Inertial Navigation. Academic Year 2008/09. Master of Science in Computer Engineering, Environmental and Land Planning Engineering

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1 Inrtal Navgaton camc Yar 8/9 Mastr o Scnc n Computr Engnrng, Envronmntal an Lan Plannng Engnrng

2 Inrtal navgaton Rrnc systms Inrtal snsors Navgaton quatons Error bugt

3 Psuo nrtal systm - orgn n th Earth barycntr; - as: ornt towars a clstal North Pol; - as: ntrscton btwn clptc an clstal quatoral plan; - as: to complt th rght-han tra.

4 Earth- systm - orgn n th Earth barycntr; - as: ornt towars a convntonal North Pol; - as: ntrscton btwn Grnwch mran plan an trrstral quatoral plan; - as: to complt th rght-han tra.

5 Navgaton systm - orgn n a gnrc pont P; - as: ornt n th East rcton; - as: ornt n th North rcton; - as: ornt as th normal to rrnc llpso, p rcton.

6 oy systm - orgn n a gnrc pont P; - as: ornt n th moton rcton; - as: prpncular to th vhcl plan an n th up rcton; - as: to complt th rght-han tra. yaw roll ptch

7 Inrtal navgaton systms Inrtal navgaton systm INS: - acclromtrs, - gyroscops, - th harwar collctng ata, - th sotwar or ral tm procssng. Each componnt s call IM Inrtal Masurmnt nt INS r. systm ornt as: - th navgaton systm by mans o srvomotors - th boy systm chapr call strap-own INS

Each componnt s call IM Inrtal Masurmnt nt INS r.

8 cclromtrs Th basc prncpl o acclromtrs s to masur th orcs actng on a proo mass. Two typs o acclromtrs: - opn loop.g. sprng bas acclromtrs masur th splacmnt o th proo mass rsultng rom trnal orcs actng on th snsor. - clos loop.g. pnulous or lctrostatc acclromtrs kp th proo mass n a stat o qulbrum by gnratng a orc that s oppost to th appl orc.

sprng bas acclromtrs masur th splacmnt o th proo mass rsultng rom trnal orcs actng on

9 Sprng acclromtrs Th systm ynamcs s scrb by: In cas o Fconst, th ollowng law hols: atr th ntal oscllatons hav cas m l kv l kl k l F F Knowng th proo mass, t s possbl to rv th acclraton a along th sprng as: F m a F m l

10 Pnulous acclromtrs M proo mass P hng S optcal snsor m magnt I currnt through th col trn orc to th pnulous F nuc orc to th magnt back systm

11 Elctrostatc acclromtrs cclromtrs wth scntc purposs an wth trmly hghr accuracs o th orr o - m/s. Mount on boar goc satllts CHMP, GRCE, GOCE or th masurmnt o th gravtatonal l.

12 cclromtr rror Th acclromtr rror can b mol as: δ a b λa c T T ν T whr b s a bas, λ s a scal actor, c T s a thrmal constant pnng on th tmpratur T an v s th masurmnt nos. Typcal valus or a commrcal nstrumnt can b: b 5 mgal λ 5 5 c T.5mGal / C σ v 4mGal / Hz namly about.6 Gal or ach obsrvaton, wth a samplng rquncy o about Hz

masurmnt nos. Typcal valus or a commrcal nstrumnt can b: b 5 mgal λ 5 5 c T.

13 Gyroscops Gyroscops or brly gyros masur th angular rat o th snsor rotaton wth rspct to an nrtal rrnc systm. Two man typs o gyroscops: - mchancal mor pnsv, sutabl or gmbal platorms - optcal chapr, mnaturzabl, only or strapown systms

Two man typs o gyroscops: - mchancal mor pnsv, sutabl or

14 Fbr Optc Gyroscops Thy ar bas on a th so-call Sagnac ct. - n obsrvr movng wth th br ss th lght to covr actly on rvoluton ϑ π - n nrtal obsrvr ss th lght to covr an angl ϑ π ϑ R ϑ π c L R R ϑ c π c I th sourc E mts anothr lght bam at th sam ntal tm t but covrng th col n th oppost rcton, thn ϑ π- ϑ. ϕ L L λ 4 λc Th masur o th phas sht allows to rv a masur o th angular rat.

lght to covr an angl ϑ π ϑ R ϑ π c L R R ϑ c π c I th sourc E mts anothr lght bam at th sam

15 Gyroscop rror Th acclromtr rror can b mol as: δ b λ c T T ν T whr b s a bas, λ s a scal actor, c T s a thrmal constant pnng on th tmpratur T an v s th masurmnt nos. Typcal valus or a commrcal nstrumnt can b: b / λ 6 ora 5 c T 5 / ora C 7 σ v 6 ra / s Hz namly about -5 ra/s or obsrvaton wth a samplng rquncy o Hz

16 Rotatons D: sn ' sn ' sn sn ' D: sn sn sn sn sn sn sn sn sn sn sn sn sn sn sn sn sn sn sn sn sn

17 Rotatons Consrng th nntsmal angls an nglctng th scon orr trms, w hav: I y nng: I t t t th tm rvatv o th rotaton matr s: I t t lm lm lm lm lm matr o angular vlocts

18 Rotatons y rcallng th rotaton matrcs proprts, t hols: I y computng th tm rvatv o ths prsson, t hols:

19 Navgaton quatons Navgaton quatons stablsh a lnk btwn th unknowns namly poston, vlocty an atttu o th vhcl an th obsrvatons o th acclromtrs an o th gyroscops an n cas o th GPS rcvrs. Two cass: - Navgaton quatons n an nrtal rrnc systm sutabl to scrb spac navgaton, or ampl th orbts o an artcal satllt - Navgaton quatons n an Earth- rrnc systm mor sutabl to scrb trrstral navgaton, somtms rqurng a urthr stp towars th local-lvl systm.

Two cass: - Navgaton quatons n an nrtal rrnc systm sutabl to scrb spac navgaton, or ampl th orbts o an

20 Navgaton quatons IRF In an nrtal rrnc systm Nwton s scon law o ynamcs can b wrttn as: F tot m stays or nrtal rrnc systm F mg g s th acclraton prouc by th gravtatonal l Knowng th mass m, th navgaton quatons rsult: g or quvalntly g

systm F mg g s th acclraton prouc by th gravtatonal l

21 Navgaton quatons IRF In strapown systms, th acclromtrs masur th spcc orc b n th boy rrnc systm. Thror: b b acclromtr masurmnts whr b s th rotaton matr rom boy to nrtal systm. Ths matr can b rv by ntgratng gyroscop masurmnts: b b b gyroscop masurmnts

22 Navgaton quatons IRF Eampl o rotaton aroun a sngl as o th rrnc systm: sn sn sn sn sn sn b b b sn sn sn sn τ τ t t t rqurng th knowlg o th ntal atttu!

23 Navgaton quatons ERF Th navgaton quatons n an Earth- rrnc systm can b rv rom Corols thorm: Corols acclraton cntrugal acclraton

24 Navgaton quatons ERF Rcallng that or Nwton s scon law : g g g Typcally gravtatonal acclraton an cntrugal acclraton ar group togthr to orm th gravty acclraton vctor: g g g b b b b b g Navgaton quatons rsult: or

25 Navgaton quatons ERF Navgaton quatons can b wrttn componnt by componnt: E E E E E Corols acclraton E cntrugal acclraton GM g GM g [ ] GM g / gravtatonal acclraton [ ] [ ] [ ] E E E E GM GM GM / / /

26 Error bugt Lt us consr a vry smpl scnaro whr th Earth s assum to b sphrcal an not rotatng, an th vhcl s movng n th surrounngs o th North Pol wth a mamum amtr o som klomtrs. GM g GM ν ν GM [ ] P I GM ν δ δ / P Th ynamcs can b splt nto two trms: lnarzng δ

27 Error bugt In th cas unr stuy R R R R R R R R R P thus obtanng: s s s ν δ δ ν δ δ ν δ δ s ra R GM s /.4 Schulr rquncy corrsponng to a pro T84.46 mnuts

28 Error bugt It s possbl to stuy how th systm ynamcs volvs as a rsult o an mpulsv nos by solvng th homognous systm o quatons: δ δ δ δ s δ s δ s Th soluton o th rst two quatons s a harmonc oscllator at th Schulr rquncy: δ t δ sn t k, k s th nos ct on th horzontal componnts os not tn to amp own but t s boun thror t s controllabl n tm. Th soluton o th last quatons s an nvrs harmonc oscllator: δ t δ p t s atr tn mnuts, th rror s alray tms largr than th ntal rror! stan-alon nrtal systm can b us or th horzontal navgaton, but not or th vrtcal on, whr th support o othr tchnqus, such as GPS, s rqur.

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