Ultra-Tight GPS/INS/PL Integration: Kalman Filter Performance Analysis

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1 Ultra-Tight GPS/INS/PL Intgration: Kalman Filtr Prformanc Analysis Ravindra Babu, Jinling Wang School of Survying and Spatial Information Systms Univrsity of Nw South Wals Abstract: Th smooth functioning of th intgration Kalman filtr is vital to th prformanc of an ultra-tight intgration systm, as th intgration filtr not only provids rror stimats of th inrtial snsor masurmnts but also provids a Dopplr fdback to th rcivr tracking loops to mitigat dynamics on GPS signals. Th I (inphas) and Q (quadratur) masurmnts from th corrlator which forms th input to th filtr ar highly non-linar, and thrfor nd to b linarisd during th masurmnt updat procss. From th drivations of I and Q signals, it can b sn that ths quadratur masurmnts ar rlatd to phas and frquncy rrors. Thrfor, linarisation of ths quadratur signals is carrid out with rspct to phas and frquncy rrors. Th prformanc of th Kalman filtr is primarily drivn by th quality of th modlling stratgis. Any mis-modlling, ithr du to lack of statistical knowldg of th signal charactristics or du to impropr assumptions, will caus th filtr to divrg. Givn th fact that th signals ar highly nonlinar, modlling plays an important rol in th dsign of th Kalman filtr. Th numbr of stats chosn for th filtr is a trad-off btwn optimal prformanc and computational complxity. Our analysis shows that a 17-stat filtr would suffic. Covarianc analysis is prformd to tst th prformanc of th filtr undr various oprating conditions. In this papr, mphasis is placd on masurmnt updat, which rlats th masurmnts and stats, as it is a uniqu charactristics compard with th loos and tight intgration mods. This papr also dscribs th prtinnt mathmatical rlationships that ar rquird to dvlop th masurmnt modl. Various trajctoris with diffrnt dynamics ar studid to valuat th modlling ffcts. Kywords. Ultra-tight Kalman filtr architctur, I, Q masurmnts, statsmasurmnts rlationship 1 Introduction Th complmntary advantags of Global Positioning Systm (GPS)/Psudolits (PL) and Inrtial Navigation Systm (INS) snsors ovrcoming ach othr s limitations hav bn th primary motivation for th intgration of ths systms. Sinc thir incption in th arly 1980 s, th GPS/INS intgratd systm has undrgon major changs in intgration architcturs, algorithms, ral-tim implmntations, hardwar/softwar, tc. Markt studis rval that th major rvnus for Satllit Basd Navigation Systms, such as GPS, GLONASS and th upcoming GALILEO, will b primarily from th commrcial sctor (Rizos, 2005). This is du to th prolifration of applications such as LBS (Location Basd Srvics) and Tlmatics. Though Ntwork Basd Positioning Tchniqus can augmnt Satllit Basd Systms, nvrthlss th infrastructur rquirmnts for this ar quit high. Thrfor low-cost INSs ar sn as an altrnativ which can augmntsatllit Basd Systms to provid robust positioning (Tittrton & Wston, 1997). In addition, th improvd prformanc of th prsntly usd GPS/INS architcturs in nonbnign nvironmnts hav popularisd thir us in both commrcial and dfnc applications. Considring GPS and INS as two indpndnt navigation systms and intgrating thir positions xtrnally in a Kalman filtr is dfind as th loosly-coupld mod. Two Kalman filtrs ar usd in a cascadd fashion in this typ of systm a navigation filtr insid th GPS rcivr, and th intgration Kalman filtr which combins both th GPS and INS outputs. Usually, th covarianc knowldg of th navigation filtr is not providd to th xtrnal intgration Kalman filtr. This lack of knowldg rsults in sub-optimal prformanc. Th othr disadvantag of such a systm is that as GPS is tratd as a navigation systm, a minimum of 4 satllits should b trackd by th rcivr to provid th coordinat/vlocity inputs to th intgration Kalman filtr. To improv upon this, th tight intgration mod was dvlopd,

2 which combins INS data with GPS psudorangs or carrir phass (Snnott, 1997; Snnott, 1999). Th biggst advantag of this systm is that it uss only on Kalman filtr, which improvs th systm prformanc gratly. Morovr, GPS is tratd as a snsor rathr than a closd navigation systm, and thrfor this systm can vn provid navigation outputs vn with lss than 4 trackd satllits (Brown & Hwang, 1997). But, as navigation systms ar incrasingly subjctd to non-bnign nvironmnts whr highr prformanc is rquird, dsignrs hav concivd of th ultra-tight intgration mod. In such a systm, th GPS masurmnts I (inphas) and Q (quadratur) from th GPS corrlator ar intgratd with th INS masurmnts (Alban t. al, 2003; Kry t. al, 2002; Poh t. al, 2000; Kim t. al, 2003). What mak this mod mor attractiv than th prvious two architcturs ar th manifold advantags it can provid, such as Jamming to Signal (J/S) ratio improvmnt, mitigating RF intrfrnc, improving GPS masurmnt accuracy, rducing non-cohrnt intgration priod in wak-signal GPS procssing and othrs. Ths advantags, in addition to th incrasing dmands of critical applications, hav mad such systm architctur attractiv. An ultra-tight systm drivs its bnfits primarily from th INS-drivd Dopplr fdback to th rcivr carrir tracking loops. This drivd Dopplr signal, which closly rflcts th Dopplr (causd du to rlativ motion btwn satllit and rcivr) on th GPS signals, whn intgratd with th tracking loop rmovs th Dopplr from th GPS signals, thrby facilitating a significant rduction in th carrir tracking bandwidth, i.. from about 12 to 18Hz to about 1 to 3Hz dpnding on th oscillator s accuracy. Unlik a stand-alon GPS rcivr whr individual channls ar controlld within th corrlator, in an ultra-tight intgratd systm th loops ar closd by th intgration Kalman filtr (Bsr t. al, 2003). A cntralisd Kalman filtr or a fdratd Kalman filtr structur can b adoptd for intgration. Though having a singl Kalman filtr rducs th modlling complxity, nvrthlss, th highr updat rquirmnts mak this unattractiv for raltim implmntation. Th masurmnt updat of th Kalman filtr rquirs a rlationship to b stablishd btwn th stats and masurmnts. Thrfor, in an ultra-tightly intgratd systm th rlationship btwn GPS masurmnts, I and Q, and INS data, position, vlocity and attitud, nds to b stablishd. It is not as straightforward as in th cas of loosly and tightly intgratd systms. This papr shows that thy ar rlatd through phas and frquncy rrors which ar xtractd from th tracking loops of th rcivr. Th implmntation of ths mathmatical rlationships in ral-tim in addition to two lvls of synchronisation, on at th intgration lvl and th othr at Dopplr synchronisation at th tracking loops, maks this systm complx. Normally, th Kalman filtr updat rat is 1 to 10Hz, whras in an ultra-tight systm as th masurmnts from th corrlator com at rats of 1000Hz th implmntation of Kalman filtr can b don in two ways run th Kalman filtr at th sam rat as th masurmnts which dmands high computational powr or dcimat th masurmnts to lowr rats. For our simulation xprimnts, a 17-stat Kalman filtr is chosn which runs at 100Hz. A Softwar GPS rcivr is usd to gnrat th I and Q masurmnts, and an INS Matlab toolbox is usd to gnrat th inrtial masurmnts. A dynamic trajctory is chosn to tst th prformanc of th ultra-tight Kalman filtr and th rsults show good prformanc. 2 Ultra-tight Intgration Masurmnts Th GPS corrlator procsss th digitisd IF signals to gnrat th 50Hz navigation data. Thr ar two loops that work in tandm to do this procss th carrir loop which locks on to th incoming carrir frquncy and masurs th apparnt Dopplr shift of th signal, ithr using a FLL or PLL or a combination of both, and th cod loop which normally is a DLL (that compars th incoming sprading cod with thr vrsions of a locally gnratd cod to dtrmin th corrlation pak, (Tsui, 2000; Ward, 1998). Th functions prformd by both ths loops ar gnrally rfrrd to as Carrir or Dopplr wip-off and cod wip-off. Mixing th incoming composit signal with th locally gnratd signals and intgrating thm ovr th pr-dtction intrval priod yilds th I and Q masurmnts. Ths masurmnts ar usd in th discriminator algorithms to gnrat th corrctions to th carrir and cod NCO s to align to th incoming signal.

3 Fig. 1 Corrlator Architctur Digitizd IF signal Carrir Mixrs Cod Mixrs I EPL,, Intgrat & Dump I EPL,, Carrir NCO E P L Cod NCO /Gnrator Q EPL,, Q EPL,, Discriminator Algorithms - FLL - PLL - DLL Cod/ carrir NCO Figur 1 shows th architctur of a typical corrlator usd in a GPS rcivr. For a dtaild xplanation th radr is rfrrd to (Kaplan, 1996). Th I and Q signals gnratd by th carrir mixrs ar convrtd to I E, PL, (in-phas arly, prompt and lat) and Q E, PL, (quadratur arly, prompt and lat) by th cod mixrs, which ar subsquntly intgratd ovr th pr-dtction intrval. Ths signals ar thn procssd by th various discriminator algorithms of th FLL, PLL and DLL to gnrat th cod and carrir NCO corrctions. 2.1 Kalman Filtr Masurmnts In a convntional rcivr, th function of th I and Q signals is to dtrmin th NCO corrctions, and to comput th signal powr 2 2 I + Q to dtrmin in which loop, i.. FLL, PLL, widband / narrowband DLL, th rcivr should oprat. But, in ultra-tight intgration ths masurmnts also calibrat th inrtial snsor rrors onc th rlationship with position, vlocity is stablishd. In th stady-stat condition, th masurmnts I P (in-phas arly) and Q P (quadratur arly) bcom th input to th intgration Kalman filtr. Lt ŵ and φˆ b th local stimats of th rcivr, k b th masurmnt poch, and T b th intgration intrval. As mntiond in th prvious sction, multiplying th local carrir stimat with th incoming signal and intgrating across th pr-dtction intrval yilds th quadratur signals: ( k T I + 1) = sin( wt ˆ + ˆ) φ + kt (1a) (1b) Expanding ths quations and avraging thm ovr th intgration priod yilds (Snnott, 1992): A E[ I] = cos( w( k + 1) T + cos( wkt + ) 2w [ A cos( w' t + φ' ) η] dt ( k = + 1) T Q cos( wt ˆ + ˆ) φ + kt [ A cos( w' t + φ' ) η] [ ϕ ϕ ] (2a) A EQ [ ] = [ sin( w( k+ 1) T+ ϕ sin( wkt + ϕ) ] 2w (2b) whr E [I] and E [Q] ar th xpctations of th I and Q masurmnts rspctivly, A is th amplitud of th signal, w is th frquncy rror btwn th masurd and stimatd signals, ϕ is th phas rror btwn th masurd and stimatd signals. Th I and Q signals of th four diffrnt PRN cods, 4, 6, 7, 10, which ar usd as input to th Kalman filtr, ar simulatd as shown in Figur 2. dt

4 Fig. 2 Kalman filtr masurmnts for Ultra-tight intgration (a) I (in-phas) signal from Corrlator (b) Q (quadratur) signal from Corrlator 2.1 Stats Masurmnt Rlationship Both th cod and carrir loops nd to b synchronisd with th incoming signal to produc th navigation data. A thrshold basd on th dynamics and signal-to-nois ratios is st to dtrmin if th loops ar lockd or not. Undr normal conditions th loops oprat in th stady-stat condition, indicating that th rrors gnratd by th loops ar within th limits. Howvr, undr high dynamic conditions, such as high acclrations and jrks, th loops may not b abl to track th incoming signal du to th transints (Cahn t. al, 1977; Jwo, 2001). In othr words, th loop filtr s bandwidth may not b sufficint to b abl to track th suddn changs in th Dopplr frquncy. To rduc th phas and frquncy rrors and maintain th loops in lock, gnrally, thr ar two options incras th tracking bandwidth (which will dgrad th raw masurmnt accuracy), or aid th tracking loops using xtrnal signals/masurmnts from an INS. INS-aiding is optimal in that it rducs th dynamic strss, and at th sam tim improving th accuracy of th masurmnts. In this intgration tchniqu, th phas and frquncy rrors ar th variabls that stablish th rlationship btwn th stats and masurmnts in th intgration Kalman filtr. In quation, w ˆ ' ˆ = w w andϕ = ϕ ϕ' ar th frquncis and phas rrors rspctivly. Ths rrors ar initially dtrmind from th apriori knowldg but vntually rach a stady-stat.

5 Fig. 3. Ultra-tight rcivr tracking loop Carrir tracking BW= 3 Hz Fig. 4. Stand-alon rcivr tracking loop Carrir tracking BW = 13Hz Figurs 3 and 4 show th frquncy and phas rrors for both th ultra-tight and stand-alon systms. It can b obsrvd that whil th ultra-tightly intgratd systm can track th signals with a bandwidth of 3Hz, th standalon rcivr could not vn track th signals with bandwidth of 13Hz, and th loops rmaind in th FLL mod and nvr ntrd th PLL mod. Ths frquncy and phas rrors, which ar th linking variabls btwn th corrlator masurmnts I and Q, and INS variabls, position, vlocity, ar dfind as: w w = V (3) c w ϕ = [ X V t ] (4) c whr w= 2π f is th angular frquncy, c is th vlocity of light, V is th vlocity rror vctor btwn th masurd and stimatd valus, and X is th position rror vctor btwn th masurd and stimatd valus. Undr stady-stat tracking conditions, i.. whn w, φ thrshold, th magnitud of I incrass and th magnitud of Q dcrass. Having dfind th rlationships btwn th I and Q, phas and frquncy rrors, position, vlocity, it is now straightforward to rlat ths using th following xprssions: { } di = { I, φ} dx + I, w dx dx {, φ } {, } dq = Q Q w dx + dx dx (5) (6) Equations (5) and (6) stablish th fundamntal rlationship btwn I, Q and P, V.

6 3 Ultra-tight Kalman filtr Architctur Th complmntary Kalman filtr structur for th ultra-tightly intgratd systm is shown in Figur 5 Fig. 5 Ultra-tight intgration Kalman filtr Error Estimats I 1, Q 1 Corrlator I n, Q n Complmntary Kalman Filtr I, Q Prdictor INS P, V, A I Prd, Q Prd As shown in Figur 5, th corrlator masurmnts ar procssd with th INS prdictd masurmnts in th intgration Kalman filtr to gnrat th inrtial rror stimats to corrct th raw INS data. Thr ar two updat stps in th Kalman filtr: masurmnt and stat updat. As th systm s apriori knowldg is usually unknown at th start of th procss, th man of th stat stimat x t0 and covarianc matrix P(0) is initialisd to zro, i.. E[ xt0 ] = 0 and T Ex [ t0xt0] = P(0) ar assumd. P (0) is a diagonal matrix corrsponding to th rror stat variancs of ach stat. Th dynamics of any linar tim invariant systm can b givn as: x = Ax + ε Procss Modl (7) z = Hx + v Masurmnt Modl (8) Whr x is th stat vctor, A is th systm matrix, H is th masurmnt matrix, z is th masurmnt vctor, and ε and v rfr to th procss and masurmnt noiss rspctivly. A 17-stat Kalman filtr is chosn for our studis. x() t = { dx, dy, dz, dx, dy, dz, ψ, ψ, ψ, x y z T x y z x y z b d a, a, a, g, g, g, c, c } (9) Th 17 stats ar: 3 inrtial rror stats ach in position, vlocity, attitud, acclromtr bias, gyro bias, 1 stat ach for clock bias and drift. Th masurmnts prsntd to th complimntary filtr ar: z = {INS prdictd masurmnts} {GPS masurmnts} { η, η } = 1: z= di + dq+ { prd, prd } i= 1: n { I η, Q η } z = I + di Q + dq (10) (11) whr di, dq ar th dviations in th INS prdictd I prd and Q prd masurmnts causd by th inrtial snsor rrors, and η I, ηq ar th quadratur nois componnts in th GPS I GPS and Q GPS masurmnts rspctivly. Suffix i in quation (11) rprsnts th numbr of channls trackd. Th prdictd INS masurmnts ar drivd from th knowldg of th currnt INS position and vlocity. Th quations prtaining to this wr introducd in th prvious sctions. 3.1 Procss Modl I Q i n GPS I GPS Q i = 1: n To start th Kalman filtr, th procss and masurmnt modls nd to b dfind. In quation (7), th dynamics matrix A is dfind using trrstrial psi-angl rror modl (Bar- Itzhack, 1988; Wang t al., 2001): dr = ρ * dr + dv dv = ( Ω * w)* dv + ψ * f dψ = w* ψ + ε (12) whr dr is th position rror vctor, dv is th vlocity rror vctor, d ψ is th attitud rror vctor, ρ is th tru fram rat with rspct to th Earth, Ω is th Earth rat vctor,

7 w is th tru coordinat systm angular rat with rspct to th inrtial fram, is th acclromtr rror vctor, f is th spcific forc vctor, and ε is th gyro drift rat vctor. 3.2 Masurmnt Modl Th masurmnt modl H is givn as (Babu & Wang, 2005): H = [{ hxi, hyi, hzi,1},{ hxi, hyi, h zi,1}] i= 1: n (13) whr n is th numbr of satllits visibl and and h h x1 x1 EI [ ] φ EI [ ] w = + φ x w x EQ [ ] φ EQ [ ] w = + φ x w x Th H matrix is updatd during th masurmnt updat procss. Th product of th prdictd stats and th H matrix ar thn diffrncd with th GPS masurmnts, and wightd by th Kalman gain to gnrat th inrtial rror stimats. 4 Simulation Exprimnts Simulation xprimnts wr prformd to tst th prformanc of th filtr. From a constant vlocity trajctory of 100m/s, th GPS and INS masurmnts wr xtractd and fd into th Kalman filtr. Th updat rat of th filtr is 10Hz. Using a Matlab-basd Softwar Rcivr, th I and Q signals for 4 channls, PRN 4, 6, 7, 10, as shown in Figur 2, wr drivd. A INS Matlab toolbox gnratd th inrtial snsor masurmnts. Th U-D covarianc factorisation algorithm was usd to nsur th positiv-dfinitnss of th covarianc matrix P. To rduc th computational burdn, th scalar updat mthod was followd. Th Kalman filtr was run for 1000 sconds and th rror stimats btwn th rfrnc and th stimatd trajctoris ar plottd in Figurs 6 and 7. Fig. 6 Attitud Error Estimats As Figur 6 shows, thr is a bias in th position stimats which may b du to synchronisation and modlling rrors. Howvr, th bias is boundd, which is critical. for th stability of th filtr. Th attitud rrors also show consistncy. Figurs 8, 9 & 10 show th accuracis for th gyro bias, th acclromtr bias and position stimats.

8 Fig. 7 Position Error Estimats Fig. 8 Covarianc of Gyro bias Fig. 9 Covarianc of Acc. bias

9 Fig. 10 Covarianc of position 5 Concluding Rmarks On of th complxitis of an ultra-tight intgration systm is th Kalman filtr implmntation. Unlik loosly and tightly intgratd systms, th mathmatical rlationships that rlat th stats and masurmnts ar complx. Th stats and masurmnts ar rlatd through th phas and frquncy rrors xtractd from th rcivr tracking loops. This papr dscribs ths rlationships, which ar critical to th undrstanding of th systm. Th tracking prformanc of a stand-alon and ultra-tight GPS rcivr illustrats th advantags of th ultra-tight systm undr dynamic conditions. It is shown that th ultra-tight intgration rcivr with a carrir tracking bandwidth of 3Hz oprats in th PLL mod vn undr high dynamic conditions. Th rror stimats and th covarianc analysis hav shown that th systm s prformanc is ncouraging. Howvr, ths ar only prliminary rsults and optimisation is still rquird in th modlling of th various paramtrs of th filtr. With rapidly volving low-cost inrtial snsors and low-cost GPS rcivrs, ultratightly intgratd systms may b poisd to captur commrcial and dfnc markts. Acknowldgmnts This rsarch is supportd by an ARC (Australian Rsarch Council) Discovry Rsarch Projct on Robust Positioning on Ultra-tight intgration of GPS, Psudolits and inrtial snsors. Rfrncs Alban, S., Akos, D., Rock, S., & Gbr-Egziobhr, D., (2003) Prformanc Analysis and Architcturs for INS-Aidd GPS tracking loops. Institut of Navigation NTM, Anahim, CA, January, Babu, R. & Wang, J., (2005) Ultra-tight GPS/INS/PL Intgration: A Systm Concpt and Prformanc Analysis. Submittd to GPS Solutions. Bar-Itzhack, I.Y., & Brman, N., (1988) Control Thortic Approach to Inrtial Navigation Systms. AIAA Journal of Guidanc, Control & Dynamics, 11, Bsr, J., Alxandr, S., Cran, R., Rounds, S., Wyman, J., & Badr, B., (2002) Trunav TM : A Low- Cost Guidanc/Navigation Unit Intgrating a SAASM-basd GPS and MEMS IMU in a Dply Coupld Mchanization. 15 th Int. Tch. Mting of th Satllit Division of th U.S. Inst. of Navigation, Portland, Organ, Sptmbr, Brown, R.G., & Hwang, P.Y.C., (1997) Introduction to Random Signals and Kalman Filtring. 3 rd dition, John Wily & Sons, NY. Cahn, R.C., Limr, D.K., Marsh, C.L., Huntowski, F.J., Laru, G.D., (1977) Softwar Implmntation of a PN Sprad Spctrum Rcivr to Accommodat Dynamics. IEEE Transactions on communications, 25 (8) pp Cox, D.B., (1982) Intgration of GPS with Inrtial Navigation Systms. Navigation, Journal of th Institut of Navigation, 1, Chakravarthy, V., Tsui, J.B.Y., & Lin, D.M., (2001) Softwar GPS Rcivr. GPS solutions, 5, Irsiglr, M., & Eissfllr, B., (2002) PLL Tracking Prformanc in th Prsnc of Oscillator Phas Nois. GPS Solutions, 5,

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