Mathematical Models of Ships and Underwater Vehicles
This entry describes the equations of motion of ships and underwater vehicles. Standard hydrodynamic models in the literature are reviewed and presented using the nonlinear robot-like vectorial notation of Fossen (Nonlinear Modelling and Control of Underwater Vehicles. PhD Thesis, Dept. of Eng. Cybernetics, Norwegian Univ. of Sci. and Techn, 1991; 1994, Guidance and control of ocean vehicles. Wiley, Chichester; 2011). The matrix-vector notation is highly advantageous when designing control systems since well-known system properties such as symmetry, skew-symmetry, and positiveness can be exploited in the design.
KeywordsKinematics Kinetics Degrees of freedom Euler angles Ship Underwater vehicles AUV, ROV Hydrodynamics Seakeeping Maneuvering
The subject of this entry is mathematical modeling of ships and underwater vehicles. With ship we mean “any large floating vessel capable of crossing open waters,” as opposed to a boat, which is generally a smaller craft. An underwater vehicle is a “small vehicle that is capable of propelling itself beneath the water surface as well as on the water’s surface.” This includes unmanned underwater vehicles (UUV), remotely operated vehicles (ROV), autonomous underwater vehicles (AUV) and underwater robotic vehicles (URV).
This entry is based on Fossen (1994, 2011), which contains a large number of standard models for ships, rigs, and underwater vehicles. There exist a large number of textbooks on mathematical modeling of ships; see Rawson and Tupper (1994), Lewanddowski (2004), and Perez (2005). For underwater vehicles, see Allmendinger (1990), Sutton and Roberts (2006), Inzartsev (2009), Anotonelli (2010), andWadoo and Kachroo (2010). Some useful references on ship hydrodynamics are Newman (1977), Faltinsen (1991), Bertram (2012).
Degrees of Freedom
A mathematical model of marine craft is usually represented by a set of ordinary differential equations (ODEs) describing the motions in six degrees of freedom (DOF): surge, sway, heave, roll, pitch, and yaw.
Seakeeping theory: The motions of ships at zero or constant speed in waves are analyzed using hydrodynamic coefficients and wave forces, which depends of the wave excitation frequency and thus the hull geometry and mass distribution. For underwater vehicles operating below the wave-affected zone, the wave excitation frequency will not influence the hydrodynamic coefficients.
Maneuvering theory: The ship is moving in restricted calm water – that is, in sheltered waters or in a harbor. Hence, the maneuvering model is derived for a ship moving at positive speed under a zero-frequency wave excitation assumption such that added mass and damping can be represented by constant parameters.
Seakeeping models are typically used for ocean structures and dynamically positioned vessels. Several hundred ODEs are needed to effectively represent a seakeeping model; see Fossen (2011), and Perez and Fossen (2011a, b).
The remaining of this entry assumes maneuvering theory, since this gives Lower-order models typically suited for controller-observer design. Six ODEs are needed to describe the kinematics, that is, the geometrical aspects of motion while Newton-Euler’s equations represent additional six ODEs describing the forces and moments causing the motion (kinetics).
x b – longitudinal axis (from aft to fore)
y b – transversal axis (to starboard)
z b – normal axis (directed downward)
Surge position x, linear velocity u, force X
Sway position y, linear velocity v, force Y
Heave position z, linear velocity w, force Z
Roll angle ϕ, angular velocity p, moment K
Pitch angle θ, angular velocity q, moment M
Yaw angle ψ, angular velocity r, moment N
The maneuvering model presented in this entry is intended for controller-observer design, prediction, and computer simulations, as well as system identification and parameter estimation. A large number of application-specific models for marine craft are found in Fossen (2011, Chapter 7).
Hydrodynamic programs compute mass, inertia, potential damping and restoring forces while a more detailed treatment of viscous dissipative forces (damping) and sealoads are found in the extensive literature on hydrodynamics – see Faltinsen (1990) and Newman (1977).
Underwater Vehicle Model
Notice that this representation of C RB (ν) only depends on the angular velocities p, q, and r, and not the linear velocities u,v, and r. This property will be exploited when including drift due to ocean currents.
The expression for D can be extended to include nonlinear damping terms if necessary. Quadratic damping is important at higher speeds since the Coriolis and centripetal terms C(ν)ν can destabilize the system if only linear damping is used.
Programs and Data
The Marine Systems Simulator (MSS) is a MATLAB/Simulink library and simulator for marine craft (http://www.marinecontrol.org). It includes models for ships, underwater vehicles, and floating structures.
Summary and Future Directions
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