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Simulation, modelling, and control of a near-surface underwater vehicle Field, Adrian Ivery 2000

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Simulation, Modelling, and Control of a Near-Surface Underwater Vehicle by ADRIAN IVERY FIELD B.Eng., McGill University, Montreal, 1997 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE IN THE FACULTY OF GRADUATE STUDIES (Dept. of Mechanical Engineering) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA May 2000 © Adrian Ivery Field, 2000 Dept. Mechanical Engineering University of British Columbia 3424 Main Mall Vancouver, B.C., V6T 1Z4 April 5, 2000 Special Collections and University Archives Main Library 1956 Main Mall Vancouver, B.C. V6T 1Z1 (604) 822-2521 E-mail: spcoll@interchange.ubc.ca Dear Sir or Madam, In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Adrian I. Field Abstract This thesis presents a theoretical study into the dynamics and control of underwater vehicles. Problems related to simulation, modelling and control are examined for a submarine operating near the free surface. Two particular issues in this study are the effects of surface waves on the submarine and the cable tension effects of a body towed by the submarine. Fundamental rigid body mechanics and hydrodynamic principles are used to construct the dynamic equations of motion for a submarine. External control, hydrodynamic, restoring and dis-turbance forces are recognized and included in the model. A specific study of the results of and the current methods for the evaluation of hydrodynamic added mass and damping for a submarine is included. The equations of motion are expressed for the particular application to the autonomous underwa-ter vehicle (AUV) DOLPHPN. A guidance system based on global positioning system (GPS) way-points is incorporated in the simulation. Two tow cable models are presented and included in the simulation model, one for straight flight and one dynamic model for tracking maneuvers. A sys-tem identification method is presented for the realization of the dynamic tow model. The superpo-sition of linear free surface waves is used to represent a given sea state for simulation. The effects of these waves on the submarine are included in simulations. Based on the information resulting from the equations of motion and simulation environment, two suitable control system algorithms are developed and compared. The open loop characteristics of the submarine are studied. The control systems designs are based on the Linear Quadratic Gaus-sian (LQG) method, and use the Loop Transfer Recovery (LTR) design process. Design based on a linear model is used as a basis, while two augmentations of the model are compared for effec-tiveness. The tracking performance for ramp and step input commands are compared. Then a turn-ing maneuver is simulated with the tow models. Finally, two long crested sea states and three relative wave directions are simulated with each of the controllers for a single commanded veloc-ity. The effects of sensor noise and the filtering of this noise are also presented. Both augmentations using the LQG control approach show satisfactory performance results for an AUV operating near the surface while towing. Actuator saturation is observed in simulations with waves. Non-linear effects degrade the performance of the compensators. Some improvements are suggested for this application. KEYWORDS: Autonomous, Underwater, Vehicle, Modelling, Control, Optimal, Linear Quadratic Gaussian, Loop Transfer Recovery, LQG, LTR, Hydrodynamics, Waves, Sea State, Submarine, Submersi-ble, Remotely Operated Vehicle, ROV, Simulation, Disturbance. Table of Contents Abstract i i Table of Contents i i i List of Tables vi List of Figures vii 1.0 Introduction 1 1.1 Background 1 1.2 Previous Work 2 1.3 Motivation 3 1.4 Approach 3 2.0 Submarine Dynamics Model Structure 5 2.1 Kinematics 5 2.1.1 Position and Orientation 5 2.1.2 Velocity and Acceleration 8 2.2 Rigid Body Dynamics 10 2.2.3 Hull Properties 10 2.2.4 Body Forces and Moments 11 2.2.5 Radiation and Restoring Hydrodynamic Forces and Moments 14 2.2.6 Incident, diffracted and environmental forces 18 2.2.7 Controlled and Additional Forces 19 2.3 Non-linear Model State Equations of Motion 20 2.4 Linear State-Space Model Equations of Motion 22 2.5 Summary 23 3.0 Hydrodynamics 24 3.1 Prediction Methods 27 3.1.1 Semi-empirical 28 3.1.2 Numerical Methods 32 3.1.3 Aerodynamics 34 3.1.4 Experiment 35 3.1.5 System Identification 35 3.2 Infinite Frequency Coefficients 36 3.3 Frequency Dependent Coefficients 37 3.4 Summary 39 4.0 Simulation 43 4.1 DOLPHIN Vehicle 43 4.2 Path Following 46 4.3 Complete 11 -State Model 49 4.4 Disturbance Modelling 52 4.4.1 Static Tow Model 54 4.4.2 Dynamic Tow Model 56 4.4.3 Wave Model 62 4.4.4 Onboard Stochastic Measurement Disturbances 70 4.5 Summary 73 5.0 Control 74 iii 5.1 Open Loop Characteristics 75 5.2 Comparison of Control Methods 83 5.2.1 Fixed Gain Compensators 84 5.2.2 Advanced Compensators 85 5.2.3 Controller Selection 85 5.3 LQG/LTR Design Methodology 86 5.3.4 Open Loop Design 86 5.3.5 Filter Design 91 5.3.6 Augmentation of Controller for Plant Input Integration 101 5.3.7 Augmentation of Controller for Disturbance Observing 105 5.3.8 Controller Design (Recovery) 109 5.3.9 Compensator Considerations 121 5.4 Summary 125 6.0 Simulation Results and Discussion 126 6.1 Initial Disturbance and Ramp Input Tests 126 6.2 Turn Maneuver Simulations 137 6.3 Simulations with Waves 145 6.4 Summary 153 7.0 Conclusions and Recommendations 155 7.1 Conclusions 155 7.2 Recommendations 156 Nomenclature 159 References 165 Bibliography 175 Appendix A: Euler Angle Transformation 179 A. 1 Translational Vector Transformation 179 A. 2 Angular Velocity Transformation 182 Appendix B: Determination of Hydrodynamic Coefficients 183 B. l Added Mass 183 B . l . l SUMAD 183 B.1.2 Aucher 187 B.2 Damping 193 B.2.3 Missile Datcom Output Data 193 B.2.4 Missile Datcom Input File 196 B.2.5 Aucher 198 B.2.6 Nondimensional Equivalent of Shupe's Results 204 B. 3 Actuator Hydrodynamic Analysis 205 B.3.7 Aucher 205 B.3.8 Whicker and Fehlner 208 B.3.9 Propulsion 211 Appendix C: Simulation Layout in Simulink 212 C. l Simulink Models 212 C.2 Matlab Functions 221 C.3 Discrete Dynamic Tow Model 239 Appendix D: Control Design Data 241 iv D. 1 Unaugmented Control Design Information 241 D. 1.1 Linear State Space Matrices 241 D. 1.2 Eigenvalues and Eigenvectors of Linear Plant 242 D. 1.3 Scaling Matrices 243 D. 1.4 Kalman Filter Gains 243 D.2 Integrator Augmented Design Information 244 D.2.5 Linear State Space Matrices 244 D.2.6 Eigenvalues and Eigenvectors of Linear Plant 246 D.2.7 Kalman Filter Gains 247 D.2.8 Control Gains 248 D.3 Disturbance Observer Augmented Design Information 249 D.3.9 Linear State Space Matrices 249 D.3.10 Eigenvalues and Eigenvectors of Linear Plant 251 D.3.11 Kalman Filter Gains 253 D.3.12 Control Gains 254 Appendix E: Simulation Results and Software 255 V List o f Tables TABLE 2.1 Vector Quantity Coordinate Descriptions for Submarine Motion 6 TABLE 3.1 Munk Coefficients of Added Mass 29 TABLE 3.2 Hull Added Mass Coefficients 40 TABLE 3.3 Hull Damping Coefficients 40 TABLE 3.4 Actuator Hydrodynamic Coefficients 41 •TABLE 4.1 Principle Particulars 44 TABLE 4.2 Appendage Particulars 45 TABLE 4.3 Dolphin Endurance 45 TABLE 4.4 Seastate Statistical Data 64 TABLE 4.5 Impinging Wave Energy Parameters 66 TABLE 4.6 Estimates of Sensor Noise Characteristics 71 TABLE 5.1 SPM Open Loop Poles 76 TABLE 5.2 Input to output gains at 1.2 rad/sec 89 TABLE 5.3 Input to output gains at 1.2 rad/sec 90 TABLE 6.1 Initial Plant Conditions 127 TABLE 6.2 Initial Condition Tracking Characteristics 133 TABLE 6.3 Summary of Turn Maneuver RMS Values 143 TABLE 6.4 Summary of Turn Maneuver Maximum Values 143 TABLE 6.5 Summary of Turn Maneuver Minimum Values 143 TABLE 6.6 Summary of Data RMS for Simulations under Sea State 3 151 TABLE 6.7 Summary of Data Maxima for Simulations under Sea State 3 151 TABLE 6.8 Summary of Data Minima for Simulations under Sea State 3 152 TABLE 6.9 Summary of Data RMS for Simulations under Sea State 4 152 TABLE 6.10 Summary of Data Maxima for Simulations under Sea State 4 153 TABLE 6.11 Summary of Data Minima for Simulations under Sea State 4 153 TABLE N. 1: Description of Roman Nomenclature and Variables 159 TABLE N.2: Description of Greek Nomenclature and Variables 162 vi List of Figures Figure 2.1: Body and Earth Fixed Reference Frames 5 Figure 2.2: Forces due to planes and propeller. 20 Figure 3.1: Hydroplane parameter illustration 27 Figure 3.2: Slender Streamlined Body 28 Figure 3.3: Discretization of a Submarine Hull into Panels 33 Figure 4.1: Overall picture of DOLPHIN Mark II Geometry with Towfish 44 Figure 4.2: Path following modes: a) line following mode; b) waypoint following mode 47 Figure 4.3: Waypoint Following Variables 48 Figure 4.4: Diagram of DOLPHIN vehicle while towing 52 Figure 4.5: Diagram of DOLPHIN vehicle in waves 54 Figure 4.6: Tow Cable Force Vector Schematic 55 Figure 4.7: Position Information for Maneuver used in Tow Identification 60 Figure 4.8: Speed over ground during maneuver. 60 Figure 4.9: Cable tension during maneuver. 61 Figure 4.10: Half difference of DOLPHIN and towfish heading 61 Figure 4.11: Wave and Submarine Relative Orientation 63 Figure 4.12: Wave Loading Regimes 67 Figure 4.13: Sensor Noise Simulation Block 72 Figure 5.1: Feedback and Feedforward 75 Figure 5.2: Linear Representation of the Non-linear Plant 76 Figure 5.3: Eigenvector Element Magnitudes for Pole 1 78 Figure 5.4: Eigenvector Element Magnitudes for Pole 2 78 Figure 5.5: Eigenvector Element Magnitudes for Pole 3 78 Figure 5.6: Eigenvector Element Magnitudes for Pole 4 79 Figure 5.7: Eigenvector Element Magnitudes for Pole 5 79 Figure 5.8: Eigenvector Element Magnitudes for Pole 6 79 Figure 5.9: Eigenvector Element Magnitudes for Pole 7 80 Figure 5.10: Eigenvector Element Magnitudes for Pole 8 80 Figure 5.11: Eigenvector Element Magnitudes for Pole 9 80 Figure 5.12: Eigenvector Element Magnitudes for Pole 10 81 Figure 5.13: Simulation Block Diagram 83 Figure 5.14: Open loop singular values of the 10 output plant 88 Figure 5.15: Singular values of the 5 output plant 89 Figure 5.16: Singular values of the scaled 5 output plant 91 Figure 5.17: Kalman Filter Structure 91 Figure 5.18: Block diagram of system 94 Figure 5.19: Singular values of unaugmented Kalman filter first design 96 Figure 5.20: Singular values of unaugmented Kalman filter loop second design 97 Figure 5.21: Singular values of unaugmented Kalman filter loop third design 98 vii Figure 5.22: Sensitivity and Complementary Sensitivity of the Kalman filter loop 99 Figure 5.23: Augmentation possibilities for loop shaping 100 Figure 5.24: MIMO loop with actuator integration augmentation 102 Figure 5.25: Singular values of the open loop integrator augmented system 102 Figure 5.26: First iteration open loop Kalman filter for integral augmented plant 103 Figure 5.27: Singular values of the open loop Kalman filter of the integrator augmented plant.... 104 Figure 5.28: Singular values of the sensitivity and complementary sensitivity of the Kalman filter for the integrator augmented plant 105 Figure 5.29: Singular values of the open loop plant with disturbance model 106 Figure 5.30: FkstiterationopenloopKalmanfdterfortheplantaugmentedwithdisturbanceobserver.... 107 Figure 5.31: Singular values of the open loop Kalman filter of the integrator augmented plant.... 108 Figure 5.32: Singular values of the sensitivity and complementary sensitivity of the Kalman filter for the integrator augmented plant 109 Figure 5.33: L Q G Compensator Block Diagram. Kalman filter is denoted by dashed box, variable is the estimated state vector. 110 Figure 5.34: Singular values of G(s)K(s) for p = 0.05 using the integrator augmented plant 112 Figure 5.35: Singular values of G(s)K(s) for p = 0.005 using the integrator augmented plant 113 Figure 5.36: Singular values of G(s)K(s) for p = 0.0005 using the integrator augmented plant.... 113 Figure 5.37: Singular values of sensitivity and complementary sensitivity of G(s)K(s) for p = 0.02 using the integrator augmented plant 114 Figure 5.38: Singular values of sensitivity and complementary sensitivity of G(s)K(s) for p = 0.002 using the integrator augmented plant 115 Figure 5.39: Poles and zeros of closed loop system augmented with actuator integrators 116 Figure 5.40: Singular values of G(s)K(s) for p = 0.08 using the disturbance observer augmented plant 117 Figure 5.41: Singular values of G(s)K(s) for p = 0.008 using the disturbance observer augmented plant 117 Figure 5.42: Singular values of G(s)K(s) for p = 0.02 using the disturbance observer augmented plant 118 Figure 5.43: Singular values of sensitivity and complementary sensitivity of G(s)K(s) for p = 0.008 using the disturbance observer augmented plant 118 Figure 5.44: Singular values of sensitivity and complementary sensitivity of G(s)K(s) for p = 0.0008 using the disturbance observer augmented plant 119 Figure 5.45: Poles and zeros of closed loop system augmented with disturbance observer.... 121 Figure 5.46: Integrator Anti-wind-up scheme 122 Figure 5.47: Block Diagram of Integrator Augmentation Compensator Overall Layout 122 viii Figure 5.48: Simulink Block Diagram of Disturbance Observer Augmentation Compensator Overall Layout 123 Figure 6.1: Step Disturbance in 'u' 128 Figure 6.2: Step Disturbance in V 128 Figure 6.3: Step Disturbance in 'w' 129 Figure 6.4: Step Disturbance in 'p' 129 Figure 6.5: Step Disturbance in 'q' 130 Figure 6.6: Step Disturbance in 'r' 130 Figure 6.7: Step Disturbance in 'z' 131 Figure 6.8: Step Disturbance in '0' 131 Figure 6.9: Step Disturbance in '0' 132 Figure 6.10: Step Disturbance in 132 Figure 6.11: States During Ramp Depth Command 134 Figure 6.12: Actuators During Ramp Depth Command 134 Figure 6.13: States During Ramp Heading Command 135 Figure 6.14: Actuators During Ramp Heading Command 135 Figure 6.15: Example of Turn Maneuver Path Results 137 Figure 6.16: Controlled State Data for Turn With Integrator Augmentation and No Tow 138 Figure 6.17: Controlled State Data for Turn With Integrator Augmentation and Dynamic Tow.... 138 Figure 6.18: Actuator Data for Turn With Integrator Augmentation and No Tow 139 Figure 6.19: Actuator Data for Turn With Integrator Augmentation and Dynamic Tow 139 Figure 6.20: Controlled State Data for Turn With Disturbance Model Augmentation and No Tow.... 140 Figure 6.21: Time Domain Data for Turn With Disturbance Model Augmentation and Dynamic Tow 140 Figure 6.22: Actuator Data for Turn With Disturbance Model Augmentation and No Tow.... 141 Figure 6.23: Actuator Data for Turn With Disturbance Model Augmentation and Dynamic Tow.... 141 Figure 6.24: Lateral Tow Cable Angle on DOLPHIN During Turn Maneuver. 142 Figure 6.25: Sea State 3 Wave Encounter Spectra at 6 m/s 145 Figure 6.26: Sea State 4 Wave Encounter Spectra at 6 m/s 146 Figure 6.27: Summary of State RMS for Wave Simulations for Sea State 3 147 Figure 6.28: Summary of Actuator RMS for Wave Simulations 148 Figure 6.29: Example of Wave Forces and Moments in Sea State 3 149 Figure 6.30: Example of Actuator Response in Sea State 3 149 Figure A. 1: Yaw Euler Rotation 179 Figure A.2: Pitch Euler Rotation 180 Figure A.3: Roll Euler Rotation 181 Figure C. 1: Overall Layout 212 Figure C.2: Navigator. 212 Figure C.3: Integrator Augmented Compensator with Feedforward and Scaling 213 Figure C.4: Disturbance Observer Augmented Compensator with Feedforward and Scaling 213 Figure C.5: Submarine Plant Model 213 Figure C.6: Submarine Plant Model with Waves 214 Figure C.7: Wave Model :? 214 Figure C.8: Submarine Plant Model with Tow Model 215 Figure C.9: Static Tow Model 215 Figure C. 10: Dynamic Tow Model 216 Figure C . l l : Sensor Suite 217 Figure C.12: Typical Noise On Measured Signals 218 Figure C.13: Analog to Digital Modelling 218 Figure C. 14: Anti-aliasing Filters 219 Figure C.15: Conversion of Error in Heading to Within -Pi to Pi 219 Figure C. 16: Conversion from Integrated Rate Gyro Angle to Heading Between 0 and360Degrees... 220 Figure C.17: Reference State Generation 220 Figure C.18: Logging of Plant Data 221 1.0 Introduction 1.1 Background The motion of submarines and other similar marine vehicles has been studied intensively since the mid 1940's when the military use of such vessels was intensified. Most of the research work done on submerged marine vehicles has subsequently been concentrated on geometries and conditions related to large cylindrical and ellipsoidal bodies of revolution, these shapes being similar to that of a submarine. Two major cases for study have been identified, the deeply submerged case [1], and the near surface case [2]. The major difference between these being the effect of waves caus-ing sinusoidal or cyclic forces on, and responses by, a body near the surface. Additional surface effects are apparent in calm water and are similar to operation near the "bottom" of the water, but these second order surface effects are less apparent. Accordingly, the control of a submersible in these two operating cases is quite different. The deeply submerged case is more predictable and easier for a planesman to keep a desired heading and orientation [3], while near surface operation provides the challenges of surface broaching, sinking out and excessive vehicle motions. This work will examine the near surface control of a submersible. In recent years, with the increased electronic capability, systems have become readily available which facilitate the use of remote and automatic control of vehicles not only in the marine envi-ronment but also in hazardous, space, and airborne environments. With such technology comes the advent of smaller, unmanned marine vehicles which eliminates the risks to human life, and the potential loss of expensive materials and construction. The widespread use of small underwater vehicles by the telecommunications, natural resources, and salvage industries brought research into this subject to the private sector. In the marine environment these unmanned underwater vehicles (UUV's) are classified as either: remotely operated vehicles (ROV's), or autonomous underwater vehicles (AUV's), depending on the extent of automation. Due to the limitations of underwater communication, ROV's are typically commanded from the water surface using an umbilical cable. These vehicles, because of their tethers, are generally slow moving (< 3 knots), use fixed or vectored thrusters for dynamic control, and are able to travel in more than one direction. The overall mission control for such vehicles is human based, and the pilot is provided feedback from various sensors on board the vehicle, the main sensors being onboard cameras. The operation of AUV's is significantly different. Typically, the control system is hierarchical [4], higher levels having greater authority. An AUV has a predefined mission plan which is pro-grammed into the onboard "high level" controller. The "high level" mission controller commands the dynamic controller of the vehicle. The dynamic controller then commands the actuator control loops, consisting of foils, thrusters, engine, and other devices that generate the forces necessary for moving the vehicle. Sensors provide all levels of control with feedback. Due to their extended range of operation, AUV's can operate at higher speeds, require a greater complement of sensors and a more complete understanding of vehicle motions for controller design. I Automatic control of vehicles emerged with the commencement of space flight and other prob-lems involving unstable or, remote flight. The field has since spread to many other industries, but the methodology and design of such systems is common. The design of a controller is based on the assumption of some type of mathematical model for the plant, the plant being the system for which control is needed. Usually, a simplified model is sufficient for most physical systems, and many control systems use classical designs based on a linear model of a given plant [5]. In this way a simple control system, in many cases, can handle complex and unstable plant systems. If the controller is designed with robust characteristics, any unmodelled dynamics will not signifi-cantly affect the performance of the controller, however, in some instances unmodelled plant char-acteristics will cause the controller and plant to become unstable and fail. Unmodelled characteristics, called disturbances can be a detriment to stability and performance. Disturbances can be taken into account using various methods, the most common being filtering. Disturbances are also modelled, and disturbance models can be used in control systems. In the case of near surface submarine control, wave forces are a major disturbance to the plant sys-tem. Other hydrodynamic, actuator, sensor and external forces, such as a tow cable, could also be disturbances. Through the complete and thorough modelling of these disturbances and their effects, knowledge is gained which can be used when designing a control system. Appropriate methods for the development of such a model will be used and a complete control system will be proposed in this work, which can then be applied to an AUV. 1.2 Previous Work Much of the work related to the present study has been done within the last 20 years, and the sub-ject has been approached from three main directions: hydrodynamics, automatic control, and application to real vehicles, largely by the navy. The most mature of these being the area of hydro-dynamics. The study of heaving cylinders in waves [6], ships in a seaway [7], and marine vehicle dynamics [8], is broad and pertains mainly to the shipping industry and oil platforms. Studies into submersible control from this perspective yields complex hydrodynamic models, yet simple con-trol schemes. The complete effects of radiation, diffraction, and incident waves are accounted for and fluid potential and viscous flow theory is used to determine the dynamic pressure at given locations on the hull. In addition, the study of stochastic distribution of waves in a seaway is used to determine overall vehicle motions for a given sea state. This approach often uses frequency domain methods and analysis. The second, largely theoretical, approach is the examination of automatic control methods. Auto-pilots were initially designed for ships but many specialists have now given attention to submarine control [9]. Most of these studies use simplified plant models, often only including deeply sub-merged hydrodynamic forces, combined with complicated control algorithms. Attention is often limited to motions in the vertical plane [10, 11, 12, 13], as it is often considered the most unstable mode in near surface operation. The work in submarine control is based mainly in the time domain, and results are obtained through computer simulation. A large thrust in this area occurred from 1984-1986 by work done at MIT [14, 15] on linear quadratic gaussian (LQG) and optimal control in application to submarines. Since then, LQG control has been a benchmark for the design of new control systems. More recently, sliding-mode control has been popular [16, 17, 9], as well as H infinity [18, 10], among other methods. 2 The third major perspective on near surface submarine control is more practical and considers application of the technology. This work is done mainly by the navy and companies interested in the realization of modern control in marine products, as large and often costly equipment is needed. Much of this work is accomplished through extensive model testing and full scale vehicle trials. Often work is particular to specific vehicles [19, 20, 21, 22], and semi-empirical [23, 24] formulas are used for the calculation of hydrodynamic forces. Reliable, basic control systems are implemented so that modification and design is time and resource efficient. Only through the combination of these three approaches can a complete study into the control of near surface submarine control be achieved, as each discipline features necessary information and understanding of the problem. It is also important to note that a series of studies has recently been completed on the near surface, or periscope depth, operation of submarines [2, 25, 3], along simi-lar lines to the research reported here. 1.3 Motivation Recent union of submarines with aircraft carrier battle groups, littoral zone operations, and increased surveillance efforts, has placed increased importance on periscope depth performance. As a result, increased knowledge and control ability is required. In the case of near surface AUV operation, vehicles may be used in severe sea states and for towing sonar arrays. In both cases the response of the controller-plant system to disturbances is critical. Altogether, complete modelling of wave and other disturbances is necessary, and a corresponding control system is sought. The desired control system should prove to be robust and applicable to an AUV. A range of maneuvers and conditions must be tested, requiring a simulation tool capable of either a regulation, fixed condition maneuver, or a tracking, changing maneuver. A time domain approach will be used so that time varying commanded trajectory information can be examined for certain maneuvers. The main fault with standard PID control systems is their inability to utilize known information about the plant system. It is desirable to use information "observed" as disturbances and differ-ences in the plant model. Accordingly, two control systems will be proposed in addition to the standard LQG approach and will be compared for minimizing errors in motions. Both systems are based on the LQG methodology, and require augmentations to the basic structure outlined in [26]. Because a submarine may operate in the deeply submerged case, the proposed controllers will be equally effective in both cases. 1.4 Approach This thesis begins with the definition of the kinematic states and a description of the dynamics of an underwater vehicle. The rigid body mechanics and hydrodynamic relations are used to con-struct equations of motion. Six different methods of determining the hydrodynamic coefficients are discussed and used based on their relative strengths. The details necessary for realistic simula-tion are determined including guidance and the addition of the appropriate disturbances. 3 After the conditions for the simulation are denned, the selection of a suitable control method is discussed. The open loop dynamics are examined for the submarine plant. Three control compen-sators are designed using the LTR design approach. Simulations are performed and discussed for a set of sea conditions, tow models, and maneuvers. The performance of the three control systems are then compared based on the simulation results. The most advantageous control design is then proposed with suggestions for future research. 4 2.0 Submarine Dynamics Model Structure 2.1 Kinematics 2.1.1 Position and Orientation In order to study the motion of an underwater vehicle, one must formulate a model of the motion of the vehicle. The nomenclature used to define the model is critical and has been suggested in various publications, some of which are specific to particular vehicles and some which are extremely general. The following will use the most accepted definitions adopted by SNAME [27]. These descriptions are better suited to development of the equations of motion for a control study [9]. Some additional nomenclature is found in Feldman [28] which is widely referenced. The pre-vious two works are the basis of the development of the equations of motion presented here. One must note that Feldman describes all velocities, angular velocities, forces and moments with respect to the body fixed frame. In the study of moving bodies, it is necessary to define a body fixed frame and an earth fixed frame, as forces can be described relative to either. The reference frames are shown in Figure 2.1. Note that body forces such as lift and drag can be described easily in the body frame while weight and buoyancy can be more easily described in the inertial frame. Body R y z Figure 2.1: Body and Earth Fixed Reference Frames. The two major reference frames shown in the figure are as follows: fE - Earth fixed (inertial) frame having axes x, y, z and origin at O. JFR - Body fixed frame having axes x', y', z' and origin at O'. 5 Variables which are described projected onto a given frame are denoted by a capital subscript labelling the frame. Other frames used in this thesis are the wave crest fixed frame, J ^ , and the translating earth fixed frame, Jj. In relation to ocean vehicles, the scalars {p, q, r} represent the quantities roll, pitch and yaw angular velocity respectively, as the translational velocity scalars {u, v, w} represent surge, sway, and heave in the body frame. A vector may be described projected onto frame using unit vectors [e i , ^  e 3] which are orthonormal and coincident with the axes of jpE, or may be described with unit vectors [e'i, e'2, e'3]T which are orthonormal and coincident with the axes of JFR- Note that all vectors and matrices are shown using an underline. The varia-bles are summarized in Table 2.1. T A B L E 2.1 Vector Quanti ty Coordinate Descriptions for Submarine M o t i o n . Reference Frames Body E a r t h Body E a r t h Body E a r t h J B J E Body E a r t h .7B Tti Description Translational Position Translational Velocity Forces F rame Uni t Vectors Axis 1 x' x u X X X E fill fii Axis 2 y ' y V y Y Y E £^ 2 £-2 Axis 3 z' z w z z Z E £ 3 £3 Description Rotational Angle Rotational Velocity Torques F rame Uni t Vectors Axis 1 Y <t> P i K K E £ 4 £4 Axis 2 a e q e M M E fi* £5 Axis 3 P ¥ r N N E £6 If the variables shown in Table 2.1 are not denoted with a capital subscript, the frame shown in the table above is assumed as a reference. The following vectors are also defined: 1] = [ B r ! l I ] T ; T ] 1 = {x,y,z}T; r, 2 = G, \|/}T Y = [ Y r Y 2 ] T ; Yi = K v . w } 7 ; Y 2 = i p , q , r } T X- = [?vh]T> T 1 = { X , Y , Z } T ; x 2 = { K , M , N } T Following the descriptions above, there exists a rotation matrix Q, which transforms a given vec-tor, r, projected onto frame JB such that it is projected onto JE i.e., 6 r.E = QiB (2.1) and, rB = Q" 1 r E (2.2) The subscripts show the frame upon which a given vector is projected and are mainly used if a vector is projected onto a frame which is different than that which is denoted by its coordinate symbols. Here, it is necessary to use notation which is consistent with coordinate transformation. The rotation matrix Q has the following properties: Q T Q = Q Q T = I; and accordingly, detQ= 1; Also, the rotation matrix can be found by: (2.3) ?'l ?i ?2 ?1 ?3 Q = ?2 ?'l 5*2 ?2 ?3 _-3 5', h ?2 ?3 5'3 (2.4) where each pair of dot multiplied vectors in this matrix must both be projected onto the same ref-erence frame. This type of rotation matrix can be factored into sets of multiple rotations, having the same end result. One such factoring is the set of 3 Euler angle rotations. Such a set of rotations is described in a specific sequence (see Appendix A). For underwater vehicles, the common formulation is as follows. Assume that the two reference frames have coincident origins due to a translation of the earth fixed frame. Then a copy of the inertial axis is rotated first about the z axis (yaw), through an angle, then about the new location of its y axis (pitch) through angle 6, and finally about the orientation of its x axis (roll) through angle ((). The final orientation of the frame is such that it is coincident with the body fixed frame. Such a set of rotations has a singularity condition for the pitch (second rotation) angle, 0 of ± 90°. Another set of kinematic variables used to describe the transformation is the set of quaternion rotations which overcome the singularity condition, and are often used for spacecraft dynamics [29]. One must note that the rotation matrix Q does not include the difference in position of the origin between one frame and another. 7 2.1.2 Velocity and Acceleration In order to determine the kinematics of a vehicle's motion, its velocity must be determined. Using relative motion, the velocity, or time derivative of a vector P can be found by, dTE 3 B E r5rB 3t (2.5) One should note that this notation implies that the time derivative in the earth fixed frame as dt and that the time derivative in the body fixed frame is ^ . If the body in question is rigid and the frame JfB is body fixed, the last term in the preceding equation becomes zero, so that the velocity at any point in the body becomes: dTE dbE (2.6) If the point of interest is actually the origin of the body frame, then the last term is zero and the velocity can be written as: dt - E di dr db OT' dt = *dt For the case in question, where the body frame origin is located at {x, y, z} in the inertial frame, d_ dt — - -x u y V z w (2.7) The variable Qj is defined in Appendix A. In this way, the rate of change of position in the inertial frame is used to find the translational velocity projected onto the body frame. (2.8) The method of transformation uses Euler angles {(j), 0, (p), and it is desirable to represent the angular rotation of the body reference frame using these parameters. In other words we desire the angular velocity C0g as a function of the Euler rates, {<j), 0, (p }. A direct vector transformation of the rotational rates is not possible in the same way as the translational rates. In order to find, (% the angular velocity of frame JFB, projected onto we must determine the time derivatives of the Euler rotations. Using eq. 2.5, it can be shown that o>B = C0E, so 8 G>B = (P, q, r} (2.9) This vector can be projected onto JE, as given in Appendix A. Giving the following relation: 0 p e q _<p_ r d_ dt Then describing the origin of the body fixed axis, combining eqs. 2.7 and 2.9, (2.10) X u y V d z Qi o w dt 4> 0 Q 2 p 0 q .9. r (2.11) In this way, the velocity seen by the vehicle, Y = [ u v w p q r ] > can be found in relation to the fixed frame pose and position vector: 11- [x y z (|) 0 <p] In the general case, for any point, r, within the body, its acceleration can be found by the time derivative of eq. 2.5: 5f X ? B I2 n at2 (2.12) Once again considering the body frame being fixed to a rigid body, 2 d r " '-E [drd.i i r r3.-i —~ = -^ -b + GO x -^b d t2 L3tLat-JBJE L-B Lat-Jj 5t~ X R B (2.13) And if r is the location of the origin of the body frame, the above equation becomes, 9 2 d r j - M s f e ] / » B * f e ] B ] E . 0-14) as r B = 0. To determine the term on the right hand side, the vector {u, v, w}T is substituted for \^-b\ . The L o t - J B angular accelerations are determined in a similar manner such that all 6 degrees of freedom are described. In this way, the acceleration of the center of mass is found using known parameters. 2.2 Rigid Body Dynamics 2.2.3 Hull Properties If one considers the distribution of mass of the vehicle in the body fixed frame, then the total mass, m, of the vehicle can be described as follows: m = J j J p s d V The density, p s shown in the above equation is obviously discontinuous and varies throughout the volume of the body. In a similar fashion, the position of the center of mass, r c m , of the body can be found by: Jem = y j " J r p s d V m V Following this, r ^ = 0 if the origin of the body frame is located at the center of mass. The moments of inertia are then found from: [xx = J J J ( y 2 + z 2 ) p s d V [ x y = I y x = J J J ( x y ) p s d V V v lyy = J J J ( X 2 + Z 2 ) P s d V [ = I = J J J ( y z ) P s d V y z z y V V [ z z = J J j ( x 2 + y 2 ) P s d V [ x z = i z x = J J J ( x ( z ) p s d V v From these definitions, a matrix I, the moment of inertia matrix can be derived as, 10 x^x ^xy ^xz - I x y 'yy _ I y z ~^xz —^yz ^zz_ Note that this matrix is assumed to be relative to and projected onto the frame JFB i.e. 1= Ig. The calculation of the moments of inertia can be achieved either through experimental methods, or detailed mass balance analysis. In a mass balance analysis, the location and magnitude of all masses in the vehicle are recorded. Generally for a submarine, I x y, I x z , I y x , I y z , I z x , and I z y, are all assumed zero due to various symmetries [9]. General estimates of the values of I x x can be found from radius of gyration of 0.2-0.5 hull diameters, I y y, and I z z, from a radius of gyration of 0.17-0.25 hull lengths, depending on hull geometry and mass distribution. 2.2.4 Body Forces and Moments In the following derivation of the equations of motion, the forces will be projected onto the body fixed frame JTB and the Newton- Euler approach will be used to solve the forces involved due to its simplicity. The Eulerian representation of Newton's second law is through the conservation of both linear and angular momentum, L and ^ "respectively, where, L = mi) —cm cm and tfm = Iw — c m _ _ c m where the variables on the right hand side of the equations have been defined previously. The sub-script 'cm' indicates the momentum or velocity of the total mass considered as a point mass located at the mass center. Using this terminology, the equations are given as, dL SF ; = T - c m (2.15) and dtf £ M . = (2.16) Here, Fj is any external force acting on the body, and Mj is any external torque acting around the mass center. According to this formulation, we require the acceleration of the mass center. Expressing the acceleration in eq. 2.14 onto the body fixed frame and combining with the Euler's equations, Newton's second law, eq. 2.15, may be written in the body frame as, = SF.l (2.17) 11 where F B is the sum of the external forces acting on the body, projected onto the body fixed frame. These will be derived in the next section. Applying similar representation of relative motion to the angular momentum, one obtains, -cm JB 0* + 1. (2.18) Where r^m is the location of the center of mass. So if the origin of B is located at the center of mass, then eqs. 2.16-2.17 become: m 31 + [ ^ B X ^ = F B , and (2.19) at + % X 1 B % = MB (2.20) These two equations define the external forces and moments about the point O' with reference to frame B. Also, \>B = v{, and <BB = v , using the notation of eq. 2.10, so combining the force and moment equations, in the body fixed frame, we obtain an expression for the rigid body motion of the form: M r b|fY + C b ( V ) V = T b Where, (2.21) M r b = m 0 '3x3 3x3 I m -m 0 0 0 m 0 0 0 m and, T = X Y Z K M N 12 Where, (X, Y, Z ) T are the components of the external forces F B projected onto the principal axes nr of the body fixed frame, JFB, and (K, M, N) are the components of M B -Shown expanded for the general case with the position of center of mass {xg, y g, zg} , the equa-tions appear as follows, 2 2 m(u + vr + wq + x (q +r ) + y (pq- r ) + z (pr + q)) = X m(v + wp + ur+x g (qp + r ) -y g (p 2 + r 2 ) + z g (q r -p ) ) = Y m(\v + uq + vp + x g(rp - q) + y g (rq + p) - z g (p 2 + q 2)) = Z yz xy + dxx " Izz)rP - (P + q r)I x v + (q2 - r 2)I X 2 + (qp - r)I = M yy^ v xx zzy xy yz 2 2 , Izz r + ( I y y - I xx)pq-(q + rP)Iyz + (q - P )IxV + (rcJ-p) Izx = N xy (2.22) (2.23) (2.24) (2.25) (2.26) (2.27) These equations are shown in expanded format and can be implemented more readily, again, using a matrix notation, redefining M r D to include the location of the center of mass offset from the origin of the body fixed frame, m 0 0 0 mz g -my g 0 m 0 -my g 0 mx g 0 0 m my g -mx g 0 0 -mz g my g x^x —^ xy mz g 0 -mx g lyy - ry Z _-myg mx g 0 -Izx Izz and the coriolis matrix: C . -_ r b 0 0 0 m(y g q + z g O - m ( x g q - w ) -m(x g r + v) -m(y g P + w) m(z g r + x G p) - m ( y g r - u ) - m ( z g p - v ) -m(z g q + u) m(x g p + y gq) -m(y g q + z g r) m(y gp + w) m ( z g p - v ) 0 - I y « q _ I x z p + I z z r . I y z r ~ I » y P + Iyy q m ( x g q - w ) -m(z g r + x g p) m(z g q + u) I y z q + I x z P - I z z r 0 ^xz^^yq + ^xP m(x g r + v) m ( y g r - u ) -m(x g p + y g q) - I y z r - I x y p + I y y q I x z r + I x y q - I x x p 13 it is possible to write the rigid body equations, in the body frame, as in eq. 2.20. These equations describe the motion of a rigid body as it would appear in a vacuum, all external forces are summed in xrb which will be addressed in the following section. 2.2.5 Radiation and Restoring Hydrodynamic Forces and Moments The external forces and moments acting on a semi-submerged vehicle are mainly hydrodynamic in nature. These effects are assumed to be linearly additive [30] and are broken down into compo-nents, radiation and restoring forces; incident, diffracted and environmental forces; and control and external forces. Radiation forces are the forces acting on the vehicle due to its own motion. In other words, the body is oscillating at a given frequency, but is not subject to any exciting or wave forces. Radia-tion potential forces are expressed using added mass, and damping coefficients. Viscous damping and restoring forces, also hydrodynamic in nature, are included. These forces add to Newton's relation in eqs. 2.15 and 2.16, d£ dmx> - c m = - - c m = V ( F + F ), and dt dl ^ - - e / d 3 f d dt dt--c™ — —e' In this case the force F e is the additional force necessary to move hull body through the fluid, whereas F would be the force necessary to move the object in a vacuum. Whereas M e is the addi-tional torque required to rotate the body through the fluid. This additional force is the radiative hydrodynamic force. As a mathematical solution of this force is extremely complex and detailed, the forces are often expressed in the form of a Taylor series expansion, such that each component of force is expressed as a function of the kinematic states, {u, v, w, p, q, r, x, y, z, <)), 0, op}, of the vehicle, for example: X = X 0 + X ).u + X i . v + . . . + X u u + X v v + . . . + X u u u u + X u v u v + . . . , (2.28) where, X„ = ^ . (2.29) Similar expansions are formed for all force and moment components. Due to hydrodynamic con-siderations, only the first order, acceleration, terms, and up to second order velocity terms are used [31]. The hydrodynamic radiation forces are categorized as added mass or damping terms depending on whether they act in phase with acceleration, so are considered as a mass, or act in phase with velocity, so are considered as damping coefficients and are multiplied by velocity. 14 Added Mass Added mass is an often misunderstood term which defines the effect of the pressure due to the inertia of the fluid on a body in motion through the fluid. The added mass of a body can be deter-mined both experimentally and theoretically, and has been found to be dependent on the fre-quency with which the body oscillates near a free surface. Generally, all the hydrodynamic force terms involving the acceleration of the body are considered added mass terms as, similar to the inertial mass, they are multiplied by accelerations to determine forces. The acceleration of the body can be expressed in the body frame such that the added mass is real-ized as [32]: X A = X . u + X. (w + uq) + X . q + Z .wq + Z . q 2 (2.30) U W q W q + X .v + X.p + X . r - Y . v r - Y . r p - Y . r 2 v p r r v p r r - X . u r - Y .wr V w + Y . v q + Z . p q - ( Y . - Z . ) q r w n p r n v q r / n Y A = X. i i + Y . w + Y . q (2.31) ^ v w q + Y . v + Y .p + Y.r + X . v r - Y . vp + X . r 2 + (X. - Z.)rp - Z . p 2 v p r r v w r r p T r pr - X . (up - wr) + X .ur- Z . wp w r u w - Z . p q + X.qr Z A = X . ( i i -wq) + Z .w + Z . q - X . u q - X . q 2 (2.32) A w v w q u q + Y . v + Z.p + Z.f + Y .vp + Y.rp + Y . p 2 w p r r v r r r pr + X up + Y .wp v r w 1 - X v v q - ( X n - Y . ) p q - X . q r K A = X . u + Z.w + K . q - X . w u + X . u q - Y . w 2 - ( Y . - Z . ) w q + M . q 2 (2.33) A p p q n v r ^ w v q r ^ r n + Y.v + K.p + K.f + Y . v 2 - ( Y . - Z . ) v r + z . v p - M . r 2 - K . r p p p r r w v q r p r r q r + X . uv - (Y. - Z . )vw - (Y. + Z . )wr- Y .wp - X .ur w x v w 7 v r q 7 p r q + (Y r + Z.)vq + K . p q - ( M Q - N r ) q r M A = X q ( u + wq) + Z .(w - uq) + M .4 - X^(u 2 - w2) - ( Z w - X u)wu (2.34) 2 2 , + Y.v + K.p + M.r + Y . v r - Y . v p - K . ( p - r ) + ( K . - N . ) r p q q r p r r r v r P r - Y . U V + X .VW w v - M . p q + K.qr Y W U V + X v v w - < X r + Z p ^ u P ~ w r > + ( Xp - ZP(WP + u r ) 15 N A = X.ii + Z.w + M.q + X . u 2 + Y . w u - ( X . - Y . ) u q - Z . w q - K . q 2 (2.35) A r r r 1 v w p q ^ p ^ q ^ + Y. v + K.p + N. r - X . v 2 - X . vr - (X. - Y. ) vp + M . rp + K . p 2 r r r r v r v p q r r q r - (X. - Y . ) u v - X . vw + (X. + Y.)up + Y.ur + Z.wp u v w q p / r r q - ( X . + Y . ) v q - ( K . - M . ) p q - K . q r q P P q / r n r n Combining all the added mass functions which multiply by rate derivatives in the body frame in matrix form, M A = -X . X . X . X . X X u v w p q Y . Y . Y . Y . Y . Y u v w p q z . z . z . z . z . z u v w p q K . K . K . K . K . K u v w p q M . M . M . M . M . M u v w p q N . N . N . N . N . N u v w p q (2.36) These functions of frequency are expressed in the body frame here, while the fluid exists in the inertial frame. To include the effects of this transformation of the acceleration, we must include coriolis components so there exist added mass coriolis terms which can be seen in eqs. 2.30 to 2.35: 0 0 0 0 - a 3 a 2 0 0 0 a 3 0 - a , 0 0 0 - a 2 a, 0 0 - a 3 a 2 0 - b 3 b 2 a 3 0 -aj b 3 0 -bj - a 2 aj 0 - b 2 bj 0 (2.37) where, a, = X . u + X .v + X .w + X . p + X.q + X.r , 1 u v w p r q ^ r (2.38) (2.39) a 9 = X . U + Y .v + Y . w + Y . p + Y . q + Y.r , z v v w p q r a, = X . u + Y . v + Z .w + Z.p + Z.q + Z.r, w w w p r q r b, = X .u +Y.v + Z.w + K.p + K . q + K.r , 1 p p p p r q ^ r b-, = X .u + Y . v + Z.w + K.p + M.q + M.r,and 1 q q q q q r b, = X . U +Y.v + Z.w + K.p + M.q + N.r. J r r r r r r (2.40) (2.41) (2.42) (2.43) (2.44) 16 In Feldman [28], these terms are omitted in the equations of motion. The coriolis and centripetal added mass terms are small for a high speed submarine because the angular rates are small but will be included in this work. The roll rate for an unmanned slender vehicle may be large which warrants the retention of these terms. The evaluation of these terms will be detailed in the follow-ing chapter. Damping The damping terms in eq. 2.27, are those terms which are multiplied by velocities [u, v, w, p, q, r}. The total damping is the sum of the viscous effects and potential effects. The viscous effects are due to vortices, and skin friction, while the radiative, potential damping forces are those forces due to surface wave development, and typically will be zero at very high and low frequencies. The total, linear damping matrix is shown in matrix form as: X u X v X w X P \ X r Y u Y v Y w Y P \ Y r D = - Z u Z v Zw Z P Z c Zr K u K v K w K P K r M u M v M w M P M q M r N u N v N w N P N r (2.45) These terms are usually highly non-linear, as shown in the following chapter. As the forces are also coupled, the matrix is shown "full", not diagonal. The potential damping forces are also extremely depth dependent as shown in the following chapter. For deeply submerged conditions, only the viscous damping terms are present. The viscous damping terms are often analysed in terms of lift and drag [33], lift being the force normal to the direction of flow and drag being colinear with the direction of flow. Restoring Forces The restoring forces on a vehicle are those which are due to hydrostatic forces, and lend to static stability. These forces, although due to the fluid properties are considered separately. When totally submerged, and neutrally buoyant (W = B) there are only restoring moments. The total weight of the vehicle is given by W = mg, where g is the acceleration due to gravity, and W acts through the center of mass. Similarly, B = -pgV, where V is the total displacement of the vehicle, g is the acceleration due to gravity and p is the density of the water. B acts through the center of buoyancy. Both these forces are referenced to the inertial frame and must be transformed to the body frame using the vector transformation matrix Qj: o 0 W , and F b = Q; 0 0 - B 17 To find the corresponding moments about the body frame origin, the forces must be cross multi-plied by their position vectors, r ^ , and rb: M =01 0 x g 0 X , and M b = w = Q, 0 x b 0 X y b - B The total external, restoring vector including both forces and torques would then be: ?bg = - (W-B)s in0 (W-B)cos9sin$ (W-B)cos6cos(() (y gW-y bB)cos9cos(j)- ( z g W - zbB)cos9sin(|) - (z gW - z bB) sin6 - (x gW - x bB) cos9cos<|> (x gW - x bB) cos 9 sin<j) + (y gW - y b B) sin 9 (2.46) One should note that for vehicles at the surface, the value of B and the vector {xb, y b, zb}, is non-linear and depends on the instantaneous surface profile and submerged hull shape. Often the effect is assumed linear and the metacentric height (distance from center of mass to center of buoyancy), is used to determine coefficients based on the state variables {z, <)), 9, cp} such as Z z , and Me-Summing the added mass terms, damping, and restoring terms we arrive at the total hydrody-namic force, Th = - M A Y - C A v - D v + T b g (2.47) The hydrodynamic matrices here are coupled, and non-linear. These forces will be acting on the rigid body as part of the non-linear equations of motion described in section 2.3. 2.2.6 Incident, diffracted and environmental forces The incident and diffracted forces are due the wave field surrounding the vehicle. This can be thought of as the forces on the vehicle due to sinusoidal incoming waves while the vehicle travels on an undisturbed trajectory. It is common practice when the beam of the vehicle is small with respect to the incident wavelength to assume that the wave field is undisturbed by the body. Under these assumptions, the reflected potential is negligible compared to the incident wave potential so no diffracted waves (reflected from the hull) exist. The incident forces are referred to as Froude-Krylov forces. The incident forces will be frequency, and wave amplitude dependent and are described further in Chapter 4. The external wave force vector will be defined by xw. It should be noted that although waves are shown to contribute to incident disturbances, the effect of waves 18 causes a difference to the fluid velocity potential which in turn affects the added mass radiation forces. Other environmental forces which impinge on a submarine could be due to currents, drift, or wind. As these effects are usually negligible with respect to other disturbances, they will not be addressed here. 2.2.7 Controlled and Additional Forces For high speed underwater vehicles (travelling faster than 3 knots) the actuator or control forces are usually provided by hydrofoils, or hydroplanes. A typical complement of foils consists of two bow planes, two stern planes and a rudder. There are numerous possibilities but only one case will be considered in this work. Forward thrust is provided by a motor or engine driven (prime mover) propulsor, typically a single screw propeller. Some of the external forces of interest are shown in Figure 2.2. Just as the hydrodynamic body forces were determined using a Taylor series expan-sion, the controlled forces xc are found in the same manner. For example the axial component of force generated due to actuators can be expressed as: X c = X 5 f p 8 f p + X 5 f s 8 f s + X 5 a p 8 a p + X 5 a s 5as + X 6 r 8 r + X 6 n 5 n + X 6 f p 5 f p 6 f p5fp + ... (2.48) X 5 f p 8 f s 8 f P 5 f s + X 8 f p 8 a p 6 f P 5 a P + -The symbols, f, a, p, s, r, n, designate foreplane, aftplane, port, starboard, rudder, and engine speed respectively, while 8 indicates change in angle, or engine speed. As a subscript, these symbols represent partial derivatives. The vector u = [8fp, 8fs, 8ap, 8as, 8r, 8n]T. Note here that no hull-plane interactions are taken into account, and that if plane-plane or plane-propeller interactions were neglected, only the first line of equation 2.48 would be considered. In linear, matrix form these coefficients take the form, X 8 f p X 8 f s X 8ap X 8as X 8 r X 8 n Y 8 f p Y 8 f s Y 8ap Y 8as Y 8 r Y 8 n B = Z 8fp Z 8fs Z 8ap Z 8as Z 8 r Z 8 n K 8 f p K 8 f s K 5ap K 8as Kgr K 8 n M8fp M 5 f s M5ap M 8 a s M 5 r M 6 „ N 5 f p N 5 f s N 5 a p N 5 a s N 8 r N 5 „ (2.49) So the controlled forces and moments on the body are: T . = Bu (2.50) 19 Figure 2.2: Forces due to planes and propeller. Extensive study has been done on plane forces and the realization of these coefficients is shown in the next chapter. The forces due to planes are generally found up to a maximum of third order by deflections, while propeller forces are found up to second order. 2.3 Non-linear Model State Equations of Motion Modern design for multi-input, multi-output control systems requires the assembly of the differ-ential equations of motion or dynamic plant equations into state-space form. The general deter-ministic state-space equations are expressed as: x = A(x, t)x + B(x, u, t)u (2.51) y = C(x, t)x + D(x,u,t)u, (2.52) Here, x is the state vector, y is the measured output vector, u, the input vector, and A, B, C, D are the plant system matrix functions, which can be time varying and non-linear. 20 The rigid body non-linear equations of motion, eqs. 2.21 to 2.26, are combined with forces on the rigid body, x = xu + x + x , (2.53) - rb - h -w _c v ' where xh represents the hydrodynamic forces on the hull, X w represents the wave forces on the vehicle, and X£ represents the actuator forces on the vehicle. Substituting into eq. 2.20, this arrives at the following equation, in matrix form: M r b v + C r b v = T h + T w + T c , (2.54) or, in expanded form, the forces in the x direction are: (2.55) 2 2 m ( u - v r + w q - x G ( q +r ) + y G (pq - r) + z G (pr + q)) = (2.56) X . u + X . v + x . w + X . p + X . q + X . r + } added mass u v w p q r X . i i + X .(w + uq) + X . q + Z . w q + Z . q 2 + X . v + X . p + X . r - Y . v r - Y . r p - Y . r 2 + u w q w q v p r v p r - x .ur - Y . wr + Y . vq + z .pq - ( Y . -Z . )q r } coriolis and centripetal v w w p q r 1 X u u + X v v + X w w + X p p + X q q + X r r + X u u u u + X u v u v + X u w u w + X u p u p + X u q u q + X u r u r + x w v v + x w w w w + X P P P P + x q q q q + x r r r r + 1 damping X 8 f p 5 f P + X 5 f s 5 f s + X 5 a p 8 a P + X 8 a s 8 a s + X 5 r 5 r + X 5 n 5 n + } Controlled x b g + x i n c 1 restoring and incident. Note here, not all the hydrodynamic terms were included as many are of negligible order, which will be determined in the next chapter. In matrix form, some of the hydrodynamic terms are included in the "left hand" side and using eqs. 2.46, 2.47 and 2.49 we get, ( M r b + M A ) v + ( C r b + C A + D)v = X b g + X w + Bu . (2.57) In addition to these body frame dynamic equations, the kinematic relations from eq. 2.11, are non-linear and contain vital state information, combined and separating the time derivatives to the left hand side, the set of equations become: v = ( M r b + M / 1 [ - ( C r b + C A + D ) v + x b g + x w + Bu] , (2.58) and, 2 1 n= Q(TJ)V. (2.59) Note that these equations are not exactly in state space form as the restoring and incident forces may not necessarily be a multiplicative function of y, Q, or u. However, if the incident and restor-ing forces can be determined explicitly, these two equations can be used in a numerical integration routine to determine the variables, v, and rj. in the time domain. 2.4 Linear State-Space Model Equations of Motion To simplify the design of the control system and facilitate analysis of the plant, eqs. 2.58-2.59, are linearized. Typically, and for this design, the equations are linearized about a mean operating state, or condition, being, x 0 = [uo 0 0 0 0 0 x 0 y o z 0 0 0 V o ] ' (2.60) This type of linear assumption is different from the hydrodynamic linear assumption, that incident wave forces are linear with wave amplitude. It is assumed here that the vehicle is travelling in "straight and level" flight, such that many of the state variables have zero magnitude, as shown in eq. 2.60. A mean forward speed is designated, U = u 0 . If the vehicle is travelling in a pose other than this, U = Ju, nents, x 0 and y 0 are a function of time 2 2 2 0 + v 0 + w 0. Also, z 0 and \|/ 0 are constant values while the position compo-x 0 = u0cos(\|/0)t, and y 0 = u0sin(v0)t Small deviations from this mean state, (2.61) (2.62) Av = v - vQ, and are used to find the linear relationship, (2.63) (2.64) Av ATI. Q o Av — AT) 0 i(2.65) The matrices are composed of terms, as opposed to non-linear functions. In this form, any steady force is not realized, only dynamic perturbations about a linearized mean operating condition. Analysis into stability in this form is performed in Chapter 5. Eq. 2.65 represents the linear equa-tions of motion in state space form. 2 2 2.5 Summary This chapter presents the notation and variables necessary to describe the equations of motion of an underwater vehicle. The various reference frames are given and explained. The Newton-Euler conservation equations are used to formulate the rigid body equations of motion. A brief explana-tion of hydrodynamic effects is given, and the structure of hydrodynamic added mass and damp-ing coefficients is presented. External buoyancy and gravity forces are included in the equations. External controlling forces are also presented in a general manner. The overall structure of the non-linear equations of motion are shown and then the linearized equations are given in matrix form explicitly. Further analysis of the external forces and moments will be given in the following chapters. 23 3.0 Hydrodynamics The most challenging part of creating a good mathematical model for an underwater vehicle is the determination of the vehicle's hydrodynamic characteristics. Good understanding of the hydrody-namics results in a more realistic model and better simulation results. Traditionally, the hydrody-namics of ships and submarines have been treated differently as ships travel on the free surface while submarines are often submerged. As the vehicle in question is a near surface craft, both areas will be examined here. The study of hydrodynamics is the analysis of the relative motion of the water around a body and the ensuing interactions of water particles and the body. The goal of hydrodynamic analysis for marine vehicles is the determination of the forces on the vehicle in question for a given operating condition or dynamic motion. These forces are due to the pressure exerted on the surface of the vehicle: F = J J (Pn)dA, (3.1) s where n is the unit vector normal to the surface of the body, P is the pressure field of the fluid sur-rounding the body, and A is area. This equation neglects shear stresses of the fluid on the hull. The complexity involved is the calculation of the pressure field for a given condition or instant in time. The fluid is considered incompressible so that, for an infinitesimal volume, conservation of mass can be written as: ^ L + |y. + ^ = 0 . (3.2) ox ay az Then, if one assumes an irrotational flow, the fluid continuum is governed by Laplace's equation: V2<J> = 0, (3.3) where it is assumed that a velocity potential function, 0(x,y,z,t), exists throughout the fluid. If the foregoing assumptions are not made, one must use one of various methods to solve the famous Navier-Stokes equations which completely describe the motion of the fluid. This is very manually and computationally intensive. As a practical approach is necessary, we continue with determin-ing the potential function. Using conservation of energy one can derive the Bernoulli equation, which, in turn, is used to determine the pressure from the known potential function. The exact solution to the continuity equation, eq. 3.3, depends on the given boundary conditions, that of the ocean floor, the immersed body, and the free surface. Numerical methods for accomplishing this are described in the follow-ing section. There are two main approaches to determining the hydrodynamic forces on a vehicle: the study of the entire body geometry in relation to the surrounding fluid, or the study of each constituent com-ponent of the body and their interactions. Both these methods use the assumption of the principle of superposition and interaction. Superposition stems from the linear property of the solution to 2 4 Laplace's equation. The potential function can be formed from the sum of constituent potentials, each due to various effects. <p = 0 1 + 0 2 + + . . . (3.4) For example, the solution for a long crested linear wave can be added to the hydrostatic solution to determine the resulting pressure field beneath the wave. As complete solutions of this manner are often quite involved, a more practical solution is the use of a Taylor expansion of the forces on the body in question (eq. 2.48), using the available kine-matic state variables, about a given operating condition. This results in a high order function describing the forces on the body, for example in the y direction: This notation is such that the subscript defines a partial derivative, for example, Although this leads to a large number of terms, it has been found that for excellent accuracy terms greater than third order may be neglected. Common practice is the use of only first and second order terms in velocity and first order terms in acceleration. It should be noted that positional or restoring forces are accounted for separately. Many terms are also neglected due to low order of magnitude. The partial derivative terms shown in eq. 3.5 are an example of hydrodynamic coefficients or hydrodynamic derivatives. As the comparison of these terms is often necessary due to evaluation by different means, or scale model testing, it is useful to define non-dimensional coefficients. There are a set of parameters by which to non-dimensionalize according to different standards. The most common standard for non-dimensionalization used in submarine analysis is that of Feldman [28]. A non-dimensional parameter is designated by a prime (') symbol. Feldman uses the dynamic pressure by vehicle length, L 0 , squared, as a basis for force non-dimensionalization, where U is the magnitude of the speed of the vehicle. The units of eq. 3.7 is Newtons if the S.I. system is used for speed and length. Y = Y u + Y . U + Y - U + ... + Y v + Y .v + Y-v + . . . Y n + ... u U U v V V (3.5) + Y u u u u + Y u v u v + ... + Y r r rr + ... (3.7) (3.8) 2 5 The speed of the vehicle, U, is considered, for non-dimensional purposes, to be the operating con-dition, U 0 . It should be noted that although coefficients may be described in non-dimensional, form, each non-dimensional parameter has been derived at a particular operating condition about which the Taylor series expansion was performed. The overall speed is used to non-dimensionalize speed components (u, v, w). So for example: v Y Y Y N y = - Y ' = — Y ' = v v Y ' = u r and N ' = p q (3 9) 1 j j2y 2' 1 T T T 2 ' * V V 1 T 2 ' u r 1 T 3 ' PI 1 T 5 ' ^ } - p U L - p U L - p L - p L - p L It should be noted that a non-dimensional derivative must be multiplied by a non-dimensional kin-ematic parameter, i.e. u/U, or pL/U, to ensure a non-dimensional force. It is also important to real-ize that in this work the linear derivatives use forward speed rather than the magnitude of the speed vector, as a non-dimensionalizing parameter. This is due to the operating condition that the Taylor series is expanded about because u=U if v and w are zero. For example: Mu) X u w vOAijJ X' = — — = — — - (3.10) ft - \ SI -N 1 TTT 2 so X w = X u w u , although, when considering an operating point having u * U , then first order coefficients must be multiplied by the magnitude of the speed vector U = {u, v, w}. Another common method of non-dimensional analysis is that used by Aucher [23]. The only minor difference with the aforementioned system is that the area parameter, vehicle length squared, used above is replaced with hull frontal area, the cross-sectional area of the hull pro-jected onto the y, z plane. The use of hydrodynamic coefficients can be applied to a hydroplane, rudder, or any other compo-nent. In the case of a foil, the lift and drag are common coefficients, and use either the chord squared or planform area of the foil in question for non-dimensionalization [33]. Surface piercing struts may use the foil thickness. Generally, lift (L) and drag (D) in Newtons, are non-dimensional by dynamic pressure by planform area, giving kg-m/s2: C L = L , and C D = D , (3.11) I 2 2 I 2 2 26 where S is the planform area (span times chord referring to Figure 3.1). Lift, L \ , — _ — U Figure 3.1: Hydroplane parameter illustration. The forces of lift and drag are given in the directions normal to the flow, and parallel to flow respectively. These forces then have to be transferred to the foil reference frame such that: C z = - C L c o s o c - C D s i n a (3.12) C x = - C D c o s a + C L since (3.13) The designation, ' C followed by a subscript of force component indicates a non-dimensional force coefficient. By calculating the sum of all body element forces transferred to the body refer-ence frame the total body forces can be found. 3.1 Prediction Methods Some of the hydrodynamic coefficients shown in the previous section (eq. 3.10) are linear with angle of attack, however, actual hydrodynamic relationships are highly non-linear and coupled. For the slender, high speed vehicle in question, most of these non-linearities will be forward speed (u component) dependent, as the relative magnitude of the other velocities will be small. Damping terms such as X u u , Z u w , and N u p would be added to construct the non-linear model, as well as con-trol terms such as X d T d T . Also it should be noted here that the added mass (MA) a n c * damping (D) terms for an object near the surface, i.e. affected by a sinusoidal forcing function (due to waves), will be frequency dependent. This frequency dependence can be converted to a function of time. This effect will not be covered here. 27 There are various methods for determining the infinite frequency and also frequency dependent hydrodynamic terms based on vehicle geometry: 3.1.1 Semi-empirical Methods based on algorithmic curve fitting of experimental data are useful for predictions of hydrodynamics [34, 35, 24]. Most of these equations are based on extensive testing. Due to the limited variation in slender submarine designs, these predictive methods are fairly accurate and tend to produce comparable results. The advantage of semi-empirical methods is that experimen-tation and relationships can be formed for individual components of a submarine geometry. Also, viscous and turbulent effects around a component can be taken into account. The difficulty with using experimental data is the matching of Reynolds numbers between cases. The viscous effects change with changes in Reynolds number so various scaling methods must be used. Unappended Hull Coefficients Generally, one begins with calculation of the hydrodynamics of the unappended hull form. For the case of submarines, hull forms are usually cylindrical or elliptical bodies of revolution having a hemi-spherical nose and conical tail section. The seminal work in this area is attributed to Munk [36, 37], for his work on dirigibles. Consider-ing a streamlined, slender body of revolution, Munk found that the moment coefficient for the body due to an angle of incidence to the fluid flow as: C M = 2k A (k z -k x )e (3.14) V where, k A = — — , (3.15) A o L o e is the angle of incidence in radians, k x is the added mass ceofficient in the x direction, k z is the added mass coefficient in the z direction, V is the displacement volume of the body, and A 0 is the frontal area. This is shown in Figure 3.2. II Figure 3.2: Slender Streamlined Body. 28 Munk's coefficients are given by the following table given in reference [36]: TABLE 3.1 Munk Coefficients of Added Mass. Slenderness Ratio K K 1 0.5 0.5 2 0.21 0.7 4 0.08 0.86 6 0.045 0.92 8 0.03 0.945 10 0.021 0.960 oo 0 1 Aucher [23] continues this approach with the following development. The frictional drag coeffi-cient on the body is defined by: 0.25 C x " = — - x _ f ( log 1 0 (Re)-2) 2 D o (3.16) where Re is the Reynolds number, and D 0 is the body diameter. The letter C is used to denote that the parameter following is non-dimensional. Then, C z ', is given by cy = - ^ + 6.6 A 0 kaft V 0.7 74 . 5 + 0 .5C x ; . (3.17) Here ocA R is the mean half angle of conicity of the tail section, and A a f t is the cross sectional area of the aftermost section of the body. Defining x A R as the axial distance the center of lift located between the point of fluid separation from the body at the tail, and the aftermost point of the tail, the following additional coefficients are found: * - - M C ' = 2kA(kz - k x) + C Z r ' x A R C M ' - - Z E ' ( X A R ) (3.18) (3.19) (3.20) The reason that the hydrodynamic coefficients are found for an angle of incidence is that for small angles, the angle of incidence can be related to the localized cross flow, w, by the approximation: 29 (3.21) As the aforementioned coefficients are for a symmetric body of revolution, the theory can be applied to an additional orthogonal axis: zw ' = Y ; , zq- = - Y ; , M W - = - N ; , and M Q ' = N ; . Appendage Coefficients Semi-empirical formulations also exist for the calculation of foil and appendage hydrodynamics [38, 39, 24, 40, 41, 42, 43, 44]. First, isolated, infinite wing forces are found, then modifications due to aspect ratio, hull interaction, end-plates, and wing to wing vortex and downwash interac-tion are applied. Effective aspect ratio, ae, is defined as: 2b a„ = — c e - (3.22) The span, from wing tip to tip is given as 2b, and the mean chord is defined as c. Typically, hydro-foils have low aspect ratios, 2<ae<3. Whicker and Fehlner [35] derive the lift and drag non-linear expressions for isolated hydrofoils as: C L = CLaa + —a\a\, and (3.23) a e C 2 C ° = C d o + 09^a"e r e s P e c t i v e l v - ( 3 ' 2 4 ) Here a is the localized angle of attack in radians similar to e in Aucher. The same small angle approximation can be used, however body motions are included such that, (for a horizontally mounted plane): a = W _ q X g a n _ e + 6 ( 3 2 5 ) To evaluate these expressions, the lift slope and crossflow drag are found by: C-Lcc -1.87ia„ 2 a 1.8 + cosy 4 + —— H (cosyp) , and (3.26) 30 C D c = 0.1 + 0.7 A.p for a rounded foil tip, or (3.27) C D c = 0.1 + 1.6Xp for a square foil tip. (3.28) Plane sweep angle is denoted by y p and the taper ratio is given by Ap. The parameter Cpo is the minimum drag value which occurs for zero lift, and is in the order of 0.01, or 0.0065 according to Whicker and Fehlner [35]. The location of these forces of lift and drag are commonly assumed to impinge axially at the 1/4 chord location (from the leading edge). Spanwise, one can assume an elliptic loading profile such that the non-dimensional spanwise location is given by: cp»"4- <3-29) from the hull. Whereas, according to Aucher [23], the lift and drag are found by: 271^1-3^ C L = C / ' a , and (3.30) cosy J— -^y + 2 + 2.6 ^(cosYp ) C D = 0.01 + 1.10a2. (3.31) This assumes that the minimum section drag value is 0.01. The parameter't' here is foil thickness from upper to lower maxima. The formulas are given for angles of incidence not exceeding 25 degrees for an aspect ratio in the order of 2. Modifications of effective aspect ratio due to endplates are described in [33], and basic effects can be realized by using modified aspect ratio: ae' = k e p a e (3.32) k e p = l + 1 . 9 i L _ 0 . 5 ( A j , (3.33) where h is the endplate height. Other appendage calculations can be found by referring to [45]. Appendage added mass values can also be found through semi-empirical methods [30]. As the added mass values for ellipsoidal bodies is well understood, the hull components are often represented by ellipsoids. The analysis 31 presented here will use other methods. Calculations using the above mentioned references for the vehicle under study are presented in Appendix B. Semi-empirical evaluation of hydrodynamics in the surge direction is often ignored in submarine simulation codes but good formulation is available [46]. The hull drag can be found by: C D f = 0.0004 + (3.34) (log 1 0 (Re)-2) 2 based on the total wetted area. The first term is a skin friction allowance and is based on the roughness of the hull. The value of 0.0004 has shown good results for new hulls. This relation is based on the Schoener relation. For added mass calculation, Lamb's coefficient of accession to inertia, kL, is found by B m k L = 0.6 (3.35) and can be used to find added mass by = k L m. (3.36) Semi-empirical formulae representing the effects of hydrodynamics are extremely useful for understanding the effects of changes in geometry and position of various components. The values generated using these methods are reputable and have good agreement with experimental data. 3.1.2 Numerical Methods There are various finite element methods for solving for the pressure distribution over a hull form, the two main approaches being those that utilize source distributions over the surface of the body and those that solve the Navier-Stokes equations. Both such methods require the sectioning of the hull surface into smaller regular shaped areas as shown in Figure 3.2. 32 Figure 3.3: Discretization of a Submarine Hull into Panels. Within the source distribution approach there exist strip [47] and panel methods. By assuming a slender body, and inviscid, irrotational flow strip theory has produced excellent predictions of ship motions and is now being used in application to submarines [48]. By assuming a slender body, the change in pressure potential in the longitudinal direction is assumed small: —0 « A<D JL<J> C3 37) 3x- dy-d - 1 } In this way, the change in pressure component normal to the hull surface in the 'x' direction is assumed negligible. By this assumption, the hull is represented by a series of 2 dimensional cross sections in the y-z plane [49]. Sources are then distributed on the segments of the two dimensional curve by methods such as the Frank close-fit method. Using the frequency domain Green function for a two dimensional source, the fluid potential is represented as a function of the line or point source. $(y> z) = ^Jo(y. z> y s. z s)?(y s> z s ) d S (y s > z s ) (3-38) s The source location is given by (xs, y s, zs), and strength by the function o. The Green functions are integrated along the surface to determine the total fluid potential. By this method the pressure on the sections are used to find sectional added mass and damping coefficients which are in turn integrated for total body added mass and damping values. The strip theory approach is used in 33 code developed by Defence Research Establishment Atlantic (DREA) in Submo [50, 51] and Shipmo [52, 53]. Strip theory is limited to low Froude number (Fr < 0.4), slender bodies (L/Bm > 4), and linear wave conditions. Although, research has shown strip theory to have good predic-tions for beamier vessels (L/Bm > 2). Panel methods are similar to strip theory in that sources are distributed on the surface of the body. The velocity potential is then solved in the three dimensional case for the discretized hull. This method is more computationally intensive and can give rise to singularities and numerical prob-lems. The representation of a body as a combination of various ellipsoidal elements is another approach to determine the added mass for a submarine hull as mentioned previously. Interference effects must be accounted for and correct fitting of the geometry to the given hull [1]. Finally, complete Navier-Stokes solutions for hulls are also possible. The advantage to a complete numerical solution is that circulation, turbulence and viscosity are all accounted for. For this solu-tion, one of several approximations must be made for the velocity profile between discrete sec-tions, and some type of turbulence model must be assumed. Navier-Stokes solutions are limited to internal flows, so in many cases a free surface profile must be assumed. Some general methods of solution include the K - E model and the Reynolds Averaged solutions. All such approximations require extraordinary computational and intellectual resources. 3.1.3 Aerodynamics It is possible to use the results of a large amount of research done on flight and apply it to sub-mersible dynamics. A popular program for use in hydrodynamic coefficient prediction is USAF DATCOM, which is now in the public domain. This program is very versatile in its acceptance of hull geometries, and has produced good results [54,55]. Factors which affect collaboration of aer-odynamic and hydrodynamic analyses are the Reynolds number and compressibility effects. However, wind tunnel experimentation and aerodynamic semi-empirical results can be readily applied to hydrodynamic conditions. The applicability of Missile Datcom [56], also in public domain, to submarine use has not yet been determined. This program is substantially easier to use and is pertinent to slender axi-sym-metric bodies having sets of planes. Plane to plane and plane to body interference factors are taken into account. Fluid properties, must be entered appropriate to water. This program is rela-tively new and much of the documentation requires revision. The code has been designed in a 'top down' fashion and the analysis of each component is performed by a separate subroutine. The basis of much of the formulation is taken from USAF DATCOM, AIAA publications, NASA pub-lications, and Hoerner [33]. The results show the contribution of various effects separately. As implementation of this code is fast, many results can be generated and compared in a short time period making the code ideal for design and prototyping work. When using aerodynamic force derivatives, it is important to note that non-dimensionalization is performed using different parameters. Results from Missile DATCOM for the vehicle in question are presented in Table 3.3 for comparison, and appropriate files are shown in Appendix B. 34 3.1.4 Experiment Using scale models in towing tank facilities to determine full scale forces is common practice in marine engineering. Non-dimensional scaling of geometry, Reynolds number, and Froude number scaling is used. It is not possible to simultaneously have Reynolds and Froude number scaling. Planar motion machines, free swimming models, and towing tanks are well suited to the analysis of different coefficients and conditions, and vary in complexity [57]. Experimentation is neces-sary for validation of theoretical results and if standards are maintained, experimental results are more accepted. The general problems associated with experimental work include test repeatability and reliability, and the resolution or accuracy of the results. Equipment must first be calibrated with known parameters in order to determine measurement linearity, hysteresis, and offset param-eters. The use of wind tunnels is also valuable in experimental research. Undoubtedly, tank testing pro-vides more applicable data for underwater vehicles although wind tunnel results are useful for flow visualization and determining relative quantitative results [58, 59, 60]. There are many flow visualization techniques for determining hydrodynamic interactions. Of these Ostafichuk [58] uses helium bubbles, yarn tufts, surface oils and smoke streaks. Each of these methods provides different information. Yarn tufts prove useful for near body flow patterns and separation. Smoke streaks collect in spiral vortices and indicate the location of the vortex. Surface oils can show sep-aration and near surface flow patterns. While helium bubbles spread in a cloud to show stream-lines in a given area. The design and use of accurate load cells and instrumentation is critical. Sting or lateral mounts interference and interaction must be accounted for by calibration measurements. In tank testing, much submarine testing is performed using a static or constrained test rig. These set-ups only per-mit translational, and limited rotational coefficient analysis. Only velocity coefficients are possi-ble. Dynamic or free swimming models are extremely costly, but allow for increased number of hydrodynamic variables. Often in experimental work, the parameter sought is measured by one or more related variables. In this way, measured results must be manipulated in order to derive the desired parameter. For the DOLPHIN vehicle, planar motion machine (PPM) testing is limited to heave and pitch information due to the partial submergence of the mast. The advantage of a PMM is that the body can be mounted at any angle and parameters related to a given plane are measured. The free sur-face effects constrain use to one plane. Much towing tank experimentation has been performed for the vehicle [61], and also full scale testing [62]. Full scale results are valuable but have large financial costs. An alternative to these methods is the Marine Dynamic Test Facility (MDTF) [63] which is under development. This six degree of freedom constrained dynamic machine allows for calculation of all dynamic coefficients to higher order through measurement of the forces in time. This also allows for free surface and frequency dependent hydrodynamic experimentation. 3.1.5 System Identification A relatively new approach in the marine industry is the use of instrumented models or full scale vehicles in parameter identification. Using the relationship between the input levels and measured 35 kinematics of the vehicle, the dynamics of the vehicle can be captured. Although stability and cost may be issues, results of this method have proved promising [64, 65]. It should also be noted that the hydrodynamic coefficients are not always found explicitly, rather, the system is identified in state-space or auto-regression with moving average ARMA form. Control of the system in these forms is then readily implemented. System dynamics identification can be implemented 'on-line' in a recursive manner [66] and incorporated into a control strategy. In such a case, stability and non-linear identification become issues, although the advantage is that of an adaptive control system. System identification may also be used in experimental testing in a 'batch' sense. Logged data is analysed after being recorded. High order linear models can be easily curve fit to the data by least-squares, or Kalman filter fitting of the parameters [67]. For any use of system identification techniques, the structure of the system must be proposed a priori. The parameters of the proposed system are then calculated using the data set. As noise and high order non-linear effects may be present, some type of filtering or averaging is usually used to ensure good results. For proposed dynamic systems which are decoupled and have a limited number of variables, identification is made simpler. For dynamic test facilities such as PPM's or the MDTF, system identification is necessary in order to determine the hydrodynamic parameters using the measured forces and moments. System identification methods are used in Chapter 4 for the analysis of a towed cable and body system. This illustrates the applicability of the technique to complex multi-component systems, where physical study of the system may be unwarranted. 3.2 Infinite Frequency Coefficients The hydrodynamic coefficients shown in eqs. 2.36-2.45 are infinite frequency coefficients. That is, the derivatives are independent of any sinusoidal motion by the vehicle causing a radiative effect, or any incident free surface waves causing diffracted and incident hydrodynamic effects. Linear coefficients of this type are useful for performing maneuvering predictions, open loop sta-bility tests, hull modifications, and hydroplane and propulsion design [68, 69, 70]. For time dependent response of the vehicle to the hydrodynamics, a more involved approach using mathe-matical maneuvering model simulations is needed. In general, maneuvering models use only infi-nite frequency coefficients [71]. This is equivalent to the case of a deeply submerged vehicle, as the fluid can be considered unbounded in all directions. Expressions for frequency dependent added mass and damping coefficients based on strip theory can be found in [47]. Using these expressions and taking the limit as frequency approaches infin-ity, the infinite frequency coefficients can be determined as shown in eqs. 3.14-3.38. The coeffi-cients for the whole vessel are calculated by integrals of the sectional values. (3.39) 36 A A 3 5 = A A 5 3 = - J ^ a 3 3 d ^ (3.40) A A 5 5 = J ^ a 3 3 d £ (3.41) A A 2 2 = Ja 2 2 d £ (3.42) A A 2 4 = A A 4 2 = J a 2 4 d £ (3.43) A A 2 6 = A A 6 2 = J ^ a 2 2 d ^ (3.44) A A 4 4 = J a 4 4 d ^ (3.45) A A 4 6 = A A 6 4 = J ^ a 2 4 d £ (3.46) A A 6 6 = J V a 2 2 d £ (3.47) Note that the infinite frequency added mass values using this method produce the same results as the limit for the expressions given by Korvin-Kroukovsky and Jacobs. Typical sectional damping values diminish at high frequency so infinite frequency damping values are considered negligible using potential flow theory. However, this does not include the effects over time of an oscillatory vehicle motion. These fluid memory terms are explained in Section 3.3. 3.3 Frequency Dependent Coefficients The presence of the free surface irregular waves causes the incident pressure field to change sur-rounding the body. Through the principle of superposition, the irregular wave profile is taken to be a summation of a series of Airy sinusoidal waves. The linear Airy wave profile satisfies Laplace's equation for the fluid. The force associated with the sinusoidal waves is assumed linear with wave height. Strip theory provides added mass and damping coefficients as a function of the frequency with which the body oscillates near the surface. The hydrodynamic coefficients account for the effects of diffraction of the incoming waves by the body, as well as the radiative effect of the vehicle's own sinusoidal motion. Generally, a frequency dependent analysis assumes that the vehicle's response frequency is the same as that of the incoming wave frequency, in regular waves, having a certain phase shift. As such, certain assumptions are made about the stability and controllability of the vehicle. Through these assumptions it is possible to define a ratio or Response Amplitude Operator (RAO) as: R A O = (3.48) 37 which is the square of the response amplitude z a to the regular wave amplitude r | w ratio. The RAO is frequency dependent and is particularly useful for design practices. It is also possible to calculate the incident wave forces using Froude-Krylov theory. Froude-Kry-lov theory also assumes a linear Airy wave profile, having a pressure field given by: n , N -ikw(xcosP + ysinP) kw(z-zs) P = (T|wpg)e e (3.49) where (3W is the relative incident wave angle given by: Pw = V - V w . ( 3 - 5 ° ) \|/w, is the wave heading, rj^ is the wave height, z s is the vertical location of the wave surface and k w is the wave number defined by: 2 k w = — = T (3.51) for deep water waves. The wavelength is defined by A, and the frequency by cow. The incident wave forces will be treated as disturbances to a calm free surface and are discussed in the following chapter. What are needed for system analysis are the radiative and diffraction components or coefficients. These values can be found using strip theory, as described above and overall frequency dependent values are found. As the coefficients are described in the frequency domain, it then becomes nec-essary to determine a time-domain equivalent through inverse Fourier transforms [72, 73]. The time domain function then accounts for the fluid 'memory' effect. The time domain forces are given as: ?m = - J [ K ( t - T ) ] { i j } d T , (3.52) —oo where elements of the kernel of the integral in the retardation function are found by [74], oo K j k(x) = ^J (A j k ( (O e )-A j k (oo ) )cosQ) e Td(O c ,or (3.53) o oo K i k ( T ) = - [ J k e s i n o y c d a v (3.54) J k It-1 CO. e e 0 Here, coe is the encounter wave frequency, A j k are the added mass components, Bj k are the wave damping components, and x is the element of integration. In order to calculate the functions 38 numerically, it is necessary to employ a method of calculating the hydrodynamic coefficients which spans the complete range of frequencies satisfactorily. From experience, strip theory deteri-orates at low frequencies. Also, from experience, equations 3.53 and 3.54 do not always produce similar results by numerical calculation. From the ongoing discussion it is apparent that the effects of free surface waves are significant and cause different results than a calm free surface condition. This is also apparent through the study of the drag coefficient or resistance of the ship in waves [75]. As the ship continues to oscil-late in the vertical plane, its forward speed component is diminished. There are various theories and methods for the calculation of wave drag [76], but it is assumed in this case that due to the submergence of the vehicle in question, that such calculations are not necessary. 3.4 Summary Several methods of computing the necessary hydrodynamic coefficients have been presented and discussed in this chapter. It is desirable that all fluid effects such as the free surface condition, vis-cosity, and changes of the pressure field all be accounted for in the calculation of added mass and damping. It is also necessary that the coefficients be pertinent to the time domain for use in simu-lation. Accordingly, various parts of each of these methods have been used according to their strengths and weaknesses. It should be noted that high order suction forces due to the vehicle operating near a surface have been neglected. Suction effects can be found using the potential flow function. Unless the vehicle is extremely close (depth/length ~ 0.2) to the surface, the effects of suction are considered negligi-ble [77]. Due to the extensive research done on hydroplanes and body geometries, semi-empirical methods have been employed for calculation of damping and lifting effects. As added mass effects are largely governed by inertial forces rather than viscous effects, numerical methods using potential theory have been used to determine added mass effects. Aerodynamic software, is employed for comparison and verification. While experimental results are used as a basis and benchmark for verification. System identification techniques will be employed in the following section to describe the towed underwater system. It is also important to note that many of the coefficients have been neglected due to their relative order of magnitude [9, 8]. Symmetry is also an important factor for the elimination or evaluation of some coefficients as per Imlay [32]. Imlay shows for a general submarine hull shape that many of the derivatives can be neglected or shown to be equivalent. A final summary of the hydrodynamic coefficients derived in Appendix B and their sources, is given in the following tables. Values in each row have been made non-dimensional by the same parameters. SUMADC is a fortran program written to use [49] and integrate the two dimensional strip theory values using [47]. The hydrodynamic terms used in the simulations in this thesis are the values which are darkened in the tables. 39 TABLE 3.2 Hull Added Mass Coefficients Hydro. Coefficient SUMADC Aucher [23] 80M [78] Shupe [79] X.' u 0.000949* 0.000798 0.000668 Y.' V 0.040812 0.0410 0.02059 0.027 Y.' P -0.0005468 -0.001433 -0.000121 Y.' r -0.001389 -0.002163 -0.000016 Z .' w 0.022559 0.0290 0.01912 0.020 z: q 0.0000510 -0.0001057 0.000127 K.' V -0.0005468 -0.001433 0.000121 K.' P 0.0001046 0.0001617 0.000010 K.' r 0.0000037 -0.000217 M .' w 0.0000510 -0.0001057 0.000127 M.' q 0.001746 0.0006432 0.001017 0.00114 N.' V -0.001389 -0.002163 -0.000016 N.' p 0.0000037 0.000217 0.000011 N.' r 0.001827 0.001149 0.00102 TABLE 3.3 Hull Damping Coefficients Hydro. Coefficient Missile Datcom [56] Aucher [23] Shupe [79] Seto/ Watt [80] ARCS 80M [78] [54] X u' -0.003496 -0.001231 -0.003151 -0.001049 y; -0.08576 -0.090 -0.099 -0.160 -0.05326 -0.049 Y P ' 0.01441 0.006773 -0.001585 y; 0.01478 0.0250 0.014 0.0356 0.01505 -0.02888 -0.0670 -0.033 -0.06798 -0.03781 0.007468 -0.006581 -0.040 -0.006974 -0.008749 Kv' 0.005240 0.004056 0.00777 0.0101 -0.001254 K P -0.001460 -0.00145 -0.0000955 40 TABLE 3.3 Hull Damping Coefficients Hydro. Coefficient Missile Datcom [56] Aucher [23] Shupe [79] Seto/ Watt [80] 80M [78] ARCS [54] K r ' 0.000013 -0.0005782 0.000086 -0.0012 -0.000121 M u ' 0.000039 0.0045 M w 0.006756 0.00120 0.015 0.007519 M q -0.006878 -0.008542 -0.004143 -0.00687 -0.005101 N ; -0.004069 -0.0230 -0.0082 -0.00500 -0.01059 -0.0011 N P 0.000528 0.000086 -0.000181 -0.005108 -0.007471 -0.001036 -0.008957 TABLE 3.4 Actuator Hydrodynamic Coefficients Hydro. Coefficient * Aucher [23] Whicker/ Fehlner [35] Shupe [79] Z 8 f - 0 . 0 0 8 9 2 5 - 0 . 0 1 3 - 0 . 0 1 1 3 Z8f8f' 0 . 0 0 4 3 0 Z 8a' - 0 . 0 1 2 - 0 . 0 1 3 - 0 . 0 1 7 3 Z8a8a' 0 . 0 0 2 8 9 Y 8 R ' 0 . 0 1 1 0 . 0 1 1 0 . 0 0 7 3 7 Y 8 R 8 R - 0 . 0 0 3 0 3 X SfSf' - 0 . 0 0 8 7 0 9 - 0 . 0 0 4 4 4 X8a8a' - 0 . 0 1 2 - 0 . 0 0 4 2 7 X 8 R 8 R ' - 0 . 0 0 9 6 6 3 - 0 . 0 0 3 8 3 Hydro. Coefficient ** Data Analysis Lewis [ 8 1 ] X 8 N ' 0 . 0 0 0 0 7 3 3 X 8 N 8 N 0 . 1 3 5 1 0 . 0 8 2 2 3 1 2 2 Nondimensionalby vehicle length and speed (=pU L ). 41 3 4 2 ** Nondimensional by propeller rps, diameter, and vehicle speed. (pD Un, and pD n respec-tively). Geometric location about which hydrodynamic coefficients are calculated differ by a small amount in the z direction based on the center of mass. The relative locations of the center of mass can be used to transform the coefficients to a new origin, but the locations are not known for the references mentioned here. This would affect mainly the representation of the K hydrodynamic coefficients. Also conditions for which coefficients were found differ in speed, between 10 to 15 knots. As the nondimensional form is shown here this of little concern. Aucher!s methods result in a 3rd order non-linear function for plane forces, only the first order coefficients are shown. The coefficients shown by Shupe were derived for a slightly different geometry (the Mark I vehicle). The added mass values shown for an 80 meter vehicle and the damping values shown for the ARCS vehicle are presented solely for an order of magnitude comparison. The basic hull shapes are all cylindrical however the associated appendages are very different. The propulsor coeffi-cients found by Lewis are based on a Series-B single screw propulsor, which is different than the complex, contra-rotating propulsor used in the DOLPHIN Mark II vehicle, but are presented for comparison. The aim of this work is not a detailed hydrodynamic study, but the study of the struc-ture of hydrodynamic effects and their incorporation into a control system design. Sternplane forces calculated by Whicker and Fehlner [35] are modified by an effectiveness factor from Lyons and Bisgood [24]. It is important to note the factors which affect each method of cal-culation in comparison. The values which were used in the generation of the mathematical model in the ensuing chapters are highlighted. From the diverse approach to determining these values, it is hoped that the best possible values have been found such that the validity of the mathematical model is maintained. 4 2 4.0 Simulation To completely examine the performance and characteristics of a given vehicle it is necessary to perform simulations using the mathematical model. In this study maneuvering control is exam-ined and time domain simulation is used. The differential equations of motion are described in state-space form. The differential equations must be integrated in time such that dynamic varia-bles are found as streams in the time domain. Data in this form allows for prediction of response which can be used as a design tool and also for analysis of the effects of certain parameters. The differential equations are modelled using the software package Matlab 5.3 and Simulink. These packages allow for rapid modelling, ease of use and graphing of results. There are many built in functions and procedures which can be used for the construction of the simulation. The structure and code are given in Appendix C. Results from the simulations are presented in Chapter 6. The submarine plant is described in continuous time and the equations of motion are integrated by a fourth order Runge-Kutta integrator. Appropriate initial conditions and simulations are described in Chapter 6. The preceding chapters describe a general approach to mathematical modelling of the vehicle, although guided to application to the DOLPHIN vehicle. The following sections and chapters are particular to the DOLPHIN vehicle and related simulations. 4.1 DOLPHIN Vehicle The DOLPHIN (Deep Offshore Logging Platform for Hydrographic Instrumentation and Naviga-tion) vehicle was conceived between 1981-1983 for the Bedford Institute of Oceanography, in Bedford, Nova Scotia. The institute is a subsidiary of the Department of Fisheries and Oceans. Development and construction of the vehicle has been done by International Submarine Engineer-ing (ISE) and, more recently, International Submarine Engineering Research (ISER). The original role of the vehicle was a stable platform for bathometric study. Travelling under the surface, the vehicle is less affected by ocean waves than a surface vessel of the same size, while able to travel at speeds up to 18 knots using a diesel engine. Related literature may be found in [79, 22, 80, 82, 83, 84, 85, 86, 62]. The vehicle is unmanned and can be launched in conditions up to Sea State 5 [82]. Between 1985 and 1990, 10 DOLPHIN vehicles were built for government and military applications. Some vehicles have been exported to the United States and developed under the name ORCA, or SEALION. The first generation vehicle (1984-1990) having a length of 24 ft is classified as the Mark I model, while the recent 28 ft DOLPHIN (1990-1998) is classified as the Mark II model. Various modifications exist based on these models. Currently, the Canadian gov-ernment is contracting development of the Dorado vehicle, which is based on the late model Mark II DOLPHIN, for the specific purpose of mine-hunting. The vehicle under study in this thesis is the Mark II vehicle, in its prototype development for the application of mine-hunting or towing of a sonar towfish. The geometry of the vehicle is shown in Figure 4.1. The hull is comprised of a cylindrical body section, a hemi-spherical nose, and an off axis, piecewise conical sectional tail. Appendages to the hull are a pair of foreplanes, a pair of aftplanes, rudder, rear stabilizer, mast, and keel. Forward and attached to the keel is a keel fairing but this will be considered part of the keel. 43 Figure 4.1: Overall picture of DOLPHIN Mark II Geometry with Towfish. A contra-rotating propeller provides forward thrust and is powered by a 260 kW caterpillar diesel engine. The propulsor has one propeller of 0.75 m diameter and one of 0.65 m diameter. Air intake for the engine is provided through the mast. Exhaust is released through the stabilizer. Fuel is stored in the nose of the vehicle and is replaced with salt water as it is used. The forward section of the hull body houses the controller, computer, and instrumentation. The electronics section is separated by a water-tight bulkhead from the engine in the rear. Principle particulars for the vehi-cle are given in Table 4.1. TABLE 4.1 Principle Particulars Description Value Units Overall Length 8.534 m Mass 4300 kg Beam 1 m Tail Length 2.35 m Operating Depth 2-5 m Ixx 1315 kg-m2 Iyy 5900 kg-m2 Izz 5057 kg-m2 Displacement 4600 kg 4 4 TABLE 4.1 Principle Particulars Description Value Units Location of C G (from nose) (-4.173,0, 0.188) m Location of CB (-4.087, 0, 0.051) m The appendage particulars are given in Table 4.2. TABLE 4.2 Appendage Particulars Semi-span Location Plane Description Type Chord (Hull CL to tip) (1/4 chord) Foreplanes NACA 0025 0.350 m 0.968 m 2.2225 m Aftplanes NACA 0025 0.350 m 1.016 m 7.4549 m Rudder NACA 0025 0.350 m 0.889 m 7.4549 m Stabilizer t/C = 0.288 0.440 m 1.183 m 7.4777 m Mast* NACA 0025 0.619 m < 5 m 3.822 m Keel t/C = 0.18 1.857 m 1.590 m 4.274 m The hydroplanes have a saturation limit of +- 25 degrees of deflection, and can travel from one limit to the other in under 1 second [62]. The planes are controlled by closed loop hydraulic servo-valves. In this study the actuator positions are assumed linear with input voltage. As well as deflection limits, actuator dynamics are included. The actuators are modelled using a first order lag system. It is assumed that the planes can travel 98% of their full range of deflection in 0.125 seconds. This occurs over a period of 4 time constants. Endplates are present on the hull and tip ends of each plane. There are various other protrusions such as bolts, lugs, and cooling bars, which may have hydro-dynamic significance but are neglected due to their relative size and proximity to the hull. The keel houses a hydraulic winch which is used for towing a smaller sonar 'towfish'. The towfish travels near the ocean floor and carries a side-scan sonar package. Using a diesel engine the vehi-cle is able to travel for extended periods of time while towing, depending on tow depth and for-ward speed. TABLE 4.3 Dolphin Endurance Speed Time Tow Depth 10 knots 16 hrs. 200 m 12 knots 14 hrs. 100 m 12 knots 24 hrs. 15 m Propeller evaluation is given by [80], and thrust calculations are given in Appendix B. The goal of the Dolphin-Towfish system is to travel a given grid or search pattern with accurate knowledge of position while minimizing surge, heave, roll and yaw perturbations of the towed fish. As such, the Dolphin vehicle must follow a given path as accurately as possible while tow-4 5 ing, this means that all disturbances on the Dolphin due to its own environment, the tow cable and the towed fish must be countered by its control system. 4.2 Path Following At the time of this study, the only positional navigation measurement is that of a Differential Glo-bal Positioning System (DGPS). Various other navigation technologies have been tested [82], some with great success, however the growing accuracy, dependability and ease of use associated with GPS systems warrants its use. The DGPS measurement gives a latitude and longitude posi-tion using a timed signal from several satellites. The location is in the earth fixed reference frame. This signal is then corrected using nearby beacons having known locations. The path or route that the Dolphin travels is presently pre-planned and not calculated autono-mously. Autonomous route planning is possible and has been accomplished by other AUV's [21]. Due to limitations of side-scan sonar, swaths must be taken side by side in order to cover a given area. Such a pattern requires turns of 100 to 300 m radius. When the turn is complete, the Dolphin vehicle must be on the correct path to continue the next pass. The Towfish then follows due to cable tension. The pre-planned GPS route is transmitted via radio frequencies to the communications unit mounted on the mast of the Dolphin. The path is a series of waypoints which mark the route. In case of erroneous data, several transmissions are compared before the path is altered and fol-lowed. In the following of waypoints there are two basic approaches for travelling from one waypoint to the next: line-following, or set-point heading [82]. In line-following mode the heading of the Dol-phin is controlled such that the vehicle follows the line connecting two adjacent waypoints. Dur-ing set-point heading mode, the vehicle simply aims directly at the upcoming waypoint. The difference between the two modes is illustrated best in the case of a cross current: 46 a) Des i red Waypoint , d b) Desi red Waypoint , d • Prev ious Waypoint , d-1 Prev ious Waypoint , d-1 Figure 4.2: Path following modes: a) line following mode; b) waypoint following mode. These illustrations do not acknowledge any complex control strategy, only PD feedback. The line following method will cause the vehicle to 'crab' or move parallel to the line while angled into the current. The set-point heading method causes the vehicle to drift off the line and approach the waypoint from an angle. As the path is mapped by a series of points, the line following mode will maintain a smooth trajectory while the set-point mode will result in a winding trajectory. How-ever, the set-point mode ensures the target waypoint will be hit while the line following mode may pass by the waypoint. The method that is proposed here is a combination of both modes which allows for adjustment and relative weighting toward either one or the other. Consider the case of the vehicle between waypoints as shown in Figure 4.3: 47 x Previous Waypoint, d-1 Figure 43: Waypoint Following Variables. In the figure, the desired heading in the earth fixed frame to the next waypoint would be: The waypoint position and heading are given by (xd, yd, \|/d) while the vehicle position and head-ing are (x, y, If the previous waypoint is available (xd_i, yd_i), then using algebra, the perpen-dicular distance from the line (x d .i , y d _ i ) , (xd, yd) to the vehicle can be found as y*. Note that y* is positive valued if the line is on the starboard side of the vehicle. Also note that y* is perpendicular to the line axis, not the vehicle body axis. In this way, the variables \|fe, and y* can be used for vehicle control states. The variables will also be used as a measure of the performance of a given controller in regard to tracking ability. In simulation, an important case to consider is the integration of the heading equations of motion. This is because the infinite integration of heading could result in angles greater than 360 degrees, which is not desirable considering the range of a typical compass. To approach this problem, the (4.1) while the total error in heading would be: Ve = V d - V (4.2) 48 integrated heading is converted to an appropriate heading between 0 and 360 degrees for data stream logging. In terms of the vehicle's sensed heading, the unbounded heading is sensed, using rate gyro inte-gration as well as the GPS system with appropriate error if simulated as mentioned in section 4.4.4. This value is then filtered if necessary. The reason that the heading is not converted before filtering is that the discontinuity between 0 and 360 degrees is difficult to consider using tradi-tional filters. If any other state calculation is necessary which uses heading, these should also be done before conversion. Then the filtered heading signal is converted to a value lying between 0 and 360 degrees. Typically, the parameter \|/ d in eq. 4.1 is found between -180 and +180 degrees using the arctan-gent function. This must be converted to a value from 0 to 360 degrees before the evaluation of eq. 4.2. The resulting error must then be converted back to an error from -180 to +180 degrees such that the vehicle turns to the correct side (port or starboard) to recover from the error. Path planning can also make a considerable difference to the performance and stability of the con-trolled vehicle. Waypoints may be considered targets, the vehicle must hit the target before contin-uing on to the next target. The target, or waypoint must have a certain area or size to account for the resolution of the sensors and the dynamics of the vehicle. Fossen [9] suggests that Rw, or the waypoint target radius should be of the order twice the vehicle length. There are several factors which affect target size selection and path planning: vehicle speed, distance between waypoints, angle between waypoint changes, vehicle dynamics, control gains and sensor resolution. One method to analyse the problem is that of a tracking or pursuit problem [87]. In this case, as the Dolphin vehicle travels within a fixed depth range, waypoint tracking is limited to two dimen-sions. The state variables used for tracking will be (v, r, y*, (p). Feedback, feedforward, integral and disturbance observer control methods will be described in the following chapters using these variables for guidance. 4.3 Complete 11-State Model Axial Force Equation: 2 2 m ( i i - v r + w q - x G ( q +r ) + y G (pq - r) + z G (pr + q)) = (4.3) 1 4 T 3 H=-[- Y .'rp - Y . ' r 2 + Z .'q2] + H ± L [ X - Y .'vr + Z . 'wq] + 2 p 1 r q * 2 u v w ^ . 2 V [ Xuu' u 2 + X 8fp8f P " 25fp8fp + X 8 f s 5 f s ' u 2 8 f s 8 f s + X 8 a p 8 a p ,'u28fp5fp + X, ,'u 8as8as] + V [ X 5R8R , u 2 8 R 8R] + p D o r x 8 n ' ^ L o n + X 8 n 8 n n 2 l - ( W - B ) s i n 6 Lateral Force Equation: 49 2 2 m( v - wp + ur + x G (qp + r ) -y G (p + r ) + z G (qr - p)) = (4.4) 4 3 2 ~ [ Y p ' p + Y. ' r - Z .'pq] + P y [X .'ur + Y .'v + Y p u p + Y r ur - Z^'wp] + 2 P y [Y v ' v + Y 5 R u 2 8 R + Y 5 R 8 R u 2 5 R 8 R ] + (W - B)cos9sin<t> Vertical Force Equation: 2 2 m ( w - u q +vp + x G ( r p - q ) + y G ( rq + p ) - z G ( p +q )) = (4.5) 4 T 3 P-=- [Z .'q] + [X .'uq + Y. ' vp + Y .'p 2 + Y.'rp + Z . 'w + Z 'uq] + 2 Py- [Z w ' u w + Z 5 f p ' u 2 8fp + Z 5 f p 8 f p ' u 2 8fp8fp ] + 2 P y [Z 8 f s u 2 8fs + Z g f s 5 f s u 2 8 f s 8 f s + Z S a p u 2 8 a p + Z 8 a p 8 a p u 2 8 a p 8 a p + Z 8 a s u 2 8 a s + Z 8 a s 8 a s u 2 8 a s 8 a s ] + (W-B)cos8cos<|> Rolling Moment Equation: 2 2 ! xxP + (!zz - ^y)^ ~ ( r + P ( l ) I xz + (r ~ ^ )Jyz + ( P r ~ <J)T x y = ( 4 - 6 ) T 5 I 4 Py- [K. 'p + K.'(r + pq) - VL'qr + N.'qr] + Py - [ Y . ' v q - Z / w r - Y .'(v - wp) + K, ' v + K p ' up + K r 'ur] + 3 P y [Z w ' vw - Y . ' v w + K w u w + K 5 f p ' u 2 8 f p + K 8 f p 8 f p ' u 2 8 f p 8 f p + K 8 f s ' u 2 8 f s] + 3 * y - t K 6 f s 8 f s ' u 2 8 f s 5 f s + K 5 a p ' u 2 5 a P + K 8 a p 8 a P ' u 2 8 a P 8 a P + K 8 a s ' u 2 8 a s + K 8 a S 8 a s ' u 2 8 a s 8 a s l + K 8 R ' u 2 8 R + K 8 R 8 R ' u 2 8 R 8 R + ( y c m W - ybB)cosecos<|> - ( z c m W - zbB)cos6sin(|) Pitching Moment Equation: lyy* + dxx - !zz)rP " (P + V)lxy + " r \ z + (IP " ^yz = ( 4- 7) 50 1 5 •> T 4 £ ^ [ K . ' p r - K . ' ( q - r 2 ) - N . ' p r + M .'q] + ^ [ Y , ' v r - Y . ' v p + Z ' (w-uq) + M 'w + M 'uq] + 2 v r r q M 2 l P r ^ q v M y ^ q M J 3 £j- [X .'uw - Z / u w + M w u w + M g f p ' u 2 8 f p + M 8 f p 8 f p ' u 2 8 f p 8 f p + M g f s ' u 2 8 f s] + 3 V [ M 8 f s 8 f s ' u 2 8 f s 5 f s + M 8 a p ' u 2 8 a P + M 8 a P 8 a p ' u 2 8 a P 8 a P + M S a s ' u 2 5 a s + M 8 a s 8 a s ' u 2 8 a s 8 a s l + - ( z c m W - z b B)sin9 - ( x c m W - xbB)cos8cos(|) Yawing Moment Equation: + ( ryy - \JW - (4 + rp) I y z + (q 2 - P 2 ) l x y + (rq - P)I Z X = (4.8) 1 5 i 4 £ i ^ [ - K . p q + M . p q + K . ( p - q r ) + N. 'p + N . ' r ] + ^ [ Y . ( v + ur) + Y .(up - vq) + Z .wp + N ,'v] + 4 3 ^ [ N p u p + N r u r ] + X .uv + Y y u v + N v ' v + N 5 R u 2 8 R + N 8 R 8 R U 2 8 R 8 R ] + ( x c m W - x bB)cos9sin0 + ( y c m W - y b B)sin9 x Position Equation: x = u[cos((p)cos(6)]-v[sin((p)cos((|))-cos((p)sin(e)sin((]))]+ (4.9) w[sin((p)sin(<|>) + cos((p)cos((|>)sin(0)] (4.10) y Position Equation: y = u[sin((p)cos(6)] + v[cos((p)cos((t)) + sin((p)sin(9)sin(()))] + (4.11) w [ - cos((p)sin((j>) + sin((p)sin((|>)sin(9)] z Position Equation: z = u[-sin(9)] + v[cos(9)sin(()))] + w[cos(e)cos(()))] (4.12) List Angle Equation: <j> = (t> + 9[sin(()))tan(8)] + {p[cos(<|))tan(9)] (4.13) Trim Angle Equation: 8 = 8[cos(<|>)]-q>[sin(<|>)] (4.14) 51 Heading Angle Equation: <j> = 9[sin(<|))/(cos(e))] + {p[(cos(<t)))/(cos(e))] (4.15) Line Offset Equation: y * = [x - xp, y - yp] x [ x d ~ V y d - _ y p ] _ ^ o r e x p l i c i t l y . ( 4 1 6 ) V( x d - x P ) 2 + ( y d - V 2 ( x - x P ) ( y d - y P ) ( y - y P ) ( x d -y , 4 1 7 . I 2 2 / 2 2 V*-1') V( x d- x P ) + ( y d - y P ) V ( x d- x P ) +(y d-y P) 4.4 Disturbance Modelling There are two main disturbance types which are considered in this thesis, those which are a result of a towed "fish" and those due to surface waves. The modelling and simulation of these distur-bances is necessary for evaluating a suitable control system. The tow forces are typically of low frequency and can sometimes be considered steady state forces while the wave effects are random and occur at frequencies within the vehicle response bandwidth. There are many types of towed sonar array devices on the commercial market. Generally, the towed sonar requires some type of depressor to ensure the requisite depth is reached [88]. The depressor wing gives a downward "lift" to compensate for the cable tension. The depressor device is either incorporated into the body of the fish or a separate depressor drogue is used. The sonar device is usually towed by a surface vessel such as a ship, which tends to have an overall mass much greater than the towed device. The large inertia of the ship is unaffected by the tension of the towed cable. In the case of the DOLPHIN vehicle, the masses of the towing vehicle and towed fish are comparable and dynamic interactions result [83]. A general diagram of the DOLPHIN and towfish complement is shown in Figure 4.4. 5 2 The towed sonar device is often a side-scan sonar device which has a beam range of approxi-mately 150 m either side of the fish for the given operating height, creating a 300 m swath. Motions resulting from cable interactions and hydrodynamic disturbances cause unwanted motions in the towfish. These motions cause smearing of the sonar image [89, 90], thus less reso-lution in the swath. There are various methods of circumventing this problem [91]. A suite of onboard sensors can be incorporated in the towfish and an active control scheme implemented [92, 5]. If accurate bottom following is required, active depth control is necessary. Also, the design of the towfish in terms of stability, tow point location and hydrodynamics can be optimized for decreased motions. If conditions are extreme, multiple swaths of the same target area are required. Modelling and simulation of the towed cable and towfish system can be numerically intensive. The major considerations in towed systems are the overall drag of the system, and the cable pro-file. The cable profile depends mainly on the buoyancy of the cable but also its hydrodynamic properties. The ratio of the scope, or length, of cable to the towfish depth setpoint can be used to minimize drag or towfish motion. The cable motions and profile are achieved by modelling the cable as a series of links [92, 93]. Typically, good results can be found by modelling the cable as a series of 10 links. The modelling complexity increases drastically for three dimensional models. Some good full scale information on the towfish and DOLPHIN motions can be found in [83]. This work is the result of extensive sea trial testing of the towed system. To minimize simulation time while retaining quality of results, two tow system models will be used in this thesis. One model is used for the regulation, or straight line simulations, which con-siders the cable tension as a force vector fixed in pose but translating in position acting on the DOLPHIN. The second model is a result of system identification of the full scale system. The tow system is approximated as a linear model which responds to the DOLPHIN speed and turning rate. The second disturbance type is that of surface waves. Ocean waves are random and can be short-crested. Wave height is affected by the energy transmitted to the surface by wind, thus the duration of a wind and the fetch or distance it travels over the surface can be used to determine the wave height. The water depth also affects the wave profile, shallow or littoral areas have steeper waves than those found offshore. Sea conditions are classified by sea state which is dictated by wind speed and wave height. Sea state information is defined from extensive data collection and statis-tical interpretation. In modelling surface waves, the sea state is generally considered long crested and uni-directional. This assumes that all the waves travel in a predominant direction. It is possible to incorporate a scatter angle but it is not in this thesis. From the seminal work of St.Denis [94], the irregular sea state is considered the sum of a series of linear waves. The particles within a linear wave follow an orbital trajectory and diminish exponentially with depth. In finite water depth this theory breaks down as the orbits become elliptic. Once the appropriate sea state is reconstructed as detailed in this Chapter, the forces of the water particles on the submarine body must be found. A diagram of the DOLPHIN vehicle in a wave field is shown in Figure 4.5. 53 Figure 4.5: Diagram of DOLPHIN vehicle in waves. Often the force on a semi-submerged body is assumed linear with wave height. The main methods of predicting the wave effects on a hull are by experiment, using Froude-Krylov theory, or using Morison's equation adding appendage forces. Other numerical methods are possible but are not appropriate for simulation work at this stage. This thesis uses forces found from Morison's equa-tion while adding appendage effects. This allows only wave forces and moments in five degrees of freedom, neglecting the surge motion. The time domain forces can either be simulated by a look up table or by calculation at each time step. The look up table uses force amplitudes found as a function of depth, speed, relative wave heading, and encounter frequency [46]. The phase of the wave is then used to find the instantane-ous force. This requires a large database of pre-calculated amplitudes, and assumes that the body is stable. The forces are found in a translating earth fixed reference frame, not allowing for large scale motions. In this thesis, forces are found by integration of perpendicular wave particle effects along the length of the hull at each instant. The correct wave amplitude and phase at each hull location is determined. The appropriate appendage cross flows are found at each plane location and are added as disturbances to the deflected hydroplane angles. This assumes that the forward speed of the vehicle is large compared to the particle velocities, and may not be true for large amplitude waves or for simulation of following seas. The disturbance models discussed here are explained in more detail in the following sections. The structure and conditions for each model are presented and included in the simulation facility. 4.4.1 Static Tow Model In analysis of this type one begins with a simple model and builds upon it. One examines the straight and level 'flight' condition, where the DOLPHIN is travelling on a fixed heading, linear course towing the Towfish at a given depth. In this case the dynamics of the DOLPHIN are con-sidered to occur at higher frequency than those of the cable. This assumption is due to the magni-tude of distubances acting on each element. If interaction forces are neglected between elements and the surface wave effects diminish exponentially with depth, this is a reasonable assumption. 54 As a result, the cable tension on the DOLPHIN will appear to have a fixed orientation in the iner-tial frame, while translating with the DOLPHIN reference frame. This will be considered the 'static' tow model. The position of attachment of the cable to the DOLPHIN is labelled rt=(xt, yt, zt). The orientation of the tow cable force is given by (o^ , Pt) as shown in Figure 4.6. Note that the cable forces are actually based in the mean operating condition as shown. Figure 4.6: Tow Cable Force Vector Schematic. The tension in the cable is designated by T t , and originally lies in the x direction. The angle of rotation about the y axis is while the subsequent rotation about the z axis is p\. As such, the force components are given in the mean translating frame as: As the cable is assumed to have low stiffness, no moment is assumed in the mean frame. One can then transform the force on the body into the body frame using the matrix defined in Chapter 2: Side View Bottom View {Ttcos(at)cos(p\), Ttcos(oct)sin(p\), -Ttsin(oct)} (4.18) - l Ttcos(at)cos(Pt) Ttcos(at)sin(Pt) -T tsin(a t) (4.19) and the moment on the body is: 55 Mt = 2 i -1 Ttcos(at)cos(Pt) Ttcos(at)sin(pt) -Ttsin(at) x (4.20) giving a total disturbance vector of: It = (4.21) This vector of forces and moments is not fixed in the body frame and causes a non-linear forcing function based on or^ , f$t, <|>, and 8 if any perturbation is encountered in body orientation. The force is, however, fixed in the inertial frame so simulations must be limited to a regulation of straight forward motion. Depth changing maneuvers are possible. This model is implemented for a turning maneuver by assuming that Pt is zero such that the tow cable is only affected by pitch and roll of the DOLPHIN. In simulation, = 186 degrees and the tension is 14905 Newtons. 4.4.2 Dynamic Tow Model Constructing a dynamic tow model involves complete analysis of the interactions between the towed vehicle, towing cable, and Dolphin vehicle. This type of analysis explores the six degrees of freedom of each vehicle, in addition to the multi degree of freedom modelling of the cable. Cable dynamics in itself is a complex study [95]. Various projects have been done in the study of towed cable systems, mainly from surface vessels [96]. Complete modelling of all interactions has also been performed for various maneuvers and configurations for the Dolphin vehicle. The work presented in this thesis uses system identification techniques for a simple analysis of the complex system. Using measured full scale data from maneuvering trials, a dynamic model is fit-ted to the data. Relationships are found between Dolphin kinematic variables, cable tension, and Towfish kinematic variables. From a study of the relative magnitude of forces on the Dolphin vehicle by Seto and Watt [83], during a turn of 357 m diameter, the cable force components differ in {X, Y, Z} by approximately {5200, 2000, 875} newtons between straight travel and the maximum value within a turn. The tow off angle (Pt) rises to about 19.4 degrees. Using the force components and equation 4.18, one can calculate the instantaneous tow off depressive angle (0^). For the data shown, the depressive angle goes from 7.9 degrees before the turn, 7.0 degrees midway through the turn, to 5.2 degrees at the end of the turn. As this is a small angle difference, the lateral and axial components change by less than 1 percent due to the change in depressive angle. The depressive force changes by -35 percent but as the overall depressive force is small, this is a negligible amount. Just as the static tow model, the depressive angle will be assumed constant for the dynamic model. This does not result in a fixed depressive force however, as overall tension magnitude is dynamic. 56 The data parameters used to determine the dynamic model will be r, U s o g , T, {pr> (|>p These are dolphin turning rate, speed over ground, cable tension, dolphin heading, and towfish heading respectively. The goal of the model construction is to identify the parameters T, o^ , p\ as a function of Dolphin kinematic variables, r and U s o g ) such that the components of the tow off vector may be constructed. In the preceding paragraph has been assumed as fixed. The lateral and axial com-ponents must then be determined. As the general form of fluid drag is of the form shown in equa-tion 3.11, the modified parameter (U s o g ) 2 is used. To identify specifically the cable tow off angle with respect to the DOLPHIN, one must model the cable dynamics. From the data sets in [83] the individual headings of each vehicle are known, but the only cable parameter known is tension. In order to construct a model, we assume that the cable remains in a single vertical plane which rotates about the z axis as a function of dolphin turning rate. Ideally the cable lateral angle would also be a function of velocity component, v. Here, v is considered small even during the turn, and the overall speed over ground velocity is used. From the data shown, the lateral tow off angle can be approximated as: P, - (4.22) From this analysis, it is now possible to use system identification for the parameters, r, (U s o g ) , T, (VD _ V F ) and , where the first two parameters are considered the inputs and the last two are the outputs of the towed system model. A batch least-squares (BLS) fit of the data is used to deter-mine the dynamic model. Such an identification method "curve fits" the output data to the input data. All such correlation methods are most effective when the input to the system is randomly varying, such that all relationships will be evident in the correlation of inputs and outputs. Accordingly, parameter estimation is usually done using "white" noise, or noise inputs that have a zero mean value and have a constant power spectrum. In reality, such an input is not possible as it requires infinite energy. The process of least-squares (LS) is widely used and can be dated back to the time of Karl Frederick Gauss (1777-1855). This method is, as most methods are, based on the minimization of some type of error criterion or loss function, in order to determine the associated parameters. For the least squares case, the input-output model is expressed as: yt(t) = O t T(t)0 t + et(t) "• (4.23) where the assumption is made that, ^e t(t) = 0 (4.24) Here, y(t) is the off-line outputs of the system, 0 are the system parameters and O is constructed from the past values of inputs (w) and outputs (y), dependent on the order of system desired. The estimated parameters are found from the equation: 57 -c T -1 T 9 t = ( 0 » > t ) <DtTyt where each variable shown is the length of the number of samples. The resulting model is a linear relationship between input and output and is given in ARMA (auto-regression moving average) form. If there are 1 output variables, m input variables, and the system is of order n: [y t l (k) y t 2 (k) ... y t p (k) ] 1 a m a u 2 ... a U p a121 a122 ••• a12p a l p l alpp. - y t l ( k - i ) - y t 2 ( k - i ) -y tp (k - i ) +... + ... a„ an21 an22 "• an2p *npl ... a„ ) - y t l ( k - n) ) - y t 2 ( k - n) ) . - y . P (k - n) + bin b i i 2 b121 b122 ' Ipl ... b, ... b, ... b lpm 1) u t 2 ( k - 1) + ... + 1 . % (k - 1) b n l l b n l 2 ••• b nlm bn21 bn22 •" b n2m L>pl ••• npm u t ] ( k - n ) u t 2 ( k - n ) u t p ( k - n ) (4.25) The order of the system must be determined before the BLS fit. The resulting model is: »t = [A t l A t n B t l (4.26) Thus using the data, it is possible to determine a discrete ARMA model for the towed system. It is desirable to transform the model into continuous time state-space form for assimilation into the simulator. The transformation from ARMA parameters which are based on input-output relations, to state-space form for a multi-input multi-output system can be quite difficult. There are various approaches to this problem, and the difficulty is in finding a state-space form which is canonical, to some extent, minimal and also controllable. In this thesis the transformation from one form to the other is done through the use of the Singular Pencil form. There exist many different forms of the Singular Pencil model, one of which uses the ARMA values, I 0 I -A,„ B t n I 0 z x t k = x t k + I - A t 0 B tq y t k (4.27) or in matrix form, 58 -z l I 0 - A t n B t n 0 . - z l . 1 - A t 0 B t 0 A t k y t k tk = 0 (4.28) It is very simple to put the parameters estimated by the previously mentioned methods into this form of the Singular Pencil model. For multi-input multi-output systems the identity matrices, 0, -zl, A, and B values are matrices instead of scalars. Once in this form, any row or column opera-tions can be performed on the matrix to transform the relationships into another form of the Sin-gular Pencil model. The state-space form of the Singular Pencil model is denned by, z * t k = 0 B t Ztk -k - - 1 D I .-tk (4.29) which can also be converted into matrix form, -z l I - A tn 0 B t 0 . - A t l - z l 0 -I 0 ~"l 1 *tk .-tk = 0 . (4.30) The only difference between these two forms, in terms of structure, is the change from - A t i for i = 0 to n, to -I and zeros. This change can be performed by simple row operations as the -A.to matrix is -I from the form of the A R M A model assumed for parameter estimation. From this point, the A t , B t , C_t, and D t sub-matrices of the Singular Pencil matrix can be extracted for state-state form. It is now a simple procedure to convert from discrete state-space form to continuous state-space form using a linear approximation. This method was used for the input and output tow model variables mentioned, and a second order system was identified. It was found for the MIMO second order system that a coupling effect existed between (U s o g) 2 and (Pt). In other words, there was a difference in tow off angle between turning to the port and turning to the starboard for the same diameter turn. From the physics of the problem, it is assumed that symmetry should exist. A latereral lifting phenomenon due to the spi-ral of the cable exists and could produce asymmetric cable responses. The spiral effect is neglected here. Two second order SISO systems were identified. The tow off angle, (p\), is assumed a second order function of turning rate, r, while the tension in the cable is assumed a sec-ond order function of the speed squared, (U s o g) . 59 Maneuver Used for Tow System Identification E 800 600 800 Latitude (m) 1000 1200 Figure 4.7: Position Information for Maneuver used in Tow Identification. Speed During Maneuver Figure 4.8: Speed over ground during maneuver. 60 Tension Model Figure 4.9: Cable tension during maneuver. Lateral Angle Model Solid- Actual Dotted- Tow model 300 Time (s) Figure 4.10: Half difference of DOLPHIN and towfish heading. 61 The maneuver used to generate the tow model is shown in Figure 4.7. The speed during the maneuver is shown in Figure 4.8. A comparison of the resulting linear predictions and the actual data is shown in Figures 4.9 and 4.10. The use of system identification assumes that all frequency modes are excited by the assumption that the inputs to the system are white noise. This is not necessarily true in this case and as a result, some of the higher frequency modes in the tow and fish system may not be represented in the linear model developed here. There are also high order interactions between the cable, tow-fish, and DOLPHIN which may not be realistically represented using this model. The two models produce reasonable results considering the given measurement errors. The tow off dynamics model allows for maneuvering simulations in all 6 degrees of freedom. The model represents the forces exerted by the cable-towfish system on the DOLPHIN and is particular to a given tow depth and cable scope. It may be possible in future work to incorporate a drag factor based on tow depth and scope, and a lateral tow off factor based on scope. It also may be possible using cable measurements to define a model to determine depressive tow angle. 4.4.3 Wave Model As the Dolphin vehicle travels near the free surface, it is affected by surface waves. In order to determine the forces on the vehicle due to surface waves, the surface profile and wave elements must be reconstructed in the simulation. The effect of the wave components on the vehicle hull and hydroplanes must then be determined. The governing equations for the fluid are given in Chapter 3. The surface profile for a single linear long crested wave is found in two dimensions using an Airy solution: The wave frequency is given by (D^ and is related to the encounter wave frequency, coe, by r | w = oc w cos ( (o w t -k x x-k y y) (4.31) toe = t o w -g cos(pw) (4.32) Where the wave number is, (4.33) where L is the wavelength of the oncoming wave and for deep water waves, L = ~2 27t (4.34) 62 The variable T w here is the period of the oncoming wave. Here, the relative angle of encounter between the wave and the ship is Pw as seen in Figure 4.11. Figure 4.11: Wave and Submarine Relative Orientation In equation 4.31, the wave number components are given by k x = kcos(\|/w), and k y = ksin(\|/w). By equation 4.31, the instantaneous wave elevation due to a single wave can be determined at the vessel center of gravity location as a function of time and the vehicle's position. In order to simulate a complete sea state of irregular, long crested waves, the principle of superpo-sition is used. Wave elevations around the world have been recorded and analysed in terms of the relationship between wave energy and wave frequency. The results of such work are wave spectra such as the Pierson-Moskowitz, Bretschneider, JONSWAP, and various others. Statistical investi-gation of the wave heights results in the following table. Statistical interpretations of wave infor-63 mation varies with the measurement location and the season for which the measurements were recorded. TABLE 4.4 Seastate Statistical Data. Reproduced from reference [S3] Sea State Number Significant Wave Height (m) Sustained Wind Speed (knots) Percentage Probability of Seastate Peak Wave Period Range Mean Range Mean Range Most Probable 0-1 0-0.1 0.05 0-6 3 0.7 - -2 0.1-0.5 0.3 7- 10 8.5 6.8 3.3 - 12.8 7.5 3 0.5-1.25 0.88 11 - 16 13.5 23.7 5.0-14.8 7.5 4 1.25-2.5 1.88 17-21 19 27.8 6.1 - 15.2 8.8 5 2.5 - 2.5 3.25 22-27 24.5 20.6 8.3 - 15.5 9.7 6 4-6 5 28-47 37.5 13.1 9.8-16.2 12.4 7 6-9 7.5 48-55 51.5 6.1 11.8-18.5 15.0 8 9-14 11.5 56-63 59.5 1.1 14.2- 18.6 16.4 >8 > 14 > 14 >63 >63 0.05 18.0-23.7 20.0 The Bretschneider spectra are defined by the equation: <. ^ B r e t S(cow) = -exp CO,,, - B B r e t A CO,, (4.35) where the constants are A B r e t = 8 . 1 x l ( T 3 g 2 , a n d B B r e t = 3.11/h? / 3. (4.36) The variable h 1 / 3 is the significant wave height in meters as shown in Table 4.4. Using the spectral energy at a given frequency, the wave height can be predicted by associating the energy across a band of frequency with the median frequency: aw(cow) = 72S(cow)dcov (4.37) The wave height and frequency are now known for the range of frequencies in the irregular sea spectrum. Using the seminal work of [94] and the theory of [97], it is possible to reconstruct the seastate as a function of time and position: r|w(x,y,f) = ^ a j c o s ( c o w j t - k x j x - k y j y + 0j) ,forj = 1...N (4.38) 64 All, j , components of the spectrum are summed. A random phase shift, 0 < 9j < 2TC, is incorporated with each frequency element. The random phase shift is different for each frequency but remains constant for that frequency in time. In other words it is merely a random initial condition for each wave. Equation 4.38 then gives the instantaneous wave height for a position in the inertial frame as a function of time. The velocity components of a long crested deep water wave component, j , can be found by: u w j = ^jr J exp(-k j z)cos(co w j t-k x j x-k y j y + ej) , and (4.39) 27ta w w j = ^r J exp(-k j z)s in(03 w j t -k x j x-k y j y + ej) (4.40) Whereas the acceleration terms corresponding to a long crested deep water wave are: 47t2cc: " w j = —2 J exp( -k j z)s in( to w j t -k x j x-k y j y+ 6j) , and (4.41) A 2 471 CC: w w j = -Y2 exp(-kjZ)cos(cowjt - k x jx - k y j y + Op (4.42) These velocity and acceleration components are given in the wave reference frame. By superposi-tion and transforming to the earth fixed frame, the overall wave velocity components in the iner-tial frame are: u,„ = v,„ = w,„ = Wj N wj COS(\J/W), sin(\j/ ),and X • N W • wj (4.43) (4.44) (4.45) Similarly, the acceleration components are summed for the overall wave particle accelerations in the inertial frame: W E = N wj wj N cos(\|/w) , sin(\j/w), and (4.46) (4.47) 65 [w w ] E = w Wj (4.48) N It is then necessary to determine the forces exerted on a body inserted into the wave field. Theory presented by Froude-Krylov, suggests that the wave pressure field be integrated on the hull sur-face. This theory however does not account for turbulent separation effects which occur. Large vortices are shed from a cylindrical body in a cross flow [98, 99, 100]. There is a similar occur-rence about a cylinder-like body such as a submarine. The significance of the separation causing the vortex can be found through inspection of the Keulegan-Carpenter number or: Kc = U T mean w (4.49) where U m e a n is the average flow velocity across the characteristic dimension or hull diameter in this case, D 0 . To find the average cross flow velocity for a sea state, the energy averaged wave amplitude, a , is used: a = 1.25 (4.50) such that using equations 4.39-4.40 based on the mean hull centre line depth and average period, -kz Kc = 27tae (4.51) Using this equation the following chart can be assembled for the DOLPHIN vehicle travelling at a depth of 3 meters: T A B L E 4.5 Impinging Wave Energy Parameters Parameter Sea State 3 Sea State 4 Sea State 5 Significant Wave Height 0.88 m 1.88 m 3.25 m Average Wave Height 0.55 m 1.18m 2.05 m Average Period 6.43 s 7.55 s 8.32 s Average Wavelength 64.5 m 89.0 m 108.0 m Average Wave Number 0.0973 m" 1 0.0706 m" 1 0.0581 m" 1 Kc 1.29 3.00 5.41 Dfj/Lo 0.0155 0.0112 0.0092 Re 1.28 x 10 5 2.54 x 105 4.17 x 10 5 The energy averaged period, T 1 ? is related to the peak period, T p , by [52]: T„ = 1.166T , . (4.52) 66 The relationship between slenderness and Kc on diffraction and separation effects was studied by [101], and show the following results: Figure 4.12: Wave Loading Regimes. The comparison between Figure 4.12 and Table 4.5, yields the fact that forces due to flow around the DOLPHIN submarine hull is governed by flow separation, rather than diffraction forces. This implies that the body is small in relation to the incoming wave and also that the wave velocity by period is large relative to the body diameter. A corresponding method of analysis for this type of force regime is the use of Morison's equation [102]: + c v P u ' v dx (4.53) This shows the lateral forces on a section, dx, of the body due to wave velocity and acceleration using the drag coefficient, C d , and the added mass coefficient C m . No axial components of force are determined, as they are using Froude-Krylov methods. The use of equation 4.53 assumes that the cross flow velocity is perpendicular to the body axis. However, studies have shown that force results are correct for cylinders at yawed angles of attack up to 50 degrees using Morison's equa-tion [100]. The drag constituent leads the acceleration constituent of the force by a phase of 90 degrees for surface waves. Typical theoretical values for C d , and C m are 1.0 and 2.0 for a sub-merged cylinder. These unsteady hydrodynamic coefficients are different from the steady flow values determined in Chapter 3. 67 Considerable research has been done in the past on the explicit determination of drag and added mass coefficients for vertical cylinders in oscillating flow. The coefficients are dependent on Rey-nolds number, Keulegan-Carpenter number and the frequency of oscillation. For high (> 1.6 x 106) Reynolds number, the values converge to 0.62 and 1.8 respectively. Experimental results for these coefficients show much scatter [102]. Using Sarpkaya's [101] review of wave forces and research in this area, for the Kc and Re numbers for the DOLPHIN operating conditions, the val-ues 0.65 and 1.95 will be used. As Kc drops below 6, converges rapidly to 2.0, while C d is highly Re dependent and approaches 0.65 for the Re values shown in Table 4.5. The steady flow value of C d for these Reynolds numbers is 1.2. Thus the lateral force constituents using Morison's equation for a circular cylinder in oscillating flow for the simulation become: Y w = / ' d - ^ - ( v w - v ) + C m — T - ( v w - v ) dx , and (4.54) J pD c ( w - w ) +C, pxtDo ( w - w ) L V dx (4.55) where the wave force components are given in the body frame as found by eqs. 4.51-4.52. The corresponding moments are: N w = J C d ^ r ( v w - V ) + C m — ; — ( v w - v ) xdx , and (4.56) M w = J P D ( ( w - w ) +C pTtDn . m ( w - w ) xdx (4.57) From eqs. 4.51-4.54 one can see that the forces and moments involve the integration along the body. Instantaneous v w , w w , v w , w w components for each x location must be found for each wave frequency. The wave velocity and acceleration components are found perpendicular to the hull centre line and at positions on the hull centre line. The eqs. 4.36-4.45 are used and are expressed in the body frame: w. = Q B W„ (4.58) In addition, the instantaneous relative motion v, w, v, w of each x location must be determined from: 68 Vj = v + rXj (4.59) Wj = w - q X j (4.60) Vj = v + rx j - wp + ur (4.61) Wj = w - q X j - u q + vp (4.62) Some of the acceleration terms have been neglected as they are small. No axial (surge) wave forces are available using this approach. As the hull is treated as a slender cylinder, the cross flow disturbance forces on the hydroplanes must also be calculated and can be determined using a semi-empirical theory [103]. As the wave induces a disturbed flow around a plane, the speed of the plane relative to the wave is given by: U - U w = 7 (u-u w ) 2 + ( w - w w ) 2 (4.63) where the wave velocities are found at the applied plane force location. The total plane angle of attack without the presence of waves is approximated by: 5 + e = atanf—M (4.64) The body angle of attack being denoted by e, giving the local plane cross flow as wf. As the plane forces due to a body angle of attack are found by the total hull hydrodynamic coefficients, we are solely interested in the relative plane angle of attack 8 « a t a n ( ^ ) . (4.65) The body angle of attack can be found from: e = atan^j (4.66) The local cross flow velocity due to the wave is ww. Combining equations 4.61-4.63, w f - w = Utan(5), (4.67) The relative plane cross flow disturbed by the wave is then Wf-w-w^ The disturbed plane angle of deflection is then found as: 69 w w - w, w atan (4.68) V In the case of this simulation, the difference U - U w is considered small as the speed of the vehicle is large, and the small angle approximation is used such that the disturbed angle induced by a wave is: The cross flow attributed to roll angular velocity is neglected due to symmetry and lower order of magnitude. The disturbed angle of attack for each plane can be found using equations 4.66-4.67 as a function of body kinematics, plane deflection, and local wave velocities. The total wave disturbances impinged on the vehicle are simulated for each sea state. The spectral energy distribution is defined and used to reconstruct the sea state. The local wave velocity ele-ments are found for the reconstructed sea state. Using Morison's equation the disturbance body forces are integrated along the hull. The effects waves on the hydroplanes are found using semi-empirical formulation. In this manner the wave effects are simulated. 4.4.4 Onboard Stochastic Measurement Disturbances The DOLPHIN vehicle uses various on-board sensing and measurement devices. Sensors measure an analog value, usually a voltage, which is then amplified and is converted to a digital signal. The conversion is performed by the analog to digital (A/D) system. These discrete signals are then fil-tered to provide a useful measurement for control of the vehicle. The following 10 states related to control are measured directly: The speed over ground is measured using doppler velocimetry and a GPS receiver. It is assumed that the forward speed, u, can be determined from these measurements. The positional and inertial kinematic variables are assumed to be sampled at a rate of 50 Hz [22]. Anti-aliasing filtering of the signals is performed using filters designed for the Nyquist frequency. The simulation assumes that there is no time lag in measurement and no skipped samples or timing errors. Time lags do occur in some applications of AUV's and cause control and stability issues [104]. These effects are generally overcome through the use of filtering and prediction methods such as a Smith pre-dictor [105]. Typically, measurement errors are modelled through the use of a white noise process. In other words, discrete time white noise is a stationary stochastic process. Its spectral density is constant for all frequencies. This can be realized by a sequence of independent random variables having gaussian normal distribution. This does not account for effects such as aliasing or sensor drift which are secondary sensor effects. If a white noise process is used, it can be described specifi-cally by its variance and mean values. As the process is assumed stationary and zeroed, it will -w w U (4.69) y = [p, q, r, x, y, z, <|>, 0, cp]T (4.70) 70 have zero mean. Thus the error can be defined by its variance. The following variances will be assumed for the measured signals, based on information gathered from full scale data: TABLE 4.6 Estimates of Sensor Noise Characteristics. State Variance Quantization u 0.0038 m 2 /s 2 0.1 m/s P 0.43 deg2/s2 0.2 deg/s q 0.25 deg2/s2 0.2 deg/s r 0.02 deg2/s2 0.1 deg/s X 2.9 m 2 1 m y 2.9 m 2 1 m z 0.0007 m 2 0.1 m 0.11 deg2 0.1 deg e 0.015 deg2 0.1 deg V 0.54 deg2 0.1 deg The sensor resolution and effects in A/D conversion result in a quantized digital, or an overall res-olution. The greater the digital word length, the higher the resolution of the digital system and the smaller the quantization steps. Actual analog resolution may vary depending on vehicle speed, temperature, and rates of motion in a non-linear fashion depending on sensor dynamics. The total quantization levels used for simulation of the respective variables are shown in Table 4.6. If the A/D system samples and holds the sampled value until the following sample, we can assume a zero-order hold discretization. The total measured signal which can be used for control is thus represented by the following system: 71 Actual Measured state state X Anti-aliasing Butterworth filter A / D X + conversion n Sensor noise Figure 4.13: Sensor Noise Simulat ion Block The analog signal is converted to digital with a random "white" noise addition with appropriate variance. There is no correlation between measurements of the different signals. The signal is then quantized to the final resolution allowable and is held until the following sample. Before the measured signals can be used for effective control, it is important to account for vari-ous other issues. Depth sensing is usually achieved through the use of a pressure sensor. At great depth, such a system provides good results, however near the free surface pressures vary with wave elevation and speed separation effects. Wave effects are apparent to a depth of half a wave-length for the deep water case. As a mean depth is necessary for regulation, some type of filtering is needed. Wave elevation errors in depth measurement are not included in this model, only wave forces. Methods to account for error in depth measurement can be found in appropriate literature The next issue in sensor management is the frequency at which the errors in measurement occur. Generally the frequencies of sensor noise are higher than the bandwidth of vehicle dynamics [9]. If there is overlap in the range of frequencies of the signal, control sensitivity and filtering become issues. It is desirable to isolate the motions of the vehicle by filtering the high frequency noise. In the case of forced wave motion, this becomes difficult and more complex methods are needed [107, 108, 12]. In the next Chapter a disturbance observer will be implemented for prediction of wave, sensor drift, component failure, or other such disturbances. The information will then be used through feedforward methods to counter these problems. As mentioned, only 9 kinematic and positional states are measured of the possible 12 states related to 6 degrees of freedom. The velocity components (u, v, w) are not measured. The predic-tion of these variables is difficult as they are attributed to the localized body frame. It is possible to use dynamic pressure measurements or accelerometer integration but both these methods pro-[106]. 72 duce considerable noise. As the pose and position (x, y, z, (j>, 8, \|/) are measured accurately, these variables will be used in a state observer. The method is described in the following chapter. This technique allows the prediction of all states, but its degree of accuracy is limited by the level of noise in the measured signals. Through this approach the measurement noise is modelled appropriately using gaussian random noise. The signal is then quantized and discretized. Observer filtering methods may then be applied to the signal to predict unmeasured values, and to improve the signal through filtering. 4.5 Summary This chapter describes details of the particular simulation and modelling of the DOLPHIN vehi-cle. The complete equations of motion with relevant hydrodynamic coefficients are given. The method of navigation is explained and the use of GPS waypoints is shown. The disturbances which are specific to near surface towing are given. The modelling of a towfish is explained and the method of simulation is described. Information regarding towed sonar devices is reviewed. The reconstruction of a given sea state is explained. The forces impinging on the near surface vehicle are derived. To ensure realism of the simulation, sensor noise is included and simulated. The details regarding sensor simulation are presented. The information in the preceding chapters is sufficient for the complete modelling and simulation of a near surface submersible vehicle. The effects of the disturbances described in this chapter are random and unpredictable, thus it is not possible to use this information in the structure of the control system specifically. It is possible, however, to use the expected maximum amplitudes and frequency ranges of these disturbances in designing the control system. 73 5.0 Control Having examined the dynamics and characteristics of vehicle motion in the previous chapters, it is possible to use the knowledge of the submarine plant model (SPM) to construct a suitable control system. An effective control system is necessary for the vehicle to complete each of its tasks reli-ably. The goals of such a controller must be emphasised. The combined submarine and controller system should: 1) Retain stability within the range of operating conditions. 2) Reject disturbances due to waves, towing, and sensor noise as well as submarine non-lineari-ties. 3) Minimize control effort and instances of actuator saturation. 4) Prevent steady state offset from reference trajectories. 5) Exhibit the desired performance characteristics. The ability to control the vehicle under study in this work has been proved [62], but advancements to include the aforementioned characteristics are desired. A suitable control system should use the existing measured variables to actuate the vehicle accordingly. Typical maneuvers can be classi-fied into two types, straight and level trajectories which require regulation control for a reference state vector and tracking control which requires the following of a changing path such as a 180 degree turn. The controller will encounter both types of trajectories and should be effective for either case. From the equations of motion, it is apparent that coupling of certain motions is present. Control techniques are possible which assume decoupled motion and separate control for vertical and hor-izontal motions. Important coupling effects are described by the off diagonal elements of the lin-ear added mass and damping matrices, as well as non-linear coupling in the equations of motion due to differences between the orientation of the body frame and the inertial frame. To account for these coupled motions, a multi-input multi-output (MEMO) approach will be used for control. The complete 6 degree of freedom system is considered with the foreplanes, aftplanes, rudder and pro-peller as the inputs, and the measured states as the outputs. This type of approach is more effective for minimizing depth and roll changes through turns and other consequences of coupling. Two general approaches are possible for control of such a system, feedback and feedforward. Feedforward uses a priori information about the SPM and reference command to instruct the vari-ous actuators. This can be accomplished through an observer which predicts future motion, or by the use of open loop control. Both such methods will be described in this chapter. The second control approach which will be used is that of feedback. Measured states are com-pared with desired reference states and the difference or error is used to command the actuators. Input from feedback is in the form: u = - K ( x - x . ) (5.1) 74 where x=(v, Tj) is the state vector, and xr is the reference trajectory, or set-point vector. Both feedback and feedforward can be used individually or in combination, as shown in Figure 5.13. The relative performance of each method will be determined in this thesis. Feedforward r K(s) Uff w r + ^ e >ack Feedt L(s) G(s) Figure 5.1: Feedback and Feedforward. 5.1 Open Loop Characteristics Before designing a control system for the vehicle, the open loop or controls fixed condition is examined. By studying the open loop system, the stability and dynamics bandwidth of the vehicle are determined. The stability can be described as the tendency of the vehicle to return to an equi-librium state of motion after a small disturbance. A great degree of stability suggests that the con-trol effort will be large for large changes in reference command whereas a less stable vehicle will respond faster for a change in reference [109]. To facilitate the study of the system, the non-linear state space equations are linearized. This line-arization is performed by assuming a steady state operating equilibrium (x l in). A constant forward speed, in straight and level flight at a depth of 3 meters, is used as a linearization state. An equilib-rium of forces is desired for this operating system. In the pitch heave motion this can be accom-plished using the vehicle trim, foreplanes, and aftplanes. If the vehicle pitch is desired to be zero, then it is possible to determine the corresponding open loop actuation levels (U]in) through itera-tion. These equilibrium open loop inputs account for non-linear buoyancy and drag effects which are not present in the linear system. As the [x, y] position states do not affect the future dynamics of the system and they are not directly controllable by the actuators, these states are neglected [110] for the compensator design, however they are measured. The linearized state space equa-tions of the submarine plant are: x = Ax + Bu + x o (5.2) y = Cx (5.3) 75 where Xo is the force offset due to the non-linear hydrodynamic forces as shown in Figure 5.2. Non-linear and Linear Force .X| o CD 60 £ = Aj-jti+Bu+x-^ —1 —ij—1 — —10 xi = f(xi,u;t) State x. Note: Figure shown for j and u fixed, actual function is two dimensional Figure 5.2: L inea r Representation of the Non-linear Plant. This is called the submarine design model (SDM). The linear model can be represented in the fre-quency domain through the Laplace transform as: Y(s) = G(s)U(s),and G(s) = C ( s l - A ) B (5.4) (5.5) Characteristics of the system can be identified by the open loop poles which are: T A B L E 5.1 S P M Open Loop Poles Speed P o l e l Pole 2 Pole 3 Pole 4 PoIe5 Pole 6 Pole 7 Pole 8 Pole 9 Pole 10 5 m/s -0.2505 0 0 -4.4933 -0.4258 -1.7079 +0.3973i -1.7079 -0.3973 -2.4508 -0.5211 -0.1125 6 m/s -0.3006 0 0 -5.5858 -0.3309 -2.0426 +0.6410i -2.0426 -0.64 lOi -2.9557 -0.0888 -0.6566 7 m/s -0.3507 0 0 -6.6425 -0.2745 -2.3759 +0.8239i -2.3759 -0.8239i -3.4587 -0.0740 -0.7853 8 m/s -0.4008 0 0 -7.6807 -0.2357 -2.7097 +0.9880i -2.7097 -0.9880i -3.9605 -0.0637 -0.9107 From Table 5.1 one can see that the poles are all located in the negative real side of the imaginary plane and only two poles show imaginary components. This indicates that the submarine is open loop stable to some degree. The pole locations are largely speed, hull angle of attack and plane deflection dependent and open loop characteristics may differ with any of these parameters. Simi-lar effects are seen in aerospace applications [111]. As the linearization speed is increased, the open loop bandwidth of the vehicle is increased. There are poles at zero which are due to the 76 direct integration of velocities to find displacements in the SPM, and do not affect the plant dynamics. In this study a linearization speed of 6 m/s is used. The highest bandwidth pole, -5.5858 rad/sec, corresponds to the roll response of the system while the slowest pole, -0.0888 is largely the static restoring response of the vehicle, this being determined by eigenvector analysis. By varying the relative locations of the center of mass and the center of buoyancy, the pole loca-tions vary tremendously. The modes and dynamics associated with each pole are determined from eigenvector or modal analysis. This is performed by assigning a new state vector, z, as: x = Tz . (5.6) Where T is a transformation matrix resulting in the equations: Tz = ATz + Bu (5.7) y = CTx . (5.8) If these equations are then multiplied through by T 1 the following transformed state space equa-tions result: z = ( T _ 1 A ) T z + (T _ 1 B)u (5.9) y = CTx (5.10) The new system can be made diagonal such that the new states represent the decoupled modes of the system. Let, A = ( T _ 1 A ) T (5.11) Then the diagonal entries of A are the eigenvalues of A. This is determined by setting, A T = T A (5.12) We see that if T is replaced by each of its constituent column vectors, ij, t2>... t 1 0, then, A t = tX{ , fori = 1 to 10 (5.13) and tj must be an eigenvector of A if ^ is an eigenvalue of A on the diagonal of A. Using this approach, the modes of the system can be found from the eigenvectors. This analysis results allows us to plot the normalized magnitudes of the eigenvectors tj to examine the modes of the system: 77 T 3 3 D) CO CD T3 O 1.20 1.00 0.80 0.60 0.40 0.20 0.00 Pole 1 Eigenvector cr Mode 0> CO Q. Figure 5 3 : Eigenvector Element Magnitudes for Pole 1 T 3 3 O) CD T3 O 1.20 1.00 0.80 0.60 0.40 0.20 0.00 Pole 2 Eigenvector cr >-Mode .c Figure 5.4: Eigenvector Element Magnitudes for Pole 2 T3 "c CO CD O 1.20 1.00 0.80 0.60 0.40 0.20 0.00 Pole 3 Eigenvector Mode Q. CO 0) CL Figure 5.5: Eigenvector Element Magnitudes for Pole 3 78 Pole 4 Eigenvector 7 3 £0 CD •o O 1.20 1.00 0.80 0.60 0.40 0.20 0.00 Mode Figure 5.6: Eigenvector Element Magnitudes for Pole 4 Pole 5 Eigenvector 1.20 •S 1.00 c 0.80 D ) J 0.60 | 0.40 ^ 0.20 0.00 Mode Figure 5.7: Eigenvector Element Magnitudes for Pole 5 1.20 •S 1.00 "! 0.80 I 0.60 •g 0.40 ^ 0.20 0.00 Pole 6 Eigenvector . I I • l Mode <D Q . Figure 5.8: Eigenvector Element Magnitudes for Pole 6. 79 1.20 1.00 0.80 0.60 | 0.40 ^ 0.20 0.00 X3 -t—* CO Pole 7 Eigenvector I I I Mode m co <u Q. Figure 5.9: Eigenvector Element Magnitudes for Pole 7. Pole 8 Eigenvector 1.20 •S 1.00 c 0.80 J 0.60 | 0.40 2 0.20 0.00 Mode Figure 5.10: Eigenvector Element Magnitudes for Pole 8. Pole 9 Eigenvector Xi co CD Q. SZ Mode Figure 5.11: Eigenvector Element Magnitudes for Pole 9. 80 Pole 10 Eigenvector 1.20 T 3 3 1.00 c 0.80 | 0.60 j 1 0.20 l I • l I 0.00 - t - * - i • • • 05 Q . Figure 5.12: Eigenvector Element Magnitudes for Pole 10. From the preceding graphs, the mode corresponding to each pole can be seen. Pole 1 is a pure decoupled surge motion. Poles 2 and 3 are pure integrators of depth and heading. These poles will change with a change of heel and trim but remain more or less fixed for small angles of excursion. Pole 4 is a mainly rolling mode. A static configuration of list, trim, and heading is shown by Pole 5. A combination of roll, sway and yaw is seen by poles 6 and 7, dominated by yawing motion. Poles 8 and 10 show a heave, pitch, and surge coupling with pole 8 dominating pitch and pole 10 dominating heave. Pole 9 shows another static pole coupling of heel and heading. One should note that these graphs show only the normalized magnitude of the eigenvector components, but they are useful for determining the response and modes of the multi-variable system. It should also be noted that the coupling effects seen from this analysis represents the terms that are included in the mathematical model. If no coupling terms are included in the model, the eigenvector analysis will show no coupling. These graphs also illustrate the level of coupling that is present in the system, including some combinations of lateral and longitudinal motions. In addition to the methods presented here, there are various semi-empirical methods for predicting the stability of submarines, based on steady state maneuvers and typical operating conditions. Comparing the steady state turning scenario using [80], by setting x = 0 in eq. 5.2: v = (m'-Y r ' )r ' -Y S r '5r Yv-CS. 14) r = ( Y v ' N 5 r ' - N v ' Y S r ' ) o r N ^ Y Z - m V Y V N / and (5.15) (5.16) Also, the maximum depth rate can be determined from: 81 Zsfp'ofp + Z S f s'8fs + Z S a p'8ap + Z 8 a s'8as + Z 5 f p'8fp + m'q' (5.17) For a 25 degree rudder deflection using the hydrodynamic coefficients found in Chapter 3, v' = 0.0153, r' = -0.221, and if = -2 degrees. This does not include the effect of fore and aft planes on roll. Thus the vehicle rolls into the turn due to the large moment provided by the rudder. The max-imum heave rate is found to be w' = 0.214 assuming no pitch rate and that the aftplanes are deflected to a level which maintains zero trim. It should be noted that combined roll and pitch angles greater than 10 degrees lead to considerable differences between the linear and non-linear models. Generally, two major factors affect stability, the location of the center of mass, and the hydrodynamic damping. Static and dynamic stability are both affected by the location of the center of mass in relation to the center of buoyancy. The larger the vertical distance between the two points, called the meta-centric height, the more stable the vehicle becomes. The second major indicator of stability is the magnitude of the diagonal elements of the hydrody-namic damping matrix. This is, of course, considered in combination with the location of the center of mass. As the center of mass is moved forward, stability is increased. For an appended cylindrical body, the damping is affected by the conicity of the tail section, and the geometry and location of the appendages. The further aft an appendage is located the greater the natural stabil-ity, assuming the appendage is positioned at the minimum drag angle. The reason that the DOL-PHIN vehicle shows stable poles is due to the larger moment arm, from the center of mass, of the aft planes in relation to the position of the foreplanes. As both sets of planes provide similar amounts of lift for a given angle of attack, a small disturbance in pitch will be stabilized by the larger moment provided by the aft planes than the foreplanes. The inverse approach to this analy-sis can be used as well; stability criteria can be used in the design process [112]. From the principle particulars of the vehicle given in Chapter 4, we see that the buoyancy of the hull is larger than the weight. Also the position of the center of buoyancy is located longitudinally forward of the center of mass. This introduces a steady state moment and lift on the hull. In order to counter these effects, steady state inputs or hydroplane deflections must be applied. This is called feed forward control, and is used in an open loop manner. An iterative procedure is used to find the necessary feed forward input magnitudes, using the non-linear model, based on the desired linear operating condition. In designing a controller for control of deviations from a non-zero reference state, the deviation state vector is stated as: A * = ( * - * l i n ) ' A * = * (5.18) substituting in eqs. 5.2 and 5.3, ( A ? - ^ i i n ) = A ( A * _ * i i „ ) + ?5 + * 0 ' a n d (5.19) V = C ( A x - x l i n ) (5.20) 82 In state space form, for x l i n, the desired equilibrium, unchanging in time, Ax = AAx + Bu + Ax.. +Bu.. +x _ — hn — h n - o y = CAx + Cx,. + y.. i — - — l i n i l i n The control u.. to maintain the reference state with zero error is found by setting - h n J Ax = 0, Ax = 0, and u = 0 , and thus, u,. = B T (B T B) _ 1 (Ax . . +x ) ,and _hn — v — — ' v lin - c r y.. = -Cx.. i l i n — h n (5.21) (5.22) (5.23) (5.24) In other words, if output regulation or tracking is used, the reference signal is simply the desired output of the system, and the feed forward input is found from the pseudo-inverse of the input matrix. The overall state space matrices remain unchanged. This approach is assumed in the fol-lowing design process. 5.2 Comparison of Control Methods As previously mentioned, control of the vehicle is performed using state feedback regulation. It should be noted that all 12 states can be used for control, as they are all measured or can be esti-mated. However, the x,y coordinates (measured by GPS) are used to generate a reference heading and perpendicular distance to the line being followed, and thus only 10 of the states are used directly. All simulations presented are based on an operating condition of 6 m/s forward speed. A block diagram of the system is shown in Figure 5.13. Figure 5.13: Simulation Block Diagram. 83 In order to control speed more effectively, a constant rpm feed-forward control term u f f is added to the command input to the plant. The simulation software implemented with MATLAB 5.0 per-forms the integration of the differential equations. This thesis only studies continuous time simu-lation and neglects actuator dynamics other than saturation, as the digital sampling and actuator response is fast in relation to vehicle dynamics. Six feedback control methods are described in the following sections. 5.2.1 Fixed Gain Compensators PD Control Proportional plus derivative control is easily implemented and uses an intuitive approach to con-troller gain selection. It is possible to dictate which planes control a desired motion e.g. bow planes may control heave alone, while stem planes control pitch. This approach is fairly robust if carefully designed, but due to unspecific pole locations, large overshoot or sinusoidal response may be apparent. This problem can be improved by checking the resulting closed loop poles. A major drawback to the PD approach is its poor steady state error. With the addition on an error integral term, resulting in a PID controller, better performance can be realized. However, the addi-tion of an integral term in a system which has actuator saturation characteristics is not desirable, and may lead to instability or oscillation. Care must be taken with the PD approach as to which states or outputs are controlled, control of body reference states can lead to deviation from the equilibrium state desired in the earth fixed frame. Pole-Placement Similar to the previous method, it is possible through pole-placement to achieve desired closed loop characteristics. Typically this method uses high gains to achieve desired poles. With a careful selection of poles, gains can be kept reasonable without a lack of performance or plane saturation. Pole placement methods used for a general MIMO design lack insight as to which modes of the system are controlled by the placed poles. As a result the complete set of poles are often placed in or about the same location. This assumes that all the modes of motion of the system have the same bandwidth and time constant which is not necessarily true. Slower modes are forced to react quickly, resulting in high gains, while faster modes are forced to react slower. Modal Control Modal control involves a MIMO application of the pole placement design method. The design system is transformed to a modal canonical form before the poles are placed. In the study of modal control for the vehicle, it is found that due to the single lateral actuator (the rudder) only 5 of 12 modes can be independently controlled. Although all the modes are stable, this method is useful for controlling those modes which show less stable characteristics, typically, roll and pitch motions. Although precise design of neglected modes is not possible. LQR Optimal Control By assuming a linear time-invariant model, it is possible, by minimizing the performance index, 84 1 J = j(z T Q*z + uTRu)dt (5.25) to to determine optimal steady-state gains. This method has shown to have good performance on a linear model [5, 22] but may not be robust enough for use in a non-linear case unless gain sched-uling is incorporated. 5.2.2 Advanced Compensators Feedback Linearization Still an area of research, the feedback linearization method uses the known non-linear relation-ships as a basis for control, and requires detailed knowledge of the vehicle hydrodynamics. Cou-pled with a basic feedback control method, this method has proved to be reliable [113] but its design must incorporate unknown dynamics and uncertainties. LQG Optimal Control A popular approach to submarine control is the linear quadratic gaussian (LQG) methodology [114, 115] which uses the combination of optimal filtering and control. The inception of L Q G design was through the study of stochastic disturbances affecting the plant and measurement noise. From the increased understanding of multivariable systems, the study of robustness became popular. An enhancing technique used with LQG is the loop transfer recovery (LTR) method. This method used with loop shaping techniques has been applied to submarines. Due to its robust char-acteristics combined with effective optimal tracking, it is often the basis of comparison for new methods and research [116]. Current work in this area is concerned with systematic loop shaping and H infinity designs [117, 118]. Other popular advanced methods which have been applied to submarine control include sliding mode control [17], fuzzy control, neural network [119], model based predictive control [120], generalized predictive control [121] and various types of adaptive control methods [108, 122]. 5.2.3 Controller Selection A wide range of control techniques can be applied to the problem of submarine guidance, each having relative merits and drawbacks. In order to specify which control alternative is most suited to the application, one must determine the basis upon which the control is to be founded. Whether detailed knowledge of disturbances is present, detailed knowledge of the plant is present, or how the plant will react in different conditions, such information must be used to define the controller most suitable for the application. Research in the area of fluid dynamics has allowed for a good understanding of the physics related to submarine hydrodynamics and the forces which impinge on a submersible. This leads to accurate simulation and modelling of underwater vehicles. For this reason model based control, specifically the LQG method, has been chosen for this study in appli-cation to the DOLPHIN AUV. Also, the wide range of non-linearities and disturbances dictate that 85 the control method must be robust. For this reason, the LTR and loop shaping methods will be applied to ensure the overall stability and robustness of the controller. 5.3 LQG/LTR Design Methodology The LQG method has been successfully applied to submarine models in the past [14, 15, 110, 123], as well as for aerospace applications [124]. Loop Transfer Recovery (LTR) is used as a design tool for ensuring robustness by studying the frequency response of the compensator. This approach to LQG design is quite different from the stochastic origins of the LQG. Traditionally, the compensator is based on the measured or determined covariance of the noise processes, and the state estimator is designed to converge much faster than the controller loop. In the LTR approach, the Kalman filter optimization procedure is simply used to determine a desired loop shape due to its stability and performance properties. The controlled process is assumed to have uncertainties in both the plant and the measurement. The robust LTR design examines the extent to which these uncertainties may exist. The combined estimator and controller loops are designed together to achieve the desired bandwidth, thus increasing the stability margins. In these works, the design plant model is augmented using various techniques to improve steady state performance and also to ensure better disturbance rejection. The present work introduces a disturbance observer augmentation of the design model. The augmentation methods are compared for two sea conditions and a turning maneuver. The actuator deflection levels and tracking error are compared for the simulated conditions. 5.3.4 Open Loop Design For an arbitrary plant in the form of eq. 5.5, there exists a matrix of transfer functions relating the inputs Uj to output yj relationships. The magnitude of the response of the output of the plant depends on the input values, a unique gain for all the input to output relationships does not exist. It is possible, however to bound the magnitudes of the responses: \\^M4 |G- l(s)y(s)| ||u(s)|| ' ^ ||y(s)|| ' using matrix norms [125]. If CL as defined by eq. 5.5 is real and has m rows or outputs and n col-umns or inputs, and m > n then the positive square roots of the eigenvalues of G_ G_, or the roots of det(sl - GTG_) = 0, are called the singular values. If n > m then the singular values are the positive square roots of the eigenvalues of GG . Thus there are the same number of singular values as the lesser of n and m. If E is the diagonal matrix of the singular values of (3, ordered from maximum to minimum, then G_ can also be written as: G = Y E V T . (5.26) 86 Where, Y and Y_ map the orthogonal row and column spaces respectively. The matrices Y and V are unitary, complex and have dimension according to the preceding paragraph. This is known as singular value decomposition. If the matrix G is considered to be a function of frequency, G(s), the there exist a set of singular values as a function of frequency, also known as the 'principle gains'. The matrix E gives an indication of the relationship between input and output of G. The singular values are denoted by Oj(s) > a2(s) > ... > 0"m i n. The maximum singular value is often denoted by o and the minimum by a. Y represents the output directions of the plant and V repre-sents the input directions. The column vectors of V are orthogonal and have unit length as do the columns of Y. Letting, y(jco) = G(j(o)u(jco) , (5.27) then, |G(jco)u(jco)| _ o((o) < I U "<q(co) IH^II . (5.28) This is proved by the following. As V is unitary, then Y T Y = I- If we multiply eq. 5.26 by V then we have: Gv. = a i Y i (5.29) Where v and u are both vectors of unit length, N l 2 = L ^ t o L 3 5 1 -If we consider an input in the direction v, and the output in the direction y, then the singular value denotes the gain of the plant, G, in due to the given input. This gives [125], = | G v . | 2 = l l j j^p (5.30) The response of the system, G, to a unit input is bounded by the singular values of G. A plot of the singular values (we will use terminology singular value plot in referring to the principle gains) for a MIMO system is often considered equivalent to a Bode plot for SISO systems. This is not exactly true but the singular value plot is a characterization of the input to output ratios for a MIMO system. The condition number is defined as the ratio of the maximum singular value to the minimum sin-gular value: 87 <J(G) 5(1)- <531) and is a measure of the parametric uncertainty of the plant. It also gives an indication of the ease of inverting the plant matrix, and subsequently may indicate the ease of controlling the plant. In this case, we are given the plant model G(s) given in eq. 5.5 which is defined in Appendix D. As this is a multi-output case, we examine the open loop singular value plot for the system as shown in Figure 5.14. Singular Va lues -400 I I ' I 10'4 10'3 10"' 10"1 10° 10* 10s 103 10* F r e q u e n c y ( r a d / s e c ) Figure 5.14: Open loop singular values of the 10 output plant. In this plot we have assumed that all states are used for output. Note that 1 of the 6 singular value curves are extremely low in magnitude. This indicates that there is no direct relationship between input and output for at least 1 of the outputs. In other words, only 5 output channels can be con-trolled independently. This does not mean that the remaining 5 states cannot be controlled, as internal compensators and loops can be used to subsequently control internal states. In order to use robust control and guarantee the independent control of given outputs, the number of output channels must be reduced. Five independent outputs must be chosen. It is also advisable to choose output signals which have similar dynamics and magnitudes. As the measurement infor-mation for inertial pose and position has less noise for the DOLPHIN case than the rate channels, we will use as many of these as possible. For this design scenario, the outputs chosen are: y = {u, z, ((), 9, \|/}T. 88 This choice of outputs allows the plant to remain observable and controllable. The rank test for observability and controllability is used. The observability and controllability matrices are shown in Appendix D. The singular value plot of the 5 output plant is shown in Figure 5.15. Figure 5.15: Singular values of the 5 output plant. Examining the singular value curves in Figure 5.15, several characteristics of the plant dynamics are immediately apparent. The low frequency limit for two of the curves have slopes of -20db/dec, while three of the curves have zero slope. The minimum singular value curve is below unity gain at low frequency, the compensator will have to deal with large steady state error and poor distur-bance rejection. The curves have varying magnitudes and four of them crossover between 1 and 6 rad/sec. Overall, the dynamics of the MIMO system are quite different. This is due to the units or dimension of the inputs and outputs. In the following design procedure, loop shaping will be used for the open loop singular value curves to achieve the desired dynamics. The most rudimentary of these methods is that of scaling. In this design input and output scaling will be used. Consider that we would like the singular value curves to have similar magnitude at crossover, and the low frequency limits should be raised. Taking the magnitude of the return ratio, ||G(jco)|| at a frequency of 1.2 rad/sec, TABLE 5.2 Input to output gains at 1.2 rad/sec. Gains dfp dfs dap das dr dn u 0.5029 0.5029 1.0567 1.0567 0 0.1350 z 0.4486 0.4486 2.1479 2.1479 0 0 89 TABLE 5.2 Input to output gains at 1.2 rad/sec. Gains dfp dfs dap das dr dn <t> 2.9657 2.9657 2.6348 2.6348 0.9558 0 0 1.3914 1.3914 2.9238 2.9238 0 0 V 0.6285 0.6285 0.5584 0.5584 1.9320 0 If we modify the inputs by multiplying by the matrix, K } = diag([i i i i i iq]) (5.32) and scaling the outputs by K o = diag([ 3 4 1 6 4]) , (5-33) then, ? P s = I p ^ i . and (5.34) C = K C . (5.35) —ps _o—p v / This gives the scaled plant return ratio magnitudes of: TABLE 5.3 Input to output gains at 1.2 rad/sec. Gains dfp dfs dap das dr dn u 1.5087 1.5087 3.1702 3.1702 0 4.0508 z 1.7945 1.7945 8.5917 8.5917 0 0 2.9657 2.9657 2.6348 2.6348 0.9558 0 0 8.3486 8.3486 17.5429 17.5429 0 0 V 2.5142 2.5142 2.2336 2.2336 7.7281 0 These magnitudes are generally much closer together at 1.2 rad/sec. These scaled values are an artificial modification of the actual plant. The scaling of pitch has been magnified to weight the pitch filtering, this will be discussed later. The actual plant singular values are approximately 30 db lower in some cases. The new modified plant is used for design of the control system and the scaling gains must be applied to the outputs and inputs of the compensator. The trim output is intentionally magnified to avoid known non-linear effects which occur as a result of high angle of attack of the hull. The resulting singular value plot for the scaled plant is shown in Figure 5.16. 90 Singular Values 10"* 10"3 10'2 10"' 10° 10' io! 103 10* Frequency (rad/sec) Figure 5.16: Singular values of the scaled 5 output plant The new singular value curves crossover between 2.8 and 9 rad/sec. All the low frequency limits are above 20 db. The scaled plant will then be used to make further loop shaping design changes. 5.3.5 Filter Design The first stage in LQG/LTR design is the design of the filter gains. The linear optimal filter approach is assumed, so the filter gains are fixed. The LTR design has been limited to 5 outputs. It is observed that the maximum number of output states which can be controlled while ensuring robustness cannot exceed the rank of the actuator matrix using the LTR method. Figure 5.17: Kalman Filter Structure. 91 Also, the plant output states to be controlled cannot be a result of linearly dependent input - output transfer functions. In other words, each output must be independently controllable to guarantee robustness. The filter is designed based on the following plant and assumed noise processes: Here the subscript 'ps' shows that the state space equations have been scaled. The process noise levels are represented by a matrix M- In the LTR method this represents a set of fictitious plant noise processes, w, of unity intensity and having covariance of W- This is different from the tradi-tional LQG design which uses actual or predicted noise characteristics. Similarly, the sensor noise levels are represented by n, a fictitious sensor noise, of covariance N.. Both w_ and n are assumed white noise processes by the traditional LQG design, E(wwT) = W > 0 , and (5.38) * = A p s x + B p s u + Mw (5.36) y = C p s x + In (5.37) E(nnT) = N>0 . (5.39) The LQG design requires the design of the filter loop and the controller loop: Control The optimal control goal is to minimize the cost function, (5.40) using the feedback relation, (5.41) One can then solve for K using, - l T (5.42) and the algebraic Riccati equation: A T P c + ? c A - P c B R _ 1 B T P c + Q = 0 . (5.43) This solution gives the steady state regulator solution to the stochastic control problem. 92 Filter In the LQG approach, a Kalman filter [126] is also designed based on the stochastic problem. The cost function function, J = E / o o J((x-x) T (x-x))dt VO J is minimized. We will define, x = x - x _e - -The Kalman filter gains are found by, (5.44) (5.45) L = P f C T N 1 (5.46) and by solving the algebraic Riccati equation, ? f a T + A ? f _ ? f ? T N _ 1 C P f + M W M T = 0 (5.47) The transfer function for the open loop, filter alone, is then: G K F (s) = C ( s I - A ) _ 1 L (5.48) Using the LTR design methodology it is advantageous in the design process to design the filter loop first, based on the desired characteristics, and then to design the controller loop. This allows preliminary shaping of the filter loop before having to complete the design to check its properties. The singular value plot of the Kalman filter is used to tune the singular values to exhibit the desired loop shape. A scalar multiplier p. can be used to increase or decrease the crossover fre-quency to the desired robustness. If (A, B) is stabilizable and (A, C) is detectable, then the follow-ing properties are guaranteed for the Kalman filter loop [110]: 1) It is closed loop stable. 2) o m i n ( I+Q K F ( s ) )>=l . Omina+GKF-1(s))>= 1/2. 3) Possible upward gain is infinite. 4) Possible downward gain is 6 dB. 93 5) The maximum phase margins are ± 60 degrees. These properties allow flexible design constraints with guaranteed performance properties based on the controller recovery. Essentially, the Kalman filter design, viewed as a feedback system has good robustness and performance characteristics and stability is guaranteed. We now turn to the problem of using the fictitious tuning parameters to shape the singular value plot of the open loop filter design. If the plant matrix, A, has distinct, non-zero eigenvalues, then the singular values can be made to match at extreme low and high frequencies using the method of [26]. However, we will look at a more 'design' oriented approach and the curves will be shaped specifically. Keeping in mind that we are designing the desired overall loop shape which we hope the control design will recover. Note that it is also possible to design the control loop first and then design, through duality, the filter loop. If we consider the plant system disturbance as acting on the input, as shown in Figure 5.18, Figure 5.18: Block diagram of system. then we can assume that, M = B . (5.49) To start the design process we choose W_ and N, in eqs. 5.36 and 5.37 as unity matrices. Solving the for the Kalman filter gains using eqs. 5.46 and 5.47 using the LQE function in matlab which solves equations 5.46 and 5.47: L f = (5.50) 1 5836 0 0054 0 0000 0 0585 0 0000 0 0000 0 0000 3 3065 0 0000 1 6923 0 0228 1 3517 0 0000 2 0317 0 0000 0 0000 0 0000 11 3218 0 0000 1 5917 0 0632 0 6154 0 0000 9 3854 0 0000 0 0000 0 0000 0 4350 0 0000 4 0047 94 0 . 0 0 4 1 0 . 7 9 3 5 0 . 0 0 0 0 0 . 2 1 4 8 0 . 0 0 0 0 0 . 0 0 0 0 0 . 0 0 0 0 4 . 7 4 7 7 0 . 0 0 0 0 0 . 3 2 0 4 0 . 0 2 9 2 0 . 1 4 3 2 0 . 0 0 0 0 1 . 7 6 2 7 0 . 0 0 0 0 0 . 0 0 0 0 0 . 0 0 0 0 0 . 0 8 0 1 0 . 0 0 0 0 1 . 4 1 2 8 Using eq. 5.48 the Kalman filter loop open loop can be examined. The singular value plot for this resulting output to estimated state ratio is found as Figure 5.19. At high frequency all the singular value curves have a slope of -20 db/dec and their respective crossover points are in the range 3.5 to 10.2 rad/sec. The minimum singular value at low frequency has a value of approximately 22 db. It is important to design the compensator based on performance requirements. Setting some design criteria, we will attempt to produce a bandwidth between 5 and 10 rad/sec for the plant and compensator loop. From the open loop singular value plots, it is known that the plant bandwidth is between 3 and 10 rad/sec. The reason for choosing a low bandwidth is due to the unknown plant non-linearities and, more importantly, the bandwidth of the compensated system must be less than that of the actuators used to control the system. If the actuators are not able to respond fast enough, instabilities will occur. This will be discussed later in this chapter. In addition, the actuators have saturation characteris-tics, which will be exceeded if high control and filtering gains are used. If the compensated sys-tem is pushed to a bandwidth higher than that of the open loop plant, high gains will typically be needed. In deciding the bandwidth of the submarine, a time constant between 2.5 and 4 seconds was considered acceptable, assuming a critically damped system. From preliminary testing in simulation [127] this appears to be a reasonable condition. Comparing with other LTR designs for large submarines [15, 123], having bandwidths of 0.05 to 0.5 rad/sec, the DOLPHIN AUV is able to maneuver much faster. Also, we will attempt to achieve a low frequency minimum singular value of 20 db. This corre-sponds, when the loop is closed, to a steady state error of less than 0.5%, using final value theo-rem. In addition, low sensitivity at low frequencies is indicated by a high open loop return ratio. Ideally, the slope of the singular value curves should be -40 db/dec at frequencies above crossover. Stability is guaranteed through the LQG design procedure assuming small deviations from the lin-ear design model. Looking at Figure 5.19, the design roll-off slope criterion cannot be met unless the system is aug-mented. However, for the sake of comparison, we will continue with the basic LQG/LTR control-ler design. 95 Singular Values 10* 10'3 10'! 10"' 10° 10' 102 103 10* Frequency (rad/sec) Figure 5.19: Singular values of unaugmented Kalman filter first design. A common design requirement is to bring the singular value curves together at crossover. To do this, we will try to make them equivalent through singular value decomposition. Taking the singu-lar value decomposition: C(jcol-A) M ^ W j = U E V (5.51) This is solved at a frequency of 5.42 rad/sec, being just below the desired frequency of 1 rad/sec for, E = diag([2.481 1.188 0.926 0.778 0.291]) (5.52) and, v = C o l u m n s 1 t h r o u g h 4 0 . 3 3 3 1 - 0 . 3 3 3 5 0 . 0 0 5 3 0 . 4 1 0 2 0 . 3 3 3 1 + O . O O O O i 0 . 3 3 3 5 + 0 . 0 0 0 0 1 0 . 0 0 5 3 - O . O O O O i - 0 . 4 1 0 2 - O . O O O O i - 0 . 6 2 3 2 - 0 . 0 2 0 4 1 - 0 . 2 9 6 3 + O . O O O O i - 0 . 0 1 1 2 + 0 . 0 0 2 4 i 0 . 3 6 4 4 + O . O O O O i - 0 . 6 2 3 2 - 0 . 0 2 0 4 i 0 . 2 9 6 3 - O . O O O O i - 0 . 0 1 1 2 + 0 . 0 0 2 4 i - 0 . 3 6 4 4 - O . O O O O i 0 . 0 0 0 0 - O . O O O O i 0 . 7 7 0 3 + 0 . 0 9 3 0 i 0 . 0 0 0 0 - O . O O O O i 0 . 6 2 6 3 + 0 . 0 7 5 6 i - 0 . 0 1 6 2 - 0 . 0 0 7 2 i 0 . 0 0 0 0 - O . O O O O i 0 . 9 7 4 4 + 0 . 2 2 4 0 i 0 . 0 0 0 0 - O . O O O O i C o l u m n s 5 t h r o u g h 6 96 0 . 6 2 3 7 0 . 6 2 3 7 + O . O O O O i 0 . 3 3 3 0 + 0 . 0 1 0 9 i 0 . 3 3 3 0 + 0 . 0 1 0 9 i 0 . 0 0 0 0 - O . O O O O i 0 . 0 0 0 3 + 0 . 0 0 1 9 i - 0 . 4 6 9 6 0 . 4 6 9 6 + O . O O O O i 0 . 5 2 8 6 + O . O O O O i - 0 . 5 2 8 6 - O . O O O O i 0 . 0 0 0 0 + O . O O O O i 0 . 0 0 0 0 + O . O O O O i We then modify W 1 / 2 by the following, 1/2 1/2 T T T W f 2 = W (I + ajreaUVjYj)) • (I + a 2 r e a l ( ^ ...(I + a 5real(v 5Vg)) (5.53) where oq is the multiplier factor for scaling each particular singular value, and V; is a column vec-tor of V. In this case, as we are attempting to bring the values together, we take, ttj = 1 / E j i - l ; i=l,2,...,5 (5.54) 1 fO 1 fO T Then, = W c (W^ ) , and we can solve for the second set of Kalman filter gains. The resulting singular values are shown in Figure 5.20. Singular Values 10 10 10 Frequency (rad/sec) Figure 5.20: Singular values of unaugmented Kalman filter loop second design. This system was found using |1 = 1, W^, M = B and N of the identity matrix. Note that the fre-quencies magnitudes are not identical at the desired crossover frequency due to the approximation of vectors Vj by the magnitude of their real components. 97 To increase or decrease the bandwidth and thus change the overall magnitude of the curves it is possible to use the design scalar factor, \L such that the filter is designed by Lf2 = LQE (AM,C,Wf2m) (5.55) recalling that M = B and N is the identity matrix. As u. is increased the bandwidth is decreased, and as \i is decreased the bandwidth is increased. For this evaluation, the magnitudes of the singu-lar values cannot be scaled down due to their low frequency limits. To achieve better low fre-quency limits, it is necessary to sacrifice going to a higher bandwidth and risk actuator saturation. Using a value of 0.5 for \i, and scaling the singular values higher, the following Kalman filter gains result: L f 2 = 2 4680 0 0024 0 0000 0 0675 0 0000 0 0000 0 0000 4 7373 0 0000 1 4583 0 0250 8 2618 0 0000 0 9451 0 0000 0 0000 0 0000 20 3259 0 0000 -2 4778 0 2324 0 9915 0 0000 4 8017 0 0000 0 0000 0 0000 -1 0247 0 0000 6 5867 0 0018 2 0265 0 0000 0 1552 0 0000 0 0000 0 0000 6 3575 0 0000 -0 4836 0 0337 0 1035 0 0000 1 2605 0 0000 0 0000 0 0000 -0 1209 0 0000 1 8107 (5.56) The resulting singular value plot is shown in Figure 5.21, Singular Values 10 10 10' Frequency (rad/sec) Figure 5.21: Singular values of unaugmented Kalman filter loop third design. 98 This small change in tuning has raised the bandwidth only slightly while achieving closer group-ing of the curves. It is necessary during the design process to have target design goals which are compatible with the plant dynamics. Large gains will be necessary to achieve dynamics which dif-fer greatly from the plant. Further shaping of this filter loop requires either augmentation or H infinity methods. The next stage in the design is to look at the sensitivity and complementary sensitivity functions. The sensi-tivity shows the effect of plant disturbances on the output, while the complementary sensitivity shows the effect of tracking or equivalently measurement disturbances on the output. The sensitiv-ity of G(s) is defined by: S(s) = I (I + G(s)) while the complementary sensitivity is defined by: T(s) = I-S(s) ,or (5.57) (5.58) T(s) = G(s) I + G(s) (5.59) Both these functions use closed loop analysis of the system at hand, one from the output side only, the other from input to output. The sensitivity and complementary sensitivity of the Kalman filter loop is shown in Figure 5.22. Singular Values Frequency (rad/sec) Figure 5.22: Sensitivity and Complementary Sensitivity of the Kalman filter loop. 99 These curves show satisfactory stability margins, as the maximum singular value of the comple-mentary sensitivity is low, but the low frequency limit of the sensitivity is large. This shows extremely poor steady state error characteristics and the tendency for disturbances to be propa-gated. Overall tracking, depicted by the complementary sensitivity low frequency sensitivity lim-its are also poor being slightly above and below 0 db. From this analysis, it is obvious that using the current design will not provide the necessary per-formance attributes. Improvements could be made on the slopes of these curves as well as the shaping of the sensitivity using types of design augmentations. It has become de rigeur to use such augmentations [26]. Details of such improvements are described in the following sections. The open loop plant loop shapes were given in Figure 5.16. If we consider that filters can be placed in certain input or output signals, then the overall loop shapes of the plant (with the filters) will change. Typically, these types of augmentations are called pre- and post- compensators as shown by E2(s) and Es(s) in Figure 5.23. The LQG loop shape with recovery design is then based on the augmented plant. In a sense, the dynamics of the plant have been artificially changed. In order to increase the overall slopes of the singular value curves, integrators or low pass filters are strategically placed before the plant. The precompensator could also include actuator dynamics models. It is possible, however to also augment the plant with disturbance model [10, 125] shown by Ei(s) and P_4(s) in Figure 5.23. Figure 5.23 shows the possible augmentations of the plant in block diagram form. Figure 5.23: Augmentation possibilities for loop shaping. The LQG block is denoted by K(s) and the plant is denoted by G(s). The augmentations are shown by P(s) functions that are assumed as part of the plant. Two of the possible augmentations are described and compared in this chapter. 100 5.3.6 Augmentation of Controller for Plant Input Integration In this approach the actual signal, u, to the actuators is a result of an integration of the commanded actuator signal, u .^ If a continuous actuator deflection is apparent, this will be integrated over time to achieve the desired output, thus no steady state error. The integrator is added artificially, and is considered part of the plant. The resulting LQG design is then based on the augmented system shown in deterministic form as: r -• u 0 0 u I — = — — - + -X B A X 0 u -c (5.60) 0 C (5.61) or in augmented form with white noise disturbances as, X i n = A i n X - + B J n U + M- W _ i n — i n _ i n _ i n _ i n y. = C- x. +n , •Lm _ i n _ m - ' where, (5.62) (5.63) x. = - i n (5.64) and Mjn i s equal to i K i B because, referring to Figure 5.23, the disturbance is added before the integrator E2(s), but this integrator is actually implemented outside the plant so disturbances actually are added after the integrator. If all disturbances were assumed to enter after the integra-tor there would be no effect of this integrator on the poles of the system. For this reason, the choice of M is crucial as it determines the zeros of the open loop system. Using the method of [26], the M matrix is actually used for loop shaping rather than W. Adding the integrators in the design creates a steeper slope in the singular value plots at frequen-cies greater than roll off, thus less sensitivity to high frequency noise. In addition, this design pro-motes less steady state error and faster response at low frequencies. The integration of the commanded signal is performed after the traditional compensator, K(s), as shown in Figure 5.23. Integration of all inputs in the same manner can cause error in some states due to coupling effects 101 which are not accounted for in this approach, as well as oscillations due to actuator dynamics and saturations. Figure 5.24: MIMO loop with actuator integration augmentation. Note that K(s) is an LQG compensator based on eqs. 5.60 and 5.61, and plant augmenting integra-tor is included after the LQG compensator. Anti-wind up reset systems must be implemented in the application as actuator saturation is a large non-linearity. In this thesis, anti-wind-up devices were used both for the integration of u in eq. 5.60. and the additional integrator shown in Figure 5.24. The Kalman filter design is achieved in a similar manner to that of the unaugmented plant explained previously. The same scaling magnitudes have been used as in eqs. 5.25 and 5.27. The open loop singular values of the augmented plant are shown in Figure 5.25. Singular Values -250 I • ' I I 10"* 10 " 3 1 0 ' 2 10 " 1 1 0 ° 10 ' 10* 1 0 3 10* Frequency (rad/sec) Figure 5.25: Singular values of the open loop integrator augmented system. The low frequency limit of the singular value curves all approach infinity, while the slopes of the curves past roll off are -40 and -60 db/dec. As in the example of the unaugmented plant, the Kalman filter gains are determined first by solving: 102 Lm ) f l = LQE(A i n , M I N , Q n . W i n , N i n]) (5.65) with W i n , and N. i n as the identity matrices for a first iteration. The open loop Kalman filter loop singular values are shown in Figure 5.26. Singular Va lues i 1111in i i 11 n m — i i i m i n — i i 11inn 0) 100 CO > 50 10 10 10 Frequency (rad/sec) 10 10 Figure 5.26: First iteration open loop Kalman filter for integral augmented plant Using the same procedure as eqs. 5.51 to 5.55, the resulting tuning parameters are: B (5.66) N i n = I, p = 0.02, and v v i n 0.5665 0.0078 -0.0576 0.3344 0.0130 -0.0020 0.0078 0.5665 0.3344 -0.0576 -0.0130 -0.0020 -0 0.0576 0.3344 0.4000 2517 0.0115 0.0010 0.3344 -0 .0576 -0 .2517 0.4000 -0 .0115 -0 .0010 0.0130 -0.0130 0.0115 -0.0115 0.1327 0.0000 (5.67) (5.68) (5.69) -0.0020 -0.0020 -0.0010 -0.0010 0.0000 0.0555 The resulting filter gains are: 103 Lin,f2 - (5.70) -0 0190 -1 8364 0 8441 0 4648 -0 1490 -0 0190 -1 8364 -0 8441 0 4648 0 1490 -0 0125 -0 9600 0 7499 -0 0715 -0 1324 -0 0125 -0 9600 -0 7499 -0 0715 0 1324 0 0000 0 0000 -0 0353 0 0000 -1 2872 8 3264 0 0268 0 0000 -0 0888 0 0000 4 0418 0 0114 0 0000 0 0302 0 0000 0 0000 0 0000 12 5104 0 0000 3 9568 0 1540 11 1855 0 0000 -0 3127 0 0000 0 0000 0 0000 68 0034 0 0000 2 4318 0 0439 0 1943 0 0000 7 9748 0 0000 0 0000 0 0000 0 7889 0 0000 14 5501 0 0086 2 3649 0 0000 -0 0011 0 0000 0 0000 0 0000 11 6595 0 0000 0 2489 0*. 0151 -0 0008 0 0000 1 6303 0 0000 0 0000 0 0000 0 0622 0 0000 2 6965 These values result in a singular value plot for the open loop Kalman filter as shown in Figure 5.27. Singular Values •100 l — — — I— — i I 10"' 10"3 10'2 10"' 10° 101 102 103 10* Frequency (rad/sec) Figure 5.27: Singular values of the open loop Kalman filter of the integrator augmented plant. 104 The resulting sensitivity and complementary sensitivity curves of the Kalman filter are shown in Figure 5.28. Singular Values 10" ' 1 0 ° 10 ' 1 0 ! 1 0 3 Frequency (rad/sec) Figure 5.28: Singular values of the sensitivity and complementary sensitivity of the Kalman filter for the integrator augmented plant. Both these filter design loop shapes show good characteristics. The overall compensator sensitiv-ity and complementary sensitivity will have these curves as a target during the recovery process of design. 5.3.7 Augmentation of Controller for Disturbance Observing The disturbance observer concept has seen reliable use in the machining and robotics fields [128]. Various approaches to its operation in submarine control have been described in literature [12]. This thesis puts forth the following approach. A disturbance process is assumed which acts by adding some value, u ,^ to the actuator signals. This could change their effective "angles of attack", or could be forces acting on the hull which is assumed, through the model, to be apparent at the plant input. The plant then reacts to the sum of the noise and actuator. Typically, a second order disturbance model is assumed [9], having poles which correspond to the peak wave spectral fre-quency. This thesis uses a first order disturbance model, but the method is presented in a general manner. The poles added to augment the model are used to construct the matrix D j^, representing the dis-turbance model. The augmented model is given in deterministic form by: - _ _ = P d 9 Hd + 0 X B A X B 105 (5.72) It should be noted here that the added poles can be tuned to any expected disturbance frequency, which could include the encounter of waypoints. As there is no process which drives the distur-bance model in the deterministic form, the states are only realized by the Kalman filter. The dis-turbance states are not directly controllable but are observed and then used to control the original 10 states of the model. The expected disturbance magnitudes, typically of the same order as actu-ator signals, are used to tune the filter. The disturbance model is used to represent the expected frequency domain spectral distribution of the noise process. From chapter 4, the wave frequencies have been evaluated to exist mainly between 0.5 and 12 rad/sec, while the waypoints are encountered every 10 m, which corresponds to approximately 1 rad/sec. Unfortunately, these frequencies coincide with the bandwidth of the plant dynamics, which increases the difficulty in filtering. However, in order to provide a repre-sentative disturbance model while maintaining a large return ratio of the open loop compensated system at low frequencies, the disturbance model will be assumed as: D d = diag([_0.45 -0.45 -0.45 -0.45 -0.45 -0.45]) (5-73) The poles of the augmented system must be representative of a stable system as its states cannot be controlled directly. The singular values of the plant augmented by the disturbance observer are shown in Figure 5.29. Singular Values -150 I •— I '— ' nl 1— I 1— I 1— I ' i ' 1 0 ' 4 10 " 3 1 0 ' 2 10" ' 1 0 ° 10 ' 1 0 ! 1 0 3 1 0 " Frequency (rad/sec) Figure 5.29: Singular values of the open loop plant with disturbance model. 106 For the plant alone, open loop case, the singular value plot is identical to that of the unaugmented plant as the added poles only affect the observed disturbances. The effects of the augmentation are not apparent until the compensator is added to the loop. As in the integrator augmentation, the Kalman filter gains are solved from: Lob,fi = LQECAob, M o b , Wob, Nob]) (5.74) with W0b> and as the identity matrices for a first iteration. The open loop Kalman filter loop singular values for the disturbance observer augmentation are shown in Figure 5.30. Singular Values 1 0 * 1 0 ' 3 1 0 ' 2 10" ' 1 0 ° 1 0 ' 1 0 2 1 0 3 10* Frequency (rad/sec) Figure 530: First iteration open loop Kalman filter for the plant augmented with disturbance observer. Again using the same procedure as eqs. 5.51 to 5.55, the resulting tuning parameters are: M o b = (5.75) N o b = I. | i = 1 , and Wob = 2.5336 1.4901 0.9897 0.9510 0.0761 1.4901 2 .5336 0.9510 0.9897 - 0 . 0 7 6 1 0.9897 0.9510 1.0356 0.0013 0.0676 0.9510 0.9897 0.0013 1.0356 -0.0676 0.0761 -0.0761 0.0676 -0 . 0676 0.6217 0.0109 0.0109 -0.0143 -0.0143 0.0000 (5.76) (5.77) (5.78) 107 0.0109 0.0109 -0.0143 -0.0143 0.0000 1.7807 The resulting filter gains are: Lob,f2: (5.79) 0 0215 -2 3919 0 9663 0 5655 -0 1773 0 0215 -2 3919 -0 9663 0 5655 0 1773 -0 0189 -1 2345 0 8584 -0 1078 -0 1575 -0 0189 -1 2345 -0 8584 -0 1078 0 1575 0 0000 0 0000 -0 0284 0 0000 -1 3786 2 5317 0 0032 0 0000 -0 0081 0 0000 1 5879 0 0040 0 0000 0 0620 0 0000 0 0000 0 0000 2 8991 0 0000 0 9533 0 0241 3 7923 0 0000 0 3540 0 0000 0 0000 0 0000 10 8277 0 0000 -0 4626 0 1305 0 3572 0 0000 2 2946 0 0000 0 0000 . 0 0000 -0 4605 0 0000 3 2322 0 0030 '1 3745 0 0000 0 0829 0 0000 0 0000 0 0000 4 6475 0 0000 -0 2369 0 0310 0 0552 0 0000 0 8723 0 0000 0 0000 0 0000 -0 0592 0 0000 1 2699 These values result in a singular value plot for the open loop Kalman filter as shown in Figure 5.31. Singular Values 10 10 10 Frequency (rad/sec) Figure 531: Singular values of the open loop Kalman filter of the integrator augmented plant. 108 The resulting sensitivity and complementary sensitivity curves of the Kalman filter are shown in Figure 5.32. Singular Values 1 0 ' ' 1 0 ° 1 0 1 10* 1 0 3 Frequency (rad/sec) Figure 532: Singular values of the sensitivity and complementary sensitivity of the Kalman filter for the integrator augmented plant. Both these filter design loop shapes show good characteristics. The overall compensator sensitiv-ity and complementary sensitivity will have these curves as a target during the recovery process of design. 5.3.8 Controller Design (Recovery) Just as crucial as the design of the filter gains is the design of the controller gains. Due to the prop-erties of duality the process is similar. The combined filter and controller system, K(s) is shown in Figure 5.33 for the LQG arrangement. 109 Figure 533: LQG Compensator Block Diagram. Kalman filter is denoted by dashed box, variable x is the estimated state vector. Relative weights were assigned to each state based on the inverse of the square of the maximum desired error during a given maneuver. These weights were then used to construct a matrix Q. The maximum desired actuator deflections were used to find a matrix R, weighting the input values. The performance index is then a function of these two variables: (xTQx + uT(pR)u)dt (5.80) Again the Ricatti equation is solved using Q and pR for the controller gains K^. The parameter p is used in LTR design to tune the amount of recovery of the compensator. As p approaches zero, the compensator approaches full recovery of the Kalman filter loop, typically using large gains. After providing feedback the compensator loop system becomes: x = ( A - L C - B K c ) x - L e so the transfer function is: K(s) = K c ( s I - A + L C + B K c ) L . Examining the complete compensator and plant loop, the system is: (5.81) (5.82) d X A " B K C dt X K c C A - L C - B K , or (5.83) no d_ dt A - ? K C B K c 0 A - L C x IX. (5.84) From this it is apparent that as long as the eigenvalues of the filter and the eigenvalues of the con-troller loops are both stable, the closed loop compensated plant should be stable, within the limits of the linear plant. Recall that the augmented plant state space equations are given by eq. 5.60. Note that the compensator is designated by K(s), the controller gains by K^ and the Kalman filter gains by L. After defining the desired cut-off frequency and singular value loop shapes in the Kalman filter design, the recovery of this design is determined. The iterative tuning procedure for recovery is summarized by: 1) Initially define Q = C £ and R = I for equal recovery of all controlled outputs. Note this does not necessarily induce any weighting for states which are not measured directly. 2) Initially define p = 1. 3) Examine singular value plot of G(s)K(s) for recovery of desired bandwidth, and roll-off. The slope of the roll of determines the amount of attenuation of high frequency uncertainties in the plant. The slope of the curves should be -20db/dec at and around crossover for good stability mar-gins. Grouping of the singular values at crossover should be achieved in recovery. 4) Examine S(s), and T(s). The maximum singular value magnitude of these plots is an indication of stability and performance margins. The -3db point of S_(s) is the considered as the bandwidth of the closed loop system. 5) Examine the resulting gains, K^ to determine conditions for actuator saturation. 6) Decrease p logarithmically until the desired amount of recovery is achieved. 7) If specific recovery requirements are necassary, modify Q and R to achieve appropriate weight-ing. In other words: once both the filter, L, and controller, K^, gains are found, the open loop system G(s)K(s) is determined. Final tuning of the controller is achieved using the singular value plot of G_(s)K(s). The closed loop feedback system (I-Q(s)K(s))_1Q(s)K(s) is examined to check the over-all tracking and disturbance propagation loop characteristics of the system. High controller gains tend to produce closed loop response closer to 0 dB at low frequencies, and tend to shift the band-width higher, increasing the amount of recovery and bringing the overall G(s)K.(s) loop shape closer to that of the open loop Kalman filter shape. This has advantages and drawbacks. The gains required for more recovery can lead to instability in the form of actuator saturation, and also decrease the steepness of the slope past the bandwidth of complementary sensitivity. However, better recovery also tends to bring about the favourable characteristics of the loop shaped filter design, and also increases the stability margin. in First, the integrator augmentation design recovery will be completed. The system defined by equations 5.60 and 5.61, are used in the controller design. Using Q a l = C a i (Kw)C ai, and R = I. Here it is necessary due to the non-linearities associated with pitch to weight the pitch control sig-nificantly by: Kw = diag ([1 1 1 100 1]) (5.85) We solve the Riccati equation 5.43, using the Matlab LQR command:' K i n = LQR(A i n , B i n , Qi,,, pR]) (5.86) The resulting open loop singular value plots of G_(s)K(s) for varying values of p, and thus varying levels of recovery are shown in the Figures: Singular Values 200 I 1 1 I I I M i l 1 1 I I I M i l 1 I 1 I 1 1 i i i i -300 I ..I I 1 0 * 1 0 ' 3 10 " 2 10' 1 0 ° 1 0 ' 1 0 2 1 0 3 10' Frequency (rad/sec) Figure 534: Singular values of G.(s)K(s) for p = 0.05 using the integrator augmented plant. 112 Singular Values 200 | | , „ Frequency (rad/sec) Figure 5.35: Singular values of Q(s)K(s) for p = 0.005 using the integrator augmented plant. Singular Values 200 | , , i , Frequency (rad/sec) Figure 5.36: Singular values of Q(s)K(s) for p = 0.0005 using the integrator augmented plant. From these figures it is obvious that the greater the level of recovery, the bandwidth is increased to approach that of the Kalman filter loop, the region of the curve having -40 db/dec and greater roll 113 off slopes are pushed to higher bandwidth. In general the shape of the curves approach those of the open loop Kalman filter loop. The trade off is that the system becomes more sensitive to high frequency disturbances due to the higher controller gains. From these figures, it appears as though the values of p = 0.05 and 0.005 show ample recovery. We now compare the sensitivity and com-plementary sensitivity for these two values. Singular Values 10 o •10 •'. -, S -20 at L" - . '. J .... ..1 .J 1 _ Mar Value: o \ \ \ -!-! !-!!-!!-C7> .E -40 CO . . . . . . . / • ; - - -• - • - -V\ -50 -60 ...y:, 7..'12. ... W ' - ' T ' J " -70 - A. W . - . r\ • -. -102 10' 10° 10' 10! 103 Frequency (rad/sec) Figure 537: Singular values of sensitivity and complementary sensitivity of Q(s)K(s) for p = 0.02 using the integrator augmented plant. Singular Values 10 M l - - t j - I I I I I U . - I - I- I- U 14II- - 4 -I -I 1 M IH • - I- I- I- I J I -111- - 4 -I -I j l-I IU -I 10'2 10' 10° 10' 102 103 Frequency (rad/sec) 114 Figure 5.38: Singular values of sensitivity and complementary sensitivity of Q(s)K(s) for p = 0.002 using the integrator augmented plant. It appears from Figures 5.37 and 5.38 that the increase in recovery lowers the maximum sensitiv-ity by 1 db and the complementary sensitivity is relatively unchanged. As the value of p = 0.02 shows the maximum T(s) value of 7 db, this shows reasonable stability margins. The resulting controller gains for the p = 0.005 value are: K i n = (5.87) Columns 1 t h r o u g h 7 11.6185 4 .9951 -4 2006 -10 0848 -0 2427 0 0060 0 0264 4 .9951 11.6185 -10 0848 -4 2006 0 2427 0 0060 0 0264 -4 .2006 -10 .0848 20 0881 14 8605 -0 2156 -0 0106 -0 0474 -10 .0848 -4 .2006 14 8605 20 0881 0 2156 -0 0106 -0 0474 - 0 . 2 4 2 7 0.2427 -0 2156 0 2156 12 9414 0 0000 0 0000 0.0060 0.0060 -0 0106 -0 0106 0 0000 11 6400 40 3426 Columns 8 t h r o u g h 14 0.2941 - 9 . 7 0 2 0 0 8686 23 5144 -1 1546 -34 5898 7 1898 - 0 . 2 9 4 1 - 9 . 7 0 2 0 -0 8686 23 5144 1 1546 -34 5898 -7 1898 0.2612 -7 .8948 0 7717 -41 9003 -1 0257 -20 0885 6 3875 -0 .2612 -7 .8948 -0 7717 -41 9003 1 0257 -20 0885 -6 3875 2 .9330 0.0000 -0 3250 0 0000 -11 2734 0 0000 -1 2925 0.0000 0.0008 0 0000 0 0603 0 0000 -0 0019 0 0000 Columns 15 t h r o u g h 16 301.1690 - 3 . 0 3 8 5 301.1690 3.0385 -518.5948 -2 .6994 -518.5948 2.6994 0.0000 -56 .2758 1.5182 0.0000 It is noted that Figures 5.34 to 5.38 use the plant modified by both scaling and an integrator for analysis. In the implementation of the LQG compensator these elements must be included outside the framework of the typical LQG system. The resulting open loop poles and zeros of the com-pensated system are: 115 Pole-zero map Figure 539: Poles and zeros of closed loop system augmented with actuator integrators. All the poles and zeros lie in the left side of the imaginary plane thus there are no unstable zeros of poles in the system. Note that the integrators are the poles which lie at the origin. We now proceed with the controller design for the augmentation involving the disturbance model. The controller gains are applied to the new state vector {u^ , x}. Following the same procedure as above, the control gains are found by solving the algebraic Riccati equation using Matlab: = LQRCA^, B^b, $ , „ , PR]) (5.88) The resulting open loop singular value plots of G(s)K(s) for varying values of p, and thus varying levels of recovery are shown in the Figures: 116 Singular Va lues Frequency (rad/sec) Figure 5.40: Singular values of G(s)K(s) for p = 0.08 using the disturbance observer augmented plant. Singular Values 150 , , , , , •200 I I 10'' 10' 3 10* 10"' 10° 10' 10* 10 3 10* Frequency (rad/sec) Figure 5.41: Singular values of fi(s)K(s) for p = 0.008 using the disturbance observer augmented plant. 117 Singular Values 10 10 10 Frequency (rad/sec) Figure 5.42: Singular values of G(s)K(s) for p = 0.02 using the disturbance observer augmented plant. From these figures, it appears as though the values of p = 0.008 and 0.0008 show good recovery. The low frequency minimum singular values are increased from 3.6 to 4.1 to 4.7 db respectively in the three situations. We now compare the sensitivity and complementary sensitivity for the p = 0.008 and 0.0008 values. Singular Values Frequency (rad/sec) Figure 5.43: Singular values of sensitivity and complementary sensitivity of G(s)K(s) for p = 0.008 using the disturbance observer augmented plant 118 Singular Values Frequency (rad/sec) Figure 5.44: Singular values of sensitivity and complementary sensitivity of G(s)K(s) for p = 0.0008 using the disturbance observer augmented plant. It appears from Figures 5.43 and 5.44 that the increase in recovery only lowers the maximum sen-sitivity by about 1.5 db and the complementary sensitivity is relatively unchanged. Although the value of p = 0.0008 shows the maximum T_(s) value of about 3 db, and this shows better stability margins than the integrator augmentation, the gains necessary are large. For this reason the value of p = 0.008 will be used. This also shows maximum T(s) which are lower than that of the integra-tor augmentation. The low frequency sensitivity is about -38 db or 0.013 which is acceptable. This is, however not as beneficial as the low frequency limit of 0 characteristic of the integrator aug-mentation. The resulting controller gains for the p = 0.008 value are: K o b = (5 -89) Columns 1 t h r o u g h 7 0.7737 0.2200 0 2446 -0 2473 0 0009 0 0001 0 0021 0.2200 0.7737 -0 2473 0 2446 -0 0009 0 0001 0 0021 0.2444 -0 .2475 0 7178 0 2807 0 0008 -0 0002 -0 0038 - 0 . 2 4 7 5 0.2444 0 2807 0 7178 -0 0008 -0 0002 -0 0038 0.0002 -0 .0002 0 0002 -0 0002 0 9952 0 0000 0 0000 0.0000 0.0000 0 0000 0 0000 0 0000 0 9868 33 3625 Columns 8 t h r o u g h 14 0.0338 - 3 . 5 3 0 1 0 3097 3 4823 -0 3774 -27 4946 5 8382 -0 .0338 - 3 . 5 3 0 1 -0 3097 3 4823 0 3774 -27 4946 -5 8382 119 0 . 0 3 0 0 - 0 . 0 3 0 0 0 . 2 5 3 2 0 . 0 0 0 0 - 2 . 0 3 5 7 - 2 . 0 3 5 7 0 . 0 0 0 0 0 . 0 0 0 0 0 . 2 7 5 1 - 0 . 2 7 5 1 - 0 . 0 6 2 3 0 . 0 0 0 0 - 6 . 0 0 7 4 - 6 . 0 0 7 4 0 . 0 0 0 0 0 . 0 0 0 8 - 0 . 3 3 5 3 0 . 3 3 5 3 - 3 . 1 5 6 9 0 . 0 0 0 0 - 1 5 . 6 2 2 0 - 1 5 . 6 2 2 0 0 . 0 0 0 0 - 0 . 0 0 0 1 5 . 1 8 6 8 - 5 . 1 8 6 8 - 1 . 4 6 3 8 0 . 0 0 0 0 C o l u m n s 15 t h r o u g h 16 2 3 4 . 3 2 5 1 2 3 4 . 3 2 5 1 -412 . 4 1 3 8 - 4 1 2 . 4 1 3 8 0 . 0 0 0 0 0 . 1 4 4 8 - 3 . 1 2 0 4 3 . 1 2 0 4 - 2 . 7 7 2 2 2 . 7 7 2 2 - 4 4 . 3 3 0 1 0 . 0 0 0 0 In comparing these gains to those of the integrator augmentation, they are similar. The informa-tion displayed in Figures 5.40 to 5.44, are based on the unaugmented plant G(s) as the disturbance model is not a part of the plant but is external. The singular values are not affected by the distur-bance model in the plant as seen by Figures 5.16 and 5.29. It is noted that if the controller gains attributed to uj are set to the identity matrix such that: u = [-1 - K (5.90) the open loop poles which were added to the compensator return to the values set in D^, however, as these states cannot be controlled, this has no effect on the closed loop poles. The effect is merely to observe the noise using the original disturbance model Dd rather than filter the noise using Dd as a filter. The closed loop poles and zeros are shown in Figure 5.45. 120 Pole-zero map 6 0 U 40 U 20 U E -20 U " -40 - , --60 - ' --80 I 1 1 1 1 1 1 1 -70 -60 -50 -40 -30 -20 -10 0 10 Real Axis Figure 5.45: Poles and zeros of closed loop system augmented with disturbance observer. Again the poles and zeros of the closed loop system show a stable configuration. The disturbance observer acts as an integrator in the closed loop system, thus the poles at zero are exhibited. All the poles shown in Figures 5.39 and 5.45 are within a damping ratio range of 0.5 and 1. The range of stability of the two compensators can be proved through checking the poles of the closed loop system using the compensators designed here, and plant models linearized about various other expected operating conditions. Much like modal control, each of the modes of the MIMO system do not necessarily have similar pole locations. The complete range of expected conditions could also be tested using a non-linear model in simulations. Further analysis of the proposed controller performance will be shown in chapter 6. 5.3.9 Compensator Considerations Implementation of the compensators designed in the previous section involves the consideration of various elements. The most important factor in application of these designs is the existence of actuator bounds. As the first augmentation presented involves integrating the actuator signals, 121 integrator wind-up effects could occur bringing instability to the system. The following anti-wind-up scheme is used: Figure 5.46: Integrator Anti-wind-up scheme. This attempts to reduce the amount of integration past the saturation point. The gains K a w affect the amount of anti-wind-up that occurs and should be based on the expected rates of change of the commanded actuation signal. This method of control does not guarantee stability necessarily, but aids in eliminating the effects of wind up. For this thesis the following diagonal matrix of anti-wind-up gains is used: K a w = diag([1000 1000 1000 1000 1000 1000]) (5.91) It is important to use this method for both the states internal to the LQG compensator as well as the plant augmenting integrators. The resulting system is shown in Figure 5.47. n Figure 5.47: Block Diagram of Integrator Augmentation Compensator Overall Layout. 122 The LQG compensator is shown in the dashed box in the figure. The plant augmenting integrator is shown to the right of the dashed box, and the actual submarine dynamics are shown by the shad-owed block G(s). This controller is based on the augmented plant shown in eqs. 5.60 and 5.61. If the compensator is applied in this manner, it is also important that feedforward input values described at the beginning of this chapter are added to the input signal after integration. If the ini-tial values of the actuator states are set to zero, then the integrators will integrate with respect to zero. If the actual initial values of the actuators are non-zero, these actual values must be used to modify the actuator limits accordingly. Actuator saturation is also taken into account in the distur-bance observer augmentation. In this compensator, the saturated signals are used in the internal feedback structure of the compensator. This illustrated in Figure 5.48. Figure 5.48: Simulink Block Diagram of Disturbance Observer Augmentation Compensator Overall Layout. This approach to the non-linear actuator saturation feeds the saturated input signal back within the LQG structure such that the state estimation is more accurate. This does not take into account any disturbance state windup however. Not only is the issue of actuator saturation problematic, but the issue of actuator dynamics also has a large effect on the design of the control system. For the designs presented here, the compen-sator is designed having a bandwidth well below that of the actuators. For the non-linear model described in Chapter 4, the actuator dynamics are assumed as first order systems, with poles at -50 and -2.58 rad/sec. The slower response is that of the diesel engine. The actual actuators are imple-mented with lower level closed loop feedback. Two augmentations of the design plant have been shown here. For the sake of discussion the design plant can additionally be augmented by an actu-ator model such as: 123' y = [o c ] . (5-93) The actuator dynamics in this example are considered first order systems having time constants represented by the matrix T^/This approach is described in more detail by [9], but is not imple-mented here. If the bandwidth of the actuators are of the same order as that of the plant, actuator dynamics must be included in the design of the LQG compensator. Higher order noise models can also be used to represent the frequency range and amplitude of the expected plant disturbances. This is explained in more detail by [78]. Typically, when using this approach, wave forces are represented by second order filtering of white noise processes. It is also apparent that all the methods of augmentation described here can be implemented in combination. It should also be mentioned that the method of loop shaping presented by [26], describes the proc-ess of "grouping" the singular value curves at crossover using the inverse of the system, A matrix and the pseudo-inverse of the output matrix C. This process limits the designer to specific loop shapes. The method described here allows the "grouping" process to be applied at any given fre-quency and can be used in an iterative process to finely tune the desired Kalman filter loops. The LTR design method used here considers the design of the state estimator loop first, followed by the recovery using the control loop. Due to the duality of the optimization procedures, the design can be accomplished in the reverse order. Stability can be a problem in the use of the LQG approach, due to large non-linearities. A non-observer based controller can be implemented using similar LTR methods. The design process is identical except when the compensator is realized the control loop is not closed. Details of this approach are explained in [129]. Actual digital realization of the LQG controller introduces many additional factors. The first necessity is that the bandwidth of the computing device must be substantially greater than that of the desired compensator. Also, time lags are introduced due to data transfer, multiplexing, and analog to digital conversion. These time lags may be apparent at the input or output channels. Some time lag effects can be taken into account using a Smith predictor model within the design of the controller [104]. The sensor sampling rate must be able to capture the dynamics of the vehicle. In the simulations presented here, the analog sensor noise is pre-filtered at the Nyquist frequency using a second order Butterworth filter. The sampling then occurs at 0.02 seconds. If the anti-aliasing pre-filter bandwidth is of the same order as that of the plant, the filter must be included in the plant dynam-ics model, and the LQG controller must be designed accordingly. 124 5.4 Summary The proceeding design procedure is based on a linear model representation of the actual non-lin-ear plant. The model is linearized about the desired operating conditions. The complete multi-input multi-output state space equations are used in design, incorporating all aspects of coupling between longitudinal and lateral dynamics. First, the natural modes of the linear plant are exam-ined. Due to the dependence of some input-output relationships, a reduced output model is used for the design process. Using knowledge of the open loop dynamics, the input and output varia-bles are scaled according to their relative magnitudes and importance. The scaled plant is then augmented by actuator integrators, and, separately, by a disturbance observer. The LQG design framework is used in conjunction with loop shaping and recovery methods to ensure a robust, sta-ble system. Actuator dynamics are then included in the compensator. It is proposed that the com-pensator design process described here is then incorporated with a gain scheduling technique to ensure the control objectives are met for the complete range of operating speeds. The differences between the actual plant and the linear model consist of many components. In the design procedure these differences are considered disturbances. In addition to modelling uncer-tainty, there are random wave forces, low frequency tow forces, measurement errors, and actuator dynamics. The inherent goal of the LQG/LTR design process is to use the "known" dynamics of the plant to design a controller which inverts the known plant dynamics but is also robust to uncer-tainties and disturbances in the model. The LQG/LTR procedure is limited by the techniques used to shape the loop. Robustness guarantees are deteriorated with the addition of non-linearities to the system. A more progressive approach to robust control is the use of H infinity methods which prescribe the desired loop shapes using measures of the stability and robustness margins. The H infinity method can be incorporated with gain scheduling methods in a one step design process. An overview of robust control methods can be found in [130]. 125 6.0 Simulation Results and Discussion The preceding chapters have described the modelling of a submersible and the design of two suit-able control systems for the vehicle based on a set operating condition. In order to compare the proposed designs, it is necessary to perform an array of simulations using the two compensators. The simulated vehicle is exposed to realistic conditions in order to determine the stability, the reli-ability, and the performance boundaries of the designs. Although the LQG framework guarantees stability margins and robustness for a linear plant, certain conditions or disturbances may cause the designed control system to fail and become unstable. The final implemented control system must be suitable for the complete operating envelope from 0 to 18 knots, and for hull angles of attack from -10 to +10 degrees, the control systems presented in Chapter 5 are developed as a sub-component of a larger control scheme. Due to the localized condition of 6 m/s at zero trim angle, it is necessary to perform the test simulations near this point. The complete operating envelope can be spanned using a series of linearized operating points, typically at 1 m/s intervals. The localized controllers are then switched using gain scheduling methods [131, 132, 133]. Ideally, a type of bumpless transfer method should be used. In aerospace applications, the dynamic pressure is used as a scheduling variable as well as hull angle of attack. The submarine plant system is highly dependent on these variables as well, and similar methods can be used. In this chapter, a series of tests will be performed for both the integrator and disturbance observer augmented controllers. Three major types of disturbances are examined, while sensor noise and waypoint following disturbances are also included in some simulations. Initially, parameter uncer-tainty tests are performed to determine the robustness of the designs and the convergence of the state estimator. Ramp trajectory commands are then tested for changes in depth and heading. This is followed by simple a turning maneuver with and without towing forces. A comparison of both steady and dynamic tow models will be performed for a straight path. Finally wave forces are con-sidered under various sea states, and relative wave headings for a straight and level commanded path. All simulations will be performed with feedback of the states {u,x,y,z,(|),0,V|/} of which x and y are used to generate the heading reference command. In simulations including measurement noise, the sensor noise will be simulated as described in Chapter 4 and analog anti-aliasing filtering using a second order Butterworth filter will be included. The cut off frequency of the anti-aliasing filter is 25 rad/sec on all channels. These simulations will provide the information necessary to judge the relative merits and draw-backs associated with the two proposed designs. The simulations will also give an indication as to the properties of robust control designs in general. The performance of the compensators will be evaluated using the error in trajectory tracking, as well as the amount of actuator deflection. 6.1 Initial Disturbance and Ramp Input Tests The LQG compensator is based on an internal linear model of the plant. The measured outputs of the plant are used to update state variable estimates. In order to ensure that the compensator is 126 robust to differences between expected plant outputs and actual plant outputs the compensator is tested using different initial conditions. The state variables of the compensator are the state errors, Ax, shown in eq. 5.18. For stability, the compensator states must converge to the plant states. This is accomplished both by the control of the plant states, as well as the update of the estimator states. In traditional L Q G designs the estimator loop is designed approximately 5 to 10 times faster than the controller loop, whereas in the robust LTR design both sub-systems are designed in conjunction. The magnitude of the measured plant uncertainties are given in Section 4.4.4. These values are the result of both measurement error and disturbances on the plant. To test the convergence of the compensator to initial disturbances the following initial disturbance values are used: TABLE 6.1 Initial Plant Conditions State u (m/s) v (m/s) w (m/s) p (rad/s) q(rad/s) Value 0.5 0.2 0.2 0.45 0.2 State r (rad/s) z(m) (j) (rad) 6 (rad) t|/(rad) Value 0.35 0.15 0.175 0.035 0.175 The values shown in this table are greater than 150% of the measured disturbances described in Chapter 4. These initial disturbances were subtracted from the linear operating condition which was used as the initial condition for the compensator. The simulations were performed for a 10 second time frame. Measurement noise was not included in the simulations. The initial conditions were only altered for one state at a time, thus a total of 10 simulations were performed. The response to multiple parameter uncertainties was not investigated but coupling is evident in the results of these tests. The time domain compensator state information is shown in Figures 6.1 to 6.10, the results for both compensators are plotted in each figure. This type of evaluation is similar to imposing a step input reference command on the system. It should be noted that the compensator has been designed as a tracking system and thus excels in following a smoothed trajectory. Typically, a smooth reference command is generated for high fidelity controllers. Imposing large step input commands will lead to actuator saturation and poor performance of the system. In order to test the tracking capability of the proposed designs ramp inputs are used. The submarine will usually not be required to track a varying roll or pitch com-mand. The potential use of these outputs will be discussed later. 127 S o l i d - Integrator Augmentation Dashed - Disturbance Model Augmentation S o l i d - Integrator Augmentation Dashed - Disturbance Model Augmentation a) 0.5 "co 1 o > -0.5 b) 20 5 1 0 co 0 Q. -10 -20 c) 50 -50 d) 10 — 5 CD r 0 a -5 -10 e) 10 -10 0 1 2 3 Time (s) 4 5 • -• -0 1 2 3 Time (s) 4 5 0 1 2 3 Time (s) 4 5 • • 0 1 2 3 Time (s) 4 5 • -• -0 1 2 3 4 5 Time (s) Figure 6.1: Step Disturbance in 'u'. Figure 6.2: Step Disturbance in V . 128 S o l i d - Integrator Augmentation Dashed - Disturbance Model Augmentation S o l i d - Integrator Augmentation Dashed - Disturbance Model Augmentation 2 3 Time (s) a ) 0 . 5 r E 0 - 0 . 5 I Figure 63: Step Disturbance in 'w\ Figure 6.4: Step Disturbance in 'p\ 129 S o l i d - Integrator Augmentation Dashed - Disturbance Model Augmentation S o l i d - Integrator Augmentation Dashed - Disturbance Model Augmentation a) 0.5 Figure 6.5: Step Disturbance in 'q'. Figure 6.6: Step Disturbance in V . 130 S o l i d - Integrator Augmentation S o l i d - Integrator Augmentation Dashed - Disturbance Model Augmentation Dashed - Disturbance Model Augmentation 20 _ 10 w O ) <D 0 c r -10 -20 • • Figure 6.7: Step Disturbance in V . Figure 6.8: Step Disturbance in '(j)'. 131 S o l i d - Integrator Augmentation Dashed - Disturbance Model Augmentation S o l i d - Integrator Augmentation Dashed - Disturbance Model Augmentation 20 -<D 0 "D o- -10 • -20 • a) 0.5 E 0 -0.5 b) 50 o 0 c) 50 co 0 d) e) Figure 6.9: Step Disturbance in '8'. Figure 6.10: Step Disturbance in '\|/\ 132 The step reference command following characteristics can be described in terms of rise time, overshoot, settling time, and steady state error. These parameters are shown collectively in Table 6.2. TABLE 6.2 Initial Condition Tracking Characteristics u v w P q r Parameter In Ob In Ob In Ob In Ob In Ob In Ob Rise Time (sec) 0.82 0.22 2.18 2.04 0.44 0.32 0.24 0.18 0.58 0.20 0.32 0.24 Overshoot (%) 71.37 17.14 0.08 0.00 65.18 61.45 43.15 35.11 56.06 29.54 65.77 47.58 Settling Time (sec) 1.48 0.74 2.18 2.04 2.20 2.78 1.16 1.38 1.08 1.94 1.38 1.72 Steady State Error 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 z 6 V Parameter In Ob In Ob In Ob In Ob Rise Time (sec) 0.96 0.82 0.40 0.34 0.22 0.30 0.50 0.42 Overshoot (%) 27.12 58.42 33.45 31.13 43.59 25.74 20.52 37.54 Settling Time (sec) 1.90 4.78 1.98 1.18 1.26 2.24 1.54 1.68 Steady State Error 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 The integrator augmentation is denoted 'In.' while the observer based augmentation is denoted by 'Ob.'. From these results, one can see that there is no steady state error for either of the compensa-tors. There is, on the other hand, a large amount of overshoot in some cases. There is generally more overshoot in the states that are not measured directly but are estimated. This is due to the time lag inherent in the state estimator. Overshoot in these simulations is caused by actuator bounds and other non-linearities that exist. Simulations which were carried out on the linear model show significantly less overshoot. Any yaw, pitch or roll angle changes the relationship between the body and inertial frame. If an actuator is not able to attain its desired position, there is a delay in the response of the vehicle. From these results we also see that the settling times are between 1 and 3 seconds, with one exception. This compares well with the expected response time which was used in the design of the compensators in the frequency domain. The two compensators perform similarly, except in the case of the depth change for which the observer based compensator is affected by a prolonged period of saturated bow planes, and produces oscillations in the recovery of this condition. The actuator time domain data for these results can be found in Appendix E. This type of effect will be less prevalent with the use of a preconditioned reference trajectory. The limiting factor in trajectory-following for a submarine is the maximum rate that can be achieved for the given motion. In other words, a depth change is limited by the maximum steady state rate of depth change, whereas a heading change is limited by the maximum steady state tam-ing rate. The calculation of these bounds are given in eqs. 5.14 to 5.17. In order to maintain a cer-tain degree of control, ramp input commands using 50% of the maximum turning rate and 20% of the maximum depth change rate will be used in the simulations. Ramp input commands are applied 0.5 seconds into the simulation. Measurement and stochastic noise is not included in the simulations. The results are shown in Figures 6.11 to 6.14. 133 S o l i d -Dashed -Integrator Augmentation Disturbance Model Augmentation S o l i d - Integrator Augmentation Dashed - Disturbance Model Augmenta-a) 7 6.5 X  6 5.5 5 5 10 Time (s) 15 d) a> S 0 5 10 Time (s) -20 • e) 20 a> ca 5. 0 T3 -20 f)1000 5 10 Time (s) 15 Figure 6.11: States During Ramp Depth Figure 6.12: Actuators During Ramp Depth Command. Command. 134 S o l i d - Integrator Augmentation Dashed - Disturbance Model Augmentation S o l i d - Integrator Augmentation Dashed - Disturbance Model Augmenta-a) 0.5 E 0 -0.5 a) b) 20 -co 0 "D a- -10 -20 I-c) 50 5 10 Time (s) 5 10 Time (s) 15 20 CD S 0 a. -20 f-b) 20 h CO 2. 0 w -20 c) _ 20 oS CD S- 0 Q. CO TJ -20 d) _ 20 " 3 co s 0 CO CO "D -20 i e) 20 V CD -20 (-f) 1000 5 10 Time (s) 5 10 Time (s) 5 10 Time (s) 5 10 Time (s) 5 10 Time (s) 15 15 15 15 Figure 6.13: States During Ramp Heading Command. Figure 6.14: Actuators During Ramp Heading Command. 135 It should be noted that there is also a non-linear effect present due to the magnitude and location of the buoyancy. As the vehicle pitches nose up, the buoyancy tends to speed up the vehicle while the actuators are set to maintain a given trim. In other words, the steady state force now acts in a differ-ent direction relative to the body. This is seen as a small scale in the linear model as seen by the eigenvectors shown in Chapter 5. Trim angle is seen from these results to have the most drastic effect on the performance of the sub-marine. Not only do non-linear effects become apparent for the hull, but also the planes. Crossflow over the hull also produces an apparent angle of attack on the planes. The worst case scenario in this case is for the vehicle to be pitched nose up, while shallow of the desired depth. The planes are in a trailing edge up position, the aftplanes producing a greater moment than the foreplanes. This causes a greater hull pitch, and due to the forward speed a greater depth error. It is also possible to achieve an equilibrium state having steady state pitch error unless feedforward or error integration is active. For this reason, it is necessary to exaggerate the pitch control. From the figures, it is seen that the depth control using the observer augmentation produces oscilla-tions in the pitch response. This is due to the saturation of the actuators. If the pitch control was not exaggerated, this would lead to an unstable condition. It is possible to remedy this scenario by using the required depth reference command to generate a reference pitch command. For a linear forward speed of: In this way, the reference depth rate and the speed setpoints can be used to generate a pitch refer-ence angle. It is also possible to filter the commanded trajectory using a first or second order filter having a bandwidth slightly below that of the vehicle open loop dynamics. More information on generation of reference trajectories using maximum rates can be found in [134]. The heading change command is well behaved and is achieved using a small rudder deflection. This is due to the large moment produced by the rudder and the damping in the lateral motion. Both augmentations show good command following for this maneuver. Also, the roll angle is small due partially to the fact that all planes independently control roll and also because the heading change is constant over time, as opposed to a step heading input. It is important to note that the observer augmentation acts as a low pass filter, and integrates the observed noise. Thus, by moving the poles of the disturbance model, performance characteristics may change. Also, during actuator saturation the noise is integrated while no integrator wind-up facility has been implemented, only saturation modelling. This is likely the cause of the oscillations seen in Figures 6.11 and 6.12. (6.1) Then, (6.2) 136 6.2 Turn Maneuver Simulations The DOLPHIN vehicle towing a sonar fish is used to survey a given area within the littoral zone. Multiple swaths have to be executed to cover the total survey area. This requires a 180 degree turn at the end of each swath [135]. The controller must be able to follow the series of waypoints defin-ing the turn while towing the towfish. The performance of the controllers for this particular maneuver is examined here. First the waypoints are followed without the addition of the tow force. The maneuver consists of an 80 meter straight section, followed by a 300m diameter semi-circle, followed by a 120 meter straight section. The maneuver is simulated for 110 seconds, for a forward speed of 6 m/s and a depth of 3 m. There is no trajectory smoothing applied for these simulations, the navigation sys-tem requires the heading toward the upcoming way point and does not use line following. This test is performed as a comparison of performance of the controllers under tow. The time domain results are shown in Figures 6.16 to 6.23. Sensor and random process noise is included in these simulations. The vehicle also has feedforward steady state values on all the actuator chan-nels to provide the necessary trim and speed based on operation without the tow. An example of the maneuver path is shown in Figure 6.15, the result of the integrator augmenta-tion with dynamic tow model is also shown. -501 1 1 1 1 1 1 I I -100 -50 0 50 100 150 200 250 300 x Position (m) Figure 6.15: Example of Turn Maneuver Path Results. 137 10k -10 k -1 • - -0 20 40 60 Time (s) 80 100 I 1 I 1 1 • 0 20 40 60 Time (s) 1 80 100 40 60 Time (s) 20 40 60 80 Time (s) 100 20 40 60 80 Time (s) 100 a) 7 6.5 1 6 5.5 5 —i r— r » » » >—v—it—V—VHr-^V—V 1- * > b) 20 40 60 80 100 Time (s) -2.8 -2.9 I-£ -3 -3.1 -3.2 —A.—*—I—K—K.— 10k r 0 °- -5 -10 d) 5 -5 e) 200 150 a 50 0 0 20 40 60 Time (s) 80 100 L i . 1. L . JL. L _ s • • . . . 0 20 40 60 Time (s) 80 100 0 20 40 60 80 100 Time (s) Figure 6.16: Controlled State Data for Turn With Figure 6.17: Controlled State Data for Turn With Integrator Augmentation and No Tow. Integrator Augmentation and Dynamic Tow. 138 a) 20 cn CD ;o 0 Q . T5 -20 0 b) 20 -oS CD 0 -CO -20 -0 c) 20 OS <D "O 0 Q . m -20 0 d) 20 -"3 CD ^ 0 • CO CO "D -20 • 0 e) 20 -at CD 3, 0 -T3 -20 -0 20 40 60 80 100 Time (s) -*—V-——V—1(—\—If—t—T—If——\l—*»-20 40 60 80 100 _ , T i m e (,s) 20 40 60 80 100 - • , T i m e ( S ' . r -V ! f V l f V V I f ' l f V l f l f , ' » 20 40 60 80 100 Time (s) rr 1500 J 0 0 0 E § 5 0 0 20 40 60 80 100 Time (s) I V V V I I V l V. I y ». 0 20 40 60 80 100 Time (s) a) 20 CD S 0 Q . •5 -20 l 0 1 — ^ n H - f - H n h H H ^ H H ^ -co -20 • 0 20 40 60 80 100 d ) , Time (s) 20 I * 0 CO cd "O -20 0 20 40 60 80 100 e) Time (s) 20 I s 0 -20 + t t t f t f f f f i 20 40 60 80 100 Time (s) , i i i e ( S ) U r 1 1 1 1 1 1500 t ^ J 0 0 0 S 5 0 0 c •D „ 20 40 60 80 100 Time (s) Figure 6.18: Actuator Data for Turn With Figure 6.19: Actuator Data for Turn With Integrator Augmentation and No Tow. Integrator Augmentation and Dynamic Tow. 139 a) 7 6.5 Y i 6 5.5 I-5 b) 0 20 40 60 80 100 Time (s) -2.8 • -2.9 -£ -3 -3.1 I--3.2 20 40 60 80 100 Time (s) d) 5 20 40 60 80 100 Time (s) 20 40 60 80 100 Time (s) 101 Oi CD S 0 .E 1 1 -5 -10 •V if ^< ^ ^ K fcn a _TWlTf|f| Figure 6.20: Controlled State Data for Turn With Figure 6.21: Time Domain Data for Turn With Disturbance Model Augmentation and No Tow. Disturbance Model Augmentation and Dynamic Tow. 140 e) 20 °> 2. 0 -20 20 40 60 80 100 _^ Time (s) Figure 6.22: Actuator Data for Turn With Disturbance Model Augmentation and No Tow. Figure 6.23: Actuator Data for Turn With Disturbance Model Augmentation and Dynamic Tow. 141 The dynamic tow model includes the modelling of lateral cable angle while the static tow model does not. Both tow models follow the mean vehicle heading in time, and include the effects of pitch and roll. An example of the tow off angle during the turn is shown in Figure 6.24 for the integrator augmented compensator. These results are specific for a towfish depth of 120m and a cable scope of 300 meters. -101 I I I I I I 0 20 40 60 80 100 120 Time (sec) Figure 6.24: Lateral Tow Cable Angle on DOLPHIN During Turn Maneuver. The angle with which the cable tension acts on the DOLPHIN causes a roll moment as well as a yaw moment. The location of the tow point relative to the center of gravity of the DOLPHIN is assumed to be {-1.263, 0, 1.265} in meters. The results of the dynamic tow model are dependent on the initial conditions used for the tow model, however the tension and angle models converge within 5 seconds. The tension is only a function of speed over ground squared while the lateral angle is a function of turning rate in the inertial frame. This combination does not take into account the change in cable tension for turning rate. It is possible to create a two input, one output model using the same system identification method to obtain more realistic results. Both tow models were assumed to have the same depressive cable tension angle of 6 degrees rela-tive to horizontal. The static cable tension model was assumed to have a constant tension of 14,905 newtons. The turning maneuver simulations result in different responses depending on the type of tow model that is used. A comparison of the compensators and the tow model results is shown in the following 142 tables. The root mean square (RMS) values for the submarine states and actuator levels are shown in Figure 6.3. TABLE 6.3 Summary of Turn Maneuver RMS Values. u (rr/s) V (rr/s) w(mfe) p(dec/s) q (deer's) r (elect's) Tew Type In a In Cb In a In Cb In Cb In a No Tow 6.000 5.999 0.042 0.058 0.003 0.016 2891 4.914 0.331 0.458 7.651 9.804 Static Tow 6.000 5.933 0.041 0.057 0.004 0.014 2721 4.866 0.321 0.463 7.670 9.820 D/narricTow 6.000 5.945 0.375 0.354 0.017 0.030 5398 7.431 0.541 0.823 6.859 9.682 V(rri) z(m) <D(deg) e(deg) V(clsg) Tew Type In Cb In Cb In Cb In Cb In Cb iNbTcw 2845 a070 aooo aooo 0.747 0.895 0.034 0.065 2745 2856 Static Tow 2844 a072 aooo aoo2 0.719 0.870 0.035 0.079 2739 2762 D/narricTow a568 3.868 aooo aoo2 1.593 1.972 0.060 0.131 a803 a519 dfp(deq) dfs(deq) dap(deq) das(deq) dr(deg) ctn(rprn) Tew Type In a In Cb In a In Cb In Cb In a NbTow a417 a6io a234 agos 1.747 1.807 1.706 2143 5.617 7.613 653379 65a379 Static Tow 1.178 1.270 1.147 2179 1.450 1.473 1.398 1.858 5626 7.626 1374.961 1374.981 rename Tow 1.528 2319 2104 asis 1.362 1.487 2093 2640 9.036 10.313 1239.879 1239.879 This table is constructed using the actual plant states which are available. The first 10 seconds of data are ignored in calculating the RMS due to the settling of the dynamic tow model. The maxi-mum measured values are shown in Table 6.4. TABLE 6.4 Summary of Turn Maneuver Maximum Values. u (nVs) V (rrVs) w (nVs) p (deg/s) q (deg/s) r(dep/s) Tow Type In Ob In Ob In Cb In Ob In Ob In Cb No Tow 6.041 6.021 0.175 0.315 0.013 0.069 8.730 27.763 0.930 0.255 39.493 49.613 Static Tow 6.050 5.958 0.175 0.313 0.015 0.056 7.591 27.808 0.849 0.201 39.708 49.988 Dynamic Tow 6.078 6.051 0.640 0.743 0.090 0.141 24.013 30.808 1.039 0.461 35.321 46.073 V(m) z(m) <D(deg) e(deq) V(deg) Tow Type In Ob In Cb In Cb In Ob In Cb No Tow 5.479 5.617 3.004 3.014 1.080 0.734 0.042 0.346 14.591 14.910 Static Tow 5.509 5.607 3.004 3.015 0.941 0.741 0.037 0.194 14.569 14.894 Dynamic Tow 6.558 6.593 3.009 3.033 0.964 0.103 0.453 0.676 18.103 18.494 The 'y' state is the perpendicular distance from the reference trajectory, while the '\|/' state is the error in heading to the next waypoint. The maximum value for 'y' is due to the radius of the target waypoint. The vehicle heads toward the next waypoint as soon as the target radius has been reached. The minimum values in this case are also important and are shown in Table 6.5. TABLE 6.5 Summary of Turn Maneuver Minimum Values. u(nVs) v(nYs) w (rrVs) p(dep/s) q (cleg's) r(deo/s) Tow Type In Cb In Cb In Cb In Cb In Cb In Cb No Tow 5.957 5.959 -0.001 -0.005 -0.016 -0.043 -0.891 -2489 -17.144 -30.855 -1.507 -2461 Static Tow 5.959 5.908 -0.001 -0.009 -0.018 -0.043 -1.051 -2462 -16.902 -30.032 -1.423 -2489 D/narricTow 5.922 5.856 0.000 0.000 -0.010 -0.091 -0.601 -5.221 -37.180 -48.891 -a210 -4.127 V (m) z(m) <t>(dea) e(deg) V(deg) Tow Type In a In Cb In Cb In Cb In Cb No Tow 0.000 0.000 2996 2984 -a386 -5.386 -0.179 -0.140 -2451 -6.434 Static Tow 0.000 0.000 2.995 2987 -3288 -5.366 -0.169 -0.179 -2508 -6404 Dynamic Taw 0.000 0.000 2979 2.975 -8.895 -11.169 -0.163 -0.333 -2481 -5.562 143 One can see that the tow cable tension component in the vehicle's axial direction is reduced dur-ing the turn due to the lateral tow off angle. Thus, the engine speed for the dynamic tow model is significantly lower for the turn, than the static model, which does not take into account a lateral tow cable angle. The increased drag due to the towed fish requires almost double the engine rpm than operating without the tow for the given speed. Several observations can be made regarding the direction of the turn. The maneuver in these sim-ulations is a turn to starboard (positive \|/). This resulted in large positive rather than negative V values, thus much of the turn was made with the nose following the trajectory rather than the nose leading the turn. This is increased in the results of the dynamic tow, likely due to the turning moment created by the tow cable and rudder. The reason that the RMS 'y' values are high is due to the following of waypoints rather than the line connecting way points. There is an increase in the RMS error in 'y' of only 1 meter for the dynamic tow. With the dynamic tow model the RMS sway crossflow is increased to between .35 and .375 m/s. This corresponds to a sideslip angle of attack of 3.5 degrees, which could induce vortex shedding over the hull causing degradation of the leeward hydroplanes. The maximum crossflow corresponds to an angle of 7 degrees. The roll angle during the turn is gready increased during the turn. Without the tow there is a greater negative roll than positive, this indicates that the vehicle rolls with the mast out of the turn. This effect is magnified with the addition of the dynamic tow model from about 3 and 5 degrees to 9 and 11 degrees for the two compensators. This is also seen by the extremely large negative val-ues of pitch rate 'q'. These results are consistent with the results found by [135]. The large values are due to the attempt to turn the vehicle using the fore and aft planes while rolling out of the turn. The static tow model does not have a large effect on the lateral motions during the turn as there is no lateral moment induced. Increased rudder RMS values are apparent as the rudder must oppose the tow force. The turn also affects the depth and pitch of the vehicle due to coupling and non-linear effects. The RMS depth values are held extremely close to the desired setpoint. It appears that the vehicle tends to pitch nose down during the torn from the minima values. The minimum pitch" angle is increased due to the nose down moment applied by the cable. On the other hand, there is a larger excursion to rather than away from the surface. The depth keeping errors are increased with the addition of the tow. It is interesting to note that the tow forces lessen the RMS deflections of all the fore and aft planes except the starboard aftplane. The decrease in the deflection of the foreplanes is due to the steady state buoyancy being opposed by the downward force of the tow cable. The difference between port and starboard plane deflections is larger during the turn to account for the increased roll caused by the tow and the rudder deflections. From Figures 6.18, 6.19, 6.22 and 6.23 one sees that the maximum deflection of the planes is large however the RMS values are well within the actua-tor boundaries. This shows that the actuators are capable of keeping the desired reference states, it would be beneficial, however, to smooth the reference such that the peak actuations requirements are lessened. 144 6.3 Simulations with Waves The DOLPHIN vehicle is able to operate in severe sea conditions. Being unmanned, this is a great advantage over conventional crafts. The desired controller must be able to perform with the pres-ence of large wave disturbances. To determine the properties of the LQG controllers in these con-ditions simulations are performed using the wave model described in Chapter 4. The presence of a local wave disturbance velocity affects not only the forces on the hull of the vehicle but also the apparent angle of attack of the hydroplanes. The frequency of the incoming waves is a function of the relative wave direction and the speed of the vehicle. The phase differ-ence between the wave forces and moments on the vehicle is also a function of the relative wave angle of attack. This investigation assumes long crested sea conditions. Simulations will be performed for sea conditions pertaining to sea state 3 and sea state 4 as detailed in Chapter 4. Three relative wave headings will be used: 90, 135 and 180 degrees. Head or oncoming seas have a relative wave heading of 180 degrees. The spectral wave information for these conditions for a speed of 6 m/s is shown in Figures 6.25 and 6.26. Wave Spectrum for Significant Wave Ht.0.88 m 0.06 r Encounter Frequency (rad/s) Figure 6.25: Sea State 3 Wave Encounter Spectra at 6 m/s. 145 0.4 Wave Spectrum for Significant Wave Ht.1.88 m 0.35 l Encounter Frequency (rad/s) Figure 6.26: Sea State 4 Wave Encounter Spectra at 6 m/s. One can compare Figures 6.25 and 6.26, to the singular value plots of the sensitivity of the com-pensator and submarine system shown in Chapter 5. A higher sea state has a peak wave amplitude at a lower frequency, in other words it exhibits longer wavelengths but higher waves. Larger rela-tive wave direction causes the peak wave amplitude to shift to a higher frequency. Note that the peak wave frequency in all cases exists within approximately 1 and 3 rad/sec. This is within the bandwidth of the submarine dynamics. For the compensator to converge fast enough to counter the disturbances due to waves, the complementary sensitivity bandwidth of the Kalman filter must be greater than the disturbance dynamics. For both the compensators designed in Chapter 5 this is the case. If the wave frequency was above the plant bandwidth, it would be possible to design the compensator bandwidth between the plant and the wave frequencies, thus filtering the wave fre-quency. The trade-off is that the controller is more sensitive to high frequency disturbances. For the wave simulations performed in this work, sensor and plant uncertainty stochastic noise was included. The reference depth command was 3.5 m. It is beneficial to travel at a greater depth in waves to reduce the forces developed. However, large wave amplitudes could cause the mast of the vehicle to become submerged. It is necessary to optimize the reference depth based on these criteria. The time domain results of the simulations are summarized in Tables 6.6 to 6.11. 146 S o l i d - Integrator Augmentation Dashed - Disturbance Model Augmentation Figure 6.27: Summary of State RMS for Wave Simulations for Sea State 3. 147 a ) ~ 4 0 CD CO T J — 2 0 b) . 0 4 0 — 2 0 <A 0 c ) — 2 0 2 0 r 1 0 0 1 0 0 1 0 0 0 5 0 0 0 S o l i d - Integrator Augmentation Dashed - Disturbance Model Augmentation 9 0 9 0 9 0 9 0 9 0 -o--e-. - -o 1 3 5 -O--B--©-1 3 5 • O -1 3 5 1 3 5 Rel. W ave Dir. (deg) O -a o -a 1 8 0 O - H o 1 8 0 -O 1 8 0 - 6 1 8 0 Figure 6.28: Summary of Actuator RMS for Wave Simulations. The figures show that the integrator augmentation produced less RMS error in for the longitudinal motions while the disturbance observer augmentation showed less RMS error for the lateral motions. The disturbance observer produced more hydroplane action shown by RMS values, almost double those of the integrator. 148 x Figure 6.29: Example of Wave Forces and Moments in Sea State 3. Figure 6.30: Example of Actuator Response in Sea State 3. 149 Figures 6.29 and 6.30 show an example of the response of the actuators for the submarine in head seas with the integrator augmented compensator. It is evident that as large waves impinge upon the vehicle, the actuators saturate in an attempt to maintain the desired depth. It can also be seen that the actuators respond to the sensor noise, causing excessive hydroplane motion. As the wave disturbance increases for a large wave the bow planes saturate initially and the stern planes attempt to both maintain pitch and also depth setting. If the depth error becomes large, both fore and aft planes will saturate and the stability of the vehicle will be compromised. The changes in the engine rotation are unrealistic compared to the actual response of large diesel engines. How-ever, these changes are due to sensor noise as the effect of waves on forward speed is not included in these simulations. Both compensators show good results as the RMS depth error is within 0.15 m of the reference while the pitch and heading RMS errors are within 3 degrees of the reference. It is interesting to note that the foreplanes show almost double the deflection RMS compared to the aftplanes for both augmentations. This is likely due to the stringent control of pitch which is largely controlled by the aftplanes in order to maintain a given depth. In other words, the aftplanes are not able to deflect to their maximum to aid in depth keeping due to the large pitching moment that would be caused. This could be remedied by enlargement of the foreplanes, or positioning of the foreplanes further forward of the center of gravity. However, placement of the foreplanes further forward would also lead to less dynamic stability. From linear wave theory, a wave of height 1.88m and wavelength 120m in deep water induces a maximum crossflow on the planes at a depth of 3 meters of 0.57 m/s. This is equivalent to a plane deflection of 5.5 degrees, at 6 m/s forward speed, which is approximately 11% of the total range of motion. When considering that the hull may also be at a trim angle of up to 10 degrees, which also induces an apparent crossflow on the planes, there may exist up to a 15 degree angle of dis-turbing on a hydroplane. This is excessive and may lead to controller instability. Also, the satura-tion angle of a hydroplane of 25 degrees may correspond to an actual angle of attack of only 10 degrees. Thus it is possible to increase the saturation bounds of the actuators, although it is sug-gested that the bounds are dynamically set based on hull angle of attack. One can see from the previous two paragraphs that there is a concession to be made as large plane sizes produce greater controlling forces but also lead to larger disturbance forces. The motion experienced by the submarine during the wave simulations is shown in the following tables. 150 TABLE 6.6 Summary of Data RMS for Simulations under Sea State 3. u(m) v(m) w (m) P (deg/s) q (deg/s) r (deg/s) Wave Dir. In Ob In Ob In Ob In Ob In Ob In Ob 90.000 135.000 180.000 6.001 5.996 6.000 5.991 6.000 5.990 0.038 0.030 0.030 0.027 0.021 0.017 0.079 0.093 0.082 0.300 0.111 0.319 4.051 2.300 4.167 3.580 3.810 3.279 1.210 1.374 2.114 4.413 2.730 4.793 5.260 5.095 5.344 5.124 5.210 5.001 V (m) z(m) <l> (deg) 9 (deg) V (deg) Wave Dir. In Ob In Ob In Ob In Ob In Ob 90.000 135.000 180.000 0.087 0.076 0.073 0.070 0.068 0.065 3.500 3.500 3.500 3.487 3.499 3.479 1.121 0.702 0.857 0.673 0.599 0.316 0.412 0.469 0.655 1.612 0.790 1.702 1.068 0.885 1.052 0.910 0.956 0.832 dtp (deg) dfs (deg) dap (deg) das (deg) dr (deg) dn (deg) Wave Dir. In Ob In Ob In Ob In Ob In Ob In Ob 90.000 135.000 180.000 5.716 10.050 7.466 20.157 9.838 21.406 5.646 9.864 7.458 20.113 9.796 21.381 3.239 5.722 3.795 9.672 4.710 10.284 3.191 5.520 3.437 9.425 4.610 10.188 5.641 9.562 5.654 9.494 5.548 9.392 659.250 659.250 663.142 663.142 671.425 671.425 From Table 6.6, we see that the roll RMS of the vehicle is double for beam seas than that for head seas, while the pitch RMS for head seas is almost double that of beam seas. These results are expected. Hydroplane activity is much larger during head seas. This is most likely due to the pitching moment that is induced by the waves in head seas, that is less apparent in beam seas. The method used to simulate wave forces does not include surge, or roll forces directly, so these effects must be considered in the analysis of the data. The heading and line following error are both good in all conditions, as the heading RMS is within about 1 degree from the desired heading in all cases. It should be noted that the mean val-ues of the vehicle states are equal to the linearized condition. This shows that both the compensa-tors achieve zero steady state error. The maxima and minima values are shown in the following Tables. TABLE 6.7 Summary of Data Maxima for Simulations under Sea State 3. u(m) v(m) w (m) P (deg/s) q (deg/s) r (deg/s) Wave Dir. In Ob In Ob In Ob In Ob In Ob In Ob 90.000 135.000 180.000 6.120 6.091 6.122 6.108 6.158 6.124 0.134 0.113 0.110 0.102 0.079 0.075 0.311 0.376 0.358 0.665 0.387 0.671 19.108 10.710 19.119 18.539 17.764 15.103 6.447 7.870 7.046 12.761 9.497 13.970 22.346 23.694 21.408 23.132 22.021 23.451 V(m) z(m) <t> (deg) 6 (deg) V (deg) Wave Dir. In Ob In Ob In Ob In Ob In Ob 90.000 135.000 180.000 0.317 0.263 0.240 0.204 0.226 0.215 3.580 3.578 3.582 3.751 3.618 3.748 3.976 2.399 3.172 2.392 2.920 1.517 1.589 2.016 2.226 4.968 2.969 5.474 5.443 5.306 4.691 4.739 4.668 4.528 151 TABLE 6.8 Summary of Data Minima for Simulations under Sea State 3. u (m) v(m) w (m) P (deg/s) q (deg/s) r (deg/s) Wave Dir. In Ob In Ob In Ob In Ob In Ob In Ob 90.000 135.000 180.000 5.888 5.810 5.883 5.832 5.870 5.834 -0.141 -0.118 -0.105 -0.120 -0.101 -0.086 -0.270 -0.349 -0.361 -0.726 -0.392 -0.725 -14.319 -13.643 -15.158 -16.850 -13.828 -17.038 -4.779 -6.161 -8.745 -14.082 -11.098 -14.700 -27.257 -28.273 -25.808 -25.508 -27.826 -26.457 V(m) z(m) * (deg) 6 (deg) V (deg) Wave Dir. In Ob In Ob In Ob In Ob In Ob 90.000 135.000 180.000 -0.240 -0.256 -0.206 -0.249 -0.185 -0.254 3.427 3.431 3.410 3.129 3.371 3.109 -4.237 -2.206 -3.050 -2.544 -2.822 -1.263 -1.343 -1.502 -1.996 -4.225 -2.438 -4.307 -4.255 -4.438 -4.535 -3.930 -4.229 -3.932 The tables above show that the range of motion above and below the condition is symmetric as expected. There is a slight tendency for more pitch in the nose down state than the nose up state. The slight asymmetry in the results is could be caused by the stochastic nature of the sensor dis-turbances, and the length of the simulation. These simulations were performed for 600 seconds each in simulated time. To obtain more realis-tic information, simulations must be performed for a period greater than 1 hour. Due to the com-plexity of the simulated system, including a non-linear plant, discrete measured variables having stochastic noise, wave effects on the planes and hull, a 16 state compensator, and a constantly updating navigation system, the simulations require up to 12 hours each for 600 seconds. In the interest of expediency, the results were obtained as described. TABLE 6.9 Summary of Data RMS for Simulations under Sea State 4. u(m) v(m) w (m) p (deg/s) q (deg/s) r (deg/s) Wave Dir. In Ob In Ob In Ob In Ob In Ob In Ob 90.000 135.000 180.000 6.000 5.994 6.000 5.990 6.000 5.989 0.065 0.050 0.049 0.042 0.022 0.018 0.125 0.130 0.145 0.348 0.178 0.374 4.365 2.941 0.081 0.091 0.071 0.061 1.525 1.901 0.050 0.089 0.064 0.108 5.381 5.237 0.096 0.092 0.091 0.087 V(m) z(m) * (deg) 6 (deg) V (deg) Wave Dir. In Ob In Ob In Ob In Ob In Ob 90.000 135.000 180.000 0.177 0.114 0.112 0.087 0.068 0.065 3.500 3.499 3.499 3.473 3.496 3.463 1.706 0.983 0.022 0.018 0.012 0.006 0.850 0.851 0.021 0.037 0.025 0.045 1.501 1.016 0.024 0.018 0.017 0.015 dtp (deg) dfs (deg) dap (deg) das (deg) dr (deg) dn (deg) Wave Dir. In Ob In Ob In Ob In Ob In Ob In Ob 90.000 135.000 180.000 6.612 14.079 9.507 21.953 12.518 22.516 6.925 13.819 9.442 21.891 12.487 22.511 4.622 7.400 4.848 10.383 5.673 10.572 4.238 7.210 4.451 10.227 5.567 10.545 6.184 10.036 5.962 9.847 5.546 9.392 661.750 661.750 671.073 671.073 683.614 683.614 The significant wave height used in simulation of sea state 4 conditions as shown by Table 4.4 is indicative of much more severe wave forces than sea state 3. The pitch, roll, depth and heading RMS values appear less for sea state 4 than sea state 3. While the depth and sway rates are much higher. This is likely due to the lower average wave encounter frequency in sea state 4. The com-pensator is able to respond fast enough to maintain the desired reference states. The encountered wavelength is higher in relation to the vehicle length and thus less pitching or yawing moment is encountered. However, as the overall magnitude of the encountered waves is greater, it is apparent that the maximum forces are greater producing larger maximum state errors. Evidence of this is seen in Tables 6.10 and 6.11. 152 TABLE 6.10 Summary of Data Maxima for Simulations under Sea State 4. u (m) v(m) w (m) P (deg/s) q (deg/s) r (deg/s) Wave Dir. In Ob In Ob In Ob In Ob In Ob In Ob 90.000 135.000 180.000 6.132 6.108 6.162 6.139 6.159 6.128 0.210 0.164 0.176 0.164 0.091 0.102 0.428 0.505 0.536 0.808 0.651 0.867 18.993 19.495 0.371 0.469 0.304 0.272 6.245 8.827 0.199 0.282 0.228 0.308 21.541 24.787 0.373 0.417 0.384 0.399 V(m) z(m) <t> (deg) 9 (deg) V (deg) Wave Dir. In Ob In Ob In Ob In Ob In Ob 90.000 135.000 180.000 0.496 0.354 0.329 0.276 0.224 0.217 3.603 3.611 3.605 3.769 3.637 3.868 5.712 3.659 3.895 4.074 2.937 1.601 2.613 3.041 4.566 6.826 5.101 7.411 5.588 5.584 5.138 4.954 4.688 4.673 TABLE 6.11 Summary of Data Minima for Simulations under Sea State 4. u (m) v(m) w (m) P (deg/s) q (deg/s) r (deg/s) Wave Dir. In Ob In Ob In Ob In Ob In Ob In Ob 90.000 135.000 180.000 5.889 5.822 5.849 5.853 5.824 5.838 -0.217 -0.171 -0.144 -0.153 -0.108 -0.090 -0.463 -0.487 -0.533 -0.885 -0.543 -0.859 -16.433 -14.758 -15.879 -24.379 -16.682 -14.694 -5.775 -9.577 -11.427 -15.750 -14.644 -18.121 -27.294 -27.799 -25.929 -26.435 -28.261 -26.210 V (m) z(m) <l> (deg) 0 (deg) V (deg) Wave Dir. In Ob In Ob In Ob In Ob In Ob 90.000 135.000 180.000 -0.492 -0.336 -0.343 -0.267 -0.182 -0.254 3.382 3.411 3.382 3.015 3.293 2.996 -5.804 -3.870 -5.643 -4.262 -3.389 -1.493 -2.445 -2.654 -3.617 -5.755 -3.904 -6.618 -5.466 -4.959 -4.703 -4.685 -4.247 -3.893 The data shows that the vehicle experienced a larger excursion toward the surface than away from the surface. This is due to the wave forces becoming larger as the vehicle approaches the surface, increasing the excursion. There also appears to be a larger pitch angle nose up than nose down. The magnitude of the maximum pitch angles overall are greatly increased for sea state 4 as are the heel angles. In comparing the results of the two compensators, we see that the integrator augmentation is able to track the depth and pitch references better in waves than the disturbance observer. The distur-bance observer is able to reduce the roll angle error. The speed and heading tracking performance is similar for the two compensators. As mentioned previously, the poor performance of the distur-bance observer in depth and pitch control is likely due to the wind-up of the disturbance observer during actuator saturation. Although saturation can cause reduction in the stability margins of the compensator, if the states are weighted in a judicious manner as to their effect on the non-linear system, it is possible to maintain good tracking in extreme conditions. It is also important to ensure an anti-wind-up system is present in the application of the compensators. 6.4 Summary This chapter described the results obtained using the compensators designed in Chapter 5 under various conditions. The effects of step and ramp inputs were initially simulated. These simula-tions showed that reference command tracking is greatly improved by smooth, ramped trajecto-ries. The maximum rate of change for a given motion should be used to determine the reference trajectory for the related states. It is also possible to use the vehicle speed and ramped depth change rate to establish a corresponding pitch trajectory. Both compensators showed good conver-gence for the initial disturbances and good tracking for the ramp inputs. The effect of actuator sat-153 uration and possibly wind-up of the observed disturbance model states caused oscillations in the response of the disturbance observer augmented controller. Simulations were also performed for the vehicle following a 300m diameter turn while towing a sonar fish. Two tow models were used, a static force in the inertial frame and a dynamic tow model. These results were also compared with the turn followed without towing. Both simulations with a tow caused an increase in engine rpm, as expected. The dynamic tow model had large effects on the sway, roll and yaw modes of the towing vehicle. There is a larger error in these states while towing. The final simulations were for a straight path in waves. Sensor noise was included for these simu-lations. Analysis shows that for large waves the actuators tend to saturate to maintain the desired depth. There is also a large amount of actuator movement due to sensor noise. Depth and pitch errors are worse in head seas for both compensators. In larger waves, the RMS errors are smaller but the maximum and minimum state errors are increased. Lateral control is unaffected by the rel-ative wave direction. There were no simulations performed to test the robustness of the compensator using different plant hydrodynamic parameters than those used to design the control system. Plant uncertainties and high order hydrodynamic effects are considered by the large stochastic, white noise process added to the measured states incorporated in the simulations in waves. Both compensators were able to control the response of the submarine in under all types of distur-bances. It is necessary, however, to tune the bandwidth of the compensators well above the wave bandwidth to ensure correct state estimation. This is due to the location of the peak sensitivity of the vehicle. Further digital pre-filtering of the sensor measurements would improve the amount of actuator flutter. In the case where only one hull angle of attack is used for compensator design, the pitch effects must be highly weighted such that the non-linear effects do not compound the effects of actuator saturation. Saturation must be dealt with in the compensator design. Both compensator designs appear robust within the designed operating framework. The effects of engine rpm response should be included in the design of the compensators for the full scale system, if the LQG design is to incorporate forward speed. 1 5 4 7.0 Conclusions and Recommendations 7.1 Conclusions Effort was made in this thesis to accurately model an underwater vehicle operating near the sur-face. Modelling was achieved using the rigid body equations of motion. Hydrodynamic forces were included to produce a non-linear set of equations. Disturbances due to waves, sensor noise, a towed system and a navigation system were then included in the model. Two compensator sys-tems have been designed based on the equations of motion linearized about a given equilibrium operating condition. The complete system was then simulated. In the development of the equations of motion it was difficult to establish a good comparison of the hydrodynamic forces. It was found that some of the hydrodynamic coefficients vary greatly depending on the method that is used to find them. The navigation system simulated here produced satisfactory results in following a path described by a series of way points, having errors in line following up to 6.6 meters during a turn. This nav-igation simply aimed the submarine at the desired way point. It is possible to use the perpendicu-lar distance from the path line as an additional parameter to determine a reference heading. Due to the dependence of both states on one actuator it is beneficial to use solely a heading reference than to track both parameters. The development of the dynamic tow model using system identification showed a good compari-son with measured data. A simple state space model was able to represent a complex system of tow and towfish. The static tow vector did not provide a true representation of the towed system. The static tow model did not induce comparable lateral excitation forces indicated by the resulting smller roll and heading deviations in the turn. Due to the diameter of the hull in relation to the average wavelength of the sea state, the Morison equation was used to determine the forces on the hull. Additional effects of the waves on the hydroplanes were included. It was found that the disturbance induced by waves on the apparent angle of attack of the planes is significant. The bandwidth of the wave frequency for the simulated sea states was less than the bandwidth of the submarine. This required the state estimation loop to be designed well above the bandwidth of the noise to ensure correct state estimation in waves. Due to the steady forces applied by the tow, buoyancy, and drag, it was necessary to include a steady feed forward actuation signal. It was also necessary to include integration, whether as a low pass filter or an integrator in either the actuator signal, the state estimate, or the observed dis-turbance. This resulted in an augmented plant system for design of the controller. The effect of the augmentation ensured complete recovery of the desired reference command after a maneuver. Because of the non-linear effects, this was essential. Actuator saturation was a major factor in the performance of the compensated system. Stability and performance margins deteriorate with actuator limits. Depth errors greater than 0.25 meters 155 can lead to actuator saturations. The saturation point depends on the control design used. The combined effect of the non-linear plant and the non-linear actuator dynamics were significant. There are methods that can be implemented to counter these effects. Anti-wind-up designs can be implemented after the design of the controller, or can be included in the design of the controller. The anti-wind-up method used here showed good results. The effect of pitch on the departure from the linear model was large. A small pitch angle causes the forward speed of the vehicle to produce a change in depth. This can be dealt with in a number of ways. In this thesis the control of the pitch state was greatly weighted. The results were satis-factory. It is also possible to use gain scheduling based on pitch. Sensor noise showed an increase in the amount of actuator activity. Performance may be compro-mised but complete analysis was not performed here. It is possible to counter this effect by using digital filtering techniques for the sensor information. Forces from the towed system had large effects on lateral motions of the towing vehicle. The error in roll, and heading were increased. There was a large sway crossflow component during the turn which increased with a tow. Depth keeping degraded minimally, both compensators were able to maintain depth to within 0.05 meters. The downward force of the tow opposed the large buoyancy of the submarine thus decreasing hydroplane RMS values. The integrator augmentation exhibited better performance in waves than the disturbance observer, having less error in states and less hydroplane motion. Both compensators showed almost twice as much foreplane RMS deflection than aftplane RMS deflection. The actuator deflections were higher in head seas than in beam seas. Simulations in sea state 4 showed depth excursions up to 0.5 m from the desired depth. The robust design of LQG controllers showed satisfactory results. The simulated system was able to track the reference command under a variety of conditions. The advantage of the model based compensator is the state estimation and filtering are combined with the controller. Stability mar-gins are guaranteed for the linear system. Knowledge of the linear hydrodynamic coefficients was needed. The performance of the submarine was dependent on the method of control system that was used. The two systems presented here show similar results, the integrator augmentation had slightly bet-ter performance. The reason for the lower performance of the disturbance observer augmentation could be the lack of anti-windup procedures on the disturbance states. 7.2 Recommendations As a study of LQG control for a small, near surface, submersible vehicle much has been learned. From this knowledge it is possible to make recommendations that are prudent in the design and control of such a vehicle. In application of a control system for this vehicle, especially a high fidelity system such as an LQG controller, it is requisite to ensure a smooth reference trajectory. This can be accomplished 156 through both filtering of the reference command and ramped reference changes. Information can be found in [134] on reference generation. It may also be possible to use the knowledge of the non-linear dynamics to generate references such as banked turns [136] which optimize the refer-ence command based on the roll requirements of the vehicle. It is obvious from this study that actuator saturation will occur if exact tracking of the reference command is required. This is exemplified in high sea states. If stability is to be ensured in such cases it may be essential to increase the force provided by the actuators. This could be done by changing the geometry of the hydroplanes. It appears that the foreplanes saturate initially, so these changes should be implemented on the foreplanes. The concession in this case is that greater dis-turbance due to wave motion will be evident. Another approach is to increase the bounds of the actuators based on hull angle of attack. Thus should be done dynamically to prevent unwanted cases of large actuation. Using the robust control method, a reduced system must be used in design. This is limited by the inter-dependence of the states and also the effect of the actuators on the states. In order to increase the robustness as well as enlarge the number of outputs used for control, the placement of the actuators could be changed. Either the incorporation of an additional lateral controller, or the use of anhedral foreplanes would allow control of an additional degree of freedom. Not only does this increase the number of outputs that can be controlled, but also the robustness of the system. If there is a fault or saturation in an actuator, other planes may be able to maintain control of the desired state. It is also possible to use alternative outputs for design of the controller. If the translational and rotational velocity states are used in the L Q G design, the pose and position states could be sepa-rately controlled, using a cascaded approach, with feedback linearization. It is imperative that augmentation of the linear model is necessary to ensure good tracking and small steady state error. This also decreases the sensitivity to noise. Both the systems described here show good results. As the bandwidth of the engine response is in the same order of magni-tude as the plant, it is recommended that a dynamics model of the engine be included in the design of the controller. The other alternative is to remove forward speed control from the MIMO LQG system and control it separately. As the forward speed coupling with other states is small this could be feasible. The LQG approach with LTR design method is based on a linear design model. In the event of speed changes or pitching, the system departs from the linear model. These effects can be coun-tered through the use of gain scheduling. In this thesis, the pitch control has been weighted heav-ily to ensure small pitch angles. In related literature, two approaches to loop shaping have been found, that of [26] and that of [125]. This thesis uses [125]. From work done using both methods, it is suggested that [26] be used to obtain better loop shaping, although [125] allows the designer more flexibility. It is possi-ble to incorporate both methods. Although the LQG compensator provides filtering of the measured outputs, sensor noise is appar-ent in the response of the vehicle. Due to the high bandwidth of the controller, it may be necessary 157 to provide digital filtering at a frequency between the bandwidth of the controller and the sam-pling bandwidth. From the literature related to this work two other types of control methods should be examined. Sliding mode and adaptive sliding mode control methods have shown good results when imple-mented with submersible vehicles. In the same vein as LQG control, H infinity methods have been applied to flight vehicles with success. Incorporation of gain scheduling and H infinity methods could prove beneficial. Identification of system parameters using system identification has shown good results. Due to the difficulty in ascertaining reliable hydrodynamic information about underwater vehicles, it may be advantageous to apply these techniques to underwater vehicles. The results and analysis in this thesis are based largely on simulation and theoretical information. Certain parameters affect the validity of these results. In order to ascertain the quality of this research it is important to perform similar testing and verification using the actual vehicle. In this way both the theory and modelling of underwater vehicles can be improved as well as the meth-ods and application used in controlling the real vehicle. Using a high fidelity control system which is designed in a similar framework, the resulting RMS, maximum and minimum values should be compared for similar conditions either using other software or full scale data. 158 Nomenclature TABLE N.l: Description of Roman Nomenclature and Variables. Symbol Description A Plant linear system matrix. A 0 Submarine frontal area a l - A3 Added mass coriolis and centripetal matrix elements. A A Added mass matrix as a function of frequency. Aaft Aftmost cross sectional area. ^Bret Bretschneider wave spectrum coefficient. a e Effective hydroplane aspect ratio. aijk Linear system model parameter elements in tow model identification. A , Tow model linear system matrix. B Buoyancy force magnitude. B_ Plant linear input matrix. b Hydroplane span. b, ... b 3 Added mass coriolis and centripetal matrix elements. BBret Bretschneider wave spectrum coefficient. bijk Linear input model parameter elements in tow model identification. fit Tow model linear input matrix. £ Plant linear output system matrix. c Hydroplane chord. Added mass coriolis and centripetal matrix. c D Hydrodynamic drag coefficient. C D0 Minimum hydroplane hydrodynamic drag value. Hydroplane hydrodynamic cross-flow coefficient. C L Hydrodynamic lift coefficient. Cpb Coefficient of spanwise center of pressure for a hydrofoil. Rigid body coriolis and centripetal matrix. c t Tow model linear output system matrix. Coefficient of hydrodynamic frictional drag (Aucher). D Hydrodynamic drag matrix. D Plant linear output matrix. D Hydrodynamic drag force. Do Submarine hull diameter. Tow model linear output matrix. £]', £2', £3'' £4' £5'' ^6' Set of unit vectors coincident with body fixed reference frame. £l. £2, £3, £4,65, fig Set of unit vectors coincident with earth fixed reference frame. 159 TABLE N.l: Description of Roman Nomenclature and Variables. Symbol Description fit Error in least squares approximation for tow model. E External rigid body force vector. J B Body fixed reference frame. Eb Buoyancy force vector. J E Earth fixed reference frame. Eg Gravitational force vector. E h Hydrodynamic force vector. E m Fluid memory effect force vector. J T Translating earth reference frame. Et Tow cable force vector. Wave reference vector. S Acceleration due to gravity. G Fluid Green function. 3£ Rotational momentum vector. h l / 3 Significant wave height. I, Rotational inertial matrix. K Roll moment in body frame. Kc Keulegan-Carpenter number. K U Fluid memory kernel of integration. k w Wave number. K Wave number in x direction. K Added mass coefficient in x direction (Aucher). k y Wave number in y direction. K Added mass coefficient in z direction (Aucher). Added mass coefficient (Aucher). £ Translational momentum vector. L Hydrodynamic lift force. L 0 Submarine hull length. M Pitch moment in body frame. M External rigid body moment vector. m, m Submarine mass, and diagonal mass matrix. M A Hydrodynamic added mass matrix. M B Buoyancy moment vector. M G Moment vector due to gravity. M „ Total hydrodynamic moment vector. 160 TABLE N.l: Description of Roman Nomenclature and Variables. Symbol Description m 0 First moment of integration of wave spectral energy function. M r b Rigid body mass matrix. M t Moment due to tow cable force. N Yaw moment in body frame. n Unit vector normal to submarine hull. P Roll rate in body frame. E Fluid pressure function. q Pitch rate in body frame. Q, Q i , Q 2 Transformation matrix from body frame to inertial frame, r Yaw rate in body frame. Position vector of center of mass. Re Reynolds number. rt Tow cable attachment position vector. R w Waypoint target radius. S Surface area of a component used in hydrodynamic analysis. S(G)e) Wave spectral energy as a function of encounter frequency. S(COw) Wave spectral energy as a function of wave frequency. 1 Hydrodynamic control matrix. T , Energy averaged wave period. T Peak wave period. *t Tow cable force and moment vector. T, Tow cable tension. T w Wave period. u Surge rate in body frame. u Actuator level vector. U Instantaneous submarine speed. U 0 Equilibrium submarine total speed. uo Equilibrium submarine surge rate. Umean Mean wave particle speed. Usog Submarine speed over ground. lit Tow model input vector. V Sway rate in body frame. v 0 Equilibrium sway rate. w Heave rate in body frame. W Submarine weight. w 0 Equilibrium heave rate. 161 TABLE N.1: Description of Roman Nomenclature and Variables. Symbol Description w f Localized foil cross flow. x Coordinate location on primary axis. X Surge force in body frame. i Plant state vector. XQ Equilibrium x position. ^ A > Y A . Z A , K A , M A , N a Added mass coriolis and centripetal force and moment terms. x-y. Axial position of fluid separation at tail of submarine. X G ' V G ' Z G Coordinate elements of center of mass of submarine. x t, y t, z t Coordinate elements of tow attachment position. y Coordinate location on secondary axis. Y Sway force in body frame. y Plant output vector. y* Perpendicular distance from submarine to line connecting waypoints. y 0 Equilibrium y position. y_t Tow model output vector. z Coordinate location in z direction. Z Heave force in body frame. z 0 Equilibrium z position. z s Position of mean water level in z direction. TABLE N.2: Description of Greek Nomenclature and Variables. Symbol Description a Rotation about y axis in body reference frame. «ar Mean half angle of conicity of the tail section. « t Tow cable depressive angle at DOLPHIN. « w Linear wave amplitude. P Rotation about z axis in body reference frame. Pt Tow cable lateral angle at DOLPHIN. Pw Relative angle of wave travel direction to vessel. 5 Hydroplane deflection angle. Sap Port aftplane deflection angle. 8as Starboard aftplane deflection angle. 5 d Apparent change in hydroplane angle due to a wave disturbance. 8fP Port foreplane deflection angle. 162 T A B L E N.2: Descript ion of Greek Nomenclature and Variables. Symbol Description 5fs Starboard foreplane deflection angle. 5n Engine rotation speed. 5R Rudder deflection angle. E Angle of attack of submarine component (Aucher). Euler roll angle. O Fluid potential function. Ot Vector of inputs and outputs for tow model identification, y Rotation about x axis in body reference frame. y p Hydroplane sweep angle. H> H i > H2 Position and pose vectors of vessel in earth reference frame. Tj Instantaneous wave height. X Surface wave wavelength. Xp Hydroplane taper ratio. Y, Yj, V 2 Translation and rotation rate vector of vessel in body reference frame. 0 Euler pitch angle. 9t Matrix of system model parameters for tow model identification. A Estimated matrix of system model parameters for tow model identification. ?t p Fluid density. p s Submarine discontinuous density function. Cj Source strength function. X, t r , T 2 Forces and moments impinging on vessel in body reference frame. X b g Gravity and buoyancy force and moment vector. Xc Control components force and moment vector. X h Hydrodynamic force and moment vector. X i n c Incident wave force and moment vector. X t Tow cable force and moment vector. X Wave force and moment vector. X) Rigid body translation rate. 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This is shown in Figure A . l . r*~ — e 2 , l / r e 2 Figure A . l : Yaw Euler Rotation. This rotation produces a set of unit vectors {e1(,, e2> i , e3 ^ } having e3 j coincident with e3. The new axis now is in the heading direction of the body x' axis. Thus, the rotation matrix is pro-duced: I i = cos(\|/) sin(\|/) 0 —sin(v|/) cos(\|/) 0 0 0 1 (A.1) 179 The new temporary orthogonal frame is then rotated about its current y axis, e^i to produce an angle of pitch. The new longitudinal axis e2,\ is n o w coincident with the longitudinal axis of the submarine as shown in Figure A .2 . 7 e.,, \ w e e l ,2 e3,2 / ^3,1 Figure A.2: Pitch Euler Rotation. This second rotation is shown by the transformation matrix: I 2 = cos(0) 0 -sin(0) 0 1 0 sin(G) 0 cos(9) (A.2) Producing another set of temporary unit vectors {ej 2, e 2 2, e 3 2 1' n a v m g e2,2 coincident with e2 1. The third and final rotation is about the ej 2 axis producing a roll angle such that the resulting orthogonal frame is that of the body fixed frame. This is shown in Figure A3. This final rotation brings all the unit vectors {ej 3, e 2 3, e 3 3 } to be coincident with those of the body fixed refer-ence frame {e,', e2', e3'}. 180 Figure A.3: R o l l Eu le r Rotat ion. The transformation matrix used to represent the roll rotation is: Is = 1 0 0 0 cos((|)) sin((()) 0 -sin(d)) cos((|)) (A.3) From this description, it is apparent that the order in which these rotations are performed is cru-cial. Once the order has been established, the convention must be followed consistently. Following the transformation shown above, any positional or translational vector in the inertial frame, t| j , can be represented in the body frame Vj by the relationship: where, (A.4) T T T Q i = l i l i l i -or, due to skew symmetry of each of the temporary transformations, QI 1 = Qi = I3I2I1. (A.5) (A.6) Expressed fully, 181 Q, = c o s c o s ( G ) - sin(\j/)cos(<|>) + cos(Y|/)sin(8)sin(<|>) sin(\|/)sin((|>) + cos(v)/)cos(()>)sin(e) sin(\)/)cos(e) cos(i|/)cos(<|>) + sin(<t>)sin(G)sin(\|0 - cos(\|/)sin(<|>) + sin(9)sin(\|/)cos((()) -sin(9) cos(9)sin((})) cos(6)cos(<|)) (A.7) A.2 Angular Velocity Transformation The angular velocity in one frame cannot be transformed in the same manner as the translational velocity as the angles used to make each of the three temporary rotations are themselves changing in time. In order to accomplish the representation of the angular velocities using Euler angles, it is found that coB = co E . Thus, the instantaneous transformation rotations can be performed while each angle is essentially considered fixed. In this way, the instantaneous rotation velocity vector in the body frame v 2 can be transformed into the inertial frame by: 4> 0 0 0 + Is e + I 3 l 2 0 0 0 Y2 = We desire to represent this series of transformations in the same manner as eq. A.4, B2 = Q2Y2 where, (A.8) (A.9) (A. 10) Then the resulting angular velocity transformation matrix is: 1 sin(cb)tan(0) cos(cb)tan(0) 0 cos(cb) -sin(<))) [0 sin((|))/(cos(e)) (cos(<|>))/(cos(8))_ (A.11) 182 Appendix B: Determination of Hydrodynamic Coefficients B.1 Added Mass B.1.1 S U M A D This uses the software code found in reference [49] and is written in Fortran. Program Saml * This program i s the main executing program for SAMUD * (Submarine Added Mass Using Drea Code) * This c a l l s a geometry f i l e having the same format as that f or * Submo3 and finds the added mass and damping for each sect i o n * The s e c t i o n a l added masses are then integrated f or t o t a l * d e r i v a t i v e s * INTEGER SEGMAX, WMAX, IMAX PARAMETER(IMAX= 85, WMAX=202, SEGMAX=30) REAL DENSITY, PI PARAMETER(PI=3.141592654,DENSITY=1025) REAL DX(IMAX), XCG, WEMIN, WEINC, LEN, WEMAX REAL ZCG, YSECT(IMAX,SEGMAX), ZSECT(IMAX,SEGMAX),ALLOF(8), & PHI2ZERO(SEGMAX-1), PHI2INF(SEGMAX-1), PHI3INF(SEGMAX-1), & PHI4INF(SEGMAX-1),AM(IMAX,WMAX), DA(IMAX,WMAX), GRAVITY, & ALPHAB2D REAL XSEC(IMAX), BSEC(IMAX), ZFS REAL AM2(IMAX), AM3(IMAX), AM4(IMAX), AM24(IMAX) REAL DA2(IMAX), DA3(IMAX), DA4(IMAX), DA24(IMAX) REAL DAM2INF, DAM3INF, DAM4INF, DAM24INF, DAM5INF, DAM6INF REAL DAM2(WMAX), DAM3(WMAX), DAM4(WMAX), DAM5(WMAX),DAM6(WMAX) REAL DDA2(WMAX), DDA3(WMAX), DDA4(WMAX), DDA5(WMAX),DDA6(WMAX) INTEGER I, W, I0LDGE0M, NSEG(IMAX), NWF COMPLEX AB2, AB24, AB3, AB4, PHI2(SEGMAX-1), PHI3(SEGMAX-1), & PHI4(SEGMAX-1) OPEN ( UNIT = 3,FILE ='geom.dat', ACTION='READ',STATUS='UNKNOWN') OPEN ( UNIT = 4,FILE ='inam.dat', STATUS = 'UNKNOWN') OPEN (UNIT = 7,FILE='geom.mod',status='unknown') OPEN (UNIT =8, FILE= 'wfamda.dat', status='unknown 1) * GRAVITY=9.81 * IOLDGEOM = 0 ALPHAB2D = 0.2 * READ IN USER DATA * WRITE(*,*) 'Enter the x, z l o c a t i o n of the eg from body a x i s : ' READ (*,*) XCG, ZCG 183 WRITE (*,*) 'Enter the z l o c a t i o n of the free surface READ ( *,* ) ZFS WRITE (*,*) 'Enter v e h i c l e length:' READ ( *,*) LEN WRITE(*,*) 'Enter wemin, weinc, wemax:' READ (*,*) WEMIN, WEINC, WEMAX * READ IN ALL GEOMETRIC DATA FOR HULL READ (3,*)NSEC ! THE NUMBER OF SEGMENTS * DO 10 1=1,NSEC READ(3,*) NSEG(I), XSEC(I), BSEC(I) DO 2 0 SEG=1,NSEG(I)+1 READ(3,*) YSECT(I,SEG),ZSECT(I,SEG) 20 CONTINUE 10 CONTINUE WRITE(*,*) 'Finished reading geom.dat' * ZCG=-ZFS+ZCG DO 15 1=1,NSEC DO 25 SEG=1,NSEG(I)+1 ZSECT(I,SEG)=-ZFS+ZSECT(I,SEG) IF (ZSECT(I,SEG).GT.0) THEN ZSECT(I,SEG)=0 YSECT(I,SEG)=0 END IF 25 CONTINUE 15 CONTINUE * * OUTPUT MODIFIED GEOM WRITE(7,*) NSEC DO 1=1,NSEC WRITE(7,*) NSEG(I), XSEC(I), BSEC(I) DO 27 SEG=1,NSEG(I)+1 WRITE(7,990) YSECT(I,SEG),ZSECT(I,SEG) 27 CONTINUE END DO 990 FORMAT (1X,2F12.6) * * CALCULATE THE INFINITE FREQUENCY AM DERIVATIVES * FOR HEAVE, PITCH, SWAY, YAW, AND YAW/ROLL DO 30 1=1,NSEC CALL B2DLIM(NSEG(I)+1, YSECT(1,1:NSEG(I)+1), * ZSECT(I,1:NSEG(I)+1), ZCG, * A2ZERO, A2INF, A24INF, A3INF, A4INF, * PHI2ZERO, PHI2INF, PHI3INF, PHI4INF) AM2(I)=A2INF*DENSITY AM3(I)=A3INF*DENSITY AM4(I)=A4INF * DENSITY AM24(I)=A24INF*DENSITY 30 CONTINUE 184 INTEGRATE OVER THE LENGTH 40 DX(1)=0.5*(XSEC(1)+(XSEC(2)-XSEC(1))) DX(NSEC)=0.5*(XSEC(NSEC)-XSEC(NSEC-1)+(LEN-XSEC(NSEC))) DO 40 1=2,NSEC-1 DX(I)=0.5*(XSEC(I+1)-XSEC(I-1)) CONTINUE DAM22 DAM33 DAM44 DAM55 DAM66 DAM24 DAM46 DAM3 5 DAM2 6 INF=0 INF=0 INF=0 INF=0 INF=0 INF=0 INF=0 INF=0 INF=0 50 DO 50 1=1,NSEC DAM22INF= DAM3 3INF= DAM44INF= DAM55INF= DAM66INF= DAM46INF= DAM26INF= DAM3 5INF= DAM24INF= CONTINUE DAM2 2INF+AM2(I)*DX(I) DAM33INF+AM3(I)*DX(I) DAM44INF+AM4(I)*DX(I) DAM55INF+AM3(I)*ABS(XCG-XSEC(I))**2*DX(I) DAM66INF+AM2(I)*ABS(XCG-XSEC(I))**2*DX(I) DAM46INF+AM24(I)*(XCG-XSEC(I))*DX(I) DAM2 6INF+AM2(I)*(XCG-XSEC(I))*DX(I) DAM3 5INF-AM3(I)*(XCG-XSEC(I))*DX(I) DAM24INF+AM24(I)*DX(I) COORDINATE CHANGE EFFECTS FROM SHIP TO SUB FRAMES DAM2 4 INF=-DAM2 4 INF DAM46INF=-DAM46INF WRITE(4,1000) 0.0, 0.0, 0.0, 0.0, 0.0, DAM22INF, 0.0, 0.0, 0.0, DAM33INF, 0.0, DAM24INF, 0.0, 0.0, 0.0, DAM35INF, 0.0, DAM2 6INF, 0.0, 0.0, 0,0, DAM24INF, 0.0, 0.0, DAM35INF, DAM44INF, 0.0, 0.0, DAM55INF, DAM46INF, 0.0, 1000 FORMAT(IX,6F11.4) NOW DO FREQUENCY DEPENDENT PARTS NWF=NINT((WEMAX-WEMIN)/WEINC)+1 DAM26INF, 0.0, DAM46INF, 0.0, DAM66INF DO 60 WVF=1,NWF DO 70 1=1,NSEC WE = WEMIN + WEINC*(WVF-1) IOLDGEOM=0 CALL BOUND2D(IOLDGEOM, NSEG(I)+1, YSECT(1,1:NSEG(I)+1) 185 70 ZSECT(I,1:NSEG(I)+1), ZCG, ALPHAB2D, AB2, AB24, AB3, AB4, AM2(I)=REAL(AB2)*DENSITY AM3(I)=REAL(AB3)*DENSITY AM4(I)=REAL(AB4)*DENSITY DA2(I)=AIMAG(AB2)*DENSITY DA3(I)=AIMAG(AB3)*DENSITY DA4 (I) =AIMAG (AB4) * DENSITY CONTINUE GRAVITY, WE, PHI2, PHI3, PHI4) D A M 2 D A M 3 D A M 4 D A M 5 D A M 6 D D A 2 D D A 3 D D A 4 D D A 5 D D A 6 (WVF (WVF (WVF (WVF (WVF (WVF (WVF (WVF (WVF (WVF 80 DO 80 I=1,NSEC DAM2(WVF DAM3(WVF DAM4(WVF DAM5(WVF DAM6(WVF DDA2(WVF DDA3(WVF DDA4(WVF DDA5(WVF DDA6(WVF CONTINUE = 0 = 0 = 0 = 0 = 0 = 0 = 0 = 0 = 0 = 0 = D A M 2(WVF ) + A M 2(I) * D X(I) = D A M 3(WVF)+ A M 3(I) * D X(I) = D A M 4(WVF)+ A M 4(I) * D X(I) =DAM5(WVF)+AM3(I)*ABS(XCG-XSEC(I))**2*DX(I) =DAM6(WVF)+AM2(I)*ABS(XCG-XSEC(I))**2*DX(I) =DDA2(WVF)+DA2(I)*DX(I) =DDA3(WVF)+DA3(I)*DX(I) =DDA4(WVF)+DA4(I)*DX(I) =DDA5(WVF)+DA3(I)*ABS(XCG-XSEC(I))**2*DX(I) =DDA6(WVF)+DA2(I)*ABS(XCG-XSEC(I))**2*DX(I) OUTPUT TO FILE WRITE(8,1010) WE, DAM2(WVF), DDA2(WVF) * DAM4(WVF), DDA4(WVF) * DAM6(WVF), DDA6(WVF) DAM3(WVF) DAM5(WVF) DDA3(WVF), DDA5(WVF), 1010 FORMAT(IX,F6.2, 12F10.4) 60 CONTINUE STOP END 186 B.1.2 Aucher Body Table I knot = 0.5145-— sec L := 8.534-m D = l m Re := V - -v 2 Ixx = 1315 kg-m 2-Ixx p := 1025 kg V := 12-knot 1.5610 6 m sec for temp. 5 deg C 7 t D Re = 3.377-10 Iyy := 5900 kg-m V O l : " D 4605 kg = 8.534 kidx p-A-L kx = 0.0276 2-Iyy kidy: = kidy = 0.024 kidz p-A-L 3 kz= 0.949 kiz = 0.851 kd Izz = 5057 kg-m 2-Izz p-A-L 2 - v o l A L CXudot = 2-kx-kd CYvdot := 2-kz-kd CNrdot '.= 2-kiz-kidz CZwdot= CYvdot CMqdot := 2-kiz-kidy CXudot = 0.074 CYvdot = 2.544 CNrdot =0.034 Xudot= CXudot Yvdot := CYvdot Nrdot := CNrdot Zwdot= CZwdot-Mqdot = CMqdot Yrdot= 0 kg-m Zqdot := 0 kg-m Kvdot := 0 kg-m Krdot := 0 kg-m2 kd = 1.341 p-A-L 2 p-A-L 2 p-A-L 3 2 p-A-L Nrdot = 8.607-103 -kg-m2 p-A-L Zwdot = 8.74-10 -kg Mqdot = 1.004-104 -kg-m2 187 Keel (0030) Table III Ro := 0.5 m b = 1.5906 m C = 1.8573 m h = 1.0906 m \2l 1 Ro Xs = 0.772 E = 0 . 3 C SI := h-C ka := 0.458 mal = p-rc - — ) -2 bka \2y mal =4.046-10 «kg Yvdotl = mal .00823 -4 xel := xel =9.644-10 8.534 at 0.1 C location xsl := xel 0.7-C Yrdotl := malxsl-L 2 2 Nrdotl := mal xsl L Ro' Yrdotl =-5.227-10 -kg-m Nrdotl =6.753-103 -kg-m2 Krdotl = mal-[b - -— j-xsl-L Krdotl =-7.493-10 -kg-m Kvdotl := mal-|b u Kvdotl =5.8-10 -kg-m Kpdotl := mal- b - Ro Kpdotl =8.314-103 -kg-m2 188 Stern Planes (NACA 0025) 2.21) b := 2.032- C := .34925-m dl = .6985-m X = — 2 C E:=.25-C S 2 = ( 2 b - d l ) . C = 2.909 at 0.8 chore xe2- 3 - 2 8 1 m xe2 =-0.384 at 0.2 chord X s 2 = x e 2 - ^ ^ xs2=-0.409 ka = .25iX - 1) 1 +-(X.- I ) 2 ka= 0.936 ma2 = p-7t - — ] -2-b-ka \ 2 i ma2= 186.786-kg Zwdot2 := ma2 Zqdot2 := ma2-xs2L Mqdot2 := ma2-xs22L2 Zwdot2= 186.786-kg Zqdot2 =-651.987-kg-m Mqdot2=2.276-103 -kg-m2 Rudder and Stabilizer Yvdot2 := Zwdot2 Yrdot2 = Zqdot2 Nrdot2 := Mqdot2 189 Bow Planes (NACA 0025) Ro=0.5m b= 0.9677 m C = .34925-m h = b X = 5.542 E=0.25-C 1.951 xe3= xe3= 0.229 8.534 at 0.1 C location o 0.7-C xs3 : = xe3 xs3 = 0.2 ka x2 .25-(X - 1) l-h(X- l ) 2 ka= 0.932 ma3= p-jt-( —] -2-b-ka \2l ma3 = 177.115-kg Zwdot3 := ma3 Zqdot3 := ma3-xs3L 2 2 Mqdot3 := ma3-xs2 L Zwdot3 = 177.115-kg Zqdot3 = 302.251 -kg-m Mqdot3=2.158-103 -kg-m2 190 Total Hydrodynamic Coefficients Yvdot = Yvdot + Yvdotl +• Yvdot2 Yvdot = 1.297-104 •kg Nrdot = Nrdot +- Nrdotl -t- Nrdot2 Nrdot = 1.764-104 i , 2 •kg-m Zwdot = Zwdot + Zwdot2 + Zwdot3 Zwdot = 9.104-103 •kg Yrdot: = Yrdot + Yrdot 1 + Yrdot2 Yrdot = -5.879-103 •kg-m Zqdot = Zqdot +• Zqdot2 + Zqdot3 Zqdot = -349.736 -kg-m Mqdot := Mqdot +• Mqdot2 + Mqdot3 Mqdot = = 1.448-104 •kg-m Nrdot = Nrdot + Nrdotl + Nrdot2 Nrdot = 2.666-104 i , 2 •kg-m Kvdot = Kvdot + Kvdotl Kvdot = 5.8-103 -kg-m Krdot = Krdot t- Krdotl Krdot = -7.493-103 V 2 •kg-m Xudot = Xudot Xudot = 254.196-kg Kpdot = Kpdotl Kpdot = 8.314-103 v 2 •kg-m Aucher Feldman Yv*=(rho/2)AVsCYv* Yv*=(rho/2)L A2VsYv* Yr*=(rho/2)ALVsCYr* Yr*=(rho/2)L A3VsYr*' Nv*=(rho/2)ALVsCNv Nv*=(rho/2)L A2VsNv' Nr*=(rho/2)AL A2Vs* Nr*=(rho/2)L A4VsNr*' 191 Non-dimensional: Yvdot Yv_= Yv_ = 0.041 0 .5p -L 2 L „ Zwdot Zw_: = Zw_ = 0.029 0.5-pL 2-L Yrdot -3 Yr_= Yr_ = -2.163-10 2 2 0.5-p-L -L Nrdot -3 Nr_ := - . Nr_ = 1.149-10 0.5-p-L2-L3) _ ^ q d o t _ = _ 1 2 8 7 . 1 0 - 4 2 2 0.5-p-L-L Mqdot -4 Mq_ := 2 Mq_ =6.24-10 0.5-p-L2-L3 Kvdot -3 Kv_= Kv =2.134-10 2 2 ~ 0.5-p-L L Krdot -4 Kr_:= Kr =-3.23-10 2 3 -0.5-p-L L Xudot -4 Xu_:= ? ^ Xu_ = 7.98-10 (o.5-p-L2-LJ Kpdot -4 Kp_ := -. ^ — K p _ = 3.584-10 4 (0.5-p-L-L 3; 192 B.2 Damping B.2.3 Missile Datcom Output Data ***** THE USAF AUTOMATED MISSILE DATCOM * REV 5/97 ***** CASE 1 AERODYNAMIC METHODS FOR MISSILE CONFIGURATIONS PAGE 32 STATIC AERODYNAMICS FOR BODY-FIN SET 1, 2, AND 3 * * * * * * * FLIGHT CONDITIONS AND REFERENCE QUANTITIES * * * * * * * MACH NO = .00 REYNOLDS NO = 3. 290E+06 /M SIDESLIP = .00 DEG ROLL = .00 DEG REF AREA = .785 M** 2 MOMENT CENTER = 4.173 M REF LENGTH 8.53 M LAT REF LENGTH = 8.53 M LONGITUDINAL -- LATERAL DIRECTIONAL ALPHA CN CM CA CY CLN CLL .00 .000 .004 .167 .000 .000 .000 1.00 .139 .008 .166 .000 .000 .000 2 .00 .281 .012 .162 .000 .000 .000 3.00 .432 .014 .156 .000 .000 .000 4.00 .593 .014 . 146 .000 .000 .000 ALPHA CL CD CL/CD X-C.P. .00 .000 .167 .000 .028 1.00 .136 .168 .808 .054 2.00 .276 .172 1.604 .041 3.00 .424 .178 2 .378 .032 4.00 .581 .187 3.103 .023' X-C.P. MEAS ***** THE USAF AUTOMATED MISSILE DATCOM * REV 5/97 ***** AERODYNAMIC METHODS FOR MISSILE CONFIGURATIONS STATIC AERODYNAMICS FOR BODY-FIN SET 1, 2, AND 3 FROM MOMENT CENTER IN REF. LENGTHS, NEG. AFT OF MOMENT CENTER CASE 1 PAGE 33 ******* FLIGHT CONDITIONS AND REFERENCE QUANTITIES ******* MACH NO = .00 REYNOLDS NO = 3.290E+06 /M SIDESLIP = .00 DEG ROLL = .00 DEG REF AREA = REF LENGTH .785 M**2 3.53 M MOMENT CENTER = 4.173 M LAT REF LENGTH = 8.53 M 193 DERIVATIVES (PER RADIAN) ALPHA CNA CMA CYB CLNB CLLB .00 7.8357 .2161 -12.6151 .2595 .0739 1.00 8.0607 .2242 -12.6156 .2596 .0732 2.00 8.4085 .1797 -12.6250 .2549 .0709 3.00 8.9209 .0633 -12.6247 .2467 .0647 4.00 9.4753 -.0724 -12.6116 .2352 .0573 PANEL DEFLECTION ANGLES (DEGREES) SET FIN 1 FIN 2 FIN 3 FIN 4 FIN 5 FIN 6 FIN 7 FIN 8 1 .00 .00 2 .00 3 .00 .00 .00 .00 ***** THE USAF AUTOMATED MISSILE DATCOM * REV 5/97 ***** CASE 1 AERODYNAMIC METHODS FOR MISSILE CONFIGURATIONS PAGE 34 BODY ALONE DYNAMIC DERIVATIVES ******* FLIGHT CONDITIONS AND REFERENCE QUANTITIES ******* MACH NO = .00 REYNOLDS NO = 3.290E+06 /M SIDESLIP = .00 DEG ROLL = .00 DEG REF AREA = .785 M**2 MOMENT CENTER = 4.173 M REF LENGTH = 8.53 M LAT REF LENGTH = 8.53 M DYNAMIC DERIVATIVES (PER RADIAN) ALPHA CNQ CMQ CAQ CNAD CMAD .00 .669 -.336 .000 1.780 .080 1.00 .669 -.336 .000 1.780 .080 2.00 .669 -.336 .000 1.780 .080 3.00 .669 -.336 .000 1.780 .080 4.00 .669 -.336 .000 1.780 .080 PITCH RATE DERIVATIVES NON-DIMENSIONALIZED BY Q*LREF/2*V 1 ***** THE USAF AUTOMATED MISSILE DATCOM * REV 5/97 ***** CASE 1 AERODYNAMIC METHODS FOR MISSILE CONFIGURATIONS PAGE 3 5 BODY ALONE DYNAMIC DERIVATIVES ******* FLIGHT CONDITIONS AND REFERENCE QUANTITIES ******* MACH NO = .00 REYNOLDS NO = 3.290E+06 /M SIDESLIP = .00 DEG ROLL = .00 DEG REF AREA = .785 M**2 MOMENT CENTER = 4.173 M REF LENGTH = 8.53 M LAT REF LENGTH = 8.53 M DYNAMIC DERIVATIVES (PER RADIAN) ALPHA CYR CLNR CLLR CYP CLNP CLLP 194 .00 .713 -.33.6 .000 .000 .000 .000 1-00 .713 - . 3 3 6 .000 .000 .000 .000 2 .00 .713 - . 3 3 6 .000 .000 .000 .000 3 .00 .713 - . 3 3 6 .000 .000 .000 .000 4 .00 .713 - . 3 3 6 .000 .000 .000 .000 YAW AND ROLL RATE DERIVATIVES NON-DIMENSIONALIZED BY R*LATREF/2*V 1 ***** THE USAF AUTOMATED MISSILE DATCOM * REV 5/97 ***** CASE 1 AERODYNAMIC METHODS FOR MISSILE CONFIGURATIONS PAGE 3 6 BODY + 3 FIN SETS DYNAMIC DERIVATIVES ******* MACH NO = SIDESLIP = REF AREA = REF LENGTH FLIGHT CONDITIONS AND REFERENCE QUANTITIES ******* .00 .00 DEG .785 M**2 3.53 M REYNOLDS NO = 3.290E+06 /M ROLL = .00 DEG MOMENT CENTER = 4.173 M LAT REF LENGTH = 8.53 M DYNAMIC DERIVATIVES (PER RADIAN) ALPHA .00 1.00 2.00 3.00 4.00 CNQ 3.690 3 .726 3 .767 3 .767 3 .724 CMQ -2.128 -2.147 -2.182 -2.193 -2.191 CAQ -.062 -.124 -.187 -.249 -.314 CNAD 2.262 2 .262 2 .262 2.262 2 .262 CMAD -.061 -.061 -.061 -.061 -.061 PITCH RATE DERIVATIVES NON-DIMENSIONALIZED BY Q*LREF/2*V ***** THE USAF AUTOMATED MISSILE DATCOM * REV 5/97 ***** AERODYNAMIC METHODS FOR MISSILE CONFIGURATIONS BODY ALONE DYNAMIC DERIVATIVES CASE 1 PAGE 37 ******* MACH NO = SIDESLIP = REF AREA = REF LENGTH FLIGHT CONDITIONS AND REFERENCE QUANTITIES ******* .00 .00 DEG .785 M**2 8.53 M REYNOLDS NO ROLL MOMENT CENTER LAT REF LENGTH 3.290E+06 /M .00 DEG 4.173 M 8.53 M DYNAMIC DERIVATIVES (PER RADIAN) ALPHA CYR CLNR CLLR CYP CLNP CLLP .00 4.707 -1.780 .077 1.057 .087 -.106 1.00 4.701 -1.778 .077 1.064 .084 -.106 2.00 4.692 -1.775 .076 1.073 .081 -.107 3.00 4.681 -1.771 .076 1.083 .078 -.107 4.00 4.669 -1.766 .076 1.094 .075 -.108 195 YAW AND ROLL RATE DERIVATIVES NON-DIMENSIONALIZED BY R*LATREF/2*V *** END OF JOB *** B.2.4 Missile Datcom Input File $REFQ LREF=8.534, XCG=4.173, RHR=250., ZCG=-.188$ $FLTCON NALPHA=5.,ALPHA=0-,1.,2.,3.,4., NMACH=1., MACH=0.0035, REN=3.29E6, VINF=5.145$ $AXIBOD LNOSE=.5, DNOSE=l., LCENTR=5.68, DCENTR=1., LAFT=2.54, DAFT=.03175,$ NACA-1-4-0025 NACA-3-4-0025 $FINSET1 SECTYP=NACA, SSPAN(1)=0.,.444, CHORD(1)=.349,.349, XLE(1)=2.143,2.143, NPANEL=2., PHIF=90.,270.,$ $FINSET2 SECTYP=HEX, ZUPPER=.0896425,.0896425, LMAXU=.06837,.06837, LFLATU=.6667,.6667, SSPAN(1)=0.,1.095, CHORD(l)=l.857,1.857, XLE(1)=3.81,3.81, NPANEL=1. , PHIF=180.,$ $FINSET3 SECTYP=NACA, SSPAN(1)=0.,.9525, CHORD(1)=.34925,.34925, XLE(1)=7.37,7.37, NPANEL=4., 196 PHIF=0.,90.,180.,270 DIM M DAMP FORMAT(20(1X.F10.4) ) WRITE SB123.1.,220. WRITE DB123.1.,400. DERIV RAD PART DUMP SB13,DB13 NEXT CASE B.2.5 Aucher Body -Cyv, CZW eqn. 2.11 -CNV,CMW eqn. 2.12 -C Y r , C Z q eqn. 2.13 -C N r , C M q eqn. 2.14 knot= 0.5145 •— L := 8.534m V := 12-knot _ , 2 for temp. 5 deg C D = lm -6 m v = 1.56-10 •— sec Re:=V-- 7t-D2 v Re = 3.377-107 4~ 2.11) „ Y f 0.25 L CXf = — CXf=0.07 (log(Re) - 2)2 D Aq := 0-m2 rq := {~j a a r : = a t a n | 2 28^  ) ' S a n ^ ' 6 °* 3 ^ c o n ' c ' t v CZe = 2-^5 +. 6.6-( 1 - rq)-aar°'5-( 1 - 0 7 = \ CZe = 0.687 A \ ^0.5 +- 4.5-CXf/ 2.12) , 4605 kg Vol Vol:= kA= kA=0.67 kx= 0.028 1025-— A L for aspect ratio 8.5 3 (Table 2) m kz=.949 Vol = 4.493'in3 xar=-.48 CMe := 2-kA-(kz- kx) + CZexar CMe = 0.905 2.13) CZq := - CZexar CZq =0.33 CMq := -CZe-xar2 CMq =-0.158 198 Keel (0030) 2.23) Ro := 0.5 m b := 1.5906 m C = 1.8573 m h := 1.0906 m \2l Xs '•- 1 - 1 ^ b Xs= 0.772 E = 0 . 3 C SI := h-C dCPel := C b Ro 1 - Xs I Xs \ ^ Ro 2 I 11/ \ + J b / 1 - 3 dCPel =4.353 dCP=dCPel xe 2.24) xel := .0835 8.534 xel =9.784-10 at 0.1 C location dCM := dCPxelxe 2.25) R := Ro dCLl := dCPel dCM= 0.043 xe dCMel := dCM dCMel =0.043 (R+.0.43-h) 2.26) xsl := xel -1 -t--0.7 C dCYrl := dCPel-xsl dCNrl := dCPel -xsl 2 dCLrl := dCLel-xsl 2.27) dCLpl Ro ? 2 h-C Xs I _ Xs\ /Ro+ .6-h 11 xe dCLl =0.376 xe dCLel = dCLl dCLel =0.376 dCYrl =-0.621 dCNrl =0.088 dCLrl =-0.054 dCLpl =0.393 199 Stern Planes (NACA 0025) b := 2.032— C := .34925 m dl = .6985 m de = 2 E = .25C S 2 . = (2-b d l ) - c de=o.844-m 4.5-CXf •D k = 2.909 2.29) dCPe2 := 2-f— D 0.75 - .055 X + 1.35-(— - 0.5 + .05-A. 2-b 1 - 3 dCPe2 =3.27 xe2 := 3 3 6 6 8 m xe2 = -0.395 at 0.2 chord xs2 = xe2 - xs2 = -0.419 at 0.8 chord dCMe2 := dCPe2xe2 dCZq2 := dCPe2xs2 dCMq2 = dCPe2-xs22 dCMe2=-1.29 dCZq2=-1.37 dCMq2 =0.574 200 Bow Planes (NACA 0025) for port plane we have: 2.23) Ro = 0.5 m Xs • Ro Rn b= 0.9677 m C = .34925 m h = b - — b Xs =2.031 E=0.25C S3 := (h)C dCPe3 := C b • H - ^ ] - i i , 2 \ 11 Ro 1 - 3 dCP := dCPe3 xe dCPe3 = 1.146 2.24) „ 1.8481 xe3= xe3 =0.217 8.534 at 0.1 C location dCM = dCPxe3xe dCM =0.248 xe 2.25) R = Ro 1 (R+0.43-h) dCMe3 := dCM dCL3 := dCPe3 1 +-R dCMe3 =0.248 xe dCL3 =0.071 xe dCLe3 = dCL3 dCLe3 =0.071 2.26) xs3 := xe3 0.7-C xs3 =0.188 dCZq3 := dCPe3xs3 dCMq3 = dCPe3xs32 dCLq3 := dCLe3xs3 2.27) d C L p 3 : = ^ H ^ W R o + . 6 - h Ro 2 } 2 11 dCZq3 =0.215 dCMq3 =0.04 dCLq3 =0.013 dCLp3 =0.089 201 Total Hydrodynamic Coefficients CZw = CZe CZw =-0.687 CYv = CZw CYv =-0.687 CMw : = CMe CMw =0.905 CNv : = -CMw CNv =-0.905 CNr := CMq CNr =-0.158 CYr := CZq' CYr = 0.33 CMq =-0.158 CZq =0.33 CLp := 0 CLr = 0 CYv := C Y v - dCPel - dCPe2 CNv := C N v - (xel-dCPel) - (xe2-CZw = C Z w - dCPe2 - 2-dCPe3 Keel dCPel =4.353 xel =9.784-10~3 xsl =-0.143 Stern Planes (1 set) dCPe2=3.27 xe2 = -0.395 xs2=-0.419 Bow Planes (1 plane) dCPe3 = 1.146 xe3 =0.217 xs3 =0.188 Assume stabilizer and rudder as a second set of stern planes. Check signs of following: CMw := (CMw + xe2dCPe2) + 2xe3dCPe3 CYr := CYr+ xsl-dCPel + xs2-dCPe2 CZq := CZq + xs2-dCPe2 + xs3-2dCPe3 CMq := CMq + xe2-xs2-dCPe2 + xe3xs3-2-dCPe3 CNr := CNr + xelxs l -dCPel + xe2xs2- dCPe2 CLv := dCLel CLr= dCLelxs l C X u = C X f (body alone) . C X u p l := .01 (planes) CYv = -8.31 CNv =-2.153 CZw =-6.249 CMw =0.111 CYr = 2.321 CZq =-0.61 CMq =-0.792 CNr =-0.693 CLv = 0.376 CLr =-0.054 CXu=0.07 Aucher Yv=(rho/2)AVsCYv Yr=(rho/2)ALVsCYr Nv=(rho/2)ALVsCN\ Nr=(rho/2)ALA2Vs Feldman Yv=(rho/2)LA2VsYv Yr=(rho/2)LA3VsYr' Nv=(rho/2)LA2VsN\ Nr=(rho/2)LMVsNr' 202 . x C N v A Zds:=dCPe 2 A Zds = 0.035 N v - : = ^ ~ Nv_=-0.023 L 2 L A „ C Z w A Zdb^dCPeS -A Zdb =0.012 Z w - = 2 ~ Z w - = -O067 ,2 L L Mds=Zds-xe2 Mds =-0.014 -Mdb=Zdbxe3 Mdb =2.676-10 3 „ C Y r A CXu-A (SI +2-S2 + 2-S3) A.U •- +• .01* " L 2 L 2 C M w A -3 Mw_= Mw =1.2-10 Yr_ = Yr =0.025 L 2 CNr-A -3 Nr_= Nr =-7.471-10 L 2 CZq A -3 Zq_=—^— Zq =-6.581-10 L 2 CMq A -3 L 2 Mq_= ^— Mq_ =-8.542-10 C L v A . -3 Lv_:= Lv =4.056-10 L 2 CLr-A -4 Lr_= Lr =-5.782-10 L 2 Xu_= 1.163-10 3 203 B.2.6 Nondimensional Equivalent of Shupe's Results L l := 3 0 0 . 0 2 5 4 LI = 7 . 6 2 A l := 8 . 3 - 1 2 2 - . 0 2 5 4 2 A l = 0 . 7 7 1 D l = 3 . 2 5 - 1 2 - . 0 2 5 4 D l = 0 . 9 9 1 rho = 1 0 2 5 A l CZw := 2 . 5 1 Zw := CZw Zw = 0 . 0 3 3 L l 2 A l C Z q = 3 . 0 0 Zq=CZq- Z q = 0 . 0 4 L l 2 CMw := 8 . 4 3 Mw := C M w A 1 D 1 Mw = 0 . 0 1 5 L l 3 D l - 3 C M q = - 2 . 4 M q = C M q - A l - Mq = - 4 . 1 4 3 - 1 0 L l 3 A l C Y v = - 7 . 4 8 Yv = C Y v Yv = - 0 . 0 9 9 L l 2 CYr = 1 . 0 4 Y r : = C Y r — | Yr = 0 . 0 1 4 ILI2] CYp := 0 . 5 1 Yp := C Y p j ^ j Yp = 6 . 7 7 3 - 1 0 ~ 3 A l - D l -1 C N v = 4 . 7 5 N v = C N v Nv = 8 . 2 - 1 0 L l 3 A l - D l -T, C N r = - 0 . 6 Nr=CNr- Nr = " 1 . 0 3 6 - 1 0 L l 3 C N p = 0 . 0 5 N p : = C N p - ^ - ° ^ N p = 8 . 6 3 2 - 1 0 ~ 5 L l 3 A l - D l - 3 CKv := 4 . 5 Kv := C K v Kv = 7 . 7 6 9 - 1 0 J L l 3 C K r = 0 . 0 5 K r : = C K r - ^ - ~ ^ Kr = 8 . 6 3 2 - 1 0 ~ 5 L l 3 A l - D l -11 C K p = - 0 . 8 4 Kp=CKp- Kp = - 1 . 4 5 - 1 0 L l 3 Zwdot Zwdot ;= 3 0 7 1 4 . 5 9 4 Zwdot= Zwdot = 0 . 0 2 0.5 - rho-Ll 3 Mqdot = 1 0 8 2 0 - 1 4 . 5 9 4 ( 1 2 - . 0 2 5 4 ) 2 Mqdot ~ M q d 0 t Mqdot = 1 . 1 1 4 - 1 0 ~ 3 0.5 - rho-Ll 5 Yvdot= 4 1 5 . 8 1 4 . 5 9 4 Yvdot = Y v d 0 t Yvdot = 0 . 0 2 7 0.5 - rho-Ll 3 204 B.3 Actuator Hydrodynamic Analysis B.3.7 Aucher Bow Planes: gamma = 1 b = 0.968 bO := 0.5 b := b- 1 bO b=0.71 c := 0.35 ae: n = 0 c D 0 := 0.01 t: p := 1025 S := ( b ) - C L = 8.534 q 0 = — h=.175 a = 0 L 2 -2-7C -ae Za 1 - 3 - 1 -c \0.5 2.6 + cos(i2)' ae 2 + -\ cos(Q) 2 c Z a = - 2 - 0 4 1 S =0.248 C Z -Czk-a-2.1-0 ^ X = ^ DO + 0.7-ae-oc k p n = 1 +- 1.9 — - 0 . 5 - — e p 2-b 2-b k e p = 1.227 z 8 f : = c Z a - q ° k e p z 8 f = 8.539-10 X g f := 0.7-ae-qO X § f = 4.842-10 Location of Center of Pressure: Cpb= rb=Cpb-b + bO rb =0.801 3-Ji 205 Stern Planes: gamma = 1 b = 1.016 bO = 0.33 b C := 0.35 p := 1025 b:=b- 1 ae: £ 2 = 0 C D 0 = 0.0065 S=(b)-c L = 8.534 qO Zoc / \2' -2-7t -ae- 1 t\ 1-3- -W . \0.5 2.6 + cos(Q)- 2 ae cos(Q) C m =-2.386 C z = C 2a a - 2.1a C ^ = C QQ + 0.7-ae-a k e n := 1 + 1 .9 -A -0 .5- — e p 2-b 2-b k m : = l - 0.0846 — - 0.843 / b ° 'I1 K k =1.178 k ^ ^ 0.858 z 5 f = c Z a q ° k e p k T i t Z 8 f=-0.011 X 8 f := km-0.7-ae-q0 X g f =6.812-10 Location of Center of Pressure: 4 Cpb= rb=Cpb-b + b0 rb =0.716 3-7t 206 Rudder: gamma = 1 b := 0.889 bO = .262 b := b- 1 bO c = 0.35 b ae := -c £ 2 = 0 C D 0 = 0.0065 p := 1025 S:=(b)-c L = 8.534 qO -2-Jt -ae Za 1 - 3 2.6 cos(fi)- 2 + ae .2 \ 0.5 cos(Q) / =-2.227 C Z = c Z a a - 2 1 a C x = C QQ + 0.7 ae a k p n = 1 + 1.9 — - 0.5 •[ — e p 2-b \2-b k e p = 1.199 k m = l - 0.0846 ™ - 0.843 / b ° k ^ ^ 0.885 zSf-=cZa^-kep-ki\t Z 8 f = -9.218-10" X 5 f = k n t 0 . 7 a e q 0 X g f = 5.605-10 -3 Location of Center of Pressure: Cpb := rb := Cpb-b +• bO rb = 0.607 3-71 207 B.3.8 Whicker and Fehlner Bow Planes: gamma = 1 bl = 0.968 c := 0.35 ae = — c bO := 0.5 Q. •= 0 C D 0 = 0.0065 p := 1025 C j-)c := 0.1 + 1.6-gamma S = ( b ) c L = 8.534 C D c = 1 7 q0=— h:=.175 L 2 a := 0 1.8-71 ae L a \0.5 C L a =2.467 1.8 + cos(£2)- 4 + cos(£2) „ *-Dc I C T : = C I j a - a + a | a Dc ae L a a ae c L a a - ° - 8 3 8 C D " C D0 + 0.9 -JC ae neglecting H.O.T. c D a a L a (0.9-Tt ae) C Daa = 1 0 6 1 k p n = 1 + 1 .9 -A -0 .5 - (— e p 2-b \2-b k e p = 1.227 Z 8 f - k e p C L a q O k ^ Z 8 f = 0.013 z 8f8f = k e p c L a a -<l 0 k tZ 8f8f=4.302-10_: x 8 i 8 f : = c D a a 1 ° k e p X S l 8 f=4.441-10 _ Location of Center of Pressure: Cpb= rb=Cpbb + b0 rb =0.801 3-Tt 208 Stern Planes: gamma: = 1 b = 1.016 bO = 0.33 b = b-| 1 - — j c := 0.35 ae = - £ 2 = 0 C D n = 0.0065 p= 1025 S=(b)-c L:= 8.534 qO = — h=.175 L 2 C D c = 0.1 + 1.6-gamma C D c = 1.7 1.8-TC -ae 2 C L« = ~ 7—F 5 C L a =2-892 1.8 + cos(£2) 4 +• \ cos(£2) 4/ c D c i i r ~ c L = C L a a + a l a l C t •- — C , -0655 ae 2 2 c D = c D 0 + — neglecting H.O.T. C D a a = — C D a a = 1.139 0.9-rc-ae (0.9-Tt-ae) k e _:= 1 + 1.9— - 0.5-f—) k e D = 1.178 e p 2-b \2-b/ e p km = 1 - 0.0846--^ - 0.843-[™j k ^ j =0.858 z 6 f : = k e p - k T i t - C L a - q 0 Z 5 f = 0.013 z S f 8 f = k e p - k n t - c L a a - q ° z 5 f 8 f = 2.891-10~3 x 8 f 8 f = k n t c D a a < l 0 X 5 S f = 4.269-10"3 Location of Center of Pressure: 4 Cpb= rb=Cpbb + b0 rb =0.716 3-71 209 Rudder: / b02\ gamma =1 b = 0.889 bO = .262 b = b- 1 - — I b 2 / c 0.35 ae = - 0 = 0 C D 0 = 0.0065 c p=l025 S=(b)-c L = 8.534 qO = — h = .175 L 2 C D c := 0.1 + 1.6 gamma C D c = 17 „ 1.8-Jt-ae C T Q , := C , „ =2.697 1.8 + cos(Q)-4 + — a e cos(Q) 4/ C J^Q Q c l = c L o c a + a a Dc -D733 L 1 X 1 ae c L a a c L a a - ° / 3 j ae C r L L a C D = C D 0 + neglecting H.O.T. c D a a = - — C D A A = 1 . 1 0 9 0.9 - 7 1 ae (0 .9 - 7 1 ae) k e n = 1 + 1 . 9 - — - 0.5-(—) k = 1 . 1 9 9 e P 2-b \2-ty e p k ^ = 1 - 0 . 0 8 4 6 - — - 0 . 8 4 3 - ( ^ | k ^ = 0 . 8 8 5 z 8 f = V k e p C L a ' q 0 Z S f = 0.011 Z 5 f 5 f = k n t k e p C L a a q 0 Z 8 j 5 f =3.034-10"3 x 8 f 8 f = V c D a a q 0 X 8 f 8 f = 3.83-10_3 Location of Center of Pressure: 4 Cpb= rb=Cpbb + b0 rb =0.607 3-7t 210 B.3.9 Propulsion LEWIS: Propeller Engine Engine Forward Diameter S p e e d S p e e d Speed Parameter Parameter Parameter Theoretical (m) (rpm) (rps) (knots) a b c Thrust (N) 0.6508 0 0 0 789 0 0 0 0.6508 500 8.333333 2.503927 789 64.211649 -5107.537 1053.5524 0.6508 1000 16.66667 5.007855 789 128.4233 -20430.15 4214.2097 0.6508 1500 25 7.511782 789 192.63495 -45967.84 9481.9719 0.6508 2000 33.33333 10.01571 789 256.8466 -81720.6 16856.839 0.6508 2500 41.66667 12.51964 789 321.05824 -127688.4 26338.811 M E A S U R E D : Drag = 0.0045 x 1/2 rho V A 2 Use data from engine rpm. Use speed over ground to estimate drag. Non-dimensional : 0 . 0 0 4 2 / ( r h o x d M x 3 6 0 0 ) = 0.0822314 0.0069 / (rho x d M x 3600) = 0.1350945 -6.884 / (rho x d A 3 x 60 x V) = -7 .38E-05 211 Appendix C: Simulation Layout in Simulink C. 1 Simulink Models Figure C . l : Overall Layout Reference States Outln |4 Navigator Sum2 In Out Heading Error S e | e c t e d 0 to 360 to 0 u ^ u t s -Pi to Pi [xinit] Outln M IC Compass Correction A/D Over/Under 360 deg toOut|-LQG/LTR Integrator Compensator PI Defl Out SPM Out InU Anti-aliasing Butterworth i n k Noise on Vector u Record DATA at .02 Figure C.2: Navigator Headng FofloMng u. v, w, p, q r Selector Selector z,pN,theta MATLAB Function MATLAB Function Two Vectors ErfY Seteaor2 NeMVteypoint NexlWky MATLAB Function |Head Vector Desired Headng Must set reference x, y to zero - » | i n O u t [ — - n to PI to 0 to 360 D .Eh 0 to 360 to -PI to PI ToWortepacol cope i~ •es-Heaoina EmrScop . CoraaetJEror S c a » * « " !<***«-r»it J 212 Figure C.3: Integrator Augmented Compensator with Feedforward and Scaling -Output Scaling Integrated Internal Plane Saturation 0 -Constant Feed Forward Su™ 5 Integrator! Augmenting Input Plane Scaling Saturation Figure C.4: Disturbance Observer Augmented Compensator with Feedforward and Scaling o -Plane Saturation Lob (filter gain) Integrator •Kin (control gains) Steady State FeedFwd. Input Scaling aeledo Xhalob w To Vforkspace Figure C.5: Submarine Plant Model OH x* = Ax+Bu y = Cx+Du Actuator Dynamics > + Plane Saturation2 213 Figure C.6: Submarine Plant Model with Waves OH )(• = Ax+Bu y = Cx+Du Actuator Dynamics Plane Velocity . Wave Force xp transport etay Integrator Out Figure C.7: Wave Model O xi ©-swplanenoise Plane Velocity Mux swnoise2 Selector W W Mux2 Waves After wave states 1 to 12 After wave state 13-14 wavedplanes To Workspace2 To Workspacel To Workspace 214 Figure C.8: Submarine Plant Model with Tow Model K 3 OH Ax+Bu y = Cx+Du Actuator Dynamics Plane SaturatJon2 Tow Model 1 Discrete SISOx2 Memory Integrator Figure C.9: Static Tow Model Tension Force Components in Earth Frame (N) Location of application in Body Frame (m) [-1.263,0,1.265] tow point location (m) Inertial Forces to Body Forces To Workspace! - K D Tow Forces Body States 2 1 5 Figure CIO: Dynamic Tow Model Q - r l • i Selector 1 ] itn*D.*mn)*Bu(n) | " ~ | >°"""^  H! I Sal actor l M U X TOW m Gains pSCTpl j— Depressive Angle Beta MATLAB Function .263,0,1.2651 | •.—J I (J Inertial Forces to Body Forces tow point location (m) 216 Figure C . l l : Sensor Suite o -Measurement Noisy Signal Subsystem 11 Measurement Noisy Signal SubsystemlO Measurement Noisy Signal Subsystem^ Measurement Noisy Signal Subsystem Measurement Noisy Signal Subsystem 1 Demux Measurement Noisy Signal Subsystem2 Measurement Noisy Signal Subsystem3 Measurement Noisy Signal Subsystem^ Measurement Noisy Signal Subsystems Measurement Noisy Signal Subsystem6 Measurement Noisy Signal Subsystem^ Measurement Noisy Signal Subsystems 217 Figure C.12: Typical Noise On Measured Signals Electronic Noise Strandard Deviation o -Measurement Noisy Signal Figure C.13: Analog to Digital Modelling pi J1!-—KD Quantization Zero-Order Hold 218 Figure C.14: Anti-aliasing Filters o-(8-ap1)(s-ap2) Figure C.15: Conversion of Error in Heading to Within -Pi to Pi. u(12) 0—Hi lonstantS Si :onstant4 ^1 Relational Switchl Operatorl 0 v :onstant6 Relational Switch2 Operator2 o-219 Figure C.16: Conversion from Integrated Rate Gyro Angle to Heading Between 0 and 360 Degrees MATLAB Function Constant!)' W\ Relational Operator! L(!2) o - -KD Figure C.17: Reference State Generation • -KD p»i 220 Figure C.18: Logging of Plant Data To Workspacel H \ Ik u IZJ * From To Workspace2 X W Over/Under 360 To Workspace E2 XY Graph Scope C.2 Matlab Functions Loadparam.m L= 8 . 5 3 4 ; %length (m) m=4390; b=4605 ; %mass; bouyancy (kg) rho=1025 ; %density of water (kg/m~3) g = 9 . 8 1 ; ^ g r a v i t y m/sA2 mx=0; my=0; mz=0; % p o s i t i o n of center of mass bx=.086; by=0; b z = - . 1 3 7 ; % p o s i t i o n of center of Bouy-ancy Jxx=1315 ; Jyy=5900 ; J z z= 5 0 5 7 ; %moment of i n e r t i a kg/m~2 Jxy=0; Jxz=0; Jyz=0; Jzx=0; Jzy=0; Jyx=0; W=m*g; B=b*g; D0=0.5*rho; D1=D0*L; D2=D1*L; D3=D2*L; D4=D3*L; D5=D4*L; D00=0.7159; %prop d i a D04=D0Cr4; A = 3 . 1 4 1 5 9 2 6 5 4 * 0 . 5 ^ 2 ; % x - s e c t i o n a l area X u _ = - 0 . 0 0 0 9 4 9 ; Xv_=0; Xw_=0; Xp_=0; Xq_=0; Xr _=0 ; Yu_=0; Yv_=-0.0408; Yw_=0; Yp _ = 0 . 0 0 0 5 4 7; Yq_=0; Y r _ = 0 . 0 0 1 3 9 ; Zu_=0; Zv_=0; Zw_= - .0226 ; Zp_=0; Zq_=-.0000510; Zr_=0 ; Ku_=0; K v _ = 0 . 0 0 0 5 4 7 ; Kw_=0; K p _ = - 0 . 0 0 0 1 0 5 ; Kq_=0; K r _ = - 0 . 0 0 0 0 0 3 7 0 ; Mu_=0; Mv_=0; Mw_= -0 .0000510; Mp_=0; Mq _= -0 .00175; Mr_=0; Nu_=0; Nv _ = 0 . 0 0 1 3 9 ; Nw_=0; N p _ = - 0 . 0 0 0 0 0 3 7 0 ; Nq_=0; N r _ = - 0 . 0 0 1 8 3 ; Xuu= - 0 . 0 0 3 1 5 ; Xuv=0; Xuw=0; Xup=0; Xuq=0; Xur=0; Yuu=0; Yuv= - 0 . 0 9 9 ; Yuw=0; Yup=0.00677; Yuq=0; Y u r= 0 . 0 2 5 0 ; Zuu=0; Zuv=0; Zuw=- .0670; Zup=0; Zuq=-.00658; Zur=0; 221 Kuu=0; Kuv=.00406; Kuw=0; Kup=-0.00145; Kuq=0; Kur=-.000578; Muu=0; Muv=0; Muw=0.00120; Mup=0; Muq=-0.00687; Mur=0; Nuu=0; Nuv=-.005; Nuw=0; Nup=0.000086; Nuq=0; Nur=-.00747; Xdfpdfp=-0.00362; Xdfsdfs=-0.00362; Xdapdap=-0.00337; Xdasdas=-0.00337; Xdrdr=-0.00342; Xdn=0.0000733; Xdndn=0.1351; Ydr=0.00917; Ydrdr=-0.00215; Zdfs=-0.010; Zdfp=-0.010; Zdfsdfs=0.00351; Zdfpdfp=0.00351; Zdap=-0.010; Zdas=-0.010; Zdapdap=0.00198; Zdasdas=0.00198; xfp=1.865 ;xfs=1.865 ;xap=-3.369; xas=xap; xr=xap ; yfp=-0.699; yfs=-yfp;yap=-0.621 ;yas=-yap ;yr=0; zfp=-0.0990-.051-.137;zfs=+0.0990+.051-.137 ;zap=-.0016-.051-.137 ;zas=+.0016+.051-.137; zr=0.528+.103-0.051-.137; %4/3pi x h +radius+ dis t of root to cb Kdfp=yfp*Zdfp; Kdfs=yfs*Zdfs; Kdap=yap*Zdap; Kdas=yas*Zdas; Kdr=-zr*Ydr; Kdrdr=-z r * Ydrdr; Kdfpdfp=yfp*Zdfpdfp; Kdfsdfs=yfs*Zdfsdfs; Kdapdap=yap*Zdapdap; Kdas-das=yas*Zdasdas; Mdfp=-xfp*Zdfp; Mdfs=-xfs*Zdfs; Mdap=-xap*Zdap; Mdas=-xas*Zdas; Mdfpdfp=-xfp*Zdfpdfp+zfp*Xdfpdfp; Mdfsdfs=-xfs*Zdfsdfs+zfs*Xdfsdfs; Mdapdap=-xap*Zdapdap+zap*Xdapdap; Mdasdas=-xas*Zdasdas+zas*Xdasdas; Mdrdr=z r * Xdrdr; Ndr= xr*Ydr; Ndfpdfp=-yfp*Xdfpdfp; Ndfsdfs=-yfs*Xdfsdfs; Ndapdap=-yap*Xdapdap; Ndasdas=-yas*Xdasdas; Ndrdr= xr*Ydrdr; 2 2 2 Massmatm % Finds the Mass matrix f or the submarine % needs x l i n save temp temp MM=SPMA(xlin, 1) ; % uO = -pinv(B)*(A*[5 0 0 0 0 0 0 0 2 . 5 0 0 0] ' + [inv(MM)*[0 0 g*(m-b) g*(my* by*b) g*(bx*b-mx*m) 0]';zeros(6,1)]); % i n t = i n p u t ( ' I t e r a t e f o r uO (y,n): ' , ' s ' ) ; %sht=input('Show i t e r a t i o n (y,n): ' , ' s ' ) ; i f int=='y', ul=[.4 .4 0.2 0.2 0 20]'; u0=[0 0 0 0 0 0]'; x0_=-[inv(MM), zeros(6,6); zeros(6,6), eye(6,6)]*SPMA(xlin,2); while abs(sum(u0(l:4)-ul(l:4)))>0.0000001, u0=ul; xl_=[inv(MM), zeros(6,6); zeros(6,6), eye(6,6)]*SPMB(xlin,uO); [A,B]=linearize(xlin,uO); delu=pinv([inv(MM), zeros(6,6); zeros(6,6), eye(6,6)]*B)*(x0_-xl_); delu(l:5)=delu(l:5)/20; i f sht=='y', ul=u0+delu else ul=u0+delu; end; end; else u0=[ -0.0552 -0.0541 -0 . 0244 -0.0256 0.0000 10.8292]; end; uO invMM= inv(MM); save temptemp invMM MM uO -append cl e a r load temptemp 223 SPMA.m function [sys] = x d o t ( x i , f l a g ) ; % % function f or f i n d i n g e i t h e r the body forces, % time d e r i v a t i v e s or mass matrix % w r i t t e n by Adrian F i e l d , 1999 • % % load a l l geometric v a r i a b l e s % u = x i ( l ) ; v=xi(2); w=xi(3); p=xi(4); q=xi(5); r=xi(6); x=xi(7); y=xi(8); z=xi(9); phi=xi(10); theta=xi(11); psi=xi(12) loadparam % transformation matrix J=Jmat(xi(10:12)); J1=J(1:3,1:3) ; J2=J(4:6,4:6) ; i f or((flag==l),(flag==3)), 0 m*mz -m*my; 0 m*mx; m*mx 0; Jxy - Jxz; Jyy -Jyz; J z z ; ] ; mi=[ m 0 0 m 0 0 0 m 0 -m*mz m*my m*mz 0 -m*mx -m*my m*mx 0 -Jxz 0 -m*mz m*my Jxx -Jxy -Jyz ma=- [ Xu_ Xv_ Xw_ Yu_ Yv_ Yw_ Yp_ Yq_ Yr_; Zu_ Zv_ Zw_ Zp_ zq_ Zr_; Ku_ Kv_ Kw_ Kp_ Kq_ Kr_; Mu_ Mv_ Mw_ Mp_ Mq_ Mr_; Nu_ Nv_ Nw_ Np_ Nq_ Nr_; ma (1:3, 1:3) =ma(1:3, 1:3) *D3; ma(1:3, 4:6) =ma(1:3, 4:6) *D4; ma(4:6, 1:3) =ma(4:6, 1:3) *D4; ma (4:6, 4:6) =ma(4:6, 4:6) *D5; ] ; MM=mi+ma; %sum of both mass matrices end; i f flag==l, sys=MM; end; i f or((flag==2), (flag==3) ) , X= -m*(-v*r+w*q-mx*(q^2+r^2)+my*(p*q)+mz* (p*r) ) +... D4*(-Yp_*r*p-Yr_*r^2+Zq_*q^2) + D3*(-Yv_*v*r+Zw_*w*q) +.. D2*(Xuu*u"2)-(W-B)*sin(theta); 224 Y= -m* (-w*p+u*r+mxMp*q)-my* (p^2+r^2)+mz* (q*r) ) +... D4*(-Zq_*p*q) + D3*(Xu_*u*r+Yup*u*p+Yur*u*r-Zw_*w*p) +... D2*(Yuv*u*v)+(W-B)*cos(theta)*sin(phi); Z= -m*(-u*q+v*p+mx*(r*p)+my*(r*q)-mz*(p~2+qA2)) +... D3*(Xu_*u*q+Yv_*v*p+Yp_*p^2+Yr_*r*p+Zuq*u*q) +... D2*(Zuw*u*w) + (W-B)*cos(theta)*cos(phi); K= -((Jzz-Jyy)*q*r-(p*q)*Jxz+(r A2-q"2)*Jyz+(p*r)*Jxy) +... D5*(-Mq_*q*r+Nr_*q*r) +... D4*(Yr_*v*q-Zq_*w*r-Yp_*(-w*p)+Kup*u*p+Kur*u*r)+... D3*(Zw_*v*w-Yv_*v*w+Kuw*u*w)+(my*W-by*B)*cos(theta)*cos(phi)-(mz*W-bz*B)*cos(theta)*sin(phi); M= -((Jxx-Jzz)*r*p-(q*r)*Jxy+(q*2-r~2)*Jxz+(q*p)*Jyz) +... D5*(Kp_*p*r-Kr_*(q~2-r^2)-Nr_*p*r) +... D4*(Yp_*v*r-Yr_*v*p+Zq_*(-u*q)+Muq*u*q)+... D3*(Xu_*u*w-Zw_*u*w+Muw*u*w) -(mz*W-bz*B)*sin(theta)-... (mx*W-bx*B)*cos(theta)*cos(phi); N= - ( (Jyy-Jxx) *p*q- (r*p) *Jyz+ (q*2-p A2) *Jxy+ (r*q) *Jzx) + . . . D5*(-Kp_*p*q+Mq_*p*q+Kr_*(-q*r))+... D4*(Yr_*(u*r)+Yp_*(u*p-v*q)+Zq_*w*p+Nup*u*p+Nur*u*r)+... D3*(-Xu_*u*v+Yv_*u*v+Nuv*u*v)+(mx*W-bx*B)*cos(theta)*sin(phi) (my*W-by*B)*sin(theta); x_= J l ( l , l : 3 ) * [ u v w]'; y_= J l ( 2 , l : 3 ) * [ u v w]'; z_= J l ( 3 , l : 3 ) * [ u v w]'; phi_= J 2 ( l , l : 3 ) * [ p q r ] ' ; theta_= J2(2,1:3)*[p q r ] 1 ; psi_= J2(3,1:3)*[p q r ] ' ; F=[X Y Z K M N x_ y_ z_ p h i _ theta_ p s i _ ] ' ; end; i f flag==2, sys=F; end; i f flag==3, xi_=[inv(MM), zeros(6,6); zeros(6,6), eye(6,6)]*F; sys=xi_; end; % end xdot 225 SPMB.m function [sys] = x d o t ( x i , u i ) ; % % function for f i n d i n g the plane forces, % % w r i t t e n by Adrian F i e l d , 1999 % % load a l l geometric v a r i a b l e s % u = x i ( l ) ; v=xi(2); w=xi(3); p=xi(4); q=xi(5); r=xi(6); x=xi(7); y=xi(8); z=xi(9); phi=xi(10); theta=xi(11); psi=xi(12); loadparam dfp=ui(1)-g*xfp/u; dfs=ui(2)-q*xfs/u; dap=ui(3)-q*xap/u; das=ui(4)-q*xas/u; dr=ui(5)-r*xr/u; dn=ui(6); X= D2*u A2*(Xdfpdfp*dfp A2+Xdfsdfs*dfs^2+Xdapdap*dap A2 +... Xdasdas*das^2+Xdrdr*dr A2)+ rho*D04*(Xdn*u/DOO*dn+Xdndn*dn*abs(dn)); Y= D2*u^2*(Ydr*dr+Ydrdr*dr*abs(dr)); Z= D2*u A2*(Zdfp*dfp+Zdfpdfp*dfp*abs(dfp)+Zdfs*dfs+Zdfsdfs*dfs*abs(dfs)+... Zdap*dap+Zdapdap*dap*abs(dap)+Zdas*das+Zdasdas*das*abs(das)); K= D3*u^2*(Kdfp*dfp+Kdfpdfp*dfp*abs(dfp)+Kdfs*dfs+Kdfsdfs*dfs*abs(dfs)+... Kdap*dap+Kdapdap*dap*abs(dap)+Kdas*das+Kdas-das*das*abs(das)+Kdr*dr+Kdrdr*dr*abs(dr)); M= D3*u A2*(Mdfp*dfp+Mdfpdfp*dfp*abs(dfp)+Mdfs*dfs+Mdfsdfs*dfs*abs(dfs)+... Mdap*dap+Mdapdap*dap*abs(dap)+Mdas*das+Mdasdas*das*abs(das)+Mdrdr*dr~2); N= D3*u~2*(Ndr*dr+Ndrdr*dr*abs(dr)); x_= 0; Y_= 0; Z _ = 0; phi_= 0; theta_= 0; psi_= 0; F=[X Y Z K M N x_ y_ z_ p h i _ theta_ p s i _ ] ' ; sys= F; % end xdot 226 swayload.m function [sys,xO,str,ts] = swayload(t,x,u,flag,route); %function [sys,xO,str,ts] = swayload(t,x,u,flag,route); % simulink waypoint loader % no state % loads matrix waypts from workspace and checks against present p o s i t i o n % u contains x, y % y outputs desired x, y, xr, yr of next and l a s t waypoint switch f l a g , %%%%%%%%%%%%%%%%%% % I n i t i a l i z a t i o n % %%%%%%%%%%%%%%%%%% case 0, [sy s , x O , s t r , t s ] = m d l I n i t i a l i z e S i z e s ( t , x, u, route) ; %%%%%%%%%% % Update % %%%%%%%%%% case 2, sys=mdlUpdate(t,x,u,route); %%%%%%%%%%% % Outputs % %%%%%%%%%%% case 3, sys=mdlOutputs(t,x,u,route); %%%%%%%%%%%%%%%%%%%% % Unexpected flags % %%%%%%%%%%%%%%%%%%%% case {1, 4, 5, 9} sys=[]; otherwise error(['Unhandled f l a g = ',num2str(flag)]); end % end sfuntmpl % %=============================================================== % m d l l n i t i a l i z e S i z e s % Return the s i z e s , i n i t i a l conditions, and sample times for the S-function. %================================================================= % function [sys,xO,str,ts]=mdlInitializeSizes(t,x,u,route); % % c a l l simsizes f o r a siz e s structure, f i l l i t i n and convert i t to a 227 % si z e s array. % % Note that i n t h i s example, the values are hard coded. This i s not a % recommended p r a c t i c e as the c h a r a c t e r i s t i c s of the block are t y p i c a l l y % defined by the S-function parameters. % . sizes = simsizes; s i z e s NumContStates = 0; siz e s NumDiscStates = 5; sizes NumOutputs = 4; sizes Numlnputs = 2; siz e s DirFeedthrough = 4; siz e s NumSampleTimes = 1; sys = si m s i z e s ( s i z e s ) ; at l e a s t one sample time i s needed % % i n i t i a l i z e the i n i t i a l conditions % l a s t = f s c a n f ( r o u t e , ' % f ' , [ l , 2 ] ) ; xO = [fscanf(route, ' % f , [1,2]), l a s t , 1]; s t r = [ ];' ts = [-1 0 ] ; % end m d l l n i t i a l i z e S i z e s function sys=mdlUpdate(t,x,u,route); rad=16; % waypoint target s i z e e=x(l:2)+u; % error i n p o s i t i o n (e(l)~2+e(2)"2)"0.5; i f (e(l) A2+e(2) A2)"0.5 > rad sys = x; end; i f (6(1)^2+6(2)^2)^0.5 <= rad refloc=fscanf(route,'% f',[1,2]); sys = [ r e f l o c , x ( l : 2 ) ' , x ( 5 ) + l ] ; x(5) end; % end mdlOutputs %==================================== % mdlOutputs % Return the block outputs. %================= = = = = = = = = = = = = = = = = = = = % function sys=mdlOutputs(t,x,u,route); sys=x(1:4); 228 scdistm function [sys,xO,str,ts] = s c d t ( t , x , u , f l a g ) ; %function [sys,xO,str,ts] = s c d t ( t , x , u , f l a g ) ; % simulink steady state disturbance vector % no state % u contains 12 states p l u s : % X, Y, Z, and rx, ry, rz (dim 6 x 1 ) switch f l a g , %%%%%%%%%%%%%%%%%% % I n i t i a l i z a t i o n % %%%%%%%%%%%%%%%%%% case 0, [sys, x O , s t r , t s ] = m d l I n i t i a l i z e S i z e s ; %%%%%%%%%%% % Outputs % %%%%%%%%%%% case 3, sys=mdlOutputs(t,x,u); %%%%%%%%%%%%%%%%%%%% % Unexpected flags % %%%%%%%%%%%%%%%%%%%% case {1, 2, 4, 9} sys=[]; otherwise error(['Unhandled f l a g = ',num2str(flag)]); end % end sfuntmpl % %==========================================^^ % m d l l n i t i a l i z e S i z e s % Return the s i z e s , i n i t i a l conditions, and sample times f o r the S-function. %========================================^  % function [ s y s , x O , s t r , t s ] = m d l I n i t i a l i z e S i z e s ; % % c a l l simsizes f o r a siz e s structure, f i l l i t i n and convert i t to a % si z e s array. % % Note that i n t h i s example, the values are hard coded. This i s not a % recommended p r a c t i c e as the c h a r a c t e r i s t i c s of the block are t y p i c a l l y % defined by the S-function parameters. % siz e s = simsizes; 229 sizes.NumContStates sizes.NumDiscStates sizes.NumOutputs sizes.Numlnputs 0; 0; 12 18 sizes.DirFeedthrough = 12; sizes.NumSampleTimes =1; % at l e a s t one sample time i s needed sys = s i m s i z e s ( s i z e s ) ; % % i n i t i a l i z e the i n i t i a l conditions % xO = [ ] ; s t r = [] ; ts = [ - 1 0 ] ; % end m d l l n i t i a l i z e S i z e s %===============================================================: % mdlOutputs % Return the block outputs. %======================================================== % function sys=mdlOutputs(t,temp,u); xi=u(7:18); Fi=u(l:3); ri=u(4:6); u = x i ( l ) ; v=xi(2); w=xi(3); p=xi(4); q=xi(5); r=xi(6); x=xi(7); y=xi(8); z=xi(9); phi=xi(10); theta=xi(11); psi=0; % % transformation matrix J=Jmat([phi,theta,psi]); %%%%%%%%%%%%%%%%%%%%%%%%%%%%% Fb=(in v ( J ( l : 3 , 1 : 3 ) ) * F i ) ; Mb=cross(ri,Fb); sys = [[Fb; Mb]; zeros(6,1)]; % end mdlOutputs 230 swplanenoise.m function [sys,xO,str,ts]=swnoise(t, x, u,flag,phiwd,wspec,wseed); % function [sys,xO,str,ts]=swnoise(t,x,u,flag,phiwd,wspec,Cd,Cm,wseed); % produces wave v e l o c i t i e s at plane locations % from wave amplitudes and freqencies found by brett.m % c a l c u l a t e s a d d i t i v e v e l , acc, height for each x, y, z l o c a t i o n % along the h u l l length and adds them % can change wave d i r e c t i o n % forces based on encounter freqency % input u = d/dt(u,v,w,p,q,r),d/dt(x,y,z,phi,theta,psi),u,v,w,p,q,r, % (x,y,z,phi,theta,psi) % outputs d/dt(u,v,w,p,q,r) disturbance i n body frame % states are Forces and Moments i n Body frame switch f l a g , %%%%%%%%%%%%%%%%%% % I n i t i a l i z a t i o n % %%%%%%%%%%%%%%%%%% case 0, [sys, x O , s t r , t s ] = m d l I n i t i a l i z e S i z e s ; %%%%%%%%%%% % Outputs % %%%%%%%%%%% case 3, sys=mdlOutputs(t,x,u,phiwd,wspec,wseed); %%%%%%%%%%%%%%%%%%% % Unhandled flags % %%%%%%%%%%%%%%%%%%% case {1', 2, 3, 4, 9 }, sys = [ ]; %%%%%%%%%%%%%%%%%%%% % Unexpected flags % %%%%%%%%%%%%%%%%%%%% otherwise error(['Unhandled f l a g = ',num2str(flag)]); end % end csfunc % %================================================================= % m d l l n i t i a l i z e S i z e s % Return the s i z e s , i n i t i a l conditions, and sample times for the S-function. %=========================================================================== % function [ s y s , x O , s t r , t s ] = m d l I n i t i a l i z e S i z e s ; s i z e s = simsizes; 231 sizes.NumContStates = 0; sizes.NumDiscStates = 0; sizes.NumOutputs = 6; sizes.Numlnputs = 12; sizes.DirFeedthrough = 6; sizes.NumSampleTimes = 1; sys = s i m s i z e s ( s i z e s ) ; xO = [ ] ; s t r = [ ] ; t s = [ .01 0] ; % end m d l l n i t i a l i z e S i z e s %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%% function sys=mdlOutputs(t,x,u,phiwd,wspec,wseed); pi=3.14159; g=9.81; rho=1025; % density of water xi=u(l:12) a=wspec(:,1)'; w2=wspec(:,2) ' ; phiwr=phiwd/18 0 * p i ; % d i r e c t i o n of wave t r a v e l 0 to 2*pi rad. (z down) xcg=4.173; % l o c a t i o n of eg xL=8.534; % length of h u l l ppos=[1.95 -0.4244*0.9677 -0.0979; 1.95 0.4244*0.9677 -0.0979; -2 -0.4244*0.9677 -.2029; -2 0.4244*0.9677 .2029; -2 0 0.4244*.889-0.188]; nx=5; % number of h u l l points for i n t e g r a t i o n rand('state',wseed); % sets the seed of the random generator offset=rand(length(w2),1)*2*pi; unit=[cos(phiwr) sin(phiwr) 0]; % uni t vector i n d i r . of phiw i n i n e r t i a l frame T=l./(w2/(2*pi)); % period of waves L= (g*T. *-2) / (2*pi) ; % wavelength (based on deep water) % Find p o s i t i o n r e l a t i v e to waves % Find phase angle % Find wave height for n=l:nx; x(n)=ppos(n,1); % body l o c a t i o n of h u l l point (nose to stern) xn(n)=x(n)*cos(xi(12))+xi(7); % i n e r t i a l x l o c . of h u l l point yn(n)=x(n)*sin(xi(12))+xi(8)+ppos(n,2); % i n e r t i a l y l o c . of h u l l point zn(n)=-xi(9)+x(n)*sin(xi(11))-ppos(n,3); % i n e r t i a l z l o c . of h u l l point xw(n)=dot([xn(n) yn(n) 0 ] , u n i t ) ; % (dot) wave loc of h u l l point theta=mod((-2*pi*xw(n)./L+w2*t+offset'),2*pi); % phase angle 232 h(n)=sum(a.*cos(theta)),-i s pos. down. % wave height at point neg. as submarine % wave v e l and acc i n i n e r t i a l frame ww(n)=-sum(pi*2*a./T.*exp(2*pi*(zn(n)./L)).*sin(theta)); vw(n)=sum(pi*2*a./T.*exp(2*pi*(zn(n)./L)).*cos(theta))*sin(phiwr); uw(n)=sum(pi*2*a./T.*exp(2*pi*(zn(n)./L)).*cos(theta))*cos(phiwr); % wave v e l and acc i n body frame J=Jmat(xi(10:12)); bodyvel(n,:)=(inv(J(l:3,1:3))*[uw(n) vw(n) ww(n) ] ' ) ' ; end; delta(l:4)=bodyvel(l:4,3); delta(5)=bodyvel(5,2); s y s = [ d e l t a 1 . / x i ( 1 ) ; 0 ] ; % end update 233 swnoise2.m function [sys,xO,str,ts]=swnoise(t,x,u,flag,phiwd,wspec,Cd,Cm,wseed); % function [sys,xO,str,ts]=swnoise(t,x,u,flag,phiwd,wspec,Cd,Cm,wseed); % produces wave forces on a c i r c u l a r h u l l shape % from wave amplitudes and fregencies found by brett.m % c a l c u l a t e s a d d i t i v e v e l , acc, height for each x l o c a t i o n % along the h u l l length and adds them % can change wave d i r e c t i o n % forces based on encounter freqency % input u = d/dt(u,v,w,p,q,r),d/dt(x,y,z,phi,theta,psi),u,v,w,p,q,r, % (x,y,z,phi,theta,psi) % outputs d/dt(u,v,w,p,q,r) disturbance i n body frame % states are Forces and Moments i n Body frame switch f l a g , %%%%%%%%%%%%%%%%%% % I n i t i a l i z a t i o n % %%%%%%%%%%%%%%%%%% case 0, [sys,xO,str,ts]=mdlInitializeSizes ; %%%%%%%%%% % Update % %%%%%%%%%% case 2, sys=mdlUpdate(t,x,u,phiwd,wspec,Cd,Cm,wseed); %%%%%%%%%%% % Outputs % %%%%%%%%%%% case 3, sys=mdlOutputs(t,x,u); %%%%%%%%%%%%%%%%%%% % Unhandled flags % %%%%%%%%%%%%%%%%%%% case {1, 2, 3, 4, 9 }, sys = [ ] ; %%%%%%%%%%%%%%%%%%%% % Unexpected flags % %%%%%%%%%%%%%%%%%%%% otherwise error(['Unhandled f l a g = ',num2str(flag)]) ; end % end csfunc % %====================================================================== % m d l l n i t i a l i z e S i z e s 234 % Return the s i z e s , i n i t i a l conditions, and sample times f o r the S-function. % = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = % function [sys,xO,str,ts]=mdl!nitializeSizes; s i z e s = simsizes; sizes NumContStates = 0; sizes NumDiscStates = 8; sizes NumOutputs = 14; sizes Numlnputs = 24; si z e s DirFeedthrough = 8; siz e s NumSampleTimes = 1; sys = s i m s i z e s ( s i z e s ) ; xO = [ 0 0 0 0 0 0 0 0 ] ; s t r = [ ] ; ts = [.01 0] ; % end m d l l n i t i a l i z e S i z e s %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%% function sys=mdlUpdate(t,x,u,phiwd,wspec,Cd,Cm,wseed); pi=3.14159; g=9.81; rho=1025; % density of water xi=u(l:12); xp=u(13:24); a=wspec(:,1)'; w2=wspec(:,2)'; phiwr=phiwd/18 0 * p i ; xcg=4.173; % xL=8.534; % D=0.5; % nx=16; % rand('state',wseed); % d i r e c t i o n of wave t r a v e l 0 to 2*pi rad. (z down) l o c a t i o n of eg length of h u l l diameter of h u l l number of h u l l points for i n t e g r a t i o n % sets the seed of the random generator offset=rand(length(w2),1)*2*pi; unit=[cos(phiwr) sin(phiwr) 0]; % u n i t vector i n d i r . of phiw i n i n e r t i a l frame T=l./(w2/(2*pi)); % period of waves L=(g*T."2)/(2*pi); % wavelength (based on deep water) % Find p o s i t i o n r e l a t i v e to waves % Find phase angle % Find wave height f o r n=l:nx; x(n)=-(xL/(nx-1)*n)+xcg; % body l o c a t i o n of h u l l point (nose to stern) xn(n)=x(n)*cos(xi(12))+xi(7); % i n e r t i a l x l o c . of h u l l point yn(n)=x(n)*sin(xi(12))+xi(8); % i n e r t i a l y l o c . of h u l l point zn(n)=-xi(9)+x(n)*sin(xi(11)); % i n e r t i a l z l o c . of h u l l point xw(n)=dot([xn(n) yn(n) 0 ] , u n i t ) ; % (dot) wave loc of h u l l point 235 theta=mod((-2*pi*xw(n)./L+w2*t+offset'),2*pi); % phase angle h(n)=sum(a.*cos(theta)); % wave height at point neg. as submarine i s pos. down. % wave v e l and acc i n i n e r t i a l frame ww(n)=-sum(pi*2*a./T.*exp(2 *pi*(zn(n) . / L ) ) . * s i n ( t h e t a ) ) ; vw(n)=sum(pi*2*a./T.*exp(2*pi*(zn(n)./L)).*cos(theta))*sin(phiwr); uw(n)=sum(pi*2*a./T.*exp(2*pi*(zn(n)./!>)) . *cos(theta))*cos(phiwr); az(n)=-sum(-4*a.*(pi./T)."2.*exp(2*pi*(-xi(9)./L)).*cos(theta)); ay(n)=sum(4*a.*(pi./T). ~2 . * e x p ( 2 * p i * ( - x i ( 9 ) . / L ) ) . * s i n ( t h e t a ) ) * s i n ( p h i w r ) ; ax(n)=sum(4*a.*(pi./T).~2.*exp(2*pi*(-xi(9)./L)).*sin(theta))*cos(phiwr); % wave v e l and acc i n body frame J=Jmat(xi(10:12)); bodyvel=inv(J(1:3,1:3))*[uw(n) vw(n) ww(n) ]'; bodyacc=inv(J(1:3,1:3))*[ax(n) ay(n) az(n ) ] ' ; % h u l l v e l and acc i n body frame wh(n)=xi(3)-x(n)*xi(5); vh(n)=xi(2)-x(n)*xi(6);\ azh(n)=xp(3)-xi(l)*xi(5)+xi(2)*xi(4)+x(n)*(-xp(5)); ayh(n)=xp(2)-xi(3)*xi(4)+xi(1)*xi(6)+x(n)*(-xp(4)); wh(n)=0; vh(n)=0; azh(n)=0; ayh(n)=0; % forces i n body frame Z(n)=Cd/2*rho*D*(bodyvel(3)-wh(n))*abs(bodyvel(3)-wh(n))+Cm*(pi*D^2/ 4)*rho*(bodyacc(3)-azh(n)); Y(n)=Cd/2*rho*D*(bodyvel(2)-vh(n))*abs(bodyvel(2)-vh(n))+Cm*(pi*D"2/ 4)*rho*(bodyacc(2)-ayh(n)); M(n)=-Z(n).*x(n); N(n)=-Y(n).*x(n); end ; Z=sum(Z)*xL/nx; Y=sum(Y)*xL/nx; M=sum(M)*xL/nx; N=sum(N)*xL/nx; X=0; K=0; sys=[X Y Z K M N h(8)theta(1)]; % end update % = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = % mdlOutputs % Return the block outputs. % function sys=mdlOutputs(t,x,u); 236 sys=[x(l:6); zeros(6,1);x(7:8)]; % end mdlOutputs 237 jmat.m f u n c t i o n [J] = Jmat(x); % [J]= J ( x ) ; % Procedure f o r f i n d i n g J ( t ) % W r i t t e n by A d r i a n I. F i e l d , 1998 % x i i s the s t a t e v a r i a b l e p h i , t h e t a , p s i % % % t r a n s f o r m a t i o n m a t r i x p h i = x ( l ) ; theta=x(2); p s i = x ( 3 ) ; T l = [ l 0 0 0 co s ( p h i ) s i n ( p h i ) 0 - s i n ( p h i ) c o s ( p h i ) ] ; T2=[cos(theta) 0 - s i n ( t h e t a ) 0 1 0 s i n ( t h e t a ) 0 c o s ( t h e t a ) ] ; T3=[cos(psi) s i n ( p s i ) 0 - s i n ( p s i ) c o s ( p s i ) 0 0 0 1]; J1=T3'*T2'*T1'; J2=[ 1 s i n ( p h i ) * t a n ( t h e t a ) c o s ( p h i ) * t a n ( t h e t a ) 0 c o s ( p h i ) - s i n ( p h i ) 0 s i n ( p h i ) / c o s ( t h e t a ) c o s ( p h i ) / c o s ( t h e t a ) ] ; % J=[J1 z e r o s ( 3 , 3 ) ; zeros(3,3) J 2 ] ; 238 C.3 Discrete Dynamic Tow Model Sampled at 0.2 seconds. Tension in lbs. as output, speed squared as input: « A s s l As s i = 0 0 -0.3334 1.0000 0 0.4151 0 1.0000 0.8959 fl B s s l B s s l = -34.4609 21.9586 14.5628 a C s s l C s s l = 0 fi Dssl Dssl = 0 Tow off angle (1/2 difference between DOLPHIN and towfish) as output, turning rate 'r' as input a Ass2 Ass2 = 0 0 0.0795 1.0000 0 0.0523 0 1.0000 0.8649 fl Bss2 Bss2 = -0.0132 0.0061 0.1394 fi Css2 239 Css2 = 0 a Dss2 Dss2 = 0 Appendix D: Control Design Data D.1 Unaugmented Control Design Information D.1.1 Linear State Space Matrices A10 = C o l u m n s 1 t h r o u g h 7 BIO C I O - 0 3006 0 0 0 0 0 0 - 1 . 7 5 3 0 0 0 2678 0 0 4902 0 0 - 1 3060 0 1 0599 0 0 - 0 . 6 5 0 5 0 - 6 2047 0 -2 2722 0 0 0 9431 0 -2 3951 0 0 - 1 . 9 4 2 5 0 0 2498 0 -2 0442 0 0 1 0000 0 0 0 0 0 0 1 0000 0 0 0 0 0 0 1 0000 0 0 0 0 0 0 1 0000 ' o l u m n s 8 t h r o u g h 10 0 0 . 4 4 9 5 0 - 0 2944 0 0 0 0 . 0 0 1 6 0 - 1 7644 0 0 0 - 0 . 1 3 3 1 0 - 0 1020 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 0 0 1 6793 3 8337 - 3 . 8 3 3 7 1 7030 - 1 7030 -2 0594 0 -2 4291 - 2 . 4 2 9 1 - 1 0601 - 1 0601 0 0 44 2547 - 4 4 . 2 5 4 7 19 6582 - 1 9 6582 - 1 3 0620 0 9 2062 9 . 2 0 6 2 - 8 3055 - 8 3055 0 0 0 2249 - 0 . 2 2 4 9 0 0999 - 0 0999 - 7 5968 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 = 0 0 0 0 0 0 o l u m n s 1 t h r o u g h 7 3 0000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5 . 0 0 0 0 0 0 241 C o l u m n s 8 t h r o u g h 10 0 0 . 0 0 0 0 0 0 0 0 0 , 5000 0 0 0 0 0 . 0 0 0 0 D10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 D.1.2 Eigenvalues and Eigenvectors of Linear Plant e = - 0 . 3 0 0 6 0 0 - 5 . 5 8 5 8 - 0 . 3 3 0 9 - 2 . 0 4 2 6 + - 2 . 0 4 2 6 -- 2 . 9 5 5 7 - 0 . 0 8 8 8 - 0 . 6 5 6 6 e v = 0 . 6 4 1 0 i 0 . 6 4 1 0 i C o l u m n s 1 t h r o u g h 4 1 . 0 0 0 0 0 0 0 . 0 0 0 0 0 0 0 - 0 . 0 6 7 4 0 0 0 0 . 0 0 0 0 0 0 0 0 . 9 7 5 8 0 0 0 0 . 0 0 0 0 0 0 0 - 0 . 1 1 0 8 0 1 . 0 0 0 0 0 0 . 0 0 0 0 0 0 0 - 0 . 1 7 4 7 0 0 . 0 0 . 0 0 0 0 0 0 1 . 0 0 0 0 0 . 0 1 9 8 C o l u m n s 5 t h r o u g h 8 0 . 0 0 0 0 - 0 . 1 8 8 8 0 . 0 0 0 0 - 0 . 2 8 3 5 0 . 0 0 0 0 0 . 1 2 1 7 0 . 0 0 0 0 + O.OOOOi 0 . 2 2 7 0 - 0 . 0 8 5 1 i 0 . 0 0 0 0 - O.OOOOi - 0 . 1 2 4 5 - 0 . 4 6 3 9 i 0 . 0 0 0 0 + O.OOOOi 0 . 0 3 9 5 + 0 . 7 3 5 1 i 0 . 0 0 0 0 - O.OOOOi 0 . 2 2 7 0 + 0 . 0 8 5 1 i 0 . 0 0 0 0 + O.OOOOi - 0 . 1 2 4 5 + 0 . 4 6 3 9 i 0 . 0 0 0 0 - O.OOOOi 0 . 0 3 9 5 - 0 . 7 3 5 1 i - 0 . 0 4 5 6 0 . 0 0 0 0 0 . 5 1 1 3 0 . 0 0 0 0 - 0 . 7 9 6 2 0 . 0 0 0 0 2 4 2 0.0000 0.8567 0.0000 -0.3677 0.0000+ O.OOOOi -0.0094+ 0 . 2 2 4 2 i 0 . 0 0 0 0 - O.OOOOi 0 .0852 - 0 . 3 3 3 2 i 0 .0000 - O.OOOOi - 0 . 1 7 3 0 - 0 . 0 0 9 4 - 0 . 2 2 4 2 i 0.0000 0.0000+ O.OOOOi 0.2694 0.0852+ 0 . 3 3 3 2 i 0.0000 Columns 9 t h r o u g h 10 0.8489 0.0000 -0 .0304 0 . 0000 - 0 . 0 3 5 5 0.0000 0.3424 0.0000 0.4000 0 . 0000 0.4835 0.0000 0.4094 0.0000 0.2514 0.0000 - 0 . 6 2 3 5 0.0000 -0 .3829 0 . 0000 D.1.3 Scaling Matrices K i = Ko = 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 10 3 0 0 0 0 0 4 0 0 0 0 0 1 0 0 0 0 0 6 0 0 0 0 0 4 D.1.4 Kalman Filter Gains Lf2 = 2 4680 0 0024 0 0000 0 0675 0 0000 0 0000 0 0000 4 7373 0 0000 1 4583 0 0250 8 2618 0 0000 0 9451 0 0000 0 0000 0 0000 20 3259 0 0000 -2 4778 0 2324 0 9915 0 0000 4 8017 0 0000 0 0000 0 0000 -1 0247 0 0000 6 5867 0 0018 2 0265 0 0000 0 1552 0 0000 0 0000 0 0000 6 3575 0 0000 -0 4836 0 0337 0 1035 • 0 0000 1 2605 0 0000 0 0000 0 0000 -0 1209 0 0000 1 8107 243 D.2 Integrator Augmented Design Information D.2.5 Linear State Space Matrices AlOin = Columns 1 through 7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1.6793 -0 3006 1.9169 -1.9169 1 7030 -1 7030 -2 0594 0 0 -1.2145 -1.2145 -1 0601 -1 0601 0 0 0 22.1273 -22.1273 19 6582 -19 6582 -13 0620 0 0 4.6031 4.6031 -8 3055 -8 3055 0 0 0 0.1125 -0.1125 0 0999 -0 0999 -7 5968 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Columns 8 through 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1.7530 0 0 2678 0 0 4902 0 -0 2944 0 -1.3060 0 1 0599 0 0 0 -0.6505 0 -6 2047 0 -2 2722 0 -1 7644 0 0.9431 0 -2 3951 0 0 0 -1.9425 0 0 2498 0 -2 0442 0 -0 1020 0 1.0000 0 0 0 0 0 0 0 1 0000 0 0 0 0 0 0 0 1 0000 0 0 0 0 0 0 0 1 0000 0 0 Columns 15 through 16 0 0 0 0 0 0 0 0 0 0 0 0 0.4495 0 244 0 0.0016 0 - 0 . 1 3 3 1 0 0 0 0 0 BlOin = 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ClOin = Columns 1. t h r o u g h 12 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Columns 13 t h r o u g h 16 0 0 0 0 4 0 0 0 0 1 0 0 0 0 6 0 0 0 0 4 DIOin = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 245 D.2.6 Eigenvalues and Eigenvectors of Linear Plant o o - 0 . 3 0 0 6 - 5 . 5 8 5 8 - 0 . 3 3 0 9 -2 .0426+ - 2 . 0 4 2 6 -- 2 . 9 5 5 7 - 0 . 0 8 8 8 - 0 . 6 5 6 6 0 0 0 0 0 0 e = 0 . 6 4 1 0 i 0 . 6 4 1 0 i Columns 1 t h r o u g h 4 0 0 0 0 0 0 0 0 0 0 0 0 1.0000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1.0000 0 0 0 0 0 0 1.0000 0 6 0 0 0 0 0 0 0 0 0 0 0 0 0 0 .0000 -0.0674 0.0000 0.9758 0.0000 -0.1108 0.0000 -0.1747 0.0000 0.0198 Columns 5 t h r o u g h 8 0 0 0 0 0 0 0 .0000 0.1888 0.0000 0.2835 0.0000 -0.1217 0.0000 0 0 0 0 0 0 0.0000+ 0.1499+ 0 .0000-0 . 4 0 4 3 -0.0000+ -0.6886+ 0.0000+ O.OOOOi 0 . 1 9 0 6 i O.OOOOi 0 . 2 5 9 2 i O.OOOOi 0 . 2 6 0 5 i O.OOOOi 0 0 0 0 0 0 0 .0000 -0 .1499 -0.0000+ 0.4043+ 0 . 0 0 0 0 --0 .6886-0 .0000-O.OOOOi 0 . 1 9 0 6 i O.OOOOi 0 . 2 5 9 2 i O.OOOOi 0 . 2 6 0 5 i 0.OOOOi 0 0 0 0 0 0 0.0456 0.0000 -0.5113 0.0000 0.7962 0.0000 0.1730 246 - 0 . 8 5 6 7 -0 .2165+ 0 . 0 5 9 0 i - 0 . 2 1 6 5 - 0 . 0 5 9 0 i 0.0000 0.0000 0 .0000- O.OOOOi 0.0000+ O.OOOOi -0 .2694 0.3677 0 . 3 4 3 3 - 0 . 0 1 9 8 i 0.3433+ 0 . 0 1 9 8 i 0.0000 Columns 9 t h r o u g h 12 0 0 0 0 0 0 - 0 . 8 4 8 9 0 . 0000 0.0304 0.0000 0.0355 0.0000 -0 .3424 0.0000 - 0 . 4 0 0 0 0 . 0000 0 0 0 0 0 0 - 0 .4835 0.0000 -0 .4094 0.0000 -0 .2514 0.0000 0.6235 0.0000 0.3829 0.0000 0.0000 0 0 0 0 0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.9467 0.0000 0.0000 -0 .3220 0 0.0000 0 0 0 0 0 .0000 0 . 0000 0.0000 0.0000 0.0000 0.0000 0.9467 0.0000 0.0000 0.3220 Columns 13 t h r o u g h 16 0 0 0 .0000 0 0 0 0 .0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.9573 0.0000 0.0000 -0.2892 0 0 0 0 .0000 0 0 0 .0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.9573 0.0000 0.0000 0.2892 0 0 0 0 0.0000 0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0 0 0 0 0 0 .1762 0.9844 0 0 0 0 0 0 0 0 0 D.2.7 Kalman Filter Gains mu = 0.0200 M I O i n = 0.5000 0 0 0 0 0 0 .5000 0 0 0 0 0 0.5000 0 0 0 0 0 0.5000 0 0 0 0 0 0.5000 0 0 0 0 0 247 W5 = Kf2 0 0 0 0 0 5 0000 0 0 0 0 0 1 6793 1 9169 -1 9169 1 7030 -1 7030 -2 0594 0 1 2145 -1 2145 -1 0601 -1 0601 0 0 2 1273 -22 1273 19 6582 -19 6582 -13 0620 0 4 6031 4 6031 -8 3055 -8 3055 0 0 0 1125 -0 1125 0 0999 -0 0999 -7 5968 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5665 0 0078 -0 0576 0 3344 0 0130 -0 0020 0 0078 0 5665 0 3344 -0 0576 -0 0130 -0 0020 0 0576 0 3344 0 4000 -0 2517 0 0115 -0 0010 0 3344 -0 0576 -0 2517 0 4000 -0 0115 -0 0010 0 0130 -0 0130 0 0115 -0 0115 0 1327 0 0000 0 0020 -0 0020 -0 0010 -0 0010 0 0000 0 0555 0 0190 -1 8364 0 8441 0 4648 -0 1490 0 0190 -1 8364 -0 8441 0 4648 0 1490 0 0125 -0 9600 0 7499 -0 0715 -0 1324 0 0125 -0 9600 -0 7499 -0 0715 0 1324 0 0000 0 0000 -0 0353 0 0000 -1 2872 8 3264 0 0268 0 0000 -0 0888 0 0000 4 0418 0 0114 0 0000 0 0302 0 0000 0 0000 0 0000 12 5104 0 0000 3 9568 0 1540 11 1855 0 0000 -0 3127 0 0000 0 0000 0 0000 68 0034 0 0000 2 4318 0 0439 0 1943 0 0000 7 9748 0 0000 0 0000 0 0000 0 7889 0 0000 14 5501 0 0086 2 3649 0 0000 -0 0011 0 0000 0 0000 0 0000 11 6595 0 0000 0 2489 0 0151 -0 0008 0 0000 1 6303 0 0000 0 0000 0 0000 0 0622 0 0000 2 6965 D.2.8 Control Gains ro = 0.0050 Koweight = 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 100 0 0 0 0 0 1 KlOin = Columns 1 through 7 248 11 6185 4.9951 -4 2006 -10 4 9951 11.6185 -10 0848 -4 -4 2006 -10.0848 20 0881 14 -10 0848 -4.2006 14 8605 20 -0 2427 0.2427 -0 2156 0 0 0060 0.0060 -0 0106 -0 Columns 8 through 14 0 2941 -9.7020 0 8686 23 -0 2941 -9.7020 -0 8686 23 0 2612 -7.8948 0 7717 -41 -0 2612 -7.8948 -0 7717 -41 2 9330 0.0000 -0 3250 0 0 0000 0.0008 0 0000 0 Columns 15 through 16 301 1690 -3.0385 301 1690 3.0385 518 5948 -2.6994 518 5948 2.6994 0 0000 -56.2758 1 5182 0.0000 0848 -0 2427 0 0060 0 0264 2006 0 2427 0 0060 0 0264 8605 -0 2156 -0 0106 -0 0474 0881 0 2156 -0 0106 -0 0474 2156 12 9414 0 0000 0 0000 0106 0 0000 11 6400 40 3426 5144 -1 1546 -34 5898 7 1898 5144 1 1546 -34 5898 -7 1898 9003 -1 0257 -20 0885 6 3875 9003 1 0257 -20 0885 -6 3875 0000 -11 2734 0 0000 -1 2925 0603 0 0000 -0 0019 0 0000 D.3 Disturbance Observer Augmented Design Information D.3.9 Linear State Space Matrices AlOob = Columns 1 through 7 -0.4500 0 0 0 0 0 0 0 -0.4500 0 0 0 0 0 0 0 -0 4500 0 0 0 0 0 0 0 -0 4500 0 0 0 0 0 0 0 -0 4500 0 0 0 0 0 0 0 -0 4500 0 0 0 0 0 0 1 6793 -0.3006 1.9169 -1.9169 1 7030 -1 7030 -2 0594 0 0 -1.2145 -1.2145 -1 0601 -1 0601 0 0 0 22.1273 -22.1273 19 6582 -19 6582 -13 0620 0 0 4.6031 4.6031 -8 3055 -8 3055 0 0 0 0.1125 -0.1125 0 0999 -0 0999 -7 5968 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Columns 8 through 14 0 0 0 0 0 0 0 249 0 0 0 0 0 0 -1.7530 0 -0.6505 0 -1.9425 0 0 0 0 0 0 0 0 0 0 0 -1.3060 0 0 .9431 0 1.0000 0 0 0 0 0 0 0 0 0 0.2678 0 -6 .2047 0 0.2498 0 1.0000 0 0 0 0 0 0 0 0 0 1.0599 0 -2.3951 0 0 0 1.0000 0 0 0 0 0 0 0 0.4902 0 -2.2722 0 -2.0442 0 0 0 1.0000 0 0 0 0 0 0 - 0 . 2 9 4 4 0 - 1 . 7 6 4 4 0 - 0 . 1 0 2 0 0 0 • 0 0 Columns 15 t h r o u g h 16 0 0 0 0 0 0 0.4495 0 0.0016 0 - 0 . 1 3 3 1 0 0 0 0 0 BlOob = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1.9169 - 1 . 2 1 4 5 22.1273 4 .6031 0.1125 0 0 0 0 0 0 0 0 0 0 0 - 1 . 9 1 6 9 - 1 . 2 1 4 5 -22 .1273 4 .6031 - 0 . 1 1 2 5 0 0 0 0 0 0 0 0 0 0 0 1.7030 - 1 . 0 6 0 1 19.6582 - 8 . 3 0 5 5 0.0999 0 0 0 0 0 0 0 0 0 0 0 .7030 .0601 -19 .6582 -8 .3055 -0 .0999 0 0 0 0 0 0 0 0 0 0 0 -2 .0594 0 -13.0620 0 -7 .5968 0 0 0 0 0 0 0 0 0 0 1.6793 0 0 0 0 0 0 0 0 0 ClOob 250 Columns 1 t h r o u g h 12 DIOob 0 0 0 0 0 0 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 .umns 13 t h r o u g h 16 0 0 0 0 4 0 0 0 0 1 0 0 0 0 6 0 0 > = 0 0 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 D.3.10 Eigenvalues and Eigenvectors of Linear Plant e = 0 0 -0 .3006 -5 .5858 - 0 . 3 3 0 9 -2 .0426+ 0 . 6 4 1 0 i - 2 . 0 4 2 6 - 0 . 6 4 1 0 i - 2 . 9 5 5 7 -0 .0888 - 0 . 6 5 6 6 - 0 . 4 5 0 0 -0 .4500 -0 .4500 -0 .4500 - 0 . 4 5 0 0 - 0 . 4 5 0 0 Columns 1 t h r o u g h 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1.0000 0.0000 0 0 0 -0 .0674 251 0 0 0 0 0 0 0 0 1.0000 0 0 0 0 0 0 1.0000 Columns 5 t h r o u g h 8 0 0 0 0 0 0 0 0 0 0 0 0 0.0000 0.0000+ O.OOOOi 0.1888 0.1499+ 0 . 1 9 0 6 i 0.0000 0 .0000- O.OOOOi 0.2835 0 . 4 0 4 3 - 0 . 2 5 9 2 i 0.0000 0.0000+ O.OOOOi - 0 . 1 2 1 7 -0 .6886+ 0 . 2 6 0 5 i 0 .0000 0.0000+ O.OOOOi - 0 . 8 5 6 7 -0 .2165+ 0 . 0 5 9 0 i 0 .0000 0 . 0 0 0 0 - O.OOOOi 0.3677 0 . 3 4 3 3 - 0 . 0 1 9 8 i Columns 9 t h r o u g h 12 0 0 0 0 0 0 0 0 0 0 0 0 - 0 . 8 4 8 9 - 0 . 4 8 3 5 0.0000 0.0000 0.0304 -0 .4094 0.0000 0.0000 0.0355 -0 .2514 0.0000 0.0000 -0 .3424 0.6235 0 .0000 0.0000 - 0 . 4 0 0 0 0.3829 0.0000 0.0000 Columns 13 t h r o u g h 16 0 0 0 0 0.0063 0 0 0.0063 0 0 0 0.0000 0 0.9758 0 0.0000 0 -0 .1108 0 0.0000 0 -0 .1747 0 0.0000 0 0.0198 0 0 0 0 0 0 0 0 0 0 0 0 0 . 0 0 0 0 - O.OOOOi 0.0456 0 .1499- 0 . 1 9 0 6 i 0.0000 0.0000+ O.OOOOi -0 .5113 0.4043+ 0 . 2 5 9 2 i 0.0000 0 .0000- O.OOOOi 0.7962 •0 .6886- 0 . 2 6 0 5 i 0.0000 0 .0000- O.OOOOi 0.1730 -0 .2165- 0 . 0 5 9 0 i 0.0000 0.0000+ O.OOOOi -0 .2694 0.3433+ 0 . 0 1 9 8 i 0.0000 0.0140 0 0 0.0140 0 0 0 0 0 0 0 0 0.6307 0.6307 0.1584 -0 .1584 0.0965 0.0965 0 .2561 - 0 . 2 5 6 1 0.0943 0.0943 0.1155 0.1155 0.2144 -0 .2144 0.5692 0.5692 0.2096 -0 .2096 0.2566 -0 .2566 0 0 0 0 0 0 0 0 0.0703 0 252 0 0 0 0 0886 -0 8240 -0 8240 0 0000 -0 9961 0 0634 -0 0634 -0 3757 0 -0 1599 -0 1599 0 0000 0 0 1025 -0 1025 -0 3791 0 -0 1232 -0 1232 0 0000 0 -0 0462 0 0462 0 0096 0 0 3553 0 3553 0 0000 0 -0 2277 0 2277 0 8424 0 0 2738 0 2738 0 0000 0 0 1027 -0 1027 -0 0214 0 D.3.11 Kalman Filter Gains mu = 0.2000 MIOob = 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 W5 = 2.5336 1.4901 0.9897 0.9510 0.0761 0.0109 Kf2 = 0.0215 0.0215 -0.0189 -0.0189 0.0000 2.5317 1.5879 0.0000 0.0241 1.4901 2.5336 0.9510 0.9897 -0.0761 0.0109 -2.3919 -2.3919 -1.2345 -1.2345 0.0000 0.0032 0.0040 0.0000 3.7923 0.9897 0.9510 1.0356 0.0013 0.0676 -0.0143 0.9663 -0.9663 0.8584 -0.8584 -0.0284 0.0000 0.0000 2.8991 0.0000 0.9510 0.9897 0.0013 1.0356 -0.0676 -0.0143 0.5655 0.5655 -0.1078 -0.1078 0.0000 -0.0081 0.0620 0.0000 0.3540 0.0761 -0.0761 0.0676 -0.0676 0.6217 0.0000 -0.1773 0.1773 -0.1575 0.1575 -1.3786 0.0000 0.0000 0.9533 0.0000 0.0109 0.0109 -0.0143 -0.0143 0.0000 1.7807 253 0 0000 0 0000 10 8277 0 0000 -0 4626 0 1305 0 3572 0 0000 2 2946 0 0000 0 0000 0 0000 -0 4605 0 0000 3 2322 0 0030 1 3745 0 0000 0 0829 0 0000 0 0000 0 0000 4 6475 0 0000 -0 2369 0 0310 0 0552 0 0000 0 8723 0 0000 0 0000 0 0000 -0 0592 0 0000 1 2699 D.3.12 Control Gains ro = 0.0080 Koweight = 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 100 0 0 0 0 0 1 KlOob = Columns 1 through 7 0.7737 0.2200 0.2444 -0.2475 0.0002 0.0000 0.2200 0.7737 -0.2475 0.2444 -0.0002 0.0000 0.2446 -0.2473 0.7178 0.2807 0.0002 0.0000 -0.2473 0.2446 0.2807 0 .7.178 -0.0002 0.0000 0.0009 -0.0009 0.0008 -0.0008 0.9952 0.0000 0.0001 0.0001 -0.0002 -0.0002 0 . 0000 0.9868 0.0021 0.0021 -0.0038 -0.0038 0.0000 33.3625 Columns 8 through 14 0.0338 -0.0338 0.0300 -0.0300 0.2532 0.0000 -3.5301 -3.5301 -2.0357 -2.0357 0 . 0000 0.0000 0.3097 -0.3097 0.2751 -0.2751 -0.0623 0.0000 3.4823 3.4823 -6.0074 -6.0074 0.0000 0.0008 -0.3774 0.3774 -0.3353 0.3353 -3.1569 0.0000 -27.4946 -27.4946 -15.6220 -15.6220 0.0000 -0.0001 5.8382 -5.8382 5.1868 -5.1868 -1.4638 0.0000 Columns 15 through 16 234.3251 234.3251 -412.4138 -412.4138 0.0000 0.1448 -3.1204 3.1204 -2.7722 2.7722 -44.3301 0.0000 254 Appendix E: Simulation Results and Software This information is provided in the accompanying CD-ROM storage media. The information is in the format of Matlab 5.3 and Simulink. A description of the files is given in file "Subsimdoc". An electronic copy of the thesis is also on the CD in Adobe Acrobat portable document format (pdf). 255 

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