# 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

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

3.2 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
13.1.3 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

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