Vol. 6, No. 1, 2007 Journal of Global Positioning Systems

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1 Vol. 6, No., 7 Jounal of Global Positioning Systems ISSN (Pint Vesion) ISSN (CD-ROM Vesion) Intenational Association of Chinese Pofessionals in Global Positioning Systems (CPGPS)

2 Jounal of Global Positioning Systems Aims and Scope The Jounal of Global Positioning Systems is a pee-eviewed intenational ounal fo the publication of new infomation, knowledge, scientific developments and applications of the global navigation satellite systems as well as othe positioning, location and navigation technologies. The Jounal publishes oiginal eseach papes, eview aticles and invited contibutions. Shot eseach and technical notes, book eviews, and commecial advetisements ae also welcome. Specific questions about the suitability of pospective manuscipts may be diected to the Editoin-Chief. Edito-in-Chief Jinling Wang The Univesity of New South Wales, Sydney, Austalia Jinling.Wang@unsw.edu.au Editoial Boad Ruizhi Chen Finnish Geodetic Institute, Finland Wu Chen Hong Kong Polytechnic Univesity, Hong Kong Doota Gene-Bzezinska Ohio State Univesity, United States Ren Da Bell Labs/Lucent Technologies, Inc., United States C.D. de Jong Fugo-Intesite B.V., The Nethelands Hans-Jügen Eule InPosition gmbh, Switzeland Yanming Feng Queensland Univesity of Technology, Austalia Yang Gao Univesity of Calgay, Canada Shaowei Han Sif, United States Editoial Advisoy Boad Junyong Chen National Bueau of Suveying and Mapping, China Yongqi Chen Hong Kong Polytechnic Univesity, Hong Kong Paul Coss Univesity College London, United Kingdom Guente Hein Univesity FAF Munich, Gemany Changdon Kee Seoul National Univesity, Koea Hansoeg Kuttee Univesity of Hannove, Gemany Jiancheng Li Wuhan Univesity, China Esmond Mok Hong Kong Polytechnic Univesity, Hong Kong J.F. Galea Monico Depatamento de Catogafia FCT/UNESP, Bazil Günthe Retsche Vienna Univesity of Technology, Austia Gethin Robets Univesity of Nottingham, United Kingdom Rock Santee Laval Univesity, Canada Buno Schezinge Applanix Copoation, Canada Gead Lachapelle Univesity of Calgay, Canada Jingnan Liu Wuhan Univesity, China Keith D. McDonald NavTech, United States Chis Rizos The Univesity of New South Wales, Austalia C.K. Schum Ohio State Univesity, United States Salah Sukkaieh The Univesity of Sydney, Austalia Todd Walte Stanfod Univesity, United States Lambet Wanninge Desden Univesity of Technology, Gemany Caiun Xu Wuhan Univesity, China Guochang Xu GeoFoschungsZentum (GFZ) Potsdam, Gemany Ming Yang National Cheng Kung Univesity, Taiwan Kefei Zhang RMIT Univesity, Austalia Guoqing Zhou Old Dominion Univesity, United States Pete J.G. Teunissen Delft Univesity of Technology, The Nethelands Sien-Chong Wu Jet Populsion Laboatoy, NASA, United States Yilin Zhao Motoola, United States IT Suppot Team Satellite Navigation & Positioning Goup The Univesity of New South Wales, Austalia Publication and Copyight The Jounal of Global Positioning Systems is an official publication of the Intenational Association of Chinese Pofessionals in Global Positioning Systems (CPGPS). It is published twice a yea, in June and Decembe. The Jounal is available in both pint vesion (ISSN ) and CD-ROM vesion (ISSN ), which can be accessed though the CPGPS website at Whilst CPGPS owns all the copyight of all text mateial published in the Jounal, the authos ae esponsible fo the views and statements expessed in thei papes and aticles. Neithe the authos, the editos no CPGPS can accept any legal esponsibility fo the contents published in the ounal. Subsciptions and Advetising Membeship with CPGPS includes subsciption to the Jounal duing the peiod of membeship. Subsciptions fom non-membes and advetising inquiies should be diected to: CPGPS Headquates Depatment of Geomatics Engineeing The Univesity of Calgay Calgay, Albeta, Canada TN N4 Fax: +(43) ounal@cpgps.og CPGPS, 7. All the ights eseved CPGPS Logo Design: Peng Fang Univesity of Califonia, San Diego, United States Cove Design and Layout: Satellite Navigation & Positioning Goup The Univesity of New South Wales, Austalia Pinting: Satellite Navigation & Positioning Goup The Univesity of New South Wales, Austalia

3 Jounal of Global Positioning Systems Vol. 6, No., 7 Table of Contents On the Feasibility of Adding Caie Phase Assistance to Cellula GNSS Assistance Standads L. Wiola, S. Vehagen, I. Halivaaa, C. Tibeius... Pecise Point Positioning Using Combined GPS and GLONASS Obsevations C. Cai and Y Gao... 3 Diffential GPS: the Reduced Diffeence Appoach A. Lannes... 3 A Robust Indoo Positioning and Auto-Localisation Algoithm R. Mautz and W. Y. Ochieng Latest Developments in Netwok RTK Modeling to Suppot GNSS Modenization H. Landau, X. Chen, A. Kipka, U. Vollath Integation of RFID, GNSS and DR fo Ubiquitous Positioning in Pedestian Navigation G. Retsche and Q. Fu Modified Gaussian Sum Filteing Methods fo INS/GPS Integation Y. Kubo, T. Sato and S. Sugimoto An Evaluation of GNSS Radio Occultation Technology fo Austalian Meteoology E. Fu, K. Zhang, F. Wu, X. Xu, K. Maion, A. Rea, Y Kuleshov, and G. Weymouth PC4 Based Low-cost Inetial/GPS Integated Navigation Platfom: Design and Expeiments D. Li, R. J. Landy and P. Lavoie... 8 An Innovative Data Demodulation Technique fo Galileo AltBOC Receives D. Magaia, F. Dovis, P. Mulassano... 89

4 Jounal of Global Positioning Systems (7) Vol.6, No.: - On the feasibility of adding caie phase assistance to cellula GNSS assistance standads Laui Wiola, Ismo Halivaaa Nokia, Inc., Finland Sanda Vehagen, Chistian Tibeius Mathematical Geodesy and Positioning, Univesity of Delft, The Nethelands Abstact. The 3GPP (Thid Geneation Patneship Poect) Release 7 of GSM and UMTS cellula standads as well as SUPL., used in IP netwoks, include mao modifications as to how AGNSS (Assisted GNSS) assistance data is tansfeed fom the netwok (cellula o IP) to the cellula teminal. Simultaneously position accuacy impovements may be intoduced. One potential option is to use caie phase -based positioning methods. This can be achieved integally in the cellula netwok o by the use of Vitual Refeence Stations and an IP netwok. The bulk of AGNSS devices will be singlefequency due to additional cost associated with two RF font-ends. Hence, this study addesses the feasibility of single-fequency caie phase-based positioning, making compaison with the dual-fequency case. The study shows that single-fequency caie phase -based positioning is feasible with shot baselines (<5 km) given that: ) eal-time ionospheic pedictions ae available and ) thee ae enough satellites available. Namely, this equies hybid-use of GPS and Galileo. Keywods. Assisted GNSS, RTK, VRS, Ambiguity Resolution, Success Rate Intoduction The annual sales of AGNSS-enabled (Assisted GNSS) handsets ae estimated to ise to 4 million units by (Stategy Analysts, 6). Cuently the size of the maket is appoximately million units annually. High gowth equies developing constantly moe efficient and capable methods to impove use expeience in tems of availability, accuacy and shot time-to-fist-fix. The assistance data available fom the netwok ae a significant facto affecting the use expeience. The advantages and benefits of assistance ae discussed in (Wiola et al., 7b). As GPS/AGPS now becomes commonplace in mobile teminals, the next step in the competition will be the ace fo accuacy. One option to achieve this is to take advantage of caie phase -measuements eadily available in GNSS eceives integated in mobile teminals. Methods utilizing caie phase -measuements include Real-Time Kinematic (RTK) as well as Pecise Point Positioning (PPP). The ecommendation given in (Nokia, 6) is that caie phase -based positioning would be added to the cellula standads in such a manne that the teminal could equest fo caie phaseassistance fom the SMLC (Seving Mobile Location Cente) and calculate the baseline vecto between the base station and the teminal. Caie phase -based positioning was fo the fist time intoduced in 3GPP (The Thid Geneation Patneship Poect) in GERAN#3 (GSM/EDGE Radio Access Netwok with GSM being Global System fo Mobile communications and EDGE being Enhanced Data ates fo Global Evolution) meeting in June 6 in Lisbon, Potugal (Nokia, 6). When the baseline implementation fo A-Galileo was ageed in GERAN#3, this featue was included in the list of items to be eviewed in the 3GPP Release 7 time fame (Alcatel et al., 6). Howeve, the featue was not included in the Release 7 due to the identified need to futhe assess the technical implementation befoe appoving the appoach. It is expected that caie phase -based positioning will be dealt with in the Release 8 of the 3GPP standads. This pape examines the feasibility of intoducing singlefequency caie phase -based positioning into cellula netwoks. The use case consideed consists of a shot baseline (<5 km) and a single-fequency eceive due to the cost easons. Howeve, the eceive may be a dual-

5 Jounal of Global Positioning Systems GNSS (GPS+Galileo) eceive. The pape includes a thoough eview of the latest eseach in the aea of caie phase-based position. The eview is complemented by simulations that ae pefomed using a state-of-the-at open-souce simulation tool developed fo the analysis of caie phase-based positioning (Vehagen, 6b). Assisted GNSS Fig. shows the high-level view of AGNSS achitectue. The coe of the achitectue is the AGNSS seve, o moe pecisely, seve centes that ae geogaphically distibuted. These centes seve the AGNSS-subscibes in each geogaphical aea. Assuming that the AGNSSteminal is to eceive assistance ove the use plane (IPnetwok) the teminal takes a data connection to the peset seve and equests fo the assistance data. The assistance data is then deliveed to the teminal as specified in the associated standads. The AGNSS seve may obtain its data fom vaious souces. These may include physical GNSS-eceives distibuted geogaphically (left hand side in Fig. ). These eceives can povide integity infomation as well as boadcast ephemeedes to the AGNSS seve fo distibution. On the othe hand, the obit and clock models (as well as othe data) can oiginate fom an extenal sevice poviding, fo instance, pecise ephemeedes and obit/clock pedictions (ight hand side in Fig. ). Such sevices include the Intenational GNSS Sevice, o IGS (Dow et al., 6). Should pedictions be available, AGNSS-enabled teminals can be povided with extended ephemeedes, in which case the teminal does not need to connect to the assistance seve in the beginning of each positioning session. This impoves use expeience due to the time saved in not having to set up a data connection and download the assistance. With longtem ephemeedes the assistance is also available, when thee is no netwok coveage (Lundgen et al., 5). Cuently it is only possible to povide assistance fo GPS L in GSM and UMTS (Univesal Mobile Telecommunications System) netwoks. In GSM the assistance is specified in the Radio Resouce LCS (Location Sevices) Potocol (RRLP, (3GPP-TS-44.3)) and in UMTS in the Radio Resouce Contol (RRC, (3GPP-TS-5.33)). Moeove, thee ae also use plane solutions, such as Open Mobile Alliance (OMA) Secue Use Plane Location (SUPL, (OMA-ULP)) potocol. It should be noted that thee ae teminological diffeences depending upon, which standad is in question. Fo instance, the mobile teminal is MS (Mobile Station) in GSM, UE (Use Equipment) in UMTS and SET (SUPL-Enabled Teminal) in SUPL. Moeove, the seve sending the assistance to the teminal is an SMLC in RRLP and RRC, while in SUPL the seve is an SLC (SUPL Location Cente). Due the upcoming changes in the GNSS infastuctue (Wiola et al., 7b), such as modenization of GPS and GLONASS as well as the intoduction of Galileo amongst othes, the 3GPP standadization body accepted a poposal which opened the way fo the addition of new GPS bands as well as othe GNSSs to the assistance standad in autumn 6 (3GPP, 6). This decision concened RRLP only, but the same solution was late appoved into RRC (3GPP, 7) as well as SUPL. (OMA, 7). GNSS boadcast GNSS eceive GNSS satellite IP netwok AGNSS-enabled teminal AGNSS seve RRLP, RRC, SUPL IP netwok Fig.. The AGNSS achitectue Extenal Sevice (such as, IGS) AGNSS intoduces common and pe-gnss elements into the standads. The supestuctue is detailed in (Syäinne et al., 6). The common elements ae GNSS-independent and include, fo instance, ionosphee model and efeence location. In the futue, fo instance, toposphee models o Eath-Oientation Paametes can be added without obstacles. The pe-gnss elements, on the othe hand, ae by definition GNSS-dependent (as well as signal-dependent) and include diffeential coections, eal-time integity, GNSS-common time elation, data bit assistance, efeence measuements as well as obit and clock models (ephemeedes). The new multi-mode navigation model capable of suppoting at least seven GNSSs is discussed in (Wiola et al., 7a) and (Wiola et al., 7b). The intoduced geneic appoach significantly educes the system complexity. 3 Caie phase -based positioning Real-Time Kinematic (RTK) techniques utilize caie phase -measuements that ae eadily obtained fom a GNSS eceive. Caie phase measuements enable centimete-level accuate baseline (i.e. distance and

6 Wiola et al.: On the feasibility of adding caie phase assistance to cellula GNSS assistance standads 3 attitude between the eceives) detemination between two (o moe) GNSS eceives. Also, if the absolute position of one eceive is known at high accuacy, the absolute position of the othe eceive can easily be deduced. The addition of caie phase -based positioning to cellula standads, theefoe, potentially enables ubiquitous cm- o dm-level positioning accuacy. The cuent commecial solutions typically utilize both GPS L and L signals fo high-pecision suveying. Moeove, with the GLONASS modenization (Klimov et al., 5), the utilization of multi-gnss is becoming eve moe attactive. Also, the ecent studies (Wiola et al., 6; Alanen et al., 6a; Alanen et al., 6b) show that single-band single-gnss RTK is feasible unde cetain cicumstances. In addition, all the Galileo as well as the modenized GPS signals can be utilized in the baseline detemination (Eisfelle et al., a; Eisfelle et al., b; Tibeius et al., ). The moe signals thee ae the moe cetain (in statistical sense) the baseline becomes (Wiola et al., 6). Caie phase -based positioning may be intoduced eithe by suppoting it in the SMLC o by utilizing an extenal sevice. In the case of an SMLC-implementation (contol plane solution in the cellula netwok), the teminal equests fo caie phase -measuements fom the SMLC. The SMLC then stats sending the measuements fom the LMU (Location Measuement Unit) to the teminal. Anothe option is to utilize Vitual Refeence Stations (VRS) as a sevice extenal to the netwok. In this case the teminal sends the AGNSS assistance seve an assistance equest that contains the appoximate position of the teminal. A VRS is ceated to this location and measuements ae steamed to the teminal most likely ove an IP-netwok. The advantage of this technology is that the baseline is always vey shot and no additional hadwae (LMUs) is equied in the netwok. The key to the high-accuacy baseline detemination is intege ambiguity esolution, fo which thee ae many algoithms available. In addition to solving the ambiguities, anothe key issue is the validation of ambiguities. Validation efes to using statistical tools to detemine, whethe the ambiguities and, hence, the fixed baseline solution can be elied on. If the ambiguities cannot be solved, somewhat less accuate option is to utilize the float solution. In this case the ambiguities ae not fixed to thei intege values, but ae consideed as eal numbes. This study concentates on discussing the vaious factos affecting the ambiguity esolution success ate and how those factos affect the feasibility of adding caie phasebased positioning to the 3GPP standads. 4 Method and analyses In the following the pefomance of the caie phase - based positioning is analyzed unde vaying cicumstances. Chapte V examines a situation, in which a set of individual measuements is exchanged between two eceives. This coesponds to Measue Position Response with Multiple Sets defined in RRLP (3GPP-TS- 44.3). Chapte VI studies a situation with peiodic epoting of measuements fom one eceive to anothe as defined in RRC (3GPP-TS-3.7). The pefomance is chaacteized in tems of the success ate fo fixing the intege ambiguities successfully. Theoetical tools fo this analysis ae given, fo instance, in (Teunissen et al., ). This wok utilizes an opensouce analysis tool called VISUAL (Vehagen, 6b), which allows fo simulating success ates in tempoal o spatial dimensions. In eal-time applications ambiguity fixing success ate can be calculated on-the-fly in ode to examine, whethe ambiguity fixing should be attempted at all. As a geneal ule, the success ate must be above 99% befoe fixing should be attempted (Vehagen, 6b). If the ambiguity solution is not available, the system can povide the use with a float solution. Baseline accuacy obtainable with a float solution is. -. metes. 5 Single-shot multiple-sets The fist set of simulations consides a case, in which one eceive makes thee measuements with 5-s spacing coesponding to the total measuement time of s. This can be consideed as a situation, in which the MS sends multiple sets of caie phase measuements to the SMLC (3GPP-TS-44.3) allowing the SMLC to calculate the baseline. Fig. shows the success ates fo Galileo E (up) and fo Galileo E+E5a (below). The paametes and assumptions of the simulation ae 5-km stationay baseline Date st Januay 8 :: UTC 5-degee elevation mask Fixed ionosphee (i.e. extenal ionosphee model used to coect the obsevations) Float toposphee (i.e. toposphee delay estimated as state) with Ifadis mapping function 3-mm STD fo caie phase obsevations 3-cm STD fo code phase obsevations 3-satellite Galileo constellation Fig. shows that single-band caie phase -based positioning using only Galileo should be consideed too uneliable fo implementation. On the othe hand, the addition of the second fequency (E5a) impoves the

7 4 Jounal of Global Positioning Systems pefomance significantly. In the dual-band case, the caie phase -based positioning is enabled and feasible globally. Conside then tempoal changes in the success ates. Fig. 3 shows the success ate as a function of time in Pais fo Galileo E (up) and Galileo E+E5a (below). The date and othe assumptions ae the same as befoe measuements (one instant). In the study seven o moe satellites wee used all the time and the baseline was in the ode of one km. Howeve, the authos epoted poblems with validating the calculated ambiguities. Finally, if GPS and Galileo ae used in hybid, the situation impoves significantly. This is shown in Fig. 4, in which the simulation shown up in Fig. 3 has been eun adding the GPS L signal. The esults show that the edundancy fom additional satellites (9-satellite GPS constellation) contibutes significantly to the success ate. Thee ae only few shot peiods duing which thee might be poblems with fixing the ambiguities. The finding is also suppoted by the liteatue. Fo instance, Vehagen (Vehagen, 6a) epots that combined dualband GPS+Galileo yields a constant success ate of >99.9%. In that case the success ate becomes almost independent of time and location. Inceased numbe of satellites is identified as the single most impotant facto fo high success ate. Howeve, thee is no infomation, how the ambiguity validation success ate behaves in a combined GPS L + Galileo E situation Success ate (ed) numbe of satellites (geen) stating epoch Fig.. Ambiguity fixing success ate fo single-shot multiple-sets. Up: Galileo E, Below: Galileo E+E5a. The simulation shows that in a single-fequency case the success ate is highly dependent upon the numbe of satellites available. In geneal, it seems that caie phase -based positioning is feasible, when thee ae at least satellites visible. Howeve, thee ae only shot peiods, when this takes place. On the othe hand, dual-band positioning does not suffe fom the lack of satellites. Only if the numbe of satellites is below seven the success ate dops below the theshold. The dual-band case clealy outpefoms the single-band case. The liteatue suppots the conclusions dawn fom the simulations. Tibeius et al. (Tibeius et al., 995) epot % ambiguity fixing ate, when using GPS L+L code and caie phase measuement and only one set of Success ate (ed) stating epoch Fig. 3. Ambiguity fixing success ate fo single-shot multiple-sets ove one day in Pais (48.5 N,. E). Red denotes success ate and geen the numbe of satellites above the elevation mask. It is assumed that all the satellites above the mask can be used in the ambiguity esolution. Up: Galileo E, Below: Galileo E+E5a numbe of satellites (geen)

8 Wiola et al.: On the feasibility of adding caie phase assistance to cellula GNSS assistance standads 5 Single-shot data delivey means that the baseline may be solved once (when the set of measuements aives), but not updated afte that. The eceiving teminal/seve may extapolate the measuements fo -3 s without losing accuacy significantly (Schüle, 6). Howeve, the baseline is lost afte this in the case the eceives (o one of the eceives) ae moving. Theefoe, the single-shot multiple-set method is useful only fo stationay eceives. Moeove, since thee is no possibility fo igoous solution quality and integity monitoing in time, baselines should be limited to shot ones. The exact length depends on the bands and GNSSs used as well as on the atmospheic conditions and also on whethe ionosphee o toposphee models ae available. Success ate (ed) stating epoch Fig. 4. Ambiguity fixing success ate fo single-shot multiple-sets ove one day in Pais (48.5 N,. E), when GPS L + Galileo E ae used. 6 Peiodic measuements Peiodic measuements efe to a case, in which one eceive peiodically sends its signal measuements to the othe eceive. This enables, fo example, monitoing the solved paametes in time and, theefoe, quality contol. Also, with multi-band eceives, filteing of ionosphee advance (as well as toposphee delay) becomes possible. Finally, longe obsevation peiods assist the validation pocess. Peiodic epoting is enabled in UMTS netwoks ove RRC. Fig. 5 shows the success ates fo Galileo E (up) and fo Galileo E+E5a (below), when one eceive steams measuements to the othe eceive - in this case signal measuement evey s fo s (in total measuements). Note that by a signal measuement one undestands a set of measuements consisting of code and caie phases fo all the obsevable satellites and signals. The othe paametes and assumptions of the simulation ae as given in chapte V. Fig. 5 shows a mao impovement in the single-band case. It appeas that the single-fequency caie phase numbe of satellites (geen) based positioning becomes feasible in many locations, when seveal epochs ae utilized in the solution. Howeve, the analysis made fo Pais fo the same situation unning ove one day (Fig. 6) shows that although thee is an impovement as compaed to the esults shown in Fig. 3, windows fo successful caie phase -based positioning ae still few. The pomising peiods ae now longe (fo instance, between epochs 4-5 s), but it can be assumed that the high vaiation in the success ate in time makes single-band positioning still vey challenging even if moe measuements ae now available Fig. 5. Peiodic epoting. Success ate fo Galileo E (up) and fo Galileo E+E5a (below). The dual band case continues to demonstate excellent pefomance globally independent of time. This can be veified fom the lowe gaphs in Figs. 5 and 6, espectively. Finally, in Fig. 4 it was shown that the combined GPS L + Galileo E shows mao impovement ove the single- GNSSs case in the single-shot situation. Repeating the same analysis fo steaming shows that inceasing the

9 6 Jounal of Global Positioning Systems numbe of available obsevations yields high success ate (above 99.9%) independent of time. The finding is suppoted by the liteatue (Vehagen, 6b). Once again, the inceased availability of signals is identified as the single most impotant facto. Success ate (ed) Success ate (ed) stating epoch stating epoch Fig. 6. Peiodic epoting. Success ate fo Galileo E (up) and fo Galileo E+E5a (below) numbe of satellites (geen) numbe of satellites (geen) 7 Measuement update ate Fom the bit consumption point of view the most impotant issue is the measuement update ate, i.e. how often the teminal is equied to epot the signal measuement to the othe eceive o seve (o vice vesa). This is analyzed by fixing the measuement peiod to s and vaying the measuement inteval. The paametes and the assumptions of the analysis ae as befoe, signals used ae Galileo E+E5a and the measuement ates in Fig. 7 a-d ae Fig 7a: a signal measuement evey s fo s (in total measuements) Fig 7b: a signal measuement evey 5 s fo s (in total 3 measuements) Fig 7c: a signal measuement evey s fo s (in total 6 measuements) Fig 7d: a signal measuement evey s fo s (in total measuements) The simulations show that the -s measuement spacing yields a constant >99% success ate. Theefoe, it is deduced that the measuement inteval shall not exceed seconds in peiodic epoting. Thee is also anothe issue suppoting this view. Once the ambiguities have been fixed, the baseline will be tacked using the solved ambiguities. The -s measuement spacing equies that in ode to be able to update the baseline continuously, the measuements fom the sending eceive must be extapolated fo seconds. Note, howeve, that this is possible only if the sending eceive is stationay. This is the case if the sending eceive is, fo example, an LMU..8.8 Success ate (ed) numbe of satellites (geen) Success ate (ed) numbe of satellites (geen) stating epoch stating epoch Figue 7a. -s spacing between measuements. Fig. 7b. 5-s spacing between measuements.

10 Wiola et al.: On the feasibility of adding caie phase assistance to cellula GNSS assistance standads Success ate (ed) numbe of satellites (geen) Success ate (ed) numbe of satellites (geen) stating epoch stating epoch Fig. 7c. -s spacing between measuements. Schüle (Schüle, 6) epots that 3-s extapolation leads to 35-mm RMS eo in the baseline as compaed to a case without extapolation. Howeve, the aticle ecommends using 5-s - -s spacing fo the best balance between bandwidth consumption and pefomance. Accepting eos of few tens of millimetes allows fo extending the spacing to -s, which was consideed maximum inteval fom the success ate point of view. Fig. 7d. -s spacing between measuements. longe widelane has on the esolution. This is shown, fo instance, in esults fo Galileo E5a+E5b. Moeove, when using widelane combinations, one must ensue that ) eal advantage can be gained by using them and that ) wideand naowlane ambiguities can be decoelated to such extent that they can be solved. Fo moe discussion see (Teunissen, 997). 8 Analysis of diffeent systems Fig. 8 shows an analysis of ambiguity fixing success ates ove one day fo single-epoch fixing attempts (i.e. only one instant of time used). The height of the ba indicates the span of the success ate ove the day and the black dot the aveage success ate. The blue bas on the left ae fo GPS, the ed bas in the middle fo Galileo and the geen bas fo GPS+Galileo hybid. The method of analysis is detailed in (Vehagen et al., 7). The assumptions fo baseline, time and othe paametes ae as befoe. Fistly, compaing the blue and ed bas in Fig. 8 shows that Galileo outpefoms GPS in single- and multi-band cases. This is attibutable to a geate numbe of satellites in the Galileo constellation as well as to highe obit altitude. Both these contibute to a geate numbe of visible satellites and, theefoe, eceivable signals. In the liteatue it is often stated that selecting fequencies close to each othe yields a longe widelane and, hence, impoved ambiguity esolution. This is evident, fo instance, in esults fo GPS L+L and L+L5, in which L is close to L in fequency than L5. Consequently, GPS L+L outpefoms L+L5. Howeve, thee is a limit to which this effect can be exploited. In all the widelane combinations noise is amplified by a facto that is dependent upon the fequencies. Now, if the fequency sepaation becomes sufficiently small, the noise amplification becomes dominant ove the effect that a Fig. 8. Single-epoch success ates ove one day. Black dot denotes the mean value and the ba the span of success ates ove the day. Blue GPS, ed Galileo, geen hybid. Anothe finding is that the dual-gnss cases clealy outpefom the single-gnss cases. This is tue acoss all the signal combinations. The main benefit fom Galileo is in fact the incease in the numbe of satellites/signals available fo caie phase -based positioning. Howeve, consideing the Galileo-only situation, (Vehagen, 6a) shows that due to constellation diffeences, Galileo E+E5a o E+E6 pefoms substantially bette at low latitudes than GPS L+L5 o L+L, but at othe latitudes no significant diffeences ae obsevable.

11 8 Jounal of Global Positioning Systems Yet anothe esult visible in Fig. 8 is that adding a thid fequency to the solution does not have significant impact on the aveage success ate, but its span deceases (minimum success ate inceases). Hence, a tiplefequency solution has impact on quality-of-sevice as well as sevice availability although the aveage success ate is not affected. Moeove, Richet (Richet et al., 5) states that the success ate fo validation impoves significantly as the thid fequency is taken into account. 9 Single-fequency field measuement esults Fixed Baseline eo (m) 4 x East Noth Up Fig. 9 shows field test esults fo GPS L taken 8th Januay 7 in Tampee, Finland (6.5 N, 3.7 E) fo 3-m and 36-m baselines, espectively. The numbe of satellites used vaied fom 8 to. The code and caie phase measuements fom two GPS measuement engines wee double diffeenced and fed to an extended Kalman filte. Intege ambiguities wee solved using the LAMBDA-algoithm using disciminato as the validato with a theshold value of 3 (Tibeius, 995). Neithe ionosphee no toposphee was modeled and no a-pioi model of atmosphee was used. In the example given the measuement ate was Hz and the time is counted fom the beginning of the session. In the beginning of the session the eceives have all the visible satellite stably in tack. It should be noted that if a success ate analysis was made fo the cuent case, the success ate would be vey high due to geat numbe of measuements ( Hz ate). In fact, in the cuent field tests the ambiguity solution conveged elatively quickly, but the solution was validated at 53 and 5 seconds, espectively. As pointed out ealie, the small numbe of signals (fequencies) makes the validation of the ambiguities challenging (Richet, 5). This was also confimed in the epoted field tests. The esults show that, when feasible, single-band caie phase -based positioning is capable of poducing cm-level baseline accuacy. On the othe hand, the esults also show that since with single-fequency measuements it is not possible to compensate fo atmosphee without an extenally supplied model, thee is a cm-level dift in the baseline coodinates. It is assumed that this is due to topospheic conditions, because the changes ae quite slow. Conside then the accuacy of the baseline, when the intege ambiguities ae not o cannot be fixed o validated. In such a case the float solution can be utilized as opposed to the fixed solution. Fig. shows data fom the 3-m baseline, which is the same case as in the uppe gaph in Fig. 9. Only the time span is shote. Fixed Baseline eo (m) Time (s) 4 x East Noth Up Time (s) Fig. 9. GPS L field test esults fo 3-m (up) and 36-m (below) baselines. Time is counted fom the beginning of the session. Validation of the solutions took 53 and 5 seconds, espectively. The uppe gaph in Fig. epesents the baseline obtained by diffeencing the standalone eceive positions. The eo is in the ode of seveal metes in all the baseline coodinates. As expected, the lagest eo occus in the up-diection (appoximately 5 metes). The lowe gaph shows the float solution. The float solution is always available (given that thee ae no cycle slips) and as shown, the eo in the float baseline is significantly smalle than in the baseline obtained by diffeencing the two positions. Afte 3 seconds fom the beginning of the session the eos in the float baseline coodinates ae aleady in the ode of cm. Hence, although ambiguity fixing is not nealy always possible in the singlefequency case, the float solution, which is eadily available, can impove accuacy significantly.

12 Wiola et al.: On the feasibility of adding caie phase assistance to cellula GNSS assistance standads 9 Position diffeence baseline eo (m) East Noth Up equiement fo the ange is that it must be geate than fou times the maximum incease (o decease) in ADR ove the maximum measuement inteval. The condition aises fom the need to identify the ADR oll-oves and as the condition is fulfilled, the eceiving end is capable of detecting the ADR oll-oves. Theefoe, the eceive is capable of econstucting the oiginal measuement by examining the two uppe bits of the pevious and cuent ADR measuements. Hence, the numbe of bits (b) equied fo epesenting the ADR measuement fulfilling the ange equiement can be given by Float Baseline eo (m) Time (s) East Noth Up Time (s) Fig.. GPS L esults fo the 3-m baseline. Up: Accuacy obtained using the diffeence of the eceive positions. Below: Accuacy of the float solution. Bandwidth equiements The data equied fo caie phase -based positioning include Time of measuements Refeence location fo the measuements Code phase measuements and uncetainties ADR measuements, uncetainties and continuities In the cuent 3GPP standad eleases thee ae fields fo tansfeing time of measuement, efeence location as well as code phase measuements fom the AGNSS assistance seve to the teminal. The missing fields ae ADR (Accumulated Delta Range, o Integated Dopple), ADR uncetainty and ADR continuity indication. ADR measuements diffe fom othe measuements in a espect that the ange equied fo the measuement depends upon the epoting inteval. This is because of the cumulative natue of the ADR measuement. The b 4 max t ADR( t) T < t (), ln( 4 max t ADR( t) T ) b = ln whee ADR(t) the time-vaying ADR measuement in metes and T the measuement inteval in seconds. Moeove, the esolution of the measuement must be (at least) mm esulting in a equiement to have additional bits ( - m < mm) fo the decimal pat. Now, if the incease (decease) ate of the ADR would depend solely on the movement of the satellite, one would have fo a static GPS-eceive on the suface of the Eath (Pakinson, 996) max ADR t m t ( ) < 93. () t s Galileo (3 km highe obit than GPS - slowe obital velocity) and QZSS (geostationay) have smalle Dopple fequencies than GPS. On the othe hand, GLONASS (~5 km lowe obit than GPS) has 3 m/s geate maximum Dopple than GPS. Hence, 97 m/s is taken as the maximum ate of incease (decease). Howeve, one must also conside ) the eceive movement and ) the eceive oscillato fequency eo. The eceive movement can be assumed to contibute at maximum 5 m/s. The eceive oscillato stability is assumed to be bette than ppm. Hence, the maximum (appaent) Dopple esulting fom this is ppm c < 6 m/s. Theefoe, the maximum absolute ADR ate of incease (decease) is set to ( ) m/s < 6 m/s. The bit consumption based on equation as a function of T taking the decimal pat ( bits) into account is summaized in table I. In addition to the ADR measuement, caie phase - based position also equies indication of the measuement continuity as well as on the quality (vaiance of the measuement). The ADR measuement continuity is defined by bit, which indicates, whethe the ADR measuement has been continuous between the cuent and the pevious measuement messages. One bit is sufficient, since the potocols used guaantee that packets aive in the coect ode and that no packets ae lost in the tansmission channel.

13 Jounal of Global Positioning Systems The measuement quality is coded accoding to the RTCM standad (RTCM, 998) using a thee-bit field and a table mapping the values to ADR measuement uncetainty. Note that it is also implicitly assumed that the ADR measuement has been coected fo the data bit polaity. Hence, thee is no need to tansfe the data bit polaity flag between the eceives. Moeove, although thee is a field fo code phase measuements, it has a esolution of appoximately 3 m. This is not sufficient fo caie phase -based positioning. Hence, additional bits ae equied to incease its esolution down to appoximately.3 m ( 3 - m). Theefoe, fom the bandwidth point of view ADR measuements add some load to the netwok, but the load can be optimized as shown. The study shows that the epoting inteval should be at maximum s, which esults in 7++3+=4 additional bits pe each signal. Consideing an exteme case of bands, GNSSs and 8 satellites pe GNSS (coesponding to 3 signals) the aveage bit ate is 3 4 b / s = 66 bps. Table I. Bits equied fo a single ADR measuement fo diffeent epoting intevals. T (s) bits model contibutes significantly on the feasibility of the single-band caie phase -based positioning. Challenges The specific challenges to be addessed befoe caie phase-based positioning can be added to the cellula standads include, amongst othes, the handoves fom one seving base station to the othe. The caie phase - measuement need to be continuous ove the hand-ove, which intoduces additional book-keeping execise to the netwok. Howeve, if a Vitual Refeence Station is used, the teminal can change the VRS without losing the baseline. This can be achieved by subscibing two VRS data steams to the teminal, solving the thee baselines (VRS-VRS and x VRS-teminal) and discading the old VRS once the baseline between the new VRS and the teminal has been established. While such an appoach is feasible in the use plane, it is difficult to implement in the contol plane of the cellula netwok. Anothe concen is the definition of the quality-ofsevice. The minimum pefomance equiements fo Assisted GPS (3GPP-TS-34.7) guide the design and implementation of the teminal. When intoducing caie phase -based positioning to the standads, it must be intoduced as a new positioning method and simila minimum pefomance equiements may be equied fo the new method. Such wok equies deep undestanding of the use cases as well as the full potential of the technology and extensive field testing. Thee is cuently no wok towads such pefomance equiements. About ionosphee modelling Caie phase -based positioning benefits significantly fom ionospheic modelling. Due to the dispesive natue of ionosphee, phase advance may be estimated, if thee ae measuements on moe than one fequency. Howeve, Richet (Richet et al., 5) epots that even in a multiband case it is still advantageous to have a-pioi estimate fo the advance fom an extenal souce. If thee is no a- pioi infomation available, the solution is potentially unstable. Moeove, Odik (Odik, ) epots that ionosphee modelling is essential fo long-baseline applications, even if using dual-band GPS measuements. The common element in the new AGNSS standad povides an oppotunity to povide the teminal with an ionospheic model (Syäinne et al., 6). Moeove, the achitectue shown in Fig. enables such a sevice by poviding an inteface to extenal sevices geneating such ionospheic pedictions. Such a souce is, fo instance, DLR (Deutsches Zentum fü Luft- ünd Raumfaht), which can povide space weathe foecasts (Jakowski et al., ). Poviding an accuate ionosphee 3 Conclusions The caie phase-based positioning has the potential to bing the positioning accuacy down to centimetes. Theefoe, it is tempting to conside adding the suppot fo caie phase-based positioning to the cellula standads. The analyses pesented in this pape show that the most significant poblem with single-fequency caie phasebased positioning is the uncetainty about its pefomance. The simulations show that duing a day thee ae bief peiods duing which the caie phasebased positioning is feasible, but at othe times the pefomance can be expected to be vey poo. The lack of measuements (satellites) is the most significant facto contibuting to the lack of pefomance. In conclusion, single-fequency caie phase-based positioning is not feasible, if thee is only one GNSS available and if ambiguities need to be fixed. Howeve, aleady the float solution, which is always available given that thee ae no undetected cycle slips, was shown to be a mao impovement ove taditional point positioning. It was

14 Wiola et al.: On the feasibility of adding caie phase assistance to cellula GNSS assistance standads also shown that the single-fequency case becomes vey inteesting with the intoduction of additional GNSSs (Galileo, GLONASS) to complement GPS. The study also shows that the full potential of Galileo lies in the use of the vaious available signals. If futue teminals ae capable of utilizing, fo instance, both GPS L + Galileo E as well as GPS L5 + Galileo E5a (since they ae in the same band, espectively) caie phase - based positioning is no doubt an attactive addition to the cuent set of positioning methods. Howeve, this equies that the teminals ae capable of multi-gnss multi-band eception and that the cellula standads/potocols suppot the peiodic epoting of ADR measuements fom the netwok to the teminal and/o vice vesa. It was also shown that the capability can be achieved with small additions to the cuent standads. The aveage additional data tansfe load was shown to be in the ode of 66 bps even when thee ae seveal GNSSs and signals available. The esulting accuacy is in the ode of centimetes in the best case and, hence, it is believed that the implementation task and additional netwok load is ustified. Refeences 3GPP-TS-3.7 Functional Stage Desciption of Location Sevices (LCS), 3GPP-TS-5.33 Radio Resouce Contol (RRC) potocol specification, 3GPP-TS-34.7 Teminal confomance specification; Assisted Global Positioning System (A-GPS), 3GPP-TS-44.3 Radio Resouce LCS (Location sevices) Potocol (RRLP), 3GPP (6) Meeting epot: Repot of TSG GERAN meeting#3, Sophia-Antipolis, Fance, 3th-7th Novembe, 3GPP (7) Meeting epot of the 36th 3GPP TSG RAN meeting, Busan, Koea, 9th May - 4th June, Alcatel, Eicsson, Nokia, Qualcomm, Siemens Netwoks, SiRF (6). GP-647 A-GNSS GERAN#3 status. Pesented in 3GPP GERAN#3, 3th-7th Octobe, Sophia Antipolis, Fance. Alanen K., Wiola L., Käppi J. and Syäinne J. (6a) Inetial Senso Enhanced Mobile RTK Solution Using Low-Cost Assisted GPS Receives and Intenet-Enabled Cellula Phones. In Poceedings of IEEE/ION PLANS 6, 5th-7th Apil, San Diego, CA, USA, pages Alanen K., Wiola L., Käppi J. and Syäinne J. (6b) Mobile RTK using low-cost GPS and Intenet-Enabled Wieless Phones. InsideGNSS, pages 3 39, May-June issue. Dow J.M. and Neilan R.E. (5) The Intenational GPS Sevice (IGS): Celebating the th Annivesay and Looking to the Next Decade. Advanced in Space Reseach, 36(3):3 36. Eissfelle B., Tibeius C., Pany T., Bibege R. Schuele T. and Heinichs G. (a) Instantaneous ambiguity esolution fo GPS/Galileo RTK positioning. Jounal fo Gyoscopy and Navigation, 38(3):7 9. Eissfelle B., Tibeius C., Pany T. and Heinichs G. (b) Real-Time Kinematic in the light of GPS Modenization and Galileo. Galileo s Wold, Autumn issue. Jakowski N., Heise S., Wehenpfennig A. and Schlüte S. () and R. Reime. GPS/GLONASS-based TEC measuements as a contibuto fo space weathe foecast. Jounal of atmospheic and sola-teestial physics, 64: Klimov V., Revnivykh S., Kossenko V., Dvokin V., Tyulyakov A. and Eltsova O. (5) Status and Development of GLONASS. In Poceedings of GNSS-5, 9th-nd July, Munich, Gemany. Lundgen D. and Diggelen F. (5) Long-Tem Obit Technology fo Cell Phones, PDAs. GPSWold, pages Octobe issue. Nokia (6) GP-65 Justification fo the addition of caie phase measuements. Discussion pape, pesented in 3GPP TGS-GERAN meeting#3, 6th-3th June, Lisbon, Potugal. Odik D. () Weighting Ionospheic Coections to Impove Fast GPS Positioning Ove Medium Distances. In Poceedings of Institute of Navigation GPS, 9th- nd Septembe, Salt Lake City, USA. OMA-ULP OMA-TS-ULP-V--579-C, Use Plane Location Potocol, OMA (7) Open Mobile Alliance Location Woking Goup meeting minutes OMA-LOC-7-9- MINUTES_Aug7Seoul, Seoul, Koea, th-4th August, Pakinson B. and Spilke J. (996) Global Positioning System: Theoy And Applications Volume I. Ameican Institute of Aeonautics and Astonautics, Inc. Washington DC, USA. Richet T. and El-Sheimy N. (5) Ionospheic modeling - The Key to GNSS Ambiguity Resolution. GPS Wold, pages 35 4, June issue. RTCM (998) Recommended Standads fo diffeential GNSS Sevice, vesion.. RTM Special Committee no 4. Januay 5th, Alexandia, Viginia, USA. Schüle T. (6) Intepolating Refeence Data - Kinematic Positioning Using Public GNSS Netwoks. InsideGNSS, pages 46 5, Octobe issue. Stategy Analysts (6) Global Handsets, GPS/A-GPS Phone Sales.

15 Jounal of Global Positioning Systems Syäinne J. and Wiola L. (6) Setting a New Standad - Assisting GNSS Receives That Use Wieless Netwoks. InsideGNSS, pages 6 3. Teunissen P. (997) On the GPS widelane and its decoelating popety. Jounal of Geodesy, 7: Teunissen P. and Tibeius C. () Bias Robustness of GPS Ambiguity Resolution. In Poceedings of Institute of Navigation GPS, 9th-nd Septembe, Salt Lake City, USA. Tibeius C. and Jonge P. (995) Fast Positioning Using the LAMBDA-Method. In Poceedings of the 4th Intenational Symposium on Diffeential Satellite Navigation Systems (DSNS), 4th-8th Apil, Begen, Noway, pages 8. Tibeius C., Pany T., and Eisfelle B. () Integal GPS- Galileo ambiguity esolution. In Poceedings of ENC- GNSS, May 7th-3th, Copenhagen, Denmak. Vehagen S. (6a) How will the new fequencies in GPS and Galileo affect caie phase ambiguity esolution?, InsideGNSS, pages 4 5, Mach issue. Vehagen S. (6b) Manual fo Matlab Use Inteface VISUAL. Delft Univesity of Technology, The Nethelands. Vehagen S., Teunissen PJG. and Odik D. (7) Caiephase Ambiguity Success-ates fo Integated GPS- Galileo Satellite Navigation. In Poceedings Of Joint wokshop WSANE7, 6th-8th Apil, Peth, Austalia. Wiola L., Alanen K., Käppi J. and Syäinne, J. (6) Binging RTK to Cellula Teminals Using a Low-Cost Single-Fequency AGPS Receive and Inetial Sensos. In Poceedings of IEEE/ION PLANS 6, 5th-7th Apil, San Diego, CA, USA, pages Wiola L. and Syäinne, J. (7a) Binging All GNSS into Line. GPS Wold, 8(9):4 47. Wiola L. and Syäinne, J. (7b) Binging the GNSSs on the Same Line in the GNSS Assistance Standads. In Poceedings of the 63d ION Annual Meeting7, 3d- 5th Apil, Boston, MA, USA, pages 4 5.

16 Jounal of Global Positioning Systems (7) Vol.6, No.: 3- Pecise Point Positioning Using Combined GPS and GLONASS Obsevations Changsheng Cai, Yang Gao Depatment of Geomatics Engineeing, Univesity of Calgay, AB, Canada Abstact. Pecise Point Positioning (PPP) is cuently based on the pocessing of only GPS obsevations. Its positioning accuacy, availability and eliability ae vey dependent on the numbe of visible satellites, which is often insufficient in the envionments such as uban canyons, mountain and open-pit mines aeas. Even in the open aea whee sufficient GPS satellites ae available, the accuacy and eliability could still be affected by poo satellite geomety. One possible way to incease the satellite signal availability and positioning eliability is to integate GPS and GLONASS obsevations. Since the Intenational GLONASS Expeiment (IGEX-98) and the follow-on GLONASS Sevice Pilot Poect (IGLOS), the GLONASS pecise obit and clock data have become available. A combined GPS and GLONASS PPP could theefoe be implemented using GPS and GLONASS pecise obits and clock data. In this eseach, the positioning model of PPP using both GPS and GLONASS obsevations is descibed. The pefomance of the combined GPS and GLONASS PPP is assessed using the IGS tacking netwok obsevation data and the cuently available pecise GLONASS obit and clock data. The positioning accuacy and convegence time ae compaed between GPS-only and combined GPS/GLONASS pocessing. The esults have indicated an impovement on the position convegence time but coelates to the satellite geomety impovement. The esults also indicate an impovement on the positioning accuacy by integating GLONASS obsevations. Keywods. GPS, GLONASS, Pecise Point Positioning, Pecise obit and Clock Intoduction Cuent Pecise Point Positioning (PPP) system developed is based on only GPS obsevations. The accuacy, availability and eliability of PPP positioning esults howeve ae quite dependent on the numbe of visible satellites. Unde envionments of uban canyons, mountains and open-pit mines, fo instance, the numbe of visible GPS satellites is often insufficient fo position detemination (Tsuii, ). Futhe, even in the open aea whee sufficient GPS satellites ae available, the PPP accuacy and eliability may still insufficient due to poo satellite geomety. One possible way to incease the availability of satellites as well as the eliability of the positioning esults is to integate GPS and GLONASS obsevations. The benefit fom such integation is obvious paticulaly fo applications in uban canyons, mountain and open-pit mining envionments. Since the Intenational GLONASS Expeiment (IGEX- 98) and the follow-on GLONASS Sevice Pilot Poect (IGLOS) conducted in 998 and espectively (Webe, 5), the pecise GLONASS obit and clock data have become available ove times. Cuently, fou oganizations can povide independent GLONASS pecise obits consistent at -5 cm level but only two centes povide post-mission GLONASS clock data (Oleynik, 6). This povides oppotunities to use GLONASS obsevations to impove pecise point positioning accuacy and eliability cuently based on only GPS obsevations. Although GLONASS achieved its Full Opeational Capability (FOC) in Januay 996 when 4 GLONASS satellites wee available fo positioning and timing, its constellation had dopped to seveal satellites by the yea of due to a decease in GLONASS budget (Zinoviev, 5). As of Nov. 9, 7, thee ae 8 GLONASS satellites in obit but only 9 of them ae opeational satellites. Howeve, the Russian govenment has appoved a long-tem plan to econstitute a GLONASS constellation of 4 satellites. 8 satellites ae expected to be opeational by 8, and a full opeational capability with 4 satellites will be achieved

17 4 Jounal of Global Positioning Systems by -. By that time, the numbe of GLONASS satellites will not be a poblem any moe. In this pape, we will investigate the integation of GPS and GLONASS obsevations fo impoved accuacy and eliability of positioning esults using PPP. The quality and chaacteistics of cuently available pecise GLONASS obit and clock poducts ae fist descibed. The positioning model fo combined pocessing of GPS and GLONASS obsevations is then pesented. IGS tacking netwok obsevation data and available pecise GLONASS obit and clock data ae used to assess the pefomance of combined GPS and GLONASS pecise point positioning. Compaisons ae also conducted on the numeical esults between GPS only and combined GPS/GLONASS pocessing. GLONASS Pecise Obit and Clock Poducts GLONASS has been on the way to its modenization. In 3, the fist GLONASS-M satellite was launched, whee M stands fo Modified. On Decembe 5, 6, thee GLONASS-M satellites (GLONASS 75, GLONASS 76 and GLONASS 77) wee launched. All the thee satellites ae placed on obit II. The GLONASS- M is a modenized vesion of the GLONASS spacecaft which suppots a numbe of new featues, such as the satellite design-lifetime inceased to 7 yeas, a second civil modulation on L signal, and impoved clock stability. The thid geneation GLONASS satellite GLONASS-K is expected to launch in 8. The satellite sevice life is futhe inceased to - yeas and a thid civil signal fequency and Synthetic Apetue Rada function will be added (Segey, 7). The GLONASS-K epesents a adical change in GLONASS spacecaft design, adopting a non pessued and modula spacecaft bus design (Kaplan, 6). The Intenational GLONASS Expeiment (IGEX-98) is the fist global GLONASS obsevation and analysis campaign fo geodetic and geodynamics applications, conducted fom Octobe 9, 998 to Apil 9, 999 and oganized ointly by the Intenational GNSS Sevice (IGS), the Intenational Association of Geodesy (IAG) and the Intenational Eath Rotation Sevice (IERS). The main obectives of the expeiment wee to collect a globally-distibuted GLONASS dataset by using dualfequency GLONASS eceives and detemine the pecise GLONASS satellite obit. IGEX-98 has a global netwok consisting of 5 stations with 9 dual-fequency and 3 single-fequency eceives. Fo the IGEX-98 campaign, an infastuctue compaable to that of the IGS was established (Habich, 999). IGEX-98 includes the poduction of pecise obits fo all the opeational GLONASS satellites (Webe, 5). The Intenational GLONASS Sevice Pilot Poect (IGLOS) is a follow-on poect of IGEX-98 that began in with the mao pupose to integate the GLONASS satellite system into the opeation of IGS. The IGLOS Pilot Poect has a global netwok consisting of about 5 tacking stations with dual-fequency GLONASS eceives. The GLONASS data ae collected continuously and achived in RINEX fomat at the IGS Global Data Centes (Webe, 5). The GPS and GLONASS obsevations ae pocessed simultaneously and theefoe the pecise obit poducts fo both systems ae given in one unique efeence fame (Webe, ). Cuently fou IGS analysis centes ae outinely poviding GLONASS pecise obit poducts. They ae CODE (Univesity Bene, Switzeland), IAC (Infomation - Analytical Cente), ESA/ESOC (Euopean Space Opeations Cente, Gemany) and BKG (Bundesamt fü Katogaphie und Geodäsie, Gemany). CODE can geneate the final GLONASS obit as well as the apid and pedicted apid obit poducts (Webe, 5; Schae, 4). The CODE obits ae expessed in the IGb efeence fame, which is the IGS ealization of the ITRF (Buyninx, 7). IAC is a depatment at MCC (Russian Mission Contol Cente) which is outinely monitoing the GLONASS pefomance. Stating fom 4, IAC stated to conduct outine obit and clock detemination based on IGS tacking netwok data. Since 5 IAC has become one of the fou IGS analysis centes who ae outinely poviding GLONASS post-mission obit and clock data including (Oleynik,6): a) the final obit and clock data with a delay of 5 days; b) the apid obit and clock data with a delay of day. ESA/ESOC began to pocess and analyze GNSS data fo pecise obit detemination in 99, fist using its GPSOBS/BAHN softwae to compute the pecise GPS obits and clock paametes and then aligning its GLONASS solution to the ITRF efeence fame using the GPS obits and tight constains on the coodinates of 7 obseving stations (Romeo, 4). BKG has pocessed and analyzed the combined GPS/GLONASS obsevations fom a netwok of global tacking stations since the beginning of the IGEX-98. Simila to ESA/ESOC, BKG fist computes GPS obits, clock estimation and eath oientation paametes and then utilizes the Benese softwae to poduce pecise GLONASS obits and station coodinates on a daily basis using double-diffeenced phase obsevations (Habich, 4). It povides GLONASS pecise obits, eceivespecific estimates of the system time diffeence between GPS and GLONASS, and the station coodinates (SINEX files).

18 Cai et al.: Pecise Point Positioning Using Combined GPS and GLONASS Obsevations 5 The independent GLONASS obits fom the above fou oganizations ae consistent at the -5cm level and have been combined to geneate the IGS final GLONASS obits using a pocedue simila to IGS final GPS obit (Webe, 5). As to pecise satellite clock data, cuently only two data analysis centes, namely IAC and ESA/ESOC, povide post-mission GLONASS clock data. The diect compaison of pecise colock data fom diffeent centes howeve can hadly be conducted due to diffeent efeence time scales used and diffeent inte-fequency biases applied to the GLONASS code measuements. The ageement between the IAC and ESOC post-mission GLONASS clock values is consideed at the level of.5ns (Oleynik, 6). 3 Combined GPS/GLONASS Data Pocessing fo PPP In the following, the positioning model fo a combined GPS and GLONASS PPP system is descibed along with mathematical equations. Based on a dual-fequency GPS/GLONASS eceive, the pseudoange and caie phase obsevables on L and L between a eceive and a satellite can be descibed by the following obsevation equations: g i P Φ g i i P Φ i = ρ + cdt + d g = ρ g mult / Pi g g i + λ N + d g i g + cdt = ρ + cdt mult / Pi = ρ + cdt i + λ N i + ε g + d cdt g Pi g cdt g mult / Φi cdt + ε + d Pi cdt mult / Φi g + d + d + ε + d + d + ε g ob g Φi ob ob Φi g ob + d + d + d g top + d top top g top + d + d d g ion / Pi d g ion / Φi ion / Pi ion / Φi whee the supescipt g and efe a GPS and a GLONASS satellite espectively, and P i is the measued pseudoange on L i (m); Φ i is the measued caie phase on L i (m); ρ is the tue geometic ange (m); c is the speed of light (m/s); dt is the eceive clock eo (s); dt is the satellite clock eo (s); d is the satellite obit eo (m); ob d is the topospheic delay (m); top d ion / L i is the ionospheic delay on i L (m); () () (3) (4) λ i is the wavelength on L i (m/cycle); N i is the intege phase ambiguity on L i (cycle); d mult / P i is the multipath effect in the measued pseudoange on L i (m); d mult ε / Φ is the multipath effect in the measued caie i phase on L i (m); is the measuement noise (m); A system time diffeence unknown paamete should be intoduced fo mixed GPS/GLONASS obsevation pocessing (Habich, 999). A eceive clock eo can be descibed as dt = t (5) t sys whee t sys denotes eithe GPS system time t GPS fo GPS obsevations o GLONASS system time t GLONASS fo GLONASS obsevations. Since the eceive clock eo is elated to a system time, the combined GPS and GLONASS pocessing includes two eceive clock offset unknown paametes, one fo the eceive clock offset with espect to the GPS time and one fo the eceive clock offset with espect to the GLONASS time. We can also descibe the GLONASS eceive clock offset as follows: dt = t t = t t = dt g GLONASS GPS + dt + t sys GPS t GLONASS which is a function of the GPS eceive clock offset and a system time diffeence between GPS and GLONASS. Applying equation (6) into equations (3) and (4) esults in the following pseudoange and caie phase obsevation equations: i P Φ i = ρ + cdt + d mult / Pi = ρ + cdt i + λ N i g + cdt + ε g + d Pi + cdt sys sys mult / Φi cdt Φi cdt + ε + d + d ob ob + d + d top top + d d ion / Pi ion / Φi Befoe GPS and GLONASS obsevations ae used fo position detemination, the GPS and GLONASS pecise obit and clock data should be fist applied to coect satellite obit eos and satellite clock offsets. The ionospheic efaction bias can be eliminated by constucting an ionosphee-fee combination of phase o pseudoange obsevable fom the L and L fequencies. Afte the application of pecise obit and clock (7) (8) (6)

19 6 Jounal of Global Positioning Systems coections, the ionosphee-fee code and phase combinations can be witten as follows: P Φ P Φ g IF g IF IF IF g = ( f P f P ) /( f f ) = ρ + cdt + d + ε g = ( f = ρ g g g g g g g g g top g PIF g g f g Φ ) /( f g f g g g g g + d top + N IF + ε Φ IF f P ) /( f f g + cdt sys + d top P Φ + cdt IF ) (9) () = ( f P ) () = ρ + cdt + ε = ( f Φ g = ρ + cdt whee f + cdt Φ sys ) /( + d f top f + N IF ) + ε Φ IF () P IF is the ionosphee-fee code combination (m); Φ IF is the ionosphee-fee phase combination (m); f i is the fequency of L i (Hz); N IF is the combined ambiguity tem (m); ε IF contains measuement noise, multipath as well as othe esidual eos. The unknown paametes of the positioning model based on the above obsevation equations include thee position coodinates, a eceive clock offset, a system time diffeence, a zenith wet topospheic delay, and ambiguity paametes equal to the numbe of obseved GPS and GLONASS satellites. The dy topospheic delay eo is fist coected using the Hopfield topospheic model and the emained zenith wet topospheic delay (ZWD) including the esidual dy delay is to be estimated as an unknown paamete. The Niell Mapping Functions have been used fo hydostatic and wet mapping functions. The positions, clock offset, system time diffeence and ZWD ae modeled as a andom walk pocess while the ambiguity paametes ae modeled as constants and ae to be estimated using a Kalman filte. The basic pocedue of PPP pocessing based on combined GPS and GLONASS obsevations is demonstated in Fig.. 4 Numeical Results and Analysis To assess the obtainable positioning accuacy based on combined GPS and GLONASS obsevations, numeical computations ae conducted and the obtained esults ae pesented in this section. At fist, the PPP pocessing esults including the positioning eo, the eceive clock offset, the zenith wet topospheic delay and the system time diffeence ae given. Then compaisons ae conducted to assess the positioning accuacy and the convegence time. Slow position convegence time is cuently an obstacle fo PPP applications and the additional obsevations fom GLONASS ae expected to educe the equied convegence time. GPS/GLONASS RINEX file Fig. PPP pocessing fo combined GPS and GLONASS Data Desciptions GPS/GLONASS obsevation datasets, collected on Apil 6 th, 7 at the IGS station HERT, GOPE and YARR, wee downloaded fom the IGS website. Each station was installed with an ASHTECH Z8 dual-fequency GPS/GLONASS eceive. Data sampling ate was 3s. The mixed GPS/GLONASS pecise satellite obit and 5- minute clock data geneated by IAC data analysis cente wee downloaded fom the IAC website. A total of GLONASS satellites wee opeational on that day. Positioning Results PPP pocessing 3-D coodinates Receive clock offset System time diffeence Zenith wet topospheic delay Pecise obit and clock data Twelve hous of obsevations acquied at the station HERT fom EPN (EUREF Pemanent Netwok) Local Analysis Centes wee fist used. The elevation mask was set to degees. The GPS only and the GPS/GLONASS obsevations wee pocessed espectively. Fig. shows the position eos ove the peiod. It can be clealy obseved that the positioning eos fo GPS only and combined GPS/GLONASS pocessing ae at a quite simila level. Table shows the mean, RMS, and standad deviation (one-sigma) of the conveged position eos based on the esults fom local time 3: to :. The diffeences in the mean, RMS and STD values fo all thee coodinate components ae less than.5 cm.

20 Cai et al.: Pecise Point Positioning Using Combined GPS and GLONASS Obsevations 7 East Eo (m) Noth Eo (m) Up Eo (m) - - GPS Only GPS/GLONASS - : : 4: 6: 8: : : GPS Time (HH:MM) MEAN STD RMS Fig. GPS only vs. GPS/GLONASS positioning eos Tab. Statistics of Position Results (m) GPS Only GPS / GLONASS East Noth..6 Up.. East.6.4 Noth.6.6 Up.. East Noth.4.7 Up.4. In addition to position detemination, PPP can also output eceive clock offset solution which has the potential to suppot pecise time tansfe applications. The estimated eceive clock offsets in both GPS only and GPS/GLONASS pocessing ae pesented in Fig. 3. The ed cuve stands fo the esults fom GPS only pocessing, which is completely ovelapped by the geen cuve fom the GPS/GLONASS pocessing esults. Since the clock offset diffeence, which has a RMS value of. ns, is vey small, the addition of GLONASS obsevations did not have a significant impact on the estimation of the eceive clock. Pesented in Fig. 4 is the estimated zenith wet topospheic delay. As can be seen, the ZWD diffeence between the GPS only pocessing and the combined GPS/GLONASS pocessing afte the position convegence is not significant with a RMS value of mm. The estimated system time diffeence is pesented in Fig. 5. The system time diffeence vaies in a ange of about 4 ns ove the twelve hous, which patially eflects the accuacy of the GLONASS system time scale. The geate vaiation befoe the GPS time : is due to the position convegence pocess. The obtained system time diffeence fom the combined GPS/GLONASS data pocessing in PPP includes not only the time diffeence between the GPS and GLONASS system times but also the eceive hadwae delay diffeences between GPS and GLONASS. Since they can not be sepaated fom each othe, the obtained value is the combined system time diffeence and eceive s inte-system hadwae delay. As a esult, the estimated system time diffeence should be consideed as only an appoximation to the tue system time diffeence and is quite dependent on the eceive used. Receive clock offset (ns) GPS Only GPS/GLONASS : : 4: 6: 8: : : GPS Time (HH:MM) Fig. 3 GPS only vs. GPS/GLONASS eceive clock offset estimates Zenith wet topospheic delay (m).3.. GPS Only GPS/GLONASS : : 4: 6: 8: : : GPS Time (HH:MM) Fig. 4 GPS only vs. GPS/GLONASS zenith wet topospheic delay estimates

21 8 Jounal of Global Positioning Systems System time diffeence (ns) : : 4: 6: 8: : : GPS Time (HH:MM) Fig. 5 Estimated system time diffeences significantly the position convegence time fo hoizontal coodinate components. East (m) Noth (m) Up (m) SVs PDOP GPS only GPS/GLO : :3 : :3 : :3 3: GPS Time (HH:MM) Fig. 6 Pocessing esults at HERT (Session ) Positioning Accuacy and Convegence Analysis Fou pocessing sessions, each with thee-hou data fom thee IGS stations, namely HERT, GOPE and YARR, wee included in the data analysis. The elevation masks wee set to degees. Fo each session, in addition to the position eos, the PDOP value and the numbe of used satellites wee also calculated. The computation of the PDOP values in GPS/GLONASS pocessing is based on the design matix coesponding to the unknowns of the thee position coodinates, the eceive clock offset and the system time diffeence. This design matix has one moe column compaed to the design matix used fo PDOP computation in the GPS only pocessing. The pocessing esults ae pesented in Figs Fig. 6 shows the positioning esults between : and 3: at HERT station. No significant PDOP impovement is found befoe the position solutions convege and as a esult, no significant convegence impovement is found. Pesented in Fig. 7 ae the pocessing esults fom the GPS time 3: to 6:. In this session, two GLONASS satellites wee utilized on aveage. Although in the beginning the PDOP value has only a slight impovement by adding GLONASS obsevations, the convegence time has been educed significantly in the east and up diections. In Fig.e 8, although PDOP has a significant impovement fom the local time 6:4 to 7:, no significant convegence impovement is found. This is because such a geomety impovement with moe visible satellites was pesent afte the position solutions have aleady conveged. Looking at the esults in Fig. 9, the PDOP impovement occued at the fist half an hou and duing the convegence pocess. As a esult, it has educed East (m) Noth (m) Up (m) SVs PDOP East (m) Noth (m) Up (m) SVs PDOP GPS only GPS/GLO 3: 3:3 4: 4:3 5: 5:3 6: GPS Time (HH:MM) Fig. 7 Pocessing esults at HERT (Session ) GPS only GPS/GLO 6: 6:3 7: 7:3 8: 8:3 9: GPS Time (HH:MM) Fig. 8 Pocessing esults at HERT (Session 3)

22 Cai et al.: Pecise Point Positioning Using Combined GPS and GLONASS Obsevations 9 East (m) Noth (m) Up (m) SVs PDOP GPS only GPS/GLO 9: 9:3 : :3 : :3 : GPS Time (HH:MM) Fig. 9 Pocessing esults at HERT (Session 4) Table shows the RMS statistics of the positioning eos at HERT station using the position esults obtained fom the last one and a half hous of obsevations fom each session. A significant accuacy impovement is found in Session whee the impovements in the east and up components each 4cm and 3cm espectively. Tab. RMS Statistics of Positioning Results at HERT (m) Session Session Session 3 Session 4 GPS Only GPS / GLONASS East..93 Noth.3.34 Up.8.9 East.9.87 Noth.9.8 Up.6.9 East Noth.4. Up.83.9 East Noth.. Up.3.3 Figs. -3 show the pocessing esults at GOPE station. No convegence impovement is found in Fig. while a slight impovement in the east component can be seen fom Fig.. Look at Fig., the convegence in the combined PPP pocessing appeas moe stable and smooth between 7: and 7:4 when compaed to the GPS-only pocessing esults. Fig. 3 indicates a slight impovement in the beginning of the convegence pocess. East (m) Noth (m) Up (m) SVs PDOP East (m) Noth (m) Up (m) SVs PDOP East (m) Noth (m) Up (m) SVs PDOP GPS only GPS/GLO : :3 : :3 : :3 3: GPS Time (HH:MM) Fig. Pocessing esults at GOPE (Session ) GPS only GPS/GLO 3: 3:3 4: 4:3 5: 5:3 6: GPS Time (HH:MM) Fig. Pocessing esults at GOPE (Session ) GPS only GPS/GLO 6: 6:3 7: 7:3 8: 8:3 9: GPS Time (HH:MM) Fig. Pocessing esults at GOPE (Session 3)

23 Jounal of Global Positioning Systems East (m) Noth (m) Up (m) SVs PDOP GPS only GPS/GLO 9: 9:3 : :3 : :3 : GPS Time (HH:MM) Fig. 3 Pocessing esults at GOPE (Session 4) Pesented in Table 3 is the RMS statistics of positioning esults at GOPE station. The maximum accuacy impovement is 3cm which can be seen in the east component of Session 3 while the accuacy degadation of cm is also found in the up component of Session. Tab. 3 RMS Statistics of Positioning Results at GOPE (m) Session Session Session 3 Session 4 GPS Only GPS / GLONASS East.8.8 Noth..8 Up.3.5 East.47.8 Noth.9.7 Up.44.3 East Noth.8.4 Up East.45.4 Noth..8 Up The pocessing esults at YARR station ae pesented in Figs A significant convegence impovement has been found in the east diection in Fig. 4 whee obsevations fom an aveage of fou GLONASS satellites ae utilized in the combined pocessing duing the peiod of : to :3. No convegence impovement is found in the othe figues by adding the GLONASS obsevations due to limited numbe of visible GLONASS satellites. This indicates a coelation between position convegence impovement and satellite geomety impovement. Table 4 shows the RMS statistics esults of the poisoning eos at YARR station. The maximum accuacy impovement of 3cm occus in the east diection of Session. East (m) Noth (m) Up (m) SVs PDOP East (m) Noth (m) Up (m) SVs PDOP East (m) Noth (m) Up (m) SVs PDOP GPS only GPS/GLO : :3 : :3 : :3 3: GPS Time (HH:MM) Fig. 4 Pocessing esults at YARR (Session ) GPS only GPS/GLO 3: 3:3 4: 4:3 5: 5:3 6: GPS Time (HH:MM) Fig. 5 Pocessing esults at YARR (Session ) GPS only GPS/GLO 6: 6:3 7: 7:3 8: 8:3 9: GPS Time (HH:MM) Fig. 6 Pocessing esults at YARR (Session 3)

24 Cai et al.: Pecise Point Positioning Using Combined GPS and GLONASS Obsevations East (m) Noth (m) Up (m) SVs PDOP GPS only GPS/GLO 9: 9:3 : :3 : :3 : GPS Time (HH:MM) Fig. 7 Pocessing esults at YARR (Session 4) Tab. 4 RMS Statistics of Positioning Results at YARR (m) Session Session Session 3 Session 4 GPS Only GPS / GLONASS East.9.74 Noth..9 Up..86 East Noth.6.6 Up.78.8 East..5 Noth.. Up..75 East.7.8 Noth.5.5 Up.47.5 In ode to compae the positioning accuacy between using GPS-only obsevations and combined GPS/GLONASS obsevations, the positioning accuacy deived fom thee-dimensional coodinate component eos is pesented in Fig. 8. As can be seen, the impovement of the positioning accuacy is obvious fo most of the position esults, and the maximum impovement eaches cm. Eo (m).3.. HERT GOPE YARR Fig. 8 Positioning accuacy compaison GPS only GPS/GLONASS 5 CONCLUSIONS A positioning model based on combined GPS and GLONASS obsevations has been poposed in this pape fo pecise point positioning. In ode to assess the positioning accuacy and convegence time impovement of the combined GPS and GLONASS data pocessing, a -hou and fou 3-hou sessions of datasets have been used in the data analysis. Compaisons have been conducted between GPS only and combined GPS/GLONASS pocessing. Based on the esults, cuent GLONASS constellation has not caused a significant impact on the positioning esults including position coodinates, eceive clock offset and zenith wet topospheic delay since only two o thee GLONASS satellites wee obseved most of time at any specific time. Moe significant impovements ae expected when with moe GLONASS satellites available in space. The eseach esults futhe indicate that even with limited numbe of GLONASS satellites the impovement of the position convegence time is dependent on the impovement level of the satellite geomety fo position detemination. The esults also indicate that the positioning accuacy can be impoved by additional GLONASS obsevations in most cases. Futhe investigation will be conducted to assess the combined GPS/GLONASS pecise point positioning in a kinematic mode. ACKNOWLEDGMENTS The financial suppots fom NSERC and GEOIDE ae geatly appeciated. The contibution of data fom the Intenational GNSS Sevice (IGS) and Infomation- Analytical Cente (IAC) is also appeciated. Based on a pape pesented at The Institute of Navigation Intenational Technical Meeting, Fot Woth, Texas, Septembe 7. REFERENCES Buyninx, C. (7). Compaing GPS-only with GPS+GLONASS positioning in a Regional Pemanent GNSS Netwok. GPS Solution, :97-6, 7. Habich, H. (999). Geodetic Applications of the Global Navigation Satellite System (GLONASS) and of GLONASS/GPS Combinations. PhD Thesis, Univesity of Bene. Habich, H., P. Neumaie, K. Fisch (4). GLONASS Data Analysis fo IGS. Poceedings of IGS Wokshop and Symposium, Univesity of Bene, 4. Kaplan, E.D., C.J. Hegaty (6). Undestanding GPS: Pinciples and Applications. nd Edition. Atech House.

25 Jounal of Global Positioning Systems Oleynik, E.G., V.V. Mitikas, S.G. Revnivykh, A.I. Sedukov, E.N.Dutov, V.F.Shiiaev (6). High-Accuate GLONASS Obit and Clock Detemination fo the Assessment of System Pefomance. Poceedings of ION GNSS 6, Fot Woth, TX, Septembe 6-9, 6. Romeo, I., J.M.Dow, R. Zandbegen, J.Feltens, C.Gacia, H.Boomkamp, J.Peez (4), The ESA/ESOC IGS Analysis Cente Repot, IGS - Technical Repot, 53-58, IGS Cental Bueau, JPL-Publication, 4. Schae, S.T., U. Hugentoble, R. Dach, M. Meindl, H. Bock, C.Uschl, G. Beutle (4). GNSS Analysis at CODE. Poceedings of IGS Wokshop and Symposium. Univesity of Bene. Segey, K., R. Segey, T. Suiya (7). GLONASS as a Key Element of the Russian Positioning Sevice. Advances in Space Reseach, 39: Tsuii, T., M. Haigae, T. Inagaki, T. Kanai (). Flight Tests of GPS/GLONASS Pecise Positioning vesus Dual Fequency KGPS Pofile. Eath Planets Space, 5: Webe, R., E. Fagne (). The Quality of Pecise GLONASS Ephemeides. Adv. Space Res. 3(), 7-79,. Webe, R., J.A. Slate, E. Fagne, V. Glotov, H. Habich, I.Romeo, S. Schae (5). Pecise GLONASS Obit Detemination within the IGS/IGLOS Pilot Poect. Advances in Space Reseach, 36: Zinoviev, A.E (5).Using GLONASS in Combined GNSS Receives: Cuent Status. Poceedings of ION GNSS 5, Long Beach, CA, Septembe 3-6, 5.

26 ÂÓÙÖÒ Ð Ó ÐÓ Ð ÈÓ Ø ÓÒ Ò ËÝ Ø Ñ ¾¼¼ µ ÎÓк ÆÓº ½ ¾ ¹ Ö ÒØ Ð ÈË Ø Ö Ù ¹ Ö Ò ÔÔÖÓ Ò Ö Ä ÒÒ ÒØÖ Æ Ø ÓÒ Ð Ð Ê Ö Ë ÒØ ÕÙ Ä¾Ë ËÙÔ Ð ÖÙ ÂÓÐ Óع ÙÖ ½½ ¾ ¹ ÙÖ¹ Ú ØØ Ü Ö Ò µ ØÖ Øº ÁÒ Ø ØÖ Ø ÓÒ Ð ÔÔÖÓ ØÓ Ö ÒØ Ð ÆËË Ø Ø ÐÐ Ø ÖÖÓÖ Ø ÖÑ Ö Ð Ñ Ò Ø Ý ÓÖѹ Ò Ø Ó¹ ÐÐ Ò Ð Ö Ò Ë µº ÇÒ Ø Ò Ø Ö Ó Ø Ö Ú Ö ÖÖÓÖ Ø ÖÑ Ý ÓÑÔÙØ Ò ÓÖ Ö Ú Ö ØÓ ÓÒ Ö Ø ÓÖÖ ÔÓÒ Ò ÓÙ Ð ¹ Ö Ò µ Ø Ö Ô Ò ØÛ Ò Ø Ò Ð Ö¹ Ò Ë µ Ò ÓÒ Ó Ø Ñ Ø Ò Ö Ö Ò º ÌÓ Ò¹ Ð Ø Ë ³ Ò ÓÑÓ Ò ÓÙ Ñ ÒÒ Ö ÓÒ Ñ Ý ÕÙ ÐÐÝ Û ÐÐ ÓÒ Ö Ø Ö Ô Ò ØÛ Ò Ø Ë ³ Ò Ø Ö Ñ Ò Ú ÐÙ º ÁÒ Ø Ô Ô Ö Ø ÒØÖ Ð Þ Ö ÒØ Ð Ø ³ Ö Ö ÖÖ ØÓ Ö Ù Ö Ò ³ Ê µº ÁÒ Ø Û Ö Ø ÆËË Ú ÒÐÙ ÓÒÐÝ ØÛÓ Ö Ú Ö Ø ÔÔÖÓ ÓÑÔÐ Ø ÐÝ ÕÙ Ú Ð ÒØ ØÓ ÓÙ Ð ÒØÖ Ð¹ Þ Ø ÓÒº³ ÅÓÖ ÔÖ ÐÝ Ø Ò ÓÖÑ Ø ÓÒ ÓÒØ Ò Ò Ø ÓÙ Ð ÒØÖ Ð Þ Ó ÖÚ Ø ÓÒ ³ Ø Ò ÑÔÐ ÒØ ¹ ÝÑÑ ØÖ ØÖ Ò Ö ÔØ ÓÒ Ó Ø Ø ÓÒØ Ò Ò Ø Ö Ù Ö Ò º Ì Ñ Ù Ø Ö Ø Ò Ö Ø ÓÒ Ð ÒÙÑ Ö Û Ö Ö Ð Ø ØÓ Ø ØÖ Ø ÓÒ Ð ÒØ Ö Ñ Ù Ø Ò Ú ÖÝ ÑÔÐ Ñ ÒÒ Öº Ì ÔÖÓÔ ÖØ Ø Ð Ò Ø Ô Ô Ö Ò Û Ð Ø ÓÒ Ø ÓÖÖ ÔÓÒ Ò Ò ÐÝ º Ì ÜØ Ò ÓÒ ØÓ ÆËË Ò ØÛÓÖ Û Ø Ñ Ò Ø Û ÐÐ ÔÖ ÒØ Ò ÓÖØ ÓÑ Ò Ô Ô Öºµ Ì ÓÖÖ ÔÓÒ Ò ÔÔÐ Ø ÓÒ ÓÒ ÖÒ Ø ÒØ Ø ÓÒ Ó ÓÙØÐ Ö Ò Ö Ð Ø Ñ º ÝÐ Ð Ô ÓÑ Ò Û Ø Ñ ÐÐ Ò ÓÙ Ë Ò Ø Ù ÐÝ ÒØ º Ã Ý ÛÓÖ º ÆËË ÈË ÒØÖ Ð Þ ÙÒ Ö ÒØ Ð Ñ Ø Ó ÊÌú Ø Ñ Ð Ø ÓÒ Á º ½ ÁÒØÖÓ ÙØ ÓÒ Ì ÐÓ Ð ÔÓ Ø ÓÒ Ò Ø Ò ÕÙ Ö ÓÒ Ø Óй ÐÓÛ Ò Ó ÖÚ Ø ÓÒ Ð ÕÙ Ø ÓÒ º ÓÖ Ö ÕÙ ÒÝ f ν ÓÖ Ö Ú Ö¹ Ø ÐÐ Ø Ô Ö (, s) Ò Ø ÔÓ t Ø Ó Ò ÖÖ Ö¹Ô Ø Ö Ö Ô Ø Ú ÐÝ Ó Ø ÓÖÑ º º Ë Øº ½ Ò ËØÖ Ò Ò ÓÖÖ ½ µ p ν,t (, s) = ρ t (, s) + c[dt ν,t () dt ν,t (s)] + ǫ ν,t (, s) ½µ φ ν,t (, s) = ρ t (, s) + c[δt ν,t () δt ν,t (s)] ¾µ + λ ν [ϕ ν, () ϕ ν, (s)] + λ ν N ν (, s) + ε ν,t (, s) ÁÒ Ø ÕÙ Ø ÓÒ Û Ö ÜÔÖ Ò Ð Ò Ø ÙÒ Ø ρ t (, s) Ø Ö Ú Ö¹ Ø ÐÐ Ø Ö Ò Ø Ø Ò ¹ ØÛ Ò Ø ÐÐ Ø s Ø Ø Ø Ñ t τ Û Ö Ø Ò Ð Ñ Ø¹ Ø µ Ò Ö Ú Ö Ø Ø Ø Ñ t Ó Ø Ö ÔØ ÓÒµº Ð ÖÐÝ Ø λ ν ³ ÒÓØ Ø Û Ú Ð Ò Ø Ó Ø ÖÖ Ö Û Ú Ø Ö Ø ÓÒ Ð ÒØ Ö N ν (, s) Ö Ø ÒØ Ö ÖÖ Ö¹Ô Ñ¹ Ù Ø º Ì Ò ØÖÙÑ ÒØ Ð Ð Ý Ò ÐÓ ÖÖÓÖ Ø Ø ÓÖ Ú Ò (ν, t) Ô Ò ÓÒÐÝ ÓÒ Ò s Ö ÐÙÑÔ ØÓ¹ Ø Ö Ò Ø Ö Ú Ö Ò Ø ÐÐ Ø ÖÖÓÖ Ø ÖÑ dt ν,t () dt ν,t (s) ÓÖ Ø Ó Ò δt ν,t () δt ν,t (s) ÓÖ Ø Ô c Ø Ô Ó Ð Øµ ϕ ν, () Ò ϕ ν, (s) Ö Ø Ò Ø Ð Ô ÜÔÖ Ò ÝÐ µ Ò Ö Ú Ö Ò Ø ÐÐ Ø s Ö Ô Ø Ú Ðݺ À Ö ÓÖ Ð Ö ØÝ Ø ÓÒÓ Ô Ö Ò ØÖÓÔÓ¹ Ô Ö Ð Ý Ö ÒÓÖ º Ø Ø ÒØÖÓ ÙØÓÖÝ Ð Ú Ð Û Ø Ù ÓÒ Ö Ø Ø Ø Ø Ú Ò ÓÖÖ Ø ÓÖ Ø Ð Ý º Ð ÖÐÝ Ø Ó Ò Ô ÖÖÓÖ ǫ ν,t (i, ) Ò ε ν,t (i, ) ÒÐÙ ÓØ ÒÓ Ò Ö Ù Ð ÑÓ Ð ÖÖÓÖ º ÓÖ ÓÙÖ ÔÖ ÒØ ÔÙÖÔÓ Û ÒÓÛ ÓÒ ÒØÖ Ø ÓÒ ÕÙ ¹ Ø ÓÒ ¾µ Ò Ø Ò Ð ¹ Ö ÕÙ ÒÝ ÑÓ φ t (, s) = ρ t (, s) + c[δt t () δt t (s)] + λ[ϕ () ϕ (s)] + λn(, s) + ε t (, s) ÁÒ Û Ø ÓÐÐÓÛ ÒÓØ Ø ÓÒ Ù a := b Ñ Ò a ÕÙ Ð ØÓ b Ý Ò Ø ÓÒº³ Ä Ø ÒÓÛ Ø Ö Ö Ò Ö ¹ Ú Ö Ò Ø Ø Ó Ø Ù Öº ÒÓØ Ý s, s,...,s n Ø Ø ÐÐ Ø ÒÚÓÐÚ Ò Ø ÈË Ú º ÕÙ ÒØ ØÝ Ù ϑ := ϑ(, s ) ϑ(, s ) µ µ Ø Ò Ö ÖÖ ØÓ Ò Ð Ö Ò Ë µ Ò ϑº Ý Ù ¹ Ò Ø ÒÓØ Ø ÓÒ ÕÙ Ø ÓÒ µ Ø Ò Ý Ð φ t = ρ t + c[δt t ( ) δt t ( )] + λ[ϕ ( ) ϕ ( )] + λa + ε t Û Ö a := N µ ÇÒ Ø Ù Ø Ö Ó Ø Ø ÐÐ Ø ÖÖÓÖ Ø ÖÑ º Ì a ³ Ö Ø ÒØ Ö Ñ Ù Ø Ó Ø Ë Ô Ø º µ

27 ¾ ÂÓÙÖÒ Ð Ó ÐÓ Ð ÈÓ Ø ÓÒ Ò ËÝ Ø Ñ ½º½ ½º½º½ ÒÓØ ÓÒ ÓÙ Ð Ö Ò ÁÒ Ø ØÖ Ø ÓÒ Ð ÔÔÖÓ ØÓ Ö ÒØ Ð ÆËË ÓÒ Ö Ø Ð Ø Ö Ö Ò Ø ÐÐ Ø º À Ö Ø Ø ÐÐ Ø ÒÓØ Ý s k º ÕÙ ÒØ ØÝ Ù ϑ k := ϑ ϑ k (i ) µ Ø Ò Ö ÖÖ ØÓ ÓÙ Ð Ö Ò µ Ò ϑ º ½µº Ý Ù ØÖ Ø Ò ÖÓÑ Õº µ Ø ÜÔÖ ÓÒ ÓÖ = k Ø ÖÑ Ý Ø Öѵ ÓÒ Ø Ò Ó Ø Ò Ø Ö Ð Ø ÓÒ φ t;k = ρ t;k + λa k + ε t;k (a k Z) µ Ì ϑ ³ Ò Ø Ö ÓÖ Ö ÖÖ ØÓ Ö Ù Ö¹ Ò ³ Ê µº ËÙ ØÖ Ø Ò ÖÓÑ Õº µ Ø ÜÔÖ ÓÒ Ò Ø ÖÑ Ó Ñ Ò Ú ÐÙ Ø ÖÑ Ý Ø Öѵ Û Ø Ò Ó Ø Ò Ø Ö Ð Ø ÓÒ Ñ Ð Ö ØÓ Õº µ φ t; = ρ t; + λa + ε t; (a Q) ½¾µ ÆÓØ Ø Ø Ø Ê Ñ Ù Ø a ³ Ö Ö Ø ÓÒ Ð ÒÙÑ Ö Ò ÒÓØ Ò Ò Ö Ð Ö Ø ÓÒ Ð ÒØ Ö µº ½º½º Ö ÒØ Ð Ó ÖÚ Ø ÓÒ Ý ÓÒ ØÖÙØ ÓÒ Ø ³ Ó Ø ÙÒØ ÓÒ ÇÒ Ø Ù Ø Ö Ó Ø Ö Ú Ö ÖÖÓÖ Ø ÖÑ º Ì a k ³ Ö ØÓ Ø ÒØ Ö Ñ Ù Ø Ó Ø ÔÖÓ Ð Ñº ϑ d ( i, s ) := { i = ÓÖ = k; ϑ k ÓØ ÖÛ º ½ µ ½º½º¾ ϑ k ϑ k ϑ º ½ ÆÓØ ÓÒ Ó ÓÙ Ð Ö Ò º Ì ÓÙ Ð Ö Ò ϑ k Ø Ú ÐÙ Ó Ø Ò¹ Ð Ö Ò ϑ Ý Ø Ò ÓÖ Ò ÓÖ Ö Ö Ò µ Ø Ú ÐÙ Ó Ø Ò Ð Ö¹ Ò ϑ k º Ê Ù Ö Ò ÁÒ Ø ÔÔÖÓ ÔÖ ÒØ Ò Ø Ô Ô Ö Û ÓÒ Ö ÓÑÓ Ò ÓÙ Û Ý Ó Ð Ñ Ò Ø Ò Ø Ö Ú Ö ÖÖÓÖ Ø ÖÑ º Ì ØÓ ÓÒ Ö Ø ÕÙ ÒØ Ø º ¾µ ϑ := ϑ ϑ Û Ö ϑ Ø Ñ Ò Ú ÐÙ Ó Ø ϑ ϑ := n ϑ n = ϑ ϑ ϑ º ¾ ÆÓØ ÓÒ Ó Ö Ù Ö Ò º Ì Ö Ù Ö Ò ϑ Ø Ú ÐÙ Ó Ø Ò Ð Ö Ò ϑ Ý Ø Ò ÓÖ Ò ÓÖ Ö Ö Ò µ Ø Ñ Ò Ú ÐÙ ϑ Ó Ø Ò Ð Ö Ò ÓÑÔ Ö Û Ø º ½µº µ ½¼µ Ð ÖÐÝ Ø ÖÝ ÒØÖ Ú ÐÙ Ò Ö Ö Ú Ö¹ ØÙ Ð Ë Ó Ø Û Ø Ú ÖØÙ Ð Ö Ö Ò Ø ÐÐ Ø s º ÓÖ Ò ØÓ Û Ðй ÒÓÛÒ ÖÝ ÒØÖ ÔÖÓÔ ÖØÝ ÓÖ ÙÖ¹ Ø Ö Ø Ð Ë Øº µ ÓÖ ÒÝ k Û Ú n ϑ ϑ k ½½µ = k Ö Ø ³ Ó Ø ÙÒØ ÓÒ ϑ( i, s )º ËÙ ÙÒØ ÓÒ Ò Ø Ö ÓÖ Ö ÖÖ ØÓ Ö ÒØ Ð Ó ÖÚ Ø ÓÒ Ð³ ǵ ÙÒØ ÓÒº ½º½º Ê Ù Ó ÖÚ Ø ÓÒ Ý ÓÒ ØÖÙØ ÓÒ Ø Ë ³ Ó Ø ÙÒØ ÓÒ { i = ; ϑ ( i, s ) := ÓØ ÖÛ º ϑ ½ µ Ö Ø Ö Ù Ö Ò Ó Ø ÙÒØ ÓÒ ϑ( i, s )º ËÙ ÙÒØ ÓÒ Ò Ø Ö ÓÖ Ö ÖÖ ØÓ Ö Ù Ó ¹ ÖÚ Ø ÓÒ Ð³ Êǵ ÙÒØ ÓÒº ½º½º ÒØÖ Ð Þ Ó ÖÚ Ø ÓÒ ÁÒ Ø ÒØÖ Ð Þ Ó ÖÚ Ø ÓÒ Ð ÔÔÖÓ ³ Ó Ë Ò À Ò ½ ¾µ ÓÒ Ø Ö Ó Ø Ø ÐÐ Ø ÖÖÓÖ Ø ÖÑ Ý ÓÖÑ Ò Ø Ò Ð ÒØÖ Ð Þ Ó ÖÚ Ø ÓÒ ϑ () c ( i, s ) = ϑ( i, s ) ϑ( i, s ) i= = ( ) i [ϑ(, s ) ϑ(, s )] = ( ) i ϑ Ì Ö Ú Ö ÖÖÓÖ Ø ÖÑ Ö Ø Ò Ð Ñ Ò Ø Ý ÓÖÑ Ò Ø ÓÙ Ð ÒØÖ Ð Þ Ó ÖÚ Ø ÓÒ ϑ () c ( i, s ) = ϑ () c ( i, s ) n = ( ) i = ( ) i ϑ ( ϑ n n = ϑ () c ( i, s ) n ϑ ) =

28 Ä ÒÒ Ö ÒØ Ð ÈË Ì Ö Ù Ö Ò ÔÔÖÓ ¾ ÇÒ Ø Ò Ð ØÓ Ý Ø Ø ¾º½º½ ÆÙ Ò Ð Ý Ô ϑ c ( i, s ) := ( ) i ϑ ½ µ ÒØÖ Ð Þ Ó ÖÚ Ø ÓÒ Ð ÙÒØ ÓÒº ÁÒ Ø Û Ö Ø ÆËË Ú ÒÐÙ ÓÒÐÝ ØÛÓ Ö Ú Ö Ø Ò ÓÖÑ ¹ Ø ÓÒ ÓÒØ Ò Ò Ø ÒØÖ Ð Þ Ó ÖÚ Ø ÓÒ Ø Ö ÓÖ ÑÔÐ ÒØ ÝÑÑ ØÖ ØÖ Ò Ö ÔØ ÓÒ³ Ó Ø Ø ÓÒØ Ò Ò Ø Ö Ù Ö Ò º Ì Ø ØÐ Ó Ø Ô Ô Ö Û Ó Ò ÓÖ Ò Ðݺ ½º¾ ÓÒØ ÒØ Ì Ø ÓÖ Ø Ð Ö Ñ ÛÓÖ Ó Ø ÓÒØÖ ÙØ ÓÒ ÔÖ ¹ ÒØ Ò Ë Øº ¾º Ð Ö Ò Ë Øº Ø Ò Ê ÔÔÖÓ ÔÖÓÚ ØÓ ÕÙ Ú Ð Òغ ÁÒ Ô ÖØ ÙÐ Ö Ð¹ Ø ÓÙ Ø Ê Ñ Ù Ø Ö Ö Ø ÓÒ Ð ÒÙÑ Ö Ø Ñ¹ Ù ØÝ ÔÖÓ Ð Ñ ØÓ ÓÐÚ Ö Ø Ñ º Ë Ø ÓÒ º¾ ÚÓØ ØÓ Ø ÔÓ Òغ Ì Ê ÔÔÖÓ ÓÙÐ ÓÛ¹ Ú Ö ÔÖ ÖÖ º ÁÒ ÓÛÒ Ò Ë Øº Ø Ö ¹ Ú Ð ÒØ Ö Ø Ò ÔÖÓÔ ÖØ Û Ú Ô Ö Ò Ø ÒØÓ Ø ÔÖÓ Ð Ñº Ì ÔÖÓÔ ÖØ Û Ö Ñ Ò Ø ÔÔÖÓ Ò Û Ð Ø ÓÒ Ø ÒØÖ Ð¹ Þ ÙÒ Ö ÒØ Ð Ñ Ø Ó Ó Ë Ò À Ò ½ ¾µº Ì Ý Ð Ó ÓÑÔÐ Ø Ø Ù Ð Ð Ö ÔÔÖÓ Ó Ä ÒÒ Ò ÙÖ Ò ¾¼¼ µº Ö ÙÐØ Ø ÕÙ Ú Ð ÒØ ÔÔÖÓ Ò Ò Ø ÖÓÑ ÓØ Öº ÓÛÒ Ò Ë Øº ÓÒ Ó Ø ÔÖÓÔ ÖØ ÔÐ Ý Ý ÖÓÐ Ò Ø Á ÔÖÓ ÙÖ Ó Ø Ø Ñ Ð Ø ÓÒ ÔÖÓ ÔÖ ÒØ Ò Ë Øº º Ì Ë ÑÓÒ Û Ø ÝÐ Ð Ô Òݵ Ò Ø Ò ÒØ Ò Ö Ð Ø Ñ º Ê Ð Ø ÓÑÑ ÒØ Ö ØÓ ÓÙÒ Ò Ë Øº º ÔÓ ÒØ ÓÙØ Ò Ø Ø Ø ÓÒ Ø Ò ÐÝ ÔÖ ÒØ Ò Ø Ô Ô Ö Ò Ö Ö Ò ÒØÖÓ ÙØ ÓÒ ØÓ Ø Ó ÆËË Ò ØÛÓÖ Û Ø Ñ Ò Ø Ä ÒÒ ¾¼¼ µº ¾ Ì ÓÖ Ø Ð Ö Ñ ÛÓÖ ÁÒ Ø ÓÒØ ÜØ Ò Ò Ë Øº ½ Ø ÒÓØ ÓÒ Ó Ó ÖÚ ¹ Ø ÓÒ Ð Ô Ò Ô ÓÐÐÓÛ º ¾º½ Ç ÖÚ Ø ÓÒ Ð Ô ÁÒ Û Ø ÓÐÐÓÛ Ø Ñ Ý ÓÒÚ Ò ÒØ ØÓ ÓÒ Ö Ø Ø ÙÒØ ÓÒ Ù ϑ(, s) Ø Ø Ú ÐÙ ÓÒ Ö Ø Ò ÙÐ Ö Ö º Ï Ò Ø ÆËË Ú ÒÐÙ ØÛÓ Ö Ú Ö Ò n Ø ÐÐ Ø Ø Ö ÒÐÙ ØÛÓ Ð Ò Ò n ÓÐÙÑÒ ϑ Ø Ò Ö Ö Ú ØÓÖ Ó Ø Ó ÖÚ Ø ÓÒ Ð Ô E := R n º Ð ÖÐÝ Ø Ú ÐÙ Ö Ø ÓÑÔÓÒ ÒØ Ó ϑ Ò Ø Ø Ò Ö Ó Eº Ì ÒÓØ Ø ÓÒ E ψ Ô Ø Ò ØÙÖ Ó Ø Ú ØÓÖ ϑ Ó E ψ = p ÓÖ Ø Ó ψ = φ ÓÖ Ø Ô º Ì Ú Ö Ò ¹ÓÚ Ö Ò Ñ ØÖ Ü Ó Ø ÓÖÖ ÔÓÒ Ò Ø Ú ØÓÖ ÒÓØ Ý V ψ V ψ Ø ÓÔ Ö ØÓÖ ÓÒ E Ò Ù Ý V ψ º ÇÒ Ø Ò Ð ØÓ Ò Ø Ó ÖÚ Ø ÓÒ Ð Ø Ô ³ Ó ØÝÔ ψ Ø Ô E + ψ Û Ø ÒÒ Ö ÔÖÓ ÙØ ϑ ϑ E + ψ := (ϑ V ψ ϑ ) E ½ µ Ð ÖÐÝ E + E + ψ Ö Ð À Ð ÖØ Ô º ÁÒ Û Ø ÓÐÐÓÛ Ø Ô E Ó ÙÒØ ÓÒ ϑ( i, s ) Ó Ø ÓÖÑ ϕ(s ) ϕ( i ) Ö ÖÖ ØÓ Ø ÒÙ Ò Ð Ý Ô Õ º ½µ ¾µ Ò º µº ÁÒ Ø Ô Ð ÙÒ Ö ÓÒ Ö Ø ÓÒ Û Ø ØÛÓ Ö Ú Ö µ Ø Ù Ô Ó E Ó Ñ Ò ÓÒ n + º ¾º½º¾ Ð Ò Ó ÖÚ Ø ÓÒ Ð Ô Ì ÓÖØ Ó ÓÒ Ð ÓÑÔÐ Ñ ÒØ Ó E Ò E ÒÓØ Ý E c Ö ÖÖ ØÓ Ø Ð Ò Ó ÖÚ Ø ÓÒ Ð Çµ Ô º³ Ì ÓÖØ Ó ÓÒ Ð ÓÑÔÐ Ñ ÒØ Ó E Ò E + E + c Ø Ò Ö ÖÖ ØÓ Ø Ç Ø Ô ³ º µº ÁÒ Ø Ô Ð ÙÒ Ö ÓÒ Ö Ø ÓÒ E c Ò E + c Ö Ó ¹ Ñ Ò ÓÒ n (n + ) = n º ÓÛÒ ÐÓÛ E c Ø Ò Ø Ô Ó ÒØÖ Ð Þ Ó ÖÚ Ø ÓÒ Ð ÙÒØ ÓÒ ³ ϑ c Ò Ý Õº ½ µº ÈÖÓÓ º ÁÒ Ø ÙÐ Ò Ô E ϑ c ÓÖØ Ó ÓÒ Ð ØÓ ÒÝ ÒÙ Ò ÙÒØ ÓÒ Ó E º ÁÒ n n [ϕ(s ) ϕ( i )]( ) i ϑ = ϑ ϕ(s ) ( ) i i= = = i= i= n ( ) i ϕ( i ) = Û Ø i= ( )i = Ò n = ϑ = º Ì ÔÖÓÔ ÖØÝ Ø Ò ÓÐÐÓÛ ÖÓÑ Ø Ø Ø Ø Ø ÙÒØ ÓÒ ϑ c ÓÖÑ Ô Ó Ñ Ò ÓÒ n º Ê Ñ Ö ¾º½º¾º Ä Ø P c Ø ÓÖØ Ó ÓÒ Ð ÔÖÓ Ø ÓÒ Ó E ÓÒØÓ E c º À Ö Ø ÒØÖ Ð Þ Ó ÖÚ Ø ÓÒ Ð ÙÒ¹ Ø ÓÒ ϑ c Ò Ý Õº ½ µ Ø ÔÖÓ Ø ÓÒ Ó ϑ ÓÒ E c ϑ c := P c ϑº ÁÒ ÓØ Ö Ø ÖÑ Ø Ð Ò Ò ÓÔ Ö ØÓÖ³ P c Ø Ò Ö Ù ØÓ Ø ÓÙ Ð ¹ ÒØÖ Ð Þ Ø ÓÒ ÓÔ Ö ØÓÖº³ Ì Ó ÒÓØ ÓÐ ÓÖ ÆËË Ò ØÛÓÖ Û Ø Ñ Ò Ø º Ì Ø Ö¹ Ñ ÒÓÐÓ Ý Û Ó Ò ÓÖ Ò Ðݺ Ô Ò Ò ÓÒ Ø ÓÒ¹ Ø ÜØ Ò Ù Ö ÔØ Ø Ò ÓÖ Ð Ò³ ÓÖ ÒØÖ Ð Þ º³ ¾º¾ Ë Ô ÒÓØ Ò Ý b := {e } n = Ø Ø Ò Ö Ó R n Ð Ø Ù ÓÒ Ö Ø Ú ØÓÖ ϑ := n = ϑ e Ò Û Ø ϑ ³ Ö Ø Ò Ð Ö Ò Ò Ò Õº µº Ð ÖÐÝ Ù Ú ØÓÖ Ò Ö Ö Ë Ú ØÓÖº ÁÒ Ø ÓÒØ ÜØ Û Ý Ø Ø F := R n Ø Ë Ô ³ º µº ¾º¾º½ Ë ÓÔ Ö ØÓÖ Ì Ë ÓÔ Ö ØÓÖ Ø ÓÔ Ö ØÓÖ ÖÓÑ E + ÒØÓ F Ò Ý Ø Ö Ð Ø ÓÒ Õº µµ Sϑ := ϑ º º (Sϑ) := ϑ ½ µ Ï ÒÓÛ ÒÓØ Ý S Ø ÓÖÖ ÔÓÒ Ò ÔÖÓ Ø ÓÒ³ ÓÔ Ö ØÓÖ º º Ø ÓÔ Ö ØÓÖ ÖÓÑ F ÒØÓ E (S ϑ)( i, s ) := ( ) i ϑ ½ µ ϑ

29 ¾ ÂÓÙÖÒ Ð Ó ÐÓ Ð ÈÓ Ø ÓÒ Ò ËÝ Ø Ñ ÓÖ ÒÝ ϑ F Ø ÙÒØ ÓÒ ϑ := S ϑ/ Ù Ø Ø Sϑ = ϑ S Ø Ö ÓÖ ÙÖ Ø Ú º ÁÒ Û Ø ÓÐÐÓÛ V ψ Ø Ú Ö Ò ¹ÓÚ Ö Ò Ñ ØÖ Ü Ó Ø Ë Ø Ú ØÓÖ ψ := Sψº ÒÓØ Ò Ý V ψ Ø ÓÔ Ö¹ ØÓÖ ÓÒ F Ò Ù Ý V ψ Û Ú V ψ = SV ψ S ½ µ Ï ÒÓÛ ÓÛ Ø Ø Ø Ó ÒØ Ó S Ú Ò Ý Ø Ö Ð Ø ÓÒ S = V ψ S Ò ÖÓÑ Õº ½ µ SS = V ψ ¾¼µ ¾½µ ÈÖÓÓ º Ý Ò Ø ÓÒ S Ø ÓÔ Ö ØÓÖ ÖÓÑ F ÒØÓ E + Ù Ø Ø ÓÖ ÒÝ ϑ E + Ò ÒÝ ϑ F Û Ú (Sϑ ϑ) F = ϑ S ϑ E + º Ð ÖÐÝ (Sϑ ϑ) F = n [ ] ( ) i ϑ ( i, s ) = i= ÖÓÑ Õº ½ µ Û Ø Ö ÓÖ Ú (Sϑ ϑ) F = i= = ϑ n ϑ ( i, s )(S ϑ)( i, s ) º º (Sϑ ϑ) F = (ϑ S ϑ) E = ( ϑ [V ψ V ψ ]S ϑ ) E º Ö ÙÐØ (Sϑ ϑ) F = ϑ V ψ S ϑ E + Ò S = V ψ S º ¾º¾º¾ Ê Ô Ò Ê Ñ Ù ØÝ Ð ØØ Ä Ø Ù ÒÓØ Ý F Ø Ô Ó Ú ØÓÖ ϑ F Û Ó ÓÑÔÓÒ ÒØ ϑ Ö ÒØ Ðº Ì ÓÖØ Ó ÓÒ Ð ÓÑÔÐ Ñ ÒØ Ó F ÒØÓ F Ø Ô º µ F := { ϑ F : n = ϑ = } F ÓÒ ¹ Ñ Ò ÓÒ Ð Ô F Ó Ñ Ò ÓÒ n º Ä Ø Q Ò Q Ø ÓÖØ Ó ÓÒ Ð ÔÖÓ Ø ÓÒ Ó F ÓÒØÓ F Ò F Ö Ô Ø Ú Ðݺ Ð ÖÐÝ Ø ÓÔ Ö ØÓÖ Ö ÜÔÐ ¹ ØÐÝ Ò Ý Ø Ö Ð Ø ÓÒ (Q ϑ) = ϑ (Q ϑ) = ϑ ϑ ¾¾µ Û Ö ϑ Ø Ñ Ò Ú ÐÙ Ó Ø ϑ ³ º Ï Ø Ö Ö ØÓ Ø Ê ÔÔÖÓ Û Ö Ø Ò Ð ØÓ Ø Õ º µ ½¼µ ¾¾µ Ò º µ ϑ := Q ϑ ¾ µ Ð ÖÐÝ Ø ÓÑÔÓÒ ÒØ Ó ϑ Ò b Ö Ø n Ö ¹ Ù Ö Ò ϑ F Ò Ø Ö ÓÖ Ö ÖÖ ØÓ Ø Ê Ô º³ ÆÓØ Ø Ø ϑ c Ö Ð Ø ØÓ ϑ Ý Ø Ö Ð Ø ÓÒ ϑ c = S ϑ / Õ º ½ µ Ò ½ µµº ϑ Ø ÔÖÓ Ø ÓÒ Ó ϑ ÓÒ E Û Ú n n n ϑ = ϑ ϑ = inf ϑ ϑ o ϑ o R = = = Ì ÔÖÓÔ ÖØÝ ÜÔÖ Ò Õº ½½µ Ö ÙÐØ ÖÓÑ Ø Ö ¹ Ð Ø ÓÒº Ì ÔÖÓ Ø ÓÒ Ó Z n ÓÒØÓ F Ð ØØ Ó Ö Ò n Ø Ê Ñ Ù ØÝ Ð ØØ ³ L º µº ÁÒ b Û ÒÓØ Ó L µ Ø ÓÑÔÓÒ ÒØ Ó ÔÓ ÒØ a Ó L Ö Ö Ø ÓÒ Ð ÒÙÑ Ö Ø n Ö Ø ÓÒ Ð Ñ Ù Ø a º Ê Ñ Ö ¾º¾º¾º Ì ÊÇ ÙÒØ ÓÒ ϑ Ò Ý Õº ½ µ ÓÖÑ Ù Ô Ó E ÒÓØ Ý E º µº Ð ÖÐÝ Ø ÊÇ Ô ³ ÑÔÐ Ò ÖØ ÓÒ Ó F Ò Eº ¾º¾º Ô Ò Ñ Ù ØÝ Ð ØØ ÁÒ Ø ÔÔÖÓ k Ò Ü ÓÒ Ð ØÓ ÓÒ Ö Ø Ù Ô Ó F º µ F d := {ϑ F : ϑ k = } Ý ÓÒ ØÖÙØ ÓÒ F d ÓÑÓÖÔ ØÓ R n º Ä Ø Q d ÒÓÛ Ø Ó Ð ÕÙ ÔÖÓ Ø ÓÒ Ó F ÓÒØÓ F d ÐÓÒ F º ÆÓØ Ø Ø Q d ÜÔÐ ØÐÝ Ò Ý Ø Ö Ð Ø ÓÒ (Q d ϑ) = ϑ ϑ k ¾ µ Ï Ö Ø Ò Ð ØÓ Ø Õº µ Ò º µ ϑ d := Q d ϑ ¾ µ Ä Ø b d := {e } k Ø Ø Ò Ö Ó F d º Ø ÓÑÔÓÒ ÒØ Ó ϑ d Ò b d Ö Ø n ÓÙ Ð Ö¹ Ò ϑ k F d Ò Ö Ö Ô º³ Ì ÒØ Ö Ø ÓÒ Ó Z n Û Ø F d Ð ØØ Ó Ö Ò n Ø Ñ Ù ØÝ Ð ØØ ³ L d º µº ÁÒ b d Ø ÓÑÔÓÒ ÒØ Ó ÔÓ ÒØ a d Ó L d Ö Ö Ø ÓÒ Ð ÒØ Ö Ø n ÒØ Ö Ñ Ù Ø a k kµº Ð ÖÐÝ L d = Q d L d Ò L d Q d Z n Ò L d Ù Ø Ó Z n µº ÙÖØ ÖÑÓÖ Q d Z n L d º Ï Ø Ö ÓÖ Ú L d = Q d Z n º L := Q Z n Ò Q = Q Q d Ø ÓÐÐÓÛ Ø Ø L = Q L d º µº Ê Ñ Ö ¾º¾º º Ì Ç ÙÒØ ÓÒ ϑ d Ò Ý Õº ½ µ ÓÖÑ Ù Ô Ó E ÒÓØ Ý E d º µº Ð ÖÐÝ Ø Ç Ô ³ Ò Ò ÖØ ÓÒ Ó F d Ò Eº ¾º Ê Ò ÓÔ Ö ØÓÖ Ì Ê ÓÔ Ö ØÓÖ Ø ÓÔ Ö ØÓÖ ÖÓÑ E + ÒØÓ F Ò Ý Ø Ö Ð Ø ÓÒ S := Q S ¾ µ ÆÓØ Ø Ø S ÙÖ Ø Ú º Ì Ö ÙÑ ÒØ Ø Ñ Ø Ø Ù ÓÖ S ºµ ÜÔ Ø Ø ÒÙÐÐ Ô Ó S ÒÓØ Ý kes µ Ø ÒÙ Ò Ð Ý Ô E º Ì ÔÖÓÔ ÖØÝ Ò ÜÔÐ ØÐÝ Ø Ð ÓÐÐÓÛ º

30 Ä ÒÒ Ö ÒØ Ð ÈË Ì Ö Ù Ö Ò ÔÔÖÓ ¾ ÈÖÓÓ º Ð ÖÐÝ E kes Û Ø dime = n + ÙØ dim(kes ) = dime dimf = n (n ) = n + Ò Ø ÔÖÓÔ ÖØݺ Ì ÓÔ Ö ØÓÖ Ø ÓÔ Ö ØÓÖ ÖÓÑ E + ÒØÓ F d Ò Ý Ø Ö Ð Ø ÓÒ S d := Q d S Ä S S d ÙÖ Ø Ú Ò kes d = E º ¾ µ ÕÙ Ú Ð Ò Ó Ø Ò Ê ÔÔÖÓ Ì Ô F d Ò F Ö ÓÑÓÖÔ º ÅÓÖ ÔÖ ÐÝ Ø Ö ØÖ Ø ÓÒ Ó Q ØÓ F d Ø ÓÔ Ö ØÓÖ ÖÓÑ F d ÒØÓ F Ò Ý Ø Ö Ð Ø ÓÒ Rϑ d := Q ϑ d ¾ µ Ñ Ô F d ÓÒØÓ F Ò L d ÓÒØÓ L º µº ÁØ ÒÚ Ö Ø ÓÔ Ö ØÓÖ ÖÓÑ F ÒØÓ F d Dϑ := Q d ϑ ¾ µ ÆÓØ Ø Ø Ø Ø ÓÒ Ó DR ÓÖÖ ÔÓÒ ØÓ Ø Ù Ú Ò Ó ÓÖ Ò ϑ k ϑ ϑ k º ½ Ò ¾µ (DRϑ d ) = (ϑ k ϑ ) (ϑ k k ϑ ) = ϑ k Ì Ú ØÓÖ e := Re kµ ÓÖÑ Ó F Û Ð Ó Ó L Ø b d := Rb d º ÁÒ Ø ¹ Ø ÓÑÔÓÒ ÒØ Ó Ú ØÓÖ ϑ Ó F Ö Ø ÓÑÔÓ¹ Ò ÒØ ϑ k Ó ϑ d = Dϑ º ÁÒ ϑ = Rϑ d = R k ϑ k e = k ϑ k Re = k ϑ k e ÁÒ Ô ÖØ ÙÐ Ö Ò Ø Û ÒÓØ ÓÖØ Ó ÓÒ Ðµ Ø ÓÑÔÓÒ ÒØ Ó ÔÓ ÒØ a Ó L Ö Ø n ÒØ Ö Ñ¹ Ù Ø a k Ó a d = Da º Ï Ö ÐÐ Ø Ø Ò Ø Ø Ò Ö Ó F Û ÒÓØ Ó L µ Ø ÓÑÔÓÒ ÒØ Ó a Ö Ø n Ö Ø ÓÒ Ð Ñ Ù Ø a º Ä Ø T ÒÓÛ Ø ÓÖØ Ó ÓÒ Ð ÔÖÓ Ø ÓÒ Ó F ÓÒØÓ F d Ö ØÖ Ø ØÓ F º µº ÓÖ ÒÝ ϑ Ò F Ò ÒÝ ϑ Ò F d Û Ú (ϑ ϑ) F = (ϑ Rϑ) F = (Tϑ ϑ) F º Ì ÓÛ Ø Ø T Ø Ó ÒØ Ó R ÓÒ F R = T º ÜÔÐ ØÐÝ (R ϑ ) = ϑ ( k); (R ϑ ) k = ¼µ D Ø ÒÚ Ö Ó R D Ø ÒÚ Ö Ó R (D ϑ d ) = ϑ k ( k); (D ϑ d ) k = k ϑ k ½µ Ä Ø V ψd ÒÓÛ Ø Ú Ö Ò ¹ÓÚ Ö Ò Ñ ØÖ Ü ÜÔÖ Ò b d µ Ó Ø Ø ψ d º Ä Û Ð Ø V ψ Ø L d L F := R n ϑ Ë ϑ,..., ϑ n F d D R T F a ϑ Ê ϑ,..., ϑ n e k F ϑd ϑ k,..., ϑ n k a d º ÓÑ ØÖ Ð Ö ÔÖ ÒØ Ø ÓÒ Ó Ø Ñ Ò Ð Ñ ÒØ Ò¹ ÚÓÐÚ Ò Ø ÕÙ Ú Ð Ò Ó Ø Ò Ê ÔÔÖÓ º À Ö e k Ø Ú ØÓÖ Ó Ø Ø Ò Ö Ó R n Ó ¹ Ø Û Ø Ø Ö Ö Ò Ø ÐÐ Ø º ÆÓØ Ø Ø ϑ k k = Ò P n = ϑ = º ½ Ò ¾ Ö Ô Ø Ú Ðݵ R Ø ÓÖ¹ Ø Ó ÓÒ Ð ÔÖÓ Ø ÓÒ Ó F ÓÒØÓ F Ö ØÖ Ø ØÓ F d R Ø Ò ÓÖ Ö ÙØ ÓÒº³ ÁØ ÒÚ Ö D Ø Ó Ð ÕÙ ÔÖÓ Ø ÓÒ Ó F ÓÒØÓ F d ÐÓÒ F µ Ö ØÖ Ø ØÓ F D Ø Ò ÓÖ Ö Ò º³ Ì Ó ÒØ Ó R Ø ÓÖØ Ó ÓÒ Ð ÔÖÓ Ø ÓÒ Ó F ÓÒØÓ F d Ö ØÖ Ø ØÓ F R = T º ÁØ ÒÚ Ö Ø Ó ÒØ Ó D D º ÙÖØ Ö Ø Ð Ò Ô ÖØ ÙÐ Ö Ø Ó ÓÒ¹ ÖÒ Ò Ð ØØ L d Ò L µ Ö ØÓ ÓÙÒ Ò Ë Ø º ¾ Ò º Ú Ö Ò ¹ÓÚ Ö Ò Ñ ØÖ Ü ÜÔÖ Ò bµ Ó Ø Ê Ø ψ º ÁÒ Û Ø ÓÐÐÓÛ V ψd Ø ÓÔ Ö ØÓÖ ÓÒ F d Ò Ù Ý V ψd º Ä Û V ψ Ø ÓÔ Ö ØÓÖ ÓÒ F Ò¹ Ù Ý V ψ º Ä Ø Q ÒÓÛ Ø Ñ ØÖ Ü Ó Q ÜÔÖ Ò bº V ψ = Q V ψ Q T = Q V ψ Q Ø ÓÔ Ö ¹ ØÓÖ V ψ Ø ÓÔ Ö ØÓÖ ÓÒ F ÜÔÐ ØÐÝ Ò Ý Ø Ö Ð Ø ÓÒ V ψ ϑ = Q V ψ ϑ (ϑ F ) ¾µ Ï Ø Ö Ö ØÓ Ø Ð Ø¹ ÕÙ Ö Ä˵ ÔÖÓ Ð Ñ ØÓ ÐØ Û Ø F d Ò F Ö Ø Ò ÕÙ ÔÔ Û Ø Ø ÒÒ Ö ÔÖÓ ÙØ Ø ÐÓÛ Ö Ô ÖØ Ó º µ ϑ d ϑ d F ψ;d+ := (ϑ d ϑ d+ ) F ϑ d+ := V ψd ϑ d µ ϑ ϑ F ψ;+ := (ϑ ϑ + ) F ϑ + := V ψ ϑ µ E + E + ψ Ö ÖÖ ØÓ Ø Ó ÖÚ Ø ÓÒ Ð Ø Ô Ó ØÝÔ ψ Û Ñ Ý Ý Ø Ø F d+ F ψ;d+ DD Ø Ô ³ Ó ØÝÔ ψº Ä Û F + F ψ;+ Ø RD Ø Ô ³ Ó ØÝÔ ψº Ï Ú V ψd = DV ψ D Ò V ψd = R V ψ R ÐÐÙ ØÖ Ø Ò Ø ÐÓÛ Ö Ô ÖØ Ó º Ø ÓÐÐÓÛ Ø Ø ϑ d+ = R ϑ + ϑ + = D ϑ d+ µ µ

31 ¾ ÂÓÙÖÒ Ð Ó ÐÓ Ð ÈÓ Ø ÓÒ Ò ËÝ Ø Ñ ÖÓÑ Õ º µ Ò µ ϑ d ϑ d Fd+ = (ϑ d R ϑ + ) F Ò ϑ d ϑ d F d+ = (Rϑ d ϑ + ) F º Ï Ø Ù Ú ϑ d ϑ d F d+ = ϑ ϑ F + Û Ö ϑ = Rϑ d Ò ϑ = Rϑ d º ÁÒ Ô ÖØ ÙÐ Ö ϑ d F d+ = ϑ F + ÓÖ ϑ = Rϑ d µ µ Ì Ò Ê ÔÔÖÓ Ö Ø Ö ÓÖ ÓÑÔÐ Ø ÐÝ ÕÙ Ú Ð Òغ³ Ì ÓÛÒ Ò Ë Øº Ò Ô ÖØ ¹ ÙÐ Ö Ê ÙÐØ º¾º¾µ Ø Ê ÔÔÖÓ Ö Ú Ð ÒØ Ö Ø Ò ÔÖÓÔ ÖØ Û Ö ÓÑÔÐ Ø ÐÝ Ò Ò ÑÓ Ê Ñ Ö º¾º¾µº ÁØ Ø Ò ÓÐÐÓÛ Ø Ø Ò S + = S V ψ ϑ + c = S V ψ ϑ = S V ψ ϑ = V ψ S V ψ ϑ º º ϑ + c = V ψϑ c+ ÖÓÑ Õ º µ Ò ¼µµº ÓÖÓÐÐ ÖÝ Õº ½ µµ ϑ + c E + = ϑ + c ϑ + c E + = ϑ + c V ψϑ c+ E + = (ϑ + c ϑ c+ ) E ϑ c Ø ÔÖÓ Ø ÓÒ Ó ϑ + c ÓÒ E c º µ Û Ú Õ º ¼µ Ò ½ µµ º½ Ç ÖÚ Ø ÓÒ Ð ÕÙ Ú Ð Ò Ù Ð ØÝ ÈÖÓ Ø ÓÒ ÓÒØÓ Ø Ç Ø Ô (ϑ + c ϑ ) c+ E = (ϑ c ϑ c+ ) E = (ϑ c S ϑ + ) E = (Sϑ c ϑ + ) F ÙØ ÖÓÑ Õº ½ µ Sϑ c = ϑ º Ö ÙÐØ Ä Ø ϑ ÓÑ ÔÓ ÒØ Ò Ø Ó ÖÚ Ø ÓÒ Ð Ô Eº ÁÒ Û Ø ÓÐÐÓÛ ϑ + c ÒÓØ Ø ÓÖØ Ó ÓÒ Ð ÔÖÓ Ø ÓÒ Ó ϑ ÓÒ Ø Ç Ø Ô E + c Ë Øº ¾º½º¾ Ò º µ ϑ + c := P + c ϑ µ Ð ÖÐÝ P + c Ø ÓÖÖ ÔÓÒ Ò ÓÖØ Ó ÓÒ Ð ÔÖÓ Ø ÓÒº Ä Ø ϑ := S ϑ ÒÓÛ Ø Ê Ú ØÓÖ Ó ϑº Ì ÓÐÙØ ÓÒ Ó Ø ÕÙ Ø ÓÒ S ϑ = ϑ Ö Ò ÙÔ ØÓ Ú ØÓÖ Ó E ϑ + c Ø ÓÐÙØ ÓÒ Û Ø Ñ ÐÐ Ø ÒÓÖÑ Ò E + º Ì ÓÔ Ö ØÓÖ Ø Ø Ñ Ô ϑ ØÓ ϑ + c Ö ÖÖ ØÓ Ø ÅÓÓÖ ¹È ÒÖÓ Ô Ù Ó ÒÚ Ö Ó S º Ì ÓÔ Ö ØÓÖ ÒÓØ Ý S + ϑ + c = S + ϑ µ Ä Û ÓÖ ϑ d = Dϑ Û Ú ϑ + c = S + d ϑ d º Ð ÖÐÝ ϑ + c Ò Ö Ö Ø ÜÔÖ ÓÒ ÓÖ ϑ ÓÖ ϑ d µ ÖÓÙ Ø ØÓ E + Ú S + ÓÖ S + d µº ÁÒ Ø ÓÒØ ÜØ Û Ò ϑ c+ ÓÐÐÓÛ Õ º µ ½ µ Ò º µ ϑ c+ := S ϑ + ¼µ Ì ÓÐÐÓÛ Ò ÔÖÓÔ ÖØÝ Ø Ò ÓÑÔÐ Ø Ø Ò ÐÝ ÔÖ ¹ ÒØ Ò Ë Øº º ÈÖÓÔ ÖØÝ º½º ÇÒ ϑ + c = V ψϑ c+ º ÓÖÓÐÐ ÖÝ ϑ + c E + = (ϑ + c ϑ c+ ) E = ϑ F + º ÈÖÓÓ º S ÙÖ Ø Ú Ø Ô Ù Ó ÒÚ Ö Ú Ò Ý Ø Ö Ð Ø ÓÒ S + = S (S S ) ÓÖ ÒÝ ϑ Ò F Û Ú Ò S = Q Sµ S ϑ = (Q S) ϑ = S Q ϑ = S Q ϑ = S ϑ Û Ö S = V ψ S Õº ¾¼µµº Ö ÙÐØ Õ º ¾½µ Ò ¾µµ S S ϑ = Q SS ϑ = Q V ψ ϑ = V ψ ϑ ϑ + c E + = (ϑ ϑ + ) F = ϑ F + º¾ Ò ÐÝ Ó ØÝÔ Ð ØÙ Ø ÓÒ ÌÓ ÐÐÙ ØÖ Ø ÓÙÖ Ò ÐÝ Û ÒÓÛ ÓÒ Ö Ø Û Ö Ø Ú Ö Ò ¹ÓÚ Ö Ò Ñ ØÖ Ü Ó Ø Ó ÖÚ Ø ÓÒ Ð Ø Ó ØÝÔ ψ Ó Ø ÓÖÑ V ψ = ( η( i, s )σψ ) ½µ Ð ÖÐÝ σψ Ö Ö Ò Ú Ö Ò ³ η(, s) ÒÓÒÒ ¹ Ø Ú Û Ø ÙÒØ ÓÒº Ì Ú Ö Ò ¹ÓÚ Ö Ò Ñ ØÖ Ü Ó Ø Ë Ø ψ := Sψ Ø Ò Ú Ò Ý Ø Ö Ð Ø ÓÒ V ψ = (η σ ψ) η := η(, s ) + η(, s ) ¾µ Ð Ö Ò Ê Ñ Ö º¾º½ Ø ÓÐÐÓÛ Ò Ö ÙÐØ Ò Û Ð Ø ÓÒ Ø ÒØÖ Ð Þ Ó ÖÚ Ø ÓÒ Ð ÔÔÖÓ Ó Ë Ò À Ò ½ ¾µº Ì Ù Ð ÔÔÖÓ Ó Ä ÒÒ Ò ÙÖ Ò ¾¼¼ µ Ð Ó Ø Ö Ý ÒÖ º Ê ÙÐØ º¾º½º ÒÓØ Ò Ý ϑ + Ò ϑ Ø ÓÑÔÓÒ ÒØ Ó ϑ + Ò ϑ Ö Ô Ø Ú ÐÝ Û Ú ϑ = + η σψ (ϑ δ ϑ) (ϑ ϑ ) Û Ö n δ ϑ := µ ϑ η µ := = n = η ÓÖÓÐÐ ÖÝ ϑ + c = ησ ψ ϑ c+ = ησ ψ S ϑ + º ÈÖÓÓ º Ý Ò Ø ÓÒ ϑ + := V ψ ϑ Õº µµº ÌÓ ÒØ Ý Ø ÒÚ Ö Ó V ψ ÓÒ F Û ÓÐÚ Ø ÕÙ Ø ÓÒ V ψ ϑ = ϑ Ò F º ÖÓÑ Õº ¾µ V ψ ϑ ÕÙ Ð ØÓ V ψ ϑ ÙÔ ØÓ Ú ØÓÖ Ó F º ÁØ Ø Ò ÓÐÐÓÛ ÖÓÑ Õº ¾µ Ø Ø Ø ÓÑÔÓÒ ÒØ Ó ϑ Ö Ö Ð Ø ØÓ Ø Ó Ó ϑ Ý Ø Ö Ð Ø ÓÒ η σψϑ = ϑ δ

32 Ä ÒÒ Ö ÒØ Ð ÈË Ì Ö Ù Ö Ò ÔÔÖÓ ¾ Û Ö δ ÓÑ ÓÒ Ø ÒØ Ò Rº Ö ÙÐØ ϑ = η σψ (ϑ δ) ϑ Ð Ò F Û Ú n = ϑ = Ò Ø ÒØ ØÝ δ δ ϑ º Ì Ö ÙÐØ Ò Ø ÓÖÓÐÐ ÖÝ Ø Ò ÓÐÐÓÛ ÖÓÑ ÈÖÓÔ ÖØÝ º½ Ò Õ º ¼ ½µº ÁØ ÑÔÓÖØ ÒØ ØÓ ÒÓØ Ø Ø Ò Ø Ô Ð Û Ö Ø Û Ø η( i, s ) Ö ÐÐ ÕÙ Ð ØÓ ÙÒ ØÝ Û Ú η = µ = /n ÓÖ ÐÐ Ò δ ϑ = º Ê ÙÐØ º¾º¾º Ì ÕÙ Ö Ó Ø ÒÓÖÑ ϑ Ò F + Ò ÜÔ Ò ÓÐÐÓÛ ϑ F + = n η σψ (ϑ δ ϑ ) = ÈÖÓÓ º ÖÓÑ ÈÖÓÔ ÖØÝ º½ Õº ½µ Ò Ê ÙÐØ º¾º½ Û Ú ϑ F + = (ϑ + c ϑ ) c+ E n = η( i, s )σψ ϑ ( c+ i, s ) = = i= = n = i= η( i, s ) η σ ψ n [η(, s ) + η(, s )] = (ϑ δ ϑ ) Ì Ö ÙÐØ Ø Ò ÓÐÐÓÛ ÖÓÑ Ø Ø Ø Ø η(, s ) + η(, s ) = η Õº ¾µº η σ ψ (ϑ δ ϑ ) Ê Ñ Ö º¾º½º ÈÖÓÔ ÖØÝ º½ ÐÐÙ ØÖ Ø Ý Ê ÙÐØ º¾º½ Ò º¾º¾ Ú Ù Ð Ò Ø³ ÒØÓ Ø ÔÖÓ Ð Ñ º µº ÓÖ Ü ÑÔÐ Ò Ø Á Ñ Ø Ó ÔÖ ÒØ Ò Ë Øº ϑ Ø [ψ ν,t ]¹ÓÑÔÓÒ ÒØ Ó Ö Ù Ð ÕÙ Ò¹ Ø ØÝ ÒÚÓÐÚ Ò ÄË ÔÖÓ Ð Ñ Ø Ø Ò Ø À Ð ÖØ ÙÑ Ó µ [ψ ν,t ]¹ÓÔ Ó F + º ÓÖ Ò ØÓ ÈÖÓÔ ÖØÝ º½ Ø Ø¹ Ò Ø ÔÖÓ Ð Ñ Ò Ø Ø Û Ý ÑÓÙÒØ ØÓ Ø Ø Ò Ø Ò Ø À Ð ÖØ ÙÑ Ó µ [ψ ν,t ]¹ÓÔ Ó E + c º Ô Ò Ò ÓÒ Ø ÓÒØ ÜØ ÓÒ Ñ Ý Ø Ù ÓÔ Ö Ø Ò Ú Ö ÓÙ ÕÙ Ú Ð ÒØ Û Ý º ÁÒ ÕÙ ÔÔ Û Ø ÔÔÖÓÔÖ Ø ÒÒ Ö ÔÖÓ ÙØ Ø Ô F + F d+ E + E d+ Ò E c+ Ö ÓÑÓÖÔ ØÓ E + c º Ä Ø Ù ÒÓÛ ÓÑ ØÓ Ø Ô Ð Û Ö Ø Û Ø η( i, s ) Ö ÐÐ ÕÙ Ð ØÓ ÙÒ Øݺ Ê ÙÐØ º¾º½ Ø Ò Ý Ð Ð Ó Õ º ½ µ Ò ½ µµ ϑ + c = ϑ c = S ϑ µ Ð ÖÐÝ Ø Ç Ø Ô E + c Ø Ò Ó Ò Û Ø Ø Ç Ô E c º µº ÓÖ Ò ØÓ Ê ÙÐØ º¾º¾ Û E E ϑ ϑd ϑ ϑ + c F F ϑ c+ ϑ d+ S ϑ c ϑ ϑd S ϑ c+ ϑ+ ϑ S d º Ù Ð Ö ÔÖ ÒØ Ø ÓÒ Ó Ø Ñ Ò Ð Ñ ÒØ Ó Ø ÔÖÓ ¹ РѺ À Ö E Ø ÒÙ Ò Ð Ý Ô Ë Øº ¾º½º½µº Ì Ù Ô Ó Ø Ó ÖÚ Ø ÓÒ Ð Ô E Ø ÒÙÐÐ Ô Ó Ø Ê Ò µ ÓÔ Ö ØÓÖ S Ò S d µ Ë Øº ¾º ϑ c Ø ÔÖÓ Ø ÓÒ Ó ϑ ÓÒ Ø ÓÖØ Ó ÓÒ Ð Óѹ ÔÐ Ñ ÒØ Ó E Ò E Ø Ç Ô E c ϑ c = P cϑº ÁÒ Ø Û Ö Ø ÆËË Ú ÒÐÙ ÓÒÐÝ ØÛÓ Ö Ú Ö ϑ c Ø ÒØÖ Ð Þ Ó ÖÚ Ø ÓÒ Ð ÙÒØ ÓÒ ϑ c Ò Ú Õº ½ µ Ê Ñ Ö ¾º½º¾º À Ö ϑ Ø Ó ÖÚ Ø ÓÒ Ð Ú Ö ÓÒ Ó # Ê Ñ Ö ¾º¾º¾µº Ä Û ϑ d Ø Ó ¹ ÖÚ Ø ÓÒ Ð Ú Ö ÓÒ Ó # d Ê Ñ Ö º ¾º¾º µº Ì Ô Ù¹ Ó ÒÚ Ö Ó S Ñ Ô F ÓÒØÓ Ø Ç Ø Ô E c + Ø ÓÖØ Ó ÓÒ Ð ÓÑÔÐ Ñ ÒØ Ó E Ò Ø Ó ÖÚ Ø ÓÒ Ð Ø Ô E + ϑ + c = S + # = P c + ϑº ÓÖ Ò ØÓ ÈÖÓÔ ÖØÝ º½º½ ÓÒ ϑ + c = V ψ ϑ c+ Û Ö ϑ c+ := S # + Û Ø # + := V # ÒÓØ Ø Ø S ϑ c+ = Sϑ c+ = SS # + = # + º Ä Û S + d Ñ Ô F d ÓÒØÓ E c + ϑ + c = S + d # d º ÁÒ Ø ÑÔÓÖØ ÒØ Ô ¹ Ð Ü Ñ Ò Ò Ê Ñ Ö º¾º½ ϑ + c Ó Ò Û Ø ϑ c Õº µµ E c + Ø Ò Ó Ò Û Ø E c º Ø Ò Ú Ð Ó Õº µµ ϑ + c E + = ϑ F + = ϑ d F d+ = E d E E + c E c F d F n σ = ψ ϑ µ Ì ÓÖØ Ó ÓÒ Ð ÔÖÓ Ø ÓÒ Ó E + ÓÒØÓ E c + Ð Ó ÐÐÝ ÒÚÓÐÚ Ò Ø Ù Ð Ð Ö ÓÖÑÙÐ Ø ÓÒ Ó Ä ÒÒ Ò ÙÖ Ò ¾¼¼ µ º Ó Ø Ö Ô Ô Öº Ì Ý Ö ¹ ÙÐØ µ ÓÑÔÐ Ø Ø Ö ÓÒØÖ ÙØ ÓÒº ÌÓ Ø Ð Ø

33 ¼ ÂÓÙÖÒ Ð Ó ÐÓ Ð ÈÓ Ø ÓÒ Ò ËÝ Ø Ñ ÔÖÓÔ ÖØÝ Ø ÙØ ÓÖ ÓÙÐ Ú Ö ÜÔÐ ØÐÝ Ò Ø Ô Ð ÙÒ Ö ÓÒ Ö Ø ÓÒ Ø Ø ÓÒ Ó Ø Ô Ù Ó ÒÚ Ö Ó ÓÔ Ö ØÓÖ S d Ø ÐÓ ÙÖ ÓÔ Ö ØÓÖ³ C Ó Ø Ö ÓÖÑÙÐ Ø ÓÒµº Ï Ø Ö Ö ØÓ ÐÐ Ø ÔÓ ÒØ Ø ÑÓÖ Ò Ö Ð Ö ÙÐØ Ø Ð Ò Ø Ø ÓÒ ÒÖ ÓØ Ø Ù Ð Ð Ö ÓÖÑÙÐ Ø ÓÒ Ó Ö ÒØ Ð ÈË Ò Ø ÒØÖ Ð Þ Ó Ö¹ Ú Ø ÓÒ Ð ÔÔÖÓ º Ê Ñ Ö º¾º¾º ÁÒ Ø Ô Ð ÙÒ Ö ÓÒ Ö Ø ÓÒ Û Ö Ø Û Ø η( i, s ) Ö ÐÐ ÕÙ Ð ØÓ ÙÒ Øݵ Ø ÒØ ØÝ ÜÔÖ Ò Ø Ö Ø¹ Ò Ó Õº µ Ò Ö ØÐÝ Ö Ú ÖÓÑ Ø ØÖ Ø ÓÒ Ð ÔÔÖÓ ØÓ ¹ Ö ÒØ Ð ÆË˺ Ì Ò ÓÛÒ ÓÐÐÓÛ º ÓÖ Ð Ö ØÝ ÓÒ Ö Ø Û Ö k = º Û ÐÐ ÒÓÛÒ Ø Ñ ¹ ØÖ Ü Ð Ñ ÒØ Ó V ψd Ö Ø Ò Ú Ò Ý Ø ÓÖÑÙÐ κ, = σψ n n = Ð ÖÐÝ ÓÖ ÒÝ ϑ Ò F d Û Ú ϑ F d+ = (ϑ V ψd ϑ) F = Ò Û ÓÖ =,...,nµ (V ψd ϑ) = Ö ÙÐØ n = = = = σ ψ σ ψ σ ψ σψ ϑ ϑ k (Vψd ϑ) = = = n = ( ϑ k n ϑ k (Vψd ϑ) n = ϑ k [ (ϑ ϑ k ) n ( ϑ n σψ = σψ = σψ = ) n ϑ ) =, {,...,n} n ] (ϑ ϑ k ) = n (ϑ ϑ k )ϑ n (ϑ ϑ k )ϑ n (ϑ ϑk )ϑ Ø Ñ Ð Ø ÓÒ Ò Ê ÑÓ ÁÒ Ø Ø Ø Ñ ÒØ Ó Ø ÐÓ Ð ÔÓ Ø ÓÒ Ò ÔÖÓ Ð Ñ Ø ÔÓ Ø ÓÒ Ú Ö Ð Ø ÔÓ t ξ t ÔÔ Ö Ú Ø Ð Ò Ö Þ ¹ Ø ÓÒ Ó Ø ÕÙ ÒØ Ø ρ t Û Ø Ö Ô Ø ØÓ Ø ÔÓ Ø ÓÒ Ú Ö ¹ Ð ξ ;t Ó Ö Ú Ö ξ ;t = ξ ;t + ξ t º ÁÒ ρ t = ρ t (, s ) ρ t (, s ) Ø Ð Ò Ö ÜÔ Ò ÓÒ Ó ρ t Ó Ø ÓÖÑ ρ t = ρ t + (d t ξ t) R 3 µ µ À Ö d t Ø ÙÒ Ø ÖÝ Ú ØÓÖ Ø Ø Ö Ø Ö Þ Ø ¹ Ö Ø ÓÒ s Ó Ø Ò Ð Ö Ú Ø ÔÓ tº Ä Ø J t Ø Ñ ØÖ Ü Û Ó Ð Ñ ÒØ Ó Ø th Ð Ò Ö Ø Ø Ö ÓÑÔÓÒ ÒØ Ó d t º ÒÓØ Ò Ý J t Ø ÓÖÖ ÔÓÒ Ò ÓÔ¹ Ö ØÓÖ Û Ø Ù Ú ρ t = ρ t + J t ξ t Ò ρ t; = ρ t; + J t; ξ t (J t; := Q J t ) µ ÁÒ Ò Ð ¹ Ö ÕÙ ÒÝ ÑÓ Ø Ø Ø Ú Ö Ð Ø ÔÓ t Ø ÐÓ Ð Ú Ö Ð x t Ø ÓÐÙÑÒ Ñ ØÖ Ü x t := (α, ξ t ) T µ Û Ø α a Ò F º Ì ÐÓ Ð Ú Ö Ð ÓÖ Ø ÔÓ t, t,... t n Ø Ò Ó Ø ÓÖÑ X := (α, ξ, ξ,..., ξ n ) T µ Û Ö ξ n ξ tn º Ð ÖÐÝ Ø Ó Ø Ñ Ù Øݳ α Ó ÒÓØ Ô Ò ÓÒ tº Ä Ø y t Ø Ê Ø Ú ØÓÖ Ø ÔÓ tµ ÑÓ Ý Ø Ø ÖÑ Ò Ù Ý Ø Ð Ò Ö Þ Ø ÓÒ y t := ( pt; ρ t; Ï Ø Ò Ú φ t; ρ t; ) y t = A t x t + ÖÖÓÖ Ø ÖÑ Û Ö A t := ( Jt; λi α J t; ) ¼µ ½µ ¾µ Ì ÔÖÓ Ð Ñ ØÓ ÓÐÚ Ò Ø Ð Ø¹ ÕÙ Ö Ò Ø Ø ÐÓ Ð Ð Ú Ðº Ï Ø Ò ÒØÖÓ Ù Ø ÓÐÙÑÒ Ñ ØÖ Ü Ë Ò n = ϑ = Ø Ø Ò ÓÐÐÓÛ Ø Ø ϑ F d+ = n σ = ψ ϑ (ϑ ϑ ) Y = (y, y,..., y n ) T Û Ö y n y tn º Ð ÖÐÝ Y = AX + ÖÖÓÖ Ø ÖÑ µ µ

34 Ä ÒÒ Ö ÒØ Ð ÈË Ì Ö Ù Ö Ò ÔÔÖÓ ½ Û Ö Ø ÐÓ Ð ÓÔ Ö ØÓÖ A Ø Ò Ó Ø ÓÖÑ J ; λi α J ; J ; λi α J ; A := J n; º½ λi α J n; Ê ÙÖ Ú Ð Ø¹ ÕÙ Ö ÐØ Ö Ò µ Ì ÓÐÙØ ÓÒ x ( α, ξ ) T Ó Ø Ò Ø ÖÓÙ Ö ÙÖ Ú Ð Ø¹ ÕÙ Ö ÊÄ˵ ÐØ Ö Ò º º Ö ½ µº Ì Ø¹ Ö Ø ÓÒ Ø ÔÓ t n Ø Ò Ó Ø ÓÖÑ x n n = x n n + K n v n Ò Û v n = y n A n x n n Û Ö µ µ x n n := ( α n, ξ n ) x n n := ( α n, ) µ Ð ÖÐÝ K n Ø ÊÄË ÐØ Ö Ø ÔÓ t n v n Ø ÔÖ ¹ Ø Ö Ù ³ ÓÖ Ø Ñ ÔÓ º Ì Ó Ø ÓÐÙØ ÓÒ α Ø Ù Ö Ò ØÓ Ø Ö Û Ø Ø Ú Ö Ò ¹ÓÚ Ö Ò Ñ ¹ ØÖ Ü V bα º º¾ Ñ Ù ØÝ Ö ÓÐÙØ ÓÒ Ø ÔÓ ÓÒ Ø Ò Ö ÓÖ Ø ÔÓ ÒØ ˇα Ó L ÐÓ Ø ØÓ α Ø Ø Ò Ò Ø Ø Ò Ù Ý Ø ÕÙ Ö ¹ Ø ÓÖÑ ÓÒ F f(ϑ) := (ϑ V bα ϑ) F º Ð ÖÐÝ ˇα = agmin α α V bα α L µ Ô Ò Ö ÐÐ Ø Ðº ¾¼¼¾µ Ø Ò Ö Ø¹Ð ØØ ¹ ÔÓ ÒØ ÔÖÓ Ð Ñ ÓÐÚ Ú Ø ÄÄÄ Ð ÓÖ Ø Ñ Ò Ð Ó¹ Ö Ø Ñ Ú Ý Ä Ò ØÖ Ä Ò ØÖ Ò ÄÓÚ Þ Ò ½ ¾º ÌÓ Ø Ø Ø ÔÖÓ Ð Ñ Ò Ø Ò Ð Ø ÖÑ Û Ø Ò Ú ØÓ ÓÓ Ö Ö Ò Ó L º Ì ÑÓ Ø Ò ØÙÖ Ð Ó ÓÖÖ ÔÓÒ ØÓ Ù b d Ë Øº µº Ì Ö Ö¹ Ò Ò Ü k Ò Ø Ò Ó Ò Ö ØÖ Ö ÐÝ ÓÖ Ü ÑÔÐ Ø Ö Ø ÓÒ ÓÖ Ø Ð Ø ÓÒ Ó Ø ÙÖÖ ÒØ Ð Ø Ó Ø Ð¹ Ð Ø º ÁÒ Ø Ø ÓÑÔÓÒ ÒØ Ó α Ö Ø ÓÑÔÓ¹ Ò ÒØ Ó α d := Dα Ò b d Ø Ò ÐÝ Ú ÐÓÔ Ò Ë Øº µº Ä Û Ø ÓÑÔÓÒ ÒØ Ó α Ò b d Ö Ø ÓÑÔÓÒ ÒØ Ó α d := D α Ò b d º ÙÖØ Ö¹ ÑÓÖ Ø Ú Ö Ò ¹ÓÚ Ö Ò Ñ ØÖ Ü Ó α ÜÔÖ Ò b d ÕÙ Ð ØÓ DV bα D º º V Dbα = V bαd º ËÓÐÚ¹ Ò Ø Ê Ñ Ù ØÝ ÔÖÓ Ð Ñ Ò L Ø Ö ÓÖ ÑÓÙÒØ ØÓ ÓÐÚ Ò Ø ÒØ Ö¹ Ñ Ù ØÝ ÔÖÓ Ð Ñ Ó Ø Ô¹ ÔÖÓ Ò L d º º Ì ÙÒ Ò ½ µ ˇα d = agmin α d α d V bα α d L d d ¼µ Ð ÖÐÝ ˇα = Rˇα d º Ì ÜÔÐ Ø Ø Ø Ñ ÒØ Ò ØÖ Ú Ð µ Ó Ø Ñ Ù ØÝ ÔÖÓ Ð Ñ Ò ÕÙ Ø ÓÒ Ø Ö ÓÖ Ø Ø Ó Ø ÔÔÖÓ º Ì Ó ÒÓØ Ñ Ò Ó ÓÙÖ Ø Ø Ø ÔÔÖÓ Ø Ø Ù Ø ÓÖ Ø Ø Ò ÐÐ Ø ÔÖÓ Ð Ñ Ó Ø Ø Ñ Ð Ø ÓÒ ÔÖÓ Ë Øº Ò Ô ÖØ ÙÐ Öµº Ø Ø Ø Ò Ð Ð Ú Ð Û Ø Ö ÓÖ ÔÖÓ ÓÐÐÓÛ º Ï Ö Ø Ø k = ÓÖ Ü ÑÔÐ º ËØ ÖØ Ò ÖÓÑ Ø Ó Ø Ê Ñ Ù ØÝ Ú ØÓÖ α Ò Ø Ú Ö Ò ¹ÓÚ Ö Ò Ñ ¹ ØÖ Ü V bα Û Ø Ò ÓÑÔÙØ Ø Ó Ø Ñ Ù ØÝ Ú ¹ ØÓÖ α d = D α Ò Ø Ú Ö Ò ¹ÓÚ Ö Ò Ñ ØÖ Ü V bαd = DV bα D º ÁÒ Ò Ð ¹ Ö ÕÙ ÒÝ ÑÓ Ø n ÓÑÔÓ¹ Ò ÒØ Ó α d Ö Ø Ö ÓÖ ÜÔÐ ØÐÝ Ú Ò Ý Ø Ö Ð Ø ÓÒ α k = α α k ( =,...,n) ½µ Ë Ñ Ð Ö ÓÔ Ö Ø ÓÒ ÓÒ Ø ÓÐÙÑÒ Ò Ð Ò Ó V bα ÔÖÓ¹ Ú Ø (n ) Ñ ØÖ Ü Ð Ñ ÒØ Ó V bαd º ËÓÐÚ Ò Ø Ñ Ù ØÝ ÔÖÓ Ð Ñ ¼µ Ø Ò ÔÖÓÚ Ø Ñ¹ Ù ØÝ Ú ØÓÖ ˇα d º ÁÒ Ò Ð ¹ Ö ÕÙ ÒÝ ÑÓ Ø Óѹ ÔÓÒ ÒØ Ó Ø Ê Ñ Ù ØÝ Ú ØÓÖ ˇα = Rˇα d Ö Ø Ò ÜÔÐ ØÐÝ Ú Ò Ý Ø Ö Ð Ø ÓÒ ˇα = ˇα k ˇα k ˇα k := n k ˇα k ¾µ Ð ÖÐÝ Û Ø Ù Ú Ô ÖÓÑ Ø Ê Ö Ñ ÛÓÖ ØÓ Ø Ö Ñ ÛÓÖ Ò Ø Ò Ú ¹Ú Ö µ ÓÒÐÝ ØÓ ÓÐÚ Ø Ø Ò Ð ÔÖÓ Ð Ñ Ò ÕÙ Ø ÓÒº º Ü Ñ Ù Ø Ï Ò Ò Ø Ø Ñ Ð Ø ÓÒ ÔÖÓ ˇα ÓÑ ÓÒ¹ Ø ÒØ Û Ø Ø ÑÓ Ð ÙÔ ØÓ Ø ÒÓ µ Ø Ñ Ù Ø Ö ØÓ Ü º Ì ÔÓ Ø ÓÒ ξ n Ö Ø Ò Ö Ò Ú Ä˳ ÐØ Ö Ò ÄË ÔÖÓ Ò Û Ø Ñ Ù Ø Ö Ü Ø Ø Ú ÐÙ º ÈÖÓ Ò Ø Ñ ÙÒ Ö ÒØ Ð Ø Ø Ö Ò Ê ÓÖ ÑÓ Ó ÓÙÖ ÔÖÓÚ Ø Ñ ÔÓ Ø ÓÒ º ÌÓ ÔÖ Ú ÒØ Ø Ø ÓÒ Ø Ë Ø ÔÖÓÔ Ø ÙÒ ¹ Ø Ø ÒØÓ Ø Ñ Ù ØÝ ÓÐÙØ ÓÒ Ò Ø ÔÓ Ø ÓÒ Ò Ö ÙÐØ Ô ÖØ ÙÐ Ö Ñ Ø Ó Ú Ò Ú ÐÓÔ º Ì Á Ñ Ø Ó Ø Ø³ Ø ÑÓ Ð ÖÖÓÖ Á ÒØ Ý³ Ø Ñ Ò Ôس Ø Ö ÙÐØ ÓÒ ÕÙ ÒØÐÝ º º Ì ÙÒ Ò ½ ¼ À Û Ø ÓÒ Ø Ðº ¾¼¼ µº Ï ÒÓÛ ÓÛ ÓÛ Ø Ö Ð Ø ÊÄË Ò ÄË ÔÖÓ ÙÖ Ò Ò Ø ÖÓÑ Ø Ê ÔÔÖÓ º

35 ¾ ÂÓÙÖÒ Ð Ó ÐÓ Ð ÈÓ Ø ÓÒ Ò ËÝ Ø Ñ Á Ñ Ø Ó Ò Ê ÑÓ ÁÒ ÊÄË ÑÓ ÓÖ Ü ÑÔÐ µ Ø ÔÖ Ò ÔÐ Ó Ø Ê Ú Ö¹ ÓÒ Ó Ø Ñ Ø Ó ÓÒ Ø Ò ÐÝ Ó Ø ¹ ØÙ Ð Ö Ù ³ w n := y n A n x n n = H n v n Û Ö ÖÓÑ Õ º µ Ò µ H n := I A n K n µ µ À Ö I Ø ÒØ ØÝ ÓÔ Ö ØÓÖº ÇÑ ØØ Ò Ù Ö ÔØ n Ò ÒÓØ Ò Ý w p Ò w φ Ø Ó Ò Ô Óѹ ÔÓÒ ÒØ Ó w Ö Ô Ø Ú Ðݵ Û Ú ÖÓÑ Ê ÙÐØ º¾º¾ Ò º º½ w := w p F p;+ + w φ F φ;+ Û Ö ÓÖ ψ = p ÓÖ φµ w ψ F ψ;+ = Û Ø n ψ = c ψ c ψ := η σψ (w ψ δ w ψ ) δ wψ := µ µ n µ w ψ µ = Ï Ò w ØÓÓ Ð Ö ÓÚ ÓÑ Ø Ö ÓÐ Ò Ý Ø Ø Ø Ð Ö Ø Ö Ë Øº º½µ Û Ø Ò Ö ØÓ ÒØ Ý ÐÓ Ð Ë Ó Ø ÓÖÑ ( ) β = β p e p, β φ e φ µ p Ω p φ Ω φ Ì ÓÙØÐ Ö Ø ³ Ω p Ò Ω φ Ö ÓÑ Ñ ÐÐ Ù Ø ³ Ó {,...,n}º Ì ÓÖÖ ÔÓÒ Ò Ë ÑÓ Ð Ø ÓÐÐÓÛ¹ Ò Ë Øº ½µ ρ + c[dt( ) dt( )] + ǫ p β p Ω p = p ÓØ ÖÛ ÓÖ Ø Ó Ò Ð Û ÓÖ Ø Ô Õº µµº Ì ÔÖÓ Ð Ñ ØÓ ÒØ Ý Ω p Ò Ω φ Û Ð ØØ Ò Ð Ø¹ ÕÙ Ö Ø Ñ Ø Ó Ø ÓÖÖ ÔÓÒ Ò β p Ò β φ º Ì Ù Ò ØÓ Ø ÓÒ Ö Ø ÓÒØÖ ÙØ ÓÒ Ó Ø ØÓ wº w = H δv = H δy Õ º µ Ò µµ Û ÑÙ Ø Ö Ø Û Ø Ø ÓÒØÖ ÙØ ÓÒ Ó Ø ØÓ yº Ø Ø Ð Ú Ð Ø ÓÖÖ Ø ÓÒ Ø ÖÑ Ò Ù Ý e p Ò e φ Ö ÒÓØ Ý z p Ò z φ z y set p := (e p, ) = y z ψ z φ := (, e φ ) µ Ð ÖÐÝ ÒÓØ Ø ÓÒ Ù a set = a+b Ñ Ò a Ø ÕÙ Ð ØÓ Ø ÙÖÖ ÒØ Ú ÐÙ Ó a + bº³ Ì ÓÑÔÓÒ ÒØ Ó Ø Ú ØÓÖ e Ö ÜÔÐ ØÐÝ Ò Ý Ø Ö Ð Ø ÓÒ Ë Øº µ e = /n ÓÖ µ e = /n Ì Ú Ö Ø ÓÒ Ó w Ò Ù Ý e p Ò e φ Ö Ø Ö ÓÖ Ö Ø Ö Þ Ý Ø ÕÙ ÒØ Ø f p Ò f φ Ò ÐÓÛ f w set p := Hz p = w Hz ψ f φ := Hz φ ¼µ Ö ÙÐØ Ø Ú Ö Ø ÓÒ Ó w Ò Ù Ý Ø ÐÓ Ð β Ö Ø Ö Þ Ý Ø Ú ØÓÖ Mβ := β p f p + β φ f φ ½µ p Ω p φ Ω φ Ï Ö Ø Ò Ð ØÓ ÓÐÚ Ò Ø Ð Ø¹ ÕÙ Ö Ò Ø ÕÙ Ø ÓÒ Mβ ³ w Ò Û Ø ÓÐÙÑÒ Ú ØÓÖ Ó M Ø f p ³ Ò f φ ³ Ú ØÓ Ø ÓÖÓÙ ÐÝ Ð Ø º Ð Ö Ò Ë Øº º½ Ø ÓÔ Ö Ø ÓÒ Ô Ö ÓÖÑ Ú Ô Ö¹ Ø ÙÐ Ö Ö Ñ¹Ë Ñ Ø ÓÖØ Ó ÓÒ Ð Þ Ø ÓÒ ÔÖÓ Û ÒØ ÖÖÙÔØ ÓÓÒ Ø ÓÖÖ Ø Ø Ö ÓÒ Ø ÒØ Û Ø Ø ÑÓ Ðº ÜÔ Ø ÕÙ Ø ÓÒ µ µ Ò µ ÔÐ Ý Ý ÖÓÐ Ò Ø ÒØ Ø ÓÒ Ó Ø ÐÓ Ð ÓÙØÐ Ö Ø Ω := Ω p Ω φ º º½ ÁÑÔÐ Ñ ÒØ Ø ÓÒ ÁÒ Ø ÔÖÓ ÙÖ Ö Ò Ø Ø ÓÒ Ø ÓÛ Ö Ñ ÓÛÒ Ò º µ θ Ø Ð Ú Ð Ó Ò Ò ÓÖ Ø ÔÖÓ ÐØÝ Ó Ð Ð ÖÑ Ó Ø ÐÓ Ð ÓÚ Ö ÐÐ ÑÓ Ð ÄÇŵ Ø Ø θ Ø Ø Ó Ø ÓÙØÐ Ö Ø Øº ½º ÒØÖ Ò ÄÇÅ Ø Ø ÓÑÔÙØ T LOM := w /m Û Ö m = (n ) 3 Ø Ö ÙÒ ÒÝ Ò Ø Ò Ð ¹ Ö ÕÙ ÒÝ µ Ø Ø ÙÖÖ ÒØ ÔÓ º Ä Ø t LOM := F θ (m,, ) ÒÓÛ Ø ÙÔÔ Ö θ ÔÖÓ ¹ Ð ØÝ ÔÓ ÒØ Ó Ø ÒØÖ Ð F ¹ ØÖ ÙØ ÓÒ Û Ø m, ¹ Ö Ó Ö ÓѺ Á T LOM < t LOM Ø ÖÑ Ò Ø Ø ÔÖÓ Ó ØÓ Ø Ô µ ÓØ ÖÛ Ø = Ø Ö ÙÖ Ú Ò Üµ Ò Ω = Π = Ø ÑÔØÝ Øµ Ø Ñ Ò Ò Ó Ø Ùܹ ÐÐ ÖÝ Ø Π Ò Ò Ø Ô ¾º¾ ÓÓÒ Ø Ò ØÓ Ù Ðصº ¾º Ê ÙÖ Ú ÒØ Ø ÓÒ Ó Ø ÓÙØÐ Ö ¾º½º ÙÖÖ ÒØ Ø Ó ÔÓØ ÒØ Ð ÓÙØÐ Ö ÓÖ ÐÐ Ø ψ / Ω ÓÑÔÙØ Ø ÓÑÔÓÒ ÒØ c ψ Ó w Ò Ø Ö Ñ Ü Ñ Ð Ú ÐÙ c max := max ψ / Ω c ψ Ì Ò Ú Ò ÓÑ ÒÓÒÒ Ø Ú ÓÒ Ø ÒØ κ ÓÖÑ Ø ÙÖÖ ÒØ Ø Ó ÔÓØ ÒØ Ð ÓÙØÐ Ö º µ Π := { ψ / Ω : c ψ κc max }

36 Ä ÒÒ Ö ÒØ Ð ÈË Ì Ö Ù Ö Ò ÔÔÖÓ c 5φ h ψ := g ψ / g ψ º Ì ÒÓÖÑ Ó Ø ÔÖÓ Ø ÓÒ ÕÙ Ð ØÓ h ψ w Ø ÓÐÙØ Ú ÐÙ Ó Ø ÕÙ ÒØ ØÝ γ ψ := g ψ w / ψ ψ := g ψ 3 p 5 p 3 φ 5 φ Ó È º ÆÓØ ÓÒ Ó ÔÓØ ÒØ Ð ÓÙØÐ Ö Ò Ê Ò Ð ¹ Ö ÕÙ ÒÝ ÑÓ µº ÓÖ Ø ÓÑÔÓ¹ Ò ÒØ c ψ ÓÛÒ Ö Ò ÓÖ κ =.5 Û Ø n = 7 Ò Ω = µ ÓÙÖ ÔÓØ ÒØ Ð ÓÙØÐ Ö Ö ÒØ 3 p 5 p 3 φ Ò 5 φ º À Ö Ø Ô ÓÙØÐ Ö 5 φ Ð ÐÝ ØÓ Ø ÓÑ Ò ÒØ ÔÓØ ÒØ Ð ÓÙØÐ Ö Ø Ô ¾º Ò Ë Øº º¾µº ¾º¾º ÓÖ ÔÓØ ÒØ Ð ÓÙØÐ Ö ψ Π È Ö ÓÖÑ Ø ÓÐÐÓÛ Ò Ù Ú ÓÔ Ö Ø ÓÒ µ Ï Ò ψ / Π ÓÑÔÙØ Ø ÓÒØ ÜØ Ó Õ º µ Ò ¼µµ (e p, ) ψ = p f ψ := H (, e φ ) ψ = φ Ì Ò Ø g ψ := f ψ Π set = { {ψ } Π = Π { ψ } ÓØ ÖÛ Ý ÓÒ ØÖÙØ ÓÒ Π Ø Ø Ó ÔÓØ ÒØ Ð ÓÙØÐ Ö ψ ÓÖ Û f ψ ÐÖ Ý Ò ÓÑÔÙØ º µ Á = Ó ØÓ Ø Ô ¾º¾º ÇØ ÖÛ Ø Ø Ð Ú Ð {gq } q< Ò ÓÖØ ÓÒÓÖÑ Ð Øº Ì Ø Ù ÐØ ÔÖÓ Ö Ú ÐÝ Ú Ø Ô ¾º ºµ Ì Ò ÓÖ ÒØ Ö q < ÓÒ Ö Ø ÒÒ Ö ÔÖÓ ÙØ Ò ÓÐÐÓÛ Õº µ Ò Ê ÙÐØ º¾º½µ ς q,ψ := gq g ψ := g q;ψ g ψ ;ψ F ψ ;+ ψ =p,φ Ì ÙÑ ÒÐÙ ØÛÓ Ø ÖÑ º Ô Ò Ò ÓÒ Û Ø ψ Ö Ö ØÓ p ÓÖ φµ g q;ψ ÒÓØ Ø Ó ÓÖ Ô ÓÑÔÓÒ ÒØ Ó gq Ò Ð Û ÓÖ g ψ ;ψ º Á ς q, ψ ÒÓØ Ò ÓÑÔÙØ Ý Ø ÓÑÔÙØ Ø ØÓÖ Ø Ò Ñ Ñ¹ ÓÖÝ Ò Ô Ö ÓÖÑ Ø Ö Ñ¹Ë Ñ Ø ÓÖØ Ó ÓÒ Ð Þ Ø ÓÒ ÓÔ Ö Ø ÓÒ set g ψ = g ψ ς q,ψ gq Ý ÓÒ ØÖÙØ ÓÒ ς q,ψ = g q f ψ º Ð ÖÐÝ Ø Ø Ò Ó ÐÐ Ø ÓÔ Ö Ø ÓÒ g ψ ÓÖØ Ó ÓÒ Ð ØÓ g q ÓÖ ÒÝ q < º µ ÓÒ Ö Ø ÔÖÓ Ø ÓÒ Ó w ÓÒ Ø ÓÒ ¹ Ñ Ò ÓÒÐ Ô Ò Ö Ø Ý g ψ º º h ψ w h ψ Û Ö ÜÔÐ ØÐÝ g ψ w := g ψ := ψ =p,φ ψ =p,φ ¾º º ÓÑ Ò ÒØ ÔÓØ ÒØ Ð ÓÙØÐ Ö g ψ ;ψ w ψ F ψ ;+ g ψ ;ψ F ψ ;+ Ý Ò Ø ÓÒ Ø ÓÑ Ò ÒØ ÔÓØ ÒØ Ð ÓÙØÐ Ö ψ Ø ÔÓØ ÒØ Ð ÓÙØÐ Ö ÓÖ Û γ ψ Ñ Ü Ñ Ð ψ := ag max ψ Π γ ψ ¾º º ÇÙØÐ Ö Ø Ø Ä Ø χ Ø ÙÔÔ Ö θ / ÔÖÓ Ð ØÝ ÔÓ ÒØ Ó Ø ÒØÖ Ð ÒÓÖÑ Ð ØÖ ÙØ ÓÒ χ := N θ/(, ) µ Á γ ψ > χ Û Ø m > Ø ÓÑ Ò ÒØ ÔÓØ ÒØ Ð ÓÙØÐ Ö Ø Ò Ö Ö Ò Ø Ú ÓÙØÐ Ö { {ω } = ω := ψ Ω set = Ω {ω } > γ := γ ω g := g ω / ω ËÙÔ Ö Ö ÔØ Ø Ò ÓÖ ÓÑ Ò ÓÙØÐ Öµº Ø Ø Ð Ú Ð Ω Ø ÙÖÖ ÒØ Ø Ó ÒØ ÓÙØÐ Ö Ω = {ω q } q= Ý ÓÒ ØÖÙØ ÓÒ {gq} q= Ò ÓÖØ ÓÒÓÖÑ Ð Ó Ø ÙÖÖ ÒØ Ö Ò Ó M q= γ q g q Ø ÔÖÓ Ø ÓÒ Ó w ÓÒ Ø Ô º Ï Ø Ö Ö ØÓ Õº ½µ Û Ø Ò Ø β := β ω f := f ω µ Ï Ò Ø ÓÑ Ò ÒØ ÔÓØ ÒØ Ð ÓÙØÐ Ö ÒÓØ ÒØ Ö Ð ÓÙØÐ Ö Û ÓÒ Ö Ø ÓÐÐÓÛ Ò ØÛÓ ØÙ ¹ Ø ÓÒ ½ γ ψ < χ Û Ø T LOM > 5 t LOM ÓÖ Ü Ñ¹ ÔÐ µº Ï Ø Ò Ö Ò Ø Ð Þ Ø ÊÄË ÔÖÓ º ¾ γ ψ < χ Û Ø T LOM < 5 t LOM Ò > ÓÖ γ ψ > χ Û Ø m = º Ï Ø Ò Ó ØÓ Ø Ô º ¾º º ÓÑÔÓÒ ÒØ Ó g Ò Ø Ó Ø f q ³ Ì ÓÑÔÓÒ ÒØ Ö ÒÓØ Ý u q, g = u q, fq q=

37 ÂÓÙÖÒ Ð Ó ÐÓ Ð ÈÓ Ø ÓÒ Ò ËÝ Ø Ñ Ì Ý Ö ÓÑÔÙØ Ú Ø ÉÊ Ö Ñ¹Ë Ñ Ø ÓÖÑÙÐ º º Ö ½ µ u q,q ς q,ω q < ω q q u q, = < q = ω ÓÖ q º Ð ÖÐÝ Ø u q, ³ Ö Ø ÒØÖ Ó Ø th ÓÐÙÑÒ Ó Ò ÙÔÔ Ö ØÖ Ò ÙÐ Ö Ñ ØÖ Ü Uº ¾º º Ë ÓÖ Ò ØÓ Õº ½µ Ø Ë β q Ö Ø ÓÑÔÓ¹ Ò ÒØ Ó q= γ q g q Ò Ø Ó Ø f q ³ γqg q = q= βqf q q= ÒÓØ Ò Ý [γ ] Ø ÓÐÙÑÒ Ñ ØÖ Ü Û Ø ÒØÖ γ q ÖÓÑ q = ØÓ µ Ò Ð Û ÓÖ [β ] Û Ú [β ] = U[γ ] Ê ÐØ Ö Ò ØÙ Ð Ö Ù w ÒØÖ Ò ÄÇÅ Ø Ø ÈÓØ ÒØ Ð ÓÙØÐ Ö ÓÑ Ò ÒØ ÔÓØ ÒØ Ð ÓÙØÐ Ö ÇÙØÐ Ö Ø Ø ÇÙØÐ Ö ÒØ Ø ÓÒ ÍÔ Ø Ë Ò w ÁÒÒ Ö ÄÇÅ Ø Ø ÔØ Ø ÓÒ Ê Ò Ø Ð Þ Ø ÓÒ Ì Ë Ö Ø Ö ÓÖ ØÓ ÙÔ Ø ÓÐÐÓÛ β q set = { β q + u q, γ q < u, γ ¾º º ÍÔ Ø w Ò w q = w set = w γ g w set = w γ ÓÖ q µ ¾º º ÍÔ Ø Ö ÙÒ ÒÝ m Ò Ø ÄÇÅ ÕÙ ÒØ Ø set m set t LOM = F θ (m,, ) = m set T LOM = w /m ¾º º ÁÒÒ Ö ÄÇÅ Ø Ø Á T LOM > t LOM ÙÔ Ø Ö ÙÖ Ú Ò Ü set = +º Ì Ò Ó ØÓ Ø Ô ¾º º ÄÓ Ð ÔØ Ø ÓÒ Ä Ø K Ω Ø Ñ ØÖ Ü Ø Ö Ò Ø ÓÐÙÑÒ Ó K ÓÖÖ ¹ ÔÓÒ Ò ØÓ Ø Ù Ú ÒØ ÓÙØÐ Ö ω,..., ω º Ì ÔØ Ø ÓÒ ÓÖÑÙÐ Ó Ø ÐÓ Ð Ø Ø Ú ØÓÖ Ø Ò ÖÓÑ Õ º µ Ò µµ x set = x K Ω [β ] [β ] = U[γ ] Ø ÔØ Ø ÓÒ Ó Ø Ú Ö Ò ¹ÓÚ Ö Ò Ñ ØÖ Ü Ó x Ø Ö ÓÖ Ú Ò Ý Ø ÓÖÑÙÐ set V bx = V bx + [K Ω U][K Ω U] T ÁÒ {g q} q= Ò ÓÖØ ÓÒÓÖÑ Ð Ø Ø Ú Ö Ò ¹ ÓÚ Ö Ò Ñ ØÖ Ü Ó [γ ] Ø ÒØ Øݺ º Ò º ÐÓÛ Ö Ñ Ó Ø Á ÔÖÓ ÙÖ Ò Ê ÑÓ º Ø Ø Ô Ó Ø ÒØ Ø ÓÒ ÔÖÓ Ø Ö Ù Ð ¹ ØÙ Ð Ö Ù ³ w Ò ÐÝÞ ÓÒ Ø ÖÓÙÒ Ó Õº ¾µ ÓÖ Ó Ø Ò Ö Ð Þ Ø ÓÒ º Õ º µ µ Ò µµº Ì ÐÐÓÛ Ø ÔÓØ ÒØ Ð ÓÙØÐ Ö ØÓ Ð Ø º Ì ÓÙع Ð Ö Ò Ø Ù ÒØ Ò Ö ÙÖ Ú Ñ ÒÒ Ö Ú Ô ÖØ ÙÐ Ö ÓÖØ Ó ÓÒ Ð Þ Ø ÓÒ Ö Ñ¹Ë Ñ Ø ÔÖÓ º Ì ÉÊ Ö Ñ¹Ë Ñ Ø ÔÖÓ Ð Ó ÔÖÓÚ Ø Ë Ò Ø Ö Ý Ø ÝÐ Ð Ô ÒÝ Ø Üصº ÁÒ ÓÖ Ö ØÓ Ø Ø ÑÓ Ð ÖÖÓÖ Ó Ø Ñ Þ Û Ø Ø Ñ ÔÖÓ Ð ØÝ Ý Ù Ò ÓØ ÄÇÅ Ò ÓÙØÐ Ö Ø Ø Ø Ö ÕÙ Ö Ø Ø ÓÖ ÓØ Ø Ø Ø Ñ Ú ÐÙ ÓÖ Ø ÒÓÒ¹ ÒØÖ Ð ØÝ Ô Ö Ñ Ø Ö ζ Ó Òº ÌÓ Ø ÖÑ Ò Ø Ø Ø Ò Ô Ö Ñ Ø Ö ÓÒ Ø Ö ÓÖ ÔÖÓ¹ ÓÐÐÓÛ º ÇÒ Ö Ø Ñ Ó ÓÖ θ Ò θ =. =.8 ÓÖ Ü ÑÔÐ µ Ì ÒÓÒ¹ ÒØÖ Ð ØÝ Ô Ö Ñ Ø Ö ζ Ó ÓØ Ø Ø Óѹ ÔÙØ ÖÓÑ Ø Ú ÐÙ º ÇÒ Ø Ò Ó Ø Ò Ø Ö Ø Ð Ú ÐÙ F θ (m,, ) Ó Ø ÄÇÅ Ø Ø Ò Ø Ö Ý θº º¾ Ü ÑÔÐ Ì Ê ÔÔÖÓ Û Ú Ð Ø Ò Ø Ö Ñ ÛÓÖ Ó ÙÖÓÔ Ò Ø ÓÒ ÒØ ØÐ ÀÈÄ º ½ Ê Ð ÈË Ø Û Ö ½ Ì ÀÈÄ À ÈÖ ÓÒ ÄÓ Ð Ð Ñ Òص ÔÖÓ Ø Û Ó¹ ÙÒ Ý Ø ÙÖÓÔ Ò ÆËË ËÙÔ ÖÚ ÓÖÝ ÙØ ÓÖ ØÝ Û Ø ÙÒ Ò ÖÓÑ Ø Ë ÜØ Ö Ñ ÛÓÖ ÈÖÓ Ö ÑÑ Ó Ø ÙÖÓÔ Ò ÓÑÑÙ¹ Ò ØÝ ÓÖ Ö Ö Ò Ø ÒÓÐÓ Ð Ú ÐÓÔÑ ÒØ ÙÖÓÔ Ò ÍÒ ÓÒ³

38 Ä ÒÒ Ö ÒØ Ð ÈË Ì Ö Ù Ö Ò ÔÔÖÓ Ø Ù ÔÖÓ Ò Ø Ù Ð Ò Ò Ð ¹ Ö ÕÙ ÒÝ ÑÓ º ÓÖ ÐÐ Ø Ø Ø Ø Á ÔÖÓ ÙÖ Û ÓÒ ÙØ Û Ø η = ÓÖ ÐÐ º ÁÒ Ø Ò Ð ¹ Ö ÕÙ ÒÝ ÑÓ ÕÙ Ø ÓÒ µ Ø Ò Ö Ù ØÓ w = n σp wp + n σφ w φ ¾µ = = Ò Ö Ð ÖÙÐ Ø Ø Ô Ó Ø Ö ÙÖ Ú ÒØ ¹ Ø ÓÒ ÔÖÓ Ø p ÓÖ φ ØÓ Ð Ø Ø ÓÑ Ò ÒØ ÔÓØ ÒØ Ð ÓÙØÐ Ö Ø Ò ÓÖÖ ÔÓÒ ØÓ Ø Ñ Ü Ñ Ð Ú ÐÙ Ó w p /σ p Ò w φ /σ φ ÓÖ =,...,n º µº ÐÐÙ ØÖ Ø Ò Ø ÓÐÐÓÛ Ò Ü ÑÔÐ Ø Ð Ó Ø Ò Ù Ð¹ Ö ÕÙ ÒÝ ÑÓ º Ï ÒÓÛ ÔÖ ÒØ Ö Ð Ø Ö ÙÐØ ÓÒ ÖÒ Ò Ø Ó ÈË Ø ÔÖÓÚ Ý Ø Ö Ò ÓÖ Ø Ø Ò ¼ ÔÓ Ø ÀÞ Ò Ù Ð¹ Ö ÕÙ ÒÝ ÑÓ Ä½¹» ľ¹Èµ Û Ø Ñ ÒÝ ÔÔ Ö Ò Ò ÔÔ Ö Ò Ó Ø ÐÐ Ø º ÇÚ Ö Ø Ø Ñ Ö Ô Ò Ò ÓÒ Ø ÔÓ Ø Ö ÒÙÑ Ö Û ÓÖ º Ì Ö Ö Ò Ò Ù Ö Ö Ú Ö Û Ö Ø Ø º Ì Ö Ð ¹ Ø Ú ÖØ Ò ÓÓÖ Ò Ø Ó Ø Ù Ö Ö Ú Ö Û Ö Ó Ø ÓÖ Ö Ó 33 Ñ Ñ Ò 38 Ѻ Ì Ø Ø Ò ÕÙ Ø ÓÒ Û Ö Ò Ø Ð Þ Ø Ø ÓÐÐÓÛ Ò ÔÓ ½¼ ½ ¼½ ¼½¼ Ò º Ì Ñ Ù Ø Û Ö Ø Ò Ü Ò ÓÒ ÓÖ ØÛÓ ÓÒ Ø ÔÓ Ø ÓÒ Ó Ø Ù Ö Ö Ú Ö Û Ø Ù Ö ØÖ Ú ÙÔ ØÓ ÓÒ ÒØ Ñ Ø Ö Ü ÔØ ÓÖ ÔÓ ½ ½¼ ½ ¼½ ½ ¼¾ ¼½¼ ¼½½ Ò ¹ ¼º Ì Ð ½ ٠й Ö ÕÙ ÒÝ Ñ Ù ¹ Ø ǎ f ν; k ÓÖ =,..., n Û Ø n = 9 Ò k = º Ø Ø ÔÓ ÙÒ Ö ÓÒ Ö Ø ÓÒ Ø Ñ Ù Ø Û Ö Ü ÓÖ Ø Ö Ê ØÖ Ò Ö ÔØ ÓÒ Ø ÜØ Ò Ë Øº º¾ Ò Ô ÖØ ÙÐ Öµº ; f ν f f ÌÓ ÐÐÙ ØÖ Ø Ø Ù Ð¹ Ö ÕÙ ÒÝ Ú Ö ÓÒ Ó Ø ÔÔÖÓ ÔÖ ÒØ Ò Ø Ô Ô Ö Û ÒÓÛ ÓÒ ÒØÖ Ø ÓÒ Ø ÔÖÓ¹ Ò ØÖÙÑ ÒØ ÓÖ ÙÒ Ò Ö Ö º Ì ÙÖÓÔ Ò ÆËË ËÙÔ Ö¹ Ú ÓÖÝ ÙØ ÓÖ ØÝ Ø ÒÝ Ò Ö Ó Ø ÑÔÐ Ñ ÒØ Ø ÓÒ Ó Ð Ð Ó ÙÖÓÔ ³ ÙØÙÖ Ø ÐÐ Ø Ò Ú Ø ÓÒ Ý Ø Ñº Ø ÔÓ 4745º Æ Ò Ø ÐÐ Ø Û Ö Ø Ò Ú Ð Ð Ø ÐÐ Ø ½ ¾½ ¾ ¾ ¼ =,...,9µº Ø Ø Ø ÔÓ Ø Ñ Ù ØÝ Ú ØÓÖ (ǎ f, ǎ f ) Û Ü º Ì ØÖ Ò Ö ÔØ ÓÒ Ó Ø ÓÑÔÓÒ ÒØ ÔÐ Ý Ò Ì Ð ½ ÓÖ k = º ÓÖ =,...,n Ò ÓÖ Ö ÕÙ ÒÝ Ø ÓÖÖ ÔÓÒ ¹ Ò Ê Ñ Ù Ø Ö Ø Ò Ú Ò Ý Õº ¾µ Û Ø k = ˇα set = ǎ f Ò ˇα set = ǎ f º ÁØ Û Ó ÓÙÖ Ú Ö Ø Ø Ø Ø Ñ Ø Ö Ð Ø Ú ÓÓÖ Ò Ø Ó Ø Ù Ö Ö Ú Ö Û Ö Ü ØÐÝ Ø Ñ Ò ÓØ ÔÔÖÓ Ñ.9 Ñ Ò Ñ Ø ÓÖÖ Ø Ú ÐÙ ÙÔ ØÓ ÓÒ Ò¹ Ø Ñ Ø Öµº Ì Á ÔÖÓ ÙÖ ÑÔÐ Ñ ÒØ Ò Ø Ø Û Ø ÄË Ù Ð¹ Ö ÕÙ ÒÝ Ú Ö ÓÒ Ó Ø Ø ÔÖ ÒØ Ò Ë Ø ÓÒ º½º Ì ÓÐÐÓÛ Ò ØÛÓ ØÙ Ø ÓÒ Û Ö Ø Ò ÓÒ Ö º Ï Ø ÓÙØ ÒÝ ÝÐ Ð Ô Ø Ö Ð Ø µ º Ï Ø Ø ÓÐÐÓÛ Ò ÝÐ Ð Ô Ù Ø ØÓ ÓÛ Ø ÒÝ Ó Ø Ñ Ø Ó µ ÝÐ Ò Ø Ö ÔØ ÓÒ Ó Ø f ¹ Ò Ð Óѹ Ò ÖÓÑ Ø ÐÐ Ø = µ ÝÐ Ò Ø Ö ÔØ ÓÒ Ó Ø f ¹ Ò Ð Óѹ Ò ÖÓÑ Ø ÐÐ Ø = 4µ º Ø Ø ÓÒ Ò ÒØ Ø ÓÒ Û Ø ÓÙØ ÒÝ ÝÐ Ð Ô ÁÒ Ø Ø Ø ÒØÖ Ò Ú ÐÙ Ó T LOM 9.3µ Û Ö Ø Ö Ø Ò Ø ÓÖÖ ÔÓÒ Ò Ú ÐÙ Ó t LOM (.6)º Ì ÓÙØÐ Ö Û Ö Ø Ò ÒØ Ò Ø ÓÐÐÓÛ Ò ÓÖ Ö ÇÙØÐ Ö T LOM (f ; p ) 5.4 (f ; 4 φ ).63 (f ; p ).63 (f ; 8 p ).6 Ì Ú ÐÙ Ò Ø Ö Ø¹ Ò ÓÐÙÑÒ Ø ÓÖÖ ¹ ÔÓÒ Ò Ö Ù Ú ÐÙ Ó T LOM º Ì Ð Ø Ú ÐÙ Ó T LOM.6µ Ñ ÐÐ Ö Ø Ò Ø ÓÖÖ ÔÓÒ Ò Ú ÐÙ Ó t LOM.3µº Ì Ø Ù ÓÙÒ Ö ÔÐ Ý Ò Ì Ð ¾º º Ø Ø ÓÒ Ò ÒØ Ø ÓÒ Û Ø ÝÐ Ð Ô Ì ÒØÖ Ò Ú ÐÙ Ó T LOM Û Ø Ò Ú ÖÝ Ð Ö 335.9µ ÑÙ Ö Ø Ö Ø Ò Ø ÓÖÖ ÔÓÒ Ò Ú ÐÙ Ó t LOM.6µº Ì ÓÙØÐ Ö Û Ö Ø Ò ÒØ Ò Ø ÓÐÐÓÛ Ò ÓÖ Ö ÇÙØÐ Ö T LOM (f ; φ ) 8.8 (f ; 4 φ ) 9.4 (f ; p ) 5.4 (f ; 4 φ ).8 (f ; p ).74 (f ; 8 p ).3

39 ÂÓÙÖÒ Ð Ó ÐÓ Ð ÈÓ Ø ÓÒ Ò ËÝ Ø Ñ Ì Ð Ø Ú ÐÙ Ó T LOM.3µ Ñ ÐÐ Ö Ø Ò Ø ÓÖÖ ¹ ÔÓÒ Ò Ú ÐÙ Ó t LOM (.34)º Ì Ø Ù ÓÙÒ Ö ÔÐ Ý Ò Ì Ð º ÐÐ ÓÚ Ö Ø Ø Ñ Ö ÙÒ Ö ÓÒ Ö Ø ÓÒ Ø Ö ÙÐØ Û Ö Ø Ñ Û Ø κ = ÓÖ κ = Ø Ô ¾º½ Ò º Ò Ë Øº º½µº ÁØ ÑÔÓÖØ ÒØ ØÓ ÒÓØ Ø Ø Ø Ó κ = Û Ò Ù ÓÑ ÈÍ ÓÚ Ö Ø Ô ¾º¾ Ò Ë Øº º½µ ÑÔÐ ØÐÝ ÓÖÖ ÔÓÒ Ø ÑÔÐ Ñ ÒØ ¹ Ø ÓÒ Ó Ø Á ÔÖÓ ÙÖ Ý Ø Ì ÙÒ Ò ÖÓÙÔ Ø Ø Ì Ò Ð ÍÒ Ú Ö ØÝ Ó Ð Ø ÌÍ µº ÁÒ Ã ÐÑ Ò ÑÓ κ ÓÙÐ Ð ÐÝ Ø ÕÙ Ð ØÓ Ñ ÐÐ Ö Ú ÐÙ Ý.5 Ò º µº Ì ÔÓ ÒØ Ö Ñ Ò ØÓ ÒÚ Ø Ø º Ì Ð ¾ Á ÒØ Ø ÓÒ Ó Ø Ó Ë º Ì β fν; ψ Ö ÜÔÖ Ò Ñ Ø Ö º Ø Ø ÔÓ ÙÒ Ö ÓÒ Ö Ø ÓÒ Ò Ò Ø ÐÐ Ø Û Ö Ú Ð Ð n = 9 Ø Üصº Ö ÕÙ ÒÝ f ψ ½ ¾ p φ.43 Ö ÕÙ ÒÝ f ψ ½ ¾ p Ì Ð Á ÒØ Ø ÓÒ Ó Ø Ó Ë ÒÐÙ ¹ Ò ÝÐ Ð Ô º Ì ØÙ Ø ÓÒ Ø Ñ Ø Ø Ò Ò Ì Ð ¾ ÙØ Û Ø ÝÐ Ð Ô º Ì Ð ØØ Ö Ö ÓÖÖ ØÐÝ Ö ØÖ Ú β f ; φ λ β f ;4 φ λ º ÆÓØ Ø Ø Ø ÒØ Ø ÓÒ ÓÖ Ö Ö Ø Ø Ø Ò Ù Ý Ø ÝÐ Ð Ô Ò Ø Ò Ø Ø ÔÐ Ý Ò Ø ÜØ Ò Ì Ð ¾µº Ö ÕÙ ÒÝ f ψ ½ ¾ p φ Ö ÕÙ ÒÝ f ψ ½ ¾ p φ.48 ÓÒÐÙ Ò ÓÑÑ ÒØ Ì Ú ÖØ Ó ÆËË Ö Ô Ö Ø Ö Ú Ö Ò Ø Ø ÐÐ Ø Ó Ø ÆËË Ú Ä ÒÒ Ò ÙÖ Ò ¾¼¼ µº ÁØ Ö Ø Ö Ú Ö¹ Ø ÐÐ Ø Ô Ö º Ì ÓÖ ¹ Ò Ð Ó ÖÚ Ø ÓÒ Ö Ò ÓÒ Ø Ë Øº ¾º½µº Ø Ó ÖÚ Ø ÓÒ Ö Ò ÙÔ ØÓ Ú ØÓÖ Ò Ø ÒÙ Ò Ð Ý Ô Ë Øº ¾º½º½ Ò º µ Ø ÓÖ¹ Ø Ó ÓÒ Ð ÓÑÔÐ Ñ ÒØ Ó Ø Ô Ò Ø Ó ÖÚ Ø ÓÒ Ð Ø Ô ÔÐ Ý Ý Ô ÖØ Ò Ø Ø Ñ Ð Ø ÓÒ ÔÖÓ¹ ÙÖ º ÁÒ Ô ÖØ ÙÐ Ö ÖÓÙ Ø ØÓ Ø ÓÖØ Ó ÓÒ Ð ÓÑÔÐ Ñ ÒØ Ø Ö Ù Ð ÕÙ ÒØ Ø ØÓ ÓÒ Ö Ò Ø Á ÔÖÓ ÙÖ Ø Ø Ö Ú ÐÙ ÓÒ Ø Ó Ø Ö Ô º ÌÓ ØÖ Û Ø ÒØ Ð Ø Ò ÐÝ ÔÖ ÒØ Ò Ø Ô Ô Ö Û Ö ØÖ Ø ØÓ Ø Ô Ð Û Ö Ø ÆËË Ö Ô ÒÐÙ ÓÒÐÝ ØÛÓ Ö Ú Ö º ÇÒ Ø ØÛÓ Ò¹ ÚÓÐÚ Ò Ø Ò Ø ÓÒ Ó Ò Ð Ö Ò Ø ÓÙ Ð ÒØÖ Ð Þ Ó ÖÚ Ø ÓÒ Ó Ë Ò À Ò ½ ¾µ Ö Ø Ò ÓÔÔÓ Ø º Ð Ö Ò Ë Øº ½º½º Ø Ò ÓÖÑ Ø ÓÒ ÓÒ¹ Ø Ò Ò Ø Ó ÖÚ Ø ÓÒ Ø Ò ÑÔÐ ÒØ ÝÑÑ Ø¹ Ö ØÖ Ò Ö ÔØ ÓÒ Ó Ø Ø ÓÒØ Ò Ò Ø Ê Ø º Ì Ò Ê ÔÔÖÓ ÔÖÓÚ ØÓ ÕÙ Ú Ð Òغ ÅÓÖ ÔÖ ÐÝ Ô Ò Ë Øº Ø Ó Ó Ø Ö ¹ Ö Ò Ø ÐÐ Ø Ò Ù Ø Ø Ó Ö Ö Ò Ó Ø Ê Ø Ô º Ì ÓÑÔÓÒ ÒØ Ó Ê Ú ØÓÖ Ò Ø Ö Ø ÓÖÖ ÔÓÒ Ò ³ º ËÓÐÚ Ò Ø ÔÖÓ Ð Ñ Ò ÑÓ Ø Ö ÓÖ ÑÓÙÒØ ØÓ ÓÐÚ Ò Ø Ò Ø º Ø ÒÝ Ø Ó Ø Ø Ñ Ð Ø ÓÒ ÔÖÓ ÙÖ ÓÒ Ñ Ý Ø Ö ÓÖ Ô ÖÓÑ Ø Ê ÑÓ ØÓ Ø ÑÓ Ò Ú ¹Ú Ö º ÁÒ Ô ÖØ ÙÐ Ö ÓÐÚ Ò Ø Ö Ø ÓÒ Ð¹ Ñ Ù ØÝ ÔÖÓ Ð Ñ Ó Ø Ê ÑÓ ÑÓÙÒØ ØÓ ÓÐÚ Ò Ò Ö Ø¹ Ð ØØ ¹ÔÓ ÒØ ÔÖÓ Ð Ñ Ó ØÝÔ Ë Øº º¾µº ÁÒ Ê ÑÓ ÐÐ Ø Ø ÐÐ Ø Ö Ò Ð Ò Ø Ñ Ñ ÒÒ Öº Ö ÙÐØ Ø ÒÙÑ Ö Ð Ó Ó Ø Ê Ø Ñ Ð Ø ÓÒ ÔÖÓ Ö ÑÓÖ Ö Ð Ø Ò Ø Ó Ó Ø Ö Ú Ö ÓÒ º ÓÖ Ü ÑÔÐ Ò Ê ÑÓ Ø Ô¹ Ô Ö Ò Ó Ø Ö Ö Ò Ø ÐÐ Ø Ó Ø ÔÔÖÓ Ò Ð Ð Ø Ø Ó ÒÝ Ø ÐÐ Ø º Ì Ø Ñ Ò ÒØ Ö Ø Ó Ø Ê ÔÔÖÓ Ð Ò Ø ÔÖÓÔ ÖØ Ö Ú Ð Ý Ø ÓÖÖ ÔÓÒ Ò Ù Ð Ò Ð¹ Ý ³ Ë Øº µº Ì ÔÖÓÔ ÖØ Û Ö Ñ Ò Ø ÔÔÖÓ Ê Ñ Ö º¾º¾µ Ò Û Ð Ø ÓÒ Ø Ç ÔÔÖÓ Ó Ë Ò À Ò ½ ¾µº ÁÒ Ô ÖØ Ù¹ Ð Ö Ê ÙÐØ º¾º¾ Ò ÜÔÐÓ Ø Ò Ø Á ÔÖÓ ÙÖ º ÖÓÑ Ø ÔÓ ÒØ Ó Ú Û ÕÙ Ø ÓÒ µ Ú ÖÝ Ò Òغ Ì ÒÓØ ÓÒ Ó ÔÓØ ÒØ Ð ÓÙØÐ Ö Ö Ú ÖÓÑ Ø ÓÖÓÐÐ ÖÝ Ø ÙÐ Ò ÕÙ Ö Ø ÓÑÔÓ Ø ÓÒ µ º º Ì ÔÖÓÔ ÖØ Ð Ó ÓÑÔÐ Ø Ø ÓÒØÖ ÙØ ÓÒ Ó Ä ÒÒ Ò ÙÖ Ò ¾¼¼ µº ÐÐ Ø Ô Ø Ö Ò ÐÝÞ Ò ÓÑÑ ÒØ Ò Ê Ñ Ö º¾º½º Ö ÙÐØ ÐÐ Ø ÕÙ Ú¹ Ð ÒØ ÔÔÖÓ Ò Ò Ø ÖÓÑ ÓØ Öº Ì Á ÔÖÓ ÙÖ Ö Ò Ë Ø ÓÒ ÓÐÐÓÛ Ø Ñ Ò Ù Ð Ò Ó Ø Á Ñ Ø Ó Ó Ø ÌÍ ÖÓÙÔ º º º Ò Ì ÙÒ Ò ½ ¼µº ÁÒ Ô ÖØ ÙÐ Ö Ø Ö ÙÖ Ú Ø Ø ÓÒ ÔÖÓ ÓÒ Ö Ñ¹Ë Ñ Ø ÓÖØ Ó ÓÒ Ð Þ Ø ÓÒ ÔÖÓ ÙÖ º Ì Ñ Ò Ò Û ÔÓ ÒØ ¹ Ö Ú ÖÓÑ Ø ÒÓØ ÓÒ Ó ÔÓØ ÒØ Ð ÓÙØÐ Ö º Ì ÓÖØ Ó Ó¹ Ò Ð Þ Ø ÓÒ ÔÖÓ ÙÖ Û ÑÔÐ Ñ ÒØ ÓÖ Ò Ðݺ Ì ÒÝ Ó Ø Á Ñ Ø Ó Ø Ù ÑÔÖÓÚ º Ì Ô Ö¹ Ø ÙÐ Ö ÑÔÐ Ñ ÒØ Ø ÓÒ Ð Ó Ò Ø ÖÓÑ Ø ÉÊ Ö Ñ¹ Ë Ñ Ø Ø Ô ¾º Ó Ë Øº º½º Ì ÉÊ ÔÔÖÓ Ó Ø È ÖØ Ø ÓÒ Ó Ì Ö Ù ½ µ Ò Ø Ö Ý Ò ÐÝ ÓÑÔÐ Ø º Ô Ò Ø Ô ¾º Ó Ë Øº º½ Ø Ë ¹ Ò Ø Ù Ö ÙÖ Ú ÐÝ Ö Ò º Ì ÒØ Ø ÓÒ Ó

40 Ä ÒÒ Ö ÒØ Ð ÈË Ì Ö Ù Ö Ò ÔÔÖÓ ÝÐ Ð Ô Ò Ô ÖØ ÙÐ Ö Ô Ö ÓÖÑ Ò Ø Û Ý Ì Ð µº Ì ÆËË Ö Ô Ñ Ý ÒÐÙ ÑÓÖ Ø Ò ØÛÓ Ö Ú Ö ÓÑ Ö Ú Ö¹ Ø ÐÐ Ø Ñ Ý Ð Ó Ñ Ò º ÁÒ Ø Ò Ö Ð ØÙ Ø ÓÒ Ø Ø Ó ÆËË Ò ØÛÓÖ Û Ø Ñ Ò Ø Ø ÑÔÓÖØ ÒØ ØÓ Ò Ø ÖÓÑ ÐÐ Ø Ö ÙÒ ÒÝ Ó Ø Ø º Ì ÒØ Ð ³ ÑÙ Ø Ø Ò Ò¹ Ø ÓÒ Ø ÓÖ Ô Ö Ó µ Û Ö Ø Ý Ô¹ Ô Öº ÌÓ ÓÐÚ Ø Ö Ð Ø ÔÖÓ Ð Ñ Ò Ò ÒØ Ñ Ò¹ Ò Ö Ø Ò Ç ÔÔÖÓ Ú ØÓ ÓÒ Ù Ø Ò Ò Ö Ð Þ Ò Ø ÔÖÓ Ø Ó ÖÚ Ø ÓÒ Ð Ö Ñ ¹ ÛÓÖ ³ Ó º º Ì Ö Ð Ø Ú ÐÓÔÑ ÒØ Û ÐÐ ÔÖ ¹ ÒØ Ò ÓÖØ ÓÑ Ò Ô Ô Ö Ä ÒÒ ¾¼¼ µº Ê Ö Ò Ö ÐÐ º Ö ÓÒ Ìº Î Ö Ý º Ò Ö Ãº ¾¼¼¾µ ÐÓ Ø ÔÓ ÒØ Ö Ò Ð ØØ º Á ÌÖ Ò º ÁÒ¹ ÓÖѺ Ì ÓÖݺ ¾¾¼½ ¾¾½ º Ö º ½ µ ÆÙÑ Ö Ð Ñ Ø Ó ÓÖ Ð Ø¹ ÕÙ Ö ÔÖÓ Ð Ñ º ËÁ ź À Û Ø ÓÒ Ëº Ä ÀºÃº Ò Ï Ò Âº ¾¼¼ µ ÄÓ Ð Þ Ð¹ ØÝ Ò ÐÝ ÓÖ ÈË» Ð Ð Ó Ö Ú Ö ÙØÓÒÓÑÓÙ ÒØ Ö ØÝ ÑÓÒ ØÓÖ Ò º Ì ÂÓÙÖÒ Ð Ó Æ Ú Ø ÓÒ ÊÓÝ Ð ÁÒ Ø ØÙØ Ó Æ Ú Ø ÓÒ ¾ ¾ º Ä ÒÒ º Ò ÙÖ Ò Ëº ¾¼¼ µ Ù Ð Ð Ö ÓÖÑÙ¹ Ð Ø ÓÒ Ó Ö ÒØ Ð È˺ º Ó º ¾¾ ¾ º Ä ÒÒ º ¾¼¼ µ ÆËË Ò ØÛÓÖ Û Ø Ñ Ò Ø ÒØ Ð Ò ÔÓØ ÒØ Ð ÓÙØÐ Ö º ÈÖÓº Æ ÆË˹¾¼¼ º ÌÓÙÐÓÙ Ö Ò º Ë Èº Àº Ò À Ò Ëº ½ ¾µ ÒØÖ Ð Þ ÙÒ Ö ÒØ Ð Ñ Ø Ó ÓÖ ÈË Ò ØÛÓÖ Ù ØÑ Òغ Ù ØÖ Ð Ò ÂÓÙÖÒ Ð Ó Ó Ý È ÓØÓ Ö ÑÑ ØÖÝ Ò ËÙÖÚ Ý Ò º ¹½¼¼º ËØÖ Ò º Ò ÓÖÖ Ãº ½ µ Ä Ò Ö Ð Ö Ó Ý Ò ÈË Ï ÐРРݹ Ñ Ö ÈÖ Å Ù ¹ Ø º Ì ÙÒ Ò ÈºÂº º ½ ¼µ Ò ÒØ Ö ØÝ Ò ÕÙ Ð ØÝ ÓÒØÖÓÐ ÔÖÓ ÙÖ ÓÖ Ù Ò ÑÙÐØ Ò ÓÖ ÒØ Ö ¹ Ø ÓÒº ÈÖÓº ÁÇÆ È˹ ¼º ÓÐÓÖ Ó ËÔÖ Ò Óй ÓÖ Ó ÍË ½ ¹ ¾¾ Ì ÙÒ Ò ÈºÂº º ½ µ Ì Ð Ø¹ ÕÙ Ö Ñ Ù ØÝ ÓÖÖ Ð Ø ÓÒ Ù ØÑ ÒØ Ñ Ø Ó ÓÖ Ø ÈË ÒØ Ö Ñ Ù ØÝ Ø Ñ Ø ÓÒº º Ó º ¼ ¾º Ì Ö Ù º ºÂºÅº ½ µ Ê ÙÖ Ú Ø ÔÖÓ Ò ÓÖ Ò Ñ Ø ÈË ÙÖÚ Ý Ò º ÈÙ Ð Ø ÓÒ ÓÒ Ó Ýº Æ Û Ö ÁËËÆ ¼½ ½ ¼ ÆÙÑ Ö º Æ Ø ÖÐ Ò Ó Ø ÓÑÑ ÓÒ Ð Øº

41 Jounal of Global Positioning Systems (7) Vol.6, No.: A Robust Indoo Positioning and Auto-Localisation Algoithm Raine Mautz Institute of Geodesy and Photogammety,Swiss Fedeal Institute of Technology, ETH Zuich Washington Y. Ochieng Cente fo Tanspot Studies, Depatment of Civil and Envionmental Engineeing, Impeial College London, London, United Kingdom, SW7 AZ Abstact. Senso netwoks that use wieless technology (IEEE standads) to measue distances between netwok nodes allow 3D positioning and eal-time tacking of devices in envionments whee Global Navigation Satellite Systems (GNSS) have no coveage. Such a system equies thee key capabilities: extaction of anges between senso nodes, appopiate suppoting netwok communications and positioning. Recent eseach has shown that the fist two of these capabilities ae feasible. This pape builds on this and develops an automatic and obust 3D positioning capability. A stategy is pesented that enables high integity positioning even in the pesence of lage mean eos in the ange measuements. This is achieved by an algoithm that geneates a tight, high-confidence uppe bound on the eo in a position estimate, given the noisy ange measuements fom the adio devices in view. As a coe featue, we pesent a novel netwok auto-localisation algoithm that fully automatically detemines the positions of all neaby fixed nodes. Results fom a eal netwok using the Cicket Indoo Location System show how all senso nodes can be detemined based on only one dynamic node. Simulations of static netwoks with nodes demonstate the impotance of solving folding ambiguities. Studies fom netwoks with impecise ange measuements have shown that it is possible to theoetically achieve a position deviation that is of the size of the anging eo. Keywods. auto-localisation, positioning algoithm, wieless senso positioning, multilateation Intoduction. Backgound Despite Global Navigation Satellite Systems (GNSS) being the most pevasive positioning systems, altenative and complementay systems ae essential because GNSS ae unsuitable fo some ad-hoc senso netwok opeational envionments. In paticula, they cannot wok indoos o in the pesence of obstacles that block the signals fom the GNSS satellites. This is commonly addessed by combining o integating GPS with deduced eckoning (DR) sensos including inetial navigation systems (INS). DR, with the aid of a gyoscope and odomete, is commonly used to bidge any gaps in GPS positioning, but its positioning eo gows apidly if not contolled by othe sensos o systems such as GPS. The use of cellula communications netwoks to assist GPS eceives in difficult envionments is efeed to as Assisted Global Positioning Sevices (A-GPS), whee GPS is integated in a mobile netwok and the pocessing is patly taken ove by the netwok. Accoding to Danell and Wilczoch () positioning accuacy of 5m indoos can be eached with A-GPS. The system poposed in this pape howeve is designed to each a decimete to centimete accuacy ( σ) indoos. The limitations of GNSS have motivated the seach fo complementay methods in addition to those above. Recently, a lage numbe of wieless positioning systems has been poposed and evaluated, e.g. Niculescu and Nath (), Savaese et al. (), Savvides et al. (3) and Smith et al. (4). Netwok positioning based on gaph theoy has been investigated extensively using a set of ange measuements between netwok nodes, e.g. by Een et al. (4) and Goldenbeg et al. (5). Wieless devices enoy widespead use in numeous divese applications including senso netwoks. The exciting new field of wieless senso netwoks beaks away fom the taditional end-to-end communication of voice and data

42 Mautz and Ochieng: A Robust Indoo Positioning and Auto-Localisation Algoithm 39 systems, and intoduces a new fom of distibuted infomation exchange. The nea futue scenaio consists of countless tiny embedded devices, equipped with sensing capabilities, deployed in all envionments and oganising themselves in an ad-hoc fashion. Knowing the coect positions of netwok nodes is essential to many applications in futue pevasive senso netwoks. Examples include usage in cime pevention, emegency and incidence espond management, poduct tacking at industial sites, wildlife habitat monitoing and home contol. Futhe applications ae use guidance, efficient outing in communication netwoks, detection of unauthoised emoval of assets and geofencing. Howeve, fo many applications, the integity of the location infomation yielded fom such a wieless senso netwok is vital. The eseach focus has been on the detemination of positions, effectively ignoing measuement noise. Little attention has been given to the fact that ange obsevations ae coupted by goss eos and also affected by measuement noise. Additionally, the coectness of the coodinate positions of ancho nodes, which know thei positions cannot be taken fo ganted fo eal wold scenaios. All these diffeent eo souces can lead to inaccuate position infomation. This pape takes these eos into account. A wieless positioning system which is used fo Safety of Life (SoL) o liability citical applications is equied to be of high eliability and integity. It is not sufficient to delive a coodinate output, even with coesponding figues of the uncetainties (in tems of a vaiance-covaiance matix). In fact, a igoous validation pocess must povide the use with eliable and complete integity infomation fo the positional data. Any patial o complete system failue needs to be fowaded immediately to the use, who is then able to ely on the system status as indicated by the system itself.. The used wieless senso platfom The system that has been used fo the expeiments in this pape in ode to obtain anging data fo positioning and tacking is called Cicket. The Cicket nodes ae tiny devices developed by the MIT Laboatoy fo Compute Science as pat of the Poect Oxygen, details ae given in Piyantha (5). A Cicket boad is shown in Fig.. A deployed Cicket location sensing infastuctue enables people o devices to detemine thei position while indoos. The Cicket unit can be pogammed as eithe as a beacon o listene. The beacons ae typically static units that ae mounted on the ceiling above the mobile listenes. The beacon unit boadcasts peiodically an ultasonic (US) pulse and at the same time a adio fequency (RF) message with its unique ID numbe. Using the time-of-flight infomation fom diffeent beacons and the tempeatue coected speed of sound measuement; the listene calculates its distance fom the beacons. Because RF tavels about 6 times faste than ultasound, the listene can use the time diffeence of aival between the stat of the RF message fom a beacon and the coesponding ultasonic pulse to infe its distance fom the beacon. The position of the listene can then be detemined based on the beacon node positions and the measued anges. Fig.. Cicket unit / RS3 cable assembly One eason to choose the Cicket system as a test bed fo the novel positioning algoithm was its flexibility and pogammability. Fo example, Cicket listenes and beacons consist of identical hadwae. Even the softwae that is unning on listenes and beacons can be the same a simple command fom the host can change a Cicket node fom a listene into a beacon and vice vesa. The embedded softwae that is unning on a Cicket device can be eplaced simply by uploading the flash memoy with modified o self-developed pogams. The open achitectue of Cickets has inspied eseaches all ove the wold to use Cicket as a platfom to develop new wieless positioning stategies and fo algoithm testing. Thee is plenty of liteatue on Cickets and applications available. The thesis of Piyantha (5) descibes the design and implementation of the Cicket indoo location system in detail. Haggag and Mehaei (6) document thei modification of the default achitectue that enables coodinated obot inteaction. Wang (4) lays the foundations fo leveaging the Cicket indoo location system to supply oientation infomation. He also demonstates end-to-end functionality of a Cicket Compass. Howeve, thee ae seveal disadvantages when choosing Cicket as a platfom fo anging and positioning. In ode to obtain anges between motes, the time of flight between an ultasonic pulse and a adio signal needs to be measued. Both, the US pulse and the RF signal sometimes suffe fom multipath effects, in paticula indoos due to eflections at walls, windows, tables o the floo. When a listene eceives a eflected signal instead of the diect signal along the line-of-sight, a too long ange is detemined. A multipath signal is paticulaly likely to occu when a beacon node is not oientated towads the listene. Typically a listene unit can detect ultasonic signals fom a beacon within a 4 degee cone. If the beacon node is not oientated diectly towads a

43 4 Jounal of Global Positioning Systems listene, the listene eceives a eflected signal instead of the diect signal. Fig. shows such a scenaio, whee a multipath signal is eceived. In ode to eliminate a goss eo due to multipath, a high edundancy of ange measuements (i.e. moe than 5 anges to each node) is necessay. Beacon Fig.. Multipath scenaio whee the signal is eflected at the ceiling Thee ae seveal moe disadvantages associated with the use of ultasound. The speed of ultasound is highly coelated to the tempeatue. Although cicket units cay tempeatue sensos on thei chip sets, it is had to obtain an accuate tempeatue along the path between sende and eceive. The speed of ultasound depends tightly on the speed of wind, which doesn t allow fo accuate positioning outdoos. With the ultasound sende not tansmitting omni-diectionally, it is almost impossible to set up a dense ad-hoc netwok with a lage numbe of Cicket units. In a lage ad-hoc senso netwok the condition that the nodes face each othe is nomally not fulfilled. Taylo (5) and Taylo et al. (6) modified a mobile cicket by attaching two additional ultasound tansduces. These tansduces moe closely simulate an omni-diectional acoustic pulse than the conic emanation of the standad cicket tansduce. His positioning algoithm uses ange measuements between sensos and a moving taget to simultaneously localize the sensos, calibate sensing hadwae, and ecove the taget's taectoy. In his expeiments he used up to 55 sensos to cove a 7 x mete oom. Ou autolocalisation algoithm howeve uses a lowe numbe of beacon nodes to pefom localisation and tacking. Ou local 3D positioning algoithm takes into account the weaknesses of cuent wieless ad-hoc positioning methods and algoithms, including the absence of quality and integity indicatos fo the positioning esults, existence of high vaiances and outlies in ange measuements, eos in ancho nodes (o even thei absence) as well as a coase positioning mode fo pooly conditioned netwoks. Positioning Ceiling Floo Listene Ou contibution to positioning addesses two diffeent netwok computation methods. While the fist section descibes a method to obtain the node positions with one mobile node the second section uses inte-beacon ange measuements to ceate a geodetic netwok that allows position detemination. Obtaining beacon coodinates by auto-localisation Auto-localisation is also known as Mobile Assisted Positioning o SLAT (Simultaneous Localization and Tacking) and efes to the poblem to obtain the coodinate positions of fixed ancho nodes which ae equied to enable tacking of mobile devices. Without the use of an auto-localisation algoithm the coodinates of the fixed beacon node positions would have to be detemined with anothe positioning system. Because GNSS is not available indoos and because the quality of the beacon nodes should be at least as good as the wieless positioning system (if not a magnitude bette), time-consuming manual positioning methods ae usually equied to obtain beacon coodinates. Typically tachymete measuements ae caied out with a positioning accuacy of 5- mm ( sigma). Howeve, it is not pactical to use a second positioning system to calibate the beacon nodes because that inceases time and effot. The auto-location stategy used fo positioning of the beacon nodes is shown in Fig. 3. A dynamic listene is slowly moved at diffeent locations in a oom theeby collecting anging data to 4 (o moe) beacon nodes that ae mounted at the ceiling. Both, the mobile and the static node positions ae unknown. Even the inte-beacon anges ae not available. This is the most challenging scenaio fo an auto-localisation task, but nevetheless the most likely scenaio to occu afte the nodes have been deployed in a oom. The descibed scenaio still allows ceating a igid netwok based on local coodinates. The deployment of static nodes in the fou cones of a oom allows the set up of a meaningful local coodinate system oientated along the fou othogonal walls. Ceiling Ceiling Beacons (static) Fig. 3. Auto-localisation of beacon nodes by a mobile node Geneally, the edundancy of the auto-localisation poblem in 3D is given by edundancy = R 3( B + P) + 6, ()

44 Mautz and Ochieng: A Robust Indoo Positioning and Auto-Localisation Algoithm 4 whee R is the numbe of obseved anges, B the numbe of fixed beacons and P the numbe of listene positions. If the diect line of sight conditions allow to obtain all combinations of anges, then R = B P holds. In Fig. 3, the mobile listene has collected 4 anges to 4 beacons at 6 diffeent locations. Assuming the 3D case, thee ae 3(B + P) = 3 unknown coodinates and B P = 4 ange measuements. Taking into account the 6 degees of feedom fo a fee 3D netwok, a solution would theoetically be possible without any edundancy. Howeve, ou esults show that a zeo o a low edundancy of the netwok causes the auto-localisation algoithm to fail unde eal field conditions. Due to the existence of outlie obsevations, bad geometic constaints and lineaization eos of the obective function lage eos in the position estimation ae likely to occu. This is paticulaly the case in scenaios with a small edundancy, lage mean o goss eos in the ange measuements. In ode to obtain the beacon coodinates the following pocedue had been caied out: a) Stepwise movement of the listene in a oom while collecting ange measuements. b) Gouping of the anges into P listene positions accoding to thei time stamps. c) Detection of goss eos by compaing timely neaby anges and testing tiangle conditions. d) Setting up a distance matix R between all nodes of the netwok with size (B + P) by (B + P), see Fig. 4. e) Filling the gaps of the distance-matix using a simple intepolation scheme. This step establishes ough appoximation of all inte-nodal anges. f) Setting up a local coodinate system based on the inte-nodal anges of fou nodes (pefeably beacon nodes). g) Computation of all coodinate positions based on multidimensional scaling (MDS), a localisation method that tansfoms poximity infomation into geometic embedding. Details of the algoithm can be found in Shang et al. (4). Altenatively, the positions can be detemined by multilateation fom fou locally defined nodes. Ou expeiments have shown that this altenative has bette pefomance than MDS. h) Refinement of the coodinates by geodetic netwok adustment. In application on eal data this step can usually not be caied out staight away. The eason is that netwok adustment involves lineaization of the obective function, which is eligible only if of good appoximate values of the unknowns ae available. Hee, the initial appoximate positions ae not pecise enough to diectly apply netwok adustment using the Gauss-Newton iteation. In ode to avoid a failue of the netwok adustment, a heuistic optimisation method is caied out that diectly uses the non-linea obective function. Using tial and eo the coodinate positions ae shifted in ode to fit the ange B measuements. The heuistic step impoves appoximate positions fo the netwok adustment typically fom mete to cm level. The disadvantage of using heuistic methods is the high computational cost. Howeve, taking into account that the auto-localisation is executed only once and not pocessed in eal-time, the usage of timely expensive heuistic methods is not citical. An insight into heuistic methods is given in Mautz (). B B B 3 B 4 P P P 3 P 4 P 5 P 6 B not obseved obseved B 3 B 4 P P P 3 obseved not obseved P 4 P 5 P 6 Fig. 4. Distance matix of the example with 4 beacons and 6 listene positions Afte the auto-localisation pocedue has been completed, the coodinates of the static beacon nodes ae available in a local system. An ove-detemined auto-location setup allows detemining quality indicatos of the coodinates. A numeical example based on mobile assisted positioning is given in the expeimental esults section.. Instant coodinate detemination in a sense netwok In case the netwok has an inte-nodal connectivity of c > 4 (o c > 3 in D), the netwok can be initialised without a multi-epochal auto-localisation pocedue. Once the anges between the beacons have been obtained and collected at a cental pocessing unit, the senso position coodinates can be detemined based on only a single epoch. Theeby it does not matte, whethe the nodes ae static o dynamic. The positioning stategy is based on the ceation of a igid stuctue: The key issue fo an ancho fee positioning is to find a globally igid gaph, o in othe wods, a stuctue of nodes and anges which has only one unique embedding, but still can be otated, tanslated and eflected. In 3D, the smallest gaph consists of five fully connected nodes in geneal position. If such an initial cluste passes statistical tests, additional vetices ae added consecutively using a veified

45 4 Jounal of Global Positioning Systems multilateation technique. Nodes that have not been able to take pat in the igid cluste ae positioned using a moe eo pone method and theeafte added to the cluste. The pocess flow of ou positioning stategy is illustated in Fig. 5. The ceation of a cluste aims to compute unique positions of vetices in a local coodinate system that can be tansfomed into a highe spatial efeence system by tanslations, otations and a eflection. A staightfowad method to detemine the position of an obect based on simultaneous ange measuements fom thee stations located at known sites is called tilateation. Manolakis (996) and Thomas and Ros (5) povide fast algebaic and numeic algoithms fo tilateation in obotics. Coope () shows that the effect of eos in the ange measuements can be paticulaly sevee when the tilateated point is located close the base plane o the thee known stations ae nealy aligned. Mooe et al. (4) show that thee is a high pobability of incoect ealisations of a D-gaph when the measuements ae noisy. The coodinate system of the cluste is conveniently defined in local coodinates based on the thee anges, 3, 3 between the nodes P, P, P 3. The coodinates ead P : (,,), P : (,,), P3 : + 3 3, ,. A foth point is added to the netwok by 3D-tilateation theeby abitaily choosing one of the two folding ambiguities and discading the othe. Howeve, as long as thee ae only 4 points involved, the flip ambiguity does not affect the inne stuctue of the geneal tetahedon which is spanned by the base plane and the tilateated point. As soon as a 5 th node is added to the cluste by tilateation fom the points in the base plane, and 3, the ambiguity poblem does matte, as thee ae two diffeent embeddings. As shown in Fig. 6, nodes 4 and 5 could be on eithe the same side of the base plane o on opposite sides. If the distance between nodes 4 and 5 is also measued, we call this gaph a quintilateal o in shot a quint since all 5 nodes ae fully linked by ange measuements to each othe. Only the additional ange measuement 45 between nodes 4 and 5 can disambiguate between these two embeddings. As can be seen in the example in Fig. 6, 45 is significantly longe than the eflected case 45, which means that if 45 is available, the coect embedding can be selected. Consequently, such a quint is igid in 3D, assuming the nodes ae not in a singula position. (). Ceation of a quint input ancho nodes failed failed. Tansfomation input anges find 5 fully connected nodes achieved volume test ambiguity test achieved assign local coodinates fee LS adustment Expansion of minimal stuctue (iteative multilateation) Meging of clustes (6-Paamete Tansfomation) ancho nodes available? yes Tansfomation into a efeence system etun efined coodinates and standad vaiations etun local coodinates Fig. 5. Positioning algoithm, which does not equie any initial appoximate coodinates Howeve, thee ae geometic constellations whee the ambiguity cannot be solved by the edundant ange 45, because the diffeence between the distances d 45 and d 45 is of the same magnitude as the anging eo. In ode to decide which of the two embeddings is coect, we compae the computed distances d 45 and d 45 with the measued distance 45. In some cases the diffeences between the measued and the calculated distances Δ 45 = 45 - d 45 and Δ 45 = 45 - d 45 may both be vey small. Assuming a mean eo of the ange measuement 45, say 5%, both diffeences Δ 45 and Δ 45 ae likely to pass the statistical test of thei null hypotheses, which means that both could be a esult of noise. Consequently, the ange 45 does not disambiguate between both embeddings. The best way to deal with this poblem is to eect such unstable point fomations. It is bette not to use a non- no

46 Mautz and Ochieng: A Robust Indoo Positioning and Auto-Localisation Algoithm 43 obust quint than ely on a stuctue with incoect intenal flips. In ou point of view it is cucial to ensue a coect embedding fo seveal easons. Fistly, the displacement caused by an incoect flip can be lage. Secondly, these eos have a negative affect on the expansion of the stuctue when additional vetices ae added late. Thidly, and most impotantly, once a folding eo has been intoduced in a netwok it is had to detect and eliminate it late. (a) Fig. 6. (a) Quintilateal, (b) a vesion whee node 5 has been mioed at the base plane Afte the quint is veified to be obust and not affected by a false flip, the next task is the expansion of the minimal igid stuctue. The emaining nodes ae added to the quintilateal individually using 3D-multilateation fom fou o moe stations at a time. Multilateation is basically a tilateation technique, whee the new node is initially detemined fom thee stations at a time. The edundant distance measuements ae used to disambiguate between two diffeent embeddings and to veify the initial computation. Multilateation allows edundant detemination of the nodes. The esulting coodinate diffeences povide essential infomation to detect false ange measuements, e.g. due to multipath effects. Howeve, thee is again a high pobability of incoect folding of a gaph when the measuements ae noisy. Fo instance, if a new node is multilateated fom points located closely to one plane and the anges ae affected by eos, a flip ambiguity may occu due to the mioing effect of that plane. These incoect gaph ealisations need to be avoided by identifying weak tetahedons with volumes smalle than a theshold which is diven by the estimated noise in the anges. Only tetahedons that have passed the test on obustness ae futhe consideed o othewise discaded. This step again eliminates the mioing ambiguity of nodes added to a igid stuctue and impoves the accuacy measues. Once a node s position is detemined, it seves as an ancho point fo the detemination of futhe unknown nodes. This way, stating fom the initial quintilateal the position (b) infomation iteatively speads though the whole netwok. The tilateation and multilateation poblem consideed so fa solves fo one single unknown point at a time. The sequential accumulation of nodes by multilateation is known as iteative multilateation (Savvides, ). Howeve, this technique is vey sensitive to measuement noise. Initially, small eos accumulate quickly while expanding the netwok. The popagation of eos in a lage netwok must be minimised as much as possible. Geodetic netwok adustment is an essential tool to evenly distibute the eos that have been accumulated by iteative multilateation. Netwok adustment povides coodinate estimates of seveal unknown nodes theeby impoving the eliability of the quality indicatos as detemined a posteioi, see Gafaend and Sanso (985). The theoy of linea Least-Squaes (LS) adustment can be found in Gafaend and Schaffin (993). Outlie obsevations distot the netwok but they cannot be isolated by pefoming a least-squaes adustment and analysing the esiduals. Thus, outlies need to be emoved in a sepaate analysis befoe the netwok is adusted. While pefoming simulations on the ancho fee stat-up, esults show that only a faction of vetices can become a membe of one single cluste. The emaining vetices ae likely to make up thei own clustes which may o may not be connected to neighbouing clustes. In case two clustes shae a sufficient numbe of vetices and/o ange obsevations between them, they can be meged using an ove-detemined 3 dimensional 6-paamete tansfomation. The outcome of clusteisation is a cluste of nodes with thei coodinates and vaiances in a local system. As this step is concluded by a fee minimally-constained leastsquaes adustment it is possible to assess the intenal consistency of the measuements. A moe elaboate discussion of the positioning algoithm and details of the mathematical backgound ae pesented in Mautz et al. (7)..3 Tansfomation into a efeence coodinate system Most applications equie the netwok nodes to be tied in a coodinate system of highe ode. With a minimum availability of fou ancho nodes, the local coodinates can be tansfomed unambiguously into the elevant taget system. This can be achieved by a 3D-Catesian coodinate tansfomation. A closed fom solution fo the detemination of tansfomation paametes using the 3D- Helmet tansfomation is given by Hon (987). Subsequent to the tansfomation, a fully constaint LS netwok adustment is pefomed that pemits all of the available ancho nodes and all ange measuements to be pocessed togethe in ode to efine all position

47 44 Jounal of Global Positioning Systems appoximates simultaneously. Additionally, the mean eo in the coodinates is epoted by the point confidence ellipse fo each node. 3 Expeimental Results In section 3. the pefomance of the localisation algoithm poposed in chapte is evaluated on eal senso data obtained fom Cicket nodes. In ode to asses the pefomance of a lage and dense netwok simulated anging data ae used in section Initialisation of a dynamic netwok In ode to assess the pefomance of netwok initialisation with the suppot of a mobile node (with unknown positions!), the following measuement setup was chosen: fou stationay nodes (= beacon nodes) wee deployed at the office ceiling. Due to the system achitectue of cickets the inte-beacon ange signals could be obtained. One dynamic node was caied though the oom and ange measuements taken at an inteval of s between the mobile node and the static nodes located in the cones. Within a time span of 5 minutes, ange measuements had been obtained with a pioi noise level of σ =.m. Now the task fo the positioning algoithm was to ecove the 3D netwok geomety without adding any supplementay infomation, e.g. geometic constaints o appoximate positional infomation. The main difficulty in ecoveing the elative node positions fo this configuation is that the connectivity gaph does not contain five nodes making up a quint. Consequently, the stategy descibed in section. could not be followed and the post-pocessing method descibed in. was used instead. This method included: setting up a ough distance matix, global optimisation of the obective function and netwok adustment. In a fist step, the ange measuements ae gouped into 34 epochs of.5 second intevals each by an implemented algoithm. Multiple ange obsevations within one goup ae aveaged if thee is a diffeence of less than cm, o discaded othewise. ange measuements ae finally taken into account to detemine 3 vitual node positions of the dynamic node and 4 beacon node positions. Consequently, the numbe of unknowns is 3 * 34 =. Accoding to () the edundancy of the system can be computed as + 6 = 4. The edundant distance constaints in the netwok could be used to detemine the system inconsistency and the empiic mean eo of the node positions. Afte step g) in section. had been caied out, the empiic mean eo was.5m. With application of the efinement step (global optimisation) the eo could be futhe educed to.5m and finally down to.96m by netwok adustment. This mean eo is within the magnitude of the obsevational noise level. Thus, the inne netwok geomety could be ecoveed successfully. Fig. 7 shows the location of the ecoveed node positions. [m] 4.5 Y Fig.7. X-Y view with 4 stationay cicket node positions in the cones of a oom (black dots) and 3 vitual positions of a mobile node (ed dots). This example shows that it is feasible to establish a local netwok in a oom without suveyed ancho nodes, any pesumptions on the node locations o any inte-node ange measuements between the static devices. Afte the mobile positioning algoithm has been caied out, all distances between the static nodes ae detemined. The gaph between the static nodes is a igid stuctue that can be used fo futhe navigation of mobile nodes. 3. Initialisation of a dense netwok In ode to assess the pefomance of the poposed positioning algoithm, a simulated netwok consisting of nodes was set up andomly in a m m m test cube. Assuming a maximum communication ange of 3.5 m between the adios, only the inte-node distances of less than 3.5 m have been ecoded into an obsevation file. Afte execution, the file contained 57 ange measuements. Based on these 57 anges, the positioning algoithm was used to ecove the node positions. As detailed in the section., the algoithm ceated quints, then lage clustes by lateation and cluste meging. points wee chosen andomly to seve as ancho nodes fo a 3D-tansfomation of the local cluste into the oiginal geodetic datum. The citeion used fo the pefomance assessment in positioning is the aveage deviation a p X [m] n i ( ˆ P ), = P (3) = n 4 i i

48 Mautz and Ochieng: A Robust Indoo Positioning and Auto-Localisation Algoithm 45 whee n is the numbe of nodes, P i the tue position vecto and Pˆ i the estimated position vecto of the localised node i. The intenal consistency of the fee netwok is assessed by the squae oot of the estimated efeence vaiance m v T v = m 3n m 3n ˆ i = i ( ) ˆ σ = (4) whee v is the vecto of esiduals containing the diffeences between the estimated distances ˆ i (obtained fom a LS-adustment) and the measued distances i. aveage deviations a p and â [m] Fig. 8. Tue position deviations a p (pluses) fo measuement noise σ between m and.m. Fo compaison, the dots show the estimated deviations â p noise level σ [m] mean eos σˆ / deviations ap [m] noise level σ [m] Fig. 9. Tue position deviations a p (pluses) in compaison with the estimated efeence mean eos shown as dots. Fig. 8 shows how the noise level in the anges σ influences the aveage position deviation a p of the localised nodes at a maximum signal ange max = 3.5m. The linea dependencies on σ and a p in Figs. 8 and 9 have appoximately an aveage popotionality facto of. This testifies that ou localisation algoithm has almost eached the best possible pefomance level, which has i been detemined by Savvides (3a) as the Came Rao Bound behaviou. This is especially tue fo noise levels with less than 3.5% eo (.m level), whee almost all nodes in all netwoks have been localised coectly. Note that a single falsely located node (e.g. due to a false folding ambiguity) causes the aveage position deviation to ise significantly. 4 Conclusions The cicket senso node system has been used to successfully apply a full 3D auto-location algoithm. Futhemoe, it could be shown that a dense netwok of inte-beacon anges can be used compute an instantaneous geodetic netwok. Expeimentation has shown that the motion of a mobile node can be exploited to automatically ceate the igid topology of the netwok nodes. Range measuements taken at vaious points in time of a mobile node have enabled positioning of all nodes in a local coodinate system. Howeve, this auto-localisation method equies a high edundancy of obsevations and a full elimination of outlies in the ange measuements, since the computation of coodinates is extemely sensitive to eos. A second expeiment based on simulation has demonstated the feasibility to detemine dense ad-hoc distance netwoks with the pesence of lage obsevation eos and poo geometic conditions. In ode to achieve a eliable positioning based on geodetic adustment even in the pesence of eos and sub-optimal geomety, a obust algoithm has been set up, that paticulaly avoids flip ambiguities in the netwok. We have studied netwoks with elatively lage measuement eos of up to 7.5% of the tue anges and shown that it is possible to achieve a position deviation that is of the size of the anging eo. Futue wok will focus on futhe moving away fom laboatoy conditions. The application of the algoithms fo othe senso hadwae (besides the Cicket System) will be the next challenge. A mao challenge will be a positioning functionality fo ill-conditioned netwoks that makes best use of available ange measuements, connectivity infomation, tempoal-spatial deivatives, tavel behaviou and GIS data. Refeences Coope I (), Reliable computation of the points of intesection of n sphees in n-space, ANZIAM Jounal, Vol. 4(E), pp. C46-477,. Danell, C and Wilczoch, C (), Real Time Positioning; Constuction and implementation of a GPS- Communicato. Maste s thesis in Contol and

49 46 Jounal of Global Positioning Systems Communication, Repot no. LITH-ISY-EX-346-, Linköping Univesity, Sweden. Een, T; Goldenbeg, D; Whiteley, W; Yang, Y; Mose, A; Andeson, B and Belhumeu, P (4), Rigidity, computation and andomization of netwok localization. In: Poceedings of IEEE Infocom 4, Hong Kong, China, Apil 4. Goldenbeg D; Kishnamuthy, A; Maness, W; Yang, Y; Young, A; Mose, A; Savvides, A and Andeson B (5), Netwok Localization in Patially Localizable Netwoks. In Poceedings of IEEE INFOCOM 5, Miami, FL, Mach 3-7, 5. Gafaend, E and Sanso, F (Editos) (985), Optimization and Design of Geodetic Netwoks. Spinge-Velag. Gafaend, E and Schaffin, B (993), Ausgleichung in lineaen Modellen, Wissenschaftsvelag, Mannheim, Leipzig, Wien Züich Haggag, H and Mehaei, G (6), Robot Inteaction Using Cicket, an Indoo Positioning System, Technical Repot of the Mayland Engineeing Reseach Intenship Teams (MERIT), Univesity of Mayland and Viginia Commonwealth Univesity. Hon, B (987), Closed-fom solution of absolute oientation using unit quatenions. Jounal of Opt. Soc. Ame., vol. A-4, pp , 987. Manolakis, D (996), Efficient Solution and Pefomance Analysis of 3-D Position Estimation by Tilateation, IEEE Aeospace and electonic systems, Vol 3, No. 4, Oct Mautz, R., Ochieng, W.Y., Bodin, G., Kemp, A. (7), 3D Wieless Netwok Localization fom Inconsistent Distance Obsevations, Ad Hoc & Senso Wieless Netwoks, Vol. 3, No. 3, pp Mautz, R. (), Solving Nonlinea Adustment Poblems by Global Optimization, Bollettino di Geodesia e Scienze Affini, Vol. 6, No., pp Mooe, D; Leonad, J; Rus, D and Telle, S (4), Robust distibuted netwok localization with noisy ange measuements, Poceedings of the ACM Symposium on Netwoked Embedded Systems, 4. Niculescu, D and Nath, B (), Ad-hoc positioning system, in: IEEE GlobeCom,. Piyantha N. B. (5), The Cicket Indoo Location System, PhD Thesis, Massachusetts Institute of Technology, June 5, 99p. Savaese, C; Langendoen, K and Rabaey, J (), Robust positioning algoithms fo distibuted ad-hoc wieless senso netwoks, in: USENIX Technical Annual Confeence, Monteey, CA,, pp Savvides, A; Han, C and Stivastava, M (), Dynamic Fine- Gained Localization in Ad-hoc Netwoks of Sensos, Poceedings of ACM SIGMOBILE, Rome, Italy, July Savvides, A; Pak, H and Sivastava, M (3), The n-hop Multilateation Pimitive fo Node Localization Poblems. MONET 8(4): Savvides, A; Gabe, W; Adlakha, S; Moses, R and Sivastava, M (3a), On the Eo Chaacteistics of Multihop Node Localization in Ad-Hoc Senso Netwoks, Poceedings of the Second Intenational Wokshop on Infomation Pocessing in Senso Netwoks (IPSN'3), Palo Alto, Califonia, Shang, Y; Ruml, W; Zhang, Y and Fomhez, M (4), Localization fom Connectivity in Senso Netwoks, IEEE Tansactions on Paallel and Distibuted Systems, vol. 5, no., pp , Nov. 4. Smith, A; Balakishnan, H; Goaczko, M and Piyantha N (4), Tacking Moving Devices with the Cicket Location System, Poc. nd USENIX/ACM MOBISYS Conf., Boston, MA, June 4. Taylo, C., Rahimi, A., Bachach, J., Shobe, H., and Gue, A. (6), Simultaneous localization, calibation, and tacking in an ad hoc senso netwok. In: Poceedings of the 5. intenational Confeence on infomation Pocessing in Senso Netwoks (Nashville, Tennessee, USA, Apil 9 -, 6). IPSN '6. ACM Pess, New Yok, NY, DOI= Taylo, C J; (5), Simultaneous Localization and Tacking in Wieless Ad-hoc Senso Netwoks, Maste Thesis of Engineeing, Depatment of Electical Engineeing and Compute Science, Massachusetts Institute of Technology. Thomas, F and Ros, L (5), Revisiting Tilateation fo Robot Localization, IEEE Tansactions on Robotics, Vol., No., pp. 93-, Febuay 5. Wang, K J (4), An Ultasonic Compass fo Context-Awae Mobile Applications, Maste Thesis of Engineeing, Depatment of Electical Engineeing and Compute Science, Massachusetts Institute of Technology.

50 Jounal of Global Positioning Systems (7) Vol.6, No.: Latest Developments in Netwok RTK Modeling to Suppot GNSS Modenization Hebet Landau, Xiaoming Chen, Adian Kipka, Ulich Vollath Timble Teasat GmbH Abstact. Global Navigation Satellite Systems like the US Global Positioning System GPS and the Russian GLONASS system ae cuently going though a numbe of modenization steps. The fist satellites of the type GPS-IIR-M with LC suppot wee launched and fom now on all new GPS satellites will tansmit this new civil L signal. The fist launch of a GPS-IIF satellite with L5 suppot is announced fo sping 8. Russia has stated to launch GLONASS-M satellites with an extended lifetime and a civil L signal and has announced to build up a full 8 satellite system by 7 and a 4 satellite system by 9. Independently of that the Euopean Union togethe with the Euopean Space Agency and othe patneing counties ae going to launch the new Euopean satellite system Galileo, which will also povide woldwide satellite navigation sevice at some time afte. As a consequence we can expect to have vey heteogeneous eceive hadwae in these efeence station netwoks fo a tansition peiod which could last until 5. Netwok seve softwae computing netwok coections will have to deal with an inceased numbe of signals, satellites and heteogeneity of the available data. The complexity but also the CPU load fo this seve softwae will incease damatically. With the inceasing numbe of signals and satellites the demands fo the netwok seve softwae is gowing apidly. The pogess on the satellite system side is going hand in hand with the tendency of the customes to opeate gowing numbes of efeence station eceives esulting in highe demands fo CPU powe. The pape pesents a new appoach, which allows us to pocess data fom a lage numbe of efeence stations and multiple signals via a new fedeated Kalman filte appoach. With the newest impovements in the GLONASS satellite system, moe and moe Netwok RTK sevice povides have stated to use GLONASS capable eceives in thei netwoks. Today, pactically all sevice povides, who ae using GLONASS, ae applying the Vitual Refeence Station (VRS) technique to delive optimized coection steams to the uses in the field. Diffeent satellite systems and geneations equie diffeent weighting in netwok seve pocessing and eceive positioning. The netwok coection quality depends vey much on the satellite and signal type. New message types have been ecently developed poviding individualized statistical infomation fo each ove on unmodeled esidual geometic and ionospheic eos fo GPS and GLONASS satellites. The use of this infomation leads to RTK pefomance impovements, which is demonstated in pactical examples. Keywods: GPS, GNSS, GNSS Modenisation, Netwok RTK. INTRODUCTION Afte its intoduction in the late 9s, Netwok RTK technology based on the Vitual Refeence Station (VRS) appoach became an accepted and poven technology, which is widely used today in a lage numbe of installations all ove the wold. Developments ove the past yeas (Chen et al., 3, 4, 5; Kolb et al., 5, Landau et al., ; Vollath et al.,, ) have esulted in a solution, which is maketed unde the name GPSNetTM since 999 (Vollath et al., ). Compaing with taditional single base RTK technology, netwok RTK emoves a significant amount of spatially coelated eos due to the toposphee, ionosphee and satellite obit eos, and thus allows pefoming RTK positioning in efeence station netwoks with distances of 4 km o moe fom the next efeence station while poviding the pefomance of shot baseline positioning. Cuently moe than 5 efeence stations ae opeating in netwoks in moe than 3 counties using the Timble GPSNet solution. Data pocessing in GPSNet utilizes the mathematically optimal Kalman filte technique to pocess data fom all netwok efeence stations. This compehends modelling all elevant eo souces,

51 48 Jounal of Global Positioning Systems including satellite obit and clock eos, efeence station eceive clock eos, multipath and paticulaly ionospheic and topospheic effects. To optimize eal-time computational pefomance, the Timble patented FAMCAR (Factoized Multi-Caie Ambiguity Resolution) methodology has been used to factoize uncoelated eo components into a bank of smalle filtes, i.e. Geomety Filte and Geomety-fee Filtes and Code-caie Filtes (Vollath et al., 4, Kolb et al., 5). This appoach esults in significantly highe computational efficiency. Howeve, due to the fact that the geomety filte still contains a lage numbe of states (seveal hundeds to thousand states depending on the numbe of stations in the netwok), GPSNet until ecently was able to pocess 5 efeence stations on a single PC seve only, lage netwoks wee divided into sub-netwoks and opeated by multi-seve solutions. In ecent yeas, moe and moe sevice povides have setup efeence netwoks to povide nation-wide o egion-wide RTK sevices. Many of them contain moe than 5 efeence stations, i.e. JENOBA, Japan (338 stations), E.ON Ruhgas AG ASCOS, Gemany (moe than 8 stations); Odnance Suvey, United Kingdom (86 stations), and many existing netwok opeatos intend to extend thei netwok to seve lage aeas. In ode to allow the pocessing of lage netwoks on one single PC, an efficient appoach Fedeated Geomety Filte has been developed and implemented in Timble s latest infastuctue softwae (GPSNet vesion.5). Speeding up the GeOMETRY FILTER Centalized Geomety Filte The geomety filte plays an impotant ole in the GNSS netwok data pocessing. It povides not only the float estimation of ionosphee-fee ambiguities fo late netwok ambiguity fixing, but also povides topospheic zenith total delay (Vollath et al, 3). This filte is usually unning as a centalized Kalman filte. The typical state vecto in the filte consists of: obseved at each station. Fo a station netwok and satellites obseved in each station, the filte has 38 states; fo a station netwok and 8 satellites obseved in each station, the filte has 47 states. With the incease in the numbe of stations in the netwok and numbe of satellites obseved on each station, the numbe of states thus pocessing time will incease damatically. Table : Numbe of states in the centalized geomety filte Stations Satellites States Fig. shows the numbe of multiplications equied fo one filte step (one epoch of data sent though the filte) fo a given numbe of stations with the assumption that satellites ae obseved on each station. As the most expensive opeation in the filte is the multiplication, this figue can be appoximately intepeted as the elationship between numbe of stations and computational load of the filte. In Fig., the blue bas give the numbe of multiplications in billions fo numbe of station fom up to. The pink line in the figue epesents the function (36x) 3, which fits pefectly to the equied multiplications. So, it is clea that the computational time inceases cubically with the numbe of stations in the netwok. Topospheic zenith total delay (ZTD) pe station Receive clock eo pe station Satellite clock eo pe satellite Ionosphee-fee ambiguity pe station pe satellite Obit eos Table shows the numbe of states in the filte with given numbe of stations and numbe of satellites

52 Landau et al.: Latest Developments in Netwok RTK Modeling to suppot GNSS Modenization 49 Num of Multiplication in Billion No. Multiplications Cubic function Numbe of stations satellite obit eo states ae estimated with a fame filte. This fame filte uses only a subset of the efeence stations in the netwok to estimate the obit eo paametes. Then the estimated obit eos ae applied diectly to obsevation pocessed in the local filtes. Fig. illustates the block diagam of a Fedeated Geomety Filte fo GNSS netwok pocessing. This appoach contains one fame filte, a bank of single station geomety filtes (one pe efeence station) and one cental fusion maste filte. Fig. : Relation between numbe of efeence stations and equied multiplications in one filte step Fedeated Geomety Filte The Fedeated Kalman filte was intoduced by N.A. Calson (99). The basic idea of fedeated filte is that: A bank of local Kalman filtes uns in paallel. Each filte opeates on measuements fom one local senso only. Each filte contains unique states fo one local senso and common system states fo all the local sensos. A cental fusion pocesso computes an optimally weighted least-squae estimate of the common system states and thei covaiance. Then the esult of the cental fusion pocesso is fed back to each local filte to compute bette estimates fo the local unique states. The main benefit of this appoach is that each local filte uns with educed numbe of states and the computation time fo the whole system inceases only linealy with the incease of the numbe of local sensos. This significantly educes the computational load compaed to the centalized filte appoach. Fo GNSS netwok pocessing, each efeence station can be teated as a local senso with unique states like ZTD, eceive clock eo and ionosphee-fee ambiguities (+n, whee n is numbe of satellites in the system), and common states like satellite clock eos and obit eos ( n + m x n, whee n is numbe of satellites in the system and m is numbe of obit eo paamete pe satellite). Theefoe the fedeated filte appoach can be applied. As thee ae still too many common states, a futhe step can be taken to futhe educe the computational load. The Fig. : Block diagam of a Fedeated Geomety Filte Pefomance Analysis Ou pefomance analysis includes two pats. One is the post-pocessing pefomance compaison between the centalized geomety filte appoach and fedeated geomety filte appoach. It is focusing on the seve pefomance availability, eliability of the netwok pocessing and pocessing time. The othe pat is the ealtime pefomance analysis focusing on the RTK ove positioning and fixing pefomance in the netwok. Post-pocessing Pefomance The post-pocessing pefomance study uses a postpocessing vesion of GPSNet. The fist test pefomed is to check the availability (pecentage of fixed ambiguities) and eliability (pecentage of coectly fixed ambiguities) with both the centalized geomety filte appoach and the fedeated geomety filte appoach. Fou days of data (days 89, 9, 9 and 3 of the yea 3) fom the Bavaian Land Suvey Depatment BLVG netwok (45 GPS stations, Gemany) wee used in the test. Table summaizes the test esults. Fo the GPS only netwok (BLVG), both appoaches give simila esults in tems of availability and eliability.

53 5 Jounal of Global Positioning Systems Table : Netwok Post-pocessing pefomance test (availability and eliability) Centalized Appoach Fedeated Appoach Availability Reliability Availability Reliability BLVG BLVG BLVG BLVG The second analysis is to check the pocessing time needed by the centalized and fedeated geomety filte appoaches. In this test, one day data of 3 efeence stations fom five Geman states [Bayen, Nodhein- Westfalen, Hessen, Thüingen and Niedesachsen] was used as shown in Fig. 3. fedeated filte appoach. Fo a 5 station netwok, the fedeated filte appoach is 8 times faste and fo a station netwok, the fedeated filte appoach is 63 times faste than the centalized filte appoach. This test poves that the fedeated filte appoach is highly computationally efficient fo lage netwoks (Table 3). Numbe of Stations Table 3: Pocessing time compaison Centalized [Minute] Fedeated. [Minute] Ratio Real Time Pefomance Fo the eal time test, two GPSNet systems wee set up in paallel. One was unning with the centalized filte appoach. Real time data steams of 45 stations fom the BLVG netwok wee used in this configuation. Anothe system was unning with the fedeated filte appoach. Real-time data steams of moe than stations fom the Geman SAPOS netwok wee used in this configuation. Two Timble 57 oves located in Timble Teasat office wee used to veify the ove positioning and fixing pefomance. The VRS data steams geneated fom these two systems wee steamed to both oves espectively. The neaest efeence station was 6 km away in both cases. Table 4: Position eo statistics Fig. 3: Test Netwok in Gemany Fom these 3 stations, we selected 5, 6, 7 up to stations to un netwok pocessing with both appoaches. The total pocessing time (including data pepaation, ionosphee modeling and netwok ambiguity fixing) of each pocess fo one day of data is summaized in Table 3. Fo a 5 station netwok, the fedeated filte appoach uses minutes to pocess the data, while the centalized filte uses 73 minutes. Fo a station netwok, the fedeated filte appoach uses 57 minutes, while the centalized filte appoach used 358 minutes (nealy.5 days) to pocess one day of data, which means it is impossible to pocess data in eal-time. Table 3 also gives the atio of pocessing time between centalized filte and Mean -Sigma RMS Centalized [m] Fedeated [m] Noth.. East Height..5 Noth.8.7 East.5.5 Height.3.3 Noth.7.7 East.8.8 Height.3.3

54 Landau et al.: Latest Developments in Netwok RTK Modeling to suppot GNSS Modenization 5 Table 4 summaizes the statistics of position eos ove one day, which indicate that the positioning pefomances fom both systems ae the same fom a statistical point of view. Anothe test conducted in eal time is to check the RTK fixing pefomance. The test setup is the same as the positioning pefomance test. Table 5 summaizes the RTK fixing pefomance duing one day in tems of mean fixing time, 68%, 9%, 95% quantiles and minimum, maximum fixing time. Though the minimum and maximum fixing times fo the ove in the system unning the fedeated filte appoach ae longe than the centalized filte appoach, othe statistics ae vey much the same. ionospheic eos at the ove. The poposed paametes and elations ae fo the ionospheic eo whee σ i = σ ic + σ id σ ic = Constant tem of standad deviation fo dispesive pediction eo σ id d = = Distance dependent tem of standad deviation fo dispesive pediction eo Distance to neaest efeence station d Table 5: Mean [s] RTK fixing pefomance 68% 9% 95% [s] [s] [s] Min [s] Max [s] Fo the non-dispesive eo we use Centalized Fedeated σ = σ c + σ d d + σ h Δh IMPROVING ROVER PERFORMANCE USING NETWORK CORRECTION QUALITY INFOR- MATION Latest developments have shown that it is possible to impove the ove positioning pefomance by using statistical infomation fo the pedicted esidual eo at the ove location. The models used in netwok RTK (e.g. ionospheic, obit and topospheic eos) ae educing eo souces damatically but they ae unable to eliminate the eos completely. Applying specific methods as descibed by Chen et al. (3) the pedicted vaiance of the geometic and ionospheic coection fo each ove location can be computed fom the available data fo each satellite individually. These pedicted values can be used in the ove to deive an optimum position solution using specific weighting mechanisms. The application of this appoach is descibed below and esults ae pesented showing the positioning pefomance due to the use of the computed statistical infomation. The VRS method geneates optimized coections fo individual ove locations. Howeve, eos cannot be completely eliminated. Based on the available data, density of the netwok and iegulaities in atmospheic conditions, diffeent esidual eos ae affecting the solution. Ou VRS netwok seve softwae GPSNet is able to pedict vaiances of esidual eos at the individual ove location fo each satellite in view. These paametes chaacteize the expected geometic and whee σ c = Constant tem of standad deviation fo non-dispesive pediction eo σ d = Distance dependent tem of standad deviation fo non-dispesive pediction eo σ h = Height dependent tem of standad deviation fo non-dispesive pediction eo d = Distance to neaest physical efeence station Δ h = Height diffeence to efeence station The distance dependent pat was intoduced to descibe the eo gowth with the distance to the neaest physical efeence station. The height dependent pat is used to descibe the eo gowth due to topospheic. Typically the eos gow with distance fom efeence stations, i.e. the estimates fo the dispesive and non-dispesive eos at the ove location will be dependent on the ove location in the netwok. As we can see in figue 4 the eo is small fo some aeas aound the efeence stations and inceasing with the distance. An altenative appoach, which is desiable, is to continuously compute the eo statistics in the netwok seve softwae fo the cuent ove position. In that case the distance and height

55 5 Jounal of Global Positioning Systems dependent pats of the equations descibing the eos will be zeo. The following figue 4 shows a typical eo behavio fo the ionospheic effect. Fig. 4: Typical ionospheic eo distibution in a VRS netwok in time peiods of stong ionosphee [values in metes] The above paametes can be used in the ove to contol the optimum weighting of Vitual Refeence Station data fo the individual satellites in the position solution and thus lead to inceased position accuacy. It can also be used to suppot the ambiguity seach pocess and the optimum combination of L and L obsevations to deive a minimum-eo position estimate. To veify this idea data fom two diffeent netwoks wee used. The fist netwok is based on Teasat owned efeence stations (Timble NetRS and NetR5 eceives) in the suounding of Munich, Gemany. The station Hoehenkichen was not pat of the netwok pocessing, it was used as a ove station only. The neaest efeence station is Gosshöhenain, which is appoximately 6 km away. An optimum VRS data steam was geneated fo a full day and this data steam was used to position the ove Hoehenkichen with the Timble RTK engine. The RTK engine was un in the standad mode and in a modified mode, in which the RTK engine made use of the statistical infomation on ionospheic and geometic esidual eos in the VRS data steam. In ode to visualize the accuacy impovement the complete day was cut in 48 ½ hou pats and the 3D RMS fo each ½ hou slot was computed and visualized. The geen bas in figue 6 epesent the RMS values fo the standad pocedue peviously used in the RTK engine while the ed bas epesent eos fo the optimized solution. The cyan bas ae showing the aveage pedicted ionospheic eos. The gaph shows that in almost all cases the optimized solution was able to educe the position eos by up to a facto of. Fo some of the ½ hou slots no impovement was eached, which will need to be the topic fo futhe eseach. The poblematic times ae mainly the ½ hou peiods with highe ionospheic esidual eos. This would be consistent with an ionosphee-fee caie phase poviding the best solution hee. Fig. 6: 3D-RMS values fo ½ hou slots fo the optimized solution in ed, standad solution in geen (iono coection sigmas in cyan) To show the individual eos in detail a ½ hou peiod was selected and the following figues show the eos fo the standad solution in blue and the optimized solution in ed in Noth, East and Height. It can be easily seen that the optimized solution povides much bette accuacy in all thee components. Fig. 5: Refeence station netwok in the suounding of Munich

56 Landau et al.: Latest Developments in Netwok RTK Modeling to suppot GNSS Modenization 53 Fig. 7: Position eos in Noth diection fo the optimized solution in ed (5 mm RMS) and the standad solution in blue (9 mm RMS) Fig. : Refeence station netwok in the suounding of Munich (mainly Land Suvey Dept. netwok stations) Fig. 8: Position eos in East diection fo the optimized solution in ed ( mm RMS) and the standad solution in blue (6 mm RMS) The distance to the neaest efeence station is appoximately 3 km. A vitual efeence station was geneated fo the position of Hoehenkichen while eceive data fom station Hoehenkichen was not used in the netwok as in the pevious example. Then the VRS data was used to position the ove. The esulting position eos ae shown in the figues below. Fig. 9: Position eos in Height diection fo the optimized solution in ed (3 mm RMS) and the standad solution in blue ( mm RMS) Fig. : Position eos in Noth diection fo the optimized solution in ed (5 mm RMS) and the standad solution in blue (6 mm RMS) The second netwok is using stations of the Bavaian Land Suvey Depatment netwok (Mainly non-timble eceives) and a ove location at the Teasat office in Hoehenkichen (Timble R8 GNSS). The distance between the efeence station is typically about 5 km.

57 54 Jounal of Global Positioning Systems SUMMARY Fig. : Position eos in East diection fo the optimized solution in ed (3 mm RMS) and the standad solution in blue (6 mm RMS) Continuing R&D on VRS technology allows us to povide solutions, which can pocess lage netwoks with moe satellites and signals and suppot multiple satellite systems. Pefomance analyses fo the fedeated filte appoach show that availability and eliability of netwok pocessing ae compaable and the ove pefomance stays the same compaed to the centalized filte appoach. Using pedicted dispesive and non-dispesive quality infomation computed fom GPSNet fo the ove location and all GPS and GLONASS satellites impoves the ove positioning pefomance consideably when using the VRS technology. We hope that this technology will be accepted soon by the industy and will be available in almost all the existing VRS netwoks. ACKNOWLEDGEMENT Fig. 3: Position eos in Height diection fo the optimized solution in ed (9 mm RMS) and the standad solution in blue (3 mm RMS) Again it can be easily seen that the position eos ae vey much smalle fo the optimized case, in which we ae using the pedicted esidual eo infomation fom the netwok. All ou tests so fa have shown that the use of the eo estimates fom the netwok have been able to impove the positioning accuacy consideably. The analysis we have done until now is a pue offline post-pocessing one, which allowed us to veify the usefulness of the appoach. The RTCM SC4 committee is cuently discussing the potential ceation of RTCM vesion 3 messages to tansmit these paametes fom the netwok seve to the use in the field fo GPS and GLONASS satellites. These new messages will allow us to impove ou RTK accuacy in futue systems. Initialization Pefomance Besides the RTK positioning accuacy the RTK initialization pefomance can also be impoved. Fist analysis of the Time To Fix pefomance fo the VRS netwoks analyzed above show that the initialization time can be educed by a facto of appoximately 3% compaed to the aleady excellent ambiguity esolution pefomance typically seen in netwoked RTK. We thank the Land Suvey depatments of Bavaia, Hessen, Nodhein-Westfalen, Niedesachsen, Baden- Wüttembeg, Thüingen and E.ON Ruhgas AG fo poviding us data and eal-time data steams fom thei netwoks duing the test and allowing us to use the data in this eseach. REFERENCES Calson, N.A. (99) Fedeated Squae Root Filte fo Decentalized Paallel Pocesses, IEEE Tan. On Aeospace and Electonic Systems, Vol. AES-6, No. 3, May, 99 Chen, X., Deking, A., Landau, H., Stolz, R., Vollath, U. (5) Coection Fomats on Netwok RTK pefomance, Poceedings of ION-GNSS 5, Sept. 5, pp Chen, X., Vollath, U., Landau, H. (4) Will GALILEO/ Modenized GPS Obsolete Netwok RTK, Poceedings of ENC-GNSS 4, May, 4, Rottedam, Nethelands. Chen, X., Landau, H., Vollath, U. (3) New Tools fo Netwok RTK Integity Monitoing, Poceedings of ION- GPS/GNSS 3, Sept. 3, pp Kolb, P.F., Chen, X., Vollath, U. (5) A New Method to Model the Ionosphee Acoss Local Aea Netwoks, Poceedings of ION-GNSS 5, Sept. 5, pp Landau, H., Vollath, U., Chen, X. () Vitual Refeence Station Systems, Jounal of Global Positioning Systems, Vol., No. : pp , Minkle, G., Minkle, J. (993) Theoy and Application of Kalman Filteing, Palm Bay: Magellan Book Company, 993.

58 Landau et al.: Latest Developments in Netwok RTK Modeling to suppot GNSS Modenization 55 Vollath, U., Deking, A., Landau, H., Pagels, C., Wagne, B. () Multi-Base RTK Positioning using Vitual Refeence Stations, Poceedings of ION-GPS, Sept., Salt Lake City, USA Vollath, U., Deking, A., Landau, H., Pagels, C. () Long Range RTK Positioning using Vitual Refeence Stations, Poceedings of the Intenational Symposium on Kinematic Systems in Geodesy, Geomatics and Navigation, Banff, Canada, June,. Vollath, U., Bockmann, E., Chen, X. (3) Toposphee: Signal o Noise?, Poceedings of ION-GPS/GNSS 3, Sept. 3, pp Vollath, U., K. Saue (4) FAMCAR Appoach fo Efficient Multi-Caie Ambiguity Estimation, Poceedings of ENC-GNSS 4, May 4, Rottedam, Nethelands

59 Jounal of Global Positioning Systems (7) Vol.6, No.: Integation of RFID, GNSS and DR fo Ubiquitous Positioning in Pedestian Navigation Guenthe Retsche and Qing Fu Institute of Geodesy and Geophysics, Reseach Goup Engineeing Geodesy, Vienna Univesity of Technology, Austia Abstact. Location detemination of pedestians in uban and indoo envionment can be vey challenging if GNSS signals ae blocked and only pseudoange measuements to less than fou statellites ae avialable. Theefoe a combination with othe wieless technologies fo absolute position detemination and dead eckoning (DR) fo elative positioning has to be pefomed. Radio Fequency Identification (RFID) is an emeging technology that can be employed fo location detemination of a mobile use in indoo and uban envionment. RFID tanspondes (o tags) can be placed at known location (so-called active landmaks) in the envionment and the use who has to be positioned can cay a RFID tansceive (o eade). Then the location of the use can be obtained using cell-based positioning o with tilateation if anges to seveal tags ae deduced. In this pape the use of active RFID in combination with satellite positioning and DR is investigated. Fo that pupose the integation with GNSS and othe wieless technologies is discussed and the deduction of anges to RFID tags is investigated. Test esults show that the anges to RFID tags can be deduced fom signal stength obsevations to tags in the suounding envionment. Two diffeent models that descibe eithe a logaithmic o linea elationship between the measued signal stength and the distance to the tag ae analyzed. In addition, if pseudoange obsevations to GNSS satellites can be measued then they can also be used with anges to RFID tags to obtain the position fx. The absolute position can then be used to update the dift ates of the DR sensos which ae used fo continuous position detemination. Diffeent scenaios fo the coection of the DR dift ae descibed in the pape. The pesented eseach is conducted in a new eseach poect at the Vienna Univesity of Technology. Keywods: Integated positioning, Active RFID, GNSS, Dead Reckonig (DR), Minimum Range Eo Algoithm (MRERA) Intoduction In a new eseach poect called Ubiquitous Catogaphy fo Pedestian Navigation (UCPNAVI) at the Vienna Univesity of Technolgy we ae cuently exploing the capabilities of poviding location based infomation and navigation via an ubiquitous envionment to enhance oute guiding in smat envionments. The eseach hypothesis that ubiquitous catogaphy, defined as a technological and social development, made possible by mobile and wieless technologies, that eceives, pesents, analyses and acts upon map data which is distibuted to a use in a emote location, enables customized oute guiding with vaious pesentation foms and theefoe optimizes the wayfinding pocess. Smat stations (in tems of active and shot-ange devices) can substitute o complement taditional positioning and infomation tansmission methods by sending infomation o coodinates of the station instead of tying to locate the use by cental seve-based solutions. Diffeent techniques and sensos ae tested and a knowledge-based multi-senso fusion model is applied (see Retsche, 5; Thienelt et al., 7) to enhance location detemination in smat envionments. Especially in complex buildings, visitos often need guidance and suppot. Studies showed that people tend to lose oientation a lot easie within buildings than outdoos, especially if not moving along windows (see e.g. Hohenschuh 4, Radoczky 3). Additionally to navigation suppot it could be beneficial to supply the use with infomation that is adapted to the cuent task, e.g. when stolling aound an aipot o tain station infomation about depating planes o tains that concen the use could be povided. Instead of passive systems that ae installed on the use s device and fequently position them as the use moves along in an indoo envionment, new technologies oiginated in ubiquitous computing could enich guiding systems by including infomation captued fom an active envionment. This would mean that the use is peceived by an ubiquitous envionment and eceives location based infomation that

60 Retsche and Fu: Integation of RFID, GNSS and DR fo Ubiquitous Positioning in Pedestian Navigation 57 is suitable fo the espective device o is supplied with helpful notes via a public display o simila pesentation tools. Additionally to the function of infomation tansmission poles, these smat stations could possibly substitute o complement taditional indoo positioning methods by sending coodinates of the station instead of locating the use. Based on the concept of Active Landmaks, which actively seach fo the use and build up a spontaneous ad-hoc netwok via an ai-inteface, a ubiquitous solution, whee an infomation exchange between diffeent obects and devices ae accomplished, is investigated fo the use in navigation. The concept fo an ubiquitous positioning solution enables a evolutionay oppotunity fo navigation systems of any kind. Within the last few yeas a lot of eseach and development has taken place concening Location-based Sevices (LBS), which could now be supplemented and expanded with the help of ubiquitous methods, and maybe in the futue they could even be eplaced. Positioning and tacking of pedestians in smat envionments function diffeently fom conventional navigation systems, since not only passive systems, that execute positioning on demand, need to be consideed. Moeove a combination of active and passive positioning methods should be the basis of a ubiquitous navigation system. Such a multi-senso system fo position detemination should theefoe be able to include both types of location detemination and as a esult lead to an impovement of positioning accuacy. In a fist step, the use of RFID (Radio Fequency Identification) fo ubiquitous positioning is investigated in the poect. Fo location detemination RFID tags can be placed at active landmaks o at known locations in the suounding envionment. If the use passes by with an RFID eade the tag ID and additional infomation (e.g. the 3-D coodinates of the tag) ae etieved. Theeby the ange between the tag and eade in which a connection between the two devices can be established depends on the type of tag. Fom measued signal powe levels the coesponding ange to the tag s location can be deduced. If anges to at least thee RFID tags ae available then the position fix can be obtained using tilateation. Navigation systems usually also employ dead eckoning (DR) sensos whee the cuent location of the use is detemined using obsevations of the diection of motion (o heading) and the distance tavelled fom a known stat position. Due to the main limitations of DR sensos, i.e., the lage dift ates of the sensos, an absolute position detemination is equied at cetain time intevals to update the DR obsevations and coect fo the senso dift. The absolute position detemination is usually pefomed with satellite positioning (GNSS). RFID positioning can povide this position updates in smat envionments whee satellite positioning is not available. In this pape the positioning of a mobile use in uban envionment based on RFID in combination with GNSS and Dead Reckoning (DR) is investigated. Use of Active RFID in Positioning Radio Fequency Identification, o RFID fo shot, is an automatic identification method. An RFID tag is a tansponde that can be attached to o incopoated into a poduct, animal, o peson fo the pupose of identification using adiowaves. Othe system components include a eade (i.e., a tansceive) with antenna. The eade is able to ead the stoed infomation of the tag in close poximity. RFID tags contain antennas to enable them to eceive and espond to adio-fequency queies fom an RFID tansceive. Passive, active and semi-passive tags can be distinguished. Passive RFID tags do not have thei own powe supply and the ead ange is less than fo active tags, i.e., in the ange of about a few mm up to seveal metes. Active RFID tags, on the othe hand, must have a powe souce, and may have longe anges and lage memoies than passive tags. Many active tags have pactical anges of tens of metes, and a battey life of up to seveal yeas. Futhe infomation about the undelying technology can be found in Finkenzelle (). To employ RFID fo positioning and tacking of obects, one stategy is to install RFID eades at cetain waypoints (e.g. entances of buildings, stoage ooms, shops, etc.) to detect an obect when passing by. Fo that pupose an RFID tag is attached to o incopoated in the obect. This concept is employed fo example in theft potection of goods in shops and in waehouse management and logistics. A second appoach fo using RFID in positioning would be to install RFID tags at known locations (e.g. at active landmaks) especially in aeas without GPS visibility (e.g. in tunnels, unde bidges, indoo envionments, etc.) and have a eade and antenna installed in the mobile device caied by the use. When the use passes by the tag the RFID eade etieves its ID and othe infomation (e.g. the location). Positioning can be pefomed using Cell of Oigin (CoO). The maximum ange of the RFID tag defines a cell of cicula shape in which a data exchange between the tag and the eade is possible. Seveal tags located in the smat envionment can ovelap and define cetain cells with a adius equal the ead ange. The accuacy of position detemination is defined by the cell size. Using active RFID tags the positioning accuacy theefoe anges between a few metes up to tens of metes. Using a configuation of the achievable ange of the RFID tags, howeve, the signal stength can be set in steps of dbm between -4 dbm and +6 dbm which coesponds in a diamete of the cell anging fom up to 5 m in aeas

61 58 Jounal of Global Positioning Systems with fee visibilty. The optimal size of the cell can then be set at 4 m. Highe positioning accuacies can be obtained using tilateation if the anges to seveal tags ae detemined and ae used fo intesection. Fo 3-D positioning ange measuements to at least thee tags ae necessay. The anges fom the antenna of the eade to the antenna of the tag is deduced fom the convesion of signal powe levels into distances. Stategies fo the convesion of the signal stength measuements into distances fo uban outdoo aeas will be discussed in the following section. 3 Signal stength to distance convesion fo RFID ange deduction To tansfom the measued signal stength fom the RFID tag into a ange between the tag and the eade a convesion model has to be employed. This convesion can be pefomed using a adio wave popagation model. Such a model is an empiical mathematical fomulation fo the chaacteization of adio wave popagation as a function of fequency, distance and othe conditions. Such models typically pedict the path loss along a link o the effective coveage aea of a tansmitte. Fo outdoo uban envionments a suitable model is the Okumua- Hata Model. The Okumua-Hata model is the most widely used model in adio fequency popagation fo pedicting the behaviou of cellula tansmissions in built up aeas. This model incopoates the gaphical infomation fom the Okumua model and develops it futhe to ealize the effects of diffaction, eflection and scatteing caused by city stuctues. This model also has two moe vaieties fo tansmission in sububan aeas and open aeas. The Hata Model pedicts the total path loss along a link of teestial micowave o othe type of cellula communications (Wikipedia, 7; Ranvie, 4). The Okumua-Hata model will be used fo modelling of the popagation of the RFID signals outside buildings. This model assumes that the eceived signal powe deceases logaithmically with the distance fom the tansponde. It can be mathematically descibed as: s T A + B log d + C () = whee s T is the total signal stength in [db], d is the distance between the RFID tag and the RFID eade in [km] and A, B, and C ae coefficients that depend on the fequency and antenna heights. The coefficients A, B, and C can be descibed as A = log ( f ) 3.8log ( h ) a( h ) () 69 c t B = log ( h ) (3) 44 t whee f is the caie fequency in [MHz], c h t is the RFID tag height above local teain height in [m], h is the RFID eade antenna height above local teain height in [m] and a ( h ) is a coection facto fo the antenna height of the RFID tag. The factos a ( h ) and C depend on the suoundings as follows: Uban aeas: - Small and medium-size cities: a( h ) = (.log ( f c ).7) h (.56log ( f c ).8) C = (4) - Metopolitan aeas: 8.9[log (.54h )]. if 5 f c MHz a( h ) = 3.[log (.75h )] 4.97 if < f c 5 MHz C = (5) Sububan aeas: C [log( f / 8)] 5.4 (6) = c Open aeas: C 4.78[log( f c )] 8.33log( f ) 4.94 (7) = c In the case of RFID the caie fequency of the employed system is f c = Mhz. The RFID tags ae installed along the oad at the same height, and the RFID eade will be caied by the use at a constant antenna height. Thus, the paametes caie fequency f c, RFID eade antenna height above local teain height h and RFID tag height above local teain height h t can be teated as constant. Fo this eason, the equations above can be simplified by using the paametes total signal stength s T and distance d between the RFID tag and the RFID eade. Also the unit fo the distance can be changed fom [km] to [m] and fo the signal stength fom [db] to

62 Retsche and Fu: Integation of RFID, GNSS and DR fo Ubiquitous Positioning in Pedestian Navigation 59 [dbm]. The elationship between the signal stength and the distance can then be expessed as: st = a + a log d (8) whee a and a ae coefficients found duing calibation using measuements on a known baseline. The distance d between the RFID tag and the RFID eade can then be detemined as follows: d st a = a [ b + b st ] = (9) with the coefficients b a = and a b =. a Fig.. Relationship between the measued signal stength and the distance along a baseline descibed by a logaithmic model of ode p = 8 Fo futhe impovement of the accuacy of the appoximation, the exponent in equation (9) can be extended by a polynomial function of ode p as descibed in the following equation: d [ b + b st + b st bp st ] = () p whee p is the ode of the polynomial function, b, b, b,, b p ae the coefficients of the polynomial function detemined fom a calibation. The convesion has been tested in sububan outdoo envionment with obstuctions caused by thee- to foustoey buildings. Fig. shows the elationship between the measued signal stength and the known distance along a baseline. The signal stength has been measued along the baseline fom m up to a distance of m fom the RFID tag with an m inteval at fou diffeent oientations of the antenna of the RFID eade. Fig. shows the measuements to one tag in diection. Fo the convesion of the signal stength to a distance the logaithmic model descibed in equation () has been employed. A good appoximation is achieved if an ode of p = 8 is chosen. In Fig. the esiduals between the logaithmic model appoximation and the measued signal stength values ae shown. As can be seen fom this figue, the standad deviation fo the convesion of the signal stength into a distance is only ±.5 m using this model with a mean value of.3 m. Fig.. Residuals between the logaithmic model and the measued signal stength along a baseline Apat fom using a logaithmic elationship between the signal stength and the distance also the use of a linea egession was investigated. Fo that pupose a polynomial function of the ode p in the fom of p d = a + a s + a s a p s () whee d is the distance to the RFID tag in [m], s is the measued signal stength in [dbm] and a p ae the unknown coefficients of the polynomial function, can be used to descibe the elationship between the signal stength and the distance. The unknown coefficients a p can be computed using a least squaes fit if the signal stength s is measued along a baseline at n

63 6 Jounal of Global Positioning Systems known egula distances. Then thee ae n equations with p+ unknowns (whee n must be > p+). The possible ode p of the polynomial function depends on the numbe of available signal stength obsevations n and the desied level of appoximation. Fig. 3 shows a linea egession model of ode p=8 fo the signal stength to distance convesion and Fig. 4 the coesponding esiduals between the polynomial model appoximation and the measued signal stength values. By compaing Fig. with 3 can be seen that both models achieve a nealy simila esult fo the signal stength to distance convesion. Using the polynomial model the esulting mean value of the esiduals is, howeve, only 7.3* -7 m and the standad deviation fo the convesion of the signal stength into a distance is a bit smalle than that of the logaithmic model (i.e., only ±.5 m) (see Fig. 4). Due to the slighty bette esults it can theefoe be ecommended to employ a polynomial model of high ode fo the distance convesion of the anges between the RFID tag and the eade. A mao advantage of the polynomial convesion model is also that it is easie to handle than the logaithmic appoach. used togethe with the ange obsevations to RFID tags to obtain the position fix. In the following sections, a GNSS algoithm fo position detemination in uban envionments with pseudoange measuements to less than fou satellites is descibed and then the integation with RFID and othe wieless technolgies is discussed. Fig. 4. Residuals between the polynomial model and the measued signal stength along a baseline 4 Opeation pinciple of the minimum ange eo algoitm (MRERA) fo GNSS positioning Fig. 3. Relationship between the measued signal stength and the distance along a baseline descibed by a polynomial model of ode p = 8 If seveal RFID tags ae located in the suounding envionment the cuent position of the RFID eade can be obtained using tilateation. Then the deduced distances to at least thee RFID tags ae needed to calculate a position fix with intesection. If moe than thee distances ae available, the position fix can be calculated using a least squaes adustment. The deduced anges to RFID tags can not only be used to obtain the position fix in indoo and uban envionments using tilateation, but also to supplement the GNSS positioning in outdoo uban envionments. Then the available pseudoange obsevations to satellites should be GNSS positioning in uban envionments can be significantly affected by signal blockage due to obstuctions and theefoe it fequently happens that less than the minimum equied numbe of fou satellites ae available to obtain a 3-D position fix. Mok and Lau () poposed an algoithm that is able to estimate positions even with thee o less satellites, i.e., the Minimum Range Eo Algoithm (MRERA). MRERA was oiginally developed fo vehicle tacking to locate vehicles in dense high-ise envionments without the use of DR sensos. But it can also be employed fo geneal geolocation and mobile positioning applications. The basic pinciple of opeation of the MRERA is illustated in Fig. 5. Conside that a seies of oad points with known coodinates in WGS84 ae stoed in a oad netwok database. When tavelling along a section of oad and continuously eceiving GPS signals fom at least one satellite, the pseudoange obsevations and the geometic ange computed fom the known satellite and oad point positiones can be obtained. Afte coection of the pseudoange obsevations fo ionospheic, topospheic, multipath and eceive clock eos, the diffeence between the geometic ange and the measued GPS pseudoange can be calculated. This ange diffeence will vay depending on the distance of the GPS eceive fom

64 Retsche and Fu: Integation of RFID, GNSS and DR fo Ubiquitous Positioning in Pedestian Navigation 6 the oad point. In othe wods, if the GPS use is tavelling towads a paticula oad point, the ange diffeence will decease. The diffeence will each its minimum value when the use is neaest to the oad point, then inceasing when the use is moving away fom the oad point. This phenomenon is illustated in Fig. 6. X, Y, Z ae the WGS84 coodinates of a satellite (with =,, 3,, M) at time t and X i, Yi, Z i ae the WGS84 coodinates of a oad point i (with i =,, 3,, N). The pseudoange R (t) fom satellite to eceive obseved at epoch t can be expessed as R = ρ + c( δ t δ t ) + ε (3) whee ρ (t) is the geometic ange fom satellite to the GPS eceive at epoch t, c is the speed of light, δ t (t) is the satellite clock bias at epoch t and δ t (t) is the GPS eceive clock bias at epoch t, ε ae othe eos such as ionospheic and topospheic biases and multipath eos. Fig. 5. Pinciple of the MRERA appoach showing a vehicle s position between oad points and (afte Mok et al., 7) Afte applying coections to the measued pseudoange R (t) the ange diffeence between the geometic ange to the oad point ρ i (t) and the measued GPS pseudoange R (t) can be calculated. The minimum of this ange diffeence is obtained when the GPS eceive is neaest to the oad point i. Then the MRERA Indicato Value (MIV) at the oad point i given as M MIV = ρ R (4) i = i Fig. 6. The minimum of the diffeence between the measued GPS pseudoange and the geometical ange to one satellite aises when the GPS eceive is neaest to the oad point s location (afte Mok et al., 7) The geometic ange ρ i (t) fom the satellite to the oad point i at epoch t can be detemined using the following equation: ρ i = ( X Xi ) + ( Y Yi ) + ( Z Zi ) () whee eaches its minimum value fo evey tacked satellite (with =,, 3,, M). In theoy, the detemination of MIV is possible with all available satellites. In confined aeas with obstuctions, howeve, the numbe of satellites M would nomally be less than thee. If moe than thee satellites ae available, MIV can also be used to veify the eceive location unde weak satellite-eceive geomety (i.e., lage DOP value). Fo futhe infomation about the basic pinciple of MRERA the inteested eade is efeed to the wok of Mok and Lau () o Mok and Xia (6). Futhe development was concentated on the supplementation of the standalone GPS mode with othe wieless technologies. An integation of ange measuements to WiFi and UWB base stations into MRERA was fist poposed by Mok and Xia (5). In simulations it could be seen that an integation of GPS pseudoanges and anges to gound tansmittes can be pefomed meaningful unde the MRERA. Apat fom WiFi and UWB also the use of anges to active landmaks equipped with active RFID has been poposed

65 6 Jounal of Global Positioning Systems (see Mok et al., 7) and will be descibed in the following section. 5 Integation of RFID and othe gound based wieless technologies with GNSS using enhanced MRERA Conside the situation that pseudoange obsevations to only one o two GPS satellites ae possible (which would not give a position fix in standalone GPS mode) and also anges to gound based tanspondes (e.g. an active landmak equipped with a RFID tag) o tansmittes (e.g. an WiFi access point o an UWB base station) at a paticula location can be obtained. Then these obsevations should be used togethe to detemine an absolute position fix. The location of the active landmak seves then in the MRERA algoithm as the oad point with known coodinates (compae Fig. 5) and the espective ange to the landmak can be used togethe with the GPS pseudoanges fo obtaining the position fix. The obtained anges to the active landmak shall be integated with othe obsevations fom adio tansmittes, e.g. pseudoange obsevations to GNSS satellites, if positioning is pefomed in uban envionment. Then the integated obsevation and positioning model does not only include the obsevation equations fo the satellite positioning systems (i.e., GPS, GLONASS, futue Galilieo), but also the ange obsevations fom gound based tansmittes (i.e., the WiFi access points o UWB base stations and RFID landmaks). This leads to the following functional model: R R R R R R G G Gk ρ l ρ l = ρ l G ρ G l G G ρ l Gk Gk ρ l m m m m m m G G Gk n n n n n n G G Gk δx δy δz δ t (5) whee R (t) ae the pseudoange obsevations fom satellite to eceive at epoch t, R Gk (t) ae the equivalent ange obsevations fom gound based tansmittes (o active landmaks) k to the cuent use s location at epoch t, ρ (t) is the ange vecto fom satellite to GPS eceive in the WGS84 coodinate fame at epoch t, Gk ρ (t) is the ange vecto fom the base station of the gound tansmitte netwok (o the active landmak) k to the mobile station in the coesponding coodinate fame at epoch t, l (t), m (t) and n (t) ae the diection cosines fom the tacked satellite to the obsevation point at epoch t, l Gk (t), m Gk (t) and n Gk (t) ae the diection cosines fom the gound tansmitte k to the mobile station at epoch t, δ X, δ Y and δ Z ae the coodinate diffeences fo point and δ is the eceive clock bias of the GPS eceive. t Theeby in equation (5) it is assumed that the ange vecto fom the base station of the gound tansmitte Gk netwok ρ (t) is given in the GNSS efeence fame, i.e., the WGS84 in the case of GPS. The unknown coodinate diffeences δ X, δ Y and δ Z ae obtained fom the least squaes adustment descibed by the wellknown fom: X = ( A PA) A PL T T δ (6) The obsevation weight matix P in equation (6) contains two pats that coesponds to the GNSS psudoanges and ange obsevations fom gound tansmitte stations o active landmaks. If σ GNSS and σ G ae used to indicate the standad deviations of the unit weight fo satellite pseudoanges and gound tansmitte netwok anges espectively, and diffeent obsevation types ae assumed to be uncoelated, it has the fom: / σ P = GNSS / σ GNSS / σ GNSS / σ G / σ G / σ G The standad deviation σ GNSS can be set accoding to the GNSS positioning mode and the standad deviation σ G accoding to the diffeent kind of the gound tansmitte netwok. Using the model descibed above anges to UWB and WiFi tansmittes o RFID tanspondes can be integated with GNSS pseudoanges to obtain the most optimal location esult. The vaied foms of obsevations can be pseudoanges, time delays, time delay diffeences o signal stengths. They can all be conveted to the (7)

66 Retsche and Fu: Integation of RFID, GNSS and DR fo Ubiquitous Positioning in Pedestian Navigation 63 geometical distance afte some tansfomation. Fo instance, the distance can be estimated fom the signal stength which is based on the elation of the signal popagation loss on the taveling path. All obsevations fom UWB, GNSS, WiFi o RFID can be used in a tightly coupled pocessing model in fom of a Kalman filte based on the obsevation domain, fom which both position and velocities ae deived (Mok and Xia, 5). An impotant consideation fo hybid positioning is the coexistence of diffeent kinds of obsevations such as data fom GNSS, UWB, WiFi, RFID o othe mobile netwoks. Each has its own quality featue that is descibed by its vaiance value of the unit weight. Theefoe in data pocessing, system pefomance evaluation based on obsevation esiduals will eflect system pefomance if the unit weight vaiance is unique. To obectively evaluate hybid location pefomance, a Helmet vaiance estimation model was poposed to optimize the quality evaluation based on an iteative estimation of the unique vaiance of the unit weight. Futhe details can be found in the papes of Mok and Xia (5) and Xia et al. (6). 6 Integation with dead eckoning (DR) Most navigation systems also employ dead eckoning (DR) sensos fo the diect obsevation of the diection of motion (o heading) and the distance tavelled fom a known stat position. In the poect at least the following DR sensos ae included: an attitude senso (i.e., a digital compass) giving the heading in combination with an inetial tacking senso (e.g. a low-cost MEMS-based Inetial Measuement Unit IMU) including a thee-axis acceleomete also employed fo tavel distance measuements as well as a digital baometic pessue senso fo altitude detemination. The mao disadvantage, howeve, of DR sensos is the accumulation of lage dift ates afte shot time if no suitable update with an absolute position is available. Depending on the envionment in which the use is cuently moving an update of the DR deived positions can be achieved with absolute positions fom the following location methods: GNSS positions in unobstucted outdoo envionments, a combination of GNSS and RFID in outdoo uban envionments whee blockage of satellite signals occus, positions detemined by RFID tilateation in aeas whee no GNSS signals ae available, pimaily fom RFID deived positions in indoo envionment as pseudoanges to GPS satellites (if available) ae usually less accuate, o a combination of RFID and othe wieless technologies such as WiFi o UWB in the case of thei availability. Fo the integated position detemination an extended Kalman filte (EKF) appoach can be employed. In a tightly coupled EKF the GPS pseudoange measuements and anges to RFID tags can be integated with compass heading, and INS-deived position and attitude infomation as well as baometic height. Anothe stategy is to use self calibation outines fo the DR sensos fo aeas whee no absolute positions at all fo the coection of the DR difts ae available. Gene- Bzezinska et. al. (7) poposed to calibate the DR obsevations (step length and step fequency fom the MEMS-based IMU as well as the heading fom the digital compass and the altitude fom the baomete) unde GPS signal blockage using the knowledge of the human locomotion model when GPS is available. In othe wods, a taining mode unde GPS availability is used to calibate the human dynamics model (step length and step fequency) as well as the digital compass and baomete with atificial neual netwoks (see e.g. Wang et al., 6) and fuzzy logic (see e.g. Abdel-Hamid et al., 6) and the calibated obsevations ae then used in the DR navigation if GPS is unavailable. Taining data that feed the atificial neual netwok o a fuzzy logic based adaptive knowledge systems ae collected fo each opeato sepaately, and functions, such as step fequency, ate of step fequency, teain slope, opeato s locomotion patten (e.g., standing, walking, ogging, spinting, climbing, etc.), as a function of senso outputs ae analyzed to fom the fuzzy ules that ae subsequently used in the actual DR navigation mode. This appoach is vey pomising, but it is still in the development stage and futhe investigations ae theefoe equied. 7 Conclusions and outlook In this pape the use of active longange RFID tags fo positioning using signal powe levels that ae conveted to distances as well as thei integation with GNSS obsevations has been discussed. Ranges have been deduced fom the measued signal powe levels using a logaithmic and polynomial convesion model. In the test expeiments it could be seen that fo the convesion of the signal stength into a distance a polynomial model with an ode of p=8 gives good esults. The tests have been pefomed along a baseline in sububan outdoo envionment fo anges of up to m fom the RFID tag. Futhe testing is equied using longe distances fom the tags in diffeent envionments.

67 64 Jounal of Global Positioning Systems The deduced anges can be used to obtain a position fix with tilateation if anges to seveal RFID tags ae measued, on the one hand, o in combination with GNSS pseudoange obsevations in obstucted aeas, on the othe. The integation algoithm fo pseudoange obsevations to GPS satellites and ange measuements to gound based tansmittes o tanspondes has been discussed in the pape. Pactical testing of this appoach will be pefomed in the nea futue and thei esults will be epoted elsewhee. Acknowledgments This eseach is suppoted by the eseach poect P9-N5 Ubiquitous Cathogapy fo Pedestian Navigation (UCPNAVI) founded by the Austian Science Fund (Fonds zu Födeung wissenschaftliche Foschung FWF). Also the suppot fom the UGC eseach gant B-QF Investigation into Seamless Indoo and Outdoo Positioning Based on UWB- GNSS Integation founded by the Hong Kong SAR Govenment is acknowledged. The authos also thank Pof. Esmond Mok fom the Depatment of Land Suveying and Geo- Infomatics fom the Hong Kong Polytechnic Univesity fo fuitful discussions and the coopeation in the above named eseach poect. Refeences Abdel-Hamid W., Abdelazim T., El-Sheimy N. and Lachapelle G. (6) Impovement of MEMS-IMU/GPS Pefomance Using Fuzzy Modeling, GPS Solutions, No. /6, pp. -. Finkenzelle K. () RFID Handbook: Fundamentals and Application in Contactless Smat Cads and Identification, Cal Hanse Velag, Munich, Gemany. Gene-Bzezinska D., Toth C. and Moafipoo S. (7) Pedestian Tacking and Navigation Using an Adaptive Knowledge System Based on Neual Netwoks, Jounal of Applied Geodesy, Vol., No. 3. Hohenschuh F. (4) Pototyping eines mobilen Navigationssystems fü die Stadt Hambug, Diploma thesis, Depatment Infomatics, Univesity Hambug, Gemany. Mok E. and Lau L. () GPS Vehicle Location Tacking in Dense High-Rise Envionments with the Minimum Range ERo Algoithm (MRERA), in: Papes pesented at the ION GPS, Septembe -4,, Salt Lake City, Utah, USA, CD-Rom Poceedings, pp Mok E. and Xia L. (5) Stategies fo Geolocation Optimization in Uban Regions, in: Papes pesented at the 5 Intenational Symposium on GPS/GNSS, Decembe 8-, 5, Hong Kong, CD-Rom Poceedings. Mok E. and Xia L. (6) Hybid GPS and Wieless System fo Geolocation Positioning in Uban Canyons, in: Papes pesented at the Intenational Wokshop on Successful Stategies in Supply Chain Management, Januay 5-6, 6, Hong Kong, pp Mok E., Retsche G. and Xia L. (7) MRERA (Minimum Range Eo Algoithm): RFID - GNSS Integation fo Vehicle Navigation in Uban Canyons, in: Papes pesented at the 5 th Symposium on Mobile Mapping Technology, May 9-3, 7, Padua, Italy, CD-Rom Poceedings, 7 pgs. Radoczky V. (3) Katogaphische Untestützungsmöglichkeiten zu Routenbescheibung von Fußgängenavigationssystemen im In- und Outdoobeeich, Diploma thesis, Institute of Catogaphy and Geo- Mediatechniques, Vienna Univesity of Technology, Austia. Ranvie S. (4) Path Loss Models, S Physical Laye Methods in Wieless Communication Systems, Postgaduate Couse on Radiocommuications, Helsinki Univesity of Technology, SMRAD Cente of Excellence, th_loss_models.pdf (Last date accessed: Apil 8). Retsche G. (5) A Knowledge-based Kalman Filte Appoach fo an Intelligent Pedestian Navigation System, in: Papes pesented at the ION GNSS 5 Confeence, Septembe 3-6, 5, Long Beach, Califonia, USA, CD-Rom Poceedings, pp Retsche G. and Fu Q. (7) Using Active RFID fo Positioning in Navigation Systems, in: Papes pesented at the 4 th Symposium on Location Based Sevices and Telecatogaphy, Novembe 8-, 7, Hong Kong, PR China. Retsche G. and Mok E. (7) UWB, RFID and GNSS Integation fo Navigation and Tacking, in: Papes pesented at the 4 th Symposium on Location Based Sevices and Telecatogaphy, Novembe 8-, 7, Hong Kong, PR China. Thienelt M., Eichhon A. and Reitee A. (7) Intelligent Pedestian Positioning in Vienna: Knowledge-Based Kalman Filteing, in: Papes pesented at the 5 th Symposium on Mobile Mapping Technology, May 9-3, 7, Padua, Italy, CD-Rom Poceedings, 7 pgs. Wang J. J., Wang J., Sinclai D. and Watts L. (6) A Neual Netwok and Kalman Filte Hybid Appoach fo GPS/INS Integation, in: Papes pesented at the IAIN/GNSS 6 Confeence, Octobe 8-, 6, Jeu Island, South Koea, Vol., pp Wikipedia (7) Hata Model Fo Uban Aeas, see Xia L., Mok E. and Xue G. (6) Optimized Hybid Location Sevice fo Supply Chain, in: Papes pesented at the Intenational Wokshop on Successful Stategies in Supply Chain Management, Januay 5-6, 6, Hong Kong, pp

68 Jounal of Global Positioning Systems (7) Vol.6, No.: Modified Gaussian Sum Filteing Methods fo INS/GPS Integation Yukihio Kubo, Takuya Sato and Sueo Sugimoto Depatment of Electical and Electonic Engineeing, Ritsumeikan Univesity, Shiga, Japan, Abstact. In INS (Inetial Navigation System) /GPS (Global Positioning System) integation, nonlinea models should be popely handled. The most popula and commonly used method is the Extended Kalman Filte (EKF) which appoximates the nonlinea state and measuement equations using the fist ode Taylo seies expansion. On the othe hand, ecently, some nonlinea filteing methods such as Gaussian Sum filte, paticle filte and unscented Kalman filte have been applied to the integated systems. In this pape, we popose a modified Gaussian Sum filteing method and apply it to land-vehicle INS/GPS integated navigation as well as the in-motion alignment systems. The modification of Gaussian Sum filte is based on a combination of Gaussian Sum filte and so-called unscented tansfomation which is utilized in the unscented Kalman filte in ode to impove the teatment of the nonlineaity in Gaussian Sum filte. In this pape, the pefomance of modified Gaussian Sum filte based integated systems is compaed with othe filtes in numeical simulations. Fom simulation esults, it was found that the poposed filte can impove tansient esponses of the filte unde lage initial estimation eos. Key wods: INS, GPS, integation, nonlinea filte Intoduction In the INS/GPS integated system, the complementay chaacteistics of INS and GPS ae exploited. INS povides position, velocity and attitude infomation at a high update ate with the continuous availability, and the long tem accuacy of position and velocity infomation of GPS pevents the gowing navigation eos of INS. In othe wods, the navigation eos of INS ae estimated and coected by using GPS measuements (Siouis, 993; Gewal, ). Fo many yeas, the extended Kalman filte (EKF) has been widely utilized as the estimato in the integated navigation systems (Maybeck, 979; Gelb, 974). Additionally, in the case of conventional navigation systems, the initialization of INS navigation states is completed pio to vehicle motion and then the odinay integated navigation is implemented. Usually, this initialization method needs 5 to minutes and the vehicle must be stopped. It is, howeve, inconvenient and impactical when thee is not enough time to stop at a stat point. Thus it is motivated to develop in-motion alignment and navigation algoithms which can povide the accuate attitude infomation while moving. Because the initial attitude of the land-vehicle is unknown, the attitude is usually assumed to be. Thus, when the initial heading eo is lage, the nonlinea chaacte of the INS eo equations is emphasized fo in motion alignment (Roges, ). Theefoe seveal nonlinea filteing methods such as Monte Calo filte (Kitagawa, 996; Doucet, ), Quasi-linea optimal filte (Sunahaa, 97), Gaussian Sum filte (Alspach, 97) and unscented Kalman filte (Julie, ), have been applied to the integated navigation systems. The pefomance compaisons of the nonlinea filtes in the integated navigation systems also have been epoted by the authos (Tanikawaa, 4; Fuioka, 5; Nishiyama, 6). Accoding to (Nishiyama, 6), although Gaussian Sum filte (GSF) woks well with lage uncetainties in the initial attitude infomation, the lineaization technique is employed similaly to the extended Kalman filte. On the othe hand, the unscented Kalman filte (UKF) has been ecently paid much attention in the aea of the integated navigation (Yi, 5; An, 5; Shin, 7). The unscented Kalman filte calculates the pedicting mean and covaiance of the state vecto fom a set of samples that ae called sigma points by means of so-called unscented tansfomation. In this pape, we ty to combine the GSF and the unscented tansfomation in ode to impove the teatment of the nonlineaity in the GSF. With this combination, it is expected that the tansient esponse of the filte can be impoved unde lage initial estimation eos. In this pape, fistly we biefly eview the algoithms of the nonlinea filtes that ae applied in this pape. Then, the modified Gaussian Sum filteing algoithm is deived

69 66 Jounal of Global Positioning Systems by utilizing the unscented tansfomation. Finally, the pefomance of EKF, GSF, UKF based and modified Gaussian Sum filte based integated systems is compaed in numeical simulations. Desciption of the system In this wok, closed-loop, tightly coupled mechanization is adopted fo the INS/GPS integation. Fig. shows the achitectue of the integation with mao data paths between the system components. The components of the system ae stapdown INS and GPS eceive. The INS contains IMU (Inetial Measuement Unit: acceleomete and gyo). Based on the measued acceleation and angula ate, the INS computes the position, velocity and attitude of the vehicle elative to thei initial value at high fequency. But thee exist unbounded position eos that gow slowly with time. The concept of the integated navigation system of Fig. is to educe the INS eos by using some extenal measuement fom a GPS eceive. In this eseach, GPS double diffeenced caie phase and undiffeenced Dopple measuements ae employed as extenal measuements to emove the INS eos. The nonlinea filte estimates the eos in the navigation and attitude infomation using the aw GPS data. towad the Geenwich Meidian. It is used fo the definition of position location such as latitude and longitude. ) The L fame ( X L, YL, Z L) is the ight-handed locally level coodinate fame. The X L - and Y L - axes ae diected towad local noth and east espectively; ZL - axis is downwad vetical at the local eath suface efeenced position location. It is used fo defining the angula oientation of the local vetical in the E fame. 3) The C fame ( XC, YC, Z C) is the ight-handed compute fame that is defined by otating the L fame aound negative Z L - axis by the wande angle α ; towad the negative YL - axis and the axis is diected towad the negative Z L ZC - - axis (upwad vetical). It is used fo integating acceleation into velocity, and used as the efeence fo descibing the stapdown senso coodinate fame oientation. 4) The B fame ( X B, YB, Z B) is the stapdown inetial senso coodinate fame (body fame). The - axis X B is diected towad the head of the vehicle; the Y B - axis is the ight-hand of the vehicle; the ZB - axis is downwad vetical to the X B -Y B plane. The fame is fixed on the vehicle and otates with the motion of the vehicle. Fig.. Desciption of the system. Coodinate systems To integate the navigation systems, definitions of coodinate systems that the navigation systems o included sensos efe to ae impotant. This section defines the coodinate fames used in this pape and epesents the angula elationship between them. The coodinate fames ae defined as follows: ) The E fame ( X E, YE, Z E) is the ight-handed eath fixed coodinate fame. It has the oigin at the cente of the eath; the Z E - axis is diected towad the Noth Pole; the X E equatoial plane, wheeby the - and YE - axes ae located in the - axis is diected X E Fig.. Coodinate fames Fig. shows the spatial image of the E, L and C fames, whee λ and ϕ epesent the longitude and the latitude espectively. In the inetial computations, the acceleation sensed with espect to the B fame have to be tansfomed onto the C fame. The velocity and position of the vehicle ae then computed with espect to the C fame. Such a tansfomation is known as the Eule angle tansfomation. We define the poduct of diection cosine C T B matix fo this tansfomation as. Then the coodinates ( xb, yb, z B) in the B fame ae tansfomed into ( xc, yc, z C) in the C fame as follows:

70 Kubo et al.: Modified Gaussian Sum Filteing Methods fo INS/GPS Integation 67 x x C B C y C T B y = B zc zb whee C T B is the diection cosine matix. 3 INS eo model () whee the dot above a lette denotes diffeentiation with L espect to time, the vecto ω E/ Lis the otation ate of the L fame with espect to the E fame in the L fame C coodinate system, and the vecto ω E / C is similaly defined. Fom equation (8), the position eo ( δ Cx,, δ Cy, ) as well as azimuth eo (δα ) equations can be deived. C The diection cosine matix T E is epesents the tansfomation fom the E fame to the C fame, and it can be decomposed as follows. T C C L E TL TE = () On the othe hand, the computed matices T L C and contain eos δ TL C and E δ T C can be fomulated as C C C E E E C L C L = TL TE TL TE C C L L L L L E δt T T = { δt [ I ( δ )] T ( δ )} T L = RT E L T E L δ T E espectively. The eo whee δ L [ δ Lx,, δ Ly,, ] T is hoizontal angula L position eo, and the elation of T E =[ I ( δ L )] TE L is used in the calculation of equation (3) with the assumption that δ Lx, and δ Ly, ae small. Also, ( a ) fo 3 vecto a = [ a x, a y, a z ] T is the skew- symmetic matix defined by az ay ( a ) az a x ay ax And R is the position eo matix defined as follows (Roges 3, ). (3) (4) δ cosα δ sinα δcy, cosδα δcx, sinδα R δ sinα δ cosα δcx, cosδα δcy, sinδα (5) δly, δlx, whee δ sinα sin( α + δα) sinα (6) δ cosα cos( α + δα) cosα (7) 3. Velocity eo model The computed velocity v C also contains the velocity eo δ v C such that vc = vc+ δ vc (9) and the velocity equation is given by v = f ( ρ + Ω ) v + g () C C C C C C whee f C is non-gavitational specific foce vecto, ρ is elative ate vecto, and Ω C is eath ate vecto. The specific foce is popotional to the inetial acceleation of the system due to all foces except gavity measued by the acceleomete. g C is the gavity vecto, positive towad the cente of the eath in the C fame. Fom equations (9) and (), the velocity eo is modelled by δvc = bc + fc δθc + vc ( δρc + δωc) ( ρc + Ω C) δvc ( δρ + δω ) δv + δg C C C C () whee δθc [ δθ Cx,, δθ Cy,, δθ Cz, ] T is the attitude eo. 3. Attitude eo model The attitude eo δθ C causes the eo of the tansfomation matix T B C. The computed matix T B C which contains the attitude eo is fomulated by C C B C B T = [ I ( δθ )] T () Theefoe, we have following attitude eo model. δθ δω δω δθ ω (3) C C C C = E/ C+ I/ E+ C I/ C+ dc whee d C denotes gyo dift. Accoding to (Roges,, 3), we have R = R ( ω ) ( ω )( + R ) + ( ω ) (8) L C C C C E/ L E/ C TL E/ C TL

71 68 Jounal of Global Positioning Systems 3.3 Senso eo model In this pape, the acceleomete bias b B and gyo bias d B ae modelled as the fist ode Makov pocesses espectively as follows: b B() t = bb() t + ub() t τb d B() t = db() t + ud() t τ d (4) whee τ b and τ d ae the coelation time constants and ub () t, ud () t ae zeo mean Gaussian white noise pocesses. 3.4 State equation In ode to implement the nonlinea filteing fo integated navigation, hee, we define the state vecto. Because the double diffeenced caie phases ae used as the measuements in this pape, the unknown intege ambiguities should be simultaneously estimated. Theefoe the state vecto is defined such that it includes the INS eos as well as the intege ambiguities as follows: x = δ Cx,, δ Cy,, δ vcx,, δ vcy,, δθcx,, δθcy,, δ hc, δ v, b, b, b, d, d, d, γ, β, Cz, Bx, By, Bz, Bx, By, Bz,,,3, ns ku, ku, ku, cδ t, N, N,, N T (5), whee N ku, denotes the double diffeenced intege ambiguity of the satellites, and the eceives k, u, and n s is the numbe of visible satellites. β and γ ae defined as follows: β cosδα γ sinδα The desciptions of the state vecto components ae listed in Table. Then, fom equations (8), (), (3) and (4), the state equation can be fomulated by x () t = f( x(),) t t +η() t (6) whee f (, t ) is the time-vaying nonlinea function, and the pocess noise η () t is assumed to be mutually independent zeo mean Gaussian white noise with covaiance matix Nt (). Table. List of states No. Symbol Eo state δ Cx, XC -axis position eo in angle δ Cy, YC -axis position eo in angle 3 δ v Cx, XC -axis velocity eo 4 δ v Cy, YC -axis velocity eo 5 δθ Cx, XC -axis tilt eo 6 δθ Cy, YC -axis tilt eo 7 γ sinδα 8 β cosδα 9 δ hc ZC -axis altitude eo δ v Cz, ZC -axis velocity eo b Cx, X B b Cy, B 3 b Cz, B -axis acceleomete bias Y -axis acceleomete bias Z -axis acceleomete bias X -axis gyo bias Y -axis gyo bias Z -axis gyo bias, N double diffeenced ambiguity 4 d Cx, B 5 d Cx, B 6 d Cx, B 7, ku, By discetizing the state equation (6), we have x( k+ ) = x( k) + f( x( k), k) Δ t+ w( k) (7) whee wk ( ) is assumed to be Gaussian white noise with zeo mean and diagonal covaiance matix Q(k), and Δ t is a sampling inteval of the measuement data. 3.5 Measuement equation In this pape, the measuements ae the double diffeenced caie phase and Dopple data. By ignoing some eos in the caie phase data such as the emaining ionospheic and topospheic delays and multipath eos, then the double diffeenced caie phase measuement can be simply modelled by yk ( ) = h ( ) + λ N ku + ε g( k) (8) E whee E is the position vecto in the E fame, the function h is the nonlinea function that indicates the distance between satellites and eceives, N ku is the ambiguity vecto, λ is the wave length, and ε g is the measuement noise. By linealizing equation (8) with the fist ode Taylo seies appoximation aound the position indicated by i INS, E ( k ), and applying appopiate tansfomations of the coodinate systems, we obtain the measuement

72 Kubo et al.: Modified Gaussian Sum Filteing Methods fo INS/GPS Integation 69 equation of the INS position eo in the C fame as follows. i yk ( ) yk ( ) h( E ( k)) (9) = Hˆ ( k ) δc ( k ) + λn ku + ε g ( k ) whee ˆ ( ) ( ) E ( ) ( ) ( ) L H k HkT kt kt kt ( k ) () and L A B C h ( E ( k)) Hk ( ) E ( k), ( ) i E k = E( k) ( Rp + h) R p + h TA, TB cos λ whee R p is the eath adius. The Dopple measuement can be modelled by the change of the distance between the eceive and the satellite in the sampling inteval Δ t (Misa, ). By using the velocity eo vecto in the C fame, δ v C, and the appopiate tansfomations similaly to the above deivations, the Dopple measuement can be fomulated as E i p { C v v ve} Du( k) = Gu( k) T ( C δ C) + c δtu + ε d( k) () whee G T u E E T E T E T T C ( TC ) ( TC ) ( TC ) p T u E i E = E T T T T ns p g g u g u g, u u, p and u denotes the distance between the eceive u and p the satellite p, v E is the velocity of the satellite p in the E fame and ε d is the measuement noise. Then, we have the following Dopple measuement equation. p E i D u( k) Du + GuvE GuT Cvc E δ vc = Gu T C + ε d c δt u () Finally, fom equations (9) and (), we have the measuement equation fo the integated navigation in geneal fom: zk ( ) = Hkxk ( ) ( ) + ε ( k) (3) 4 Nonlinea filteing Nonlinea filteing techniques ae applied to the integated INS/GPS system in ode to estimate the state vecto (the eos of INS descibed above). In this section, fistly, we biefly eview the filte algoithms of the GSF and the UKF. Then the modified Gaussian Sum filte (MGSF) algoithm is deived. 4. Gaussian Sum filteing Let Z k be the set of the measuement such that Z = {() z, z(),, z()} k (4) k In the GSF (Alspach, 97), a posteioi pobability density pxk ( ( ) Z k ) is fomed by the convex combination of the outputs of seveal Kalman filtes pocessed in paallel. The a pioi density pxk ( ( ) Zk ) is assumed that it is fomulated by the sum of seveal nomal distibutions as follows: pxk ( ( ) Z ) = γ ( k k ) k m = N μ ( k k ), Pμ ( k k ) (5) whee m is the numbe of distibutions, and γ is the weight fo the -th distibution such that m = γ ( k k ) =, γ ( k k ) And N [ θ,σ ] denotes the nomal pobability density function with mean θ and covaiance matix Σ. Then, by the Bayesian ecusion elations, a posteioi density can be fomulated by m p( x( k) Zk ) = γ ( k k) N μ ( k k), Pμ ( k k) (6) whee = μ ( k k) = μ ( k k ) + K ( k) z( k) H( k) ( k k ) μ ( μ ) P ( k k) = P ( k k ) K ( k) H( k) P ( k k ) μ μ μ μ K ( k) = P ( k k ) Hk ( ) μ μ T HkP ( ) μ ( k k ) Hk ( ) + Rk ( ) and the weight γ ( k k) is given by T whee T T T zk ( ) [ y ( k) D u( k)]. γ ( k k) = γ ( k k ) β ( k) m l l l = { γ ( k k ) β ( k)} (7)

73 7 Jounal of Global Positioning Systems whee γ ( k k ) = γ ( k k ) β ( k) = N ν ( k k ), P νν ν ( k k ) z( k) H( k) μ ( k k ) T P H( k) P ( k k ) H( k) + R( k) νν μ Theefoe, we have the filteed estimato m xˆ( k k) = γ ( k k) μ ( k k) (8) = The a pioi density pxk ( ( + ) Z k ) can be ewitten with the same algoithm as the EKF as follows. pxk ( ( + ) Z) = γ ( k+ k) whee k m = N μ ( k + k), Pμ ( k + k) (9) μ ( k+ k) = f( μ ( k k)) (3) T P ( ) ( ) ( ) μ k+ k = F k Pμ k k F ( k) + Q( k) (3) f( x) F ( k) = x x μ = ( kk ) γ ( k+ k) = γ ( k k) 4. Unscented Kalman filte (3) In the UKF, the pedict mean xˆ( k+ k) and covaiance Pk ( + k) ae calculated fom a set of samples which is called the sigma points. This method is called the unscented tansfomation (Julie, ). Unde the assumption that the system noise is independent and additive, the pedict mean and covaiance ae computed as following steps. Step: choose the sigma points χ ( k k) which is associated with the n-dimensional state vecto x( k ) as follows. κ χ ˆ ( k k) = x( k k), W = n + κ χ ( k k) = xˆ ( k k) + ( n+ κ) P( k k) χ + n ( k k) = xˆ ( k k) ( n+ κ) P( k k) W = W+ n =, ( =,,, n) ( n + κ) Step: compute a set of tansfomed samples though the pocess model equation (7), χ ( k + k) = f( χ ( k k), k) Step3: compute the pedicting mean and covaiance as follows n xˆ( k+ k) = W χ ( k + k) = ( + n T ) = χ χ + ( ) = Pk k W Qk whee χ ( ) ˆ χ k + k x( k + k) W is the weight of the -th point and κ is a scaling paamete. ( ( n+ κ) P( k k)) is the -th column of the matix squae oot of ( n+ κ ) P( k k). Then, once the obsevation zk+ ( ) is obtained, xˆ( k+ k) and Pk ( + k) ae updated to xk ˆ( + k+ ) and Pk ( + k+ ) as follows. Z ( k + k) = H( k) χ ( k + k) n = zk ˆ( + k) = W Z ( k+ k) (33) n T νν ( + ) = Z Z + ( ) = P k k W R k (34) n T xν ( + ) = χ = P k k W Z (35) whee Z Z ( k+ k) zˆ ( k+ k) K k+ = P k+ k P k+ k (36) ( ) ( ) xν νν ( ) xˆ( k+ k + ) = xˆ( k+ k) + K( k)( z( k) zˆ ( k + k)) (37) Pk+ k+ = Pk+ k KkP k+ kk k (38) T ( ) ( ) ( ) νν ( ) ( ) Since the measuement equation (3) is linea in this navigation poblem, above equations (33)-(35) can be simply expessed by zk ˆ( + k) = Hkxk ( ) ˆ( + k) (39) T P ( k k) H( k) P( k k) H ( k) R νν + = + + (4) P k k P k k H k T xν ( + ) = ( + ) ( ) (4) 4.3 Modified Gaussian Sum filte In the Gaussian Sum filteing algoithm, we can see fom equations (3)-(3) that the lineaization technique is

74 Kubo et al.: Modified Gaussian Sum Filteing Methods fo INS/GPS Integation 7 employed similaly to the extended Kalman filte. In this pape, we popose the modified Gaussian Sum filte by applying the unscented tansfomation algoithm to the time updating algoithm of the GSF, equations (3)-(3). Step: similaly to the step of the UKF, fo - th ( =,,, N) density in GSF, choose the sigma points and weights as follows. ( ) ( ) κ χ k k k k W n = μ, = + κ thee exist no eos in the othe initial estimates. Theefoe, in the EKF and UKF, the initial estimate x ˆ( ) is set to, and P ( ) and Q ae configued fom the nominal equipment specifications in Table. In this case, the states elated to the azimuth eo, i.e. 7th and 8th components of the state vecto have 6 [deg] initial estimation eo espectively. χ ( k k) = μ ( k k) + ( n+ κ) P ( k k) χ l l+ n μ μ l ( k k) = μ ( k k) ( n+ κ) P ( k k) Wl = Wl+ n =, ( l =,,, n) ( n + κ) Step: compute a set of tansfomed samples though the pocess model equation (7), l χ ( k + k) = f( χ ( k k), k) l l Step3: compute the -th pedicting mean and covaiance as follows. n μ ( k+ k) = Wl χ l ( k + k) (4) l = n T ( + ) = l l( l) + ( ) l = Pμ k k W χ χ Q k (43) whee χ l χ l( k + k) μ ( k+ k) In the MGSF, the oiginal time updating algoithm of equations (3) and (3) ae substituted by (4) and (43) espectively. 5 Expeimental esults The expeiments of the INS/GPS In-Motion Alignment and navigation algoithms descibed above wee caied out by using simulated INS and GPS data. In the expeiments, we assume the vehicle uns at a speed of aound 5 [km/h] fo about minutes. The speed at the stat point was [km/h], and the initial azimuth angle was 6 [deg]. The test taectoy in the local level hoizontal plane is shown in Fig. 3. The data wee obtained by utilizing the Matlab6.5 and INS Toolbox. (GPSoft LLC.) at 5 [Hz] ate fo IMU and at [Hz] ate fo GPS. Fou types of filtes, i.e. EKF, GSF, UKF and MGSF ae used in the expeiments and compaed. The nonlineaity of the INS usually occus when thee exist lage attitude eos. So in the expeiments, the initial state estimates ae set to have lage azimuth eo. And we assume that Fig. 3. Test taectoy Table. Senso eo specification Acceleomete Specification Bias 8 [ μ G] ( σ ) Scale facto 5 [ppm] ( σ ) Random eo.3 [m/s] Gyoscope Specification Bias [deg/h] ( σ ) Scale facto 5 [ppm] ( σ ) Random eo.6 [deg/ h ] In the GSF and MGSF, thee nomal distibutions ae utilized, i.e. m = 3, and Pμ ( ), =,, 3 ae set to the same value of the EKF and UKF, i.e. Pμ ( ) = P( ). The initial estimates μ ( ), = 3,, ae also set to except fo the 7th and 8th components of the state vecto (see Table ), β and γ, that epesent the azimuth eo. They ae assumed to have the initial azimuth eo estimates such that δα = 6,, + 6 [deg]. The pocessing esults ae shown in Fig. (4)-(7). Figs. (4) and (5) show the esults of the positioning and compaison of the positioning eos. Table 3 also shows RMS (Root Mean Squae) values of the position eos. Fom Fig. (5) we can see that the MGSF shows faste convegence than the othes, and the GSF and MGSF show bette pefomances than EKF and UKF. Theefoe, the GSF and MGSF can wok well when thee exist lage

75 7 Jounal of Global Positioning Systems azimuth eo because they can teat lage azimuth eo by assuming multiple initial eo distibutions. Fom Table 3, we can also see that the MGSF achieves the best pefomance in this simulation. Fig. 6 and Table 4 show the velocity eos and thei RMS values espectively. Fom these figues and tables, the all filtes show almost same pefomance, wheeas the UKF and MGSF show slightly bette pefomance than the EKF and GSF. Finally, Fig. 7 shows the esults of the azimuth eos. Fom Fig. 7, we can see that all filtes show almost same esults afte [sec], but the MGSF shows faste convegence in its tansient esponse fom to [sec]. Theefoe, fom these esults of the simulation, we can conside that the UKF and MGSF can achieve bette pefomance than the EKF and GSF, and the MGSF can wok well when thee exist a lage initial azimuth eo. Fig. 4. Positioning esults Fig. 6. Velocity eos Fig. 5. Positioning eos Table 3. Root mean squae of position eos Noth eo [m] East eo [m] EKF.8.6 UKF.9. GSF..3 MGSF..9 Table 4. Root mean squae of velocity eos Noth eo [m/s] East eo [m/s] EKF.36.5 UKF.34.9 GSF.36. MGSF Conclusions Fig. 7. Azimuth eos In this pape, the modified Gaussian Sum filteing algoithm was deived by applying the unscented tansfomation to the Gaussian Sum filte, and it was applied to the GPS/INS integated system. The algoithm was tested and compaed with the EKF, GSF and UKF by using simulated data. Fom the expeimental esults, it was found that the deived MGSF show the quick tansient esponse fo azimuth eo estimation. Theefoe the MGSF has an ability to impove the navigation pefomance, when thee ae lage initial azimuth eos.

76 Kubo et al.: Modified Gaussian Sum Filteing Methods fo INS/GPS Integation 73 Refeences Alspach D. L. and Soenson H. W. (97) Nonlinea Bayesian Estimation Using Gaussian Sum Appoximations, IEEE Tans. on Automatic Contol, Vol. AC-7, No. 4, pp , 97. An D. and Liccado D. (5) A UKF Based GPS/DR Positioning System fo Geneal Aviation, Poc. of the Institute of Navigation, ION GNSS 5, pp , Long Beach, CA, 5. Doucet A., Godsill J. and Andieu C. () On sequential Monte Calo sampling methods fo Bayesian filteing, Statistics and Computing, Vol. 3, pp. 97-8,. Fuioka S., Tanikawaa M., Nishiyama M., Kubo Y. and Sugimoto S. (5) Compaison of Nonlinea Filteing Methods fo INS/GPS In-Motion Alignment, Poc. of the Institute of Navigation, ION GNSS 5, pp , Long Beach, CA, 5. Gelb A. (974) Applied Optimal Estimation, MIT Pess, Massachusetts, 974. Gewal M. S., Weill L. R. and Andews A. P. () Global Positioning Systems, Inetial Navigation, and Integation, John Wiley & Sons, New Yok,. Julie S., Uhlmann J. and Duant-Whyte H. F. () A New Method fo the Nonlinea Tansfomation of Means and Covaiances in Filtes and Estimatos, IEEE Tans. On Automatic Contol, Vol. 45, No. 3, pp ,. Kitagawa G. (996) Monte Calo Filte and Smoothe fo Non-Gaussian Nonlinea State Space Models, Jounal of Computational and Gaphical Statistics, Vol. 5, pp. -5, 996. Maybeck P. S. (979) Stochastic Models, Estimation and Contol (Mathematics in Science and Engineeing), Academic Pess, New Yok, 979. Misa P. and Enge P. () Global Positioning System -- Signals, Measuements, and Pefomance, Ganga-Jamuna Pess, Massachusetts,. Nishiyama M., Fuioka S., Kubo Y., Sato T. and Sugimoto S. (6) Pefomance Studies of Nonlinea Filteing Methods in INS/GPS In-Motion Alignment, Poc. of the Institute of Navigation, ION GNSS 6, pp , Fot Woth, TX, 6. Roges R. M. () Lage Azimuth INS Eo Models fo In- Motion Alignment Land-Vehicle Positioning, Poc. of the Institute of Navigation National Technical Meeting, pp. 4-4, Long Beach, CA,. Roges R. M. (3) Applied Mathematics in Integated Navigation Systems, nd edition, AIAA, Viginia, 3. Shin E.-H. and El-Sheimy N. (7) Unscented Kalman Filte and Attitude Eos of Low-Cost Inetial Navigation Systems, Navigation: Jounal of The Institute of Navigation, Vol. 54, No., pp. -9, Sping, 7. Siouis G. M. (993) Aeospace Avionics Systems A Moden Synthesis, Academic Pess, San Diego, 993. Sunahaa Y. (97) An Appoximate Method of State Estimation fo Nonlinea Dynamical Systems, Jounal of Basic Engineeing, Tans. ASME, Seies D, Vol. 9, No., pp , 97. Tanikawaa M., Asaoka N., Oiwa M., Kubo Y. and Sugimoto S. (4) Real-Time Nonlinea Filteing Methods fo INS/DGPS In-Motion Alignment, Poc. of the Institute of Navigation ION GNSS 4, pp. 4-4, Long Beach, CA, 4. Yi Y. and Gene-Bzezinka D. (5) Nonlinea Bayesian Filte: Altenative To The Extended Kalman Filte In The GPS/INS Fusion Systems, Poc. of the Institute of Navigation, ION GNSS 5, pp. 39-4, Long Beach, CA, 5.

77 Jounal of Global Positioning Systems (7) Vol.6, No.: An Evaluation of GNSS Radio Occultation Technology fo Austalian Meteoology Eiang Fu, Kefei Zhang, Falin Wu, Xiaohua Xu and Kaye Maion Suveying, Positioning and Navigation (SPAN) eseach goup, RMIT Univesity Anthony Rea, Yuiy Kuleshov, and Gay Weymouth Austalian Bueau of Meteoology, Austalia Abstact. Eath atmospheic infomation has been pimaily obseved by a global netwok of adiosonde weathe obsevation stations fo global weathe foecasting and climatologic studies fo many yeas. Howeve, the main disadvantage of this method is that it can not sufficiently captue the complex dynamics of the Eath s atmosphee since its limited and heteogeneous geogaphic distibution of launching stations. Since the fist low eath obit (LEO) satellite equipped with a GPS eceive was launched in ealy 99s, thee ae moe than a dozen of GPS eceives onboad LEO satellites used fo Eath atmospheic obsevation. Recent eseach has shown that the Global Navigation Satellite System (GNSS) adio occultation (RO) deived atmospheic pofiles have geat potentials to ovecome many limitations of existing atmospheic obsevation methods. Constellation Obseving Systems fo Meteoology, Ionosphee, and Climate (COSMIC) etieved atmospheic pofiles ae investigated using adiosonde measuements at 4 collocated stations in the Austalian egion. Statistical esults show that the diffeence in aveage tempeatue is about.5 C with a standad deviation of.5 C and the diffeence in aveage pessue is -.6 hpa with a standad deviation of.9 hpa. This eseach has also demonstated that the GNSS RO deived atmospheic pofiles have good ageement with the adiosonde obsevations. Keywods. Radio Occultation, COSMIC, Radiosonde, GNSS.. Intoduction A netwok of about 9 adiosonde weathe obsevation stations located globally is cuently poviding the maoity of atmospheic ai infomation fo global weathe foecasting and climatologic studies. Radiosonde weathe sensos attached to balloons measue atmospheic popeties (i.e., tempeatue, pessue and elative humidity) fom the Eath s suface up to about 3 km altitude of atmosphee. One key poblem of the adiosonde obsevation netwok is its limited and heteogeneous geogaphic distibution of stations due to, fo example, the difficulty to establish obsevation stations ove lage ocean aeas. Moeove, the high cost of station opeation and equipment of adiosonde obsevation limits the coveage of the netwok and obsevation fequency. In addition, the diect adiosonde measuements have senso-icing poblem in the uppe atmosphee (ove km height) because of the vey low tempeatue (Wicket 4). Theefoe, the atmospheic infomation cuently deived by the adiosonde can not adequately epesent the complex dynamics (in both space and time) of the Eath s atmosphee. GPS RO technique has demonstated exciting potentials fo weathe foecasting and climate studies since the fist LEO satellite equipped with a GPS eceive launched in the ealy 99s (Foelsche et al. 3; Kichengast ; Kusinski et al. 997; Steine et al. ). This technique has a numbe of advantages such as global coveage, high vetical esolution, high accuacy, all weathe capability and calibation-fee. Due to these unique advantages, it has opened new oppotunities fo vaious meteoological and climate elated applications and fo bette undestanding of the Eath s atmosphee. Fo example, GNSS RO data can be used to (Anthes et al. ; Kichengast 999; Zhang et al. 7 (b)): Impove foecast accuacy of numeical weathe pediction and climate system studies; Povide accuate geopotential heights; Reveal the height and shape of the topopause on a global scale (an impotant goal in atmospheic and climate eseach);

78 Fu et al.: An Evaluation of GNSS Radio Occultation Technology fo Austalian Meteoology 75 Detemine the global distibution of gavity wave enegy fom the uppe toposphee to statosphee; Investigate Ĕl Nino events; Investigate the global wate vapo distibution and map the atmospheic flow of wate vapo; Impove the global suface pessue fields; and Study the electon density iegulaities in the ionosphee. The oint effot of Stanfod Univesity and JPL in the ealy 96s pioneeed the eseach of (planetay) RO technique. Cuently, moe than a dozen of GPS eceives ae equipped onboad LEO satellites on Eath s obit (see Fig. fo a histoical oveview) and they povide thousands of GPS RO Eath s atmospheic obsevations evey day. This new valuable data souce can enhance ou knowledge of both Eath atmospheic stuctue and pocesses (Pavelyev et al. ; Schmidt et al. 4). With the apid development of the new GNSSs (e.g., Euopean Galileo and Chinese Beidou systems) and the inceasing numbe of GNSS eceives onboad LEO satellites available, the global coveage and tempoal esolution of such GNSS RO sounding obsevations will be impoved significantly (Schmidt et al. 4). Futhemoe, continuing impovements in LEO satellite obits detemination, space-bone GNSS eceives design, GNSS signal pocessing algoithms, data etieval and data assimilation methods ae boosting the eliability and applicability of the GNSS RO meteoology technique. In the nea futue, the GNSS-based emote obsevation method can povide a obust altenative to monito and ecod in eal time the Eath s atmospheic dynamic infomation with sufficient accuacy, esolution, and high spatiotempoal coveage. The Austalian Bueau of Meteoology has been investigating in the GNSS RO technique and its applications in Austalia since 4. Ealy studies have demonstated good ageements of tempeatue and wate vapou between the CHAMP GPS RO etievals and adiosonde obsevations ove fou adiosonde weathe stations in Westen Austalia (Zhang et al. 7 (a)). Good esults have also been shown in anothe case study that evaluates CHAMP RO deived atmospheic pofiles ove the whole Austalian egion using US National Centes fo Envionmental Potection (NCEP) numeical weathe model (Fu et al. 7). These studies wee conducted with few compaison pais because of the limited numbe of GNSS RO etievals available fom one-satellite-constellation CHAMP, especially fo Austalian egional studies. Since the successful launch of the COSMIC mission with a constellation of six LEO satellites in Apil 6, about 5 daily GPS RO events globally (see Fig. (Occultation Locations fo COSMIC, 6)) and ove daily RO events in Austalian egion have been obtained. Such a lage numbe of etievals bing unpecedented oppotunities fo moe detailed egional evaluation studies of the GNSS RO etievals. Theefoe, a futhe compaison eseach was conducted to evaluate the COSMIC GPS RO etieved atmospheic pofiles with adiosonde measuements fo the whole Austalian aea. A total numbe of 4 coincidences of COSMIC deived atmospheic pofiles and adiosonde obsevations ae identified using a adial buffe of km and a tempoal buffe of -hou duing a thee-month peiod (between Januay and Mach 3, 7). This pape pesents this evaluation study and its coesponding esults. Fig.. Histoical developments of the LEO satellite pogams with GPS eceives fo GPS adio occultation meteoology eseach Although the GNSS RO meteoology technique has many advantages ove the taditional adiosonde obsevation technique, the chaacteistics of the GNSS RO etievals eo have to be evaluated popely. Compehensive assessment of the GNSS RO etievals will contibute to bette applications of the new technique and also povide helpful infomation fo the assimilation of the new data souces into the cuent meteoological model systems. A collaboation eseach team between Suveying, Positioning and Navigation (SPAN) eseach goup and Fig.. A global distibution of typical daily GPS RO events (geen dots) obseved by COSMIC, location of adiosonde in ed. GPS adio occultation technique and COSMIC GPS RO events happen when a GPS satellite sets o ises behind the Eath s atmospheic limb elated to a LEO satellite and the LEO onboad GPS eceive captues the delayed signals that tavese the Eath s atmospheic limb. Fig. 3 illustates the geomety of the RO event that

79 76 Jounal of Global Positioning Systems occus between the LEO satellite and GPS satellite () pai. The ay path and tangent point can be detemined to a high accuacy based on pecise locations of both GPS and LEO satellites. Tangent point adius г and asymptotic ay miss-distance α can be obtained using the knowledge of pecise tangent point location, and the bending angle а can be then calculated. A pofile of bending angles can be pocessed to a efactivity index pofile by applying an Abel tansfomation (Wae et al. 996). The efactivity is a function of the electon density in the ionosphee, tempeatue, pessue and wate vapou in the atmosphee. Futhe pocessing of the efactivity index povides valuable infomation on the pofile of tempeatue, wate vapo and pessue in the neutal atmosphee, and electon density in the ionosphee. COSMIC data poducts ae classified into fou classes (Levels, A, B and ). Level is aw GPS measuement data while Level is the final poducts including atmospheic efactivity, tempeatue, pessue and wate vapou pessue pofiles. Intemediate poducts (Levels A and B) consist of atmospheic phase delay, signal amplitude and atmospheic Dopple shifts and bending angles espectively. In this study, 4 coincidences of the adiosonde atmospheic ecods and the COSMIC deived atmospheic pofiles (Level data) ae identified using km adial distance buffe and hous tempoal buffe based on the adiosonde station locations and ecods. 3. Numeical analysis and discussion GPS () GPS () Tangent Point а LEO α г α Eath Fig. 3. A schematic demonstation of GPS adio occultation geomety The COSMIC mission is designed fo weathe and space weathe foecast, climate monitoing, and atmospheic, ionospheic and geodesy eseach by via a constellation of six LEO satellites equipped with GPS eceives. Its final obits ae between 75-8 km with an inclination of 7 in 6 planes spaced 4 apat (Wu et al. 5). Thee gound stations ae established in Alaska, Sweden and Taiwan and one multiple task station in Taiwan is fo communicating and contolling satellites. Each COSMIC satellite is equipped with a GPS RO payload fo tacking GPS signals, a tiny ionospheic photomete payload fo measuing ionospheic total electon content (TEC) fom the satellite s nadi diection, and a ti-band beacon payload fo geneating high esolution satellite to gound-station TEC (Rocken et al. ). The COSMIC GPS eceive is geneated fom NASA/JPL Black Jack space-bone GPS eceive which was used by CHAMP pogams. The integated GPS eceive system consists of five units, namely a scientific gade GPS RO eceive, dual occultation antennas, dual pecision obit detemination antennas, payload contolle and solid state ecode. With the obust space-bone GPS eceive system, the LEO satellites pecise obit can be well detemined and both the ising and setting GPS RO events can be captued. Fig. 4. Distibutions of the adiosonde stations (big dots) and COSMIC RO events (small dots) The Austalian egional atmospheic infomation is pimaily obtained fom 38 adiosonde weathe stations (big dots in Fig. 4). On the othe hand, with a window of latitude [-,-7 ] and longitude [75, 7 ] (coves all the 38 stations), 4,638 COSMIC RO events (small dots in Fig. 4) wee ecoded duing a 3-month peiod (fom Januay to Mach 3 7). In ode to detemine the compaable pais, km adial distance buffe and hous tempoal buffe based on the adiosonde ecods ae employed and 4 coincidences ae identified. 3. Data pe-pocessing and oveview esult Data pe-pocessing is a necessay step fo data analysis. Meteoological infomation, especially fo the uppe-ai pofiles, is extemely dynamic in both space and time. Consequently, its database is extemely lage and complicated. The thee-month COSMIC data sets include millions of measuements. Fo an effective management and analysis of infomation in such a lage database, all the textfile-based data (i.e., each atmospheic pofile is stoed in one TEXT file) is tansfeed into Oacle database management system and oganized using logical tables and views. SQL (Stuctued Quey Language) database functions ae designed fo automatically data pocessing, such as data input and unit convesion.

80 Fu et al.: An Evaluation of GNSS Radio Occultation Technology fo Austalian Meteoology 77 Intepolation of the atmospheic pofiles is necessay since the pofiles measued by adiosonde and COSMIC ae diffeent in heights. COSMIC has a much bette vetical esolution (about metes) than CHAMP (about 3 metes) due to the impovements of the onboad GPS eceives. The adiosonde data acquied fom The Austalian Bueau of Meteoology has a vetical esolution of about -mete. Hence, COSMIC data is intepolated to match with adiosonde data set. Tempeatue (C) Latitude Tempeatue Pessue (mba) Latitude Pessue Fig. 5. Tempeatue (uppe plot) and pessue (lowe plot) mean diffeences of the 4 coincidences against diffeent latitudes Tempeatue and dy pessue deived fom both adiosonde and COSMIC ae compaed at the 4 selected coincidence samples in the altitude ange ~3km. A good ageement between the two data souces has been found. Thee ae 88% matches that have less than C mean diffeences in tempeatue. The diffeence in mean aveage tempeatue is about.5 C with a standad deviation of.5 C. Fo pessue, 9% samples have less than hpa mean diffeences and the aveage of mean diffeences is -.6 hpa with a standad deviation of.9 hpa. Fig. 5 shows the tempeatue (uppe plot) and pessue (lowe plot) mean diffeences of the 4 pais against with thei latitudes. Most samples have negative pessue mean diffeences which suggest that COSMIC esults have geneal lage values than adiosonde. It also can be seen that some lage eos appea in the middle latitudes. Howeve, this is not conclusive since the limited numbes of samples ae in the lowe and highe altitude egions. 3. Homoscedasticity method Fig. 6 shows the diffeences of both tempeatue (uppe plot) and pessue (lowe plot) against the heights with a 95% statistical confidence level. These estimates wee obtained by tansfoming the esponse vaiable in each case so that the assumptions equied by the odinay least squaes estimation pocedue fo egession models (in paticula the equiement of homoscedasticity fo the esiduals of the model) wee satisfied. Polynomials of sufficiently high degee wee fitted, the pediction intevals calculated (that is, the confidence intevals fo the individual esponse) and the invese tansfomation applied to the fitted cuve and the associated pediction intevals. These gaphs pesent eos chaacteistics and pattens along thei heights. The andom eos of the COSMIC GPS RO tempeatue etievals along altitude ae appaent since the mean diffeence line is close to zeo and nealy paallel to the 95% confidence inteval lines. Fo pessue, the eos in lowe heights ae much geate than those in uppe heights and the mean diffeence line is always unde the zeo standad line which again indicates the smalle COSMIC pessue etievals compaing with adiosonde measuements. Fig. 6. A gaph shows 95% confidence inteval of the tempeatue diffeences (uppe plot) and pessue diffeences (lowe plot): The diffeences (dak dots), the means (middle line) and the 95% confidence intevals (between uppe and lowe lines).

81 78 Jounal of Global Positioning Systems 3.3 Spatial and tempoal chaacteistics Spatial and tempoal chaacteistics of the new data souces ae vital fo meteoological eseach and pactical applications. Spatial epesentation is an effective way to illustate spatial pattens to undestand the eos spatial chaacteistics of the COSMIC GPS RO technique. In Fig. 7, the tempeatue eo ange is between -.9 C and.98 C. Many sites (ed dots) have small eo, which is less than.5 C; a few have negative and less than -.5 C (geen dots), and only a couple of sites have geate than C (dake blue dots). Fom this gaph, no spatial patten can be identified. Fig. 7. Tempeatue diffeences between adiosonde measuements and COSMIC deived values Similaly, Fig. 8 is a map of pessue diffeences between adiosonde and COSMIC. Those sites in ed dots have less than.5 C diffeences. Pink dots in the map epesent those values between C and -.5 C and geen dots ae those fom - C up to -.94 C. Fom this figue, it can be seen that the diffeences between the COSMIC and adiosonde data ae smalle in the highe latitude egions. Howeve, the conclusion cannot be ustified based on the limited data. Futhe eseach applying moe data will be conducted. 4. Conclusions Continuous and accuate measuements of atmospheic pofiles with good spatial and tempoal esolution ae impotant fo numeical weathe pediction analysis and climate elated studies. GNSS RO deived atmospheic pofiles have been consideed as good data souces fo atmospheic elated eseach. In this study, the quality of the COSMIC data is assessed with detailed statistical methods and the outcome of this study shows a vey good ageement with the Austalian egional adiosonde data. Such a lage volume of steam-in high esolution atmospheic pofiles will have a temendous impact on meteoological studies and applications. Most impotantly, the GNSS RO deived atmospheic pofiles ae not esticted by the geogaphic locations unlike the adiosonde technique (only 38 stations in Austalia). Theefoe, the new data souces deived fom the GNSS RO technique has a geat potential to fill up the gaps in cuent gound-based weathe station netwoks. Many counties, such as U.S., Geman, Austia, Russia, Finland, Italy, Denmak, Agentina, Bazil and South Afica, ae investigating GNSS RO technique fo meteoological puposes. The impotance of applying the GNSS RO meteoological technique in Austalia is clea since Austalia has lage but unpopulated aeas (limited weathe obsevation stations), dy continent (bette etieval esults in toposphee) and lage aeas suounded by oceans. SPAN goup at RMIT is cuently collaboating with scientists fom The Austalian Bueau of Meteoology, UNSW, Wuhan Univesity, Canada and Taiwan to identify key issues fo a long-tem eseach effot in ode to exploit the potential and full benefits of this emeging and enabling technology fo the Austalian community. Futhe eseach with newly eleased COSMIC data will be employed fo long-tem and moe detailed evaluation studies. Reseach on the coe data etieval techniques that tansfe GPS measuements to atmospheic pofiles ae being implemented now. Acknowledgements The Austalian Bueau of Meteoology patially funded this eseach and povided adiosonde data. The COSMIC adio occultation data ae collected fom NASA. The ealy vesion of this pape was submitted to ION confeence. Fig. 8. Pessue diffeences between adiosonde measuements and COSMIC deived values Refeences Anthes, RA, Rocken, C & Kuo, YH () Applications of COSMIC to meteoology and climate, Teestial, Atmospheic and Oceanic Sciences, vol., no., pp

82 Fu et al.: An Evaluation of GNSS Radio Occultation Technology fo Austalian Meteoology 79 Foelsche, U, Kichengast, G & Steine, A (3) Global climate monitoing based on CHAMP/GPS adio occultation data, pape pesented to the confeence of the Fist CHAMP Mission Results fo Gavity, Magnetic and Atmospheic Studies. Fu, E, Wu, F, Zhang, K, Xu, X, Rea, A, Kuleshov, Y & Biadeglgne, B (7) Validation of GNSS Radio Occultations' Pefomance Using NCEP Data in Austalia, pape pesented to Intenational Global Navigation Satellite Systems, Sydney, Austalia, Decembe 4-6, 7. Kichengast, G (999) A simple analytical atmospheic model fo adio Occultation applications, Inst. Meteool. Geophys., Univ. of Gaz, Austia. Kichengast, G () Climate change monitoing by adio occultation: Fom simulation studies via CHAMP to COSMIC and ACE+ constellations, Pepint COSMIC Radio Occ. Science Wokshop, pp Kusinski, ER, Ha, G, Schofield, JT, Linfield, RP & Hady, K (997) Obseving Eath's atmosphee with adio occultation measuements using the Global Positioning System, Jounal of Geophysical Reseach, vol., no. D9, pp Occultation Locations fo COSMIC (6), etieved Mach, 8 fom ml Pavelyev, AA, Liou, YA, Reigbe, C, Wicket, J, Igaashi, K, Hocke, K & Huang, CY () GPS adio hologaphy as a tool fo emote sensing of the atmosphee, mesosphee, and teestial suface fom space, GPS Solutions, vol. 6, no. -, pp. -8. Rocken, C, Kuo, YH, Scheine, W, Hunt, D, Sokolovskiy, S & McComick, C () COSMIC system desciption, Teestial, Atmospheic and Oceanic Sciences, vol., no., pp. -5. Schmidt, T, Heise, S, Wicket, J, Beyele, G & Reigbe, C (4) GPS adio occultation with champ: monitoing of climate change paametes, Atmospheic Chemisty and Physics, vol. 4, pp Steine, AK, Kichengast, G, Foelsche, U, Konblueh, L, Manzini, E & Bengtsson, L () GNSS Occultation Sounding fo Climate Monitoing, Physics and Chemisty of the Eath, Pat A: Solid Eath and Geodesy, vol. 6, no. 3, pp Wae, R, Exne, M, Feng, D, Gobunov, M, Hady, K, Heman, B, Kuo, YH, Meehan, TK, Melboune, W, Rocken, C, Scheine, W, Sokolovskiy, S, Solheim, F, Zou, X, Anthes, RA, Businge, S & Tenbeth, K (996) GPS sounding of the atmosphee fom low Eath obit: Peliminay esults, Bulletin of the Ameican Meteoological Society, vol. 77, no., p. 9 Wicket, J (4) Compaison of Vetical Refactivity and Tempeatue Pofiles Fom CHAMP With Radiosonde Measuements DMI, Copenhagen. Wu, BH, Chu, V, Chen, P & Ting, T (5) FORMOSAT- 3/COSMIC Science Mission Update, GPS Solutions, vol. 9, no., pp. -. Zhang, K, Biadeglgne, B, Wu, F, Kuleshov, Y, Rea, A, Hodet, Gd & Fu, E (7a) A Compaison of Atmospheic Tempeatue and Moistue Pofiles Deived fom GPS Radio Occultation and Radiosone in Austalia, pape pesented to Wokshop fo Space, Aeonautical and Navigational Electonics, Peth, Austalia, Apil. Zhang K, Fu, E, Wu, F, Xu, X, Rea, A, Kuleshov, Y & Biadeglgne, B (7b) GNSS Radio Occultation fo Weathe and Climate Reseach - A Case Study in Austalia, pape pesented to Intenational Global Navigation Satellite Systems 7, Sydney, Austalia, Decembe 4-6, 7.

83 Jounal of Global Positioning Systems (7) Vol.6, No.: 8-88 PC4 Based Low-cost Inetial/GPS Integated Navigation Platfom: Design and Expeiments Di Li, René J. Landy and Philippe Lavoie LACIME, Depatment of Electical Engineeing École de Technologie Supéieue (ÉTS), Univesity of Quebec, Monteal, Quebec, Canada, H3C K3 Abstact. The integation of Global Positioning System (GPS)/Inetial Navigation System (INS) has become vey impotant in vaious navigation applications. In the last decade, with the apid development of Mico Electo Mechanical Sensos (MEMS), geat inteest has been geneated in low cost integated GPS/INS applications. This pape pesents a PC4 based low cost GPS/INS integated navigation platfom. The platfom hadwae consists of low cost inetial sensos and an assembly of vaious PC4 compatible peipheals, such as data acquisition cad, GPS eceive, Ethenet cad, mothe boad, gaphic cad, etc. The platfom softwae including inetial/gps data acquisition, inetial navigation calculation and integated GPS/INS Kalman filte is implemented with Simulink, which can be diectly loaded and pocessed in the PC4 mothe boad with the aid of Matlab Real-Time Wokshop (RTW) utility. This platfom is totally self-embedded and can be applied independently o as pat of a system. Simulation and eal data expeiments have been pefomed to validate and evaluate the poposed design. A vey low cost MEMS inetial senso was utilized in the expeiments. The efeence is the navigation solution deived fom a tactic gade Inetial Measuement Unit (IMU). Test esults show that PC4 navigation platfom delives the integated navigation solutions compaable to the efeence solutions, which wee calculated with a conventional laptop compute, howeve with less powe consumptions, less system volume/complexity and much lowe ove-all costs. Moeove the platfom hadwae is compatible to vaious inetial sensos of diffeent gades by configuing the elated paametes in the system softwae. Keywods: PC4, GPS, MEMS, Kalman filte, Realtime Wokshop, xpc Taget Intoduction As an independent means of navigation, GPS is capable of deliveing position and velocity infomation with timeindependent pecision, while the pefomance becomes uneliable howeve when the system is exposed to high dynamics, intefeence fom communication equipments and intentional/non-intentional amming, etc. Compaed with GPS, INS poviding position, velocity, and attitude infomation via the measuements fom inetial sensos has vaious advantages, such as totally autonomous, high dynamic esponse, good shot-tem accuacy and obust pefomance when exposed to intefeence and o amming. Howeve its usage as a stand-alone navigation system is limited due to time-dependent gowth of the inetial senso bias/noise. Because of the afoementioned complementay chaacteistics, GPS and INS ae commonly coupled by Kalman filte to augment the oveall pefomance by ovecoming the shotcomings of each individual system. A high pecision integated GPS/INS system equies expensive inetial sensos that have exceptional long tem bias stability. The senso cost limits such kind of integated navigation systems to vey expensive applications (Haywad et al., 997). Ove the past decade low cost MEMS ae expeiencing apid impovements in tems of pecision, obustness, size, high dynamic esponse and so on. With the quick gowth in demand fo low cost navigation systems fo geneal aviation, unmanned automotive vehicles, locating pesonnel, mobile mapping systems, athletic taining and monitoing, and compute games, etc, it has become impotant to develop low cost integated navigation systems. This pape intoduces the development of a low cost inetial/gps integated navigation platfom. The platfom hadwae is constucted on the basis of a PC4 compute and an assembly of PC4 peipheals, such as data acquisition cad, gaphic cad, Ethenet cad, powe

84 Li et al.: PC4 Based Low-cost Inetial/GPS Integated Navigation Platfom: Design and Expeiments 8 supply boad and a PC4 compatible GPS eceive, etc. Matlab Simulink s modulaity and gaphical design make it convenient fo point-wise impovements and facilitate the amp-up knowledge of futue contibutos (Gioux, 5). Also one of the Simulink s poweful assets is the possibility to do apid eal-time testing though the RTW and xpc Taget. Theefoe the system softwae compising the data acquisition, stapdown inetial mechanization and integated Kalman filte is implemented by the Matlab Simulink. This Simulink based platfom softwae can be diectly compiled into executable code fo PC4 compute by the RTW. The Simulink based design scenaio has vaious advantages, such as apid eal-time pototyping and fast edesigning/debugging which is paticulaly helpful in the ealy stage of developing a eal-time navigation system. The achitectue of PC4 navigation platfom is depicted in Fig.. Hadwae platfom and softwae design. Hadwae platfom The hadwae platfom consists of a PC4 compute and an assembly of PC4 peipheals, such as data acquisition cad, gaphic cad, Ethenet cad, powe supply boad and a PC4 compatible GPS eceive. All the component pats ae stacked up togethe though the PC4 bus. This configuation makes the individual peipheal independent fom each othes as well as povides a good synchonization fo the data tansmission. Fo example, the platfom utilizing MEMS inetial senso is depicted in Fig.. Fig. Achitectue of PC4 navigation platfom This pape is oganized as follows. Section pesents an oveview of the platfom, the devices utilized, the softwae used to communicate and the algoithm stuctue. Section 3 descibes the low cost MEMS eal time test esults followed by a pefomance evaluation. Finally, futue wok and potential enhancement of the poect ae discussed in the conclusions. Fig.. PC4 navigation platfom pototype On the top laye thee is a MEMS inetial senso which geneally is not PC4 compatible but the infomation is sent via an extenal bus to the data acquisition cad. This laye povides the specific foce and angula ate measuements given by the MEMS. On the next laye, thee is a PC4 compatible GPS eceive which obtains and tansmits the position and velocity obsevations to the algoithm. Following the GPS eceive, thee is a 6- bit data acquisition cad which collects and convets the MEMS inetial analog signal to digital data. If the digital inetial data ae available in MEMS senso, e.g. a USB inteface is povided by the newly acquied MEMS senso in the lab (nimu MEMSense TM ) o a RS3 inteface used in the tactic IMU, this cad can be excluded. The digital data should be diectly connected to the PC4 compute USB/RS3 inteface, unde which a PC4 compatible powe boad povides the powe supply to each cad. Then, a PC4 compatible netwok cad is added to build a TCP/IP connection to a host compute. This connection is used fo caying out seveal impotant tasks: fist, to download the pogams to the mothe boad, second to setup the contol paametes and ecod the calculated navigation esults, then it enables the platfom in the emote-contol mode though Intenet. This platfom is also equipped with a gaphic cad to display all the data/paametes/esults esiding in the platfom in eal time on an additional sceen. On the bottom laye, thee is the PC4 motheboad whee the algoithm is loaded and executed.

85 8 Jounal of Global Positioning Systems. Platfom softwae The poposed system softwae package consists of stapdown inetial mechanization, the integated Kalman filte and the softwae inteface. The scheme of the system softwae is shown in Fig. 3. Fig. 3. Scheme of the system softwae Stapdown inetial mechanization Thee ae many appoaches to implementing the stapdown mechanization, which ae geneally divided into two categoies, i.e. a multi-speed digital design including accuate coning, sculling, scolling compensations fo attitude/velocity calculation and a simplified single speed continuous design without any attitude, velocity o position compensation algoithm (Savage, ). Both algoithmic designs have been implemented and investigated by Li et al. (7). By utilizing Matlab Simulink s capabilities to diectly evaluate the diffeential equations in the continuous mode, a simple single high speed inetial calculation algoithm stuctue based on the INS analytically continuous diffeential equations is implemented in this study. The attitude, velocity and position solutions ae deived by evaluating the ate equations as follows: v& N C& = a L B N SF = C ( ω ) ( ω ) C + g L B N P B IB N ( ωen E C& N + = C L L IL B N N ωie ) v E N N ( ω EN ) ω = C L IL g h& L N P N = v ( ω + ω ) N IE IE N EN = g ω ( ω R) L B whee CB is the attitude matix; ω IB is the skewsymmetic matix of the angula ate vecto in the body L fame (B-fame), ω IL is the skew-symmetic matix of the angula ate vecto caused by the tanslational motion in the L fame, ω N EN is the angula ate of N Fame elative to E Fame, v N is the velocity vecto, a N SF is the specific foce vecto, g P is the plumb-bob gavity, g is the standad gavity, R is the position location vecto fom the eath cente. Fo example the Matlab implementation of the attitude solution is depicted in Fig. 4. IE () Integated Kalman filte Accoding to the afoementioned advantages, the Kalman filteing is applied to combine the inetial/gps data in this study. The oles of the Kalman filte in ou application ae to estimate/coect the eos in navigation paametes, e.g. position, velocity and attitude, and also to estimate the inetial senso bias/dift which enables the in-motion calibation of inetial senso aw measuements. The measuements in the poposed Kalman filte ae fomed fom the compaison between the INS calculated and GPS eceive deived position/velocity data. Such measuements ae deived using the estimated attitude, velocity, position and MEMS senso eo states. The Fig. 4. Attitude implementation in Simulink design of the Kalman filte dynamic model is based on the INS eo model. In ou system, the so-called psiangle eo model is applied, which defines the eos in attitude, velocity and position paametes ( Ψ,δ V, δr ) (Savage, ): Ψ& δv& δr& N = C δω N N N B = C δa N B = δv N B IB B SF ω Ψ + a N EN N IN N SF Ψ ω δr N N N + δg N Mdl N N N ( ω IE + ω EN ) δv () whee, Ψ,δ V, δr ae the eos in attitude, velocity and N position paametes, C B is the attitude matix expessed B in the navigation fame (N-fame), δω is the angulaate eo vecto in the B-fame, asf, δa ae the specific IB N B SF foce vecto in the N-fame and the specific foce eo

86 Li et al.: PC4 Based Low-cost Inetial/GPS Integated Navigation Platfom: Design and Expeiments 83 N vecto in the B-fame, δ g Mdl is the plump-bob gavity N N eo, ω IE, ω ae the eath otation ate vecto and the EN tanspot ate vecto in the N-fame, espectively; ω N IN the N-fame otation ate in the inetial fame (I-fame). The eo paametes in attitude, velocity and position ae epesented as the eo states which ae popagated in the Kalman filte though the dynamic model. As in the application of low cost inetial sensos, the senso noises ae the dominant tems causing attitude, velocity and position eos, i.e. many eo tems in the INS eo model ae negligible when compaed with aw measuements eos B B δ asf and δω. Theefoe those IB tems can be emoved fom the eo model which in tun educes the numbe of the Kalman filte eo states, emakably deceasing the computing load (Li et al., 7). The simplified eo equation is given as: Ψ& δv& δr& N = C δω N N N B N B N = C δa = δv B IB B SF + a This continuous Kalman dynamic model should be discetized to build Kalman state tansition matix (Phi- Matix) and the integated dynamic noise matix (Q matix). The discetizing ate is chosen as the available maximum senso data ate, which is senso dependent, typically vaying fom Hz to Hz. The Matlab implementation of the Kalman Dynamic model is depicted in Fig. 5. N SF Ψ N (3) Softwae inteface Fo eal time pocessing, the Simulink implemented navigation algoithm can be e-witten in C, complied as executable codes and then downloaded to the hadwae platfom by the RTW. Simulink s xpc Taget Libay povides the necessay dives fo the diffeent peipheals. The connection to the monito, keyboad and the host compute to display, ead and command the platfom in the eal time is ceated by the dives fom xpc Taget Libay. Fo example, the data acquisition of the aw inetial measuement implemented in Simulink by xpc Taget is depicted in Fig. 7. In this study, the Diamond MM 6 AT cad is utilized by the platfom. The configuation of the Diamond MM6AT Acquisition Dive Module is shown in Fig. 6. Fig. 5. Simulink implementation of Kalman state tansition matix Fig. 6. Configuation of data acquisition dive The settings fo this module ae Numbe of Channels: Numbe of diffeent outputs; Range Vecto: Maximum Amplitude of the Output; Input Coupling: Numbe of input fo the ADC; Sample Time: Output Sampling Time; Base Addess: Output Addess. In this platfom, the data ae encoded with 6 bits pecision fo the 5 volts ange. Theefoe the esolution is V/bit which means a esolution of 76.3 μg fo the acceleometes as the sensitivity is mv/g and a esolution of º/sec fo the gyoscopes as the sensitivity is.5 mv/º/sec. 3 Expeiments Simulation and eal data expeiments wee pefomed to validate and evaluate the poposed design. The simulation test was to validate the designed navigation algoithm. Following the simulation, the eal data expeiment tests the platfom hadwae and softwae. A low cost MEMS and a low cost tactic IMU wee utilized in the eal data expeiments.

87 84 Jounal of Global Positioning Systems 3. Simulation test Fig. 8 depicts the taectoy pocessed in the simulation. The ed cuve is the efeence taectoy geneated by the Flight Simulato (Micosoft FS4) and the blue one coesponds to the solution calculated by the algoithm. Fig. 7. Simulink implementation of data acquisition softwae Velocity: Compaed with the efeence solutions, the velocity eos depicted in Fig. neve exceeds % of the velocity efeence pofile, indicating that the estimated velocity values ae accuate. Lat E[deg] x Position Eos Fig. 8. 3D taectoy simulation vs. efeence Position: The Kalman filte povides the updated coections of the position eos evey ms. The maximum eo located on an axis is about 5 metes. This peiod coesponds to the high dynamic peiod duing the flight. It can be seen fom the Fig. 9 that the position eos emain small and stable duing the simulation. P matix: i.e. the state covaiance matix gives the infomation about Kalman filte estimation pefomance. Its convegence means that the estimates ae close to the eality. The P matix of this expeiment shows that all the paamete estimates convege quickly as shown in Fig.. Long E[deg] Alt E[m] 5 x Time[s] Fig. 9. Position eos Simila to the position and velocity esults, the attitude eos shown in Fig. ae small and stable. Fom the above esults it can be concluded that the simulation test validates the poposed navigation algoithms.

88 Li et al.: PC4 Based Low-cost Inetial/GPS Integated Navigation Platfom: Design and Expeiments 85 P POS xyz[m ] P Vel enu[(m/s) ] vel n E[m/s] vel e E[m/s] vel u E[m/s] P Att RPH[deg ] Time [s] Time [s] Fig.. Convegence of Kalman estimation Velocity Eos 4 Time [s] Time [s] Roll E[deg] Pitch E[deg] Heading E[deg] 5 Fig.. Velocity eos Attitude Eos Time [s] Fig.. Attitude eos 3. Real data test The eal data expeiment was pefomed to test the poposed navigation platfom using a vey low cost MEMS inetial senso (MEMSense TM s AccelRate3D). Only analog inetial measuements ae available fom AccelRate3D MEMS senso, which wee collected and conveted to digital data in PC4 data acquisition cad (Diamond MM 6 AT). The compiled navigation softwae was downloaded to the platfom fom host compute though the PC4 Ethenet cad. The contol commands and navigation softwae paametes wee setup by an independent keyboad. Moeove the contol commands and the eal-time navigation solution may also be displayed by the monito with the aid of PC4 gaphic cad. The navigation solutions (NovAtel SPAN TM Best PVA) deived fom a tactic gade IMU (Honeywell HG7 IMU TM ) wee employed as the efeence. The inetial devices ae shown in Fig. 3 and Fig. 4. IMU specifications ae povided in Table and Table espectively. Fig. 3. AccelRate3D MEMS inetial senso Fig. 4. Refeence SPAN IMU Table MEMS senso specs (MEMSense AccelRate3D) AccelRate3D Dynamic Range Noise Acceleomete ± (g) 35(µg/ Hz) Angula Rate Senso ±3 (º/s).(º/h/ Hz) SPAN IMU Table Refeence IMU (NovAtel SPAN) Dynamic Range Noise(Random Walk) Acceleomete ±5(g) 34(µg/ Hz) Angula Rate Senso ± (º/s).5(º/ h)

89 86 Jounal of Global Positioning Systems Fist of all, a stand alone test was made with this vey low-cost MEMS senso. The pupose of this test was to test its pefomance without any aiding fom GPS. The taectoy was made inside THE ETS building. Since the positions ae know pecisely in the testing coido, the efeence can be achieved accuately as depicted in Fig. 5. Longitude (ad) Taectoy (Latitude-Longitude) MEMS SPAN Fig. 5. Coido taectoy The data wee pocessed by the navigation platfom, and the esults ae shown in Fig. 6. The detailed esults ae shown in Table 3. Due to the MEMS senso s high noise chaacteistics, the test of seconds is faily long fo MEMS standalone application. Hence thee ae significant time-dependent difts in the navigation solutions as shown in Fig MEMS Taectoy Latitude (ad) Fig. 7. Taectoy MEMS vs. SPAN Second, the integated navigation test was pefomed on the platfom. The taectoy was made outside the building with the good acquisition of GPS signals. The GPS and MEMS inetial data wee acquied and pocessed by the PC4 platfom and the esults wee logged. The efeence solutions wee calculated in the laptop compute. The taectoy, i.e. the integated MEMS/GPS solution vs. the efeence, is depicted in Fig. 7. The duation of the test was 7s. It can be seen that the MEMS/GPS integation solutions stat to divege fom the efeence at the end of the test due to the MEMS IMU eo gowing much faste than that of the high quality tactic gade SPAN IMU senso. Compaed with the efeence solutions, the maximum position solution eos fom the platfom shown in Fig. 8 ae about 5-7 adian (- metes in Catesian coodinates). longitude(deg) Latitude(ad) Longitude(ad) Position Eos (MEMS Vs SPAN) x x latitude(deg) Fig. 6. MEMS standalone solution Table 3 Taectoy vs. MEMS esults Refeence MEMS Total Distance 5.97m 8.3m Latitude Dist 8.3m 5.3m Longitude Dist. 3.67m 6.36m Test Duation.36s Altitude(m) t (s) Fig. 8. Position solution eos The velocity solution eos ae depicted in Fig. 9. The eos emained small duing the test, while the vetical velocity stated to divege at the end. One of the easons causing this divegence is the time dependent biases in MEMS acceleomete aw measuements. It can be seen that the aw specific foce measuements of the MEMS IMU ae much noisie than those of the tactic gade IMU as shown in Fig..

90 Li et al.: PC4 Based Low-cost Inetial/GPS Integated Navigation Platfom: Design and Expeiments 87 East Vel (m/s) Noth Vel(m/s) Up Vel(m/s).5 Velocity Eos (MEMS Vs SPAN) t (s) Fig. 9. Velocity solution eos Compaed with the efeence solutions, the attitude eos ae 5.7º,.9º and º espectively in oll, pitch and heading angle, as shown in Fig.. These eos ae caused by the high noise contaminating the MEMS angula ate measuements, and moe impotantly thee is no diect attitude obsevation available fom GPS. The heading eo gows ove the time, which is the essential eason causing the slight divegences in velocity and position solutions. X (m/s ) Y (m/s ) Z (m/s ) Raw Acceleometes data (m/s ) SPAN MEMS time (s) Fig.. Raw specific foce measuements MEMS vs. SPAN Roll(ad) Pitch(ad). Attitude Eos (MEMS Vs SPAN) Heading(ad) t (s) Fig. Attitude solution eos Similaly the aw angula ate measuements fom the MEMS IMU and the efeence IMU ae depicted in Fig.. Although measuing the same angula dynamics, the aw angula ate measuements of the MEMS IMU ae much noisie than those of the efeence IMU. Hence it can be concluded the MEMS navigation solutions ae geatly impoved by the integation of the IMU with GPS. X (ad/s) Y (ad/s) Z (ad/s). Raw Gyoscopes data (ad/s) SPAN MEMS time (s) Fig.. Raw angula ate measuements MEMS vs. SPAN 4 Conclusions This pape pesents the design & expeiment fo a low cost inetial/gps integated navigation platfom based on PC4 computes. The navigation system softwae has been specifically designed in Simulink fo low cost inetial senso applications. The simulation expeiment validates the poposed hadwae and softwae designs. The eal-time/data test has demonstated that the PC4 navigation platfom can delive the integated navigation solutions compaable to the efeence solutions, which ae calculated in the conventional desktop compute system, wheeas with less powe consumptions, less system volume/complexity and much lowe ove-all costs. Futhemoe the platfom hadwae is compatible with vaious inetial sensos of diffeent gades. Although thee ae vaious advantages in the Simulink based softwae design poposed by this study, many PC4 hadwae devices ae cuently not suppoted by the Simulink xpc Taget. Moeove the C-pogamming based digital mode navigation softwae compising attitude coning and velocity sculling compensation algoithms is moe appopiate fo a high pefomance navigation system. Hence fully functional C- pogamming based platfom softwae is cuently unde development. By applying a newly acquied USB-based digital MEMS nimu (MEMSense TM ) instead of the analog AccelRate 3D, a complete digital hadwae/softwae PC4 navigation platfom design

91 88 Jounal of Global Positioning Systems with a C pogamming high pefomance navigation softwae will be implemented. Refeences Haywad, R., Gebe-Egziabhe, D., Schwall, M., and Powel, J.D. (997) Inetially Aided GPS Based Attitude Heading Refeence System (AHRS) fo Geneal Aviation Aicaft, Poceeding of Institute of Navigation ION-GPS Confeence,Page Savage, P.G. () Stapdown Analytics Pat I, Stapdown Association, Maple Plain, Minnesota. Savage, P.G. (b) Stapdown Analytics Pat II, Stapdown Association, Maple Plain, Minnesota. Li, D. and Landy, R. (7) MEMS IMU Based INS/GNSS Integation: Design Stategies and System Pefomance Evaluation, the Navigation Confeence & Exhibition NAV7, London, UK Li, D., Landy, R., and Lavoie, P. (7) Validation and Pefomance Evaluation of Two Diffeent Inetial Navigation System Design Appoaches, Intenational Global Navigation Satellite Systems Society Symposium 7, Sydney, Austalia MEMSense AccelRate3D. Gioux R., Landy R.J., Leach B. and Goudeau R. (3), Validation and Pefomance Evaluation of a Simulink Inetial Navigation System Simulato, Jounal of Canadian Aeonautics and Space, Vol. 49, No. 4, p Gioux, R., Goudeau R. and Landy R.J. (5) Extended Kalman filte eal-time implementation fo low-cost INS/GPS Integation in a Fast-pototyping Envionment, 6th Canadian Navigation Symposium, CASI, Toonto, Canada, 6-7 Apil. Titteton, D. and Weston, J. (4) Stapdown Inetial Navigation Technology: Second Edition. AIAA. Flight Simulato 4

92 Jounal of Global Positioning Systems (7) Vol.6, No.: An Innovative Data Demodulation Technique fo Galileo AltBOC Receives Davide Magaia and Fabio Dovis Electonics Depatment, Politecnico di Toino, Italy Paolo Mulassano Navigation Lab, Istituto Supeioe Maio Boella, Italy Abstact. This pape descibes an innovative solution that can be used to ecove the navigation data fom Altenative Binay Offset Caie (AltBOC) modulated signals, a modulation scheme foeseen fo the Galileo satellite navigation system to tansmit fou channels in the E5 band (64-5 MHz). In this pape a novel data demodulation appoach, called Side-Band Tanslato (SBT), suitable to coheent dual band AltBOC eceive achitectues, is intoduced and validated fom the analytical point of view. This patented appoach is based on the idea to pefom a tanslation opeation : this means that the two sepaate in-phase components of the AltBOC signal, containing the navigation data, ae ecoveed fom the eceived signal with a pope signal pocessing, moving the infomation fom the side lobes of the AltBOC spectum to the baseband. The innovative aspects of this demodulation technique ae pointed out in the pape, highlighting the main advantages with espect to aleady poposed techniques. Keywods. Data demodulation, Side-Band Tanslato, E5, AltBOC, Galileo. modulation, in ode to exploit the wideband featues of the signals (e.g. in tems of multipath obustness), with an affodable complexity of the eceives achitectues as fo example in (Dovis et al. 7). In spite of the fact that the featues of the AltBOC modulated signals and the potential pefomance of futue AltBOC eceives ae discussed in seveal papes in liteatue, the ecoveing of the navigation data (demodulation of the two data channels) is not exhaustively examined. Only few patents claim eceive achitectues fo eceiving and pocessing AltBOC modulated signals: they popose some demodulation stategies that show some dawbacks, in tems of implementation complexity and intefeence vulneability. The pape is oganized as follows: in Section a bief eview of the AltBOC signal is povided, and in Section 3 cuent poposals fo data demodulation ae eviewed. Section 4 will intoduce the AltBOC eceive and Section 5 will focus on the poposed Side Band Tanslato. The impact on the implementation is analyzed in Section 6, and then Section 7 will daw some conclusions. Oveview of the AltBOC modulation Intoduction The futue Galileo system, a new Global Navigation Satellite System (GNSS) developed by the Euopean Commission and the Euopean Space Agency (ESA) and foeseen to be opeational in 3, will use the novel Altenative Binay Offset Caie (AltBOC) modulation scheme to tansmit fou channels in the E5 band (64-5 MHz). Seveal papes in the liteatue have addessed the design of acquisition schemes and tacking stages fo this Fou channels (e E5a-I, e E5a-Q, e E5b-I and e E5b-Q ) will be tansmitted in the E5 band by each Galileo satellite taking advantage of a novel modulation and multiplexing scheme, the AltBOC modulation. Two of fou E5 channels ae the so-called data channels (e E5a-I and e E5b-I ), since they cay navigation data, wheeas the othe two (e E5a-Q and e E5b-Q ) ae called pilot channels and ae not data modulated. A eceive will be able to distinguish the fou channels since fou diffeent quasi-othogonal Pseudo-Random Noise (PRN) codes (c E5a-I, c E5a-Q, c E5b-I and c E5b-Q ) will be

93 9 Jounal of Global Positioning Systems used fo each satellite of the Galileo system. In this way it is possible to ecognize the two data channels (e E5a-I and e E5b-I ) in the eceived signal and to demodulate thei navigation data. It must be noticed that the fou codes tansmitted by one satellite ae synchonous, without elative bias o elative chip-slip. In paticula fo the data channels the edge of each data symbol coincides with the edge of a code chip: peiodic speading codes stat coincides with the stat of a data symbol. A detailed desciption of the geneation of the Galileo AltBOC modulated signal s 5( t E ) can be found in the Galileo Open Sevice Signal In Space Inteface Contol Document (GAL OS SIS ICD/D., 6). The analytical expession of the s 5( t E ) signal is epoted hee with the notation used in the Galileo OS SIS ICD (baseband complex envelope epesentation): s = E5 [ e E5a I + ee5a Q ] [ sce5 S sce5 S ( t TS, E5 / 4) ] [ ee5b I + ee5b Q ] [ sce5 S + sce5 S ( t TS, E5 / 4) ] [ ee5a I + ee5a Q ] [ sce5 P sce5 P ( t TS, E5 / 4) ] [ ee5b I + ee5b Q ] [ sce5 P + sce5 P ( t TS, E5 / 4) ] In Equation () the two data channels ( () ee5 a I and ee5 b I ) ae shown with bold types. They ae defined with the following expessions: [ ce a I, i de5a I, [ i] ectt ( t i TC, E a I ] + E5a I = 5 ) L E a I DCE a I C, E 5a I i= e () [ ce b I, i d E5b I, [ i] ectt ( t i TC, E b I ] + t = ) E5b I ( ) 5 E b I DC E b I C E b I i= L, e (3) whee ectt is the ectangle function, which is equal to fo < t < T and it is equal to elsewhee. In Equation () and Equation (3) the two PRN codes codes ( ce5 a I and ce5 b I ) and the two navigation data steams ( d E5 a I and d E5 b I ) ae pointed out. e The othe two channels ( E5 a Q and E5 b Q ), the socalled pilot channels, do not cay navigation data, as shown in Equations (4) and (5): e e [ ce5a Q, i ectt ( t i TC, E5a Q ] + E5a Q(t) = ) L E a Q C, E 5a Q i= e 5 (4) [ ce5b Q, i ectt ( t i TC, E5b Q ] + E b Q(t) = ) 5 E b Q C E b Q i= L, 5 5 (5) It must also be noticed that the AltBOC modulation allows to use the E5 band as two sepaate sidebands, conventionally denoted as E5a ( MHz) and E5b ( MHz). In this way, a single data channel (equivalent to a BPSK signal) and a pilot channel (anothe BPSK signal) will be tansmitted in each sideband. Accodingly, this modulation scheme can be teated as to two sepaate QPSK modulations, placed espectively aound the E5a and the E5b cente fequency. The demodulation of the navigation data fom the eceived signal is then a cumbesome task that must be caied out by futue AltBOC eceives, since the two channels e E5a-I and e E5b-I ae tansmitted in two adacent sidebands. 3 Existing AltBOC Demodulation Techniques At time of witing, only few patents (Geein, 5 and De Wilde et al 6) claim eceive achitectues fo eceiving and pocessing AltBOC modulated signals, consideing some diffeent implementations of the complex coelation opeations needed fo the coheent tacking of the entie E5 band (coheent dual band Galileo AltBOC eceive achitectue). In detail only in (Geein, 5) a possible solution fo the data demodulation is poposed. In (De Wilde et al 6) the tem demodulation is impopely used, since in this document the ecoveing of the navigation data is not discussed, but only some methods and devices fo tacking the pilot channels ae pesented. The demodulation stategy poposed in (Geein, 5) shows some dawbacks, concening the implementation complexity and intefeence vulneability. In this case a not staightfowad solution is used to ecove the navigation data. Fist, two eplicas of the PRN codes used in the data channels ( ce5 a I and ce5 b I, called espectively c and c in the patent) and the coesponding squae wave subcaies ae locally geneated and combined. The obtained local signals ae coelated with the eceived signal, aiming to obtain the eal and imaginay components of the sum ( R + R) and the diffeence ( R R ) between the coelation functions of the two codes. Futhe signal pocessing is equied to ecove the navigation data fom ( R + R ) and ( R R ), using a look-up table appoach. Moe details and the complete demonstation can be found in (Geein, 5). This demodulation technique fo AltBOC signals shows the following dawbacks: cumbesome signal pocessing is equied, since complex local signals must be geneated and combined and, afte the coelation opeations,

94 Magaia et al.: Data Demodulation Technique and Device Suitable to Galileo AltBOC Receives 9 futhe calculations ae equied to decode the navigation data (look-up table); the eceive pefomance is degaded by coelation losses: this is due to the fact that the subcaies locally geneated in (Geein, 5) ae diffeent fom those used by the Galileo satellites and this implies a coelation loss, as stated in (Soellne and Ehad, 3). In paticula in (Geein, 5) the Complex- BOC and the Complex-LOC modulations ae consideed as appoximations of the AltBOC eceived signals. But the tue AltBOC modulation that will be used fo the Galileo E5 band diffes fom the Complex-LOC and the Complex-BOC essentially fo the pesence of additional tems in the modulated signal expession (the so-called poduct signals) and fo a diffeent shape of the subcaie wavefoms (GAL OS SIS ICD/D., 6); this demodulation technique is vulneable, since the two data channels ae ointly demodulated, taking advantage of ( R + R ) and ( R R ) coelation esults. In this way an eo on one data bit (e.g. caused by an intefeing signal on a single sidelobe of the E5 band) can affect also the coect demodulation of the othe channel; it is not possible to tempoaily demodulate only one data channel (e.g. in a cetain condition whee the navigation data of the othe channel ae not necessay), switching off the demodulation section of the othe channel o eusing its dedicated hadwae o softwae esouces. 4 Poposed Galileo AltBOC Receive Achitectue A modified achitectue fo an AltBOC eceive, based on the coheent eception and pocessing of the entie Galileo E5 band, is depicted in Fig.. This eceive is simila to the ones poposed in (Geein, 5) and (De Wilde et al 6), but an innovative despeading and demodulation section, tailoed to the AltBOC modulation, is used. In Fig. a high level block diagam of the eceive is pesented: it is only intended to simply explain the functioning of the eceive. The implementation details about the complex coelation and discimination opeations and the possible optimizations that can be pefomed in the achitectue of the eceive (e.g. see Geein, 5 and De Wilde et al 6) ae not epoted hee, due to the fact that ae consideed backgound. Afte the Radio Fequency (RF) font end and the Intemediate Fequency (IF) section, the eceived signal is pocessed by the PLL, the DLL and the demodulation sections that ae the most impotant functional blocks of the eceive. In fact the main diffeences between a conventional GPS eceive and the AltBOC eceive can be noticed in the opeations pefomed by these blocks: the Phase Locked Loop (PLL) is used to coheently tack the cental caie of the E5 band (located at MHz), sepaating the in-phase and the quadatue components of the eceived signal (I and Q); the Delay Locked Loop (DLL) is necessay in ode to ecove speading code synchonism and then data symbol synchonism. In fact, as peviously noticed, the fou E5 channels of each Galileo satellite ae coheently tansmitted, without elative bias o elative chip-slip. The DLL functioning is based on the tacking of the two pilot channels ( ee5 a Q and ee5 b Q ). This is done geneating local eplicas of the PRN codes used fo the pilot channels ( c E5 a Q and ce5 b Q ) and of the subcaie wavefoms ( sce 5 S and sce5 S ( t TS, E5 / 4) ). These local signals ae used to pefom complex coelation opeations with the I and Q eceived samples, as discussed in (Sleewaegen et al, 4). It must be pointed out that the tacking opeations can be pefomed taking advantage of diffeent kinds of disciminato: in Fig. the simplest one, the Ealy-Late disciminato, is used fo sake of simplicity; the demodulation section ecoves the navigation data fom the two data channels ( ee5 a I and ee5 b I ), taking advantage of the synchonism ecoveed by the DLL. In paticula it is necessay to pefom the despeading, with local eplicas of the PRN codes used fo the data channels ( ce5 a I and ce5 b I ), and the data detection. It must be noted that the demodulation section in the eceive achitectue in Fig. shows emakable diffeences with espect to the achitectue poposed in (Geein, 5). In fact a diffeent demodulation technique, based on an innovative device called Side- Band Tanslato, is used.

95 9 Jounal of Global Positioning Systems Fig. : Block diagam of a modified coheent dual band eceive achitectue fo Galileo AltBOC signals 5. The Side-Band Tanslato (SBT) The sideband tanslato is an innovative subsystem within the AltBOC eceive that can be used to demodulate the navigation data included in the wideband AltBOC signal. This solution has been patented (Magaia, Mulassano and Dovis, 7), and it is based on the idea to pefom a tanslation opeation : this means that the two sepaate in-phase components, containing the navigation data ( e and E5 a I ee5 b I ), ae ecoveed fom the eceived signal 5( t) s E, peviously descibed in Equation (). To undestand the opeations pefomed by the SBT, it is useful to conside a simple situation, as in the case of a BOC eceive. With a BOC modulation, the signal to be tansmitted is multiplied with a ectangula subcaie: this opeation causes a fequency shift that leads to the two typical sidelobes of the BOC spectum (simila to the spectum of the E5 AltBOC signal). To demodulate this split-spectum signal, once the eceived signal is coectly tacked by the DLL and the PLL of the BOC eceive (the local PRN code is synchonized), a possible appoach is to multiply the eceived BOC signal again with a local eplica of the ectangula subcaie that can be geneated with the synchonism ecoveed by the DLL. This opeation

96 Magaia et al.: Data Demodulation Technique and Device Suitable to Galileo AltBOC Receives 93 tanslates the two sidebands of the BOC signal again to the baseband: in this way, the signal becomes again a baseband signal and the infomation contained in it could be easily ecoveed with a BPSK data detecto, afte the despeading with the local PRN code. Accodingly, with a BOC modulation the sideband tanslation opeation coesponds to a simple multiplication with a local ectangula subcaie that e-convets the eceived signal in a baseband signal. Howeve, with the AltBOC modulation this opeation is moe complex, because thee ae fou channels tansmitted in the E5 band (instead of only one, as in the pevious example) and the fequency shifts of these channels to the two sidebands ae pefomed taking advantage of complex exponentials. In detail the SBT selects the two in phase data channels ee5a I and ee5b I and moves them fom the sidebands of the AltBOC spectum to the baseband, as highlighted by the ed aows in the scheme in Fig.. modulated signal expession, epoted again hee fo sake of claity: s = E5 [ ee5a I + ee5a Q ] [ sce5 S sce5 S ( t TS, E5 / 4) ] [ ee5b I + ee5b Q ] [ sce5 S + sce5 S ( t TS, E5 / 4) ] [ ee5a I + ee5a Q ] [ sce5 P sce5 P ( t TS, E5 / 4) ] [ ee5b I + ee5b Q ] [ sce5 P + sce5 P ( t TS, E5 / 4) ] (6) The dashed tems contained in the last two lines of Equation (6) can be neglected, because they coespond to the so-called poduct signals: these tems ae multiplied by sce5 P subcaie wavefom, with smalle amplitude than sce5 S, and they do not cay useful infomation. The poduct signals ae only needed to obtain a constant envelope modulated signal. Moe details can be found in (GAL OS SIS ICD/D., 6), (Ries L. et al, ), (Ries L. et al, 3) and (Soellne and Ehad, 3). Fig. 3: Theoetical scheme of the sideband tanslato Fig. : Illustation of the fequency spectum of the E5 AltBOC modulated signal and the opeations pefomed by the sideband tanslato block Accodingly, the sideband tanslato block needs to use complex exponential multiplications to move these channels to the baseband, pefoming two sepaate fequency shifts, and then it must choose the coect channels (only the in-phase channels, containing the navigation data), selecting only the eal pat of the obtained signals as shown in Fig. 3. Finally the SBT povides the two ecoveed in-phase channels ( ee5 a I and ee5 b I ), that ae passed to subsequent despeading and BPSK data detecto blocks. In this way the navigation data ae ecoveed by means of a staightfowad signal pocessing opeation, simple than the appoach used in (Geein, 5). The opeations pefomed by the sideband tanslato block can be undestood consideing the AltBOC It is then possible to decompose the modulated signal s E in its eal and imaginay components, ( ) 5 t neglecting the poduct signals: s s E5I E5Q s = se5i + se5q( ) (7) E5 t [ e + e ] E5a I + [ e e ] sc ( t T / 4) E5a Q E5b I E5b Q sc [ e + e ] E5a Q E5 S E5 S + S, E5 [ e e ] sc ( t T / 4) E5b I E5b Q E5a I sc E5 S E5 S S, E5 (8) (9) The two components s E I ( ) and s E Q ( ) can be 5 t 5 t consideed as the ideal eceived signals in the I and Q

97 94 Jounal of Global Positioning Systems banch of the eceive in Fig.. In fact, assuming the coect synchonization of the eceive (PLL and DLL coectly locked) and neglecting the noise, the distotions and othe popagation effects, the eceived signal s E 5( t) is downconveted to the baseband and is patitioned in the I and Q banch of the eceive, sepaating its eal and imaginay pats. It must be emaked that taking advantage of the E5 AltBOC modulation, the fou channels e ( ) E5a I t, ee5a Q, ee5b I and ee5b Q ae tansmitted in the two sidebands of the E5 band. This is achieved using the subcaie wavefom sce5 S, that esembles a sampled cosine, and its delayed vesion sce5 S ( t TS, E5 / 4), simila to a sampled sine. The two subcaie wavefoms ae pesented in detail in (GAL OS SIS ICD/D., 6). In the following, fo sake of simplicity, the second function is denoted as sc off E5 S. In the fist two lines of Equation (6) these two wavefoms ae used like complex exponentials: The fist subcaie exponential is obtained with off the tem [ sce5 S sce5 S ]. It pefoms a simila opeation in the fequency domain than the complex exponential exp( πf subt), whee f sub is the subcaie fequency f sub = R S, E5 = MHz. This exponential opeates a downshift fo the two E5a channels and in this way ee5a I and ee5a Q ae shifted fom the baseband to the left sidelobe of the AltBOC spectum (E5a sideband); In a simila way, the second subcaie exponential off [ sce5 S + sce5 S ] coesponds to the complex exponential exp( + πf subt) and it upshifts the two E5b channels ee5b I and e ( ) E5b Q t. The sideband tanslato takes advantage of this idea, pefoming the opposite opeation: with a pope use of the two exponentials, the two in-phase channels ee5a I and ee5b I can be extacted fom the baseband eceived signal s E ( ). 5 t To obtain the ee5a I channel it is necessay to opeate an upshift of the eceived signal in the fequency domain, multiplying it fo the second exponential. In this way the ee5a I signal becomes centeed to the baseband and it can be ecoveed selecting the in-phase (eal) component of the esult of the multiplication, as shown in the following equations: e E5a I e e E5a I [ se5i + se5q ] Re off [ sce5 S + sce5 S ] off [ s E5I sce5 S se5q sce5 S ] + Re off + [ se5i sce5 S + se5q sce5 S ] () () off se5i sce5 S se5q( t) sce5 S ( ) () E5a I t Similaly to that done fo the ee5a I channel, it is possible to ecove the ee5b I signal, downshifting the eceived signal s E 5( t) with the following opeations: e E5b I e e E5b I [ se5i + se5q ] Re off [ sce5 S sce5 S ] off [ s E5I sce5 S + se5q sce5 S ] + Re off + [ se5q sce5 S se5i sce5 S ] (3) (4) off se5i sce5 S + se5q sce5 S ( ) (5) E5b I t Equations () and (5) then define the functioning of the sideband tanslato and allows to simply ecove the two data channels ee5a I and ee5b I. 6 Implementation of the SBT Functional Block A possible implementation of the sideband tanslato is pesented in Fig. 4. In this functional block the two opeations descibed by Equation () and Equation (5) ae diectly implemented in the discete time domain, with multiplications and sums between the samples of the eceived signal and the locally geneated subcaie wavefoms. As shown in the block diagam, the esults of the two equations could be filteed, with two baseband lowpass filtes, in ode to educe the intefeence and the coss-coelation caused by the adacent channels. The shape and the bandwidth of the filtes must be optimized, because a naow band filteing can educe the pefomance of the demodulation section, wosening the coelation popieties of the two data channels, but also a filte too wide could be an issue in pesence of noise and intefeences.

98 Magaia et al.: Data Demodulation Technique and Device Suitable to Galileo AltBOC Receives 95 Fig. 4 Block diagam of the sideband tanslato In conclusion, the sideband tanslato functional block povides as two sepaate outputs the two data channels ee5a I and ee5b I, extacted fom the eceived signal. In this way it is possible to subsequently ecove the navigation data fom the two outputs of the SBT, pefoming two sepaate despeading opeations and two BPSK data detections, as peviously epesented in Fig.. 7 Conclusions In this pape an innovative appoach has been pesented as a valid solution in ode to demodulated the navigation data fom an AltBOC modulated signal. o The two data channels ee5a I and ee5b I of the Galileo E5 band ae ecoveed taking advantage of the idea to opeate two fequency shifts on the eceived signal; o The fequency shifts ae pefomed using eal signals, obtained with local eplicas of the AltBOC subcaie wavefoms sce5 S and ( ) sc off E5 S t ; o The two signals ecoveed with these fequency shifts can be sepaately filteed, in ode to educe intefeences and coss-coelations with adacent channels; o Finally, the navigation data ae sepaately ecoveed as two BPSK signals, pefoming the despeading and the demodulation opeations. The poposed demodulation appoach shows seveal diffeences with espect to the solution in (Geein, 5), since a diffeent signal pocessing is used. This leads to the following advantages: A simple signal pocessing that implies a saving in hadwae and softwae esouces. In fact the navigation data ae diectly ecoveed fom the two outputs of the sidebands tanslato and futhe calculations to decode the data fom thei sum and diffeence, as in (Geein, 5), ae not necessay; A bette eceive pefomance, avoiding coelation losses in the demodulation section; in fact in the poposed eceive achitectue (see Fig. ) the coect subcaie wavefoms sce5 S and sc off E5 S ae locally geneated and used by the sideband tanslato to pefom the fequency shifts; An impoved obustness of the demodulation section, since an eo in a data bit of one channel (e.g. caused by an intefeing signal on the E5a sideband) does not affect the coect demodulation of the othe data channel; in fact the two data channels ae sepaately downconveted and demodulated, taking advantage of the SBT; A bette intefeence eection, because the two low-pass filtes in the SBT allow to educe out-ofband intefeing signals and coss-coelations caused by PRN codes of adacent channels; Moe flexibility fo the functioning of the demodulation section; in fact it is possible to tempoaily demodulate only one data channel (e.g. in a cetain condition whee the navigation data of the othe channel ae not necessay), switching off the demodulation of the othe channel (powe saving) o eusing its dedicated hadwae o softwae esouces. Refeences Dovis F., Mulassano P., Magaia D. (7), Multiesolution Acquisition Engine Tailoed to the Galileo AltBOC Signals, in Poceedings of ION GNSS 7, Fot Woth, TX (USA), Sept. 4-8, 7 De Wilde W. et al (6), A Method and Device fo Demodulating Galileo Altenate Binay Offset Caie (AltBOC) Signals, Euopean Space Agency (Pais, FR), Intenational Patent (WIPO) No. WO 6/74 A, 6 Mach 6. GAL OS SIS ICD/D. (6), Galileo Open Sevice Signal In Space Inteface Contol Document (OS SIS ICD), Daft, Euopean Space Agency / Galileo Joint Undetaking, 3 May 6. Geein N. (5), A Hadwae Achitectue fo Pocessing Galileo Altenate Binay Offset Caie (AltBOC)

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