae− (k−k′+1)δt

Gt R (A.16)

= e−δt/Gte−(k−k′+1)δt/GtR (A.17)

= e−(k−k′+2)δt/GtR (A.18)

which concludes the proof.

118

A.3. Thrust and ESF control: Flight Path Angle Control

O(n− 1)

O(n)

γ

Γ(n− 1)

Figure A.1: Final descent

A.3. Thrust and ESF control: Flight Path Angle Control

The model for setting the thrust and flight path angle (or ESF) in Sections 2.4.1 and 2.4.2,is adequate for most of the flight, but may be unacceptable for aircraft in final approach, orin cases where the ATC explicitly commands an aircraft to track a specific ROCD due totraffic conditions in the area. For these cases specialized controllers need to be developed.We present these controllers for the case of final approach; the procedure is the same forall controlled ROCD situations.

We assume that aircraft measure the local wind speed and use it in their attempt totrack a given ROCD. The computation is done assuming constant wind instantaneously.Figure A.1 shows the motion in the vertical plane, as an aircraft approaches the runway.We assume that O(n) is the last way point of the flight plan which coincides with thebeginning of the runway and O(n − 1) is the penultimate way point. The angle this lastsegment of the flight plan makes with the horizontal plane is given by

Γ(n− 1) = tan−1

(Z(n) − Z(n− 1)

‖(X(n), Y (n)) − (X(n− 1), Y (n− 1))‖),

where X(n), Y (n), Z(n) denote the coordinates of the O(n) way-point. Let also Γw denotethe angle the wind, WI = (wX , wY , wZ), makes with the horizontal plane

Γw = tan−1

(wZ

‖(wX , wY )‖).

We determine the flight path angle, γ, that will result in the aircraft descending at anangle Γ(n− 1) in the presence of wind wI , by the sine rule

‖w‖sin(Γ(n− 1) − γ)

=V

sin(Γw + Γ(n− 1)),

119

A. Model Dynamics

which leads to the following setting for the flight path angle

γ = Γ(n− 1) − sin−1

(‖wI‖V

sin(Γw + Γ(n− 1))

).

To ensure that the aircraft also tracks the nominal speed, Vnom, we set the thrust accordingto

T =CDSρ(zI)

2V 2

nom +mg sin

(Γ(n− 1) − sin−1

(‖wI‖V

sin(Γw + Γ(n− 1))

)).

120

Appendix BCurriculum Vitae

Ioannis LymperopoulosBorn on April 11th, 1981 in Athens, Greece

ETH Zürich - Zurich, SwitzerlandAutomatic Control Laboratory

2007–2010 Doctoral Studies at the Department of Electrical Engineeringand Information Technology

University of Patras, GreeceSystems and Measurements Laboratory

2006–2007 Doctoral Studies at the Department of Electricaland Computer Engineering

University of Patras, Greece1999–2006 Diploma in Electrical and Computer Engineering

3d Lyceum Tripolis, Greece1996–1999 High School Attendance

123

A ship in a harbor is safe, but that is not what a ship is built for.

Grace Hopper