SPIRALING MOTION OF AN UNDERWATER GLIDER: DYNAMIC MODELING

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1 OL. 11, NO., NOEBER 16 SSN ARPN Journal of Engneerng and Appled Scences 66 Asan Research Pulshng Network (ARPN). All rghts resered. SPRALNG OON OF AN UNDERWAER GLDER: DYNAC ODELNG uhaad Yasar Jaad 1, ark Ons 1, Fakhruldn B Hash 1, Ad aun, Yasser. Ahed, 3 and Barkat Ullah 1 1 Departent of echancal Engneerng, Unerst eknolog Petronas, Bandar Ser skandar, Perak, alaysa arne echnology Centre, Faculty of echancal Engneerng, Unerst eknolog alaysa, Johor Bahru, alaysa 3 Departent of Naal Archtecture and arne Engneerng, Faculty of Engneerng, Alexandra Unersty, Alexandra, Egypt E-al: yasar48@gal.co ABSRAC An underwater glder s a class of autonoous underwater ehcles. Whle these glders typcally oe n a sawtooth pattern, a spral oton, whch ay ore effecte for specfc applcatons, s consdered here. he spral oton of glder ay extend ts possle applcatons such as delery or recoery equpent for susea nstallaton. n ths paper, a spral glde path for the glder s consdered, and the correspondng dynac odel ased on Lagrangan prncple and analytcal expressons deterned. he steady-state spralng equatons were dered and soled recursely usng the fsole algorth. he results copare well wth sulaton results ased on Newton s ethod. he spralng oton s hghly aneuerale, wth less than 1 turnng radus. eywords: underwater glder, dynac odel, spralng oton. NRODUCON n 1989, Henry Stoel [1] pulshed a reolutonary artcle aout uoyancy dren floats for oceanography. Snce then, arous underwater glders such as ALBAC [], Slocu [3], Spray [4], Seaglder [5]were deeloped for arne applcatons at dfferent depths of ocean. An underwater glder s a specal type of autonoous underwater ehcles (AU) that s prarly used n oceanographc sensng and data collecton. n these applcatons, they are attracte ecause of ther low cost, autonoy and capalty for long-range, extended duraton deployents. hese glders are uoyancy dren low energy consupton autonoous underwater ehcles wth fxed wngs and rudder, a cylndrcal hull and nternal ong asses [6, 7]. he operatng prncple of an underwater glder s that at deployent t s negate uoyant and therefore tends to de, durng whch ts wngs conert downward oton nto the horontal plane, thus producng a forward force. Once a predeterned depth s reached, the ehcle changes ts uoyancy to ecoe neutrally uoyant [8, 9]. hor. Fossen [1, 11] deeloped a nonlnear 6-DOF dynac odel for arne ehcles ased on Euler LaGrange syste ncludng control otons of nternal coponents. Leonard and Graer [1, 13] dered a generale dynac odel of underwater glder ased on frst prncples ncludng nonlnear couplng etween nternal ong ass and glder. Zhang [14] dered the dynac odel of gldng rootc fsh ased on Newton prncples y splfyng ts dynacs oton at sagttal plan. Zhang et al [157] were proposed the spral oton of gldng rootc fsh y deflectng ts tal. Leonard and Bhatta [18] proposed the nuercal sulaton of spral oton y changng the poston of nternal ong asses. t s requred ore coplcated control systes to anage the poston of nternal ong asses durng spral oton. Howeer, he spral oton approach that used y Zhang [15, 16] s ore approprate to acheed hgh aneueralty. hs work s an extenson of Zhang s, wth ephass on the dynacs of glders n a spral glde path. hs paper s organed as follows: Secton outlnes the deraton of a coplete non-lnear dynac odel of an underwater glder ased on Euler LaGrange prncple wth an nternal ong ass n a undrectonal oeent to control the ptch angle of glder. n Secton the splfed nonlnear dynac odel for a steady state 6-DOF spralng oton of glder s dered and soled usng the fsole recurse algorth. n Secton, the results are aldated wth preously pulshed experental and sulaton results y Zhang [15, 16]. DYNAC ODELNG OF UNDERWAER GLDER A splfed pont ass dynac odel of a glder wth nternal ong ass for ptch control s used to descre the glders 6-DOF oton. n ths work, an underwater glder, ncludng all nternal ong ass oents and external forces, s consdered as a rgd ody. he poston of nternal ong ass wll control the glde angle and speed. he ass dstruton an underwater glder s shown n Fgure. Fgure. Glder ass dstruton [19]. 136

2 OL. 11, NO., NOEBER 16 SSN ARPN Journal of Engneerng and Appled Scences 66 Asan Research Pulshng Network (ARPN). All rghts resered. he total glder ass or ody ass can e expressed as = h + w + +. Where h represents a unfor glder hull ass, w pont ass wth dsplaceent r w to the fxed center of graty and uoyancy, the oale ass wth ector poston r p control the ptch angle durng gldng and the arale allast ass wth respect to geoetry center (GC). he ass s the ass of dsplaced flud. he glder s neutrally uoyant f the s poste (float) and ce ersa. NEACS ODEL wo fraes of reference, one ody frae and other nternal frae are requred to defne the oton of rgd ody as shown n Fgure-. Let e 1, e, e 3 denote ody frae related to, j, k nertal frae of references respectely, as shown n Fgure-. e 1 and e les n the horontal plane and s perpendcular to the graty along the wngs of the glder respectely. he k axs s poste downwards and les n the drecton of graty. to where s s sne and c s cosne. [θ, φ, ] represents the ptch angle, roll angle and yaw angle respectely. DYNAC ODEL he dynac odel s dered ased on the Lagrangan prncple nstead of the Newton-Euler forulaton adopted y Zhang [14]. he Lagrangan forulaton s ased on the energy of a dynac syste []. he Lagrange s prncple n general coordnates s d dt dl d dl Q dx L=.E-P.E (4) Where, L=netc Energy (.E) Potental Energy (P.E) and Q s external forces. he general knetc energy expresson of dynac syste s (3) 1 1.E (5) Fgure-. Body frae axs of [1] n order to odel the kneatcs of the glder atheatcally, the center of graty (CG), center of uoyancy (CB), rotatonal atrx R, generaled poston and elocty ust e dentfed. he sx degrees of freedo kneatc equatons for the glder are descred n [14] Let the poston of the glder fro the nerta frae to the orgn of ody frae e x y and orentaton R φ s x y θ. he glders longtudnal elocty and relate angular elocty x y. hus, the kneatc equatons are. R R (1) R () cθ cφ sφ *sθ * c cφ *s cφ *sθ * c sφ *s R cθ * c cφ * c sφ *sθ *s sφ * c cφ *sθ *s sθ sφ * cθ cφ *cθ he frst ter n Equaton 5 s the knetc energy due to translaton elocty and the second ter s the knetc energy due to angular elocty ω of the dynac syste. Underwater glders work under the nfluence of graty force and uoyancy. he graty force acts n a downward drecton along the poste the axs of the glder. he knetc energy of underwater glder s.e 1 1 (6) r Where, r f added ass respectely, f r f s rgd ody ass and rgd ody nerta r f ass and added nerta respectely. he potental energy of dynac syste due to graty force can e expressed as P.E gz (7) where the g s the graty force and Z s the poston of center of ass of dynac syste n nertal frae of reference. Fro equaton 7 the potental energy of syste n ody fxed coordnates s P.E g R Z (8) 137

3 OL. 11, NO., NOEBER 16 SSN ARPN Journal of Engneerng and Appled Scences 66 Asan Research Pulshng Network (ARPN). All rghts resered. where Z R s represented as []. Usng the alues of.e and P.E n Equaton 4 L 1 1 g R Z (9) Dfferentatng Equaton. (9) wth respect to translatonal elocty and angular elocty and then dfferentatng wth respect to te. d dl d d 1 1 g R Z (1) dt d dt d After splfcaton, the translaton elocty of the glder s gr k F (11) Here, F s total external force actng on the dynac syste. F gr k gr k F u (1) ext Here, k s the unt ector along the graty n Z- axs drecton, F ext s the external hydrodynac forces actng on the glder ody, expressed n the ody-fxed frae, and u s the total force exerted on the oale ass y the glder structure, expressed n the ody fxed frae. s glder ass actng along the graty of the glder. he control nput force u for oeale ass s u gr k u (13) 1 gr k F u (14) ext Here r f Slarly, the equaton of angular oents s J (15) J J he external oent can e descred ncludng oale ass and external force as gr (R k) r u (16) ext w w p he uoyancy ass and nternal ong ass control force u s u g 1 (J J R k u ext gr (R k) r u p p u s uoyancy control. oltage appled to the pup. gr w w (R k) (17) u wll e splfed as the HYDRODYNAC ODEL n order to study the hydrodynac ehaors, all the elocty fxed coordnates are frst transferred to the ody frae of reference. For ths purpose, the rotatonal atrx R s used. CS CS S R S C (18) SC SS C where s s sne and c s cosne. 1 Where α s the angle of attack α tan ( / x ) and β s the sdeslp angle sn 1 ( y /). he hydrodynac forces (drag, lft and sde force) and oent (roll oent x, ptch oent y and yaw oent ) are transferred fro elocty frae to ody frae of reference. Hence, F R D SF L (19) ext R () ext x y he hydrodynac forces and oents are generally dependent on the angle of attack, sdeslp angle and elocty. δ α α δ D D D D (1) δ δ SF sf sf () α α D L L (3) x R q1 x (4) 138

4 OL. 11, NO., NOEBER 16 SSN ARPN Journal of Engneerng and Appled Scences 66 Asan Research Pulshng Network (ARPN). All rghts resered. y α q y y q3 y (5) (6) Here δ s the rudder deflecton,,, are q1 q q3 the rotatonal dapng coeffcents. he coeffcents are ased on the CFD sulaton [6, 17, 1]. SEADY-SAE SPRALNG OON n ths study, the steady spralng oton of underwater glder s nestgated nuercally through an terate ethod. A nuercal algorth ased on the fsole recurse algorth was used to sole the 6-DOF dynac equatons of glder. he spralng oton of glder wll e anpulated y three control nputs ( δ,, ) deflecton of rudder, poston of nternal ass to control the ptch angle and net uoyancy rate respectely. he hydrodynac angles (α, β) are consderaly effected y the hydrodynac forces and oents durng the oton of glder under the nfluenced of fxed control nputs. he angular elocty along -axs n ody fxed coordnates s snθ φcos θ φcos θ R R sn (7) cos he translatonal elocty n ody fxed coordnates s R (8) αcos β β αcos β cos sn (9) sn he steady state spralng oton dynac equatons were otaned y settng the derates and all control forces n Equaton. 14 and Equaton. 17 to ero. gr k F (3) ext J gr R k ext w w gr R k p he 6-DOF dynac equatons at steady state condtons are otaned y susttutng all the alues n the dynac odel. y sn gsn sn gsn x gcos βcos φcos θ snαcos βsnφcos θ θ Dcos αcos β SFcosα snβ Lsnα...( 31) αcos βsnθ cosα cos βcos φcos θ x φcos θ Dsnβ SFcosβ ( 3) αcos βsn φcos θ snβsn θ cos θcos φ Dsnαcos β SFsnα snβ Lcosα ( 33) (J - J 3 3 )sn x cos x )cos gr p )sn φcos θ cosφ βsn αcos β - w gr w snφcos θ + αcos β - cosαsn β - snα...( 34) = (J x - J )cos y )cos + x sn y φsnθ cos θ y + ( αsnα cos β - w gr w snθ - cosφcos θ snβ cosβ....(35) = (J y - J x )sn + x φsnθ cos θ + ( - + y βcos αsnβ + gr p snφcos θ αcos β - snαsnβ cosα...(36) y + ( x - + he sx states: ptch angle, headng angle, angular elocty along -drecton, glder elocty, angle of attack and sde angle θ, φ,,, α,βwas soled nuercally to defne the spralng oton of underwater glder for three nputs forces (nternal ong ass poston, net uoyancy and rudder deflecton). he dynac of spralng otons s hghly nonlnear due to trgonoetry functons noled n these equatons. We wrte these equatons n steady state functon to otan the soluton nuercally. n ths study, the spralng otons of a prototype gldng rootc fsh-lke underwater glder [17] wth paraeters n ale 1 are consdered. he spralng oton of glder noles two paraeters. he rotatonal oton around the -drecton wth radus R and glde speed n ertcal plane ertcal - ertcal sn(θ α) (37) R cos(θ α)/ (38) RESULS AND DSCUSSONS θ,φ, ntal alues.1, were 3, α,β chosen for the fsole teraton algorths. he nuercal 139

5 OL. 11, NO., NOEBER 16 SSN ARPN Journal of Engneerng and Appled Scences 66 Asan Research Pulshng Network (ARPN). All rghts resered. soluton of the gen states and ntal condtons are shown n ale. he glde elocty ncreases wth ncreasng net uoyancy whle at fxed uoyancy and poston of oeale ass wth change of ruder deflecton s decreased the turnng radus of spral otons. he nuercal sulaton results are consstent wth the Newton teraton ethod and experental results y Fetan [157] as shown n Fgure-3 and Fgure-4. he Newton ethod requres the Jacoan/derates for the ealuaton of the nonlnear equatons, whch s te consung and dffcult for 6- DOF nonlnear equatons [], although a Jacoan ay e desrale for a poorly scaled prole. Howeer, Newton s ethod cannot e used f the Jacoan s a sngular atrx. On the other hand, the atla s fsole functon usng Leenerg-arquardt Algorth s coputatonally effcent and does not requre a Jacoan to sole nonlnear equatons. ale. Glder paraeters Paraeters alues Paraeters alues x 3.88kg y 9.9kg 5.3kg.8kg D.45 - SF rad L /rad r 5 /rad y - kg. - kg D 1.5 SF rad rad rad L.5 - kg /rad P -. /rad y q s/rad q -.5 -s/rad q s/rad S.1 and longtudnal axs of the glder. On the other hand, the steady spralng oton of glder s also functon of net uoyancy. As the uoyancy ass ncreases, the ertcal elocty ncreases, ut the radus of the spral oton decreases. he poston of nternal ptch control ass ncreases or decreases the sdeslp angle whch nfluences the glde speed and turnng radus. Fgure-3. urnng radus of spralng oton erse net uoyancy at fxed poston of nternal ptch control ass. he relatonshp etween the spral turnng radus R and the elocty on ertcal plan ertcal s nersely proportonal, as the turnng radus and elocty decreases wth an ncrease of tal deflecton angle. When the turnng radus s ncreased, the sde slps angle β also ncreases, whch nfluence the elocty of the glder. he sdeslp angle, β s an angle etween the elocty ector Fgure-4. urnng radus of a spralng oton erse tal deflecton at fxed poston of nternal ptch control ass (5c) and a net uoyancy of 3 gra. 131

6 OL. 11, NO., NOEBER 16 SSN ARPN Journal of Engneerng and Appled Scences 66 Asan Research Pulshng Network (ARPN). All rghts resered. he experental results and sulaton results dffer -8%. he experental results ay e sujected to easureents error. hese errors ay also e due to the glder not fully acheng steady state condtons [17, 3] e.g. n a transent phase. Experental data acqured through the caera s susceptle to errors. CONCLUSONS n ths study, a atheatcal odel of a glder ased on Euler-Lagrangan ethod s dered. hs odel s a splfcaton of Graer and Leonard [1, 13] nonlnear dynac odel and consders all external forces. he steady state spralng oton equatons were dered and nuercally soled ased on the fsole recurse algorth. he results are n close agreeent wth those ased on Newton s ethod as well as experental results. he dered odel s an alternate to that ased on Newton s ethod, as t can e soled wthout coputng the Jacoan. ACNOWLEDGEENS Authors are thankful to Unerst eknolog PERONAS for prodng the resources requred for ths work. REFERENCES [1] H. Stoel, "he slocu sson," Oceanography, ol., pp. -5, []. awaguch, Y. ooda, H. oayash, and. Ura, "Deelopent and sea trals of a shuttle type AU" ALBAC"," n nternatonal Syposu On Unanned Untethered Suersle echnology, 1993, pp [3] D. C. We, P. J. Sonett, and C. P. Jones, "SLOCU: An underwater glder propelled y enronental energy," Oceanc Engneerng, EEE Journal of, ol. 6, pp , 1. [4] J. Sheran, R. E. Das, W. Owens, and J. aldes, "he autonoous underwater glder," Oceanc Engneerng, EEE Journal of, ol. 6, pp , 1. [5] C. C. Erksen,. J. Osse, R. D. Lght,. Wen,. W. Lehan, P. L. San, et al., "Seaglder: A long-range autonoous underwater ehcle for oceanographc research," Oceanc Engneerng, EEE Journal of, ol. 6, pp , 1. [6] S. A. Jenkns, D. E. Huphreys, J. Sheran, J. Osse, C. Jones, N. Leonard, et al., "Underwater glder syste study," 3. [7]. sa,. Arshad, and S. shak, "A hyrd-dren underwater glder odel, hydrodynacs estaton, and an analyss of the oton control," Ocean Engneerng, ol. 81, pp. 1119, 14. [8] G. Grffths, C. Jones, J. Ferguson, and N. Bose, "Undersea glders," Journal of Ocean technology, ol., pp , 7. [9]. Y. Jaad,. Ons,. Nagarajan, and F. B. Hash, "Underwater Glders: A Reew," n AEC We of Conferences, 14, p.. [1].. Fossen, Gudance and control of ocean ehcles ol. 199: Wley New York, [11].. Fossen, arne control systes: gudance, nagaton, and control of shps, rgs and underwater ehcles,. [1] J. G. Graer, "Underwater glders: Dynacs, control and desgn," Cteseer, 5. [13] N. E. Leonard and J. G. Graer, "odel-ased feedack control of autonoous underwater glders," Oceanc Engneerng, EEE Journal of, ol. 6, pp , 1. [14] F. Zhang, J. hon, C. hon, and X. an, "nature underwater glder: Desgn, odelng, and experental results," n Rootcs and Autoaton (CRA), 1 EEE nternatonal Conference on, 1, pp [15] F. Zhang, "odelng, desgn and control of gldng rootc fsh," chgan State Unersty, 14. [16] F. Zhang, F. Zhang, and X. an, "Steady spralng oton of gldng rootc fsh," n ntellgent Roots and Systes (ROS), 1 EEE/RSJ nternatonal Conference on, 1, pp [17] F. Zhang, F. Zhang, and X. an, "al-enaled Spralng aneuer for Gldng Rootc Fsh," Journal of Dynac Systes, easureent, and Control, 14. [18] P. Bhatta and N. E. Leonard, "Nonlnear gldng stalty and control for ehcles wth hydrodynac forcng," Autoatca, ol. 44, pp. 145, 8. [19] F. Zhang, J. hon, C. hon, and X. an, "nature Underwater Glder: Desgn and Experental Results,"

7 OL. 11, NO., NOEBER 16 SSN ARPN Journal of Engneerng and Appled Scences 66 Asan Research Pulshng Network (ARPN). All rghts resered. [] G. Antonell and G. Antonell, Underwater roots: Sprnger, 14. [1] P. Jagadeesh and. ural, "RANS predctons of free surface effects on axsyetrc underwater ody," Engneerng Applcatons of Coputatonal Flud echancs, ol. 4, pp , 1. [] W. Y. Yang, W. Cao,.-S. Chung, and J. orrs, Appled nuercal ethods usng ALAB: John Wley and Sons, 5. [3] S. Zhang, J. Yu, A. Zhang, and F. Zhang, "Spralng oton of underwater glders: odelng, analyss, and experental results," Ocean Engneerng, ol. 6, pp. 13,