Calculation of GDOP Coefficient

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1 Calclaton of DOP Coffcnt Ing. Martna Mronovova, Ing. Hynk Havlš Sprvsor: Prof. Ing. Frantšk Vjražka, CSc., Prof. Ing. Jří Bíla DrSc. Abstrakt Účlm této prác j odvodt kofcnt DOP (omtrcal Dlton of Prcson, přdstavjící chyb rční polohy žvatl na povrch Změ, a kázat jho výpočt na praktcké aplkac. Úkolm j rozmístt až 45 držc po obloz tak, aby s kofcnt DOP mnmalzoval a dával tak poloh hldaného objkt na povrch Změ s co njmnší chybo. Pro optmální rozmístění držc na obloz j požto gntckého algortm. Abstract h prpos of ths work s to drv a DOP coffcnt (omtrcal Dlton of Prcson, that dtrmns th rror of th poston of obsrvr on th Earth srfac, and to show a calclaton of th coffcnt n practcal applcaton. h task s to dstrbt p to 45 satllts abov th sky so that th DOP coffcnt s mnmzd and dtrmns poston of th sarchd objct on Earth srfac wth mnmal rror. ntc algorthm s sd to dtrmn optmal dstrbton of all satllts abov th sky. Kywords: DOP, satllt navgaton, gntc algorthm. Introdcton h man motvaton for ths work s a drvaton of DOP coffcnt, practcal xampl of ts calclaton and s of DOP coffcnt for dtrmnng postons of arbtrary nmbr of satllts abov th sky. MALAB softwar s sd for th calclaton and smlaton of satllts postons and for DOP coffcnt calclaton. h whol task s dvdd nto thr sctons. h frst part s thortcal and shows th drvaton of DOP coffcnt. In th scond part th s of a gntc Fgr : Imagnary pyramds wth for satllts n cornrs of th bas and sr placd at th tp. Pyramd on th lft shows satllt dstrbton that ylds a lowr val of DOP coffcnt and s bttr than confgraton of satllts dstrbtd accordng to th pyramd on th rght that shows wors (hghr val of th coffcnt.

2 algorthm s shown. h algorthm s sd to fnd th most optmal stp of satllt postons so that th lowst val of DOP s rachd. h thrd part nclds calclaton of th coffcnt d to postons of satllts prvosly calclatd by a gntc algorthm. A DOP coffcnt (omtrc Dlton of Prcson s sd n satllt navgaton and postonng and rprsnts a rato of th poston rror to th rang rror []. h coffcnt rflcts th dlton of prcson n poston n thr dmnsons (PDOP and dlton of prcson n tm (DOP. o compt ths for dmnsons poston n x,y,z and tm for satllts ar ndd. h rcvr poston s comptd from satllts postons, th masrd psdo-rangs and rcvr poston stmat. Lt s magn that a sqar pyramd s formd by lns jonng for satllts wth th rcvr placd at th tp of th pyramd (s Fgr. h volm of th shap dscrbd by th nt-vctors from th rcvr to th satllts sd n a poston fx s nvrsly proportonal to DOP. h largr s th volm of th pyramd, th bttr (lowr th val of DOP coffcnt s. vrsly, th smallr volm of th pyramd s, th wors (hghr th val of DOP wll b. Smlarly, th gratr nmbr of satllts s sd for poston stmaton, th bttr th val of DOP coffcnt s [], []. h mportanc of satllt dstrbton can b also sn n Fgr that shows thr cass how two satllts and thr psdo-rangs dtrmn th ara of possbl occrrnc of th rcvr (sr. Each satllt has a psdo-rang rprsntd by sctor of a crcl. al poston of th sr s n th ntrscton of ral satllt rangs (crcl wth a cntr n ach satllt. Howvr, n ralty ths rangs shap not jst Fgr : Dstrbton of two satllts abov th rcvr (sr - top lft and rght mag show bad dstrbton wth larg psdo-rangs, bottom mag dsplays optmal psdo-rang.

Fgr 3: omtry of vctors for poston dtrmnaton on pont n th plac whr thy cross, bt an ara nstad. hs psdo-rang can vary accordng to th poston of satllts abov th sky.

3 Fgr 3: omtry of vctors for poston dtrmnaton on pont n th plac whr thy cross, bt an ara nstad. hs psdo-rang can vary accordng to th poston of satllts abov th sky. Whl th angl btwn satllts s too wd, th psdo-rang whr sr can b prsnt s larg rathr long. h smlar staton occrs whl th angl btwn satllts s too small. Optmally a smallst possbl sqar ara s rqrd.. Drvaton of DOP Coffcnt For th poston of th sr th followng qaton appls (s Fgr 3 [3]: = D (. poston vctor of th sr (nknown cntr of Earth - sr vctor from th Earth s cntr to th satllt D vctor from th sr to th satllt h qaton (. s mltpld by vctor, that s a nt vctor n drcton of : = D (. For th vctor, ts absolt val and a nt vctor n ts drcton th followng proposton s vald: = (.3 Now for th qaton (. qaton (.3 s appld, and ths opraton gans followng: = D (.4 Vctor D magntd (ts absolt val s a dstanc sr - satllt. h qaton (.4 conforms (.3 qaton btwn nmbrs, not vctors. For th dstanc D th dfnton appls: D = ρ-b-b (.5 In ths qaton: ρ - s so calld psdo-rang (psdo-dstanc sr satllt that corrsponds to th draton of sgnal propagaton from th satllt to th sr. rm psdo-rang mans that ths dstanc s not dtrmnd by th draton of sgnal propagaton from th satllt to th sr. It s d to th non-constant proprts of th atmosphr (prmttvty, prmablty and/or d to th rason of non-propagaton of th sgnal to th sr (boncs from th bldngs, montans, tc..

4 B s a sr s tm offst from th thortcally corrct tm B s a satllt tm offst from th thortcally corrct tm h qaton (.5 s stablshd nto (.4 and th followng qaton s obtand: = B ρ + B = ρ + B + B (.6 h qaton (.6 appls for on sr and thortcally for any satllt (arbtrary nmbr of satllts. Lt s mark coordnats of th drctonal vctor from th sr to th th satllt as,, 3. h qaton (.6 can b transcrbd for mor satllts. It wll b shown that for th xplct dtrmnaton of poston t s ncssary to hav thr satllts. If mor satllts ar avalabl, th optmal solton wll b fond n accordanc by a last sqar mthod. If th sr tm offst B s ncssary to dtrmnat, at last for satllts wll b ndd. h qaton (.6 can b transcrbd for mor satllts: B B B B = = = = B + B + B + B ρ ρ ρ 3 ρ 4 (.7 If s dtrmnd thn th poston of th sr s known and th task s bng solvd. h systm (.7 s obvosly possbl to rwrt and solv ot sng a matrx form. It can b markd that: A = 3 4 = = B [ B B ] S =... ρ = [ ρ ρ ρ...] 3 ρ4 Matrx dmnsons ar followng (n = nmbr of satllts, n 4. <n 4> <4 > 3

5 A <n 4n> S <4n > ρ <n > sng ths assmptons th qaton (.7 can b r-wrttn n a matrx form as follows: = AS ρ (.8 Sarchd poston of th sr and th tm offst of hs rcvr ar ncldd n vctor, and th rsolton of th systm (.8 for s or task. If w had xactly for satllts th solton of (.8 wold b followng: = ( AS ρ (.9 Provdng that th nvrs of matrx xsts f all satllts ar dstrbtd corrctly. It wold not b calclatd whl all satllts ar n on poston or n on ln consctvly. Both statons ar nvrsally rratonal. Postons of satllts obvosly nflnc th accracy of sr poston dtrmnaton, bcas psdorangs ar brdnd by random rrors. If th sgnal rcvd coms from mor than for satllts, th qaton (.8 can b convrtd by addng vctor of nknown rrors so that qatons ar algbracally corrct: + = AS ρ (. For th rror vctor appls th followng: ρ = AS (. h systm (. can b calclatd sng a last sqars mthod mnmzng th sm of qadrat of rrors,.. sm of qadrats of vctor lmnts. h sm of qadrats of rrors s gvn by a scalar prodct: P = = ( AS ρ ( AS ρ (. h qaton (. can b modfd sng an oprator for transpos and by mltplcaton: P = ((A S ρ P = (A S ρ (A S ρ (A S ρ (A S ρ (A S ρ + Whl applyng a transpos fncton th followng qaton was sd: (AB = B A (. D to th prvosly gvn dmnsons of vctors and matrcs n qaton (. t s obvos that mmbrs (AS ρ and (AS ρ ar scalars (nmbrs and ar qal. sng ths assmpton th qaton (. can b r-wrttn as follows: P + = (AS ρ (AS ρ (AS ρ (.4 In ths momnt w ar lookng for sch so that th sm of qadrats of rrors P s mnmal. W can drv th qaton (.4 by and st t qal to zro n ordr to fnd optmal :

6 P = - (A S ρ + In drvaton w hav sd followng matrx qatons: ( x Ay = Ay, ( x Ax = Ax + A x x x Now w st th drvaton qal to zro and calclat. = - (A S ρ = = ( (A S ρ + (A S ρ (.5 (.6 Arrangmnts ar procdd on condton that nvrs of matrx xsts. From now on w wll consdr abot th rror n dtrmnaton of poston DOP from satllts sng th qaton (.6. Covaranc matrx of poston rrors s dfnd as follows: covδ = E(( E ( E (.7 whr E s th oprator of th man val. W assm that th man val of th poston rror s qal to zro. Wth ths assmpton th qaton (.7 can b smplfd: covδ = E( (.8 whl knowng that qaton for s known (.6. h qaton (.6 can b sbstttd nto (.8: covδ = E(( (AS ρ(as ρ ( (.9 Matrcs consst of masrd vals and that s why w can mpt t bfor th oprator of man val: = ( ( E(A S ρ(a S ρ covδ (. Bcas w ar ntrstd n calclatng th poston rror from th gomtry dstrbton of satllts (DOP, w can dfn th followng: E(A S ρ(as ρ = I (. In th qaton abov, I rprsnts th nt matrx 4 4. ndr ths assmpton for covaranc matrx of rrors, th followng qaton s vald: cov δ = ( ( = ( (. Bcas th frst two matrcs ar mtally nvrs: ( = I Bcas th covaranc matrx s symmtrcal, th rst of th qaton (. can b smplfd as follows: cov δ = ( = ( (.3

7 hs covaranc appars as: ( xx Y = yx Z zx m tx xy yy zy ty Y Z xz yz zz tz m xt yt zt tt (.4 h dagonal vals n matrx (.4 rprsnt th varanc of th stmatd sr poston n ach axs and n th sr tm offst. h ndvdal factors of DOP ar gvn as follows: HDOP = + (.5 xx yy VDOP = zz (.6 PDOP = + + (.7 xx yy zz DOP = tt (.8 DOP = (.9 xx yy zz tt Estmats of rrors n sr poston or n sr tm ar gvn as a prodct of DOP factors and stmats of rrors n rang masrmnts. 3. Fndng Postons of Satllts For th calclaton of DOP coffcnt tslf t s ncssary to know postons of satllts on orbt arond th Earth. Bcas from th thory and drvaton from th Chaptr t appars that th mnmal val of th coffcnt DOP s whl all satllts ar dstrbtd qally abov th sky thr mtal dstancs ar qal [3]. h sky can b approxmatd as a hmsphr wth a cntr placd n th cntr of Earth. h stmatd poston of th objct s placd on th Earth s srfac, howvr, satllts ar placd on gostatonary orbt wth a rads approxmatly 7-tms gratr than rads of Earth. hs th abov lstd assmpton can b appld. Snc th satllt cannot b placd xactly abov th horzon, bcas th mttd wavs from th mttr may not rach th rcvr d to th srfac asprty, t s ncssary to st p th mnmal lvaton angl. ndr ths angl th satllt cannot b placd. Practcally ths lvaton angl qals to 5º. A schmatc sktch dsplayd n Fgr 4 shows th stmaton of total lvaton angl. D to th approxmaton dscrbd abov t s ncssary to add a corrcton of 8.7º.

8 Fgr 4: Schmatc sktch dsplayng th stmaton of total lvaton angl qal to 4º (dgrs approxmatly. For qal dstrbton of satllts abov th sky mltpl hrstcs can b sd from th planary covrag of th ara, throgh random slcton p to algorthms of artfcal ntllgnc [4]. In ths work a mthod of gntc algorthms was slctd. hs mthod s nsprd by Darwn s thory of dscnt. h algorthm ss a grop of soltons btwn whch t slcts th bst ons. hs slctd soltons ar combnd btwn ach othr (so calld crossovr and ths lads to nwly ncrrd soltons. In th mantm th wk soltons (worst ons ar bng lmnatd. In ordr to prvnt th grop of soltons to b rstrand jst to combnaton of alrady xstng soltons, wth known probablty th mtaton of slctd spcmns occrs n vry stp of th algorthm. h ffort s to nhanc th poplaton n vry stp of th algorthm that can lad to th rtrval of optmal solton, or solton clos to th optmal on. For th stmaton of th qalty of th solton so calld ftnss fncton can b sd. hs fncton can tak arbtrary form. In ths xampl th ftnss fncton s a fncton of dstancs btwn two closst satllts and s spcfd by followng formla:, j j h am s to maxmz ths fncton. ( arccos ( x x + y y z z d = mn + Postons of satllts of concrt solton ar for th prpos of algorthm codd nto so calld gnom, or chromosom. Sch a chromosom contans of gns and ach gn rprsnts on satllt. Vals of th gn rprsnt th sphrcal coordnats of th satllt. At th bgnnng of th algorthm ths sphrcal coordnats ar slctd randomly for ach spcmn. h poplaton matrx gos throgh for phass n th algorthm. Frstly th ltsm s appld whr k spcmns wth th bst val of ftnss fncton ar slctd from th poplaton. hs spcmns ar solatd from th poplaton and prsrvd to nsr th consrvaton of th prsnt bst solton. In th scond phas of th algorthm th crossovr of two slctd gnoms occrs. h plac of th crossovr procss s slctd randomly and a nw spcmn orgnats. hs nw spcmn pcks j j j

Fgr 5: Procss of crossovr of two spcmns and orgnaton of a nw spcmn p part of th nformaton from th frst spcmn and part of nformaton from th scond spcmn. hs procss s shown n Fgr 5.

9 Fgr 5: Procss of crossovr of two spcmns and orgnaton of a nw spcmn p part of th nformaton from th frst spcmn and part of nformaton from th scond spcmn. hs procss s shown n Fgr 5. In th mtaton phas th random chang of on gn n th gnom occrs. Drng th lmnaton phas th worst spcmn s sspndd from th poplaton. Aftr ths phas th spcmns that wr at th bgnnng prsrvd (ltsm phas ar rtrnd back nto th poplaton. All for phass rpat agan, as s shown n th followng psdo cod: Poplaton = nrat_random_oplaton(poplaton_sz whl nd condton not satsfd do // Prsrv lt Elt = Pop_lt_from_poplaton(lt_sz // prodcton Indv_ = t_random_ndvdal_from_poplaton( Indv_ = t_random_ndvdal_from_poplaton( Nw_ndv = Apply_crossovr(Indv_, Indv_ Add_ndv_to_poplaton(Nw_ndv // Mtaton Mtatat = andom_nmbr f Mtat <= Mtaton_probablty Mtatd = t_random_ndvdal_from_poplaton( Mtatd = Apply_mtaton(Mtatd nd // Elmnaton Elmnat_worst(lmnaton_cont // stor lt trn_lt_to_poplaton // mmbr bst solton Crrntly_bst = t_bst_ndvdal( f Evalat(Crrntly_bst > Evlaat(Bst bst = Crrntly_bst nd nd rtrn Bst; Algorthm s trmnatd whn all k tratons rn ovr. In ordr to obtan good rslts t s ncssary to slct sffcnt amont of tratons or slct th trmnaton condton so that th val of ftnss fncton wll not chang mor than s th dfnd crtan val.

Fgr 6: Postons of satllts drng thr dstrbton (top lft: traton, top rght: 5 tratons, bottom:, tratons For th calclaton of th algorthm dscrbd abov and for th vsalzaton of th rslts th MALAB applcaton can

10 Fgr 6: Postons of satllts drng thr dstrbton (top lft: traton, top rght: 5 tratons, bottom:, tratons For th calclaton of th algorthm dscrbd abov and for th vsalzaton of th rslts th MALAB applcaton can b sd. Fgr 6 dsplays thr plots n ach thr s an Earth dsplayd wth th magnary hmsphr wth rads of gostatonary orbt. On ths magnary hmsphr ndvdal satllts ar dsplayd. h fgr shows th procss of th algorthm whn th solton rachs ts optmm. Frstly th systm starts from th random confgraton of satllts that ar bng slowly dstrbtd n th way that th mnmm dstanc fncton s maxmzd. Aftr, tratons satllts ar dstrbtd qally abov th hmsphr. Nmbr of satllts and lvaton angl can b changd arbtrarly. Also th st p of th algorthm can b changd (nmbr of tratons, nmbr of rprodcd spcmns, sz of th poplaton, mtaton probablty. 4. Calclaton of DOP Coffcnt Calclaton of DOP coffcnt was dscrbd n dtal n Chaptr. ntc algorthm dscrbd n Chaptr 3 allows obtanng postons of satllts abov th sky. For DOP coffcnt calclaton MALAB and data obtand from prvos calclatons by gntc algorthm can b sd. h otpt of ths part of comptr program s a DOP coffcnt and ts paramtrs: PDOP (dlton of prcson n poston n thr dmnsons, HDOP (dlton of prcson n two horzontal dmnsons and DOP (dlton of prcson n tm. MALAB program nvronmnt allows sr to optmz ts own graphcal sr ntrfac (I for work facltaton wth a comptr program. In th top lft cornr of th applcaton wndow of I (plas s Fgr 7 t s possbl to ntr thr

11 varabls npt paramtrs.. nmbr of satllts abov th sky, lvaton angl ovr th horzon and nmbr of tratons of gntc algorthm. A pshbtton labld Start výpočt rns th comptr program whch accordng to ntrd npt paramtrs calclats th optmal dstrbton of satllts and dtrmns vals of coffcnt. hs vals ar dsplayd n corrspondng txt flds n I wndow. As an addtonal otpt two plots ar dsplayd on th rght hand sd of th I wndow. op plot shows a polar chart of satllts abov th sky, th bottom on dsplays th sam satllts dstrbtd n 3-D vw. Fgr 7 dsplays th applcaton wndow and rslts for dstrbton of 45 satllts ncldng DOP vals. From ths xampl t can b obsrvd that satllts ar dstrbtd vnly abov th hmsphr, howvr to rach sch a prformanc at last 6, tratons ar ncssary to rach a good dstrbton at th xpns of calclaton tm. 5. slts Exampls n prvos chaptrs show that program sng a gntc algorthm s abl to dstrbt varabl amont of satllts vnly on th hmsphr and that th dstrbton qalty s dpndnt on nmbr of tratons sd. h hghr th nmbr of satllts, th hghr th val of tratons s ndd n ordr to obtan a good solton. Fgr 7: Inpt and otpt paramtrs and rslts for dstrbton of 45 satllts

In ths cas 4 satllts ar bng dstrbtd vnly, th procss s shown n pctrs and rspctv vals of comptd DOP ar dsplayd bllow ach stp.

12 Nr. of tratons: 5 DOP: 4.34 Drvd calclaton of DOP coffcnt was ntgratd nto th comptr program and val of DOP can b comptd for any stp of satllts. In ordr to show how ths coffcnt changs d to th satllts dstrbton s shown on th followng xampl n Fgr 8. In ths cas 4 satllts ar bng dstrbtd vnly, th procss s shown n pctrs and rspctv vals of comptd DOP ar dsplayd bllow ach stp. h comptr program was tstd for varos nmbrs of satllts, dffrnt vals of lvaton angl ovr th horzon and for varos nmbrs of tratons of gntc algorthm. abl shows npt paramtrs and calclatd DOP coffcnts for thr xampls 4, and 45 satllts. abl : slts and vals of coffcnts and npt paramtrs for 4, and 45 satllts Nr. of satllts [-] Angl abov th horzon [º] Nr. of tratons [-] DOP [-] PDOP [-] HDOP [-] DOP [-] 4 5, , , h modl prsntd n ths papr s consdrably smplfd. h Earth s consdrd as a rglar sphr wth no nvnnss as montans, bldngs, trs, clods tc. All ths obstacls spawn dtroraton n qalty of satllt vsblty and sgnal transmsson. In ralty t s possbl to rcv sgnal from 4 to satllts at th tm [5], ths vals for and 45 satllts shown n abl ar not corrspondng to ralty, howvr t s shown that gntc algorthm s abl to dstrbt vn hgh amont of satllts abov th hmsphr and that wth mor satllts sd th bttr val of DOP s obtand. h qalty of DOP coffcnts s valatd accordng to th catgorzaton shown n abl [6]. abl : DOP ratngs DOP atng Idal 3 Excllnt 4 6 ood 7 8 Modrat 9 Far 5 Poor Nr. of tratons: 5 DOP:.87 Nr. of tratons: 5 DOP:.7 Fgr 8: Progrss of satllt dstrbton and vals of rspctv DOP coffcnts

13 6. Conclson In th papr th drvaton of DOP coffcnt and comptd xampls wr shown. h calclatons prov that coffcnt s dpndnt on th dstrbton of satllts rlatvly to th sr s poston. Mor vn th dstrbton s th bttr val (lowr of DOP coffcnt s obtand. Also mor satllts ar sd for navgaton th bttr val of DOP s rachd. h optmal dstrbton of satllts was rachd sng a gntc algorthm. Frthr work wll b focsd on bttr approxmaton of condtons that wold approach a ral staton. Also th possblty of s of nral ntworks for DOP approxmaton wll b xamnd. Acknowldgmnt h task was assgnd and dvlopd wthn th cors of Modrn systms of CNS (Control, Navgaton and Srvllanc ndr th sprvson of Prof. Ing. Frantšk Vjražka, CSc., profssor of Dpartmnt of adolctroncs, Faclty of Elctrcal Engnrng, C n Prag. frncs: [] Dana, P.: lobal Postonng Systm Ovrvw, avalabl on-ln [March 9, ] < [] Zrar, S., Canalda, P., Sps, F.: Modllng And Emlatons Of An Extndd DOP For Hybrd And Combnd Postonng Systm [3] Mllkn,.J., Zollr, C.J.: Prncpl of Opraton of NAVSA and Systm Charactrstcs [4] Majda, F.: Vyžtí mělé ntlgnc v opračním výzkm. sarch projct, FJFI C n Prag, 9 [5] Bao, J., s, Y.: Fndamntals of lobal Postonng Systm cvrs: A Softwar Approach. John Wly & Sons,, lctronc ISBN [6] Kaya, F., Sartas, M.: A Comptr Smlaton of Dlton of Prcson n th lobal Postonng Systm sng Matlab. az nvrsty, Faclty of ngnrng and archtctr

Lecture 08 Multiple View Geometry 2. Prof. Dr. Davide Scaramuzza

Lecture 08 Multiple View Geometry 2. Prof. Dr. Davide Scaramuzza Lctr 8 Mltpl V Gomtry Prof. Dr. Dad Scaramzza sdad@f.zh.ch Cors opcs Prncpls of mag formaton Imag fltrng Fatr dtcton Mlt- gomtry 3D Rconstrcton Rcognton Mltpl V Gomtry San Marco sqar, Vnc 4,79 mags, 4,55,57