A Simple Satellite Exclusion Algorithm for Advanced RAIM

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1 A Smple Satellte Excluson Algorthm for Advanced RAIM Juan Blanch, Todd Walter, Per Enge Stanford Unversty ABSTRACT Advanced Recever Autonomous Integrty Montorng s a concept that extends RAIM to mult-constellaton and dual frequency that could provde worldwde coverage of vertcal gudance [1], [2]. A baselne algorthm for the user recever has been descrbed n [3]. Ths algorthm ncludes the computaton of the Protecton Levels, the Effectve Montor Threshold, the accuracy, and the descrpton of a prelmnary excluson algorthm, whch was refned n [4]. The excluson functon helps mantan avalablty through the duraton of a fault or to meet the contnuty requrements. In our prevous work [4], we had proposed an excluson algorthm for ARAIM whose goal was to provde a set of condtons under whch a fault (satellte or group of satelltes, for example) can be excluded and treated as f t was flagged by the ground. Ths property allows the recever to treat the satelltes remanng after excluson as an all-n-vew stuaton, whch greatly smplfes the processng. To acheve ths goal, that algorthm used excluson confrmaton tests. The thresholds of the excluson confrmaton tests were ted to the Protecton Level equaton. There were two mnor drawbacks n ths algorthm. Frst, the excluson confrmaton tests, although not computatonally expensve, requre a complex descrpton and a somewhat complcated logc. Second, when the excluson confrmaton tests do not pass, (that s, n practce when there are other consstent sets,) there can be a momentary loss of performance due to the ambguty n whch s the correct excluson. In ths paper we present an evoluton of the excluson scheme presented n [4] that, whle retanng ts qualtes, solves the frst ssue and mtgates the second one. To address the frst one, the logc of the algorthm has been smplfed: t now uses the same functon to compute the Protecton Levels whether excluson has been performed or not. The second ssue, that s, the loss of performance, s mtgated by explotng the ambguty as to whch satellte must be excluded. In ths new algorthm, the excluson confrmaton tests are now only an output of the all-n-vew Protecton Level calculaton. We wll present a step by step descrpton of the algorthm, and the assocated analytcal proof of ntegrty. To show the performance of the algorthm n a realstc settng, we wll test the algorthm usng GPS and GLONASS measurements (both wth real faults and nected faults). INTRODUCTION The purpose of the excluson functon for ARAIM descrbed n ths paper s to recover avalablty after a detecton event, wth the followng lmtatons: - The recever s not aware of the Alert Lmt - The recever does not need to characterze each of the excluson optons pror or after fault detecton In addton, t would be desrable that: - There s a neglgble mpact on the detecton only performance - The protecton level after detecton and excluson s equal to the protecton level as f the excluded measurements had been flagged by the ground montorng (or as f t was an outage) f the fault s unambguous (ths wll be defned) We note that the frst two lmtatons prevent the recever from guaranteeng performance when a fault s present. A performance guaranteed to a gven probablty would requre the recever to characterze all excluson optons and outage condtons. Ths s done by makng sure that the probablty that there s at least one subset that s consstent s below the requrement, and by accountng

2 for all the excluson optons when computng the ntegrty rsk. Ths approach was mplemented n [5], (although t dd not account for outage condtons other than faults). EXCLUSION ALGORITHM Fndng excluson canddates The excluson algorthm conssts n fndng a subset of measurements that s consstent. (A subset s determned to be consstent f t passes the soluton separaton tests [3]). As shown n [3], t s possble to avod testng all possble subsets by checkng the ch-square statstc of each of the subsets. It s defned by: where: T 1 T T q y W W G G W G G W y y s the vector of measurement resduals as defned n [3] W s the weghtng matrx for subset correspondng to the ntegrty error model [3] G s the geometry matrx as defned n [3] The set of canddates s gven by q Texcluson, the threshold T excluson, wll be defned below. Selectng measurements or fault mode to be excluded (1) where Among the subsets that pass the test, we fnd one subset that also passes the consstency checks. There may be several subsets that pass the consstency check. In that case, the recever can ether choose the frst one that passes the consstency check, or compute all possbltes and choose the one provdng the lowest protecton level. PL where:, excluson PL, baselne, max max PL T xˆ xˆ q T 0, baselne 0, excluson, (2) PL,excluson s the protecton level calculated after subset has been excluded. PL,baselne s the protecton level calculated assumng the set of measurements s unavalable (0 means no excluson). It s defned as n [3]. T 0, s the threshold for the soluton separaton test on fault mode as defned n [3] T excluson, s the threshold for the ch-square statstc xˆ s one of the coordnates of the poston fx that s not affected by fault mode [3] The nterpretaton of ths equaton s the followng: the set of measurements has been selected for excluson, but the subsets excludng are also consstent, whch could mean that the fault mght be on the set of measurements. Snce the estmate xˆ s free of the error n case the fault s present, we make sure that the protecton level ncludes t (as well as a margn). As the error grows, there wll be fewer sets wthn the excluson statstc. If there are none left, the protecton reverts to the baselne protecton level, as f there was an outage. Ths protecton level s defned for each coordnate. The HPL can be obtaned by combnng the PL n the East and North coordnates as explaned n [3]. Threshold computaton POST EXCLUSION PROTECTION LEVEL EQUATION Let us suppose that fault mode has been selected for excluson. We show that the followng protecton level meets the ntegrty requrements: The excluson thresholds must be chosen such that: where: Nfault modes 1 p T P 2 fault, k excluson, HMI, excluson (3) N fault modes s the number of fault modes as defned n [3]

3 p s the probablty of fault mode P HMI, excluson s the ntegrty rsk budget accountng for the excluson. Ths parameter should be set so that nomnal performance s not affected. In ths paper we used a prelmnary value of If q T excl,, we mpose the constrant: PL, PL ˆ ˆ 0, T0, x x excluson baselne Ths nequalty mples: (8) 2 k s the dstrbuton of the ch-square statstc assumng that there s no fault. One way to choose the thresholds T excluson, to fulfl Equaton s to set: T 2 HMI, excluson excluson, k Nfault_modes p P INTEGRITY PROOF The ntegrty rsk ncludes all the possble excluson optons. The equaton can be wrtten as: Integrty Rsk = (4) P xˆ x PL, excluson,excluson of,fault (5) The protecton level PL,excluson can have dfferent values dependng on the results of the excluson tests, so we wrte: xˆ x xˆ xˆ PL T 0, baselne 0, As a consequence, we have: ˆ 0, baselne 0, (9) x x PL T (10) We therefore have the followng nequalty: Integrty rsk from fault q Texcl, P or fault p xˆ x PL0, baselne T 0, (11) We can now remove the unon over the ndex, as the event s the same for all ndces. We fnally have: Integrty rsk from fault q Texcl, P or fault p xˆ x PL0, baselne T 0, (12) Integrty rsk from fault = xˆ x PL,excluson,excluson of, q T excl, P or fault p xˆ x PL,excluson,excluson of, q T excl, (6) We have the upper bound: Integrty rsk from fault q T excl, P or fault p xˆ x PL,excluson, q T excl, (7) An upper bound of the probablty of havng ether of these two events s gven by the sum of the probabltes of each event: Integrty rsk from fault P q T fault p P xˆ x PL T fault p 0, baselne 0, fault, excl, fault, (13) The second term n Equaton (13) s the ntegrty rsk when no fault has been excluded. Therefore, the ntegrty rsk added by the excluson functon s gven by: Integrty rsk from excluson = Nfault modes 1 p P q T fault, excluson, (14)

4 The statstc q s ndependent of the fault I, so as long as the thresholds fulfl Equaton (3), we wll have: Integrty rsk from excluson P HMI, excluson (15) GPS Mask All 1 All 1 P const, P sat, RESULTS WITH REAL DATA The man property of ths scheme s that, as long as the error s large enough, the protecton level wll revert to the baselne protecton level wth very lttle penalty (over the fault detecton only performance). The examples shown here are not exhaustve and are only meant to llustrate ths property. The frst example (Fgure 1) was generated wth smulated data. The nomnal nose s fxed. A ramp error s nected n one of the measurements. At frst the fault s too small to be detected. Then, the fault s detected, but there s some uncertanty as to whch measurement should be excluded. Fnally, when the error s large enough, the VPL reverts to the VPL obtaned by assumng that the excluded satellte s not avalable. α URA, 1 1 α URE, 1 1 b nom, Table 1. Integrty Support content assumed n the GPS GLONASS ARAIM prototype Fgures 2 through 6 shows both the Horzontal Poston Error (HPE) and the HPL. In Fgure 2, no fault was nected. In Fgures 3 through 5, a constellaton wde fault consstent wth an East West rotaton of 25, 50, and 100 m was nected n all GLONASS measurements. Fgure 6 shows the results assumng that GLONASS s not avalable. As the error grows, there are fewer perods were the fault was undetected (or not excluded correctly). When the error s large enough (Fgure 5), the fault s always detected and excluded, and the HPL reverts to the GPS only HPL (ths can be seen by comparng Fgure 5 wth Fgure 6). In all cases the HPL was above the HPE. Fgure 1. VPL (n red) and Vertcal Poston Error (n blue) when a fault s nected. In Fgures 2 through 5 we examne the behavor of a L1- L2 GPS GLONASS ARAIM prototype (as descrbed n [6]) wth nected faults. We assumed the followng Integrty Support Message content [3]: Fgure 2. HPE and HPL of the L1-L2 GPS GLONASS prototype (no fault).

5 Fgure 3. HPE and HPL of the L1-L2 GPS GLONASS prototype wth an nected constellaton wde GLONASS fault (25 m) Fgure 5. HPE and HPL of the L1-L2 GPS GLONASS prototype wth an nected constellaton wde GLONASS fault (100 m) Fgure 4. HPE and HPL of the L1-L2 GPS GLONASS prototype wth an nected constellaton wde GLONASS fault (50 m) Fgure 6. HPE and HPL of the L1-L2 GPS GLONASS prototype assumng that GLONASS measurements are not avalable SUMMARY We have presented a smple excluson algorthm and the assocated protecton levels for Advanced RAIM. Ths algorthm does not requre the characterzaton of all excluson optons. The protecton level after excluson depends on outputs of the functon that computes the alln-vew protecton level and the protecton level computed assumng that the fault s actually an outage. Ths property makes the excluson computatonal load

6 low. When the fault magntude s large enough, the protecton level s smply the protecton level that would have been obtaned f the fault had been an outage. ACKOWLEDGEMENTS Ths work was sponsored by the FAA GPS Satellte Product Team (AND-730). REFERENCES [1] Workng Group C, ARAIM Techncal Subgroup, Interm Report, Issue 1.0, December 19, Avalable at ent.cfm?doc_d=7793 and workng-group-c/araim-report-1.0.pdf [2] Blanch, J., Walter, T., Enge, P., Wallner, S., Fernandez, F., Dellago, R., Ioanndes, R., Pervan, B., Hernandez, I., Belabbas, B., Spletter, A., and Rppl, M., Crtcal Elements for Mult-Constellaton Advanced RAIM for Vertcal Gudance, to appear n Navgaton, Journal of the Insttute of Navgaton. [3] Blanch, J., Walter, T., Enge, P., Lee, Y., Pervan, B., Rppl, M., Spletter, A., "Advanced RAIM user Algorthm Descrpton: Integrty Support Message Processng, Fault Detecton, Excluson, and Protecton Level Calculaton," Proceedngs of the 25th Internatonal Techncal Meetng of The Satellte Dvson of the Insttute of Navgaton (ION GNSS 2012), Nashvlle, TN, September 2012, pp. -. [4] Blanch, J., Walter, T., Enge, P., "Excluson for Advanced RAIM: Requrements and a Baselne Algorthm," Proceedngs of the 2014 Internatonal Techncal Meetng of The Insttute of Navgaton, San Dego, Calforna, January 2014, pp [5] Joerger, M., Stevanovc, S., Chan, F-C., Langel, S., Pervan, B., "Integrty Rsk and Contnuty Rsk for Fault Detecton and Excluson Usng Soluton Separaton ARAIM," Proceedngs of the 26th Internatonal Techncal Meetng of The Satellte Dvson of the Insttute of Navgaton (ION GNSS+ 2013), Nashvlle, TN, September 2013, pp [6] Walter, Todd, Blanch, Juan, Cho, Myung Jun, Red, Tyler, Enge, Per, "Incorporatng GLONASS nto Avaton RAIM Recevers," Proceedngs of the 2013 Internatonal Techncal Meetng of The Insttute of Navgaton, San Dego, Calforna, January 2013, pp

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