Analysis of Iridium-Augmented GPS for Floating Carrier ... · undetected hazardous navigation...

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Analysis of Iridium-Augmented GPS for Floating Carrier Phase Positioning MATHIEU JOERGER, LIVIO GRATTON, and BORIS PERVAN Illinois Institute of Technology, Chicago, Illinois, 60616 CLARK E. COHEN Coherent Navigation, Inc., San Mateo, California, 94404 Received December 2009; Revised March 2010 ABSTRACT: Carrier-phase ranging measurements from Global Positioning System (GPS) and low-Earth-orbit- ing Iridium telecommunication satellites are integrated in a precision navigation system named iGPS. The ba- sic goal of the system is to enhance GPS positioning and timing performance, especially under jamming. In addition, large satellite geometry variations generated by fast-moving Iridium spacecraft enable rapid estima- tion of floating cycle ambiguities. Augmentation of GPS with Iridium satellites also guarantees signal redun- dancy, which enables Receiver Autonomous Integrity Monitoring (RAIM). In this work, parametric models are developed for iGPS measurement error sources and for wide-area corrections from an assumed network of ground reference stations. A fixed-interval positioning and cycle ambiguity estimation algorithm is derived and a residual-based carrier-phase RAIM detection method is investigated for integrity against single-satellite step and ramp-type faults of all magnitudes and start-times. Predicted overall performance is quantified for various ground, space, and user segment configurations. INTRODUCTION The iGPS navigation and communication system combines ranging measurements from the Global Positioning System (GPS) and low Earth orbit (LEO) Iridium telecommunication satellites. iGPS is a Boeing initiative now under contract from the U.S. Navy aimed at enhancing GPS timing and positioning performance, especially under jam- ming. In addition, while many details of the iGPS system are not public, assumptions are made herein indicating that iGPS opens the possibility for rapid, robust, and accurate floating carrier- phase positioning over wide areas (without need for local reference stations). The system’s promise for real-time high-accuracy positioning perform- ance makes it a potential navigation solution for demanding precision applications such as autono- mous terrestrial and aerial transportation. The ability of iGPS to harness the exacting precision of carrier phase emanates from the concept of geo- metric diversity. The algorithms and analyses developed in this paper are founded on two core principles that exploit the addition of fast-moving LEO Iridium space vehicles (SVs): large changes in satellite geometry for rapid cycle ambiguity esti- mation and satellite redundancy for fault-detec- tion. Historical Background Observations from the fast-moving LEO space- craft Sputnik were at the origin of the first satel- lite radio-navigation system, known as the Navy Navigation Satellite System or Transit, which became operational in 1964 [1]. The Transit con- stellation was comprised of 4-7 LEO SVs in nearly circular, polar orbits, which broadcast radio-fre- quency signals with encoded orbital parameters and time corrections. Users could determine their position by tracking the apparent compression and stretching of the carrier wavelength due to space- craft motion over 10–20 min passes. Each location in sight of the satellite observed a unique Doppler shift curve (defined as the difference between sig- nal frequencies at the transmitter and at the re- ceiver). As a result, Doppler-based position fixes were achievable several times a day (at 100 min intervals at mid-latitudes) with better than 70 meters of accuracy, which met the requirements originally intended for slow moving military vessels and The peer review of this paper was independently handled by Professor John Crassidis, Associate Editor for the AIAA Journal of Guidance, Control, and Dynamics. NAVIGATION: Journal of The Institute of Navigation Vol. 57, No. 2, Summer 2010 Printed in the U.S.A. 137

Transcript of Analysis of Iridium-Augmented GPS for Floating Carrier ... · undetected hazardous navigation...

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Analysis of Iridium-Augmented GPS forFloating Carrier Phase Positioning

MATHIEU JOERGER, LIVIO GRATTON, and BORIS PERVANIllinois Institute of Technology, Chicago, Illinois, 60616

CLARK E. COHENCoherent Navigation, Inc., San Mateo, California, 94404

Received December 2009; Revised March 2010

ABSTRACT: Carrier-phase ranging measurements from Global Positioning System (GPS) and low-Earth-orbit-ing Iridium telecommunication satellites are integrated in a precision navigation system named iGPS. The ba-sic goal of the system is to enhance GPS positioning and timing performance, especially under jamming. Inaddition, large satellite geometry variations generated by fast-moving Iridium spacecraft enable rapid estima-tion of floating cycle ambiguities. Augmentation of GPS with Iridium satellites also guarantees signal redun-dancy, which enables Receiver Autonomous Integrity Monitoring (RAIM). In this work, parametric models aredeveloped for iGPS measurement error sources and for wide-area corrections from an assumed network ofground reference stations. A fixed-interval positioning and cycle ambiguity estimation algorithm is derived anda residual-based carrier-phase RAIM detection method is investigated for integrity against single-satellite stepand ramp-type faults of all magnitudes and start-times. Predicted overall performance is quantified for variousground, space, and user segment configurations.

INTRODUCTION

The iGPS navigation and communication systemcombines ranging measurements from the GlobalPositioning System (GPS) and low Earth orbit(LEO) Iridium telecommunication satellites. iGPSis a Boeing initiative now under contract from theU.S. Navy aimed at enhancing GPS timing andpositioning performance, especially under jam-ming. In addition, while many details of the iGPSsystem are not public, assumptions are madeherein indicating that iGPS opens the possibilityfor rapid, robust, and accurate floating carrier-phase positioning over wide areas (without needfor local reference stations). The system’s promisefor real-time high-accuracy positioning perform-ance makes it a potential navigation solution fordemanding precision applications such as autono-mous terrestrial and aerial transportation. Theability of iGPS to harness the exacting precision ofcarrier phase emanates from the concept of geo-metric diversity. The algorithms and analysesdeveloped in this paper are founded on two core

principles that exploit the addition of fast-movingLEO Iridium space vehicles (SVs): large changes insatellite geometry for rapid cycle ambiguity esti-mation and satellite redundancy for fault-detec-tion.

Historical Background

Observations from the fast-moving LEO space-craft Sputnik were at the origin of the first satel-lite radio-navigation system, known as the NavyNavigation Satellite System or Transit, whichbecame operational in 1964 [1]. The Transit con-stellation was comprised of 4-7 LEO SVs in nearlycircular, polar orbits, which broadcast radio-fre-quency signals with encoded orbital parametersand time corrections. Users could determine theirposition by tracking the apparent compression andstretching of the carrier wavelength due to space-craft motion over 10–20 min passes. Each locationin sight of the satellite observed a unique Dopplershift curve (defined as the difference between sig-nal frequencies at the transmitter and at the re-ceiver). As a result, Doppler-based position fixeswere achievable several times a day (at 100 minintervals at mid-latitudes) with better than 70 metersof accuracy, which met the requirements originallyintended for slow moving military vessels and

The peer review of this paper was independently handled byProfessor John Crassidis, Associate Editor for the AIAA Journalof Guidance, Control, and Dynamics.

NAVIGATION: Journal of The Institute of NavigationVol. 57, No. 2, Summer 2010Printed in the U.S.A.

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submarines [2]. It was often used in conjunctionwith inertial navigation systems, which wereemployed to correct for the added uncertainty dueto user motion and to bridge gaps between infre-quent position updates.

In the 1990s, Transit was superseded in bothmilitary and civilian applications by GPS, whichdirectly utilizes range instead of range rate. Codesmodulated on GPS signals provide instantaneousand absolute measurement of the travel timebetween the satellite transmitter and user receiver.In addition, the GPS medium Earth orbit constel-lation ensures that at least four SVs are continu-ously visible anywhere on Earth. This enablesreal-time determination by trilateration of theuser’s receiver clock deviation and three-dimen-sional position within about 10–20 m of accuracy[3]. The ultimate in GPS performance is obtainedusing measurements of the signal’s carrier phase.Its tracking error is lower than the code’s by twoto three orders of magnitude, but it requires thatan unknown constant cycle ambiguity be deter-mined (receivers can only track the carrier phasemodulus 2p). An efficient solution for this problemis to exploit the bias observability provided by sat-ellite motion. Unfortunately, the large amount oftime for GPS spacecraft to achieve significantchanges in line of sight (LOS) precludes its use inmost real-time applications.

In contrast, angular variations from LEO satel-lites quickly become substantial. Therefore in thiswork, the geometric diversity of GPS ranging sour-ces is enhanced using additional carrier phasemeasurements from fast moving Iridium satellites.In fact, carrier phase observations are equal tointegrated Doppler shift, so that the underlyingconcepts of utilizing spacecraft motion to resolvecycle ambiguities and of Transit’s Doppler position-ing are equivalent. Combined with GPS, real-timeunambiguous carrier-phase based trilateration ispossible without restriction on the user’s motion.In the longer term future, integration of GPS andIridium signals with additional ranging sourcesfrom other global navigation satellite systems(GNSS) such as GLONASS and Galileo will lead tounprecedented levels of navigation performance.

Related Work

The Integrity Beacon Landing System (IBLS)devised in the early 1990s was an explicit imple-mentation of this principle for aircraft precisionapproach and landing [4, 5]. GPS signal transmit-ters serving as pseudo-satellites (‘pseudolites’)placed on the ground along the airplane’s trajec-tory provided additional ranging sources and alarge geometry change as the receiver’s downward-looking antenna flew over the installation. The

efficiency of IBLS was demonstrated in 1994 as itenabled 110 successful automatic landings of aBoeing 737 [4] and in 2003 for the high angle-of-attack Extremely Short Takeoff and Landing(ESTOL) of the Navy/Boeing/EADS X-31 aircraft[6]. However, pseudolite placement constraints andoff-field operations and maintenance access costhas dampened wider application of the system.

By 2000, Rabinowitz et al. designed a receivercapable of tracking carrier-phase measurementsfrom GPS and from GlobalStar (another LEO tele-communication constellation) [7]. Using GlobalStarsatellites’ large range variations, precise cycle am-biguity resolution and positioning was achievedwithin 5 min. Numerous practical issues relativeto the synchronization of GPS and GlobalStar data(without modification of the SV payload) had to beovercome to obtain experimental validation results.Such considerations are outside the scope of thispaper, but Rabinowitz’s work offers a proof of con-cept for the precision applications of iGPS.

Measurement Error Models

The treatment of measurement errors plays acentral part in the design of the iGPS navigationsystem. Error sources include uncertainties in sat-ellite clocks and positions, signal propagationdelays in the ionosphere and troposphere, user re-ceiver noise, and multipath (unwanted signalreflections reaching the user antenna). The iono-sphere, a non-uniform upper layer of the atmos-phere, is the largest source of measurement error.It is a dispersive medium where refraction varieswith signal frequency. Dual-frequency implementa-tions effectively eliminate the ionospheric errorthus improving the positioning accuracy, but willnot be widely available for civilian applicationsbefore 2025 [8].

Instead, differential corrections can help miti-gate satellite-dependent and spatially-correlatedatmospheric errors. In differential GPS, measure-ments collected at ground reference stations arecompared with the known distance between thesestations and the satellites. The resulting correctionaccuracy varies with user-to-ground-station sepa-ration distance. In the aforementioned pseudoliteand GlobalStar-augmented GPS research, theshort baseline-distance from the differential refer-ence station to the user (1–5 km) was instrumentalin achieving high performance. In Rabinowitz’swork in particular, residual measurement errorsover short baselines could be modeled reliablyenough to allow for integer cycle ambiguities to befixed.

iGPS however aims at servicing wide areas withas little ground infrastructure as possible, andtherefore relies on long-range, lower-accuracy

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corrections similar to the ones generated by theWide Area Augmentation System (WAAS). The lat-ter has been operational since 2003 and produces a95% positioning accuracy better than 10 m for sin-gle-frequency code-phase GPS users across theUnited States [9]. When using long-range correc-tions, the unpredictability of atmospheric effectsmakes it difficult to capture residual errors withhigh levels of confidence.

Hence, in this work, a conservative approach isadopted for the derivation of new parametric mea-surement error models. They account for the in-stantaneous uncertainty at signal acquisition(absolute measurement error) as well as variationsover the signal tracking duration (relative errorwith respect to initialization). Unlike existing GPSmeasurement models used in WAAS and in theLocal Area Augmentation System (LAAS), theydeal with large drifts in ranging error for LEO sat-ellite signals moving across wide sections of theatmosphere. Published data and experimentalresults help establish an initial knowledge of themeasurement error probability distributions. Theyalso show that the dynamics of the errors can bereliably modeled over short time periods [10].

Algorithms and Analysis

Thus, two conflicting considerations are shapingthe carrier-phase iGPS estimation and detectionprocesses that are devised in this paper: rangingmeasurements must be tracked for as long as pos-sible to draw maximum benefit from changes insatellite geometry, but as this filtering durationincreases, the robustness of the measurement errormodel decreases. In response, an upper limit on fil-ter duration is set to ensure the model’s validity,and a fixed-interval filtering algorithm is carriedout for the simultaneous estimation of user posi-tion and real-valued (floating) carrier-phase cycleambiguities. It is compatible with real-time imple-mentations provided that sufficient memory is allo-cated to the storage of a finite number of pastmeasurements and LOS coefficients.

In addition, Iridium and GPS code and carrier-phase observations collected within the filteringinterval are all vulnerable to rare-event integritythreats such as user equipment and satellite fail-ures. In this regard, the augmentation of GPSwith Iridium offers an advantage in guaranteeingredundant measurements, which enables ReceiverAutonomous Integrity Monitoring (RAIM) [11].Indeed, if five or more satellites are available, theconsistency of the over-determined position solu-tion is verifiable. The accuracy of carrier-phaseobservations further allows for an extremely tightdetection threshold while still ensuring a very lowfalse-alarm probability. To protect the system

against faults that may affect successive measure-ments, an innovative least-squares residual-baseddetection routine is developed. It is similar to aninstantaneous or ‘snapshot’ RAIM algorithm [12],but is applied to batches of code and carrier phasemeasurements, whose time-correlated error distri-butions are bounded.

Potential applications for iGPS are investigated,including ground and aerial transportation. Specif-ically, the ability of iGPS to meet a predefined ver-tical alert limit (VAL) [13] is investigated here.Since transportation involves safety-of-life opera-tions, special emphasis is placed on integrity. Themost stringent known requirements proposed forsuch operations specify that no more than oneundetected hazardous navigation system failure isallowed in a billion approaches [13].

Performance evaluations are structured aroundthese example requirements. They measure theimpact of Iridium’s near-polar orbits on positioningand monitoring for a series of satellite geometries,thus providing detailed insights into observabilitymechanisms. Finally, the multidimensionality ofthe algorithm and the multiplicity of system pa-rameters make the design of the navigation archi-tecture particularly complicated. A sensitivityanalysis is proposed that compares the relativeinfluence of individual system parameters on theoverall end-user output. The methodology singlesout system components likely to bring about sub-stantial performance improvement and establishesrecommendations on possible orientations forfuture design iterations.

Outline of the Paper

The second section of this paper introduces aconceptual architecture for the iGPS space,ground, and user segments. In the third section,measurement error models applicable to short timeperiods are derived. They are employed in a posi-tioning and cycle-ambiguity estimation process,which serves as basis for the residual-based RAIMdetection algorithm, as developed in the fourthsection. Next, the benchmark aircraft precisionapproach mission and its requirements are pre-sented, which set the framework for the simula-tions. In the sixth section, fault-free (FF) integrityis evaluated by covariance analysis, and residual-based detection is tested for a set of canonical stepand ramp-type single-satellite faults (SSF) of allmagnitudes and of multiple start-times. In the lastsection, a sensitivity analysis of the combined FFand SSF performance brings dominant system pa-rameters to the foreground, investigates alterna-tive system configurations, and assesses the poten-tial of iGPS to provide nationwide high-integritypositioning in a near-term and longer-term future.

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IGPS SYSTEM DESCRIPTION

As part of this work, a nominal navigation sys-tem configuration is proposed. It was iterativelyrefined as a result of simulated performance sensi-tivity analyses (final section of the paper), so thatiGPS could fulfill the intended civilian applicationrequirements.

GPS and Iridium Satellite Constellations

In this work, a baseline GPS constellation is con-sidered [14]. 24 satellites follow near-circular orbitsat about 20,000 km altitude. They are arranged insix orbital planes of four spacecraft each, with 55deg inclination angles, and with an orbital period,TGPS, of 1/2 sidereal day. One distinctive feature ofGPS satellites is that they are equipped withhighly-stable atomic cesium and rubidium clocks(long-term stability on the order of 10�13 [15]),which are essential to the system’s precise syn-chronization on a common time-reference for directtransit time measurements.

The Iridium satellite constellation’s primaryfunction is to provide telecommunication capabil-ities to users worldwide, particularly in remoteplaces where other communication means areunavailable. Messages are exchanged betweenusers and satellites and satellite-to-satellite cross-links enable uninterrupted communications so thatany two points on the globe are connected [16]. Tele-communication satellite clocks do not meet theatomic standard, but it has been demonstratedthat their clock drift can be effectively modeledand corrected using GPS measurements at theuser receiver [17]. Continuous global coverage isrealized using 66 spacecraft orbiting at an altitudeof 780 km, which is much lower than the 20,000km GPS orbit altitude. As a result, a spacecraftspends on average 10 minutes in view of a givenlocation on Earth’s surface, and circles the earth ina period, TIRI, of 100 min 28 s [18]. As illustratedin Figure 1, the 66 satellites are distributed amongsix planes in near-circular orbits at 86.4 deg incli-nation. A 31.6 deg angle separates each co-rotatingorbital plane, and the remaining 22 deg angleseparates the two planes at the seam of the con-stellation, where spacecraft are counter-rotating[19]. Each plane contains 11 regularly spaced satel-lites, whose positions are offset from SVs in the ad-jacent co-rotating plane by one-half of the satellitespacing.

As a consequence of the constellation design, thesatellite density is much higher near the polesthan at lower latitudes; for example, the averagenumber of Iridium SVs visible at any instant is 2.2in Chicago and 1.8 in Miami. In addition, thespacecraft trajectories generate larger North-South

LOS variations relative to a ground observer thanEast-West. Accordingly, the horizontal positioningperformance is heterogeneous and higher precisionis generally achieved for the North coordinate.

In this work, the possibility of expanding theIridium space segment was considered with amodified 88 Iridium satellite constellation. In thiscase, eight orbital planes are separated by a 23deg angle, leaving 19 deg at the seam of the con-stellation. The resulting increased average space-craft numbers over Chicago and Miami are respec-tively 3 and 2.4.

Finally, nominal GPS and Iridium orbits are pic-tured together in Figure 2(a). The constellations’orbital planes are quasi-stationary in an Earth-centered inertial (ECI) frame, whose origin is thecenter of the Earth and whose axes are fixed withrespect to the stars. Satellites travel along theorbits, while the Earth rotates about its North-South axis. Figure 2(b) shows (from the point ofview of a user at the Miami location) the large dif-ference in accumulated angular variations betweenGPS and LEO satellites over a 10 minute period.In parallel, for the same location and duration, anazimuth-elevation sky plot underscores again thedifference in spacecraft motion, and the North-South directionality of Iridium satellites.

Assumed Ground Segment

The GPS ground-based Operational Control Seg-ment (OCS) makes satellite position and time syn-chronization information available to users. Space-craft dynamics are modeled using observationsfrom six ground monitoring stations spread around

Fig. 1–Iridium Satellite Coverage

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the world. Orbit ephemeris parameters are com-puted at a master control station, uploaded to thesatellites, and broadcast to users as part of thenavigation message modulated on the GPS signal.The six monitoring stations are equipped withatomic clocks to establish satellite clock correctionsalso transmitted in the navigation message. Addi-tional functions fulfilled by the OCS include moni-toring satellite health and commanding occasionalSV station-keeping maneuvers. A similar architec-ture is assumed for Iridium satellites. Althoughthey cannot be continuously tracked by ground sta-tions, they are visible several times a day.

Precision navigation requires that additional in-formation be transmitted, in particular to correctfor errors due to refraction in the ionosphere.Unlike clock errors that have a similar impact forall ground stations within a satellite’s footprint,the effects of ephemeris errors and, to a greaterextent, of ionospheric disturbances vary with re-ceiver location. Indeed, the LOS to the receiverdetermines the section of the atmosphere crossedby the signal. Two differential approaches to miti-gate ionospheric effects have been the subject ofextensive studies over the past decade. The firstapproach, used in LAAS, aims at providing correc-tions at selected airport locations, based on meas-urements from ground receivers experiencingerrors that are very similar to the aircraft’s within

a limited broadcast radius. The second approach,implemented in WAAS, consists of sampling theionosphere over wide areas using observationsfrom a network of ground stations. In WAAS, 25wide area reference stations (WRS) spread acrossthe United States collect dual-frequency measure-ments, which are used at a wide area master sta-tion (WMS) to compute ionospheric delays for a5 deg 3 5 deg latitude-longitude grid of locations[14]. The WAAS ionospheric delay estimates (aswell as additional satellite ephemeris and clockcorrections) are broadcast via geostationary satel-lites to users who can interpolate corrections atthe location of interest. The precision of these esti-mates decreases in coastal areas due to depletedground station coverage.

In this work, the conceptual iGPS ground seg-ment consists of a network of ground reference sta-tions (illustrated in Figure 3), whose density deter-mines the accuracy of ionospheric corrections. In afirst attempt to determine the overall system per-formance, iGPS ground stations are assumed co-located with the WRS, whose correction accuracyhas been documented over the past five years [9].WAAS-like ionospheric delay estimates areassumed to be derived at a master station (fol-lowing the algorithm given in reference [20]) andbroadcast to the user via Iridium communicationchannels. Moreover, Iridium’s communication capa-bility expands the potential to transmit data,which is limited both for GPS and for WAAS (50bits per second (bps) for the GPS navigation mes-sage, 250 bps for WAAS corrections). This featureis exploited in the third section of this paper byassuming precise Iridium clock and ephemerisdata (assuming more numerous, more frequentlyupdated orbital and clock parameters).

User Segment

The user segment is composed of all GPS/Irid-ium receivers. Receivers are equipped with low-cost quartz oscillator clocks that are unstable overlong durations (10�6 � 10�9 over a day [15]). Thedeviation from GPS time introduces a nuisance pa-rameter that can be solved for if four or more sat-ellites are available.

The iGPS concept described in this workassumes civilian users, who can collect single-fre-quency L-band code and carrier ranging observa-tions (centered at 1575 MHz for GPS (L1) and at1624 MHz for Iridium). Users also have access tonavigation messages for each constellation andmeasurement corrections. In the perspective ofGPS III, a modernization of GPS which plans toprovide L1 and L5 (1176 MHz) signals to civiliansby 2025 [8], dual-frequency GPS measurementsare considered in the sensitivity analysis for

Fig. 2–Joint GPS and Iridium Constellations, (a) Orbits, (b)from a User’s Perspective

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longer-term future implementations. In addition,Iridium satellites are equipped with Ka-band (19.5GHz) transmitters [19]. Dual-frequency Iridiumsignals are therefore simulated as well, althoughKa signals might be attenuated by heavy rain.

Finally as mentioned in the introduction, userequipment is assumed to include the necessarycomputational and memory resources to processcurrent and past-time observations collected withina fixed filtering period, TF, which is limited toensure the validity of the measurement errormodels.

NOMINAL MEASUREMENT ERROR MODELS

Wide area differential corrections are insufficientto make residual errors negligible with respect tocarrier phase tracking errors. Therefore, a conserv-ative approach is adopted for the derivation of newparametric error models under nominal fault-freeconditions. In the absence of actual Iridium meas-urements and of detailed information on the iGPSsystem (which is not publicly available), furtherassumptions are made on the accuracy of Iridiumclock and orbit ephemeris corrections. Theseassumptions can in turn be considered as interme-diary ground and space system requirements neces-sary to achieve the targeted end-user performance.

Residual Satellite Orbit Ephemeris andClock Errors

Individual GPS satellite clocks, in spite of theirhigh stability and corrections provided by the OCS,exhibit a slow but significant drift with respect totrue GPS system time. During the short filteringinterval, the remaining ranging error can be over-bounded for a satellite, s, by:

• an undetermined clock bias, sbC, at the timethe satellite first comes in sight, which is con-stant over TF,

• plus a ramp over time with an unknown butconstant gradient, sgC, accounting for linearvariations from the initial value over TF.

Based on several years of GPS data [21, 22], theinitial uncertainty on sgC is assumed normally dis-tributed with zero mean and standard deviation,rCG, of 4�10�4 m/s. The following notation is usedin the rest of the paper:

sgC � N 0; rCGð Þ:

The same gradient model is applied to Iridium sat-ellites. The satellite clock bias is modeled as sbC �N(0,rCB), where rCB�GPS is approximately 1.5 mfor GPS [15]. The corresponding value for Iridiumis addressed later when combined with the ephem-eris bias.

Fig. 3–Conceptual Overview of the Assumed iGPS Architecture

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Another primary source of error stems from theorbit ephemeris parameters computed by theground segment. In reference [23], several years ofbroadcast GPS ephemeris data were compared todecimeter-level precision post-processed satellite(truth) positions: from 1997 to 2003, daily root-mean-square ranging errors due to orbit parametererrors remain around 1.1 m. In addition, Reference[24] investigates the sensitivity over 24 hour peri-ods of computed satellite positions to individualephemeris parameter errors. It shows that themost sensitive parameter is the inclination angle,which causes satellite position deviations to varyperiodically with the orbital period, TGPS. This isfurther corroborated in the 24-hour broadcastephemeris error plots of References [22, 23].

For short filtering durations, TF (10 minutes orless) relative to TGPS, changes in orbit errors arelinear [10]. In this work, a worst slope approxima-tion is used to evaluate the ephemeris gradient,sgE, assuming periodic variations with frequency2p/TGPS and normally distributed amplitudemE�GPS (with zero mean and standard deviationrE�GPS of 1.1 m [23]):

sgE�GPS ¼ smE�GPS2p

TGPS;

so that sgE�GPS � N(0,rEG�GPS) with rEG�GPS ¼rE�GPS2p/TGPS. In addition, the GPS ephemerisbias is modeled as sbE�GPS � N(0,rEB�GPS) (withrEB�GPS ¼ rE�GPS). Since Iridium benefits fromhigher communication data rates, more numerousand more frequently updated orbital parameterscan be exploited. The proposed orbit error model forIridium is similar to GPS, with rEG�IRI ¼ rE�IRI2p/TIRI. A lower sigma value is assumed (rE�IRI ¼ 0.1m). The value chosen is consistent with that realis-tically achievable in near-real-time using GPSreceivers onboard the LEO spacecraft [25, 26]. Deci-meter-level performance is also assumed for resid-ual Iridium satellite clock errors and can be consid-ered as a correction accuracy requirement on theiGPS ground and space segments.

The combined ephemeris and clock bias is mod-eled as sbEC � N(0,rECB) [15, 22], and different val-ues are allocated to rECB for GPS and Iridium

models (respectively,ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2

CB�GPS þ r2EB�GPS

qand 0.1

m). Finally, the combined ephemeris and clock gra-dient is defined as:

sgEC ¼ sgC þ sgE;

so that sgEC � N(0,ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2

CG þ r2EG

q). Altogether at epoch

k of the filtering interval, for a satellite, s, that hasbeen visible over a period Dtk (from filter initiationat t0 to the sample time of interest tk), GPS and Irid-ium SV-related errors are expressed as:

seSAT;k ¼ sbEC þ Dtk � sgEC: (1)

Residual Ionospheric Error

The ionosphere is a layer of the atmosphereextending from an altitude of 50 km to 1000 kmabove Earth. It is composed of electrons and chargedatoms and molecules of gases that have been ionizedby solar ultraviolet radiation. The resulting non-uni-form density of electrons causes changes in the satel-lite signal propagation speed that vary with geomag-netic latitude, time of day, season, and level of activ-ity in the 11-year long solar cycle.

The ionosphere is the largest source of uncer-tainty in SV ranging observations. It generates adelay in code measurements and an advance ofequal magnitude in carrier-phase data, which areproportional to the total electron content in thepath of the signal, and to the inverse square of thecarrier’s frequency. As mentioned in the introduc-tion, this frequency-dependence is exploited in dual-frequency architectures to eliminate ionosphericdisturbances. Approximately 50% of the error forsingle-frequency users can be removed using Klobu-char’s empirical model, whose parameter values arebroadcast by GPS [27].

The residual ionospheric error model imple-mented in this work hinges on three majorassumptions. First, consider an Earth-centeredsun-fixed (ECSF) frame whose x-axis is pointingtoward the sun and whose z-axis is Earth’s axis ofrotation. Under anomaly-free conditions, the iono-sphere’s slow dynamics in the mid-latitude temper-ate zones justifies that it be assumed constant overthe short mission duration in an ECSF frame [10,28, 29]. In Figure 4, the varying thickness of theegg-shaped grey area surrounding Earth repre-sents the non-uniform electron density in the iono-sphere, and is fixed in ECSF.

Second, the peak electron density occurs between250 km and 400 km above Earth’s surface. A spheri-cal thin shell approximation is typically adopted tolocalize the effect of the ionosphere. An ionosphericpierce point (IPP) is defined as the intersection

Fig. 4–Three Assumptions for the Ionospheric Error Model

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between the satellite-user LOS and the thin shell atan altitude, hI, of 350 km. IPP displacement inECSF coordinates is due to the relative motions ofthe SV, of the user close to Earth’s surface, and ofthe Earth itself. In most precision applications(including aircraft final approach with a relativelyslow vehicle velocity of 70 m/s), GPS IPPs movemostly westward in ECSF (especially for high-eleva-tion satellites) because the rotation of the Earth isthe dominant factor (surface velocity larger than200 m/s at latitudes lower than 65 deg). IridiumIPPs in contrast move along a North-South axis dueto fast satellite motion. The effect of Earth’s rotationis highlighted in Figure 5: the IPP displacement rel-ative to some initial time is plotted over 10 minutes,for a user at a Miami location, in Earth-Centered-Earth-Fixed (ECEF) and ECSF reference frames.

Third, dual-frequency GPS data as well as exten-sive LAAS and WAAS-motivated research reportedin Appendix I suggest that the vertical ionosphericdelay varies linearly with IPP separation distances(actually ‘great circle distances’ or GCD) of up to2000 km (it levels off for larger distances) [30–33].The distribution of the corresponding slope can bebounded by a Gaussian model [30, 31].

The equivalent delay or advance is therefore mod-eled as an initial vertical ionospheric bias, bVI,associated with a ramp, whose slope over IPP dis-placement, dIPP (in an ECSF frame), is the verticalionospheric gradient, gVI. A single gradient per SVaccounts for ramps along one direction only, whichmeans that the model assumes pierce points traceapproximately straight paths along the great circle,with little or no lateral motion (as seen in Figure 5).An obliquity factor, cOI, adjusts this error for thefact that the LOS does not pierce the ionosphereperpendicularly, but with a slant angle function ofthe satellite elevation angle, sh. As a result,

seI;k ¼ scOI;k � sbVI þ dIPP;k � sgVI

� �(2)

with

scOI;k ¼ 1. ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1� RE cos shkð Þ= RE þ hIð Þ½ �2q

;

where RE is the radius of the Earth (6378 km).

Initial uncertainties on the bias and gradient aremodeled as:

sbVI � N 0; rVIBð Þ and sgVI � N 0; rVIGð Þ:

Values allocated to these parameters are justifiedin Appendix I using published data [9, 10, 15, 30–34], experimental results, and a covariance analy-sis of the WAAS algorithms [20]. Cases with andwithout corrections from the iGPS network ofground reference stations are considered. Hence,values of 1 mm/km and 4 mm/km for the correctedand uncorrected rVIG, respectively, and of 1–2 mand 5 m for the corrected and uncorrected rVIB areimplemented.

Finally, the maximum GCD traveled by an Irid-ium IPP when occasionally crossing the sky withnear-zero azimuth amounts to 3300 km. With abounding rVIG value, linearity for distances beyondthe suggested 2000 km limit makes the modeloverly conservative (in the sense that the rangingerror due to ionosphere is over-bounded), but hassmall consequences on the overall performanceresults. Therefore, Eq. (2) is used to model iono-spheric errors affecting both GPS and Iridium sig-nals. A second order spatial correlation model maybe considered for future work. As mentioned in Ap-pendix I, further analysis of the ionosphere usingexperimental data is in progress.

Residual Tropospheric Error

Signal refraction in the troposphere, the lowerpart of Earth’s atmosphere, delays the transmis-sion of SV measurements. The troposphere is madeof electrically neutral gases not uniform in compo-sition, including dry gases whose behavior islargely predictable, and water vapor, which is gen-erally treated as random but represents a muchsmaller fraction of the error. The majority of thedelay can therefore be removed by tropospheremodeling [35].

The residual uncertainty is modeled as a zenithtropospheric delay, bZT (i.e., associated with a hy-pothetical signal coming from 90 deg elevation),which is constant over the time interval TF. Inaddition, user motion causes variations relative tothis initial value, which are captured by a LAAS-like residual tropospheric error model [13]expressed as a function of the local air refractivityindex, Dn, so that the total zenith troposphericerror is:

eZT;k ¼ bZT þ 10�6h0 1� e�Dhk=h0

8: 9; � Dn:

Here, Dhk designates the difference in height thatthe user (e.g., aircraft) experiences from the startFig. 5–IPP Displacement

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of the filtering interval to epoch k. A fixed value of15 km is assigned to the tropospheric scale height,h0. Notations are simplified as follows:

eZT;k ¼ bZT þ cT;k � Dn;

where

cT;k ¼ 10�6h0 1� e�Dhk=h0

� �:

An obliquity factor, scOT,k, is applied [13] (scOT,k ¼(0.002 þ sin2(shk))�1/2) because a larger segment ofthe signal’s path travels through the troposphereat lower elevations:

seT;k ¼ scOT;k bZT þ cT;k � Dn� �

: (3)

The parameters bZT and Dn are not satellite-de-pendent because they characterize the environ-ment surrounding the airplane (Eq. (3) applies toGPS and Iridium signals). They are modeled asrandom constants over the time interval TF suchthat bZT � N(0,rZTB) [14] and Dn � N(0,rDn) [13],where the nominal standard deviations are givenin Table 1.

Receiver Noise and Multipath

Receiver noise and multipath errors depend onthe signal structure, signal to noise ratio, antennadesign, and receiver electronics. A signal can typi-cally be tracked to within approximately 1% of acycle, which explains the difference of two ordersof magnitude for the receiver measurement noiseof GPS code (300 m chip length) and carrier-phase(19 cm wavelength). The code and carrier-phase re-ceiver noise (seRN-q,k and seRN-/,k) for both GPS andIridium are modeled as normally distributed whitesequences with standard deviations, rRN-q and rRN-

/, respectively [36].In addition, multipath error caused by unwanted

signal reflections reaching the user receiverdepends on the satellite geometry, on the environ-ment surrounding the antenna, and on theantenna technology. Signals reaching the antennaat elevations lower than five degrees are not con-sidered (which eliminates low-elevation reflectionsand accounts for minor changes in user dynamics).To account for the time-correlation it introduces,multipath is modeled as a first-order Gauss-Mar-kov Process (GMP) with time constant, TM, stand-ard deviation, rM-//q [36], and driving noise, mM,k:

seM;kþ1 ¼ e�TK=TM � s eM;k þ mM;k

with

mM�/=q;k � N 0; rM�/=q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� e�2TK=TM

p� �:

Large azimuth-elevation variations generate fastchanges in the directions of signal reflections forIridium. The multipath time-constant for Iridium,TM,IRI, was therefore computed by multiplying theangular rate ratio between GPS and Iridium satel-lites (approximately 0.017) with the time constantfor GPS, TM,GPS (conservatively estimated at 60 sin a dynamic environment [37]). Current receiversare capable of tracking signals with high updaterates (higher than 1 Hz). Within the filtering inter-val, TF, there is limited benefit in using all theavailable measurements both because of their cor-relation in time, and because the contribution ofgeometry change to the estimation process far out-weighs that of redundant measurement averaging.Therefore past measurements within TF are con-sidered at regular intervals, TK, of 30 s, whichgreatly decreases the computational burden.Finally, the combined receiver noise and multipath(RNM) is expressed as:

seRNM;k ¼ seRN;k þ seM;k (4)

In summary, the complete linearized GPS and Irid-ium carrier-phase measurement equation for a

Table 1—Summary of Error Parameter Values

Parameter Description (s: standard deviations,a.f.i.: at filter initialization) Nominal

sCG : residual satellite clock gradient a.f.i. 4�10�4 m/ssCB�GPS : residual GPS satellite clock bias

a.f.i.1.5 m

sE�GPS : amplitude of GPS ephem. errorvariations a.f.i.

1.1 m

sE�IRI : amplitude of Iridium ephem.error variations a.f.i.

0.1 m

sECB�IRI : residual Iridium SV clock andephemeris bias a.f.i.

0.1 m

sZTB : residual zenith tropospheric biasa.f.i.

0.12 m

sDn : residual refractivity index a.f.i.(unit-less)

30

sVIB : vertical ionospheric bias a.f.i.(corrected)a

1 m

sVIG : vertical ionospheric gradient a.f.i.(uncorrected)a

4 mm/km

sRN�r : code-phase receiver noiseb 0.3 msRN�f : carrier-phase receiver noiseb 0.003 msM�r : code-phase multipath noiseb 1 msM�f : carrier-phase multipath noiseb 0.01 mTM,GPS : GPS multipath time constant 1 minTM,IRI : Iridium multipath time constant 1 sTK : sampling interval 30 s

a for dual-frequency at f1 and f2 (GPS: L1/L5, Iridium:L/Ka): the terms in bVI and gVI are eliminated

b for dual-frequency (f1, f2), these terms are multipliedby ([f 2

1/(f 21 � f 2

2)]2 þ [f 22/(f 2

1 � f 22)]2)1/2

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satellite, s, at epoch, k, can be written as (usingEq. (1–4)):

s/k ¼ sgTk uk þ sN þ sbEC þ Dtk � sgEC

þ scOT;k bZT þ cT;k � Dn� �

� scOI;ksbVI þ sdIPP;k � sgVI

� �þ seRNM�/;k ð5Þ

where uk ¼ [xE xN xU s]Tk is the user position (in a

local East-North-Up or ENU coordinate frame) andreceiver clock deviation, and N is the cycle ambigu-ity bias. The vector, sgk, is a function of the unitLOS vector between user and satellite (noted sek)such that sgT

k ¼ [�seTk 1]. The equation for the

code-phase measurement, sqk, is identical exceptfor the absence of sN, a positive sign on the iono-spheric error, and the receiver noise, seRNM-q,k,which replaces seRNM-/,k.

The assertion that error models are conservative isonly true if the Gaussian models over-bound the cu-mulative distribution functions of each error sources’ranging errors [38]. The next phase of this researchwill involve establishing probability distributions forthe error parameters, and verifying the fidelity of thedynamic models to experimental data. Alternatively,error parameter values may be considered asrequirements that ground corrections should meet inorder to achieve the desired system performance.

Nominal values for all error parameters arelisted in Table 1. They constitute the prior knowl-edge acquired from experimental observations ofphysical phenomena, and are a crucial input to theestimation algorithm.

ESTIMATION AND DETECTION ALGORITHMDERIVATION

Position and Cycle Ambiguity Estimation

The satellite clock and ephemeris as well as theionosphere and the troposphere error modelsderived previously all assume that measurementsare collected over a short, limited duration. In ordernot to exceed this period of validity, optimal positionand cycle ambiguity estimation is performed usinga fixed-interval filtering algorithm. Continuousreal-time operation requires that measurementsand LOS coefficients be stored over the interval TF:at each new epoch, tk, incoming data is updated andthe oldest data at tk�1 � TF can be erased frommemory. Current-time optimal state estimates areobtained from iteratively feeding the stored finitesequence of observations (from tk � TF to tk) into aKalman filter (KF). The KF also provides an indica-tion of the estimation uncertainty in the form of astate covariance matrix, which serves as the basisfor the upcoming analysis. State augmentation isused to integrate the dynamics of the multipathGMP [39]. Practical implementation of the KF also

necessitates that rows and columns for all vectorsand matrices (including covariance matrices) beadded and removed as satellites come in and out ofsight (which is frequent for Iridium).

In addition, measurement faults, whether theyaffect recent or older observations within the inter-val, are just as likely to generate hazardous current-time positioning information. In anticipation of theRAIM-type residual-based fault detection intro-duced in the next subsection, a fixed-intervalsmoothing (instead of filtering) process is used; itcan be efficiently realized using, for example, a for-ward-backward iterative smoother, or a Rauch-Tung-Striebel algorithm [40]. Although smoothing iscomputationally more intensive than filtering, thecomputation time is negligible with respect to thespecified one-second time-to-alarm limit [13].Finally, in applications where timing and computa-tional load are not of primary concern, measure-ments can be processed as a batch, which is themethod presented below for clarity in exposition.Batch processing produces results identical to theKF for the current time as well as optimal estimatesfor past epochs that are used for residual generation.

Consider first the vector of carrier-phase obser-vations for a satellite, s, in view between epochs kO

and kF:

su ¼ s/kO� � � s /kF

� �T:

These epochs are generally the first and last of thesmoothing interval for GPS signals, but not forIridium satellites, whose passes are often shorterthan TF � Let 0n 3 m be an n 3 m matrix of zeros.State coefficients defined in Eq. (5) are arranged inmatrices that are needed in later steps, so that forspacecraft s:

sG ¼

sgTkO

0134

. ..

0134sgT

kF

2664

3775;

sDt ¼ 0 Dtk1� � � DtkF

� �T;

scOT ¼ scOT;kO� � � s cOT;kF

� �T;

scT ¼ 0 scOT;k1cT;k1

� � � s cOT;kF� cT;kF

� �T;

scOI ¼ scOI;kO� � � s cOI;kF

� �Tand

scI ¼ 0 scOI;k1� s dIPP;k1

� � � s cOI;kF� s dIPP;kF

� �T:

Carrier-phase observations for all nS Iridium andGPS satellites are then stacked together:

u ¼ 1uT � � � nS uT� �T

;

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and

u ¼ Huxþ mu; (6)

where mu designates the carrier-phase measure-ment noise vector and Hu will be defined below.The state vector is

x ¼ uTkO� � � uT

kFNT bT

EC gTEC bZT Dn bT

VI gTVI

h iT;

where kO and kF (subscripts of u) are now the firstand last epochs of the smoothing interval. Boldface characters for parameters other than uk desig-nate vectors of states for all satellites, for example:

N ¼ 1N � � � nS N� �T

;

The dynamics of the user position and clock devia-tion vector, uk, are unknown; uk is therefore allo-cated different states at each time step, as opposedto the other parameters that are modeled as con-stants over interval TF.

The observation matrix, Hu, is constructed byblocks:

Hu ¼ G BN BECB BECG BZTB BDn BVIB BVIG½ �;

each block corresponds to a state parameter, andcontains coefficients for all spacecraft, for theentire sequence of measurements. Let nK(s) be thenumber of samples for satellite s (which generallydiffers for Iridium SVs), and 1n be an n 3 1 col-umn-vector of 10s:

G ¼ 1GT � � � nS GTh iT

;

BN ¼ BECB ¼

1nK 1ð Þ 0

. ..

0 1nK nSð Þ

2664

3775;

BECG ¼

1Dt 0

. ..

0 nSDt

264

375;

BZTB ¼

1cOT

..

.

nScOT

2664

3775; BDn ¼

1cT

..

.

nScT

2664

3775

BVIB ¼ �

1cOI 0

. ..

0 nScOI

2664

3775;

and BVIG ¼ �

1cI 0

. ..

0 nScI

2664

3775:

A measurement equation similar to (6) is estab-lished for the code-phase observation vector, q. Inthis case, the sign on the ionospheric coefficients,BVIB and BVIG, is positive. Also, the columns of onesin BN corresponding to the cycle ambiguity vector,N, are replaced by zeros; this explains why statevectors, N and bEC, have to be distinguished, eventhough columns of BN and BECB are linearly de-pendent for carrier-phase measurements. Since noprior knowledge is assumed for N, the system’s per-formance sensitivity to bEC (investigated later)reflects the influence of code-phase measurements.In addition, since the integer nature of the constantcycle ambiguities, N, is not exploited, potential nui-sance parameters on carrier-phase data that areconstant over time (such as inter-frequency biasesfor dual-frequency signals) can be lumped togetherwith N without affecting the results.

The complete sequence of code and carrier-phasesignals for all satellites over the smoothing inter-val are included into a batch measurement vector:

zB ¼ uT qT� �T

;

and

zB ¼ HBxþ vB: (7)

The measurement noise vector, vB, is utilized tointroduce the time-correlated noise due to multi-path modeled as a GMP. Its covariance, VB, isblock diagonal, each block corresponding to obser-vations from the same SV over time. Within eachblock, the time-correlation between two measure-ments originating from the same satellite s at sam-ple times ti and tj is modeled as r2

M-q// � e�Dtij/TM,where Dtij ¼ |ti � tj|. The quantities r2

RN�q andr2

RN-/ are also added to the diagonal elements toaccount for code-phase and carrier-phase uncorre-lated receiver noise, respectively.

Finally as mentioned in the prior section, valua-ble information is gained from the study of thephysical phenomena causing measurement distur-bances. This prior knowledge of the error parame-ters is expressed in terms of bounding values ontheir probability distributions. It can be includedas a vector of pseudo-measurements, zP, that pro-vide direct a priori observations to the correspond-ing states. Let nS, nE, and nK be respectively thenumbers of available satellites, of error states andof samples over TF; in addition, let In be an n 3 nidentity matrix. HP is defined as [0nE3(4nKþnS) InE

],so that:

zP ¼ HPxþ vP: (8)

The covariance matrix, VP, of the pseudo-measure-ment noise vector, vP, is diagonal, with values of

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the initial conditions on the error states, bEC, gEC,bZT, Dn, bVI, and gVI. In other words, the diagonalvector of VP is:

h1T

nS� r2

ECB 1TnS� r2

ECG r2ZTB r2

Dn 1TnS� r2

VIB 1TnS� r2

VIG

i:

The elements of zP are the mean values of the errorstates (zeros in this case). The vector zP can be addedto the system by direct augmentation of zB, in whichcase z ¼ [zT

B zTP]T, H ¼ [HT

B HTP]T, and the covariance

matrix, V, of the measurement noise, v, is block diag-onal with VB and VP on its diagonal. It can also beincorporated using the equivalent reduced-orderform proposed in Appendix II. An alternative deriva-tion based on the method of Lagrange multipliersand without introducing the concept of pseudo-measurements is available in Reference [40]. Thetotal measurement equation becomes

z ¼ Hxþ v: (9)

The weighted least squares state estimate is definedas x ¼ S z, where S is the weighted pseudo-inverseof H:

S ¼ HTV�1H� ��1

HTV�1; (10)

and the state covariance matrix is:

Px ¼ HTV�1H� ��1

:

The diagonal element of Px corresponding to the cur-rent-time vertical position covariance (noted r2

U) isused in the next sections to determine the position-ing performance under fault-free conditions. Thefocus is on the Up-coordinate, both because of thetighter requirements in this direction (see the nextsection) and because of the generally poorer satellitegeometry due to the absence of open LOS below thehorizon.

RAIM-Type Single-Satellite Fault (SSF) DetectionAlgorithm

State estimation is based on a history of observa-tions, all of which are vulnerable to equipmentfaults (satellite clock excessive acceleration, cor-rupted ephemeris parameter, user receiver cycleslip, etc. [13]) or unusual atmospheric phenomena.To protect the system against abnormal eventspotentially affecting successive observations, aRAIM-type detection algorithm is derived based onthe batch least-squares residual.

The residual-based RAIM methodology is articu-lated around two dimensions. First, let sT

U;kFbe the

row of S corresponding to the vertical position at

the last epoch of the filtering interval kF. The cor-responding positioning error due to a measurementfault vector, f (of the same dimension as z), whosenon-zero elements introduce deviations from nor-mal FF conditions, is such that:

dxU � N sTU;kF

f ; rU

� �:

The system is said to produce hazardous informa-tion if a failure causes a vertical position errorthat exceeds a vertical alert limit, VAL:

dxUj j > VAL;

Second, a failure may be detected using a residualvector, r, defined as the difference between a possi-bly faulty measurement z and the best currentestimate of this measurement z, so that:

r ¼ z�z:

The residual vector corresponding to a failure, f,can be written as [12] (see Appendix II for areduced-order form that includes prior knowledgeon the error states):

r ¼ I�HSð Þf :

The norm of r weighted by the measurement noisematrix, V, is noted krkW. krkW is chi-square distrib-uted with nZ � (4nK þ nS) degrees of freedom (nZ

is the dimension of zB) and non-centrality parame-

ter,ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifTV�1ðI�HSÞf

q[41]. A detection threshold,

RC, is set in compliance with a continuity require-ment (2 � 10�6) to limit the probability of falsealarms [42]. As a result, a measurement failure isundetected if:

rk kW< RC:

The influence of an SSF on both of these dimen-sions can be represented on a plot of dxU versuskrkW (Figure 6). The upper left quadrant delimitedby VAL and RC corresponds to the missed-detection(MD) area, where failures are both hazardous andundetected. The probability of missed detection,PMD, is defined as a joint probability:

PMD ¼ P dxUj j > VAL; rk kW< RC

� �:

Therefore, within the MD area, PMD is the productof the cumulative probability distribution functionsof |dxU| and krkW. The normal and chi-square dis-tributions of dxU and krkW, respectively, explainthe ovoid shape of the isoprobability contours par-tially depicted in Figure 6 for an example failuremode f.

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The failure mode slope, FMS, defined as the ra-tio of |dxU| over krkW, is a useful concept that isindependent of the fault’s magnitude; the resultingfailure mode curve for all magnitudes is a line ofslope FMS passing through the origin. On the fail-ure mode plot of Figure 6, as the magnitude isincreased from zero (fault-free case) to some largervalue, the point moves along the line from the ori-gin towards the right of the plot. The detectionperformance of the integrity threat search algo-rithm can therefore be evaluated by finding thesteepest FMS for a set of failures, and then vary-ing magnitudes along the corresponding line insearch of the highest missed-detection probability,PMD (corresponding to the worst-case fault).

A set of canonical failure modes is injected intothe system for evaluation. At this stage of theresearch, simulated faults are limited to satellitefailures because they are the only types of faultsfor which the failure rate is reliably known. Simu-lated faults of arbitrary magnitude, of numerousstarting and stopping times, are easily constructedusing blocks of truncated triangular matrices. Forexample, for a satellite s available over threeepochs, the set of failures is:

sf ¼0 0 3 2 00 1 2 1 11 1 1 0 2

24

35:

The first two columns represent step faults, thenext two are descending ramps, and the last one isan ascending ramp. For efficiency, all redundantfailure modes were eliminated (e.g., ascending anddescending ramps over the entire duration, TF,have identical effects). Similar matrices are gener-ated for each satellite over the entire smoothingduration and applied to code and carrier-phasemeasurements individually as well as simultane-

ously, one SV at a time. The number of failureshence simulated exceeds 1,500 for a ten-minutesmoothing interval. Future work includes testingfor failures with multiple breakpoints and affectingmultiple satellites at a time.

FRAMEWORK FOR THE PERFORMANCEANALYSIS

Example Mission Definition and Requirements

A preliminary performance analysis is structuredaround a benchmark application of an aircraft pre-cision approach. As illustrated in Figure 7, an air-plane equipped with a GPS/Iridium receiver is fol-lowing a simplified straight-in trajectory: it is flyingat a constant speed of 70 m/s with a 3 deg glide-slope angle towards the runway until touchdown(TD) where lateral and vertical requirements apply(symbolized by a rectangle). As mentioned earlier,the focus in this work is on the vertical positioncoordinate. A VAL requirement of 10 m is assumedfor the purpose of this analysis to apply from a 200ft altitude to TD. The filtering interval (whichequals 5 min in the example of Figure 7) is simu-lated for position estimation at TD. Protection lev-els (PL) are defined below.

In this work, system performance is measured interms of availability of a high-integrity verticalposition solution. The integrity risk requirement,or probability of hazardous misleading information(HMI), noted PHMI, is defined as the limit probabil-ity of any information sent by iGPS resulting inout-of-specification position error without timelywarning [13]. It is subdivided here into twohypotheses, so that the integrity budget allocatedto normal fault-free conditions (FF) is aPHMI, andthe fraction for rare-event single-satellite failures(SSF) is (1 � a)PHMI. The coefficient a rangesbetween 0 and 1 and is selected to maximize the

Fig. 6–Failure Mode Plot

Fig. 7–Final Approach Simulation Description (Case s in Fig. 9)

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combined FF and SSF performance. Cases of mul-tiple simultaneous measurement faults (consideredindependent events, hence having an even lowerprobability of occurrence) are neglected in thisphase of the analysis, but will be addressed in thefuture.

Under normal conditions, the vertical protectionlevel, VPL, a statistical over-bound on the position-ing error in the Up-direction, is defined as a func-tion of the standard deviation of the vertical posi-tion coordinate, ru:

VPL ¼ jFF � rU ;

where the probability multiplier, jFF, correspond-ing to aPHMI is a confidence-level coefficient (it isin fact the value for which the cumulative distribu-tion function of a normal distribution equals 1 �aPHMI/2). Nominal simulation parameters are sum-marized in Table 2. An approach or a geometry isdeemed available under FF conditions if and onlyif

VPL < VAL: (11)

Rare-event faults such as equipment and satellitefailures (whose rate FR is assumed to be 10�4/hr[43]) become significant threats when aiming atensuring an integrity risk, (1 � a)PHMI, on theorder of 10�9 [13]. The SSF-availability criterion,established here using a RAIM methodology, speci-fies that:

PMD < 1� að ÞPHMI= FR � TFð Þ: (12)

When testing the detection algorithm, if the proba-bility of missed-detection, PMD, for any one of thesimulated faults violates Eq. (12), the approach isconsidered SSF-unavailable. The method presented

in the previous section aims at directly finding thefault causing the highest PMD.

Equations (11) and (12) are the expressions ofFF and SSF binary criteria that either validate ornullify availability for an approach. In the follow-ing sections, approaches starting at regular inter-vals are simulated for sequences of satellite-usergeometries, over a period TAV defined below. Ulti-mately, the percentage of available approaches isthe measure of iGPS FF and SSF performance. Itis evaluated for a nominal iGPS configuration, con-servatively selected to produce reliable availabilityresults, and to exhibit performance variations thatcan be analyzed in the next section. This nominalnavigation architecture is also used in the finalsection as a reference for comparisons when evalu-ating the sensitivity of individual system parame-ters.

Period for the Availability Analysis

When combining information from GPS and Irid-ium constellations, the duration over which avail-ability simulations are carried out (TAV) shouldenable sampling of a complete set of satellite geo-metries. For GPS, TAV is one sidereal day (theEarth’s rotation period in an ECI frame, whichequals 2TGPS). It corresponds to the time-periodthe constellation needs to completely repeat itselfwith respect to the Earth.

The orbital period for Iridium, TIRI, is 6028 s.The combined GPS/Iridium constellation repeat-ability period with respect to the Earth can be eas-ily calculated if secular variations due to Earth’soblateness are neglected. Secular effects prove tobe very small both for GPS and Iridium duerespectively to the high-altitude and the high-incli-nation of their orbits. Hence, it takes 1,507 side-real days (more than 4 years) for the geometrybetween the earth, GPS, and Iridium satellites tocompletely repeat itself. Simulating the algorithmsover 1,507 days is computationally too intensive.

Fortunately, an approximated duration represen-tative of a large number of geometries can be uti-lized. In fact, Iridium satellites circle the earthexactly 43 times in 3 solar days and 4 seconds.Also, the remainder of the closest integer numberof intervals, TIRI, within n � 2TGPS, where n is a se-ries of consecutive integers, exhibits a 3 to 4 daycyclic trend. Finally, Figure 8 demonstrates thatthe computed cumulative FF-availability averagesout after a few days, for the nominal system config-uration. Indeed, the maximum deviation relativeto the accumulated value after 3 days does notexceed 0.03% at the Miami location over onemonth of simulation. Concurrently, it is importantthat the interval between simulated approaches beselected short enough as illustrated with the 30 s

Table 2—Summary of Simulation Parameters

Parameter Description Nominal

PHMI : integrity risk 10�9

a : FF integrity risk coefficient 10�5

Continuity risk 2�10�6

FR : failure rate 10�4/hrVAL : vertical alert limit 10 mTAV : availability simulation period 3 daysTF : filtering period 10 mina

Location : near-worst case (25.5degNorth, �81.1deg East) Miami

GPS constellation 24 SVsIridium constellation 66 SVsSignals : single-frequency (SF) or

dual-frequency (DF) SF

a 5min is used in the FF-availability analysis (first sub-section of Sec. VI)

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and 2 min interval curves at the Miami location.In view of these results, approaches are simulatedevery 30 s over a period, TAV, of 3 days.

FF AND SSF ALGORITHM ANALYSIS

FF Availability Analysis

Under FF conditions, the recursive KF or forwardsmoother that results in an optimal position esti-mate at TD is illustrated in Figure 7 for the aircraftprecision approach mission. The covariance ellipsesrepresent the protection levels (PLs) in the Northand Up coordinates at different time steps. Positionestimate standard deviations at each epoch aremultiplied by jFF (to get PLs), and inflated by a con-stant scaling factor for the clarity of the plot. Theirshape and size change with geometry (mainlybecause of Iridium satellite motion). A gray area forthe corresponding vertical position covariance enve-lope along the aircraft trajectory is projected in thebackground. This example illustrates the incremen-tal improvement in positioning accuracy within thefiltering interval, TF, and the dramatic enhance-ment of GPS/Iridium over non-augmented GPS, forwhich the position error barely changes. In thiscase, the FF integrity requirements are met, mean-ing that the ellipse at TD is fully contained insidethe specified rectangle of alert limits.

The estimation algorithm performance is furtheranalyzed to understand the impact of satellite geo-metries on carrier phase positioning. Cycle ambi-guity estimation for mobile users requires SV re-dundancy as well as variations in user-to-satellitelines of sight, both of which are provided by aug-menting GPS with Iridium [44]. In general, posi-tion and cycle ambiguity estimates improve as thechange in LOS angle increases [5, 17]. The exam-ple geometry depicted in Figure 2(b) underscoresthe sharp contrast between an Iridium satellitewhose angular variation over 10 minutes exceeds130 deg, and GPS satellites whose LOS rotationbarely reaches 5 deg.

A similar argument on observability helpsexplain the influence of range variations on indi-

vidual position coordinates. Approaches are simu-lated at regular 30-second intervals over the periodTAV for a nominal configuration. Nominal parame-ters summarized in Tables 1 and 2 describe systemconditions for a single-frequency aircraft userattempting a precision approach in Miami (near-worst location for the conterminous United States,or CONUS) and receiving WAAS-like ionosphericcorrections. In this FF analysis, in order to investi-gate cases of FF unavailable geometries, the filter-ing period, TF, is set to 5 min.

In Figure 9(a), a 15-hour period is extracted outof the total 72-hour simulation period, TAV, in orderto better visualize the variations in vertical protec-tion level (VPL). Solid gray vertical lines indicatecases where the VAL is exceeded. The averagenumber of Iridium satellites in view over TF (i.e.,the sum of visible Iridium SVs at each epoch di-vided by the number of epochs) for each simulatedapproach is presented beneath, over the same 15-hour period. In the following discussion, parallelsare established between these two plots, whichshed light on underlying estimation mechanisms.These observations are consistently repeated dur-ing TAV, and have been established for a widerange of system parameters. The first finding isthat GPS satellite measurements have a lesserinfluence on positioning. Neither the number ofGPS satellites nor the vertical dilution of precision(not shown for conciseness) influence VPL resultsas significantly as Iridium spacecraft, which havea major impact.

In this regard, three regions are identified inFigure 1 to distinguish different areas of Iridiumsatellite visibility at Miami’s latitude. Region A islocated at the seam of the constellation, where theorbital plane separation-angle is smaller, so thatthe number of satellites in view at any instant isexpected to be higher than elsewhere. Regions ofType B are located in-between co-rotating orbitalplanes and benefit from the coverage of satellitesfrom both planes. Finally, regions of Type C desig-nate areas around orbital planes, covered by onlyone or two satellites from a single plane at a time.Moving along a parallel at Miami’s latitude,regions of Types B and C succeed each other, pre-senting respectively 1–3 and 1–2 visible satellitesat a time.

Over time, the Earth rotates about its axis whileSVs move in their orbital planes, which remainquasi-stationary in an ECI-frame. So, regions ofTypes B and C in Figure 1 correspond to intervalsof time in Figure 9(a), which presents an alternat-ing of Type B periods over which about 2.2 satel-lites are visible (on average over TF) and Type Cperiods of about 1.3 satellites. The number of satel-lites increases every half sidereal day during TypeA intervals. This increase is more pronounced on

Fig. 8–Determination of the Period of the Availability Analysis(TAV)

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one end of the seam than on the other (around the15-hour point) because of the less-than-90 deg or-bital plane inclination. Finally, users at Miami’slatitude all experience patterns similar to Figure9(a), which are shifted in time depending on theirlongitude.

The first obvious parallel between the two curvesin Figure 9(a) is that during intervals of Type A,numerous Iridium measurements logically producelower VPLs. A close look at the high-frequency var-

iations of both curves shows that peaks in VPLs,especially within phases of Types A and B, corre-spond to valleys in average number of visible Irid-ium satellites over TF (this again, regardless ofGPS geometry). Another strong observation is thatall unavailability cases occur around the beginningor the end of Type B intervals. This is verified overthe 3-day TAV period for all 50 unavailableapproaches (out of 8620 total simulated cases).Now what seems paradoxical is that sharp drops

Fig. 9–FF Availability Analysis: a) VPL at TD and Average Number of Visible Iridium Satel-lites vs. Time; b) Azimuth-Elevation Sky Plots and Range Variations in the Position-Domain

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in VPLs are achieved during intervals of Type C,where the average number of Iridium satellites isthe lowest.

To further examine this point, three characteris-tic cases are investigated: a ‘good’ case (noted g), a‘bad’ case (b), and a ‘standard’ case (s). For eachcase, Figure 9(b) introduces azimuth-elevation skyplots of the SV trajectories over the filtering pe-riod, TF, and Iridium satellite LOS variations alongeach of the three local position coordinates. Case(g), the best of the three cases, occurs in themiddle of a Type C interval: one LEO satellite istraveling directly overhead the user, so that thevariation in LOS coefficients corresponding to thevertical coordinate is the largest. Case (b), theworst of the three, is at the beginning of a Type Bphase, and a single Iridium SV is visible over thelargest part of the interval. Two mechanisms havebeen identified that explain the poor performance.First, the amount of angular variation with respectto the vertical direction is decreased relative to theprevious case. Second, an observability issueappears that prevents estimation: unlike the twoother cases, the LOS variation profiles in the Eastand Up coordinates have become difficult to distin-guish so that the corresponding states remainundetermined, hence causing VPL to exceed theVAL. Finally in the standard case (s), in the middleof a Type B interval, additional satellites from theadjacent orbital plane come into sight, slightlyaugmenting the cumulated amount of angular var-iations and above all, resolving the East-Up ambi-guity.

In summary, excellent performance is obtainedwhen a satellite crosses the sky directly overheadthe user, where angular variations with respect tothe vertical axis are the most substantial. As timepasses, the user’s location drifts away from the or-bital plane due to Earth rotation, causing verticalLOS variations to decrease with Iridium satelliteelevation. On top of that, for a few spacecraft tra-jectories, occasional observability issues arise fromthe difficulty to distinguish variations in the Eastand Up directions. Additional low-elevation satel-lites come into sight as the user location ap-proaches the adjacent plane, which solves theobservability problem, and provides sufficient accu-mulated geometric diversity to meet the VALrequirement.

Practical lessons learned from this exercise are,for example, that if Iridium satellites were addedfor navigation purposes, the constellation shouldbe rearranged to include extra orbital planes withtighter separation angles, rather than launchingmore SVs in the existing planes. Results for an 88-Iridium-satellite constellation (with eight orbitalplanes instead of six) are presented in the next sec-tion. Another costless and straightforward way to

improve FF performance is to increase the filteringperiod, TF, in order to include extra range varia-tions. An upper limit of 10 minutes is imposed inorder to ensure the validity of the error models.Using an interval, TF, of 10 minutes, 100% avail-ability is achieved over TAV at the Miami location,under FF conditions.

Undetected Failures Analysis

The RAIM-type single satellite fault detectionprocess is tested against measurement steps andsingle-breakpoint ramps of all magnitudes and ofmultiple starting times, for simulated approachesrepeated every 30 seconds over the 3 day TAV-period. A small value is assigned to coefficient a inorder to maximize the fraction of the integrity riskallocated to SSF (see Table 2). The 10 minutesmoothing period, TF, enables 100% FF availabilitybut does not prevent SSF unavailable approaches,using the nominal parameter values listed inTables 1 and 2.

Fault-modes causing unavailability can be iden-tified. First, results show that step-type faults aresystematically detected. They generate huge resid-uals, even for low failure magnitudes, which trans-late into a gentle slope on a failure mode plot (i.e.,Figure 6), and reaffirm the ability of the systemto estimate constant biases. Then, after inspection,all 96 unavailable approaches (out of 8620 simu-lated cases) are due to ascending or descendingramp-type faults on Iridium carrier-phase meas-urements. Carrier-phase observations are the mostsensitive because they carry by far the mostweight in the estimation algorithm as opposed tocode-phase measurements, which if corrupted butundetected, are typically not hazardous to the finalposition estimate. For the same reason, faults onIridium signals have a more dramatic impact onthe vertical position estimate error than faults onGPS measurements.

Further examination shows that the overwhelm-ing majority of undetected faults affect satellitesvisible for short periods (for only part of the filter-ing interval), and the corresponding ramps stretchover the entire time the spacecraft is in view. Ingeneral, ramps whose starting or ending pointsoccur sometime within the satellite pass are easilydetected. Indeed, it was verified that elements ofthe residual corresponding to measurements col-lected just before and just after the breakpoint ex-hibit sharp variations, hence inflating krkW. Rareexceptions, where faults that include a breakpointgo undetected, are addressed below.

Uninterrupted ramps over the entire pass consti-tute the main cause of unavailability. Analysis ofthe residuals suggests that these ramps might beindistinguishable from fault-free behaviors. Faults

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causing missed-detection were therefore comparedto the signature profiles of individual terms in themeasurement equation, and turn out to match thesatellite range variations. Another clue that con-firms this idea is simply that the longest unde-tected failures do not stretch over more than 6minutes, which is about half the maximum dura-tion of an Iridium satellite pass (e.g., from rise tomaximum elevation), and beyond which its rangevariation is no longer ramp-like (see the range var-iation curve in Figure 2). Even more revealing arethe aforementioned exceptions, where the unde-tected fault includes a breakpoint. In all of theserare cases, the failure-mode is a ramp stretchingover most of the pass, and a constant over one ortwo epochs. Now all undetected cases apply to sat-ellites whose passes are approximately truncatedin half by the filtering interval and whose peaksatellite elevation (where range variations leveloff) occurs either at the beginning or at the end ofthe filtering period, so that the fault’s breakpointmatches to the change of slope in range variation.Here again, the fault is masked by an FF behavior.Interestingly, the same ramp-type geometric sensi-tivity in range that is providing observability forcycle ambiguity bias estimation is also sometimesmaking ramp-type failures undetectable.

In some applications, operational system require-ments include a limit on the period of navigationservice outage, i.e., the maximum period of timeover which high-integrity positioning is unavail-able. As illustrated in the previous sub-section,cases of poor geometries are isolated, and do notlast long. Accordingly, SSF availability resultsshow that outages last on average 2.4 min, with amaximum of 4 min, meaning that users wouldnever have to wait more than 4 min to recover therequired performance.

Besides, in addition to verifying SSF availability(by finding the fault magnitude that generates thehighest PMD), the boundaries of the integritythreat space can be identified by determining theminimum and maximum fault magnitudes forwhich PMD violates Eq. (12). Simulations indicatethat the slopes of undetected hazardous ramp-likefailures range from 7.7 to 33 mm/s over 2–6 minperiods. Whether physical phenomena actuallyexist that cause such faults and how likely theyare to actually occur is as yet unknown, but wouldbe a useful subject for future work.

The analysis results described above provide apreliminary evaluation of the system’s detectionperformance for a variety of faults that are encoun-tered in actual operating environments. For exam-ple, steps model receiver cycle slips and rampsaccount for certain types of excessive satellite clockdeviations and erroneous ephemeris parameters.However, it is understood that ultimate validation

of fault detection performance must be verifiedagainst a comprehensive set of all real fault modes,as well as unusual ionospheric disturbances.

Now that the root causes for loss of availabilityare understood, the performance sensitivity to indi-vidual system parameters can be investigated.

COMBINED FF-SSF AVAILABILITY SENSITIVITYANALYSIS

Parameter values for the nominal configuration(Tables 1 and 2) are used here as a reference forcomparisons, and were conservatively selected todescribe a system architecture implementable inthe short term. Results are presented in terms of‘combined availability,’ which is only granted foran approach if both the FF and SSF criteria aresatisfied. As mentioned earlier, the SSF criterion isthe driving factor for loss of availability. The nomi-nal combined availability performance is 98.89% atthe Miami location, which is a near-worst-caselocation for CONUS.

Parameters of the Measurement Error SourceModels

The performance sensitivity to individual errormodel parameters is investigated for realisticranges of values in Figure 10. More precisely, thecombined FF-SSF availability at the Miami loca-tion, for a fixed smoothing interval, TF, of 10minutes, is plotted for each parameter’s nominalstandard deviation, rNOM (listed in Table 1; e.g.,for the vertical ionospheric bias, bVI, rNOM is rVIB),inflated by a scaling factor given on the x-axis(ranging between 0.2 and 1.8). As expected for allparameters, values lower than rNOM produce bet-ter results than the nominal case, and converselyavailability decreases for higher values.

Three parameters stand out as being the mostinfluential. First, the GPS ephemeris and clock

Fig. 10–Sensitivity to Parameters of the Measurement ErrorSource Models at Worst-Case Miami Location

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bias parameter, rECB�GPS, generates the largestperformance variations. As noticed earlier, theimpact of rECB�GPS reveals the importance of GPScode-phase measurements to the overall systemperformance. The nominal 1.86 m parameter valuefor rECB�GPS, representative of GPS ephemeris andclock errors after OCS correction, was selectedbased on multiple years of data [22, 23]. Furtherdata analysis is required before taking credit forWAAS-like ground corrections. Segments of thecurve label, bEC�GPS, to the left-hand side of thenominal point show the improvement such correc-tions could bring about. The corresponding param-eter for Iridium, rECB�IRI, only produces minutevariations, and was not included for clarity of theplot.

Receiver noise and multipath are the secondlargest cause of performance variations. Values forrRN and rM depend on user receiver technology,and may vary with satellite elevation dependingon the antenna. Thus the corresponding result inFigure 10 describes the sensitivity of iGPS per-formance to user equipment. The nominal valuesselected in Table 1 are typical of aircraft equip-ment.

Third, availability performance is very sensitiveto the standard deviation of the vertical iono-spheric bias, rVIB. Assuming WAAS-like groundcorrections under non-anomalous ionosphere condi-tions at mid-latitudes, a 1-meter nominal value forrVIB was selected [9]. The need for ionospheric cor-rections determines in large part the scale of theground infrastructure, which motivates furtheranalysis below.

Finally, in view of the remaining results, biases(bVI and bEC) have a more significant impact thangradients (gVI and gEC) and than the troposphereparameters, bZT and Dn. Obviously, rZTB is rela-tively small, and the accumulated error for thegradient-terms, cOIdIPPgVI, Dt � gEC and cOTcTDn, overthe short smoothing interval is not nearly as large asthe bias-terms, bEC and cOIbVI.

Locations and System Configurations

Combined FF and SSF availability (for the nomi-nal configuration) is presented for a 5 deg 3 5 degand a 4 deg 3 4 deg latitude-longitude grid of loca-tions, respectively, over CONUS and over Europein Figure 11. As expected, results improve athigher latitudes, as the density of Iridium satel-lites increases. The map shows that the availabil-ity performance can drop below 99% at 25 degreesof latitude but reaches 100% for all locations at lat-itudes higher than 40 degrees.

Since the performance is driven by Iridium SVmotion, and since a large part of the variations inlongitude averages out over the 3-day simulation

period, availability is plotted versus latitude inFigure 12, for an example longitude of �80 deg.Six different system configurations are considered,including the nominal case. The bottom curvepresents the worst performance, obtained withoutionospheric correction, assuming a rVIB of 5 m.This is evidence that single-frequency iGPS with-out corrections from a sizeable network of groundstations (e.g., WAAS-like) is not sufficient to enableapplications that require high accuracy and integ-rity, such as aircraft precision approach. Still, the

Fig. 11–Combined FF-SSF Availability Maps for the NominalConfiguration

Fig. 12–Sensitivity to System Configurations

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system produces 99% combined availability for lati-tudes higher than 45 deg.

The addition of ionospheric corrections for thevertical ionospheric bias, bVI, is decisive as under-scored with the nominal case. Furthermore, theexisting WAAS ionospheric delay estimation algo-rithm presented in Reference [20] not only com-putes corrections for the bias, bVI, but for the gra-dient, gVI, as well. Significant enhancement wouldbe obtained if these corrections were also broad-cast, increasing the availability at 24 deg latitude,�80 deg longitude, from 98.9% for the nominalcase to 99.9%. The curve labeled ‘DF Iridium’shows that substantial improvement could also beobserved if the second Iridium broadcast frequency(Ka-band) could be reliably tracked by the user.

More promising results are obtained for futurelong-term evolutions. First, extending the Iridiumconstellation by adding two additional orbitalplanes, as suggested earlier, produces maximumavailability. Then, modernization of GPS (sched-uled over the next two decades) will provide civil-ians with dual-frequency (DF) signals, which arefree of ionospheric error. There is also potential toadd DF capability to the next generation of Iridiumsatellites. Thus, ranging measurements would onlyrequire corrections for satellite orbit and clockrelated errors. This considerably reduces the needfor densely spaced ground stations. Results gener-ated for the dual-frequency GPS and Iridiumimplementation are excellent at all latitudes.

Potential Near-Future iGPS Performance

To refine the sensitivity analysis, the emphasisis placed on two of the most influential parame-ters: the smoothing interval, TF, and the standarddeviation of the residual vertical ionospheric bias,rVIB � TF is limited to a maximum of 10 minutes toensure robustness of the error models. rVIB

depends on the accuracy of ionospheric corrections,which is determined by the ground segment. Thus,this preliminary analysis investigates what invest-ment in ground infrastructure is needed to achievehigh-integrity positioning.

The system’s potential near-future performanceis evaluated for a single-frequency GPS and Irid-ium architecture. As a reminder, the nominal con-figuration assumes that users are provided withGPS ephemeris and clock data from the OCS aswell as precise Iridium satellite orbit information.Moreover, the uncorrected instantaneous iono-spheric error is such that rVIB ranges between 5 mand 10 m. After correction from a WAAS-like net-work of 25 reference stations spread across theU.S., this number drops to 0.5–1.5 m. In fact, aone-sigma root-mean-square value of 0.51 m was

computed using quarterly 95% ionospheric errorindeces for all locations and all GPS satellitestabulated in the WAAS performance analysisreports [9] from spring 2002 to spring 2008.

Combined FF-SSF availability results for theMiami location are given in Figure 13 versus rVIB

and TF. Contours of constant availability show thatthe performance sensitivity to filtering perioddecreases at low rVIB values, especially in thegrey-shaded area corresponding to WAAS-like iono-spheric corrections. The 100%-availability domainranges from rVIB of 0.5 m and lower, and filteringperiods, TF, longer than 9 min. Further enlarge-ment of this range is immediately achievable usingcorrections for the gradient, gVI (dashed contours),transmitted here via Iridium communication chan-nels.

Finally, the superposition of dashed and solidlines in Figure 13 highlights the multidimensional-ity of the problem. In the next step of this naviga-tion system design process, similar analyses aswell as further error model validation will be car-ried out for the satellite ephemeris and clock pa-rameters, whose impact on the performance resultshas been illustrated earlier in Figure 10. Ulti-mately, quantifying the influence on the overallend-user performance of parameters such as iono-spheric or satellite-related corrections helps estab-lish and prioritize recommendations on individualsystem components.

CONCLUSION

This paper investigates the potential for Iridium-augmented GPS to enable rapid, robust, and accu-rate navigation at continental scales. Iridium’s largesatellite motion over short periods of time opens the

Fig. 13–Sensitivity to Filtering Period and Ionospheric Correc-tions

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possibility to perform floating carrier-phase basedpositioning in real-time applications over extendedgeographic areas. The resulting bias observability iseffectively exploited in the optimal position andcycle ambiguity estimation algorithm developed inthis work. The detection of faults affecting sequen-ces of time-correlated code and carrier phase signalsin the presence of bounded measurement errors isachieved using a batch least-squares residual-basedRAIM procedure also derived in this paper.

A methodology is established to evaluate theavailability of high-integrity positioning solutionsunder fault-free and single-satellite fault condi-tions. Single-frequency iGPS without ionosphericground corrections was shown to offer practical util-ity at latitudes of 40 deg and above. Below 40 deg oflatitude, the influence of nominal ionospheric errorson more challenging geometries was shown to belargely mitigated using ground corrections broad-cast by a network of ground reference stations.

Early performance analysis results over CONUSand Europe show promising results for navigationwith high accuracy and integrity. However, adetailed integrity analysis, including the effects ofactual satellite fault modes and atmosphericanomalies, is still needed to prove sufficiency forhigh-integrity applications. In particular, the sys-tem’s robustness under stormy ionospheric condi-tions has yet to be evaluated. Future evolutionsincluding dual-frequency architectures yield aneven more decisive impact for Iridium-augmentedGPS, as they may relax the requirements onground infrastructure while extending the avail-ability of high-integrity carrier-phase positioning.

ACKNOWLEDGMENTS

This work was sponsored in part by The BoeingCo., Phantom Works Division. In particular theauthors gratefully acknowledge the efforts Dr.David Whelan and Mr. Bart Ferrell of Boeing fortheir early recognition of the significance ofembarking on this research and their subsequentsupport.

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APPENDIX I IONOSPHERIC ERROR MODELASSUMPTIONS

Extensive analysis of ionospheric data is in pro-gress and has been the subject of significant priorresearch. Assumptions made in this paper arebased on existing literature and a limited set ofnew experimental data.

Literature Review

As mentioned in the fourth section, the iono-spheric delay varies greatly with sun exposure.Under normal conditions, a rVIB of 10–20 m is con-sidered [15] and empirical ionosphere modeling(e.g., using the Klobuchar model, whose parametervalues are broadcast by GPS) helps decrease thisnumber by approximately 50% [27]. In addition, animportant amount of work motivated by LAAS andWAAS aims at determining the vertical ionosphericgradient, gVI, under anomaly-free conditions.Researchers agree on a rVIG of 1 mm/km for quietdays [34, 45] and 4 mm/km for active (but non-stormy) ionospheric days. The results in [29–34]strongly suggest that the linearity and boundingGaussian assumptions are valid for IPP separationdistances of up to 2000 km.

Dual-Frequency GPS Data Processing

Dual-frequency GPS data (L1 and L2) were col-lected in Chicago on one winter and one summerday (12/01/06 and 07/10/07). Measurements wereprocessed using a ‘time-step method’ [34], whichexploits the dispersive nature of the ionosphere,and the fact that most unwanted quantities areconstant biases over time (cycle ambiguities, inter-frequency biases, and part of the ionosphericdelay). The noise of the carrier-phase double-differ-enced measurement (used in this method) is smallwith respect to the effect of the vertical ionosphericgradient, gVI. Experimental data confirm that theprobability distribution of the gradient, gVI (afterde-trending using Klobuchar’s model), can bebounded by a Gaussian with a 1 mm/km rVIG valueduring quiet days. A line-fitting algorithm isemployed to verify the linearity assumption; thecoefficient of recession remains close to 1 for largeperiods of time (up to an hour). Further data proc-essing is ongoing, but the initial results supportthe assumptions made in this paper.

WAAS Data Analysis and Simulation

Dual-frequency measurements from redundantreceivers are collected at 25 wide area referencestations (WRS) and processed at the Wide areaMaster Stations (WMS) to compute ionospheric

corrections for a grid of locations over CONUS.According to [20] and [32], for each grid point(IGP) location, a planar fit is applied to all WRSionospheric delay measurements contained withina certain radius. The resulting IGP vertical delayestimates (IVDE) are sent to the user.

In our work, IVDEs were recorded for the afore-mentioned summer and winter days, for a grid of40 IGP locations over CONUS. The time intervalbetween bVI corrections (2–5 min) does not allowfor efficient gVI corrections (using values of bVI

over time). An estimate of large-scale variations ofthe gradient, gVI, can be computed by differencingthe IVDEs at different IGPs. A maximum of 1.5mm/km was observed.

In addition to the bias, bVI, corrections for thegradient, gVI, are computed at the WMS (the slopesof the plane fit), but are not broadcast due to thelow data transmission rate (250 bps) and becausethey are not needed in aviation applications cur-rently serviced by WAAS. A covariance analysisreplicated from [20] establishes that a rVIG aftercorrection of 0.5 mm/km is achievable. Further-more, the additional data bandwidth enabled byIridium provides ample margin for conveying newaiding data to the user. Finally, a one meter valueis assigned to the corrected rVIB based on numer-ous WAAS Performance Analysis Reports [9] docu-menting the daily and quarterly 95% ionosphericerror for multiple locations since 2002.

APPENDIX II REDUCED-ORDERWEIGHTED RESIDUAL EQUATION WITHPRIOR KNOWLEDGE

Pseudo-measurements (Eq. (8)) can be incorpo-rated to the batch observation (Eq. (7)) by augmen-tation:

z ¼ HB

HP

� xþ vB

vP

� ; (A:1)

in order to get an equation of the same form as Eq.(9):

z ¼ Hxþ v:

HP is subdivided to distinguish states with no priorknowledge (u and N) from the error stateswhose initial uncertainty can be bounded: HP ¼[0nE3(4nKþnS) InE

]. The same subdivision is per-formed on HB, so that HB ¼ [HuN HE]. Theweighted least squares state covariance matrix is:

Px ¼ HTV�1H� ��1

; where V ¼ VB 00 VP

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Px ¼HT

uNV�1B HuN HT

uNV�1B HE

HTEV�1

B HuN HTEV�1

B HE þV�1P

� �1

which can be written as

Px ¼0 00 V�1

P

� þHT

BV�1B HB

8>>: 9>>;�1

: (A:2)

The last expression is actually a KF measurementupdate covariance equation, in which the firstterm of the addition is the information matrix atmission initialization, the inverse of which isreferred to as the a priori state estimate covari-ance matrix in [40].

The following notation is then defined:

Px ¼PuN PuNE

PTuNE PE

� :

The augmented residual (r ¼ [rTB rT

P]) subject to afailure, f, is:

rB

rP

� ¼ I�HSð Þ f

0

� :

With the above notations, HS is subdivided intofour blocks. Based on Eq. (10) describing S, compu-tations of rB and rP result in:

rB ¼ f � HuNPuNHTuN þHEPT

uNEHTuN

�þ HuNPuNEHT

E þHEPEHTE

�V�1

B f

which reduces to:

rB ¼ I�HBPxHTBV�1

B

� �f (A:3)

and

rP ¼ �HPPxHTBV�1

B f : (A:4)

Finally, the weighted norm of the residual krkWcan be expressed as:

rk k2W¼ rTV�1r ¼ rT

BV�1B rB þ rT

PV�1P rP: (A:5)

If Eqs. (A.2) and (A.5) are implemented ratherthan the augmented system (A.1), the dimension ofthe residual vector, rB, decreases by an average of50 elements (for the nominal configuration) withrespect to r, and brings about substantial computa-tional gain when determining krkW.

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