The Clock-Aided Method for GPS Receiver Positioning in an Urban Environment

Transcription

1 Internatonal Journal of Comuter and Electrcal Engneerng, Vol. 3, No. 3, June The Cloc-Aded Method for GPS Recever Postonng n an Urban Envronment Yunlong Teng, Ybng Sh, and Zh Zheng Abstract As the sgnals of GPS satelltes are suscetble to obstructons n an urban envronment wth many tall buldngs around, the number of vsble satelltes s usually less than four. In the crcumstances, the normal ostonng method can not be used for ostonng calculaton. In order to obtan contnuous ostonng results under three satelltes, the seres of recever cloc bas (RCB) s consdered as one vsble satellte n ths aer. A combned model for redctng the RCB seres based on the theory of tme seres analyss s resented frstly. And then the model s used to ad the normal ostonng method for ostonng calculaton under the condtons of three satelltes. From both statc and dynamc exerments, t s shown that the redcton model s ft for redctng the RCB seres, and the cloc-aded ostonng method adng of the redcton model can mlement the functon of three-dmensonal ostonng calculaton even f there are only three vsble satelltes. Index Terms Urban envronment, Postonng calculaton, Recever cloc bas (RCB), Combned redcton model I. INTRODUCTION As a hgh-recson ostonng technology, the Global Postonng System (GPS) lays an mortant role n our daly lves []. Normally, there must be at least four satelltes n vew so that the ostonng results of the GPS recevers can be obtaned [,3]. However, n certan alcatons, for examle n an urban envronment wth many tall buldngs around, there are stuatons wheren the requrement of a mnmum of four satelltes s not satsfed because of the obstructons to recever-to-satellte lne of sghts. In ths case, the tradtonal method cannot be used for calculatng the oston of GPS recever anymore. How to deal wth ths bad-condtoned roblem so that the GPS recever can rovde contnuous and relable ostonng nformaton s the man theme of ths aer. In order to obtan the contnuous and relable ostonng nformaton of GPS recevers under bad condtons, several methods have been resented. Jan solved the roblem from three satelltes and altmeter [4], Hong combned GPS Manuscrt receved January, ; revsed Arl 8,.Ths wor was suorted n art by the Fundamental Research Funds for the Central Unverstes (ZYGX9J9) and the Aeronautc Scence Foundaton of Chna (983). Yunlong Teng s wth the School of Automaton Engneerng, and Research Insttute of Electronc Scence and Technology, Unversty of Electronc Scence and Technology of Chna, Chengdu 673, Schuan, Chna (corresondng author, e-mal: tyluestc@ 63.com). Ybng Sh s wth the School of Automaton Engneerng, Unversty of Electronc Scence and Technology of Chna, Chengdu 673, Schuan, Chna (e-mal: ybsh@uestc.edu.cn). Zh Zheng s wth the Research Insttute of Electronc Scence and Technology, Unversty of Electronc Scence and Technology of Chna, Chengdu 673, Schuan, Chna (e-mal: zhengzh-uestc@sohu.com). recever wth the Inertal navgaton system []. Lu made use of the addtonal nformaton rovded by seudolte [6]. If a road ma database exsts, the ma-matchng technque can also be utlzed [7]. The common characters of these methods need external hardware. The above mentoned methods may be useful, but not effcent. In fact, the cost of them has great mact on ther oularzaton. Actually, besdes the oston nformaton, the cloc bas between GPS recever and GPS system s gven when the functon of ostonng calculaton s executed normally. The bas s called recever cloc bas and short for RCB n the aer. If the cloc frequency ees steady, a redcton model of the RCB seres can be obtaned by usng a length of hstory data [8]. Hence, the RCB seres can be regarded as one vsble satellte, and the redcton model on t s then ntroduced nto GPS recever for ostonng calculaton under the condtons of only three satelltes. Comared wth other auxlary methods, the cloc-aded ostonng method needs no external equments. The advantages of the cloc-aded ostonng method are clear and obvous. There are many comlcated factors whch have an mact on the RCB seres, ncludng the atmoshere and onoshere delays, the orbtal errors of GPS satelltes, etc. Under the nfluence of these factors, the RCB seres can be consdered to be nonlnear tme seres. Generally, redcton models for nonlnear tme seres can be groued nto two tyes, sngle models and combned models. Combned models ntegrate several sngle models to mrove the redcton recson. Most combned models are decomoston based. They decomose the orgnal tme seres nto several solated comonents frstly. And then they emloy a sngle model to redct each of them, and lastly they ntegrate the results to gan the fnal redcton value. Because of the aroxmate of one-fold roerty of each comonent, ther redcton wll be mroved [9]. A combned redcton model can cature many more of these roertes than a sngle model, and thus mght mrove the redcton recson. Consdered the nonlnear characters of the RCB seres, a combned model for redctng t based on the theory of tme seres analyss s resented n the aer. The basc deas and detaled stes of the combned redcton model are dscussed resectvely. Then the redcton model s ntroduced nto GPS recever for ostonng calculaton together wth three vsble satelltes. Test results are nvestgated to demonstrate the feasblty and valdty of the method n the end. II. DESIGN SCHEME OF RCB PREDICTION MODEL The combned redcton model for the RCB seres on the bass of the theory of tme seres analyss ncludes three 389

2 Internatonal Journal of Comuter and Electrcal Engneerng, Vol. 3, No. 3, June hases. Frstly, the RCB seres were decomosed nto a low- frequency comonent (LFC) and several hgh-frequency comonents (HFCs) by wavelet transform. Then the LFC was redcted through olynomal model (PM), and the HFCs were redcted by usng auto-regressve (AR) models. The redcted results were the suermoston of the resectve redcton. Based on these rocedures, Fg. llustrates the flowchart of the combned redcton model. Fg. Flowchart of the combned redcton model. A. Wavelet Transform of RCB Seres In the combned redcton model, the decomoston method used s wavelet transform for ts smlcty. There are contnuous wavelets transforms and dscrete wavelet transforms. Snce the RCB seres are dscrete, the dscrete wavelet transform s adoted. The Mallat algorthm [] based on mult-resoluton reresentaton s a fast dscrete wavelet transform algorthm, and t has shown romsng results n varous areas [,]. On the bass of the characters of the Mallat algorthm, t s utlzed to decomose the RCB seres. Let { C() t }( t =~ N ) denote the orgnal RCB seres, the decomoston rocedure can be descrbed as follows. c = C c+ = Hc, =,,,( -) () d+ = Gc In (), =logn denotes the maxmal decomoston level. H and G are the wavelet flters for decomoston n the tme doman, that s to say, H s the low-ass flter and G s the hgh-ass flter. c and d are the LFC and HFC at the corresondng level, resectvely. Through teraton, the RCB seres could be decomosed nto one LFC lus a set of HFCs []. The LFC reresents the trend term, and the HFC s the statonary comonent wth the zero mean. After the wavelet decomoston, the lengths of the LFC and HFCs are halved comared wth the orgnal seres. So they have to be reconstructed by usng (). D+ = H C + G D, = ( -),, () Where H and G are the dual oerators of H and G. By means of usng (), c ( d ) can be reconstructed as ( ) C D, whose lengths equal to the one of the RCB seres. And they C t = C t + D t + + D t. satsfy () () () () B. The Predcton of LFC The LFC reveals the develoment trend of the RCB seres wth resect to tme. Related exerments have demonstrated that when the cloc frequency of GPS recever ees stable, the RCB seres can be modeled and ftted by a olynomal model [8]. Accordngly, the aer taes advantage of the olynomal model to dstll the trend of the RCB seres, and obtan the redcted seres C () t. C. The Predcton of HFC As dscussed above, the HFCs ( D () t, D () t ) obtaned from the wavelet transform have the statstcs characters, and can be regarded as statonary seres. The AR model s the basc and mortant tme seres model, whch s mostly used for modelng and redctng statonary tme seres. Hence, AR model s establshed for redctng the above HFCs. In ths secton, we tae examle for D () t, and the modelng rocedures of other HFCs ( D () t, D () t ) are smlar. For convenent for further research, D () t s short for x() t n ths secton. The mathematcal exresson of AR model for redctng x() t can be descrbed as x() t = φ x( t ) + μ() t (3) = Where s the order of ths model, and t can be determned by emloyng the AIC method. The arameters φ can be calculated by usng the Least-square method n (4). T ( ) - φ φ = V V VR (4) Where x( ) x( ) x( + ) x( +) x( ) x( ) V = +, R = () x( N ) x( N ) x( N) On the bass of the above exressons, we can get the redcted results of x() t. (6) = () = φ ( ) x t x t D. Predcton of RCB Seres Ths ste s dedcated to calculate the redcted results of the RCB seres. By means of combnng the redcted results of all the comonents, we can obtan the underlyng redcted results of the RCB seres. () () () C t = C t + D t (7) = III. GPS POSITIONING USING CLOC-ADIED METHOD The GPS ostonng method adng of the RCB redcton model s gven n ths secton. We begn wth showng the normal ostonng method of GPS recever. Then the clocaded ostonng method under the condtons of three vsble satelltes s dscussed artcularly. A. Normal Postonng Method Suose at (,, ) denotes the seudorange from GPS recever. After the modeled errors are roerly comensated, the seudorange between x yz to GPS satellte at ( x, y, z ) 39

3 Internatonal Journal of Comuter and Electrcal Engneerng, Vol. 3, No. 3, June them can be wrtten as = ( ) + ( ) + ( ) + + x x y y z z c ω (8) Where c denotes the RCB value exressed n unts of length, and ω s the measurement error. In (8), x, yzand, c are four arameters need to be estmated. Durng the course of ostonng calculaton, (8) should be lnearzed through Taylor s seres around the aroxmate x, yz, and neglectng the hgher order terms. coordnate ( ) Defnng can obtan Where as at the aroxmate coordnate, then we Δ = = axδ x+ ayδ y+ azδ z (9) Δ ς = ς ς, ς = xyz,, ς ς aς =, ς = x, y, z () = ( x x) + ( y y) + ( z z) Let M reresent to the number of vsble satelltes, then we can obtan the lnear measurement equaton n a comact matrx formulaton as Y = AX + V () In (), the measurement vector Y, the measurement matrx A, and the four-dmensonal estmated vector X are defned by Δ ax ay az Δx a x ay az Δ Δy Y =, A=, X = () Δ z Δ axm aym azm M c Vector X ncludes four arameters. It s clear that fewer than four satelltes wll not rovde enough nformaton for an accetable oston of GPS recever. Unfortunately, for car navgaton n an urban envronment wth many tall buldngs around, t frequently occurs that there are only three vsble satelltes. In the crcumstances, the normal ostonng method cannot solve the four arameters. B. GPS Postonng Usng Cloc-Aded Method In order to solve the roblem of GPS ostonng under such bad condtons, the RCB redcton model s consdered to be one vsble satellte and used to comlete the ostonng rocess by exandng the lnear measurement equaton. When the cloc-aded ostonng method s adoted to calculate the oston of GPS recever, the redcted value of the RCB seres C( N + ) s used to relace the actual one. When there are three vsble satelltes, the exended equaton for ostonng calculaton can be exressed by the followng equaton Y = AX + V (3) In (3), vector X only ncludes three modfed tems to the aroxmate coordnate ( Δx, Δy, Δ z). The exanded measurement vector Y and the exanded measurement matrx A are descrbed n (4). ( ) ( ) ( ) Δ C N + a a a Y = C N A a a a x y z Δ +, = x y z 3 C N ax3 ay3 a z3 (4) Δ + From () and (3), t can be seen that the estmated arameters decrease from four to three after ntroducng the RCB redcton model. Accordngly, the cloc-aded method can obtan the oston of GPS recever even f there are only three vsble satelltes. When the nverse of A exsts, the soluton of X and ts modulus can be got from (). ( ) X = A Y () X = ( Δ x) + ( Δ y) + ( Δz) Once the modulus s obtaned, t wll be used to comare wth a threshold to judge whether the rocess of ostonng calculaton s acheved or not. If t s larger than the threshold, the aroxmate coordnate should be modfed accordng to the underlyng exressons. ς = ς +Δ ς, ς = x, yz, (6) After obtanng the modfed oston, we consder t as the new aroxmate oston and mae use of t to erform the functon of ostonng calculaton through teraton. Untl the modulus of the estmated vector s lower than the gven threshold, the unnterruted ostonng results can be fxed from three satelltes. Based on the above mentoned rocedures, Fg. dects the flowchart of the GPS ostonng usng the cloc-aded method. Fg. Flowchart of GPS ostonng usng cloc-aded method. IV. EXPERIMENTAL RESULTS AND DISCUSSIONS In ths secton, we nvestgate the ostonng erformance of the resented method n redctng RCB seres and fxng the oston of GPS recever by some exerments. In these exerments, the statc and dynamc tests wth GPS recever are dscussed resectvely. The orgnal data were obtaned from an OEM board wth samlng frequency Hz, and there were four vsble satelltes. Furthermore, the ostonng results brought out by the normal ostonng method are consdered as the correct ones. In an urban envronment wth many tall buldngs around, the stuatons wheren only three vsble satelltes only last for a few seconds. Accordngly, the method ntroduced n the aer can redct the subsequent values of RCB seres by 39

4 Internatonal Journal of Comuter and Electrcal Engneerng, Vol. 3, No. 3, June usng the hstorcal data frstly, and then the redcted values are ntroduced nto GPS recever to erform the functon of ostonng calculaton wth three vsble satelltes. A. Statc Exerment Performance of the Method Ths secton analyzes the erformance of the resented method under statc exerment. In ths art, GPS recever et statonary, and t lasted for s. Frstly, the four estmated arameters of GPS recever were calculated by the normal ostonng method. Secondly, the numbers of vsble satelltes decreased from four to three between 9s and s. In ths case, the 9 hstorcal data n the RCB seres were adoted to buld the redcton model, and then the 3 subsequent redcted values were obtaned through the model. After obtanng the redcted values, the ostonng nformaton by usng the cloc-aded method s gven. ) Predcton Results of RCB seres By usng the combned redcton model, Fg.3 llustrates the redcted errors of the RCB seres. It s observed that the maxmal value among the redcted errors s less than 8 meters. In addton, the redcted errors cannot ncrease wth the ncrement of redcton nterval on the whole. That s to say, the combned redcton model n the aer s sutable for redctng the RCB seres. In fact, the combned redcton model resented n the aer decomoses the RCB seres nto several comonents ncludng a LFC and several HFCs frstly. Based on the artcular characterstc of dfferent comonents, the PM and AR models are establshed n them resectvely. Fnally, the redcted values from the above sngle models are ntegrated together for obtanng the fnal redcted values of the RCB seres. The combned redcton model can cature the characterstcs and advantages of these sngle models, and thus may mrove the redcton recson. Predcted Errors(m) Fg.3 The redcted errors of the RCB seres. ) Postonng Performance of the Cloc-aded Method After the redcted values of the RCB seres are obtaned, they are used for calculatng the oston of GPS recever through the cloc-aded method n the aer. The threedmensonal ostonng errors (ErrX, ErrY and ErrZ) of GPS recever are llustrated n Fg.4. It s seen from Fg.4 that the ostonng errors n three dmensons are less than meters. That s to say, the cloc-aded ostonng method can not only mrove the contnuty and relablty of GPS ostonng, but also can mlement the functon of threedmensonal ostonng calculaton under the condtons of only three vsble satelltes. ErrX(m) ErrY(m) ErrZ(m) Fg.4 The three-dmensonal ostonng errors. When the condton of the normal ostonng method can not be satsfed, the cloc-aded ostonng methodology resented n the aer can obtan the ostonng results by consderng the redcted RCB values as one vsble satellte. Consequently, the ostonng erformance of GPS recever s related wth the redcton recson of the RCB seres. In order to analyze the relatons between them, Fg. llustrates the absolute redcted errors of the RCB seres together wth the ostonng errors of GPS recever. It s shown that the more accurate the redcted values of the RCB seres, the smaller the ostonng errors of GPS recever. Predcted Errors(m) Postonng Errors(m) Fg. The redcted errors and ostonng errors. The reason of ths henomenon s that the cloc-aded ostonng method taes advantage of the redcted RCB value to calculate the exended measurement vector, and then the exended measurement vector s utlzed for the ostonng calculaton. In the above rocess, the redcton recson of the RCB value lays an mortant art n the recson of the exended measurement vector, and then nfluences the ostonng erformance of GPS recever as a result. When the redcton error of the redcted RCB value s smaller, the recson of the exended measurement vector s hgher. In the crcumstances, the ostonng results of GPS recever usng the cloc-aded method rovded by these values are more accurate. B. Dynamc Exerment Performance of the Method A dynamc exerment s nvestgated to demonstrate the caablty of the resented method. In the exerment, the GPS recever moved along wth a straght lne and ts seed was m/s. In order to assess the erformance of the resented method under the condtons of the dynamc exerment, ths secton maes use of the 9 hstorcal data of the RCB seres to obtan the 3 subsequent redcted values n the RCB seres. the redcted errors of the RCB seres and the 39

5 Internatonal Journal of Comuter and Electrcal Engneerng, Vol. 3, No. 3, June ostonng errors of GPS recever are shown n Fg.6. From ths fgure, we can see that: () When the length of the stuatons wth only three vsble satelltes s shorter than 3s, the redcted errors are less than meters. Consequently, the whole ostonng errors of GPS recever are less than 8 meters and n the range of normal ostonng recson. () Smlar to the statc exerment, the ostonng recson accords wth the redcton recson of the RCB seres. The lower redcted errors of the RCB seres mean that there are better ostonng results of GPS recever. Otherwse, the ostonng recson becomes worse. The reason of ths henomenon s that the redcted error n the RCB seres s the domnant error whch wll decrease the ostonng recson after other errors have been roerly modeled and comensated. (3) Comared wth the statc status, the redcton recson of the RCB seres and the ostonng recson of GPS recever under the condtons of the dynamc status are lower than the corresondng ones. Predcted Errors(m) - 9 Postonng Errors(m) Fg.6 Forecastng errors of recever cloc bas. From the exerments n ths secton, we can come to a concluson that the combned redcton model n the aer s sutable for redctng the RCB seres, and the cloc-aded ostonng method can rovde unnterruted and relable ostonng results whch satsfy the recson requrement of GPS ostonng n an urban envronment. Addtonally, the resented methodology does not need extra equments, and t s much more feasble and effectve. Equed wth the cloc-aded ostonng method n the aer, the ostonng contnuty and relablty of GPS recever can be ensured for a few seconds even f there are only three satelltes. The research wor n the aer aves the way for ostonng calculaton, and should be valuable n the roject alcaton for mrovng the ostonng contnuty and relablty of GPS recever n an urban envronment. V. CONCLUSION In the aer, the three-dmensonal ostonng roblem of GPS recever n an urban envronment has been dscussed. Based on the characters of the RCB seres, the combned redcton model on the bass of the theory of tme seres analyss s establshed and the cloc-aded ostonng method s resented to mlement the functon of ostonng calculaton. Exermental results demonstrate the method resented n the aer can rovde a relable and contnuous ostonng result wthout extra equments under the condtons of only three vsble satelltes. Furthermore, the redcton recson of the RCB seres has mortant effect on the ostonng erformance of GPS recever. How to mrove the redcton model and how to evaluate the relatons between the ostonng recson and redcton recson quanttatvely are two mortant ssues worthy of further research. 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Coggns, Multresoluton forecastng for future tradng usng wavelet decomoston. Neural Networs, IEEE Transactons on Neural Networs, vol., no.4, July, Yunlong Teng receved the M.S. degree n Oeratonal research and cybernetcs from Unversty of Electronc Scence and Technology of Chna n 8. He s currently worng towards the Ph.D. degree n Measurng and Testng Technology & Instruments at the same unversty. Hs research nterests are focused on test theory and technology, GPS navgaton and ostonng. Ybng Sh receved the B.S. degree n Rado Engneerng, M.S. degree n Crcut and System, and Ph.D. degree n Measurng and Testng Technology & Instruments, all from the Unversty of Electronc Scence and Technology of Chna, n 98, 988, and, resectvely. Now he s worng as a rofessor and Ph.D. suervsor at the same unversty. Hs research nterests nclude comuter-aded test of VLSI and electronc systems, the theory and desgn of electronc measurement nstruments, the research on well loggng and the nternet-based test. Zh Zheng receved the M.S. degree n Crcut and Systems from Unversty of Electronc Scence and Technology of Chna n 7. Currently, he s ursung the Ph.D. degree n Communcaton and Informaton systems at the same unversty. Hs research nterests are focused on array sgnal rocessng, smart antenna and wreless communcaton. 393