SSRG International Journal of Electronics and Communication Engineering (SSRG-IJECE) volume 2 Issue 4 April 2015

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1 SSRG International Journal of Electronics an Communication Engineering (SSRG-IJECE) volume Issue April 0 Estimation of GPS User Position ug Bancroft Algoritm Prof B.ari kumar, S.Venkatesara Rey, A.Leelaar, Y.Dines Kumar Rey Prof & Hea, Senior U.G Stuent, ECE Department, Brinavan Institute of ecnology & Science, Kurnool, Anra Praes, Inia. ABSRAC: e paper escribes etermination of GPS receiver position ug Bancroft algoritm. e position is etermine by obtaining pseuo ranges from seven ifferent GPS satellites. e measure ranges o not represent te true ranges as te signal coming from a GPS satellite ill be affecte by various errors like Epemeris error; Propagation error in te form of Ionosperic an osperic elays; Satellite an Receiver clock biases it respect to GPS ime (GPS); Multipat error etc. Most of tese errors can be estimate accurately an can be accounte for. In tis paper some stanar mitigation tecniques ave been applie for te estimation of GPS errors an user position is ten etermine ug Bancroft algoritm tecnique after making appropriate corrections. GPS ata of Citrakut station available in RIEX (Receiver Inepenent Excange) format ave been use for tis purpose Keyors: Pseuo range, GPS, GPS, RIEX, SV I. IRODUCIO A GPS user estimates an apparent or pseuo range to eac SV (Satellite Veicle) by measuring te transit time of te signal. Ug te pseuo ranges, user position in -D (latitue, longitue an eigt) an te time offset beteen te transmitter an receiver clock can be estimate after making appropriate corrections for te observe pseuoranges. If te unknon coorinates of te user position are represente by x u, y u an z u an te knon positions of Satellite Veicles are it x, y, z, (ere =,,,) in ECEF coorinate system, te user position (in -D) an time offset t u are obtaine by simultaneously solving te nonlinear equations given belo. x xu y yu z zu ct u ; =,,, () Were c is te free space velocity of electromagnetic signals in m/s. e ranges measure from satellites are calle pseuo ranges ce biases in te receiver clock prevent te precise measurement of actual ranges (true ranges). o etermine te receiver position accurately, all te errors ave to be estimate an compensate for. In tis paper, te ionosperic elay is estimate ug Klobucar moel []. Hopfiel moel as been use for te estimation of troposperic elay. Satellite clock bias an te Relativistic effects also ave been estimate an correcte. II. a) Epemeris errors: GPS ERRORS o compute te pseuo range of te user it respect to a satellite, te user soul kno teposition of te satellite. is information transmitte by te satellite as part of te navigation message. e position of te satellite is ynamic one an is influence by gravitational fiel, solar pressures. e estimation of te position by a groun master control station is subect to errors ue to clock rifts an procesg elays in te monitoring stations. e estimate positions of te satellites are uploae perioically to te respective ones. e errors in te estimation of te satellites position results into te pseuo range errors an are to be correcte at `te user level. e rms ranging error attributable to epemeries error is about.m b) Multipat effects: Multipat error results from te combination of signals from more tan one propagation pat, istorting te signal caracteristics. Sometimes te reflecte signal is as strong as te irect signal. e magnitue of multipat error epens on te strengt of te reflecte signal an te ability of user s antenna to resist te reflecte signal. e strengt of te reflecte signal is epenent on a reflection coefficient of te reflecting surface tat varies not only it a ifference in te groun coarseness, but also it a ifference in groun plant quills. A reuction of multipat effect may also be acieve by igital filtering, ieban antennas, an raio frequency absorbent antenna groun planes. Multipat error ranges from.m to.7m. ISS: 8 89.internationalournalssrg.org Page

2 SSRG International Journal of Electronics an Communication Engineering (SSRG-IJECE) volume Issue April 0 c) Satellite Clock Error: Altoug all te satellites carry atomic clocks tat are very accurate, tese can eviate from GPS system time (GPS), ic results in pseuo range error. GPS ime (GPS) is a composite time scale erive from te times kept by clocks at GPS monitor stations an aboar te satellites an is use as te reference time. Ranging errors inuce by clock errors are of te orer of.0m. e Master Control Station etermines te clock error of eac satellite an transmits clock correction parameters to te satellites for rebroacast of tese in te navigation message. e receiver ug tese parameters implements correction. e SV clock correction for te C/A coe pseuo range observation is given by te expression: t s =af 0 + af * (t - t oc ) + af * (t - t oc ) + t r () af 0 = clock bias (sec) af = clock rift (sec/sec) af = frequency rift (sec/sec ) t oc = clock ata reference time (sec) t = current time epoc (sec) t r = correction ue to relativistic effects (sec) ) Relativistic Effects: Bot general an special teories of relativity affect te satellite clock. ee for general relativistic corrections arise en te signal source (satellite) an signal receiver (GPS receiver) are locate at ifferent gravitational potentials. Special relativity relativistic corrections arise any time te signal source or te signal receiver is moving it respect to te cosen isotropic frame, ic in te GPS system is te ECI frame. In orer to compensate for bot of tese effects, te satellite clock frequency is auste to MHz prior to launc. e frequency observe by te user at sea level oul be 0. MHz, ence te user oes not ave to correct for tis effect. But te user as to make a correction for anoter relativistic effect tat arises because of te sligt eccentricity of te satellite orbit. Wen te satellite is at perigee, te satellite velocity is iger an te gravitational potential is loer because of ic te satellite clocks run sloer. Wen te satellite is at apogee, te satellite velocity is loer an te gravitational potential is iger an te satellite clocks run faster. is effect can be compensate as follos t r = Fe a SinE k () F= x 0 0 sec/m e = satellite orbital eccentricity a = semi maor axis of te satellite orbit E k = eccentric anomaly of te satellite orbit Due to rotation of te Eart uring te time of signal transmission, a relativistic error is introuce ic is calle te sagnac effect. During te propagation time of te SV signal transmission, a clock on te surface of te eart ill experience a finite rotation it respect to te resting reference frame at te geocenter. If te user experiences a net rotation aay from SV, te propagation time ill increase an vice versa. e) Atmosperic Errors e satellite signals propagate troug atmosperic layers as tey travel from te satellite to te receiver. o layers are generally consiere en ealing it GPS: te Ionospere, ic extens from a eigt of 70 to 000 km above te Eart, an te ospere ic stretces to about 6 kms. Above te equator an 9 kms. Above te poles from te surface of te eart. As te signal propagates troug te ionospere, te carrier experiences a pase avance an te coes experience a group elay. In oter ors, te GPS coe information is elaye resulting in te pseuo ranges being measure too long as compare to te geometric istance to te satellite. (Hofmann-Wellenof, 99). e extent to ic te measurements are elaye epens on te otal Electron Content (EC) along te signal pat, ic is a measure of te electron ensity. Furter tis is epenent on tree factors: te geomagnetic latitue of te receiver, te time of ay an te elevation of te satellite. Significantly larger elays occur for signals emitte from lo elevation satellites (ce tey travel troug a greater section of te ionospere), peaking uring te aytime an subsiing uring te nigt (ue to solar raiation). In regions near te geomagnetic equator or near te poles, te elays are also larger (Parkinson, 996). e ionosperic elay is frequency epenent an can terefore be eliminate ug ual frequency GPS observations; ence te to carrier frequencies in te GPS esign. Single frequency users, oever, can partially moel te effect of te ionospere ug te Klobucar moel (Hofmann-Wellenof, 99). Eigt parameters of tis moel are transmitte along it oter navigation ata from te satellites. ese parameters epen on te time of ay an te geomagnetic latitue of te receiver. e moel results in an estimate of te vertical ionosperic elay, ic is ten combine it an obliquity factor, epenent on satellite elevation, proucing a elay for te receiver-satellite line of sigt. e final value provies an estimate itin 0% of te true elay an prouces elays ranging from m (nigt) to 0m (ay) for lo elevation satellites an -m for ig elevation satellites at mi latitues. e tropospere causes a elay in bot te coe an carrier observations. Since it is not frequency epenent (itin te GPS L ban range) it cannot be cancele out by ug ual frequency measurements but it can, oever, be successfully moele. e tropospere is split into to parts: te ry component, ic constitutes about 90% of te total refraction, an te et part, ic constitutes te remaining 0%. Values for ISS: 8 89.internationalournalssrg.org Page

3 SSRG International Journal of Electronics an Communication Engineering (SSRG-IJECE) volume Issue April 0 temperature, pressure an relative umiity are require to moel te vertical elay ue to te et an ry part, along it te satellite elevation angle, ic is use it an obliquity/mapping function. Moels put forar by Hopfiel, Black an Saastamoninen are all successful in preicting te ry part elay to approximately -cm an te et part to cm. Ionosperic an troposperic elays ave been escribe in etail in te subsequent sections. Ionosperic Delay: e vertical time elay for te coe measurement is given by ta iono v A ACos A () Were A = nano secons A m m m IP IP IP A = local time A m m m IP IP IP e values for A an A are constant, te coefficients i, i, i =, --- are uploae to te satellites an broacast to te user. e parameter t in Eq. () is te local time of te ionosperic point IP an may be erive from IP t t U () Were IP is te geomagnetic longitue positive to east for te ionosperic point in egrees an t U is te observation epoc in Universal ime. is te sperical istance beteen te geomagnetic pole an te ionosperic point. osperic Delay: e troposperic pat elay is efine as m IP n- s (6) In general, instea of te refractive inex n te refractivity is use. Were te refractivity 0 6 n- -6 Equation becomes 0 s (7) e troposperic elay can be estimate ug several moels ic are escribe belo: Hopfiel (969) sos te possibility of separating into ry an et components as (8) Were te ry part results from te ry atmospere an te et part from te ater vapor. Corresponingly te relations become -6 0 s an -6 0 s (9) -6 0 s s (0) About 90% of te troposperic refraction arise from te ry component an 0% from te et component. ere are a number of moels ic give us te ry an et refractivity at te surface of te eart (e.g., Essen an froom 9). Dry component of refractivity is given by p -, o c, c 77.6 K mb () Were p is te atmosperic pressure in millibar (mb) an is te temperature in Kelvin (K). Wet component is given by e e -,o c c, c.96 K mb an - c.78*0 K mb () Were e is te partial pressure of ater vapor in mb. Accoring to Hopfiel moel te ry refractivity as a function of te eigt above te surface is given by (),o () Were = ( 7.6) m. Substituting te above equation results in 6 0,o s () e integral can be solve if te elay is calculate along te vertical irection an if te curvature of te signal pat is neglecte. us, for an observation site on te surface of te eart (i..e., =0), te above equation becomes 6 0,o ( - ) () 0 Were te constant enominator as been extracte. After integration, 6 0,o - ( ) 0 (6) is obtaine. e evaluation of te expression beteen te brackets gives so tat 6 0,o (7) is te ry portion of te troposperic zenit elay. e et portion is muc more ifficult to moel because of te strong variations of te ater vapor it respect to time an space. everteless, ue to te lack of an appropriate alternative, te Hopfiel moel assumes te same functional moel for bot te et an ry components. us, ISS: 8 89.internationalournalssrg.org Page

4 SSRG International Journal of Electronics an Communication Engineering (SSRG-IJECE) volume Issue April 0 ( ), 0 mean value, 000m ; Were te (8) is use. Sometimes oter values suc as, 000m ave been propose. Unique values for an cannot be given because of teir epenence on location an temperature. e integration of above equation results in 6 0,o (9) erefore, te total troposperic zenit elay is 0 6,o,o (0) Wit te imension in meters. e moel in its present form oes not account for an arbitrary zenit angle of te signal. Consiering te line of sigt, an obliquity factor must be applie ic, in its simplest form, is te proection from te zenit onto te line of sigt. Frequently, te transition of te zenit elay to a elay it arbitrary zenit angle is enote as te application of a mapping function. Introucing te mapping function te above equation becomes Were (E) 0 6 ( ),o m E,o m an ) m ( E) () m (E are te mapping functions for te ry an te et part an E (expresse in egrees) inicates te elevation at te observing site (ere te line of sigt is simplifie as straigt line). Explicitly, m (E) = m (E) = E 6. E. () () Are te mapping functions for te Hopfiel moel. e above equation can be represente as (E)= (E) + (E) () 0 6 = ( E).96.78(0 [ ](000) E. ) e () Measuring p,,e at te observation location an calculating te elevation angle E, te total troposperic pat elay is obtaine in meters by te equations. III. BACROF ALGORIHM e set of nonlinear equations ic represent te relationsip beteen te unknon user position u = [x u, y u, z u ] an te knon satellite positions s = [x, y, z ] can be ritten as b x x y y z z u u u (6) Were b is substitute for ct u Expaning te above equation leas to x y z x x y y z z b x y z b 0 u u u u u u Defining te Lorentz inner prouct as u, v u v u v u v u v Equation (7) can be ritten as s, s s, u u, u 0 ; (7) (8) Equation (8) can be represente as a Bu e 0 an B u a e (9) Were s, s s, s a s, s, e s, s x y z - x y z - B x y z - x y z - an, uu If more tan satellites are observe, matrix B ill not be a square matrix. e solution can be obtaine ug te least square meto as given belo (Bancroft, 98). u B ( a e) (0) Were B ( B B) B IV. RESULS AD COCLUSIOS RIEX ata from Citrakut station (ear II Kanpur) is use for tis purpose. e observation ata of st January 006 at (0rs. 0 min. 0 sec.) Have been use. Eigt satellites (SV PR. os )are observe at te epoc time. Algoritms ave been implemente to sort out te epemeris ata into matrix format an for te etermination of satellites position at te epoc time. By ug clock correction parameters ic are available as ISS: 8 89.internationalournalssrg.org Page

5 SSRG International Journal of Electronics an Communication Engineering (SSRG-IJECE) volume Issue April 0 part of te avigation message, te satellite clock bias an error ue to relativistic effect ave been obtaine. e Ionosperic elay as been estimate ug Kloubucar moel. All te eigt coefficients for te implementation of Kloubucarmoel are available as part of avigation message. e osperic elay as been estimate ug Hopfiel meto. e estimate errors an te correcte ranges ave been represente in able. All te calculations ave been carrie out by riting programs in MALAB. e resultsare summarize belo for Eigt observe satellites it PR nos. of at 0 ours, 0 minutes an 0 secons of st January 006are foun to be Sv. o Az. (eg) ABLE ESIMAIO OF GPS ERRORS AD CORRECIO OF PSEUDORAGES El. (eg) Observe Pseuoranges Sv. clock+ relativistic Error Ionospe ric elay os peric Delay Correcte Pseuoranges Ug te correcte pseuo ranges user position is etermine an te results are given belo: User Position as per te observation ata: X u = m, Y u = an Z u = m. User position by linearization tecnique: X u = m, Y u = 707.9m an Z u = m. User position by Bancroft algoritm: X u = m, Y u = m an Z u = m REFERECES []. Bancroft. S, An algebraic solution of te GPS equations, IEEE ransactions on Aerospace an Electronic Systems (98) 6 9. []. B.HofmannWellenof,.Lictenegger an J.Collins, GPS eory an Practice, Springer-Verlag Wien, e York, 99. []. Klobucar J, Design an caracteristics of te GPS ionosperic time elay algoritm for gle frequency users, Proceeings of PLAS 86, Las Vegas, evaa, ovember -7, pp []. Hopfiel HS, o quartic troposperic refractivity profile for correcting satellite ata, Journal of Geopysical researc, 7(8): []. Strang, G. an Borre, K., Linear Algebra, Geoesy, an GPS, Wellesley Cambrige, - Wellesley, MA, 997. [6]. ttp://ome.iitk.ac.in/~rames/gps ISS: 8 89.internationalournalssrg.org Page