BUILT-IN SELECTION OF THE BEST ADAPTATION MECHANISM FOR INS ERROR MODEL IDENTIFICATION

Transcription

1 European Congress on Compuaional Mehods in Applied Sciences and Engineering ECCOMAS 2004 P. Neiaanmäki, T. Rossi, S. Koroov, E. Oñae, J. Périaux, and D. Knörzer (eds.) Jyväskylä, July 2004 BUILT-IN SELECTION OF THE BEST ADAPTATION MECHANISM FOR INS ERROR MODEL IDENTIFICATION Innokeni Semoushin, Andrew Yurjev and Arem Nikonorov Ulyanovsk Sae Universiy 42 L Tolsoy Sree, Ulyanovsk, Russia SemushinIV@ulsu.ru, web page: hp://saff.ulsu.ru/semoushin Samara Sae Aerospace Universiy 34 Moscow Road, Samara, Russia, fascode@fromru.com Key words: Sysem Faul Deecion, Kalman Filering, Sochasic Conrol. Absrac. The paper is devoed o he exension o avionics of he main ideas developed in deecion and idenificaion heory for creaing design ools ha are equipped wih a buil-in means of selecion of adapaion mechanism for he Inerial Navigaion Sysem (INS) error model idenificaion. I is anicipaed ha such ools will enable o design he novel ypes of INS which will ouperform comparable sysems for accuracy and reliabiliy hus affording a quanum leap forward in fligh safey. The ulimae goal of he paper is o see how he oal INS error can be divided up so ha conribuions from he separae error sources can be evaluaed. The seps for reaching his goal are he following. Firs, we formulae a mahemaically racable exended INS error model ha would include all he separae error sources conribuing o he resuling INS readous. Second, we consruc a real daa model o replace he real sysem for simulaed ess. Third, we inroduce a simplified INS error model on which he exended numerically sable Kalman filer is o be based. Fourh, we deermine he way o make his filer insensiive o uncerainy inheren o he INS error model by selecing he bes adapaion mechanism. Fifh, he sofware (named Adapive Sysem Parameer Idenificaion) implemening he above soluions, is developed and verified hrough he execuion of a wide range of compuaional experimens. Finally, we address he efficiency issue of he ASPID in design pracice. 1

2 1 INTRODUCTION 1.1 Relevance of he opic for he area of CE Today, o creae aircraf sysems ha will mee higher safey sandards for lower coss means a susained evoluion of design -cieolsson. Changes in design could no have been possible wihou process ools like Concurren Engineering (CE). CE, as i sands oday [2] and [3], is shaping up as an inegraed, highly cooperaive, eam-oriened, compuersuppored ool communicaing on he nework for oal produc realizaion, incorporaing no only he design phase, bu he operaing (real-life) phase, oo. Tradiionally, he design of an effecive avionics navigaion sysem enails a complex ieraive process of proposing alernaive designs hrough physical insighs, uning each, and rading off performance capabiliies and coss. The process is saggered in he following racks: (1) Selec and design he sysem componens. In doing so, Inerial Navigaion Sysems (INS) are ofen chosen o be he cenral componen. The design of INS involves arranging he seleced pars (acceleromeers, gyros, orquer moors, and gimbals) ogeher wih elecronics, in he form of a single measuremen uni. Because of characerisics and he very physical naure, every par is a poenial source of errors in INS readous, so he following rack is bound o occur. (2) Propose opimal aiding INS by oher navigaion sensors. These exernal daa sources, EDS, come in hree ypes: Posiion DS, Velociy DS and Aliude DS. Typical posiion daa sources are radars, radio-navigaion aids, navigaion saellies, sar sighings and landmarks, radiomeric picure of errain, and laser ranging. Doppler radar and airspeed sensors are velociy daa sources. Alimeers may be baromeric, radar or laser. (3) Generae a Kalman filer implemenaion in conjuncion wih he INS and exernal aids. (4) Creae a specimen copy (or pilo specimen) of INS and make i run he bench-es. (5) Analyze he bench-es resuls and ierae he above racks o aain he desired performance indices. (6) Finalize he specimen copy of INS and give i a fligh rial. (7) Analyze he fligh rial resuls and refine upon he arifac. To shoren he design erm and minimize coss, engineers seek o make he racks as concurren as possible. Even if he racks have saggered saring poins, each should have pars running in parallel. Simulaion and modelling are useful ools in aaining he aims. 1.2 Relevance of he opic for he area of RSS Noe ha wihin he design process, once a paricular design of INS has been proposed, an error budge should be esablished o elici he navigaion errors due o an every single source of error inheren o INS. This informaion provides an effecive means of improving he design and conducing a radeoff analysis among he various proposed designs. For example, if he error budge indicaes navigaion errors dominaed by gyro drif rae and acceleromeer non-lineariy characerisics, he mos effecive hardware improvemen would be an INS wih a lower drif rae gyro and a beer precision acceleromeer. 2

3 To esablish he error budge, a special-purpose model-based sofware should be developed and simulaion ess mus be conduced a he early sages of design. The use of his ool would help no only o find he unfailing way of sysem design bu also o deermine some special he mos effecive for error idenificaion pahs for fuure rial flighs and, by doing so, o minimize coss. This CAD ool would also be useful in early developing algorihms of aiding and correcing INS for fuure real-life operaing. However, he more deailed knowledge of navigaion sysem errors is needed, he more sophisicaed CAD ools mus be a engineers s disposal a each designing area: flighor bench- or simulaion-es area. While developing such a complex CAD ool for he aeronauics indusry, design engineers encouner a number of mahemaical and pracical problems. One of he problems addressing o uncerainy is selecion of he bes adapaion approach and, correspondingly, he bes adapaion mechanism which could be buil in he anicipaed CAD ool. The maer is ha here exis many adapive filering and idenificaion mehods, some of hem are rigorously subsaniaed, ohers are only heurisically se up, ha could be employed, bu someimes i is very difficul o selec and recommend he bes one. Since convenional numerical mehods are no capable of coping wih his problem, he more flexible mehods of Replicaion-Selecion Sysems (RSS) and geneic algorihms (GA) may be used. The paper describes a se of INS models, gives a generalized overview of adapive model idenificaion mehods, and presens he simulaion-based buil-in selecion of he bes way o confer adapaion properies on he filer 2 CONCEPTUAL MODELS 2.1 Exended INS error model, EINSEM This model includes 15 consan values o be esimaed as facors of separae error sources, as well as 15 variable values of 9 error sae variables and 6 random inpus modeled as firs order Gauss-Markov processes [4], [5] and [6]. The fifeen consan values are classified ino 5 groups of 3 each, aken along he axes x, y, z of a gyro-sabled plaform, as follows. 1. n Gx, n Gy, n Gz he gyro consan drif rae. 2. K Ax, K Ay, K Az non-lineariy facors of acceleromeer scaling coefficiens. 3. K Gx, K Gy, K Gz he gyro characerisic firs-order non-lineariy facors due o nonsymmeric cener of mass posiion. 4. l Gx, l Gy, l Gz he gyro characerisic second-order non-lineariy facors due o nonequal gimbal rigidiy along he GSP axes. 5. K DMx, K DMy, K DMz he acuaor (gyro moor) characerisic non-lineariy facors along he axes. 3

4 These values are referred o as parameers and inroduced here so ha design engineers could esimae any offending componen conribuing o he oal INS error. The nine error sae variables are described by he following sochasic differenial equaions. Errors in indicaed posiion Errors in indicaed velociy ϕ = ν x /r, λ = ν y /r, h = ν z ν x = f zβ + f y δ + m Ax + f x K Ax ν y = f zα f x δ + m Ay + f y K Ay ν z = f y α + f x β + m Az + f zk Az The wo angular errors (α, β) in he indicaed verical (α being he angular deflecion of he verical in he eas/wes direcion and β being he angular deflecion of he verical in he norh/souh direcion) α = r 1 ν y + ω z β ω y δ + m Gx + n Gx + f x K Gx + f x f zl Gx + (ω x ω x0 )K DMx β = r 1 ν x ω z α + ω x δ + m Gy + n Gy + f y K Gy + f y f zl Gy + (ω y ω y0 )K DMy The angular error δ in he indicaed azimuh (azimuhal deflecion) δ = ϕω cos ϕ + ω y α ω x β + m Gz + n Gz + f zk Gz + f y f zl Gz + (ω z ω z0 )K DMz In he above equaions, r is he Earh s large half-axis; Ω is he Earh s angular velociy; g is he graviy acceleraion; f x, f y, f z are he projecions of he vehicle acceleraion on he plaform axes and f z = f z g; ω x, ω y, ω z are he projecions of Ω on he plaform axes; ω x0, ω y0, ω z0 are he iniial values of ω x, ω y, ω z (a ime = 0 ); and ϕ is he laiude. The six random inpus wih variances σi 2 and correlaion inervals γ 1 i are assumed o be muually independen and modeled by he equaions m i + γ i m i = σ i 2γi w i ; i = Ax, Ay, Az, Gx, Gy, Gz where w i are muually independen sandard whie Gaussian noises. I is worh of noing ha if he variances and correlaion inervals are assumed o be known, he whole sysem of equaions is linear in he unknown parameers and he maximum dimension of i is 30. However, if hese values are unknown and are ye o be esimaed, he sysem becomes non-linear and he dimension increases o Real daa mahemaical model, RDMM While conducing he simulaed ess, i is necessary o have an acual daa mahemaical model o replace he above real sysem. The model mus include: 1. A vehicle moion model ha would generae he geographical componens of vehicle velociy, vn, v E and v H, for a kinemaical model of INS. 4

5 Table 1: Sage 1 Takeoff (iniial heading =45 ) Phase Iniial poin (s) Final poin (s) Duraion (s) Velociy (m/s) Saring Pause Takeoff Roll Pull-up Climbing Speeding-up Giving a Level Fligh Level Fligh Acceleraion Sraigh-Line Fligh The Kinemaical INS Model ha would generae f x, f y, f z and ω x, ω y, ω z and ϕ (laiude) along wih λ (longiude) and χ (he bearing of he GSP axis) for an INS error model. 3. The INS Error Model. INSEM is required o be of desired (pre-seleced) composiion/dimension. An easy possibiliy mus be pu a engineer s disposal o formulae his model on he basis of he exended INS error model (see Secion 2.1) by including or excluding a selecion of parameers and/or variables a his own will. 2.3 Vehicle Moion Model, VMM We simulaed wo sages of fligh named by convenion as Sage 1 Takeoff and Sage 2 Manoeuvring. Sage 1 is he sequence of 9 phases named as follows: (1) Saring Pause, (2) Takeoff Roll, (3) Pull-up, (4) Climbing, (5) Speeding-up, (6) Giving a Level Fligh, (7) Level Fligh, (8) Acceleraion, and (9) Sraigh-Line Fligh (Table 1). Sage 2 ha is defined as a sequence of fligh legs, each leg beginning wih a urn and ending wih a sraigh-line fligh, follows his sage. Each urn can be a simple one (righ, lef, up, down) or a composie one (righ-and-up, righ-and-down, lef-and up, lef-anddown). The possible fligh legs are no limied in number or in he numerical values of parameers of each leg so ha a vehicle-manoeuvring program can be readily prepared. The numerical values of 9 phases a Sage 1 can be varied o sui he invesigaor, as well. The vehicle moion model is consruced in such a manner ha helps o deermine he manoeuvring program ha is mos favorable for separae INS error idenificaion and hen o recommend his program for a rial-run (fligh) es. Examples of such program are shown in Tables 2, 3 and 4. 5

6 Table 2: Sage 2 Manoeuvring (varian 1) Phase Iniial poin (s) Final poin (s) Duraion (s) Righ Turn hrough Sraigh-Line Fligh Table 3: Sage 2 Manoeuvring (varian 2) Phase Iniial poin (s) Final poin (s) Duraion (s) Righ Turn hrough Sraigh-Line Fligh Lef Turn hrough Up Turn hrough Down Turn hrough Sraigh-Line Fligh Lef Turn hrough Up Turn hrough Down Turn hrough Sraigh-Line Fligh Table 4: Sage 2 (var. 2): The sequence of fligh legs Iniial heading Final heading Iniial pich Final pich Duraion of he sraigh-line fligh, s

7 2.4 Kinemaical INS Model, KINSM While he vehicle is in moion, he rihedral xyz ied o he GSP is moving relaive o he rihedral ied o he Earh. This is described by he well-known Euler angles expressed via he following values [4], [5] or [6]: b 11 = cos χ cos ϕ, b 13 = cos χ sin ϕ cos λ sin χ sin λ, b 22 = sin χ sin ϕ sin λ + cos χ cos λ, b 31 = sin ϕ, b 12 = cos χ sin ϕ sin λ sin χ cos λ b 21 = sin χ cos ϕ b 23 = cos χ sin λ sin χ sin ϕ cos λ b 32 = cos ϕ sin λ b 33 = cos ϕ cos λ The whole algorihm of he KINSEM is as follows. (1) Solve he differenial equaions for b ij : ḃ 11 = u z b 21 u y b 31, ḃ 12 = u z b 22 u y b 32, ḃ 13 = u z b 23 u y b 33 ḃ 21 = u z b 11 + u x b 31, ḃ 22 = u z b 12 + u x b 32, ḃ 23 = u z b 13 + u x b 33 ḃ 31 = u y b 11 u x b 21, ḃ 32 = u y b 12 u x b 22. ḃ 33 = u y b 13 u x b 23 (2) Compue χ = arcan(b 21 /b 11 ), ϕ = arcan(b 31 / b b 2 21), λ = arcan(b 32 /b 33 ) (3) Solve he differenial equaion for h (aliude): ḣ = v z (4) Compue he vehicle geographical angular velociies: u N = ve/(r + h), u E = vn/(r + h), u H = 0 (5) Compue he vehicle angular velociies relaive o he GSP axes: u x = u E sin χ + u N cos χ, u y = u E cos χ + u N sin χ u z = u H (6) Compue he vehicle velociies relaive o he GSP axes: vx = ve sin χ + vn cos χ, v x = vx + 2(R r)[(b 2 31/2 b 2 11)u y + u x b 11 b 21 ] vy = ve cos χ + vn sin χ, v y = vy + 2(R r)[(b 2 21 b 2 31/2)u x u y b 11 b 21 ] v z = vh 7

8 where r is he Earh s small half-axis. (7) Compue Ω in projecions on he GSP axes: Ω x = Ωb 11, Ω y = Ωb 21, Ω z = Ωb 31 (8) Compue he vehicle angular velociies relaive o he GSP axes: ω x = Ω x + u x, ω y = Ω y + u y, ω z = Ω z (9) Compue he vehicle seeming acceleraion relaive o he GSP axes: 2.5 INS error model, INSEM f x = v x + (2Ω y + u y )v z (2Ω z + u z )v y f y = v y + (2Ω z + u z )v x (2Ω x + u x )v z f z = v z + (2Ω x + u x )v y (2Ω y + u y )v x + g The INS Error Model is required o be of desired (pre-seleced) composiion/dimension. An easy possibiliy mus be pu a engineer s disposal o formulae his model on he basis of he exended INS error model (see Secion 2.1) by including or excluding a selecion of parameers and/or variables a his own will. The INSEM is so consruced ha i is capable o: provide flexible means for choosing characerisics, allow for adjusmen of he model complexiy, include modeling of oal sysem error measuremens and hus sui differen users invesigaing sysem parameer idenificaion mehods in his applicaion. In he full dimension, he INSEM is defined by he following equaions x +1 = Φx + Ψa + Ξb + Γc c +1 = Λc + w, w N(0, I) wih a +1 = a, b +1 = b where he las expression, w N(0, I), means ha he random vecor w is aken from he Gaussian disribuion wih he mean equal 0 (zero) and covariance equal I (ideniy marix). The remaining vecors are defined by x = ( ϕ, λ, h, ν x, ν y, ν z, α, β, δ) T a = (n 4, n 5, n 6, K 1, K 2, K 3, K 4, K 5, K 6 ) T b = (l 4, l 5, l 6, K 7, K 8, K 9 ) T c = (c i ) T, i = 1,..., 6; c i = m i /(σ i 2γi τ) Numerical indexing here and symbolical indexing of he same values in Subsecion 2.1 are equivalen o each oher, i.e. 1 Ax, 2 Ay, 3 Az, 4 Gx, 5 Gy, 6 Gz, 7 DMx, 8 DMy, and 9 DMz. Parameer τ is he sampling period when a digial compuer performs compuaions using daa samples from he coninuous-ime dynamic sysem described in Secion 2.1, or from he mahemaical model of i being described here. 8

9 The sae x ransiion marix Φ, he parameer a inpu marix Ψ and parameer b inpu marix Ξ have he form: Φ Ψ Ξ I Φ I Φ 23 0 B Φ 31 Φ 32 Φ 33 A 0 B C D where 3 3 marices Φ 12, Φ 23, Φ 32, Φ 33 are Φ 12 Φ 23 φ φ τ 0 φ 4 φ 3 φ 4 0 φ 2 φ 3 φ 2 0 Φ 32 Φ 33 0 φ 1 0 φ φ 8 φ 7 φ 8 1 φ 6 φ 7 φ 6 1 φ 1 = τ/r, φ 2 = τf x, φ 3 = τf y, φ 4 = τf z, φ 5 = τωcos φ, φ 6 = τω x, φ 7 = τω y, φ 8 = τω z, in Φ 31 is only one non-zero elemen φ 5, and A, B, C and D are diagonal: A = diag {τ, τ, τ} B = diag {τf x, τf y, τf z} C = diag {τf x f z, τf y f z, τf y f z} D = diag {τ(ω x ω x0 ), τ(ω y ω y0 ), τ(ω z ω z0 )} and 0 is 3 3 zero marix, and I is he uni marix. The random process c inpu marix Γ = [γ ij ] is defined by is enries: { τσj 2γj τ, i = j + 3 γ ij = 0, oherwise The random process c ransiion marix Λ is diagonal: Sensor equaions are: Λ = diag {(1 τγ i )}, i = 1,..., 6 z = Hx + v, v N(0, R ) R = diag {r (k) }, k = 1,..., m wih he measuremen marix H = [h ij ] defined by: 1, i = j & i = 1, 2, 3 (case 1) h ij = 1, i = j & i = 1,..., 6 (case 2) 0, oherwise 9

10 for one of wo cases: 1 or 2. In case 1, he measuring process for he sysem is represened by 3 indicaed posiion errors (m = 3). In case 2, i includes, addiionally, 3 errors in indicaed velociy (m = 6). 2.6 Filer INS error model, FINSEM I is clear ha o esimae parameers a and/or b of RDMM (and, correspondingly, EINSEM), he powerful echniques of opimal and adapive esimaion heory should be used. A he same ime, i is necessary o operae on he fac ha he filer-esimaor remains linear unil he random inpu characerisics, variances σi 2 and correlaion inervals, are known. Bu if a leas one of hem is considered unknown and is ye o be esimaed, he filer becomes non-linear and he discree algorihms of invarian imbedding are advisable. In eiher case, he algorihm has he form of exended filer since he filer sae vecor is augmened wih he esimaes of unknown parameers. The oher requiremen is ha he INSEM used as a basis for filer consrucion (we call i FINSEM) mus have he possibiliy o be simplified o one exen or anoher agains he EINSEM (Secion 2.1) and RDMM (Secion 2.5). For insance, random inpus may be considered in FINSEM consrucion as whie noises of unknown variances no o be esimaed. Thus he filer sae vecor can be formed wihou blind copying of he RDMM sae vecor. For example, FINSEM of dimension n F = 18 is described by he filer augmened sae vecor y T = (x T a T ) subjec o he following equaions: γ 1 i y +1 = Φ a y + Γ a w, w N(0, I) Φ a [ Φ Ψ 0 I 9 9 ] The filer operaion needs wo consequen seps. (i) Time propagaion ( = 0, 1,...): Γ a diag {τσ i }, i = 1,..., ŷ +1 = Φ a ŷ +, P +1 = Φ a P + Φ T a + Γ a Γ T a + Gq G T where G = diag {g i }, i = 1,..., 18 is a pre-seleced marix and q is a covariance of a ficiious noise [7]. To ensure numerical sabiliy, square roo mechanizaion for error covariance marices P = S (S ) T, P + = S + (S + ) T is used: [ ] [ ] (S +1 ) T (S + = T ) T Φ T a 0 B T, B = diag {b i } 10

11 where S and S + are low riangular marices and T denoes modified Gram-Schmid riangularizaion. The diagonal enries of B a ime are chosen o be: [ ] gi q, i = 1, 2, 3, 10,..., 18 {b i } = τ 2 σ 2 i 3 + g2 i q, i = 4,..., 9 (ii) Scalar measuremen updae ( = 1, 2,...). 1. Se iniial values: ŷ (0) = ŷ, Ŝ (0) = Ŝ, δ (0) = 0 2. For k = 1, 2,..., m where m is a dimension of he measuremen vecor z, h (k) is he k-h row of marix H and z (k) is he k-h elemen of z, compue: f (k) = S (k 1) (h (k) ) T α (k) = 1/[(f (k) ) T f (k) γ (k) = 1/[1 + α (k) K (k) 3. Obain resuls of Sep (ii): = S (k 1) f (k) α (k) S (k) = S (k 1) γ (k) + r (k) ] r (k) ] K (k) (f (k) ) T ν (k) = z (k) (h (k) ) T y (k 1) ŷ (k) = ŷ (k 1) δ (k) = δ (k 1) + K (k) ν (k) + (ν (k) ) 2 α (k) δ = (1/m)δ (m) 1, ŷ + = ŷ (m), S + = S (m) The new wha has been made in his filer is an adapive mechanism o deermine value q in Sep (i). Adapive filer mechanism Compue wo values δ = (1/) j=1 α j δ j, s = m/(2) j=1 α j δ j where α (0, 1), and aferwards obain q according o one of he following 15 formulae where N AD sands for Number of adapaion formula : 11

12 NAD = 1 : NAD = 2 : NAD = 3 : NAD = 4 : NAD = 5 : NAD = 6 : NAD = 7 : NAD = 8 : NAD = 9 : NAD = 10 : NAD = 11 : NAD = 12 : NAD = 13 : NAD = 14 : NAD = 15 : q = γ δ τ q = γ δ { γ δτ if s q = τ η { q 1 oherwise γ δ if s q = τ η q 1 oherwise q = γ s τ q = γ s q = α q τ + γ( δ τ δ τ T ) q = α q τ + γ δ τ sign ( δ τ δ τ T ) q = α q 1 + γ( δ δ 1 ) q = α q τ + γ δ τ sign ( s τ s τ T ) q = α q 1 + γ δ sign ( s s 1 ) q = α q τ + γ( s τ s τ T ) q = α q 1 + γ δ sign ( δ δ 1 ) q = α q 1 + γ( s s 1 ) q = q 1 ( no adapaion) Some parameers of hese formulae have o be chosen experimenally. We have chosen α = 0.99, γ = 0.1, τ = ( mod T ), T = 500, and η = 1 The bes formula also has o be seleced. We do his by using a simple decision generaor considered in [8] and by execuing a wide range of compuaional experimens wih his applicaion problem. 3 SOFTWARE AND COMPUTATIONAL EXPERIMENTS We have developed a special CAD sofware and conduced many differen simulaion ess wih i. Sofware is buil in modular fashion and has he following main feaures. Purpose: An inegraed sofware package for compuer aided design and performance analysis of INS under consrucion. Tile: Adapive Sysem Parameer Idenificaion - ASPID. Composiion: 33 separae modular programs. Size: Toaling 3942 FORTRAN lines. Modes of operaion:1. Simulaed es (1 basic mode and 8 supplemenary checking modes). 2. Trial-run es (1 basic and 3 supplemenary modes). 3. Reconciliaion es (4 modes). Remark: Modes 2 and 3 are under developmen. Some experimenal resuls are shown in Fig. 1. The figures show he percen errors in esimaes of parameers. They enable a comparison beween algorihms N AD = 1 o N AD = 15 of inroducing he ficiious noise ino adapive filer-esimaor, including 12

13 14 adapive algorihms and 1 non-adapive. All 15 experimens were made for n F = 18 and Manoeuvring consising of one urn. Random inpus m i in RDMM were modeled as muually independen Gauss-Markov processes. However, hey were assumed o be whie Gaussian noises wihin he FINSEM. The experimens were direced owards he selecion of he bes algorihm among he proposed ones. I has been found ha he mos promising mehods are NAD = and NAD = 9. There exis mehods close o each oher: similar resuls appeared for four pairs of cases, namely, NAD = 2 works like NAD = 9, NAD = 3 works like NAD = 14 (hey are shown in Fig. 1, middle); NAD = 5 works like NAD = 12 and alhough (5) works like (12), and NAD = 8 works like NAD = 10, corresponding figures are omied as showing he wors convergence. 4 CONCLUSIONS Idenificaion of he exended INS error model is invesigaed as an applied adapive filering problem allowing o deec and esimae a variey of consrucional sysem flaws. The main problem solved in his work is he answer o he quesion: Wha adapaion formula should be recognized as he bes one for INS error sources idenificaion a he design sage? Two approaches o adapive filering sae augmening and characerisic maching have been seleced o be used ogeher, he laer wih a ficiious noise inpu. An exensive program for compuaional experimens wih he sofware is implemened. Foureen heurisically formulaed adapaion formulas are examined for adjusing he roo mean square (RMS) of he ficiious noise inroduced ino he filer o impar some adapiviy properies o i, and he bes mehod is seleced and recommended for use in acual experimenal condiions. The works conduced in his paper allows us o draw he following key conclusions: 1. The mos suiable way o adapively esimae he INS errors is he combinaion of he Exended Model Approach and he Covariance Maching Approach, he laer using he ficiious noise of covariance q. 2. The mos efficien way o une he ficiious noise RMS value q o opimaliy for he exended filer-esimaor is defined by wo formulae: (a) δ = α δ 1 + (δ α δ 1 ), α 0.98 (b) q = γ δ, γ 0.1 where he non-opimaliy index δ is compued in he filer (Secion 2.6) and δ is he exponenially smoohed version of δ wih he coefficien α. 3. The vehicle rajecory has proven o have a profound impac in idenificaion, however he idea of parly freezing he esimaes ha have become good enough does no work. 13

14 120 NAD=1 120 NAD=2 100 NGY KAX KGY 100 NGY KAX KGY Percen Error (%) Percen Error (%) Time (sec) Time (sec) 120 NAD=3 (~NAD=14) 120 NAD=4 100 NGY KAX KGY 100 NGY KAX KGY Percen Error (%) Percen Error (%) Time (sec) Time (sec) 120 NAD=6 120 NAD= Percen Error (%) Percen Error (%) NGY KAX KGY Time (sec) 40 NGY KAX KGY Time (sec) Figure 1: Parameers n Gy, K Ax, K Gy esimaion (n F = 18). Top NAD = 1 and NAD = 2. Middle NAD = 3 and NAD = 4. Boom NAD = 6 and NAD =

15 REFERENCES [1] U. Olsson. Aeronauics for Europe. European Commission, [2] B. Prasad. Concurren Engineering fundamenals: Inegraed produc and process organizaion. Prenice Hall. Vol. 1, 1996, Vol. 2, [3] B. Prasad. Concurren Engineering fundamenals: Inegraed produc developmen. Prenice Hall. Vol. 2, [4] C. Broxmeyer. Inerial navigaion sysems. McGraw-Hill Book Co., [5] V.D. Andreev. Theory of inerial navigaion: Auonomous sysems. Nauka, [in Russian] [6] P.V. Bromberg. Theory of inerial navigaion sysems. Nauka, [in Russian] [7] H. Kaufman and D. Beaulier. Adapive parameer idenificaion. IEEE Trans. on Auoma. Conr., 17(5), , [8] I.V. Semoushin, A.D. Yurjev and A.E. Kondraiev A Simple decision generaor for deecion/selecion problems in linear sochasic sysems. This book of Proceedings of he ECCOMAS