Optimization-based Alignment for Strapdown Inertial Navigation System: Comparison and Extension

Transcription

1 Opiizaio-ased Alige for Srapdow Ierial Navigaio Syse: Copariso ad Exesio Lui Chag, Meer, IEEE, Jigshu Li, ad Kailog Li Asrac I his paper, he opiizaio-ased alige (OBA) ehods are ivesigaed wih ai focus o he vecor oservaios cosrucio procedures for he srapdow ierial avigaio syse (SINS). The coriuios of his sudy are wofold. Firs he OBA ehod is exeded o e ale o esiae he gyroscopes iases coupled wih he aiude ased o he cosrucio process of he exisig OBA ehods. This exesio rasfors he iiial alige io a aiude esiaio prole which ca e solved usig he oliear filerig algorihs. The secod coriuio is he coprehesive evaluaio of he OBA ehods ad heir exesios wih differe vecor oservaios cosrucio procedures i ers of coverge speed ad seady-sae esiae usig field es daa colleced fro differe grades of SINS. This sudy is expeced o faciliae he selecio of appropriae OBA ehods for differe grade SINS. Idex Ters Aiude esiaio, ierial avigaio, iiial alige, velociy iegraio forula I. INTRODUCTION As a dead-recoig avigaio ehod, he perforace of he srapdow ierial avigaio syse (SINS) Mauscrip received Deceer 9, 23; revised April 25, 24, Augus 23, 24; released for pulicaio May, 26. DOI. No..9/TAES Refereeig of his coriuio was hadled y A. Depser. This wor was suppored, i par, y he Naioal Naural Sciece Foudaio of Chia (63424, , ad 4442). Auhors addresses: Depare of Navigaio Egieerig, Naval Uiversiy of Egieerig, Chia, Jiefag Road 77, Qiaoou Disric,Wuha, 4333 People s Repulic of Chia. Correspodig auhor is L.Chag, E-ail: (chaglui@63.co).

2 relies largely o he accuracy of he iiial codiios which are deeried y he so called iiial alige [, 2]. Up o his day, he iiial alige has ee oe of he os researched opics for SINS ad various ehods have ee proposed o address his prole [3]. Typically, he iiial alige ca e achieved hrough wo cosecuive sages: coarse alige ad fie alige. Regardig he fie alige, cosesus has ee reached ha he Kala filerig ased opial esiaio schee is he sadard ehod. The Kala filerig ased fie alige is fouded o he liearized syse error odels ad ecessiaes he coarse alige o provide roughly ow iiial aiude o guaraee he validiy of such liearizaio [4, 5]. Uforuaely, he radiioal aalyic coarse alige ehods are o loger applicale whe he carrier is ovig or aeuverig. Recely, a aiude arix decoposiio ased coarse alige ehod has ee proposed for he carrier uder swayig or aeuverig codiios [6-]. By iroducig soe fixed referece fraes ow as ierial fraes, he aiude arix ca e decoposed io hree pars: aiude arix as a fucio of he carrier s agular rae, aiude arix as a fucio of he Earh ad raspor raes ad a cosa aiude arix ecodig he rasforaio ewee he ody ad avigaio fraes a he very sar of he iiial alige. The firs wo aiude arices ca e oaied hrough he aiude updae procedure accordig o he correspodig agular raes. The cosa aiude arix ca e derived usig soluios o Waha s prole ased o he cosruced vecor oservaios. To his respec, he aiude alige prole has ee rasfored io a coiuous aiude deeriaio prole [6]. I [6, 8], such iiial alige ehod is ered as opiizaio-ased alige (OBA) ecause he Waha s prole is virually a cosraied leas square prole ad all he soluios are devoed o iiizig he correspodig cos fucio i a opial aer. Sice ay fruiful algorihs ca e readily used o address he aiude deeriaio prole, say Davepor s q ehod used i [6, 8, ], he ey of deeriig he cosa aiude arix has ee he cosrucio of he vecor oservaios. Sice he cosa aiude arix ecodig he rasforaio ewee he fixed ody ad avigaio fraes a he very sar of he iiial 2

3 alige, he vecor oservaios expressed i he correspodig referece fraes are cosruced ased o he specific force equaio. The specific force equaio ecodes he relaioship ewee differe ids of acceleraio iforaio ad herefore, acceleraio iforaio ased vecor oservaios are firsly cosruced ad used i he OBA ehod i [6]. However, here is uch exeral disurace ihere i he acceleraio iforaio, which ca degrade he perforace of he OBA ehod sigificaly. Alhough he low-pass filer desiged i [6] ca resrai he disurace o a cerai exe, he paraeers desig for he low-pass filer ased o differe aeuverig codiios is a roulesoe process. Meawhile, he disurace of he acceleroeer oupus has o ee ae io accou i [6]. A sraighforward ad effecive ehod o resrai disurace is o iegrae he acceleraio iforaio over cerai ie ierval, which is jus he case i [8, ].Regardig he legh of he iegral ie ierval, here are aily wo iegraio procedures: oe is he iegraio procedure over he whole alige ie ierval i [8] ad he oher is he iegraio procedure over cerai ie ierval wih fixed legh i []. I is well ow ha resraiig effec o he disurace ecoes eer ad eer whe he iegral ie ierval icreases, which seeigly favors he iegraio procedures i [8]. Moreover, aig use of ore acceleraio iforaio ca accelerae he coverge speed of he OBA ehod. However, whe he iegral ie ierval icreases, he ierial sesors iases will also cause accuulaive errors i he cosruced vecor oservaios ad i ur, degrade he perforace he OBA ehod. To his respec, he iegral ie ierval should o e oo log, which favors he iegraio procedures i [] o he oher way. The aforeeioed coflicig coclusios deosrae ha ay issues eed o e furher fully addressed for he OBA ehods. Moivaed y he aforeeioed discussio, his paper is devoed o evaluaig he perforaces of he OBA ehod wih differe vecor oservaios cosrucio procedures usig field es daa colleced fro differe grades of SINS. Sice he ierial sesors iases, especially he gyroscopes iases, have a uch egaive effec o he precisio of he calculaed aiude, he exisig OBA ehods will e o loger applicale for he low-grade SINS, say icro-elecroechaical syse (MEMS) 3

4 echology ased low-cos SINS [2, 3], due o heir iailiy o esiae ayhig oher ha he aiude. Wih his cosideraio, his paper exeds he exisig OBA ehods ad ivesigaes a aiude esiaio ased alige ehod which ca esiae he gyroscopes iases coupled wih he aiude. Sice he ivesigaed aiude esiaio ased alige ehod is used o esiae he aiude a curre ie ha is o cosa for he i-oio alige, we er he ivesigaed alige ehod as dyaic OBA ehod. I coras, he radiioal OBA ehods are used o deerie he cosa aiude arix a he very sar of he iiial alige ad ca e viewed as saic OBA ehod. The aforeeioed differe iegraio procedures have also ee ivesigaed ad evaluaed i he cosrucio of he vecor oservaios for he developed dyaic OBA ehod. The res of he paper is orgaized as follows. Secio II aheaically aes he saic OBA ehod saee wih ai focus o he vecor oservaio cosrucio ehods wih differe iegraio procedures. I Secio III, he dyaic OBA ehod ale o esiae he gyroscopes iases coupled wih he aiude is developed. Meawhile, he explici filerig procedure is also preseed o solve he esalished aiude esiaio prole. Secio IV evaluaes he aforeeioed OBA ehods coprehesively i ers of coverge speed ad seady-sae esiae usig field es daa colleced fro differe grades of SINS. Fially, coclusios are draw i Secio V. II. STATIC OBA METHOD A. Forulaio of he Saic OBA Mehod For he OBA ehod, he aiude ieaics equaio is used direcly ad here is o eed o derive he correspodig agle error odels. Therefore, he OBA ehod ca e see as a aalyical ehod. The igeiousess of he OBA ehod aily aifess i he reapplicaio of he avigaio (aiude, velociy ad posiio) rae equaios i a ew aer. For he radiioal applicaio, he aiude ieaics equaio i ers of aiude arix is give y 4

5 C C () where T T i i i ie e C C (2) The defiiios of he ivolved referece fraes i his paper are he sae wih hose i [8]. C deoes he aiude arix fro he ody frae o he avigaio frae, deeriig he values of which prior o he avigaioal copuaio is jus he purpose of he iiial alige. deoes he ody agular rae easured y gyroscopes i he ody frae, ie he earh roaio rae expressed i he avigaio frae ad e he agular rae caused y he liear oio of he carrier. Deoe yhe ierially o-roaig frae ha is aliged wih he avigaio frae a, y ierially o-roaig frae ha is aliged wih he ody frae a, y he avigaio/ody frae a. Accordig o he chai rule of he aiude arix, he aiude arix C i ca e decoposed as C C C C (3) Sice he aiude arices C adc ca e deeried hrough he aiude updae procedure ased o he agular raes ad, respecively as i i he C C C i C i (4) he hear of deeriig C is rasfored io deeriig he cosa arix ], he calculaio ofc C. Accordig o [6, 8, is virually a coiuous aiude deeriaio prole usig ifiie vecor oservaios which are cosruced ased o he specific force equaio. The specific force equaio expressed i he avigaio frae is give y v C f 2 v g (5) ie e 5

6 where v deoes he groud velociy, ad g he graviy vecor. Susiuig (3) io (5) gives MuliplyigC f he specific force easured y acceleroeers i he ody frae v C C 2 C f ie e v g (6) o oh sides ad recogizig he resulig equaio yield which ca also e rewrie i a copac for as wih The deeriaio of he cosa arix C C C f C v v g (7) 2 ie e a a C (8) a C f (9a) 2 a ie e C v v g (9) i (8) ased o he vecor oservaios i (9) is well ow as he Waha s prole [4]. Accordigly, ay fruiful algorihs have ee developed for such prole, such as he hree axis aiude deeriaio (TRIAD) ased ehod, Davepor s q ehod, he quaerio esiaor (QUEST) ased ehod ad so o. More deails aou hese aiude deeriaio ehods ca e foud i he excelle survey paper [4] ad he refereces herei. I [9], he TRIAD ased ehod is applied o deerie he cosa arix C while i [6, 8,, ], he Davepor s q ehod is applied. Sice ay effecive aiude deeriaio ehods are readily o e used, he cosrucio ehods for he vecor oservaios are crucial o such prole. Alhough he vecor oservaios i (9) ca e used direcly, he severe disuraces ihere i he acceleraio iforaio, say f ad v, have a uch egaive effec o he perforace of he aiude deeriaio. A sraighforward way o aeuae he egaive effec of exeral disuraces is o iegrae he acceleraio iforaio hrough cerai ie ierval. Accordig o he 6

7 differe iegraio procedures, here are aily wo differe vecor oservaios cosrucios ehods for he developed aiude deeriaio prole, which will e preseed deailedly i he followig secio. B. Velociy Iegraio Forula for Vecor Oservaios Cosrucio Deoe y he curre ie isa durig he iiial alige process. I [8], he vecor oservaios i ers of acceleraio i (9) are iegraed over he ie ierval,. The resulig for of he aiude deeriaio prole associaed wih he vecor oservaios i ers of velociy is give y wih v v C () v C f d (a) C v 2 v g d () v ie e Siilarly o [8, 5], deoe he curre ie isa i is discree-ie for as M wih deoig he ie duraio of he updae ierval,,,,2,, M. Accordigly, he iegraio of () ca e calculaed usig he velociy iegraio forula developed i [8, 5] as v M C C f d (2a) v M ie C v C C ω v d M C C g d (2) The illusraio of such iegraig procedure is show i Fig.. I ca e see fro Fig. ha, as he ie goes o, he legh of he iegraio ie ierval is icreasig ad, i ur, he aeuaig effec of he exeral disuraces will e ore rearale. Moreover, aig use of such iegraio procedure ca guaraee he uers of he cosruced vecor oservaios wihou loss of ierediae GPS daa saples. Sice he acceleraio iforaio of he acceleroeer ad he GPS a cerai ie isa ca e used 7

8 repeaedly for ay ies, he coverge speed of he aiude deeriaio is expeced o e uch faser. The dowside of such iegraio procedure is ha, he sesor iases will cause accuulaive errors i he cosruced vecor oservaios as is show i (2a). I [8, ], he OBA ehod is regarded as a coarse alige ehod which eeds uch shorer ie ierval, so he accuulaive errors have a igored effec o he avigaioal or ore precise SINS. However, for he low-cos SINS, say MEMS ased SINS, he egaive effec of hese accuulaive errors will e uch rearale ad eve ivalidae he OBA ehod. f f f2 f M f M 2 M M Fig.. Iegraio procedure i () A sraighforward way o aeuae he egaive effec of hese accuulaive errors is o iegrae he acceleraio ased vecor oservaios i (9) over ie ierval wih,,. The so called ierleaved iegraig ehod proposed i [] is jus wih such caegory. I his case, he correspodig aiude deeriaio forulaio wih he associaed vecor oservaios i ers of velociy icree is give y wih C (3) v v v C f d (4a) C v 2 v g d (4) v ie e Here, he suscrip v is used o deoe he velociy icree over ie ierval, ad o sig differece wih he velociy deoaio v used i (). Alhough he ierval of he ierleaved iegraio is se o e he GPS-saple ie i [], i ca e exeded o a ore geeral for wih arirary legh. We ca 8

9 deerie he legh of he ierleaved iegraio ierval ased o he cosideraio of he rade-off ewee he coverge speed ad he aeuaig he egaive effec of accuulaive errors. Tha is o say, he ie ierval should e as log as possile o guaraee he coverge speed of he OBA ehod o oe had, while o he oher had, he ie ierval should e as shor as possile o aeuae he egaive effec of accuulaive errors. The ierleaved iegraig procedure ca also guaraee he uers of he cosruced vecor oservaios wihou loss of ierediae GPS daa saples, which is jus he oivaio of he derivaio of he ierleaved iegraio i []. I [], here is o explici iegraio forula o calculae he iegraio i (4). However, he velociy iegraio forula developed i [8, 5] ca e readily applied as v M C C f d (5a) v M ie C v C C ω v d M C C g d (5) where. The illusraio of he ierleaved iegraio procedure is show i Fig. 2. f f f 2 f f Slidig Fixed-ierval iegraio Fig. 2. Iegraio procedure i (4) I should e oed ha all he exisig OBA ehods are all uale o esiae ayhig oher ha he aiude. Sice for he low-cos SINS he sesors iases have uch egaive effec o he aiude calculaio, i is desired o exed exisig OBA ehods ad ae he e ale o esiae he sesors iases coupled wih he aiude for he iiial alige prole. Wih his cosideraio, a dyaic OBA ehod is 9

10 developed i his paper, which is used o esiae he aiude a curre ie raher ha he cosa aiude a he very sar of he iiial alige. III. DYNAMIC OBA METHOD A. Aleraive Perspecive o Ierleaved Iegraio Before geig io he develope of he dyaic OBA ehod, we firs give a aleraive perspecive o he ierleaved iegraig procedure, which will faciliae he derivaio of he developed dyaic OBA ehod. Specifically, accordig o he chai rule of he aiude arix, we have C C C C C C (6) Susiuig (6) io (7) ad iegraig he resulig equaio over ie ierval, yield C C C f d 2 ie e C C v v g d (7) For he iegraig operaor d, he ie isa is cosa ad so as he correspodig aiude aricesc adc. To his respec, (7) ca e rewrie as C C C f d 2 ie e C C v v g d (8) MuliplyigC o oh sides ad icorporaig he correspodig aiude arices yield C C f d 2 ie e C v v g d (9) where

11 C C C C (2) Eq. (9) ca also e wrie i he for of aiude deeriaio prole as wih v v C (2) v C f d (22a) C v 2 v g d (22) v ie e The relaioship ewee (4) ad (22) is give as v v C C v v (23) Regardig (2), C ca e viewed as a dyaic cosa arix. Here, he er cosa i dyaic cosa eas ha he aiude arix vecor oservaios i followig ie ierval (2) shares he sae procedure as ha of C C a ie isa is regarded as a cosa for he. I his respec, he deeriaio of dyaic i dyaic cosa eas ha he aiude arix, C ased o ased o () ad (3). O he oher had, he er C is variaioal o he whole ie ierval of he iiial alige as he ie isa is chaged fro o. The illusraio of he iegraio procedure i (22) is show i Fig. 3. All he aiude arices a he ared ie isa are jus he aforeeioed dyaic cosa arices. The avigaio ad ody fraes a ie isa, i.e. ad fraes for he correspodig fraes a he followig ie ierval,, ca also e viewed as ierial jus he sae as he ad for he correspodig fraes a ie ierval,. Sice he ie isa is chaged fro o, he correspodig fraes ad are oly he eporary ierial fraes. I he followig secio, he

12 cocep of eporary ierial fraes will also e used o derive he dyaic OBA ehod. f f f f f f f f f f f f f Afer he dyaic cosa arix ca e readily oaied as where Fig. 3. Iegraio procedure i (22) C eig derived, he aiude arix a he curre ie isa C C C C (24) C C C C C C (25) All he aiude arices a he righ had of (25) ca e calculaed accordig o (4). I should e oed ha he aiude deeriaio procedure (2)-(25) is o a ew OBA ehod ad i is he sae as ha of he ierleaved iegraio ased ehod. B. Dyaic Aiude Esiaio Model I his secio par, we will develop he dyaic OBA ehod wih cosideraio of esiaig he sesors 2

13 iases. The derivaio of such ehod is ased o he aforeeioed coceps dyaic cosa arix ad eporary ierial fraes. For he aleraive perspecive o he ierleaved iegraio, he fraes a he ie isa are reaed as ierial for he correspodig fraes a he followig ie ierval,, while i his secio, he fraes a ie isa are reaed as ierial for he correspodig fraes a he previous ie ierval. Specifically for ay ie isa, Susiuig (26) io (5) yields,, accordig o he chai rule of he aiude arix, we have C C C C (26) 2 v C C C f v g (27) ie e I should e oed ha all he quaiies aove are fucios of ie ad heir ie depedeces o are oied for reviy. Muliplyig C Iegraig (28) o oh sides over he ie ierval o oh sides of (27) ad reorgaizig he resulig equaio yield 2 C C f C v ie e v g (28), C C f d 2 ie e C v v g d (29) For he iegraig operaor d,he ie isa is cosa ad so as he correspodigc respec, (29) ca e rewrie as wih dv dv. To his C (3) dv C f d (3a) C v 2 v g d (3) dv ie e 3

14 I is show ha we have ade use of he siilar eporary ierial cocep i goig fro (7) o (8) ad (29) o (3). The referece fraec correspodig o he eporary ierial fraes ca also e viewed as a dyaic cosa arix. I las secio, he cosa characerisic of he dyaic cosa C is used for he saic aiude deeriaio. I coras, he dyaic characerisic of he arix dyaic cosa arix C aiude arix C will e used for he followig dyaic aiude esiaio. Whe he is reaed as a dyaic value, he gyroscopes iases ca e easily icorporaed io he dyaic aiude updae odel. Cosiderig he coaiaed ias ad oise, he direc oupu of he gyroscopes is odeled as i i i gv where is he gyro drif derived fro a rado wal process. ad gu gu gv (32) are idepede zero-ea Gaussia whie-oise processes wih specral desiies. Reorgaizig (32) ad susiuig he resul io () give he followig aiude odel T C C C (33) i i gv Eq. (33) ad (3) cosiue he ypical dyaic aiude esiaio odel wih (33) eig he process odel ad (3) he easuree odel. Wih he developed dyaic odel, ay exisig dyaic aiude esiaio ehods ow as filerig ehods ca e readily applied. Before geig ivolved i explici deails of filerig algorihs ha will e preseed i ex secio, we firs exed he velociy iegraio forula developed i [8, 5] o solve he iegraio (3). The velociy iegraio forula developed i [8, 5] ca o e applied i (3) i is curre for. Nex, we will show how he iegraio i (3) ca e calculaed usig he velociy iegraio forula. The iegraio i (3a) ca e expressed as 4

15 C f d C C f d C C f d (34) As i is show, he iegraio o he righ had of (34) is he sae as ha i (4a). Therefore, he iegraio i (3a) ca e calculaed as M dv The firs par of he iegraio (3) ca e developed as C C C f d (35) i C v d C v C v d v C v C v d i (36) Susiuig (36) io (3) yields dv v C v ie C v d C g d (37) Siilar o he procedure i (34), he firs iegraio o he righ had of (37) ca e expressed ad calculaed as ie ie C v d C C v d M ie C C C ω v d (38) The secod iegraio o he righ had of (37) ca e expressed ad calculaed as C g d C C g d M C C C g d (39) Susiuig (38) ad (39) io he righ side of (37) yields dv v C v M M C C C ω ie v d C C g d (4) 5

16 The illusraio of he developed dyaic aiude deeriaio prole ad he associaed vecor oservaios (3) are show i Fig. 4. I Fig. 4, x is he sae of he dyaic odel, which icludes aiude ad gyroscopes iases. I should e oed ha, alhough he vecor oservaios are cosruced ased o he ierleaved iegraio procedure, hey ca also e cosruced ased o he iegraio procedure i (). Le, he iegraio procedure i (3) is jus he sae as ha i (). The properies of he wo iegraio procedures discussed i he las secio for he OBA ehod are also suiale for he dyaic aiude esiaio prole. f f f 2 f y 2 y y x 2 x x Fig. 4. Iegraio procedure i (3) for he dyaic odel (32) C. Modified USceed QUaerio Esiaor Sice he aiude is usually paraeerized i ers of quaerio for o-copuer applicaio, os of he exisig filerig ehods for aiude esiaio are devoed o hadle he quaerio oralizaio cosrai i he filerig recursio, such as he uliplicaive exeded Kala filer [4], he USceed QUaerio Esiaor (USQUE) [6, 7] ad is odified versio (MUSQUE) [8, 9]. I his paper, he MUSQUE is used. Cosiderig clariy ad copleeess, he explici filerig procedure of MUSQUE for he ivesigaed dyaic aiude esiaio prole is preseed i his secio par. Paraeerizig he aiude i ers of quaerio, he aiude ieaics equaio (33) i is discree-ie for is give y 6

17 i i q ff q,,, (4) Eq. (4) is a geeral for of he discree-ie aiude ieaics equaio i ers of quaerio. The discree-ie aiude ieaics equaio is virually he o-copuer calculaio procedure of he aiude. Various fruiful algorihs have ee developed for he o-copuer aiude calculaio, say sigle saple algorih, doule saple algorih ad so o. We ca selec he appropriae oe ased o he cosideraio of he algorih precisio ad coplexiy. Sice his paper focuses o he iiial alige ehod develope, he explici procedures of hese calculaio algorihs are o preseed ad oly a geeral for as show i (4) is preseed. The MUSQUE is virually a exesio of he usceed Kala filer (UKF) [2, 2] wih cosideraio of he quaerio or cosrai. Sice he or cosrai of he quaerio ca e easily desroyed y he weighed-ea copuaio ihere i he UKF, he aiude quaerio ca o e used direcly as he filerig sae. I order o overcoe his prole he MUSQUE uses he quaerio for gloal osigular aiude represeaio ad ucosraied geeralized Rodrigues paraeer (GRP) for local aiude T represeaio ad filerig. Give he error quaerio q q 4, where is he vecor par of he quaerio q, he correspodig GRP represeaio p is give y T p f a q4 (42) where a is a paraeer fro o, ad f is a scale facor. I he MUSQUE, he sae is seleced as x ;. Give he sae esiae xˆ a ie isa, he correspodig sae covariace P ad he quaerio esiae qˆ, he ai of MUSQUE is o deerie he sae esiae xˆ ad quaerio esiae qˆ a ie isa. The explici filerig procedure is give as follows Tie Updae 7

18 Geerae he siga pois as p ˆ x P (43) where is a ue paraeer which is usually se o3 i he UKF o capure soe higher order iforaio p of he disriuio wih deoig he sae diesio [9]. The i-h copoe of is deoed as p T i ad is quaerio for is deoed as q i i q i accordig o he reverse for of (42) as 4, T which ca e calculaed q i 2 2 p p 2 2 a i f f a i i 4, 2 p 2 f (44a) The correspodig quaerio ased siga pois p i f a q i i 4, (44) q i are give as i q i qˆ (45) q Propagae he quaerio ad gyroscopes ias ased siga pois hrough he process odel (4) q q i ff i, i, i, i (46) Sice he gyroscopes ias is viewed as cosa, he propagaed gyroscopes ias ased siga pois are give as i i (47) The predicio of he aiude quaerio is give y he quaerio averagig algorih as [22] qˆ arg ax q T Aq (48) 3 qs where 2 A i i 2 q q T (49) i 8

19 The prediced quaerio error ased siga pois are give y T Deoe qˆ i i q i y 4, q q i i qˆ (5) T, he correspodig prediced GRP ased siga pois are give p i i f a q i 4, (5) T pt T Deoe, he prediced ea ad covariace of he sae are give y whereq is he covariace of he process oise. Measuree Updae Regeerae he siga pois as xˆ 2 i (52a) 2 i 2 P i x i x Q 2 T ˆ ˆ (52) i * p * xˆ P * (53) Deoe he quaerio for of * p i as T * * * q i i q i 4, which ca e calculaed accordig o he reverse for of (42) as T q 2 2 * p 2 2 * p a i f f a i * 4, i 2 2 * p f i p i f a q i i * * * 4, (54a) (54) 9

20 * The quaerio ased siga pois correspodig o q i is give y i q i qˆ (55) * q * Propagae he quaerio ased siga pois hrough he easuree odel (3) as wherec * q i C i (56) * q * q i deoes he aiude arix correspodig he quaerio i The ea ad covariace of he easuree are give y dv,. yˆ 2 i (57a) 2 i 2 P i y i y R T ˆ ˆ (57) y, 2 i where R is he covariace of he easuree oise. The cross-correlaio covariace of he sae ad easuree is give y 2 P i x i y T ˆ ˆ (57c) xy, 2 i The sae esiae ad correspodig covariace are give y dv, x ˆ x ˆ K y ˆ (58a) P P K P K (58) T y, where Aiude Updae Deoe xˆ T K P P (58c) xy, y, T T T pˆ ˆ. The quaerio correspodig o pˆ is deoed as qˆ q 4, which ca e calculaed as T 2

21 q ˆ 2 a pˆ f f a p 4, 2 2 f pˆ (59a) (59) ˆ f a q4, i p The quaerio esiae is give y qˆ qˆ qˆ (6) Rese he firs hree elees (GRP) of xˆ o zero ad go o he ex filerig cycle. IV. PERFORMANCE EVALUATION THROUGH FIELD TEST I his secio, he vecor oservaios cosrucio procedures for oh he saic ad dyaic OBA ehods are evaluaed usig field es daa. Specifically, he followig four iiial alige ehods are evaluaed: q ehod: full iegraio q ehod: parial iegraio filer: full iegraio filer: parial iegraio The er full iegraio eas he iegraio procedure i () ad parial iegraio he ierleaved iegraio procedure i (4) or (3). The er q ehod is correspodig o he OBA ehod as he Davepor s q ehod is used o calculae he cosa aiude arix. The er filer is correspodig o he dyaic OBA ehod as he MUSQUE algorih is used o calculae he dyaic aiude arix. I he parial iegraio procedure, ierleaved saples are used, i.e. for (4) ad (3). As is discussed i he las secio, he wo differe iegraio procedures aily affec he coverge speed ad seady-sae esiae of he correspodig aiude deeriaio ehods. I his secio, we will evaluae he wo characerisics of he aforeeioed four iiial alige ehods usig car-oued field es daa colleced o SINS wih differe levels of precisio. Oe is he avigaio-grade SINS equipped wih 2

22 a riad of rig laser gyroscopes ad acceleroeers. The oher is he low-grade SINS equipped wih a riad of MEMS gyroscopes ad acceleroeers. The specificaios of he avigaio-grade ad low-grade ierial sesors are lised i Tales I ad II, respecively. TABLE I.SPECIFICATIONS OF THE NAVIGATION-GRADE INERTIAL SENSORS Gyro Acceleroeer Dyaic 4 s 3g Updae rae Hz Hz Bias Bias sailiy.5 h 5g. h 5g TABLE II. SPECIFICATIONS OF THE LOW-GRADE INERTIAL SENSORS Gyro Acceleroeer Dyaic 5 s g Updae rae Hz Hz Bias Bias sailiy.5 s.5g.2 s.g There is also a GPS aea isalled ouside he cai o he op of he car ad he SINS/GPS iegraed avigaio resul is used as he referece daa o copare wih. The raw easurees ( Hz) of GPS were liearly ierpolaed o oai he velociy ad posiio a oh eds of he updae ierval. A. Tesig Resuls Usig he Navigaio-grade SINS Accordig o [8], he OBA ehod ca reach a saisfacory perforace for he avigaio-grade SINS afer oly s, herefore, we also ae use of s colleced daa o es he ivolved iiial alige ehods i his paper. The esig resuls of he four iiial alige ehods usig he avigaio-grade SINS daa are preseed i Fig. 5-, respecively. Fig. 5 ad 6 plo he pich agles esiaes ad esiae error, Fig. 7 ad 8 he roll agles, Fig. 9 ad he yaw agles, respecively. The sead-sae aiude error 22

23 suary of he four iiial alige ehods is lised i Tale III for copariso. I is show ha he full iegraio procedure ca reach a uch faser coverge speed ad eer sead-sae esiae perforace ha he parial iegraio procedure. This is ecause ha i he full iegraio procedure he saplig daa colleced fro he ierial sesors is used ore ies ad, i ur, ore adequaely ha ha i he parial iegraio procedure. The iprovee i ers of coverge speed ad seady-sae precisio y he full iegraio procedure is guaraeed y he high-grade specificaios of he avigaio-grade ierial sesors. Tha is o say, aig use of he saplig daa colleced fro he avigaio-grade ierial sesors repeaedly will o cause accuulaive errors. Therefore, for he OBA of he high-grade SINS, he full iegraio procedure will e ore celeraed. Moreover, wih he full iegraio procedure he dyaic ehod developed i his paper is a lile eer ha he saic ehod. However, he copuaioal cos of he dyaic ehod is uch larger ha ha of he saic ehod. All he iiial alige ehods are ipleeed usig Mala o a copuer wih a 2.66G CPU, 2.G eory ad he Widows7 operaig syse. I our experie, he cos ie of processig he s daa is aou52s for dyaic OBA ehod wih full iegraio ad 8s for saic OBA ehod wih full iegraio. If he OBA ehods are oly regarded as coarse alige ehods which are used o provide rough aiude for he followig fie alige, he saic OBA ehod will e ore celeraed due o is less copuaioal cos. The proposed dyaic OBA ehod shows o uch iprovee over he radiioal ehod is aily due o he iailiy of he proposed ehod o esiae he acceleroeer iases. The acceleroeer iases aily deerie he uliae precisio of he iiial alige. If he egaive effec of he gyroscopes iases is o so ovious as for he avigaio-grade SINS, he seady-sae alige precisio is aily deeried y he acceleroeer iases. Sice oh he saic ad dyaic OBA ehods are uale o esiae he acceleroeer iases, heir perforace is siilar o each oher i his case. Advaced ehod ha is ale o esiae he acceleroeer iases ca e expeced o furher iprove he alige perforace of he avigaio-grade SINS. 23

24 Pich / arci Pich / deg IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 52, NO. 4 AUGUST True q ehod: full iegraio q ehod: parial iegraio filer: full iegraio filer: parial iegraio / s Fig. 5. Pich agles esiaes q ehod: full iegraio q ehod: parial iegraio filer: full iegraio filer: parial iegraio / s Fig. 6. Pich agles esiae error 24

25 Roll / arci Roll / deg IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 52, NO. 4 AUGUST True q ehod: full iegraio q ehod: parial iegraio filer: full iegraio filer: parial iegraio / s Fig. 7. Roll agles esiaes 5 q ehod: full iegraio q ehod: parial iegraio filer: full iegraio filer: parial iegraio / s Fig. 8. Roll agles esiae error 25

26 Yaw / arci Yaw / deg IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 52, NO. 4 AUGUST True q ehod: full iegraio q ehod: parial iegraio filer: full iegraio filer: parial iegraio / s Fig. 9. Yaw agles esiaes q ehod: full iegraio q ehod: parial iegraio filer: full iegraio filer: parial iegraio / s Fig.. Yaw agles esiae error Tale III. ATTITUDE ERROR SUMMARY USING NAVIGATION-GRADE INERTIAL SENSORS (ui: arci) Pich Roll Yaw q ehod: full iegraio q ehod: parial iegraio filer: full iegraio

27 filer: parial iegraio The ai advaage of he developed dyaic ehod is is ailiy o esiae he gyroscopes iases coupled wih he aiude. The gyroscopes iases esiae resuls wih differe iegraio procedures are preseed i Fig. ad 2, respecively. Meawhile, he gyroscopes iases esiae resuls usig he SINS/GPS iegraio are also ploed i Fig. ad 2 o copare wih. For he SINS/GPS iegraio, he iiial gyroscopes iases are assued o e zero, which is he sae as ha i he dyaic OBA ehods. However, all he aiude, velociy ad posiio are properly iiialized. Tha is o say, he SINS/GPS iegraio is acually carried ou afer he aiude alige. As i is show, he gyroscopes iases esiae resuls usig he full iegraio procedure are ore close o he resuls of he SINS/GPS iegraio ha ha of he parial iegraio procedure. However, he resuls of all he hree esiae procedure are o so well. This is ecause ha for he high-grade SINS, he oservailiy of he iases errors is feele ad uch loger ie is eeded o coverge he esiae resuls [6, 7] SINS/GPS X SINS/GPS Y SINS/GPS Z parial X parial Y parial Z.5 / /h / s Fig.. Gyroscopes iases esiaes of he filer: parial iegraio ad SINS/GPS 27

28 SINS/GPS X SINS/GPS Y SINS/GPS Z full X full Y full Z / /h / s Fig. 2. Gyroscopes iases esiaes of he filer:full iegraio ad SINS/GPS B. Tesig Resuls Usig he Low-grade SINS I his es, he car wih low-grade SINS was ovig i a large circle aroud he spors grouds wih a good sy view of he GPS aeas. Two circles of es daa wih differe velociies are colleced o evaluae he alige ehods. I he firs circle, 25s were ae for he car o ae oe circui aroud he spors grouds. The esig resuls of he four iiial alige ehods usig he firs daa sege are preseed i Fig. 3-8, respecively. Fig. 3 ad 4 plo he pich agles esiae ad esiae error, Fig. 5 ad 6 he roll agles, Fig. 7 ad 8 he yaw agles, respecively. The sead-sae aiude error suary of he four iiial alige ehods is lised i Tale IV for copariso. I is show ha oh he wo saic OBA ehods are o loger applicale. This is ecause ha he gyroscopes iases have a uch egaive effec o he accuracy of he aiude ad he saic OBA ehods are uale o esiae he iases durig he aiude alige. For he dyaic OBA ehods developed i his paper, he parial iegraio procedure is effecive while he full iegraio procedure is oeffecive jus as he saic ehods. This is ecause ha he ierial sesors iases will cuulae i he cosruced vecor oservaios ad whe he iegraig ierval icreases wih ie jus as he full iegraio procedure he cuulaed errors will e larger ad larger, resulig i uch degraded vecor oservaios ad, i ur, uch degraded aiude esiae. The 28

29 effeciveess of he dyaic OBA ehod wih parial iegraio procedure lies i is ailiy o esiae he gyroscopes iases. The gyroscopes iases esiae resuls wih parial iegraio procedure alog wih he resuls of he SINS/GPS iegraio are preseed i Fig. 9. I is show ha iases esiae resuls of he dyaic OBA ehod wih parial iegraio procedure are coparale wih ha of he SINS/GPS iegraio. For copariso, he gyroscopes iases esiae resuls wih full iegraio procedure alog wih he resuls of he SINS/GPS iegraio are preseed i Fig. 2. I is show ha iases esiae resuls of he dyaic OBA ehod wih full iegraio procedure are uch degraded. Therefore, he dyaic OBA ehod wih parial iegraio procedure is os celeraed for he low-grade SINS. I is show ha he esiae of he gyroscopes ias i he verical direcio has a uch slower covergece speed ha ha of he gyroscopes iases. This is aily due o he differe degrees of oservailiy. For ore horough deails aou he oservailiy of he iiial alige, he reader is referred o he syseaical ad coprehesive sudy [7] for his prole. Alhough i appears ha he es ie is a lile shor o oai a clear resul for he verical gyroscopes ias, i does o affec he deosraio of he validiy of he proposed dyaic OBA ehod. This is ecause ha he perforace of he proposed dyaic OBA ehod i esiaig he gyroscopes iases is coparale wih ha of he SINS/GPS iegraio which is he accredied ehod i esiaig he sesor iases. However, i should e oed ha he SINS/GPS iegraio is fouded o he liearized syse error odels ad ecessiaes he iiial alige o provide roughly ow iiial aiude o guaraee he validiy of such liearizaio. If he aiude ca o e aliged appropriaely, he perforace of he SINS/GPS iegraio will e uch degraded. Therefore, he ivesigaed iiial alige ehod is sill ecessary i pracical applicaio. I ca e see i Fig. 3-8 ad Ta. IV ha he dyaic OBA ehod wih parial iegraio procedure ca alig he aiude quie well. Therefore, he suseque sage of SINS/GPS iegraio ca e readily carried ou afer he dyaic OBA. Meawhile, he aiude alige ie should o e oo log due o soe special iliary requirees i ers of aeuverailiy. I his respec, he es ie is appropriae for he aiude alige. The caliraio of he 29

30 Pich / deg Pich / deg IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 52, NO. 4 AUGUST 26 sesor iases ca e furher carried ou i he suseque SINS/GPS iegraio sage True -6 q ehod: full iegraio q ehod: parial iegraio -7 filer: full iegraio filer: parial iegraio / s Fig. 3. Pich agles esiaes (firs circle) q ehod: full iegraio q ehod: parial iegraio filer: full iegraio filer: parial iegraio / s Fig. 4. Pich agles esiae error (firs circle) 3

31 Roll / deg Roll / deg IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 52, NO. 4 AUGUST True q ehod: full iegraio q ehod: parial iegraio filer: full iegraio filer: parial iegraio / s Fig. 5. Roll agles esiaes (firs circle) q ehod: full iegraio q ehod: parial iegraio filer: full iegraio filer: parial iegraio / s Fig. 6. Roll agles esiae error (firs circle) 3

32 Yaw / deg Yaw / deg IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 52, NO. 4 AUGUST True q ehod: full iegraio q ehod: parial iegraio filer: full iegraio filer: parial iegraio / s Fig. 7. Yaw agles esiaes (firs circle) q ehod: full iegraio q ehod: parial iegraio filer: full iegraio filer: parial iegraio / s Fig. 8. Yaw agles esiae error (firs circle) Tale IV. ATTITUDE ERROR SUMMARY USING LOW-GRADE INERTIAL SENSORS (firs circle, ui: deg) Pich Roll Yaw q ehod: full iegraio q ehod: parial filer: full iegraio

33 filer: parial iegraio SINS/GPS X SINS/GPS Y SINS/GPS Z parial X parial Y parial Z / /h / s Fig. 9. Gyroscopes iases esiaes of he filer: parial iegraio ad SINS/GPS (firs circle) 5 5 SINS/GPS X SINS/GPS Y SINS/GPS Z full X full Y full Z / /h / s Fig. 2. Gyroscopes iases esiaes of he filer: full iegraio ad SINS/GPS (firs circle) I he secod circle, 5s were ae for he car o ae oe circui aroud he spors grouds. The esig resuls of he four iiial alige ehods usig he secod daa sege are preseed i Fig. 2-26, respecively. Fig. 2 ad 22 plo he pich agles esiae ad esiae error, Fig. 23 ad 24 he roll agles, Fig. 25 ad 26 he yaw agles, respecively. The sead-sae aiude error suary of he four iiial alige ehods is lised i Tale V for copariso. The gyroscopes iases esiae resuls of he dyaic 33

34 Pich / deg IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 52, NO. 4 AUGUST 26 OBA ehod wih parial iegraio procedure alog wih he resuls of he SINS/GPS iegraio are preseed i Fig. 27. The gyroscopes iases esiae resuls of he dyaic OBA ehod wih full iegraio procedure alog wih he resuls of he SINS/GPS iegraio are preseed i Fig. 28. We ca draw he sae coclusios fro hese resuls wih hose fro he las experieal resuls. I should e oed ha he gyroscopes iases i he wo experies are o he sae wih each oher. This is ecause ha he gyroscopes iases are always o he sae afer each applicaio of power, which is ow as he ias isailiy. Adiedly, his oservaio suggess ha he gyroscopes iases should e esiaed i each applicaio, which ca furher give proiece o he sigificace of he proposed dyaic OBA ehod for he aiude alige of he low-grade SINS True -3 q ehod: full iegraio q ehod: parial iegraio -4 filer: full iegraio filer: parial iegraio / s Fig. 2. Pich agles esiaes (secod circle) 34

35 Roll / deg Pich / deg IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 52, NO. 4 AUGUST q ehod: full iegraio q ehod: parial iegraio filer: full iegraio filer: parial iegraio / s Fig. 22. Pich agles esiae error (secod circle) True q ehod: full iegraio q ehod: parial iegraio filer: full iegraio filer: parial iegraio / s Fig. 23. Roll agles esiaes (secod circle) 35

36 Yaw / deg Roll / deg IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 52, NO. 4 AUGUST q ehod: full iegraio q ehod: parial iegraio filer: full iegraio filer: parial iegraio / s Fig. 24. Roll agles esiae error (secod circle) True q ehod: full iegraio q ehod: parial iegraio filer: full iegraio filer: parial iegraio / s Fig. 25. Yaw agles esiaes (secod circle) 36

37 Yaw / deg IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 52, NO. 4 AUGUST q ehod: full iegraio q ehod: parial iegraio filer: full iegraio filer: parial iegraio / s Fig. 26. Yaw agles esiae error (secod circle) Tale V. ATTITUDE ERROR SUMMARY USING LOW-GRADE INERTIAL SENSORS (secod circle, ui: deg) Pich Roll Yaw q ehod: full iegraio q ehod: parial iegraio filer: full iegraio filer: parial iegraio SINS/GPS X SINS/GPS Y SINS/GPS Z parial X parial Y parial Z / /h / s Fig. 27. Gyroscopes iases esiaes of he filer: parial iegraio ad SINS/GPS (secod circle) 37

38 2 5 SINS/GPS X SINS/GPS Y SINS/GPS Z full X full Y full Z 5 / /h / s Fig. 28. Gyroscopes iases esiaes of he filer: full iegraio ad SINS/GPS (secod circle) I should e oed ha he ierleaved saples used i he parial iegraio is oly a special case. The legh of iegraio ie ierval has a rearale effec o he perforace of he developed dyaic OBA ehod. For he avigaio-grade SINS, aig use of he saplig daa ore adequaely ca reach a eer perforace i ers of coverge speed ad seady-sae precisio. For he low-grade SINS, however, he siuaio is a lile coplex. O he oe had, oo less saplig daa icorporaed io he iegraio ca o guaraee he appropriae coverge speed. O he oher had, oo ore saplig daa icorporaed io he iegraio ay cause accuulaive error due o he ihere ierial sesors error. I his respec, i is desired o deerie he legh of iegraio ie ierval uder differe codiios. However, here is o uiversal ehod o achieve such ed ad i ca oly e deeried hrough experie sudy. Regardig he uilizig procedure of he saplig daa over cerai ie iervals, he ivesigaed ehod has a reasoale siilariy o he fiie ipulse respose (FIR) filer. Recely, Shaliy ad Sio have ivesigaed uiased FIR filer for he sae-space odel ased proles ad preseed a uified fors for Kala ad FIR filerig [24, 25]. I [24], i is poied ou ha he opial widow size ca e easily esiaed experieally y iiizig a elaorae cos fucio wihou ay rue referece. The siilariy ewee he Kala ad FIR filerig oives us o deerie he legh of iegraio ie ierval accordig o he proposed ehod i [24], 38

39 which is he furher research regardig he ivesigaed dyaic OBA ehod. V. CONCLUSION The ewly developed OBA ehod has ee prove o e a igeious ehod for he SINS iiial alige uder swayig or aeuverig codiios. Typically, here are aily wo iegraio procedures for he cosrucio of he vecor oservaios used i he OBA ehods. However, he coflicig properies of he wo iegraio procedures ecessiae a furher ivesigaio of he OBA ehods. Meawhile, he exisig OBA ehods are all uale o esiae ayhig oher ha he aiude ad ca e viewed as saic ehods, which sulifies he for he low-grade SINS. Based o he cosrucio process of he saic OBA ehods, a dyaic OBA ehod is developed o esiae he gyroscopes iases coupled wih he aiude, which is expeced o e applicale for he low-grade SINS. The exisig saic OBA ehods ad he developed dyaic OBA ehod wih differe iegraio procedures are coprehesively evaluaed i ers of coverge speed ad seady-sae esiae usig field es daa fro oh avigaio-grade ad low-grade SINS. The copariso aalysis shows ha for he avigaio-grade SINS he saic OBA ehod is os celeraed due o is less copuaioal cos ad appropriae seady-sae esiae precisio. I his case, aig use of ore acceleraio iforaio ca resul i a uch faser coverge speed wih o degradaio i seady-sae esiae due o he high-grade specificaios of he avigaio-grade ierial sesors. For he low-grade SINS, he saic OBA ehods are o loger applicale due o he accuulaive errors i he cosruced vecor oservaios caused y he large ierial sesors iases. I his case, he developed dyaic OBA ehod wih ierleaved iegraio procedure is os celeraed sice i ca esiae he gyroscopes iases appropriaely ad hus resul i a uch eer aiude alige perforace. Maig use of ore acceleraio iforaio fro he low-grade ierial sesors will cause uch accuulaive errors i he cosruced vecor oservaios, which ca eve sulify he dyaic OBA ehod. A disadvaage of he proposed algorih is ha i ca o esiae he acceleroeer iases. Quie recely, 39

40 Wu e al. [23] propose a olie cosraied-opiizaio ehod o siulaeously esiae he aiude ad he ierial sesor iases. This olie cosraied opiizaio ehod perfors quie well i esiaig he aiude ad he acceleroeer iases. However, is perforace of esiaig he gyroscopes iases is coproised due o he severe oise ad disurace. Therefore, his olie cosraied opiizaio ehod is cofied o hose accurae applicaios wih avigaio-grade SINS easurees [23]. I his respec, he olie cosraied opiizaio ehod ad he proposed dyaic OBA ehod is copleeary o each oher i joi esiaio of he gyroscopes/ acceleroeer iases. I he fuure wor, we will ry o hadle he gyroscopes/ acceleroeer iases siulaeously y coiig he virue of he olie cosraied-opiizaio ehod ad he proposed dyaic OBA ehod. ACKNOWLEDGMENT The auhors would lie o ha Gogi Ya, ad Zogwei Wu for providig he field es daa. REFERENCES [] D. H. Tiero ad J. L. Weso, Srapdow Ierial Navigaio Techology: he Isiue of Elecrical Egieers, Lodo, Uied Kigdo, 2d Ed., 24. [2] P. D. Groves, Priciples of GNSS, Ierial, ad Mulisesor Iegraed Navigaio Syses: Arech House, 28. [3] J. L. Crassidis, Siga-Poi Kala Filerig for Iegraed GPS ad Ierial Navigaio, IEEE Trasacios o Aerospace ad Elecroic Syses, vol. 42, o. 2, pp , 26. [4] Y. F. Jiag, Error aalysis of aalyic coarse alige ehods, IEEE Trasacios o Aerospace ad Elecroic Syses, vol. 34, o., pp , Ja [5] J. C. Fag, ad D. J. Wa, A fas iiial alige ehod for srapdow ierial avigaio syse o saioary ase, IEEE Trasacios o Aerospace ad Elecroic Syses, vol. 32, o. 4, pp.5-55, 4

41 996. [6] M. P. Wu, Y. X. Wu, X. P. Hu, ad D. W. Hu, Opiizaio-ased Alige for Ierial Navigaio Syses: Theory ad Algorih, Aerospace Sciece ad Techology, vol. 5, pp. -7, 2. [7] Y. X. Wu, H. L. Zhag, M. P. Wu, X. P. Hu, ad D. P. Hu, Oservailiy of SINS alige: A gloal perspecive, IEEE Trasacios o Aerospace ad Elecroic Syses, vol. 48, o., pp. 78 2, 22. [8] Y. Wu, ad X. Pa, Velociy/posiio iegraio forula, Par I: Applicaio o i-fligh coarse alige, IEEE Trasacios o Aerospace ad Elecroic Syses, vol. 49, o. 2, pp. 6-23, 23 [9] D. Q. Gu, N. El-Sheiy, T. Hassa ad Z. Syed, Coarse alige for arie SINS usig graviy i he ierial frae as a referece, i Proc. IEEE/ION PLANS, Moerey, CA, May 6 8, 28, pp [] P. M. G. Silso, Coarse Alige of a Ship's Srapdow Ierial Aiude Referece Syse Usig Velociy Loci, IEEE Trasacios o Isrueaio ad Measuree, vol. 6, o. 6, pp , 2. [] T. Z. Kag, J. C. Fag ad W. Wag, Quaerio-Opiizaio-Based I-Fligh Alige Approach for Airore POS, IEEE Trasacios o Isrueaio ad Measuree, vol. 6, o., pp , 22. [2] Z. Wu, M. Yao, H. Ma ad W. Jia, Iprovig Accuracy of he Vehicle Aiude Esiaio for Low-Cos INS/GPS Iegraio Aided y he GPS-Measured Course Agle, IEEE Trasacios o Iellige Trasporaio Syses, vol. 4, o. 2, pp , 23. [3] Z. Wu, M. Yao, H. Ma, W. Jia ad F. Tia, Low-Cos Aea Aiude Esiaio y Fusig Ierial Sesig ad Two-Aea GPS for Vehicle-Moued Saco-o-he-Move, IEEE Trasacios o vehicular echology, vol. 62, o. 3, pp , 23. [4] J. L. Crassidis, F. L. Marley, ad Cheg Y, Survey of Noliear Aiude Esiaio, Joural of Guidace, Corol, ad Dyaics, vol.3, o., pp. 2-28, 27. 4

42 [5] Y. Wu ad X. Pa, Velociy/Posiio Iegraio Forula (II): Applicaio o Srapdow Ierial Navigaio Copuaio, IEEE Trasacios o Aerospace ad Elecroic Syses, vol. 49, o. 2, pp , 23 [6] J. L. Crassidis ad F. L. Marley, Usceed filerig for spacecraf aiude esiaio, Joural of Guidace, Corol, ad Dyaics, vol. 26, o. 4, pp , 23. [7] J. L. Crassidis ad F. L. Marley, Usceed filerig for spacecraf aiude esiaio, Joural of Guidace, Corol, ad Dyaics, vol. 26, o. 4, pp , 23. [8] L. B. Chag, B. Q. Hu, ad G. B. Chag, Modified USceed QUaerio Esiaor ased o Quaerio Averagig, Joural of Guidace, Corol, ad Dyaics, vol.37, o., pp , 24. [9] L. B. Chag, B. Q. Hu, S. Y. Che ad F. J. Qi, Coes o A Quaerio-Based Mehod for SINS/SAR Iegraed Navigaio Syse, IEEE Trasacios o Aerospace ad Elecroic Syses, vol. 42, o. 2, pp , 23. [2] S. J. Julier, ad J. K. Uhla, Usceed filerig ad oliear esiaio, Proc. IEEE, vol. 92, o. 3, pp , 24. [2] L. B. Chag, B. Q. Hu, A. Li, ad F. J. Qi, Trasfored usceed Kala filer, IEEE Trasacios o Auoaic Corol, vol. 58, o., pp , 23. [22] F. L. Marley, Y. Cheg, J. L. Crassidis, ad Y. Osha, Averagig Quaerios, Joural of Guidace, Corol, ad Dyaics, vol. 3, o. 4, pp , 27. [23] Y. X. Wu, J. L. Wag ad D. W. Hu, A New Techique for INS/GNSS Aiude ad Paraeer Esiaio Usig Olie Opiizaio, IEEE Trasacios o Sigal Processig, vol. 62, o., pp , 24. [24] Y. S. Shaliy, Suopial FIR filerig of oliear odels i addiive whie Gaussia oise, IEEE Trasacios o Sigal Processig, vol. 6, o., pp , 22. [25] D. Sio, ad Y. S. Shaliy, Uified fors for Kala ad fiie ipulse respose filerig ad 42

43 soohig, Auoaica, vol. 49, o., pp , 23. Lui Chag (M 3) received B.Sc. ad Ph.D. degrees i avigaio fro he Depare of Navigaio, Naval Uiversiy of Egieerig, Wuha, P.R. Chia i 29 ad 24, respecively. He is currely a Lecurer wih he Naval Uiversiy of Egieerig. His curre research ieress iclude ierial avigaio syses, ierial-ased iegraed avigaio syses, ad sae esiaio heory. Jigshu Li received he B.Sc. ad M.Sc. degrees i avigaio fro he Depare of Corol Egieerig, Aviaio Uiversiy of Air Force, Chagchu, P.R. Chia, i 28 ad 2, respecively, ad he Ph.D. degree i avigaio fro he Depare of Navigaio, Naval Uiversiy of Egieerig, Wuha, P.R. Chia i

44 He is currely a Lecurer wih he Naval Uiversiy of Egieerig. His curre research ieress iclude ierial avigaio syses, ad corol algorihs. Kailog Li received B.Sc. ad Ph.D. degrees i avigaio fro he Depare of Navigaio, Naval Uiversiy of Egieerig, Wuha, P.R. Chia i 2 ad 25, respecively. He is currely a Lecurer wih he Naval Uiversiy of Egieerig. His curre research ieress iclude ierial avigaio syses, aiude esiaio, ad iegraed avigaio. 44

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