Dual-space Ambiguity Resolution Approach Theory and Application

Transcription

1 Journl of Globl Postonng Systems (00) Vol., No. : 8-6 Dul-spce Ambguty Resoluton Approch heory nd Applcton Guorong Yu nd Jngnn u GPS Engneerng Reserch Center, Wuhn Unversty Receved: /0/00 / Accepted: //00 Abstrct. In rel tme nemtc (RK) GPS postonng the reference stton(s) s (re) sttc, nd the movng recevers must not be fr from the reference stton(s). But n some cses, such s formton flyng, stellte-tostellte orbt determnton, etc, t s dffcult to estblsh sttc reference stton. GPS nemtc-to-nemtc postonng (KINRK) wll meet such requrements. he ey wor of mbguty resoluton for KINRK s to obtn n mbguty flot soluton rpdly. he flot soluton cn be estmted usng ether the Geometrybsed (GB) or Geometry-free (GF) pproch, requrng the constructon of GB or GF mbguty serch spce. hese two spces re dfferent but hve the sme true nteger mbguty result. Serchng n two spces t the sme tme, referred to here s Dul-spce Ambguty Resoluton Approch (DARA), wll be fster thn n the ndvdul spces becuse only few mbguty cnddtes meet the condtons of both spces smultneously. It cn be shown tht DARA cn drmtclly reduce mbguty cnddtes even f the C/A-code pseudo-rnge observbles re used. he results of vehcle test confrm tht our pproch s promsng, resultng n mllmeter-level msclosure of the KINRK run. Key words: GPS, Ambguty Resoluton, RK Introducton On-the-fly GPS postonng reltve to movng reference, referred to s KINRK n ths pper, hs mny pplctons, such s formton flyng, stellte-tostellte orbt determnton, nd others, where sttc reference s dffcult to estblsh (e.g., Hermnn et l., 995, Kwno et l., 00]. Quc nteger crrer phse mbguty estmton plys n mportnt role n KINRK. Once the mbgutes re resolved, centmeter-level ccurcy of KINRK cn be cheved (e.g., Hermnn et l., 995). In the GPS lterture there re two of the mny dfferent pproches proposed for nteger mbguty estmton, whch hve drwn much nterest. he two pproches dffer n the model used for nteger mbguty estmton. In the frst pproch, whch s the common mode of operton for most surveyng pplctons, n explct use s mde of the vlble reltve recever-stellte geometry (e.g., Chen & chpelle, 995, Fre & Butler, 990, eunssen, 995) nmed the GB (GB) model by eunssen (e.g., eunssen, 997). Integer mbguty estmton s lso possble however, when one opts for dspensng wth the reltve recever-stellte geometry (e.g., Euler et l., 99, Htch, 98, Horemuž, et l., 00, Sjöberg, 998), nd s nmed the GF (GF) model. In fct from the conceptul pont of vew, ths s the smplest pproch to nteger mbguty estmton. he pseudo-rnge dt re drectly used to determne the unnown nteger mbgutes of the observed phse dt (e.g., eunssen, 997). hese two models hve ther dvntges nd dsdvntges. Combnng ther dvntges ths pper proposes n mbguty resoluton lgorthm bsed on the ntegrton of both mbguty resoluton pproches. In our dscusson below we ssume tht dul-frequency pseudo-rnge nd crrer phse observbles re vlble. In ths pper the unt mtrx of order p s denoted s E p nd the p-vector hvng ll ones s entres s denoted s e p. Furthermore the cnoncl unt vector hvng the one s ts th entry s denoted s c. vec s the opertor tht trnsforms mtrx nto vector by stcng the columns of the mtrx one underneth the other. he symbol denotes the Kronecer product (eunssen, 997):

2 Guorong Yu nd Jngnn u: Dul-spce Ambguty Resoluton Approch 9 B A B = M n B nb M B nn () only slowly wth tme. he mtrx B s therefore only wely dependent on tme. In our further nlyss t wll therefore be ssumed tht B s tme-nvrnt mtrx, B = B. he epochs lner system of DD observton equtons re: Observton Model. Stellte Geometry-bsed Model et us ssume tht double-dfferenced phse nd pseudornge observbles re vlble. For ech observed stellte (except for the one selected s the reference stellte), four observton equtons cn be wrtten: R = ρ + I + δr R = ρ + ri + δr λ φ = ρ I + λ + δφ λ φ = ρ ri + λ + δφ where R, R, φ, ndφ re phse nd pseudornge observbles of the sgnl wth wvelength λ nd frequency f, nd of the second sgnl wth wvelength λ nd frequency f respectvely; ρ s the geometrcl dstnce between recever nd stellte; nd re nteger mbgutes; I s the onospherc errors of frequency; r = f / f ; nd δ re rndom observton errors. In the short bselne cse, the onospherc bs cn be gnored (e.g., Kleusberg, 986). he lnerzed observton equtons t the th epoch of m stelltes re: [ Em A] δ () l = Bb + + () ρ where B = A = c E b l [ ] = R R λ φ λ φ Rm Rm λ φm λ φm (4) [ λ λ λ ] = λ m m b denotes the bselne vector of epoch. he geometry of the double-dfferenced (DD) reltve recever-stellte confgurton s contned wthn the (m-)x mtrx B. It s well nown tht due to the hgh lttude orbts of the GPS stelltes, the recever-stellte geometry chnges (5) [ e ( E A ] δ vec = E B) vecb + ) + where l ( m = [ l ] b = [ b b ] l. Stellte Geometry-free Model In ths model the observton equtons re not prmetrzed n terms of the bselne components. Insted, they remn prmetrzed n terms of the unnown DD recever-stellte rnges. hs mples tht the observton equtons remn lner nd tht the recever-stellte geometry s not explctly present n these equtons. Hence the model permts both recevers to be ether sttonry or movng. [ Em e4 ] ρ + [ Em A] δ b (6) l = + (7) where [ ] ρ = ρ ρ ρm DD observton equtons re: vec = [ E + ( E m e )] vecρ [ e ( E A) ] + δ m where ρ [ ρ ρ ρ ] 4. he epochs system of =. For further detls we refer reders to eunssen (e.g., eunssen, 997). Shpe nd Orentton of Ambguty Serch Spce In our nlyss we ssume tht the vrnce mtrx of the observbles s gven by: Σ = E P. Solved by the lest squres djustment method, the norml equtons re: (8) NX= U (9) E B PB N = e A PB e B PA N = A PA N N N (0)

3 0 Journl of Globl Postonng Systems E ( B P) U = vec e ( A P) nd the vrnce of the mbgutes s: ( ΣQ ) Q, Q U = ( N N N N) = U ( Σ ) U () = () = A P A A P B ( B P B ) = A P l A P B ( B P B ) B P A () B Pl (4) where the subscrpt ndex mens the epoch. he mtrx B of the GB model s the lnerzed mtrx of equton (), whle B of the GF model s B = E m e 4. Assume σ =0.009m, σ =0.004m, σ CA =.0m, σ P =0.m, nd selectng 5 stelltes nd 50 epochs of nemtc observbles, we get the vrncecovrnces of the mbgutes of the GB model: Q 0.5 = SYS (5) he vrnce-covrnces of the mbgutes for the GF model re: Q 0.0 = SYS (6) Of prtculr nterest here s the shpe nd orentton of the ellpsod of stndrd devton for the mbgutes (e.g., ec, 995). he sem-mnor nd sem-mjor xes nd the orentton re gven n b. to 8. b Sem-mjor xes for the GB model (m) b Sem-mnor xes for the GB model (m) b Sem-mjor xes for the GF model (m) b 4 Sem-mnor xes for the GF model (m) b 5 he orentton for the GB model (Degree) b 6 he orentton for the GF model (Degree)

4 Guorong Yu nd Jngnn u: Dul-spce Ambguty Resoluton Approch b 7 he orentton for the GB model wth 00 epochs (Degree) b 8 he orentton for the GB model wth 50 epochs (Degree) he orentton for the GF model does not chnge s the number of epochs ncreses. he orentton of nd for the sme stellte s exctly 7.9,.e., the slope s = λ / λ, nd the sem-mnor xes for the GB re equl to those of the GF model, whle the sem-mjor xes for the GF model re longer thn those of the GB model, whch mens tht the ellpse for the GF model lwys contns the ellpse for the GB model (see Fg.). he geometry s the sme for every epoch soluton, s long s the stochstc model remns unchnged. of - s lmost strght lne wth slope of = λ / λ. If (, ) s the nerest pont to the lne, the next nerest ponts re ( ± 9, ± 7), whch mens f (, ) s n mbguty cnddte, the next cnddtes re exctly ( ± 9, ± 7). Becuse the sem-mnor xes re very short, the ellpse nerly only contns ponts such s ( ± *9, ± *7), =0,, herefore, the ey ts for mbguty resoluton s to obtn the flot mbguty soluton. 4 ner Combnton of wo Frequency Observtons et s consder lner trnsformton (e.g., Hn & Rzos, 996, eunssen, 995): φ mφ + nφ m, n = (7) If we set ( m, n ) = (-7,9), ll ponts le ( ± *9, ± *7), =0,, wll be mpped to one pont (, ). In other words, the sem-mjor xes (or the ellpse becuse of very short sem-mnor xs) s mpped to pont (see Fg.). But f the nteger soluton ( m, n hs been obtned, the soluton ( (, ( ) cn not be determned by n nverted trnsform. So let s consder nother lner trnsform: φ +, j = φ jφ (8) As ( ± 5, ± 4) re the next ponts ner to the sem-mjor xes, we choose (, j ) = (+4,-5). Usng these two trnsforms t the sme tme: Fg. he ellpse of - he orentton for the GF model does not chnge for every epoch, whle the orentton of two dfferent stelltes for the GB model s closer to the xes s the number of epochs ncreses, whch mens tht these two ellpses overlp ech other prtly (see Fg.). φ, j φ m, n = m the mbgutes re:, j m, n j φ n φ φ = H (9) φ = H (0) Fg. he ellpse of SV-SV As cn been seen from b. to 8 nd Fg., the ellpse Fg. shows the sngle epoch DD mbguty flot soluton for stelltes to of rover versus rover (refer to Fg.8) processed usng the GB model. he horzontl xs s GPS Seconds of Wee (GSW), the vertcl xs s the flot soluton, nd 0 s the nteger mbguty soluton. Fg.4 shows the sngle epoch mbguty flot soluton processed usng the GF model. From Fg. nd Fg.4, we cn see tht the flot soluton undultes round the fxed

5 Journl of Globl Postonng Systems soluton. 5 Ambguty Serch Strtegy et us ssume tht the observbles φ, φ, R CA, R P re not correlted, nd tht the stndrd errors of the observbles re σ =0.009m, σ =0.004m, σ CA =.0m, σ P =0.m, then the stndrd errors of the lner combnton mbgutes re σ =0.5, σ, j m, n =0. wth (, j ) = (+4,-5) nd ( m, n ) = (-7,9), nd the rto of sem-mjor xs to semmnor xs s 5:, whch mens tht the re of the serch ellpse s lmost equl to tht of the confdence ntervl determned by ( â ± σ ), see Fg.5. â Fg. Sngle epoch mbguty flot soluton processed by the GB model Fg. 5 Dfference between the confdence ellpse nd the confdence ntervl of the mbguty prmeters We obtn the mbguty serch spce of the GF model (GF) by estmtng, j nd m, n. On the other hnd, we cn construct the mbguty serch spce of the GB model (GB) by processng the orgnl observbles φ, φ. When the, j nd m, n re fxed, nd cn be determned from equton (0), nd vce vers. But other cnddtes my not eep the mp relton,.e. f we mp cnddte of GB usng equton (0), t would not belong to GF, nd vce vers, due to these two spces beng constructed usng two dfferent models. Assume the GF s (, j, m, n ) F, F= ( â ± σ â ), =4; nd GB s (, ) B, B=[- 0,0] [-0,0],.e. 44 cnddtes. hen we get b. 9. b 9 he reltonshp between GF nd GB, j m, n Fg.4 Sngle epoch mbguty flot soluton processed by the GF model In b.9, s the dstnce of n mbguty cnddte (, ) to the sem-mjor xs. For exmple: Assume the nteger mbguty soluton s:

6 Guorong Yu nd Jngnn u: Dul-spce Ambguty Resoluton Approch ( (, ( )=(-0548, ) nd the flot soluton processed by the GF model s: ˆ, ˆ m, n )=( , 0.609) (, j whose nerest ntegers re: (-697, 0) the cnddtes wthn the confdence ntervl F= ( â ± σ â ), =4 become: stelltes, eght double-dfferenced observbles re formed, nd the number of mbguty serch loops s four, not eght. he next step s to consder the serch ellpses of dfferent stelltes. he selected cnddte whch s processed by the GF model should be locted n the serch ellpse of the GB model. In other words, the dstnce of the cnddte to the sem-mjor xs of the serch ellpse of the GB should be not longer thn the length of the sem-mnor xs of the serch ellpse, see Fg.7. (-697 ±, 0), (-697, 0), (-697 ±, 0), (-697, 0) Whle the flot soluton processed by the GB model s: ( ˆ, ˆ )=( , ) whose nerest ntegers re: (-064, ) nd ssumng the confdence ntervls re: (-064 ±, ± ), =-0,-9 -,0,+,,+9,+0 then the cnddtes selected by b. 9 become: ( , ), ( , ), ( , ), (-0648, ) Only four cnddtes need to be consdered further, see Fg. 6. Fg. 6 Ambguty cnddtes b. 9 lso mens tht mbgutes nd or, j nd m, n should be serched n prs n one loop, not two ndvdul loops! In other words, f there re fve Fg. 7 he dstnce of the cnddte to the sem-mjor xs. 6 est nd Results 6. Vehcle est A vehcle nemtc test ws performed (see Fg.8). he followng re detls of the experment: Ste: MngZu Rod, Wuhn Cty, Chn me: 9--00, 5:00-6:0 (ocl tme); Recevers: hree rmble 4700; Smple: second; nd Elevton: 5 degree. One recever ws set s sttc reference (nmed bse stton ), the other two were mounted on two vehcles, (nmed rover nd rover ). Frst, 0 mnutes of sttc observtons were collected by the three recevers. hen rover moved fst nd rover followed. 0 mnutes lter these two recevers stopped movng nd fve-mnute perod of sttc observtons were collected. hen bc to the strtng plce, nd nother fve mnutes of sttc observtons were gn collected.

7 4 Journl of Globl Postonng Systems Rover Rover 6.5Km Mngzu Rod Rover Rover Rover Rover Bse Stton () Fg.8 Knemtc test scenro 6. Ambguty Resoluton he nemtc test dt were processed by the proposed DARA (Dul-spce Ambguty Resoluton Approch) method,.e., serchng mbgutes n two spces t the sme tme, lso usng lest squres (S) mbguty serch pproch,.e., serchng mbgutes only n the GB mbguty spce (nd vldtng by the Rto test). b. 0 s summry of the mbguty resoluton results. Bselne b 0 Summry of mbguty resoluton results Epochs rue S Wrong Success Rte rue DARA Wrong Success Rte Rover Rover Bse Rover (b) Fg.9 Dstnces between recevers Fg.0 () shows the comprson of sttc-nemtc to nemtc-nemtc. he sold lne s the dstnce between rover nd rover, whle the dotted lne s clculted from the two sttc-nemtc bselnes,.e. bse stton-rover nd bse stton-rover. Fg.0 (b) shows the dfference between the two lnes of Fg.6 (). Bse -Rover Postonng Fg.9 () shows the sttc-nemtc results, the sold lne s rover to bse stton nd the dotted lne s rover. Fg.9 (b) shows the nemtc-nemtc results,.e. rover s movng reference. As we cn see, the three recevers strted observng t 8450 GSW, rover nd rover begn movng t 8690 GSW, reltve sttc (not sttc) t GSW, then bc t 880, then reltve sttc t 8960, nd the test ws stopped t 8968 GSW. () (b) Fg.0 A Comprson of sttc nd nemtc results

8 Guorong Yu nd Jngnn u: Dul-spce Ambguty Resoluton Approch 5 Fg. shows the msclosure of the three bselnes. he results show tht mllmeter-level msclosure ws cheved. () 7 Concluson Crrer phse mbguty resoluton s the ey to fst nd hgh precson GPS prmeter estmton. he flot soluton cn be estmted by two nds of models whch my hve the reltve recever-stellte geometry ncluded (GB) or excluded (GF), nd used to construct GB or GF mbguty serch spces. hese two models hve ther dvntges nd dsdvntges. Further nlyss shows these two spces re dfferent but hve the sme true mbguty resoluton result. A new concept for mbguty resoluton, clled Dul-spce Ambguty Resoluton Approch (DARA), nvolvng serch n both spces t the sme tme, s proposed. In ths pper the vehcle test shows tht the DARA performed well, resultng n mllmeter-level msclosure. Acnowledgments We re grteful to Prof. Chrs Rzos for useful comments nd for hs help n the Englsh expresson. We lso le to thn Dr. u Xngln for hs revews of ths pper. (b) References Hermnn, B.R., Evns. A.G., w. C.S., et l. (995). Knemtc on the fly GPS Postonng Reltve to Movng reference. NAVIGAION: Journl of the Insttute of Nvgton, 4(): Kwno, I., Mouno, M., Ks,., et l. (00) Frst Autonomous Rendezvous Usng Reltve GPS Nvgton By ES_VII. NAVIGAION: Journl of the Insttute of Nvgton, 48(): (c) Chen, D., chepelle, G., (995) A Comprson of he FASF nd est Squres Serch Algorthms For On-he-Fly Ambguty Resoluton. NAVIGAION: Journl of the Insttute of Nvgton, 4(): (d) Fg. Msclosure of the three bselnes Fre, E., Beutler, G., (990) Rpd Sttc Postonng Bsed On he Fst Ambguty Resoluton Approch FARA : heory nd Frst Results. Mnuscrpt Geodetc, 5(6): eunssen, P.J.G., (995) he est-squres Ambguty Decorrelton Adjustment: A Method for GPS Integer Ambguty Estmton. Journl of Geodesy. 70(-): eunssen, P.J.G., (997) GPS Double Dfference Sttstcs: Wth nd Wthout Usng Stellte Geometry. Journl of Geodesy, 7(): 7-48.

9 6 Journl of Globl Postonng Systems Euler, H. J., nd God, C.C., (99) On Optml Flterng of GPS Dul Frequency Observtons Wthout Usng Orbt Informton. Bull.Geod, 6(): 0-4. Htch, R., (98) he Synergsm of GPS Code nd Crrer Mesurements. In Proceedngs of the hrd Interntonl Geodetc Symposum on Stellte Doppler Postonng,MA, s Cruces, New Mexco,. Horemuž, M., nd Sjöberg,.E., (00) Rpd GPS Ambguty Resoluton For Short And ong Bselnes[J]. Journl of Geodesy, 76(6-7), 8-9. Sjöberg,.E.,(998) A new method for GPS phse bse mbguty resoluton by combned phse nd code observbles. Surv Revew, 4(68): 6-7. Kleusberg, A., (986). Ionospherc Propgton Effects In Geodetc Reltve GPS Postonng. Mnuscrpt Geodetc, Vol., No.4, ec, A., (995) GPS Stellte Surveyng, second edton John wley & sons.nc. New Yor, 560pges. Hn, S., nd Rzos, C., (996) Improvng the computtonl effcency of the mbguty functon lgorthm. Journl of Geodesy, 70(6): 0-4. eunssen, P.J.G., (995) he nvertble GPS mbguty trnsformtons. Mnuscrpt geodetc, 0(6):