MOTION CONTROL STRATEGY OF AN UNDERWATER GLIDER IN THE PRESENCE OF EXTERNAL DISTURBANCES

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1 VOL., NO., NOVEMBER 6 ISSN Asin Reserch Publishing Nework (ARPN). All righs reserved. MOION CONROL SRAEGY OF AN UNDERWAER GLIDER IN HE PRESENCE OF EXERNAL DISURBANCES Brk Ullh, Mrk Ovinis, Msri B Bhro, og D. Seiwn, Syed Sd Azhr Ali nd M. Y. vid Depren of Mechnicl Engineering, Universii eknologi Perons, Bndr Seri Iskndr, Perk, Mlysi Depren of Elecricl & Elecronic Engineering, Universii eknologi Perons, Bndr Seri Iskndr, Perk, Mlysi E-Mil: rk_ovinis@perons.co.y ABSRAC his pper describes oion conrol sregy of n underwer glider in he presence of exernl disurbnces bsed on se feedbck nd full order observer feedbck. In his pper, we will be concerned only wih he phugoid oion in he vericl plne. A conrol sregy for he glider ws ipleened wih fs nd sble convergence. Siulion resuls show h he open loop syse is conrollble nd observble. he resuls sugges he use of well-known guidnce for fs rcking nd sbiliy conrol. Keywords: conrol sregy, underwer gliders, liner qudric regulor, siulion. INRODUCION Underwer gliders re ype of uonoous underwer vehicle (AUV) h oves by odifying heir buoyncy nd cenre of ss hrough n inernl oving ss. herefore, oion conrol is chieved by vrying hese wo preers, which re ypiclly conrolled hrough feedbck. o speed up convergence nd iprove perfornce, feed forwrd coponen y be used. However, n underwer glider is subjeced o vrious environenl disurbnces such s curren disurbnce. For exple, wer curren he surfce cn slide he glider fro is pre-se rjecory, which requires ddiionl conrol effor o bring i bck o desired ph [-3]. In order o reduce he energy consued on ph correcion during sedy gliding, i is very iporn o sbilize he desired ph. However, lile work hs been done on gliding oion sbiliy involving curren disurbnces which cn esily shif he glider fro is desired rjecory. However, lile work hs been done on gliding oion sbiliy involving curren disurbnces which cn esily shif he glider fro is desired rjecory. Leonrd [4] nd Nin [5] proposed LQR nd PID conroller respecively for oion conrol, bu boh of hese pproches were bsed on linerized dynic odel of glider nd he pplicion ws liied o equilibriu condiions. Bh [6] proposed nonliner conroller involving orque nd buoyncy conrol which is bsed on he ssupion of equilibriu condiion. Noh e l [7] de perfornce coprison beween LQR nd PID conroller for glider pich conrol using SISO conroller nd he effec of exernl disurbnces ws negleced. his pper copres nd conrss he use of liner nd nonliner LQR observer o esie he syse ses for n open loop nd closed loop syse. A dynic odel is used o design full se feedbck observer nd cobin linerizion pproch is used o check he sbiliy nd conrollbiliy of he odel. Figure-. Underwer glider sw ooh profile. UNDERWAER GLIDER MOION CONROL MECHANISM In his secion, we describe he heicl odel of underwer glider using longiudinl plne dynics. he glider is considered s n ellipsoidl rigid body wih uniforly disribued ss. he wings nd rudder re fixed nd lerl disurbnces re negleced. he forces on he glider re grviy, hydrodynic forces, lif, drg nd piching oen. Buoyncy nd dded ss effecs re lso considered in he heicl odelling such s presened by [8]. he oion of underwer glider is conrolled by vrying buoyncy nd inernl oving sses bou is cenre of grviy. Figure- shows he oion profile of he underwer glider. he longiudinl fre of he body is he x-xis while he z-xis is righ ngles o he x-xis θ Figure-. Underwer glider fre ssignen. he relionship beween grviy W nd buoyncy B is: W g, nd B g where g is grviy, is wer densiy nd is volue of he glider body. 33

2 VOL., NO., NOVEMBER 6 ISSN Asin Reserch Publishing Nework (ARPN). All righs reserved. he glider is designed o be posiively buoyn ( B W) such h i will surfce uoiclly. he cenre of buoyncy is considered in he origin O due o syery of he glider. Considering he ssupions of Grver[] nd Zhng [9]h ovble ss is fixed origin O nd sionry sses re uniforly disribued, such h he cenre of buoyncy will coincide wih he cenre of grviy. he dded sses re ssued o be equl ( = 3). he grviy nd buoyncy forces in ers of conrol he pich of he glider fro origin O o xiu lengh of buoyncy nk. he ne buoyncy ss is. = G- w. Where G is he ol ss of he glider which is he su of unifor hull ss h, fixed poin ss o blnce he cener of grviy. he ss of he displced wer is w, blls ss is b which is used o conrol buoyncy, fixed oving ss for pich conrol long he nose x-xis. G h w b. he scheic digr for uniforly disribuion of inernl sses is shown in Figure-3. where V represens he gniude of he velociy nd K L, K L, K D, K D, K M nd K M re hydrodynic coefficiens. is he ngulr velociy of he pich nd K q is he piching dping coefficien [8]. he hydrodynic coefficiens re deerined for UP glider in [] for vrious ngle of cks nd for consn fluid speed nd shown in ble-. ble-. Hydrodynic preers. Preers Vlues K D.393 K D 3.56 K L.7 K L 6.6 K M.575 K M.44 MAHEMAICAL MODEL Zhng e l in [3] nd Leonrd nd Bh in [8] used he equions of oion for underwer glider for buoyncy conrol s represened in equions 5-8 in polr coordines. he velociy V represens he ol velociy vecor of he glider wih respec o gliding ngle γ. he siplified dynic odel for four ses [V, θ g, α, ω] of underwer glider is shown s: V ( g sin g D ) (5) Figure-3. Inernl ss disribuion of glider. Le be he oen of ineri in e direcion pssing hrough he cenre of grviy of he glider. he pich ngle is θ nd he gliding ngle is θ g. he ngle of ck is α. he relion beween he pich ngle nd gliding ngle is g () Hydrodynic forces he lif L, Drg D nd hydrodynic oen M for underwer glider wih wings nd rudder re dependen on he ngle of ck nd velociy of he gliders. Lif of he glider is vrying linerly wih ngle of ck while drg of he glider is prbolic funcion of ngle of ck s follows: L K K V ( L L) () D K K V ( D D ) (3) g ( g cos g L ) V (6) (7) g ( ) KM KM Kq V (8) where is ol ineri bou y-xis, V & θ g re he ol velociy vecor nd gliding ngle respecively. Equions (5) - (8) re nonliner equions. For sedy glides, he preers in equions (5) - (8) re deerined by using equilibriu vlues e. he se vribles equilibriu hve he following relionships s clculed in [3]. K K e e De M n ( ), e ( ) K Le K M g, V ( ) e KDe KLe De D De, Le L Le Where K K K K K K M K K K V ( M M q) (4) 33

3 VOL., NO., NOVEMBER 6 ISSN Asin Reserch Publishing Nework (ARPN). All righs reserved. Conroller design In his secion he perfornce of oion conrol of underwer glider is checked by pplying he heicl odel nd equilibriu vlues in previous secion o wo differen conrollers. he velociy of he glider is nlysed by super iposing he vehicle relive velociy wih esied nd esured ocen curren. he ocen curren is iposed in he dynic odel nd in conroller in he for of exernl noise. A generlized syse given in se spce for cn be sed s x () Ax () Bu () Bv () (9) v y() Cx() () wo ypes of conrollers re nlyzed for his syse. he syse is considered o be linerized where u() is he inpu signl which is he force required o regule he blls nk nd v() is disurbnce signl. LQR design Liner Qudric Regulor (LQR) is designed bsed on he glider dynic odel in vericl plne. Nonliner equions of oion re linerized nd hen he closed loop syse wih LQR conroller for sbilizion is siuled. LQR sbilizes conrol lw h cn be iniized using cos funcion which cn be defined s: () LQR x Qx () u () Rud () () Q nd R re weighing rices for se vribles nd conroller for inpu vribles. I is ssued h ll he glider ses re esurble nd vilble for se feedbck. he gin (rix) K is deerined by firs solving he lgebric Ricci equion K. PA A P PBQ u B P Q x P is posiive sei definie funcion nd K cn be clculed fro LQR Mlb cond. Q u nd Q x re clculed fro digonl rices king ll oher eleen o zero. K r is clculed fro he syse odel. As here re sensors used o clcule he deph of glider so here y hve chnces of unceriny h will rise he sedy se error. In order o bring he sedy se error o zero, n inegrl cion hs been ugened in he se feedbck syse rix. his ehod is siilr o PID conroller. he conroller cn copense for sll oupu deviions fro he reference signl. he ugened rix is described s: x () A x () B u () r () z () C z () I () u () Kx () (3) he vlue of K increses if is incresed. Le Q is 4-by-4 rix nd R is consn. he vlue of Q is se o: Q C C (4) By ipleening he vlues of Q nd R, he rix K is copued s follows: K (5) he block digr of se feedbck syse is shown in Figure-4. Figure-4. Block digr for se feedbck design Assue h ll ses of he syse re esured, he conrol lw for ugened rix in equion () cn be wrien s u Kx() K r() (6) r he closed loop poles re deerined by considering he closed loop syse chrcerisic polynoil. he closed loop syse is described by: x () ( ABK) x() BK r() B v() (7) r In equion (7), A BK is syse rix nd corresponding polynoil chrcerisic equion which decides he closed loop poles is de( SI A BK). In equion 7, he consn K r does no ffec he sbiliy of he syse however i ffecs he sedy se soluion. he consn gin K r is clculed in he following equion. K r ( ) C ABK B v (8) OBSERVER DESIGN o esie he desired vlue of pich ngle which is seleced s he oupu of he syse, liner full se observer is designed. he block digr of he liner observer is shown in Figure-5. Due o ugened rix, he new conrol lw is: 333

4 VOL., NO., NOVEMBER 6 ISSN Asin Reserch Publishing Nework (ARPN). All righs reserved. B KMV C Figure-5. Observer design block digr. he rel se rix of he syse fro equions (5) -(8) is (9) X V he esied se rix is Xˆ Vˆ ˆ ˆ ˆ () Consider liner ie invrin coninuous syse fro Zhng e l [7]. X ˆ AXˆ BUˆ L ( ) () ˆ CXˆ () L is observer gin rix nd he rices A, B nd C re he linerized syse rices which re obined hrough coben linerizion is given by: 3 3 A KM Y() 4 Where ^ KqY() gcos(y()) - (Y()(K +K Y(3) )), +K Y(3)) Y() D D K DY() Y(3) (Y()(K 3, L L ^ he se spce rices re used for LQR nd observer design nd he resuls for gliding ngle differen buoyncy res re copred in he nex secion. he preer of he observer for gliding ngle nd velociy re obined by uning he Qs nd Rs rices of LQR in Mlb/Siulink. he eigenvlues of observer re obined by following he robus pproch of pole plceen design. he observer gin is designed wih eigenvlues of -.7, -., i, which show h he syse hs wo rel roos nd wo iginry roos nd is rginlly sble. Figure-6 shows he gliding ngle vrious pup res by using designed observer. I shows h he pich iniil vlue hs been se o -35 o using LQR nd higher pup re he convergence is fs. On he oher hnd, he lower vlue hs slower response. According o he clculions of conrollbiliy nd observbiliy, he rnk is 4 which is equl o he nuber of se vribles. hus glider dynic syse is conrollble nd observble. SIMULAION RESULS In order o verify he effeciveness of full se observer, he gliding ngle nd velociy of he glider is checked by dding disurbnce s esureen noise v() wih norl rndo signl wih en µ= nd sndrd deviion σ =. ws dded o he inpu for gliding ngle nd velociy. he resuls re copred wih disurbnce nd wihou disurbnce. Figure-7 shows he gliding ngle vrious buoyncy res. he glider sbilizes quickly fer s higher buoyncy res. Figure-6. Inpu signl for esureen noise 8 sple res. gsin(y()) =, Y() 3 KY() L Y() (Y()(K +K Y(3)) gsin( Y()) -, = Y() Y() L L 3 3 ^ KY() L Y()(K M +K MY(3)) 33=-, 4= Y() Figure-7. Open loop response of gliding ngle wih differen pup res. g As shown in he Figures-8 nd 9, he high observer gin kes he syse very sensiive. 334

5 VOL., NO., NOVEMBER 6 ISSN Asin Reserch Publishing Nework (ARPN). All righs reserved..45 Gliding ngle o =6g Velociy (/s) Gliding ngle ( o ) Figure-. Response of he gliding ngle using full se observer. Rel se Esied se ie (s) Gliding ngle ( o ) ie (s) Figure-8. Velociy of he glider wih full se feedbck conroller. he siulion is crried ou using close loop se feedbck syse wih observer esiion. he djusble inpu ne buoyncy keep he vlue for he firs.5 seconds is iniil pek vlue nd hen he puping re grdully decreses. he glider oion is conrolled hrough he chnge of ne buoyncy using liner cuors. Buoyncy of he gliders is checked differen vlues of puping re pek vlues i.e. (g, 4g, 6g, 8g) wih respec o liner cuor s shown in Figure 4. he oher se vribles such s gliding ngle nd ngle of ck is chnging s he puping re is chnged. his ide is ken fro SLOCUM glider in which he pich is djused by chnging he cener of grviy ie (s) Figure-9. Response of he gliding ngle using full se feedbck Figure-. Closed loop response of gliding ngle sndrd deviion σ =. nd en µ=. Figure-. Response of velociy in he presence of noise. However, differen reserchers hve used differen puping re ccording o is desired rjecory which is enioned in ble-. 335

6 VOL., NO., NOVEMBER 6 ISSN Asin Reserch Publishing Nework (ARPN). All righs reserved. Gliders ble-. Differen underwer gliders wih in specificions. Pylod (kg) Slocu (bery) 3-4 Spry [] Seglider [] 5 Deep-glider [] [] Slocu (herl) [3] Liberdde XRAY glider [4] 5 85 Speed (/s) Rnge (k) Weigh (kg) Ne buoyncy (g) Mjor liiion nd chllenges here re severl hings h us be given creful reen. he in ep is o choose conroller h cn gurnee fs convergence nd rck he desired ph wih iniu overshoo vlue in he presence of exernl disurbnce. An pproprie conrol sregy is required h will follow desired rjecory wih iniu error for vrying pylods. ACKNOWLEDGEMENS Auhors re hnkful o Universii eknologi PERONAS for providing he resources required for his work. REFERENCES []. G. Grver, R. Bchyer, N. E. Leonrd, nd D. M. Frnoni, "Underwer glider odel preer idenificion," in Proceedings of he 3h Inernionl Syposiu on Unnned Unehered Subersible echnology, 3. [] K. Is nd M. Arshd, "Modeling nd oion conrol of hybrid-driven underwer glider," IMS, vol. 4, pp , 3. [3] F. Zhng nd X. n, "Nonliner observer design for sbilizion of gliding roboic fish," in Aericn Conrol Conference (ACC), 4, 4, pp Figure-3. Gliding ngle differen pup res equilibriu. CONCLUSION AND FUURE WORK Observer bsed full se feedbck conroller ws developed o invesige he dynics of he syse including blls re s n inpu nd gliding ngle ws esied hrough observer. Apr fro his, he response of he ngle of ck gins he ne buoyncy is lso observed. Bsed on he observer bsed conroller, furher invesigion cn be crried ou o esie he pich ngle when response is ken fro on-bord sensors of underwer glider during he glide operion nd hen he rel se nd esied se cn be copred using nonliner observer syse. Furherore, he nlysis cn be iproved by ipleening dvnced conroller echniques such s dpive sliding ode conrollers. In fuure work, nonliner observer bser bsed closed loop response by using robus conrol echnique will be exined. [4] N. E. Leonrd nd. G. Grver, "Model-bsed feedbck conrol of uonoous underwer gliders," Ocenic Engineering, IEEE ournl of, vol. 6, pp ,. [5] N. Mhoudin,. Geisber, nd C. Woolsey, "Dynics nd conrol of underwer gliders I: sedy oions," echnicl Repor, Virgini Polyechnic Insiue nd Se Universiy9. [6] P. Bh nd N. E. Leonrd, "Sbilizion nd coordinion of underwer gliders," in Decision nd Conrol,, Proceedings of he 4 s IEEE Conference on,, pp [7] M. M. Noh, M. R. Arshd, nd R. M. Mokhr, "Deph nd pich conrol of USM underwer glider: perfornce coprison PID vs. LQR," Indin 336

7 VOL., NO., NOVEMBER 6 ISSN Asin Reserch Publishing Nework (ARPN). All righs reserved. ournl of Geo-Mrine Sciences, vol. 4, pp. -6,. [8] P. Bh nd N. E. Leonrd, "Nonliner gliding sbiliy nd conrol for vehicles wih hydrodynic forcing," Auoic, vol. 44, pp. 4-5, 8. [9] F. Zhng, X. n, nd H. K. Khlil, "Pssiviy-bsed conroller design for sblizion of underwer gliders," in Aericn Conrol Conference (ACC),,, pp []. NAGARAAN nd U. BARKA, "Sudy on Wing Aspec Rio on he Perfornce of Gliding Roboic Fish," Applied Mechnics & Merils, vol. 786, 5. [] S. Wood, "Auonoous underwer gliders," Underwer Vehicles, pp , 9. [].. Osse nd C. C. Eriksen, "he Deepglider: A Full Ocen Deph Glider for Ocenogrphic Reserch," in OCEANS 7, 7, pp. -. [3] C. ones, D. Webb, S. Glenn, O. Schofield,. Kerfoo,. Kohu, e l., "Slocu glider exending he endurnce. Durh, NH, Augus 3-6 9," in he 6h Inernionl Syposiu on Unnned Unehered Subersible echnology (UUS9). [4] M. F. bin Ibrhi, M. Ovinis, nd K. bin Shehbuddeen, "An Underwer Glider for Subse Inervenion: A echnicl Fesibiliy Sudy," Applied Mechnics nd Merils, vol. 393, pp ,