Noncommutativity Error Analysis of Strapdown Inertial Navigation System under the Vibration in UAVs

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1 Ieraoal Joural of Advaced Roboc Sye ARTICLE Nocouavy Error Aaly of Srapdow Ieral Navgao Sye uder he Vbrao UAV Regular Paper Jzhou La,*, P Lv, Jaye Lu ad B Jag College of Auoao Egeerg, Najg Uvery of Aeroauc ad Aroauc, Najg, Cha * Correpodg auhor E-al: lajz@uaa.edu.c Receved Ju ; Acceped 7 Aug DOI:.577/5758 La e al.; lceee ITech. Th a ope acce arcle drbued uder he er of he Creave Coo Arbuo Lcee (hp://creavecoo.org/lcee/by/3.), whch per urerced ue, drbuo, ad reproduco ay edu, provded he orgal work properly ced. Abrac Nocouavy error of a rapdow eral avgao ye (SINS) a uaed aeral vehcle (UAV) vbrao evroe aalyed. The radoal aaly of ocouavy error baed o a cog oo odel, whch coe wh a UAV vbrao evroe. I h paper he UAV vbrao for dcued ad odelled a a uodal agular vbrao ad a rado agular vbrao. The, SINS oo odel uder hee wo for of vbrao are bul up ad he forula for he ocouavy error are derved eparaely. I addo, he effec of a ul aple algorh explored, whch a effecve ehod for copeag for ocouavy error cae of cog oo. Fally, he UAV vbrao evroe ulaed ad dcaed ha he ulao reul of he SINS ocouavy error are coe wh heorecal aaly. Keyword UAV vbrao, Srapdow Ieral Navgao Sye (SINS), Nocouavy error, rado vbrao. Iroduco The avgao ye upple he eceary forao for he corol of UAV. Recely, reearch o UAV avgao ha efed coderably hak o he growh of cvl ad lary ere UAV []. SINS oe of he o popular avgao ehod for UAV whch ca eaure he vehcle poo, velocy ad aude whou gal rao o exeral equpe []. The eral eauree u (IMU), whch co of gyro ad acceleroeer, a pora copoe of he SINS ad error ca grealy affec avgaoal accuracy. The vbrao evroe of UAV coplex, flueced by he acuag u ad aerodyac force [3]. Sce IMU are drecly alled o UAV, hey are eve o vehcle vbrao [4]. Specal uppor [5] or daper [6] for he SINS are uually eeded o reduce he fluece o he IMU of vehcle vbrao, bu he IMU cao be copleely olaed fro vbrao by hee ea. Moreover, oe vbrao reduco ehod cao be appled o all UAV due o he lao o wegh ad Jzhou La, P Lv, Jaye Lu ad I B J Adv Jag: Roboc Nocouavy Sy,, Vol. Error 9, 36: Aaly of Srapdow Ieral Navgao Sye uder he Vbrao UAV

2 ze. The SINS affeced by vbrao wo way: o oe had, oe brough o he gyro ad acceleroeer [7 8], leadg o decreaed IMU accuracy; o he oher had, ocouavy error are roduced due o he roaoal ocouavy heory of a rgd body [9] ad he SINS oluo accuracy decreaed. The vehcle uually odelled a havg a cog oo a radoal ocouavy error aaly of he SINS []. I h codo, aude error pread over e ad avgao accuracy degraded gfcaly by ocouavy error. A ul aple algorh, whch creae he aplg frequecy of he IMU, ha a good copeag effec for ocouavy error. Meawhle, ocouavy error uder a wofrequecy cog oo [] ad a geeral cog oo [] are uded, whch are ore oral ha clacal cog oo. However, he oo odel above do o coder he characerc of a UAV vbrao. I h paper, he for of a UAV vbrao aalyed, ad a aaly ehod of he SINS ocouavy error brough abou by he UAV vbrao propoed. I addo, he effec of he radoal algorh appled o ocouavy error copeao explored he cae of he UAV vbrao. Fr, he vbrao evroe of UAV dcued Seco II. I poed ou ha uodal vbrao ad rado vbrao are he a vbrao for of UAV. The a ocouavy error forula of a SINS uder uodal agular vbrao derved Seco III ad he effec of a ul aple algorh aalyed. I Seco IV he expreo of a SINS oo a e doa uder rado agular vbrao deduced fro power pecral dey (PSD) ad he he ae reearch doe a Seco III. SINS uder uodal vbrao ad rado vbrao are ulaed Seco V ad he ulao reul verfy he effecvee of he propoed heory.. Vbrao evroe of UAV Ege oe ad ar durbace are he a ource of UAV vbrao [3]: ege oe characerc are cloely relaed o he ege ype, whch depeded o he UAV ype ad wll be dcued ubeque par of h eco; ar durbace flueced by eeorologcal codo oude he UAV, uch a wd velocy ad weaher, whch have rogly rado characerc ad ca be decrbed by rado vbrao. Elecrc oor [4] ad po ege [5] are uually ued a acuag u cro ad all UAV, whch are aly for hor dace recoaace. The vbrao for of uch ype of UAV cloe o uodal vbrao [5]. Suodal vbrao perodc oo whoe dplacee chage over e he for of a e fuco. Vbrao aplude, vbrao frequecy ad phae are deoed by L, f ad repecvely, ad vbrao dplacee gve by: x) L f () Je ege are uually foud edu ad large UAV [6] whch are aly ued for log dace recoaace or rke. Thee UAV vbrao are caued by he je ege exhau, ad he vbrao for dffer fro uodal vbrao ad decrbed a rado vbrao [7]. Rado vbrao expreed by vbrao PSD. Fgure how a agular acceleroeer PSD of a X ax gyro a SINS a rado vbrao expere. The PSD draw ug gyro oupu ad obvou ha he PSD pree rado characerc uder he expereal codo. Led o he aple frequecy of gyro, he upper cu off frequecy of he PSD oly 5 Hz. However, he frequecy badwdh of real rado vbrao uually wder ad ca be up o everal houad Hz. PSD (rad / 3 ) Frequecy(Hz) Fgure. Agular acceleroeer PSD of X ax gyro of a SINS a rado vbrao expere IMU are alled o UAV wh uppor. Due o he dapg effec uppled by he uppor vbrao aplude are degraded whe raed o he IMU. However, he vbrao for of IMU are alo he ae a he UAV bode. So uodal vbrao ad rado vbrao are choe o decrbe he IMU oo uder he UAV vbrao. Sce ocouavy error are roduced oly by agular oo [8] he dcuo h paper led o agular vbrao ad he vbrao codered oly oe degree of freedo for plcy. 3. Nocouavy error of SINS uder uodal agular vbrao Suodal vbrao he a vbrao for for UAV wh elecrc oor or po ege (uch a Quadcoper ad all uaed helcoper). Nocouavy error expreo of he SINS I J Adv Roboc Sy,, Vol. 9, 36:

3 derved h eco ad he effec of a ul aple algorh explored. 3. Nocouavy error expreo Body coordae frae b aued o udergo uodal vbrao relave o referece coordae frae r. Vbrao aplude ad vbrao frequecy are deoed by L ad repecvely ad phae aued o be whou lo of geeraly. Sce vbrao gle degree of freedo he u drecoal vecor gve by[αcoβ coαcoβ β], ad he referece o he body quaero of uodal vbrao ca be wre a: Q co L ω / ) = αcoβ L ω / coαcoβ L ω / β L ω / Q ) ad Q ) ad for he referece o he body quaero a e ad repecvely ad - = T. q, ) he updaed quaero bewee Q ) ad Q ), ad, ) he correpodg roao vecor. Through quaero ulplcao we ca ge: Q ) Q ) q, ) () (3) where Φ T, ). Sce Φ T a all value, (Φ T / ) Φ T ca be approxaed a /. The, ) ca be derved by coparg Eq. (4) ad Eq. (5):, ) αcoβ L ω T / co ω T / coαcoβ L ω T / co ω T / β L ω T / co ω T / ) deoe he agular velocy bewee b frae ad r frae. The roao vecor dffereal equao ca be wre a [9]: he ) : (6) Q ) Q ) ) / (7) * ) Q ) Q ) αcoβlωco(ω) coαcoβlωco(ω) βlω co(ω ) I ca be ee ha all copoe of ) pree coe er fro Eq. (8). Aug ha here are N gyro aple bewee ad, he gyro aplg e h = T / N. The he gyro aple () ad he accuulaed gyro oupu fro o ca be derved fro Eq. (8): (8) ad q, ) ca be derved a: * q, ) Q ) Q ) co L ωt / coω T / αcoβ L ωt / coω T / = coαcoβ L ω T / co ω T / β L ωt / coω T / where Q ). (4) * Q ) he cojugae quaero of () +h +()h )d αcoβl ω h / co ω + h - h / coαcoβl ω h / co ω + h - h / βl ω h / co ω + h - h / )d N = () αcoβl ω T / co ω + T / coαcoβl ω T / co ω + T / βl ω T / co ω + T / (9) () Accordg o he relao bewee he updaed quaero q, ) ad he roao vecor, ), we have: Φ (Φ / ) q T T, ) co, ) Φ (5) T where =,,..., N. I he above aaly Eq. (6) he heorecal value of he roao vecor, ) whch approxaed by he accuulaed gyro oupu durg he SINS calculao e. Arg fro he approxao, ocouavy error are a follow: Jzhou La, P Lv, Jaye Lu ad B Jag: Nocouavy Error Aaly of Srapdow Ieral Navgao Sye uder he Vbrao UAV 3

4 , ) αcoβ L ω T / co ω T / L ω T / co ω T / coαcoβ L ω T / co ω T / L ω T / co ω T / β L ωt / co ω T / L ωt / co ω T / () I ca be ee ha all he copoe of he ocouavy error uder a uodal agular vbrao are perodc, whch do o accuulae over e, leadg o all aude error. However, for cog oo oly wo copoe of he ocouavy error are perodc ad oe copoe ha a coa value, whch produce a drf rae error he updaed quaero Q ) ad degrade he SINS aude error. 3. Effec of he ul aple algorh Accordg o he ul aple algorh, gyro are apled ulple e per aude updae, ad ocouavy error ca be copeaed by he cro produc of he gyro aple. However, fro Eq. (9) we ee ha he cro produc of he gyro aple are all zero. Th ea ( ) ( j), o ca be cocluded ha he ul aple algorh ha o copeaoal effec for uodal agular vbrao. 4. Nocouavy error of SINS uder rado agular vbrao Rado vbrao he a vbrao for for UAV wh je ege (uch a hgh alude log edurace UAV). I h eco he oo odel of a SINS aalyed, a ocouavy error expreo derved ad he effec of a ul aple algorh explored. 4. SINS oo odelg Body coordae frae b aued o do rado vbrao relave o referece coordae frae r. The agular accelerao dcree PSD deoed a S a (ω), aplg e T, oal e T, he he reoluo of he PSD f = / T. Sce a SINS oo odel relaed o agular dplacee, requred o rafor he agular accelerao PSD he followg forula: S a (ω) o a agular PSD S d (ω) by 4 S (ω) = S (ω) / ω () d a Accordg o he PSD aaly heory he relao bewee he PSD ad he frequecy pecru kow for led daa. The he expreo of rado vbrao he e doa ca be wre a: (3) = θ) = L co(ω + φ ) where L = ΔfS d (Δf), ω = πδf, φ he phae correpodg o each frequecy copoe. However, he PSD oly coa he aplude forao of he gal ad he phae ukow, o he gal expreo he e doa cao be reored oly by he PSD. Referrg o he proceg ehod he vbrao able [], all phae are aued o be a eve drbuo wh [-, ]. 4. Nocouavy error expreo The u drecoal vecor of rado vbrao [αcoβ coαcoβ β] ad he referece o body quaero ca be wre a: Q r co θ) / αcoβ θ) / ) = coαcoβ θ) / β θ) / r, ) he roao vecor fro o lar o uodal vbrao ad ca be gve by: (4), ωt T -αcoβ L ω + φ = ωt T r, ) = -coαcoβ L ω + φ = ωt T -β L ω + φ = (5) Aug ha N gyro aple are ake bewee ad, he gyro aplg e h = T / N. Led o he gyro badwdh, hgh frequecy vbrao cao be eed. If f b deoe he gyro badwdh, f h deoe hghe frequecy of vbrao, he he gyro aple () ad he accuulaed gyro oupu fro o are: 4 I J Adv Roboc Sy,, Vol. 9, 36:

5 (j) ω h -αcoβ L ω + h - h / + φ = ω h -coαcoβ L ω + h - h / + φ = ωh -β L ω + h - h / + φ = (6) N j= (j) ωt T -αcoβ L ω + φ = ωt T -coαcoβ L ω + φ = ωt T -β L ω + φ = (7) r, ) ωt T ωt T -αcoβ L ω + φ L ω + φ = = ωt T ωt T -coαcoβ L ω + φ L ω + φ = = ωt T ωt T -β L ω + φ L ω + φ = = (8) where j =,,..., N, = fb / fh. Eq. (4) he heorecal value of he roao vecor r, ) ad error are produced whe approxaed by he accuulaed gyro oupu : Alhough every frequecy copoe of he ocouavy error perodc, he perodcy of he whole expreo ukow ce all he phae are rado value. Th ea ha cocluo cao be drecly draw fro he aalycal expreo, bu ulao aaly ca be doe ug he above ep. 4.3 Effec of he ul aple algorh Slar o uodal vbrao he cro produc of he gyro aple are alo zero accordg o Eq. (6). So ca be cocluded ha he ul aple algorh ha o copeag effec for rado agular vbrao. 5. Sulao reul ad dcuo Suodal agular vbrao ad rado agular vbrao for SINS are ulaed o verfy he cocluo h eco. Nocouavy error ad ul aple algorh are boh dcued. 5. Sulao uder uodal agular vbrao Sulao codo are e a follow: he e vbrao frequecy Hz ad he agular vbrao aplude deg; α = 3deg, = 6deg ; oal e ad he ulao ep.. Sgle aple algorh ad fouraple algorh are ued for he SINS aude oluo repecvely. The ocouavy error are how Fg. ad Fg.3. he SINS aude error are how Fg.4. X-ax (deg/h) Y-ax(deg/h) Te() Te() Te() Fgure. The ocouavy error of he gle aple algorh uder uodal agular vbrao X-ax(deg/h) Y-ax(deg/h) Z-ax(deg/h) Te() Te() Te() Fgure 3. The ocouavy error of he four aple algorh uder uodal agular vbrao Z-ax(deg/h) Jzhou La, P Lv, Jaye Lu ad B Jag: Nocouavy Error Aaly of Srapdow Ieral Navgao Sye uder he Vbrao UAV 5

6 x -7.5 x -7 3 x Pch Error(") Roll Error(") Head Error(") -5 5 Te() Te() Te() Fgure 4. Aude error uder uodal agular vbrao Fro fgure, ca be oberved ha all he copoe of ocouavy error are perodc ad he aplude dfferece are caued by he drecoal vecor. By coparg Fg. ad Fg. 3 foud ha he ocouavy error of he gle aple algorh are equal o hoe of he four aple algorh, whch ea he four aple algorh ha o copeag effec. Fgure 4 how he aude error uder uodal agular vbrao, whch are he order e o he egave eve arc ecod. Thee value are very all ad ca be oed pracce. Th becaue he ocouavy error are perodc ad he aude error do o accuulae over e. 5. Sulao uder rado agular vbrao Sulao codo are e a follow: fgure 5 how he agular accelerao PSD of a rado vbrao, whch bewee 5Hz ad Hz; α = 3deg, = 6deg ; oal e, ad he ulao ep.. The ocouavy error of he gle aple algorh ad he four aple algorh are how Fg.6 ad Fg.7. The SINS aude error are how Fg X-ax(deg/h) Y-ax(deg/h) Te() Te() Te() Fgure 6. The ocouavy error of he gle aple algorh uder rado agular vbrao Fro Fg. 6 clear ha all he copoe of he ocouavy error are rado ad whou perodcy. By coparg Fg.6 ad Fg. 7, foud ha he four aple algorh ha o copeag effec for ocouavy error, whch lar o uodal agular vbrao. Fgure 8 how ha aude error uder rado agular vbrao are he order e arc ecod. Aude error pree ocllao for ad are o dverge. The aplude of he aude error are relavely large ad eed o be codered durg he SINS avgao proce. X-ax(deg/h) Y-ax(deg/h) Te() Te() Te() Fgure 7. The ocouavy error of he four aple algorh uder rado agular vbrao Z-ax(deg/h) Z-ax(deg/h) PSD (rad / 3 ) Pch Error(") Roll Error(") Head Error(") Frequecy (Hz) Fgure 5. Agular accelerao PSD of rado vbrao Te() Te() Te() Fgure 8. Aude error uder rado agular vbrao 6 I J Adv Roboc Sy,, Vol. 9, 36:

7 6. Cocluo SINS ocouavy error a ew oo for are aalyed. Th parly baed o a rado oo odel ad a beer decrpo of UAV vbrao ha he radoal deerc odel. Through heorecal aaly ad ulao we ca draw he followg cocluo: ) Nocouavy error pree perodcally uder gle degree of freedo uodal agular vbrao. Aude error are all ad ca be oed pracce. ) Aude error are flueced by boh ocouavy error ad gyro badwdh uder gle degree of freedo rado agular vbrao. Uually PSD ued o decrbe rado vbrao ad doe coa phae forao of he vbrao. So a rado vbrao expreo a e doa cao be derved hrough PSD ad ocouavy error cao be foud hrough heorecal aaly. 3) The ul aple algorh ued ocouavy error copeao uder cog oo o uable for gle degree of freedo uodal agular vbrao ad rado agular vbrao. Bede, a ulao ehod propoed o aalye SINS ocouavy error uder rado agular vbrao, whoe prcple ad ep are preeed. By eg he PSD of rado agular vbrao ad he gyro badwdh, he ocouavy error ad aude error ca be aalyed hrough ulao. 7. Ackowledge Th work uppored by Naoal Naural Scece Foudao of Cha(969, 67497), he Fudg of Jagu Iovao Progra for Graduae Educao (CXLX_) ad he Fudg for Ouadg Docoral Derao NUAA (BCXJ 4). 8. Referece [] X.G. Peg ad D. Xu, Iellge Ole Pah Plag for UAV Adveraral Evroe, Ieraoal Joural of Advaced Roboc Sye, Vol.9, No.3, pp.,. [] J. Xog, J. Lu, J. La ad B. Jag, A argalzed parcle fler al alge for SINS, Ieraoal Joural of Iovave Copug, Iforao & Corol, Vol.7, No.7, pp ,. [3] J. Srwog ad R.W. Sullva, Expereal Vbrao Aaly of a Copoe UAV Wg, Mechac of Advaced Maeral ad Srucure, Vol.9, No. 3, pp.96 6,. [4] A. Nera ad N. Aouf, Robu INS/GPS Seor Fuo for UAV Localzao Ug SDRE Nolear Flerg, IEEE Seor Joural, Vol., No.4, pp ,. [5] Z. Tuo, D. Hu, R. L ad J. We. Dapg deg of rapdow eral avgao ye, Joural of Chee Ieral Techology, Vol. 7, No.6, pp , 9. [6] J.I. Lahha, D.J. Wge ad A.L. Colea, Tued uppor rucure for rucure bore oe reduco of eral avgaor wh dhered rg laer gyro (RLG), Poo Locao ad Navgao Sypou, USA, pp.49 48,. [7] Y. Yukel, N. El Shey ad A. Noureld, Error odelg ad characerzao of evroeal effec for low co eral MEMS u, IEEE/ION PLANS, USA, pp.598 6,. [8] P. Aggarwal Z. Syed, X. Nu ad N. El Shey, A Sadard Teg ad Calbrao Procedure for Low Co MEMS Ieral Seor ad U, Joural of Navgao, Vol.6, No., pp , 8. [9] M. E. Pelkau, Roao Vecor Aude Eao, Joural of Gudace, Corol, ad Dyac, Vol.6, No.6, pp , 3. [] P.G. Savage, Cog Algorh Deg by Explc Frequecy Shapg, Joural of Gudace, Corol, ad Dyac, Vol.33, No.4, pp.3 3,. [] K. K, T.G. Lee, Aaly of he wo frequecy cog oo wh SDINS, AIAA Gudace, Navgao ad Corol Coferece ad Exhb, Caada, pp. 5,. [] H. Muoff ad J. H. Murphy, Sudy of Srapdow Navgao Aude Algorh, Joural of Gudace, Corol, ad Dyac, Vol.8, No., pp.87 9, 995. [3] J. Rohac, M. Ree ad K. Draxler, Daa Proceg of Ieral Seor Srog Vbrao Evroe, The 6h IEEE Ieraoal Coferece o Iellge Daa Acquo ad Advaced Copug Sye: Techology ad Applcao, Prague, pp.7 75,. [4] A. Sachez, L. R. García Carrllo, E. Rodo, R. Lozao ad O. Garca, Hoverg Flgh Iprovee of a Quad roor M UAV Ug Bruhle DC Moor, Joural of Iellge & Roboc Sye, Vol.6, No. 4, pp.85,. [5] Y.C. La ad S.S. Ja, Aude Eao Baed o Fuo of Gyrocope ad Sgle Aea GPS for Sall UAV uder he Ifluece of Vbrao, GPS Soluo, Vol.5, No., pp.67 77,. [6] C. Kog. Ege Tred Moorg of a Log Edurace UAV Ug Lear Regreo ad Fuzzy Logc, Ieraoal Joural of Turbo ad Je Ege, Vol.6, No.3, pp.87, 9. [7] MIL STD 8G, Depare of Defece Te Mehod Sadard: Evroeal Egeerg Coderao ad Laboraory Te, 8. [8] K.M. Rocoe, Equvalecy Bewee Srapdow Ieral Navgao Cog ad Scullg Iegral/Algorh, Joural of Gudace, Corol, ad Dyac, Vol.4, No., pp. 5,. Jzhou La, P Lv, Jaye Lu ad B Jag: Nocouavy Error Aaly of Srapdow Ieral Navgao Sye uder he Vbrao UAV 7

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