free and constrained relative kinematic positioning with low cost ...

POLITECNICO DI MILANO

Doctoral Programs in Geodesy and Geomatics

goGPS free and constrained relative kinematic positioning with low cost receivers Ph.D. thesis by

Eugenio Realini

Tutor:

Prof. Maria Antonia Brovelli

Co-tutor:

Dr. Mirko Reguzzoni

Coordinator:

Prof. Fernando Sansò

Como, February 2009

goGPS - free and constrained relative kinematic positioning with low cost receivers Eugenio Realini Ph.D. Thesis Geodesy and Geomatics, XXI cycle Politecnico di Milano

Department of Environmental, Hydraulic, Infrastructures and Surveying Engineering Dipartimento di Ingegneria Idraulica, Ambientale, Infrastrutture Viarie, Rilevamento

Geomatics Laboratory Laboratorio di Geomatica Politecnico di Milano – Polo Regionale di Como via Valleggio, 11 – 22100 Como (CO) Italy http://geomatica.como.polimi.it February, 2009

Abstract goGPS is a software package designed to enhance the accuracy of standalone GPS receivers by exploiting networks of GNSS permanent stations to apply real-time relative positioning, extended Kalman filtering techniques to better model the kinematics of a roving GPS receiver, digital terrain model observations to mitigate the GPS weakness in the vertical direction and, if it is known in advance that the receiver is moving along a predefined path (e.g. a railway), linear paths to constrain the positioning. The principal innovation introduced by goGPS is the possibility to apply kinematic relative positioning in an effective way on low cost single frequency GPS receivers, enhancing their accuracy from the usual 2-4 m up to some decimeters. Though this kind of receivers is the main target for goGPS, also double frequency receivers are supported. goGPS positioning capabilities have been assessed by testing it under different conditions of sky visibility, signal degradation and dynamics of the roving receiver. Since goGPS needs GPS raw observations (i.e. code pseudorange, phase measurement, signal-to-noise ratio, etc.) the u-blox AEK-4T evaluation kit was chosen as a roving receiver, since it provides them. goGPS performance using AEK-4T were compared both with other low cost instruments (eBonTek eGPS 597, TomTom MKII) and with high level professional receivers (Leica GS20 and Leica GPS System 1200). The results show that goGPS managed to get higher accuracy than low cost receivers during all the tests, in some cases obtaining accuracy levels of the same order of magnitude of those obtained by the single frequency professional receiver (Leica GS20). goGPS is developed in a MATLAB environment and it can run either in real-time mode, receiving the low cost receiver data stream on a USB port and the master station data stream through the Internet, or in post-processing mode, reading master and rover RINEX files or goGPS data saved during a real-time session.

i

Summary

Abstract .......................................................................................................................................... i Acknowledgements ....................................................................................................................... iv List of acronyms .............................................................................................................................v Index of figures............................................................................................................................. vii Index of tables ................................................................................................................................x Introduction .................................................................................................................................. 1 Chapter 1.

Basics of GPS positioning ...................................................................................... 3

1.1.

Overview on GPS signals and receivers......................................................................... 5

1.2.

Absolute positioning ..................................................................................................... 8

1.2.1.

Code observation .................................................................................................. 8

1.2.2.

Phase observation ............................................................................................... 10

1.2.3.

Point positioning by code observations .............................................................. 11

1.2.4.

Point positioning by phase observations ............................................................ 12

1.2.5.

Ephemerides and reference frame of the positioning ........................................ 13

1.3.

Relative positioning ..................................................................................................... 14

1.3.1.

Single differences ................................................................................................ 15

1.3.2.

Double differences .............................................................................................. 16

1.3.3.

Networks of permanent GNSS stations .............................................................. 17

Chapter 2.

Low cost GPS receivers........................................................................................ 18

2.1.

Receiver architectures ................................................................................................ 18

2.1.1.

Antenna ............................................................................................................... 19

2.1.2.

RF front-end ........................................................................................................ 21

2.1.3.

Acquisition/tracking module ............................................................................... 21

2.1.4.

Embedded processor/memory ........................................................................... 22

2.1.5.

GPS software ....................................................................................................... 23

Chapter 3.

Kalman filtering in goGPS .................................................................................... 25

3.1.

Generic KF design ........................................................................................................ 28

3.2.

goGPS KF design .......................................................................................................... 31

3.2.1.

The state variables .............................................................................................. 33

3.2.2.

The dynamic model ............................................................................................. 34

3.2.3.

goGPS observation modules ............................................................................... 36

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Summary

3.2.4.

Initialization ......................................................................................................... 44

3.2.5.

Satellite configuration changes ........................................................................... 45

3.2.6.

Cycle slips ............................................................................................................ 49

3.3.

Generic line-constrained KF design............................................................................. 50

3.4.

goGPS line-constrained KF design ............................................................................... 52

3.4.1.

Line-constrained initialization ............................................................................. 54

Chapter 4.

goGPS test system design ................................................................................... 55

4.1.

Rover segment ............................................................................................................ 56

4.2.

Master segment .......................................................................................................... 57

4.3.

Core segment .............................................................................................................. 59

4.4.

Communication formats ............................................................................................. 63

4.4.1.

UBX-RXM-RAW through USB COM port.............................................................. 63

4.4.2.

RTCM 3.1 via NTRIP through TCP/IP port............................................................ 66

4.5.

Stream synchronization .............................................................................................. 70

Chapter 5. 5.1.

Tests and results.................................................................................................. 72

Instrumentation .......................................................................................................... 72

5.1.1.

Leica GPS System 1200........................................................................................ 72

5.1.2.

Leica GS20 ........................................................................................................... 73

5.1.3.

eBonTek eGPS 597 .............................................................................................. 73

5.1.4.

TomTom MKII ...................................................................................................... 74

5.2.

Short baseline test ...................................................................................................... 74

5.2.1.

Double differenced AEK-4T code quality ............................................................ 77

5.2.2.

Double differenced AEK-4T phase quality........................................................... 78

5.2.3.

Elevation and C/N0 weight calibration ............................................................... 81

5.3.

Basketball field tests ................................................................................................... 87

5.4.

Road tests.................................................................................................................... 89

5.4.1.

Test 1: good sky view, slow velocity, flat terrain ................................................ 93

5.4.2.

Test 2: bad sky view, slow velocity, flat terrain .................................................. 96

5.4.3.

Test 3: Normal driving test .................................................................................. 97

Conclusions ............................................................................................................................... 100 References................................................................................................................................. 102

iii

Acknowledgements First of all, I would like to thank Mirko for all the efforts and time he has spent to make everything function: his technical skills and his passion in programming have been both encouraging and inspiring for me. goGPS would have never seen the light of day without him, and I sincerely hope I will be able to continue working with him on goGPS even when GOCE will finally blast off! Then I want to thank Maria for having led me up to here during these three years of doctorate studies, after having tutored me also for my master’s degree and bachelor’s degree theses. She has been not only my tutor, but also a good friend for me. A “thank you all” also to my friends of the Geomatics Laboratory in Como for having put up with me for all these looong three years. I just wonder who will decide where to go to eat when I will not be there with you anymore… I think emi does not need to read this page to know that I’m grateful to her for every single day we are spending together and for all the patience she has demonstrated during these months of goGPS testing and thesis writing. Anyway, just in case: I really hope that goGPS will allow me to take you to Japan for more than a short vacation! Finally, I thank my family for having given me the opportunity to get to this point and for the continuous support throughout the years.

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List of acronyms A-GPS AGC C/N0 CPU DD DGPS DLL DOP DTM ECEF EGNOS EKF FLL FKP FTP GLONASS GNSS GPRS GPS GSM HDOP HSDPA IC IGS INS IREALP ITRF KF LBS LiDAR LNA NGA NIMA NMEA NTRIP

Assisted GPS Automatic Gain Control Carrier-to-Noise Ratio Central Processing Unit Double Differences Differential GPS Delay-Locked Loop Dilution Of Precision Digital Terrain Model Earth-Centered Earth-Fixed European Geostationary Navigation Overlay Service Extended Kalman Filter Frequency-Locked Loop Flächenkorrekturparameter (area correction parameters) File Transfer Protocol GLObal’naya NAvigatsionnaya Sputnikovaya Sistema (GLObal NAvigation Satellite System) Global Navigation Satellite System General Packet Radio Service Global Positioning System Global System for Mobile communications Horizontal Dilution Of Precision High-Speed Downlink Packet Access Integrated Circuits International GNSS Service Inertial Navigation System Istituto di Ricerca per l’Ecologia e l’Economia applicate alle Aree Alpine (Research Institute for Ecology and Economy applied to Alpine Areas) International Terrestrial Reference Frame Kalman Filter Location Based Services Light Detection And Ranging Low Noise Amplifier National Geospatial-intelligence Agency National Imagery and Mapping Agency National Marine Electronics Association Networked Transport of RTCM via Internet Protocol

v

MATLAB PDA PLL PRN RAM RF RINEX RMSE RTCM RTK SA SAW SBAS SNR SPI TCP/IP UART UKF UMPC UMTS VRS WAAS WGS84

MATrix LABoratory Personal Digital Assistant Phase-Locked Loop Pseudo Random Noise Random-Access Memory Radio Frequency Receiver INdependent EXchange format Root Mean Square Error Radio Technical Commission for Maritime services Real-Time Kinematic Selective Availability Surface Acoustic Wave Satellite Based Augmentation System Signal-to-Noise Ratio Serial Peripheral Interface Transmission Control Protocol / Internet protocol Universal Asynchronous Receiver/Transmitter Unscented Kalman Filter Ultra-Mobile PC Universal Mobile Telecommunications System Virtual Reference Station Wide Area Augmentation System World Geodetic System 1984

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Index of figures FIGURE 1.1 EXAMPLE OF HIGH COST GPS RECEIVERS ............................................................................... 4 FIGURE 1.2 EXAMPLE OF LOW COST GPS RECEIVERS................................................................................ 5 FIGURE 1.3 GPS MINIATURIZATION ...................................................................................................... 5 FIGURE 1.4 GPS-ENABLED CONSUMER DEVICES...................................................................................... 5 FIGURE 1.5 SCHEMATIC REPRESENTATION OF GPS SIGNALS ...................................................................... 6 FIGURE 2.1 SCHEMATIC DESIGN OF A STANDARD GPS RECEIVER .............................................................. 19 FIGURE 2.2 PATCH ANTENNA............................................................................................................. 20 FIGURE 2.3 HELIX ANTENNA .............................................................................................................. 20 FIGURE 2.4 TOMTOM HELIX ANTENNA ................................................................................................ 20 FIGURE 2.5 MINIATURIZED GPS MODULE WITH HELIX ANTENNA ............................................................. 20 FIGURE 2.6 SCHEMATIC DESIGN OF A STANDARD GPS RECEIVER INTERFACED WITH ITS HOST SYSTEM............. 23 FIGURE 2.7 SCHEMATIC DESIGN OF A PROCESSOR-LESS GPS RECEIVER INTERFACED WITH ITS HOST SYSTEM .... 24 FIGURE 3.1 TYPICAL KALMAN FILTERING CYCLE ..................................................................................... 26 FIGURE 3.2 KF EVOLUTION: INITIALIZATION – FIRST STEP – GENERIC ITERATION ......................................... 31 FIGURE 3.3 LINEAR CONSTRAINT MODELED AS A BROKEN LINE................................................................. 50 FIGURE 4.1 GOGPS TEST SYSTEM SCHEMA ........................................................................................... 55 FIGURE 4.2 GOGPS TEST SYSTEM ARCHITECTURE .................................................................................. 56 FIGURE 4.3 U-BLOX AEK-4T RECEIVER WITH ANN-MS ANTENNA ........................................................... 57 FIGURE 4.4 THE GNSS PERMANENT STATION IN COMO ......................................................................... 58 FIGURE 4.5 GPSLOMBARDIA NETWORK OF PERMANENT GNSS STATIONS................................................. 59 FIGURE 4.6 GOGPS FUNCTIONING FLOW CHART (REAL-TIME MODE) ........................................................ 60 FIGURE 4.7 GOGPS MAIN PANEL........................................................................................................ 62 FIGURE 4.8 GOGPS POSITIONING PLOTTED ON GOOGLE EARTH .............................................................. 63 FIGURE 4.9 UBX MESSAGE STRUCTURE ............................................................................................... 64 FIGURE 4.10 RTCM MESSAGE STRUCTURE .......................................................................................... 67 FIGURE 4.11 GOGPS STREAM INITIALIZATION ...................................................................................... 71 FIGURE 4.12 GOGPS STREAM DETAILS ................................................................................................ 71 FIGURE 5.1 LEICA GPS1200 (LEFT) AND LEICA GS20 (RIGHT) ................................................................ 73 FIGURE 5.2 EBONTEK EGPS597 (LEFT) AND TOMTOM MKII (RIGHT) ...................................................... 73

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FIGURE 5.3 AEK-4T ANTENNA POSITIONED FOR SHORT BASELINE TEST ..................................................... 74 FIGURE 5.4 EXAMPLE OF NOISE ANALYSIS ON DD CODE OBSERVATIONS .................................................... 75 FIGURE 5.5 OUTLIERS IN DD CODE ERRORS DUE TO VERY LOW C/N0 ....................................................... 76 FIGURE 5.6 DD CODE ERRORS AFTER OUTLIER REMOVAL ........................................................................ 77 FIGURE 5.7 DD CODE ERROR DISTRIBUTION ......................................................................................... 78 FIGURE 5.8 DISCONTINUITIES OF DD PHASE ERRORS ............................................................................. 79 FIGURE 5.9 EXAMPLE OF DECREASING TREND AFTER DD PHASE ERROR DISCONTINUITY REMOVAL ................. 79 FIGURE 5.10 EXAMPLE OF INCREASING TREND AFTER DD PHASE ERROR DISCONTINUITY REMOVAL ................ 80 FIGURE 5.11 DD PHASE ERROR TIME SERIES FOR SATELLITES 3,6 ............................................................. 80 FIGURE 5.12 DD PHASE ERROR COVARIANCE FUNCTION FOR SATELLITES 3,6 ............................................. 80 FIGURE 5.13 PLOT OF SHORT BASELINE RMSE VALUES COMPUTED BY C/N0 CLASSES ................................. 82 FIGURE 5.14 EXAMPLES OF C/N0 WEIGHTING FUNCTION ...................................................................... 83 FIGURE 5.15 WEIGHTING SURFACE FUNCTION AND RMSE VALUES (BLUE) ................................................ 84 FIGURE 5.16 PLOT OF SHORT BASELINE RMSE VALUES COMPUTED BY ELEVATION CLASSES .......................... 85 FIGURE 5.17 WEIGHTING FUNCTION BEHAVIOR FOR DIFFERENT ELEVATION ANGLES ................................... 86 FIGURE 5.18 WEIGHTING FUNCTION BEHAVIOR FOR DIFFERENT COMBINATIONS OF C/N0 AND ELEVATION ANGLE ............................................................................................................................................ 86 FIGURE 5.19 GOGPS 2D BASKETFIELD RESULTS (NO DTM) .................................................................... 88 FIGURE 5.20 GOGPS 2D BASKETFIELD RESULTS (WITH DTM) ................................................................. 88 FIGURE 5.21 EBONTEK 2D BASKETFIELD RESULTS ................................................................................. 88 FIGURE 5.22 GOGPS 3D BASKETFIELD RESULTS (NO DTM) .................................................................... 88 FIGURE 5.23 GOGPS 3D BASKETFIELD RESULTS (WITH DTM) ................................................................. 88 FIGURE 5.24 EBONTEK 3D BASKETFIELD RESULTS ................................................................................. 89 FIGURE 5.25 GOGPS 2D BASKETFIELD RESULTS (CONSTRAINED) ............................................................. 89 FIGURE 5.26 GOGPS 3D BASKETFIELD RESULTS (CONSTRAINED) ............................................................. 89 FIGURE 5.27 THE INSTRUMENTATION MOUNTED ON THE CAR ROOF......................................................... 90 FIGURE 5.28 ROAD TEST: DEVICE DISPOSITION ..................................................................................... 90 FIGURE 5.29 EXAMPLE OF CAR ATTITUDE AT EACH EPOCH...................................................................... 92 FIGURE 5.30 INSTRUMENTS PREDICTED POSITIONS ............................................................................... 92 FIGURE 5.31 GOGPS (BLACK), GS20 (RED) AND EBONTEK (BLUE) OBSERVED POSITIONS WITH RESPECT TO PREDICTED POSITIONS (EMPTY SQUARES) .................................................................................... 92

FIGURE 5.32 THE ROUTE DRIVEN DURING TEST 1 (IMAGES TAKEN FROM GOOGLE EARTH) ........................... 94

viii

FIGURE 5.33 TEST 1 3DCQ HISTOGRAM ............................................................................................. 94 FIGURE 5.34 TEST 1 SKYPLOTS WITH DIFFERENT CUTOFF LEVELS .............................................................. 95 FIGURE 5.35 TEST 2 3DCQ HISTOGRAM ............................................................................................. 96 FIGURE 5.36 TEST 2 GOGPS OUTLIERS (BLACK) .................................................................................... 97 FIGURE 5.37 THE ROUTE DRIVEN DURING TEST 3 (IMAGES TAKEN FROM GOOGLE EARTH) ........................... 98 FIGURE 5.38 TERRAIN HEIGHT VARIATION DURING TEST 3 ...................................................................... 98 FIGURE 5.39 TEST 3 3DCQ HISTOGRAM ............................................................................................. 99

ix

Index of tables TABLE 1.1 IGS PRECISE EPHEMERIDES ................................................................................................. 14 TABLE 4.1 UBX DATA TYPES .............................................................................................................. 64 TABLE 4.2 DESCRIPTION OF THE UBX-RXM-RAW MESSAGE .................................................................. 65 TABLE 4.3 RTCM 3.1 MESSAGES (1001 – 1020) ................................................................................ 67 TABLE 4.4 RTCM 1002 MESSAGE STRUCTURE ..................................................................................... 68 TABLE 4.5 RTCM 1019 MESSAGE STRUCTURE ..................................................................................... 69 TABLE 5.1 OVERALL SHORT BASELINE DD AEK-4T CODE QUALITY ........................................................... 77 TABLE 5.2 SHORT BASELINE DD AEK-4T PHASE QUALITY (ROUGH ESTIMATION) ........................................ 81 TABLE 5.3 SHORT BASELINE RMSE VALUES COMPUTED BY C/N0 CLASSES [M] .......................................... 82 TABLE 5.4 SHORT BASELINE NUMBER OF SAMPLES BY C/N0 CLASSES [M].................................................. 82 TABLE 5.5 SHORT BASELINE RMSE VALUES COMPUTED BY ELEVATION (E) CLASSES ..................................... 84 TABLE 5.6 BASKETBALL FIELD RESULTS ................................................................................................ 87 TABLE 5.7 TEST CHARACTERISTICS AND USED INSTRUMENTATION ............................................................ 93 TABLE 5.8 TEST 1 RESULTS ................................................................................................................ 95 TABLE 5.9 TEST 1 RESULTS (WITH HIGHER CUTOFF) ............................................................................... 95 TABLE 5.10 TEST 2 RESULTS (A) ......................................................................................................... 96 TABLE 5.11 TEST 2 RESULTS (B) ......................................................................................................... 97 TABLE 5.12 TEST 3 RESULTS .............................................................................................................. 99 TABLE 5.13 TEST 3 RESULTS (WITH HIGHER CUTOFF) ............................................................................. 99

x

Introduction

This thesis deals with accurate GPS navigation with low cost devices. Increasing the positioning accuracy of this kind of devices requires taking some steps beyond geodesy, entering the neighboring fields of telecommunications, informatics and geomatics. Modern navigation techniques and tools in fact have to interact with wireless internet connections, web services, road networks and various kinds of geographic data in order to exploit at best the available information sources. The latest efforts in technological advancement are gathering more and more computational capabilities, wireless connection tools and storage capacity on small devices such as smartphones, personal digital assistants (PDAs), netbooks and ultra-mobile personal computers (UMPCs). Nowadays most of these devices incorporate also GPS chipsets, that allow users to interact spatially with the Web 2.0 by geolocating features and events; this merging between geographical and digital worlds is giving birth to what is commonly known as the GeoWeb (Leclerc, 2001). One of the main reasons that drive hardware producing companies to embed GPS chipsets on more and more mobile devices is to provide users with Location Based Services (LBSs) where wireless connectivity is available. These embedded GPS chipsets are usually defined “low cost” in the geodetic community because they are compared to high level (and high cost) professional receivers, which generally provide accuracies ranging from 1 m to some centimeters in real-time navigation, while low cost ones usually range from 10 m to 1 m. Anyway, apart from the cost issue, high level GPS receivers could not be embedded in mobile devices due to physical limitations that usually require miniaturization, while high precision antennas are often bigger of the mobile devices themselves. Therefore, in order to increase the localization accuracy of mobile devices, low cost receivers have to be

1

Introduction

exploited at their maximum extent, also by supporting them with information coming from external sources (Alanen et al., 2006). Currently low cost GPS receivers are not taking full advantage of the growing wireless internet access with all the implications and new possibilities that it entails. Especially low cost chipsets embedded in modern 3G smartphones or PDAs can rely on (nearly) always available wireless Internet connection that, nowadays, is often broadband too. At the moment the only benefit that some GPS chipsets draw from the availability of Internet connections are A-GPS (Assisted GPS) services. These services can aid the GPS positioning in the sense that they can make it start up faster by providing broadcast ephemeris through the Internet when the receiver is powered on, but this does not increase the GPS precision or accuracy at all. One element that can help improving the accuracy of low cost receivers once they have an Internet connection available are positioning services such as networks of permanent GNSS stations. By receiving real-time data coming from one or more reference stations low cost receivers can perform double difference relative positioning that can correct most of the bias associated to the absolute positioning they usually perform. In order to apply double differences, raw measurements (i.e. pseudorange and carrier phase) must be available for both the receiver (rover) and the permanent station (master). goGPS is a software package we are developing at the Geomatics Laboratory of Politecnico di Milano, Campus Como, specifically designed to improve the positioning accuracy of low cost devices in real-time GPS navigation. Currently developed in MATLAB, goGPS applies a Kalman Filter (KF) on double difference observations, receiving data from a low cost receiver through a COM port and from a master station, belonging or not to a network of permanent GNSS stations, through the Internet via NTRIP protocol. goGPS can also exploit DTM data, if available, in order to improve its accuracy on the vertical direction and it is also able to force its estimated position on linear paths. At the moment goGPS is designed to obtain GPS raw data from the low cost receiver u-blox AEK-4T by decoding its proprietary binary stream (UBX format) and from the master GNSS permanent station in RTCM 3.1 format. goGPS can run in real-time, synchronizing the rover and master data streams and managing temporary outages or permanent data losses, applying a free or constrained KF to obtain positions epoch-byepoch and displaying data either on a MATLAB figure or on Google Earth.

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Chapter 1. Basics of GPS positioning

The Global Positioning System (GPS) was created in the 70s by the U.S. Department of Defense (DOD). It is the only fully functional Global Navigation Satellite System (GNSS) among those currently operational (the others are the Russian GLONASS and the Chinese regional geostationary BEIDOU). Other GNSS are being developed by the European Union (GALILEO), China (COMPASS or BEIDOU 2), India (IRNSS) and Japan (QZSS). The GPS is composed by three parts: the space segment, the control segment and the user segment. The space segment is currently formed by a constellation of 32 satellites, orbiting at an altitude of approximately 20000 km on six orbital planes, approximately tilted of 55° with respect to Earth’s equatorial plane. The system was designed to always guarantee the visibility of at least 4 satellites from every point on the Earth surface with clear sky view. The control segment includes a Master Control Station located near Colorado Springs and several monitoring stations located around the world. The purpose of the control segment is to monitor the satellite transmissions continuously, to predict the satellite ephemerides, to calibrate the satellite clocks and to periodically update the navigation information that the satellites carry onboard. The user segment includes all the receivers used to acquire the satellite signals and compute positions. While initially GPS was used exclusively for military applications, during the 80s some of the GPS signals were opened to civilian use, though the DOD maintained the possibility of 3

Chapter 1

Basics of GPS positioning

intentionally degrading the publicly available signals (i.e. Selective Availability - SA; for more details see Rocken, 1991) for strategic needs, introducing errors of some tens of meters in the positioning of civilian receivers. This situation induced civilian users, who started promptly to use GPS for research and work purposes, to conceive new techniques to reduce the loss of accuracy due to SA, which resulted in the first applications of Differential GPS (DGPS) positioning (Brown, 1989). Throughout the 80s and the 90s the civilian use of GPS increased greatly. Both the continuous refining of the theory of GPS positioning and the advances in electronic components and computer technology allowed to build smaller and more capable receivers, expanding the pool of users from highly trained experts to professionals from various disciplines. Additionally, throughout the 90s the U.S. military was pressured to disable SA permanently: this was done on May 1, 2000. Since the late 90s and up to now the use of GPS among professionals has been growing also thanks to the improvements made to professional receivers (Figure 1.1), which have been made more and more usable. While still keeping a high cost (thousands of Euros), they achieve centimeter-level accuracies (or better).

Figure 1.1 Example of high cost GPS receivers

In the last decade a true revolution in the diffusion of GPS devices happened as they were getting smaller and cheaper (at the expense of accuracy, dropping to some meters). These devices, that reached the consumer market, are either wired or wireless and are addressed as low cost receivers since they usually cost less than 100 Euros (Figure 1.2).

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Basics of GPS positioning

Figure 1.2 Example of low cost GPS receivers

Nowadays miniaturized low cost GPS receivers (Figure 1.3) have reached the mass market since they are embedded in car navigation equipment and various kinds of palmtop devices, ranging from netbooks to smartphones (Figure 1.4).

Figure 1.3 GPS miniaturization

Figure 1.4 GPS-enabled consumer devices

These solutions are obviously challenging for both signal acquisition capabilities and performances of receivers, but they are experiencing so much success and diffusion (see online references [2] and [3]; anyway a simple visit to any electronic store will confirm this statement) that the efforts put in their development are huge.

1.1. Overview on GPS signals and receivers GPS positioning is based on the measurement of the distance between the receiver and the satellites, derived by the flight time of a signal, which in turn is obtained by correlating the received signal and an internal replica generated by the receiver itself. Of course satellite positions have to be known, and they are computed from the available ephemerides. All GPS satellites transmit at frequencies derived from the fundamental frequency of 10.23 MHz, made available by onboard atomic clocks. There are two main signals broadcast by satellites, the L1 carrier phase at 1575.42 MHz (154 x 10.23 MHz) and the L2 carrier phase at 1227.60 MHz (120 x 10.23 MHz). Signals coming from different satellites can be distinguished

5

Chapter 1

Basics of GPS positioning

because both carriers are modulated by pseudorandom noise (PRN) sequences, different for each satellite. Two binary sequences are used: coarse/acquisition (C/A) code with a chipping rate of 1.023 MHz and precision (P) code with a chipping rate of 10.23 MHz. The C/A codes belong to the family of Gold codes, which characteristically have low cross-correlation between all members of the family. This property make them particularly apt to distinguish the signals received simultaneously from different satellites (Leick, 1995). In particular the L1 carrier is modulated by both codes, while the L2 carrier is modulated just by the P code. The P code is often called P(Y) because it can be encrypted to Y code for U.S. military needs. There is also a navigation message (D), modulated on both carriers at a chipping rate of 50 bps, that contains information on the ephemerides of the satellites, GPS time, clock behavior and system status messages. See Figure 1.5 for a schematic representation of the signals.

Figure 1.5 Schematic representation of GPS signals

Either codes or carrier phases can be used to measure the satellite-receiver distance (i.e. pseudorange). Code measurements have a higher noise than phases, namely about 1 m for C/A code and 0.3 m for P code while for phases it is some millimeters, but the corresponding pseudorange is easily obtainable with one receiver acquiring the signal from one satellite over a single epoch. Phase measurements require more complex solutions because of integer ambiguities: what is measurable is the phase shift between the received signal and the replica

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Basics of GPS positioning

over the last cycle; the total number of integer cycles excluding the last one is unknown and it must be estimated, either as a float or an integer value. Commonly float solutions are used when it is not possible to obtain integer solutions. When integer solutions are obtained, they are “fixed” so that ambiguities are known and the full potential of phase measurements can be exploited. Temporary losses of signal between the receiver and a satellite cause the corresponding number of integer cycles to be lost, requiring their new estimation: this event is known as a “cycle slip”. Taking into account that each additional satellite requires its own ambiguity to be solved, the resolution of ambiguities requires multiple observations over multiple epochs (e.g. 2 epochs for 5 satellites without cycle slips). Civilian receivers can be split into two main categories on the basis of which signals they are able to receive. Those that can receive both the L1 and L2 phases (together with C/A and P codes) are known as double frequency or geodetic receivers and, since their price is of the order of some tens of thousands of Euros, they are used mainly by professionals or universities. They have high quality electronic components, usually not forced into small size designs, that allow the precise measurement of both codes and phases. This results in the possibility of obtaining high accuracy positioning, of the order of some millimeters. Single frequency receivers can acquire only the L1 phase (with C/A code) and their price spans from some thousands of Euros down to some tens of Euros on the basis of the type and quality of their electronic components. In particular, receivers with lower cost electronics are usually affected by more phase measurement errors, because it is more difficult to measure and track correctly phases than codes. This generally leads to a lower accuracy compared to geodetic receivers, ranging from some centimeters to some meters. The expression “low cost receiver” usually refers to a single frequency receiver that costs less than 100 Euros on the consumer market. They are designed to be small enough to be easily carried around during everyday use or even to be embedded into palmtop devices. Ever growing miniaturization needs require GPS receivers designers to face more and more challenges in order to maintain the signal measurements as accurate as possible.

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Basics of GPS positioning

1.2. Absolute positioning GPS positioning is based on the hypothesis that the signal travel time from a satellite to the receiver is known. This would imply that the signal transmission and reception epochs with respect to a reference GPS time are known exactly. Nevertheless, since a perfect synchronism between either satellite or receiver clocks and GPS time is impossible, the offset between each clock and GPS time must be taken into account. For satellite and receiver clocks respectively the offset is defined as 𝑑𝑑𝑡𝑡 𝑠𝑠 = 𝑡𝑡 𝑠𝑠 − 𝑡𝑡𝐺𝐺𝐺𝐺𝐺𝐺

𝑑𝑑𝑡𝑡𝑟𝑟 = 𝑡𝑡𝑟𝑟 − 𝑡𝑡𝐺𝐺𝐺𝐺𝐺𝐺

(1.1) (1.2)

where 𝑡𝑡 𝑠𝑠 is the satellite clock, 𝑡𝑡𝑟𝑟 is the receiver clock and 𝑡𝑡𝐺𝐺𝐺𝐺𝐺𝐺 is the reference GPS time.

Once the flight time of the signal is known, positioning can be performed by using code or phase observations. Observation equations slightly differ in the two cases, therefore they will be presented separately in the next two sections.

1.2.1. Code observation Each satellite broadcast a unique code PRN sequence that is used to identify the different satellites. When a satellite signal is received, it is correlated with an internal replica of it produced by the GPS receiver. In this way the receiver can identify the visible satellites and it can measure the delay between the real signal and its replica. Let Δ𝑇𝑇𝑟𝑟𝑠𝑠 (𝑡𝑡) be the delay

observed at epoch 𝑡𝑡 between the signal coming from the satellite 𝑠𝑠 to the receiver 𝑟𝑟. Then the observation equation is given by Δ𝑇𝑇𝑟𝑟𝑠𝑠 (𝑡𝑡) = 𝑡𝑡𝑟𝑟(𝑟𝑟) − 𝑡𝑡 𝑠𝑠(𝑠𝑠)

(1.3)

where 𝑡𝑡𝑟𝑟(𝑟𝑟) is the receiving epoch recorded by the receiver 𝑟𝑟 clock and 𝑡𝑡 𝑠𝑠(𝑠𝑠) is the starting

epoch recorded by the satellite 𝑠𝑠 clock. Since both 𝑡𝑡𝑟𝑟(𝑟𝑟) and 𝑡𝑡 𝑠𝑠(𝑠𝑠) are not exactly synchronized

with GPS time, they can be expressed explicitly with the respective offsets as follows 𝑡𝑡𝑟𝑟 (𝑟𝑟) = 𝑡𝑡𝑟𝑟 + 𝑑𝑑𝑡𝑡𝑟𝑟 (𝑡𝑡)

(1.4)

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Chapter 1

Basics of GPS positioning

𝑡𝑡 𝑠𝑠(𝑠𝑠) = 𝑡𝑡 𝑠𝑠 + 𝑑𝑑𝑡𝑡 𝑠𝑠 (𝑡𝑡)

(1.5)

where 𝑡𝑡𝑟𝑟 and 𝑡𝑡 𝑠𝑠 are respectively the starting and receiving epochs in GPS time, while 𝑑𝑑𝑡𝑡𝑟𝑟 and 𝑑𝑑𝑡𝑡 𝑠𝑠 are the receiver and satellite clock offsets - see equations (1.1) and (1.2).

Thus equation (1.3) becomes

Δ𝑇𝑇𝑟𝑟𝑠𝑠 (𝑡𝑡) = 𝑡𝑡𝑟𝑟 − 𝑡𝑡 𝑠𝑠 + 𝑑𝑑𝑡𝑡𝑟𝑟 (𝑡𝑡) − 𝑑𝑑𝑡𝑡 𝑠𝑠 (𝑡𝑡) = 𝜏𝜏𝑟𝑟𝑠𝑠 + 𝑑𝑑𝑡𝑡𝑟𝑟 (𝑡𝑡) − 𝑑𝑑𝑡𝑡 𝑠𝑠 (𝑡𝑡)

(1.6)

where 𝜏𝜏𝑟𝑟𝑠𝑠 is the true signal traveling time from the satellite to the receiver.

By multiplying Δ𝑇𝑇𝑟𝑟𝑠𝑠 (𝑡𝑡) by the signal propagation velocity in vacuum 𝑐𝑐 the code observation equation (i.e. pseudorange) is obtained as

𝑃𝑃𝑟𝑟𝑠𝑠 (𝑡𝑡) = 𝑐𝑐Δ𝑇𝑇𝑟𝑟𝑠𝑠 (𝑡𝑡) = 𝑐𝑐𝜏𝜏𝑟𝑟𝑠𝑠 + 𝑐𝑐�𝑑𝑑𝑡𝑡𝑟𝑟 (𝑡𝑡) − 𝑑𝑑𝑡𝑡 𝑠𝑠 (𝑡𝑡)�

(1.7)

Since the signal does not travel in vacuum but it passes through the atmosphere, also tropospheric and ionospheric delays must be taken into account. Let 𝑇𝑇𝑟𝑟𝑠𝑠 (𝑡𝑡) be the tropospheric

delay and 𝐼𝐼𝑟𝑟𝑠𝑠 (𝑡𝑡) the ionospheric delay; the code observation equation, taking into account also

the measurement noise 𝜈𝜈𝑟𝑟𝑠𝑠 (𝑡𝑡), becomes

𝑃𝑃𝑟𝑟𝑠𝑠 (𝑡𝑡) = 𝜌𝜌𝑟𝑟𝑠𝑠 (𝑡𝑡) + 𝑐𝑐�𝑑𝑑𝑡𝑡𝑟𝑟 (𝑡𝑡) − 𝑑𝑑𝑡𝑡 𝑠𝑠 (𝑡𝑡)� + 𝐼𝐼𝑟𝑟𝑠𝑠 (𝑡𝑡) + 𝑇𝑇𝑟𝑟𝑠𝑠 (𝑡𝑡) + 𝜈𝜈𝑟𝑟𝑠𝑠 (𝑡𝑡)

(1.8)

where the geometric distance 𝜌𝜌𝑟𝑟𝑠𝑠 (𝑡𝑡) as a function of satellite and receiver positions in Cartesian coordinates is defined as

𝜌𝜌𝑟𝑟𝑠𝑠 (𝑡𝑡) = �(𝑋𝑋𝑟𝑟 (𝑡𝑡) − 𝑋𝑋 𝑠𝑠 (𝑡𝑡))2 + (𝑌𝑌𝑟𝑟 (𝑡𝑡) − 𝑌𝑌 𝑠𝑠 (𝑡𝑡))2 + (𝑍𝑍𝑟𝑟 (𝑡𝑡) − 𝑍𝑍 𝑠𝑠 (𝑡𝑡))2

(1.9)

The atmospheric delays can be described by standard models, for example the Klobuchar ionospheric model (Klobuchar, 1987) and the Saastamoinen tropospheric model (Saastamoinen, 1972, 1973). Summaries can be found on Wellenhof et al. (1992) and Seeber (1993). Since the satellite position (𝑋𝑋 𝑠𝑠 , 𝑌𝑌 𝑠𝑠 , 𝑍𝑍 𝑠𝑠 ) is assumed to be known from the ephemerides, the

satellite clock offset (𝑑𝑑𝑡𝑡 𝑠𝑠 ) is modeled and sent in the navigation message (D) and the

atmospheric delays 𝐼𝐼𝑟𝑟𝑠𝑠 (𝑡𝑡) and 𝑇𝑇𝑟𝑟𝑠𝑠 (𝑡𝑡) are estimated from standard models, there are four unknowns: the receiver position coordinates 𝑋𝑋𝑟𝑟 , 𝑌𝑌𝑟𝑟 , 𝑍𝑍𝑟𝑟 and the receiver clock offset 𝑑𝑑𝑡𝑡𝑟𝑟 .

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1.2.2. Phase observation The phase observation is obtained by measuring the offset between the received carrier phase and a sinusoid with the same frequency internally generated by the receiver. The observation equation is defined as follows Φ𝑟𝑟𝑠𝑠 (𝑡𝑡) = Φ𝑟𝑟 (𝑡𝑡) − Φ𝑟𝑟 ( 𝑟𝑟𝑠𝑠 )(𝑡𝑡)

(1.10)

where Φ𝑟𝑟𝑠𝑠 (𝑡𝑡) is the measured offset, Φ𝑟𝑟 (𝑡𝑡) is the internally generated replica and Φ𝑟𝑟 ( 𝑟𝑟𝑠𝑠 )(𝑡𝑡) is the phase of the satellite 𝑠𝑠 signal received by the receiver 𝑟𝑟 at epoch 𝑡𝑡.

The satellite phase received by 𝑟𝑟 at the epoch 𝑡𝑡 is equal to the phase emitted by the satellite 𝑠𝑠

at the emission epoch, namely Φ𝑟𝑟 ( 𝑟𝑟𝑠𝑠 )(𝑡𝑡) = Φ 𝑠𝑠 (𝑡𝑡 − 𝜏𝜏𝑟𝑟𝑠𝑠 )

(1.11)

where 𝜏𝜏𝑟𝑟𝑠𝑠 is the signal traveling time. For a stable oscillator with a frequency 𝑓𝑓 = (1.11) can be expressed by a first-order Taylor expansion as follows

Φ 𝑠𝑠 (𝑡𝑡 − 𝜏𝜏𝑟𝑟𝑠𝑠 ) = Φ 𝑠𝑠 (𝑡𝑡) −

𝑑𝑑Φ 𝑠𝑠 𝑠𝑠 𝜏𝜏 𝑑𝑑𝑑𝑑 𝑟𝑟

= Φ 𝑠𝑠 (𝑡𝑡) − 𝑓𝑓𝜏𝜏𝑟𝑟𝑠𝑠 + 𝑁𝑁𝑟𝑟𝑠𝑠 (𝑡𝑡)

𝑑𝑑Φ , 𝑑𝑑𝑑𝑑

equation

(1.12)

where 𝑁𝑁𝑟𝑟𝑠𝑠 is the integer number of cycles between the emission epoch and the reception

epoch. 𝑁𝑁𝑟𝑟𝑠𝑠 cannot be observed.

Using equations (1.11) and (1.12), equation (1.10) becomes

Φ𝑟𝑟𝑠𝑠 (𝑡𝑡) = Φ𝑟𝑟 (𝑡𝑡) − Φ 𝑠𝑠 (𝑡𝑡) + 𝑓𝑓𝜏𝜏𝑟𝑟𝑠𝑠 − 𝑁𝑁𝑟𝑟𝑠𝑠 (𝑡𝑡)

(1.13)

Taking into account the offsets of receiver and satellite clocks, Φ𝑟𝑟 (𝑡𝑡) and Φ 𝑠𝑠 (𝑡𝑡) are expressed

as follows

Φ𝑟𝑟 (𝑡𝑡) = Φ(𝑡𝑡) + 𝑓𝑓𝑓𝑓𝑡𝑡𝑟𝑟 (𝑡𝑡)

Φ 𝑠𝑠 (𝑡𝑡) = Φ(𝑡𝑡) + 𝑓𝑓𝑓𝑓𝑡𝑡 𝑠𝑠 (𝑡𝑡)

(1.14)

(1.15)

where Φ(𝑡𝑡) is the phase of an ideal oscillator synchronized with the GPS time and 𝑑𝑑𝑡𝑡𝑟𝑟 , 𝑑𝑑𝑡𝑡 𝑠𝑠 the clock offsets.

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Basics of GPS positioning

Using equations (1.14) and (1.15), equation (1.13) becomes Φ𝑟𝑟𝑠𝑠 (𝑡𝑡) = 𝑓𝑓𝜏𝜏𝑟𝑟𝑠𝑠 + 𝑓𝑓�𝑑𝑑𝑡𝑡𝑟𝑟 (𝑡𝑡) − 𝑑𝑑𝑡𝑡 𝑠𝑠 (𝑡𝑡)� − 𝑁𝑁𝑟𝑟𝑠𝑠 (𝑡𝑡)

(1.16)

By multiplying it by the signal wavelength, introducing the ionospheric and tropospheric delays and taking into account the phase measurement noise 𝜂𝜂𝑟𝑟𝑠𝑠 (𝑡𝑡), the phase observation equation is obtained, namely

𝜆𝜆Φ𝑟𝑟𝑠𝑠 (𝑡𝑡) = 𝜌𝜌𝑟𝑟𝑠𝑠 (𝑡𝑡) + 𝑐𝑐�𝑑𝑑𝑡𝑡𝑟𝑟 (𝑡𝑡) − 𝑑𝑑𝑡𝑡 𝑠𝑠 (𝑡𝑡)� − 𝐼𝐼𝑟𝑟𝑠𝑠 (𝑡𝑡) + 𝑇𝑇𝑟𝑟𝑠𝑠 (𝑡𝑡) − 𝜆𝜆𝜆𝜆𝑟𝑟𝑠𝑠 (𝑡𝑡) + 𝜂𝜂𝑟𝑟𝑠𝑠 (𝑡𝑡)

(1.17)

𝑐𝑐 𝜆𝜆

where 𝑓𝑓 = .

The known and unknown terms in the phase observation equation recall those of the code observation equation (see the final part of section 1.2.1), apart from the additional unknown 𝜆𝜆𝜆𝜆𝑟𝑟𝑠𝑠 , which is called integer ambiguity. 1.2.3. Point positioning by code observations The estimation of the receiver position (i.e. its coordinates in an Earth-Centered Earth-Fixed (ECEF) reference frame), just based on the signals received from the satellites, with no addition of external information, is usually called point positioning. As seen in section 1.2.1, the code observation equation retains four unknowns: the three coordinates 𝑋𝑋𝑟𝑟 , 𝑌𝑌𝑟𝑟 , 𝑍𝑍𝑟𝑟 and the clock offset 𝑑𝑑𝑡𝑡𝑟𝑟 . At each epoch of measurement, point positioning can thus be performed by having at least four observations, which means receiving

signal from at least four satellites. Since it is possible to estimate the receiver position by using single epoch observation, point positioning by code observations can be effectively used in kinematic positioning for navigation purposes. The system of code observation equations is non linear, therefore it is required to linearize it around an approximate value of the receiver coordinates in order to solve it. An approximation of the receiver position can be computed by means of deterministic geometric approaches such as the Bancroft algorithm (Bancroft, 1985).

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Chapter 1

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The linearization of the code observation equation with respect to the receiver coordinates for a single epoch yields 𝑃𝑃𝑟𝑟𝑠𝑠 =

𝑋𝑋�𝑟𝑟 −𝑋𝑋� 𝑠𝑠 �𝑋𝑋𝑟𝑟 �𝑟𝑟𝑠𝑠 𝜌𝜌

− 𝑋𝑋�𝑟𝑟 � +

𝑌𝑌�𝑟𝑟 −𝑌𝑌 𝑠𝑠 �𝑌𝑌𝑟𝑟 �𝑟𝑟𝑠𝑠 𝜌𝜌

𝜀𝜀(𝑋𝑋 𝑠𝑠 , 𝑌𝑌 𝑠𝑠 , 𝑍𝑍 𝑠𝑠 , 𝑑𝑑𝑡𝑡 𝑠𝑠 , 𝐼𝐼𝑟𝑟𝑠𝑠 , 𝑇𝑇𝑟𝑟𝑠𝑠 ) + 𝜈𝜈𝑟𝑟𝑠𝑠

− 𝑌𝑌�𝑟𝑟 � +

𝑍𝑍�𝑟𝑟 −𝑍𝑍� 𝑠𝑠 �𝑍𝑍𝑟𝑟 �𝑟𝑟𝑠𝑠 𝜌𝜌

− 𝑍𝑍�𝑟𝑟 � + 𝑐𝑐𝑐𝑐𝑡𝑡𝑟𝑟 + �𝜌𝜌�𝑟𝑟𝑠𝑠 − 𝑐𝑐𝑐𝑐𝑡𝑡̃ 𝑠𝑠 + 𝐼𝐼̃𝑟𝑟𝑠𝑠 + 𝑇𝑇�𝑟𝑟𝑠𝑠 � +

(1.18)

where �𝑋𝑋�𝑟𝑟 , 𝑌𝑌�𝑟𝑟 , 𝑍𝑍�𝑟𝑟 � are the coordinates of the receiver approximate position, �𝐼𝐼̃𝑟𝑟𝑠𝑠 , 𝑇𝑇�𝑟𝑟𝑠𝑠 � are

computed from standard atmospheric models, �𝑋𝑋� 𝑠𝑠 , 𝑌𝑌� 𝑠𝑠 , 𝑍𝑍� 𝑠𝑠 , 𝑑𝑑𝑡𝑡̃ 𝑠𝑠 � are derived from the navigational message and 𝜀𝜀(𝑋𝑋 𝑠𝑠 , 𝑌𝑌 𝑠𝑠 , 𝑍𝑍 𝑠𝑠 , 𝑑𝑑𝑡𝑡 𝑠𝑠 , 𝐼𝐼𝑟𝑟𝑠𝑠 , 𝑇𝑇𝑟𝑟𝑠𝑠 ) is a term containing the effects of the

various model approximations.

The system of linearized observation equations for 𝑚𝑚 satellites can be solved by least squares adjustment for 𝑚𝑚 ≥ 4.

1.2.4. Point positioning by phase observations Compared to the code observation equation, the phase observation equation (1.17) contains one additional unknown, the initial phase ambiguity. This term must be estimated for each satellite in view, therefore it is necessary to increase the number of observation epochs in order to solve the system with respect to the receiver coordinates. For example, with five satellites in view over two observation epochs there are ten unknown terms (three receiver coordinates if the receiver is still, two receiver clock offsets and five initial phase ambiguities) and ten observations, which makes the system solvable. Phase observation noise is much lower than that of the code, thus it should provide higher accuracy results. Anyway, in order to correctly observe phases, high quality receivers are needed: usually this means high cost receivers provided with good antennas, high precision oscillators and, in general, sophisticated electronic components. Moreover, point positioning by phase observations is not feasible in single epoch solutions, unless a first estimation of the phase ambiguities is performed (e.g. on-the-fly estimation, see Abidin, 1993). For these reasons, real-time navigation with low cost receivers is generally based on code observations.

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Nevertheless, low cost receivers are able to acquire the L1 phase, even if moderately degraded; generally they exploit this information by applying the so-called “phase-smoothed code” positioning. This technique (Westrop et al., 1989) is used to enhance code observations by updating them using the variation of phase observations. In the single frequency case the phase smoothing algorithm is expressed by the following equation (assuming that there are no cycle slips) 𝑃𝑃𝑟𝑟𝑠𝑠 (𝑡𝑡)𝑠𝑠𝑠𝑠 = 𝑃𝑃𝑟𝑟𝑠𝑠 (0) + 𝜆𝜆[Φ𝑟𝑟𝑠𝑠 (𝑡𝑡) − Φ𝑟𝑟𝑠𝑠 (0)]

(1.19)

where 𝑃𝑃𝑟𝑟𝑠𝑠 (𝑡𝑡)𝑠𝑠𝑠𝑠 is the smoothed pseudorange at epoch 𝑡𝑡, 𝑃𝑃𝑟𝑟𝑠𝑠 (0) is the initial code pseudorange

and Φ𝑟𝑟𝑠𝑠 (𝑡𝑡) is the phase measurement at epoch 𝑡𝑡.

1.2.5. Ephemerides and reference frame of the positioning The reference frame of the receiver estimated coordinates strictly depends on the reference frame of the satellite positions, thus on the used ephemerides. There are two kinds of ephemerides: the broadcast and the precise ones. Broadcast ephemerides are predicted by the NGA (National Geospatial-intelligence Agency, formerly NIMA - National Imagery and Mapping Agency) and correspond to those transmitted by the satellites. They are available in real-time and therefore they are used for real-time navigation. Their accuracy is about 2 m. Precise ephemerides are computed a posteriori and provided via the Internet by various groups. The precise ephemerides internationally used as a reference are those computed and provided by the IGS (International GNSS Service). IGS provides three kinds of precise ephemerides via the Internet: ultra-rapid, rapid and final ones (see Table 1.1 for more details; source: [7]).

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Table 1.1 IGS precise ephemerides

GPS Satellite Ephemerides / satellite & station clocks

Accuracy

orbits

~160 cm

sat. clocks

~7 ns

Ultra-rapid

orbits

~10 cm

(predicted half)

sat. clocks

~5 ns

Ultra-rapid

orbits

=5 : PR+DO+CP OK