Design and Development of Low Cost Variable Buoyancy System for the Soft Grounding of Autonomous Underwater Vehicles

Transcription

1 Design and Development of o Cost Variable oyancy System for the Soft ronding of Atonomos Underater Vehicles Jeffery S. Riedel, Anthony J. Healey, David. arco and ahadir eyazay Naval Postgradate School Center for AUV Research onterey, CA voice, fax {jsriedel, healey, marco, bbeyazay}@me.nps.navy.mil ASTRACT To provide a vehicle ith the ability to hold position in a coastal environment reires a significant amont of onboard poer. This poer reirement either forces the vehicle size to increase to allo for sitable mission dration or redces the amont of time the vehicle has to condct its mission. To relax the poer reirement, e propose to develop vehicles that can employ a bottom-sitting or soft gronding behavior. To obtain this behavior reires vehicles that have the capability to selfballast. y optimally positioning itself and sitting on the bottom, the AUV can be placed in a sleep mode, ith only monitoring sensors aake, thereby conserving poer. In this paper e present the preliminary ork condcted in the areas of simlation, design and testing of a Variable oyancy System VS for an Atonomos Underater Vehicle AUV. This boyancy system ill be integrated into the ne NPS AUV hich is crrently nder constrction, to spport the pcoming joint operations ith the University of isbon's ARIUS vehicle. We ill discss the tradeoffs and analysis that ent into the design of the system, as ell as the challenges associated ith the integration of sch a behavior and system into the vehicle. INTRODUCTION Energy storage is limited in AUV s. To assist ith energy management, data gathering missions have been proposed here the vehicle shold sit on the bottom and gather acostic/video/chemical data over extended periods of time. In this gronding scenario, thrsters may be sed. Hoever, there are to disadvantages for this method: high energy consmption and restricted se close to the ocean bottom. The motivation for this paper is to stdy a lo cost, simple soft gronding capability for a sbmersible vehicle sing controllable ballast. For simplicity, ater ballast is considered. The design of the control system is based on the NPS Phoenix AUV. The ballast system is designed to control the eight addition into or ot of the to ballast tanks. allast control of vehicles is not a ne sbject and e can find many examples beginning in the 9 s, the non-rigid airships are very good examples of ballast control. One of the most important elements of a non-rigid airship is the ballonet-system. A ballonet as seen in Figre is an airbag one or to of them inside the envelope, hich is provided ith air from a bloer or directly from the engine nit. The air cold be removed from the ballonet throgh the valves. If the airship has a front and aft ballonet then the height and pitch of the airship can be steered. For example, if the aft bag is filled ith more air, then the airship ill become heavier in the rear part of the envelope and the ship ill incline increasing the altitde of the ship by sing the engines. As Figre depicted, the airship can also be statically trimmed []. Control as manal. For most nderater vehicles, the depth / pitch control is normally provided by hydroplanes. As an example, consider the NPS Phoenix AUV, the IT Odyssey and the WHOI Rems. At lo speed hoever, the control srfaces provide redced control athority and the ballast control problem is very complex de to nonlinear, time-varying, ncertain hydrodynamics. Inherent lags arising from the integration of ballast ater flo rate commands into eight change makes the control difficlt to stabilize. There are some designs that sed a bang-bang control system []. The ARPA s Unmanned Undersea Vehicle UUV employed a fzzy logic ballast controller hich as claimed to be comparable ith the performance that can be obtained from standard control technies, bt does not reire traditional linear or nonlinear design methods. Figre. Sectional elevation of the Parseval-Airship "P VI", 9.

2 Report Docmentation Page Form Approved O No Pblic reporting brden for the collection of information is estimated to average hor per response, inclding the time for revieing instrctions, searching existing data sorces, gathering and maintaining the data needed, and completing and revieing the collection of information. Send comments regarding this brden estimate or any other aspect of this collection of information, inclding sggestions for redcing this brden, to Washington Headarters Services, Directorate for Information Operations and Reports, 5 Jefferson Davis Highay, Site, Arlington VA -3. Respondents shold be aare that notithstanding any other provision of la, no person shall be sbject to a penalty for failing to comply ith a collection of information if it does not display a crrently valid O control nmber.. REPORT DATE 5. REPORT TYPE 3. DATES COVERED -. TITE AND SUTITE Design and Development of o Cost Variable oyancy System for the Soft ronding of Atonomos Underater Vehicles 5a. CONTRACT NUER 5b. RANT NUER 5c. PRORA EEENT NUER 6. AUTHORS 5d. PROJECT NUER 5e. TASK NUER 5f. WORK UNIT NUER 7. PERFORIN ORANIATION NAES AND ADDRESSES Naval Postgradate School,Center for AUV Research,onterey,CA, PERFORIN ORANIATION REPORT NUER 9. SPONSORIN/ONITORIN AENCY NAES AND ADDRESSES. SPONSOR/ONITOR S ACRONYS. DISTRIUTION/AVAIAIITY STATEENT Approved for pblic release; distribtion nlimited 3. SUPPEENTARY NOTES The original docment contains color images.. SPONSOR/ONITOR S REPORT NUERS. ASTRACT To provide a vehicle ith the ability to hold position in a coastal environment reires a significant amont of onboard poer. This poer reirement either forces the vehicle size to increase to allo for sitable mission dration or redces the amont of time the vehicle has to condct its mission. To relax the poer reirement, e propose to develop vehicles that can employ a bottom-sitting or soft gronding behavior. To obtain this behavior reires vehicles that have the capability to selfballast. y optimally positioning itself and sitting on the bottom, the AUV can be placed in a sleep mode, ith only monitoring sensors aake, thereby conserving poer. In this paper e present the preliminary ork condcted in the areas of simlation, design and testing of a Variable oyancy System VS for an Atonomos Underater Vehicle AUV. This boyancy system ill be integrated into the ne NPS AUV hich is crrently nder constrction, to spport the pcoming joint operations ith the University of isbon s ARIUS vehicle. We ill discss the tradeoffs and analysis that ent into the design of the system, as ell as the challenges associated ith the integration of sch a behavior and system into the vehicle. 5. SUJECT TERS 6. SECURITY CASSIFICATION OF: 7. IITATION OF ASTRACT a. REPORT nclassified b. ASTRACT nclassified c. THIS PAE nclassified 8. NUER OF PAES 9a. NAE OF RESPONSIE PERSON Standard Form 98 Rev Prescribed by ANSI Std 39-8

3 In another fzzy logic control model, a 5 state Kalman filter as developed to provide estimates of the motion variables and the applied lift and tore acting on the UUV. The control la decided beteen three possible control actions; pmp ater in both tanks, pmp ater ot of both tanks and trn both pmps off. The fzzy inpt state space as composed of depth error and depth rate, and each is divided into partitions. The fzzy controller interpolated beteen the partitions alloing the control to vary smoothly as the states move from one partition to another. These movements of states ere provided by on and off of ballast pmps [3]. In this paper e otline the development of a depth controller sing sliding mode control technies for a netrally boyant vehicle. The sliding mode controller is designed on the basis of the simplified for degrees of freedom vertical plane eations of motion. A linear adratic reglator QR proportional approach is then tilized for the design of the ballast controller, hich prodces flo rate commands, alloing the vehicle to have a soft gronding behavior. These to controllers se a logic based depth reglator to provide realistic simlation of the vehicle s flight and gronding operations in a single mission. VEHICE ODEIN AND EQUATIONS OF OTION We ill deal ith only vertical plane variables; i.e., heave, pitch, and srge. The vertical plane stability analysis involves heave and pitch motions. Hoever, the srge eation coples into pitch and heave throgh the offset, z. This is a dynamic copling, and cold be eliminated by redefining hydrodynamic coefficients ith respect to the ship s center of gravity instead of its geometric center. Restricting the motions of the vehicle to the vertical dive plane, the only significant motions that mst be incorporated to model the vehicle in the dive plane are, the srge velocity, the heave velocity, the pitch velocity, the pitch angle θ and the global depth position. θ mx + I + U + U s y + α b ρ mx U mz nose x CDb x x xw xcosθ zw zsinθ U sinθ + cosθ s tail 3 xdx These eations can be linearized for a level flight path hen the folloing are obtained : θ m mx + mx + I y θ U + m U U + z z W U mx U U U For the cases considered dring this ork, the vehicle has also to ballast tanks hich ere designed to be sed dring the gronding. These ballast tanks can be seen in Figre. allast Tank pmp pmp allast Tank Figre. The ocation of allast Tanks in the AUV The ne forces are and and since the ballast tanks are not located in the same distance from the center of gravity of the vehicle there ill be also to moments, and. There ill be also small change in moment of inertia. So all these changes can be listed as; W Wo + + Wo + + m, g and the ne eations of motion become, m mx mx I y +

4 3 + U mx U m z W z A o U U d d o s U iables state θ _ var here x - A o x + - o CONTRO SYSTE DESIN Flight Control Flight control and eight control compose to main sbsystem of a soft gronding system. In flight control the vehicle is kept netrally boyant and the plane angles are the control inpts. Hoever in eight control, the flo rates for both ballast tanks are controlled ith zero forard velocity and plane angle. These to components of the designed control system ere explained in folloing sections. The dynamics of nderater vehicles are described by highly nonlinear systems of eations ith ncertain coefficients and distrbances that are difficlt to measre. An atomatic controller for this kind of vehicle mst satisfy to conflicting reirements: First, it mst be sophisticated enogh to perform its mission in an open ocean environment ith ever-changing vehicle /environment interactions. Second, it mst be simple enogh to achieve real-time control ithot nonessential comptational delays. Sliding mode control theory yields a design that flfills the above reirements. It provides accrate control of nonlinear systems despite n-modeled system dynamics and distrbances. Frthermore, a sliding mode controller is easy to design and implement. A very effective sliding mode controller can be developed from the linearized eations of motion for an nderater vehicle [7], here the control la becomes [7], / sgn φ η x s s kx T T sat 3 The gain vector k can be fond easily by sing atlab. The atlab command place accepts as inpts the A and matrices along ith a vector of the desired closed loop poles and retrns the vector k. Weight Control The vehicle s gronding behavior can be simlated by adding eight proportionally to both tanks at constant flo rate and by sing zero plane angle. It is needed to add eight proportionally to eliminate the moment effect since these ballast tanks are not located in the same distance from the center of gravity. As it can be seen from the Figre, >. To get zero moment, Since ballast control is achieved throgh commanded pmp flo rate, to more state eations are added to those existing for states, f f here i represents change of eight in tank i and f i represents flo rate of pmp i. So the ballast control eations of motion become : z I mx mx m y ω θ + + f f z U mx z W z U m o ω θ Flo rates for ballast tanks are control inpts. Weight Control With inear Qadratic Reglator The system above is given as, Ax x + and the closed loop optimal control la can be fond by Sx R T here S can be fond by solving the algebraic Riccati eation for the positive-definite S, Q S SR SA S A T T + + In atlab, the command lra,,q,r

5 gives the continos-time, linear, adratic reglator problem and the associated Riccati eation. This command calclates the optimal feedback gain matrix K for control la hich minimizes the ell knon performance index. ROUNDIN WITH VERTICA THRUSTERS The bladed thrsters are the essential elements of improved vehicle positioning systems. With atomatic position control, the thrsters enable important scientific and indstrial tasks sch as atomatic docking, station keeping, precise srveying, inspection, sample gathering and maniplation. Incorporating precise models of thrster dynamics into the feedback control systems of marine vehicles promises improved vehicle positioning [9]. ost small-to-medim sized nderater vehicles are poered by electric motors driving propellers monted in dcts. The propeller is monted in a dct or shrod in order to increase the static and dynamic efficiency of the thrster. Thrsters are sbject to serios degradation de to axial and cross flo effects. Axial flo effects can be reasonably approximated by the modeling of the thrster nit alone, the velocity of the flid entering the thrster shrod effectively changes the angle of attack of the propeller, ths altering the force prodced. Cross flo effects are mch more difficlt to model and are highly dependent on the position of the thrster on the vehicle. The amont of force prodced by the thrster ill redce the overall gain of a control system nless these effects are specifically in the controller design []. ρ mx ρ nose tail nose x C Db x dx + x + I y x C b x x 3 3 xdx thrster mx U mz D tail xwo xcosθ zw zsinθ + + cosθ thrster + U here thrster and thrster are thrster force and thrster moment respectively. verthrst horthrst verthrst Figre 3 Horizontal and Vertical Thrsters of NPS Phoenix AUV RESUTS AND DISCUSSION horthrst For this ork, NPS Phoenix vehicle is taken as an example. Figre 3 shos the locations of vertical and horizontal thrsters on the vehicle. Those for tbes represent the thrster shrods. In Figre 3, the vertical thrster tbes can be seen throghot the vehicle. Thrster blades are located close to the bottom of those tbes. Thrster moment and force eations ere developed by Whitcomb and Yoerger [9], amongst others. In that paper, the control system for these thrsters as also discssed. t in this stdy, these thrster force and moments ere assmed as some constant parameters and also no control la as developed to control them. Since the main element for gronding is the eight control, the thrsters ere jst sed as axiliary elements of this procedre in order to increase depth rate. y sing thrsters in addition to the eight control, folloing changes shold be made to heave and pitch rate eations, m + mx m + U + mz + U + + In previos chapters, the mathematical models of the control system ere developed. To prove the validity of the flight and eight controllers, the system as simlated by sing the parameters of NPS Phoenix AUV. First the AUV as controlled by a flight controller sliding mode control dring its diving from srface to a commanded flight depth. Second, the flight controller and the different cases of eight controller ere simlated on the vehicle. Flight Control In the NPS Phoenix vehicle, there are for vertical control planes poered by servo motors. Using the parameters of NPS Phoenix AUV and a nominal speed of ft/sec, A and matrices become A cosθ

6 y choosing the poles as p [-, -., -., ], the vector k as calclated, k [ , -.7, , ] 5.3 A C is calclated from A C A k A C The eigenvector of A T C for the pole at the origin is the sliding srface s [ ; -.7 ; -.69 ;.87] as a reslt ith φ., the control la becomes θ.7ω θ.7ω satsgn /...87 z z com 5 The motion of the vehicle is restricted to the vertical plane. The motion profiles for depth and pitch have been specified sing sliding mode control. For the manever, the commanded depth as ft and the vehicle as originally at the srface. As can be seen from the Figre, dring the flight, maximm pitch angle becomes. rad ~.5 degrees. When the vehicle reaches the commanded depth as seen in Figre 5, the pitch angle becomes zero as expected. The controller prodces the dive plane angle command according to the depth error. In the beginning the depth error is large, so the system prodces higher vales of plane angle command in order to eliminate this error. With the fll state feedback, the vehicle responded very ell to these commands. Adding Weight To oth Tanks Withot Control After completion of the flight to the commanded depth, the vehicle gets ater to both tanks in order to become heavy. Figre 6 shos the eight increase in the tanks. Dring gronding, the planes kept at zero degrees as depicted in Figre 7. To keep depth rate ithin limits, the maximm eight pmped in as limited at 5 lb. for each tank. With this additional eight, the vehicle sat on the grond ft. ith.6 ft/sec depth rate. Even thogh the eight as added proportionally / to get zero moment effect, there is still some moment becase of the vehicle motion. As a reslt, the pitch angle increases since there is no control on either depth rate or pitch angle. As seen on Figre 8 and Figre 5-6, at the end of a 6 ft. drop, the pitch angle becomes.6 rad. 35 degrees. This method can be sed for very short gronding depths -3 ft, bt for other cases, it is not recommended since the system is completely nstable. Pitch Angle Rad. Depth ft Pitch angle vs time Time sec Figre. Pitch Angle Plot Dring The Flight Depth vs time Time sec Figre 5. Depth Change As A Fnction Of Time Dring The Flight. Weight in balast tank lb Weight in balast tank vs time Figre 6. Weight Addition Dring ronding With No Control On Flo Rates 5

7 Plane angle rad Plane angle vs time Figre 7. Plane Angle Dring Flight And ronding Depth vs time Depth ft Figre 8. Depth Change Dring The Flight And ronding Pitch angle vs time Pitch angle rad Figre 9. Pitch Angle Change For Flight And ronding Weight Control With inear Qadratic Reglator In order to keep the depth rate and pitch angle ithin limits, the linear adratic reglator technie as sed in designing a eight control. The parameters of NPS Phoenix vehicle ere sed for simlation. With these knon parameters and. ft/sec forard velocity, A and matrices become, A In practice, high vales of pitch angle > 5 degrees are not desired for a safe and stable gronding of the vehicle. So the designed control la shold not tolerate large pitch angles. Choosing larger elements of Q for the pitch angle compared ith the others can provide improved control. So, the Q and R matrices ere chosen as follos: Q 3 R y sing lr command in atlab, the control gain matrix K can be obtained, as K Since gronding to the ocean floor from a certain depth desired, the command matrix shold be x com [ z gr ] here z gr grond depth - the depth here gronding is started. So the control la becomes, here -Kx error x error x - x com The simlation of the system ith this control la can be seen in Figre throgh Figre 7. Positive flo rate represents ater inlet to the ballast tanks, and negative flo represents the opposite. The pmps are not alloed to pmp ot hen there is no ater in ballast tanks. Figre and Figre sho hen the eight in a ballast tank and the pmp flo rate become zero. This is provided by a simple controller hich compares the eight in ballast tank and flo rate f. If is eal to zero and flo rate is a negative nmber, than the control inpt f of that pmp becomes zero. 6

8 At the commanded depth, the speed control nit slos don the vehicle to an almost zero forard velocity U. ft/sec. The speed controller's other dty is to control the longitdinal position. Dring the flight, the speed control nit compares the vehicle s location X ith commanded location X com hich is a longitdinal distance from the original position. When the vehicle is at the commanded depth of flight, a deceleration procedre starts. A simple algorithm as sed to calclate the minimm distance needed for deceleration to reach the commanded location at the end of the gronding. The change of forard velocity de to the depth change can be seen in the Figre 6. The depth rate as seen in Figre is very lo in this method becase of the command given to the eight control. The eight control prodces its control vales de to the errors that are the differences beteen the commanded and the actal states. In the above case, only the depth command has a vale, the commands for other states are zero. At the end of the simlation, the pitch angle becomes almost zero as seen in Figre 5. In the first half of the gronding, Figre and Figre 3 sho an increase in eight for both tanks, bt in the second half, the system tries to make the vehicle netrally boyant as expected. QR With Positive Weight Command So hen the vehicle reaches the grond, there ill be almost no ater in ballast tanks. t for the stability of the gronding, the vehicle shold be heavier. For this reason, in addition to the depth command, the eight can also be commanded to increase the depth rate or the eight of the vehicle at the end of the gronding. So, the command vector, x com is changed to, x com [,,, z grond, com, com ] here com are some positive nmbers and represent the command for additional eight and the system as simlated ith these ne parameters. Figre 8 throgh Figre 3 sho the plot of this simlation. In this case, pitch angle reaches a maximm vale of.8 rad.3 degrees hich is times greater than the previos simlation. With increasing pitch angle, there is also an increase in the depth rate. The depth rate becomes.35 ft/sec hich is again almost times greater than the previos simlation. Since the commanded depth for the control la is the depth of the grond, the system tries to make the vehicle netrally stable to keep the vehicle on that depth by pmping ater ot of ballast tanks. After gronding, pmps shold pmp ater in ballast tanks ntil they are fll. ecase this additional eight is needed to keep the vehicle sitting on the grond against the crrent. 7 Another method for increasing the depth rate is to command ith a depth vale that is greater than the actal grond depth. ecase in the beginning the error ill be higher, the eight controller ill prodce higher vales of flo rate. Flo rate of pmp- vs time Flo rate for pmp lb/sec Figre. Flo Rate As A Fnction Of Time For Pmp- With Depth Command Only Flo rate for pmp- vs time Flo rate for pmp- lb/sec Figre. Flo Rate As A Fnction Of Time For Pmp- With Depth Command Only

9 .6 Weight in tank- vs time. Pitch angle vs time.5 Weight in tank- lb..3.. Pitch angle rad Figre. Weight Change In allast Tank- With Depth Command Only Weight in tank- vs time Figre 5. Pitch Angle Change With Depth Command Only Change in depth and forard velocity Velocity Weight in tank- lb Depth Figre 3. Weight Change In allast Tank- With Depth Command Only Depth vs time Figre 6. Comparison of Depth and Forard Velocity Change Distance from the original position vs depth - Depth ft 6 ft Figre. Depth Change Dring Flight And ronding X ft Figre 7. Response Of The Vehicle To The ongitdinal Position Command 8

10 .35 Flo rate for pmp - vs time.8 Weight in tank - vs time.3.6 Flo rate for pmp- lb/sec Weight in tank - lb Figre 8. Flo Rate For Pmp- With Weight And Depth Commands Flo rate for pmp - vs time Flo rate for pmp - lb/sec Figre 9. Flo Rate For Pmp- With Weight And Depth Commands Weight in tank- vs time Weight in tank - lb Figre. Weight Change In allast Tank- With Depth And Weight Commands Depth vs time Depth ft Figre. Depth Change With Weight And Depth Commands Pitch angle vs time Pitch angle rad Figre. Weight Change In allast Tank- With Depth And Weight Commands Figre 3. Pitch Angle As A Fnction Of Time With Weight And Depth Commands 9

11 ronding With Thrsters In Addition To Weight Control NPS Phoenix AUV s cross-body thrsters consist of a 3 in. ID alminm tbe ith a centrally located blade brass propeller. A spr gear is monted arond a 3 in. diameter propeller and driven by a pinion connected to a Vdc motor giving a.5: gear redction. The tist of the propeller blade is symmetric enabling bi-directional operation delivering approximately. pond of bollard pll force in either direction []. The system as simlated ith the ne state eations and the same eight control conditions defined in section C of this chapter. Figre throgh Figre 9 sho the reslt of this simlation. The depth rate becomes.5 ft/sec and the pitch angle reaches a higher vale since there is no control on thrsters. ecase the thrsters are dominant in this case, the eight controller losses most of its effect on pitch control. ottom Stability The grond also affects the vehicle closing to the bottom. The theory explained by Hoerner and orst [], predicts that roghly belo C.5 C represents lift coefficient, lift ill be increased in proximity of the grond. Even thogh no experimental data is provided for NPS Phoenix AUV, it can be assmed that lift coefficient ill be less than.5. For stdy of the bottom stability, to different cases of gronding ere considered. Figre 3 shos these to cases. In the case that the vehicle s stern toched to the bottom first, the lift ill decrease the eight and increase the angle of attack. This redces the stability and makes gronding more difficlt. t this featre can be very helpfl hen leaving the grond. In the other case, the bo of the vehicle toches the grond first. This time lift makes the vehicle heavier and decreases the angle of attack providing more stable gronding. After the completion of gronding process, the vehicle sits on the bottom ith no lift since the lift coefficient, C is assmed zero becase of the symmetric shape of the NPS Phoenix AUV. Figre 3 shos the ocean crrent that the vehicle can stand ith different ballast eights sitting on soil ith.7 friction coefficient. When both ballast tanks are filled ith ater completely ~3. lb. ater in each tank, the vehicle can keep its position against.5 m/sec ~3 knots of crrent. Flo rate for pmp- lb/sec Flo rate for pmp- vs time Figre. Flo Rate For Pmp- With Thrsters And Weight Control Flo rate for pmp- vs time Flo rate for pmp- lb/sec Figre 5. Flo Rate For Pmp- With Thrsters And Weight Control Weight in tank- vs time Weight in tank- lb Figre 6. Weight Change In allast Tank- With Thrsters And Weight Control

12 .35 Weight in tank- vs time +.3 V C Weight in tank- lb W + + V C Figre 7. Weight Change In allast Tank- With Thrsters And Weight Control Pitch angle vs time W Figre 3. Forces Acting On The Vehicle In To Cases Of ronding Weight in Each allast Tank vs. Crrent 5 Pitch angle rad Weight in allast Tank lb Figre 8. Pitch Angle As A Fnction Of Time With Thrsters And Weight Control Depth vs time Depth ft Time -sec Figre 9. Depth Change Dring Flight And ronding With Thrsters And Wt. Cont..5.5 Crrent m/sec Figre 3. The Crrent That Vehicle Can Keep Its Position REFERENCES. Schenkenberger, J., Jens Schenkenberger s ftschiff Homepage. [ Frankfrt. netsrf.de/jens.schenkenberger/] otyka, P., and ergmann, E., The Design of a Control System for the allast and Trim of an Unmanned Sbmersible, Proceedings of the 98 American Control Conference, San Diego, California, v.3, pp , 6 Jne Deitetto, P.A., Fzzy ogic for Depth Control of Unmanned Undersea Vehicles, IEEE Jornal of Oceanic Engineering, v., no.3, pp. -7, Jly 995. Healey, A.J., Dynamics and Control of obile Robotic Vehicles, ectre Notes, chp., pp. 3-9, 995

13 5. Riedel, J.S., Pitchfork ifrcations and Dive Plane Reversal of Sbmarines at o Speeds, Engineer s Thesis, Naval Postgradate School, onterey, California, Jne Papolias, F.A., Dynamics of arine Vehicles, ectre Notes, chp., pp 8-6, Hakinson, T.D., ltiple Inpt Sliding ode Control for Atonomos Diving and Steering of Underater Vehicles, aster s Thesis, Naval Postgradate School, onterey, California, December Papolias, F.A., odern Control Systems, ectre Notes, chp. 6, pp 95-, Whitcomb,.., and Yoerger, D.R., Comparative Experiments in the Dynamics and odel-ased Control of arine Thrsters, Proceedings of the 995 IEEE Oceans Conference, San Diego, California, v., pp 9-8, October 995. Yoerger, D.R., Cooke, J.., and Slotine, J.E., The Inflence of Thrster Dynamics on Underater Vehicle ehavior and Their Incorporation Into Control System Design, IEEE Jornal of Oceanic Engineering, v. 5, no. 3, pp 67-78, Jly 99. arco, D.., Atonomos Control of Underater Vehicles and ocal Area anevering, Ph.D. Dissertation, Naval Postgradate School, onterey, California, September 996. Hoerner, S.F., and orst, H.V., Flid Dynamic ift, chp. -, rs. iselotte A. Hoerner, 975 ACKNOWEDEENTS The athors ish to thank the Office of Naval Research for financial spport of the NPS AUV program nder Work Reest N98WR375. This ork is based on the SE thesis of TJ ahadir eyazay, Trkish Navy,