NAVAL POSTGRADUATE SCHOOL THESIS

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1 NAVAL POSTGRADUATE SCHOOL MONTEREY, CALIFORNIA THESIS CONTOUR TRACKING CONTROL FOR THE REMUS AUTONOMOUS UNDERWATER VEHICLE by Alan Robet Van Reet June 2005 Thesis Adviso: Anthony Healey Appoved fo public elease; distibution is unlimited

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3 REPORT DOCUMENTATION PAGE Fom Appoved OMB No Public epoting buden fo this collection of infomation is estimated to aveage 1 hou pe esponse, including the time fo eviewing instuction, seaching existing data souces, gatheing and maintaining the data needed, and completing and eviewing the collection of infomation. Send comments egading this buden estimate o any othe aspect of this collection of infomation, including suggestions fo educing this buden, to Washington headquates Sevices, Diectoate fo Infomation Opeations and Repots, 1215 Jeffeson Davis Highway, Suite 1204, Alington, VA , and to the Office of Management and Budget, Papewok Reduction Poject ( ) Washington DC AGENCY USE ONLY (Leave blank) 2. REPORT DATE 3. REPORT TYPE AND DATES COVERED June 2005 Maste s Thesis 4. TITLE AND SUBTITLE: Contou Tacking Contol fo the 5. FUNDING NUMBERS REMUS Autonomous Undewate Vehicle 6. AUTHOR(S) Alan Van Reet 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) Naval Postgaduate School Monteey, CA SPONSORING /MONITORING AGENCY NAME(S) AND ADDRESS(ES) N/A 8. PERFORMING ORGANIZATION REPORT NUMBER 10. SPONSORING/MONITORING AGENCY REPORT NUMBER 11. SUPPLEMENTARY NOTES The views expessed in this thesis ae those of the autho and do not eflect the official policy o position of the Depatment of Defense o the U.S. Govenment. 12a. DISTRIBUTION / AVAILABILITY STATEMENT 12b. DISTRIBUTION CODE Appoved fo public elease; distibution is unlimited 13. ABSTRACT (maximum 200 wods) In the inteest of enhancing the capabilities of autonomous undewate vehicles used in US Naval Opeations, contolling vehicle position to follow depth contous pesents exciting potential fo navigation. Use of a contou tacking contol algoithm in lieu of pepogammed waypoint navigation offes distinct advantages within new challenges. The difficult natue of this poblem lies in the non-tivial connection between the necessay coective action and the feedback eo used in taditional contol methods. Stated simply, moden vehicle contol algoithms sepaate hoizontal and vetical plane navigation. The autonomous vehicle senses heading eo and applies udde to stee the vehicle to a desied heading. Simultaneously, the vehicle might sense altitude and apply sten plane angles to maintain a safe height above gound. This thesis eseach examines the new poblem of sensing depth and altitude in the vetical plane while steeing the vehicle hoizontally to find a specified bathymety contou. While moe emains to undestand, this eseach poves the existence of a solution and suggests simila appoaches may facilitate tying vehicle navigation to othe indiect sensos. This thesis pesents two contou tacking contol algoithms and examines the pefomance of each by simulating the esponse of the REMUS undewate vehicle to ideal and eal-wold bathymety models. 14. SUBJECT TERMS 15. NUMBER OF PAGES Contou Tacking, Autonomous Contol, REMUS, Undewate Vehicle, AUV PRICE CODE 17. SECURITY CLASSIFICATION OF REPORT Unclassified 18. SECURITY CLASSIFICATION OF THIS PAGE Unclassified 19. SECURITY CLASSIFICATION OF ABSTRACT Unclassified 20. LIMITATION OF ABSTRACT NSN Standad Fom 298 (Rev. 2-89) Pescibed by ANSI Std UL i

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5 Appoved fo public elease; distibution is unlimited CONTOUR TRACKING CONTROL FOR THE REMUS AUTONOMOUS UNDERWATER VEHICLE Alan R. Van Reet Ensign, United States Navy B.S., United States Naval Academy, 2004 Submitted in patial fulfillment of the equiements fo the degee of MASTER OF SCIENCE IN MECHANICAL ENGINEERING fom the NAVAL POSTGRADUATE SCHOOL June 2005 Autho: Alan Van Reet Appoved by: Anthony J. Healey Thesis Adviso Anthony J. Healey Chaiman Depatment of Mechanical and Astonautical Engineeing iii

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7 ABSTRACT In the inteest of enhancing the capabilities of autonomous undewate vehicles used in US Naval Opeations, contolling vehicle position to follow depth contous pesents exciting potential fo navigation. Use of a contou tacking contol algoithm in lieu of pepogammed waypoint navigation offes distinct advantages within new challenges. The difficult natue of this poblem lies in the non-tivial connection between the necessay coective action and the feedback eo used in taditional contol methods. Stated simply, moden vehicle contol algoithms sepaate hoizontal and vetical plane navigation. The autonomous vehicle senses heading eo and applies udde to stee the vehicle to a desied heading. Simultaneously, the vehicle might sense altitude and apply sten plane angles to maintain a safe height above gound. This thesis eseach examines the new poblem of sensing depth and altitude in the vetical plane while steeing the vehicle hoizontally to find a specified bathymety contou. While moe emains to undestand, this eseach poves the existence of a solution and suggests simila appoaches may facilitate tying vehicle navigation to othe indiect sensos. This thesis pesents two contou tacking contol algoithms and examines the pefomance of each by simulating the esponse of the REMUS undewate vehicle to ideal and eal-wold bathymety models. v

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9 TABLE OF CONTENTS I. INTRODUCTION...1 A. BACKGROUND...1 B. MOTIVATION AND RELEVANCE...2 C. SENSOR BASED CONTROL Contou Tacking Related Reseach...6 D. SCOPE OF THESIS Contou Tacking Contol The REMUS Vehicle...8 E. THESIS STRUCTURE...10 II. VEHICLE MODEL AND SIMULATION...11 A. INTRODUCTION...11 B. MATHEMATICAL VEHICLE MODEL Equations of Motion Hydodynamic Coefficients...14 C. SIMULATION Vehicle Motion Ocean Bathymety...16 III. DIRECT, LOGIC-BASED RUDDER CONTROL...19 A. INTRODUCTION...19 B. DESIGN Logic Feedback Gadient Appoximation...21 a. Linea Gadient Appoximation (LGA)...21 b. Estimated Local Gadient (ELG)...26 C. SUMMATION...29 IV. HEADING-STABILIZED LOGIC CONTROL...31 A. INTRODUCTION...31 B. DESIGN Heading Command fom Logic Feedback Steeing Contol...33 C. SIMULATION RESULTS Contolle Pefomance Design Limitations...39 V. CONCLUSIONS AND RECOMMENDATIONS...45 A. CONLUSIONS...45 B. RECOMMENDATIONS FOR FUTURE WORK...46 APPENDIX A: SEA FLOOR SIMULATION...49 APPENDIX B: DIRECT-RUDDER CONTROL; LINEAR GRADIENT...51 vii

10 APPENDIX C: DIRECT-RUDDER CONTROL; ESTIMATED LOCAL GRADIENT.55 APPENDIX D: HEADING-STABILIZED LOGIC CONTROL...59 APPENDIX E: WATER COLUMN DEPTH SENSOR...63 LIST OF REFERENCES...65 INITIAL DISTRIBUTION LIST...67 viii

11 LIST OF FIGURES Figue 1. Waypoint Navigation with Obstacle Avoidance (Fom: Fodea and Healey, 2003)...4 Figue 2. REMUS Vehicle (Fom: Hust)...9 Figue 3. Discetized Bathymety Data Fom Monteey Bay...17 Figue 4. Block Diagam: Logic Feedback...21 Figue 5. Staight Contou Response: Diect Contol, LGA...23 Figue 6. Cuved Contou Response: Diect Contol, LGA...24 Figue 7. Real Contou Response: Diect Contol, LGA...25 Figue 8. Staight Contou Response: Diect Contol, ELG...27 Figue 9. Cuved Contou Response: Diect Contol, ELG...28 Figue 10. Real Contou Response: Diect Contol, ELG...29 Figue 11. Block Diagam: Heading-Stabilized Logic Feedback.32 Figue 12. Staight Contou Response...34 Figue 13. Staight Contou Response: Stating Deep...35 Figue 14. Staight Contou Response: Stating Shallow...36 Figue 15. Real Contou Response...37 Figue 16. Real Contou Response: Stating Deep...38 Figue 17. Real Contou Response: Stating Shallow, ψ nominal = Figue 18. Real Contou Response: Stating Shallow, ψ nominal = Figue 19. Effect on Pefomance of ψ nominal...42 Figue 20. Cuved Contou Response: ψ nominal = Figue 21. Cuved Contou Response: ψ nominal = Figue 22. Contou Tacking of Minima/Maxima...46 Figue 23. Block Diagam: Dynamically-Updated Heading- Stabilized Logic Contol...47 ix

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13 LIST OF TABLES Table 1. Chaacteistics of REMUS AUV...10 Table 2. REMUS Hydodynamic Coefficients fo Equations of Motion in the Hoizontal Plane...15 Table 3. Logic States Requied fo Diect Rudde Contol...21 Table 4. Logic Requied fo Heading-Stabilized Contol...32 xi

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15 ACKNOWLEDGMENTS I would like to thank my thesis adviso, Distinguished Pofesso Anthony J. Healey, fo lending me his expetise, guidance, help, and most of all, time. While I leaned much fom my effots, and managed to enjoy my stuggles, I would not be able to claim success without his help. I would paticulaly like to thank my wife, Bobbi Van Reet, fo suppoting and encouaging me. Not only has all he wok given me the time to devote to this endeavo, but also she still manages to push me to succeed and to do so in a timely manne. Finally, I would like to thank all my family and fiends whose suppot and undestanding has helped to do eveything thus fa. xiii

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17 I. INTRODUCTION A. BACKGROUND Unmanned vehicles povide both civilian and militay uses with geate access to the vaied envionments on this planet and beyond. Unmanned vehicles typically ente aeas that pesent conditions impossible fo humans to endue, that pose a isk to human life geate than the intended benefit, o that ae simply to expensive to each with a similaly equipped manned vehicle. In the ai, on land, o in the sea, specific missions ae bette suited to cetain vehicle types and cetain contol pogams. If thee is one absolute tuth egading unmanned vehicles, it is that no single platfom will be able to fulfill evey possible mission equiement. Though multiple machines will be necessay, designing each obot to pefom in as many situations as possible not only keeps vehicle pogams simple and cost-effective, but also inceases suvivability when encounteing unpedictable events. Unmanned Undewate Vehicles (UUVs) employed in naval applications fall into two basic classifications, Remotely Opeated Vehicles (ROVs) and Autonomous Undewate Vehicles (AUVs). While neithe type caies people onboad, ROVs still equie manned contol. As the name implies, ROVs must eceive continuous contol input, o piloting, fom a emote use making decisions based on output fom the vehicle s sensos, usually video feed. As opposed to ai and land vehicles that can easily eceive adio wave signals, vehicles undewate cannot yet tansmit clea, high-speed signals though thei medium ove any appeciable distances. Due to this limitation, ROVs emain 1

18 connected to the host ship via physical tethes. These tethes have advantages by poviding ample powe supplies and lage communications bandwidths. Convesely, tetheing the vehicle limits the distance it can ventue fom the host ship, and emote opeation, while potecting pesonnel fom hostile envionments, does not fee individuals to pefom othe tasks o significantly educe man-hous. AUVs essentially pesent opposing capabilities to those of ROVs. AUVs ae self-contained units that un off contol pogams stoed in onboad memoy. They may communicate with thei host as well as othe vehicles but not continuously. AUVs execute thei stoed missions without constant attention fom an opeato and wok without inteuption ove any distance o duation allowed by onboad battey stoes. To maximize battey stoes, AUVs must be designed efficiently. This design extends beyond pat selection to using vehicle motion efficiently. AUVs manipulate hydodynamic foces athe than using thustes as ROVs do, hence they ae not suited fo station keeping and do not easily stop o back tack. AUV contol design must anticipate actions and manage each on the fly in ode to maximize pefomance based upon the capabilities of these platfoms. B. MOTIVATION AND RELEVANCE FoceNet objectives, pat of the United States Navy s doctines set foth by the Chief of Naval Opeations in SeaPowe 21, dictate the need fo collecting vast amounts of infomation fom a netwok of all available sensing platfoms. Assembling this data into a single intelligence model accessible by all combat platfoms ensues that evey 2

19 fighting membe enjoys the full infomation supeioity of the entie foce (Clak, 2002). AUVs pimaily suppot US Naval Opeations by pefoming suveillance and econnaissance missions such as beach suveys, ocean sampling, and covet opeations in the littoal aea. AUV eseach began in the 1960s, with the fist pototypes emeging in the 1980s (Blidbeg, 2001). Pesent AUV capabilities allow vehicles to follow pe-planned flight paths in ode to execute specific mission objectives. Most AUVs navigate by tacking paths constucted linealy between pe-planned waypoints. Waypoint navigation essentially uses Line of Sight guidance to constantly point the vehicle at its pesent position diectly towad the desied waypoint. Othe algoithms, such as Coss Tack Eo, compensate fo factos such as steady cuents. In ode to make the vehicle follow the tacking algoithm s commands, any of a numbe of contol methods may be used to achieve the desied tacking pefomance. Waypoint tacking algoithms ae well developed and the contol methods ae poven. One basic limitation of waypoint navigation is the equiement fo enough advance knowledge to appopiately locate waypoints duing mission planning. The idea of having advance knowledge of an aea negates the eason fo conducting suveillance of the aea. As a mino note, when paths equie following cuves, tacking staight lines between waypoints is not the most efficient way. Though pesently useful, AUVs need to opeate successfully while eacting to unknown conditions sensed in eal-time in ode allow moe obust platfoms to moe effectively fulfill vaious mission equiements without the need fo sepaate 3

20 vehicles. Cuent wok hopes to ende vehicles fully autonomous in unknown suoundings and centes aound adding obstacle avoidance algoithms to planned path navigation. An example of such wok is shown in figue 1. Figue 1. Waypoint Navigation with Obstacle Avoidance (Fom: Fodea and Healey, 2003) The motivation fo this thesis is to find an altenative to waypoint navigation that satisfies the need fo adapting to the unknown. Navigation is accomplished by tacking a featue of the ocean floo, specifically a specified contou of constant depth. Tacking equies steeing the vehicle in the hoizontal plane to follow the depth contou. Though an altenative to waypoint navigation, contou tacking would also benefit fom simila obstacle avoidance capabilities. Waypoint navigation and contou tacking ae each suited to diffeent mission stuctues, and some missions may benefit fom the use of both in conjunction. 4

21 C. SENSOR BASED CONTROL Contou tacking poblems belong to a type on contol theoy known as Senso Based Contol. Othe Senso Based Contol poblems include tacking plumes in the ocean, following ocean tempeatue gadients, o steeing towads the location of geatest communications signal stength. The name Senso Based Contol seems somewhat misleading, as all contol theoy utilizes feedback of eithe senso outputs o eos deived fom these outputs. Pehaps the concept might moe appopiately be called Indiect Senso Contol. The distinction comes fom an inceased complexity in the contol method due to the natue of the feedback signal and its indiect elationship with the coective contol action. As an example, steeing to a commanded heading is a compaably simple poblem because the feedback loop ties output fom a compass, which measues heading, to the heading command. When the heading eo indicates that the vehicle is pointed left of the desied heading, a ight tun is clealy in ode, and a ight udde command can be easily manipulated mathematically. 1. Contou Tacking As will be discussed late, autonomous undewate vehicle contols ae simplified by sepaating the effects of motion in the hoizontal plane fom those in the vetical plane. The pevious example of tacking a commanded heading falls into the simple categoy lagely because both the contol model and the senso output lie in the same plane of motion. In tacking a depth contou, the poblem design equies steeing the vehicle, a hoizontal 5

22 plane contol, based upon sensos measuing wate column depth, a vetical plane output. Compae this poblem to the pevious example. Rudde action and vehicle speed detemines position at any point in time. At a given position, wate depth is detemined by the ocean floo s geomety. While the ocean floo is assumed to emain fixed fo elevant peiods of time, the location of the contou elated to the vehicle s tuns may change constantly. Though the vehicle may sit to the ight of the commanded depth contou, tuning left towad the contou s local position may not be ideal if the contou cuves ight towads the vehicle s position close ahead. The poblem is futhe complicated because the vehicle knows neithe futue tends of the contou no local tends when only single senso values ae available, as is the case with the REMUS vehicle. 2. Related Reseach Cuently, eseach is undeway to contol vehicles to tack numeically computed gadients fo use in following zones of constant tempeatue in the ocean in thee dimensions. This poblem elates diectly to tacking depth contous in that the gadient at a point in the field foms the basis fo the diection and/o magnitude of the contol command. Contous of constant value ae by definition othogonal to the gadient, which points in the diection of steepest ascent, making the tacking of eithe featue mathematically elated to the othe. Pofesso Naomi Leonad of Pinceton Univesity has co-authoed much of the elated eseach in this field. Paticulaly elevant to this discussion ae effots 6

23 enabling multiple AUVs to climb (Ogen, Fioelli, and Leonad, 2004) and descend (Moeau, Bachmaye, and Leonad, 2003) gadients, and fo the fist time this yea to use multiple AUVs to tack and plot tempeatue contous (Zhang and Leonad, 2005). The initial wok by Moeau, Bachmaye and Leonad in 2003 focused on tacking the diection of the negative gadient. When each vehicle has enough sensos to measue the full gadient, the closed-loop system becomes Lagangian. This eseach allows the calculation of the gadient with only a single senso vehicle; howeve, multiple sensos ae still equied though the use of multiple single senso vehicles acting togethe in a single fomation. In 2004, Ogen, Fioelli, and Leonad woked on the elated poblem of tacking the diection of the positive gadient. These effots still use multiple single senso vehicles to constuct a single multi-senso fomation; howeve, in this evision the fomation can be econfigued on the fly without hindeing tacking ability. Finally, eseach conducted by Zhang and Leonad duing the same time peiod as this thesis allows the vehicles to tack contous and fom contou plots based on collected data. The fomation still tacks as a single unit optimally shaped to minimize eos in the gadient calculation. The goup may consist of as few as fou single senso vehicles, but tacking still equies the use of multiple sensos and a full numeical gadient calculation. The elated eseach in this field beas elevance to the effots of this thesis, yet the methods used diffe distinctly. The algoithms used fo REMUS focus on 7

24 appoximating gadients fom data at the cuent and pevious locations allowing a single senso vehicle to successfully tack contous without pefoming numeical gadient calculations. D. SCOPE OF THESIS 1. Contou Tacking Contol As peviously mentioned, the fundamental goal of this thesis is to develop contol algoithms that successfully allow AUVs to tack constant depth contous. The lessons leaned fom this wok povide insight into the geneal poblems of Senso Based Contol. While the contol algoithms developed ae applicable to all AUVs that move by manipulating hydodynamic foces via uddes and planes, these algoithms ae specifically tailoed to the REMUS vehicle with only cuently available sensos in mind. Fo this eason, a bief discussion of the REMUS vehicle, as used in the Naval Postgaduate School s Cente fo Autonomous Undewate Vehicle Reseach, is in ode. 2. The REMUS Vehicle Remote Envionmental Monitoing Units (REMUS) ae lowcost, lightweight autonomous undewate vehicles oiginally developed by the Oceanogaphic Systems Laboatoy at Woods Hole Oceanogaphic Institution. The vehicles opeate with a laptop compute and simplify launching and ecovey opeations due to thei compact size and weight. As a package, REMUS incopoates a wide ange of onboad sensos and includes an upgadeable payload fo the addition of unique senso packages (Hust). All of these factos make 8

25 REMUS an attactive platfom fo US Navy missions. Futhemoe, eseach tailoed to the REMUS platfom has the distinct advantage of being diectly elevant to a vehicle aleady in poduction and pesently deployed by US Navy vessels. REMUS geneally deploys in the Vey Shallow Wate zone defined by wate depths anging between 40 and 100 feet (Fodea, 2002). In standad use, REMUS can un fom 8 up to 20 hous when taveling at 5 and 3 knots, espectively (Hust). Figue 2 shows REMUS in a basic configuation along with its impact esistant case, which allows it to be caied o shipped as conventional baggage. Table 1 lists moe detailed chaacteistics of REMUS physical featues and functional capabilities. Figue 2. REMUS Vehicle (Fom: Hust) Of the many sensos aleady caied by REMUS, two ae elevant to this thesis. REMUS simultaneously senses its depth unde the suface of the wate and uses its RDI Dopple sona to detect its altitude above the ocean floo. Fo tacking depth contous, summing these two values povides the wate column depth at the pesent position, which also educes the output fom two sensos to a single value useful fo feedback. As stated in the pevious section on elated eseach, as few as fou sensos can poduce accuate gadients calculations. The difficulty of 9

26 gadient tacking with only single senso feedback waants the effots of this thesis to allow existing REMUS vehicles to pefom contou tacking without equiing 4 additional expensive, powe consuming sensos. Table 1. Chaacteistics of REMUS AUV PHYSICAL/FUNCTIONAL AREA CHARACTERISTIC Vehicle Diamete 7.5 in Vehicle Length 62 in Weight in Ai 80 lbs Extenal Ballast Weight 2.2 lbs Opeating Depth Range 10 ft to 66 ft Tansit Depth Limits 328 ft Typical Seach Aea 875 yds X 1093 yds Typical Tansponde Range 1640 yds Opeational Tempeatue Range +32F to +100F Speed Range 0.5 knots to 5.6 knots Maximum Opeating Wate Cuent 2 knots Maximum Opeating Sea State Sea State 2 Battey 1 kw-h intenally echageable Lithium-ion Enduance 20 hous at 3 knots; 9 hous at 5 knots E. THESIS STRUCTURE The objective of this eseach is to develop a stable, obust algoithm fo tacking contous of constant wate depth. The algoithm is developed to suit the REMUS autonomous undewate vehicle and is tested by simulating the motions of REMUS in a vitual ocean envionment. Chapte II explains the necessay motion and ocean models. Chapte III discusses an attempt to use logic feedback to diectly contol the vehicle s udde. Chapte IV details the method of using logic to command heading to a stable steeing contolle. Finally, Chapte V povides geneal conclusions and ecommendations fo futue wok. 10

27 II. VEHICLE MODEL AND SIMULATION A. INTRODUCTION The pupose of this thesis is not to deive equations modeling undewate vehicle motion, no to calculate the specific hydodynamic foces expeienced by the REMUS vehicle. Both issues have been adequately addessed pio to this eseach. The notes fom (Healey, 2003) contain complete deivations of the equations in this section, and the thesis by (Fodea, 2002) contains an additional discussion. The thesis by (Pesteo, 2001) calculates the pecise values of the hydodynamic coefficients needed to model a REMUS vehicle with these equations. Though full deivations ae not pat of this thesis, in ode to adequately undestand the wok pesented, a discussion of the elevant equations and assumptions is in ode. Futhemoe, the methods used to numeically simulate vehicle motion and ocean floo data ae included to enhance the eade s compehension. B. MATHEMATICAL VEHICLE MODEL 1. Equations of Motion Vehicle motion is fully modeled by six equations of motion that elate foce inputs to esulting motions in thee tanslational and thee otational degees of feedom. Thee easonable assumptions must be made in ode to epesent motion by these six equations. The fist assumes that the vehicle behaves as a igid body despite acceleations. The second assumes that acceleation tems can neglect the effects of the eath s 11

28 12 sideeal ate. The thid assumption consides only inetial and gavitational foces esulting fom thust, hydostatic effects, and hydodynamic lift and dag. The six equations descibe suge, sway, heave, oll, pitch and yaw motions and ae shown, espectively, in equations 1 though 6 below. (6) )sin ( sin )cos ( )] ( ) ( [ ) ( ) ( ) ( ) ( (5) )sin ( cos )cos ( )] ( ) ( [ ) ( ) ( ) ( ) ( (4) sin )cos ( cos )cos ( )] ( ) ( [ ) ( ) ( ) ( ) ( (3) cos )cos ( )] ( ) ( ) ( [ (2) sin )cos ( )] ( ) ( ) ( [ (1) )sin ( )] ( ) ( ) ( [ f B G B G G G xz yz xy x y z f B G B G G G xz yz xy z x y f B G B G G G xz yz xy y z x f G G G f G G G f G G G N B y W y B x W x q w v u y p w u v m x p q I q p I q p I pq I I I M B z W z B x W x q w v u z p v q u w m x p I pq I p q I p I I q I K B z W z B y W y p w u v z p v q u w y m pq I q I q p I q I I p I Z B W q p z p q y q p x p v q u m w Y B W p q z p y pq x p w u m v X B W q p z pq y q x q w v m u = = = = = = θ φ θ θ φ θ φ θ φ θ φ θ φ θ θ & & & & & & & & & & & & & & & & & & & & & & & & EQUATION YAW EQUATION PITCH EQUATION ROLL EQUATION HEAVE EQUATION SWAY EQUATION SURGE

29 One additional assumption geatly simplifies contol calculations. When developing AUV contols, motion in the hoizontal plane is sepaated fom that in the vetical plane. Although designed aound two-dimensional contol planes, thee-dimensional vehicle contol can be achieved by simply unning the hoizontal and vetical contol algoithms simultaneously. Fo steeing in the hoizontal plane, only the suge, sway, and yaw equations ae impotant, educing a sixdimensional poblem to just thee dimensions. Assuming constant speed only in the fowad diection, and eiteating that all vetical plane motions ae ignoed equations 7, 8, and 9 show the simplified foms of the thee hoizontal plane equations of motion. Equations 10 and 11 compute changes in the vehicle s Catesian hoizontal plane position based on linea velocity and angula tun ates. Fo this model, vehicle speed is assumed constant in the fowad diection ( u = U 0 ) and zeo cuent is consideed ( U U = 0 ). cx = cy mv& = mu + Y 0 f ( t) (7) I zz & = N f ( t) (8) ψ& = (9) X& = U 0 cosψ v sinψ + U cx (10) Y& = U 0 sinψ + v cosψ + U cy (11) Finally, in ode to model the specific behavio of the REMUS, o any othe, vehicle submeged in wate and esponding to inputs fom contol sufaces, the associated 13

30 lineaized fluid foces ae epesented in the equations of motion by coefficient tems multiplied with the appopiate individual motions o contol suface angles. The final equations of motion, with focing effects, ae detailed below in matix equation 12. Equations 10 and 11 fo Catesian position emain unchanged. As peviously stated, the equations fo this model wee obtained fom Distinguished Pofesso Anthony Healey s deivations as found in (Healey, 2003). m Yv& N v& 0 I Y zz & N 0 & v& Yv 0 & = N v 1 ψ& 0 0 Y mu 0 N 0 0 v Yδ 0 N + δ δ ( t) 1 ψ 0 (12) 2. Hydodynamic Coefficients The coefficient tems in the pevious equations of motion ae called hydodynamic coefficients, and they epesent the magnitude of the effects of vaious populsive and maneuveing foces on vehicle motion, assuming that the effects ae linealy elated. The hydodynamic tems above, which ae elevant to hoizontal plane motion, epesent the following foces: Y v & = coefficient of added mass in sway Y & = coefficient of added mass in yaw Y v = coefficient of sway foce induced by side slip Y = coefficient of sway foce induced by yaw N v & = coefficient of mass moment of inetia in sway N & = coefficient of mass moment of inetia in yaw N v = coefficient of sway moment fom side slip N = coefficient of sway moment fom yaw Y δ = coefficient of udde moment N = coefficient of udde moment δ 14

31 Table 2 gives actual values fo the hydodynamic coefficients that accuately model the REMUS vehicle s maneuveing chaacteistics. These numeical values wee obtained fom eseach found in the thesis by (Pesteo, 2001). As an exception, LT Lynn Fodea modified the udde moment coefficient values afte obseving that the Pesteo model did not agee with expeimental esults (Fodea, 2002). Table 2. REMUS Hydodynamic Coefficients fo Equations of Motion in the Hoizontal Plane Y v & -3.55e01 kg Y & Y v 1.93 kg m/ad -6.66e01 kg/s (Same as Zw) Y 2.2 kg m/s (Same as Zq) N v & 1.93 kg m N & kg m 2 /ad N kg m/s v N kg m 2 /s (Same as Mq) Y δ -3.46e01/3.5 kg m/s 2 N δ 5.06e01/3.5 kg m/s 2 15

32 C. SIMULATION 1. Vehicle Motion The equations of motion ae odinay diffeential equations. Simulation of vehicle motion esults fom integating the equations ove time and adding a unique initial condition. This simple mathematical concept computes the value of evey system state at any moment in the integated time. Numeical methods ae employed to integate of the diffeential equations. While any numeical integation method would wok, the model in this thesis uses simple Eule integation fo compute coding simplicity, and a sufficiently small time step assues easonable accuacy in the numeical solution. 2. Ocean Bathymety Running the contol simulation equies the ceation of a vitual ocean envionment. Thee ocean models wee developed fo this thesis. Two models simulate staight and cuved contous using depth data fom simple fist and second ode equations, espectively. Ceating bathymety data fom low ode equations esults in an ideally smooth ocean floo model. The ideal models seve as initial measues of the contou tacking algoithm s pefomance. The final test of an algoithm s usefulness uses a vitual bathymety model constucted fom eal-wold, sampled data. Figue 3 shows the eal-wold ocean floo model used. The data in this model comes fom actual REMUS sampling uns pefomed in Monteey Bay by the Naval Postgaduate School s Cente fo AUV Reseach. This section of ocean floo featues geneally staight contou 16

33 lines with local deviations and one significant dogleg tun. When using eal data, the oughe natue of the floo pesents a geate challenge fo stability concens. Figue 3. Discetized Bathymety Data Fom Monteey Bay Actual vehicles cove continuous ocean floo eceiving discete senso feedback at the sampling ate. In ceating a vitual model, the ocean floo becomes a discete field. Though discete sampling can also be simulated, placing two discetized signals in seies compounds advese effects, so continuous feedback of a discete signal is used instead. The assumption has been made that this switch does not significantly affect the pefomance of the simulations, and it is moe than easonable to assume that it has no effect on the stability of the algoithms. Appendix A contains MATLAB code fo Bathymety simulation. The colo scaling in these figues and the Monteey Bay bathymety data ae saved in files attached to the electonic vesion of this thesis. The code in Appendix E simulates senso output as a single wate column depth value. 17

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35 III. DIRECT, LOGIC-BASED RUDDER CONTROL A. INTRODUCTION In ode to successfully constuct an autonomous contol, one must ceate a stable, closed-loop system using contol feedback. When consideing this contou tacking poblem, constucting this feedback loop pesents such a challenge because the vetical plane depth eadings cannot be diectly conveted to hoizontal steeing contols though taditional mathematical elations, such as constant feedback gains. Having multiple depth eadings at a given time povides enough infomation to mathematically compute the gadient of the wate column field. Gadient computation mathematically links the depth eadings to a heading, which is exactly the fom of feedback suited to hoizontal steeing contol. Without having multiple sensos available, o when the gadient calculations ae computationally budensome, single depth eadings simply cannot elate to steeing commands in any simila way. The fist attempt to close the loop with a single depth senso uses logic states to detemine contol actions based on cetain conditions in the depth field along the vehicle s path. Although the appoach theoetically ovecomes the feedback obstacle, the algoithm is ultimately unstable when tested with eal-wold data, whee oughness in the eal ocean bottom amounts to noise in the depth senso signal. With diect logic contol, elatively small noise levels esult in lage contol equiements and lage motions, which by definition is unstable. The esults of this algoithm will be pesented only biefly because the method did not ultimately succeed; howeve, it 19

36 meits discussion because its limitations hold valuable lessons and the method s success with ideal data highlights elements of the algoithm that may benefit futue wok. B. DESIGN 1. Logic Feedback Logic feedback selects set contol actions fom the condition of paticula states, athe than mathematically elating the actions to the states. The state of the vehicle s path though the depth field sepaates into two pats. The cuent depth eo can sufficiently tell the vehicle which way it must tun to each the commanded contou. Moe infomation is needed because once the vehicle points geneally towad the contou, futhe tuning would cause the vehicle eithe to evese diection o to cicle indefinitely without eaching the contou. Clealy a vehicle sensing shallow wate should initially tun away fom shoe, and once the vehicle is moving towad deepe wate, it can continue fowad without tuning. The additional infomation needed by the algoithm is the cuent tend in the depth eo. The tend essentially eplaces a numeically calculated gadient with a vey geneal appoximation. The tend state comes fom compaing the cuent depth eo with pevious depth eos held in memoy to detemine whethe the vehicle is moving into deepe o shallowe wate. Figue 4 visually epesents this diect logic contol algoithm in block diagam fom. Table thee details the logic-based elationship between thee possible contol actions the states defined by depth eo and tend. 20

37 Figue 4. Block Diagam: Logic Feedback Table 3. Logic States Requied fo Diect Rudde Contol Logic State Vehicle s Condition Requied Contol Action 0 On Contou Rudde Amidships 1 Too Deep & Getting Deepe Tun Towads Shoe Too Deep but Getting Shallowe Too Shallow & Getting Shallowe Too Shallow but Getting Deepe Rudde Amidships Tun Away fom Shoe Rudde Amidships 2. Gadient Appoximation a. Linea Gadient Appoximation (LGA) With the depth eo easily calculated and the necessay logic states established, all that emains to implement this contol algoithm is specifying a method by which the eo tend is calculated. The simplest tend calculation compaes the cuent and last depth eos and assumes the tend is exactly the diffeence between the 21

38 two. Tends computed with this method ae linea appoximations, which should have easonable accuacy assuming that the elapsed time between the depth eo values used is sufficiently shot. The MATLAB code attached in Appendix B simulates diect logic-based udde contol using a linea gadient appoximation. With the tend calculated accoding to this linea gadient appoximation (LGA) method, the algoithm successfully tacks ideal depth contous whethe staight o cuved. Figue 5 shows the algoithm s pefomance when tacking staight contous, and figue 6 shows the same method tacking a cicula contou. Thoughout this thesis, the figues of tacking pefomance show thee pieces of infomation. The cental image plots the vehicle s path though the vitual wate column field. The heavy black line indicates the vehicle s path and labels claify the stat and end points of the un simulation. The bottom left image shows the wate column depth histoy at evey moment in time duing simulation. This infomation has use in detemining the vehicle s deviation fom the commanded contou and also shows the discete natue of the depth feedback. Finally, the bottom ight image shows the contol command histoy duing simulation. Fo the diect logic contol simulations, the contol histoy shows the logic states, which ae elated to udde commands as peviously specified in table 3. 22

39 The diect logic contol un using the linea gadient appoximation tend, simulated in figue 5, shows that the linea method is well suited to staight-line contous. In all staight-line simulations, the tacking contol commands constant 15-mete depth. In this un, the vehicle begins in wate just slightly deepe than the command with an initial heading 10 degees towads deepe wate. The vehicle tuns towad the desied contou, and tacks the emainde of the un with elatively little contol action and depth eo. Figue 5. Staight Contou Response: Diect Contol, LGA 23

40 Figue 6 shows the tacking esponse ove cuved contous. In all cuved contou simulations, the tacking contol commands 10-mete wate depth in ode to tack the longest path though this vitual envionment. The vehicle stats in wate 1 mete too shallow, again with the initial heading not paallel to the local diection of the contou. Tacking the cuve equies significantly inceased contol action, and the inefficient sinusoidal path esults fom the bang/bang action of logic contol. Bang/bang means the contol action is full on even when feedback eos ae small. This type of contol does not eliminate steady state eo causing inefficient tacking of ideal contous. Figue 6. Cuved Contou Response: Diect Contol, LGA 24

41 Testing the contol s esponse to eal bathymety data povides the most meaningful analysis. In all ealdata simulations, the tacking contol commands 15-mete wate depth. Figue 7 shows that the LGA diect logic contol fails to tack with eal data. Although elatively staight, tacking the contou is difficult because the floo slopes gently nea the 15 mete depth contou esulting in noisy data as local floo oughness changes much faste than the tend of the geneal slope. Though the vehicle deviates only a mete fom commanded depth, this elates to significant lateal deviations due to the gentle slope, and the eo gows with inceasing time. Figue 7. Real Contou Response: Diect Contol, LGA 25

42 b. Estimated Local Gadient (ELG) To check whethe the unstable behavio of the diect logic contol is due to the contol s design o caused by epesenting the gadient by a simple linea tend appoximation, an altenative tend appoximation is developed. By definition, the linea appoximation assumes the vehicle tavels in a staight line between depth eadings, and that the tend between these eadings matches the tend ahead of the vehicle along its pesent heading. Clealy, this appoximation does not account fo any tuning that occus between the eadings, o the fact that the vehicle may continue to tun ahead. Appoximating the tend with the estimated local gadient (ELG) method seeks to account fo the effects of tuning. The estimated local gadient consides not only the wate depth eadings at two locations, but also the vehicle s heading at those locations. Using diffeences in the two headings, if any, the method ties to distinguish the tend in the x-diection fom the tend in the y- diection, which moe appopiately appoximates an actual gadient. The tend appoximation is still fist ode, but the appoximation is two-dimensional instead of onedimensional. The MATLAB code attached in Appendix C simulates diect logic-based udde contol using the estimated local gadient appoximation. Figues 8 and 9 show that using the ELG appoximation with diect udde contol slightly impoves efficiency when tacking cicula contous but actually hindes efficiency when tacking staight contous. Both figues pove that the diect logic contol design tacks ideal data egadless of the appoximation method. Figue 26

43 10 shows an unstable esponse when tacking eal data, despite calculating tend with the new ELG appoximation method. Figue 8. Staight Contou Response: Diect Contol, ELG 27

44 Figue 9. Cuved Contou Response: Diect Contol, ELG 28

45 Figue 10. Real Contou Response: Diect Contol, ELG C. SUMMATION It is appopiate to conclude that the diect-logic contol design is esponsible fo the failue of the tacking contol because it cannot tack eal data using eithe tend appoximation. Fom this, it is clea that logic feedback alone is insufficient to ceate a stable closed-loop system when noise is pesent in the eo feedback. What logic feedback offes is the ability to geneate contol commands fom indiectly elated senso output. This obsevation leads to the next contol algoithm, which feeds these commands to a sepaate stabilized closed-loop contol. 29

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47 IV. HEADING-STABILIZED LOGIC CONTROL A. INTRODUCTION Afte leaning fom the esponse of the fist contol attempt and econsideing the poblem at hand, it appeas that diect logic-based udde contol attempts to einvent the wheel so to speak, in tems of contol theoy. The pimay objective in solving this poblem is to elate indiect senso output to contol commands when taditionally mathematical elations ae not pactical. Inventing a new fom of stable closed-loop contol feedback need not be pat of this eseach. Taditional contol theoy has aleady solved the poblem of autonomously steeing a vehicle to tack heading commands. The pimay obsevation fom the logic feedback attempt is that the model can solve the indiect elationship poblem. Using logic to geneate appopiate heading commands athe than udde commands allows the vehicle to use existing steeing autopilots to tack these commanded headings. This geatly simplifies the contol poblem, and this method effectively geneates heading without gadient computations. B. DESIGN 1. Heading Command fom Logic Feedback The logic used in the algoithm is simila to the pevious logic but simple. When the vehicle is in wate eithe too deep o shallow, it must point towads o away fom shoe, espectively. Because heading changes ae used instead of udde deflections, limiting heading changes 31

48 about a nominal heading value, athe than adding moe logic states can solve the poblem of the vehicle evesing diection. The logic states sufficient fo stable contol ae listed in table 4. Table 4. Logic Requied fo Heading-Stabilized Contol Logic State Vehicle s Condition Requied Contol Action 0 Too Deep Tun Towads Shoe 1 Too Shallow Tun Away fom Shoe The logic feedback ceates an oute heading command loop suounding the autopilot, which is a contol loop issuing udde commands. Figue 11 visually epesents the contol stuctue in block diagam fom. The inne/oute loop stuctue is appaent, as ae the unchanged vehicle and depth senso models. The MATLAB code fo this model is attached in Appendix D. Figue 11. Block Diagam: Heading-Stabilized Logic Feedback 32

49 2. Steeing Contol The steeing contolle eceives heading commands within a seach cone extending 57.3 degees to eithe side of the nominal heading. Any stable contol design could be used that can successfully tack the heading command. The thesis by Fodea, which was the souce fo the REMUS vehicle model used in this eseach, uses a sliding-mode contolle fo obstacle avoidance. Fo this eseach, state-feedback contol is used because it is quite simple to implement in MATLAB code and it has desiable pefomance chaacteistics. The state-feedback contol law pulls the obsevable vehicle states, in this model v,, and ψ, and multiplies each state by an individual gain calculated to place the closed-loop poles at locations design to meet specified pefomance goals. In this method, the contol law calculates udde commands mathematically elated to each vehicle state. As was the case in the pevious design, logic feedback behaves as a bang/bang contolle. Whateve the natue of the autopilot inne loop contol, the oveall vehicle motion should have sinusoidal steady-state eo because it eceives bang/bang contolle commands. C. SIMULATION RESULTS 1. Contolle Pefomance The vehicle motion model pedicts REMUS behavio in the vitual envionment that closely agees with the pevious design expectations. The algoithm s esponse to 33

50 ideal, staight line contous, figue 12, shows vey quick acquisition of the taget depth followed by tight, stable tacking. As pedicted, the contolle does not eliminate steady state eo, and the oscillatoy motion associated with the bang/bang contol is evident. To note, all simulation figues fo the emainde of this discussion pesent vehicle heading in the lowe ight image athe than the udde commands peviously shown. Figue 12. Staight Contou Response 34

51 To ensue the contolle exhibits stable behavio in all situations, the vehicle s initial stat point is petubed significantly away fom the taget contou. As seen in figue 13, stating the vehicle in wate appoximately 8 metes too deep distanced almost 30 metes away fom the contou esults in stable tacking. As the vehicle acquies the taget contou, it tacks the contou without any geate depth eo than is seen with the bette stat point. Figue 13. Staight Contou Response: Stating Deep 35

52 Figue 14 shows that stating the vehicle in wate 10 metes too deep and 40 metes away fom the desied contou still esults in successful tacking. The vehicle s tacking pefomance emains as desiable as that achieved with eithe the deep o best stats. Figue 14. Staight Contou Response: Stating Shallow 36

53 The most impotant esult fom the heading-stabilized contol poves that the algoithm successfully guides REMUS to tack depth contous in simulation with eal ocean data. Depth eo now emains less than half a mete, and lateal deviation is educed to the ode of one o two metes along a 40-mete un. Figue 15. Real Contou Response 37

54 Petubing the vehicle nealy 40 metes away fom the pimay location of the contou towads deepe wate, figue 16 shows the algoithm coectly finds and tacks not only the taget depth contou, but also tacks the dogleg in the contou seen along the westen edge of the bathymety sample. Figue 16. Real Contou Response: Stating Deep 38

55 2. Design Limitations One facto is not immediately appaent in the pevious tacking simulations. As stated in the contol design section, heading commands ae calculated within a finite span about a specified nominal heading. This fact causes poblems when the contous points in a diection outside of the seach cone. This limitation has one significant implication. The motivation fo using contou tacking is to eliminate the need fo advance knowledge of an opeating aea. Having to choose an appopiate nominal heading does not fulfill this objective; howeve, the amount of advance knowledge equied fo mission planning has been geatly educed. Futhemoe, it is not necessay to exactly match the nominal heading to the contou diection. The contol will tack the contou as long as it lies geneally inside the nealy 120-degee zone coveed by the seach cone. Simple modifications suggested in chapte V should eliminate this issue altogethe. Figue 17 shows one example of this limitation. In this situation, the depth contou diection lies well within the seach cone. As seen in the pevious two simulations, the vehicle is moe than capable of tacking this ocean floo model. In this simulation, stating the vehicle in shallow wate equies that the vehicle move noth to acquie the taget contou. Because the contou diection points oughly 20 degees noth of the nominal heading, the 57.3 degee seach cone limitation does not allow the vehicle to acquie the contou in any easonable peiod of time. This simulation coves 90 seconds of vehicle un time. Aguably, the vehicle would eventually each the contou and then successfully tack it; howeve, 39

56 taking this much time to do so is unacceptable. Convesely, the opposing agument suggests that if the contou diection cuves appeciably to the noth ahead, the vehicle would neve catch its moving taget. Figue 17. Real Contou Response: Stating Shallow, ψ nominal =90 40

57 Choosing a moe appopiate nominal heading leads the vehicle to each the contou much moe quickly. Figue 18 shows the same vehicle simulation as figue 17 with identical initial conditions. With the 90-degee nominal heading used in the pevious simulation, the vehicle neve eached the contou duing the un. Changing the nominal heading to 45 degees, which still does not match the contou diection, the vehicle now acquies the contou in almost as little time as possible, then successfully tacks the contou fo the emainde of the un, as seen in figue 18. Figue 18. Real Contou Response: Stating Shallow, ψ nominal =45 41

58 One additional obsevation aises fom the change in nominal heading. Figue 19 compaes the tacking of eal ocean data fom a best-case initial condition using the 90 degee nominal heading (left) o the 45 degee nominal heading (ight). By using the moe appopiate nominal heading, it appeas that the vehicle tacks the contou moe tightly, chaacteized by the path following moe localized contou cuvatue with less lateal deviation ove the majoity of the un. Figue 19. Effect on Pefomance of ψ nominal 42

59 Figue 20 povides anothe example of the seach cone limitation. Stating only slightly displaced fom the taget contou, the vehicle quickly acquies and tacks the contou, exhibiting the sinusoidal motion expected with ideal data. In the situation pesented in this simulation, the contou fist lies well within the seach cone, and then cuves continually until it points almost othogonal to the nominal heading. This un shows that as the contou diection appoaches the seach cone limit, the vehicle takes longe to each it, and once the contou points outside the seach cone, the command fom the contolle satuates. Figue 20. Cuved Contou Response: ψ nominal =90 43

60 Figue 21 depicts the ideal data simulation showing the impovement associated with selection of a moe appopiate nominal heading. Duing the potions of the un when the nominal heading and contou diection do not agee, the expected slow vehicle esponse is appaent. Fo the majoity of the un, the contou lies well within the seach cone and the vehicle s tacking pefoms emakable well. Thoughout this potion of the un, depth eo emains mostly below 0.2 metes, and lateal deviation emains less than appoximately 3 metes despite the oscillation induced by the bang/bang logic commands. Figue 21. Cuved Contou Response: ψ nominal =135 44