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UBC Theses and Dissertations

AUV hydrodynamics and modelling for improved control Ostafichuk, Peter Michael 2004

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AUV HYDRODYNAMICS AND MODELLING FOR IMPROVED CONTROL by PETER MICHAEL OSTAFICHUK B.Sc, The University of British Columbia, 1997 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Department of Mechanical Engineering) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA April 2004 © Peter Michael Ostafichuk, 2004 A B S T R A C T The performance and management of control surfaces on a near-surface autonomous underwater vehicle (AUV) were examined in terms of hydrodynamics, modelling, and control. Experiments were conducted using a one-quarter scale physical model of the International Submarine Engineering (ISE) Mark II DOLPHIN AUV. The experi- ments involved extensive tests in wind tunnels and a tow tank and included both force measurement and flow visualization studies. The experiment results were used to mathematically describe the performance of A U V control surfaces for use in simula- tion. Additionally, based on the new control surface hydrodynamics information, enhancements were made to the vehicle controller. For the various vehicle and con- troller configurations, the overall vehicle performance was evaluated through simula- tion using representative manoeuvres and operating conditions. In straight and level flight, the performance of the forward control surfaces (bow- planes) was well described through existing semi-empirical predictions. Trailing vor- tices shed by the forward control surfaces significantly affected the performance of the aft control surfaces (sternplanes). This interaction was a strong function of vehicle orientation and was accurately predicted by a simple potential flow model. Signifi- cant changes were noted to bowplane performance when the vehicle was oriented with trim or yaw. The influence of waves on the planes, including the dependence on vehi- cle speed and depth, agreed well with analytical predictions. In simulation, the effect on vehicle performance of changes in the control surface modelling, control surface configuration, and the controller design were studied. Although significantly different mathematical models of control surface performance were developed, they were found to have similar effects on the overall vehicle perfor- mance. Changes to the control surface configuration had a very significant effect on performance. In particular, increasing the bowplane span and adding dihedral to the bowplanes or sternplanes were both found to improve manoeuvring and minimize the effect of waves on the vehicle. Only minor differences in performance were noted between different PD controller implementations. The PD controllers were sensitive to modeling errors and exhibited unstable behavior in one instance. A linear quadratic gaussian controller with loop transfer recovery (LQG/LTR) was more robust and was further improved with the incorporation of sliding mode control. University of British Columbia Peter M. Ostafichuk April 2004 Table of Contents iii T A B L E O F C O N T E N T S Abstract ii List of Figures viii List of Tables xvi List of Symbols xviii Preface xxx Acknowledgements xxxi Chapter 1 Introduction 1 1.1 Background 2 1.1.1 Applications for AUVs 2 1.1.2 AUV Operation 3 1.1.3 The Dolphin AUV 5 1.2 Motivation 7 1.3 Survey of Previous Work 8 1.4 Objectives 10 1.5 Approach and Rationale 11 1.5.1 Hydrodynamics 11 1.5.2 Modelling and Simulation 11 1.5.3 Control 12 Chapter 2 Control Surface Hydrodynamics - Theory 13 2.1 Conventions 14 2.2 Control Surface Geometry 18 2.2.1 Chord 18 2.2.2 Span 19 2.2.3 Foil Section 19 2.2.4 Planform 20 2.2.5 Aspect Ratio 20 2.2.6 Endplates 21 2.3 A U V Control Surface Layout 23 2.3.1 Anhedral and Dihedral 23 2.3.2 Tail Control Surface Configurations 24 2.3.3 Dolphin-Specific Control Surfaces 24 2.4 Control Surface Performance Prediction 27 2.4.1 Isolated Control Surface 27 2.4.2 Hull Influence on Bowplanes 29 2.4.3 Hull Influence on Sternplanes 31 2.5 Control Surface Interaction 35 2.5.1 Downwash to Aircraft Elevators 35 2.5.2 Interaction Between Warship Stabilizer Fins 35 2.5.3 Control Surface Interaction Predictions on Dolphin-like Vehicles 36 University of British Columbia Peter M. Ostafichuk April 2004 Table of Contents iv 2.6 Free Surface Influence 38 2.6.1 Calm Water Free Surface Considerations for A U V s 38 2.6.2 Wave Theory 38 2.6.3 Sea State and Wave Spectra 38 2.6.4 Wave Considerations for A U V s 39 Chapter 3 Control Surface Hydrodynamics - Experiments 41 3.1 Model and Facilities 42 3.1.1 Similarity and Scaling 42 3.1.2 Scale Model 43 3.1.3 Load Cells and Signal Conditioning 48 3.1.4 Data Collection Protocol 53 3.1.5 Testing Facilities 53 3.2 Plane Performance 54 3.2.1 Isolated Plane 54 3.2.2 Straight and Level Flight 55 3.2.3 Non-zero Trim 62 3.2.4 Non-Zero Yaw 68 3.3 Control Surface Interaction 73 3.3.1 Straight and Level Flight 73 3.3.2 Non-Zero Trim 81 3.3.3 Non-Zero Yaw 87 3.4 Free Surface Interaction 89 3.4.1 Wave Geometries Tested 89 3.4.2 Wave and Control Surface Force Spectra 89 3.4.3 Wave Geometry Influence 90 3.4.4 Depth Influence 93 3.4.5 Speed Influence 95 3.4.6 Bowplane Position Influence 97 3.5 Summary 99 Chapter 4 Control Surface Hydrodynamics - Modelling 100 4.1 Background 101 4.2 Reynolds Number Effect Modelling 102 4.2.1 Change in stall angle with Reynolds number 102 4.2.2 Scaling experimental data 103 4.3 Control Surface Performance Modelling 107 4.3.1 Hydrodynamic Derivatives 107 4.3.2 Bowplane Performance Prediction 107 4.4 Trim Effect Modelling 110 4.4.1 Experimental Results 110 4.4.2 Analytical Prediction of Trim Influence I l l 4.5 Yaw Effect Modelling 113 4.5.1 Semi-Empirical Relations for Bowplane Performance in Yaw 113 4.5.2 Analytical Prediction of Yaw Influence on Bowplanes 115 4.6 Control Surface Interaction Modelling 119 4.6.1 Potential Flow Model of Control Surface Interaction 119 4.6.2 Modelling Bowplane-Sternplane Interaction in Trim and Yaw 127 University of British Columbia Peter M. Ostafichuk April 2004 Table of Contents v 4.7 Free Surface Interaction Modelling 129 4.7.1 Variation of Wave-Induced Plane Force with Depth 129 4.7.2 Variation of Wave-Induced Plane Force with Forward Speed 131 4.8 Implementation 133 4.9 Summary 136 Chapter 5 Control System Design 137 5.1 Control B ackground 138 5.1.1 Feedback Control 138 5.1.2 PID Control 138 5.1.3 Linear Quadratic Control with Loop Transfer Recovery 139 5.1.4 Adaptive Control Techniques 140 5.1.5 Fuzzy Logic 141 5.2 Control System Implementation 143 5.2.1 State Space Representation 143 5.2.2 Feed-forward Compensation 144 5.2.3 Series compensation 145 5.3 PD Controllers 146 5.3.1 Basic PD 146 5.3.2 PD Control With Gain Scheduling 147 5.3.3 PD Control With Fuzzy-Tuned Series Compensation 151 5.4 LQG/LTR Controller 159 5.4.1 Basic LQG/LTR 159 5.4.2 Sliding Mode LQG/LTR Controller 164 5.4.3 LQG/LTR Controller with Fuzzy-Tuned Series Compensation 169 5.5 Summary 170 Chapter 6 Simulation 171 6.1 Simulation Background 172 6.2 Simulation Environment and Design 173 6.2.1 Simulation Procedure 173 6.2.2 Simulator Layout 173 6.2.3 Simulation Programming 176 6.3 Simulation Test Cases and Evaluation 177 6.3.1 Evaluation Manoeuvres 177 6.3.2 Evaluation Criteria 178 6.3.3 Simulated Cases 182 6.4 Results: Influence of Modelling Improvements 184 6.4.1 Changes to the Plant Model 184 6.4.2 Changes to the Controller Model 185 6.5 Results: Evaluation of Plane Configuration 188 6.5.1 Plane Geometry 188 6.5.2 Plane Location 190 6.6 Results: Evaluation of Control System 196 6.6.1 PD Controller Performance 196 6.6.2 LQG/LTR Controller Performance 198 6.6.3 Comparison of PD and LQG/LTR Control Strategies 200 University of British Columbia Peter M. Ostafichuk April 2004 Table of Contents vi 6.7 Summary 206 Chapter 7 Conclusion 208 7.1 Summary 209 7.1.1 Overview 209 7.1.2 Influence of Control Surface Performance Model 210 7.1.3 Influence of Control Surface Configuration 210 7.1.4 Influence of Controller Design 211 7.2 Recommendations 213 7.2.1 Hydrodynamics and Modelling 213 7.2.2 Control and Evaluation 213 7.3 Conclusions 214 Appendices 215 A - Supplementary Calculations 216 A . l Bowplane Upwash Calculations 217 A.2 Change in Stall Angle With Aspect Ratio 221 A. 3 Change in Maximum Lift Coefficient with Reynolds Number 222 A.4 Analytical Prediction of Yaw Influence on Bowplanes 224 A. 5 Control Surface Interaction Potential Flow Model 227 Bowplane Tip Vortex Model 227 Bowplane Vortex Interaction with Sternplanes 230 B - Dolphin Scale Model Details 232 B. l Scale Model Engineering Drawings 233 B.2 Load Cell Design 270 B. 3 Boundary Layer Tripping 274 C - Numerical Calculations 277 C l Slope Error Estimation 278 C. 2 Radius of Curvature Calculation 279 D - Controller development 281 D. l State Space Equations 282 D. 2 L Q G / L T R Controller Design 285 Determination of Win 285 Tuning of LQG/LTR Parameters 285 E - Matlab / Simulink Program 288 E. l M A T L A B Simulation m.file® Summary 289 Overview 289 Batchfile.m 290 CalcuO 292 flqgdesign.m 293 fuzrepid 295 gsrepid 296 Jmat 297 Linearize 298 Loadparam 304 Siminit (PID) 307 rid.m 308 ridrow.m 308 University of British Columbia Peter M. Ostafichuk April 2004 Table of Contents vii smlqgdesign 309 smlqggainset 312 SPMA 314 SPMB 316 SPMB_S 319 swnoise.m 322 E. 2 Simulink® Block Diagrams 325 F - Simulation Results 332 F. l Simulations Conducted 333 F.2 Evaluation of Modelling Influence 336 Influence of Plant Model Improvements 336 Influence of Controller Model Improvements 337 F.3 Evaluation of Vehicle Geometry 338 Plane Geometry 338 Plane Location 344 F.4 Evaluation of Control System 350 PD Controller Augmentations 350 LQG/LTR Controller Augmentations 356 References 362 Index of Authors 374 Subject Index 377 University of British Columbia Peter M. Ostafichuk April 2004 List of Figures viii L I S T O F F I G U R E S Figure 1-1 Typical geometry of an AUV 4 Figure 1-2 AUV degrees of freedom 4 Figure 1 -3 ISER DOLPHIN Mark II AUV 5 Figure 2-1 Typical geometry of an autonomous underwater vehicle 14 Figure 2-2 Sign conventions 15 Figure 2-3 Control surface coordinate system (shown on starboard side plane) 18 Figure 2-4 Control surface parameters (shown on port side plane) 19 Figure 2-5 Variation lift coefficient with aspect ratio 21 Figure 2-6 Inboard and outboard endplates 22 Figure 2-7 Span-wise bowplane inclination angle: (a) dihedral, (b) anhedral 23 Figure 2-8 Tail configurations: (a) cruciform; (b) Y-tail; (c) inverted Y-tail; (d) X-tail 25 Figure 2-9 Dolphin-specific control surface locations: (a) keel planes; (b) articulated mast 26 Figure 2-10 Incident flow angle to planes 28 Figure 2-11 Flow angle near inclined hull 30 Figure 2-12 Vortex pattern on the leeward side of the hull in yaw 32 Figure 2-13 Fin area used in Lyons and Bisgood's tail efficiency correction factor 33 Figure 2-14 Fin area used in Dempsey's tail efficiency correction factor 34 Figure 2-15 Fin vortex model of Lloyd 36 Figure 3-1 Quarter-scale model exploded view 44 Figure 3-2 Cut-away view of plane showing load cell 45 Figure 3-3 Plane Mounting Locations 46 Figure 3-4 Region of tail boundary layer influence 48 Figure 3-5 Boundary Layer Trip Mounting Locations 49 Figure 3-6 Wind Tunnel Mount Showing Beam and Base Plate 50 Figure 3-7 Towing tank mount showing height adjustable structure 51 Figure 3-8 Load cell mounting to hull (the coupling and motor are specific to bowplanes) 52 Figure 3-9 Load cell arrangement for bowplane showing direct connection to stepper motor 52 Figure 3-10 Isolated standard geometry plane 0.916 scale, Rec = 280 000 (equivalent ReL = 7 million).54 Figure 3-11 Oil film sample image - suction side of port bowplane at +16 degrees deflection. The separation line is shown in dashed black and arrows indicate flow direction in attached region 56 Figure 3-12 Separation lines on suction side of bowplane at level trim; plane deflection is shown left hand side 57 Figure 3-13 Bowplane force to deflection relationship compared with higher Reynolds number isolated plane relationship (standard bowplane geometry) 58 University of British Columbia Peter M. Ostafichuk April 2004 List of Figures ix Figure 3-14 Bowplane aspect ratio influence on bowplane force production: (a) versus bowplane deflection; (b) versus bowplane deflection 58 Figure 3-15 Bowplane geometry influence on bowplane force production for large planes: (a) versus bowplane deflection; (b) versus bowplane deflection 59 Figure 3-16 Endplate influence on short bowplane performance 60 Figure 3-17 Bowplane position effect 61 Figure 3-18 Keelplane compared to standard bowplane and isolated plane 62 Figure 3-20 Bowplane root incident flow angle versus trim angle from yam tuft visualization 63 Figure 3-19 Sample yam tuft visualization images: (a) high bowplane installed, trim angle = +6 degrees; (b) no bowplane installed, trim angle = +6 degrees 63 Figure 3-21 Oil film visualization: bowplane suction side, =2.25 million, trim = +6 degrees 64 Figure 3-22 Oil film visualization: bowplane suction side, =2.25 million, trim = -6 degrees 65 Figure 3-23 Trim influence on bowplane force production 66 Figure 3-24 Trim influence on bowplanes when expressed using effective plane deflection 66 Figure 3-25 Trim influence on anhedral bowplanes 67 Figure 3-26 Trim influence on anhedral bowplanes using effective plane deflection 68 Figure 3-27 Trim influence on keel plane force production 69 Figure 3-28 Trim influence on keel plane force production expressed using effective deflection 69 Figure 3-29 Trim influence on stemplane force 70 Figure 3-30 Yam tuft visualization of body at +20 degrees yaw: (a) leeward (starboard) side; (b) windward (port) side 71 Figure 3-31 Effect of yaw on port bowplane performance (Re = 2.25 million) 72 Figure 3-32 Helium bubble path sample image; trim: +6 degrees, deflection: +25 degrees, Reynolds number: 2.5 million, model inverted 74 Figure 3-33 Oblique stem helium bubble path sample 74 Figure 3-34 Helium bubble path vortex mapping side view: -6 degrees trim, +16 degrees bowplane deflection, 2.5 million Reynolds number 75 Figure 3-35 Standard configuration straight and level flight schematic vortex path diagram 75 Figure 3-36 Bowplane and stemplane force for the standard configuration in straight and level flight 76 Figure 3-37 Comparison of bowplane and stemplane interaction force 77 Figure 3-39 Stemplane span influence on stemplane force interaction: (a) versus bowplane deflection; (b) versus bowplane deflection 78 Figure 3-38 Stemplane interaction force dependence on bowplane span: (a) versus bowplane deflection; (b) versus bowplane deflection 78 Figure 3-41 Stemplane geometry influence on stemplane force interaction: (a) versus bowplane deflection; (b) versus bowplane deflection 79 Figure 3-40 Bowplane geometry influence on stemplane force interaction for large planes: (a) versus bowplane deflection; (b) versus bowplane deflection 79 Figure 3-42 Influence of endplates on the short bowplanes on bowplane-sternplane interaction 80 University of British Columbia Peter M. Ostafichuk April 2004 List of Figures x Figure 3-43 Helium bubble path schematic for anhedral bowplanes in straight and level flight 81 Figure 3-44 Bowplane-sternplane interaction dependence on bowplane position 82 Figure 3-45 Bowplane-sternplane interaction dependence on sternplane position 83 Figure 3-46 Standard configuration vortex path: (a) +6 degrees trim; (b) -6 degrees trim 84 Figure 3-47 Anhedral configuration vortex path for (a) +6 degrees trim; (b) -6 degrees trim 85 Figure 3-48 Interaction force on standard sternplane as a function of trim angle 86 Figure 3-49 Ratio of bowplane-sternplane interaction curve slope to bowplane lift curve slope 86 Figure 3-50 Interaction force from anhedral bowplane on sternplane as a function of trim angle 87 Figure 3-51 Slope ratio of averaged sternplane interaction to bowplane lift curves versus yaw angle 88 Figure 3-52 Dependence of wave orbital velocity on depth and wavelength 90 Figure 3-53 Wave height and bowplane force trend chart 91 Figure 3-54 Wave height and non-dimensional bowplane force FFT spectra 92 Figure 3-55 Wave induced plane force dependence on wave amplitude 93 Figure 3-56 Plane performance dependence on hull depth in calm water 94 Figure 3-57 Wave induced force amplitude as a function of hull depth 95 Figure 3-58 Wave induced force on bowplanes versus Froude number 96 Figure 3-59 Influence of bowplane position on wave induced plane forces 97 Figure 3-60 Ratio of wave induced force normalized by horizontal planform area on anhedral bowplanes versus standard bowplanes 98 Figure 4-1 Control surface stall angle as a function of Reynolds number 103 Figure 4-2 Isolated appendage force data extrapolated from a Reynolds number of 7 million to 35 million 104 Figure 4-3 Bowplane force data extrapolated from a Reynolds number of 2.5 million to 35 million.... 105 Figure 4-4 Experimental to semi-empirical comparison of isolated control surface performance 108 Figure 4-5 Experimental to semi-empirical comparison of bowplane performance 109 Figure 4-6 Effective shift in bowplane deflection as a function of trim angle 110 Figure 4-7 Predicted effective shift in bowplane deflection as a function of trim angle 112 Figure 4-8 Bowplane stall angle as a function of yaw 113 Figure 4-9 Port bowplane lift curve slope as a function of yaw 114 Figure 4-10 Normalized bowplane lift curve slope dependence on yaw angle 115 Figure 4-11 Normalized bowplane lift curve slope dependence on yaw angle 116 Figure 4-12 Description of flow velocity components on the bowplane in yaw 117 Figure 4-13 Analytical prediction of port bowplane lift curve slope dependence on yaw angle 118 Figure 4-14 Vortex system used in control surface interaction calculations 120 Figure 4-15 Vortex system for potential flow model of control surface interaction 121 Figure 4-16 Starboard side vortex system at aft plane; port side vortices also contribute to induced velocity but are not shown • 121 University of British Columbia Peter M. Ostafichuk April 2004 List of Figures Figure 4-17 Plane geometry effect on fore-aft plane interaction comparison for: (a) standard configuration; (b) short bowplanes; (c) long bowplanes; (d) short sternplanes; (e) long sternplanes; (f) large sternplanes. Measured data with error bounds are shown with points, values from the numerical prediction are shown with solid lines 122 Figure 4-18 Plane location effect on fore-aft plane interaction comparison for: (a) centreline bowplanes; (b) high bowplanes; (c) dihedral bowplanes; (d) anhedral bowplanes; (e) Y-tail; (f) Inverted Y-tail. Measured data with error bounds are shown with points, values from the numerical prediction are shown with solid lines 123 Figure 4-19 Reynolds number effect on fore-aft plane interaction comparison for: (a) ReL = 7.5 million; (b) ReL = 4.2 million; (c) ReL = 2 million; (d) ReL = 1.5 million; (e) ReL = 1.0 million; (f) ReL = 0.5 million. Measured data with error bounds are shown with points, values from the numerical prediction are shown with solid lines.: 125 Figure 4-20 Body trim angle effect on fore-aft plane interaction comparison for: (a) -8 degree trim; (b) -4 degree trim; (c) -2 degree trim; (d) +2 degree trim; (e) +4 degree trim; (f) +8 degree trim. Measured data with error bounds are shown with points, values from the numerical prediction are shown with solid lines.: 126 Figure 4-21 Interaction curve fit of stemplane to bowplane lift curve slope ratio in trim 127 Figure 4-22 Interaction curve fit of stemplane to bowplane lift curve slope ratio in yaw 128 Figure 4-23 Modelled flow over plane 130 Figure 4-25 Modelled flow over plane 131 Figure 4-24 Analytical prediction of force on bowplanes as a function of depth 131 Figure 4-26 Analytical prediction of wave-induced bowplane force as a function of speed 132 Figure 4-27 Comparison of modelled and actual stall behavior at 3 m/s and 20 degrees yaw 134 Figure 5-1 Basic feedback control system schematic 138 Figure 5-2 Feed forward compensator 145 Figure 5-3 Feedback controller with series compensator 145 Figure 5-4 Basic PD controller block diagram 147 Figure 5-5 Gain scheduling PD controller schematic 148 Figure 5-6 Tuner schematic used in gain scheduling PD controller 150 Figure 5-7 Fuzzy-tuned series compensator block diagram 151 Figure 5-8 Fuzzy-tuner schematic 152 Figure 5-9 Input (yaw) membership function for and 153 Figure 5-10 Output membership function for and 154 Figure 5-11 Fuzzy logic control surface for 154 Figure 5-12 Fuzzy logic control surface for 155 Figure 5-13 Input (yaw) membership function for 156 Figure 5-14 Input (trim) membership function for 157 Figure 5-15 Output membership function for 157 Figure 5-16 Fuzzy logic control surface for 158 Figure 5-17 LQG/LTR controller block diagram 159 University of British Columbia Peter M. Ostafichuk April 2004 List of Figures xii Figure 5-18 Singular value plot for the plant 161 Figure 5-19 Singular value plot for the scaled plant 162 Figure 5-20 Filter singular values 163 Figure 5-21 Filter sensitivity and complimentary sensitivity singular values 163 Figure 5-22 Open loop singular value plot for plant and controller 165 Figure 5-23 System sensitivity and complimentary sensitivity singular values 166 Figure 5-24 Bumpless switching example for one input variable 167 Figure 5-25 Bumpless switching with two input variables 167 Figure 5-26 Switching surface representation in two dimensions for the sliding mode controller 168 Figure 6-1 Simulation process schematic 174 Figure 6-2 Simulator block diagram 175 Figure 6-3 Simulator plant block diagram 175 Figure 6-4 Simulator sensor and filter block diagram 176 Figure 6-5 Manoeuvre used for performance evaluation 177 Figure 6-6 Fictitious scanning operation used in performance evaluation .„ 179 Figure 6-7 Influence of the plant model on PD controller performance in pitch 186 Figure 6-8 Depth history in calm water for various plane geometry combinations 189 Figure 6-9 Roll history in calm water for various plane geometry combinations 190 Figure 6-10 Heave response based on bowplane geometry for sea state three (head seas) 191 Figure 6-11 Bowplane deflection based on plane geometry for sea state three (head seas) 192 Figure 6-12 Trajectory in x-y plane for alternate plane locations 193 Figure 6-13 Roll history in calm water for alternate plane locations 193 Figure 6-14 Roll history in calm water for Y-tail with increased roll error sensitivity 194 Figure 6-15 Bowplane deflection spectra based on plane location for sea state three head seas 195 Figure 6-16 Pitch history comparison for PD controller with and without fuzzy compensator 197 Figure 6-17 Depth response spectra for PD controller augmentations is sea state three head seas 198 Figure 6-18 Heave response for LQG/LTR augmentations in sea state three, head seas 200 Figure 6-19 Depth history for Gain Scheduled PD and Sliding Mode LQG/LTR in calm water 201 Figure 6-20 Pitch history for Gain Scheduled PD and Sliding Mode LQG/LTR in calm water 202 Figure 6-21 Roll history for Gain Scheduled PD and Sliding Mode LQG/LTR in calm water 203 Figure 6-22 Keel planes correcting for depth error 203 Figure 6-23 Stability in depth for PID and LQG/LTR controllers with keel planes 204 Figure 6-24 Heave response for Sliding Mode LQG/LTR and PD (sea state 3 head seas) 205 Figure 6-25 Deflection spectra for Sliding Mode LQG/LTR and PD (sea state 3 head seas) 205 Figure A-1 Vortex system used in upwash calculations 217 Figure A-2 Vortex system for forward planes (freestream flow into page) 218 University of British Columbia Peter M. Ostafichuk April 2004 List of Figures xiii Figure A-3 Vortex system for forward planes (freestream flow into page) 219 Figure A-4 Upwash comparison for isolated and body-mounted planes 220 Figure A-5 Change in Stall Angle With Aspect Ratio for Clark Y Airfoil 221 Figure A-6 Effect of Reynolds number on (Barlow et al., 1999) 222 Figure A-7 Effect of Reynolds number on for NACA 0025 airfoil 223 Figure A-8 Normalized bowplane lift curve slope dependence on yaw angle 224 Figure A-9 Description of flow velocity components on the bowplane in yaw 226 Figure A-10 Vortex system for forward planes (freestream flow into page) 227 Figure A-11 Vortex system for forward planes (freestream flow into page) 229 Figure A-12 Starboard side vortex system at aft plane (freestream flow into page) 231 Figure B-l Model assembly (ISE98-0011) 235 Figure B-2 Fore hull section (ISE98-0001) 236 Figure B-3 Hull nose (ISE98-0002) 237 Figure B-4 Mid-hull section (ISE98-0003 sheet 1 of 2) 238 Figure B-5 Mid-hull section (ISE98-0003 sheet 2 of 2) 239 Figure B-6 Aft hull section (ISE98-0007) 240 Figure B-7 Tail section (ISE98-0005) 241 Figure B-8 Mast connection (dorsal) (ISE98-0004) 242 Figure B-9 Mast tube (ISE98-0010) 243 Figure B-10 Mast assembly (ISE98-0012) 244 Figure B-l 1 Strain gauge assembly to mast (ISE98-0021) 245 Figure B-12 Mast fairing assembly (ISE98-0023) 246 Figure B-l3 Keel assembly (ISE98-0026) 247 Figure B-14 Sternplane deflection control assembly (ISE98-0018) 248 Figure B-15 Sternplane deflection control lead screw nut (ISE98-0015) 249 Figure B-16 Sternplane deflection control arm components (ISE98-0013) 250 Figure B-17 Sternplane deflection control motor casing (ISE98-0019) 251 Figure B-l 8 Bowplane stepper motor connector (ISE98-0014) 252 Figure B-l9 Bowplane load cell shaft (ISE98-LC-1) 253 Figure B-20 Sternplane load cell shaft (ISE98-LC-2) 254 Figure B-21 Load cell sleeve (ISE98-LC-3) 255 Figure B-22 Load cell sub-assembly (ISE98-LC-5) 256 Figure B-23 Bowplane load cell strain gauge installation (ISE98-LC-7) 257 Figure B-24 Sternplane load cell shaft (ISE98-LC-8) 258 Figure B-25 Bowplane load cell assembly (ISE98-LC-10) 259 Figure B-26 Sternplane hull insert (ISE98-LC-6) 260 University of British Columbia Peter M. Ostafichuk April 2004 List of Figures xiv Figure B-27 Force balance setup in wind tunnel (ISE98-B-1) 261 Figure B-28 Force balance assembly (ISE98-B-2) 262 Figure B-29 Force balance beam (ISE98-B-3) 263 Figure B-30 Balance pitch mechanism load bearing pin (ISE98-B-4) 264 Figure B-31 Balance pitch mechanism adjustment rod (ISE98-B-7) 265 Figure B-32 Main balance beam assembly (ISE98-B-12) 266 Figure B-33 Balance beam clamp for mast (ISE98-B-13) 267 Figure B-34 Force balance beam strain gauge installation (ISE98-B-14) 268 Figure B-35 Power and instrumentation wiring diagram (ISE98-E-1) 269 Figure B-36 Load cell deflection for maximum design load 270 Figure B-37 Load cell microstrain for maximum design load 271 Figure B-38 Sample load cell calibration data 272 Figure B-39 Boundary Layer Trip Mounting Locations 274 Figure B-40 Boundary Layer Trip Terminology 275 Figure D-l Depth response spectra for various LQG/LTR parameters (sea state three head seas) 286 Figure D-2 Pitch response spectra for various LQG/LTR parameters (sea state three head seas) 287 Figure D-3 Bowplane spectra for various LQG/LTR parameters (sea state three head seas) 287 Figure E-l Generic controller layout 325 Figure E-2 Navigation Module 326 Figure E-3 Heading correction 326 Figure E-4 Compass Correction 327 Figure E-5 Write data 327 Figure E-6 Noise 328 Figure E-7 Anti-aliasing filter 329 Figure E-8 Gain scheduling PD controller 330 Figure E-9 Controller tuner 330 Figure E-10 Controller with series compensation 331 Figure F-l Sway response based on control surface geometry in sea state three head seas 340 Figure F-2 Heave response based on control surface geometry in sea state three head seas 340 Figure F-3 Roll response based on control surface geometry in sea state three head seas 341 Figure F-4 Pitch response based on control surface geometry in sea state three head seas 341 Figure F-5 Yaw response based on control surface geometry in sea state three head seas 342 Figure F-6 Bowplane deflection based on control surface geometry in sea state three head seas 342 Figure F-7 Sternplane deflection based on control surface geometry in sea state three head seas 343 Figure F-8 Sway response based on control surface position in sea state three head seas 346 Figure F-9 Heave response based on control surface position in sea state three head seas 346 University of British Columbia Peter M. Ostafichuk April 2004 List of Figures xv Figure F-10 Roll response based on control surface position in sea state three head seas 347 Figure F-l 1 Pitch response based on control surface position in sea state three head seas 347 Figure F-l2 Yaw response based on control surface position in sea state three head seas 348 Figure F-l3 Bowplane deflection based on control surface position in sea state three head seas 348 Figure F-14 Stemplane deflection based on control surface position in sea state three head seas 349 Figure F-15 Sway response for PD augmentations in sea state three head seas 352 Figure F-16 Depth response spectra for PD augmentations in sea state three head seas 352 Figure F-17 Roll response spectra for PD augmentations in sea state three head seas 353 Figure F-l8 Pitch response spectra for PD augmentations in sea state three head seas 353 Figure F-l9 Yaw deflection spectra for PD augmentations in sea state three head seas 354 Figure F-20 Bowplane deflection spectra for PD augmentations in sea state three head seas 354 Figure F-21 Stemplane deflection spectra for PD augmentations in sea state three head seas 355 Figure F-22 Sway response spectra for LQG augmentations in sea state three head seas 358 Figure F-23 Depth response spectra for LQG augmentations in sea state three head seas 358 Figure F-24 Roll response spectra for LQG augmentations in sea state three head seas 359 Figure F-25 Pitch response spectra for LQG augmentations in sea state three head seas 359 Figure F-26 Yaw response spectra for LQG augmentations in sea state three head seas 360 Figure F-27 Bowplane deflection spectra for LQG augmentations in sea state three head seas 360 Figure F-28 Stemplane deflection spectra for LQG augmentations in sea state three head seas 361 University of British Columbia Peter M. Ostafichuk April 2004 List of Tables xvi L I S T O F T A B L E S Table 1-1 Dolphin specifications 6 Table 1-2 Summary of research into AUV modelling, simulation and control 8 Table 2-1 Variable conventions for translation 15 Table 2-2 Variable conventions for rotation 15 Table 2-3 Variable conventions for translation 16 Table 2-4 Location of Hull Vortices in Yaw 32 Table 2-5 Open ocean sea state occurrences of North Atlantic and North Pacific 39 Table 3-1 Key Dimensionless Parameters 42 Table 3-2 Plane Geometries 45 Table 3-3 Plane positions 46 Table 3-4 Test facility specifications 53 Table 3-5 ANOVA for wave force to wave geometry 93 Table 4-1 Comparison of hydrodynamic derivatives 108 Table 5-1 Fuzzy logic calculation methods 153 Table 5-2 Fuzzy rule-base for 153 Table 5-3 Fuzzy mle-base for 155 Table 5-4 Fuzzy mle-base for 156 Table 5-5 Sliding mode controller 166 Table 6-1 Sensor noise characteristics 176 Table 6-2 Body position and orientation corresponding to one sonar segment error 180 Table 6-3 Cases considered in simulation 183 Table 6-5 Effect of model improvements on performance in calm water with PD control 185 Table 6-4 Effect of model improvements on performance in calm water with LQG/LTR control 185 Table 6-6 Effect of controller model on performance in calm water with LQG/LTR control 186 Table 6-7 Effect of plane geometry on performance in calm water 188 Table 6-8 Effect of plane geometry on performance in sea state three 190 Table 6-9 Effect of plane location on performance in calm water 192 Table 6-10 Effect of plane location on performance in sea state three 194 Table 6-11 Effect of PD control structure on performance in calm water 196 Table 6-12 Effect of PD control structure on performance in sea state three 198 Table 6-13 Effect of LQG/LTR control structure on performance in calm water 199 Table 6-14 Effect of LQG/LTR control structure on performance in sea state three 200 Table 6-15 Gain scheduled PD to Sliding Mode LQG/LTR comparison for calm water 201 University of British Columbia Peter M. Ostafichuk April 2004 List of Tables xvii Table 6-16 Gain scheduled PD to Sliding Mode LQG/LTR comparison for sea state three 204 Table A- l Stall Types for Selected Airfoils 222 Table B-l Scale model component list 233 Table D-l LQG/LTR tuning parameter combinations 286 Table E-1 Summary of simulation m.file® 289 Table E-2 Simulink® Block Diagrams 325 Table F-l Cases considered in simulation 334 Table F-2 Effect of hydrodynamic effects modelled in plant on performance in calm water 336 Table F-3 Effect of hydrodynamic effects included in controller on performance in calm water 337 Table F-4 Effect of plane geometry on performance in calm water 338 Table F-5 Effect of plane geometry on performance in sea state 3 339 Table F-6 Spectral analysis for plane geometry effect on performance in sea state 3 339 Table F-7 Effect of plane location on performance in calm water 344 Table F-8 Effect of plane location on performance in sea state 3 345 Table F-9 Spectral analysis for plane location effect on performance in sea state 3 345 Table F-10 Effect of PD controller augmentation on performance in calm water 350 Table F-l 1 Effect of PD controller augmentation on performance in sea state 3 351 Table F-l2 Spectral analysis for PD controller performance in sea state 3 351 Table F-13 Effect of LQG/LTR augmentation on performance in calm water 356 Table F-14 Effect of LQG/LTR augmentation on performance in sea state 3 357 Table F-l5 Spectral analysis for LQG/LTR controller performance in sea state 3 357 University of British Columbia Peter M. Ostafichuk April 2004 List of Symbols xviii L I S T O F S Y M B O L S Symbol Description Application A a B b b' beP ^exp *i» c c Cr c Dc c DO Cr C L bp c L sp C* sp c La plant matrix wave amplitude of surface wave A matrix is LQG/LTR integrator implementation state space input matrix plane span spanwise location of plane tip vortex nondimensionalized by b spanwise location of inboard endplate exposed plane span B matrix in LQG/LTR integrator implementation state space state-to-output matrix control surface mean chord Coriolis added mass matrix coefficient of drag CD = D 1 >U2l2 crossflow drag coefficient minimum coefficient of drag value C matrix in LQG/LTR integrator implementation L coefficient of lift CL = Wl2 coefficient of lift of general bowplane coefficient of lift of general stemplane coefficient of lift of stemplane nondimensionalized by bowplane area L. C* „ = sp Lsp I oU2S bp coefficient of lift curve slope CLc dCL da control hydrodynamics hydrodynamics control control hydrodynamics hydrodynamics hydrodynamics hydrodynamics control control hydrodynamics all hydrodynamics hydrodynamics hydrodynamics control hydrodynamics hydrodynamics hydrodynamics hydrodynamics hydrodynamics University of British Columbia Peter M. Ostafichuk April 2004 List of Symbols Symbol Description Application Co Cr crb ct cx cz D D D d Ddist e F critical Fr Frc S H h K I J J. ct In constant value from curve fitting control surface root chord Coriolis matrix control surface tip chord coefficient of force in x -direction C = — CQCOSCC + C^sincc coefficient of force in z-direction C z = - C^cosa - C^sina state space control output to output matrix hull diameter drag depth damping matrix disturbance input vector to plant error vector e = v| ,-r J ^measured statistical value determined in ANOVA procedure critical F value determined in ANOVA procedure U Froude number Fr = Froude number based on chord Frc = U 'gC gravitational acceleration shape factor in Thwaite's method endplate height exposed endplate height h! = h — t Identity matrix LQG controller performance index simulation performance index for complete trajectory scanning error at time-step n Vn-y, In = simulation performance index for response significant amplitude e-e. hydrodynamics hydrodynamics all hydrodynamics hydrodynamics hydrodynamics control all hydrodynamics all all control control modelling modelling all hydrodynamics all hydrodynamics hydrodynamics hydrodynamics control control simulation simulation simulation University of British Columbia Peter M. Ostafichuk April 2004 List of Symbols Symbol Description Application simulation performance index for straight and level segments simulation K controller gain matrix control K moment about x -axis all fC non-dimensional K moment fC = all sliding mode LQG/LTR controller (positive trim and positive yaw) control x+. sliding mode LQG/LTR controller (positive trim and negative yaw) control K+Q sliding mode LQG/LTR controller (positive trim and zero yaw) control sliding mode LQG/LTR controller (negative trim and positive yaw) control K sliding mode LQG/LTR controller (negative trim and negative yaw) control sliding mode LQG/LTR controller (negative trim and zero yaw) control Ko+ sliding mode LQG/LTR controller (zero trim and positive yaw) control Ko- sliding mode LQG/LTR controller (zero trim and negative yaw) control Koo sliding mode LQG/LTR controller (zero trim and zero yaw) control K{a) matrix relating desired to actual plane deflections in gain scheduling control K Kj-C matrix element for bowplane-sternplane interaction control Kd PID controller derivative gain matrix control KP endplate aspect ratio correction factor hydrodynamics Kfc Fuzzy tuned compensator matrix control kfP Kj-C matrix element for change in port bowplane force with yaw control kfs Kj-C matrix element for change in starboard bowplane force with yaw control "gs controller gain matrix used in gain scheduling control Ki PID controller integral gain matrix control inboard K -moment on a plane hydrodynamics K i n LQG/LTR controller with integration controller gain matrix control *o outboard A"-moment on a plane hydrodynamics KP PID controller proportional gain matrix control University of British Columbia Peter M. Ostafichuk April 2004 List of Symbols Symbol Description Application *« hull curvature correction factor hydrodynamics K control error weighting matrix used in LQG/LTR control control \t tail efficiency correction factor hydrodynamics L LQG/LTR controller filter gain matrix control L lift force hydrodynamics I vehicle length all Lin LQG/LTR controller with integration filter gain matrix hydrodynamics M State-space process noise level matrix control M moment about y -axis all M non-dimensional M moment M = ——— all m mass all MA added mass matrix all Min M matrix for LQG/LTR integrator implementation control Mrb rigid body mass matrix all N noise input vector control N moment about z -axis all N total number of time-steps in simulation modelling /V N non-dimensional N moment ./V = all n state space noise vector control n current time-step in simulation modelling n propeller rotation rate all "in N matrix for LQG/LTR integrator implementation control P matrix determined in solution of Riccati equation for LQG control P angular velocity about x -axis all Q rotational transformation matrix all Q LQG control state error weighting matrix control q angular velocity about y -axis all University of British Columbia Peter M. Ostafichuk April 2004 List of Symbols xxn Symbol Description Application R LQG control control effort weighting matrix control R hull radius all r reference trajectory vector control r angular velocity about z -axis all r\/l hull radius at 1/3 chord position of plane hydrodynamics Re Reynolds number Re = — v all Ret Reynolds number at transition point hydrodynamics S control surface planform area hydrodynamics s generalized coordinate in boundary layer trip calculations hydrodynamics S(X) shear stress correlation factor in Thwaite's method hydrodynamics bowplane planform area hydrodynamics 5exp exposed control surface area hydrodynamics SJ response spectrum of scanning error simulation stemplane planform area hydrodynamics sy sensitivity of sonar scanning error to error in sway in simulation simulation sz sensitivity of sonar scanning error to error in heave in simulation simulation se • sensitivity of sonar scanning error to error in pitch in simulation simulation s * sensitivity of sonar scanning error to error in roll in simulation simulation s v sensitivity of sonar scanning error to error in yaw in simulation simulation T load cell calibration matrix, force from voltage hydrodynamics r load cell calibration matrix, voltage from force hydrodynamics t control surface thickness hydrodynamics t time all hrip boundary layer trip thickness hydrodynamics t XX load cell calibration slope vx versus X hydrodynamics lxz load cell calibration slope vx versus Z hydrodynamics Ux load cell calibration slope vz versus X hydrodynamics lzz load cell calibration slope vz versus Z hydrodynamics University of British Columbia Peter M. Ostafichuk April 2004 List of Symbols xxiii Symbol Description Application u total velocity U = Ju2 + v2 + w2 all u controller output control u velocity in x -direction all U convective acceleration hydrodynamics " l controller output in series compensation implementation control "2 compensator output in series compensation implementation control uc LQG/LTR controller output before integration control Uff feed-forward control signal control uw wave orbital velocity hydrodynamics wave velocity in x -direction all V velocity in y -direction all vx load cell voltage for x -axis (drag direction) hydrodynamics Vz load cell voltage for z -axis (lift direction) hydrodynamics w covariance of process noise in state space control W white noise vector with covariance W for process in state space control w velocity in z -direction all Win W matrix for LQG/LTR integrator implementation control W w wave velocity in z -direction all X force in x -direction all X non-dimensional X force X = all X state vector control X forward direction all *'ap X force of port aft plane (sternplane) all x„„ ap x -position of port aft plane (sternplane) from centre of gravity all ^as X force of starboard aft plane (sternplane) all xas x -position of starboard aft plane (sternplane) from centre of gravity all Xbp X force of general bowplane all University of British Columbia Peter M. Ostafichuk April 2004 List of Symbols xxiv Symbol Description Application Xfp X force of starboard fore plane (bowplane) all xfp x -position of port fore plane (bowplane) from centre of gravity all X force of starboard fore plane (bowplane) all Xfs x -position of starboard fore plane (bowplane) from centre of gravity all Xin state vector for LQG/LTR integrator implementation x( = u X control xplane x -position of general plane all xr X force of rudder all Xr x -position of mdder from centre of gravity all X force of general stemplane all ^bap first derivative of X non-dimensional force with 8 deflection all haphap second derivative of X non-dimensional force with 5ap deflection all ^has first derivative of X non-dimensional force with & a s deflection all ^ hashas second derivative of X non-dimensional force with 8as deflection all X¥P first derivative of X non-dimensional force with 8^ deflection all xhfp&fP second derivative of X non-dimensional force with 8y deflection all Xhfs first derivative of X non-dimensional force with 5 f c deflection all X hfshfs second derivative of X non-dimensional force with 8^ deflection all first derivative of X non-dimensional force with 8 r deflection all ^hrhr second derivative of X non-dimensional force with 8 r deflection all Y force in y -direction all Y Y non-dimensional Y force Y = all y control system output vector control y lateral direction all Y ap Y force of port aft plane (stemplane) all y ap y -position of port aft plane (stemplane) from centre of gravity all University of British Columbia Peter M. Ostafichuk April 2004 List of Symbols xxv Symbol Description Applicat ion Y ' as Y force of starboard aft plane (sternplane) all y as y -position of starboard aft plane (sternplane) from centre of gravity all Y force of general bowplane all y cp spanwise centre of pressure for a horizontal plane all Y force of starboard fore plane (bowplane) all Y force of starboard fore plane (bowplane) all yfp V -position of port fore plane (bowplane) from centre of gravity all V -position of starboard fore plane (bowplane) from centre of gravity all y in plant output vector for LQG/LTR integrator implementation y in ^inxin control yn y -position for time-step n simulation yplane y -position of general plane all Y Y force of rudder all yr y -position of rudder from centre of gravity all yr commanded (reference) y -position from navigation module simulation Y sp Y force of general sternplane all yv y -position of hull vortex in yaw measured from hull centreline all ^"hap first derivative of Y non-dimensional force with 8ap deflection all 1 haphap second derivative of Y non-dimensional force with b~ap deflection all ^Sas first derivative of Y non-dimensional force with 5as deflection all 1 SasHas second derivative of Y non-dimensional force with 8as deflection all rSfp first derivative of Y non-dimensional force with 8j-p deflection all Ybfphfp second derivative of Y non-dimensional force with bj- deflection all r¥s first derivative of Y non-dimensional force with 5j-s deflection all ^Sfsdfs second derivative of Y non-dimensional force with 5^ deflection all first derivative of Y non-dimensional force with 5 r deflection all V 5r5r second derivative of Y non-dimensional force with br deflection all University of British Columbia Peter M. Ostafichuk April 2004 List of Symbols xxvi Symbol Description Application Z force in z -direction all T Z all non-dimensional Z force Z = IpC/2/2 z vertical direction; depth all Zap Z force of port aft plane (stemplane) a ^ zap 2 - P o s i n o n 0 1 P o r t 3 1 1 plane (stemplane) from centre of gravity all Zas Z force of starboard aft plane (stemplane) a u zas i z -position of starboard aft plane (stemplane) from centre of gravity all Zfrp Z force of general bowplane a ^ Zj-p Z force of starboard fore plane (bowplane) a ^ Zj- z -position of port fore plane (bowplane) from centre of gravity all Zfs Z force of starboard fore plane (bowplane) a u Zj-S z -position of starboard fore plane (bowplane) from centre of gravity ah Z1 Zwave 311 8 ^wave nondimensionalized using gravity Zg = zn z -position at time-step n simulation 2plane z-position of general plane all Zr Z force of rudder a n zr z -position of rudder from centre of gravity all zr commanded (reference) z -position from navigation model simulation Zsp Z force of general stemplane a u zf boundary layer parameter in Thwaite's method zt = 0.25 - X( hydrodynamics z v z-position of hull vortex in yaw measured from hull centreline a l l Zwave Z force due to waves a l l Zgap first derivative of Z non-dimensional force with & a p deflection a u Z 5 a p 5 a p second derivative of Z non-dimensional force with 8 deflection a n Z§as first derivative of Z non-dimensional force with 8as deflection a n University of British Columbia Peter M. Ostafichuk April 2004 List of Symbols X X V l l Symbol Description Application '-' hashas second derivative of Z non-dimensional force with 8as deflection all Z¥P first derivative of Z non-dimensional force with 5̂ , deflection all Z6fp&fp second derivative of Z non-dimensional force with 5^ deflection all Zhfs first derivative of Z non-dimensional force with 8ys deflection all Z &fs&fs second derivative of Z non-dimensional force with 8^ deflection all Zhr first derivative of Z non-dimensional force with 8 r deflection all second derivative of Z non-dimensional force with 8 deflection all a incidence angle to plane all abp incidence angle to bowplane all alocal local change in the incidence angle to the planes hydrodynamics asp incidence angle to sternplanes all aw local incidence angle due to waves hydrodynamics P dihedral angle all r circulation hydrodynamics Y angle of tip vortex to freestream hydrodynamics AC Z increase in C z due to hull curvature hydrodynamics 5 plane deflection all 8 0 boundary layer thickness hydrodynamics «1 boundary layer displacement thickness hydrodynamics s2 boundary layer momentum thickness hydrodynamics port aft plane (port sternplane) deflection all 8« starboard aft plane (starboard sternplane) deflection all V / effective (bow)plane deflection all 5eff,sp effective sternplane deflection all ^eff, keel effective keelplane deflection all *fp port fore plane (port bowplane) deflection all he commanded plane deflections in fuzzy-compensated implementation control University of British Columbia Peter M. Ostafichuk April 2004 List of Symbols xxviii Symbol Description Application starboard fore plane (starboard bowplane) deflection all K rudder deflection all 8 downwash angle hydrodynamics vertical displacement of trailing vortex from longitudinal horizontal axis through fin/fin outreach hydrodynamics non-dimensional spanwise location on plane from root hydrodynamics e angular displacement about y -axis all e„ angular displacement about y -axis at time-step n simulation 0, commanded angular displacement about y -axis from navigation simulation ® local w ~ w w ~ a x local trim angle 6 / o c f l / - y all A Karman-Pohlhausen parameter used in Thwaite's method hydrodynamics X control surface taper ratio hydrodynamics X wavelength of surface wave hydrodynamics X* parameter used in determining hull influence X* = ^1 - (jf^J hydrodynamics \ boundary layer parameter used in Thwaite's method hydrodynamics fluid dynamic viscosity hydrodynamics p- LQG/LTR tuning parameter control V fluid kinematic viscosity hydrodynamics P fluid density all P recovery factor in LQG/LTR controller control wall shear stress hydrodynamics angular displacement about x -axis all •„ angular displacement about x -axis at time-step n simulation commanded angular displacement about x -axis from navigation simulation X non-dimensional position behind plane non-dimensionalized by span hydrodynamics angular displacement about z -axis all v» angular displacement about z -axis at time-step n simulation commanded angular displacement about z -axis from navigation simulation University of British Columbia Peter M. Ostafichuk April 2004 List of Symbols xxix Symbol Description Application Q control surface quarter-chord sweep angle hydrodynamics CO angular frequency simulation University of British Columbia Peter M. Ostafichuk April 2004 XXX P R E F A C E Westheimer's Rule states that to predict the time it takes to do a task, estimate the time you think it should take, multiply that by two, and change to the next highest unit of measure; thus, two weeks should be allocated for a one day task. My experiences though this project lead me to believe that, if anything, Westheimer's Rule is too conservative in its prognostication. The original description of this project was to simply measure and describe forces on an underwa- ter vehicle through scale model testing. Unfortunately, a working, instrumented model was not provided as initially planned and by rough estimates over 2500 person hours were required to pro- duce one. Additionally, a two week period of preliminary wind tunnel testing intended to validate model performance was greatly expanded to over one year to include system debugging, a full force measurement study, three different forms of flow visualization, and research by a second graduate student (namely, the study of yaw by Rodolfo Dominguez). Before experimentation began in the tow tank, many additional months were needed to design and construct mounting hardware and to seal and reconfigure the model for water. Midway through this work, a transfer was made from the Master of Applied Science program to a doctoral program. Along with the transfer came an expansion of the project scope to include work in modelling and control. Several months of unplanned modification to a simulator originally developed by Adrian Field was required in order to perform the added work and, even then, unex- pected results required even more time to confirm and validate. Still, with each delay and each additional task something was learned and the resulting quality of the project was improved. Throughout all the hardships and countless hours invested in this project, it remains one of the most educational and rewarding experiences of my life. -December 11,2003 In the time between when this thesis was sent for review and when it was defended there was a very sad good-bye. Dr. Dale Cherchas passed away after a courageous battle with cancer. Dr. Cherchas co-supervised this thesis and he worked extremely hard to see its completion, all the while fighting his illness. I cannot say enough about how much Dale's contribution means to me and I am left wishing that Dale were still here so I could express my gratitude in person. Thank you Dale. - April 22, 2004 University of British Columbia Peter M. Ostafichuk April 2004 XXXI A C K N O W L E D G E M E N T S The author would like to thank all the people that have contributed to this project, either through engineering insight and technical assistance, financing, or emotional support. In no particular order, much deserved thanks are owed to Josephine Lee, Olga and Joe Ostafichuk, Dr. Sander Calisal, Dr. Dale Cherchas, Dr. Mae Seto, Adrian Field, Gord Wright, Shu Oshika, Dr. Gary Schajer, Dr. Ian Gartshore, Dr. Sheldon Green, Doug Yuen, Dave Camp, Dr. George Watt, Matthais Borstad, Ayhan Akinturk, Rodolfo Dominguez, Ben Kaplan, Kevin Lam, Tom Takahashi, Kevin Kondra, Ben Triplett, and Dr. Raphael Cohen. The author would also like to thank the Science Council of BC, the Advanced Systems Institute (ASI), the Center for Integrated Computer Systems Research (CICSR), the University of British Columbia Graduate Student Society, and ISE for generous financial contributions. Portions of this work were performed under U B C Project Grant Number PG11R56901. University of British Columbia Peter M. Ostafichuk April 2004 X X X 1 1 To Josephine, for your unlimited love and support and To Dale, for working so hard to see this through while facing your own challenges University of British Columbia Peter M. Ostafichuk April 2004 Chapter 1 - Introduction 1 I I M T R O D U G T I O Ocean: A body of water occupying about two-thirds of a world made for man - who has no gills. - Ambrose Bierce The origins of underwater vehicles can be traced back over two millennia. Alex- ander the Great is said to have descended into the ocean in 332 B C E using a prim- itive diving bell and Leonardo da Vinci is credited with creating a wooden-frame submersible covered in goatskins with oars for propulsion (Clancy, 1993). Begin- ning in the early 1900s, submarines evolved into one of the most feared weapons in naval warfare. At times, these vessels ruled the sea in wolf packs; now their nuclear-powered descendants lurk undetected in the oceans, surfacing only to restock food and supplies. Recent advancements in electronics and automatic control have resulted in an explosion in the development of small, unmanned vehicles for more benevolent uses. Even with the long history of underwater vehicles, there is much to learn. In par- ticular, the design and usage of control surfaces-the topic that forms the basis for this research effort-has great potential for advancement. The remainder of this chapter provides an introduction to relevant background information on autono- mous underwater vehicles and control surfaces, a brief outline of previous research, and a summary of the motivation and approach used in this thesis. Sub- sequent chapters contain information relating to the hydrodynamics (Chapter 2), experimental studies (Chapter 3), and modelling (Chapter 4) of typical control surfaces on a near-surface underwater vehicle. The development of control strate- gies that make use of improved control surface hydrodynamic information is out- lined in Chapter 5 and an analysis of vehicle performance through control and dynamics simulation is detailed in Chapter 6. The final chapter includes a sum- mary and discussion of results including recommendations for future research efforts. University of British Columbia Peter M. Ostafichuk April 2004 Chapter 1 - Introduction 2 1.1 BACKGROUND Autonomous underwater vehicles, or AUVs, are unmanned, self-contained systems designed to carry out tasks in the marine environment. AUVs typically have one axial propulsor providing thrust with a complement of movable wing-like appendages, called control surfaces, which pro- vide manoeuvring forces. Unlike remotely operated vehicles (ROVs), which are teleoperated using a power and communications umbilical cable, AUVs are almost completely self-sufficient. Operating with on-board power and sensors, command decisions are made with a minimum of human intervention. The lack of an umbilical allows AUVs to operate at ranges unattainable by ROVs; missions of over 1000 kilometres are possible even in areas fully covered by ice (see Fer- guson and Pope, 1995, and Ferguson et al., 1999, for example). The purchase and maintenance costs for even elaborate AUVs are far below those associated with a manned launch (Shupe and McGeer, 1987). Likewise, the absence of an on-board human operator permits development of simpler, smaller, and more cost effective vehicles suitable for use in even the most remote and hos- tile environments. 1.1.1 Applications for AUVs The possible applications for AUVs are virtually limitless. The traditional role of an A U V is in track-line surveys of static oceanographic features such as bathymetry (An et al. 2001), sea floor magnetism (Willcox et al. 2001), and offshore petroleum deposits (Lorentz and Yuh 1996) for example. An A U V also provides a covert platform for remote minehunting tasks, virtually elimi- nating risk to both personnel and equipment (Watt et al. 1997; Gilbert 1990). Currently in the development of mine countermeasure systems, multiple coordinated A U V are being developed to detect both proud and buried targets over very large coverage areas (LePage and Schmidt 2002; Edwards et al. 2001). Although well suited to track-line survey missions, the true potential of AUVs is realized in the study of dynamic oceanographic phenomena. The wide range of temporal and spatial variability found in many oceanographic processes makes surveying by conventional means problematic or impossible (Willcox 2001). AUVs on the other hand are ideal for the study of phenomena such as oceanic circulation as it relates to climate change (Craven et al. 1999) as well as heat and green- house gas transport mechanisms (Huggins and Packwood 1995). Similarly, for predictive models of tides, hydrodynamics and bio-optical properties affecting the visibility and buoyancy of coastal waters, AUVs provide a means by which to effectively collect the necessary initial and boundary data (Carder et al. 2001). Numerous other applications are equally well suited to study by AUVs: cable inspection (Asakawa et al. 2002); temperature and salinity profiling (Levine et al. 1997); vertical turbulent velocity and flux measurement (Hayes and Morrison 2002); and oceanic convec- tion measurement (Zhang et al. 2001a; Zhang et al. 2001b), are but a few. The low-cost, portability, and self-sufficiency of AUVs permits operation in environments unsuited to ROVs and manned vessels. As an example, in 1996 the International Submarine Engi- neering (ISE) Theseus A U V successfully completed a 320 km under ice transit during a fibre-optic cable laying mission (Ferguson et al. 1999). This feat would not have been possible with an ROV and a human crew would have been at risk had the mission been carried out using a manned vessel. Through cost and portability considerations, Laval et al. (2000) selected AUVs for temperature profiling in small lakes. The AUVs offered additional benefits over conventional temperature pro- filing techniques as they allowed data collection along arbitrary paths and the characterization of University of British Columbia Peter M. Ostafichuk April 2004 Chapter 1 - Introduction 3 larger phenomena. In a similar fashion, Kumagai et al. (2002) developed a small A U V for the study of plankton distribution and water quality in lakes. From safety and size considerations, AUVs have also demonstrated great potential for use in coastal environments where very shallow water and significant wave effects makes other vessels impractical or unsafe (see Carder et al., 2001, Peterson et al., 1994, and Reidel and Healey, 1998, for example). Perhaps the most exciting example in the potential future of AUVs lies some 600 billion kilome- ters away on the Jovian moon, Europa. This satellite, roughly the same diameter as the Earth's moon, is believed to have an ocean of liquid water covered by vast sheets of ice. Akiyama et al. (1999), in addition to many others, suggest sending a probe capable of drilling through the ice to release an A U V to search for signs of life in Europa's waters. It is believed that life may be present surrounding thermal vents on the sea floor, at depths un-reachable by ROVs. It will be many years before manned space travel to even Mars is attempted, let alone through the oceans of a moon in the outer solar system; however, the technology is within reach to send an A U V to Europa. The cost and remoteness of such a mission-signals would take close to an hour to travel from Europa to Earth-reinforces the need for AUVs that are very well designed, intelligent, and completely self-sufficient. 1.1.2 AUV Operation AUVs fall under the category offlight vehicles as they require forward motion to manoeuvre. The typical geometry for an A U V is a streamlined shape with an axially-mounted propeller, as shown in Figure 1-1. ROVs, in contrast, are generally box-like and manoeuvre with thrusters, which allow those vehicles to hover but restrict speeds to about 3 knots (1.5 m/s). It is worth noting that several hybrid A U V / R O V concepts are currently being considered (see Underwater, 2000, Cancil- liere, 2001, and McFarlane et al., 2001, for example) so the distinctions between A U V and ROV are not entirely rigid. However, these hybrid vehicles are generally designed for very specific environments or tasks; as such the inherent complexity generally makes them impractical for widespread use. Manoeuvring of an A U V is achieved through adjustment of the control devices shown in Figure 1-1; specifically these include the propeller, bowplanes, sternplanes, and rudder. Except in very special circumstances, an A U V is a six degree-of-freedom (DOF) system capable of three translational motions and three rotational motions (as shown in Figure 1-2). In general, these dif- ferent motions are non-linear and strongly coupled. Various strategies have been devised for dis- tributing the responsibility for control of each motion between the various control devices. Typically, surge control is restricted to the propeller and no attempt is made to control sway directly. The rudder is generally used for heading (yaw) control but in the process introduces an unwanted roll moment due to the rudder distance from the hull longitudinal axis. The bowplanes and sternplanes are used for control of pitch and depth (heave), as well as for compensation of the roll effects introduced by the rudder. In some instances, authority for pitch and depth control is assigned separately to the sternplanes and bowplanes respectively (see Watt, 1997, or Shupe and McGeer, 1987). This strategy is well suited to vehicles that have bowplanes positioned close to the longitudinal centre of mass as in this position they do not impart a large pitch moment. University of British Columbia Peter M. Ostafichuk April 2004 Chapter 1 - Introduction 4 Figure 1-1 Typical geometry of an AUV Figure 1 -2 AUV degrees of freedom University of British Columbia Peter M . Ostafichuk April 2004 Chapter 1 - Introduction 5 1.1.3 The DOLPHIN AUV The DOLPHIN is a near-surface A U V developed by International Submarine Engineering Research (ISER). The initial development of DOLPHIN began in 1981 in conjunction with the Canadian Hydrographic Service. Since that time, a number of different variants have been produced (includ- ing the Mark I, Mark II, and Dorado). The work in this thesis is based on the Mark II DOLPHIN shown in Figure 1-3; key features are identified in the figure. Figure 1-3 ISER DOLPHIN Mark II A U V The unique characteristic of DOLPHIN is the semi-submersible design. The vehicle operates with the hull centreline at depths of 3 to 4.5 hull diameters (Williams et al., 2000) in conditions up to sea state 5 (Seto and Watt, 1998). Throughout operation, the top of the mast remains above the waterline and is used to provide air to the diesel powerplant and facilitate data communication. This vehicle design allows the stability and covertness of a submarine vehicle with the endurance and communications benefits of a surface ship. In particular, although the DOLPHIN is less than University of British Columbia Peter M. Ostafichuk April 2004 Chapter 1 - Introduction 6 8.5 m in length, in waves it behaves similarly to a much larger surface vessel. The greatly reduced above-water profile also makes the DOLPHIN much more covert that even similarly sized surface craft. DOLPHIN operates with a diesel engine which provides both increased power and endurance in comparison to typical submarines. Similarly, the surface piercing snorkel provides a mounting point for global positioning system and real-time communications hardware, a feature not found on conventional underwater vehicles. The snorkel is fitted with self-aligning fairings to minimize the effects of cross flow on vehicle roll. The specifications of the Mark II DOLPHIN (obtained from Watt et al. (1997) and Seto and Watt (1998) are given in Table 1-1. Table 1-1 DOLPHIN specifications Parameter Value Overall length Hull diameter Dry mass Variable ballast Power Top speed3 Enduranceb 8.534 m 1 m 4500 kg 907 kg 350 horsepower 18 knots/9.3 m/s 16 hours a. with towfish housed at the keel b. while towing The DOLPHIN was developed as a platform for instrumentation with typical applications including oceanographic surveys, seafloor mapping, and mine reconnaissance, for example. The common requirement in these operations is the ability to maintain heading and orientation while travelling through potentially adverse seas. In comparison to other underwater vehicles, the disturbances that DOLPHIN is subjected to are particularly large due to the proximity to the surface. Likewise, the permissible range of operating depths and roll angles is comparatively small for DOLPHIN as the vehicle cannot continue to operate if the mast should become fully submerged. These condi- tions and restrictions make the DOLPHIN an ideal candidate vehicle for examining the control sur- face effectiveness and improved control strategies for underwater vehicles in general. University of British Columbia Peter M. Ostafichuk April 2004 Chapter 1 - Introduction 7 1.2 MOTIVATION The performance of autonomous underwater vehicles is governed by the effectiveness of the vehi- cle control system. The control system is comprised of the controller (which is responsible for low-level command decisions) and the plant (the AUV). Control actions are administered by the propulsor and control surfaces. The command decisions made by the controller are based on knowledge or estimates of the position and orientation of the vehicle. In order to compensate for errors in position or pose, the controller determines required hydrodynamic forces on the control surfaces necessary for corrective action. Estimates of control surface performance are used to determine the required positions of each of the control surfaces in order to generate the desired forces. The effectiveness of such a system is dependent not only on the performance of the controller and the control surfaces but also on their inter-relationship. It is essential that the control surfaces be capable of generating forces that are sufficient in magnitude and oriented in suitable directions. Likewise, the control surface forces—hence the control surfaces themselves—must be positioned at appropriate locations on a vehicle. The behavior of the control surfaces is a function of the con- trol surface configuration, the vehicle orientation, and the operating conditions and can be compli- cated, non-linear, and highly coupled. Furthermore, the controller must make appropriate decisions regarding the management of control surfaces in the presence of environmental distur- bances based on measurements that may be contaminated by noise. Without a sound understand- ing of the hydrodynamic performance of the control surfaces under all conditions, controller decisions will likely be sub-optimal and, in some cases, may threaten vehicle stability. Addition- ally, with improved knowledge of control surface behavior, more intelligent choices can be made regarding the control methodologies and strategies employed. As the development of an A U V is both costly and time consuming, the use of modelling and simu- lation are essential to the design process. There is a need for mathematical descriptions of control surface performance that are accurate and complete while at the same time not overly complex so as to be difficult to implement. Existing control surface models used in A U V development and simulation are based on simple, linear representations (see, for example, Triantafyllou and Hover, 2002, Prestero, 2001a, or Doucy et al., 2000) or second order representations (see Oh et al., 2002, or Field, 2000); these models do not account for changes in control surface performance due to vehicle orientation, flow field interactions, or viscous effects such as flow separation. It is impor- tant to fully understand the hydrodynamic behavior of the vehicle control surfaces but it is also important to know how much detail is required in mathematical models used for simulation and control. Models of control surface performance should be as simple as possible while retaining key performance characteristics. Specifically for the DOLPHIN AUV, ISER identified the need to conduct a detailed investigation of the performance of the bowplanes and sternplanes. During the development of the first DOLPHIN, past experience, field tuning, and "good guesses" were used for determining the vehicle geometry (den Hertog 1997). ISER speculated that the arrangement of control surfaces on the Mark II DOL- PHIN was not ideal based on both geometric and operational considerations (Seto, 1997). Further- more, small scale preliminary testing by university student groups (Han et al. 1998) and simple analytical models (Watt et al. 1997) suggested flow based interactions between fore and aft control surfaces (unaccounted for in the controller strategy) may significantly compromise depth and roll control as well as stability. University of British Columbia Peter M. Ostafichuk April 2004 Chapter 1 - Introduction 8 1.3 SURVEY OF PREVIOUS WORK A summary of work by others on underwater vehicle modelling, control, and simulation is pro- vided below in Table 1-2 (based in part on work by Lea et al. (1999)). None of the previous work focussed specifically on the performance of control surfaces, either in terms of hydrodynamics or controller design. The research vehicle studied in each is noted along with the variables controlled and the control methodology. The predominant method of research is indicated as either experi- mental (Exp) or simulation (Sim). (Bibliographic references are provided for each entry in the table.) Table 1-2 Summary of research into A U V modelling, simulation and control Variables Researchers Year Controlled3 Type Controlb Vehicle Bystrom 1988 z,e Sim LQ Submarine Rodriguez & Dobeck 1989 y,z,9,tj/ Exp GS LSV Cristi, Papoulios & Healey 1990 z Sim A-SM NPSAUVII Dougherty & Woolweaver 1990 Exp SM MUST Healey & Marco 1992 x,y,z,v|/,u Exp SM NPSAUVII Venugopal, Sudhakar & Pan- 1992 Sim NN Ocean Voy- dya ager Healey & Lienard 1993 x,y,z,v|/,u Sim SM NPSAUVII Fryxell et al. 1994 x,y,z Sim MARIUS Hills & Yoerger 1994 z,6,y Exp SM LDUUV Jalving 1994 z,9 ,v|/ ,u Exp PID NDRE-AUV Smith, Rae, Anderson & Shein 1994 z,e,v/ Sim FL Torpedo Peterson, Nguyen & Rodriguez 1994 z, e Sim LQR CSS Xu & Smith 1994 z Sim FL Torpedo DiBiterro 1995 z Sim FL ARPA UUV Liceaga-Castro & van der 1995a z Sim GS Submarine Molen Liceaga-Castro & van der 1995b z Sim Submarine Molen OO Lorentz & Yuh 1996 z Sim& Exp NN ODIN Perrier & Canudas-de-Wit 1996 - Sim& Exp PID, PID-NL VORTEX Lea 1997 V|/,U Sim& PID Subzero II Exp Lea, Allen & Merry 1997 u Sim GS, FL, SM Autosub University of British Columbia Peter M. Ostafichuk April 2004 Chapter 1 - Introduction 9 Table 1-2 Summary of research into A U V modelling, simulation and control (Continued) Variables Researchers Year Controlled3 Type Control6 Vehicle Silvestre, Pascoal & Healey 1997 z,u Sim NPSAUVII Suto & Ura 1997 z Exp NN Manta-Cere- sia Riedel & Healey 1998 Z,(|) ,0 ,u Sim SM NPS PHOE- NIX An & Smith 1998 z,e Exp SM-FL Ocean Explorer Barlow, Harris & Ranzenbach 1998 z,e Sim& Exp PID Generic AUV Field, Cherchas & Calisal 2001a z,<() ,6 ,u Sim LQG/LTR DOLPHIN Field et al. 2001b z, (p ,9 ,\(/ ,u Sim A- LQG/LTR DOLPHIN Song, An & Smith 2002 e,\\i Sim& Exp SM-FL Ocean Explorer a. see Section 2.1 on page 14 for a definition of the variables used b. Control Methodology Legend: A - adaptive, FL - fuzzy logic; GS - gain scheduling; H^ - H-infmity; LQG/LTR - linear quadratic gaussian with loop transfer recovery; LQR - linear qua- dratic regulator; NN - neural network; PID-PID or three-term; NL - non-linear; SM - sliding mode. Two traditional control strategies combined to form a hybrid are separated by a hyphen; multiple control schemes examined independently are separated by a comma. In the survey of previous work, sixteen of the twenty-eight cases were based solely on simulation. Without physical measurement of control surface performance or validation of simulation results, the complicated and non-linear behavior of the control surfaces is unlikely to manifest itself in the results. O f the remaining twelve cases, seven relied exclusively on experimentation which is not only costly and impractical for typical A U V development, but is difficult to generalize and not well suited to the study of control surface modelling. With the five cases that use both simulation and control, in each case motion is restricted to two degrees of freedom. As the performance of the control surfaces is known to depend on body orientation, restricting the permissible vehicle motions results in flow conditions and control surface performance that are not representative of a typical AUV. The work of this thesis is intended to address the shortcomings in the previous research in regard to hydrodynamic, modelling, and control aspects of A U V control surface performance. Experi- mentation is used to accurately determine control surface hydrodynamic characteristics; modelling is used to represent the experimental results in a compact mathematical form; and simulation is used to study the effect of control surface modelling and control on overall vehicle performance. University of British Columbia Peter M. Ostafichuk April 2004 Chapter 1 - Introduction 10 1.4 OBJECTIVES The primary objective of this research was to improve A U V performance through improvements to vehicle control surface effectiveness. The factors affecting the performance of control surfaces are interdisciplinary in nature—spanning hydrodynamics, modelling, and control—and this was reflected in the primary objective. To improve control surface effectiveness, it was necessary to fully describe the hydrodynamic performance of control surfaces for the test vehicle, improving the accuracy and completeness of existing performance predictions where possible. The control surface hydrodynamic behavior was required to be modelled in a practical mathematical form that retained all key performance characteristics. It was also necessary to determine the significance of including or ignoring various aspects of control surface performance in performance models. Finally, understanding the consequence of incorporating new control surface hydrodynamic effects in control system design, as well as the investigation of enhanced control methodologies to account for any new hydrodynamic effects, was needed. A secondary objective of this research was to provide ISER with guidelines and strategies for use in development of control surfaces for their AUVs. In particular, an assessment of the current con- trol surface configuration on the Mark II D O L P H I N was needed in addition to recommendations for improvements to the control surface layout. Additionally, it was important to establish the validity of previous simulations conducted for ISER on controller development that were conducted with less complete control surface hydrodynamic models (Field, 2000). Lastly, DOLPHiN-specific improvements to the control strategy based on the new control surface hydrodynamic information were sought. University of British Columbia Peter M. Ostafichuk April 2004 Chapter 1 - Introduction 11 1.5 APPROACH AND RATIONALE Improvements to control surface effectiveness on near-surface underwater vehicles was considered from three main approaches: hydrodynamics, modelling and simulation, and control. The under- standing of the physics regarding control surface performance is furthered through a systematic series of hydrodynamic experiments on several scale models. The resulting information of control surface behavior can be directly applied to underwater vehicle control surface designs and layouts. However, with the enhanced understanding of control surface hydrodynamics, improved control schemes are also developed through modelling and simulation. 1.5.1 Hydrodynamics Two main aspects of the physics of control surface performance were studied in multiple testing facilities. The first area of study was the influence of operating factors (such as the body orienta- tion and the free surface) on the performance of control surfaces. The second was the effect of control surface geometry and mounting location on the control surface effectiveness. Both wind tunnels and towing tank facilities were used for experimental testing with findings pre- sented in Chapter 3. The wind tunnels provided a means by which to quickly investigate the behavior over a wide range of operating conditions and were also conducive to a wide variety of flow visualization techniques. Generally, the time required per test is much lower in a wind tunnel compared to a towing tank (Watt et al. 1993) or other water based facility (such as a flume or water tunnel for example). However, wind tunnels are incapable of fully duplicating the operating envi- ronment for near-surface underwater vehicles and the wind speeds required exceed the capabilities of most facilities. For these reasons, preliminary control surface force measurement tests were conducted in the wind tunnels covering a wide range of conditions. Several flow visualization techniques were also employed to complement the wind tunnel force measurement work. Based on the findings from the wind tunnel, additional testing was conducted in a towing tank for the most important cases to improve the modelling realism. 1.5.2 Modelling and Simulation The hydrodynamic information obtained through testing was modelled numerically. The numeri- cal modelling (detailed in Chapter 4) was used to describe the control surface behavior in as con- cise and simple a representation as possible without loss of realism. The influence of operating parameters such as Reynolds number, body orientation, and the free surface condition, for exam- ple, were treated separately and then linearly combined. The numerical representation of control surface performance was incorporated into a simulation package (outlined in Chapter 6). Simulations were conducted to examine the importance of the various improvements in control surface modelling. By comparing simulation results with changes to only specific aspects of the control surface modelling, the importance of those changes could be quantified. The influence of modifications to control surface geometry and layout were examined in the same manner as were improvements to the control methodology (Chapter 5). A single performance parameter criterion, based on sonar image smearing in a representative case, was developed for comparison between simulations. University of British Columbia Peter M . Ostafichuk April 2004 Chapter 1 - Introduction 12 1.5.3 Control The performance of the existing LQG/LTR control scheme was examined using the control and dynamics simulation package mentioned above. Augmented control methodologies (as discussed in Chapter 5) were studied by incorporating additional modules into the simulation package. A base-line PD controller was also developed to represent the existing controller in the D O L P H I N . Due to the scope of this work, limited time and resources were available for the study of vehicle control. Accordingly, the intention of the work in controller augmentation was only a "proof-of- concept". Extensive validation and further development would be required prior to implementa- tion of the control strategies on a vehicle; in particular, a rigorous analysis of stability would be required. University of British Columbia Peter M. Ostafichuk April 2004 Chapter 2 - Control Surface Hydrodynamics - Theory 13 2 U R F A A M I O Diving [a submarine] is a carefully controlled and balanced pro- cedure that resembles a ballet danced by an elephant. -Tom Clancy on the USS Miami Naval Submarine The manoeuvrability and stability of a submarine, as well as the ability to com- pensate for disturbances, are dependent on the hydrodynamic forces generated by the control surfaces. Underwater vehicles are frequently unstable (Barlow et al., 1999) and would be unusable without active operation of the control surfaces. Accurate knowledge of the control surface performance is essential in all stages of underwater vehicle development including initial design and specification, model- ling, simulation, and control system development. Within this chapter, the background information relating to control surface hydro- dynamics is presented. Following a description of conventions and terminology, a brief summary of semi-empirical findings is presented for both isolated control surfaces and control surfaces mounted on a hull. (In the literature on underwater vehicle technology, the term "semi-empirical" is used to describe analytical rela- tionships derived from experimental data.) The chapter concludes with a discus- sion of the interaction between fore and aft control surfaces as well as with the free surface. University of British Columbia Peter M. Ostafichuk April 2004 Chapter 2 - Control Surface Hydrodynamics - Theory 14 2.1 CONVENTIONS The components of an A U V (or any underwater flight vehicle) that are central to this research were highlighted in Figure 1-1 and are reproduced in Figure 2-1 below. propeller Figure 2-1 Typical geometry of an autonomous underwater vehicle Of particular importance are the horizontal control surfaces identified as the bowplanes and stern- planes. These wing-like appendages are also referred to as fore planes and aft planes respectively, or in general as simply planes. Typically the planes and the rudder are all-moveable, but in some cases they may be fixed to the body with a moveable flap on the trailing edge. Other concepts are possible, such as rotating cylinders on fixed fins (den Hertog and Modi, 2001), but the all-movable design is by far the most studied and most commonly employed. The conventions in this thesis for describing position, pose, motion and forces relevant to an A U V are the same as those proposed by Feldman (1979). The sign conventions are shown in Figure 2-2 and the symbols used are summarized in Table 2-1 and Table 2-2 for translation and rotation respectively. University of British Columbia Peter M . Ostafichuk April 2004 Chapter 2 - Control Surface Hydrodynamics - Theory 15 Figure 2-2 Sign conventions Table 2-1 Variable conventions for translation Symbol used Force Force Quantity Displacement Velocity (dimensional) (non-dimensional) surge X u X X sway y V Y r heave z w Z z Table 2-2 Variable conventions for rotation Symbol used Angular Angular Moment Moment Quantity Displacement Velocity (dimensional) (non-dimensional) roll • P K fC pitch e <l M M yaw v r N iV University of British Columbia Peter M . Ostafichuk April 2004 Chapter 2 - Control Surface Hydrodynamics - Theory 16 Where practical, forces are non-dimensionalized using the dynamic pressure and the vehicle length squared. For example, in the x -direction, the force is non-dimensionalized according to X = X (1) where p is the fluid density, U is the speed of the freestream flow, and / is the vehicle length. The dynamic pressure specifically is the combination ^ p U2. Moments are non-dimensionalized by dynamic pressure and vehicle length cubed. For example, the moment about the x -axis, K, is non- dimensionalized as K K •oU2!3 (2) In some instances, such as with isolated control surfaces, the forces are non-dimensionalized by a reference area, S, and expressed as a force coefficient. For example, the lift force, L, on an iso- lated plane with exposed planform area, 5 e x p , is described by the coefficient of lift, CL by C, L (3) The variables describing the actuator deflections (and propeller speed) are summarized in Table 2-3 and use the sign conventions of Figure 2-2. Table 2-3 Variable conventions for translation Control Surface Deflection Name Alternate Name Symbol port bowplane port fore plane hp starboard bowplane starboard fore plane port sternplane port aft plane starboard sternplane starboard aft plane Ks rudder - K propeller - n University of British Columbia Peter M. Ostafichuk April 2004 Chapter 2 - Control Surface Hydrodynamics - Theory 17 Derivatives of force and moment with respect to control surface deflections—referred to as hydro- dynamic derivatives—are expressed using subscript notation. For example, the first derivative of the X force with respect to the port bowplane is while the second derivative of the X force with respect to port bowplane deflection is d2 -—-X'=X'5fp6fp (5) °°fP University of British Columbia Peter M. Ostafichuk April 2004 Chapter 2 - Control Surface Hydrodynamics - Theory 18 2.2 CONTROL SURFACE GEOMETRY The geometric and sign conventions used with isolated control surfaces in this work are shown in Figure 2-3. Note that the coordinate system for the isolated appendage is fixed to the appendage while the coordinate system for a plane on a body is fixed to the body (see Figure 2-2). In both cases, the angle of incidence, a, of the flow onto the plane is defined by the angle between the incoming flow vector, U, and the chord line as shown. The main parameters that define control surface geometry are summarized in Figure 2-4. Details regarding each of these parameters are discussed in turn below. 1 z, Z. w Figure 2-3 Control surface coordinate system (shown on starboard side plane) 2.2.1 Chord The chord is denoted by C and, for a given two-dimensional section, it is defined as the distance from the leading edge to trailing edge. The chord is measured parallel to the section at the root, or inboard portion, of the control surface. In general, the chord can vary along the span (defined below), in which case the geometric mean chord, C, is used in computations unless noted. For this work, trapezoidal shaped control surfaces are used and C is defined based on Figure 2-4, as Lift, CL x, X, u Drag, CD y (6) University of British Columbia Peter M. Ostafichuk April 2004 Chapter 2 - Control Surface Hydrodynamics - Theory 19 Figure 2-4 Control surface parameters (shown on port side plane) 2.2.2 Span The span, denoted by b, is a measure of the distance from the control surface root to tip along the line perpendicular to the root section. Mathematically, there are several ways in which the span is commonly defined: the root-to-tip distance for an isolated plane; the distance from the body cen- treline to the plane tip; the tip-to-tip distance for two planes mounted on opposite sides of a body; or twice the root-to-tip distance for a plane mounted on a large body. For the first two cases, the measurement indicated is sometimes also referred to as the semi-span. For this work, unless noted otherwise, the "twice root-to-tip distance" method (as shown in Figure 2-4) is used. 2.2.3 Foil Section The geometry of a control surface is mainly defined by the two-dimensional foil section(s) used and the planform. The foil section is identified by the shaded region in Figure 2-4; it can take almost any geometry although virtually all forms have a rounded leading edge and a sharp trailing edge. Symmetric, four-digit National Advisory Committee for Aeronautics (NACA) sections are most often used with underwater vehicles. For symmetric foils, the four-digit N A C A designation is "NACA OOXX" where the "XX" represents the thickness expressed as a percentage of the foil chord. In practice, the same section is generally used throughout a plane although a transition between sections is possible along the control surface span. In this work, a N A C A 0025 section, University of British Columbia Peter M. Ostafichuk April 2004 Chapter 2 - Control Surface Hydrodynamics - Theory 20 corresponding to a thickness-to-chord ratio of 0.25, was used except where noted otherwise; the planes and rudder on the Dolphin vehicle all use the N A C A 0025 section. 2.2.4 Planform While the section describes a two-dimensional slice through a control surface, the planform identi- fies the layout of sections that form the three-dimensional plane. Essentially, the planform describes the chord as a function of the position along the span. In most cases, simple trapezoidal planforms are used that can be defined by four main parameters: chord, span, taper ratio, and sweep. The taper ratio, X, is defined for a trapezoidal planform as the ratio of tip chord to root chord. The sweep can be defined is several ways; for this work the quarter-chord sweep angle, Q, as given in Figure 2-4 is used. A rectangular planform has a taper ratio of unity and sweep angle of zero. Complicated planforms, such as the elliptical one made famous on the RAF Spitfire, can offer slight performance benefits but are not often used due to added difficulty in manufacture. Further information about low aspect ratio swept wings can be found in Aronson and Lekander (1986) and van den Berg et al. (1977). The planform area, S, is often used in reference to control surfaces. The definition of S can be based on either the exposed area of a single control surface, as with Aucher (1981), or on two con- trol surfaces, as with Whicker and Fehlner (1958). In either case, the mathematical definition is S = bC (7) where the difference in interpretation of S is included within the definition of b. 2.2.5 Aspect Ratio The aspect ratio is based on the planform and represents one of the main parameters determining control surface performance. Denoted by ae in this work, and sometimes denoted by AR by oth- ers, aspect ratio is a dimensionless value that describes the planform slenderness. As with the other parameters, various definitions of aspect ratio exist but typically for underwater vehicles it is defined as ae - j (8) For this work with rectangular and trapezoidal control surfaces, this is equivalent to ae = I (9) C with b and C as defined in Figure 2-4 and equation (6). Thus, a large value of ae corresponds to a slender wing. University of British Columbia Peter M. Ostafichuk April 2004 Chapter 2 - Control Surface Hydrodynamics - Theory 21 Control surfaces are considered to be of low aspect-ratio when the span is of the same order as the mean chord, that is for values of ae less than about 2 or 3 (Aucher 1981). In general, a reduction in aspect ratio results in reduced force, generation for a given angle of incidence but is also associ- ated with delayed stall and an increase in the range of operation (see White, 1986, or Talay, 1975). For illustration, Figure 2-5 shows the variation in the lift coefficient with aspect ratio for a Clark-Y airfoil (reproduced from Zimmerman, 1933, and Ffoerner and Borst, 1975). Note that from the source, the data were adjusted to zero lift at zero degrees incidence. 1.4 , Angle of Incidence [deg] Figure 2-5 Variation lift coefficient with aspect ratio 2.2.6 Endplates Endplates, sometimes called fences, are added to the ends of control surfaces in order to minimize leakage from the suction to pressure side. Typically, endplates are added to the tip of an append- age but may also be added to the root (see Figure 2-6). Endplates reduce the spanwise flow near ends of the plane and thereby increase the effective aspect ratio. According to Talay (1975) the endplates also reduce the strength of the tip (trailing) vortices shed by a plane. In terms of performance prediction, Mackay (1998) provides the following endplate factor *«p = 1 + 14-05(s;)2 forS-<2-5) (10) University of British Columbia Peter M. Ostafichuk April 2004 Chapter 2 - Control Surface Hydrodynamics - Theory 22 Inboard endplate Outboard endplate Figure 2-6 Inboard and outboard endplates The endplate factor is used to adjust the effective aspect ratio of the planes. The endplate height is given by, h, and the plane span, b, is measured to the hull centreline. For outboard (tip) endplates, the effective aspect ratio becomes ae => Kpae (11) while for inboard (root) endplates, the effective aspect ratio is (12) where bBn is the spanwise location of the endplate measured to the hull centreline. ep University of British Columbia Peter M. Ostafichuk April 2004 Chapter 2 - Control Surface Hydrodynamics - Theory 23 2.3 AUV CONTROL SURFACE LAYOUT The effectiveness of control surfaces is determined in part by the control surface geometry but also by the placement of the control surfaces on a vehicle. Typically, control surfaces are mounted as shown in Figure 2-1 with the rudder in the vertical plane and bowplanes and sternplanes in the hor- izontal plane. There may be a vertical offset between the bowplanes and sternplanes—the bow- planes on the DOLPHIN Mark II, for example, have a vertical position just over 0.01 body lengths lower than sternplanes. More elaborate options include adding dihedral (anhedral) to the bow- planes, modifying the orientation of control surfaces on the tail, or repositioning planes to loca- tions vehicle-specific locations. The size and location of control surfaces is often limited by practical considerations. Increased span (width of horizontal control surfaces from the vehicle) may be advantageous in terms of increased force and moment generation but may be undesirable due to increased vulnerability to damage during launch and recovery. Likewise, low appendages may be at risk in shallow water or in towing operations (Watt et al., 1997). 2.3.1 Anhedral and Dihedral Dihedral refers to the span-wise inclination angle of a wing or control surface from the horizontal plane and about the longitudinal axis of a vehicle. The dihedral angle is measured positive as shown in Figure 2-7 (a); negative dihedral is called anhedral as shown in Figure 2-7 (b). In the lit- erature, the symbols r or 9 are often used for dihedral but for this work, 6 is chosen as T and 0 are reserved to denote circulation and pitch angle respectively. (a) (b) Figure 2-7 Span-wise bowplane inclination angle: (a) dihedral, (b) anhedral Dihedral is often added to aircraft wings to alter lateral stability characteristics. For underwater vehicles, dihedral has several additional benefits: Firstly, the bowplanes tips generate significant flow disturbances that can have adverse effects on the sternplanes; inclining the bowplanes with University of British Columbia Peter M. Ostafichuk April 2004 Chapter 2 - Control Surface Hydrodynamics - Theory 24 dihedral tends to move the bowplane tips to a position less in-line with the sternplanes, thus poten- tially reducing the flow interactions. Secondly, a bowplane with non-zero dihedral generates a control force with both horizontal and vertical components. This allows for greater redundancy in the control of yaw (otherwise controlled solely by the rudder) and makes active control of sway possible. Thirdly, the use of anhedral specifically (not dihedral) orients the bowplanes in a posi- tion less likely to be influenced by free surface disturbances such as waves. 2.3.2 Tail Control Surface Configurations The common configurations for the stern control surfaces (the sternplanes and rudder) are shown in Figure 2-8. These include the standard (cruciform) tail and several variations with the stern- planes at non-zero inclination angles. In the case of the Y- and inverted Y-tail there is a vertical rudder with two inclined planes while with the X-tail all four planes are inclined at non-zero dihe- dral. The benefits of the alternate tail geometries presented in Figure 2-8 (b) through (d) are similar to those for dihedral bowplanes. Namely, the sternplanes are positioned such that flow from bow- planes is less likely to result in adverse interactions and there is increased redundancy in the con- trol of pitch and yaw. The drawback of these alternate control surface layouts is the increased complexity in relating control actions to vehicle responses. 2.3.3 DoLPHiN-Specific Control Surfaces Other control surface configurations also possible for this work include options specific to the D O L P H I N AUV. Two such examples are shown in Figure 2-9. Figure 2-9 (a) shows a configura- tion with the bowplanes relocated to the vehicle keel. The advantage of such a configuration is that the bowplanes are moved away from the hull centre line where considerable flow disturbances are expected in yaw and pitch. Additionally, the control surfaces are moved further below the free surface thus reducing the influence of waves. Lastly, as the keel thickness is less than the hull diameter, it is possible to use planes of greater span and higher aspect ratio—and hence higher effi- ciency—while maintaining the same tip-to-tip distance. This configuration is considered in this work but has limited opportunity for application to AUVs of more typical geometry. Figure 2-9 (b) illustrates another concept that has been considered for the D O L P H I N in which a ver- tical control surface is added to the mast. Such a design would allow for improved effectiveness in roll control although additional roll (and sway) loading on the mast would result from the replace- ment of the self-aligning fairings. This configuration is presented for completeness and is not con- sidered in the current work; further details regarding the performance of such an articulated mast can be found in Williams et al. (2000). University of British Columbia Peter M. Ostafichuk April 2004 Chapter 2 - Control Surface Hydrodynamics - Theory 25 University of British Columbia Peter M. Ostafichuk April 2004 Chapter 2 - Control Surface Hydrodynamics - Theory 26 (a) (b) Figure 2-9 DOLPHIN-specific control surface locations: (a) keel planes; (b) articulated mast University of British Columbia Peter M. Ostafichuk April 2004 Chapter 2 - Control Surface Hydrodynamics - Theory 27 2.4 CONTROL SURFACE PERFORMANCE PREDICTION Control surface performance prediction is discussed below in turn for isolated planes, bowplanes, and sternplanes. 2.4.1 Isolated Control Surface Method of Whicker and Fehlner The performance of isolated control surfaces, based mainly on N A C A 0015 airfoil sections, was studied by Whicker and Fehlner (1958). This semi-empirical study provided the approximation of the lift-curve slope of an isolated low aspect ratio wing as reproduced in equation (13). 1.8710, CLa = J = = (13) 1.8 + cosQ 4 + A| C O S 4 Q Here ae is the effective aspect ratio and Q is the angle of the quarter chord. The second-order lift coefficient (that is, CL as shown in Figure 2-3 on page 18) is then determined from equation (14) by using the localized angle of incidence, a , the lift-curve slope, and the crossflow drag coeffi- cient, CDc. CL = CLaa + ̂ a\a\ (14) As noted by Field (2000), the angle of incidence for a plane can be expressed as w-qx i « = 8 + JJ-— + a/0Cfl/ (15) where 8 is the deflection of the plane relative to the hull, the middle term gives the angle between the freestream flow and the velocity of the point on the hull where the plane is mounted, and the term alocal includes local changes in the flow direction about the hull. The angle of incidence is shown graphically in Figure 2-10 and further information regarding alocal provided in Section 2.4.2.) The crossflow drag is a function of the plane tip geometry only; Whicker and Fehlner (1958) pro- vide the following suggested values based on the taper ratio, X. 0.1 + 0.7 A, for a round tip [0.1 + 1.6A, for a square tip cnc = <• — ; : ~ " d6) University of British Columbia Peter M. Ostafichuk April 2004 Chapter 2 - Control Surface Hydrodynamics - Theory 28 a flow vector Figure 2-10 Incident flow angle to planes The drag coefficient is determined as For the minimum drag value, CD0, 0.0065 can be used for a N A C A 0015 plane. The force coefficients, Cx and C z , are related to the lift and drag by a coordinate transformation from the frame of the free stream flow to that of the submarine axes: Cx = - C^cosa + Q s i n a (18) Cz = - C^cosa - C D s i n a (19) Method ofAucher Aucher (1981) provides an alternate estimate of force prediction on low aspect ratio wings based on experience and theoretical calculations. C z (a) = Z a + 2.1a 3 (20) University of British Columbia Peter M. Ostafichuk April 2004 Chapter 2 - Control Surface Hydrodynamics - Theory 29 where - is the thickness ratio and ae is the aspect ratio, defined for rectangular or trapezoidal Aucher notes that equation (20) is valid only for taper ratios less than about 2 or 3 and angles of incidence not exceeding 25 degrees. 2.4.2 Hull Influence on Bowplanes The flow field surrounding control surfaces mounted on a body are, at times, substantially differ- ent from that of isolated appendages. In the case of the bowplanes, there is a change in lift curve slope due to the cylindrical hull. For non-zero trim angles, there is acceleration and significant redirection to the incoming flow to the planes, particularly near the root. Hull Curvature Correction (Mackay) As noted by Mackay (1998), an appendage has a lower effective aspect ratio when mounted on a simple convex surface as compared to a plane surface since the appendage image has reduced span. Mackay modifies the aspect ratio by the factor kss, where R is the radius of curvature of the surface (hull) and bexp is the exposed bowplane span. With this formulation, a£ in equation (13), for example, would be replaced by kssae. Hull Curvature Correction (Aucher) Aucher (1981) also considers the effect of a curved surface to which the plane is mounted. To begin, he suggests that the clearance between a plane and a flat wall does not significantly affect the validity of equation (20) provided that the clearance does not exceed 0.02 times the length of the adjacent chord. In the presence of a curved wall, he recommends augmenting the result of equation (20) by wings as: 2b (21) (22) a (23) where (24) University of British Columbia Peter M. Ostafichuk April 2004 Chapter 2 - Control Surface Hydrodynamics - Theory 30 and again R is the hull radius and b, C and t are the plane geometric parameters (span, chord, and thickness, respectively). Influence of Bowplane Location Mackay (1998), notes that bowplanes may be located in a high, mid, or low position with respect to the hull centreline. Quoting Lawerence and Flax (1954) and Schlichting (1949) as sources, he goes on to say that the plane lift is generally insensitive to this location within a moderate range. Unfortunately, Mackay does not quantify what this range is but does say that location cannot be ignored for bowplanes sufficiently smaller than the hull radius, particularly when mounted in an extremely high or low position. Flow Angle Correction (Trim) When the hull trim angle differs from zero with respect to the free stream flow, localized changes in flow velocity along the hull centreline are expected. Aucher (1981) as well as Hoerner and Borst (1975) show that the local flow angle against the hull at the centreline is equal and opposite to the hull trim angle (see Figure 2-11). Figure 2-11 Flow angle near inclined hull The local incident flow angle, a , diminishes with radial distance, y , from the A U V centreline; Aucher gives the relationship where 0 is the hull trim angle measured from the incoming flow and a is measured from the hull centreline. In contrast, Mackay (1998) suggests that the incident flow angle relationship is which also agrees with Figure 2-11 but decays more slowly with radial distance. In either case, during a manoeuvre with the hull at a trim angle of 0, there will be a distribution of angle of inci- dence across the span of a plane. In the limit of 0 = 90 degrees, the equation suggested by (25) (26) University of British Columbia Peter M. Ostafichuk April 2004 Chapter 2 - Control Surface Hydrodynamics - Theory 31 Mackay follows the R2/y2 behavior expected in the velocity field about a circular cylinder inclined perpendicular to a uniform flow (White, 1986). Hull Vortices (Yaw) It is known that two counter rotating vortices form on the lee side of a body of revolution inclined to a freestream flow (Allen and Perkins, 1951). Using flow visualization, Kubota et al. (1992) identified the position of the vortices at various locations along the length of an inclined cylindri- cal body with length to diameter ratio, l/D = 6. In Figure 2-12, a submersible with a cylindrical hull is shown oriented at a yaw angle, \\i. As shown in the figure, the two counter rotating vortices follow the longitudinal axis of the hull rather than the free stream flow. From the flow visualiza- tion images of Kawamura and Aihara, the locations of the vortices at 25 degrees yaw were mea- sured and are summarized in Table 2-4. As shown in the table, along the length of the hull there is no appreciable change in the lateral position of the two vortices and only a minimal increase in the vertical separation (2zv/R ). 2.4.3 Hull Influence on Sternplanes Several sources are available for predicting the combined lift effectiveness of low aspect ratio wings located on the aft portion of streamlined bodies. Lyons and Bisgood (1950) examined the lift slope of small aspect ratio wings on aircraft. Dempsey (1977) used a systematic series of stern control surfaces on a representative streamlined body of revolution to determine the forces due to angle of attack. Lyons and Bisgood Lyons and Bisgood (1950) suggest treating sternplanes and rudders as carried through to the hull centreline as shown in Figure 2-13. The modified appendages are treated as isolated and perfor- mance is computed using one of the methods above. A tail efficiency correction factor is then applied based on a span parameter defined by r , / 3 , the hull radius at the 1/3 chord position of the (actual) appendage root: f 1 -0.267(r, /,/Z>)-0.7455(r, /V6)2 for a bluff tail ^ [l-0.0846(r 1 / 3/6)-0.843(r 1 / 3/Z>) 2 for a tapered tail In practice, the tail efficiency factor would be applied as follows Zsp = knt-Zsp\ a a (28) sp t]t ^ I extended where ZSD\ is the estimated force of the hypothetical stemplane of Figure 2-13, treated as " I extended an isolated appendage for the purpose of calculation. University of British Columbia Peter M . Ostafichuk April 2004 Chapter 2 - Control Surface Hydrodynamics - Theory 32 (b) Figure 2-12 Vortex pattern on the leeward side of the hull in yaw Table 2-4 Location of Hull Vortices in Yaw Station Lateral Location Vertical Location -X/L yv/R V * 0.243 088 +/- 0.35 0.388 1.12 +/- 0.37 0.426 0.98 +/- 0.42 Average 0.99 +/- 0.38 University of British Columbia Peter M. Ostafichuk April 2004 Chapter 2 - Control Surface Hydrodynamics - Theory 33 actual fm Lyon and Bisgood's hypothetical stemplane stemplane tail 1/3 Figure 2-13 Fin area used in Lyons and Bisgood's tail efficiency correction factor Dempsey Dempsey proposes that the leading and trailing edges should be extended to the tail centreline as shown in Figure 2-14. Again, an efficiency factor is applied in the same manner, this time using the maximum hull radius, R, in the span parameter. Dempsey suggests the efficiency factor knt = -0.3644 + 1 .238o|-0.3728^) * (29) for 0.734<^< 1.426 R and = I_°J2556 j(^]2- 0.1612 - 0 . 6 3 6 6 S U 1 - 1 f \ 0.4015 V R J (30) for 0.4015 <^< 0.734 and 1.426 < ̂  < oo R R As noted by Mackay (1998), the negative root for b/R < 0.4015 represents a practical lower limit for submarine and submersible appendages. The main drawback of the method presented by Dempsey is that the results were obtained using appendage Reynolds numbers based on chord between 0.32 and 1.05 million with no guidance for scaling results to other Reynolds numbers. On the other hand, the use of the local hull radius in the span parameter in the method of Lyons and Bisgood implies very high Reynolds number flow (Mackay 1998). This corresponds to a small hull boundary layer and fully attached flow on the University of British Columbia Peter M. Ostafichuk April 2004 Chapter 2 - Control Surface Hydrodynamics - Theory 34 fill |||§§|§§ Dempsey's extended fin Flow tail Figure 2-14 Fin area used in Dempsey's tail efficiency correction factor appendages. For sternplane efficiency prediction in this thesis, the method of Dempsey has been used since the full-scale chord-based Reynolds numbers on the D O L P H I N were only 1.4 million, close to the upper limit of those in the experiments by Dempsey. University of British Columbia Peter M. Ostafichuk April 2004 Chapter 2 - Control Surface Hydrodynamics - Theory 35 2.5 CONTROL SURFACE INTERACTION Limited information exists regarding the interaction between fore and aft low aspect ratio control surfaces on slender bodies. Given the significant longitudinal separation of typical control sur- faces, interaction is essentially limited to flow disturbances generated by the bowplanes and car- ried aft with the bulk flow to the sternplanes. Previous work includes a study of downwash to elevators on aircraft, the interaction between roll stabilizer fins and bilge keels on warships and a preliminary investigation of induced roll effects from bowplanes to the tail of a D O L P H I N vehicle. 2.5.1 Downwash to Aircraft Elevators Munk (1923), demonstrated that the angle of downwash at various locations along the span of an aircraft wing was not uniform. Munk related the downwash angle, e , to various wing parameters by the relationship: C,S e = g O ) - ^ 5 7 . 3 ° (31) where y, CL, S and b are used as defined in Section 2.2. The values of the function g are deter- mined by experiment as they varied not only by the arrangement of the wing but also point to point on the wing. For the limited range of conditions considered by Munk, the values of g were varied from 1.2 to 2.2. This range of variation was observed not only between wings, but sometimes at different points on the same wing. Unfortunately, the reliance on wing specific experimental information for the parameter g makes the application of Munk's equation difficult for generalized use. 2.5.2 Interaction Between Warship Stabilizer Fins The interference between roll stabilizer fins and bilge keels on warships was studied by Lloyd (1974). Lloyd modelled the stabilizer fin trailing vortex as a line vortex with strength determined by the circulation about the fins. The ship hull was modelled as an infinite vertical plane by the use of an image vortex mirrored about the fin root (as shown in Figure 2-15). The downwash localized at the inception of the trailing vortex (near the fin tip) was computed using the induced flow of the image vortex. The vortex trajectory was assumed linear and oriented at an inclination angle given by ratio of the computed local downwash velocity to freestream velocity. Using the linearly approximated position of the fin trailing vortex in the longitudinal plane of the bilge keel, the downwash due to the trailing vortex and associated image vortex were computed. The incident flow on the bilge keel was thereby determined as the ratio of downwash to freestream velocity. Lloyd gives the downwash angle, s, at any point aft of a fin by C L f r} + V ( n - b') 2naeb' .C2 + ( n + b')2 C2 + (n -b')2. (32) University of British Columbia Peter M . Ostafichuk April 2004 Chapter 2 - Control Surface Hydrodynamics - Theory 36 1 Flow fin trailing vortex b b' \ \ \ \ image fin \ j image trailing vortex Figure 2-15 Fin vortex model of Lloyd where a e is the effective aspect ratio, 2b/C, V is the spanwise position of the trailing vortex, n is the non-dimensional spanwise position, For the system examined by Lloyd, C, represents the vertical displacement of the trailing vortex from the longitudinal horizontal axis through the fin tip. Several key features of the system studied by Lloyd are similar to those found on submarine-like vehicles. Most notably, stabilizer fins are of similar aspect ratio to submarine control surfaces and interaction effects are predominantly due to the fin trailing vortices. Nonetheless, the fm to body dimension ratio is essentially zero in Lloyd's analysis which means it is not directly applicable to slender body vehicles (where the fin to body width ratio may approach unity). Likewise, Lloyd's development does not permit the trailing vortex trajectory to have a transverse component (that is, lie outside the x-z plane). This is of concern in any case where the fin tip is vertically offset from the hull image vortices, such as with a dihedral plane or a plane vertically offset from the hull cen- treline. In such situations, there will be a transverse component to the vortex trajectory. 2.5.3 Control Surface Interaction Predictions on DoLPHiN-like Vehicles Hopkin et al (1990) used a combination of numerical modelling and experiments to examine downwash effects on a towed underwater vehicle. They were able to show that a single vortex, shed from forward wing tips, was sufficient to predict the loading on the aft wings. Unfortunately, (33) and C, is defined in terms of the point of interest aft of the fin, x, as follows (34) University of British Columbia Peter M. Ostafichuk April 2004 Chapter 2 - Control Surface Hydrodynamics - Theory 37 the geometry studied by Hopkin was somewhat different than that for typical AUVs such as D O L - PHIN. As such, the exact findings of Hopkin cannot be directly applied in this research but the gen- eral observations about interaction modelling can be used. Watt et al. (1997) used similar methods to Lloyd to provide a simple demonstration of counter- intuitive flow interaction effects due to bowplane tip vortices on a D O L P H I N vehicle. They exam- ined the effect of differentially deflected bowplanes on the aftward portion of the D O L P H I N Mark I vehicle. A potential flow model consisting of the bowplane tip vortices with appropriate image vortices to represent the cylindrical hull was used. It was suggested that the two equal bowplane tip vortices and a single coalesced hull vortex on the vehicle centreline would generate rolling moments on the keel, rudder, vertical stabilizer, and sternplanes of the vehicle. Specifically, the induced rolling moment was suggested to be opposite in direction and have a strength 1.4 times that generated by the bowplanes. In other words, a roll moment of K generated by the bowplanes would result in a net vehicle roll of -OAK. Considering only the bowplane-sternplane roll interac- tion, Watt suggested that an interaction level of 30% was present in straight and level flight (for bowplane roll moment of K the induced roll moment on the sternplanes would be 0.3K). The analysis of Watt et al. provides some indication of the interaction between the bowplanes and sternplanes. Although the work only considered differentially deflected bowplanes, when the methods are applied to bowplanes deflected in the same direction, the difference in theoretical induced flow on the sternplanes was reduced by only 4.8%. Therefore, this model suggests that the bowplane-sternplane interaction for a DOLPHlN-like vehicle is approximately 25% to 30%. The limitation of the model of Watt et al. for general use includes the condition that bowplanes and sternplanes must be horizontal positioned and on the hull centreline. Furthermore, the model does not account for downwash influencing the trajectory of the trailing vortices; that is, the transverse and vertical position of the bowplane tip vortices does not shift from their inception point. University of British Columbia Peter M. Ostafichuk April 2004 Chapter 2 - Control Surface Hydrodynamics - Theory 38 2.6 FREE SURFACE INFLUENCE 2.6.1 Calm Water Free Surface Considerations for AUVs In calm water, Wadlin et al. (1955) experimentally determined the lift and drag of various plates over a range of angles of incidence. The depths studied ranged from 0.07 to 0.85 chord lengths and the Froude numbers based on chord ranged from 1.1 to 5.2. The Froude number based on chord is defined as (35) With a plate of aspect ratio 1.00 at a 12 degree deflection, Wadlin et al. found that the most signif- icant change in lift occurred for depths of 0.5 chord lengths and less. Ventilation of the plate at 12 degrees deflection was not observed for depths greater than 0.07 chord lengths. Wadlin et al. concluded that for depths greater than 0.85 chord lengths, the surface effect is negligible. 2.6.2 Wave Theory In deep water, waves travelling along a surface in one direction cause cyclical motion in a plane. With a wave travelling in the ^-direction, a particle in the water traces a circular path in the x-z plane. When the wavelength, k, is small compared to the depth of the body of water, the velocity of a particle is given by the following two equations (see Lamb, 1997, or Newman, 1977): where a is the wave amplitude, d is the depth from the calm water surface, k is the wavelength, and x is the position in the x-direction. Thus particles trace out circles with a constant speed of Clearly, the wave influence diminishes with depth. The case where the wavelength was not small compared to the depth of the body of water was not considered for this thesis; however, further information regarding wave behavior in this regime can be found in Lamb (1997). 2.6.3 Sea State and Wave Spectra The random, irregular wave pattern that can be found in open water is described using statistically derived spectra. The forms of the spectra depend on the seaway being modelled but in general, the wave energy density is determined as a function of wave frequency. Following Field (2000), this (36) (37) (38) University of British Columbia Peter M. Ostafichuk April 2004 Chapter 2 - Control Surface Hydrodynamics - Theory 39 thesis uses the Bretschneider spectrum which can be considered an average spectrum suitable for open sea areas (Journee, 2001). The Bretschneider spectrum is described by 172.8 • Hi/-, ~T4 4 S^) = r 4 5 W 3 e r ' ffl (39) where is the energy density in m2/s, co is the wave frequency in s~l, Hxn is the significant wave height in m , and T{ is the average wave period, in 5 . The relation of sea state to H]/3 and T, is given in Table 2-5 Table 2-5 Open ocean sea state occurrences of North Atlantic and North Pacific H l / 3 North Atlantic North Pacific Sea State Number Mean Range 7, Probability Probability 0-1 0.05 m 0-0.1 m - 0% - 0% 2 0.3 0.1-0.5 7.5 s 7.2 7.5 4.1 3 0.88 0.5-1.25 7.5 22.4 7.5 16.9 4 1.88 1.25-2.5 8.8 28.7 8.8 27.8 5 3.25 2.5-4 9.7 15.5 9.7 23.5 6 5.0 4-6 12.4 18.7 13.8 16.3 7 7.5 6-9 15.0 6.1 15.8 9.1 8 11.5 9-14 16.4 1.2 18.0 2.2 By the principle of superposition, waves of differing amplitude and frequency (wavelength) are combined to form the overall sea state. The response of a vehicle in irregular waves can similarly be described by the summation of the responses to each of the distinct wave frequencies (Journee, 2001). In other words, to find the general dynamic response of a vessel in irregular seas of a known spectrum, it is sufficient to determine the response at distinct wave frequencies and to scale those responses by the wave spectrum. 2.6.4 Wave Considerations for AUVs Little work has been conducted on the study of the free surface influence to control surface opera- tion on AUVs although there is a demonstrated need for active wave disturbance compensation. An and Smith (1998) have shown that even at depths of up to three hull diameters, depth following abilities are significantly compromised at sea state 2 and above. Not surprisingly, An and Smith stress the importance of plane and controller design for near-surface depth keeping performance. The influence of waves on control surfaces is often neglected as the magnitudes of wave induced forces on the hull are much greater (see for example (Peterson et al., 1994) and (Ananthakrishnan and Zhang, 1998)). In examining the effect of wave-induced motion on A U V imaging sensor performance, Peterson et al. (1994) show that active control of bowplanes could eliminate the majority of unwanted pitch University of British Columbia Peter M. Ostafichuk April 2004 Chapter 2 - Control Surface Hydrodynamics - Theory 40 and heave motions. In a related topic, Riedel and Healey (1998) have demonstrated that on-board sensors on a typical low-cost A U V can provide an estimate of wave induced disturbances. Fur- thermore, through simulation, they demonstrated that is was possible to compensate for the distur- bances from shallow water waves resulting in an improvement in position and heading station keeping. For his work on the DOLPHIN A U V , Field (2000) estimated wave-induced forces using Morrison's equation for the hull and using wave trajectory approximations for the planes. In particular, the wave disturbance force on the planes was computed by augmenting the plane incident flow angle given in equation (15) by a wave angle, aw, as defined by = -JJ (40) where ww is the vertical component of the wave orbital velocity and U is the vehicle forward speed. Components of the wave orbital velocity in the axial direction of the vehicle (x-direction) were ignored as they were small in comparison to the forward speed of the vehicle. Note that for an A U V capable of large depth excursions or pitch angles, this approximation is no longer valid and the complete wave orbital trajectory must be considered. The complete angle of incidence on a plane, by incorporating the wave disturbance effects of equation (40) to equation (15), is a = 5 + ^ . - - - . - ^ + a i o a i a 5 + »-». -*•„- . + a _ ( 4 1 ) The local variation in angle of incidence due to the hull, a l o c a l , given by equation (26) for exam- ple, is computed with the wave induced velocity included. That is, the trim angle, localized at the bowplanes, is determined from a - ,1T1-1 W ~ W - ~ q X P ^ „ W ~ W W - 1Xplane °local ~ S m TJ = JJ V*l> The resulting force is computed using the formulation presented above beginning with equation (13). University of British Columbia Peter M. Ostafichuk April 2004 Chapter 3 - Control Surface Hydrodynamics - Experiments 41 3 C O N T R O L S U R F A C E H Y D R O D Y N A M I C S - E X P E R I M E N T S Measurements are not to provide numbers but insight. - Ingrid Bucher In the design of an underwater vehicle, manoeuvrability and controllability must be examined in advance, preferably with the use of a mathematical model (Kim, 2002). The validity of the model, as well as the findings it yields, depends on how accurately the operational characteristics of the vehicle are represented. There are several techniques commonly employed in determining mathematical models for AUVs, including analytical, computational, system identification, and empirical methods (see Chapter 4 for further information). Each of these techniques has specific benefits and shortcomings, but empirical and semi-empirical methods still form the basis for most modelling of AUVs. Empirical techniques rely on experiments to characterize vehicle performance. Experiments are most often conducted with a scale model in a research facility but at times may use a full-scale vehicle in open water (Hopkin and den Hertog, 1993). Towing tank tests with captive models or with planar-motion-mechanisms are commonly employed to determine submarine performance characteristics (Feldman, 1995). In some advanced testing facilities, a model can also be made to follow a prescribed path in three dimensions (Perron et al., 1998) Alternatively, with proper treatment of scaling effects, wind tunnels can been used (Barlow et al., 1998, Huggins and Packwood, 1995, and Watt et al., 1993); the benefit of working in the air environment is a ten-fold or more increase in the rate of testing. This chapter summarizes experiments examining control surface performance that were conducted for this thesis. It begins with a discussion of the equipment and facilities used followed by experimental findings regarding the performance of isolated planes, bowplanes, and sternplanes. The chapter concludes with results from experiments examining the interaction of the free surface (waves) with the control surfaces. In Chapter 4, these experiment results are used to develop the mathematical models describing underwater vehicle performance. University of British Columbia Peter M. Ostafichuk April 2004 Chapter 3 - Control Surface Hydrodynamics - Experiments 42 3.1 MODEL AND FACILITIES 3.1.1 Similarity and Scaling Dimensional analysis provides procedural techniques in which to simplify a physical process in terms of non-dimensional groups. Similarity, a key result of dimensional analysis, guarantees that if all the relevant variables for a process are included, there is a set of dimensionless groups—less than or equal in number to the number of original variables—that completely describes the pro- cess. Additionally, similarity states that the number of dimensionless groups is less than or equal to the number of original variables. Further information regarding dimensional analysis and simi- larity can be found in (White, 1986), (Potter and Wiggert, 1997), or (Avallone and Baumeister, 1996), for example. In reference to experiments with underwater vehicles, similarity means that model results can be scaled to the actual vehicle as long as the dimensionless groups have been maintained. There are several key dimensionless groups that are used including the Reynolds number, the Froude num- ber, and the force coefficients; a simplified list is summarized in Table 3-1, for a complete list the reader is referred to (Feldman, 1979). The Reynolds number and Froude number are independent parameters controlled by adjusting the model speed in the fluid during testing. The force (and moment) coefficients are dependent parameters measured through the course of the experiments. Table 3-1 Key Dimensionless Parameters Parameter Symbol Definition Notes Reynolds number Re Re = — V Always relevant Froude number Fr Fr = — Relevant near the free surface Isolated control surface force CL c - L Force non-dimension- alized by plane area Control surface (on body) force Z Force non-dimension- alized by body length squared For model testing in the wind tunnel, the Reynolds number was used as the main scaling parame- ter. Due to the increased kinematic viscosity of air compared to water, compounded by the reduced scale of the model, it was not possible to match Reynolds number between the model and full-scale vehicle. As such, additional precautions were required to account for the Reynolds num- ber discrepancy (as detailed in Section 3.1.2 on page 47). In the tow tank, ideally both Froude number and Reynolds number similarity should be maintained; however, without complete control of fluid properties (that is, v), simultaneous Froude and Reynolds number similarity is not possi- ble. As flows near the free surface are dominated by Froude number (White, 1986), Froude num- ber similarity is generally used for vessels with shallow depth in the tow tank. For deeply submerged submarines and for control surfaces on near-surface vehicles, Reynolds number is pre- ferred as the governing parameter (Barlow et al., 1999). For this thesis, testing was conducted University of British Columbia Peter M. Ostafichuk April 2004 Chapter 3 - Control Surface Hydrodynamics - Experiments 43 over a range of speeds up to the facility maximum; at maximum speed, Froude number scaling was achieved and coincided with a three-fold increase in Reynolds number compared to the wind tun- nel. The same Reynolds number corrections as in the wind tunnel were applied for the tow tank. 3.1.2 Scale Model A 1/4-scale model of the ISE Mark II D O L P H I N was constructed for both wind tunnel and towing tank testing. In the original project description, ISE was responsible for the construction of a water-tight, self-propelled model complete with mounts and instrumentation (Seto, 1997). ISE was unable to fulfill this obligation and model development was added to the scope of this thesis research. The model development represented a significant portion of the experimental work; over 20 persons (from both the University and ISE) were involved in the process and, by rough estimates, over 2500 person-hours were invested. Details regarding the general model design, the control surfaces geometry and actuation, scaling considerations, and mounting hardware are dis- cussed below. Model Design The philosophy adopted for the model design was to maintain a high level of flexibility with a sim- ple, cost-effective construction. The scale of one-to-four was chosen partly because several com- ponents from an existing 1/4-scale Mark I D O L P H I N model were available, thus reducing fabrication time and costs. The hull was assembled from five main segments with the addition of a keel and mast, as illustrated in Figure 3-1. The nose and tail from the Mark I model were used and the remaining components were designed and fabricated specifically for this work. Hull segments were machined from aluminum tube, the mast was constructed from stainless steel tube and bar, and the fairings and keel were constructed from syntactic foam. All joints were fitting with o-rings or seals such that the hull was water tight. Based on the one-quarter scale, the model length was approximately 2.13 m and the hull diameter was 0.248 m. Complete engineering drawings for the scale model are provided in Appendix B . l . For a vehicle with control surfaces located at the tail, Barlow et al. (1999) recommend including in the model a propeller that is operated at the correct advance coefficient. In wind tunnel tests of submarine propulsion, Watt and Fournier (1995) found that it was unnecessary to model propul- sion while measuring the overall hydrodynamic forces; however, they suggest that propulsion should be modelled in the study of tailplanes. Unfortunately, through space, cost, and time con- straints, it was not possible to incorporate a propulsion system to the D O L P H I N 1/4-scale model. The implications of the absence of propulsion on the current model are believed to be a local reduction in the flow velocity and reduced resistance to boundary layer separation at the tail. These factors would tend to cause deterioration in stemplane performance on the model. The con- cerns are partially addressed by the model design in that the movable (measuring) portions of the sternplanes were located outside of the maximum hull diameter (discussed further on page 47 in regards to Reynolds number considerations). As a final note, although Barlow et al. (1999) advise against omitting propulsion on the model, Barlow (1998) uses a static model without propulsion for wind tunnel tests of a submarine with sternplanes located completely within the maximum hull diameter. University of British Columbia Peter M. Ostafichuk April 2004 Chapter 3 - Control Surface Hydrodynamics - Experiments 44 Figure 3-1 Quarter-scale model exploded view Control Surfaces Control surfaces of various geometries were constructed from composite materials. Basic plane forms were hot-wire cut from high-density foam insulation using two-dimensional templates. The foam core was coated in layers of carbon fibre and room temperature cure two-part epoxy. Planes were painted and sanded to a smooth finish. The planes were symmetrical, N A C A 0025 sections with no span-wise twist and aspect ratios ranging from about 2.3 to 4.1. The short planes were also used with endplates similar to those on the full size vehicle. The endplate height, h, was 3.95 cm (0.35 times the exposed span of the short planes) and the shape approximated a N A C A 0045 foil. The endplate thickness was approx- imately 1.6 mm (1.86 chord lengths). The complete range of plane geometries constructed is detailed in Table 3-2. The planes were attached to the vehicle through load cells (discussed further below) that passed along the quarter chord location. A 0.75" diameter span-wise hole drilled in the foam core was University of British Columbia Peter M. Ostafichuk April 2004 Chapter 3 - Control Surface Hydrodynamics - Experiments 45 used to fasten the planes to a sleeve through an interference fit (see Figure 3-2). The sleeve was in turn fastened to the load cell with a sliding fit secured with a cap screw. Table 3-2 Plane Geometries Exposed Mean Aspect Exposed Taper 1/4 chord Plane Name Span 3 Chord Ratio Area b Ratio sweep [deg.] Short 0.0530 0.0399 2.66 0.00211 1.0 0 Std Bowplane 0.0613 0.0399 3.07 0.00245 1.0 0 Long 0.0810 0.0399 4.06 0.00323 1.0 0 Std Sternplane 0.0483 0.0399 2.42 0.00193 1.0 0 Swept 0.0613 0.0488 2.51 0.00299 0.691 11.4 "Large Swept 0.0613 0.0539 2.27 0.00333 0.588 18.7 Large 0.0702 0.0601 2.34 0.00422 1.0 0 a. exposed span and mean chord are non-dimensionalized by vehicle length; exposed span = b/2 b. exposed plane area has been non-dimensionalized by vehicle length squared Load cell Figure 3-2 Cut-away view of plane showing load cell In total, fifty different combinations of bowplane and sternplane mounting position were possible with the model; Figure 3-3 shows the various plane positions superimposed on the D O L P H I N body. Details regarding each of the plane locations are provided in Table 3-3. For reference, the vehicle centre of gravity is located at x/l = 0.489, y/l = 0.000, and z/1 = 0.022 as measured from the nose and non-dimensionalized by the vehicle length, / . Control Surface Actuation Each bowplane and sternplane was outfitted with a stepper motor actuator to control deflection. The bowplanes were connected directly to the motors while the sternplanes were connected through a lead screw. The corresponding bowplane deflection step size was 0.9 degrees and the sternplane step-size was better than 0.05 degrees. University of British Columbia Peter M. Ostafichuk April 2004 Chapter 3 - Control Surface Hydrodynamics - Experiments 46 Figure 3-3 Plane Mounting Locations Table 3-3 Plane positions 1/4 Chord Position (non-dimensional) Bowplane Location x a z b Dihedral Angle 0 Standard 0.261 0.010 0 degrees Centreline 0.261 0 0 High 0.261 -0.010 0 Dihedral 0.261 0 30 Anhedral 0.261 0 -30 Forward Standard 0.216 0.010 0 Forward Centreline 0.216 0 0 Forward High 0.216 -0.010 0 Forward Dihedral 0.216 0 30 Forward Anhedral 0.216 0 -30 Keel 0.557 0.225 0 Stemplane Location X z Dihedral Angle Standard 0.906 -0.002 0 Forward 0.848 -0.002 0 Low 0.906 0.022 0 Y-tail 0.882 0.008 45 Inverted-Y 0.882 0.008 -45 a. measured from the 1/4 chord line to the nose and non-dimensionalized by the vehicle length b. determined by the distance to the hull centreline when the 1/4 chord line is extended to the x-z plane c. measured in the positive roll direction for port planes and negative roll direction for starboard planes University of British Columbia Peter M. Ostafichuk April 2004 Chapter 3 - Control Surface Hydrodynamics - Experiments 47 A basic three-axis stepper motor controller was modified and software was written to allow control of the motors via a personal computer. Additional circuitry was constructed to increase the power output of the motors and hence the torque to the control surfaces. The software permitted the con- trol of the motors in a single step (jog) mode or in a pre-programmed sequence. The pre-pro- grammed deflection sequence was used extensively in the wind tunnel due to the absence of restrictions on measurement duration. In particular, for the majority of wind tunnel testing the bowplanes were deflected in 3.6 degree increments from 0 degrees to +28.8 degrees, then to -28.8 degrees, and back to 0 degrees. In the towing tank, the planes were held fixed for each trial due to the limited measurement duration. Reynolds Number Considerations The Reynolds number is the governing parameter for the boundary layer on the vehicle and control surfaces. Unless otherwise noted, the Reynolds number based on vehicle length has been used Re = ^ (43) In regions with a strong adverse pressure gradient, such as near the tail and trailing edges, flow separation characteristics may be strongly dependent on Reynolds number. In particular, a turbu- lent boundary layer (associated with large Reynolds number) is known to exhibit greater resistance to separation than a laminar boundary layer (associated with lower Reynolds numbers). Ideally, in wind tunnel and tow tank tests the model would be scaled matching the Reynolds number to the full scale vehicle thus ensuring similar flow characteristics. Using.a 1/4 scale model, the maxi- mum Reynolds numbers based on length in the wind tunnel and tow tank were 2.5 million and 7.5 million respectively; the full scale vehicle operates in the Reynolds number range of 30 to 50 million. In terms of control surface performance, separation at lower angles of attack, and conse- quently lower maximum control forces, could be expected from the reduced Reynolds numbers in the model tests. However, these concerns were mitigated by the positioning of the planes on the model and the use of boundary layer stimulators. For the experimental work, the model was designed such that forces on the sternplanes were mea- sured on the movable portion of the planes (which extended from radially from roughly the maxi- mum hull diameter outward as shown in Figure 3-4). The influence on the experimental data of premature flow separation behind the tail due to the lower Reynolds numbers was therefore mini- mized. Hull flow separation near the bowplanes was not of concern; it was expected that the hull boundary layer thickness near the bowplanes would change slightly with Reynolds number but not enough to significantly influence the results. In order to minimize the Reynolds number dependence of boundary layer transition, boundary layer trips of a style recommended by Torres et al. (1999) were installed in various locations on the model. The function of the trips was to cause a transition from laminar to turbulent flow in the boundary layer by placing an obstruction in the flow. As noted above, the (artificially tripped) tur- bulent boundary layer would be more resistant to flow separation than the otherwise laminar boundary layer. The regions in which boundary layer trips were located are identified in Figure 3-5. In particular, the trips were positioned on the nose, mast fairings, keel (two positions), rudder, stabilizer, and on all of the control surfaces. Detailed information regarding boundary University of British Columbia Peter M. Ostafichuk April 2004 Chapter 3 - Control Surface Hydrodynamics - Experiments 48 F l o w Figure 3-4 Region of tail boundary layer influence layer transition on wings can be found in (Eppler, 1990, and Eppler, 1999); further details regard- ing the trip design and associated calculations for this thesis can be found in Appendix B.3. Mounting Hardware For both wind tunnel and towing tank tests, specially designed mounting hardware was con- structed. In the wind tunnel, the model was mounted upside-down by the mast as shown in Figure 3-6. Underneath the wind tunnel, the mast was connected to a mounting beam which was in turn attached to a heavy steel base. Both the mast and the mounting beam were instrumented with strain gauges to allow measurement of body forces. The trim angle of the submarine was changed using a lead screw mechanism connected to the mounting beam. In research conducted by others, yaw was achieved by rotating the model and base using a low-friction pivot mechanism connected to the base (not shown in the figure). In the towing tank, the same basic mounting hard- ware was used except in this case the model was mounted right side-up and the steel base was replaced by a height adjustable frame (see Figure 3-7). The mounting hardware was manually rotated and re-fastened in order to adjust the yaw angle in the tow tank. 3.1.3 Load Cells and Signal Conditioning Each plane was outfitted with a load cell located along the 1/4 chord line. The load cells were mounted to the body as shown in Figure 3-8 and key features are identified in Figure 3-9. In par- ticular, eight strain gauges were affixed to hollow stainless steel cylindrical beams in two longitu- dinal positions. The diameter was reduced in the regions where the strain gauges were applied in order to locally increase the strain without compromising the overall stiffness. Each half-bridge circuit measured bending moment—two corresponding to lift and two corresponding to drag—and in post-processing the moments were used to estimate lift and drag. University of British Columbia Peter M. Ostafichuk April 2004 Chapter 3 - Control Surface Hydrodynamics - Experiments 49 Boundary Layer Trips (typical of 18 lengths total) Figure 3-5 Boundary Layer Trip Mounting Locations Considering either lift or drag, with moments on the inboard and outboard gauges of KI and K0 respectively, and a longitudinal separation distance of the gauges of Ay, the corresponding force is Z = K: ~ Kn A y (44) This represents a shear-beam style arrangement and is insensitive to the point of force application. The span-wise center of pressure—that is, the location along the plane span at which the force, Z , could be assumed to act—was also computed from the moment measurements by y cp K, Ki ~ Ko A y (45) where y. is the span-wise position of the inboard gauges. Further information regarding the design and calibration of the load cells can be found in Appendix B.2. The task of measuring, conditioning, and recording the load cell voltage signals was handled by an IOTech PC-based data acquisition system. A 12-bit DAQBoard 1 0 0 / A A / D card was used with ten University of British Columbia Peter M. Ostafichuk April 2004 Chapter 3 - Control Surface Hydrodynamics - Experiments 50 Figure 3-6 Wind Tunnel Mount Showing Beam and Base Plate Peter M. Ostafichuk University of British Columbia April 2004 Chapter 3 - Control Surface Hydrodynamics - Experiments 51 Figure 3-7 Towing tank mount showing height adjustable structure DBK16 strain gauge signal conditioning cards, two DBK 8 general purpose voltage cards, and one DBK32A auxiliary power card. Eight of the DBK16 cards, one DBK8, and the DBK32A were mounted in a specially designed rack and placed inside the model hull. The signal conditioning hardware was located inside the model to minimize the length of cable carrying low-level voltage signals from the load cells. As low-level signals are much more susceptible to noise, the signal-to- noise characteristics of the measurements was greatly improved in this manner. Special precau- University of British Columbia Peter M. Ostafichuk April 2004 Chapter 3 - Control Surface Hydrodynamics - Experiments 52 Outboard strain gauges(4)in two-half bridge circuits Exit hole (inside hull) for strain gauge lead wires Inboard strain gauges (4) in two-half bridge circuits Flat for alignment Typical Signal Wire Typical Signal Wire Access hole for strain gauge lead wires (typ. of 4) Bearing surface (hull location) Stepper motor Figure 3-9 Load cell arrangement for bowplane showing direct connection to stepper motor tions were taken with the model to ensure the electronics housed inside the hull were not damaged by water during tow tank tests: o-rings and other seals were used on joints; numerous small cracks in the hull introduced during manufacture were sealed; and the hull was pressurized to approxi- mately 70 kPa (see Figure 3-7) to both serve as a leak indicator and to prevent inflow of water in the event of a leak. The remaining two DBK16 and the DBK8 cards were used for body force mea- surements and were located in close proximity to the personal computer. Under static calibration, the load cells demonstrated exceptional overall performance. All load cells were highly linear and showed minimal hysteresis; the typical correlation coefficients were 0.99999 and the largest residuals from a linear best-fit were approximately 0.2% of full scale. Cross-talk between the lift and drag axes on the order of 6.5% was noted with similar linearity in University of British Columbia Peter M. Ostafichuk April 2004 Chapter 3 - Control Surface Hydrodynamics - Experiments 53 the coupling as with the primary measurement axis. A 2x2 calibration matrix was used to convert the measured voltages to lift and drag forces while at the same time removing the coupling effects. Overall, the load cell accuracy was approximately 0.5% of full scale when including all sources of error. 3.1.4 Data Collection Protocol Raw voltage data was recorded in the ASCII text file format during the experiments and was later post-processed to determine hydrodynamic quantities. Initial readings of quiescent loads (zeros) were taken in each case followed by the measurement of interest. The data collection rate for both the wind tunnel and towing tank tests was 50 samples per second and measurement duration varied by experiment. Specialized data analysis software written in Visual Basic was used to extract the relevant data from the ascii text files. In particular, the software coordinated data windowing with measurement initiation and termination, as well as changes in control surface deflection (as outlined on page 45), in order to avoid transient flow effects. Over a minimum window of 15 seconds (750 samples) the analysis software computed the mean voltage for each strain gauge bridge— with quiescent loads removed—and the standard deviation in the measurements. 3.1.5 Testing Facilities Tests were conducted in two different wind tunnels and a towing tank, all located on the campus of the University of British Columbia. The Boundary Layer Wind Tunnel (BLWT) was used for the majority of tests with the scale model. A smaller, faster wind tunnel (the Parkinson Wind Tunnel) was used for high Reynolds number tests on an isolated control surface. Finally, the BC Research Inc. towing tank was used with the fully instrumented model to study wave and free surface effects. Table 3-4 summarizes the basic specifications for each of the test facilities. Table 3-4 Test facility specifications Specification Boundary Layer Wind Tunnel Parkinson Wind Tunnel Towing Tank Description Open-circuit wind tun- nel with upstream fan Closed-circuit wind tunnel Rectangular freshwa- ter basin with cantile- ver carriage and wavemaker Dimensions 2.4 x 1.6 m (w x h) 0.91 x0.69m(wxh) 3.7 x 2.4 x 60 m (w x h x 1) Maximum Speed 20 mis 35 m/s 3.5 m/s Turbulence Intensity <2% unknown negligible University of British Columbia Peter M. Ostafichuk April 2004 Chapter 3 - Control Surface Hydrodynamics - Experiments 54 3.2 PLANE PERFORMANCE In addition to isolated control surfaces, the performance of body-mounted planes was examined using various techniques over a range of orientations. Namely, oil film and yarn tuft visualization were employed along with direct force measurement for different yaw and trim angles; the major- ity of tests were conducted with yaw and trim angles of zero (straight and level flight) as that rep- resents the most common operating condition. 3.2.1 Isolated Plane The performance of an isolated control surface was examined in the Parkinson wind tunnel with a 0.916 scale plane fitted with a load cell (as described in Section 3.1.3). The Parkinson tunnel and larger plane were used to achieve higher Reynolds numbers than those possible with the 1/4-scale model in the BLWT. The plane was mounted near the wind tunnel floor with a gap of approxi- mately 5 mm. Tests were conducted at 13.6 m/s for deflections from -32.4 to +32.4 degrees in 1.8 degree increments. The corresponding Reynolds number based on chord was 280 000; at this scale, the equivalent Reynolds number based on vehicle length is 7 million. The resulting force to deflection curve is shown in Figure 3-10. I Error Scale Stall = 17deg -40 -30 -20 -10 0 10 20 30 40 Deflection [deg] Figure 3-10 Isolated standard geometry plane 0.916 scale, Re c = 280 000 (equivalent Re L = 7 million) University of British Columbia Peter M. Ostafichuk April 2004 Chapter 3 - Control Surface Hydrodynamics - Experiments 55 The force, Z, has been non-dimensionalized according to Section 2.1. Namely, Z = — ^ — (46) where p is the air density, U is the freestream velocity, and / is the vehicle length. The stall angle was approximately 17 degrees but is expected to increase with Reynolds number; a detailed examination of the effect of Reynolds number on stall can be found in Section 4.2. 3.2.2 Straight and Level Flight In testing the bowplanes in straight and level flight conditions, both oil film visualization and direct force measurement were employed. In the force measurement tests, a configuration similar to the Mark II D O L P H I N was used (the standard configuration) as well as configurations with dif- ferent bowplane geometry and location on the hull. Oil Film Visualization Oil film visualization is a common technique used to identify various surface flow phenomena such as separation, boundary layer transition, and unsteadiness. This technique is well docu- mented (the reader is referred to Barlow et al. 1999, Merzkirch 1987, Maltby and Keating 1962, and RWDI 1989 for further information). In this research, oil film was used to detect regions of flow separation on horizontal control surfaces where the influence of gravity was less significant. Merzkirch (1987) has shown that the distance from the leading edge of an airfoil to the line of sep- aration is changed by not more than 2% due to the presence of an oil sheet making the oil film technique ideal for this application. A variety of different oil mixtures were tested but best results were obtained with the traditional "China clay" mixture. The exact mixture used consisted of 16 parts kerosene, eight parts kaolin, two parts oleic acid and one part artists' black oil paint. The Kerosene was selected due to the low viscosity and relatively fast evaporation under wind-on conditions. This was important as particu- late matter was deposited on the model during a test as the oil evaporated thus allowing sufficient time for photography without concern for movement of the oil streaks. The oil film technique worked best when the oil mixture was evenly applied to the entire control surface by sponge brush. The wind tunnel was then quickly brought to speed to minimize prema- ture drying of the oil mixture. After approximately one minute at speed, the wind was turned off and photos were taken from directly above and below the planes. A sample image from the oil film technique is provided in Figure 3-11 for the suction side of a bowplane at 16 degrees deflection. The leading edge is at the top of the figure and a boundary layer trip is visible. Arrows have been added to indicate the flow direction in the attached region and the separation line is indicated with a dashed black line. Separation is delayed for the out- board region of the plane, likely due to momentum introduced through leakage from the pressure side of the plane. University of British Columbia Peter M. Ostafichuk April 2004 Chapter 3 - Control Surface Hydrodynamics - Experiments 56 Figure 3-11 Oil film sample image - suction side of port bowplane at +16 degrees deflection. The separation line is shown in dashed black and arrows indicate flow direction in attached region. A series of images such as Figure 3-11 were used to form a composite image to illustrate the trends in separation. Figure 3-12 shows the composite image for the standard bowplanes in straight and level flight. The approximate size of the separated flow for moderate angles is roughly in agree- ment with wind tunnel tests of the N A C A 0025 airfoil by N A C A [Bullivant 1941]. In the N A C A tests, a large aspect ratio airfoil was tested and showed a span-wise symmetrical stalling pattern. In the current tests, hull and tip influence are apparent due to the non-symmetrical stalling pattern at larger angles of attack. Standard Configuration Force Measurement Force measurement results are shown in Figure 3-13 for the standard configuration in straight and level flight. The experiments were conducted at the maximum sustainable wind speed in the BLWT (16m/s) which corresponded to a length-based Reynolds number of 2.25 million. For comparison in the figure, the force to deflection curve is also shown for an isolated plane of the same geometry but tested at a Reynolds number of 4.0 million in the Parkinson Wind Tunnel. In general, the isolated plane showed more linear behavior and delayed stall compared to the bow- plane. This can be partially attributed to the difference in Reynolds number (see Section 4.2 for information regarding Reynolds number scaling effects). It was also shown in Section 2.4.2 that the presence of the hull influences the effective geometry of the bowplane as well as the incident University of British Columbia Peter M. Ostafichuk April 2004 Chapter 3 - Control Surface Hydrodynamics - Experiments 57 Leading Edge Tip Trailing Edge Root Figure 3-12 Separation lines on suction side of bowplane at level trim; plane deflection is shown left hand side flow angle. Additionally, the gap between the plane and the hull changes with deflection whereas an isolated plane operates with constant gap size. Increased gap size would promote additional leakage flow near the plane root thus causing the body-mounted plane to behave like a lower aspect ratio plane. As shown in Section 2.2.5 on page 20, there tends to be decreased magnitude and increased curvature in the lift to deflection behavior as the aspect ratio of a plane is decreased. The most dramatic change in the behavior of the force to deflection behavior for the bowplane occurred in the region of maximum lift force for angles between 12 and 18 degrees. These deflec- tions correspond to the range over which the largest changes were observed in the separation pat- tern analysis from the oil film visualization (Figure 3-12). As noted by Talay (1975), the largest increase in the zone of separation occurs very close to the angle of maximum lift. Plane Geometry Influence In the same manner that the performance of the standard bowplane was measured, experiments were conducted to examine the influence of plane geometry on bowplane performance. Specifi- cally, the effect of aspect ratio, size, and sweep and taper were considered. In Figure 3-14 (a) the bowplane force versus deflection is shown for bowplanes of three different aspect ratios. In each case the chord was constant and only the span was changed (see Table 3-2 on page 45). As would be expected, the bowplane force and lift curve slope increased with the plane span (plan- form area). To account for the change in planform area, in Figure 3-14 (b) the force is non-dimen- sionalized by exposed plane area instead of vehicle length squared. With this representation, the University of British Columbia Peter M. Ostafichuk April 2004 Chapter 3 - Control Surface Hydrodynamics - Experiments 58 University of British Columbia Peter M. Ostafichuk April 2004 Chapter 3 - Control Surface Hydrodynamics - Experiments 59 performance between the different planes was more similar, particularly for the two planes with larger span. Following typical behavior for finite wings, the lower aspect ratio planes generated marginally less force for a given deflection but also showed less abrupt stall. (Note that the curves in Figure 3-14 (b) have been shifted vertically to pass through the origin in order to more clearly illustrate the differences between the planes.) In addition to the planes of varying aspect ratio, three other planes were considered: two tapered swept wings and one large-scale plane. As with the other planes, details regarding geometry can be found in Table 3-2 on page 45. Figure 3-15 shows the force to deflection behavior for these additional planes using both nondimensionalization by vehicle length (Figure 3-15 (a)) and by exposed plane area (Figure 3-15 (b)). Considerable differences were noted between the planes in the vehicle length non-dimensionalization, as would be expected. However, when forces were non-dimensionalized by planform area, for deflections prior to stall the resulting curves agreed to within 3% of the force range. Significant differences were still noted beyond stall. Of particular interest was the large plane which demonstrated stall that was both the most gradual and occurred at the highest angle of attack. 2.5 2.0 1.5 1.0 0.5 3 ; oo 3 -0.5 -1.0 -1.5 -2.0 -2.5 I Error scale Std BP (ae = 3.07) - A — Large (ae=2.34) -o— Swept (ae = 2.51) —B— Large Swept (ae = 2.27) -10 0 10 Bowplane Deflection [deg] 0.8 0.6 0.4 0.2 0.0 O -0.2 -0.4 -0.6 Plane Error - » - S t r J B P ( a e = 3.07) 1 - f i - Large (ae = 2.34) I —o— Swept (ae = 2.51) I —B— Large Swept (ae = 2.27) j -10 0 10 Bowplane Deflection [deg] (a) (b) Figure 3-15 Bowplane geometry influence on bowplane force production for large planes: (a) Zbp versus bowplane deflection; (b) CL b p versus bowplane deflection Finally, tests were conducted with endplates installed on the short bowplanes and data were com- pared to the previous tests without endplates (square tips). As shown in Figure 3-16, the planes with the endplates show very similar performance to the square tips although there is a small but consistent increase in force (2.5% on average before stall). The significance of this finding is lim- ited given that the difference is within the measurement error of the system. For comparison, through the endplate factor (from equation (10) on page 21), for ( - < 2.5 (47) University of British Columbia Peter M. Ostafichuk April 2004 Chapter 3 - Control Surface Hydrodynamics - Experiments 60 1.5 1.0 0.5 -1.0 H .-I.5-I , , 1 , , 1 -30 -20 -10 0 10 20 30 Bowplane Deflection [deg] Figure 3-16 Endplate influence on short bowplane performance it is predicted that there should be an increase of 8.6% in the bowplane force. The significant dif- ference between prediction and measurement may lie in the fact that equation (47) does not take into account the plane thickness. Clearly, there should be no endplate correction as the endplate height approaches the plane thickness (there would however be a minor influence from the increase in plane aspect ratio due to the non-zero endplate thickness). For the configuration tested, the thickness of the N A C A 0025 control surface was 0.25 chord lengths while the endplate height was 0.47 chord lengths. If, instead, the exposed endplate height is used in equation (47), a 3.4% increase in force is predicted (the exposed endplate height, H, is given by H = h-t where h is the endplate height and t is the plane thickness). This prediction compares more favorably with the measured difference of 2.5% and suggests that H should be used in place of h in equation (47). Plane Location Influence The effect of the bowplane location on the hull was studied using the standard geometry bow- planes in six different locations on the hull. Horizontal planes were used at different elevations for the standard, high and centreline positions. In the anhedral and dihedral positions, the bowplanes were inclined to the horizontal plane at 30 degrees downward and upward respectively. Finally, keel planes were examined but only in a limited number of tests as deflections had to be adjusted manually (due to space constraints in the keel, it was not possible to use the stepper motors required for computer controlled articulation). With the exception of the keel planes, all planes were located at the same longitudinal position on the hull. Complete details regarding each of the bowplane positions can be found in Table 3-3 on page 46. University of British Columbia Peter M. Ostafichuk April 2004 Chapter 3 - Control Surface Hydrodynamics - Experiments 61 Figure 3-17 shows the resulting force to deflection behavior of the standard bowplanes in the five positions on the hull described above (the keel planes are considered separately). For moderate deflections (up to stall), the differences between the curves are of similar order to the error scale. Interestingly, both the high and anhedral planes showed delayed stall for negative deflections but earlier stall for positive deflections. Overall, bowplane position does not appear to be a major fac- tor influencing bowplane performance for straight and level flight. 1.5 0 . 5 x 0 .0 - 0 . 5 - 1 .0 I Error Scale - • - S t d - • -High -*— Dihedral -A - Anhedral —•— Centerline - - 3 0 - 2 0 - 1 0 0 1 0 2 0 3 0 Bowplane Deflection [deg] Figure 3-17 Bowplane position effect The performance of the keel planes is compared to the standard bowplanes and an isolated plane in Figure 3-18. The data for the isolated plane was taken from Section 3.2.1. The performance of the keel planes was more similar to the isolated appendage which is to be expected since the keel planes are mounted against a relatively large flat surface (the keel). Between stall angles, the force to deflection behavior of the keel plane closely matches that of the isolated plane with all measure- ments agreeing within experiment error. Differences were noted in the stall behavior of the keel plane and isolated plane; however, the Reynolds number from the isolated plane tests was larger than that from the keel plane tests making comparison difficult (see Section 4.2 for further infor- mation about the effects of Reynolds number). The stall angles were similar on the keel planes and standard planes. University of British Columbia Peter M. Ostafichuk April 2004 Chapter 3 - Control Surface Hydrodynamics - Experiments 62 2.0 1-5 H -2.0 -30 -20 -10 0 10 20 30 Bowplane Deflection [deg] Figure 3-18 Keelplane compared to standard bowplane and isolated plane 3.2.3 Non-zero Trim Various techniques were used to study how vehicle trim influenced plane performance. Yarn tuft visualization was used to estimate incident flow angles on the body (at the root of the planes), oil film visualization was once again employed to identify stall patterns on the planes, and force mea- surement was used to quantify performance. Yarn Tuft Visualization The yam tuft technique is an excellent method for quickly obtaining preliminary flow field data. Short lengths of material are attached on one end to the body and align with the flow to indicate flow direction. In unsteady conditions, the tufts exhibit a certain degree of motion and in separated flow they often lift completely off the surface or hang limp. Further information on this tech- niques can be found in many fluid mechanics texts including (Barlow et al. 1999), (Merzkirch 1987), and (Pope and Harper, 1966), to name a few. The main use of tufts for this research was to determine flow orientation and condition for various regions about the hull. Specifically, the incident flow angle for the root portion of various poten- tial bowplane mounting locations was examined. Best results were found with a high-quality wool yam due to the low weight, low stiffness, and large thickness. Figure 3-19 shows sample images from the yam tuft study both with and without a bowplane installed. The flow angles for the various regions of interest on the hull were determined through measure- ments of digitized photographs using a computer. Corrections were applied for parallax, camera University of British Columbia Peter M. Ostafichuk April 2004 Chapter 3 - Control Surface Hydrodynamics - Experiments (a) (b) Figure 3-19 Sample yarn tuft visualization images: (a) high bowplane installed, trim angle = +6 degrees; (b) no bowplane installed, trim angle = +6 degrees angle, and the effect of gravity on the tufts (adjusted by location on the hull). The resulting tuft angles as a function of hull trim angle for the centreline, anhedral, and keel bowplane locations (see Table 3-3 on page 46 for details) are shown in Figure 3-20. The tuft angles were measured with respect to the hull. The relationship for the centreline location was in agreement with the lit- erature (Aucher, 1981) in that the tuft angles were roughly equal and opposite to the hull trim angle. For both the anhedral and keel locations the trim influence on the incident flow angle at the plane root was less than for the centreline location as noted by the reduced slope; however, the influence of the keel on the anhedral plane position, and possibly the keel plane position as well, is apparent due to the lack of symmetry across the v-axis. 15 10 ~ , 5 D) O ;o a; c < o -10 -15 Center Line Keel Anhedral -15 -10 10 15 Trim Angle [deg] Figure 3-20 Bowplane root incident flow angle versus trim angle from yarn tuft visualization University of British Columbia Peter M. Ostafichuk April 2004 Chapter 3 - Control Surface Hydrodynamics - Experiments 64 Oil Film Visualization The oil film technique was used on the standard bowplanes in the standard position for trim angles of +6 and -6 degrees. The composite separation line images are shown in Figure 3-21 and Figure 3-22 respectively. In this case, two major differences were noted to the separation patterns of Figure 3-12 on page 57: the separation lines were considerably more uniform across the plane span and the angle at which large scale separation was first noted was significantly different. As noted in Section 2.4.2, in trim the hull is responsible for a significant change in the flow angle across the bowplane span. For +6 degree trim (Figure 3-21), the majority of the suction side of the plane showed attached flow at deflections of even 24 degrees. For comparison, approximately 50% of the plane showed signs of flow separation at 16 degrees for straight and level flight. With -6 degrees trim (Figure 3-22), flow on the suction side of the plane was almost fully separated at only 6 degrees deflection. Leading Edge Figure 3-21 Oil film visualization: bowplane suction side, ReL =2.25 million, trim = +6 degrees Trim Influence on Bowplane Performance The measured forces are shown Figure 3-23 for the standard bowplanes in the standard position as a function of body trim angle. The predominant effect of trim was to introduce a horizontal shift in the bowplane force to deflection curves. Based on a least squares best fit, the curves were shifted horizontally to pass through the origin (for details regarding the shift of the bowplane curves, see Section 4.4). The result was equivalent to defining an effective plane deflection of the form 8 e f f = 8 + 1.59 (48) University of British Columbia Peter M. Ostafichuk April 2004 Chapter 3 - Control Surface Hydrodynamics - Experiments 65 Leading Edge +20 +12 T i P Trailing Edge Root Figure 3-22 Oil film visualization: bowplane suction side, ReL =2.25 million, trim = -6 degrees In other words, with non-zero trim the planes behaved as if in straight and level flight but at a deflection augmented by 1.5 times the trim angle. Recall from equation (26) on page 30 that trim induces an incidence angle of 29 at the root of the planes and 6 very far from the body. The bow- plane force data in trim expressed using the effective plane deflection is shown in Figure 3-24. As evident in the figure, the eleven individual force to deflection curves essentially collapse to a sin- gle curve with reasonable agreement even in the regions beyond stall. Bowplane Location Effects in Trim Limited tests were conducted with alternate bowplane configurations in trim. The force on the bowplanes in the anhedral and keel configuration was measured for trim angles of -6, 0, and +6 degrees. As shown in Figure 3-25, the effect of trim on the anhedral bowplanes is similar to that with the standard configuration in that a horizontal shift in the curves was observed. When the effective deflection from equation (48) is used, the data again approaches a single curve as shown in Figure 3-25. This result suggests that equation (48) reasonably approximates the effect of trim angle on bowplanes on the hull and that the effect is not highly sensitive to the bowplane location or orientation. The keel planes behaved more like isolated appendages, as was noted in Section 3.2.2. As shown in Figure 3-27, the effect of vehicle trim on the force to deflection behavior was again a lateral shift in the curves; however, the magnitude of the shift in this case was equal to the trim angle. University of British Columbia Peter M. Ostafichuk April 2004 Chapter 3 - Control Surface Hydrodynamics - Experiments 66 o o CD U o I i . N 2.0 1.5 1.0 0.5 0.0 -0.5 -1.0 -1.5 -2.0 Trim angle fdegl —x— -10 —m—2 \ S-K — ° — -8 • 4 / \ \ A — -6 —A— 6 V \ —o—-4 — 8 X \ \ - a — - 2 -+—10 -30 -20 -10 0 10 Plane Deflection [deg] 20 30 Figure 3-23 Trim influence on bowplane force production o o o X CD o 1_ o N 2.0 1.5 1.0 0.5 0.0 -0.5 -1.0 -1.5 -2.0 • Trim angle fdegl x -10 • 2 o -8 • 4 A -6 A 6 o -4 • 8 • -2 + 1 0 I I I x % ^ 0 xe* I I I -40 -30 -20 -10 0 10 20 Effective Plane Deflection [deg] 30 40 Figure 3-24 Trim influence on bowplanes when expressed using effective plane deflection University of British Columbia Peter M. Ostafichuk April 2004 Chapter 3 - Control Surface Hydrodynamics - Experiments 67 2.0 -| 1.5 - 1.0 - o o o 0.5 - X 0> u 0.0 -o u. a. -0.5 - N -1.0 - -1.5 - -2.0 - Trim angle Tdegl —A— 6 -30 -20 -10 0 10 Plane Deflection [deg] 20 30 Figure 3-25 Trim influence on anhedral bowplanes That is, the effective deflection for the keel planes was defined not as in equation (48), but rather by Jeff, keel = 8 + e (49) The shifted keel plane data is shown in Figure 3-28. In general, the keel plane force curves col- lapse reasonably well but not with the same agreement as observed with the standard planes. This is likely because the body geometry in the proximity of the keel planes is not symmetric for posi- tive and negative trim angles and only about 50% of the keel plane span extended beyond the lat- eral extent of the hull. Trim Influence on Sternplane Performance The study of the influence of trim on sternplane performance was limited to force measurements. It was not practical to test over a complete range of sternplane deflections for each trim angle but measurements were conducted at zero degrees sternplane deflection for trim angles between -10 and +10 degrees. The results are shown in Figure 3-29; the measured force on the sternplanes is indicated using points and the estimated force due to a sternplane deflection equal to the trim angle (with the vehicle straight and level) is indicated by the solid line. Unlike the influence of trim on the bowplanes (see equation (48)), the effect of trim on the sternplanes was essentially lim- ited to a change in deflection due to body inclination. That, is § s p , e f f = s + e (50) University of British Columbia Peter M. Ostafichuk April 2004 Chapter 3 - Control Surface Hydrodynamics - Experiments 68 3.2.4 Non-Zero Yaw Yaw can result from a number of conditions such as turning and transverse currents and waves. The effect may be uniform along the hull (as with a current) or it may be localized (as with rotation in the horizontal plane). Under typical conditions, the total yaw angle at the bowplanes can reach as high as 20 degrees (Dominguez 2000). With non-zero yaw, the port and starboard bowplanes are subjected to different flow fields. Experiments examining the effect of yaw on plane perfor- mance were conducted by Dominguez (2000) as part of his Master of Engineering degree require- ments. Dominguez used yarn tuft visualization and force measurement in the wind tunnel with the one-quarter scale D O L P H I N model; the key findings are reproduced below. Yarn Tuft Visualization With yarn tufts positioned uniformly about the hull, the body influence in yaw was observed. In Figure 3-30 (a) the starboard side of the vehicle is pictured for +20 degrees yaw; hereafter, the side of the vehicle in the wake of the hull is referred to as the leeward side. The port side of the vehicle for +20 degrees yaw, or windward side, is shown in Figure 3-30 (b). It is well documented that two counter rotating vortices form on the leeward side of a finite length inclined cylinder in uniform flow (see, for example, Kawamura and Aihara (1992) or Allen and Perkins (1951)). Considering the yarn tuft images, the flow divides over the windward side of the hull as noted by the chevron pattern in the tufts. However, as the flow reaches the leeward side of the hull it does not appear to fully separate as the yarn tufts also form a relatively uniform chevron University of British Columbia Peter M. Ostafichuk April 2004 Chapter 3 - Control Surface Hydrodynamics - Experiments 69 O o o 2.0 1.5 1.0 0.5 8 o.o 2 -0.5 N -1.0 -2.0 Trim angle |degl ^A—6 -30 -20 -10 0 10 Plane Deflection [deg] 20 30 Figure 3-27 Trim influence on keel plane force production o o o x CD o N 2.0 1.5 1.0 0.5 0.0 -0.5 -1.0 -1.5 -2.0 A • A 2\. Trim angle fdegl A -6 A 6 ^ A J^^N. >W / A A * A -30 -20 -10 0 10 Effective Plane Deflection [deg] 20 30 Figure 3-28 Trim influence on keel plane force production expressed using effective deflection University of British Columbia Peter M. Ostafichuk April 2004 Chapter 3 - Control Surface Hydrodynamics - Experiments 70 1.5 1 H -1 H I Force due to Equivalent Deflection - 1 . 5 -12 -8 -4 0 4 8 1 2 Trim Angle [deg] Figure 3-29 Trim influence on sternplane force pattern. The flow pattern in Figure 3-30 (a) suggests the presence of two counter rotating vortices (located approximately as shown in the figure). Unfortunately, further attempts to verify the exist- ence and location of such vortices-helium bubble flow visualization and a vortex probe were used-were only partially successful. Baseline Configuration Force Measurements The forces on the bowplanes were measured over a range of deflections and yaw angles. The forces measured on the port bowplane are shown in Figure 3-31. In a more general sense, the curves with positive yaw angle apply to the windward side plane while negative yaw correspond to the leeward plane. There are significant differences in the bowplane performance for changing yaw angle. Specifi- cally, the magnitude of the force and the stall angle both increased for increasing yaw on the lee- ward bowplane. Stall was not observed at even 25 degrees deflection for 20 degrees yaw (compared to stall at approximately 14 degrees for straight and level flight). Conversely, for the windward side plane the force magnitude and stall angle decreased with increasing yaw. The force measurement results are consistent with the suggestion of two counter-rotating vortices located on the leeward side of the body. Such vortices would tend to draw momentum from the freestream and increase the local flow velocity near the planes. As such, the planes would effec- tively be operating in a higher velocity and higher Reynolds number flow which would increase both the magnitude of forces as well as the stall angle. In contrast, with increasing yaw angle, the windward planes would experience greater span-wise crossflow and obstruction from the body and hence decreased chord-wise flow velocity. University of British Columbia Peter M. Ostafichuk April 2004 Chapter 3 - Control Surface Hydrodynamics - Experiments 71 (b) Figure 3-30 Yarn tuft visualization of body at +20 degrees yaw: (a) leeward (starboard) side; (b) windward (port) side University of British Columbia Peter M. Ostafichuk April 2004 Chapter 3 - Control Surface Hydrodynamics - Experiments 72 University of British Columbia Peter M. Ostafichuk April 2004 Chapter 3 - Control Surface Hydrodynamics - Experiments 73 3.3 CONTROL SURFACE INTERACTION Flow induced interaction from the bowplanes to the sternplanes was examined as part of the wind tunnel and towing tank research. Wind tunnel flow visualization was used to identify the location of trailing vortices shed from the bowplanes and their proximity to the sternplanes. In both the wind tunnel and tow tank, the force measurement techniques used previously with the bowplanes were used for the sternplanes to measure the resultant forces due to the interaction. 3.3.1 Straight and Level Flight Helium Bubble Flow Visualization For straight and level flight, helium bubble flow visualization was used to identify and map the tip vortices shed from the bowplanes. These vortices are believed to adversely influence the perfor- mance of the sternplanes. With the helium bubble technique, helium-filled soap bubbles of neutral buoyancy are injected into the freestream flow upstream from the model. The bubbles track the flow due to their very low mass and deviations from the fluid pathlines are generally quite small (Merzkirch 1981). The main advantage of bubbles over a smoke streakline technique is that where smoke tends to dissi- pate in high speed and regions of turbulence, helium bubbles persist. With appropriate lighting, film, and shutter settings on a camera, the bubbles appear as streaks when photographed. For fur- ther information regarding the helium bubble path technique, the reader is referred to Barlow et al. (1999), Hale et al. (1971), and Ostafichuk et al. (1999). A sample image from the helium bubble path technique is shown in Figure 3-32. The photograph shows the starboard bowplane with flow passing from left to right (the model is inverted). The streaks in the image are the trails left from individual bubbles. To the right of the plane the bub- bles follow a helical path and quickly migrate to the core of the bowplane tip vortex; the vortex core is identified by the thick horizontal white line. A second sample image for the helium bubble path technique is shown in Figure 3-33. In this case the tail of the vehicle is shown with the helium bubble streaks digitally enhanced. Once again, the vortex core is easily identified by the thick white line running along the length of the hull. The helical motion of the individual bubbles is also evident as they migrate to the vortex core. For the purpose of mapping the bowplane tip vortex path, a wide angle lens was used with a long exposure time. As shown in the sample image in Figure 3-34, the contrast of individual streaks was reduced in the process but the path of the trailing vortex was maintained. In the image, the model is shown as mounted in the wind tunnel (inverted) and a dashed white outline has been added. For various body orientations and bowplane configurations, the relative proximity of the bowplane tip vortex to the sternplane was measured using similar vortex path images. For straight and level flight, the strongest bowplane trailing vortices are shown schematically in Figure 3-35. The bowplane deflections associated with the two vortices are +12 and -12 degrees which also coincide with the approximate deflection with maximum force from Figure 3-13. From the schematic it is apparent that the bowplane tip vortex is closer in proximity to the sternplane for +12 degrees deflection compared to -12 degrees deflection. University of British Columbia Peter M. Ostafichuk April 2004 Chapter 3 - Control Surface Hydrodynamics - Experiments 74 Vortex^Core Figure 3-32 Helium bubble path sample image; trim: +6 degrees, deflection: +25 degrees, Reynolds number: 2.5 million, model inverted Figure 3-33 Oblique stern helium bubble path sample Baseline Configuration The force measured on the bowplanes and sternplanes for the standard configuration in straight and level flight is shown in Figure 3-36. The bowplane deflection is used as the independent vari- able and the stemplane deflection was fixed at zero degrees. The flow induced interaction between the bowplane and stemplane is apparent given the change in stemplane force with bow- plane deflection; without interaction, the stemplane force would be a horizontal line. For a qualitative comparison, in Figure 3-37 the interaction force measured on the sternplanes from Figure 3-36 was scaled and plotted against the bowplane force. A scale factor of -4.5 was used and, as shown in Figure 3-37, the scaled stemplane force very closely matches the bowplane University of British Columbia Peter M. Ostafichuk April 2004 Chapter 3 - Control Surface Hydrodynamics - Experiments 75 Figure 3-35 Standard configuration straight and level flight schematic vortex path diagram force over bowplane deflections of approximately -7 to +14 degrees. In other words, over this range of bowplane deflections, there is an interaction force on the sternplanes of approximately 22% (1/4.5) of the force generated on the bowplanes. As shown in both Figure 3-36 and Figure 3-37, the interaction force on the sternplanes is smaller for negative bowplane deflections; this is consistent with the closer proximity of the bowplane tip vortex to the stemplane for positive bowplane deflections (noted in Figure 3-35). For deflections beyond stall on the bowplanes, the interaction force measured on the sternplanes is proportionally smaller. This effect would be University of British Columbia Peter M. Ostafichuk April 2004 Chapter 3 - Control Surface Hydrodynamics - Experiments 76 1.5 0.5 -0.5 -1.0 I Error Scale —•— Bowplane force • Sternplane force -30 -20 -10 0 10 20 30 Bowplane Deflection [deg] Figure 3-36 Bowplane and sternplane force for the standard configuration in straight and level flight expected due to the diminished bowplane tip vortex strength and large scale turbulence in the wake of bowplanes as a result of flow separation. The role of the bowplane tip vortex in the interaction with the sternplanes is examined in greater detail in Section 4.6. Plane Geometry Influence In addition to the standard configuration, the bowplane-sternplane interaction force was examined for a variety of different plane geometries (both bowplane and sternplane). In Figure 3-38 and Figure 3-39 the effect of plane span is examined for the bowplanes and sternplanes respectively. The influence of other geometric changes (including sweep, taper, and size) to the bowplanes and sternplanes are considered respectively in Figure 3-40 and Figure 3-41. In each figure, the inter- action force measured on the sternplanes is plotted as a function of bowplane deflection. Addi- tionally, for each figure, the sternplane force is shown non-dimensionalized in the standard form using vehicle length squared, Z, well as using the bowplane planform area, C*L C* sp sp 2PUlSbp, exp (51) The intent of the latter normalization is to account for differences in bowplane force associated with changes to bowplane geometry since it was established in Figure 3-37 that there is a strong correlation between the force on the bowplanes and the interaction force on the sternplanes. University of British Columbia Peter M. Ostafichuk April 2004 Chapter 3 - Control Surface Hydrodynamics - Experiments 77 1 .5 Plane Error x N o o o •0 0 0 1 1 1 5 -30 -20 -10 0 10 20 30 Bowplane Deflection [degrees] Figure 3-37 Comparison of bowplane and sternplane interaction force Bowplane span has a moderate effect on the interaction with the sternplanes as shown in Figure 3-38 (a). Surprisingly, the short bowplanes (with an exposed span of 0.0530 vehicle lengths) show the largest absolute interaction force for positive deflections and the least interaction for negative deflections. Considering the force normalized by bowplane area in Figure 3-38 (b), over all deflections, the long bowplanes have the least effect on the sternplanes. As will be shown in Section 4.6, the reason for the reduced interaction appears to be the relative position of the bow- plane tip vortex to the sternplanes. The effect of sternplane span in the bowplane-sternplane interaction appears to be similar in mag- nitude to that of bowplane span. Figure 3-39 (a) shows the measured interaction force normalized by vehicle length squared. The standard sternplanes exhibit the greatest force due to interaction with the bowplanes; however, the differences are of the same order as the experimental error. When the interaction force is expressed in non-dimensional form using the sternplane area (Figure 3-39 (b)), the relative difference between the various sternplanes increases (with the stan- dard sternplanes again showing the largest interaction). This second form of representation, CL s p , is not to be confused with C*L s p above. The effects of changes to the bowplane geometry other than aspect ratio (span) are shown in Figure 3-40. The swept plane had the same span as the standard plane with slightly larger plan- form area while the large and large swept planes had considerably larger area (see Table 3-2 on page 45). Not surprisingly, the bowplanes with the largest planform area developed the largest University of British Columbia Peter M. Ostafichuk April 2004 Chapter 3 - Control Surface Hydrodynamics - Experiments 78 - • — Std BP (ae = 3.07) • Short {ae = 2.66) * Long (ae a 4.06) Error scale -10 0 10 Bowplane Deflection [deg] 0.20 0.15 0.10 0.05 •" 0.00 u -0.05 -0.10 -0.15 -0.20 -Std BP (ae = 3.07) - Short (ae = 2.66) J - Long (ae = 4.06) T -10 0 10 Bowplane Deflection [deg] (a) (b) Figure 3-38 Sternplane interaction force dependence on bowplane span: (a) Zsp versus bowplane deflection; (b) CL* versus bowplane deflection force (see Figure 3-15 on page 59) and also showed the greatest interaction force with the stern- planes (Figure 3-40 (a)). When the interaction force was expressed in non-dimensional form in terms of bowplane planform area (Figure 3-40 (b)), the differences in interaction resulting from bowplane geometry were statistically insignificant based on measurement error for deflections between deflections of approximately -7 to 11 degrees. Beyond this range, differences were noted but can be largely attributed to the different performance of the bowplanes near and after stall (Figure 3-15). In short, the bowplane-sternplane interaction does not appear to be strongly depen- dent on bowplane shape for planes of similar span; however, the size of the bowplanes, as mea- sured by planform area, does have a significant effect due to the associated increase in bowplane force. University of British Columbia Peter M. Ostafichuk April 2004 Chapter 3 - Control Surface Hydrodynamics - Experiments 79 As with changes to bowplane geometry, the interaction force measured on the sternplanes increased with increased stemplane planform area. As shown in Figure 3-41 (a), the interaction force for the larger planes (swept, large, and large swept) was greater than the standard plane; however, when the interaction force was non-dimensionalized by stemplane planform area, the largest interaction was found with the standard plane and the smallest interaction with the large plane. In other words, although the bowplane-sternplane interaction force increased with stem- plane size, the increase was not as large as the near proportional increase noted for the bowplanes. 0.4 0.3 0.2 0.1 o o \ 0 0 -0.1 -0.2 -0.3 -0.4 —•—Standard • Large —ft— Large Swept —ft— Swept J Error Scale -10 0 10 Bowplane Deflection [deg] 0.15 0.10 0.05 0.00 -0.05 -0.10 -0.15 - I -0.20 Plane Error —Standard J - •— Large X —ft— Large Swept \ -ft— Swept \ -10 0 10 Bowplane Deflection [deg] (a) (b) Figure 3-41 Stemplane geometry influence on stemplane force interaction: (a) Zsp versus bowplane deflection; (b) CL s p versus bowplane deflection Finally, the effect on bowplane-sternplane interaction of the endplates added to the short bow- planes was examined. As was shown in Figure 3-16 on page 60, the endplates had limited effect on the force production by the bowplanes. The interaction force on the sternplanes is shown in University of British Columbia Peter M. Ostafichuk April 2004 Chapter 3 - Control Surface Hydrodynamics - Experiments 80 Figure 3-42 to which the bowplane endplates appear to have limited influence other than a local- ized decrease in interaction near the bowplane stall angle for negative deflection. Thus, the reduc- tion in bowplane tip vortex strength as claimed by Talay (1975) appears to be a local phenomenon. 0 .3 0 . 2 H -0.4-1 , , 1 , , 1 - 3 0 - 2 0 - 1 0 0 1 0 2 0 3 0 Bowplane Deflection [deg] Figure 3-42 Influence of endplates on the short bowplanes on bowplane-sternplane interaction Plane Location Influence The bowplane-sternplane interaction was more strongly influenced by changes to the location of the planes on the hull than by the geometry of the planes. Limited flow bubble path visualization was used to evaluate alternate bowplane configurations followed by extensive direct force mea- surement for a wide range of plane arrangements. The helium bubble path flow visualization technique was used for the anhedral bowplanes in sev- eral trim angles. Figure 3-43 shows the path of the bowplane tip vortices for the anhedral bow- planes in straight and level flight. In comparison to the standard bowplanes (Figure 3-35 on page 75), there is greater separation between the bowplane tip vortices and the sternplane with the anhedral configuration. Similarly, while the tip vortex map for the standard configuration bow- planes suggests stronger interaction for positive bowplane deflections, Figure 3-43 suggests that the interaction for the anhedral planes will be stronger for negative deflections. The plane locations considered in force measurement testing included bowplanes in the standard, high, centreline, anhedral, and dihedral positions and sternplanes in the standard, Y-Tail, and Inverted-Y positions (with reference to Table 3-3 on page 46). As indicated previously, with changes to bowplane position the sternplanes were held in the standard location and visa versa. University of British Columbia Peter M. Ostafichuk April 2004 Chapter 3 - Control Surface Hydrodynamics - Experiments 81 Figure 3-43 Helium bubble path schematic for anhedral bowplanes in straight and level flight The interaction force plotted against bowplane deflection is shown in Figure 3-44 for the different bowplane configurations. Most notably, the anhedral and dihedral configurations show signifi- cantly reduced interaction compared to the three configurations near the centreline (standard, high, and centreline). As was suggested by the helium bubble path visualization, the standard configu- ration shows greater interaction compared to the anhedral configuration and the largest interac- tions in each case occur for positive and negative deflections respectively. In addition to providing detailed information regarding the performance of specific bowplane locations these results dem- onstrate the value of the simple tip vortex mapping in qualitatively predicting bowplane-sternplane interaction. Figure 3-45 shows the bowplane-sternplane interaction for two alternate stemplane locations. The Y-tail and Inverted-Y were similar to the anhedral/'dihedral bowplanes in alignment and also resulted in significant reduction in bowplane-sternplane interaction. Although flow visualization was not conducted in either of these cases, the bowplane tip vortex path from Figure 3-35 on page 75 suggests that the reduced interaction is due to increased separation between the stemplane and the vortex. 3.3.2 Non-Zero Trim Analysis of the bowplane-sternplane interaction for non-zero trim was conducted in the same man- ner as for straight and level flight. Helium bubble path visualization was again used for several plane configurations while direct force measurements were used for a more comprehensive selec- tion of configurations. Helium Bubble Path Visualization Schematic diagrams of the bowplane tip vortex path for the standard configuration are shown in Figure 3-46 (a) and (b) for trim angles of+6 and -6 degrees respectively. Through the proximity University of British Columbia Peter M. Ostafichuk April 2004 Chapter 3 - Control Surface Hydrodynamics - Experiments 82 0.3 0.2 0.1 o o x 0.0 * * * \ ^ — - z t r ^ /* — • • T —•—Std Error Scale - • — H i 9 n J _ A Dihedral —A—Anhedral o Centerline -30 -20 -10 0 10 20 30 Bowplane Deflection [deg] Figure 3-44 Bowplane-sternplane interaction dependence on bowplane position of the tip vortex to the sternplanes, the flow visualization demonstrates that there is still potential for adverse bowplane-sternplane interaction even at trim angles as large as 6 degrees. Similar schematic diagrams are shown for the anhedral configuration at +6 and -6 degrees in Figure 3-47 (a) and (b) respectively. The flow visualization suggests that interaction will remain small for anhedral bowplanes with negative trim angles but the interaction may increase for posi- tive trim angles approaching 6 degrees. Baseline Configuration Force Measurement With the planes in the standard location, the bowplane-sternplane interaction was measured for changing vehicle trim angle. Figure 3-48 shows the sternplane force for trim angles ranging from -10 to +10 degrees in 2 degree increments. As the incident flow onto the sternplanes changes with trim, the various curves are shifted vertically; however, since the force is plotted as a function of bowplane angle, the interaction for each curve is characterized by the change in force with bow- plane deflection. It is apparent that the interaction is less significant for larger trim angles as the variation from the average value of each curve becomes increasingly smaller. This result is more clearly demonstrated in Figure 3-49 where the ratio of the bowplane-sternplane interaction curve slope to the bowplane lift curve slope is plotted versus trim angle. The slopes were computed over the range of bowplane deflections from -7.2 to +7.2 degrees. The largest interaction between the planes is measured for +2 degrees trim (not zero) which can be attributed to the below-centreline position of the bowplanes. As shown in the figure, the interaction quickly diminishes with trim University of British Columbia Peter M. Ostafichuk April 2004 Chapter 3 - Control Surface Hydrodynamics - Experiments 83 -0.3 -30 -20 -10 0 10 Bowplane Deflection [degrees] 20 30 Figure 3-45 Bowplane-sternplane interaction dependence on sternplane position angle and reaches approximately 5% or less of the bowplane lift curve slope for trim angles of approximately 10 degrees. University of British Columbia Peter M. Ostafichuk April 2004 Chapter 3 - Control Surface Hydrodynamics - Experiments 84 (b) Figure 3-46 Standard configuration vortex path: (a) +6 degrees trim; (b) -6 degrees trim University of British Columbia Peter M. Ostafichuk April 2004 Chapter 3 - Control Surface Hydrodynamics - Experiments 85 (b) Figure 3-47 Anhedral configuration vortex path for (a) +6 degrees trim; (b) -6 degrees trim University of British Columbia Peter M. Ostafichuk April 2004 Chapter 3 - Control Surface Hydrodynamics - Experiments 86 1.5 Trim Angle -*—10 - e - - 8 - A — - 6 —e—4 - B - - 2 - * - 0 -m-2 —*-4 -*—6 - » - 8 —i—10 Figure 3-48 Interaction force on standard stemplane as a function of trim angle 0.35 0.30 0.25 H 0.20 N N 0.15 H 0.10 0.05 0.00 1 / \ / \ ^ ^ —4 • -15 -10 0 Trim Angle 10 15 Figure 3-49 Ratio of bowplane-sternplane interaction curve slope to bowplane lift curve slope University of British Columbia Peter M. Ostafichuk April 2004 Chapter 3 - Control Surface Hydrodynamics - Experiments 87 Plane Location Influence With the anhedral bowplanes in place of the standard bowplanes, the interaction force measured on the sternplanes as a function of trim angle is shown in Figure 3-50. The strongest interaction is noted for +6 degrees trim and there is comparatively little interaction at -6 degrees trim; these find- ings correspond to the bowplane tip vortex mapping that was shown in Figure 3-47. — t * > i t t t , ^ • I Error Scale > Anhedral 0 deg trim • Anhedral +6 deg trim A Anhedral -6 deg trim , . * A * A A A A A ^ * A * * * : -30 -20 -10 0 10 20 30 Bowplane Deflection [deg] Figure 3-50 Interaction force from anhedral bowplane on stemplane as a function of trim angle 3.3.3 Non-Zero Yaw The final aspect of the bowplane-sternplane interaction that was examined was the influence of vehicle yaw. Experimental data was collected by Dominguez (2000) through wind tunnel testing with the 1/4-scale model. The interaction force to bowplane deflection slope, normalized by the slope for straight and level flight, is shown as a function of yaw angle representative of the port stemplane in Figure 3-51. Data from both sternplanes was used to form the curve; negative yaw angles from the starboard stemplane were averaged with the corresponding positive angles from the port plane. The error bars are based on the standard error in the computation of the slopes. Similar to the observation with trim, the bowplane-sternplane interaction quickly diminished with increasing yaw. With the standard configuration, the strongest interaction was observed for zero degrees yaw. Interestingly, for yaw angles greater than roughly 8 degrees in magnitude, the inter- action force and the force on the bowplanes were oriented in the same direction (under normal conditions, the bowplane and interaction force on the sternplanes are oriented in opposite direc- tions). University of British Columbia Peter M. Ostafichuk April 2004 Chapter 3 - Control Surface Hydrodynamics - Experiments 88 A 1 f 1 1 i K \ \ \ NT K T -25 -20 -15 -10 -5 0 5 10 15 20 25 Yaw Angle [deg] Figure 3-51 Slope ratio of averaged sternplane interaction to bowplane lift curves versus yaw angle University of British Columbia Peter M. Ostafichuk April 2004 Chapter 3 - Control Surface Hydrodynamics - Experiments 89 3.4 FREE SURFACE INTERACTION The wave conditions encountered by an A U V in practice are random in nature and classified by a statistically determined sea state. By the principal of superposition (see, for example, Tipler, 1991, or Journee, 2001), it is possible to replicate the irregular sea conditions by summing a series of reg- ular waves (that is, sinusoidal waves with well defined frequency and amplitude). In order to determine the effect of the free surface and waves on plane performance, tests were conducted in the towing tank using the instrumented quarter-scale model in regular waves. Time series data were examined using fast Fourier Transform (FFT) analysis. The relationship between wave geometry and the force on the control surfaces was examined in terms of vehicle depth, speed, and plane position. 3.4.1 Wave Geometries Tested Wave frequencies of 0.4, 0.6, 0.8, and 1.0 Hz were used and corresponded to wavelengths of 4.57 /, 2.00 /, 1.14 /, and 0.73 / respectively (where the wavelengths have been non-dimensional- ized by vehicle length). Nominal wave heights included 0.011 /, 0.024 /, 0.0417 /, and 0.0714 / (corresponding to 2.5, 5, 9, and 15 cm respectively). These wavelengths and amplitudes roughly span the range of well formed regular waves obtainable in the tow tank facility. For the maximum wavelength of 4.57 /, only the 0.011 / amplitude waves were used as other amplitudes resulted in non-sinusoidal waves. The wavelengths tested at the tow tank are substantially below those encountered in sea state 3. (From Section 2.6.3 on page 38, the most probable wave period is 7.5 seconds which corresponds to a wavelength of 88 m in full scale, or 22 m for the 1/4-scale model.) As vehicle depth is deter- mined using pressure sensors (Field, 2000), at wavelengths greater than several vehicle lengths the vehicle will track the wave profile rather than maintain constant depth with respect to the calm water free surface. Therefore, in terms of influence on the control surfaces on DOLPHIN, the most significant waves are those of length of the same order of magnitude or less than the vehicle and with maximum wave orbital velocity at the depth of the control surfaces. The wave orbital veloc- ity is a function of both depth and wavelength (see Section 2.6.2) and is illustrated in Figure 3-52 for 0.011 / amplitude waves. In the figure, the range of wavelengths considered for this thesis is indicated and, as shown, roughly covers the maximum expected wave influence over operating depths. 3.4.2 Wave and Control Surface Force Spectra In the tow tank tests, force data was collected from each of the planes along with carriage speed and wave height (measured using a capacitance-style wave probe). Typical data for the bowplane force and wave amplitude is shown in Figure 3-53 over a seven second window for a forward speed of 1 m/s and a depth of one hull diameter (to the centreline). Note that the force and wave amplitude share the same graph but use different vertical axes. The general agreement between the signals is good—they have similar overall responses with similar dominant frequencies—although the force signal shows more noise and a phase delay of approximately one-quarter cycle. The phase delay can be largely attributed to differences in longitudinal position and time response of the force transducer and wave probe. University of British Columbia Peter M. Ostafichuk April 2004 Chapter 3 - Control Surface Hydrodynamics - Experiments 90 Wavelength / L Figure 3-52 Dependence of wave orbital velocity on depth and wavelength The time series data is shown in spectral form in Figure 3-54 from the FFT analysis (again using different axes for the wave amplitude and plane force). The peaks from the wave amplitude and plane force are sharp, very well defined, and almost indistinguishable (both indicating a dominant frequency of 1.26 Hz). The peak frequency amplitudes of 0.484 for Zbp and 0.0155 for the wave height also agree well with visual observations from Figure 3-53. By using spectral analysis, sig- nals were effectively averaged over a large number of cycles and the complications introduced by phase shift and noise observed in the time series data were avoided. Subsequent findings are based on the peak values obtained from the dominant frequencies found in the FFT analysis. 3.4.3 Wave Geometry Influence The effect of the wave amplitude and wavelength on the oscillatory forces on the planes was mea- sured using the quarter-scale model held stationary in the tow tank. The vehicle depth during these experiments was 1.5 hull diameters (0.174 1) as measured from the calm water line to the hull cen- treline. A 3 x 3 test matrix consisting of three wavelengths and three wave amplitudes, plus one additional point, was used. Wavelengths of 0.73 /, 1.14 /, and 2.00 / with nominal wave heights of 0.024 /, 0.0417 /, and 0.0714 / were used. An additional test point was added with 4.57 / wave- length and 0.011 / wave amplitude; again this combination represented the low frequency limit of the wave generation equipment at the tow tank. In all cases, the exact wave amplitudes were mea- sured using a capacitance wave probe with the F F T method outlined above. University of British Columbia Peter M. Ostafichuk April 2004 Chapter 3 - Control Surface Hydrodynamics - Experiments 91 O o o X N 1.5 1.0 0.5 0.0 -0.5 -1.0 -1.5 0.03 0.02 0.01 • Bowplane Force •Wave Height CD I 0 > CO -0.01 -0.02 -0.03 10 11 Time [sec] 12 13 14 Figure 3-53 Wave height and bowplane force trend chart The results of the wave geometry tests are shown in Figure 3-55. The wave amplitude, a, has been non-dimensionalized by the vehicle length and the plane forces, Zwave have been nondimen- sionalized by density, p, gravity, g, and the vehicle length as: Z' (52) Furthermore, so as to compare all data on a single graph, the stemplane forces have been scaled by the bowplane to stemplane planform area ratio. Data is not shown for the bowplanes with the 15 cm nominal wave amplitude test due to a fault that occurred in the instrumentation. A least squares best-fit line has been added to the figure and suggests that the oscillatory force on the planes is primarily a linear function of the wave amplitude with little influence from wavelength for the range of wave slopes considered. For comparison to previous data, the maximum force a bowplane would generate when moving at 1 m/s in water corresponds to Zg « 50 x 10~6. The linear dependence of the plane force on wave length and amplitude was examined using a sta- tistical analysis of variance (ANOVA). The key parameter of the A N O V A output is the F-ratio, or simply F, which indicates whether a trend in data is significant enough to be related to a particular variable or if it can simply be attributed to measurement error. Based on the number of samples, a critical F value is computed; if the F of a particular variable is greater than F critical, then the University of British Columbia Peter M. Ostafichuk April 2004 Chapter 3 - Control Surface Hydrodynamics - Experiments 92 0.6 0.5 0.4 o o o x 0.3 J3 N 0.2 0.1 0 • Bowplane Force •Wave Amplitude T^iirlhiAin 2 3 Frequency[Hz] 0.02 0.015 0.01 a) XJ 3 "Q. E < > 0.005 Figure 3-54 Wave height and non-dimensional bowplane force FFT spectra effects of the variable are considered to be significant, otherwise effects are attributed to noise. Further information regarding the A N O V A technique can be found in any standard statistical anal- ysis text (such as Hogg and Ledolter, 1992, or Speigel, 1992). Considering plane force as a function of wave length and wave amplitude, the A N O V A results are given in Table 3-5 as based on a 95% level of confidence. The F-value for the plane force depen- dence on wavelength is below the critical value (4.4 < 5.7) while the F-value for wave amplitude is well above the critical value (154.9 » 5.7). In other words, there is a very strong relationship between wave amplitude and the force on the planes while the influence of wavelength is statisti- cally insignificant. This is not to suggest that in general wavelength does not affect the force on the planes—there is a clear relationship in Figure 3-52—but rather it indicates that there was little change in performance for the range of wavelengths considered in the experiments. Likewise, the University of British Columbia Peter M. Ostafichuk April 2004 Chapter 3 - Control Surface Hydrodynamics - Experiments 93 0.09 Wave Amplitude: a/L Figure 3-55 Wave induced plane force dependence on wave amplitude F-value associated with the interaction between wavelength and wave amplitude is well below the associated critical value indicating that the interaction is also statistically insignificant. Table 3-5 A N O V A for wave force to wave geometry Source F F critical wavelength 4.4 5.7 wave amplitude 154.9 5.7 interaction 1.2 4.1 3.4.4 Depth Influence The influence of depth was examined in both calm water and for various wave conditions. The calm water tests were conducted at speed while the tests with waves were conducted with the model held stationary. Calm Water Tests Due to the proximity of the hull to the free surface, the flow field about the control surfaces is not necessarily like that for deep water (equivalent to the wind tunnel tests). Tests were conducted in calm water at various depths in order to determine the surface influence at shallow depths. Depths University of British Columbia Peter M. Ostafichuk April 2004 Chapter 3 - Control Surface Hydrodynamics - Experiments 94 of 1, 2, and 3 hull diameters measured from the free surface to the hull centreline were considered. The resulting force to deflection curves for the bowplanes are shown in Figure 3-56. 0.8 0.4 N • d/D = 1 — • — d/D = 2 — * - d/D = 3 J_ Error Scale i V -30 -20 -10 0 10 20 30 Bowplane Deflection [deg] Figure 3-56 Plane performance dependence on hull depth in calm water Unfortunately, after the tests were conducted, it was discovered that boundary layer trips were not correctly installed on the planes. The result was a much more gradual stall with earlier onset; in short, the force to deflection curves were different from those measured previously in the wind tunnel and subsequently in the tow tank. However, on average, the three curves agree with each other to better than 0.75% of the full-scale range. Furthermore, the maximum difference between any two measurements was only 0.07 units of Z (3.8% of full-scale) while the measurement accu- racy was 0.04 units of Z . This similarity within the curves from Figure 3-56 strongly suggests that the calm water free surface effect on the planes is negligible over typical operating depths. These findings agree with results for submerged rectangular flat plates reported by Wadlin et al. (1955) and provided in Section 2.6.1 on page 38. Wadlin et al. concluded that for depths greater than 0.85 chord lengths, the surface effect is negligible. For comparison with the DOLPHIN, a depth of 1 hull diameter corresponds to 2.9 chord lengths. University of British Columbia Peter M. Ostafichuk April 2004 Chapter 3 - Control Surface Hydrodynamics - Experiments 95 Wave Influence at Depth The influence of vehicle depth on the wave-induced force on the planes was examined primarily using a stationary model in the tow tank. The wave amplitude for these tests was approximately 0.042 / and the wavelength was approximately 1.14 / (corresponding to a frequency of 0.8 Hz). Figure 3-57 shows the dependence of the wave force amplitude on the bowplanes in terms of vehi- cle depth, d, measured from the hull centreline to the calm water line. The depth has been non- dimensionalized by the hull diameter, D. The best-fit curve shows the expected exponential form: 4itrf Z = C0e x (53) For the wave geometry considered, there was over a four-fold reduction in the wave force at 3.5 hull diameters depth compared to that at 1.5 hull diameters depth. The data for the depth of d/D = 0.5 did not agree well with the exponential fit however significant interference was intro- duced by waves breaking over the exposed hull at that depth. The comparison between experi- mentally and analytically derived wave forces is examined in detail in Section 4.7. d / D Figure 3-57 Wave induced force amplitude as a function of hull depth 3.4.5 Speed Influence The influence of forward speed of the vehicle on the wave induced force on the bowplanes was examined for the speed range obtainable in the tow tank. For all tests, waves with 0.016 / ampli- tude and 1.14/ wavelength were used. Figure 3-58 shows the force amplitude plotted against University of British Columbia Peter M. Ostafichuk April 2004 Chapter 3 - Control Surface Hydrodynamics - Experiments 96 Froude number for depths of 1.5 and 3.5 hull diameters. The wave force amplitude is nondimen- sionalized in the standard form using forward speed: Z = (54) where the Froude number is defined as: Fr = U 4gl (55) Figure 3-58 Wave induced force on bowplanes versus Froude number There is a dramatic reduction in the wave influence with speed. This is to be expected since the wave orbital velocity (constant in this case) represents a smaller proportion of the total velocity on the planes at higher forward speeds. For Froude numbers of 0.33 and 0.77, tests were conducted at both 1.5 and 3.5 hull diameters depth. The wave induced force at the 3.5 hull diameter depth for these two speeds was 26% and 25% respectively of that at the 1.5 hull diameter depth. For the stationary model tests (Section 3.4.4), the ratio of plane force for 3.5 to 1.5 hull diameters depth was 23%. This favor- able comparison suggests that the best fit form of wave induced force versus depth from the sta- tionary tests (Figure 3-57) is valid over typical speeds. University of British Columbia Peter M. Ostafichuk April 2004 Chapter 3 - Control Surface Hydrodynamics - Experiments 97 3.4.6 Bowplane Position Influence The effect of relocating the bowplanes was studied using the stationary model with waves of vary- ing wavelength and 9 cm (0.0417 I) nominal amplitude. Tests were conducted using the standard bowplane configuration along with the bowplanes in the anhedral position (angled 30 degrees downward.) Figure 3-59 shows the wave induced force amplitude on the bowplanes as a function of the incoming wavelength. The maximum wavelength examined was extended beyond that of previous experiments; however, the wave amplitude in those cases was necessarily decreased to prevent generation of non-linear waves. The non-dimensionalized plane force was subsequently normalized by the non-dimensional wave amplitude, a/1, to account for wave amplitude variation between trials. The normalization justified by the near-linear behavior of plane force with wave amplitude (as demonstrated in Figure 3-55) 200 0 \ , , , r- _] 0 1 2 3 4 5 Wavelength / L Figure 3-59 Influence of bowplane position on wave induced plane forces For the range of wavelengths studied, the anhedral bowplanes consistently show less wave induced force as compared to the standard bowplanes. This can be partially attributed to the downward inclination of the anhedral planes in that the exposed plane area is reduced when viewed from above. Figure 3-60 shows the ratio of wave amplitude force on the anhedral planes to that on the standard planes with a correction for horizontal planform area. That is, the forces have been non-dimensionalized by the exposed control surface area in the x-y plane such that the wave induced forces on the anhedral planes from Figure 3-59 have been increased by l/(cosP) (where p is the anhedral angle). This is approximately equivalent to an increase in the span of the anhedral planes by l/(cosP); even for this condition, the wave induced forces remain smaller on the anhedral planes than the standard planes for all free surface conditions that were used in the experiments. University of British Columbia Peter M. Ostafichuk April 2004 Chapter 3 - Control Surface Hydrodynamics - Experiments 98 1.25 (A .2 g IS « CC Q- % I 3 ca ~ -o p * 5 TJ o CO o ~~ o | 0.75 iS ^ a. c < 1 4- 0.5 - Anhedral planes area normalized to same as standard planes. i i i 1 1 . 0.5 1.5 2 2.5 3 Wavelength / L 3.5 4.5 Figure 3-60 Ratio of wave induced force normalized by horizontal planform area on anhedral bowplanes versus standard bowplanes University of British Columbia Peter M. Ostafichuk April 2004 Chapter 3 - Control Surface Hydrodynamics - Experiments 99 3.5 SUMMARY In this chapter, the experimental findings from both wind tunnel and towing tank tests were dis- cussed. Following a discussion of the experimental hardware and facilities, the following observa- tions were presented: Straight and Level Flight •a body mounted plane has similar (though slightly less linear) force versus deflection behav- ior compared to an isolated appendage • oil film visualization indicates a gradual increase in the separated flow region on bowplanes for deflections well below stall angle and a sudden increase in separation near the plane root for deflections close to stall • plane geometry was found to influence force generation but the effect was associated mainly to the change in the control surface planform area • the mounting location of the bowplanes on the hull was found to have little effect on their performance in straight and level flight in calm water Bowplane Sternplane Interaction • bowplane tip vortices were identified through bubble path visualization and found to pass very near sternplanes in some cases •bowplane tip vortices were found to interact with the sternplanes resulting in a force on the sternplanes of 22% of the force on the bowplanes (for the standard configuration) • the bowplane-sternplane interaction was somewhat dependent on plane geometry and highly dependent on plane location (both bowplane and sternplane) Non-Zero Trim and Yaw • yam tuft visualization indicated significant changes in flow angle on the hull with trim •the effect of trim on the bowplanes was to shift the force to deflection curves horizontally • in yaw, the leeward planes showed improved performance while the performance of the windward planes deteriorated • with increasing yaw or trim, the bowplane-sternplane interaction quickly diminished Free Surface Effects • there was negligible effect from the calm water free surface on the plane force based on tests over a range of depths • there was a measurable effect from waves on the planes; the wave influence decayed expo- nentially with depth and increased almost linearly with wave amplitude • the wave effect was less significant with increasing forward speed as the wave induced force represented a smaller component of the total force University of British Columbia Peter M. Ostafichuk April 2004 Chapter 4 - Control Surface Hydrodynamics - Modelling 100 4 C O N T R O L S U R F A C E H Y D R O D Y N A M I C S - M O D E L L I N G Everything should be made as simple as possible, but not simpler. - Albert Einstein The design and development of an A U V is a costly and time consuming process and the use of modelling and simulation to predict and evaluate system perfor- mance is essential. The relevance of any such simulations depends on the accu- racy and completeness of the models used. In some cases, (as noted by Watt, 1993, for example) small errors in modelling coefficients may significantly affect outcomes depending on how simulations are conducted. Field (2001) suggests that accurate determination of the hydrodynamic values is essential and represents the most demanding area of submarine modelling. In general, as the closed-loop bandwidth of a control system is increased, greater precision is required in the model (Palm, 1986); conversely, the requirements on model accuracy are relaxed if the closed-loop bandwidth is decreased (Astrom and Wittenmark, 1997). The need for mathematical descriptions that are accurate and complete suggests models of ever-increasing complexity; however, extremely complex models are cumbersome and difficult to implement. Thus, one of the goals of modelling is to represent a given system in a way that is as simple as pos- sible while retaining all key performance characteristics. For this thesis, emphasis was placed on modelling the DOLPHIN using vehicle-specific experimental data. This chapter builds on the experimental results from Chapter 3. It begins with a presentation of background information relevant to A U V modelling. The remain- ing sections use various analytical and numerical techniques to validate, predict, and mathematically describe the experimental findings from Chapter 3. In partic- ular, attention is given to Reynolds number scaling effects, control surface perfor- mance prediction, the influence of trim and yaw on bowplanes, bowplane- sternplane interaction, and the interaction with waves and the free surface. The models developed will be used for closed-loop A U V control (Chapter 5) and sim- ulation (Chapter 6). University of British Columbia Peter M. Ostafichuk April 2004 Chapter 4 - Control Surface Hydrodynamics - Modelling 101 4.1 BACKGROUND There are several techniques commonly employed in A U V modelling including analytical meth- ods, computational methods, system identification, and empirical methods. Perhaps the simplest of these are analytical methods which include development from physical laws and first principles, as given by (Nahon, 1996) or (Solberg, 1992) for example. Computational methods can also be employed, such as strip theory (see, for example, McTaggart et al., 1993), panel methods, (see Conway and Mackay, 1990, or Barlow et al., 1998), and even computational fluid dynamics approaches (McDonald and Whitfield, 1997, and Huyer, 2001). Due to the complexity, nonlinear- ity, and coupling in actual A U V systems, such approaches are often only, useful for rudimentary analysis. Quite often, these techniques are employed initially and then later validated by experi- ment (such as with Hopkin, 1990.) Others, such as Caccia and Gianmarco (2001) use a combina- tion of modelling and system identification with the general Newton-Euler motion equation for derivation of the dynamic model of an AUV. System identification involves treating the plant as a black box and attempts are made to correlate the input-output behavior. There are various limitations in using system identification with AUVs including the presence of significant disturbances as well as the coupled, nonlinear plant behavior. As Cellier et al. (1996) note, identification of nonlinear systems in even noise-free environments is difficult to say the least. Recent advances in identification, such as nonlinear observers (Kim, 2002) and neurofuzzy identification (Bossley et al., 1999) show promise for use with AUVs and other nonlinear systems. Nonetheless, input-output information for a given A U V is usually not readily accessible either because it is not possible to consider all aspects of system performance or because the A U V being studied does not yet exist. As such, empirical and semi-empirical methods form the basis for most modelling on AUVs. As was outlined in Chapter 3, experiments may be conducted in towing tanks, wind tunnels, or in open water. The data from the experiments is most often condensed, simplified, and generalized to some degree for use in simulation and prediction. For example, in the D R E A Submarine Simula- tion Program (DSSP), plane forces and moments are expressed in terms of hydrodynamic coeffi- cients (see Feldman, 1995); for DSSP, these coefficients were evaluated experimentally or derived from other semi-empirical sources (Mackay, 1993). Additionally, it is common practice to sim- plify modelling by considering the vehicle as a set of decoupled subsystems consisting of forward speed, roll, vertical plane motions, and horizontal plane motions (Lea et al., 1999). The unmod- elled coupling effects that remain are assumed to be small and are treated as disturbances in appli- cation (see, for example, Healey and Lienard, 1993, or Jalving, 1994). University of British Columbia Peter M. Ostafichuk April 2004 Chapter 4 - Control Surface Hydrodynamics - Modelling 102 4.2 REYNOLDS NUMBER EFFECT MODELLING As mentioned in Section 3.1, it was not possible to conduct experiments with the same Reynolds number for the model as for the full-scale vehicle. The major change in plane performance with Reynolds number was the angle at which stall occurred. Experimental data is examined below fol- lowed by the procedure used to scale this data to full-scale. 4.2.1 Change in stall angle with Reynolds number The stall angle was determined from experimental data for a range of different Reynolds numbers. For this work, the control surface stall angle was defined as the smallest magnitude control surface deflection for which the first derivative of the Z force with respect to deflection became zero. In general, the stall angles for positive and negative deflection were not the same as the influence of the hull was not symmetric (the bowplanes were located below the centreline). The Reynolds num- bers considered, based on vehicle length, ranged from 1 to 7 million; these correspond to chord- based Reynolds numbers of approximately 40 000 to 280 000. Measurements were conducted for bowplane deflections ranging from -28.8 to +28.8 degrees in increments of 3.6 and 1.8 degrees. The uncertainty in the stall angle was assumed to be one half of the deflection increment angle, that is, 1.8 and 0.9 degrees respectively. All experimental data are based on the standard vehicle configuration in straight and level flight. Data were compared to published results of wind tunnel tests on standard N A C A 0012, 0015, 0018, and 0021 sections (Jacobs and Sherman, 1937, and Stack, 1931); unfortunately, only the 0021 section had finite aspect ratio and the others were two-dimensional sections. Figure 4-1 shows a comparison on the experimental data on the N A C A 0025 section as well as the other N A C A sections from the literature. For each foil, a logarithmic best-fit is included to indicate the trend of stall angle versus Reynolds number. As expected, the thicker sections of the same aspect ratio tend to show larger stall angles for a given Reynolds number (see White, 1986, for example). The stall angle to Reynolds number dependence for all three sections agree quite well with the logarithmic trends suggested. The high Reynolds number stall angle performance for the 0025 section has been extrapolated from the experimental data. The logarithmic curve fit to the N A C A 0025 data in Figure 4-1 for the stall angle in degrees and radians respectively are given by: astall[deg] = 6.221 InRe- 78.24 (Re > 1 x 106) (56) and astall[rad] = 0.1091n/fe- 1.366 (Re > 1 x 106) (57) To confirm that the logarithmic extrapolation of the N A C A 0025 data gave a reasonable prediction of stall behavior, high Reynolds number data from a two-dimensional N A C A 0025 section was used. The two-dimensional airfoil data was based on N A C A experiments conducted at a Reynolds number of 5 million (Bullivant, 1941). The stall angle of the equivalent low-aspect ratio section was predicted based on a relationship derived from Clark Y airfoils of different aspect ratio (pro- University of British Columbia Peter M. Ostafichuk April 2004 Chapter 4 - Control Surface Hydrodynamics - Modelling 103 O) 0) •a 0) O) c < CO 40 35 30 25 20 15 • NACA 0025 (ae = 1.54) o NACA 0025 (predicted from 2D) / ° o NACA 0021 (ae = 6) o NACA 0018 (2D) / - A NACA 0015 (2D) / x NACA 0012 (2D) - X- - - • • 1 1 0.1 1 10 100 Reynolds Number Based on Vehicle Length (Millions) 1000 Figure 4-1 Control surface stall angle as a function of Reynolds number vided in Appendix A.2). The prediction of the N A C A 0025 performance at high Reynolds number is shown in Figure 4-1 and agrees to better than 7% with the logarithmic extrapolation. The lower bound on Reynolds number corresponds to a full-scale vehicle speed of 0.12 m/s, well below typical operating speeds. At a forward speed of 6 m/s (Re =51 million), the predicted stall angle is 32.2 degrees; this is sufficient for the maximum obtainable range of the planes on DOL- PHIN used in practice (±25 degrees). The results from this analysis were used to predict the stall angle as a function of Reynolds number in vehicle simulations. 4.2.2 Scaling experimental data Reynolds number scaling for the experimental plane force versus deflection data was performed using a method outlined by Barlow et al. (1999). Specifically, the steps of the scaling procedure are: 1. The linear portion of the lift curve from experimental data is extended with the same slope 2. The maximum lift (full scale), CL m a x is estimated (detailed below) 3. The curved portion of the experimental lift curve is raised until it has the proper value of CL m a x and is shifted laterally until it joins the linear part of the constructed full-scale lift curve University of British Columbia Peter M. Ostafichuk April 2004 Chapter 4 - Control Surface Hydrodynamics - Modelling 104 As Barlow notes, this method results in a full-scale lift curve that has the proper lift at zero deflec- tion, slope, and CL m a x but likely has an angle of maximum lift that is too great and a stall that is too gentle. Isolated Planes Scaled experimental data for an isolated plane are shown in Figure 4-2 using the above method. The experimental data were collected at a vehicle length based Reynolds number of 7 million and scaled to 35 million (Reynolds number of 0.28 to 1.4 million based on chord). The sign conven- tions and force non-dimensionalization outlined in Section 2.1 on page 14 are used. Specifically, in terms of the dimensional force, Z , the fluid density, p , the freestream velocity, U, and the vehi- cle length, / , the non-dimensional force is: (58) Two methods were used to predict the maximum full-scale lift (step 2 above). The first method was provided by Barlow et al. (1999) in which the change in section maximum lift coefficient, ACL m a x , was determined from tabulated semi-empirical data (see Appendix A.3). The second method was to predict the stall at full-scale Reynolds number using an extrapolation from the data in Section 4.2.1 and then to shift the model data along the lift curve to achieve the predicted stall angle. Both methods predict similar performance and are shown with the original experimental data in Figure 4-2. 0.003 0.002 0.001 N 0 -0.001 -0.002 -0.003 ----x----Experimental Data — » — S c a l e d Data (change in CLmax, Barlow) —•—Scaled Data (stall angle extrapolation) »L x -45 -35 -25 -15 -5 5 Deflection [deg] 15 25 35 45 Figure 4-2 Isolated appendage force data extrapolated from a Reynolds number of 7 million to 35 million University of British Columbia Peter M. Ostafichuk April 2004 Chapter 4 - Control Surface Hydrodynamics - Modelling 105 Bowplanes With the bowplanes, the above scaling method was used with maximum lift determined through an extrapolation of the stall angle. As the force to deflection curve of the bowplanes was less linear than that of an isolated appendage (see Figure 3-13 on page 58), the curve was extended based on the best linear curve fit to the data between positive and negative stall angles. Between the stall angles, the curve was scaled uniformly for both force and deflection; beyond stall, the curve was not scaled. The bowplane force curves, both scaled and original, are shown in Figure 4-3. 3.0 o o x 0.0 a Si N -3.0 • Bowplane scaled to Re = 35 million • Bowplane at Re = 2.5 miilion -35 -25 -15 -5 5 15 Bowplane Deflection [deg] 25 35 Figure 4-3 Bowplane force data extrapolated from a Reynolds number of 2.5 million to 35 million As a result of the 'stretching' of the bowplane force to deflection curve, the localized second deriv- ative was less in the scaled curve than in the original. As a result, for similar deviations about a given deflection, the scaled curve had increased linearity. As noted by Jacobs and Sherman (1937), the variation in lift curve slope with Reynolds number is very small; however, as the Reynolds number increases, the lift curve slope does become more linear. Given the significant difference in Reynolds number between experiment and full-scale, caution must be exercised when using the data for absolute deflections greater than 15 degrees. Recall from Section 4.2.1 on page 102, that a 7% difference was noted between the stall angle determined through the extrapolation procedure (used to generate Figure 4-3) and the stall angle approximated University of British Columbia Peter M. Ostafichuk April 2004 Chapter 4 - Control Surface Hydrodynamics - Modelling 106 from two-dimensional tests on a N A C A 0025 section. The error bars shown in Figure 4-3 for the scaled curve represent ±7% accuracy. Note that the stall angle comparison in Section 4.2.1 was made at a Reynolds number based on vehicle length of 125 million (significantly larger than the full-scale Reynolds number). University of British Columbia Peter M. Ostafichuk April 2004 Chapter 4 - Control Surface Hydrodynamics - Modelling 107 4.3 CONTROL SURFACE PERFORMANCE MODELLING The performance of both bowplanes and sternplanes in straight and level flight was modelled using experimental data from the wind tunnel and towing tank. The performance of isolated con- trol surfaces was considered first and hull interaction effects were added subsequently. The plane performance was described by a second order polynomial for use in the simulations; the coeffi- cients of the polynomial are the hydrodynamic derivatives. The effects of plane stall were consid- ered separately and were used to establish the maximum permissible plane deflection in operation. 4.3.1 Hydrodynamic Derivatives The force to deflection curves of the control surfaces was represented using second order hydrody- namic derivatives of the form: Here the coefficients Z g and Z g 5 are first and second derivatives respectively of the force to deflection curve about the origin (see Section 2.1 for further information); For positive deflec- tions, experimental plane data is shown in Figure 4-4. The curve has been scaled to a vehicle length Reynolds number of 35 million and a second order best fit curve was added for the portion of the curve prior to stall. Also shown in the figure are performance predictions from Whicker and Fehlner (1958) and Aucher (1981); detailed information regarding these performance estimates can be found in Section 2.4. The lift curves from the experimental best fit and the two semi-empirical predictions all show good agreement up to stall although the curve from Whicker and Fehlner slightly over predicts the force on the plane. The RMS errors for the method of Whicker and Fehlner and Aucher, compared against the experimental data, were respectively 1.24 x 10~4 and 8.3 x 10~5 , in units of Z . These RMS errors respectively represent 2.6% and 1.7% of the force range between stall angles. The resulting second-order hydrodynamic derivatives for each method are shown in Table 4-1. Aucher predicts greater curvature in the lift to deflection behavior compared to the experimental data while Whicker and Fehlner predict a more linear curve. Overall, the method of Aucher appears to more accurately predict the performance of isolated appendages. 4.3.2 Bowplane Performance Prediction The isolated appendage performance prediction methods above were used as the basis for estimat- ing the performance of the bowplanes. Corrections to account for the presence of the curved hull, as outlined in Section 2.4.2, were applied. For the isolated appendage force prediction of Aucher (1981), the suggested curvature correction was used (reproduced below from equation (23) on page 29): Z = Z g 5 + Z 5 8 5|5 | (59) a (60) University of British Columbia Peter M. Ostafichuk April 2004 Chapter 4 - Control Surface Hydrodynamics - Modelling 108 -0.0005 -0.001 N -0.0015 -0.002 H -0.0025 -0.003 15 20 25 Deflection [deg] Figure 4-4 Experimental to semi-empirical comparison of isolated control surface performance Table 4-1 Comparison of hydrodynamic derivatives Method 2*8 Experiment best fit -0.0057 -0.0018 Whicker & Fehlner -0.0063 -0.0003 Aucher3 -0.0046 -0.0043 a. a second-order fit was used; the method of Aucher is based on a third-order polynomial In the case of isolated appendage curve from Whicker and Fehlner, a correction was used to account for the cylindrical hull. The correction, derived from equation (22) on page 29, was applied to the effective aspect ratio, ae, of bowplanes: 4- R exp' C (61) Additionally, for the estimate from Whicker and Fehlner, the upwash to the plane was modelled using lifting line theory (see, for example, Ashley and Landahl (1965) or von Mises (1959) for more information). As flow incident to the hull was redirected as discussed in Section 2.4.2, the University of British Columbia Peter M. Ostafichuk April 2004 Chapter 4 - Control Surface Hydrodynamics - Modelling 109 incidence angle, a(y), was augmented using the computed upwash at each span-wise location, y, according to: AaCy) = a u p w a s n ( y ) ( * ) (62) Compared to the upwash of a plane mounted against a flat wall (an isolated appendage), it was computed that there was a reduction in upwash due to the hull image vortices (see Appendix A. 1 for details). The resulting bowplane force to deflection predictions are shown in Figure 4-5 along with the experimental data (adjusted to full-scale Reynolds number). The agreement between the experimental data and both semi-empirical methods is good. For the deflection range between positive and negative stall, the RMS error in the prediction from Aucher to the experimental data was 0.000 33 (units of Z ) while the RMS for the prediction from the modified Whicker and Fehl- ner method was 0.000 18. When expressed in terms of the force range between stall, these errors are 7.2% and 4.0% respectively. Considering only the range between +/- 22 degrees, the RMS errors are 3.8% and 2.9%. In short, both the method given by Aucher and the method of Whicker and Fehlner (modified as described above) provide good estimates of the bowplane force as a function of deflection. Whereas the method of Aucher more accurately predicted the performance of an isolated plane, the modified method of Whicker and Fehlner appears to provide better perfor- mance predictions for a bowplane. 0.003 0.002 0.001 -0.001 -0.002 j \ \ N \ . X ^ X . X \ . x _ •̂"xVs ^ W X • Experimental Data Whicker & Fehlner (modified) Aucher Vs. x . _ vv. X ^ v J ^ S ^ ^ X >v | T s \ 37 X ^ X X \ . X X v X x N -30 -20 -10 0 10 20 30 Deflection [deg] Figure 4-5 Experimental to semi-empirical comparison of bowplane performance University of British Columbia Peter M. Ostafichuk April 2004 Chapter 4 - Control Surface Hydrodynamics - Modelling 110 4.4 TRIM EFFECT MODELLING It is known that there is a distribution of angles of incidence across the span of the bowplanes for non-zero trim angles (see Section 2.4.2). Furthermore, the experimental data showed that the effect of trim on the bowplanes was to cause a horizontal shift in the bowplane force to deflection curves. This horizontal shift in the data is quantified below and then compared with theoretically predicted shift based on an analytical expression of the incidence flow angle. 4.4.1 Experimental Results In Section 3.2.3, experimental results were provided describing the performance of bowplanes in trim. The effect of trim was to shift the bowplane force to deflection curves horizontally. The amount of shift, measured from the origin, is shown as a function of trim angle, 6 , in Figure 4-6. 20 -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 Trim Angle [degrees] Figure 4-6 Effective shift in bowplane deflection as a function of trim angle Based on a linear least squares fit in the above figure, the force on the bowplanes was described by the effective plane deflection 8 e f f = 5+ 1.59 (63) In regions very far from the body, the influence of the hull is negligible and the effective deflection in that region is Oeffoo = 5 + 0 (64) University of British Columbia Peter M. Ostafichuk April 2004 Chapter 4 - Control Surface Hydrodynamics - Modelling 111 In other words, very far from the body, the effective angle is simply the sum of the plane angle rel- ative to the body and the body angle relative to the freestream. Close to the body, it is known that there is a change in the flow angle around the hull when the vehicle has non-zero trim (see Section 2.4.2, for example). From the yarn tuft visualization, the incident flow angle at the root with respect to the freestream was equal and opposite to the trim angle. Therefore, the effective deflection for this region is W o t = 5 + 29 (65) which is in agreement with both equations (25) and (26) on page 30. Coincidentally for the DOL- PHIN plane and body geometry considered, the effect of the hull on the entire plane, equation (63), is midway between the effect at the plane root, equation (65), and the effect at very large distances, equation (64). Although equation (63) is directly dependent on the bowplane and hull geometry considered, the same result was found for both the standard and anhedral bowplane configurations suggesting the effect is not highly sensitive to bowplane location or orientation. 4.4.2 Analytical Prediction of Trim Influence The influence of trim on the bowplanes was estimated through predicting the change in incidence angle on the planes due to the hull. The incidence was estimated using equation (26) on page 30, reproduced below The effect of the hull-induced change in incidence angle was averaged across the bowplane span, from y = R to y = b = 2.06R. The result was equivalent to a net increase in the angle of inci- dence of 0.490; in addition to the geometric angle to freestream of 0 due to the vehicle trim, the resulting effective plane angle is which compares very favorably with the best fit of 8 e / / = 5 + 1.500 to the experimental data as given in equation (63). The resulting shift predicted in the bowplane force curves as a function of trim angle is shown in Figure 4-7 (the predicted shift is shown against the measured shift from Figure 4-6). (66) 8 'eff = 8+ 1.499 (67) University of British Columbia Peter M. Ostafichuk April 2004 Chapter 4 - Control Surface Hydrodynamics - Modelling 112 -20 -I , , , , , 1 , , , , , j -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 Trim Angle [degrees] Figure 4-7 Predicted effective shift in bowplane deflection as a function of trim angle University of British Columbia Peter M. Ostafichuk April 2004 Chapter 4 - Control Surface Hydrodynamics - Modelling 113 4.5 YAW EFFECT MODELLING In modelling the influence of yaw on the bowplanes, two aspects were examined. First, empirical relations were developed to mathematically describe the influence of yaw on the bowplanes for the purpose of simulation. Second, simplified analytical models were used to verify whether the pres- ence of two lee-side hull vortices could account for the bowplane behavior described in Section 3.2.4. 4.5.1 Semi-Empirical Relations for Bowplane Performance in Yaw Based on the data of Section 3.2.4, and in particular Figure 3-31 on page 72, semi-empirical rela- tions were developed to simplify bowplane models for the purpose of simulation. The dependence of stall angle and of lift curve slope with yaw were both considered. Stall Angle Dependence on Yaw As noted in Chapter 3, the influence of yaw on the bowplanes was to increase the stall angle on the leeward plane and decrease the stall angle on the windward plane. The stall angle (measured to the nearest 3.6 degree increment) was determined from Figure 3-31 and is shown for the port bow- plane in Figure 4-8 as a function of the yaw angle relative to the straight and level conditions. Data are expressed in terms of the ratio of stall angle in yaw to the stall angle for straight flight where the yaw angle is equal to zero. This ratio is hereafter referred to as the 'yaw stall scaling factor' and was modelled using a polynomial curve fit. 2 T 1.75 - 1.5 - 1.25 - °w/(y) 1 a^ f l//(°) 0.75 - 0.5 - 0.25 - 0 - -25 -20 -15 -10 -5 0 5 10 15 20 25 Leeward Yaw [deg] Windward Figure 4-8 Bowplane stall angle as a function of yaw University of British Columbia Peter M. Ostafichuk April 2004 Chapter 4 - Control Surface Hydrodynamics - Modelling 114 The yaw stall scaling factor is represented by the polynomial h(\\i) where the equation describing h(\y) for the port bowplane with yaw measured in degrees is h W = a*'al^/a\ * 1 ~ 6 ' 8 6 x 1 0 _ V ~ 6 - 9 4 4 x 10" V + 3.403 x 1 0 " V (68) For the starboard bowplane, y is substituted with -vy in equation (68). Lift Curve Slope Dependence on Yaw Again using on the results of Section 3.2.4, the dependence of the lift curve slope on yaw was computed and is shown for the port bowplane in Figure 4-9. As was noted in Chapter 3, for larger yaw angles there is a greater lift curve slope magnitude for the leeward bowplane and a reduced slope magnitude for the windward plane (note that, as shown in the figure, the slope is negative). The computation of the slope was based on a linear regression using measured Zbp values from - 10.8 to +10.8 degrees. The error bars shown in the figure are based on the standard error in the slope (computed as shown in Appendix C l ) . -0.05 -0.11 -I , , , 1 1 , , , , 1 -25 -20 -15 -10 -5 0 5 10 15 20 25 Leeward Yaw [deg] Windward Figure 4-9 Port bowplane lift curve slope as a function of yaw For the purpose of simplifying the results with the intent of using them in simulation, the influence of yaw was converted to a format where the lift curve slope for a given yaw angle was normalized University of British Columbia Peter M. Ostafichuk April 2004 Chapter 4 - Control Surface Hydrodynamics - Modelling 115 by the lift curve slope for zero yaw. In Figure 4-10, this 'yaw scaling factor', / i » , is shown in terms of yaw angle. The third-order best fit curve of Figure 4-10, with yaw in degrees, is 2 bph _ 2 1 - 0.00384V)/ - 2.28 x l O ' V ^ 5 l v = o (69) The above approach of modelling the lift curve slope dependence on yaw was used in simulation as opposed to attempting to model the curve in Figure 4-9 directly due to the simplicity in resulting expression. (The fifth-order polynomial for the curve in Figure 4-9 requires six coefficients whereas the curve in Figure 4-10 requires three, as given by equation (69).) 1.4 1.3 1.2 1.1 'bph 71 1 -\ 0.9 0.8 0.7 0.6 • Experimental Data 3rd Order Approximation -25 -20 -15 -10 Leeward -5 0 5 Yaw Angle [rad] 10 15 20 25 Windward Figure 4-10 Normalized bowplane lift curve slope dependence on yaw angle 4.5.2 Analytical Prediction of Yaw Influence on Bowplanes An analytical analysis of the effect of the two hull vortices due to yaw on the bowplanes was con- ducted. The objective of the analysis was not to predict bowplane performance but simply to determine if the increased lift curve slope observed on the leeward planes could be attributed to the hull vortices. The vortex pattern in yaw was presented in Figure 2-12 on page 32 and is shown in Figure 4-11 below in a slightly different form. University of British Columbia Peter M. Ostafichuk April 2004 Chapter 4 - Control Surface Hydrodynamics - Modelling 116 vortex (b) Figure 4-11 Normalized bowplane lift curve slope dependence on yaw angle In order to predict the effect of yaw on the bowplanes, the two hull vortices were idealized as line vortices as shown in part (b) of Figure 4-11. Each vortex induces a velocity for every point on the bowplane. As the two hull vortices are symmetric about the x-y plane, the vertical components of the induced velocity cancel and the result is a net induced velocity along the plane span, as shown in Figure 4-11 (a). Using rough estimates of the strength and location of the hull vortices (details are provided in Appendix A.4), the induced velocity across a leeward bowplane was predicted. At each point, p, along the plane span, the net velocity, t7 l a n e , and the flow angle with respect to the -x-axis, y p i a n e , were computed (see Figure 4-12). As noted by Hoerner (1939), lift force on a rectangular wing of low aspect ratio (that is, ae« 1 to 5) decreases roughly in proportion to the cosine of the yaw angle. Therefore, each element along the span was treated as an airfoil at a yaw angle of H / p | a n e in University of British Columbia Peter M. Ostafichuk April 2004 Chapter 4 - Control Surface Hydrodynamics - Modelling 117 a freestream flow with velocity f/ p l a n e. For a given point, p, described by the vector, r, the net induced velocity on the plane span is approximated by v'induced^) * 3 . 2 1 7 „ s i n v | / ( £ - z v ) r s £ (70) In where the induced velocity is directed on the j-axis towards the body. Complete details regarding the assumptions and computations of the hull vortex effect on the bowplanes can be found in Appendix A.4. Figure 4-12 Description of flow velocity components on the bowplane in yaw Using the above model, the ratio of force on the leeward bowplane to a bowplane at zero yaw was computed. Since the two hull vortices were shown not to induce a change in the local angle of attack of the bowplane—the vortices are symmetric about the hull centreline—the change in bow- plane force was computed from the local increase in velocity on the plane modified by the yaw correction of Hoemer given above: 2 ' bp — — « average Z h » (C/2lane(»C0SV l a n e ^ V (71) Expressed in terms of the ratio of bowplane lift curve slope for a given yaw angle to that at zero yaw angle, the estimated yaw influence is presented in Figure 4-13. For most yaw angles there is surprisingly good agreement between the estimate developed above and experimental data (repro- duced from Figure 4-10). At large yaw angles, the estimate does not fully match the observations; this is to be expected given the assumptions used and the simplicity in the model. Nevertheless, the purpose of the model above was not as a predictive tool but rather to verify whether the vorti- ces from the hull could be responsible for the increase in bowplane lift curve slope. The results strongly suggest that the hull vortices indeed may be responsible for the increased force measured on the leeward bowplanes in yaw. University of British Columbia Peter M. Ostafichuk April 2004 Chapter 4 - Control Surface Hydrodynamics - Modelling 118 1.3 H 1-2 H Z bpb V = 0 1 0.8 0.7 • Experimental Data Estimated Yaw Influence Yaw Correction (Hoerner) -25 -20 -15 -10 -5 0 5 Yaw Angle [deg] 10 15 25 Figure 4-13 Analytical prediction of port bowplane lift curve slope dependence on yaw angle In terms of the windward bowplanes, the yaw angle correction from Hoerner (1939) was applied and is shown in Figure 4-13 (for positive yaw angles). Given the poor comparison to experimental data, the orientation of the windward planes to the freestream flow alone is not enough to account for the decrease in the performance of those planes. University of British Columbia Peter M. Ostafichuk April 2004 Chapter 4 - Control Surface Hydrodynamics - Modelling 119 4.6 CONTROL SURFACE INTERACTION MODELLING To characterize the interaction between bowplanes and sternplanes, a potential flow model was developed and the results were compared with the experimentally measured interaction. For the purpose of simulation, empirical relations regarding the dependence of the bowplane-sternplane interaction on body orientation were determined from experimental data. 4.6.1 Potential Flow Model of Control Surface Interaction A simple numerical model was used to predict the interaction between fore and aft planes on a cylindrical body. Planes were modelled with a system of bound and trailing vortices as shown in Figure 4-14. Vortex strength, T , was predicted using two-dimensional lifting line theory (see Talay, 1975, or Jacob, 1995, for example). With the experimentally determined bowplane force, CL, the vortex strength was approximated by As an aside, this method of approximating the tip vortex strength neglects the span-wise variation in circulation predicted on a finite aspect ratio wing. Specifically, Prandtl proposed an elliptical distribution for the circulation (see von Mises, 1959) such that where r0 is the maximum circulation at the wing centreline. Since the circulation is approximated from the measured lift on the planes and since all bound vorticity is assumed to roll-up into the tip vortex, the same prediction of tip vortex strength is obtained regardless of whether constant or dis- tributed circulation is assumed. Returning to the potential flow model, as shown in Figure 4-15, image vortices were added to sat- isfy wall boundary conditions of the body (excluding the control surface). To predict the position of a trailing vortex at the longitudinal location of the sternplanes, the velocity components induced by the opposite trailing vortex and the image vortices were added vectorially to the freestream. The trailing vortex was assumed to follow a linear path in the direction of this resultant velocity vector (directed at an angle of y to the freestream as shown in Figure 4-14). To satisfy the wall boundary conditions of the tail, image vortices were added and the resulting induced flow was computed from the freestream and the trailing and image vortices (see Figure 4-16). The local incident -flow velocity to the aft planes, along with the previously measured lift curve slope, was used to compute the interaction force on the aft planes. Complete details in the development of the potential flow model can be found in Appendix A.5. A comparison of results from the numerical interaction model and experimental data is provided below. A representative set of cases has been chosen including the examination of the influence of plane geometry, plane location, Reynolds number, and trim angle. r = (72) (73) University of British Columbia Peter M. Ostafichuk April 2004 Chapter 4 - Control Surface Hydrodynamics - Modelling 120 Ml Figure 4-14 Vortex system used in control surface interaction calculations Plane Geometry Effects A comparison of results from the numerical model and experiments showing the effect of plane geometry is given in Figure 4-17. Results are shown for variations about a baseline configuration. That is, in each case either the fore or the aft plane geometry, but not both, is changed from the standard configuration. The changes to geometry include short and long bowplanes and stern- planes and large bowplanes that roughly maintain the same aspect ratio of the standard planes. The geometric specifications of each of the planes can be found in Table 3-2 on page 45. The agreement for each of the cases considered is very good, particularly considering the simplifi- cations and assumptions made in the potential flow model. The predicted interaction was least accurate for the short bowplanes; the validity of the line vortex model in this case is compromised as the bowplane tip vortex would be expected to burst on the stemplane—the bowplane tip lies inboard of the stemplane tip in this case. For the other cases, the agreement between experiment and prediction is exceptionally good for the portions of the curves between stall. University of British Columbia Peter M. Ostafichuk April 2004 Chapter 4 - Control Surface Hydrodynamics - Modelling 121 Figure 4-15 Vortex system for potential flow model of control surface interaction z t Figure 4-16 Starboard side vortex system at aft plane; port side vortices also contribute to induced velocity but are not shown Plane Location Effects The interaction resulting from various fore and aft plane locations on the body was studied. Figure 4-18 shows results for both the experimental and numerical methods. Included in the figure are bowplanes located along the centreline and high positions as well as both anhedral and dihe- dral bowplanes and sternplanes. Detailed specifications of each of the plane locations were pro- vided in Table 3-3 on page 46. The baseline (standard) configuration is shown in Figure 4-18 (a). Note the change in the vertical axis scale for cases (c) to (f) in Figure 4-18. Once again, the agree- ment between the experimental data and the computational data is exceptionally good. University of British Columbia Peter M. Ostafichuk April 2004 Chapter 4 - Control Surface Hydrodynamics - Modelling 122 (e) (f) Figure 4-17 Plane geometry effect on fore-aft plane interaction comparison for: (a) standard configuration; (b) short bowplanes; (c) long bowplanes; (d) short sternplanes; (e) long sternplanes; (f) large sternplanes. Measured data with error bounds are shown with points, values from the numerical prediction are shown with solid lines. University of British Columbia Peter M. Ostafichuk April 2004 Chapter 4 - Control Surface Hydrodynamics - Modelling 123 (C) 0.2 0.1 -0.1 -0.2 "1 1 -20 -10 0 5bP [deg] 10 20 0.2 0.1 ^ 0 x N -0.1 -0.2 -20 (d) -10 0 Sbp [deg] 10 20 (e) (f) Figure 4-18 Plane location effect on fore-aft plane interaction comparison for: (a) centreline bowplanes; (b) high bowplanes; (c) dihedral bowplanes; (d) anhedral bowplanes; (e) Y-tail; (f) Inverted Y-tail. Measured data with error bounds are shown with points, values from the numerical prediction are shown with solid lines. University of British Columbia Peter M. Ostafichuk April 2004 Chapter 4 - Control Surface Hydrodynamics - Modelling 124 Reynolds Number Effects The influence of Reynolds number on the interaction between fore and aft planes in the baseline configuration is summarized in Figure 4-19. Higher body length-based Reynolds numbers of 7.5 million (Figure 4-19 a) and 4.2 million (Figure 4-19 b) were obtained during towing tank tests at a depth of three hull diameters. The water surface was ignored in the numerical model as justified by the findings of Section 3.4.4 on page 93. Lower Reynolds numbers based on body length, rang- ing from 2.0 million (Figure 4-19 c) to 0.5 million (Figure 4-19 f) were obtained through wind tun- nel testing. At higher Reynolds numbers, the agreement of the computational result to experimental data is excellent, both in the shape and magnitudes of the curves. As the Reynolds number is reduced, the agreement becomes less favorable. The model under predicts the maximum force at a Reynolds number of 1.0 million and under predicts the maximum and minimum forces at a Reynolds num- ber of 0.5 million. This disagreement at lower Reynolds number can partly be attributed to several factors: the reduced bowplane tip vortex strength and coherence at lower speeds; the reduction in stall angle at low Reynolds number; and the increased non-linearity in the bowplane force to deflection curve (which is used as an input to the potential flow model). From an operational point of view, the low Reynolds number data is the least important for an A U V such as DOLPHIN (that operates in the 35 to 50 million Reynolds number range); the numerical results are included to identify limitations in the modelling technique. Body Trim Angle Effects Figure 4-20 shows the change in fore-aft interaction with body trim. Results are shown for trim angles of +8, +4, +2, -2, -4, and -8 degrees for the baseline configuration. The zero trim case can be found in Figure 4-17 (a). Even though the model is very simple and does not take into account the effect of the hull on the freestream flow, the bowplane-sternplane interaction is still well pre- dicted for all trim angles up to +8 and -8 degrees. University of British Columbia Peter M. Ostafichuk April 2004 Chapter 4 - Control Surface Hydrodynamics - Modelling 125 (e) (f) Figure 4-19 Reynolds number effect on fore-aft plane interaction comparison for: (a) Re L = 7.5 million; (b) Re L = 4.2 million; (c) Re L = 2 million; (d) Re L = 1.5 million; (e) Re L =1.0 million; (f) Re L = 0.5 million. Measured data with error bounds are shown with points, values from the numerical prediction are shown with solid lines.: University of British Columbia Peter M. Ostafichuk April 2004 Chapter 4 - Control Surface Hydrodynamics - Modelling 126 (e) (f) Figure 4-20 Body trim angle effect on fore-aft plane interaction comparison for: (a) -8 degree trim; (b) -4 degree trim; (c) -2 degree trim; (d) +2 degree trim; (e) +4 degree trim; (f) +8 degree trim. Measured data with error bounds are shown with points, values from the numerical prediction are shown with solid lines.: University of British Columbia Peter M. Ostafichuk April 2004 Chapter 4 - Control Surface Hydrodynamics - Modelling 127 4.6.2 Modelling Bowplane-Sternplane Interaction in Trim and Yaw The change in bowplane-sternplane with body orientation was modelled using curve fits to empir- ical data. In each case, the slope through the origin of the stemplane force to bowplane deflection was used as the dependent variable. Interaction Dependence on Trim As a function of trim angle, the slope of the interaction force on the sternplanes was considered in Section 3.3.2. In Figure 4-21, the interaction curve slope is shown non-dimensionalized by the corresponding bowplane force to deflection slope. 0.35 T 0.30 - 0.25 - Q. -Q I 0.20 - N 1 0.15 - N 0.10 - 0.05 - 0.00 -- -15 -10 -5 0 5 10 15 Trim Angle Figure 4-21 Interaction curve fit of stemplane to bowplane lift curve slope ratio in trim A polynomial best fit to the data was established using the least squares method. The expression defining the curve is S P 0 D P ( 9 ) ~ 0.28 - 0 .03606|9 - 1 . 5 5 | 1 5 + 0.008715|9 - 1.55| 2 (74) •Z'bp 8bp where 0 is measured in degrees. In general, the polynomial fit shows good agreement with the data; however, the local minima at a trim angle of-9 degrees suggests that the polynomial approx- imation is only suitable for trim angles between approximately -10 and +11.5 degrees. This range is adequate for the work in this thesis since a practical limit for the range of trim angle in operation for the DOLPHIN vehicle is +1-6 degrees (Seto, 1998). University of British Columbia Peter M. Ostafichuk April 2004 Chapter 4 - Control Surface Hydrodynamics - Modelling 128 The 1.55 degrees subtracted from the trim angle in each term in equation (74) is due to the maxi- mum interaction occurring at a trim greater than zero. As was noted previously, this is due to the position of the standard bowplanes below the hull centreline. For comparison, considering only the relative position of the planes in the standard configuration, maximum interaction would be anticipated for a trim angle of 1.1 degrees (in which case the sternplanes are directly downstream of the bowplanes). Interaction Dependence on Yaw As above with trim, the dependence of the bowplane-sternplane interaction on the vehicle yaw was examined. In Section 3.3.3, the slope of the sternplane interaction force versus bowplane angle, normalized by the slope at zero yaw, was presented for a range of yaw angles. A polynomial best fit curve was added to the experimental data and is shown in Figure 4-22. 1.2 1 - 0.8 - o -• -0.2 - .0.4 -I , , , • , 1 , , r= , 1 -25 -20 -15 -10 -5 0 5 10 15 20 25 Yaw Angle [deg] Figure 4-22 Interaction curve fit of sternplane to bowplane lift curve slope ratio in yaw The curve shown in the figure is defined by Z 'spO) a x _ 0.07063|vp|2 + 0.02741lq/|2-5 - 0.00274|q/|3 (75) Z S P | V = O There are local maxima in equation (75) for yaw angles of approximately +/-20 degrees; these val- ues are believed to represent the limit for the local yaw angles to which the bowplanes are sub- jected (Dominguez, 2000). University of British Columbia Peter M. Ostafichuk April 2004 Chapter 4 - Control Surface Hydrodynamics - Modelling 129 4.7 FREE SURFACE INTERACTION MODELLING In Section 3.4, experimental results were presented regarding the effect of waves on the planes. Using simplified analytical models, the form and magnitude of the wave induced forces on the planes are predicted below for varying vehicle depth and speed. Only sinusoidal waves (with a single, well defined frequency and amplitude) are considered; the principle of superposition (see, for example, Tipler, 1991) can be used to reconstruct the complete wave spectrum. 4.7.1 Variation of Wave-Induced Plane Force with Depth Surface waves induce rotational velocity components as a function of depth. The magnitude of the wave orbital velocity is given by U. - afi*.™ ™ where a is the wave amplitude, g is the acceleration due to gravity, X is the wavelength, and d is the depth (Lamb, 1997). In order to predict the form of the wave induced force on the planes of the stationary hull, the max- imum Z force was assumed to result from flow in the +z or -z direction (that is, straight up or down). This was justified through a comparison of the force coefficient in lift and drag. The max- imum measured lift force, expressed in terms of CL, was approximately 0.6 for the standard bow- planes while a flat plate of the proportions of the bowplane (exposed aspect ratio of 1.5) has a maximum drag coefficient, CD, of approximately 1.18 (White, 1986). In other words, throughout one wave cycle the largest positive and negative forces on the plane are expected to result from drag when the angle of incidence is +/-90 degrees to the plane. With a wave induced velocity directed vertically, the cylindrical hull experiences cross-flow and the velocity profile across the plane (shown in Figure 4-23) is predicted using experimental results reported by White, (1991). The maximum wave-induced force on the bowplanes was estimated using the velocity profile described by Figure 4-23 and equation (76) with the force coefficient, CD, given above. For a given plane element with area dS and position y, the force to depth relationship has the form dZwave(d) = C D i p w 2 ( y , d)dS (77) Integrating this expression, with reference to equation (76) and Figure 4-23, yields University of British Columbia Peter M. Ostafichuk April 2004 Chapter 4 - Control Surface Hydrodynamics - Modelling 130 calm water free surface b Figure 4-23 Modelled flow over plane Finally, collecting all the parameters not relating to wave geometry into the variable, C0, the wave force dependence on wave parameters can be expressed as Zwave = C 0 ^ a 2 i ~ (79) In other words, the force is predicted to increase linearly with the square of wave amplitude (con- trary to what was observed in Figure 3-55 on page 93) and decay exponentially with d/k. The difference in the predicted behavior with wave amplitude may be a result of interference with the hull as well as a consequence of the limited range of amplitudes considered in testing. The amplitude of the wave induced force was computed for conditions corresponding to experi- ments; specifically, a wave amplitude of a = 0.043/ and a wavelength of X = 1.14/ were used. The computed wave-induced velocity ranged from approximately 0.5 m/s (at a depth of 0.5 hull diameters) to 0.07 m/s (at 3.5 hull diameters depth). Over this range of depths, the estimated force was computed using equation (78) and was non-dimensionalized according to 7' = -z g i (80) The resulting predicted wave force on the bowplanes is compared to experimental results in Figure 4-24. Except for the shallowest depth, the predicted wave influence on the bowplanes shows good agreement with the experimental data. For the d/D = 0.5 depth, hull immersion was observed and, in general, diffraction forces are very significant for small values of d/D. The gen- eral form of the predicted wave force to depth curve closely matches that from experiments. The methods outlined above appear suitable for the prediction of wave induced forces on underwater vehicle control surfaces. University of British Columbia Peter M. Ostafichuk April 2004 Chapter 4 - Control Surface Hydrodynamics - Modelling 131 d/D Figure 4-24 Analytical prediction of force on bowplanes as a function of depth 4.7.2 Variation of Wave-Induced Plane Force with Forward Speed With the same formulation as above, the effect of vehicle forward speed on the wave-induced force was computed. In this case, the largest velocity component was along the vehicle axis due to forward motion. The wave orbital velocity was vectorially combined with the forward speed and the resulting incidence angle was used with the findings of Section 4.3 to estimate the force on the bowplane. The largest force was found when the wave induced velocity was directed orthogonally to the incoming flow, creating the maximum incidence angle (see Figure 4-25). Figure 4-25 Modelled flow over plane University of British Columbia Peter M. Ostafichuk April 2004 Chapter 4 - Control Surface Hydrodynamics - Modelling 132 For a wave amplitude of a = 0.016/ and a wavelength of X = 1.14/, the wave induced velocity ranged from 0.018 m/s to 0.066 m/s for depths of 3.5 and 0.5 hull diameters respectively. Forward speeds of 0.5 to 3.5 m/s were considered in 0.5 m/s increments; these speeds corresponded to Froude numbers based on vehicle length of approximately 0.1 to 0.75. The resulting force on the bowplanes, non-dimensionalized in the standard form to Z, is shown in Figure 4-26 against the experimental data from Section 3.4.5. 1 -r 0.9 - 0.8 - 0.7 - o 0.6 - o X 0.5 -a. & N 0.4 - 0.3 - 0.2 - 0.1 - 0 - — Predicted Value • Experimental Data 0.1 0.2 0.3 0.4 0.5 Froude Number 0.6 0.7 0.8 Figure 4-26 Analytical prediction of wave-induced bowplane force as a function of speed Overall, the agreement between the predicted and measured oscillatory force on the bowplanes is good. With the exception of the point at maximum Froude number and 1.5 hull diameter depth, the predicted force lies within or very close to the error bands of the experimental data. Further- more, the variation in wave induced force with depth is also accurately predicted as shown by the data at 3.5 hull diameters. University of British Columbia Peter M. Ostafichuk April 2004 Chapter 4 - Control Surface Hydrodynamics - Modelling 133 4.8 IMPLEMENTATION For the purposes of control and simulation, the implementation of the above hydrodynamic model- ling is summarized below. The hydrodynamic forces on the planes were computed from the flow incidence angle and the vehicle speed and orientation. Specifically, the flow incidence angle, a , was used with the sec- ond-order hydrodynamic derivatives to establish the force on single control surfaces in straight flight. For example, the force computed for a generic plane is given by 27(6,0) = Z ' 8 a + Z , 5 5 a | a | (81) where a is the effective incidence angle on the plane due to deflection, vehicle motion, wave induced velocity, and the trim effects (outlined in Section 4.4.1). In reference to equation (41) on page 40, a is given by A - 8 TJ —  LOCAL where a l o c a l is the local change in flow angle due to the hull. For the bowplanes, a l o c a l , is the effect of trim described by equation (63) on page 110 and the resulting flow incidence angle is w-ww-qxhn a b p = o+\.5 W-U^R (83) For the sternplanes, local hull effects were not observed (see equation (50) on page 67) thus giving alocal = 0 a n d w-w w-qx asP = 8 + fj (84) Based on the vehicle speed, U, the Reynolds number was calculated and the stall angle computed from equation (56) on page 102 astall[deg] = 6.221 InRe- 78.24 (Re> 1 x 106) (85) This was compared to the stemplane incidence angle; if the stemplane incidence angle given by equation (84) exceeded the stall angle (in equation (85)) then the stemplane force was computed in equation (81) using the stall angle. The procedure was the same for the bowplanes except the stall angle from equation (85) was scaled by the yaw stall scaling factor (h(y) from equation (68)) prior to comparison with the incidence angle of equation (83). To illustrate the minimal error introduced in this approximation, the actual and modelled forces are shown for the worst case with a windward bowplane in Figure 4-27. The local yaw angle consid- ered is 20 degrees and the speed is 3 m/s (this represents 50% of the typical operating speed in the simulations). Over the range of maximum physical plane deflections, the error in the modelled force remains less than 5% of the force range. For comparison, in the simulations presented in University of British Columbia Peter M. Ostafichuk April 2004 Chapter 4 - Control Surface Hydrodynamics - Modelling 134 Chapter 6, the speed remained above 4.7 m/s and local yaw remained below 4 degrees under all cases; for these conditions, the minimum predicted stall angle is 29 degrees, well above the physi- cal plane deflection limit of 25 degrees. 3.0 o o o 2.0 i 1.0 S 0.0 N -1.0 -2.0 H -3.0 - Actual Force • Modelled Force maximum physical deflection -30 -20 -10 0 10 Plane Deflection [deg] 20 30 Figure 4-27 Comparison of modelled and actual stall behavior at 3 m/s and 20 degrees yaw The force from equation (81) was then adjusted to account for the yaw influence the bowplanes and the bowplane influence on the sternplanes. The bowplane force was scaled according to Section 4.5.1; specifically the bowplane force was scaled by the yaw scaling factor, f(\y), first given in equation (69) on page 115 / ( » = l-0.220v)/- 4.295 V)/3 The force on the port and starboard bowplanes respectively was (86) (87) (88) The force on the sternplanes was augmented by the interaction force due to the bowplanes. The interaction was characterized by the factor, g(8, u/), which was a function of trim and yaw. Specif- ically, g(0, v)/) was the product of equation (74) and equation (75) g(6,v|/) = (0.28-0.03606|9-1.55| 1 5+ O.OO8715|0-1.55|2) x (1 -0.07063|vp|2 + 0.027411 2 5 - 0.00274|vy|3) (89) University of British Columbia Peter M. Ostafichuk April 2004 Chapter 4 - Control Surface Hydrodynamics - Modelling 135 The resulting force on the port and starboard sternplanes respectively was ^ 9 ^ ) = ^ e ' 0 ) L b o w P , a n e + ^ ^ Finally, the force for each plane was re-dimensionalized by multiplying by the dynamic pressure, ^ p U2, and the vehicle length, / , squared, to give Z = Z^pU2!2 University of British Columbia Peter M. Ostafichuk April 2004 Chapter 4 - Control Surface Hydrodynamics - Modelling 136 4.9 SUMMARY In this chapter, various analytical and numerical techniques were used to validate, predict, and mathematically describe the experimental findings from Chapter 3. The topics considered included: Reynolds number scaling effects, control surface performance prediction, the influence of trim and yaw on bowplanes, bowplane-sternplane interaction, and the interaction from the free surface. The key outcomes of the control surface modelling are outlined below. Reynolds Number Scaling Effects • the stall angle was found to increase with the logarithm of Reynolds number • several methods for scaling plane force to deflection curves with Reynolds number were found to give similar results Control Surface Performance Prediction • the semi-empirical control surface performance prediction techniques of Aucher were found to agree closely with experimental data for an isolated appendage. Similar methods of Whicker and Fehlner showed a slight over prediction but generally reasonable agreement with the measured force • when the hull corrections were applied to the isolated appendage predictions, both Aucher and Whicker and Fehlner yielded estimates that closely matched experimental data for bow- planes Trim and Yaw Influence • for both trim and yaw, simple analytical models were able to predict the change in bowplane performance with changing vehicle orientation • in yaw, polynomial curve fitting was used to describe the change in stall angle and lift curve slope of the bowplanes Bowplane-Sternplane Interaction • the interaction between the bowplanes and sternplanes was accurately described using a sim- ple potential flow model • the agreement between the potential flow model and experimental data was exceptional for changing plane geometry and location as well as changing Reynolds number and trim angle Free surface Interaction • the influence of depth and forward speed on the wave induced force on the planes was exam- ined with a simple analytical model • the magnitude and form of the wave influence predicted from the model matched data from experiments University of British Columbia Peter M. Ostafichuk April 2004 Chapter 5 - Control System Design 137 5 C O N T R O L S Y S T E M D E S I G N A good scientist is a person with original ideas. A good engineer is a person who makes a design that works with as few original ideas as possible. - Freeman Dyson Active control is required to operate the DOLPHIN as it has been shown to be unstable in pitch and heave (Shupe and McGeer, 1987); as noted by Field (2000), this instability is related to the hydrodynamic drag of the vehicle when in motion. In general, the highly nonlinear behavior of autonomous underwater vehicles (AUVs), in addition to the noisy operating environments, places challenges on controller design. Furthermore, with the exception of thrust from the propeller, A U V control forces are only generated when the vehicle is in motion. For these reasons, Song et al. (2002) suggest that from the outset of development, a robust control law must be considered. Recently, through significant effort, advances have been made in the development of controllers suitable for underwater vehicles but only minimal emphasis is placed on the treatment of control surfaces by the controller. Based on the experimental data from Chapter 3 and the modelling relationships developed in Chapter 4, improvements to the design of control systems for under- water vehicles is considered in this chapter. In total six controllers are considered including three variants on a PD controller and three variants on an LQG/LTR controller. In addition to a conventional design with the PD controller, fuzzy- tuned series compensation and gain scheduling implementations are also devel- oped. Likewise, the basic LQG/LTR controller is augmented with sliding mode control and fuzzy-tuned series compensation. This chapter begins with background information on various feedback control strategies including PID, LQG/LTR, adaptive control and fuzzy logic. The imple- mentation of the control system is then discussed including information on state space representation and control system block diagrams. Finally, three PD con- troller variants and three LQG/LTR controller variants are introduced for use in simulation in Chapter 6. University of British Columbia Peter M. Ostafichuk April 2004 Chapter 5 - Control System Design 138 5.1 CONTROL BACKGROUND Various feedback control strategies were considered in this thesis for use with the DOLPHIN A U V These included P I D control, linear quadratic gaussian control with loop transfer recovery (LQG/LTR), various adaptive control techniques, and fuzzy logic. Background information is pro- vided below for each of these methodologies following a discussion of the fundamentals of feed- back control. 5.1.1 Feedback Control The general feedback control system for use with underwater vehicles is shown in block diagram form in Figure 5-1. The controller acts upon the submarine model (or plant) using feedback that has been measured and filtered from the plant output, y. The difference between the reference tra- jectory, r, determined by a navigation module and the measured output gives the error, e. The error is used as an input to the controller from which the controller determines the required control action, « . Physically, the control action represents the commanded plane deflections or propulsor speed given to the plant. Disturbances, Ddist, (from waves, for example) act upon the plant and the measured plant output is also contaminated by noise, N. Navigation Module Controller Submarine Model Sensors and Filters Figure 5-1 Basic feedback control system schematic Specific details regarding the controller block are described beginning in Section 5.3 following background information on control strategy; details regarding the implementation of the remaining blocks for simulation are outlined beginning in Section 6.2.2 on page 173. Two further augmenta- tions to the basic feedback control structure, including feed-forward compensation and series com- pensation, were also considered and are described in Section 5.2. 5.1.2 PID Control P I D (proportional-integral-derivative) control and its variants are by far the most commonly used strategy in automatic control. This is largely due to the ease of implementation and intuitive nature of the control methodology. P I D control is well described in any elementary control text University of British Columbia Peter M. Ostafichuk April 2004 Chapter 5 - Control System Design 139 (see for example, Dutton et al., 1997, Palm, 1986, or Astrom and Wittenmark, 1997). The control action, u(t), is determined from the state error, e(t) according to u(t) = Kpe(i) + Kt \e(t) + K^e(t) (90) where Kp, Ki, and Kd are the gain matrices for the proportional, derivative, and integral compo- nents of the control law. The proportional term produces a control signal that is proportional to the error. Stable response with large steady-state error is associated with low values of Kp. For higher values of Kp, the steady-state performance improves but at the expense of transient performance (overshoot and set- tling time increase). If Kp is too large, system instability can result. With the use of integral action, steady state error can be reduced. The integral term acts like an automatic reset to the controller in the presence of disturbances. The integral action does not take effect instantly but continuously corrects for error over time. The drawback of integral action is a reduction in stability as well as increases in oscillation and settling time. In cases where actuators have limited operating range (that is, they can be saturated), special steps must be taken to avoid integrator wind-up which occurs when the commanded control output exceeds the physical limits of an actuator. Finally, derivative action increases system damping and reduces overshoot and oscillation. Inclu- sion of a derivative term also allows increases in the proportional and integral gains and has no effect on steady-state behavior. When inputs to the controller are noisy, undesirable behavior can result as the controller inputs may have artificially high rates of change. PID control has been applied to AUVs in the past and has been used with D O L P H I N (see Shupe and McGeer, 1987, and Butler, 1990); however, it must be implemented with care for several reasons. The highly nonlinear and coupled nature of underwater vehicles results in sub-optimal and poten- tially unstable behavior with PID controllers. Additionally, tuning of the PID controller gains remains a significant problem and requires extensive knowledge about the vehicle performance characteristics and adequate field testing. Recent efforts, by Pierre and Canudas-de-Wit (1996) for example, have shown success in handling some nonlinear characteristics by adding an outer non- linear control loop to a standard underwater vehicle PID controller; however, tuning of the original PID controller gains, in addition to the new non-linear components, remains difficult. 5.1.3 Linear Quadratic Control with Loop Transfer Recovery L Q G is a model-based form of optimal control which has been applied to underwater vehicles in the past. In general, linear quadratic (LQ) controllers, operate by minimizing a quadratic perfor- mance index, J. 00 J= $(xTQx + uTRu)dt (91) o University of British Columbia Peter M. Ostafichuk April 2004 Chapter 5 - Control System Design 140 where x is the error in the outputs that are controlled and u is the control effort. An attractive fea- ture of L Q controllers is that as long as the performance index is well formed, the resulting LQ controller will be stable (Dutton et al., 1997). The L Q design process does not produce a unique controller since the matrices Q and R can be adjusted to weight the output error or control effort respectively. The controller is determined by solving the algebraic Riccati equation A TP + PA - PBRXBTP + Q = 0 (92) where, as will be discussed in Section 5.2.1, A and B are matrices that represent the system for which the controller is being designed. In general, the Riccati equation is not solvable except for by numerical means (Dutton et al., 1997). The value of P is determined in the solution of equation (92) and is in turn used to determine the controller K = R lBTP (93) The resulting controller, K, is implemented in a feedback control loop according to u = -Kx (94) As opposed to traditional linear quadratic controllers, the L Q G control scheme is designed to oper- ate in a noisy environment and as such uses feedback with state estimates in lieu of actual states. The result is a potential loss in robustness over the original L Q design; loop transfer recovery (LTR) provides a means by which to modify the design in order to regain some of the robustness lost from using state estimates (Tay and Moore, 1991). In the LTR approach, the frequency response of the controller is shaped to achieve desired stability and performance properties. The Kalman filter optimization procedure is used for this loop shaping and the combined estimator and controller are designed simultaneously to achieve the desired bandwidth. Further information regarding LQG/LTR, the reader is referred to Stein and Athans (1987) or Kulcsar (2000). LQG/LTR control was employed by Field for his work on the D O L P H I N A U V (Field, 2000); other researcher that has used this control scheme on underwater vehicles includes (Triantaflyllou and Grosenbaugh, 1991), (Juul, 1994), and (Naeem et al., 2003). Although LQG/LTR is known to be robust for linear systems, the application of this methodology to non-linear systems often leads to unnecessary conservatism in the controller; additionally, the control robustness is valid only in the neighborhood of the operating point around which the sys- tem was linearized (Ghalia, 1997). In the case of an AUV, several new control surface hydrody- namic features identified in Chapter 3 and Chapter 4 depend on body orientation (such as the trim and yaw effects on the bowplanes and stall angle dependence on Reynolds number). Since the most suitable operating point for the vehicle is straight and level flight, an LQG/LTR controller designed for this case will not take into account effects that only become apparent at non-zero trim and yaw angles. 5.1.4 Adaptive Control Techniques The control strategies described thus far are based on a stationary (unchanging) submarine plant model. Adaptive control accounts for either changes or uncertainty in plant behavior by adjusting control parameters during operation. In practice, many parameters describing the submarine University of British Columbia Peter M. Ostafichuk April 2004 Chapter 5 - Control System Design 141 dynamics are functions o f one or more operating conditions, such as the trim and yaw effects on the bowplanes, as mentioned above. Model l ing errors, and hence sub-optimal performance, are introduced by failing to allow changes in control system parameters to account for physical changes in the plant. Wi th adaptive control, it is possible to operate over a wider range o f condi- tions without sacrificing control optimality. Many forms of adaptive control exist, such as self tuning control, learning control, and model-ref- erenced control, to name a few. Sliding mode control and switching control, in which the control- ler shifts between different modes depending on operating conditions, is a form of adaptive control. Sliding mode control increases robustness in the presence o f nonlinearity, model uncer- tainty, disturbances, and parameter variation (de Silva, 1995) and has been shown to be effective for the control o f A U V s in the dive plane (Cristi et al., 1990). In reference to the transformation of existing Navy platforms, Cancilliere (2001) recommends a system that uses an adaptive nonlinear controller for unmanned underwater vehicles; nonlinear sliding mode control methodology, aug- mented with real-time model adaptation through neural networks, is used to maintain robust con- trol of a vehicle with a single axial propulsor for velocity control and four small control fins for attitude control. Gain scheduling techniques are similar to sliding mode control except that the measured changes in operating condition are used to adjust the gain or other parameters in a controller rather than to switch between different controllers. Nonetheless, switching and gain scheduling should produce similar results because the error, controller gains, and controller output should be the same within a given set-point for similar controllers; it is only during sudden transition that the two methods differ (Field et al., 2001). Gain scheduling is widely used in flight control systems and is most applicable to situations where the plant parameters change due to changes in plant load (Dutton et al., 1997). A n attractive feature of gain scheduling is that it allows the application o f proven linear design methods to nonlinear problems. In traditional gain scheduling techniques, a series of controllers is designed for the various local equilibrium points and then they are combined. However, this limits operation to the near equilibrium conditions and the dynamic characteristics of a controller with gain-scheduling can be strongly dependent on the manner in which the local (equilibrium) control- lers are combined (Leith and Leithead, 1997). Gain scheduling representations based on fuzzy logic and neural network approaches have shown success in addressing this issue (see Leith, 2000, and Lee et al., 2001). 5.1.5 Fuzzy Logic The roots of fuzzy logic can arguably be traced back as far as to the time of Lao-tze and the Budda (Kosko, 1993). In conventional or bivalent logic, quantities are represented in a crisp sense with discontinuous boundaries. In contrast, fuzzy logic uses a more continuous form by expressing quantities in matters o f degree. Conventional logic is akin to black and white while fuzzy logic is analogous to 'shades of grey'. Moreover, while a quantity is strictly true or false in conventional logic, in fuzzy logic that same quantity can simultaneously be both true and false in differing degrees. In control applications, fuzzy logic is particularly suitable in situations where the plant is complex or ill-defined (de Silva, 1995). Fu l l fuzzy logic control has been attempted with underwater vehi- University of British Columbia Peter M. Ostafichuk April 2004 Chapter 5 - Control System Design 142 cles in the past; for example, Lea et al. (1999) compared data from the Subzero //test A U V operat- ing with fuzzy, sliding mode, and classical control schemes. Performance was satisfactory with all three controllers but each had drawbacks. The fuzzy logic controller in particular required exten- sive tuning through simulation and even then performance was not as good as with the sliding mode controller. For pitch and heading, automated design of fuzzy logic controllers for AUVs has shown comparable performance to more conventional methodologies (Song and Smith, 2000a). As noted by de Silva (1995), although fuzzy logic can be implemented independently as a low- level controller, it is better suited to tuning and other high-level tasks. With such a scheme, fuzzy logic operates much like a human supervisor, intelligently adjusting control parameters in the pres- ence of imprecise, general, or uncertain information. For example, Craven et al. (1999) employed a fuzzy inference system for tuning of an A U V autopilot which in turn sends commanded trajecto- ries to a low-level multivariable controller. Research has developed recently in the application of fuzzy logic to adaptive control; particular areas of research include model reference (Banerjee et al., 2001), gain scheduling (Tzafestas et al., 2001), sliding-mode control (Tzafestas and Rigatos, 1999), and sliding-mode control guidance laws (Lin and Hsu, 2000). By implementing the same controller on different types of vehicles without a significant change in performance, Song and Smith, (2000b) demonstrated robustness benefits of a fuzzy logic sliding mode controller. University of British Columbia Peter M. Ostafichuk April 2004 Chapter 5 - Control System Design 143 5.2 CONTROL SYSTEM IMPLEMENTATION The state space representation used for modelling and simulation in this thesis is introduced below. Additionally, a description of the basic feedback control system design is provided along with information on feed-forward compensation and series compensation augmentations that were uti- lized. 5.2.1 State Space Representation In state space, the submarine model is represented by the following equations x = Ax + Bu + Mw (95) y = Cx + Du + In (96) where x is the vehicle state vector x = \ u v w p q r x y z § Q \ \ > \ > (97) u is the controller output vector given by the commanded plane deflections, rudder deflection, and propeller speed « = [KP 5fi KP Ks K «]T W and y is the vehicle output y = [ « z | 8 j r (") The construction of the plant matrix, A , for a six degree-of-freedom submarine is complicated; it is easiest to visualize the 12 x 12 matrix as the union of four 6 x 6 sub-matrices as shown in equation (100). A = -(Mrb + MA)~ l (Crb + CA + Dd) -{Mrb + M,)-' G Q o (100) The terms that combine to form the plant matrix are: the rigid body mass matrix, Mrb ; the Coriolis matrix, Crb; the added mass matrix, MA; the Coriolis added mass matrix, CA; the damping matrix, Dd; and the rotational transformation matrix, Q. Further information regarding the com- ponents that from the plant matrix is provided in Appendix D. 1 and complete details are provided by Field (2000). University of British Columbia Peter M. Ostafichuk April 2004 Chapter 5 - Control System Design 144 The input matrix, B, is a linear description of the forces generated by the control surfaces and pro- peller for a given deflection or engine speed. Specifically, B = XSfp Xhfs X5ap XSas X&r X n Yhfp Y5fs Y&ap Y&as Y5r Yn Z6fp Z5fs Zhap Zhas Zhr Zn K5fp K5fs KSap K&as KSr K n MhfP Mhfs M&ap M8as M&r Mn NhfP N5fs N5ap NSas N5r Nn_ (101) Examining the relationship between vehicle state and the output, the remaining matrices in this implementation are C = 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 (102) and D is the zero matrix or D = 0-1 The process noise levels are represented by the matrix M with white noise w of unit intensity and covariance W. The sensor noise, « , is also assumed to be white noise of unit intensity with cova- riance N. 5.2.2 Feed-forward Compensation In some instances, non-zero control output is desirable even when the error signal input to the con- troller is zero. As an example with DOLPHIN, there is a non-zero pitching moment that changes with speed and depth as a result of the vehicle asymmetry about the horizontal plane; likewise, there is a hydrostatic pitching moment that changes as fuel is consumed and replaced by seawater ballast. Rather than allow these effects to act as unmodelled disturbances, the control signals nec- essary to maintain level trim can be automatically added to the controller output. These feed-for- ward control signals, are determined using a feed-forward compensator as shown in Figure 5-2. For this thesis, the feed-forward control is computed based on the initial conditions and thereafter remains constant; it is included in all further controller augmentations whether stated explicitly or not. University of British Columbia Peter M. Ostafichuk April 2004 Chapter 5 - Control System Design 145 5.2.3 Series compensation It is not always possible to adjust controller gains in order to satisfy all performance requirements; in such cases, one option is to add a compensator to the system to alter the original controller response. In addition to the feed-forward method discussed above, there are several common forms in which compensation is implemented including series, parallel, and disturbance compen- sation (see, for example, Palm, 1995). In this thesis, several controllers are augmented with series compensation (shown in Figure 5-3). As an input, the compensator takes the commanded plane deflections from the controller and adjusts these based on operating conditions. The form of the compensator is tuned during operation (discussed in detail in sections Section 5.3.3 and Section 5.4.3). Feed Forward Compensator Navigation Module -H 2 Controller 1 * Submarine Model Sensors and Filters Figure 5-2 Feed forward compensator Navigation Module Controller Compensator Submarine Model Sensors and Filters X Figure 5-3 Feedback controller with series compensator University of British Columbia Peter M. Ostafichuk April 2004 Chapter 5 - Control System Design 146 5.3 PD CONTROLLERS A simple P D controller was developed based on previous work on the DOLPHIN. Two augmenta- tions to the basic controller methodology were also considered in an attempt to better compensate for the newly identified control surface hydrodynamic effects outlined in Chapter 3 and Chapter 4. In one instance, gain scheduling is used to change the controller gains during operation based on the predicted control surface performance. In a second augmentation, a series compensator is added to the controller to adjust the controller outputs again based on predicted control surface performance. 5.3.1 Basic PD A basic PD controller was developed based on a design reported by Shupe and McGeer (1987). The form of the controller is shown in Figure 5-4 where the controller is represented by the 6 x 12 matrix, K. In state space, the control signals are generated according to u = -Ke In expanded form this becomes (103) 0 0 0.25 0 0 0 0 0 0.125 0 0 0 0 0 0.25 0 0 0 0 0 0.125 0 0 0 0 0 0 - 4 4 0 0 0 0 -3.6 3.57 0 0 0 0 4 4 0 0 0 0 3.6 3.57 0 0 0 0 0 0 4 0 0 0 0 0 4 1 0 0 0 0 0 0 0 0 0 0 0 (104) Depth control is provided exclusively by the bowplanes; port and starboard planes are coupled (that is, deflected by the same amount in the same direction as opposed to being controlled differ- entially). Depth control is based on the depth error, ez, as well as depth rate error, ew (these repre- sent the proportional and derivative elements of the PD control respectively). Roll and pitch control authority are assigned to the sternplanes. In this case the port and starboard planes are cou- pled in pitch control and have equal and opposite commanded deflections in roll control. For the roll and pitch errors, the sternplanes operate with a proportional component (based on and e e ) and a derivative component (based on ep and e ). The rudder is solely responsible for heading (yaw) control based on the heading error, , and the heading rate error, er. Finally, the propeller University of British Columbia Peter M. Ostafichuk April 2004 Chapter 5 - Control System Design 147 speed, n, is controlled proportionally, based on the error in surge, eu. Evident by the columns in K with only zero entries, there is no control of the error in v, x, or y. •4 I e K u Submarine w w Model Sensors and Filters <7> Figure 5-4 Basic PD controller block diagram The controller gains, K, were tuned through simulation (Chapter 6) using initial values provided by Shupe and McGeer for the Mark I DOLPHIN. In the simulation, controller gains were iteratively modified using an evaluation manoeuvre until the gain matrix with the best performance was determined. This is not unlike the procedure used in the initial development of DOLPHIN where controller parameters were tuned in the field (den Hertog, 1997). More systematic methods of tun- ing PID controllers exist—the most common of these tuning methods being that of Ziegler- Nichols (1941) (see also, Dutton et al., 1997, Palm, 1986, or Astrom and Wittenmark, 1997)— however, due to the strong coupling in the plant equations, it was not possible to implement this type of tuning in this case. 5.3.2 PD Control With Gain Scheduling Gain scheduling was incorporated in one implementation of the PD controller to account for change in bowplane force with body orientation. In this case, a crisp (that is, non-fuzzy) tuner was used to alter the values of the gain matrix, Kgs, during operation. As shown in Figure 5-5, the tuner used the measured vehicle state as input from which it adjusted the controller accordingly. Adjusted Bowplane Deflection Based on the commanded port bowplane deflection, 5y , the approximate commanded port bow- plane force (to the first order) is: Z f p = Z b f p b f p -/(0) = Z h f p b f p (105) Where /(0) is the yaw scaling factor introduced in Section 4.5; in this case it is applied for the ini- tial conditions with a yaw angle of zero, in which f(0) is equal to unity. However, the first order University of British Columbia Peter M. Ostafichuk April 2004 Chapter 5 - Control System Design 148 Tuner Navigation Module 1 e U Submarine w Model Sensors and Filters Figure 5-5 Gain scheduling PD controller schematic approximation of the actual force, Zfpta) (for which the yaw angle may not be zero) includes the yaw scaling factor, /(vu), giving: Zfpia) = ZbfpSMayAv) (106) As the actual force should be equal to the commanded force, equation (105) and equation (106) can be equated to give the desired plane deflection, : (107) fp(a) - / ( v | / ) Similarly, for the starboard bowplane, the desired plane deflection is: 5fs{a)~A-v) Adjusted Sternplane Deflection For the port sternplane, the force based on commanded deflection is: Z ap = Zsap&ap • force includes the bowplane-sternplane interaction factor, g(9, vy): (108) (109) The actual port sternplane Zap(a) = Zhap§ap'a) + g(9, HO ' Zfpfa) Substituting the port bowplane force from equation (106) gives Zap(a) = Z5ap8ap(a) + g(Q, v|/) • Z¥p8fp(a) -/(V|/) (110) (111) University of British Columbia Peter M. Ostafichuk April 2004 Chapter 5 - Control System Design 149 Again, the actual force should be equal to the commanded force and therefore equation (109) and equation (111) can be equated to give and the desired port sternplane deflection is 5 f l p ( f l ) s8 e / , -g(e,i | /) .y( V ) .^-8 ' Sap (112) (113) Likewise, the desired starboard sternplane deflection is as(a) — as Sfs •5 ' has 'Ma) (114) Changes to Controller Gain Matrix The desired plane deflections can be expressed in a relationship to the commanded deflections as: where K,a^ is based on the above empirically-derived relationships giving: (115) K(a) - 1 Aw) o o o 0 1 ' Sap -g(e,y)/(-v(y)-^ 0 ' Sap 0 0 (116) In the Basic PD control implementation, the control output was determined according to u = -Ke. To determine the desired control action, « ( a ) , that accounts for the new control surface hydrody- namics, u must be pre-multiplied by Kld) (as given in equation (115)). The new control law is therefore described by "(a) = ' K ( a ) K e = ~ K g s e (117) University of British Columbia Peter M. Ostafichuk April 2004 Chapter 5 - Control System Design 150 where JRT is the new gain matrix for use with gain scheduling and is given by K. 0 0 0 0 0.25 Xv) 0.25 / ( - V ) 0 0 0 0 0 0 0 0 0 0 0.125 0.125 fiv) 0 0 0 0 0 0 0 0 -g(Q, v|/)/(v|/)-^0.25 -4 4 0 0 0 -g(Q, VJ/>/TH»)—Ŝ O. 125 -3.6 3.57 0 'S/i '5/s 'S/i 0 0 -^(6, -u/)yr-u/)—^0.25 4 4 0 0 0 -g(9,-v|/)/(-v|/)-^0.125 3.6 3.57 0 '5/s '5/s 0 0 0 0 4 0 0 0 0 0 0 0 0 0 4 0 0 0 (118) Implementation The gain scheduling matrix, K G S , and the tuner are implemented as shown Figure 5-5. The tuner design is shown in Figure 5-6. Demultiplexer u,w,q Compute Local Yaw {y) Compute Local Trim (0) 9(e,v|/) Reassign K Figure 5-6 Tuner schematic used in gain scheduling PD controller The measured state, x, is separated into the elements of u, v, w, q, and r and then the local trim and yaw at the bowplanes are computed. The trim, 9, is given by 9 = atan w u J (119) where xb is the ^-coordinate of the bowplanes measured from the centre of gravity. Likewise, the local yaw is given by \\i - atan v + x "fp (120) University of British Columbia Peter M. Ostafichuk April 2004 Chapter 5 - Control System Design 151 With the local values of 9 and \\i, the yaw scaling effect on the bowplanes, f(\\i), and the bow- plane-sternplane interaction factor, g(9, are computed. These values are used to recompute the gain matrix, K , as given in equation (114); the value of K is then updated to the controller. 5.3.3 PD Control With Fuzzy-Tuned Series Compensation A second method of correcting for the control surface performance characteristics outlined in Chapter 4 was devised using a series compensator and fuzzy logic. In principle, the design of series compensation augmenting was similar to that of the gain scheduling; commanded outputs from the controller were adjusted to account for the known control surface hydrodynamic effects. The block diagram for the control system with the fuzzy-tuned series compensator is shown in Figure 5-7. Fuzzy Tuner Navigation Module s Controller 1 1 D 1 1 • 1 Compensator % Submarine (K,c) w Model Sensors and Filters 2 H- Figure 5-7 Fuzzy-tuned series compensator block diagram Fuzzy Tuner Layout The schematic of the fuzzy tuner is shown in Figure 5-8. Similar to the gain scheduling controller of Section 5.3.2, the local yaw and trim is computed based on the measured states. These two val- ues are used as inputs to three fuzzy logic input-output modules. One module accounts for the change in port bowplane force with yaw angle (kj- ), a second accounts for the change in starboard bowplane force with yaw angle (k^s), and the final module accounts for the change in bowplane- sternplane interaction as a function of trim and yaw (ka). University of British Columbia Peter M. Ostafichuk April 2004 Chapter 5 - Control System Design 152 Demultiplexer u,w,q Compute Local Yaw (y) Compute Local Trim (9) Fuzzy Control (port BP) Fuzzy Control (starboard BP) Fuzzy Control (BP-SP interaction) Assigned to Figure 5-8 Fuzzy-tuner schematic The output from the fuzzy modules is assigned to the series compensation matrix, Kjc shown in Figure 5-7 and given by: Kfc = kfp 0 0 0 0 0 0 kfi 0 0 0 0 K 0 1 0 0 0 0 K 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 (121) Determination of Tuning Parameters The tuning parameters kj- , kjs, and ka were computed using the local yaw and trim as inputs for functions developed off-line. The functions were developed using fuzzy logic and were based on the control surface behavior determined in Chapter 3 and Chapter 4. The deflection for the port bowplane is increased by a factor kj- to account for changes in the per- formance of that plane in yaw. The fuzzy rule-base used in determining the value of kj is given in Table 5-2. The three conditions of the yaw input refer to positive yaw ( P Y ) , zero yaw ( Z Y ) , and negative yaw ( N Y ) ; three conditions of the kj output refer to negative kjp ( N O ) , zero kjp ( Z O ) , and positive kj- ( P O ) . In short, if the yaw is zero then there is no correction; for positive yaw kjp is increased; and for negative yaw kjp is decreased. The fuzzy logic calculation methods used are summarized in Table 5-1. The membership functions of the yaw input and kjp output are shown in Figure 5-9 and Figure 5-10 respectively. After defuzzification, the resulting fuzzy control surface—as used in the conventions of fuzzy logic and not to be confused with the planes on the vehicle—is given in Figure 5-11. University of British Columbia Peter M. Ostafichuk April 2004 Chapter 5 - Control System Design 153 Table 5-1 Fuzzy logic calculation methods Method Type And method min Or method max Implication min Aggregation max Defuzzification centroid Table 5-2 Fuzzy rule-base for k Rule Input Case Output Case 1 i f Yaw is P Y then kfP is PO 2 if Yaw is Z Y then kfP is zo 3 if Yaw is N Z then kfP is NO a 0.8 !E 12 Q) . Q E „ „ a 0.6 S o a> £ g> 0.4 a 0.2 YN YZ YP -% - N \ / \ / \ / \ / \ /' \ / \ / \ / \ /' \ / \ / \ / \ / \ /' \ / \ / \ / \ / / \ / \ / \ / \ / \ / \ S \ / \ / \ / s / s / \ ^ / V / s • s — —. — • — r — — — —-i 1 -0.35 -0.25 -0.15 -0.05 0.05 0.15 0.25 0.35 Yaw [rad] Figure 5-9 Input (yaw) membership function for kf and & University of British Columbia Peter M. Ostafichuk April 2004 Chapter 5 - Control System Design 154 Figure 5-10 Output membership function for kj- and kj-s 1.4 1.3 1.2 1.1 0.9 0.8 0.7 -0.35 -0.25 -0.15 -0.05 0.05 0.15 0.25 0.35 Yaw [rad] Figure 5-11 Fuzzy logic control surface for kj-p University of British Columbia Peter M. Ostafichuk April 2004 Chapter 5 - Control System Design 155 For the starboard bowplane, the membership functions for kfs are the same as those for kfp above. The only difference is in the rule-bases (shown in Table 5-3) which results in a control surface for kjs that is a mirror image of that for kf (see Figure 5-12). Table 5-3 Fuzzy rule-base for k Rule Input Case Output Case 1 if Yaw is YP then kfs is NO 2 if Yaw is YZ then kfs is zo 3 if Yaw is YN then kfs is PO 1.4 0.7 J , , 1 1 , , , 1 -0.35 -0.25 -0.15 -0.05 0.05 0.15 0.25 0.35 Yaw [rad] Figure 5-12 Fuzzy logic control surface for kp For the bowplane-sternplane interaction (corrected using ka) both yaw and trim are used as inputs in generating the fuzzy control surface. The rule-base for this case is given in Table 5-4 where the output, ka, is now either small and positive (SPO), zero (ZO), small and negative (SNO), or large and negative (LNO). The correction factor, ka, is largest for straight and level flight where trim and yaw are both zero, there is no correction when both trim and yaw are large, and there are small corrections otherwise. The membership functions for the yaw input, trim input, and ka output are University of British Columbia Peter M. Ostafichuk April 2004 Chapter 5 - Control System Design 156 given in Figure 5-13, Figure 5-14, and Figure 5-15 respectively. The resulting fuzzy logic control surface for ka is given in Figure 5-16. Table 5-4 Fuzzy rule-base for ka Rule Input 1 Case Input 2 Case Output Case 1 if Yaw is YN and if Trim is TN then K is ZO 2 if Yaw is YN and if Trim is TZ then K is SPO 3 if Yaw is YN and if Trim is TP then K is ZO 4 if Yaw is YZ and if Trim is TN then K is SNO 5 if Yaw is YZ and if Trim is TZ then K is LNO 6 if Yaw is YZ and if Trim is TP then K is SNO 7 if Yaw is YP and if Trim is TN then K is ZO 8 if Yaw is YP and if Trim is TZ then K is SPO 9 if Yaw is YP and if Trim is TP then K is ZO University of British Columbia Peter M. Ostafichuk April 2004 Chapter 5 - Control System Design 157 University of British Columbia Peter M. Ostafichuk April 2004 Chapter 5 - Control System Design 158 Yaw Figure 5-16 Fuzzy logic control surface for ka University of British Columbia Peter M. Ostafichuk April 2004 Chapter 5 - Control System Design 159 5.4 LQG/LTR CONTROLLER Three different LQG/LTR controller implementations were developed to examine potential bene- fits to control surface performance. A basic LQG/LTR controller developed by Field (2000) is presented first, followed by a sliding mode LQG/LTR controller and an LQG/LTR controller with fuzzy-tuned series compensation. 5.4.1 Basic LQG/LTR Complete details regarding the development of the LQG/LTR controller was provided by Field (2000); the key steps in the controller development are reproduced below. Controller Layout The block diagram of the LQG/LTR controller used is shown in Figure 5-17. The Kalman filter (shown in the dashed box) has a gain matrix Lin and takes the error, e, scaled by the matrix K0 as input. The errors from the different states have different ranges and different units; by applying the scaling gain K0, the states are artificially adjusted to have similar range. K0 was a square matrix with diagonal elements [2.5 8 4 8 5] • The controller is contained in the gain matrix Kin and is applied to the state estimate output of the Kalman filter, x. Following the design of Field (2000) the controller output, uc, is integrated and then multiplied by a second scaling matrix, Kt. This second scaling matrix is used to adjust the output to the propeller and is diagonal with ele- ments [ 1 1 1 1 1 5 ] - U c s-< W Figure 5-17 LQG/LTR controller block diagram University of British Columbia Peter M. Ostafichuk April 2004 Chapter 5 - Control System Design 160 Both integrators in the controller had saturation limits (that is, they include anti-wind up). In order to facilitate the integrator implementation, a change in the use of state variables was required in the controller design. Specifically, a modified system was considered with revised state equations u 0 0 u + I X B A x 0 y = [o c] (122) (123) where uc is the commanded control signal prior to integration, u is the integrated control signal, and x is the state vector as given in equation (97). The benefit of this approach is that steady-state error due to continuous plane deflection can be eliminated. In the notation of Field, this system can be rewritten with noise, n, and disturbances, w, (both assumed to be white noise in the design process) as Xi. = A i n x i n + B i n u + M i n W yin = Cinxin + n where, (124) (125) u X (126) and B (127) Open Loop Plant Behavior The open loop plant behavior for the five controlled states is described by the singular value plot in Figure 5-18. Of particular concern is the yaw singular value curve that is below unity gain at low frequency; as Field (2000) notes, in operation this would be reflected by large steady state error and poor disturbance rejection properties. Also noted is the differences noted in the magnitudes and crossover frequencies of the curves which is a consequence of the differences in dynamic behavior of the different states. When the input and output scaling gain matrices, K0 and Kt, are included in the computation of the plant singular values, the above concerns are alleviated. As shown in Figure 5-19, the singular values of the scaled plant are much more similar and have similar crossover frequencies. University of British Columbia Peter M. Ostafichuk April 2004 Chapter 5 - Control System Design 161 60 140 '• ' 1 1 l - l - l - l - U - i - L . . L . . L L U i 1 L...l....i....Lli.i X 1 1 L.J...L.U....... i I i .....L.J......U _ 10"2 1 0 " 1 1 0 ° i o 1 1 0 2 1 0 " Frequency (rad/sec) Figure 5-18 Singular value plot for the plant Filter Design With the LQG/LTR controller, both the filter, Lin , and controller, Kin must be designed. Due to the properties of duality in the system, the process is similar for each and either one can be designed first. Following the work of Field (2000), the filter was designed first and the controller design (recovery) was completed once a suitable filter was realized. The filter was determined by solving the algebraic Riccati equation AinP + PAfn-PCfn(vNin)-'CinP + MinWinMTn = 0 (128) where u. is a design scaling factor; increasing u. decreases the bandwidth and decreasing u. increases the bandwidth. Nin is a 5 x 5 identity matrix and Win was determined by Field (2000) through singular value decomposition so as to bring the singular value curves together at cross- over. The procedure of determining Win is outlined in Appendix D.2.1. The filter gains, Lin, are determined from equation (128) by Lin = PCjn(vNin)-\ (129) University of British Columbia Peter M. Ostafichuk April 2004 Chapter 5 - Control System Design 162 150 -200 * X ' ' ' • i J - L i J • i _ l _ J _ l _ L U I I. ( L . J . . L . U J __t i i I..J....U..U i 1 i L..i...i..U,j - 2 - 1 0 1 ? 10 10 10 10 10' 10 Frequency (rad/sec) Figure 5-19 Singular value plot for the scaled plant where P is solved as part of the Riccati equation. The determination of the filter gains is accom- plished with the MATLAB® command L i n = L Q E ( A i n , M i n , C i n , W i n , u N i n ) The resulting singular value plot for the filter based on u. = 0.0005 is shown in Figure 5-20. This value of u. was determined through results presented by Field (2000) and through tuning by simu- lation. (Details regarding the simulation tuning are provided in Appendix D.2.2.) The sensitivity and complementary sensitivity functions for the filter are shown in Figure 5-21. The sensitivity function dictates disturbance rejection and tracking performance of the system and the complementary sensitivity function governs the noise rejection. Ideally both quantities should be small but this is not possible and a trade-off between the two functions is required. Since noise tends to be a high frequency phenomenon, it is desired that the complementary sensitivity function is small at high frequencies and the sensitivity function is small at low frequencies. The resulting sensitivity functions in Figure 5-21 show a reasonable compromise and are similar to those of Field (2000). University of British Columbia Peter M. Ostafichuk April 2004 Chapter 5 - Control System Design 163 140 .0 ' > > i...J...J...i.J..i.J i > > i...i...i..!..u i ' i [-J.-a-i-.i-s i i i i..J...i.J..iJ-.. i ' i-J-J-u.i 1 0 " Z 1 0 " ' 1 0 ° 1 0 1 1 0 ^ 1 0 3 Frequency (rad/sec) Figure 5-20 Filter singular values Frequency (rad/sec) Figure 5-21 Filter sensitivity and complimentary sensitivity singular values University of British Columbia Peter M. Ostafichuk April 2004 Chapter 5 - Control System Design 164 Controller Design Having determined the filter, it remains to determine the controller gain matrix, Kt . A similar procedure was used as for the filter design and is described in detail by Field (2000). The objec- tive was to minimize a cost function of the form: where the matrices Q and R are used to weight the state error and actuator deflections. The parameter p is used to tune the amount of recovery in the control. As p approaches zero, the con- troller approaches full recovery of the Kalman filter; however, small values of p are associated with large gains and thereby introduce a risk of increased actuator saturation. The values Q = Cj„KC. and R = I were used by Field where J5T was a 6x6 diagonal matrix with ele- in w in * w *—' ments h \ \ \ \QQ \ . Specifically, Kw was used to weight pitch error due to significant non-lin- earities in that state. Like the filter, the controller design was based on solving the Riccati equation and in this case was implemented in MATLAB® using K i n = L Q R ( A i n , B i n , Q , uR) Field found that values for p of 0.05 to 0.005 performed adequately. For this thesis, unless stated otherwise, a value of p = 0.05 was used in the LQG/LTR controller. This value was tuned along with u. through simulation based on actuator performance in conditions with disturbances (see Appendix D.2.2 for further information). The resulting singular values for the plant with the LQG/LTR controller are shown in Figure 5-22. The four states controlled by the planes (depth, roll, pitch, and yaw) have similar responses and almost identical crossover frequencies. The low frequency limits of the singular value curves all approach infinity suggesting that steady state error and low-frequency disturbance rejection prop- erties of the system are good. The sensitivity and complementary sensitivity functions are shown for the complete system in Figure 5-23. 5.4.2 Sliding Mode LQG/LTR Controller The second augmentation of the LQG/LTR controller used sliding mode to account for changes in performance of the control surfaces based on body orientation. In short, the LQG/LTR controller is designed based on linearization about a setpoint; by allowing that setpoint to change based on operating condition, the errors due to linearization are reduced. In this implementation, nine con- trollers in total were designed and spanned a range of trim and yaw angles encountered during typ- ical simulated manoeuvres (see Table 5-5). For this thesis, the nine controllers were designed by changing only the control surface hydrody- namic effects in each instance. In all other regards, the body was assumed to remain at zero trim J = $(xTQx + uTpRu)dt <0 (130) University of British Columbia Peter M. Ostafichuk April 2004 Chapter 5 - Control System Design 165 150 100 10" 10 Frequency (rad/sec) 10 Figure 5-22 Open loop singular value plot for plant and controller and yaw for the controller design such that any differences in performance observed could be attributed solely to the controller treatment of the control surfaces. Bumpless switching was used to transition from one controller to another based on operating con- dition. Similar to the fuzzy membership functions introduced in Section 5.3.3, there was a distri- bution of conditions over which each controller was applied. As a simplified example considering trim only, Figure 5-24 shows the proportion of three trim controllers applied based on trim angle. At a trim angle of zero, only the KQ controller would be applied; as the trim angle is increased, the KQ and K + controllers would be averaged with increasing weight given to the K + controller and decreasing weight to the K 0 controller; and for positive trim angles greater than 2.3 degrees, the K + controller would be used exclusively. For the this thesis, the sliding mode controller used both trim and yaw as inputs and therefore the curves of Figure 5-24 were replaced by surfaces as shown in Figure 5-25. For simplicity, only the KQQ and K__ controllers are shown in the figure. The K__ surface continues indefinitely in the negative trim and yaw directions. The complete set of nine surfaces used in the sliding mode control are shown in two-dimensional form in Figure 5-26. In each case, the base of the surface is identified by the solid line and vertices in the surfaces are indi- cated with a point (corresponding to 100% implementation of the controller as in Figure 5-24). University of British Columbia Peter M. Ostafichuk April 2004 Chapter 5 - Control System Design 166 Frequency (rad/sec) Figure 5-23 System sensitivity and complimentary sensitivity singular values Table 5-5 Sliding mode controller Name Trim Yaw Koo Odeg Odeg K+0 2.3 0 K-o -2.3 0 Ko+ 0 3.4 Ko+ 0 3.4 K+ + 2.3 3.4 K+_ 2.3 -3.4 K.+ -2.3 3.4 K.. -2.3 -3.4 University of British Columbia Peter M. Ostafichuk April 2004 Chapter 5 - Control System Design 167 T r i m [ d e g ] Figure 5-24 Bumpless switching example for one input variable K Trim Figure 5-25 Bumpless switching with two input variables University of British Columbia Peter M. Ostafichuk April 2004 Chapter 5 - Control System Design 168 5 4 3 2 ; 1 I — 0 I E : -1 -2 -3 -4 -5 5 4 3 2 h ' , ' 0 -2 -3 -4 -5 N -3 -1 1 Yaw (deg) ( a ) K . -1 1 Yaw [deg] 5 4 3 2 •3 1 <D 2 0 E P-1 -2 -3 -4 -5 -1 1 Yaw [deg] 5 4 3 2 •3 1 <a 2-0 E E-1 -2 -3 -4 -5 -1 1 3 Yaw [deg] (b)Ko + (c )K+ + ( d ) K , (e) Koo (f) K+o 5 4 3 2 . 1 I — 0 E -2 -3 -4 -5 s / s -1 1 Yaw [deg] ( g ) K . 5 4 3 2 o 1 a 0 E £-1 -2 -3 -4 -5 -1 1 3 Yaw [deg] (h)K<>. 5 4 3 2 = ! 1 CD 2. 0 E £-1 -2 -3 -4 -5 -1 1 3 Yaw [deg] (0K+- Figure 5-26 Switching surface representation in two dimensions for the sliding mode controller University of British Columbia Peter M. Ostafichuk April 2004 Chapter 5 - Control System Design 169 5.4.3 LQG/LTR Controller with Fuzzy-Tuned Series Compensation The LQG/LTR controller with series compensation, tuned using fuzzy logic, was developed in the same manner as the equivalent PD controller augmentation described in Section 5.3.3. The only difference was that the controller of Figure 5-7 was replaced with the basic LQG/LTR version detailed above. University of British Columbia Peter M. Ostafichuk April 2004 Chapter 5 - Control System Design 170 5.5 SUMMARY The development of controllers for use in simulation with the DOLPHIN A U V was discussed in this chapter. Following a discussion of background information on various control methodologies and implementations, six controllers were presented in total. PD Controller Variants • A basic PD controller was presented based on a publication regarding the DOLPHIN A U V by Shupe and McGeer (1987). • The first augmentation of the PD control strategy used gain scheduling in an attempt to account for changes to control surface performance with body orientation. The controller was designed such that the gain matrix was actively modified during operation based on results from Chapter 4. • The second augmentation of the PD control strategy used a series compensator to adjust the output from the controller based on vehicle orientation. The series compensator consisted of a square 6x6 matrix that modified the 6 controller outputs before they reached the plant. The series compensator was tuned using fuzzy logic with rules and membership functions based on the data of Chapter 3. LQG/LTR Variants • Following the work of Field (2000) an LQG/LTR controller was developed. The controller parameters used to tune performance (u and p) were selected based on the findings of Field and the results of simulation (discussed in Chapter 6). • The first augmentation of the LQG/LTR controller used sliding mode control to switch between different LQG/LTR controllers designed (off-line) for different operating condi- tions. The changes in the different controllers were restricted to the treatment of control sur- faces. Depending on vehicle trim and yaw angle, the various controllers were combined using bumpless switching. • The second augmentation of the LQG/LTR controller was a fuzzy-tuned series compensator identical in form to that designed for the PD controller. University of British Columbia Peter M. Ostafichuk April 2004 Chapter 6 - Simulation 171 6 S I M U L A T I O N Physicists like to think that all you have to do is say, these are the conditions, now what happens next? - Richard Feynman For complex nonlinear plants, such as A U V s , modelling and simulation are cen- tral to the design of the control systems. The validity of such simulations depends and the accuracy and completeness o f the model used in simulation. Nahon (1996) suggests that the greatest errors in the simulation of an underwater vehicle arise from uncertainty in the hydrodynamic characteristics. Short of building a dedicated test platform, simulation is often the only way to examine plant and controller behavior. This chapter uses the experimental results from Chapter 3—modelled according to Chapter 4—to improve the realism of A U V control and dynamics simulations. Fol lowing a presentation of background information regarding A U V simulation, the mechanics of the MATLAB®-based simulator are outlined. The remaining sections present simulation results in three main areas: the differences in simula- tion outcome depending on the accuracy and completeness of control surface hydrodynamic modelling; the effect of the configuration of vehicle control sur- faces on performance; and potential control strategies for accounting for the previ- ously unmodelled control surface hydrodynamic characteristics. In terms of control strategy, several possible enhancements to basic P D and L Q G / L T R con- trollers, as outlined in Chapter 5, are considered. University of British Columbia Peter M. Ostafichuk April 2004 Chapter 6 - Simulation 172 6.1 SIMULATION BACKGROUND If a model is nonlinear, time-variant, or complex, closed form solutions are almost certainly not available and numerical methods must be used to evaluate system performance. Simulation is the term given to the process of using numerical methods to predict the response of a system (Palm, 1986). Complex, nonlinear models can be used in simulation and are necessary for investigating performance of a control system designed based on a linearized model (such as PD and LQG/LTR controllers). However, simulation results must be used with caution as it is not always possible to investigate all conditions that are unfavorable in terms of stability or observability (Astrom and Wittenmark, 1997). Likewise, it is difficult to decide when the nonlinear model is sufficiently accurate and to know when simulations have examined all conceivable modes of plant operation (Dutton et al., 1997). Development of underwater vehicles, including changes to the geometry or control of existing vehicles, can be an expensive, time-consuming process. Simulation is essential for validation and evaluation of the system design and can minimize unsafe or costly errors in the field. Song et al. (2002), for example, recommend extensive in-lab simulation of the control subroutines and entire control structure long before conducting sea trials. Of direct relevance to this thesis, Field (2000) developed a six degree-of-freedom simulation pack- age specifically for the DOLPHIN AUV. Field's simulator included several augmentations of a lin- ear submarine model for study and design of control compensators. Field invested considerable effort in formulating the equations of motion for a near-surface AUV. In addition to an extensive analysis of the rigid body dynamics, his model included forces arising from radiation (added mass and damping), restoring forces, incident and diffracted forces (due to the wave field) and control forces. The majority of these quantities are hydrodynamic in origin and Field used several meth- ods to estimate the values. In particular, Field employed semi-empirical methods—such as those of Whicker and Fehlner (1958), Dempsey (1977), and Lyons and Bisgood (1950) outlined in Chapter 2—numerical methods using strip theory (McTaggart, 1996), and aerodynamics software (Nahon, 1993). Simulink®, in conjunction with the MATLAB® programming environment, was used for the development a of control and dynamics simulator. Although his work was general in nature, Field's simulator was based on the DOLPHIN A U V making it ideal for the current research. To study control surface effectiveness, the formulation developed by Field is used with modelling improvements as outlined in Chapter 4. University of British Columbia Peter M. Ostafichuk April 2004 Chapter 6 - Simulation 173 6.2 SIMULATION ENVIRONMENT AND DESIGN A basic simulator of the control and dynamics of the DOLPHnx A U V was developed by Field (2000) and was modified for this work. Based on the experiments and modelling presented in Chapter 3 and Chapter 4, improvements were made to the simulator in terms of the representation of the control surface hydrodynamics. Simulations were developed in the Simulink® environment of MATLAB® in block diagram form; specialized routines were programmed using native MAT- L A B ® m-file® code. Others (such as Naeem et al., 2003, Prestero, 2001b, and Carreras et al., 2000, for example) have also used Simulink® in the investigation of A U V control and dynamics. Further details regarding the development of the simulator is discussed in the following sections with emphasis on the simulation procedure, simulator layout, and specialized m-file® program- ming. 6.2.1 Simulation Procedure There were three main stages to the overall simulation procedure: initialization, execution, and post processing. The simulation steps are shown in greater detail in Figure 6-1. In the figure, spe- cialized m-file® routines are shown in an alternate font (such as S i m i n i t X , for example) where 'JT is used to designate the controller (such as PD or LQG/LTR). The simulations were run over a predetermined interval, until the stop time, T, was reached. The time step size was 0.02 seconds (corresponding to 50 Hz) and the results were refined at two intermediate points during each time- step. (In other words, the simulations were run as if the time-step was equivalent to 0.00667 seconds (150 Hz) but with data only recorded for every third point or 0.02 seconds). In cases where a tuner was included in the control architecture, it was operated at a frequency less than that of the overall simulation. Unless noted otherwise, computations by the tuner and the resulting tuning actions were performed once every 0.1 seconds (10 Hz, denoted by Ttune in Figure 6-1). A fourth-order Runge-Kutta continuous solver with variable step size (ode45 in MATLAB®) was used in all simulations (further information regarding Runge-Kutta can be found in a standard numerical methods text such as Hoffman, 1992, or Boyce and DiPrima, 1992). During simulation, the allowed relative error tolerance for each state was 0.1% of the state value and the maximum allowed absolute error was 1 x 10~6 . The maximum error in simulation at each time step for each state was the maximum of the relative and absolute error tolerance. The real-time duration required to run a 70 second simulation (3500 time-steps) on a 1.6 GHz Athlon processor with 1 GB of R A M was approximately 20 minutes. 6.2.2 Simulator Layout The overall simulator layout in Simulink® (see Figure 6-2) closely matched that of the control system block diagram. The navigation module, controller, plant, and sensors sub-systems where constructed using nested block diagrams. In Field's simulator, the navigation module computed reference states based on interpolation from a set of user supplied waypoints. For this thesis, the commanded trajectories in depth and heading were assigned step input changes and all of the states were held constant—speed had a constant non-zero reference while all other states were held at zero. The exact layout of the controller block depended on the control strategy used; specific University of British Columbia Peter M. Ostafichuk April 2004 Chapter 6 - Simulation 174 Initialization SiminitX Run Simulation Sim(X,T) Post Processing r < v. r < r < Compute Steady State Control Surface Deflections CalcuOX Design Controller Xdesign Store Environment Variables t = T ' '* n Compute control actions and plant response Frequency Analysis FFT(x,u) Write Data File wklwrite (X , t,x,u) Simulation complete *. Tune Controller Tuner(X) Figure 6-1 Simulation process schematic University of British Columbia Peter M. Ostafichuk April 2004 Chapter 6 - Simulation 175 information for each controller can be found in Chapter 5. The same design for the plant block was used in all cases and is shown in Figure 6-3. Wave Disturbances Navigation Module ref. trajectory E )—error- Controller — control_ signal measured output Sensors and Filters Submarine Model ( I \« •output- Data Recorder Noise Figure 6-2 Simulator block diagram 8C K Actuator Plane J ^ Dynamics Saturation *\ Wave Influence on Flow Angle Plane Forces Wave Forces on body Transport Delay Body Influence on Flow Angle Inverse Mass Matrix d2x Integrator Body Forces Figure 6-3 Simulator plant block diagram In the plant block, the commanded plane deflections, 5C, were adjusted to account for the dynam- ics of the vehicle actuation system (the rate at which planes could change deflections) as well as the physical saturation limits (the maximum obtainable deflections). The resulting actual deflec- tions, 8Q, were combined with the change in incidence angle due to the vehicle orientation and motion—including wave effects where applicable—in order to determine the net incidence angle on the planes, a . The plane forces were computed according to Section 4.8 and were combined with the hydrodynamic, hydrostatic, gravitational, and wave induced body forces to determine the net force on the vehicle. By multiplying the net body force by the inverse mass matrix, the vehicle acceleration was determined which in turn was integrated to give the vehicle state. Finally, the sensor and filter block (in Figure 6-2) added measurement noise independently to each state and then processed signals with an anti-aliasing filter and an analog-digital converter (as shown in Figure 6-4). The noise was represented as stochastic and stationary with a gaussian nor- University of British Columbia Peter M. Ostafichuk April 2004 Chapter 6 - Simulation 176 mal distribution characterized by the variance as shown in Table 6-1. The analog to digital conver- sion was modelled using a zero-order hold system with the quantization also shown in the table. Actual State Anti-aliasing Butterworth Filter A/D Conversion ^ Noise Sensor and Filter Block Measured State Figure 6-4 Simulator sensor and filter block diagram Table 6-1 Sensor noise characteristics State Variance Quantization u 0.0038 m2/s2 0.1 m/s P 0.43 deg2 A 2 0.2 degA q 0.25 deg 2 A 2 0.2 degA r 0.02 deg2 A 2 0.2 degA X 2.9 m 2 1 m y 2.9 m 2 1 m z 0.0007 m2 0.1 m 0.11 deg2 0.1 deg e 0.015 deg2 0.1 deg 0.54 deg2 0.1 deg Further information regarding simulator development (including complete details of the system design and operation, the plant model, and noise and disturbance characteristics) can be found in (Field, 2000). Simulink® block diagrams for all systems are provided in Appendix E.2. 6.2.3 Simulation Programming Specialized MATLAB® code was required to design and operate the Simulink® system. Where possible, built-in MATLAB® commands were used but the majority of code was custom written. For both the PD and LQG/LTR control methodologies, gains were determined and assigned using MATLAB® routines called during the initialization phase of the simulations (see Figure 6-1). In cases where a tuner was used with the controller, function calls were made from Simulink® blocks during simulation; simulation processing was paused while MATLAB® routines recomputed and reassigned controller parameters. To determine the body forces, plane forces, and wave effects in the plant block, similar function calls from Simulink® were made to m-file® programs during the simulations. The code used in the simulations is provided in Appendix E. 1. University of British Columbia Peter M. Ostafichuk April 2004 Chapter 6 - Simulation 177 6.3 SIMULATION TEST CASES AND EVALUATION 6.3.1 Evaluation Manoeuvres AUVs can be designed to perform a wide variety of tasks but are typically employed as platforms for instrumentation. In such applications, improved disturbance rejection and manoeuvrability are in general highly beneficial. In a mapping operation for example, excessive motion of the plat- form may result in smearing or distortion of the sonar image. Likewise, in cases where a towfish containing instrumentation is pulled by an AUV, there is significant interaction between the vehi- cles (Seto and Watt, 1998); for acceptable performance, a high degree of towfish stability is required so, again, the A U V must remain stable. To examine the manoeuvrability and disturbance rejection of the D O L P H I N in simulation, a repre- sentative manoeuvre was selected. This manoeuvre, shown in Figure 6-5, is characterized by a 180 metre segment of straight and level flight with a depth change of 0.5 metre at the 60 metre mark. A 180 degree turn to starboard is then initiated and the simulation is run out through a final segment of straight and level flight. Simulations were conducted with a constant commanded for- ward speed of 6.0 m/s for a duration of 70 seconds. The length of the run-out at the end of the sim- ulation was approximately 145 metres but depended on how well the vehicle could maintain the commanded speed throughout the manoeuvre as well as how quickly it could perform the turn. 60 m 120 m • • x depth change 2R ~145 m mn-out y X 3.0 m 3.5 m z Figure 6-5 Manoeuvre used for performance evaluation University of British Columbia Peter M. Ostafichuk April 2004 Chapter 6 - Simulation 178 In addition, for frequency analysis of the vehicle response, a second manoeuvre consisting of straight and level flight at 3.5 m depth and 6.0 m/s commanded forward speed was used. This sim- ulation was conducted for 81.92 seconds such that the total number of steps was equal to a power of two (81.92 seconds with 0.02 second step size is equivalent to 4096 steps = 2 1 2). 6.3.2 Evaluation Criteria The criteria for performance evaluation of the vehicle and controller were threefold. First and foremost, the stability of the vehicle was considered; configurations in which unstable behavior was observed were deemed to be unusable. Secondly, cases were identified that violated opera- tional constraints; specifically, these cases included complete submersion of the snorkel—the snorkel is required for communication as well as to provide air to the diesel engine—and hull breach of the free surface. Considering the 3.5 metre operating depth, the independent limits on depth, roll, and pitch were +1.06/-3.0 metres, ±39.8 degrees, and ±39.8 degrees respectively (for combinations of these motions, the limits were even more stringent). Finally, three performance indices were developed to analyze behavior of different configurations in a fictitious seafloor map- ping exercise. The performance indices were defined in terms of the percentage of data lost during a hypothetical seafloor scanning operation using the trajectories above. The vehicle was assumed to be operating in water of 100 metre mean depth with a downward facing sonar beam of 50 metre transverse width (see Figure 6-6). In this idealized environment, the sonar beam width was divided into 1000 segments (pixels) and the sonar resolution (pixel size) was assumed to be 0.05 metres in all direc- tions. With reference to the commanded trajectory, error in vehicle position and pose resulted in inaccuracy in sonar beam placement on the sea floor. In Table 6-2, the sensitivity of sonar beam inaccuracy to errors in body position and orientation is shown. All three performance indices quantify the error in the sonar beam position, in pixels, relative to the total number of pixels scanned. One index, Jct, is used for the complete trajectory of Figure 6-5, Jsl, is used for the straight and level portions of Figure 6-5 only; and the final index, Jsa, is the significant amplitude from the response spectra (explained in detail below) in straight and level flight. Al l three indices are strictly positive quantities with an improvement in perfor- mance associated with a decrease in the performance index; for each index, a value of zero indi- cates error-free performance. During the turn of Figure 6-5, the lateral position, y, and the yaw angle, y , were not included in the performance indices—there were no reference values for these quantities during the turn—but depth, z, roll, dp, and pitch, 9 were included in the Jct index. The turn was considered to begin at x=180 metres and to last for 16 seconds; this period was sufficient for a vehicle travelling at the commanded speed of 6 m/s to complete a 180 degree turn with a 30.5 metre radius. In this approach, configurations which were less effective in turning or could not maintain speed during a turn were penalized. University of British Columbia Peter M. Ostafichuk April 2004 Chapter 6 - Simulation 179 A Figure 6-6 Fictitious scanning operation used in performance evaluation University of British Columbia Peter M. Ostafichuk April 2004 Chapter 6 - Simulation 180 Table 6-2 Body position and orientation corresponding to one sonar segment error Description Parameter Error Sensitivity lateral position y s = 0.05 m/pixel vertical position z sz = 0.05 m/pixel roll angle = 0.028 deg/pixel trim angle e sQ = 0.028 deg/pixel yaw angle = 0.229 deg/pixel Jct Performance Index The J t performance index is a measure of the ability of a vehicle to track a complicated trajectory and it is applied both in calm water and in waves. The exact definition of the performance index, y c t , i s : Jct N . P N z n = 0 yn-yr\ n< 1500 n > 2300 otherwise (131) The total number of pixels scanned is the product of the sonar beam width, P, in pixels, and the number of time-steps, N. As the time-step size used in the simulations was 0.02 seconds, the beginning of the turn (x=180m) for a vessel travelling at 6 m/s corresponded to time-step n = 1500. The end of the turn, 16 seconds after the initiation, corresponded to n = 2300. At each time-step, the difference between the actual and reference value for each state (y, z, § , 0 and \\i) was computed and scaled by the sensitivity factor from Table 6-2. In equation (131), the reference values generated by the Navigation Module are denoted with subscript r and the actual values at each time-step are denoted with n. Jsl Performance Index The J t index is a more reasonable measure of performance for a vehicle intended for track-line survey operations where the primary function is to maintain a straight and level course and quickly University of British Columbia Peter M. Ostafichuk April 2004 Chapter 6 - Simulation 181 recover from changes in direction or depth. As with Jct, the Jsl performance index was applied in both calm water and waves. The definition of this second performance index is J., = s l (N-Nt)P n = 0 \yn-y, n < 1500 n > 2300 otherwise (132) In this case, only the number of time-steps during the commanded straight and level flight was used in the computation of the total number of pixels. In other words, the number of time-steps during the turn, Nt = 800, was removed from the total number of steps, N, in computing the total number of pixels. JSA Performance Index The third performance index, Jsa was the significant amplitude from the scanning error spectrum. It is a measure of the wave disturbance rejection qualities of a particular configuration of vehicle and controller and is applied only in waves. To determine Jsa, simulations were conducted for 4096 time-steps (81.92 seconds) in straight and level flight in head seas of sea state three. At each time-step, n, the scanning error, , was defined as In yn-y, + + + e.-e. + (133) A fast Fourier transform (FFT) analysis was performed in order to obtain the response spectrum the scanning error, 5-(n). (The reader is referred to a numerical analysis text such as Burden and Faires, 1997, for further information on the FFT procedure.) In keeping with convention, the spec- trum Sj was expressed in terms of the square of the response amplitudes (see, for example, Journee, 2001). The performance index, Jsa, was the significant amplitude which represents the mean value of the highest one-third portion of the amplitudes; as given by Journee (2001), this is mathematically defined as I j"S(co)cfo (134) University of British Columbia Peter M. Ostafichuk April 2004 Chapter 6 - Simulation 182 As such, JSA is given by 14096 14096 J, sa = 2 £ 5,.(©)Aoo = 2 5}(Aco • «)Aco (135) Where, Aco is the size of the frequency bands in the computed spectra and is given by Aco 1 = 0.01227/z (136) 81.925 Control Effort As a final note, control effort was not included in the performance indices as the main impact of large control effort was only a slight increase in the overall body drag. Appendages (including the planes, rudder, and rear stabilizer) represent only 10% of the total drag on a lone DOLPHIN (Watt et al., 1997). The contribution to total drag drops to approximately 5% under typical towing condi- tions (Seto and Watt, 1998). Therefore, even significant changes in plane control effort (and hence drag) would have a very limited impact on the overall vehicle energy consumption. 6.3.3 Simulated Cases The cases examined by simulation were grouped into three main areas: the effect of accurate and complete hydrodynamic modelling of control surfaces; the effect of control surface configuration; and the effect of the control system design. The hydrodynamic modelling examined the effect of improving the plant model in simulation while using the existing (conventional) model in control- ler design; likewise, the effect of using the improved model in both plant and controller was ana- lyzed. The study of control surface configuration used both the control surface geometries and layouts that showed the most promise based on experimental data of Chapter 3. Finally, the inves- tigation of the control system design was based on the three PD and three LQG/LTR controller augmentations summarized in Chapter 5. The simulations were conducted in straight and level flight and using the manoeuvre of Figure 6-5. Likewise, both calm water and sea state three based on a Bretschneider spectrum were used. The simulations are summarized in brief in Table 6-3; a comprehensive summary of the simulations is given in Appendix F. 1. University of British Columbia Peter M. Ostafichuk April 2004 Chapter 6 - Simulation 183 Table 6-3 Cases considered in simulation Parameter changed Vehicle Model Controller Path Sea Plant model Standard Varied LQG/LTR Figure 6-5 Calm Model for controller design Standard Varied LQG/LTR Figure 6-5 Calm Plane geometry Varied New LQG/LTR Figure 6-5 Calm Varied New LQG/LTR Figure 6-5 Sea State 3 Varied New LQG/LTR Straight Sea State 3 Plane mounting location Varied New LQG/LTR Figure 6-5 Calm Varied New LQG/LTR Figure 6-5 Sea State 3 Varied New LQG/LTR Straight Sea State 3 Controller Standard New Varied Figure 6-5 Calm Standard New Varied Figure 6-5 Sea State 3 Standard New Varied Straight Sea State 3 University of British Columbia Peter M. Ostafichuk April 2004 Chapter 6 - Simulation 184 6.4 RESULTS: INFLUENCE OF MODELLING IMPROVEMENTS The influence of the various improvements to control surface modelling from Chapter 3 and Chapter 4 were studied through the simulation. In addition to improvements in the estimates of the control surface hydrodynamic derivatives, the effect of trim (Sections 3.3.2 and 4.4) and yaw (Sections 3.2.4 and 4.5), plane stall (Sections 4.2.1 and 4.5.1), and bowplane-sternplane interac- tion (Sections 3.3 and 4.6) were included. All modelling improvement simulations were con- ducted using the standard plane configuration with the LQG/LTR control system. LQG/LTR control weightings of u = 5.0 x 10~6 and p = 0.005 were used in this series of tests. Although these weightings did not produce an ideal controller for any one case, they did produce a working controller for every case. The control weightings of u = 0.0005 and p = 0.05 determined in Chapter 5 generally resulted in better performance but could not be used for the examination of modelling improvements since in some situations the changes to the plant resulted in non-minimal modes in the Kalman filter design. 6.4.1 Changes to the Plant Model Simulations were conducted to examine the effect of control surface modelling errors on system performance. The simulations were based on a controller designed for the conventional plant model but acting upon improved plant models that more accurately represented the true control surface hydrodynamic behavior. In short, the simulations quantified the influence of neglecting plant hydrodynamic information in the controller development and instead treating it unmodelled disturbances. Five cases were considered using the standard vehicle configuration and basic LQG/LTR controller. In addition to the conventional model, originally developed by Field (2000), the features modelled were the bowplane-sternplane interaction effects, the effect of yaw on the bowplanes, the effect of trim on the bowplanes, and the effect of Reynolds number and yaw on the stall angle. The simulation results for the commanded trajectory of Figure 6-5 in calm water are summarized in Table 6-4 in terms of the performance indices. A complete summary of tabulated simulation data is provided in Appendix F.2.1. There is very little difference between the Jct performance indices of the various models—from highest to lowest, the difference is 3.1%. Interestingly, the performance actually improves when plane interaction and trim influence are included in the plant model but neglected in the control strategy. In terms of the controller, these effects are essentially unmodelled disturbances that happen to have a beneficial influence. Absolute differences between the J [ performance indices are even less than those with the Jct index. University of British Columbia Peter M. Ostafichuk April 2004 Chapter 6 - Simulation 185 Table 6-4 Effect of model improvements on performance in calm water with LQG/LTR control Performance Index Effects Modelled in Plant Jct Jsl None (original plant) 0.1105 0.0026 Plane interaction only 0.1070 0.0022 Yaw influence only 0.1106 0.0025 Trim influence only 0.1073 0.0024 Plane stall influence only 0.1093 0.0025 All effects 0.1078 0.0023 Only a minor difference in roll was observed for the four cases—RMS roll error ranged from 6.9 degrees to 7.1 degrees—and there was no observed difference in the RMS error of depth, pitch, and yaw (see Appendix F.2.1 for complete data). Although these findings appear to suggest that the simulation outcome does not strongly depend on whether the hydrodynamic effects are included in the plant, with a less robust controller there were differences noted in the simulation output. Using a simple PD controller, simulations were conducted both with the original plant model and with all the control surface hydrodynamic improvements added. Unlike with the LQG/LTR con- troller, there was a large decrease in performance effectiveness when the complete plant model was used (see Table 6-5). Specifically, the unmodelled bowplane-sternplane interaction resulted in a steady state pitch error as shown in Figure 6-7; a similar steady state error in depth was also observed. Without integral action, the PD controller was unable to account for steady state error. (The influence of controller methodology and design on vehicle performance is considered in detail in Section 6.6.) Table 6-5 Effect of model improvements on performance in calm water with PD control Performance Index Effects Modelled in Plant Jct Jsl None (original plant) 0.0124 0.0096 All effects 0.0184 0.0167 6.4.2 Changes to the Controller Model Simulations were also conducted with the new hydrodynamic information incorporated in the con- troller as well as the plant; these tests complement those above in which the new hydrodynamics were included in the plant only. In this case, only the most complete plant model (with the trim, yaw, stall, and plane interaction effects) was used. In the models used for controller design, the different effects were included both individually and as a group. The performance is summarized in Table 6-6 and the complete data is provided in Appendix F.2.2. University of British Columbia Peter M. Ostafichuk April 2004 Chapter 6 - Simulation 186 o 0.02 0.015 0.01 =5" 0.005 (0 -0.005 -0.01 -0.015 •Original Plant -All Effects 50 100 150 200 250 Distance over ground [m] 300 350 400 Figure 6-7 Influence of the plant model on PD controller performance in pitch Table 6-6 Effect of controller model on performance in calm water with LQG/LTR control Performance Index Effects Modelled in Controller J c t J si None (original controller) 0.1078 0.0023 Plane interaction only 0.1120 0.0024 Yaw influence only 0.1078 0.0023 Trim influence only 0.1078 0.0023 Plane stall influence only 0.1078 0.0023 All effects 0.1120 0.0024 The performance for the straight and level portions of the manoeuvre, characterized by Jsl, was virtually unaffected by the changes to the model used for the controller. For the entire manoeuvre, the inclusion of trim, yaw, and stall effects in the LQG/LTR controller had no significant effect (the values of Jsl and Jct, as well as other simulation outputs, remained unchanged). This is because the LQG/LTR controller was designed based on linearization about an operating point with trim and yaw equal to zero (for which there are no trim or yaw effects on the planes). This linearized controller was then used for all conditions; specifically, the controller remained unchanged for non-zero trim and yaw angles. University of British Columbia Peter M. Ostafichuk April 2004 Chapter 6 - Simulation 187 Bowplane-sternplane interaction—which is almost at maximum for straight and level flight—does have an effect on performance when incorporated in the controller. When the new hydrodynamic effects are included in both controller and plant, the controller compensates for them and the sys- tem operates very similarly to the original model (that is, the case where interaction effects are not included in either the controller or plant). Parallel to what was observed in Section 6.4.1, the inter- action effects appear to have a somewhat beneficial influence as the vehicle performance is better when they are not included in the controller. Although neglecting the interaction effects may appear to have some benefits in this case, the effects are unmodelled disturbances that change the forces on control surfaces from those commanded by the controller. In general, with a robust controller the difference in simulation with and without the control sur- face hydrodynamic model improvements were small but this is not to suggest that these effects should be omitted. A limited number of trials were conducted for this thesis and there may be cases in which simulation output may more strongly depend on the control surface model used. In particular, if the controller is unable to adequately compensate for the added hydrodynamic effects in some situation—due to limitations in control surface response rate or saturation limits, for example—the simulation outcome could change significantly. University of British Columbia Peter M. Ostafichuk April 2004 Chapter 6 - Simulation 188 6.5 RESULTS: EVALUATION OF PLANE CONFIGURATION Simulations were conducted to investigate the effect of control surface configuration on perfor- mance. The modelled experimental data of Chapter 3 and Chapter 4 for plane geometry and posi- tion were used. Al l simulations were conducted with the LQG/LTR control system with control weightings u, = 1.21 x 10"5 and p = 0.05 . The control weightings of u = 0.0005 and p = 0.05 from Chapter 5 could not be used for the examination of the plane configuration since the changes to the plant resulted in non-minimal modes in the Kalman filter design in some instances. 6.5.1 Plane Geometry Four different combinations of plane geometry were examined through simulation. In addition to the standard configuration, the long and short bowplanes were used as were the long sternplanes (see Table 3-2 on page 45 for physical specifications). The changes in geometry were limited to the plane span as forces on the planes were found to be primarily a function of planform area (for the range of geometries studied, as noted in Chapter 3). Calm Water For the test manoeuvre of Figure 6-5 in calm water, the resulting performance indices are summa- rized for each of the plane combinations in Table 6-7. For the entire manoeuvre (characterized by the Jct index) the short bowplane configuration showed the best performance while long bow- planes were least effective. In contrast, for the performance over the straight and level positions of the commanded trajectory, the long bowplanes were by far the most effective and the short bow- planes were the least effective. Complete performance information, including maximum and RMS errors, turn radii, and plane saturation statistics can be found in Appendix F.3.1. Table 6-7 Effect of plane geometry on performance in calm water Performance Index Plane Geometry (Bowplane - Sternplane) Jct Jsi Standard - Standard 0.0957 0.0057 Long - Standard 0.0999 0.0028 Short - Standard 0.0886 0.0062 Standard - Long 0.0960 0.0074 For each of the position and pose states, the long bowplane configuration generally had smaller errors over the entire trajectory. For illustration, the depth as a function of horizontal distance trav- elled is shown in Figure 6-8 and is typical of most states in terms of how the various plane config- urations ranked. In almost all respects, the long bowplanes performed better than the standard bowplanes which in turn performed better than the short bowplanes. The relatively poor Jct per- formance index for the long bowplanes is predominantly due to an increased average roll angle during the turn (see Figure 6-9). The long sternplanes ranked poorly in terms of both Jct and Jsl. As a further measure of performance, short bowplanes operated in saturation for 4.2% of the tra- jectory compared to 2.6% for the long sternplanes, 2.5% for the standard bowplanes, and only University of British Columbia Peter M. Ostafichuk April 2004 Chapter 6 - Simulation 189 1.6% for the long bowplanes. In other words, the long bowplanes achieved generally better per- formance and used less control effort in the process. 3.5 - 3.4 - 3.3 - ][ 3.2 - JE *~> a. 3.1 - o o 3 - 2.9 - 2.8 - 2.7 - Reference • Standard-Standard • Long-Standard - Short-Standard • Standard-Long 50 100 150 200 250 300 Distance over ground [m] 350 400 Figure 6-8 Depth history in calm water for various plane geometry combinations Sea State Three With the same simulations as above conducted in waves of sea state three and with sensor noise, the short bowplanes were the least effective in all regards (see Table 6-8). Over the entire trajec- tory, the standard configuration performed best (measured using Jct) followed closely by the long bowplanes. In the straight and level portions of the manoeuvre with a turn, again the long bow- planes were most effective; considering only wave disturbance rejection (measured by Jsa) the long bowplanes were by far the most effective. To further examine the frequency response, spectra and significant amplitudes were computed for sway, depth, roll, pitch, yaw, bowplane deflection and stemplane deflection (all are provided in Appendix F.3.1). The depth spectra are shown in Figure 6-10 (with the steady state component removed). As would be expected from Jsa, the long bowplane configuration had the best wave compensation as shown by the lowest response curve. The bowplane deflection spectra are shown in Figure 6-11; the short bowplanes show the most plane activity and the long bowplanes show the least. These findings are consistent with those above; overall, the long bowplanes best track the commanded trajectory while rejecting wave disturbances and using the least control effort. University of British Columbia Peter M. Ostafichuk April 2004 Chapter 6 - Simulation 190 0.05 0 -0.05 •] -0.1 18 -0.15 o -0.2 -0.25 -0.3 -0.35 -0.4 Reference Standard-Standard Long-Standard Short-Standard Standard-Long 50 100 150 200 250 300 Distance over ground [m] 350 400 Figure 6-9 Roll history in calm water for various plane geometry combinations Table 6-8 Effect of plane geometry on performance in sea state three Plane Geometry (Bowplane - Sternplane) Performance Index 'si Standard - Standard 0.0970 0.0154 0.479 Long - Standard 0.0984 0.0126 0.222 Short - Standard 0.1030 0.0204 0.397 Standard - Long 0.0997 0.0141 0.308 6.5.2 Plane Location In addition to plane geometry, four plane location configurations were considered in simulation. The configurations were those that showed potential for improved performance based on the experimental and modelling analysis (Chapter 3 and Chapter 4 respectively); they included the standard configuration, anhedral bowplanes, a Y-tail configuration, and keel planes. Specifica- tions for each of these configurations were provided in Table 3-3 on page 46. Calm Water The performance indices for each of the configurations in calm water are given in Table 6-9 with complete data summarized in Appendix F.3.2. Similar to observations with plane geometry, the University of British Columbia Peter M. Ostafichuk April 2004 Chapter 6 - Simulation 191 Figure 6-10 Heave response based on bowplane geometry for sea state three (head seas) best performing configuration in straight and level conditions (Y-tail) was least effective over the entire trajectory; the keel planes were poor strictly in the straight and level portions but effective over the entire trajectory. Significant differences were noted in the x-y path of the different configurations as shown in Figure 6-12. The average turn radius of the Y-tail was 26% less than that of the standard configu- ration (22.5 metres compared to 30.5 metres). In the Y-tail configuration, where sternplanes had dihedral, a component of the associated control force acts in the ^-direction thus assisting the rud- der in turning. The anhedral bowplanes, mounted ahead of the centre of gravity of the vehicle, appear to have an opposite effect in that the average radius of curvature of the turn increases to 31.2 metres. The criteria of the manoeuvre used in evaluation was to change heading by 180 degrees rather than follow a set of waypoints; it is expected—although it has not been veri- fied—that the anhedral bowplanes would perform better in waypoint tracking as they allow con- trol of sway directly without a change in heading. Finally, the keel planes had an average turn radius of 30.4 metres, similar to the standard configuration, but there was an offset in v-position introduced during the depth change. There was no corrective action taken in this case since the trajectory is based on heading not ̂ -position. The fast, tight turn of the Y-tail configuration is accompanied by large roll (see Figure 6-13) which was the primary source of error. The maximum roll angle of Y-tail configuration was approxi- mately 28.7 degrees corresponding to an increase in submergence of the snorkel of 0.49 metres (in calm water at 3.5 metres depth). At this roll angle, 0.82 metres of snorkel remain above the water University of British Columbia Peter M. Ostafichuk April 2004 Chapter 6 - Simulation 192 0.02 0.015 • Standard-Standard Long-Standard - Standard-Long • Short-Standard •a re o ••P u a> <p o o a c a a. § CQ 0.01 0.005 H i 0 -1 1 1 1 1 1 1 r - 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Frequency [Hz] Figure 6-11 Bowplane deflection based on plane geometry for sea state three (head seas) Table 6-9 Effect of plane location on performance in calm water Performance Index Plane Position Jct Standard 0.0957 0.0057 Anhedral Bowplanes 0.0900 0.0077 Y-tail 0.1015 0.0018 Keel Planes 0.0912 0.0154 line; for comparison, at minimum 1.1 metres of snorkel remain above the calm water line for the standard configuration during the turn. As an aside, by increasing the weighting to roll error in the controller, the turn was performed with reduced roll as shown in Figure 6-14. In this case, the roll error was artificially scaled by a factor of ten in the controller; not only was the physical roll error reduced substantially but the Jct and Jsl performance indices improved to 0.0481 and 0.0016 respectively (both significantly lower than any other configuration). Once again considering the original control error weighting, considerable differences were noted with respect to control effort over the course of the manoeuvre. The standard and anhedral bow- planes operated in saturation for 2.5% and 2.4% of the time, respectively, over the trajectory. With the Y-tail, the bowplanes were in saturation 8.7% of the time, predominantly in trying to cor- University of British Columbia Peter M. Ostafichuk April 2004 Chapter 6 - Simulation 193 University of British Columbia Peter M. Ostafichuk April 2004 Chapter 6 - Simulation 194 0.1 -0.1 e -o.2 o O) 1-0.3 o or -0.4 -0.5 -0.6 J - ^ Y-tail , ! ! Y-tail with 10x roll error weighting 1 ! 1 10 20 30 40 Time [sec] 50 60 70 Figure 6-14 Roll history in calm water for Y-tail with increased roll error sensitivity rect for the large roll during the turn. The keel planes operated in saturation 12.5% of the time both as a result of pitch error during the depth change and roll error during the turn. Sea State Three The performance summary for the alternate plane locations in sea state three is given in Table 6-10. The Y-tail performs best over the straight sections of the turning manoeuvre while the anhedral bowplanes are best over the complete trajectory and show the best wave disturbance rejection characteristics. The keel planes are effective at rejecting wave disturbances but rank poorly compared to other configurations through all portions of the path following manoeuvre. Table 6-10 Effect of plane location on performance in sea state three Plane Position Jct Performance Index Jsa Standard 0.0970 0.0154 0.479 Anhedral Bowplanes 0.0839 0.0160 0.231 Y-tail 0.1046 0.0136 0.430 Keel Planes 0.1049 0.0338 0.338 University of British Columbia Peter M. Ostafichuk April 2004 Chapter 6 - Simulation 195 The differences in bowplane usage were even more dramatic in waves as compared to calm water; the proportion of the trajectory with bowplane saturation for the standard, anhedral, Y-tail, and keel planes was 2.1%, 4.2%, 5.3%, and 23.4% respectively. The bowplane deflection spectra are shown in Figure 6-15. In addition to operating in saturation most often, the keel planes were also the most active as shown by the pronounced peak at frequencies near 0.5 Hz. The Y-tail and stan- dard configuration showed similar frequency response spectrum, significantly lower in magnitude than both the keel and anhedral planes. 0.02 0 -I 1 1 1 1 1 1 r- 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Frequency [Hz] Figure 6-15 Bowplane deflection spectra based on plane location for sea state three head seas Finally, when considering the sway and yaw significant amplitudes of each configuration, the anhedral planes showed improvements in wave disturbance rejection in those states of 74% and 23% compared to the standard planes. As was alluded earlier, the downward inclination of the anhedral planes is expected to provide improved performance in sway and yaw due to the redun- dancy in the control those states. The improvements in sway and yaw were with the anhedral planes were offset by increases in the significant amplitudes in heave and pitch by 32% and 17% respectively. Complete results, including all tables and spectra, are provided in Appendix F.3.2. University of British Columbia Peter M. Ostafichuk April 2004 Chapter 6 - Simulation 196 6.6 RESULTS: EVALUATION OF CONTROL SYSTEM In addition to the effect of the control surface configuration on vehicle performance, the influence of the controller and controller improvements were examined. The majority of work was based on the LQG/LTR controller developed by Field (2000) but a simple PD controller modelled after one reported by Shupe and McGeer (1987) was also considered. The main objective was to determine if it was possible to improve performance by accounting for the new control surface hydrodynamic information in the controller design. For both the PD and LQG/LTR controllers, the basic control strategy was augmented in an attempt to account for the new control surface hydrodynamic effects added to the plant. Al l simulations were conducted using the standard plane configuration. 6.6.1 PD Controller Performance Three augmentations of a PD controller were considered: a basic PD controller, a PD controller with gain scheduling, and a PD controller with a series compensator tuned using fuzzy logic. Details regarding the development of each of the controllers were provided in Section 5.3. For each controller, simulations were conducted with the trajectory of Figure 6-5 in both calm water and in sea state three with waves moving in the -x direction (that is, in head seas for the start of the manoeuvre). Various response spectra were also computed by considering straight and level flight in sea state three head seas for 3.5 metres depth. Calm Water For the three PD controller augmentations, the performance indices for the evaluation manoeuvre in calm water are summarized in Table 6-11. In general, there are only minor differences between the controllers. For the complete trajectory, the PD controller with gain scheduling has a perfor- mance equivalent to the basic PD controller; the PD controller with a fuzzy-tuned series compen- sator has a slightly poorer performance index (roughly 3.5% higher). Over the straight and level portions of the trajectory, both the gain scheduled PD controller and the PD controller with series compensation showed an improvement in the Jsl performance index of over 5%. Table 6-11 Effect of PD control structure on performance in calm water Performance Index Control Structure Jct Js, PD 0.0114 0.0098 PD with Gain Scheduling 0.0114 0.0093 PD with Fuzzy Compensation 0.0118 0.0093 The improved straight and level performance of the controllers with gain scheduling and fuzzy compensation is largely due to reduced pitch error. The maximum pitch errors for these control- lers were 0.7 and 0.6 degrees, respectively, compared to 1.0 degrees for the basic PD controller. However, the two PD controller augmentations also had slightly larger RMS depth errors and larger average bowplane deviations compared to the basic PD controller. Complete details regard- ing the performance of the PD controllers are provided in Appendix F.4.1. University of British Columbia Peter M. Ostafichuk April 2004 Chapter 6 - Simulation 197 As an aside, for the PD controller with, series compensation, simulations of Section 6.4.1 on page 184 were repeated where the improved control surface hydrodynamic model was used for the plant but the conventional model was used in controller development. The resulting pitch history is shown in Figure 6-16; the steady state error previously observed is greatly reduced as is the maximum pitch error during the turn. The steady state error that remains is due to the slight dis- crepancy between the plant and the compensator in the representation of the bowplane-sternplane interaction. The pitch error during the turn with the compensator and improved plant is even smaller than the basic controller and original plant. A similar improvement in steady state depth error was also noted. 0.02 0.015 0.01 TJ 0.005 o <*-• CL -0.005 -0.01 -0.015 •PD control - original plant - PD control - improved plant • PD control with series compensation - improved plant 50 100 150 200 250 Distance over ground [m] 300 350 400 Figure 6-16 Pitch history comparison for PD controller with and without fuzzy compensator Sea State Three In sea state three, the basic PD controller showed comparable performance to the gain scheduling and series compensation augmentations (see Table 6-12). With the wave disturbances, both the gain scheduling controller and the controller with fuzzy tuned series compensation had 8% lower maximum pitch error compared to the basic controller; however, the maximum depth errors were 1 % and 6% higher respectively. Complete details for all three controller augmentations are sum- marized in Appendix F.4.1. The response spectra for depth are shown in Figure 6-17 below. The PD controller with series compensation had a lower response than the basic PD controller while the gain scheduling imple- mentation had higher response. Similar results were observed for other motions and in terms of University of British Columbia Peter M. Ostafichuk April 2004 Chapter 6 - Simulation 198 Table 6-12 Effect of PD control structure on performance in sea state three Control Structure Jct Performance Index Js, Jsa PD 0.0233 0.0222 0.125 PD with Gain Scheduling 0.0235 0.0223 0.135 PD with Fuzzy Compensation 0.0236 0.0222 0.124 plane deflection, both augmentations to the PD controller showed less plane activity than the basic controller. Complete frequency response spectra plots are provided in Appendix F.4.1. 0.0015 0.0000 -I i 1 1 1 1 1 - T - 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Frequency [Hz] Figure 6-17 Depth response spectra for PD controller augmentations is sea state three head seas 6.6.2 LQG/LTR Controller Performance Three augmentations of the LQG/LTR controller were considered: a basic LQG/LTR controller developed by Field (2000), a sliding mode LQG/LTR controller, and an LQG/LTR controller with fuzzy-tuned series compensation. Details regarding the development of each of the controllers were provided in Section 5.4. Similar to with the PD controllers, simulations were conducted with the trajectory of Figure 6-5 in both calm water and in sea state three, as well as for straight and level flight in sea state three head seas. The standard vehicle configuration was used for all tests and control weightings of u = 0.0005 and p = 0.05 were used in LQG/LTR controller. University of British Columbia Peter M. Ostafichuk April 2004 Chapter 6 - Simulation 199 Calm Water For the three LQG/LTR controllers considered, the performance indices for the evaluation manoeuvre in calm water are summarized in Table 6-13. For both the complete trajectory and the straight and level portions of the trajectory, the sliding mode LQG/LTR controller had the best per- formance. The Jct and Jsl performance indices were approximately 12% and 60% better respec- tively than the basic LQG/LTR controller. The basic LQG/LTR controller and the LQG/LTR controller with series compensation showed similar performance. Table 6-13 Effect of LQG/LTR control structure on performance in calm water Performance Index Control Structure Jsl Basic LQG/LTR 0.1014 0.0058 Sliding Mode LQG/LTR 0.0891 0.0023 LQG/LTR with fuzzy tuned 0.1018 0.0059 series compensation The improved performance of the sli