# Lecture Notes

Triantafyllou, Michael S., and Franz S. Hover. Maneuvering and Control of Marine Vehicles. (Full text available here (PDF - 1.6 MB); also available by chapter below)

## Contents

1. Kinematics of Moving Frames (PDF)
   1.1 Rotation of Reference Frames
   1.2 Differential Rotations
   1.3 Rate of Change of Euler Angles
   1.4 Dead Reckoning
2. Vessel Inertial Dynamics (PDF)
   2.1 Momentum of a Particle
   2.2 Linear Momentum in a Moving Frame
   2.3 Example: Mass on a String
   2.3.1 Moving Frame Affixed to Mass
   2.3.2 Rotating Frame Attached to Pivot Point
   2.3.3 Stationary Frame
   2.4 Angular Momentum
   2.5 Example: Spinning Book
   2.5.1 x-axis
   2.5.2 y-axis
   2.5.3 z-axis
   2.6 Parallel Axis Theorem
   2.7 Basics for Simulation
3. Nonlinear Coefficients in Detail (PDF)
   3.1 Helpful Facts
   3.2 Nonlinear Equations in the Horizontal Plane
   3.2.1 Fluid Force X
   3.2.2 Fluid Force Y
   3.2.3 Fluid Moment N
4. Vessel Dynamics: Linear Case (PDF)
   4.1 Surface Vessel Linear Model
   4.2 Stability of the Sway/Yaw System
   4.3 Basic Rudder Action in the Sway/Yaw Model
   4.3.1 Adding Yaw Damping through Feedback
   4.3.2 Heading Control in the Sway/Yaw Model
   4.4 Response of the Vessel to Step Rudder Input
   4.4.1 Phase 1: Accelerations Dominate
   4.4.2 Phase 3: Steady State
   4.5 Summary of the Linear Maneuvering Model
   4.6 Stability in the Vertical Plane
5. Similitude (PDF)
   5.1 Use of Nondimensional Groups
   5.2 Common Groups in Marine Engineering
   5.3 Similitude in Maneuvering
   5.4 Roll Equation Similitude
6. Captive Measurements (PDF)
   6.1 Towtank
   6.2 Rotating Arm Device
   6.3 Planar-Motion Mechanism
7. Standard Maneuvering Tests (PDF)
   7.1 Dieudonné Spiral
   7.2 Zig-Zag Maneuver
   7.3 Circle Maneuver
   7.3.1 Drift Angle
   7.3.2 Speed Loss
   7.3.3 Heel Angle
   7.3.4 Heeling in Submarines with Sails
8. Streamlined Bodies (PDF)
   8.1 Nominal Drag Force
   8.2 Munk Moment
   8.3 Separation Moment
   8.4 Net Effects: Aerodynamic Center
   8.5 Role of Fins in Moving the Aerodynamic Center
   8.6 Aggregate Effects of Body and Fins
   8.7 Coefficients Zw, Mw, Zq, and Mq for a Slender Body
9. Slender-Body Theory (PDF)
   9.1 Introduction
   9.2 Kinematics Following the Fluid
   9.3 Derivative Following the Fluid
   9.4 Differential Force on the Body
   9.5 Total Force on a Vessel
   9.6 Total Moment on a Vessel
   9.7 Relation to Wing Lift
   9.8 Convention: Hydrodynamic Mass Matrix A
10. Practical Lift Calculations (PDF)
    10.1 Characteristics of Lift-Producing Mechanisms
    10.2 Jorgensen's Formulas
    10.3 Hoerner's Data: Notation
    10.4 Slender-Body Theory vs. Experiment
    10.5 Slender-Body Approximation for Fin Lift
11. Fins and Lifting Surfaces (PDF)
    11.1 Origin of Lift
    11.2 Three-Dimensional Effects: Finite Length
    11.3 Ring Fins
12. Propellers and Propulsion (PDF)
    12.1 Introduction
    12.2 Steady Propulsion of Vessels
    12.2.1 Basic Characteristics
    12.2.2 Solution for Steady Conditions
    12.2.3 Engine/Motor Models
    12.3 Unsteady Propulsion Models
    12.3.1 One-State Model: Yoerger et al
    12.3.2 Two-State Model: Healey et al
13. Electric Motors (PDF)
    13.1 Basic Relations
    13.1.1 Concepts
    13.1.2 Faraday's Law
    13.1.3 Ampere's Law
    13.1.4 Force
    13.2 DC Motors
    13.2.1 Permanent Field Magnets
    13.2.2 Shunt or Independent Field Windings
    13.2.3 Series Windings
    13.3 Three-Phase Synchronous Motor
    13.4 Three-Phase Induction Motor
14. Towing of Vehicles (PDF)
    14.1 Statics
    14.1.1 Force Balance
    14.1.2 Critical Angle
    14.2 Linearized Dynamics
    14.2.1 Derivation
    14.2.2 Damped Axial Motion
    14.3 Cable Strumming
    14.4 Vehicle Design
15. Transfer Functions and Stability (PDF)
    15.1 Partial Fractions
    15.2 Partial Fractions: Unique Poles
    15.3 Example: Partial Fractions with Unique Real Poles
    15.4 Partial Fractions: Complex-Conjugate Poles
    15.5 Example: Partial Fractions with Complex Poles
    15.6 Stability in Linear Systems
    15.7 Stability ⇔ Poles in LHP
    15.8 General Stability
16. Control Fundamentals (PDF)
    16.1 Introduction
    16.1.1 Plants, Inputs, and Outputs
    16.1.2 The Need for Modeling
    16.1.3 Nonlinear Control
    16.2 Representing Linear Systems
    16.2.1 Standard State-Space Form
    16.2.2 Converting a State-Space Model into a Transfer Function
    16.2.3 Converting a Transfer Function into a State-Space Model
    16.3 PID Controllers
    16.4 Example: PID Control
    16.4.1 Proportional Only
    16.4.2 Proportional-Derivative Only
    16.4.3 Proportional-Integral-Derivative
    16.5 Heuristic Tuning
    16.6 Block Diagrams of Systems
    16.6.1 Fundamental Feedback Loop
    16.6.2 Block Diagrams: General Case
    16.6.3 Primary Transfer Functions
17. Modal Analysis (PDF)
    17.1 Introduction
    17.2 Matrix Exponential
    17.2.1 Definition
    17.2.2 Modal Canonical Form
    17.2.3 Modal Decomposition of Response
    17.3 Forced Response and Controllability
    17.4 Plant Output and Observability
18. Control Systems - Loopshaping (PDF)
    18.1 Introduction
    18.2 Roots of Stability - Nyquist Criterion
    18.2.1 Mapping Theorem
    18.2.2 Nyquist Criterion
    18.2.3 Robustness on the Nyquist Plot
    18.3 Design for Nominal Performance
    18.4 Design for Robustness
    18.5 Robust Performance
    18.6 Implications of Bode's Integral
    18.7 The Recipe for Loopshaping
19. Linear Quadratic Regulator (PDF)
    19.1 Introduction
    19.2 Full-State Feedback
    19.3 The Maximum Principle
    19.4 Gradient Method Solution for the General Case
    19.5 LQR Solution
    19.6 Optimal Full-State Feedback
    19.7 Properties and Use of the LQR
    19.8 Proof of the Gain and Phase Margins
20. Kalman Filter (PDF)
    20.1 Introduction
    20.2 Problem Statement
    20.3 Step 1: An Equation for ∑
    20.4 Step 2: H as a Function of ∑
    20.5 Properties of the Solution
    20.6 Combination of LQR and KF
    20.7 Proofs of the Intermediate Results
21. Loop Transfer Recovery (PDF)
    21.1 Introduction
    21.2 A Special Property of the LQR Solution
    21.3 The Loop Transfer Recovery Result
    21.4 Usage of the Loop Transfer Recovery
    21.5 Three Lemmas
22. Appendix 1: Math Facts (PDF)
    22.1 Vectors
    22.1.1 Definition
    22.1.2 Vector Magnitude
    22.1.3 Vector Dot or Inner Product
    22.1.4 Vector Cross Product
    22.2 Matrices
    22.2.1 Definition
    22.2.2 Multiplying a Vector by a Matrix
    22.2.3 Multiplying a Matrix by a Matrix
    22.2.4 Common Matrices
    22.2.5 Transpose
    22.2.6 Determinant
    22.2.7 Inverse
    22.2.8 Trace
    22.2.9 Eigen values and Eigen vectors
    22.2.10 Modal Decomposition
    22.2.11Singular Value
    22.3 Laplace Transform
    22.3.1 Definition
    22.3.2 Convergence
    22.3.3 Convolution Theorem
    22.3.4 Solution of Differential Equations by Laplace Transform
    22.4 Back ground for the Mapping Theorem
23. Appendix 2: Added Mass via Lagrangian Dynamics (PDF)
    23.1 Kinetic Energy of the Fluid
    23.2 Kirchhoff's Relations
    23.3 Fluid Inertia Terms
    23.4 Derivation of Kirchhoff's Relations
    23.5 Nomenclature
    23.5.1 Free versus Column Vector
    23.5.2 Derivative of a Scalar with Respect to a Vector
    23.5.3 Dot and Cross Product
24. Appendix 3: LQR via Dynamic Programming (PDF)
    24.1 Example in the Case of Discrete States
    24.2 Dynamic Programming and Full-State Feedback
25. Further Robustness of the LQR (PDF)
    25.1 Tools
    25.1.1 Lyapunov's Second Method
    25.1.2 Matrix Inequality Definition
    25.1.3 Franklin Inequality
    25.1.4 Schur Complement
    25.1.5 Proof of Schur Complement Sign
    25.1.6 Schur Complement of a Nine-Block Matrix
    25.1.7 Quadratic Optimization with a Linear Constraint
    25.2 Comments on Linear Matrix Inequalities (LMI's)
    25.3 Parametric Uncertainty in A and B Matrices
    25.3.1 General Case
    25.3.2 Uncertainty in B
    25.3.3 Uncertainty in A
    25.3.4 A and B Perturbations as an LMI
    25.4 Input Nonlinearities