Instant Active Positioning with One LEO Satellite

Transcription

1 Instant Active Positioning with One LEO Satellite NADAV LEVANON el Aviv Univesity, el Aviv, Isael Received Decembe 997; Revised June 999 ABSRAC: Autonomous position detemination in the Globalsta satellite communication system is discussed. he two-way communication between a use teminal on the eath s suface a single low eath obit ( LEO) satellite makes it possible to deive ange ange-ate detemine the use teminal position instantly. Expected accuacy is pesented, a simple diect solution is given. INRODUCION Globalsta is a satellite communication system designed to povide voice low-ate data communication to use teminals Ž Us. on the eath. he satellite constellation includes 48 low eath obit Ž LEO. satellites at a height of 400 km, aanged in eight obits with an inclination of 5 deg. he Globalsta satellite acts as a bent-pipe elay between a egional gateway Ž GW. the U. he obit of the satellite is known accuately. Pimaily fo opeational easons, befoe a U phone call is connected, the system needs to know the U position with a hoizontal accuacy of bette than 0 km. Connection of the call cannot be delayed fo moe than about 3 s, a call should go though even if the U sees only one Globalsta satellite. Hence thee is a need fo instant coase positioning using the system s own satellite signals. While two-satellite, two-dimensional Ž -D., active positioning is found in othe opeational systems 3, 4, single-satellite instant positioning is unique. his pape theefoe concentates on this aspect of Globalsta positioning. Single-satellite positioning is based on measuements of ange ange ate. he ange between the satellite the U is deived fom a ound-tip delay measuement, fom which the known GW-tosatellite leg is emoved. Globalsta uses a code division multiple access Ž CDMA. concept, which utilizes a wideb spead-spectum signal, as in GPS. Hence the delay measuement esolution is elatively high. Range ate is deived fom Dopple measuements. In a passive satellite Dopple navi- NAVIGAION: Jounal of he Institute of Navigation Vol. 46, No., Summe 999 PintedintheU.S.A. gation system, such as RANSI 5, the tue Dopple cannot be sepaated fom the U oscillato fequency offset; hence only Dopple diffeences Ž athe than absolute Dopple. can be measued. When the U is active Ž eceives tansmits. uses the same maste oscillato in its eceive tansmitte, two Dopple measuements one at the satellite Ž in pactice at the GW. one at the U Ž communicated to the GW. povide enough infomation to sepaate the tue Dopple fom the U fequency offset Ž see Appendix A.. his pape deives the expected accuacy of -D positioning based on one ange one ange-ate measuement to a single satellite. In the thid dimension, the U is assumed to be on the eath s suface. he hoizontal eo esulting fom an elevation eo is also deived. Finally, a simple diect solution is pesented that can be used by itself o to povide a fist estimate fo an iteative positioning algoithm. POSIIONING BASED ON RANGE AND RANGE RAE he positioned U is located at one of the two intesections between thee sufaces Ž see Figue.: the ange sphee, the ange-ate Ž Dopple. cone, the eath s suface. he ange sphee is centeed at the satellite antenna, which is also whee the apex of the cone is located. he cone axis of symmety is the velocity vecto. Delay Ž ange. Dopple Ž ange-ate. eos cause an eo in the detemined U position. he coss-tack positioning eo is magnified damatically when the intesection is nea the satellite subtack on the eath s suface. An eo in the U assumed height above 87

2 the ange ate by vž y vt. Rt Ž. ' x Ž y vt. Ž h H. Fig. hee Sufaces Defining the U Position the efeence eath suface causes futhe cosstack positioning eo, which also inceases as the U gets close to the subtack. he pefomances of such a positioning system can be analytically deived fo a simplified model that assumes a flat eath a staight-line obit. he validity of this assumption is demonstated by compaing analytic esults using the flat eath model with numeical esults using a spheical eath model. he satellite-u geomety in a flat eath model is depicted in Figue. At t 0, the satellite is at the Ž 0, 0, H. coodinate moving in the y diection with a velocity v. Both H v ae known accuately. he stationay U is at Ž x, y, h.. A flat efeence eath suface is assumed, epesented by the z 0 plane. he ange to the U is given by ' Ž. Rt x y vt h H Ou simple analysis assumes that the ange ange ate, deived fom the delay Dopple measuements, ae fee fom bias eos, but suffe fom elatively small, independent, om eos with zeo mean stad deviations of R Ṙ, espectively. In eality, bias eos do exist. In Globalsta s single-satellite positioning, om measuement eos dominate the position eo. he bias eos become a dominant facto in twosatellite positioning. Yet it should be noted that even though the eo contou maps in this pape wee deived assuming om, zeo-mean measuement eos, aeas with lage sensitivity to om measuement eos ae also sensitive to bias measuement eos. he U location Ž x, y. is detemined on the efeence suface Ž z 0 plane., assuming eoneously that h 0. Late it is shown that a deviation of the tue elevation fom the assumed elevation Ž zeo. esults in a bias eo in the estimated coss-tack Ž x. coodinate of the U. LOCAION ERROR AS FUNCION OF MEASUREMEN ERROR he om eos in ange ange ate cause a U position eo. he position eo is sepaated into coss-tack along-tack components, with stad deviations of x y, espectively. he tansfomation of eos fom the measuement domain to the geometical domain is done fo t 0, equies the patial deivatives of the measuements with espect to the location paametes. At Fig. Satellite-U Geomety 88 Navigation Summe 999

3 t 0, the patial deivatives ae given by R x x R Ž 3. R y y R Ž 4. R xyv Ž 5. x R 3 ž / R v y Ž 6. y R R R R x y H Ž 7. R R x y hey ae aanged as a matix of patial deivatives: he eos in ange ange ate ae assumed to be independent. his assumption is well justified in Globalsta, whee the two measuements delay Dopple ae nealy uncoelated. he eo covaiance matix is theefoe diagonal. ŽCoela- tion would esult in nonzeo off-diagonal elements.. Its invese, also a diagonal matix, is temed the weight matix: 0 R W Ž 8. 0 Ṙ he eo vaiance of the estimated U coodinates, x y, can be obtained fom H W 6: x H WH, 9 y H WH, 0 Symbolic solutions of equations 9 0 yield ž / R y y x R R x v R R y y R R v R Figues 3 4 ae contou maps of the cosstack eo, x, the along-tack eo, y,e- ( ) Fig. 3 Coss-ack Location Eo in kilometes Vol. 46, No. Levanon: Instant Active Positioning with One LEO Satellite 89

4 ( ) Fig. 4 Along-ack Location Eo in kilometes spectively, in kilometes, fo the following typical scenaio: H 400 km, v 7 km s, R 30 m, Ṙ 3m s. he most pominent featue is the lage coss-tack eo in positioning Us located nea the satellite subtack Ž coss-tack 0.. his featue implies lage geometic dilution of pecision Ž GDOP. nea the subtack. he poo GDOP stems fom the fact that on the subtack, the cicle dawn on the eath s suface by the intesection with the ange sphee Ž Figue. is tangential to the hypebola dawn on the eath s suface by the intesection with the ange-ate cone. Any small eo in ange o ange ate will cause a lage coss-tack eo in the calculated location. As the U location gets fathe away fom the subtack, the GDOP singulaity is eplaced by anothe poblem ambiguity. Single-satellite positioning based on one pai of ange ange-ate measuements always yields two symmetical solutions on the two sides of the subtack Ž Figue.. When the tue U location is fa fom the subtack, the two solutions ae fa fom each othe. In that case, the tue solution can be identified using infomation fom seveal satellite antenna beams. When the tue U location is close to the subtack, the two solutions ae close to each othe may both fall within the illumination aea of the same antenna beam. In that case, the ambiguity cannot be esolved. Figues 3 4 ae plots of the analytic esults of equations Ž., espectively, which wee deived using a flat eath model. o demonstate thei validity fo the eal eath, a numeical eo analysis was pefomed using a spheical eath model a cicula obit. he esult fo the cosstack eo is pesented in Figue 5. he excellent ageement between Figues 3 5 Ž simila ageement, not plotted, with espect to the alongtack eo. poves that the use of a flat eath model is justified, that equations apply to moe ealistic eath models. ELEVAION-INDUCED ERROR Equations indicate that in both the ange the ange-ate measuement, the U s elevation, h, coss-tack coodinate, x, ae cou- pled togethe in the expession x Ž h H.. Hence an eo, h, in the assumed elevation must ceate an eo, x, in the coss-tack coodinate. 90 Navigation Summe 999

5 ( ) Fig. 5 Coss-ack Location Eo in kilometes Deived Numeically Using Spheical Eath Model o keep the expession unchanged, the following elationship must hold H h H x h h Ž 3. x x Accoding to equation 3, when the coss-tack distance is appoximately equal to the satellite height, the elevation-induced coss-tack eo is equal to the elevation eo. Clealy the eo inceases as the U gets close to the subtack. In the actual Globalsta positioning algoithm, the hoizontal position is fist solved using sea-level elevation. hen the teain elevation fo the solved position is obtained fom a topogaphic map used instead of sea level, an impoved position is obtained. Unless the teain is vey ough, convegence is expected afte two o thee iteations. he elevation-induced hoizontal positioning eo can be incopoated in the eo contou maps by declaing the elevation obtained fom the topogaphic map to be a measuement with eo stad deviation E. he patial deivative weight matix ae inceased in size to 3 3: R R R x y h R R R H Ž 4. x y h h h h x y h R 0 0 W 0 0 Ž 5. Ṙ 0 0 E Vol. 46, No. Levanon: Instant Active Positioning with One LEO Satellite 9

6 whee the additional elements ae R h H Ž 6. h R R Ž h Hyv. h R 3 Ž 7. h h h 0, 0, x y h Ž 8. ž / R y y Ž h H. x v R x x R R E 9 R y Ž 0. y R R v R h E he coss-tack, along-tack, height eo vaiances ae deived fom the thee diagonal ele- Ž ments of the matix H WH., yielding Note that the along-tack eo expession Žequa- tion Ž 0.. is identical to the -D case Žequation Ž..; hence Figue 4 holds fo the 3-D case as well. he coss-tack eo Žequation Ž 9.. diffes fom equation in the additional tem pedicted by equation Ž 3.. he slightly degaded coss-tack eo contou map assuming an elevation measuement eo stad deviation of E 00 m is pesented in Figue 6. hee is no need to plot the stad deviation of the eo in the estimated height since it must eveywhee be equal to E,as indicated by equation Ž.. OHER ERRORS As the vaious eo contou maps indicate, with the expected om measuement eo, typical positioning eos ae between km Žex- cluding the aea nea the satellite subtack.. his pefomance level can be consideed coase positioning, but it meets the system s needs. Othe eo souces, such as ionospheic effects ephemeis eos, that ae consideed in moe accuate positioning systems can be neglected in ou eo analysis. he main emaining souce of eo is U velocity. U velocity geneates additional Dopple, which effectively inceases the ange-ate eo. Because we ae using a single Dopple measuement, the fact that it is a bias athe than a om eo makes no diffeence, the effect of ( ) Fig. 6 Coss-ack Location Eo with Contibution fom Elevation Eo in kilometes 9 Navigation Summe 999

7 U velocity can be incopoated into the eo analysis by inceasing the ange-ate om eo. POSIIONING ALGORIHM he opeational positioning algoithm applies to all possible scenaios, including the two- thee-satellite cases. he latte ae ovedetemined systems, which invites a least-squaes solution. he geneal algoithm thus uses an iteative weighted least-squaes solution. Such an algoithm equies an initial estimate of the U location. A simple appoach to obtaining a good fist estimate is to solve explicitly the minimal set of two equations Ž measuements.: the ange ange ate to a single satellite. One explicit solution is descibed in Appendix B. In a single-satellite situation, the explicit solution can eplace the iteative solution. CONCLUSIONS Autonomous positioning in Globalsta may necessitate detemining the U position when only one satellite is available. he om eo analysis pesented in this pape demonstates that with the expected measuement accuacy, coase position can be detemined instantly fo Us eveywhee except nea the satellite subtack. ACKNOWLEDGMEN he autho gatefully acknowledges the suppot of Qualcomm Inc., San Diego, Califonia. APPENDIX A SEPARAING DOPPLER FROM U OSCILLAOR OFFSE o detemine the satellite-u ange ate, it is necessay to sepaate the U oscillato offset fom the Dopple shift Ž in the satellite-u leg.. In an unpublished epot, Steven A. Kemm suggested a method that applies when the U uses the same local oscillato fo both tansmit eceive. he method is explained with the help of Figue A. the index below. It is assumed that the Dopple shifts in the GW-satellite leg ae pefectly known emoved. Ṙ ange ate of the satellite-to-u leg C popagation velocity Ž speed of light. f F fowad link nominal caie fequency Ž 500 MHz. f R evese link nominal caie fequency Ž 600 MHz. f off nomalized fequency offset of U s f 0 oscillato wo measued fequencies ae available at the GW: the epoted measuement fom the U / Ṙ f off f f Ž A-. meas, U F ž C f 0 Fig. A. Relationship Between Fequencies Used to Sepaate Dopple fom Fequency Offset Vol. 46, No. Levanon: Instant Active Positioning with One LEO Satellite 93

8 the measuement pefomed at the GW itself / Ṙ f off f f Ž A-. meas, GW Rž C f 0 Adding subtacting equations A- A- yields both the U offset the ange ate: ž / C fmeas, GW fmeas, U ž f f / R F foff fmeas, GW fmeas, U Ž A-3. f f f APPENDIX B DIREC SOLUION 0 R F R Ž A-4. he diect solution assumes a smooth ellipsoid eath model uses eath-centeed, eath-fixed Catesian coodinates. hee ae many efeences to diect solutions fo geolocation using time-diffeence o fequency-diffeence measuement 7. Howeve, the diect solution becomes vey simple when absolute measuements ae available. Definitions he U Catesian coodinates Žeath-centeed, eath-fixed. ae u xyz Ž B-. Initially, the U is assumed to be on a sphee of adius Ž c whose value is abitaily selected be- tween the equatoial eath adius E the pola eath adius. P. hat assumption yields the fist equation: u u x y z B- c Six satellite paametes at the measuement epoch ae known: the position coodinates the velocities he adius of the satellite obit is s s s s x y z B-3 x y z v vvv B-4 ss x y z B-5 s s s s he two available measuements ae ange ' s s s R s u Ž x x. Ž y y. Ž z z. ange ate B-6 Ž s u. v Ṙ Ž x s xv. x Ž ys yv. y Ž zs z. vz Ž B-7. Outline Define xs ys A Ž B-8. v v x y b R Ž B-9. s c c s v RR xv yv zv RR B-0 s x s y s z Obtain b z A s Ž B-. c vz he z coodinates of the U s tue mio solutions ae obtained fom 'Ž. Ž.Ž c. z Ž B-. ˆ, 94 Navigation Summe 999

9 the emaining coodinates of the two solutions ae given by ˆx z ˆ Ž B-3.,, ˆy z ˆ Ž B-4.,, Resolving which of the two is the tue solution equies additional infomation. hat infomation can be obtained, fo example, fom the satellite antenna beam though which the U signal was eceived. Explanation he ange ange-ate equations Ž B-6. Ž B-7. can be ewitten as s v u b c B-5 whee b c ae as defined in equations Ž B-9. Ž B-0.. Fom equation Ž B-5., x y can be expessed as functions of z: x z B-6 y z B-7 whee,,, ae as defined in equation Ž B-.. Inseting equations Ž B-6. Ž B-7. in equation Ž B-. yields a quadatic equation of z c Ž z. Ž z. z Ž B-8. whose two solutions ae given by equation Ž B-. above. Using the two solutions fo z back in equations Ž B-6. Ž B-7. yields the coesponding two solutions fo x y, as given in equations Ž B-3. Ž B-4. above. Ellipsoid Eath Model he fist impovement to the above simple solution is to assume that the U is on an oblate ellipsoid that has a pola adius P an equatoial adius E. Fom the Catesian coodinates deived assuming a spheical eath, we find sine cosine tems of the appoximate geocentic latitude : z x y sin ; cos Ž B-9. c Using the ellipsoid model the appoximate geocentic latitude yields a moe accuate distance to ' c the cente of the eath: P c p sin ž E / Ž B-0. cos his local eath adius Žfo the appoximate U location. is used in equation Ž B-9., the diect solution is epeated to yield moe accuate Catesian coodinates of the U location. When the diect solution is used to obtain a fist estimate fo the iteative solution, the accuacy achieved thus fa is sufficient. If the diect solution is used by itself, an elevation coection is equied. o obtain the elevation, the Catesian coodinates must be conveted into longitude geodetic latitude 8, the suface elevation fo that location is obtained fom a digital teain map, the distance to the eath cente is adjusted accodingly, the diect solution is epeated. REFERENCES. Hishfield, E., he Globalsta System, Applied Micowave & Wieless, Summe 995, pp Levanon, N., Quick Position Detemination Using o LEO Satellites, IEEE ansactions on Aeospace Electonic Systems, Vol. AES-34, No. 3, July 998, pp Ames, W. G., A Desciption of Qualcomm Automatic Satellite Position Repoting Ž QASPR. fo Mobile Communications, Poceedings of the nd Intenational Mobile Satellite Confeence, Ottawa, Ontaio, Canada, June 990, pp Colcy, J. N., G. Hall, R. Steinhause, Euteltacs: he Euopean Mobile Satellite Sevice. Electonics & Communication Engineeing Jounal, Apil 995, pp Pakinson, B. W.,. Stansell, R. Bead, K. Gomov, A Histoy of Satellite Navigation, NAVI- GAION, Jounal of he Institute of Navigation, Vol. 4, No., 995, pp Soenson, H. W., Paamete Estimation, Pinciples Poblems, New Yok: Macel Dekke Inc., 980, Ch.. 7. Ho, K. C. Y. Y. Chan, Geolocation of a Known Altitude Object fom DOA FDOA Measuements, IEEE ansactions on Aeospace Electonic Systems, Vol. AES-33, No. 3, July 997, pp Olson, D. K., Conveting Eath-Centeed, Eath-Fixed Coodinates to Geodetic Coodinates, IEEE ansactions on Aeospace Electonic Systems, Vol. 3, No., Jan. 996, pp Vol. 46, No. Levanon: Instant Active Positioning with One LEO Satellite 95