A Dedicated ARAIM Ground Monitor to Validate the Integrity Support Message

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1 Dedcated RM Ground Montor to Valdate the ntegrty Suort Message Yawe Zha Shahrar arash Mchael amoom llnos nsttute of echnology Matheu oerger he Unversty of rzona ors Pervan llnos nsttute of echnology OGRPHY Yawe Zha obtaned a achelor s degree n Mechancal Engneerng from Qngdao Unversty of Scence and echnology Chna n. He s currently a Ph.D. canddate and Research ssstant n Mechancal and erosace Engneerng at the llnos nsttute of echnology (). Hs research focuses on advanced recever autonomous ntegrty montorng (RM) fault detecton and excluson (FDE) usng mult-constellaton global navgaton satellte systems (GNSS). Shahrar arash s senor undergraduate student n erosace Engneerng wth a mnor n Comuter Scence at. He s exected to receve hs.s n May 8. He was also dean s student lst nomnee n 5. He oned the Navgaton Lab n 7 and s assonate about GNSS and navgaton feld. He s currently comletng an nternsh at Contnental G. Dr. Mchael amoom receved hs.s. n stronautcal Engneerng from the US r Force cademy n 997 M.S. n eronautcs and stronautcs from the Massachusetts nsttute of echnology n 999 and Ph.D. n Mechancal and erosace Engneerng from the llnos nsttute of echnology n 6. From 997 to 8 he served actve duty n the r Force rmarly as a C-5 nstructor Plot. From 6-8 he was the FOEC C-X est Drector where he develoed the oeratonal test lan for the r Force s ar refuelng taner relacement. From 8- he was a Mltary Oeratons nalyst for Systems Plannng and nalyss nc. where he erformed oeratons research analyss for the ont Staff n the Pentagon. From to the resent he serves n the r Force Reserves as a C-5 nstructor Plot wth over 9 hours of flyng exerence. He s also currently a Senor Research ssocate n Mechancal and erosace Engneerng at the llnos nsttute of echnology () Navgaton Lab focused on Unmanned rcraft System Detect and vod research. Dr. amoom s a member of and of the nsttute of Navgaton (ON). Dr. Matheu oerger obtaned a Dlôme d ngéneur n Mechatroncs from the Ecole Natonale Suéreure des rts et ndustres de Strasbourg France n and a M.S. and a Ph.D. n Mechancal and erosace Engneerng from the llnos nsttute of echnology () n and 9 resectvely. He s the 9 recent of the nsttute of Navgaton (ON) Parnson award and the 4 recent of the ON s Early chevement ward. He s currently an assstant rofessor at he Unversty of rzona worng on mult-sensor ntegraton sequental fault-detecton for mult-constellaton navgaton systems and relatve and dfferental recever autonomous ntegrty montorng (RM). Dr. ors Pervan s a Professor of Mechancal and erosace Engneerng at where he conducts research on advanced navgaton systems. Pror to onng the faculty at he was a sacecraft msson analyst at Hughes rcraft Comany (now oeng) and a ostdoctoral research assocate at Stanford Unversty. Prof. Pervan receved hs.s. from the Unversty of Notre Dame M.S. from the Calforna nsttute of echnology and Ph.D. from Stanford Unversty. He s an ssocate Fellow of the a Fellow of the nsttute of Navgaton (ON) and Edtor-n-Chef of the ON ournal NVGON. He was the recent of the Sgma X Excellence n Unversty Research ward ( ) Ralh arnett Mechancal and erosace Det. Outstandng eachng ward (9 ) Mechancal and erosace Det. Excellence n Research ward (7) Unversty Excellence n eachng ward (5) EEE erosace and Electronc Systems Socety M. arry Carlton ward (999) RC Wllam E. acson ward (996) Guggenhem Fellowsh (Caltech 987) and lbert. Zahm Prze n eronautcs (Notre Dame 986).

2 SRC Future dual-frequency mult-constellaton advanced recever autonomous ntegrty montorng (RM) s exected to brng sgnfcant navgaton erformance mrovement to cvl avaton. he RM user algorthm whch ncludes fault detecton and excluson (FDE) functons s autonomously executed at the arborne recever. o acheve secfc ntegrty and contnuty requrements the real-tme FDE rocess requres assertons on the sgnal-n-sace (SS) erformance and ths nformaton s carred n the ntegrty suort message (SM). hs aer descrbes the desgn analyss and evaluaton of the offlne ground montor whch ams at valdatng the SM broadcast to users. o acheve ths GNSS satellte ts and clocs must be estmated. here are many sohstcated t determnaton rocesses such as the one used by the nternatonal GNSS servce (GS) whose erformance s secfed n terms of accuracy. n contrast the roosed offlne RM archtecture s manly ntended for safetycrtcal avaton alcatons n whch ntegrty s of the rmary concern. hs montor emloys a straghtforward aroach to estmate satellte t/cloc whch ams at facltatng SM generaton and valdaton. t taes advantage of the exstng satellte based augmentaton system (SS) ground nfrastructure. n ths aer a worldwde networ of sarsely dstrbuted reference statons s consdered and arametrc satellte tal models are emloyed n the estmators whose dervaton and mlementaton are descrbed ste by ste. wo searate analyses covarance analyss and model fdelty evaluaton are carred out to resectvely assess the mact of measurement errors and of resdual model errors on the montor s estmated t/cloc. We have nvestgated dfferent t models (GPS legacy versus CNV tal model) and reference staton cloc models (quadratc model versus no model). he results ndcate the standard devaton of the montor s t/cloc estmaton error s on the order of cm whch s adequate for SS erformance valdaton. NRODUCON he man challenge when usng global navgaton satellte systems (GNSS) n safety-crtcal cvl avaton alcatons s to mtgate the ntegrty threats caused by measurement faults ncludng satellte and constellaton falures []. For the sngle constellaton case usng the global ostonng system (GPS) only fault detecton (FD) can be mlemented usng recever autonomous ntegrty montorng (RM). RM has been mlemented snce the md-99s as a bacu navgaton tool to suort arcraft en-route flght usng GPS only [-4]. he core rncle of RM s to exlot redundant measurements to acheve self-contaned FD at the user recever [5]. fter decades of worldwde develoment two ey elements have been foreseen n future GNSS. Frst nomnal measurement errors wll be sgnfcantly reduced usng dual-frequency sgnals whch wll remove the largest source of error onosherc delay. Second four constellatons ncludng GPS (U.S.) GLONSS (Russa) Galleo (Euroean Unon) and edou (Chna) are exected to fnsh ther modernzatons and/or full deloyments n the near term [6] whch wll rovde many more satelltes n vew than we have avalable today usng GPS alone. hese revolutonary develoments n GNSS together wth mortant advancements n the RM concet wll oen the ossblty to ndeendently suort arcraft navgaton usng GNSS from enroute flght towards fnal aroach to landng wth mnmal nvestment n ground nfrastructure. herefore consderable effort has recently been exended esecally n the Euroean Unon and n the Unted States [7 8] to develo new dual-frequency mult-constellaton advanced RM (RM) fault detecton and excluson (FDE) methods. he erformance of RM/RM s measured ether n terms of ntegrty rs or n terms of rotecton level (PL). t s hghly deendent on the assumed GNSS nomnal sgnal-n-sace (SS) error models and on the a-ror fault robabltes. n current conventonal RM mlementatons ths nformaton s defned by the GPS constellaton servce rovder (CSP) commtments and s hardcoded n the recever. s an evoluton of RM RM wll (a) addtonally use constellatons that are not as mature as GPS and (b) see to rovde assured navgaton for vertcal gudance. o rovde flexblty n the evoluton of RM to multconstellaton RM and to mnmze nvasveness nto the avoncs RM wll nclude an ntegrty suort message (SM). he SM wll carry nformaton defnng SS error and fault statstcs ncludng nomnal measurement bases standard devatons of the ehemers and cloc errors ror robabltes of satellte faults and ror robabltes of constellaton-wde faults [7]. he SM arameters wll be generated and valdated at the ground and udated to users as needed. Varous methods of SM dssemnaton are resently beng consdered ncludng on-arcraft databases and data broadcast through geosynchronous satelltes or one or more of the GNSS core constellatons. he methods and result descrbed are alcable regardless of method of dssemnaton.

3 o valdate the SM onlne and offlne RM archtectures have been under nvestgated [9 ]. Offlne archtectures have generally been erceved as referable because they do not requre a real-tme communcaton ln between users and ground segment and therefore elmnates the connectvty rs [9]. n offlne RM montor would rely on ost-rocessed GNSS measurements to bound errors n CSP broadcast navgaton message on a long-term bass [9]. o do ths ror research has taen truth satellte ostons and cloc bases from the nternatonal GNSS servce (GS) networ [-]. However gven that RM s ntended to oerate over several decades montor deendence on external systems or organzatons wth lttle or no stae n cvl avaton must be carefully consdered and deally avoded. Most mortantly RM wll serve safety crtcal alcatons any otental safety rss must be roerly accounted for and quantfed. he GS the natonal geosatal ntellgence agency (NG) and others currently rovde hgh-accuracy satellte t/cloc roducts. ut none of these agences mae secfc commtments on the relablty of ther roducts or on the rocesses used to obtan those roducts. Further data gas exst n those roducts esecally durng satellte fault events whch are crucal to RM. n resonse ths aer develos a new aroach to defne and valdate SM arameters by desgnng a dedcated RM offlne ground montor archtecture. hs new montor emloys a smle and transarent t determnaton rocess to estmate satellte ts/clocs. Measurements are collected from a worldwde networ of sarsely dstrbuted reference statons (RS). arametrc t model s used to ft data over tme. ecause most satellte faults are caused by cloc anomales the standard quadratc cloc model s not aled for satellte clocs. n contrast an RS can be equed wth redundant atomc clocs so any recever cloc faults can be detected removed n ost-rocessng. herefore as an alternatve to drectly estmatng nstantaneous RS clocs states at each tme-eoch we also nvestgate the beneft of usng quadratc RS recever cloc models. wo estmators wll be consdered: a batch leastsquares method and a alman Flter (F). n both cases satellte t satellte clocs and RS clocs wll be smultaneously estmated over a four-hour ft nterval suorted by both the GPS legacy and GPS CNV t models. Satellte t and cloc errors are determned by dfferencng the ostonng and tmng soluton derved from the GNSSbroadcast navgaton message and the estmates generated by the offlne montor. hs data wll rovde the means to evaluate the SM arameters. On the one hand we desre that these arameters nclude enough margn to bound the actual constellaton erformance over the secfed erod of SM valdty (e.g. one month). On the other hand the SM arameters should not be set too conservatvely or RM avalablty wll degrade. herefore t s essental that the ground montor be caable of roducng accurate satellte oston and cloc estmates. n ths wor a two-ste evaluaton of the montor s t and cloc estmate accuracy s carred out: frst the mact of measurement errors at RS recevers s quantfed by covarance analyss and second the fdelty of the t model s valdated by comutng the resdual errors relatve to nown satellte ostons. Four scenaros are consdered n erformance evaluatons: (a) GPS legacy versus CNV t models and (b) quadratc model versus no model for RS recever clocs. n regard to (a) usng the more sohstcated CNV model can reduce resdual fttng errors but t may also ncrease the standard devatons of the estmated satellte ostons (because more arameters need to be estmated) [4]. n regard to (b) drect estmaton of nstantaneous RS clocs elmnates cloc model fdelty errors and rovdes robustness aganst RS cloc faults but standard devatons of estmaton errors may ncrease because each cloc estmate s nstantaneous n tme (earler measurements can t be used). n the analyss the worst-case SS rangng error (SSRE) s derved to conservatvely quantfy the resultng estmate errors n the along-trac cross-trac radal and cloc comonents. REQURED OFFLNE MONOR PERFORMNCE Usng an offlne montor the udate rate of the SM may vary from a month to year deendng on the need [9]. large amount of data wll be rocessed at the ground to obtan the SM whch s exected to bound the SS erformance untl the next udate. o acheve ths dfferences between broadcast ehemers and the montor s estmated satellte t/cloc wll be frst evaluated over tme and then these wll be used to valdate (or modfy f needed) the SM arameter values. n ths aer we focus manly on roducng satellte t and cloc estmates to valdate b nom and σ UR whch secfy a Gaussan bound on the SS erformance under nomnal condtons (.e. no faults). Snce the montor s t/cloc estmaton errors wll drectly contrbute to our ablty valdate SM t s necessary to frst defne the requred accuracy of the montor s t/cloc estmator. Equaton () shows the relatonsh among the actual standard devaton of broadcast satellte t/cloc error σ CUL (actual σ UR ) the standard devaton of the montor s estmated t/cloc error σ MONOR and the valdated standard devaton of the satellte t/cloc error σ VLDE (valdated σ UR ).

4 VLDE () CUL MONOR Fgure lots σ VLDE versus σ MONOR over dfferent σ CUL values. For examle the fgure shows that for σ UR =.4 m whch s the current mnmum value that the GPS CSP s wllng to commt to havng σ MONOR s less than.7 m s suffcent for valdaton. ccordng to the most recent study on GPS SS erformance to suort RM the maxmum actual σ UR s observed on sace vehcle number (SVN) 6 where σ CUL s.65 m and for most satelltes t s much smaller []. he red curve s of most nterest because t s exected future erformance of the GPS constellaton wll rovde σ CUL of m or less. nd dualconstellaton RM avalablty smulatons have revealed localzer erformance wth vertcal gudance (LPV)- aroach can only be suorted when σ CUL of both constellatons are aroxmately m [9]. he fgure shows slow growth of σ VLDE as σ MONOR ncreases. Even when σ MONOR reaches half of σ CUL at the end of the curve the achevable valdated σ UR s stll around. m. Fg. σ VLDE vs. σ MONOR over Varyng σ CUL Values n addton to σ MONOR the estmated satellte t/cloc error may have a non-zero mean denoted by b MONOR. hs term s accounted as one comonent of the valdated SM arameter b nom. However usng baselne soluton searaton RM user algorthm [7] the absolute value of b nom s addtve for each measurement. hs causes the ntegrty rs bound to become loose as the number of measurements ncreases thereby degradng avalablty erformance [7]. herefore t s desrable to mtgate any contrbutons of b MONOR to the valdated b nom. From the analyss and dscusson above t can be seen that the requred accuracy of the offlne montor s sgnfcantly lower than the recse satellte t/cloc roducts by GS or NG. nstead t s the relablty of the montor s estmator outut that s ey. n other words even though the montor s satellte t/cloc estmates may have larger errors ther stable erformance and consstent avalablty enables valdate SMs wthout mssng data gas. herefore n the estmator desgn t s not necessary to ursue a comlcated t determnaton rocess. We can nstead the smle transarent aroach descrbed n the next secton. OFFLNE MONOR RCHECURE hs secton descrbes the offlne montor archtecture ste-by-ste. networ of worldwde sarsely dstrbuted RS s emloyed to collect code and carrer measurements over tme. n the selecton of stes for the RS we tae advantage of the satellte based augmentaton system (SS) ground nfrastructures snce they are already exstng and are desgned to suort cvl avaton alcatons. Fgure (left) shows all of the exstng SS RS from dfferent countres or regons and Fgure (rght) shows the networ of RS used n ths wor. o obtan a roughly unform global dstrbuton fve non-ss RS stes are added. hs networ ensures that each satellte can contnuously be traced by at least two reference statons but does not allow reverse

5 ostonng (four RS smultaneously observng a sace vehcle (SV) are requred to drectly estmate the satellte oston). nstead the offlne montor uses arametrc t models to determne SV traectores. 45 N 45 N 8 W 5 W 9 W 45 W 45 E 9 E 5 E 8 E 45 S WS EGNOS MSS GGN SDCM 8 WS W 5 W 9 W 45 W 45 E 9 E 5 E 8 E EGNOS MSS 45 S GGN SDCM NG USF Fg. ll Exstng SS Statons (Left) and Examle Networ of RS Used n hs Wor (Rght) wo canddate GPS t models legacy and CNV are nvestgated. oth models are vald for medum earth t (MEO) SVs wth 5 t arameters beng used for legacy model and 7 arameters for CNV model [5]. hroughout ths aer we assume no satellte maneuvers occur because the t models wll not be vald whle the sacecraft s thrustng. Maneuvers occur rarely and they can be handled by smly ntatng a new estmator after the maneuver. However these detals of ractcal mlementaton are beyond the scoe of ths wor and wll not be consdered further. Let be the 5 (or 7 deendng on whch t model s used) vector of t model arameters for SV and let g be the non-lnear functon that determnes SV t. he true SV oston x of SV at tme eoch can be exressed as: x g ( ν () ) where ν s the devaton of the model outut from the true oston x. ν reresents the model s nablty to erfectly cature the true t and ths t wll be further analyzed n later sectons. he GPS t model s vald over a four-to-sx-hour tme nterval noted F [5]. Senstvty to F has been evaluated n our ror wor n [4]. We use a four-hour fttng nterval whch s the most common value for GPS ehemers. oth RS and SV are equed wth atomc clocs and a quadratc olynomal s usually emloyed to model ther nomnal errors. However most SV faults are caused by ther clocs and for the montor to clearly observe them no assumton can be made on the SV cloc dynamcs. s for the RS whose cloc faults can be detected at the ground t s feasble to aly quadratc cloc model. o assess the otental beneft of usng such a model for the RS clocs we analyze two cases: (a) alyng a quadratc olynomal to model RS cloc errors and (b) mang no assumton on RS clocs (.e. unctually estmate RS cloc states). Under the scenaro of (a) let b to be the cloc bas of RS at tme ; t can be modeled as: cl cl b gcl ( ) ν () cl Smlar to equaton () n equaton () s the vector of cloc arameters ncludng a f a cl f a f [5] and ν s the cloc model error. g cl ( cl ) s a lnear functon (n the cloc arameters) that can be exressed as: g cl cl ( ) a f a f ( t tref ) a f ( t tref ) (4) where t and t ref are resectvely the true tme and reference tme.

6 Dual-frequency onoshere-free code and carrer measurements for SV from RS at tme eoch are exressed by the followng two equatons: x x (5) b troo RNM x x (6) b troo RNM where x : nown locaton of RS for examle n an Earth-Centered Earth-Fxed (ECEF) reference frame x : unnown locaton of SV at tme τ : unnown cloc offset of SV at tme b : unnown cloc offset of RS at tme η : unnown constant carrer hase cycle ambguty for SV at RS : Eucldean norm oerator n ths case rovdng the dstance between RS and SV. troo he error terms n equatons (5) and (6) account for resdual troosherc delay noted and recever nose and multath (RNM) errors denoted by RNM for code and RNMφ for carrer. We assume raw code and carrer measurement RNM error standard devatons of.5 m and. m resectvely. hese standard devatons are multled by.588 to account for the onoshere-free combnaton at L and L5 frequences. o account for the temoral and satal correlaton of zenth troosherc deal (ZD) a frst order Gauss-Marov rocess wth a two-hour correlaton tme s aled. ssumng the RS have access to data from weather statons we use the value of.5 m as the standard devaton of resdual ZD. hs value s scaled for lower elevaton satelltes usng the troosherc mang functon gven n [7 9]. o estmate the SV t arameters and clocs the measurement equatons (5) &. (6) and the t/cloc model equatons () & (4) need to be lnearzed and ncororated nto one flter. For llustratve uroses we wll only show code measurement n the followng dervatons. he code measurement equaton (5) s frst lnearzed at x : e δx (7) b where δ : devaton from nomnal values e.g. δx = x x : lne of sght (LOS) vector between SV and RS at tme n ECEF e : code measurement error ncludng troo and RNM. n the next ste the t model s lnearzed at where x = g ( ). Substtutng the lnearzed t model equaton and the cloc error model equaton (4) nto the code measurement equaton (7) becomes: e δ (8) cl cl where : acoban matrx for the SV t model n equaton () whch s 5 for the GPS legacy t model and 7 for the CNV model. t s comosed of numercally-derved artal dervatves of the oston coordnates of SV at tme x = [x y z ] wth resect to the t arameters = [ 5] :

7 z y x z y x cl : vector that exress equaton (4) n a more comact form: ) ( ) ( ref ref cl t t t t ESMOR DESGN Once the lnearzed sngle measurement equaton (8) s derved the estmator can be establshed. he arameters to be estmated or states nclude SV t arameters δ SV cloc τ RS cloc model arameters cl and cycle ambgutes η. ll of these states can be smultaneously estmated usng ether a batch and alman flter mlementaton. he states are estmated over F of 4 hours usng a 4 mn samle erod to avod modellng correlaton between samles due to RS multath. n addton to obtan observablty all the cloc error states ncludng both SV and RS clocs are measured wth resect the cloc of RS. atch Estmator he batch estmator s establshed by stacng all code and carrer measurements over the ft nterval F. Measurements for all SVs from to are collected by all RS from to at all tme eochs from to. he resultng estmaton equaton s wrtten as: cl cl cl cl cl cl cl τ τ φ φ (9) Four grous of states are dstngushed by the dashed thn lnes. he notatons are worth clarfyng: a b s an a b matrx of zeros a b s an a b matrx of ones and a a s an a a dentty matrx. n addton the roduct of LOS vector and acoban matrx n equaton (8) s defned as.e. = e. Snce measurements at all tme eochs are ncororated n equaton (9) we use catal to denote the tme sequence. herefore τ and () t s obvous from equaton (9) that ths estmator s humongous due to the substantal number of measurements n the ft nterval.

8 t s mortant to note that not all SVs wll be vsble to all RS durng the fttng nterval. o lghten the notatons these cases are not exlctly exressed n equatons (9) and (). ut the corresondng rows must be removed whenever the measurements are unavalable. Let us smlfy equaton (9) by defnng z H s and to resectvely be the measurement vector observaton matrx state vector and error vector. hen equaton (9) becomes z = Hs + and the states can be evaluated usng a weghted least-squares estmator: z V H H V H s ˆ () where V s the covarance matrx of the measurement error vector. he SV t arameters δ and cloc τ can be extracted from the full-state estmate vector s. Furthermore the SV oston s obtaned by substtutng the estmated t model arameters nto equaton (). alman Flter roach n addton to the batch estmator we also rovde a F aroach. he F mlementaton s more comutatonally effcent and t allows us to cross-chec wth the results from batch estmator. t one tme eoch the measurement equaton s exressed as: RNM RNM RNM RNM RNM RNM RNM ZD troo ZD troo cl cl cl cl cl cl cl c c c c c c c φ φ τ τ () Comared wth equaton (9) the addtonal states troo ZD n equaton () reresent the resdual ZD at RS and c s the oblquty factor. We use state augmentaton to account for tme correlaton of the troosherc ZD error [6]. ecause troo ZD s modeled as a frst order Gauss-Marov rocess (FOGMP) the assocated dynamc model of the F aroach s exressed as: ZD troo ZD troo ZD troo ω e s s () n equaton () the frst row accounts for the states n equaton (9) whch do not change over tme. he second row shows the dynamcs of FOGMP where s the samle erod and μ s the tme constant [7]. troo ZD s obtaned by stacng troo ZD of all RS and the varance of ω troo ZD s ( e (/μ) )σ ZD [8]. s a result wth both the measurement equaton and dynamc model avalable the nomnal F stes [6] can be mlemented to estmate SV t/cloc. U untl now the offlne montor s t and cloc estmator has been fully establshed usng two aroaches and the estmaton rocess has been descrbed n detal. he dervatons of both aroaches were done for the case where the legacy t model s

9 aled and RS cloc errors are modeled usng a quadratc olynomal. For cases when the CNV t model s used or RS clocs are not modeled the exressons of equatons (9) and () need to be modfed to accommodate the corresondng scenaro. here wll be two error sources contrbutng the montor s SV t/cloc estmaton error: (a) measurement error and (b) resdual model error. o resectvely address ther macts on the montor s erformance we carry out two searate analyses: covarance analyss and model fdelty analyss. COVRNCE NLYSS n ths secton covarance analyss used to nvestgate the contrbuton of measurement error on σ MONOR. For llustratve uroses we wll emloy the batch estmator equatons to derve the covarance matrx of SV t/cloc states. ccordng to equatons (9) and () the covarance matrx of the full-states s can be comuted by: Σ H V H (4) o evaluate the oston and cloc estmate error covarance of SV at tme eoch we frst buld the covarance matrx of the SV t arameters and cloc D by extractng corresondng elements from Σ. herefore D s 6 6 f a legacy t model s emloyed. hen the 4 4 covarance matrx P LL of satellte oston and cloc n the local-level (LL) reference frame (alongtrac cross-trac radal) can be evaluated by: R LL R LL P LL C D C (5) where R LL s the ECEF to LL rotaton matrx and C s defned as: C (6) o nvestgate the SS n range doman we evaluate the maxmum SSRE standard devaton at the worst-case user locaton. hs s acheved by roectng P LL along LOS for all locatons wthn SV footrnt []. nd the worst-case SSRE standard devaton σ SSRE s defned as the maxmum roecton. n Fgure the roecton regon s shaded n lght blue and the blac dashed lne s one examle roecton lne G m where m s the ndex of the lnes. herefore σ SSRE s evaluated by: SSRE m... LL m LL m max G P G (7) Fg. Covarance Matrx Proectons along LOS for ll Locatons

10 o exlore the otental benefts of emloyng t/cloc models to the estmaton rocess the covarance analyses are erformed under 4 scenaros: CSE : legacy t model quadratc RS cloc model CSE : legacy t model free RS cloc states CSE : CNV t model quadratc RS cloc model CSE 4: CNV t model free RS cloc states n the analyses we tae all measurements over the fttng nterval F as one data set wth each data set startng at regular onehour ntervals over a day. SV ostons are comuted from a GPS almanac whch s tycally aled n covarance analyss [7 9]. Fg. 4 Examle Profles of Error Devatons along One Fttng nterval for Case (Left) and Case 4 (Rght) Fgure 4 shows standard devaton rofles of an examle satellte (PRN 5) over one fttng nterval. hree lght blue lnes are the t error devatons of t along-trac cross-trac and radal. he blac thn lnes reresent the satellte cloc error devaton. he red thc lnes are of greatest nterest because they show the worst-case SSRE standard devaton evaluated usng equaton (7). n both cases shown n the fgure the t radal and cloc comonents are hghly correlated whch results n sgnfcantly smaller σ SSRE than each ndvdual term. Comarng the two fgures even though the t radal and cloc terms are ncreased for case 4 there s no notceable dfference between the two red lnes. We tae the central two-hour values as the fnal results of each fttng nterval. herefore only the data between Hour and Hour n Fgure 4 wll be consdered. Fg. 5 Maxmum σ SSRE over One Day (Left) and her Dstrbuton (Rght) for Case 4

11 he left lot of Fgure 5 shows the maxmum σ SSRE rofles over a one-day erod. Each column reresents the largest σ SSRE values of that fttng nterval and satelltes are dstngushed by dfferent colors. he dstrbuton of σ SSRE s catured by the hstogram on the rght whch shows that a large maorty σ SSRE values are well below.5 m and the maxmum σ SSRE s. m. able. Maxmum σ SSRE Values under Four Scenaros Quadratc Model for RS Cloc Free RS Cloc Legacy Orbt Model CSE :.77m CSE :.87m CNV Orbt Model CSE :.8m CSE 4:.m able summarzes the covarance analyss results by resentng the maxmum σ SSRE values of the four cases. t the frst sght of ths table t can be observed that there are only small varatons among those values. So from the ersectve of mtgatng the contrbuton of measurement error on σ MONOR no sgnfcant beneft can be obtaned by usng dfferent t/cloc models. s exected case 4 results n the largest σ SSRE because t has the largest number of unnown arameters to be estmated. ased on these results we recommend not usng a quadratc model for RS cloc error snce there s lttle beneft from t and not usng t smlfes the estmator and allows for seamless erformance n the event of an RS cloc fault (because cloc errors are estmated at each eoch ndvdually). herefore cases and are no longer nvestgated and so t only remans to address the mact of t model error on the montor s erformance. OR MODEL FDELY NLYSS n ths secton the resdual model error s evaluated for two canddate t models: legacy and CNV. o do ths recse GPS t data s frst fed as measurements nto equaton () and then the t arameters are estmated usng a non-lnear least squares algorthm [9]. GS data x GS was emloyed as the truth t and the CSP broadcast ehemers arameter vector was used as the ntal guess n the estmaton rocess. Once s estmated the resdual error can be comuted by subtractng x GS wth the model s outut. herefore for SV at tme eoch the ECEF resdual s: r GS x g (ˆ ) (8) where s the estmated t arameter vector. Fnally r can be exressed n the LL frame as: r (9) LL R LL r [ r rc rr ] n equaton (9) the three comonents of the last vector reresent the resdual error n three dmensons: along-trac cross-trac and radal. o evaluate the rangng error from r LL multle analytcal exressons can be found n the lterature [] [-]. n ths aer we consder the followng equaton for t-only SSRE: r SSRE.98rR.4sgn( rr ) r rc () Equaton () s very smlar to the one n [] excet the SV cloc term s removed. he dervaton can be found n the reference and wll not be restated here. hs analyss utlzes GS data of the frst ten days of 6.e. //6 - //6. Smlar to the covarance analyss the data s ft to the model over F of 4 hours and each fttng nterval starts at hours over a day. he samle erod s 5 mns and only the -hour central results are recorded.

12 Fg. 6 est Ft Resdual SSRE Orbt over One Day Fgure 6 shows the resdual SSRE error for an examle SV (PRN 5) on /6/6. n each ndvdual fgure the color-code hels dstngush ft ntervals. Vertcal t oe -lnes ndcate the tme of ehemers whch s at the center of each -hour wndow. he results show that the resdual SSRE of CNV t model s sgnfcantly smaller than that of legacy t model. Fg. 7 Resdual SSRE Orbt Ranges Fg. 8 Resdual SSRE Orbt Overbound Fgures 7 and 8 rovde a more general vew of the resdual SSRE rofle by rocessng all of the GS data. he SSRE ranges n Fgure 7 are exressed n terms of GPS SV blocs. Usng the GPS legacy t model the maxmum SSRE s.5 m and SSRE

13 root-mean-square (RMS) s.89 m. hese two values are resectvely reduced to. m and. m wth the CNV model. n Fgure 8 the data s overbounded usng a Gaussan dstrbuton wth mean at b ob and standard devaton of σ ob. y comarng the two t models t can be observed that emloyng the CNV model wll dramatcally reduce the resdual error. he CNV model fdelty error can be quantfed usng a Gaussan bound wth aroxmately zero-mean cm standard devaton whch does not cause a sgnfcant contrbuton to the montor s t/cloc error. Gven that contrbuton of CNV model error s neglgble comared to the effect of measurement error shown n able and that the same s not true for the legacy model t s recommended to use the CNV t model for the RM offlne montor s t estmator. CONCLUSON hs aer descrbes the desgn analyss and evaluaton of an RM offlne ground montor whch ams at valdatng the SM broadcast to the users. he montor emloys a networ of worldwde sarsely dstrbuted reference statons (RS) and a arametrc t model to smultaneously estmate SV ts SV clocs and RS clocs. wo estmators are derved and ther mlementaton s descrbed n detal n the aer. Covarances and model fdeltes are analyzed to assess the mact of RS measurement error and model fttng error on the montor s erformance: the most robust mlementaton wth the overall lowest SV t/cloc errors s acheved by mang no assumtons on SV or RS cloc dynamcs and by usng the CNV t model. hs montor s redcted to acheve a sgnal n sace range error standard devaton σ MONOR of about. m whch wll allow valdaton RM user range accuraces σ UR of.5 m f the actual σ UR s m. n the next stes of ths wor we wll use exermental data to valdate the desgn of the offlne montor. CNOWLEDGEMEN he authors would le to than the Federal vaton dmnstraton for sonsorng ths wor. However the vews and onons exressed n ths aer are those of the authors and do not necessarly reflect those of any other organzaton or erson. REFERENCES [] Pervan. Navgaton ntegrty for arcraft recson landng usng the global ostonng system Ph.D. Dssertaton Det. of eronautcs and stronautcs Stanford Unv. Stanford C 996. [] Lee Y. C. nalyss of Range and Poston Comarson Methods as a Means to Provde GPS ntegrty n the User Recever Proceedngs of the 4nd nnual Meetng of he nsttute of Navgaton Seattle W [] Parnson. W. and xelrad P. utonomous GPS ntegrty Montorng Usng the Pseudorange Resdual NVGON: ournal of the nsttute of Navgaton Vol. 5 No do:./ tb955.x [4] RC Secal Commttee 59 Mnmum Oeratonal Performance Standards for rborne Sulemental Navgaton Equment Usng Global Postonng System (GPS) RC/DO-8 uly 99. [5] oerger M. Chan F.-C. and Pervan. Soluton Searaton Versus Resdual-ased RM NVGON: ournal of the nsttute of Navgaton Vol. 6 No. 4 Wnter do:./nav.7 [6] Gbbons G. Munch Summt Charts Progress of GPS GLONSS Galleo edou GNSSes nsde GNSS March. [7] lanch. Walter. Enge Per. Lee Y. Pervan. Rl M. Sletter. and ro V. aselne dvanced RM User lgorthm and Possble mrovements EEE ransactons on erosace and Electronc Systems Vol 5 anuary do:.9/es.4.79 [8] Phase of the GNSS Evolutonary rchtecture Study February. URL:

14 htts:// ESPhase_Fnal.df [9] EU-U.S. Cooeraton on Satellte Navgaton Worng Grou C RM echncal Subgrou Mlestone Reort February 5. URL: htt:// [] lanch. et al rchtectures for dvanced RM: Offlne and Onlne Proceedngs of the 7th nternatonal echncal Meetng of he Satellte Dvson of the nsttute of Navgaton (ON GNSS+ 4) ama Florda Setember [] Walter. lanch. eynote: Characterzaton of GNSS Cloc and Ehemers Errors to Suort RM Proceedngs of the ON 5 Pacfc PN Meetng Honolulu Hawa rl [] Perea S. Meurer M. Rl M. elabbas. and oerger M. UR/SS nalyss for GPS and Galleo to Suort RM NVGON: ournal of the nsttute of Navgaton Vol. 64 No. Summer do:./nav.99. [] Walter. Gunnng. lanch. eynote: Valdaton of the Unfaulted Error ounds for RM Proceedngs of the ON 7 Pacfc PN Meetng Honolulu Hawa May [4] oerger M. Zha Y. Pervan. Onlne Montor ganst Cloc and Orbt Ehemers Faults n RM Proceedngs of the ON 5 Pacfc PN Meetng Honolulu Hawa rl [5] Global Postonng System Drectorate Systems Engneerng and ntegraton nterface Secfcaton S-GPS- Revson H. URL: htt:// [6] rown R. G. Hwang P. Y. C. ntroducton to Random Sgnals and led alman Flterng 4th Ed Wley. SN: [7] Gelb. led Otmal Estmaton the M Press. SN: [8] oerger M. Carrer Phase GPS ugmentaton Usng Laser Scanners and Usng Low Earth Orbtng Satelltes Ph.D. Dssertaton Det. of Mechancal Materals and erosace Engneerng llnos nsttute of echnology Chcago L 9. [9] Crassds. unns. Otmal Estmaton of Dynamc Systems Chaman & Hall/CRC 4. SN: X [] Heng L. Safe Satellte Navgaton wth Multle Constellatons: Global Montorng of GPS and GLONSS Sgnal-n-Sace nomales Ph.D. Dssertaton Det. of eronautcs and stronautcs Stanford Unv. Stanford C. [] Malys S. et al he GPS ccuracy mrovement ntatve Proceedngs of the th nternatonal echncal Meetng of the Satellte Dvson of he nsttute of Navgaton (ON GPS 997) ansas Cty MO Setember [] ssstant Secretary of Defense for Command Control Communcatons and ntellgence. Global Postonng System Standard Postonng Servce Performance Standard. Washngton DC 8. URL: htt://