Integrity Monitoring Techniques in GPS/Galileo

João Pedro Duque Duarte

Thesis to obtain the Master of Science Degree in

Aerospace Engineering

Supervisor: Professor Fernando Duarte Nunes

Examination Committee

Chairperson: Prof. João Manuel Lage de Miranda LemosSupervisor: Prof. Fernando Duarte NunesMember of the Committee: Prof. José Eduardo Charters Ribeiro da Cunha Sanguino

May 2015

ii

Acknowledgments

To me, this thesis is a major milestone as it marks the end of a journey at Instituto Superior Técnico.

The journey was not always perfect or easy but I am very glad and proud to achieve many of the goals

I had when I enrolled in my master degree in Aerospace Engineering. Getting to this point was only

possible thanks to the support and motivation from my family and friends.

I would like to express my appreciation and gratitude to Prof. Fernando Duarte Nunes, my supervisor

in this thesis for his support, advice, help and availability during this master thesis.

To my colleagues, and friends for their help and friendship that made the journey easier.

To Deimos Engenharia for their support providing me precious time to complete this thesis, and good

fellowship.

And finally to my family, specially my parents for all the emotional and financial support, patience

and motivation without which nothing of this would have been possible.

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iv

Resumo

A utilização de sistemas de navegação em aviação requer um elevado nível de confiança na solução a ser

usada. Define-se integridade como o nível de confiança que se tem na exactidão da informação fornecida

pelo sistema de navegação. Existem várias arquitecturas que permitem verificar o nível de integridade

da solução apresentada pelo sistema. Uma dessas arquitecturas são as chamadas técnicas de RAIM que

consistem em algoritmos implementados a nível do receptor que permitem medir esse nível de integridade

e em caso de existência de falha num satélite essa falha não só é detectada como o satélite com defeito

pode ser excluído.

Existem vários algoritmos de integridade, entre os quais o Least-Squares-Residuals e o Range-Comparison-

Method que foram os analisados nesta tese. Os algoritmos de monitorização de integridade vêem o seu

desempenho afectado pela geometria dos satélites. Interessa portanto comparar os dois algoritmos con-

siderados quanto ao seu desempenho em casos de diferentes geometrias.

Para testar o desempenho dos algoritmos foi simulada uma constelação de GPS e uma trajectória do

receptor cuja solução de posição foi calculada com recurso a um filtro de Kalman. Foram consideradas

duas geometrias distintas e ambos os algoritmos foram testados para as mesmas condições. Foi testada

não só a capacidade de detecção de falha como também a de exclusão.

Os resultados mostram que ambos os algoritmos têm desempenhos muito idênticos pelo que a maior

diferença é mesmo o nível de complexidade dos algoritmos que é consideravelmente superior no caso do

Range-Comparison-Method.

Palavras-chave: GPS, Galileo, Integridade, RAIM

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Abstract

The use of navigation systems in aviation requires a high level of trust in the solution to be used. Integrity

is defined as the measure of the trust that can be placed in the correctness of the information supplied by a

navigation system. There are several architectures that allow to compute the integrity levels on a solution

provided by the system. One of those architectures are the integrity monitoring techniques called RAIM

and these consist in algorithms implemented at the receiver and allow to measure the integrity level and

in case of a faulty satellite that fault is not only detected but the faulty satellite may be excluded.

There are several integrity algorithms, namely the Least-Squares-Residuals and the Range-Comparison-

Method that were analysed in this thesis. Integrity monitoring algorithms have their performance affected

by satellites geometry. Thus, it is important to compare both the considered algorithms as its performance

in different geometry cases.

To test the algorithm’s performance a GPS constellation was simulated along with a trajectory whose

position solution was obtained using a Kalman filter. Two distinct geometries were considered and both

algorithms were tested for the same conditions. It was tested not only the ability to perform fault

detection as also exclusion.

The results show that both algorithms have similar performances whereby the major difference is at

the algorithms complexity level, which is considerable higher in the Range-Comparison-Method case.

Keywords: GPS, Galileo, Integrity, RAIM

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Contents

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Resumo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi

Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii

1 Introduction 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Goals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Structure and proposed approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.4 State of the art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Global Navigation Satellite Systems 5

2.1 GPS overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 GPS Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2.1 Space Segment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2.2 Control Segment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2.3 User Segment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3 Galileo overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.4 Galileo Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.4.1 Space Segment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.4.2 Control Segment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.4.3 User Segment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Fundamentals 11

3.1 Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.1.1 ECI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.1.2 ECEF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.1.3 ENU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2 Coordinate System Conversions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.2.1 Cartesian to Geodetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

ix

3.2.2 Geodetic to Cartesian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2.3 ECEF to ENU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.3 Receiver Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.4 Measurement Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.4.1 Satellite Clock Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.4.2 Ephemeris Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.4.3 Atmospheric Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.4.4 Receiver Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.4.5 Multipath . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.4.6 Pseudorange Error Budget . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.5 DOP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4 Position Determination 25

4.1 Least-Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.2 DOP calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.3 Position solution using Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.3.1 Dynamics models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.3.2 Clock state model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.3.3 Extended Kalman filter: dynamics models . . . . . . . . . . . . . . . . . . . . . . . 33

4.3.4 Extended Kalman filter: observations model . . . . . . . . . . . . . . . . . . . . . . 36

5 Integrity 41

5.1 SBAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

5.1.1 SBAS around the world . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5.1.2 WAAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5.1.3 EGNOS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.2 GBAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5.3 RAIM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.3.1 FD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.3.2 FDE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.3.3 RAIM algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.3.4 RAIM Availability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.4 Integrity in Galileo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5.4.1 Integrity parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5.4.2 Galileo user integrity algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

6 Computer Simulation 65

6.1 Computer Simulation Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

6.2 Computer Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

6.2.1 Error calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

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6.2.2 RAIM Algorithms Performance Analysis . . . . . . . . . . . . . . . . . . . . . . . . 69

7 Conclusion 83

Bibliography 87

A YUMA Almanac definitions 89

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List of Tables

2.1 Positioning performances for Galileo SoL Service [3] . . . . . . . . . . . . . . . . . . . . . 8

3.1 Fundamental parameters of WGS 84 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.2 Pseudoranges error table for SPS C/A code [17] . . . . . . . . . . . . . . . . . . . . . . . . 23

3.3 DOP value description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.1 Typical values of h0 and h−2 depending on the oscillator used [40] . . . . . . . . . . . . . 32

5.1 ICAO requirements for common phases of flight . . . . . . . . . . . . . . . . . . . . . . . . 42

5.2 Subsets of the available 5 satellites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5.3 Subsets of the available 6 satellites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.4 Normalized thresholds for Pfa = 115000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

6.1 Experimental Pfa for Least-Squares-Residuals Method . . . . . . . . . . . . . . . . . . . . 69

6.2 Experimental Pfa for Range Comparison Method . . . . . . . . . . . . . . . . . . . . . . . 70

6.3 SLOPE values for first epoch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

6.4 GDOP values for the specified subsets for first epoch . . . . . . . . . . . . . . . . . . . . . 73

6.5 SLOPE values for the various subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

6.6 SLOPE values for second epoch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

6.7 GDOP values for the specified subsets for second epoch . . . . . . . . . . . . . . . . . . . 77

6.8 Mean first detection time for LSR method (ramp 1 m/s) . . . . . . . . . . . . . . . . . . . 78

6.9 Mean first detection time for RCM method (ramp 1 m/s) . . . . . . . . . . . . . . . . . . 79

6.10 Mean first detection time for LSR method (ramp 0.1 m/s) . . . . . . . . . . . . . . . . . . 80

6.11 Mean first detection time for RCM method (ramp 0.1 m/s) . . . . . . . . . . . . . . . . . 81

A.1 YUMA almanac definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

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List of Figures

2.1 Illustration of GPS architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.1 Definition of ECI reference frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.2 Definition of ECEF reference frame [42] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.3 Latitude Φ and Longitude λ on an ellipsoid . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3.4 Definition of ENU reference frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.5 ENU reference frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.6 Generic receiver architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.7 Example of an acquisition process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.8 Ephemeris Error Illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.9 Example of a worldwide TEC map [21] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.10 GNSS signals path deviation due to Atmospheric Effects . . . . . . . . . . . . . . . . . . . 22

3.11 Multipath effect Illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.12 Good DOP geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.13 Bad DOP geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.1 Illustration of position calculation with 4 satellites . . . . . . . . . . . . . . . . . . . . . . 26

4.2 Recursive Least-Squares Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.3 PV model for each coordinate using an integrated Brownian motion . . . . . . . . . . . . 31

4.4 PVA model for each position coordinate using a Gauss-Markov process for the acceleration 31

4.5 Clock state model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.6 Kalman filter flowchart applied to the resolution of the navigation equation . . . . . . . . 38

5.1 SBAS systems worldwide [44] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

5.2 EGNOS map of Ground Segment [43] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5.3 Probability Density Functions for fault and no fault cases . . . . . . . . . . . . . . . . . . 52

5.4 False Alarm probability versus of λσ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

5.5 False Alarm and Missed Detection probabilities versus λσ . . . . . . . . . . . . . . . . . . . 53

5.6 Decision rule for the Range-Comparison Method when 5 satellites are used . . . . . . . . 58

5.7 GPS and other GNSS system visibility [36] . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5.8 Number of visible GPS satellites at receiver position during the simulation day . . . . . . 61

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6.1 RAIM algorithm flowchart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

6.2 Simulated Trajectory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

6.3 3D RMS position error for Kalman Filter Solution method . . . . . . . . . . . . . . . . . . 68

6.4 Estimeted trajectory using Kalman filter vs Simulated trajectory . . . . . . . . . . . . . . 68

6.5 SLOPE method illustration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

6.6 Detection probability versus error for LSR method . . . . . . . . . . . . . . . . . . . . . . 71

6.7 Theorical and simulated detection probability for satellite ID 5 . . . . . . . . . . . . . . . 72

6.8 Time of first detection for sat ID 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

6.9 Detection probability versus error for RCM method . . . . . . . . . . . . . . . . . . . . . . 74

6.10 Detection probability difference between RCM and LSR method for the first epoch . . . . 75

6.11 First exclusion simulation for LSR method . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

6.12 First exclusion simulation for RCM method . . . . . . . . . . . . . . . . . . . . . . . . . . 76

6.13 Second detection simulation for LSR method . . . . . . . . . . . . . . . . . . . . . . . . . 77

6.14 Second detection simulation for RCM method . . . . . . . . . . . . . . . . . . . . . . . . . 78

6.15 Test statistic variation over a ramp error of 1 m/s . . . . . . . . . . . . . . . . . . . . . . 79

6.16 Test statistic variation over a ramp error of 0.1 m/s . . . . . . . . . . . . . . . . . . . . . 80

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Glossary

AoD Ages of Data.

CDMA Code Division Multiple Access.

DoD Department of Defence.

ECEF Earth Centered, Earth Fixed.

ECI Earth Centered Inertial.

EGNOS European Geostationary Navigation Overlay

System.

ENU East North Up.

ESA European Space Agency.

EU European Union.

FDE Fault Detection and Exclusion.

FD Fault Detection.

GA Ground Antennas.

GBAS Ground-Based Augmentation System.

GCC Galileo Control Centre.

GCS Ground Control Segment.

GDOP Geometric Dilution of Precision.

GEAS GNSS Evolutionary Architecture Study.

GEO Geostationary Earth Orbit.

GMS Ground Mission Segment.

GNSS Global Navigation Satellite System.

GPS Global Positioning System.

GSS Galileo Sensor Station.

GUS Ground Upper Station.

HAL Horizontal Alarm Limit.

HDOP Horizontal Dilution of Precision.

HPL Horizontal Protection Level.

ICAO International Civil Aviation Organization.

IF Integrity Flag.

LLA Latitude Longitude Altitude.

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LSR Least-Squares-Residuals.

MCS Master Control Station.

MEO Medium Earth Orbit.

MS Monitor Stations.

OS Open Service.

PDOP Position Dilution of Precision.

PPS Precise Positioning Service.

PRN Pseudorandom Noise.

PVA Position Velocity Acceleration.

PVT Position Velocity Time.

PV Position Velocity.

RAIM Receiver Autonomous Integrity Monitoring.

RCM Range-Comparison-Method.

SARPs Standards and Recommended Practices.

SA Selective Availability.

SBAS Satellite-Based Augmentation System.

SISA Signal-In-Space-Accuracy.

SISE Signal-In-Space-Error.

SISMA Signal-In-Space Monitoring Accuracy.

SIS Signal-In-Space.

SPS Standard Positioning Service.

SSE Sum of Squared Errors.

SV Service Vehicle.

SoL Safety of Life.

TDOP Time Dilution of Precision.

TEC Total Electron Content.

UERE User Equivalent Range Error.

VAL Vertical Alert Level.

VDOP Vertical Dilution of Precision.

VPL Vertical Protection Level.

WAAS Wide Area Augmentation System.

WGS-84 World Geodetic System datum 1984.

WRS Wide-Area Reference Station.

xviii

Chapter 1

Introduction

This chapter provides an overview of the work, including a brief description of the motivation, goals,

structure of the thesis and the state of the art.

1.1 Motivation

Nowadays various Global Navigation Satellite Systems (GNSS) are used worldwide providing the user with

its position. Examples of GNSS systems are the Global Positioning System (GPS) from United States and

Galileo from Europe. These systems consist in a space segment composed by a satellite constellation, a

ground segment constituted by a network of ground stations that perform the necessary tasks to maintain

the proper operation of the system and a user segment that is composed by users equipped with a receiver

capable of using signals from the space segment. The applications of such systems have been increasing

in the past years and now these technologies play a fundamental role in modern society, being present in

almost every life situation like aviation, car navigation, agriculture, railways, emergency services, global

communications services, etc. As there is a wide range of applications not all have the same requirements.

Some just need a rough estimate of the user position while others require significantly more accuracy and

integrity.

Integrity is the measure of the trust that can be placed in the correctness of the information supplied by

a navigation system [30]. Several architectures have been proposed to provide integrity of the information

to the aviation community, for which integrity is a critical requirement as any failure could lead to

catastrophic consequences.

One of the techniques used to provide integrity is the Receiver Autonomous Integrity Monitoring

(RAIM). A RAIM algorithm can be included in the satellite navigation airborne equipment and these

algorithms compare the pseudorange measurements among themselves to ensure that they are all consis-

tent. If there is lack of consistency a fault will be declared and, depending on the number of satellites

being used, the faulty satellite may be excluded.

In this thesis two different RAIM algorithms used to provide integrity in GPS will be studied and their

performance will be compared. The integrity concept of Galileo will also be explained and an algorithm

1

that takes advantage from the integrity data provided by Galileo services will be presented.

1.2 Goals

A variety of RAIM algorithms have been developed and studied. All of these techniques, however are

sensitive to the geometry provided by the redundant satellite measurements [6].

In cases of poor satellite geometry the performance of the integrity algorithms degrade and large

navigation error can occur before they are detected. Therefore, it is important to simulate and compare

the RAIM algorithms studied, namely the Least-Squares-Residuals and the Range-Comparison Method,

to test its performance, in case of a faulty measurement, for detection or detection and exclusion purposes.

These methods will also be compared in terms of complexity of implementation being each one analysed

for their advantages and disadvantages.

1.3 Structure and proposed approach

This thesis is composed by 7 chapters, including the present one, which are organized as follows.

In Chapter 2 the GPS and Galileo systems studied in this thesis are presented. An overview of the

systems is introduced and their architecture is explained.

In Chapter 3 the fundamental concepts involved in this thesis are presented and explained. This

chapter starts with an overview of the coordinate systems and coordinate systems conversions followed

by an explanation of the error sources that can corrupt the measurements from the satellites. The chapter

ends with a brief description of the DOP concept and its importance to the position solution.

In Chapter 4 the equations that allow the receiver to calculate its position are presented and explained.

The solution of these equations can be obtained from several methods and two of them are presented and

explained.

The Chapter 5 explains the definition of integrity and several developed architectures to provide

integrity are presented. The RAIM algorithms studied in this thesis are also explained in detail in this

chapter and the Galileo integrity concept is shown.

Finally, in Chapter 6 the computer simulation is presented and the results obtained from the simula-

tions are shown. In this chapter the comparison from the RAIM algorithms is performed for the various

test cases considered.

Chapter 7 summarizes the results obtained from the simulations and conclusions from those results

are drawn.

1.4 State of the art

There are different types of RAIM algorithms: conventional RAIM, Relative RAIM, and Advanced RAIM

[25].

2

Conventional RAIM algorithms focus on pseudorange measurements using those measurements and

position noises, along with the false alarm and missed detection probability to compute the corresponding

protection levels for the position solution. These techniques can be decomposed in two phases. The first

is the detection and exclusion phase where the detection threshold is computed based on the noise level

and required false alarm probability. The detection is made when the detection threshold is exceeded by

the test statistic, followed by an exclusion from the faulty satellite whenever the number of satellites in

use allows it. The second phase may be optional, depending on the purpose, and consists in calculating

the protections levels based on the geometry and confirm if the necessary protection limits are available

according to the application.

Relative RAIM (RRAIM) uses precise carrier phase measurements to propagate older pseudorange

based positions solutions forward in time. RAIM is performed on the carrier trajectory to ensure integrity

and the new protection levels are calculated based upon the original values and the accumulated uncer-

tainty over time. RRAIM concept shares the integrity task between the aircraft, the GNSS constellation

and the external monitors which provide the a priori set of valid measurements [25].

The Advanced RAIM (ARAIM) scheme was introduced by the GNSS Evolutionary Architecture Study

(GEAS) to guarantee LPV-200 (localizer performance with vertical guidance) operation worldwide. This

algorithm uses multi constellations data and takes advantage from dual frequency capabilities as well as

an integrity support message.

3

4

Chapter 2

Global Navigation Satellite Systems

Navigation is the science of getting a person or an equipment from one place to another and it has been

used daily in our lives for many centuries.

Centuries ago explorers used to navigate by primitive charts and observations of the sun and stars

to determine directions. The twentieth century brought important advances to navigation with radio

beacons, radars, gyroscope compass and, ultimately, satellite navigation systems.

2.1 GPS overview

The Global Positioning System is a satellite based navigation system originally made up of 24 satellites,

placed into orbit by the U.S Department of Defence. Initially it was meant for military applications, but

in the 80’s the government of the USA made the system available for civilian use. The system became

fully operational in 1995 when the 24 satellites constellation was in place and the testing was complete

[17].

GPS was originally a military project but intended for military and civilian applications. Nowadays

its applications are far beyond the usual positioning of planes or boats, and the GPS system is used for

cartography, automated vehicles, fleet tracking, surveillance, agriculture, etc.

As a dual system, the service provided to military PPS (Precise Positioning Service) is intended for US

authorized military and selected government agency users, and its access is controlled through cryptog-

raphy. The SPS (Standard Positioning Service) is available to all users worldwide free of direct charges.

At the moment SPS is the predominant satellite navigation service used all over the world. Civilian use

of the PPS is permitted but only with special US DOD (Department of Defence) approval. This system

is controlled by two cryptographic features known as antispoofing (AS) and selective availability (SA).

AS has the objective of protecting the signal against any data replication by the adversary which would

deceive the victim’s receiver. SA policy added controlled errors to the signal available to unrestricted

use, degrading the resultant signal, with a significant loss of accuracy. SA was deactivated on 2000 which

represented an important milestone for civil users of GPS.

5

2.2 GPS Architecture

The GPS navigation system currently consists of three major segments. These are the space segment, a

control segment and an user segment.

2.2.1 Space Segment

The space segment is composed by the orbiting GPS satellites, and its main functions are to transmit

radio-navigation signals and to store and retransmit the navigation message sent by the Control Segment.

The GPS constellation was initially constituted by 24 space vehicles distributed in six orbital planes

with an inclination of 55 degrees in relation to the equator and are separated 60 degrees from each other.

Orbits are near circular and have a semi-major axis of 26560 Km, meaning their altitude is 20200 Km.

The satellites have a velocity around 3.9 Km/s making their period almost half of a sidereal day (11 h

58 m) so that the satellites pass over the same location every day. The present configuration allows users

to have a simultaneous observation of at least 6 satellites in view worldwide. Nowadays there are 32

satellites in the GPS constellation. These additional satellites improve the precision of the GPS receiver

by providing redundant measurements.

2.2.2 Control Segment

The control segment consists in a network of ground stations that perform a series of tasks in order

to maintain a proper operation of the GPS system. The control segment is composed by a network of

Monitor Stations (MS), a Master Control Station (MCS), a backup of the MCS and the Ground Antennas

(GA). The MCS processes the measurements received by the MS to estimate satellite orbits (ephemeris)

and clock error and generate the navigation message. These corrections and the navigation message are

uploaded to the satellites through the GA, which are located in the MS.

The summary main tasks performed by the control segment are:

• Monitoring and control of satellite orbital parameters;

• Monitoring health status of the satellite subsystems;

• Activation of spare satellites;

• Update parameters in the navigation message;

• Resolving satellite anomalies.

2.2.3 User Segment

The user segment includes all the users equipped with an L-band radio receiver/processor and antennas

which receive GPS signals in order to obtain their coordinates and provide an accurate time. This segment

is composed of hundreds of thousands of US and allied military users of the secure GPS PPS and tens of

millions of civil, commercial and scientific users of the SPS.

6

Figure 2.1: Illustration of GPS architecture

2.3 Galileo overview

Galileo is the GNSS that is being developed by the European Union (EU) and the European Space Agency

(ESA) and promises to provide highly accurate global position service under civilian control. Galileo will

provide highly accurate autonomous navigation and positioning services but it will also be interoperable

with other systems like GPS and GLONASS, the Russian GNSS.

Galileo program is thought to be very important because of today’s dependence of satellite navigation

in our daily lives. As GPS and GLONASS are military systems under military control, these services can

be switched off or made less precise when needed for the civil service which can affect several businesses

including transportation, aviation, communications and many others.

The Galileo system will have five main services [17] [23] [15]:

• Open Service (OS): This service targets the generality of the users. It will provide PVT (position,

velocity and time) and will be suitable for applications like in-car navigation, mobile-phones location

services, etc. This service only requires the user to have a compatible receiver and no authorization

is needed. This service can be used for applications that require higher accuracies, as it can be

complemented with services of GNSS augmentation, but that do not need integrity as it is not

provided in this service.

• Safety-of-Life Service SoL: This service is intended to satisfy the needs of safety critical users,

such as civil aviation. In particular, it is intended to provide a guarantee of service, and to com-

ply with legislation applicable to all considered domains and existing standards (Standards and

Recommended Pratices-SARPSs- by ICAO).

7

The SoL service will be open with the ability to digitally authenticate signals (so as to assure users

that the received signal is the actual Galileo signal) [33].

Its key feature is to provide integrity information at global level. The applicable performances are

specified in table 2.1 and include two levels of risk exposure [33]:

– Critical level: corresponds to time-critical operations such as operations with vertical guidance

in aviation domain (such as approach and landing phases).

– Non-critical level: corresponds to less time-critical operations such as open sea navigation.

Safety-of-Life Service

Type of receiverCarriers Dual frequency

Computes Integrity YesIonosphericcorrection Based on dual frequency measurements

Coverage GlobalCritical Level Non-critical level

Accuracy (95%) H: 4 mV: 8 m H: 200 m

IntegrityAlert limit H:12 m, V:20 m H: 556 m

Time-to-Alert 5.2 seconds 10 secondsIntegrity Risk 2× 10−7/150s 10−7/h

Continuity Risk 8× 10−5/15s 10−4/h−10−8/h

Certification/Liability Yes

Availability of Integrity 99.5%Availability of Accuracy 99.8%

Table 2.1: Positioning performances for Galileo SoL Service [3]

• Commercial Service (CS): This service is intended for applications that require higher perfor-

mance than the one offered by the open service. This service uses a combination of two encrypted

signals for higher data throughput rate and higher accuracy authenticated data. These will be

developed by service providers, that will buy the rights to use the two commercial signals from the

Galileo operator. The user of the CS will access these added value services by paying a fee. The

CS does not offer integrity information.

• Public Regulated Service (PRS): The PRS is a restricted service to government and authorized

users for sensitive applications that require high level of service continuity, such as security and

strategic infrastructures like energy, communications or finances. This service will be encrypted,

featuring anti-jamming mechanisms and reliable problem detection.

• Search and Rescue Service (SAR): Galileo satellites will be able to pick up signals of emergency

from emergency beacons carried on ship, planes or personal equipments and help to forward distress

signals to rescue coordination centres. With this information rescue centres can know the precise

location of an accident.

8

2.4 Galileo Architecture

The Galileo system that assures these services is divided in three major segments. These are the space

segment, a control segment and an user segment.

2.4.1 Space Segment

The Galileo space segment is composed of MEO satellites, with the following characteristics:

• nearly circular orbits with a semi-major axis of 29600 km (altitude 23222km)

• orbital inclination of 56

• three equally spaced orbital planes

The satellites will be organized as a Walker constellation [29]. A Walker constellation is defined by

three parameters T/P/F . T is the total number of satellites in the constellation, P is the number of

orbital planes and F defines the relative phase spacing between the satellites in the adjacent orbital

planes. Galileo constellation will be a Walker 27/3/1 comprising a total of 30 MEO satellites, of which 3

are spares. Galileo satellites have a velocity around 3.7 Km/s making their period equal to 14 hours and

7 minutes.

2.4.2 Control Segment

Galileo ground segment is composed by two sub-segments which are:

• the Ground Control Segment (GCS), whose purpose is to monitor and control the constellation

through the use of Telemetry Tracking and Command (TT&C) uplinks

• the Ground Mission Segment (GMS), whose purpose is to compute and disseminate the information

required for navigation (orbit determination and clock synchronization) and integrity monitoring

(SISA, SISMA, IF).

The second sub-segment is composed by:

• the Galileo Control Center (GCC),which includes all control and processing facilities

• the network of Galileo Sensor Stations (GSS) which collect navigation data from the satellites and

meteorological data

• Uplink Stations to disseminate the computed information

• A global area network linking all system elements

2.4.3 User Segment

The Galileo User Segment consists in the users equipped with receivers. Like in the GPS User Segment the

main functions are to receive Galileo Signals, determine pseudoranges and solve the navigation equation

to obtain their coordinates and provide accurate time synchronization.

9

10

Chapter 3

Fundamentals

In this chapter the subjects being discussed are: Coordinate Systems where a brief description of the

several referential used is done and some important conversions are presented, the structure of a generic

GNSS receiver is discussed as well as its signal acquisition process, the main errors that affect measure-

ments are presented and their summarized effect in terms of error bias is shown and finally the DOP

concept is presented.

3.1 Coordinate System

It is now necessary to choose a reference coordinate system in which both satellite and receiver can be

represented. In this chapter the most common coordinate frames used will be discussed.

3.1.1 ECI

Earth Centered Inertial (ECI) coordinate system has its origin at the center of the Earth. This reference

frame is called inertial because its axes are pointing in fixed directions with respect to the stars. This

reference frame is commonly used for objects in space, like GNSS satellites as it is simpler to describe

their motion in a non-rotating frame. ECI reference frame is also useful to specify directions towards

celestial objects [17].

The ecliptic is the plane of the Earth’s mean orbit about the sun [13] and it does not coincide with

Earth’s equatorial plane. The angle between the Earth’s mean equator and ecliptic is called the obliquity

of the ecliptic ε [13] and its value is about 23.5 . As this angle is not constant and changes with time to

make this frame truly inertial, the orientation of the axis is defined at a specific instant in time.

One common ECI frame used for GPS satellites is denoted as J2000. This frame uses the orientation

of the equatorial plane at 12.00 hours UTC on January 1, 2000.

The xx axis as its origin in Geocenter (Earth centre of mass) and its direction is fixed towards the

mean equinox (at J2000), zz axis is defined by the direction of the Earth mean rotation pole and yy axis

is rotated 90 East about celestial equator so the system is right handed [37].

11

Figure 3.1: Definition of ECI reference frame

3.1.2 ECEF

For the purpose of calculating the user position using a GPS receiver or its velocity it is more convenient

to use an Earth-centered, Earth-fixed (ECEF) referential which rotates in inertial space in order to remain

fixed with respect to the surface of the Earth. ECEF uses three-dimensional XYZ coordinates (in meters)

to describe the location of a GPS user. The term “Earth-centered”, like in the ECI referential, comes

from the fact that the origin of the axes (0,0,0) is located at the centre of mass of the Earth. The xx axis

as its origin in the Earth center and points in the direction of 0 longitude (Greenwich meridian), the

yy axis points in the direction of 90 East longitude, passing the equatorial plane and zz axis is defined

by the direction of the Earth mean rotation pole. Since the xx axis is fixed to Greenwich meridian this

reference frame will rotate along the zz axis with the Earth angular velocity.

Figure 3.2: Definition of ECEF reference frame [42]

12

WGS84

It is much more common to express geodetic mapping coordinates in Latitude, Longitude and Altitude

(LLA). Latitude, Φ, is a geographic coordinate that specifies the North-South position of a point on the

Earth surface. Latitude is defined by an angle which ranges from 0 at the Equator to 90 at North or

South poles. Longitude, λ, is a geographic coordinate that specifies the East-West position of a point on

the Earth surface. Similarly to Latitude, Longitude is also defined by an angle ranging from 0 to 180 .

Altitude or height expresses the minimum distance between the user and the reference ellipsoid and is

measured in meters.

Figure 3.3: Latitude Φ and Longitude λ on an ellipsoid

Because the Earth has a complex shape, a method to approximate Earth’s shape is required, and for

that we can use a reference ellipsoid which allows for the conversion of the ECEF coordinates (x, y, z) to

LLA coordinates. A reference ellipsoid can be described by a series of parameters that define its shape.

The standard physical model of the Earth used for GPS applications is the World Geodetic System 1984

(WGS 84) [5].

WGS 84 is an ECEF system and a geodetic datum. WGS 84 is based on a consistent set of constants

and model parameters that describe the Earth’s size, shape, gravity and magnetic fields. The parameters

that define this reference frame are listed on the table below [1]:

Parameter ValueSemi-Major axis (a) 6378137.0 meters

Flattening of the Earth (f) 1298.257223563

Nominal Mean Angular Velocity of the Earth (ω) 7.2921151467× 10−5rad/s

Geocentric Gravitational Constant(Mass of the Earth Atmosphere included) (GM) (µ) 3.986004418× 1014m3/s2

Table 3.1: Fundamental parameters of WGS 84

3.1.3 ENU

For some applications or calculations a local reference frame is more practical than using ECEF Cartesian

or geodetic coordinates. A possible local referential is the East, North, Up (ENU) reference frame. The

ENU reference frame is defined at a specific location at the Earth surface, usually the receiver or user

13

position and the E-N plane often coincides with the Earth surface at the origin point. This reference

frame is illustrated in the next figure.

Z

Y

X

North

East

Up

ecef

ecef

ecef

φ

λ

Figure 3.4: Definition of ENU reference frame

3.2 Coordinate System Conversions

Sometimes it is useful to convert from one coordinate system to other. In this section the coordinates

systems conversions used in this thesis will be presented.

3.2.1 Cartesian to Geodetic

Using the constants presented on table 3.1, the conversion from Cartesian to geodetic coordinates can be

done using the following method [32]:

By iteration for φ and h. There is quick convergence for h N .

λ = arctan 2(Y

X

)(3.1)

Starting with h0 = 0

φ0 = arctan(

Z

p (1− e2)

)(3.2)

Iterate φ and h

Ni = a√1− e2 sin2 φi

(3.3)

hi+1 = p

cosφi−Ni (3.4)

14

φi+1 =

arctan Z

p

(1− e2 Ni

Ni + hi+1

) (3.5)

where the auxiliary values are,

p =√X2 + Y 2 (3.6)

e =√a2 − b2

a2 (3.7)

b = a(1− f) (3.8)

3.2.2 Geodetic to Cartesian

Geodetic coordinates can be converted to Cartesian using:

X = (N + h)× cosφ cosλ (3.9)

Y = (N + h)× cosφ sinλ (3.10)

Z =(N(1− e2) + h

)sinφ (3.11)

where

φ : latitude

λ : longitude

h : height above the ellipsoid (meters)

N : radius of curvature (meters) defined as: a√1− e2 sin2 φ

3.2.3 ECEF to ENU

It is possible to convert ECEF (cartesian) coordinates of a point P = (x, y, z) to a local ENU reference

frame. As stated before this ENU reference frame will have its origin at the user location, in this case

Pu = (xu, yu, zu) as shown in figure 3.5.

The transformation equation are given by [35], [32]:

x′

y′

z′

=

− sinλu cosλu 0

− sinφu cosλu − sinφu sinλu cosφucosφu cosλu cosφu sinλu sinφu

x− xuy − yuz − zu

(3.12)

where φu and λu are respectively latitude and longitude of Pu. The coordinates (x′, y′, z′) are the new

coordinates of P relative to the new ENU reference frame.

15

P

U

N

E

εα

Pu

Figure 3.5: ENU reference frame

For z′(t) > 0 the point P is above the skyline. The azimuth angle α and the elevation angle ε of point

P can be obtained from:

tanα = x′

y′(3.13)

sin ε = z′√(x′)2 + (y′)2 + (z′)2

(3.14)

3.3 Receiver Structure

GNSS receivers determine the user position, velocity and precise time (PVT) by processing the signals

broadcasted by satellites. Because the satellites are in motion, the receiver has to continuously acquire

and track signals from the satellites in view, in order to compute an uninterrupted solution.

The receiver’s architecture is adjusted to the different GNSS systems available and the different

applications, but the basic building blocks of a generic GNSS receiver are shown in figure 3.6 [24].

Signal Antenna Front EndBasebandProcessing

ApplicationProcessing PVT

Figure 3.6: Generic receiver architecture

The receiver’s components have the following functions:

• Antenna: captures the GNSS signals, noise and possible interferences.

16

• Front End: The front-end typically down-converts, filters, amplifies and digitizes the incoming

signals.

• Baseband signal processing: Several signal processing routines are used to acquire and track

the different signals. The tracking mode is carried out by two types of closed loops: the phase-lock

loop (PLL) and the delay-lock loop (DLL) that estimates the code of the received signals.

• Applications processing: Depending on the application, the receiver performs different tasks

with the resulting GNSS information, and provides meaningful results to the user. The main task

is the determination of the navigation equation solution.

Most of the GNSS systems use Code Division Multiple Access (CDMA) techniques to multiplex

several satellite signals onto the same frequency. The basic concept behind the CDMA scheme is that

each satellite is assigned with a pseudorandom noise code (PRN code) that modulates the transmitted

signal [24]. Besides, the PRN codes have properties such that autocorrelation function is at maximum

when they are completely aligned and decays rapidly to zero in case of misalignment. GNSS receivers

have prior knowledge of each satellite’s PRN code, and correlate the incoming signals with local code

replicas to determine if a given satellite is visible or not. This task is performed by the baseband signal

processing component. An example of the acquisition process showing the Doppler/code delay search

space and correlation peak is shown in figure 3.7.

Figure 3.7: Example of an acquisition process

In their most common architecture, GNSS receivers assign a dedicated channel to each signal being

tracked and, for the case of multi-frequency receivers, several signals from each satellite can be pro-

17

cessed independently. In order to ensure tracking of the signals in each processing channel, receivers are

continuously estimating and correcting two parameters:

• Code delay: quantifies the misalignment between the local PRN code replica and the incoming

signal. The code delays permit to determine the pseudoranges that are used to solve the navigation

equation.

• Carrier phase (or its derivative, the Doppler frequency): reflects the relative motion between the

satellite and the user.

To determine these parameters the receiver uses tracking loops that form the core of the signal

processing and continuously track the incoming satellite signal in order to generate the code and

carrier phase measurements. Each estimate of the code delay and the carrier phase is used to

regenerate the local PRN code replica, which is then correlated again with the incoming signal.

The result of this operation is then re-assessed at the receiver to re-estimate these parameters,

in a continuous loop. After synchronization with the incoming signals and demodulation of the

navigation message, the data is returned to the application processing block and the receiver uses

the information from the tracking loops for different purposes like determining pseudoranges to

each satellite and computing a navigation solution, performing time transfer or simply collecting

data to be post-processed in ground stations.

3.4 Measurement Errors

There are a number of error sources that can corrupt the measurements from each satellite. The errors

are often categorized as noise or bias. Noise generally refers to a quickly varying error that averages to

zero over a short period of time. A bias tends to persist over a period of time.

These sources can be grouped according to their origin, and can be categorized in three groups [32]:

• Errors in the parameters values broadcast by a satellite in its navigation message;

• Uncertainties associated with propagation medium which affect the travel time of the signal from

a satellite to the receiver (includes ionospheric and tropospheric effects);

• Receiver noise which affects the precision of a measurement, and interference from signals reflected

from surfaces in the proximity of the antenna (includes thermal noise and multipath effects).

In this section the main error sources will be emphasized and a final GPS error budget will be defined.

3.4.1 Satellite Clock Error

GPS satellites contain atomic clocks that control on-board timing operations, including broadcast signal

generation. Although these clocks are highly stable they can experience noise and clock drift errors. To

account for these errors the navigation message contains corrections and estimates of the atomic clock

accuracy. As these parameters are computed using a curve-fit to predicted estimates of the actual clock

18

errors [17] some residual errors still remain as they may not indicate the clock’s current state. These

errors tend to be very small but may add up to few meters of inaccuracy, typically varying from 0.3 - 4m

[17].

Range errors due to residual clock errors slowly degrade over time until next upload. These errors are

recorded and analysed by user equipment that is tracking all visible satellites with ages of data (AoD)

varying from 0 to 24 hours, defined as the typical time between two data uploads [17]. Averaging over

AoD, a nominal 1σ clock error for the constellation in 2004 was 1.1 meter based on data from [38], [14].

3.4.2 Ephemeris Error

The ephemeris error corresponds to the differences between the real orbit and the estimated orbit from

the ephemeris. Ephemerides are estimated and uplinked for each satellite which rebroadcasts them to

the user. As with satellite clock errors these ephemeris parameters are estimated using a curve-fit [17]

of the control’s segment best prediction of each satellite’s position at the time of uplink. An illustration

of this effect is shown in figure 3.8. These residual errors have an effective error on measurements in the

order of 0.8 meters (1σ) [41].

Figure 3.8: Ephemeris Error Illustration

3.4.3 Atmospheric Effects

From the satellites to the receiver signals travel through different mediums that have different indices of

refraction which have effects on signal propagation. These effects include the "excess" of GNSS signal

path due to the variation of the refraction index in the atmosphere and, mainly, to the signal delays or

advances due to its propagation velocity in the atmosphere.

The index of refraction of the signals is given by

19

n = c

v(3.15)

where

c : light speed in vacuum

v : light speed in the medium

As the Earth atmosphere is constituted from several layers it is important to understand how these

layers can influence signal propagation and therefore the measurements error.

The two atmospheric layers that most influence signals propagation are ionosphere and troposphere

[17] so it is important to understand the contribution from each of these effects in the measurements

error.

Ionospheric Effects

The ionosphere is the zone of the atmosphere that extends approximately from 70 to 2000 kilometres

above the Earth’s surface. This layer contains a partially ionized medium resulting from X and UV rays

of Solar radiation and the incidence of charged particles.

The propagation speed of the GNSS signals in the ionosphere depends on its electron density. A map

of the worldwide Total Electron Content can be seen in figure 3.9.

Figure 3.9: Example of a worldwide TEC map [21]

A medium where the angular frequency ω and the wave number k are not proportional is a dispersive

medium (i.e the refractive index depends on the frequency). This is the case with the ionosphere where

ω and k are related by [12]:

ω2 = c2k2 + w2p (3.16)

20

and

wp = 2πfp

fp = 8.98√Ne

(3.17)

where Ne is the Electron Density in e/m3

The TEC influences the ionospheric refraction [22] which depends on user location as seen in figure

3.9. Also, as the ionosphere is a dispersive medium, GNSS signals refraction depends on its frequency.

This dependence on the signal frequency allows us to remove its effect using two frequency measurements.

Single frequency receivers have to apply an ionospheric prediction model (such as the Klobuchar model)

to remove this effect. A typical 1σ value for residual ionospheric delays, averaged over the globe and over

elevation angles is 7 meters [31].

Tropospheric Effects

Troposphere is the atmospheric layer below ionosphere, from the Earth surface to an altitude of about 70

kilometres. The effect of the troposphere on GNSS signals appears as an extra delay in the measurement

of the signal [22] travelling from the satellite to the receiver. This delay depends on temperature, pressure,

humidity and satellite and receiver position according to [22]:

∆ =∫straight line

(n− 1)dl (3.18)

it can be written as

T =∫

(n− 1)dl = 10−6∫Ndl (3.19)

where n is the refractive index of the air and n = 10−6(n − 1) is the refractivity. The refractivity can

be divided in hydrostatic (dry gases) and wet (water vapour). Each of these components has different

effects on GNSS signals.

Troposphere is a non dispersive media with respect to electromagnetic waves up to 15 GHz, so the

tropospheric effect are not frequency dependent for GNSS signals. A consequence of this is that delay in

the troposphere can not be removed by combination of dual frequency measurements. The only way to

mitigate this effect is to use models or estimate it from observational data.

The longer the path of the GNSS signals through the troposphere the higher is the delay (3.18), (3.19),

so lower elevation angle satellites will have their measurements most affected by this error. An average

of the troposphere delay 1σ is 0.2 meters [17].

3.4.4 Receiver Noise

Receiver noise also affects measurements, though, in a small proportion compared with other error sources.

The dominant source of measurements errors on the receiver are due to thermal noise and the effect of

interferences [17]. Typical modern receivers have a 1σ value for noise and resolution in the order of 0.1

meters.

21

Figure 3.10: GNSS signals path deviation due to Atmospheric Effects

3.4.5 Multipath

Multipath interference is generated when signal arrives by two or more paths to the antenna. Since the

path travelled by a reflection is always longer than the direct path, multipath signals arrivals are delayed

relative to the direct path. When this delay is large the receiver can resolve the multipath. As long as

the receiver tracks the direct path these resolvable multipath have little influence. If multipath reflections

are from near surfaces, these interferences can arrive with a short delay after the direct path which can

produce more considerable errors in position, velocity and time [17]. Signals from satellites with low

elevations are also more likely to suffer from multipath effects [22].

Typical 1σ value for noise from multipath is in the order of 0.2 meters.

Figure 3.11: Multipath effect Illustration

3.4.6 Pseudorange Error Budget

As discussed in this section several parameters can influence the measurement error from GNSS mea-

surements.

The combination of these errors is known as User Equivalent Range Error (UERE) and corresponds

22

to the root-sum-squared of the components that contribute to measurements errors. These errors are

considered independent Gaussian random variables [17], thus the UERE can be identically distributed

by the satellites.

The summary of these errors for SPS single frequency C/A code receiver is summarized in table 3.2.

Segment Source Error sorce 1 σ Error (m)

Space Broadcast clock 1.1L1 P(Y)-L1 C/A group delay 0.3

Control Broadcast ephemeris 0.8

UserIonospheric delay 7

Receiver noise and resolution 0.1Multipath 0.2

System UERE Total 7.1

Table 3.2: Pseudoranges error table for SPS C/A code [17]

3.5 DOP

Dilution of Precision is a term used to specify the multiplicative effect of the visible satellites constellation

geometry on the solution precision. This effect can be easily understood in the following figures.

Figure 3.12: Good DOP geometry

Figure 3.13: Bad DOP geometry

From figure 3.13 we can understand that when visible satellites are close together its geometry is poor

and their respective DOP value is high. When the satellites are separated in the sky their geometry is

good (figure 3.12) and their respective DOP value is low.

Depending on the applications some solutions may have to be discarded due to the poor satellite

geometry and respective DOP value. In table 3.3 the DOP values and their meanings are presented.

23

DOPvalue Rating Description

1 Ideal Highest possible confidence level to be used in for applications demanding thehighest precision.

1-2 Excellent Positional measurements are considered accurate enough for most sensitive ap-plications

2-5 Good Minimum level appropriate for business decisions.5-10 Moderate Positional measurements could be used for calculations but a more open view of

the sky is recommended.10-20 Fair Represents a low confidence level. Positional measurements should only be used

to have a rough estimate of the current location>20 Poor At this level the solution should be discarded

Table 3.3: DOP value description

24

Chapter 4

Position Determination

Let’s consider

Di =√

(Xi − xR)2 + (Yi − yR)2 + (Zi − zR)2 (4.1)

the distance between the satellite i with coordinates (Xi, Yi, Zi), and the receiver with coordinates

(xR, yR, zR).

GPS receivers receive and decode signals from satellites with information about their positions, en-

abling the receivers to compute their own position [27].

Let’s consider tR is the nominal time (time indicated by the clock) on the receiver at the time of signal

reception and ti the nominal time on the satellite at time of signal output. The time on the satellites is

obtained by the set of atomic clocks while the time on the receiver is obtained by a quartz clock with a

much lower accuracy. The relations between nominal time and real time ti and tR are

tR = tR + ∆tR (4.2)

ti = ti + ∆ti (4.3)

where ∆ti and ∆tR are respectively the clock errors of the satellite and receiver. Generally |∆ti| |∆tR|.

Pseudorange is a distance measured between the satellite and the receiver

ρi = (tR − ti)c (4.4)

where c is the light speed.

The pseudorange ρi would be equal to the geometric distance Ri if the propagation medium was

vacuum and if there were no clock errors or other disturbances to the signal propagation. As this is not

the case these distances are not coincident. These errors were discussed in more detail in chapter 3.4.

To determine the user position (xu, yu, zu) and clock offset tu, pseudorange measurements are made

25

relatively to M satellites, with M ≥ 4, resulting in [27]:

ρ1 =√

(X1 − xu)2 + (Y1 − yu)2 + (Z1 − zu)2 + ctu + ε1

ρ2 =√

(X2 − xu)2 + (Y2 − yu)2 + (Z2 − zu)2 + ctu + ε2

...

ρM =√

(XM − xu)2 + (YM − yu)2 + (ZM − zu)2 + ctu + εM

(4.5)

where

ρi : pseudorange

(Xi, Yi, Zi) : coordinates of satellite i

(xu, yu, zy) : coordinates of the user

εi : measurement error associated with ρi

Figure 4.1: Illustration of position calculation with 4 satellites

Generally these errors are considered independent, Gaussian, zero mean and equal variance.

As the equations are nonlinear there are several methods that can be used to obtain a solution, namely

[17]:

1. Bancroft algorithm;

2. Iterative solutions;

3. Kalman filtering.

Some of these solutions, like 1 and 2, carry little information to the next measurement, namely, they

carry no information about user dynamics. Kalman filtering solution overcomes this kind of limitation

[27]

26

4.1 Least-Squares

Considering equations (4.5) in the form:

ρi =√

(xi − xu)2 + (yi − yu)2 + (zi − zu)2 + τu (4.6)

where εi was not taken into account and τu = ctu.

Considering an approximate position location (xu, yu, zu) and a receiver clock offset τu:

ρi =√

(xi − xu)2 + (yi − yi)2 + (zi − zi)2 + τu (4.7)

The unknown user position and receiver clock offset consists of an approximate component and an

incremental component:xu = xu + ∆xu

yu = yu + ∆yu

zu = zu + ∆zu

τu = τu + ∆τu

(4.8)

Let us consider the following Taylor series expansion: [27]

ρi = ρi + ∂ρi∂xu

∆xu + ∂ρi∂yu

∆yu + ∂ρi∂zu

∆zu + ∂ρi∂τu

∆τu + . . . (4.9)

The expansion is truncate after first-order partial derivatives to eliminate nonlinear terms. The partial

derivatives evaluate [27]:∂ρi∂xu

= −xi − xuri

∂ρi∂yu

= −yi − yuri

∂ρi∂zu

= −zi − zuri

∂ρi∂τu

= 1

(4.10)

where

ri =√

(xi − xu)2 + (yi − yu)2 + (zi − zu)2 (4.11)

Substituting (4.10) in (4.9) [27]:

ρi − ρi = xi − xuri

∆xu + yi − yuri

∆yu + zi − zuri

∆zu −∆τu (4.12)

27

For convenience:∆ρi = ρi − ρi

axi = xi − xuri

ayi = yi − yuri

azi = zi − zuri

(4.13)

Rewriting (4.12):

∆ρi = axi∆xu + ayi∆yu + azi∆zu − c∆τu (4.14)

For four satellites and in matrix notation:

∆ρ =

∆ρ1

∆ρ2

∆ρ3

∆ρ4

H =

ax1 ay1 az1 −1

ax2 ay2 az2 −1

ax3 ay3 az3 −1

ax4 ay4 az4 −1

∆v =

∆xu∆yu∆zu∆τu

(4.15)

Finally we obtain:

∆ρ = H∆v (4.16)

which leads to the solution:

∆v = H−1∆ρ (4.17)

Once the unknowns are computed the user position and the receiver clock offset can be calculated

using (4.8).

The solution (4.17) assumes that 4 satellites are being used. If the number of satellites is bigger than

4 the matrix H takes the form:

H =

ax1 ay1 az1 −1

ax2 ay2 az2 −1... · · ·

......

axn ayn azn −1

(4.18)

and solution (4.16) cannot be (4.17). In this case the least squares method can be used:

∆v = K∆ρ (4.19)

where

K = (HTH)−1HT (4.20)

User coordinates are

vk = vk−1 + ∆vk = vk−1 + (HTk Hk)−1HT

k ∆ρ (4.21)

In figure 4.2 the flowchart represents the recursive Least-Squares scheme.

28

Computation of Hk and ρk

Computation of vk

k → k − 1

Figure 4.2: Recursive Least-Squares Scheme

4.2 DOP calculation

DOP parameters can be defined as the quotient between the RMS positional error and the σUEREparameter.

DOP = ErrorRMS

σUERE(4.22)

Taking now into account (4.19) and (4.20), and assuming that ∆ρ is a Gaussian vector of independent,

zero-mean errors with equal variances σ2UERE , we have

E∆v∆vT = σ2UERE(HTH)−1 (4.23)

Let

E∆v∆vT =

σ2xu σ2

xuyu σ2xuzu σ2

xuctu

σ2xuyu σ2

yu σ2yuzu σ2

yuctu

σ2xuzu σ2

yuzu σ2zu σ2

zuctu

σ2xuctu σ2

yuctu σ2zuctu σ2

ctu

(4.24)

The most generic parameter, GDOP (Geometric DOP), can be calculated from:

GDOP = 1σUERE

√σ2xu + σ2

yu + σ2zu + σ2

ctu (4.25)

If we write matrix (HTH)−1 in the form

(HTH)−1 =

h11 h12 h13 h14

h21 h22 h23 h24

h31 h32 h33 h34

h41 h42 h43 h44

with hij = hji (4.26)

we can obtain for (4.25)

29

GDOP =√trace(HTH)−1 =

√h11 + h22 + h33 + h44 (4.27)

Several other DOP parameters in common use are useful to characterize the accuracy of various

components of the position/time solution. These are termed Position DOP (PDOP) and time DOP

(TDOP). These can be calculated from:

PDOP = 1σUERE

√σ2xu + σ2

yu + σ2zu (4.28)

or

PDOP =√h11 + h22 + h33 (4.29)

TDOP = σctuσUERE

(4.30)

or

TDOP =√h44

c(4.31)

Other parameters like Horizontal DOP (HDOP) and Vertical DOP (VDOP) can be calculated similarly

but these parameters are both dependent on the coordinate system used. In this case an ENU reference

frame must be used. The expressions are

HDOP = 1σUERE

√σ2xu + σ2

yu (4.32)

V DOP = σzuσUERE

(4.33)

Assuming a matrix similar to (4.26) with hij we obtain

HDOP =√h11 + h22 (4.34)

or

V DOP =√h33 (4.35)

Generally the receivers use an algorithm based on GDOP parameter to select the best satellite set

among the satellites in view.

4.3 Position solution using Kalman Filter

Extended Kalman Filter is frequently used as an alternative to Least Squares to obtain a solution for the

navigation equation. The observables are incorporated in discrete time intervals, in this case of 1 second,

and the observation model is linearised relatively to the best state estimate. The xk state has five, eight

or eleven components (depending on the model chosen) including two components for the receiver clock

30

model [35].

4.3.1 Dynamics models

The number of space components depends on the dynamics model chosen. This model is usually chosen

according to the type of trajectory expected.

For a fixed receiver the P model is usually enough. The state vector dimension in this model is 5

(3 position coordinates plus 2 clock components). When we have a receiver in motion it is convenient

to use a PV model (position + velocity) with a state vector dimension of 8 (3 position coordinates + 3

velocity coordinates + 2 clock components). If the receiver suffers big accelerations in its trajectory the

PVA model is recommended, with a state vector dimension of 11 (3 position coordinates + 3 velocity

coordinates + 3 acceleration coordinates + 2 clock components). In this thesis it was chosen to use the

PV and PVA model.

In the PV model each coordinate is modelled as an integrated Brownian motion as illustrated in 4.3

gaussian whitenoise

1s

1s

positionvelocityX2

Figure 4.3: PV model for each coordinate using an integrated Brownian motion

In the PVA model we could model each coordinate as an integrated Brownian motion but it is more

appropriate to use a first order Gauss-Markov process to characterize the acceleration [35].

gaussian whitenoise

α

s+ β1s

1s

positionaccelerationX3

velocityX2 X1

Figure 4.4: PVA model for each position coordinate using a Gauss-Markov process for the acceleration

In addition to the models PV and PVA considered it is necessary to characterize the clock offset as

a state model.

4.3.2 Clock state model

A simple clock model is defined as a state vector with dimension two, in which frequency and phase show

variations of Brownian motion in reasonable time intervals. In figure 4.5 the state model considered in

[35] is shown.

The quantities uφ(t) and uf (t) are independent white noises, with zero mean and are characterized

by the covariance matrix:

Qu =

qφ 0

0 qf

(4.36)

in which qf and qφ are, respectively, the spectral density of power of uf (t) and uφ(t).

31

Gaussian whitenoise

1s

+ 1s

xφ(t) = ∆Txf (t) y(t)

frequency

uf (t)

Phase

Gaussian whitenoise

uφ(t)

Figure 4.5: Clock state model

The dynamics equation is xφ(t)

xf (t)

=

0 1

0 0

xφ(t)

xf (t)

+

uφ(t)

uf (t)

(4.37)

and the equation in discrete time is

xφ,k+1

xf,k+1

=

1 ∆t

0 1

xφ,k

xf,k

+

uφ,k

uf,k

(4.38)

where the covariance noise matrix is

Qk =∫ tk+1

tk

Φ(tk+1, τ)QuΦT (tk+1, τ)dτ

=

qφ∆t+ qf (∆t)3

3qf (∆t)2

2qf (∆t)2

2 qf∆t

≈ ΦQuΦT∆t(4.39)

The entries of matrix (4.36) are given by [35]

qφ ≈h0

2qf ≈ 2π2h−2

(4.40)

where h0 and h−2 are Van Allan variance parameters [40].

The values of h0 and h−2 depend on the clocks used in the GPS receivers. The typical values are

shown in table 4.1 and correspond to the clock offsets measured in seconds.

As these values will be used for clock offsets measured in meters, they have to be multiplied by squared

speed of light.

Type of oscillator h0 h−2

Temperature compensated crystal 2× 10−19 2× 10−20

Crystal oven 8× 10−20 4× 10−23

Rubidium 2× 10−20 4× 10−29

Table 4.1: Typical values of h0 and h−2 depending on the oscillator used [40]

32

For the simulation performed in this thesis it is required to generate the values of

uφ

uf

with

correlated components of matrix 4.39. The values of uφ and uf can be obtained from [27]:

uφ = aW1

uf = bW1 + cW2

(4.41)

where W1 and W2 are independent, zero-mean Gaussian r.v. with zero mean variance, that is

W1 ∼ N(0, 1)

W2 ∼ N(0, 1)(4.42)

thus

σ2uφ

= Eu2φ = a2EW 2

1 = a2 (4.43)

σ2uf

= Eu2f = Eb2W 2

1 + c2W 22 + 2bcW1W2

= b2 + c2 + 2bc× EW1W2 (4.44)

Euφuf = EaW1(bW1 + cW2) = ab (4.45)

Qk =

σ2uφ

σ2uφuf

σ2uφuf

σ2uf

=

a2 ab

ab b2 + c2

(4.46)

So from 4.39

a2 = qφ∆t+ qf (∆t)3

3

ab = qf (∆t)2

2b2 + c2 = qf∆t

(4.47)

Solving for a, b and c yields

a =√qφ∆t+ qf (∆t)3

3

b = qf (∆t)2

2ac =

√qf∆t− b2

(4.48)

4.3.3 Extended Kalman filter: dynamics models

The dimensions of the dynamics models depend on the model used. As stated before in this thesis the

models PV and PVA were implemented, therefore they will be described in detail.

33

PV model

The state space equation of the continuous model for the xu coordinate is: x1(t)

x2(t)

=

0 1

0 0

x1(t)

x2(t)

+

0

uv(t)

(4.49)

where noise vector covariance is E[0uv(t))]T [0uv(τ)] = Qδ(t− τ) and the covariance matrix is

Q = qv

0 0

0 1

(4.50)

For a discrete time model we obtain x1,k+1

x2,k+1

=

1 ∆t

0 1

x1,k

x2,k

+

u1,k

u2,k

(4.51)

where the covariance noise matrix is

Qk = E[u1,ku2,k]T [u1,ku2,k] = qv∆t

(∆t)2

3∆t2

∆t2 1

(4.52)

If xk = [x1,k . . . x8,k]T is the state vector in which x1,k and x2,k are the position and velocity of the xuuser coordinate, x3,k and x4,k are the position and velocity of the yu user coordinate and x5,k and x6,k

are the position and velocity of the zu user coordinate, the dynamics model in discrete time is given by

x1,k+1

x2,k+1

x3,k+1

x4,k+1

x5,k+1

x6,k+1

x7,k+1

x8,k+1

=

1 ∆t

1 0

1 ∆t

1

1 ∆t

1

0 1 ∆t

1

︸ ︷︷ ︸

Φk

x1,k

x2,k

x3,k

x4,k

x5,k

x6,k

x7,k

x8,k

+

u1,k

u2,k

u3,k

u4,k

u5,k

u6,k

u7,k

u8,k

(4.53)

34

where Φk is the dynamics matrix and the elements of the noise covariance matrix Qk are

qii = qv(∆t)3

3

qi,i+1 = qi+1,i = qv(∆t)2

2qi+1,i+1 = qv∆t, i = 1, 3, 5

q77 =[qφ∆t+ qf (∆t)3

3

]c2

q78 = q87 = qf (∆t)2

2 c2

q88 = qfc2∆t

(4.54)

and the other elements are zeros.

The qv quantity can be calculated using (4.51) yielding

acceleration ≈ x2,k+1 − x2,k

∆ t = u2,k

∆t (4.55)

so

qv = Eu22,k ≈ ∆t2(acceleration)2 (4.56)

PVA model

In this model positions xu, yu and zu are modelled as Gauss-Markov processes doubly integrated. So,

if x1, x2 and x3 are the state variables corresponding to position, velocity and acceleration of the xucoordinate, we can obtain

x1(t)

x2(t)

x3(t)

=

0 1 0

0 0 1

0 0 −β

x1(t)

x2(t)

x3(t)

+

0

0

α

u(t) (4.57)

where u(t) is a Gaussian white noise process with unity power spectrum.

The acceleration model is given by x3(t) = −βx3(t) + αu(t), corresponding to a low-pass Gaussian

process defined in the frequency by

j2πfX3(f) = −βX3(f) + αU(f) (4.58)

with transfer function

H(f) = X3(f)U(f) = α

β + j2πf (4.59)

The cutting frequency of x3(t) at −3dB is fc = β

2π and power is given by

∫ +∞

−∞

α2df

β2 + 4π2f2 = α2

2β (4.60)

35

The state transition matrix corresponding to (4.57) is

Φk =

1 ∆t 1

β2 (e−β∆t + β∆t− 1)

0 1 1β

(1− e−β∆t)

0 0 e−β∆t

(4.61)

The covariance matrix of the dynamics noise of discrete time corresponding to (4.57) is

Qk =

q11 q12 q13

q12 q22 q23

q13 q23 q33

(4.62)

where

q11 = α2

β5

[−e−2β∆t

2 − 2β∆te−β∆t − β2∆t2 + 12 + β∆t

]q12 = α2

β4

[e−2β∆t

2 + β∆te−β∆t − e−β∆t + β2∆t2

2 − β∆t+ 12

]q13 = α2

β3

[−e−2β∆t

2 − β∆te−β∆t + 12

]q22 = α2

β3

[−e−2β∆t

2 + 2e−β∆t + β∆t− 32

]q23 = α2

β2

[e−2β∆t

2 − e−β∆t + 12

]q33 = α2

2β[1− e−2β∆t]

(4.63)

To obtain the dynamics model in discrete time we use a state vector xk = [x1,k . . . x11,k] and the

process is the same as for the PV model, where x1,k, x2,k, x3,k are the position, velocity and acceleration

for the xu coordinate, respectively, and the same for yu and zu coordinates. The noise covariance matrix

Qk (11× 11) is block diagonal Qk = diagQk, Qk, Qk, Qk

4.3.4 Extended Kalman filter: observations model

The dynamics equations are linear for the two dynamics models implemented, but the observation’s

equation

zk = h[x(tk)] + vk (4.64)

is nonlinear. In (4.64) zk = [ρ1,k . . . ρn,k]T , with n ≥ 4, is the measured pseudoranges vector and

h[x] =

√

(x1 − xa)2 + (y1 − xb)2 + (z1 − xc)2 + xd...√

(xn − xa)2 + (yn − xb)2 + (zn − xc)2 + xd

(4.65)

36

where xi, yi and zi are the satellite coordinates of the i satellite, with i = 1, . . . , n and xa, xb, xc and xdare the state vector components of xu, yu, zu and xφc. The covariance noise matrix of the observations

is

Rk =

σ2

1,UERE 0

σ22,UERE

...

0 σ2n,UERE

(4.66)

If all the variances of the several pseudoranges are equal to σ2UERE , Rk will be given by

Rk = σ2UERE × I (4.67)

The observation’s matrix of the extended Kalman filter is

Hk =[∂hi[x(k | k − 1)]

∂xj

](n×l)

(4.68)

where n is the satellite number and l is the state vector dimension, depending on the dynamics model

adopted. For the PV model l = 8

Hk = −

ax1 0 ay1 0 az1 0 −1 0...

......

......

......

...

axn 0 ayn 0 azn 0 −1 0

(4.69)

For the PV A model l = 11

hk = −

ax1 0 0 ay1 0 0 az1 0 0 −1 0...

......

......

......

......

......

anx 0 0 ayz 0 0 azn 0 0 −1 0

(4.70)

where

axi = xi − xuri

ayi = yi − yuri

azi = zi − zuri

(4.71)

with

ri =√

(xi − xu)2 + (yi − yu)2 + (zi − zu)2 (4.72)

and

[xu yu zu]T = [xu(k | k − 1) yu(k | k − 1) zu(k | k − 1)]T (4.73)

In figure 4.6 the flowchart represents the Kalman filter applied to the resolution of the navigation

37

equation

Gain calculation:Kk = P−k H

Tk [HkP

−k H

Tk +Rk]−1

Estimation updatex+k = x−k +Kk(zk − zk)

Error covariance updateP+k = (I −KkHk)P−k (I −KkHk)T +KkRkK

Tk

Predictionx−k+1 = Φkx+

k

P−k+1 = ΦkP+k ΦTk +Qk

k + 1⇒ k

Initial conditionsx−0 P

−o

Observationsz0, z1, . . .

Estimatesx+

0 , x+1 , . . .

Figure 4.6: Kalman filter flowchart applied to the resolution of the navigation equation

In figure 4.6:

x−k = x(k | k − 1) x+k = x(k | k)

P+k = P (k | k) P−k+1 = P (k + 1 | k)

zk = h[x(k | k − 1)]

zk = [ρ1,k . . . ρn,k]T

For the initialization of the algorithm in this thesis the following parameters were used for P and x,

considering the PV model

P (1 | 0) =

p11 0

p22. . .

0 p88

(4.74)

pkk →∞ (108considered)

38

x(1 | 0) =

x1

x2

x3

x4

x5

x6

x7

x8

→

→

→

→

→

→

→

→

best x position estimate

0

best y position estimate

0

best z position estimate

0

0

0

(4.75)

For PVA model initialization is similar but the model has dimension 11.

39

40

Chapter 5

Integrity

The positioning estimates obtained with a GNSS system are not absolutely accurate. Besides the errors

discussed in section 3.4 some other malfunctions can lead to bigger errors that could compromise the

navigation solution.

In civil aviation strict requirements are imposed on levels of precision, integrity, continuity and avail-

ability of the service. Integrity is the measure of the trust that can be placed in the correctness of the

information supplied by a navigation system and includes the ability of the system to provide timely

warnings to the user when the service should not be used [30]. Integrity is defined by the integrity risk,

time to alert and alert limit requirements [33]:

• Integrity Risk: Is the probability of an undetected failure of specified accuracy. It is expressed

per hour or per operation.

A position failure is defined to occur whenever the position solution error exceeds the applicable

xPL or xAL (if the equipment is aware of the navigation mode). xPL stands for Horizontal or

Vertical Protection Level and xAL stands for Horizontal or Vertical Alert Limit.

• Time to Alert: It is the maximum allowable time interval between system performance ceasing to

meet operational performance limits and the appropriate integrity monitoring subsystem providing

an alert.

• Alert Limits: For each phase of flight, to ensure that position error is acceptable, alert limits are

defined that represent the largest position error which result in a safe operation.

While horizontal alert limits (HAL) requirements are defined for all the phases of flight, vertical

alert limits (VAL) are only defined for phases of flight under Non-Precision Approach. ICAO

requirements for common phases of flight are shown in table 5.1.

Integrity is one of the most essential aspects in navigation as abnormal positioning results would

reflect in safety. Integrity anomalies are a rare occurrence, accounting only for a couple of times a year,

but can be critical especially in aviation. The three main sources of errors that could lead to integrity

problems are: satellite clocks, ephemeris errors and faults of main control station [27].

41

Typical operations(s) Accuracy 95%Integrity

Time to alert Alert limit Integrity riskEn-route H 3.7 km 5 min H 7.4 km 10−7/h

En-route terminal H 0.74 km 15 s H 1.85 10−7/h

Initial approach,Intermediate approach,

NPA, departureH 220m 10 s H 0.6 km 10−7/h

APV-I H 16 mV 20 m 10 s H 40 m

V 50 m 2× 10−7/app

APV-II H 16 mV 8 m 6 s H 40 m

V 20 m 2× 10−7/app

CAT I H 16 m4 ≤V≤ 6 m 6 s H 40 m

10 ≤V≤ 15 m 2× 10−7/app

Table 5.1: ICAO requirements for common phases of flight

Ground Segment of GNSS systems controls the health status of the satellites to ensure that messages

do not degrade beyond specified operational tolerances. However not all satellites are visible all the time

by control stations so an anomaly in one of the satellites could take up to a few hours to be identified

and disseminated by the control segment [25]. To overcome these limitations there are several integrity

architectures to provide integrity to critical applications like aviation. Three different architectures have

been proposed to provide integrity to the aviation community [16]:

• SBAS - Satellite-Based Augmentation System

• GBAS - Ground base augmentation System

• ABAS- Aircraft-Based Augmentation System

– RAIM - Receiver Autonomous Integrity Monitoring

– AAIM - Airborne Autonomous Integrity Monitoring

5.1 SBAS

SBAS is a satellite-based augmentation system that works as a safety critical system for civil aviation that

supports wide-area or regional augmentation, through the use of geostationary satellites that broadcast

augmentation information [17]. This augmentation method works by providing integrity and correction

information to the system. As external information is incorporated in the calculus process not only

integrity is assured but also accuracy is improved, with position errors bellow 1 meter (1σ) [25].

Some GNSS systems, like GPS, were not designed to provide integrity data required by civil aviation

for safety needs. An Augmentation System like SBAS can overcome this limitation. SBAS is a wide-area

differential augmentation system constituted by a network of ground stations at known positions over

the SBAS service area to monitor signals from the satellite constellation. These ground stations collect

42

and process the information and provide corrections to the original navigation information of the primary

constellation, and integrity information over that region [25].

5.1.1 SBAS around the world

SBAS system is available in many parts of the world, provided by a collection of interoperable systems.

Worldwide SBAS coverage is continuing to grow.

To ensure seamless operation each SBAS system has been developed to the same standard as defined

by [16].

The systems available at the moment are listed below [11]:

• Wide Area Augmentation System (WAAS)

– Serves North America with benefits that extend into Central and South America over the

Atlantic and Pacific oceans.

• European Geostationary Navigation Overlay Service (EGNOS)

– Commissioned for Safety-of-Life in March 2011;

– Serves Europe and northern portion of Africa.

• Multi-functional Transport Satellite (MSAT) Satellite Augmentation System (MSAS)

– Serves Japan and surrounding areas;

– Provides lateral navigation service.

• GPS Aided Geostationary Earth Orbit (GEO) Augmented Navigation (GAGAN)

– Still under development;

– Will serve India and surrounding area.

• System of Differential Correction and Monitoring (SDCM)

– Still under development;

– Augmentation for GPS and GLONASS;

– Will serve Russia and the surrounding area.

5.1.2 WAAS

The Wide Area Augmentation System (WAAS) is the United States Satellite Based Augmentation Sys-

tem. The development started in 1992 and the system is specially developed for civil aviation community.

The system was declared operational since 2003 and its service area includes US, Alaska, Canada and

Mexico.

43

Figure 5.1: SBAS systems worldwide [44]

WAAS Services

WAAS has the objective of providing improved accuracy, integrity, and continuity of the GPS service.

The WAAS provides signal-in-space to WAAS users to support en route through precision approach

navigation. WAAS GEO satellites have ranging capabilities, so their satellites can be used as extra GPS

satellites to enhance the performance achieved in the user location because of the additional statistics

and improved geometry. Additionally, WAAS signals broadcast augmentation information that corrects

GPS ephemeris and ensures integrity.

Presently WAAS supports en-route, terminal and approach operations for the WAAS service area.

WAAS supports the following flight procedures [25]:

• LNAV (Lateral Navigation)

• LNAV/VNAV (Lateral Navigation / Vertival Navigation)

• LP (Localizer Performance)

• LPV (Localizer Performance with Vertical guidance).

WAAS Architecture

WAAS works by processing GPS data collected by a network of reference stations to generate the SBAS

message which is uploaded to the GEO satellites. The GEO satellites broadcast this information to the

user receivers, which compute the aircraft positioning information on potential alert messages [17].

WAAS architecture includes:

• WAAS Ground Segment: The WAAS ground segment includes reference stations (Wide-Area Ref-

erence Stations WRS), master stations (WAAS Master Stations WMS), uplink stations (Ground

Uplink Stations GUS) and operative centers. The WRS collect signals transmitted by GPS satel-

lites and send that information to the WMS. The WMS process all data to build an augmentation

44

message that will contain corrections to the GPS data and confidence limits to those corrections.

The message generated is uplinked from the GUS to the WAAS GEO satellites which will broadcast

them to the users.

• WAAS Space Segment: The space segment is composed by multiple geosynchronous communication

satellites (GEO) that broadcast the WAAS augmentation messages generated by the WMS to the

User Segment.

• User Segment: The user segment is constituted by any user, usually an aircraft, equipped with

approved WAAS avionics.

5.1.3 EGNOS

European Geostationary Navigation Overlay System (EGNOS) is the European SBAS system that com-

plements existing satellite navigation systems like GPS, GLONASS and Galileo. EGNOS reports on the

reliability and accuracy of the positioning data from such systems.

EGNOS Services

EGNOS system supports the following systems:

• Open service (OS) - The open service consists of a set of signals for timing and positioning for general

propose applications. Its main objective is to obtain enhanced positioning accuracy by correcting the

error contributions that affect GPS signals, related with satellite clocks, satellite signal distortions,

satellite positioning uncertainties and ionospheric delays. Effects like tropospheric delay, multipath

and receiver contributions cannot be corrected using a SBAS system. To access this open service the

user only needs a GNSS/SBAS compatible receiver and does not require any specific certification

nor authorization.

• Safety of Life Service (SoL) - EGNOS Safety of Life Service consists of an augmentation system

intended for most transport safety critical applications. This service is intended to support domains

where the degradation in the navigation system performance without a warning would endanger

lives. To have access to SoL service the user should be equipped with an EGNOS certified receiver

and located within the EGNOS SoL Service area. Usually an authorization is needed to use this

service as defined in document [10].

• Commercial Data Distribution Service (CDDS) - EGNOS Commercial Data Distribution Service

provides additional data for professional users not provided by EGNOS signal broadcast by geo-

stationary satellites but by other distribution channels. Access to CDDS is offered on a controlled

access basis and is intended for ground based users who require enhanced performances.

EGNOS Architecture

The EGNOS system is composed by the following systems [10]:

45

• Ground Segment - The Ground Segment is composed by 34 Ranging and Integrity Monitoring

Stations (RIMS) that receive signals from GPS satellites, 4 Mission Control Centers (MCC) that

process data, and 6 Navigation Land Earth Stations (NLES) responsible for accuracy and reliability

data sending to geostationary satellite transponders to allow user devices to receive them.

• Space Segment - EGNOS Space Segment is composed by the three GEO satellites that broadcast

EGNOS signal over the service area.

• User Segment - Composed by the user receivers and user terminals

SOF

GOL

WRS

LAP

KIR

TRO

TRD

GVL

GLG

EGIRKK

CRK

BURSWA

ALB

BRN

LAN

ZUR

ROM

CIA

FUC

CTN

SCZ

PAR

TLS

AUS

PDM

TOR

SDC

ACR

LSB

MLG

MON

KOU

MAD

CNR

NOU

DJA

ALY

HBK

MCC

NLES

EGNOS

RIMS

Figure 5.2: EGNOS map of Ground Segment [43]

5.2 GBAS

A Ground-Based Augmentation System (GBAS) is a civil-aviation system that supports local augmenta-

tion of the primary GNSS constellation by providing enhanced levels of service that support all phases of

approach, landing, departure and surface operations. Mainly its purpose is to provide integrity assurance

but it also increases accuracy with position below 1 meter (1σ) [25].

The GBAS is intended primarily to support precise approach operations. The GBAS system is

constituted by GBAS ground subsystem and a GBAS aircraft subsystem. The ground subsystem can

support an unlimited number of aircrafts within the GBAS coverage volume.

The GBAS ground system is constituted by one or more GNSS receivers that collect pseudo-ranges for

all the primary GNSS satellites in view and computes and broadcasts differential corrections and integrity

information for them based on its own position. The corrections are transmitted from the ground station

via very high frequency (VHF). In the case of GPS these techniques are known as differential GPS

(DGPS).

46

5.3 RAIM

RAIM is a user algorithm that determines the integrity of the GNSS solution. When more satellites are

available than needed to have a position fix (satellite number > 4), the extra pseudoranges should be

consistent with the computed position. If the pseudorange from one satellite differs significantly from

the expected value some fault may be associated with it or with another signal integrity problem. A key

assumption usually made in RAIM algorithms for civil aviation is that only one satellite may be faulty,

mainly because the probability of multiple satellite failures is negligible [25].

In order for a receiver to use a RAIM algorithm it is necessary to have a minimum of five visible

satellites with a good geometry. With five satellites available we can use an algorithm called Fault

Detection (FD). If six or more satellites are available we can use a more sophisticated algorithm called

Fault Detection and Exclusion (FDE).

5.3.1 FD

Fault detection algorithm can be used with only five satellites visible. With five satellites available we can

make five subsets of four satellites. The position solutions obtained by the various subsets are analysed

for consistency, and an alert is provided if that consistency check fails. With this technique the receiver

only detects an anomaly but the satellite in question is not identified.

Exemplifying, let’s suppose 5 satellites are available: Satellites (1, 2, 3, 4, 5). Let’s assume that satellite

1 is faulty. With these 5 satellites, 5 subsets of 4 are possible:

Subset index Subset Solution1 1,2,3,4 Faulty2 1,2,3,5 Faulty3 1,2,4,5 Faulty4 1,3,4,5 Faulty5 2,3,4,5 Correct

Table 5.2: Subsets of the available 5 satellites

If satellite 1 is faulty only the last subset would allow to compute the correct user position but RAIM

algorithms cannot be used in these subsets, as at least 5 satellites are needed because with 4 satellites

there would be no redundancy, so no exclusion can be performed. The navigation solutions obtained

with 5 subsets are not consistent among themselves: they lead to different results. However, the RAIM

algorithm is not able to decide which is the right solution.

5.3.2 FDE

A more sophisticated algorithm is FDE. To use this algorithm a minimum of six visible satellites are

needed, not only to detect a fault but also exclude the faulty satellite from the solution. In fact the

big difference from the FD algorithm is that, as the faulty satellite is excluded, navigation can continue

without interruption.

Exemplifying, let’s suppose 6 satellites are available: Satellites (1, 2, 3, 4, 5, 6). Let’s assume that

satellite 1 is faulty. With these 6 satellites 6 subsets of 5 are possible:

47

Subset index Subset Solution1 1,2,3,4,5 Faulty2 1,2,3,4,6 Faulty3 1,2,3,5,6 Faulty4 1,2,4,5,6 Faulty5 1,3,4,5,6 Faulty6 2,3,4,5,6 Correct

Table 5.3: Subsets of the available 6 satellites

If satellite 1 is faulty, again, only the last solution would allow to compute the right position in a

consistent way. To get to this conclusion the algorithm should analyse the (n − 1) subsets and search

for subsets without fault detection condition, using the test statistics for the subsets. Since the faulty

satellite is included in all subsets but one, only one subset, (2, 3, 4, 5, 6), is free from the error of the

faulty satellite and leads to 5 consistent navigation solutions. Concluding this, the satellite missing from

the subset that does not have the fault condition is identified as the faulty satellite and exclusion can be

performed [18].

5.3.3 RAIM algorithms

There are a variety of RAIM algorithms but all those algorithms include the following functions:

1. Have an observable discriminator called test statistic, which shows the effect of a faulty measurement

with a bias error;

2. Know the likely noise in the system and their interactions with the test statistic, so that the

relationship between the test statistic and the faulty measurement can be statistically described;

3. Establish a fault-free limit for the test statistic which will only, rarely, be exceeded by the observed

test statistic when there is no faulty measurement. This fault-free limit, called the detection thresh-

old, is typically based on a specific false alarm probability for the particular RAIM application;

4. Perform the detection test by comparing the observed test statistic against the detection threshold,

meaning:

• if the test statistic is less than the limit, then declare that no fault is present;

• if the test statistic is equal to or greater than the limit, then declare that a fault has been

detected and issue an integrity alarm.

5. In case of a fault, depending on the satellites available, two techniques can be applied:

• If only 4 satellites are available integrity cannot be provided as at least 5 satellites are needed;

• If 5 satellites are available fault detection can be performed;

• If 6 or more satellites are available fault detection and exclusion can be performed.

6. Compute the protection levels (This step is optional and not taken into account in this thesis).

48

The two methods that are being analysed and performance tested in this thesis are the Least-Square

Residuals and the Range Comparison Method. These methods will be described in detail in the following

sections. The analysis is partially based on [26].

Least-Squares-Residuals (LSR)

Considering noisy pseudoranges from n satellites to be given by

y =

U1...

Un

+

ε1...

εn

(5.1)

and the pseudorange from the satellite i is given by

Ui =√

(Xi − xu)2 + (Yi − yu)2 + (Zi − zu)2 + c∆T (5.2)

where (xu, yu, zu) is the receiver position, (Xi, Yi, Zi) is the satellite position, c∆T is the receiver clock

bias and εi is the measurement error which includes the receiver noise, imprecise knowledge of satellite

position, satellite clock error, vagaries in propagations and possibly unexpected errors due to satellite

malfunctions.

Considering (5.1) and (5.2) the corresponding incremental equations is

∆y = G∆x+ ε (5.3)

with ε = [ε1 . . . εn].

The incremental solution corresponding to (5.3) is

∆x = (GTG)−1GT∆y (5.4)

Thus, the least squares solution is

xls = xpred + (GTG)−1GT (y − ypred) (5.5)

where xpred and ypred are respectively the predicted position and corresponding pseudorange vector, and

G is the linearised measurement connection matrix which consists of the line-of-sight (LOS) vectors to

the satellites, with ones in the fourth column corresponding to the clock bias [8]. That is

G =

∂U1

∂x

∂U1

∂y

∂U1

∂z1

......

......

∂Un∂x

∂Un∂y

∂Un∂z

1

(5.6)

49

The pseudorange vector may be reconstructed from (5.5) as

y = ypred +G(xls − xpred)

= ypred +G(GTG)−1GT (y − ypred)(5.7)

and the residuals w ≡ y − y result from

w = y − ypred −G(GTG)−1GT (y − ypred)

= [I −G(GTG)−1GT ](y − ypred)(5.8)

where P = G(GTG)−1GT is the projection matrix as it projects any vector into the subspace formed by

the columns of G and S = I − P is the projection matrix associated with the complementary subspace

[39]. Assuming convergence of the least-squares algorithm we have ypred ≈ U and y − ypred ≈ ε due to

(5.1). Thus

w = [I −G(GTG)−1GT ]ε (5.9)

We can obtain the sum of the squared errors from the residuals by

SSE = wTw = εTS2ε = εTSε (5.10)

because S is an n× n idempotent matrix (S2 = S).

Considering the test statistic

t =√SSE

n− 4 (5.11)

an orthogonal transformation K can be found which diagonalizes S to a n×n matrix with n−4 diagonal

elements equal to 1 and 4 diagonal elements equal to 0. The sum of the squares of the range residual

errors (SSE) can be expressed as [4]

SSE = wTw = trace(wwT )

= εTSε = trace(SεεTS)

= uT diag(1, . . . , 1, 0, . . . , 0)u

= u21 + . . .+ u2

n−4

(5.12)

where

u = Kε (5.13)

If the measurement errors are independent, zero-mean Gaussian random variables, with variances σ2,

that is, εi ∼ N(0, σ2), then the ui’s are also normally distributed with the same mean and variance, and

u1, . . . , un−4 are independent. Thus

t =

√(u1√n− 4

)2+ . . .+

(un−4√n− 4

)2(5.14)

50

and the pdf of the test statistic is [34]

pt(r) = (n− 4)n−42 rn−5

2n−62 σn−4Γ

(n−4

2)exp [− (n− 4)r2

2σ2

], r ≥ 0 (5.15)

with

Γ(p) = (p− 1)!, p = 1, 2, . . .

Γ(

12

)=√π, Γ

(32

)=√π

2

Γ(n+ 1

2

)= 1 · 3 · 5 · 7 . . . (2n− 1)

2n√π

(5.16)

If the measurement errors are independent, normally distributed random variables, with nonzero

means, εi ∼ N(µi, σ2), then SSE is chi-squared distributed with n − 4 degrees of freedom and non-

centrality parameter [4]

(ui)2 + . . .+ (un−4)2 = µTSµ (5.17)

with µ = [µ1 . . . µm]T . Thus

t2 =(

u1√n− 4

)2+ . . .+

(un−4√n− 4

)2(5.18)

is chi-squared distributed with non-centrality parameter

ρ2 =(

u1√n− 4

)+ . . .+

(un−4√n− 4

)2= µTSµ

n− 4 (5.19)

and the corresponding pdf is [34]

pt(r) = (n− 4)r n−42

σ2ρn−6

2exp

[−(n− 4)r

2 + ρ2

2σ2

]In

2−3

[(n− 4)rρ

σ

2]

(5.20)

where In() is the nth order modified Bessel function of the first kind.

The probability density functions pt(r) for the no-fault case and the pt(r) for the fault case, with

σ = 7.1m and n = 6 visible satellites are shown in figure 5.3.

Consider that a fault affects the ith satellite which corresponds to add a bias b to εi. Thus

µ = [0 0 . . . b . . . 0]T (5.21)

with µk = 0, k 6= i and µi = b. Also

ρ2 = b2Siin− 4 (5.22)

where Sii is the element (ii) of matrix S.

Fault detection is based on the hypothesis testing where the decision variable t is tested against an

51

0 1 2 3 4 5 6 70

0.02

0.04

0.06

0.08

0.1

0.12

0.14

r/σ

Pro

babi

lity

dens

ity fu

nctio

n

pt (r) fault

pt (r) no-fault

σ = 7.1m

ρ/σ = 3

n = 6

Figure 5.3: Probability Density Functions for fault and no fault cases

alert threshold λ. The decision criteria is t ≥ λ→ fault

t < λ→ no fault(5.23)

The test is characterised by the probability of false alarm (Pfa) and probability of missed detection

(Pmd)

The probability of false alarm is

Pfa =∫ ∞λ

pt(r)dr = 1−∫ λ

o

pt(r)dr (5.24)

Expressing (5.24) in terms of the normalized threshold λσ we get

Pfa = 1− (n− 4)n−42

2n−62 Γ

(n−4

2) ∫ λ

σ

0x(n−5) exp

[− (n− 4)x2

2

]dx (5.25)

The probability of false alarm versus λσ is plotted in figure 5.4 for n = 6 and σ = 7.1m.

Intuitively and from figure 5.4, we conclude that for a low value of λ we have high values of false

alarm. Increasing the threshold λ the probability of false alarm decreases.

For a Pfa = 115000 [27] we get the normalized thresholds λ

σ in table 5.4

Number of Satellites Degrees of Freedom λσUERE

5 1 3.996 2 3.107 3 2.718 4 2.479 5 2.31

Table 5.4: Normalized thresholds for Pfa = 115000

52

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

λ/σ

Pfa

n=6

σ=7.1m

Figure 5.4: False Alarm probability versus of λσ

Given this we are able to compute the threshold λ for the decision criteria.

The probability of missed detection is given by

Pmd =∫ λ

0pt(r)dr (5.26)

The expression (5.26) can be expressed in terms of normalized quantities of λσ

and ρ

σresulting:

Pmd = n− 4( ρσ

)n2−3

∫ λσ

0exp

[−(n− 4)

x2 +(ρσ

)22

]In

2−3

[(n− 4) ρ

σx]dx (5.27)

1 1.5 2 2.5 3 3.5 4 4.5 510

−15

10−10

10−5

100

λ/σ

Pro

babi

litie

s

Pfa

ρ/σ = 6ρ/σ = 5ρ/σ = 4

n = 6

σ = 7.1

Figure 5.5: False Alarm and Missed Detection probabilities versus λσ

From figure 5.5 we conclude that with higher values of λ we reduce the probability of false alarm but

on the other hand we are raising the missed detection probability. Also the higher the ρ the lower the

53

probability of missed detection as higher ρ values mean higher bias in the measurements (equation (5.22)).

Note that the probability of missed detection depends also on the constellation geometry through the

parameter Sii (equation (5.22)). Thus, faulty satellite with large values of Sii tend to be better detected.

Range-Comparison Method (RCM)

Another method to provide integrity considered in this thesis is the Range comparison method. With

two methods one can validate the results and compare their performance.

Consider we use n satellites measurements

y =

y

· · ·

y

(5.28)

where

y =

y1...

y4

, y =

y5...

yn

(5.29)

In this method the position is solved for the first four measurements y where the order of the n equa-

tions is immaterial [8]. The resulting solution is then used to predict the remaining n− 4 measurements

and the predicted values are compared with the actual measured y. If the n − 4 residuals are small, we

will have near-consistency in the measurements and the algorithm will declare no failure. On the contrary

if one or several residuals are large, it will declare failure [8].

Let matrix G in (5.6) be partitioned as

G =

G

· · ·

G

(5.30)

with G(4× 4) and G((n− 4)× 4). Then according to (5.8)

y1...

y4

=

y1,pred

...

y4,pred

+ G(xls − xpred) (5.31)

and y5...

yn

=

y5,pred

...

yn,pred

+ G(xls − xpred) (5.32)

54

As

xls = xpred + G−1

y1...

y4

−y1,pred

...

y4,pred

(5.33)

then y5...

yn

=

y5,pred

...

yn,pred

+H

y1...

y4

−y1,pred

...

y4,pred

(5.34)

where H((n− 4)× 4) is defined as

H = GG−1 (5.35)

The residuals of y5, . . . , yn are given by

w1...

wn−4

≡y5 − y5

...

yn − yn

=

y5...

yn

−y5,pred

...

yn,pred

−Hy1 − y1,pred

...

y4 − y4,pred

=

ε1...

εn

−Hε1...

ε4

(5.36)

or

w = −Hε+ ε (5.37)

where

ε = [ε1 . . . ε4]T , ε = [ε5 . . . εn]T (5.38)

Assume a no-fault scenario where the measurement errors are independent, zero-mean Gaussian ran-

dom variables, with variances σ2, that is, εi ∼ N(0, σ2). Thus, w is a Gaussian vector with mean

Ew = 0 and covariance matrix

EwwT = HEεεT HT + EεεT

= σ2[HHT + I](5.39)

55

Consider now a scenario where the kth satellite (with k = 1, . . . , n) has a malfunction such that

Eε =

0

0...

b...

0

← position k

(5.40)

In this method two cases can be distinguished:

1. The faulty satellite belongs to the set used to solve the equation of navigation, this is, 1 ≤ k ≤ 4;

2. The fault satellite does not belong to that set, that is 5 ≤ k ≤ n.

In case 1 the expected value of w is

w = −H

0

0...

b...

0

= −b

h1k

h2k...

hn−4,k

(5.41)

and we have the following diagonal matrices

EεεT =

σ2 0

σ2

. . .

σ2 + b2

. . .

0 σ2

← position k

(5.42)

and

EεεT = σ2

1 0

. . .

0 1

(5.43)

56

According to (5.39)

EwwT = σ2[HHT + T ] +H

0 0 . . . 0...

...

0 . . . b2 . . . 0...

...

0 . . . 0 0

HT

= σ2[HHT + I] + b2

h2

1k h1kh2k . . . h1kh(n−4),k

h2kh1k h22k . . . h2kh(n−4),k

......

. . ....

h(n−4),kh1k h(n−4),k . . . h2(n−4),k

(5.44)

where hij is the element (i, j) of H. The covariance matrix of w is given by

E(w − w)(w − w)T = Ewwt − wwT

= σ2[HHT + I](5.45)

In case 2 the expected value of w is

w = b

0

0...

1...

0

← position k-4

(5.46)

and ones have the following diagonal matrices

EεεT = σ2

1 0

. . .

0 1

(5.47)

and

EεεT =

σ2 0

ffl σ2

. . .

σ2 + b2

. . .

0 σ2

← position k-4

(5.48)

57

so

EwwT = σ2[HHT + I] + b2

0 0 . . . 0...

...

0 . . . 1 . . . 0...

...

0 . . . 0 0

(5.49)

and the covariance matrix of w is still given by (5.45).

Solution for n = 5 satellites

Let us now consider the case when 5 satellites are used in the RCM method. In this case matrix G is

(4× 4) and G is (1× 4). Given (5.35), matrix H is defined by

H =[h11 h12 h13 h14

](5.50)

In this case, the residuals vector w is reduced to a scalar w1 and taking (5.45) into account we have

σ21 = Ew2

1 = σ2(HHT + 1) = σ2

(1 +

4∑i=1

h21i

)(5.51)

The probability density function of w1 in the absence of any satellite malfunction is

pw(w1) = 1√2πσ2

1exp

(− w2

12σ2

1

)(5.52)

Let us now consider the decision rule that divides the real axis in three distinct regions, |w1| < λ1,

corresponding to the hypothesis of no-failure, and |w1| ≥ λ1, corresponding to the hypothesis of satellite

failure. This decision rule is illustrated in figure 5.6.

−λ1 0 λ1w1

Fault No Fault Fault

Figure 5.6: Decision rule for the Range-Comparison Method when 5 satellites are used

The probability of false alarm is

Pfa = 1−∫ λ1

−λ1

pw(w1)dw1 = 2Q(λ1

σ1

)(5.53)

where Q is the error function

Q(x) = 1√2π

∫ ∞x

exp(−y

2

2

)dy (5.54)

58

In the presence of a satellite malfunction the probability density function of w1 is given by

pw(w1) = 1√2πσ2

1exp

[− (w1 − w1)2

2σ21

](5.55)

where w1 is the expected value of w1, given by (5.41) or (5.46) depending whether the faulty satellite

belongs or not to the subset of 4 satellites used to find the receiver coordinates.

Solution for n = 6 satellites

Let us now consider the case when 6 satellites are used in the RCM method. In this case matrix G is

(4× 4) and G is (2× 4). Given (5.35), matrix H is defined by

H =

h11 h12 h13 h14

h21 h22 h23 h24

(5.56)

In this case the vector of residuals is given by w =[w1 w2

]. In a no-fault scenario the covariance

matrix is

EwwT = σ2[HHT + I] =

σ21 ρσ1σ2

ρσ1σ2 σ22

(5.57)

where

σ21 = σ2

(1 +

4∑i=1

h21i

)(5.58)

σ22 = σ2

(1 +

4∑i=1

h22i

)(5.59)

and the correlation coefficient is

ρ =∑4i=1 h1ih2i√

1 +∑4i=1 h

21i

√1 +

∑4i=1 h

21i

(5.60)

The joint probability density function of w is [34]

pw(w1, w2) = 12πσ1σ2

√1− ρ2

exp− 1

2(1− ρ2)

[(w1

σ1

)2− 2ρw1w2

σ1σ2+(w2

σ2

)2]

(5.61)

The integral of pw(w1, w2) over an area A, bounded by the ellipse

(w1

σ1

)2− 2ρw1w2

σ1σ2+(w2

σ2

)2= α2(1− ρ2) (5.62)

is given by [19] ∫ ∫A

pw(w1, w2)dw1dw2 = 1− exp(−α

2

2

)(5.63)

Let us now consider a decision rule that divides the plane into two distinct regions, one hypothesis

corresponding to no failure and other the to failure. A common way to choose the decision boundary is to

59

let it be an equal probability density contour, conditioned to the assumption of no satellite malfunction,

according to (5.62) [8].

The probability of false alarm is

pfa = 1−∫ ∫

A

pw(w1w2)dw1dw2 = exp(−α

2

2

)(5.64)

such that the failure/no failure decision boundary is given by (5.62), that is

(w1

σ1

)− 2ρw1w2

σ1σ2+(w2

σ2

)2= −2(1− ρ2)ln(Pfa) (5.65)

5.3.4 RAIM Availability

Another important concept is RAIM availability because as said before a minimum of five satellites with

a good geometry have to be available. In figure 5.7 the number of available satellites from the different

satellite navigation according to latitude systems is shown.

Figure 5.7: GPS and other GNSS system visibility [36]

Since RAIM requires a minimum of five visible satellites in order to perform fault detection and a

minimum of six for fault detection and exclusion, RAIM and FDE will have a lower availability than the

navigation function [17].

In figure 5.8 the number of visible GPS satellites from the user position is shown for the day of the

simulations. We can see that a minimum of 6 satellites were available so the number of visible satellites

was no limitation.

Besides the minimum of 5 satellites needed, availability in RAIM is determined by comparing the

60

0 1 2 3 4 5 6 7 8 9

x 104

6

7

8

9

10

11

12

Time (seconds)

Num

ber

of V

isib

le S

atel

lites

Number of Satellites

Figure 5.8: Number of visible GPS satellites at receiver position during the simulation day

value of Horizontal Protection Level (HPL) to the maximum alert limit for the intended operation [17].

HPL is the horizontal position error that the fault detection and exclusion algorithm guarantees will not

be exceeded without being detected by the fault detection function, in accordance with the missed alert

and false alert probability requirements [18]. It is a function only of the visible satellites, user geometry

and expected error characteristics.

Define A and B as

A = (GTG)−1GT (5.66)

B = G(GTG)−1GT = GA (5.67)

After A and B have been defined a quantity called SLOPE is computed for each satellite in view:

SLOPE(i) =√

(A21i +A2

2i)n− 4

1−Biifor i = 1, 2, . . . , n (5.68)

defining SLOPEmax as

SLOPEmax = Maxi[SLOPE(i)] (5.69)

There are different methods to calculate the HPL. The method followed in this thesis is one of the

methods used in [18] and [7]. This method states that using the satellite with SLOPEmax the value of

the bias in the measurement of that satellite should be such that the missed detection probability is the

specified value. This critical bias value is called pbias.

HPL = SLOPEmax × pbias (5.70)

61

5.4 Integrity in Galileo

The integrity concept introduced by Galileo consists in providing the users with data enabling them to

monitor their integrity level.

Galileo has the capability to monitor the satellite behaviour through its complex global distributed

ground network consisting of more than 30 sensor stations. Taking these measurements into account,

satellite failures (due to orbit or clock inaccuracies) can be detected and alerts can be disseminated to

the user.

The system takes care of always monitoring the constellation and to broadcast to the user information

about the health status of each satellite through a three states flag (Integrity Flag) related to each satellite

(e.g a "Not OK" flag if something is wrong with the satellite, or an estimated signal in space accuracy

otherwise). This monitoring process is performed by a stations network (GSS stations) located around

the globe. This network carries out pseudorange measurements from every satellite and, through an

inverse navigation algorithm, estimates the pseudorange error relevant to each satellite (SISE, Signal in

Space Error) [9]. The estimation carried out (estimated-SISE) represents the range error contribution

due to satellite contribution, which impacts on the user solution.

5.4.1 Integrity parameters

The SoL service is particularly studied since it is the one to provide users with integrity information. The

nature of the provided information consists in three parameters per satellite:

• Signal-In-Space Accuracy (SISA)

• Signal-In-Space Monitoring Accuracy (SISMA)

• Integrity Flag (IF)

These three parameters relate to Signal-In-Space Error (SISE) and are defined as follows [28] [2]:

• SISE: SISE is related to a satellite and is the maximum error (over the user position) of the SIS

in the range domain caused by the SV, its payload and the navigation message

• SISA: for purposes of integrity monitoring the SISE is estimated and its consistency with SISA

value is checked. As the SISE distribution may not be Gaussian distribution, the methodology of

overbounding is applied to describe the SISE distribution with an overbounding Gaussian distribu-

tion.

The SISE distribution is characterized by SISA which is a prediction of the minimum standard

deviation of a Gaussian distribution that over-bounds the SISE distribution for a fault-free SIS

• SISMA: As the SISE cannot be measured directly, one has to estimate the SISE from measure-

ments. The estimation of SISE results in an estimated SISE (SISE).

The diference between SISE and (SISE) is the SISE estimation error has a distribution. This distri-

bution must be over-bounded by a Gaussian distribution with a standard deviation called SISMA.

62

So, SISMA is defined as the minimum standard deviation of the unbiased Gaussian distribution

which overbounds the error distribution of the estimation of SISE as determined by the integrity

monitoring system

• IF: if the (SISE) for SIS is larger than the integrity flag threshold, the integrity flag for this SIS is

set to "NOT OK". The integrity flag threshold can be computed from the overbounds of the SISE

distribution, the distribution of the difference between SISE and (SISE), and the allowed False

Alarm probability.

Each alert is coded with four bits and therefore 16 different states can be disseminated to the user

[33]:

– "Not OK"

– "NOT MONITORED"

– 14 states for "OK" with corresponding SISMA value in the range of 30cm to 520cm.

If the computed SISMA value is larger than 520cm, the satellite is declared "NOT MONITORED"

5.4.2 Galileo user integrity algorithm

A particular algorithm has been introduced in [29] to make best use from the integrity data provided in

the Safety-of-Life service. The analysis of this topic follows essentially [33].

The algorithm proposed in [29] differs from usual integrity monitoring algorithms, like SBAS, mainly

because:

1. It does not provide primarily protection levels but rather directly computes the user’s integrity risk

at the alert limit. Protection levels can be computed with an additional algorithm.

2. There is no allocation between vertical and horizontal errors. Thus, there is only one threshold

value in the equation.

3. It anticipates a non-declared failure on the satellite with highest impact.

System availability is obtained by comparing the algorithm’s output to the integrity risk requirement

corresponding to the current phase of flight, in table 5.1.

The algorithm is based on the classic assumption that positioning errors distributions are independent

and can be overbounded by centred Gaussians. The independence assumption is realistic because all

measurements are performed at the same epoch. This allows one to take into account the high time

corrections of the measurements errors.

The algorithms inputs are:

• horizontal alert level;

• vertical alert level;

• the SISA of all available satellites;

63

• the SISMA of all available satellites;

• the prediction of the standard deviation of the residual measurement noise for each line-of-sight.

The role of the SISMA is to quantify the maximum error due to a faulty satellite.

The final expression of the global (i.e taking into account vertical and horizontal errors, in both

fault-free and faulty cases) integrity risk PIR at the alert limits HAL and VAL is [33]:

PIR(HAL, V AL) = PIR,V + PIR,H

= 1− erf(

V AL√2σV,FF

)+ exp

−HAL2ξ2FF

+ 12

N∑j=1

PSatFail,j

((1− erf

(V AL+ µV,j√

2σV,j,FM

))+(

1− erf(V AL− µV,j√

2σV,j,FM

)))

+N∑j=1

PSatFail,j

(1− Sχ2

2,δj

(HAL2

ξ2FM

))(5.71)

where

PIR,V : are respectively the vertical and horizontal components of the integrity risk

pSatFail,j : probability satellite j to fail

σV,FF , σV,j,FM : are the standard deviations of the vertical positioning error respectively in fault-free

mode and with satellite j failing

µV,j : is the worst-case vertical error bias due to the failing of satellite j

ξFF , ξFM : are the semi-major axes of the horizontal error ellipse respectively in fault-free and

faulty modes

erf : error function

In equation (5.71) Sχ22,δj

stands for the cumulative distribution function of the chi-squared distribution

with two degrees of freedom.

Sχ22,δj

(x) =∫ x

0

12 exp

[−1

2(t+ δ)] ∞∑j=0

tjδj

22j(j!)2 dt (5.72)

The PIR(HAL, V AL) expression in equation (5.71) amounts to:

horizontal risk with no satellite failure

+ horizontal risk with one failing satellite

+ vertical risk with no satellite failure

+ vertical risk with one failing satellite

64

Chapter 6

Computer Simulation

In this chapter the simulation results from the integrity algorithms described in chapter 5 will be pre-

sented and analysed. For this, a series of Monte Carlo simulations were performed under the considered

trajectory. The comparison of these integrity algorithms is performed for the various simulation scenarios.

6.1 Computer Simulation Procedure

To perform the computer simulations a GPS constellation was simulated with parameters used from

YUMA almanac file format described in Appendix A. After the definition of the GPS constellation and

after the user position is defined, the satellites with an elevation angle above 10 are considered visible.

To be able to perform detection and exclusion a set of 6 satellites is chosen with the lowest GDOP

value, to achieve lower error in position results. After the selection of the 6 satellites a trajectory is

simulated and the user position is calculated, the test statistic t is calculated and compared to the

threshold λ. The results from this comparison can be grouped according:

• No faulty satellites

1. If t < λ we have a normal operation with probability of 1− Pfa

2. If t ≥ λ we have a false alarm with probability Pfa.

• Presence of a faulty satellite

1. If t ≥ λ we have an integrity alarm with probability 1− Pmd

2. If t < λ we have a missed detection with probability of Pmd.

A flowchart of the RAIM algorithm is shown in figure 6.1.

6.2 Computer Simulation Results

As refereed in chapter 4 there are several methods that can be used to obtain a position solution. Two

methods were implemented in this thesis:

65

GPS Constellation Simulation

User Position

Determination of visible Satellites

Subsets of 6 Satellites andselection of lowest available DOP

Trajectory Simulation andposition determination

Test statistic t computation

Threshold λ computation

t< λ?

Faulty Satellite?

Missed Detection Normal Operation

Faulty Satellite?

Integrity fault detection False alarm

n− 1 combinations

Test statistic t computation forn− 1 combinations

Check for fault-free subcombinationand excluse faulty satellite

t→ t+ 1

Yes No

Yes No Yes No

Figure 6.1: RAIM algorithm flowchart

66

• Least-Squares (section 4.1);

• Kalman Filter Solution (section 4.3).

To compute the errors in both methods a trajectory was simulated and the position error performance

was compared for both methods. The trajectory model simulated an aircraft circling around the north

tower of Instituto Superior Técnico, in Lisbon, with a constant speed of 300 Km/h and 2000 m radius

during 240 seconds. The trajectory can be visualized in figure 6.2.

−2000−1000

01000

2000

−2000

0

2000−2000

0

2000

Z p

ositi

on (

met

ers)

X position (meters)Y position (meters)

Figure 6.2: Simulated Trajectory

6.2.1 Error calculation

The 3D RMS position error can be determined taking into account the three dimension components

x, y, z. This 3D RMS error is defined as :

3DRMS =

√√√√1t

t∑k=1

(xk − xk)2 + (yk − yk)2 + (zk − zk)2 (6.1)

and, as in conventional GPS navigation, the final solution accuracy is function of the measurement error,

σUERE , and the satellite geometry or dilution of precision. The theoretical 3D RMS position error is

given by [32] [6]:

3DRMSposition error = σUERE × PDOP (6.2)

The 3D RMS error associated with the position calculation when using the Kalman filter, is shown

in figure 6.3.

The mean value of the 3D RMS error for position calculated with Kalman Filter, not taking into

account the first 10 seconds of simulation, is 14.74 meters (using equation (6.1)).The same trajectory was

used to obtain the error with the Least-Squares method but the error value with this method was higher

than with the Kalman filter.

67

0 50 100 150 200 2500

20

40

60

80

100

Time (seconds)

Err

or (

met

ers)

Error ValueError Mean

Figure 6.3: 3D RMS position error for Kalman Filter Solution method

The position solution for these results was computed from a subset of 6 satellites with value of PDOP

of 2.08. It is assumed that this PDOP value is an average value and that it does not change significantly

during the 240 seconds of trajectory. With this assumption, according to equation (4.22) and equation

(6.2), the theoretical expected value of error for this simulation is 2.08×7.1 = 14.77 meters which is very

close to the practical value obtained.

As for the trajectory considered the solution using Kalman filter has a lower error, this method was

selected in the simulations of this thesis. The estimated trajectory vs the simulated trajectory is shown

in figure 6.4.

−2000−1000

01000

2000

−2000−1000

01000

2000−2000

−1000

0

1000

2000

X position (meters)Y position (meters)

Z p

ositi

on (

met

ers)

ReferencePredicted Trajectory

Figure 6.4: Estimeted trajectory using Kalman filter vs Simulated trajectory

68

6.2.2 RAIM Algorithms Performance Analysis

To measure and compare the performance of the RAIM algorithms discussed in section 5.3.3 a series

of Monte Carlo simulations are performed. Monte Carlo simulations consists of repeating a single test

procedure subject to random conditions. In this case the random condition is noise. The performance of

the RAIM algorithms is investigated in terms of probability of error detection versus the error induced

to the measurements.

To investigate the probability of error detection, two types of error can be induced to the measurements

[18]:

• A slow ramp: an example of a slow ramp error is a satellite clock oscillator failure that results in a

slow measurement drift;

• A step error: this type type of failure results in a large and instantaneous measurement jump, and

could be caused by a bad ephemeris upload. Also the GPS receiver clock may change abruptly

because of the oscillator, a clock bias roll-over or other causes.

To test both algorithms, two epochs were used. The first epoch was on 22/01/2015 at 16:40 and the

second on 22/01/2015 at 19:20. The purpose of using two epochs is to test two distinct geometries to see

how the two algorithms perform under different test scenarios.

For the first epoch, the detection detection test was performed with step and ramp errors and exclusion

was also performed. For the second epoch only the detection test was carried out as there would be no

added value in performing another exclusion as no new conclusions would be extracted.

For the detection tests the trajectory was initiated and at second 50 the step error was induced for

the rest of the trajectory, with the process being repeated 50 times, counting a total of 9500 tests for

each error value, for each satellite. For the exclusion test, each time detection occurs, the algorithm tests

if is able to perform the exclusion successfully.

As for the ramp error test, the error is introduced at the beginning of the trajectory and is incremented

each second according to its slope. When detection occurs the time of detection and respective error value

are then analysed.

Performance analysis with no error

To test the false alert requirement the algorithms should run without any positioning failure induced,

since the false alarm probability assumes that no real positioning failure is present.

The threshold is set defining the Pfa required, as referred in section 5.3.3. With no error induced to

the measurement the algorithms should agree with the Pfa.

The results of this test for the Least-Squares-Residuals method are summarized in table 6.1.

Number of Runs Number of Detections Experimental Pfa Theoretical Pfa

1330000 88 881330000 ≈ 6.6165× 10−5 1

15000 ≈ 6.6666× 10−5

Table 6.1: Experimental Pfa for Least-Squares-Residuals Method

69

From the analysis of the results we can conclude that the Pfa from simulation results is very close

to the expected theoretical value. The small discrepancy found can be due to the finite number of runs

which can influence the final result.

The same performance test was performed for the Range Comparison method. The results from this

test are summarized in table 6.2.

Number of Runs Number of Detections Experimental Pfa Theoretical Pfa

1330000 82 821330000 ≈ 6.1654× 10−5 1

15000 ≈ 6.6666× 10−5

Table 6.2: Experimental Pfa for Range Comparison Method

Performance analysis with error induced

To test the detection capability of the algorithms an error is induced to the measurements as referred in

section 6.2.2.

The first test to be performed is the detection test with several values of step errors. To perform

this test two epochs are analysed and each satellite in the subset will have a step error induced to their

measurements at a time.

In cases of poor satellite geometry DOP values get large, and as stated by equation (4.22), the

navigation accuracy degrades. A similar effect occurs with RAIM algorithms [6]. When the satellite

geometry is poor the performance of the integrity monitoring algorithms degrade and large navigation

errors can occur before they are detected, so it is important to understand how the detection capabilities

of the algorithms varies depending on the satellite affected by the error.

Various criteria have been used for evaluating the quality of the satellite geometry for detection

purposes [8]. The method used in this thesis is the SLOPEmax method. The calculation of SLOPEmaxis given by equations (5.68) and (5.69) from section 5.3.4. With this method one can anticipate which

satellite(s), if any, will be the most difficult to detect. This means that if the noise level is the same for

all satellites in the combination, the satellite with the highest value of SLOPE will be the most difficult

to detect. Also, the SLOPEmax satellite is the one that yields to the smallest test statistic for a given

position error. An illustration of this method is shown in figure 6.5.

SLOPEmax

Position Error

Test Statistic

Figure 6.5: SLOPE method illustration

The first simulation test was performed on 22/01/2015 at 16:40. At this time satellites 5, 13, 15, 21,

70

24, 28, 30 were visible and the subset of 6 satellites with lowest GDOP value was 5, 13, 15, 21, 24, 30

with a correspondent GDOP of 2.3793. The SLOPE values at the beginning of the trajectory for these

satellites are shown in table 6.3.

Sat 1 (ID=5) Sat 2 (ID=13) Sat 3 (ID=15) Sat 4 (ID=21) Sat 5 (ID=28) Sat 6 (ID=30)

SLOPE 1.4005 0.8740 0.2022 3.2774 2.7196 12.0709

Table 6.3: SLOPE values for first epoch

From the analysis of the table 6.3 the SLOPEmax corresponds to the last satellite, so it is expected

that satellite 6 (corresponding to satellite ID=30) will be less sensitive to errors, thus making more

difficult the error detection by the RAIM algorithms.

Let us now analyse the results with LSR method for this test. These results are shown in figure 6.6.

0 50 100 150 200 2500

0.2

0.4

0.6

0.8

1

Error (meters)

Det

ectio

n P

roba

bilit

y

Satellite 1Satellite 2Satellite 3Satellite 4Satellite 5Satellite 6

Figure 6.6: Detection probability versus error for LSR method

From the analysis of figure 6.6 we conclude that satellite 6 (corresponding to satellite ID 30) is the

least sensitive to errors. This is easily noticeable because for the same amount of error induced in the

various satellites of the subset, satellite 6 is the one that requires a larger error to reach 100% probability

of detection. This result is in accordance with what was expected from the analyses of the values of

SLOPES.

From equation (5.27) the theoretical values for the probability of missed detection can be obtained.

With these values we can check the simulation results for consistency. As equation (5.27) is computa-

tionally very time consuming this verification will be performed just for this test and just for satellite

1 (satellite ID 5). If the results are consistent for this particular satellite in this particular test we can

assume that for the other satellites and for the other tests it would also be consistent as the algorithms

remain unchanged.

In figure 6.7 the theoretical probability of detection is shown. The value of this probability is obtained

71

by Pdetection(theoretical) = 1− Pmd.

0 50 100 150 200 2500

0.2

0.4

0.6

0.8

1

Error (meters)

Det

ectio

n P

roba

bilit

y

Satellite 1Satellite 1( theoretical)

Figure 6.7: Theorical and simulated detection probability for satellite ID 5

From the analysis of figure 6.7 for the theoretical probability of detection we can confirm that the

values are according with the values that were obtained from the simulation, as the results consistency is

almost perfect. The small discrepancy in the values may be due to the limited number of runs that were

performed in the test.

From figure 6.8 we can also analyse that the time of first detection for each satellite for each value of

induced error.

From the analysis of figure 6.8 we conclude that for small induced values of error the time of first

detection can vary much and doesn’t have a real important meaning because as the error is so small for

the algorithm to detect, the few detections that are recorded can happen at any time of the trajectory.

As the induced error increases the time of first detection starts to decrease meaning that after the error

is introduced to a satellite the detections occurs each time in less time. Finally with higher error values,

the algorithm detects the failures immediately after the errors are induced.

During the simulations it was also noted that the satellite that required a higher induced error for

detection to occur corresponds to the satellite that is more important to the DOP value of the subset.

This means that when that satellite is removed from the subset the DOP value raises significantly more

than when any other satellite is removed from the subset. An example of this situation is shown in table

6.4 for this test.

From the analysis of table 6.4, and as stated before, we can note that when satellite 6 (ID 30) is

removed from the original subset the GDOP value becomes much higher than when any other satellite is

removed from the original satellite.

Lets now analyse the results with Range Comparison Method for this same test. These results are

shown in figure 6.9.

72

0 50 100 150 200 25040

60

80

100

120

140

160

180

200

220

240

Error (meters)

Tim

e of

Firs

t Det

ectio

n

Satellite 1

Figure 6.8: Time of first detection for sat ID 5

ID Subset GDOP valueOriginal (5, 13, 15, 21, 24, 30) 2.3792

(5, 13, 15, 21, 24) 6.1576

(5, 13, 15, 21, 30) 2.9769

(5, 13, 15, 24, 30) 2.6440

(5, 13, 21, 24, 30) 2.6634

(5, 15, 21, 24, 30) 2.9914

(13, 15, 21, 24, 30) 2.6439

Table 6.4: GDOP values for the specified subsets for first epoch

73

0 50 100 150 200 2500

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Error (meters)

Det

ectio

n P

roba

bilit

y

Satellite 1Satellite 2Satellite 3Satellite 4Satellite 5Satellite 6

Figure 6.9: Detection probability versus error for RCM method

From the analysis of figure 6.9 we can see that the results are very similar for all the satellites. This

lead us to conclude that the two methods are equally sensitive to induced errors in the measurements

and that the methods are also equally sensitive to satellite geometry.

In figure 6.10 the probability of detection difference between the two methods is shown.

It’s easily understandable that the differences are marginal and that the two methods have a very

similar performance in the detection test. Also, these small differences that are verified may be associated

with the limited number of runs that each test used.

After introducing these errors to test the algorithms for detection capabilities, a test of exclusion of

the faulty satellites was performed for the first test case. The method behind the exclusion of the faulty

satellite was previously discussed in section 5.3.2. The results from this test are shown in figure 6.11.

From the analysis of figure 6.11 we understand that the values of error that lead us to an exclusion

level of 100% are substantially higher than the ones needed for detection only. This is due to the fact

that for a satellite to get successfully excluded, that satellite has to be declared as faulty in all subsets,

as explained in section 5.3.2. Actually, some subsets may have such geometries that large errors are

needed for a detection to happen and that leads to large errors for the exclusion process. In table 6.5 the

SLOPE values for the various subsets are presented.

Analysing table 6.5 we can verify that in subset 1 and subset 6, satellite 5 (ID 24) have a high SLOPE

value meaning that in these two particular subsets the error needed for detection will be considerably

higher. The same can be verified in satellite 1 (ID 5) for subset 2. This lead us to expect that these

two satellites in these particular subsets will be very hard to detect for fault leading to a large error of

exclusion for both these satellites. Such can be verified in 6.11 where we see that both satellites 1 and 5

are, by far, the most difficult to exclude.

Let us now analyse the same test with the Range Comparison Method. The results from this test are

74

0 50 100 150 200 250−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

Error (meters)

Det

ectio

n pr

obab

ility

dife

renc

e

Satellite 1Satellite 2Satellite 3Satellite 4Satellite 5Satellite 6

Figure 6.10: Detection probability difference between RCM and LSR method for the first epoch

0 1000 2000 3000 4000 5000 60000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Error (meters)

Exc

lusi

on P

roba

bilit

y

Satellite 1Satellite 2Satellite 3Satellite 4Satellite 5Satellite 6

Figure 6.11: First exclusion simulation for LSR method

Satellite 5 Satellite 13 Satellite 15 Satellite 21 Satellite 24 Satellite 301st subset (5, 13, 15, 21, 24) 1.03 4.99 7.18 5.15 23.86 –

2nd subset (5, 13, 15, 21, 30) 77.73 0.84 1.10 1.59 – 2.83

3rd subset (5, 13, 15, 24, 30) 1.70 0.95 2.99 – 1.74 10.36

4th subset (5, 13, 21, 24, 30) 1.52 2.37 – 1.33 0.38 4.65

5th subset (5, 15, 21, 24, 30) 3.29 – 2.23 1.13 1.13 3.18

6th subset (13, 15, 21, 24, 29) – 1.86 0.58 2.68 43.47 5.38

Table 6.5: SLOPE values for the various subsets

75

shown in figure 6.12.

0 1000 2000 3000 4000 5000 60000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Error (meters)

Exc

lusi

on P

roba

bilit

y

Satellite 1Satellite 2Satellite 3Satellite 4Satellite 5Satellite 6

Figure 6.12: First exclusion simulation for RCM method

From the analysis of figure 6.12 we conclude that, as expected from the detection tests seen before,

the results are very similar to results from exclusion with Least-Squares-Residuals method. There are

some small differences that can be explained due to the random noise conditions and due to the limited

number of runs.

For the test carried out on 22/01/2015 at 19:20, where conditions are slightly similar among the

satellites, detection tests are performed for both methods. In table 6.6 the SLOPE values can be

analysed for this epoch.

Sat 1 (ID=13) Sat 2 (ID=15) Sat 3 (ID=17) Sat 4 (ID=22) Sat 5 (ID=24) Sat 6 (ID=25)

SLOPE 1.3502 4.512 5.094 2.874 2.055 3.517

Table 6.6: SLOPE values for second epoch

From the analysis of table 6.6 all SLOPE values are close, meaning that all these satellites produce

a similar positioning error for a given test statistic, so it is expected that in this case all the satellites

from the subset require a similar value of error to obtain a similar value of detection probability.

The results from detection test for the LSR method are shown in figure 6.13.

From figure 6.13 we can confirm that in this case all satellites require similar values of error for the

same detection probability to be obtained. The DOP values of the subsets was also analysed and the

results are presented in table 6.7.

Also, as expected and unlike the epoch 1, for this epoch all the subsets have very similar GDOP values

meaning none of the satellites of the original subset has a major role in the original subset geometry.

This same test was repeated for range comparison method and the results from this detection test are

shown in figure 6.14.

76

0 20 40 60 80 100 1200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Error (meters)

Det

ectio

n P

roba

bilit

y

Satellite 1Satellite 2Satellite 3Satellite 4Satellite 5Satellite 6

Figure 6.13: Second detection simulation for LSR method

ID Subset GDOP valueOriginal (13, 15, 17, 22, 24, 25) 2.0861

(13, 15, 17, 22, 24) 2.6178

(13, 15, 17, 22, 25) 2.6582

(13, 15, 17, 24, 25) 2.5799

(13, 15, 22, 24, 25) 2.7981

(13, 17, 22, 24, 25) 2.5259

(15, 17, 22, 24, 25) 2.3473

Table 6.7: GDOP values for the specified subsets for second epoch

77

0 20 40 60 80 100 1200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Error (meters)

Det

ectio

n P

roba

bilit

y

Satellite 1Satellite 2Satellite 3Satellite 4Satellite 5Satellite 6

Figure 6.14: Second detection simulation for RCM method

As expected from previous comparatives between these two methods the results are very similar, and

some of the residual differences may be due to the limited number of runs. This result increases our

confidence in saying that the two methods show the same sensitiveness to errors and satellite geometry

as for epoch 1 the results were also very similar between the two methods.

Let us now analyse the results from an induced ramp error to the measurements. These tests will be

relative to first epoch.

Two tests will be performed: one with a ramp with a slope of 1m/s and another with a slope of

0.1m/s.

Starting with the higher slope ramp the results in terms of time of first detection are shown in table

6.8.

Mean first time detection Standard deviationSatellite 1 36.29 7.38Satellite 2 34.34 6.58Satellite 3 30.31 6.12Satellite 4 34.92 5.6Satellite 5 36.96 7.03Satellite 6 80.53 15.79

Table 6.8: Mean first detection time for LSR method (ramp 1 m/s)

From table 6.8 we can understand that for the first 5 satellites the first detections occur, in mean, a

few moments after 30 seconds. As the slope of the error is 1m/s this corresponds to a mean value of error

of about 30 meters. These values are according to the expected, as in figure 6.6 the first magnitude of

errors that are being detected are errors of about 30 meters for the first 5 satellites. For the last satellite

the mean value of first detection comes much later in time as for this satellite in this specific geometry a

78

much larger error is needed for it to be detected. The standard deviation values around the first detection

gives us one idea of how much centered about the first detection time the detection is.

In figure 6.15 an illustration of the test statistic variation with this ramp error is shown.

0 50 100 150 200 250 300 350 400 4500

20

40

60

80

100

120

140

160

180

Time (seconds)

Tes

t Sta

tistic

t

Figure 6.15: Test statistic variation over a ramp error of 1 m/s

The same test was performed for Range Comparison Method and the results are presented in table

6.9.

Mean first time detection Standard deviationSatellite 1 37.16 6.32Satellite 2 34.2 5.83Satellite 3 29.16 5.58Satellite 4 35.2 6.7Satellite 5 39.14 6.98Satellite 6 79.73 14.27

Table 6.9: Mean first detection time for RCM method (ramp 1 m/s)

From the analysis of table 6.9 we can conclude that, as expected from the results from previous tests,

results are identical to those obtained with the LSR method. Slightly variations on the values reflect the

limited number of simulations performed.

For the ramp bias with a slope of 0.1 m/s the two methods were also tested. The results for LSR are

shown in table 6.10.

The results from table 6.10 show an obvious higher time to first detection as the slope is now 0.1 m/s.

While with a slope of 1 m/s the magnitude of error for first detection was about 30 meters, for this slope

we have a lower value of about 23 meters (∼ 230s× 0.1m/s). This can be explained because as the slope

is lower more points of the test statistic will be near the threshold, so is more probable that any point

near it, due to random noise, can break the threshold counting for a earlier detection when compared to

79

Mean first time detection Standard deviationSatellite 1 260.5 60.22Satellite 2 223.6 48.79Satellite 3 201.4 44Satellite 4 239.5 55Satellite 5 243 54Satellite 6 513.4 139.4

Table 6.10: Mean first detection time for LSR method (ramp 0.1 m/s)

the higher slope. This fact is illustrated in figure 6.16.

0 100 200 300 400 500 600 700 800 9000

5

10

15

20

25

30

35

40

45

Time (seconds)

Tes

t Sta

tistic

t

Figure 6.16: Test statistic variation over a ramp error of 0.1 m/s

This same test was also performed for the Range Comparison Method. For this method the results

are shown in table 6.11.

The values of table 6.11 show us very similar results comparing to the equivalent test performed with

LSR which validates, once again, that the two methods have very similar performances even for slope

errors.

80

Mean first time detection Standard deviationSatellite 1 258.1 64.87Satellite 2 226.3 53.84Satellite 3 201.5 40.67Satellite 4 234 57.3Satellite 5 241 57.3Satellite 6 502.9 125.7

Table 6.11: Mean first detection time for RCM method (ramp 0.1 m/s)

81

82

Chapter 7

Conclusion

The objectives of this thesis were to study and make a comparison of two algorithms implemented to

provide receiver autonomous integrity. To make this study a GPS constellation was simulated from

YUMA almanac parameters. The position solution from that trajectory was obtained using a Kalman

filter. To test the performance of the algorithms a subset of 6 satellites with the best GDOP parameter

was chosen among the visible satellites at the time of simulation. Two epochs were considered to test

different geometries and to study how those differences could affect the algorithms performance.

The two algorithms studied have different approaches to achieve the solution. The Least-Square-

Residuals uses the estimated position solution from all satellites in the subset to predict the six measure-

ments. The sum of squares of residuals is calculated and used as test statistic. In the range comparison

method with 6 satellites in the subset, the first 4 are used to obtain a solution and then that solution is

used to predict the remaining 2 measurements.

Regardless of the different approaches of the two methods studied, the results from the simulation

showed that the performance of both methods are almost identical in the various tests for the different

geometries studied. The small differences verified are negligible and besides they can be also affected by

the limited number of runs performed in each test.

The Least-Squares-Residuals RAIM method is specially simple in its implementation because the test

statistic is a scalar regardless of the number of satellites in use. For the Range-Comparison-Method such

is not the case and the implementation complexity raises as we use more satellites in the subset as the

decision rule divides the plane in more regions to test the failure hypotheses.

As the Range-Comparison-Method showed no advantages in the tested cases, the Least-Squares-

Residuals method is recommended because its performance is identical and the implementation is con-

siderably less complex.

Future Work Carry out some simulations with the Galileo integrity algorithm in order to understand

the real performance of the presented Galileo integrity concept.

83

84

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Appendix A

YUMA Almanac definitions

ID PRN of the SVNHealth 000=usable

Eccentricity This shows the amount of the orbit deviation fromcircular (orbit). It is the distance between the foci di-vided by the length of the semi-major axis (our orbitsare very circular).

Time of Applicability The number of seconds in the orbit when the almanacwas generated. Kind of a time tag. It is reset at thestart of the GPS week.

Orbital Inclination The angle to which the SV orbit meets the equator(GPS is at approximately 55 degrees). Roughly, theSV’s orbit will not rise above approximately 55 degreeslatitude. The number is part of an equation: # = π

180= the true inclination.

Rate of Right Ascension Rate of change in the measurement of the angle of rightascension as defined in the Right Ascension mnemonic.√

A Square Root of Semi-Major Axis This is defined as the measurement from the center ofthe orbit to either the point of apogee or the point ofperigee.

Right Ascension at Time of Almanac (TOA) Right Ascension is an angular measurement from thevernal equinox Ω0.

Argument of Perigee An angular measurement along the orbital path mea-sured from the ascending node to the point of perigee,measured in the direction of the SV’s motion.

Mean Anomaly Angle (arc) travelled past the longitude of ascendingnode (value = 0 ± 180 degrees). If the value exceeds180 degrees, subtract 360 degrees to find the meananomaly. When the SV has passed perigee and head-ing towards apogee, the mean anomaly is positive. Af-ter the point of apogee, the mean anomaly value willbe negative to the point of perigee.

af0 SV clock bias in seconds.af1 SV clock drift in seconds per seconds.Week GPS week (0000–1023), every 7 days since 1999 Au-

gust 22.

Table A.1: YUMA almanac definitions

89