Navigation Laboratory Master of Science in Computer Engineering, Environmental and Land Planning Engineering Politecnico di Milano Campus Como

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1 Navgaon Laoraory Masr o Scnc n Compur ngnrng, nvronmnal an Lan Plannng ngnrng Polcnco Mlano Campus Como LCTR NOTS ON INRTIL NVIGTION

2 Rrnc sysms rrnc sysm s a s o ruls an masurs ha allow us o answr h ollowng qusons: whr s a pon?, whn an vn happn?. Hsorcal an nsrumnal rasons, as wll as rn applcaons, hav l o h non o rn rrnc sysms o go-rrnc pons an o m-rrnc vns. Gvn a rrnc sysm, s possl o n a coorna sysm ha unvocally assocas pons n h spac wh a ra,, o ral numrs or ampl Carsan coornas. n nral sysm s a rrnc sysm whr vry maral pon, whch s no sujc o any rnal orcs, says a rs or movs wh a unorm rclnar moon, as posula y h rs law o ynamcs. Th sars can ra as maral pons whn comparng hr mnson wh hr hug rlav sancs; nglcng hr gravaonal aracon, hy can consr sola pons an so hy can us o n a rrnc sysm or whch h rs law o ynamcs s sas on a m-scal o som yars. cually, or phnomna wh spaal n o m an m n smallr han ay, a hlocnrc sysm can rasonaly consr an nral sysm. Howvr, any coorna sysm assoca o hs hlocnrc rrnc sysm woul qu nconvnn o scr a vhcl navgaon on or aroun h arh. sysm ransla no h arh cnr s cranly much asr o us, u unorunaly canno consr an nral sysm anymor, manly u o h rvoluon moon o h arh aroun h Sun ha mpls an acclraon o h arh sysm. Nvrhlss, h mpac o h Sun on h moon o a pon clos o h arh surac pns only on h rn aracon ha h Sun sl proucs on h consr pon compar o h arh arycnr. Ths rnc, whch s call, s qu small an can nglc or moll n cas o long osrvaon pros. Th sam can on or h Moon. psuo-nral sysm s a rrnc sysm wh orgn n h arh arycnr an wh as paralll o h ons o an nral sysm ypcally n y asronomcal osrvaons. In such a sysm, Nwon s law hols wh h convnon ha h gravy acclraon nclus also h al rms u o lunar-solar aracon. Psuo-nral sysm : - orgn n h arh arycnr; - as: orn owars a clsal Norh Pol nsananous arh roaon as; - as: nrscon wn h clpc an h clsal quaoral plan; - as: o compl h rgh-han ra. clpc: h apparn pah o h Sun moon as sn rom h arh

3 psuo-nral sysm can usul o scr h sall ors aroun h arh, u s no convnn o rprsn h moon on a vhcl on h arh. Ths s u o h ac ha h arh roas aroun s own as. s a consqunc s usul o n a rrnc sysm whch s movng jonly wh h arh. Such a sysm s also call arh-. arh- sysm : - orgn n h arh arycnr; - as: orn owars a convnonal Norh Pol; - as: nrscon wn h Grnwch mran plan an rrsral quaoral plan; - as: o compl h rgh-han ra. arh sysm No ha h arh roaon as changs n m an hror h as o h psuo-nral sysm s n gnral rn rom h on o h arh- sysm. Morovr h arh angular vlocy s sujc o vry small varaons an h arh sl s ac y ormaons u o gologcal an al orcs. In prncpl hs phnomna shoul an no accoun; howvr, or h purpos o navgaon, apar rom avalal an compual corrcons, h arh can consr as a rg oy, unormly roang aroun a as wh a pro o 4 hours. ll ncssary corrcons ar prov y IRS Inrnaonal arh Roaon an Rrnc Sysms Srvc. Som ampls o arh- rrnc sysms ar WGS84 an ITRF. In h cas o arh- sysms, h raonal Carsan coornas can ovously us; howvr, n orr o loca pons on h arh surac, s mor convnn o us h so call goc coornas ϕ, λ, h, namly h lau, h longu an h llpsoal hgh wh rspc o a gocnrc llpso. Thror s rasonal o n a urhr rrnc sysm, sual o scr h moon o a vhcl on h arh surac; hs s call navgaon or locallvl rrnc sysm an s n as ollows. Navgaon n or local-lvl sysm: - orgn n a gnrc pon P no ncssarly on h rrnc llpso; - as: orn n h as rcon; - as: orn n h Norh rcon; - as: orn as h normal o h rrnc llpso, n h p rcon;

4 Navgaon sysm Whl h navgaon sysm s sual o scr h moon o a vhcl, h nsrumns ar ypcally consran o h vhcl sl an hror hy rqur h nroucon o a urhr rrnc sysm. Vhcl or oy sysm: - orgn n a gnrc pon P; - as: orn n h moon rcon; - as: prpncular o h vhcl plan an orn n h up rcon; - as: o compl h rgh-han ra. Th sysm n an h sysm hav h sam orgn n h pon P, u hr as ornaon s gnrally rn. In parcular, s n yaw h roaon aroun h as, pch h roaon aroun h as an roll h roaon aroun h as. No ha hr s nsrumns ha ar always orn l h sysm n y mans o Caran jons an srvomoors. Howvr, srapown sysms ha ollows h vhcl ornaon ar mor common an y ar lss pnsv. yaw roll pch oy sysm an roaons 4

5 Inral nsrumns n nral navgaon sysm INS s compos y a plaorm whr ypcally s snsors ar rgly moun. Thr o hs snsors masur h acclraons acng on h plaorm sl acclromrs, whl h ohr hr snsors masur h angular ra wh rspc o an nral rrnc sysm gyroscops. Th harwar collcng aa an h sowar procssng hm n ral m ar par o h sysm. ach componn o an INS s call IM Inral Masurmn n. Somms hs acronym s us as a synonym o INS. Th rrnc sysm mplcly n y h INS, whch s consr as a rg oy, s maralz y a ra o orhogonal as concng o h masurmn as o h acclromrs. Th gyroscops ar moun n such a way ha hr masurmn as ar paralll o h as o h acclromrs. Two major yps o INS conguraons s. In h mos pnsv cas, h plaorm s moun on gmals ha consran s moon an, hans o a sysm o srvomoors, p h INS rrnc sysm orn as h local-lvl sysm n or a las paralll o h horzonal plan. Th ohr yp o nsrumns, call srapown sysms, p h plaorm rgly consran o h movng vhcl, so ha h INS rrnc sysm concs o h oy sysm. Schm o nral navgaon sysm cclromrs Th asc prncpl o acclromrs s o masur h orcs acng on a proo mass. Thr ar wo major cagors o acclromrs,.. opn-loop an clos-loop snsors. In h ormr cas, h acclromrs rcly masur h splacmn o h proo mass rsulng rom rnal orcs acng on h snsor. In h lar cas, h acclromrs p h proo mass n a sa o qulrum y gnrang a orc ha s oppos o h appl orc. Ths s achv y mans o an lcrc or magnc ac sysm. Th man avanag o h clos-loop archcur s h r lnary o h masur wh rspc o h appl orcs. In s mos lmnary conguraon, an acclromr s compos y a chasss ha consrans h proo mass o mov along a rcon an y a sprng ha lns h proo mass o h chasss. Th ynamcs o such a sysm s scr y h ollowng lnar rnal quaon: m l v l l F whr m s h proo mass, l s splacmn o h proo mass corrsponng o h sprng srchng, v s h vsy cocn, s h la lascy cocn o h sprng an F s h orc acng on h snsor. Th homognous soluon o hs quaon namly h soluon whn 5

6 no rnal orcs ar appl s a harmonc oscllaor, whch s amp rag s prsn. In cas o Fcons, ar h nal oscllaons hav cas, h ollowng law hols: l F manng ha s possl o rv h appl orc y masurng h sprng srchng h lascy cocn s nown. Snc rom Nwon s law hols: F m a nowng h proo mass, s possl o rv h acclraon a along h sprng as. F m Opn-loop acclromr prncpl mor ralsc clos-loop sysm s h pnulous acclromr, whr h moon o h proo mass m s consran o osclla aroun a hng P, whch s a sysm wh a unqu gr o rom. Whn an acclraon s appl o h proo mass m, h pnulous sars oscllang; howvr h poson varaon o h proo mass s c y opcal snsor S whch n urns conrols a col. Ths col nucs a orc F o a magn ha s connc o h proo mass, hus pung ac h proo mass m n rs s gur. Th orc nuc y h col whch s conroll y a ac sysm s proporonal o h rnal orc an can asly masur caus pns on h currn I passng hrough h col. l m ac sysm Clos-loop acclromr prncpl 6

7 cclromrs wh scnc purposs an wh rmly hghr accuracs ar or nsanc nsall on oar goc salls l CHMP, GRC an GOC or h masurmn o h gravaonal l. Thy ar gnrally lcrosac acclromrs, whr h planum proo mass s plac n a lcrc surroun y lcros. ach movmn o h proo mass proucs a varaon o h capacy masur y h lcros. Ths varaon s c, ampl an hn us no a ac sysm o pu ac h proo mass n s orgnal poson, hus rsorng h orgnal capacy. Th volag appl o h lcros rprsns h oupu o h acclromr s gur. lcrosac acclromr or goc sall mssons Th acclromr rror can mol as: δ λa c T T ν a T whr s a as, λ s a scal acor, c T s a hrmal consan pnng on h mpraur T an v s h masurmn nos. Typcal valus or a commrcal nsrumn can : 5 mgal 5 λ 5 c T.5mGal / C σ v 4 mgal / Hz namly aou.6 Gal or ach osrvaon, wh a samplng rquncy o h orr o Hz Gyroscops Th gyroscop or rly gyro s an nsrumn ha can masur h angular ra o h snsor roaon wh rspc o an nral rrnc sysm. Traonally h gyroscops wr mchancal nsrumns, ascally compos y a rg oy wh a crcular symmry roor ha s al o roa aroun s prncpal symmry as spn-as. In h las yars h us o hs mchancal snsors s sgncanly mnsh n avour o opcal snsors, whch ar characrz y a hgh accuracy, as o unconng an rlavly low snc hy o no rqur hgh prcson mchancal componns. Morovr hy or h possly o mnaurz. Th opcal nsrumns clusvly wor on srapown sysms, rnly rom h mchancal ons ha ar also sual or gmal plaorms. 7

8 mong h opcal snsors, h r opc gyroscop FOG s noworhy; s as on a wll nown rlavsc c, h so call Sagnac c, rom h nam o s scovrr n 9. Consr a crcular r opc col, unormly roang on s plan wh an angular ra wh rspc o an nral rrnc sysm; a lgh am propagas along h r n h sam rcon o h roaon. Ths lgh am s m y a sourc a m an s collc y h sam snsor a m. Thror an osrvr whch s jonly movng wh h r ss h lgh covrng acly on rvoluon ϑ π h suprscrp ncas a quany sn y an osrvr whch s movng oghr wh h r, whl or an nral osrvr h lgh am covrs an angl ϑ π ϑ s gur. Sagnac c Tang no accoun ha an nral osrvr ss h lgh am movng a a vlocy c, npnnly rom h vlocy o h lgh sourc, h angular rnc rsuls: R ϑ π c whr R s h raum o h r col an R s h vlocy o h sourc. Ths angl corrspons o h opcal pah: R L R ϑ c π c whr s h ara mrac y h col. I now h sourc ms anohr lgh am a h sam nal m u covrng h col n h oppos rcon, s asy o unrsan ha h nral osrvr ss h am comng ac o h sourc ar an angl ϑ π- ϑ. y assumng ha h wo lgh ams hav h sam rquncy an ar m n phas, s possl o masur h phas sh a rcpon y mang h wo ams nrr, namly: ϕ L L λ 4 λc whr L L - ar h wo opcal pahs n h wo oppos rcons. Th masur o h phas sh allows o rv a masur o h angular ra. Snc h acor 4/λc rprsns h snsly o h nsrumn, n orr o mprov s prormanc h opc r s wrapp up so o oan N cols, wh a oal lngh o m an yon. 8

9 Smlarly o an acclromr, h rror o a gyroscop can mol as: δ λ c T T ν T whr s a as, λ s a scal acor, c T s a hrmal consan pnng on h mpraur T an v s h masurmn nos. Typcal valus or a commrcal nsrumn can : / λ 6 hour 5 c T 5 / hour C σ v 6 7 ra / s Hz namly aou -5 ra/s or ach osrvaon, wh a samplng rquncy o h orr o Hz Roaons an angular vlocs L us rly rcall h man concps aou roaons o rrnc sysms. Dnng, h Carsan coornas o a gnrc vcor n h rrnc sysm an, h coornas o h sam vcor n a rrnc sysm roa y an angl wh rspc o, h ollowng rlaon hols: ' ' sn or n vcor noaon: ' wh sn sn sn No ha h sam rsul can oan y png h rrnc sysm an roang h vcor y an angl -, namly R. In h r-mnsonal cas, any roaon can always oan y comnng hr sngl roaons aroun ach Carsan as, namly: sn sn,, sn sn sn sn sn sn sn sn sn sn sn sn sn sn sn sn sn sn sn 9

10 In orr o rv h rlaon wn roaon mar an angular vlocs o us n h masurmns o h gyroscops, l us consr h nnsmal angls,,, so ha, sn. Th hr roaon marcs com:,, an h jon roaon, nglcng h scon orr rms, can wrn as: I y ang no accoun hs mar an y nng h roaon a m as: h roaon a m, wh an nnsmal, can prss as h comnaon o wo conscuv roaons, namly: I Th m rvav o h roaon mar can hn wrn as ollows: I lm lm lm whr h mar o h angular vlocs s n as: lm lm Ths prsson can urhr wor ou. y rcallng h roaon marcs proprs, hols: I manng ha wo oppos roaons cancl ou. y compung h m rvav o hs prsson, hols:

11 Fnally, y mulplyng oh ss o h rnal quaon y h mar, wh som algra on gs: Navgaon quaons Th navgaon quaons salsh a conncon wn h unnowns o h prolm namly h poson, h vlocy an h au o h vhcl m y m wh h avalal osrvaons rom h acclromrs an h gyroscops an n cas rom h GPS rcvrs. ccorng o h chosn rrnc sysm, h navgaon quaons can mor or lss complca rom h analycal pon o vw. In h ollowng w wll consr h cas o nral rrnc sysm sual o scr spac navgaon, or ampl h ors o an arcal sall an h cas o arh- rrnc sysm mor sual o scr rrsral navgaon, somms rqurng a urhr sp owars h local-lvl sysm. Navgaon n an nral rrnc sysm In an nral rrnc sysm Nwon s scon law o ynamcs can wrn as: F o m whr F o rprsns h ovrall orc acng on h oy o mass m along h hr as o h rrnc sysm an s h acclraon prouc y hs orc. I h oy s sujc o a gravaonal l, h prvous quaon can prss as: m F mg whr g s h gravaonal acclraon. y vng oh rms o h quaon y m, on gs h unamnal rlaon: g whr s h non-gravaonal spcc orc, namly h orc or un mass, appl o h mass m. Snc h sa o h movng vhcl s gnrally scr n rms o poson an vlocy

12 an n cas au, h prvous hr mnsonal rnal quaon can spl no a sysm o wo rnal quaons always n hr mnsons, namly: g I s sraghorwar o s ha h rmnaon o poson an vlocy rqurs h nowlg or h molzaon o h arh gravaonal l. In h common cas o srapown sysms, h acclromrs masur h spcc orc n h oy rrnc sysm. Thror, n orr o us h prvous rlaon s ncssary o apply h ollowng roaon: whr s h roaon mar rom h oy sysm o h nral sysm. Ths mar changs n m an can rv rom h masurmns o h gyroscops y numrcally ngrang h rnal quaon: No ha n h cas o a roaon aroun a sngl as o h rrnc sysm, or ampl: sn sn h prvous rnal quaon coms: sn sn sn sn sn sn sn sn so ha:

13 τ τ rqurng h nowlg o h nal au. Navgaon n an arh- rrnc sysm Th navgaon quaons n an arh- rrnc sysm can rv rom Corols horm, whch scrs vlocy an acclraon o a pon, osrv rom wo rn rrnc sysms n rg moon on o h ohr. ny poson vcor can scr hr n an nral sysm or n an arh- sysm y smply applyng an nsananous roaon an n cas an orgn ranslaon wn h wo sysms, namly: To ma h noaon smplr, h m pnnc s om alhough concrns all vcors an marcs nvolv n h ollowng quaons. Compu a rs rnaon n m, oanng: an hn a scon rnaon n m, namly: y mulplyng oh rms o h quaon y an rcallng ha I on gs h ynamcs o h pon wh rspc o a roang rrnc sysm: L us analyz ach rm o h quaon wh parcular anon o h apparn acclraons acng on h movng pon n h arh- rrnc sysm: - acclraon o h pon as sn n h arh- rrnc sysm; - acclraon o h pon as sn n h nral rrnc sysm; - acclraon o h orgn o h nral rrnc sysm. Snc h nral sysm an h arh- sysm hav h sam orgn h cnr o h arh hs rm s qual o namly h ransormaon wn h wo rrnc sysms s only a roaon whou any ranslaon; - nsananous roaon mar rom h nral rrnc sysm o h arh- rrnc sysm. Th roaon s aroun h as, whch s h sam or oh rrnc sysms;

14 - mar o h angular ras wn h wo rrnc sysms, whch ar roang on wh rspc o h ohr. Snc h arh roas aroun h as, assumng a unorm crcular moon, hols, π 4 hours cons ; - Corols acclraon ha appars whn h oy s movng wh rspc o a roang rrnc sysm. Du o hs acclraon h movng oy prncs a lcon owars h rgh wh rspc o h vlocy rcon n h norhrn hmsphr an a lcon owars h l n h souhrn hmsphr. Ths acclraon s prpncular o h plan orm y h oy vlocy an h roaon as o h rrnc sysm. I s cancll ou whn h oy vlocy s orn as h roaon as. I has a unamnal rol n h amosphrc crculaon; - cnrugal acclraon u o h roaon o h arh- sysm wh rspc o h nral sysm. I s orn n h ouwar rcon, prpncular o h roaon as o h arh. - angnal acclraon u o h angular ra varaon o h arh- rrnc sysm. Snc h arh vlocy s assum o consan, hs rm s qual o. y applyng all h propr smplcaons, h quaon o h ynamcs coms: Rcallng ha or Nwon s scon law h acclraon n h nral rrnc sysm s gvn y: g on gs: g g whr g an rspcvly rprsn h gravaonal acclraon an h non-gravaonal spcc orcs appl o h movng oy, prss n h arh- rrnc sysm. Typcally gravaonal acclraon an cnrugal acclraon ar group oghr o orm h gravy acclraon vcor: g g In hs way, h nal prsson o h ynamcs n an arh- rrnc sysm s gvn y: g Tang no accoun ha h spcc orcs ar masur n h nrnal rrnc sysm o h vhcl oy sysm, hols: 4

15 whr s h roaon mar rom h oy sysm o h nral sysm ha can rv rom h masurmns o h gyroscops, whl s h nown roaon mar rom h nral sysm o h arh- sysm, whch pns on h osrvaon poch, namly: sn sn assumng ha h nral sysm an h arh- sysm ar algn a m. Snc h sa o h movng vhcl s gnrally scr n rms o poson, vlocy an au, h navgaon quaons o h vhcl ar prss y mans o h ollowng sysm o rnal quaons: g whr an ar osrv quans, g, an ar mol quans as a uncon o h osrvaon m an nally, an ar h unnowns o h prolm. Th rnal quaon ha scrs h ynamcs n h arh- rrnc sysm can v no a sysm o hr rnal quaons, on or ach componn; n ac, y rcallng ha: hols: Corols acclraon 5

16 cnrugal acclraon Furhrmor, as a rs appromaon, h arh gravaonal l can assum qual o h l prouc y a pon-mass loca n h cnr o h arh an wh a mass qual o h arh mass. In hs way, or h law o unvrsal gravaon, hols: g g whr G s h unvrsal gravaon consan an M s h arh mass. In rms o ach componn, h prvous rlaon can wrn as g / I has o srss ha hs sphrcal appromaon o h arh gravaonal l nroucs unaccpal rrors n h navgaon, hror s prrr o us an appromaon ha a las accouns or h prvalng llpsoal shap o h arh plan. Thr also s gloal an local mols o h arh gravaonal l ha allow o urhr ruc h rror u o hs rm, o cours a h o a hghr compuaonal urn. y susung h prsson o ach componn o h gravy acclraon, h navgaon quaons n h arh- rrnc sysm ar gvn y: / / / 6

17 paron o h rnal quaon no s rn componns woul rqur h nroucon o h Rorguz mar an h quarnon noaon; howvr hs sujcs ar no par o hs cours. sng hs urhr suvson h prolm o rmnng h sa o h movng vhcl coul spl no wo spara prolms: - rs h au o h vhcl s rmn on h ass o h osrvaons o gyroscops an npnnly rom h poson an h vlocy o h vhcl sl; - hn h sysm o rnal quaons wh unnown poson an vlocy s solv, assumng ha h au s nown an roang h spcc orcs masur y h acclromrs no h arh- rrnc sysm. rror ug In orr o suy h prormanc o nral sysms, l us consr a vry smpl scnaro whr h arh s assum o sphrcal an no roang hror w wll us an nral rrnc sysm wh a gravaonal l pnng only on h sanc rom h arh cnr. Morovr s assum ha h as passs hrough h cnr o h nrs ara, manng ha h vhcl s movng n h surrounngs o h Norh Pol wh a mamum amr o som lomrs. Wh such hypohss s possl o assum ha, as a rs appromaon, h an coornas scr h horzonal moon, whl h coorna rprsns h vrcal moon. Th ynamcs o such an approma sysm can scr as: g Now h non-gravaonal orcs ar osrv rom h acclromrs on oar h vhcl wh a cran masurmn rror, namly: ν so ha s possl o wr: ν y lnarzng h oan quaon wh h am o suyng h nos propagaon, on gs: J ν whr 7

18 J Jacoan mar Whn compung h paral rvavs, wo rn cass ar prsn: / / / / / / / / / / h h h h wh h In ohr wors h Jacoan mar can wrn as: / P I J whr h raal projcor P s n as: P 8

19 Th ynamcs can spl no wo rms: δ, h ormr scrng h rrnc rajcory u o h applcaon o h osrv spcc orcs: h lar scrng h nos c: P I ν δ δ / whr δ In h cas o h consr smpl scnaro, rcallng h man arh raus R678 m, h ollowng appromaons hol: R, R, R so ha, nglcng rrors o h orr o -, s possl o consr h ollowng appromaons: R R R R R R R P hus oanng h ollowng sysm o rnal quaons or h nos ynamcs: s s s ν δ δ ν δ δ ν δ δ whr s ra R s /.4 s call Schulr rquncy an corrspons o a pro T84.46 mnus. 9

20 I w apply a masurmn nos only a a cran m poch, s possl o suy how h sysm ynamcs volvs as a rsul o hs mpulsv npu y solvng h corrsponng homognous sysm o quaons. In ohr wors, h sysm s asympocally sal, h c o h mpulsv nos ns o amp own ar a cran ranson pro. s an alrnav, oscllang or vrgn havours can happn. Th homognous sysm can wrn as: δ s δ δ s δ δ s δ O cours hr s rval soluons,, corrsponng o h cas whn no mpulsv nos s appl o h sysm an hror h vhcl s p sll; hs soluons ar ovously ou o our nrs. Th rs consraon ha on can nr s ha h Carsan componns hav a sjon ynamcs; n h cas o mor compl mols hs s only parally vr, n h sns ha h wo horzonal componns, ar connc o ach ohr hn or ampl o h Corols acclraon whn h arh roaon s mol. Th scon consraon s ha h horzonal componns, hav a ynamcs ha s complly rn rom h on o h vrcal componn. Th soluon o h rs wo rnal quaons s a harmonc oscllaor a h Schulr rquncy, namly: δ δ sn, s whr δ s h mamum amplu o h oscllaon. Ths mans ha h nos c os no n o amp own h sysm s no asympocally sal, u s oun an hror s conrollal n m. Th sam os no happn or h hr rnal quaon ha scrs h vrcal moon, caus s soluon s an nvrs harmonc oscllaor: δ δ p s whr δ s h nal vrcal splacmn a m u o h nal nos mpuls. r a m nrval o n mnus, h rror s alray hr ms largr han h nal rror. Ths s h rason why h san-alon nral sysm can us or h horzonal navgaon, u no or h vrcal on, whr h suppor o ohr chnqus, such as GPS, s rqur. S MTL scrp or rror smulaons. lograa. Homann-Wllnho, K. Lga, M Wsr. Navgaon. Prncpls o posonng an guanc. Sprngr-Vrlag. F. Sansò 6. Navgazon goca rlvamno cnmaco. Polprss L. ag. ppun l corso Msur Goch. hp://gomaca.como.polm./cors/msur_goch/