Mathematical Models of Ships and Underwater Vehicles
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Abstract
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 robotlike 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 matrixvector notation is highly advantageous when designing control systems since wellknown system properties such as symmetry, skewsymmetry, and positiveness can be exploited in the design.
Keywords
Kinematics Kinetics Degrees of freedom Euler angles Ship Underwater vehicles AUV, ROV Hydrodynamics Seakeeping ManeuveringIntroduction
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.
Hydrodynamics

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 waveaffected 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 zerofrequency 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 Lowerorder models typically suited for controllerobserver design. Six ODEs are needed to describe the kinematics, that is, the geometrical aspects of motion while NewtonEuler’s equations represent additional six ODEs describing the forces and moments causing the motion (kinetics).
Notation

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
Kinematics
Singularities can be avoided by using unit quaternions instead (Fossen 1994, 2011).
Kinetics
Formulae (8) and (9) together with (4) are the fundamental equations when deriving the ship and underwater vehicle models. This is the topic for the next sections.
Ship Model
Other nonlinear representations are found in Fossen (1994, 2011).
The maneuvering model presented in this entry is intended for controllerobserver design, prediction, and computer simulations, as well as system identification and parameter estimation. A large number of applicationspecific 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
CrossReferences
Bibliography
 Allmendinger E (ed) (1990) Submersible vehicle systems design. SNAMEGoogle Scholar
 Anotonelli G (2010) Underwater robots: motion and force control of vehiclemanipulator systems. Springer, HeidelbergGoogle Scholar
 Bertram V (2012) Practical ship hydrodynamics. Elsevier, AmsterdamGoogle Scholar
 Faltinsen O (1990) Sea loads on ships and offshore structures. Cambridge University Press, Cambridge, UKGoogle Scholar
 Fossen TI (1991) Nonlinear Modelling and Control of Underwater Vehicles. PhD Thesis, Dept. of Eng. Cybernetics, Norwegian Univ. of Sci. and Techn.Google Scholar
 Fossen TI (1994) Guidance and control of ocean vehicles. Wiley, ChichesterGoogle Scholar
 Fossen TI (2011) Handbook of marine craft hydrodynamics and motion control. Wiley, ChichesterCrossRefGoogle Scholar
 Inzartsev AV (2009) Intelligent underwater vehicles. ITech Education and Publishing Open Access: http://www.intechweb.org/
 Lewanddowski EM (2004) The dynamics of marine craft. World Scientific, SingaporeGoogle Scholar
 MSS (2010) Marine systems simulator. Open Access: http://www.marinecontrol.org/
 Newman JN (1977) Marine hydrodynamics. MIT Press, Cambridge, MAGoogle Scholar
 Perez T (2005) Ship motion control. Springer, LondonGoogle Scholar
 Perez T, Fossen TI (2011a) Practical aspects of frequencydomain identification of dynamic models of marine structures from hydrodynamic data. Ocean Eng 38:426–435CrossRefGoogle Scholar
 Perez T, Fossen TI (2011b) Motion control of marine craft, Ch. 33. In: Levine WS (ed) The control systems handbook: control system advanced methods, 2nd edn. CRC Press, Boca RatonGoogle Scholar
 Rawson KJ, Tupper EC (1994) Basic ship theory. Longman, New YorkGoogle Scholar
 SNAME (1950) Nomenclature for treating the motion of a submerged body through a fluid. SNAME, Technical and Research Bulletin No. 1–5, pp 1–15Google Scholar
 Sutton R, Roberts G (2006) Advances in unmanned marine vehicles (IEE Control Series). The Institution of Engineering and Technology, HertfordshireGoogle Scholar
 Wadoo S, Kachroo P (2010) Autonomous underwater vehicles: modeling control design and simulation. Taylor and Francis, London/New YorkGoogle Scholar