The Marine Systems Simulator State-Space Model Representation for Dynamically Positioned Surface Vessels

Transcription

1 The Marine Systems Simulator State-Space Model Representation for Dynamically Positioned Surface Vessels Øyvind N. Smogeli, Tristan Perez, Thor I. Fossen, and Asgeir J. Sørensen ) Department of Marine Technology ) Department of Engineering Cyernetics Norwegian University of Science and Technology NO-749 Trondheim, Norway oyvind.smogeli@ntnu.no tristan.perez@ntnu.no fossen@ieee.org asgeir.sorensen@ntnu.no ABSTRACT: This paper presents a nonlinear state-space model suitale for numerical simulation and ship motion control system testing for dynamic positioning (DP) operations. This model represents a typical application of the tools included in the Marine Systems Simulator (MSS), and illustrates the modelling philosophy of the MSS. This simulator is a Matla r /Simulink r -ased toolox specially developed at The Norwegian University of Science and Technology (NTNU) for education, rapid prototyping and evaluation of marine control systems. The model presented here is ased on results of standard potential theory tools, which are used to otain radiation and wave excitation forces. The uid memory e ects associated with the radiation forces are expressed as a reduced-order state-space model. This is a key issue, which allows one to replace the traditional low-/wave-frequency motion superposition principle y a uni ed state-space model in ody- xed coordinates. The resulting model is suitale for simulation of marine operations requiring station-keeping and low-speed ship manoeuvring in a seaway, and can incorporate viscous forces, wave drift, current loads, wind loads, multi-ody interactions, and ship motion control systems. INTRODUCTION Numerical simulators have ecome an indispensale tool for control system design and analysis. In the case of ships performing marine operations, simulation scenarios can easily ecome complex when including models of the environment, vessels, mooring systems, risers, cranes with suspended loads, pipe-line equipment, ship motion control systems, etc. At NTNU, there has een a continuous research focus on marine control system simulation tools during the past years. Recently, the Marine Cyernetics Simulator (Sørensen et al., 3) and the Guidance Navigation and Control toolox (Fossen, ) have een merged under the so-called Marine Systems Simulator (MSS, 4). This simulator is as a Matla r / Simulink r -ased toolox specially developed for rapid prototyping and evaluation of marine control systems. MSS includes a set of Matla functions and several Simulink liraries that cover a range of simulation scenarios ased on model detail, user expertise and various applications of marine control systems for course-keeping, path-following, dynamic positioning, roll stailization, etc. This paper presents a particular application that illustrates the modeling philosophy of MSS: a nonlinear state-space model for dynamically positioned surface vessels. A key aspect of this approach is the state-space formulation of the equations of motion in terms of ody- xed coordinates adopted for MSS. Traditional models used for control system design and testing use data generated from standard hydrodynamic computer programs, which compute hydrodynamic coe cients and motion response transfer functions due to waves (RAO response amplitude operators). These models together with data otained from captive model testing in calm water (zerofrequency coe cients) provide one with the traditional framework that uses the superposition of the wave-frequency motion and the low-frequency motion. This approach is suitale for studying wavevessel interactions for single vessels. In marine operations, however, vessels interact with other physical systems; thus, these types of operations requires models that includes energy exchange etween the interacting systems. Motivated y the work of Bishop and Price (98) and Bailey et al. (998) on a uni ed model for manoeuvring in a seaway, an approach enaling this consists of using force transfer functions to represent the rst-order wave excitation forces and express the uid memory e ects associated with the radiation forces in a state-space form. By doing so, other e ects like viscous forces, wave drift, current, and wind loads, multi-ody interactions, and ship motion control systems can e incorporated into the model y means of force superposition. This modular approach is the core philosophy of the MSS. To facilitate this, the model presented y Fossen and Smogeli (4) is here extended to incorporate current, wind and second-order wave drift forces, as well as external forces from interacting systems. The coordinate transformations from the reference

2 frames used in hydrodynamics to the ody- xed coordinates used for control and the conversion of the convolution integrals associated with potential damping terms into a state-space form are put into a rigid mathematical framework, and simulation results for a particular vessel are used to illustrate the use of the models in the MSS. EQUATIONS OF MOTION This section presents the 6 degrees-of-freedom (DOF) equations of motion using vectorial mechanics. Emphasis is placed on keeping the kinematics nonlinear while linear theory is assumed for the hydrodynamic forces and moments. In the following, a linear velocity vector in the point O decomposed in reference frame n is denoted as vo n ;and the angular velocity of frame with respect to frame n decomposed in frame n and are denoted! n n and! n respectively.. Coordinate Systems Three orthogonal coordinate systems are used to descrie the vessel motions in 6DOF; see Figure. North-East-Down frame ( n-frame): The n- frame X n Y n Z n is assumed xed on the Earth surface with the X n -axis pointing North, the Y n - axis pointing East, and the Z n -axis down. It is considered inertial. The n-frame position p n = [n; e; d] > (north, east, down) and Euler angles = [; ; ] > (roll, pitch, yaw) are de ned in terms of the generalized coordinate vector (Fossen, ): = [(p n ) > ; > ] > =[n; e; d; ; ; ] > : () Hydrodynamic frame ( h-frame): The hydrodynamic forces and moments are de ned in a steadily translating hydrodynamic coordinate system X h Y h Z h moving along the mean path of the ship with the constant speed U with respect to the n-frame. For DP, U = : The X h Y h -plane is parallel to the mean water surface, the Z h -axis is positive downwards, the Y h -axis is positive towards staroard, and the X h -axis is positive forwards, coinciding with the time-average yaw angle of the vessel. The h-frame is considered inertial, and the ship carries out oscillations aout the steadily translating frame X h Y h Z h : The h-frame origin is denoted W while the h-frame generalized position vector (descriing the incremental displacement and rotations of the vessel with respect to the h-frame) is: = [ ; ; 3 ; 4 ; 5 ; 6 ] > : () Note that the incremental heave position 3 = d. The incremental Euler angles, which take the h-frame into the orientation of the -frame, are de- ned in terms of the Euler angles, ; and according to: " # " # " # 4 = 5, = ; (3) 6 Figure : De nitions of coordinate systems with origins: W (h-frame), O (-frame), and N (n-frame). T is the draught. such that represents the wave-frequency oscillations aout a slowly-varying (constant for straightline motion) yaw angle. Body- xed frame ( -frame): The -frame X Y Z is xed to the hull. The coordinate origin is denoted O and is located on the center line a distance LCO relative to L pp = (positive ackwards) and a distance VCO relative to the aseline (positive upwards), where L pp is the length etween the perpendiculars. The center of gravity G with respect to O is located at r g = [x g ; y g ; z g ] > ; while the h-frame origin W with respect to O is located at r w = [x w ; y w ; z w ] > : The X -axis is positive toward the ow, the Y -axis is positive towards staroard, and the Z -axis is positive downwards. Consequently, the ody- xed -frame carries out oscillations aout the steadily translating h-frame. The -frame translational velocities vo = [u; v; w] > (surge, sway, heave) in O and angular velocities of the -frame with respect to the n-frame expressed in the -frame! n = [p; q; r]> (roll, pitch, yaw) are de ned in terms of the generalized velocity vector (Fossen, ): = [(v o )> ; (! n )> ] > = [u; v; w; p; q; r] > : (4) Generally, the speci c hydrodynamic program used to calculate the hydrodynamic properties of the vessel de nes a local reference frame with axes di erent from the h-frame. Consequently, the data sets from this program must e transformed to the h-frame y using rotation matrices.. Kinematics ( -frame to n-frame) From Fossen (), the translational velocity transformation etween the - and n-frame is: v n o = R n ()v o; (5) where v n o = _p n, and the Euler angle rotation matrix (zyx-convention) etween the n- and -frame R n () R33 is de ned as: R" n () = c c s c + c ss s s + c cs s c s c c + sss cs c s + ss c cc # : (6)

3 Here, s = sin() and c = cos(). Note that the rotation matrix R a etween any two frames a and (from a to ) has the special properties that (R a) = (R a) > = R a : The Euler angle rates satisfy: _ = T ()! n; (7) where T () R 33 is the Euler angle attitude transformation matrix: " # st ct T () = c s ; 6= s=c c=c ; (8) and t = tan(). Consequently: _ = J(); (9) where J() R 66 is the generalized velocity transformation matrix: R n J() = () 33 ; 6= 33 T () : ().3 Kinematics ( -frame to h-frame) The transformation of a generalized velocity in the -frame to a generalized velocity _ = [(vw) h > ; ( _ ) > ] > in the h-frame is done in two steps: a rotation and a translation. The translational velocity vw of W and the angular velocity of the -frame with respect to the h-frame! h decomposed in the -frame are given y: v w = v o +! n r w; ()! h =! n: () where r w is the vector from O to the mean position of W. The angular velocity of the h-frame with respect to the n-frame! nh is approximately zero, since the h-frame is considered inertial (only a slowly-varying yaw angle is assumed). The vector cross product is de ned in terms of the matrix S(r w) R 33 such that: where:! n r w, S(r w)! n = S(r w) >! n; (3) S(r w) = S > (r w) = " zw y w z w x w y w x w # : (4) Introducing the screw transformation H(r w) R 66 : H(r w), I33 S(r w) > ; (5) 33 I 33 it is clear that: v w! h = H(r w) v o! n : (6) The rotation matrix from the -frame to the h- frame is de ned as R h ( ) such that: v h w = R h ( )v w: (7) Similarly, the angular velocity transformation from the -frame to the h-frame ecomes: _ = T ( )! h: (8) Note that _ 6=! h h, since! h h = R h ( )! h. From (6), (7), and (8) it follows that: v h w R _ = h ( ) 33 v w 33 T ( )! (9) h R = h ( ) T ( H(r ) v o w)! : n The generalized velocity transformation etween the h and frames then ecomes: _ = J ( ); () where J ( ) R 66 is a generalized velocity transformation matrix: J ( R )= h ( ) R h ( )S(r w) > 33 T ( : () ) For most applications, the roll and pitch oscillations will e small, i.e., such that: R h ( ) I 33 ; (a) T ( ) I 33 ; () J, J () = H(r w ); (c) where now J is a constant matrix. Assumption A The oscillations of the - frame with respect to the h-frame are small such that () holds. Note that assumption A only applies to the transformation of forces and matrices etween the -frame and the h-frame, while the nonlinear kinematics etween the n-frame and the -frame (9) are preserved and hence valid for large Euler angles..4 Kinetics The generalized forces acting on the vessel are found y formulating Newton s nd law in -frame coordinates. Since the hydrodynamic forces and moments are computed in h-frame coordinates these will e transformed to the -frame. Note that using () and assumption A, a generalized force (containing forces and moments) in the - frame can e transformed to the generalized force in the h-frame y: = (J ) : (3) 3

4 The 6DOF rigid-ody equations of motion in the -frame are (Fossen, ): M RB _ + C RB () = H + ; (4) where H R 6 contains generalized hydrodynamic forces, R 6 is a generalized vector containing environmental, propulsion and external forces, C RB R 66 is the rigid-ody Coriolis and centripetal matrix, and M RB R 66 is the rigid-ody system inertia matrix: mi33 ms(r M RB = g) ms(r ; (5) g) I o where m is the vessel mass, r g is the vector from the center of gravity to the ody frame origin, and I o = I > o > is the inertia tensor. The generalized force vectors and inertia matrix are computed with respect to the coordinate origin O. Assumption A The Coriolis and centripetal forces due to a rotating -frame and added mass are neglected in the following. Assumption A is justi ale in low-speed applications like DP, where these terms are dominated y damping and feedack. However, they may e easily included in the nal model if desired. Under assumption A and A (i.e. C RB = 66 and small), the -frame equations of motion can e transformed to the h-frame y using (): = J _; (6) Pre-multiplying y (J ) >, (4) can e written: (J ) > M RB (J ) = (J ) > ( H + ): (7) The generalized forces in the h-frame are de ned as H = (J ) > H and = (J ) > such that: M RB = H + ; (8) where M RB, (J ) > M RB (J ). 3 HYDRODYNAMIC FORCES The generalized hydrodynamic forces in the h- frame can e expanded according to Faltinsen (99): H = HS + R + W ; (9) where HS is the generalized hydrostatic force, R is the generalized radiation force representing uid memory e ects, and W is the generalized rstorder wave excitation force, consisting of Froude- Krylov and di raction forces. The generalized hydrostatic forces, contriuting only in heave, roll, and pitch, are written as: HS = g (): In the traditional hydrodynamic models it is assumed that the response of the vessel is linear. Thus superposition can e applied, and the hydrodynamic forces can e computed in the frequency domain. Then, for sinusoidal excitation and motion, the generalized radiation forces can e written as R = A (!) B (!)_; (3) where A (!) R 66 is the frequency dependent added mass and B (!) R 66 is the frequency dependent potential damping. Note that under assumption A, B (!) is de ned for zero forward speed such that the Coriolis and centripetal terms due to added mass are zero and B = B. Inserting (3) in the equations of motion (8), the h-frame representation ecomes: (M RB + A (!)) + B (!) _ + g () = W + : (3) Hydrodynamic programs like WAMIT (4), ShipX (VERES) (Fathi, 4), SEAWAY (Journée and Adegeest, 3), etc., can e used to compute A (!) and B (!) in terms of coe cient tales, and the generalized rst-order wave excitation force W in terms of force transfer functions. These terms are all computed in the h- frame. The radiation-induced forces and moments R are functions of frequency and time. In the next section another form of the equations of motion, valid for more general excitation forces, is presented. 3. Time-Domain Representation From Cummins (96) and Ogilvie (964), the frequency dependent terms A (!) and B (!) can e removed from (3) y writing the equations of motion in the following form: (M RB + A ()) + B () _ + +g () = W + ; (3) where A () = A () R 66 and B () are constant matrices evaluated at the in nite frequency, and is a potential damping term de ned as:, Z t K (t ) _ (t)d: (33) K () R 66 is a matrix of retardation functions, and may e expressed as: K () = Z (B (!) B ()) cos(!)d!: (34) Note that K () can e computed o -line using the B (!) data set. For details, including how to add viscous damping and additional numerical considerations, consult Fossen and Smogeli (4). Kristiansen and Egeland (3) have proposed a state-space formulation for the frequency dependent damping term in (3). Since for causal systems K (t ) = for t <, (33) is rewritten 4

5 as: (t) causal = Z t K (t ) _ (t)d: (35) If _ is a unit impulse, then (t) given y (35) will e an impulse response function. Consequently, (t) can e approximated y a linear state-space model: _ = A r + B r _ ; () = ; = C r + D r _ ; (36) where (A r ; B r ; C r ; D r ) are constant matrices of appropriate dimensions. Transforming (3) to the -frame, the equations of motion ecomes: M _ + D + + g() = W + ; (37) where the following transformations ased on assumption A are used: M = M RB +J > A ()J ; D = J > B ()J ; = J > ; g() = J > g (); W + = J > ( W + ); (38a) (38) (38c) (38d) (38e) Notice that the properties M = M > > and _M = also holds for this model since the generalized added mass matrix A () is constant and symmetric. The resulting nonlinear state-space model from (9), (37), and (36) is: _ = J(); (39a) M _ = D J > (C r + D r J ) {z } g() + W + ; (39) _ = A r + B r {z} J ; () = : (39c) _ 4 ENVIRONMENTAL MODELS 4. Waves Irregular waves are commonly descried y a wave spectrum S(!; ) = S(!)D(), where the frequency spectrum S(!) descries the energy distriution of the sea state over di erent frequencies!, and the spreading function D() descries the distriution of wave energy over directions in the n-frame. Common frequency spectra and spreading functions may e found in e.g. Ochi (998). For simulation, the sea state is realized as a superposition of harmonic components extracted from the wave spectrum, where the harmonic component j is de ned in terms of the amplitude j, frequency! j, direction j, and random phase j. 4. Wind The wind is commonly parameterized y the velocity U w (z; t) = U w (z) + U g (t) and the direction w in the n-frame, where U w (z) is a height dependent mean value and U g (t) is a uctuating gust component de ned from a wind gust spectrum. Commonly used gust spectra are the Harris wind spectrum (Det Norske Veritas, ) and the NORSOK spectrum (NORSOK, 999). 4.3 Current For surface vessels, only the surface current is of any importance. If the current is given y magnitude U c and direction c in the n-frame, the current velocity vector relative to the vessel v c may e written as v c = U c [cos( c ); sin( c ); ] : (4) 5 LOAD MODELS This section details the implementation of the rst-order wave loads W, and introduces additional environmental loads E from current, wind, and second-order wave drift forces. How to include external forces from interacting systems I (mooring systems, risers etc.) is also considered. Propulsion forces P are not discussed further, since accurate propulsion models are outside the scope of this paper. The generalized force vector in (4) is therefore rewritten according to: = E + P + I : (4) 5. Transformation tools The inclusion of the forces in the -frame equations of motion (39) typically requires transformations of positions, velocities, accelerations, and forces from one coordinate to another. Following the approach in Section.3, the position x n p, velocity v p and acceleration _v p of a point p located at r p in the -frame can e expressed as: x n p = + R n ()r p; (4) v p = H(r p); (43) _v p = H(r p) _; (44) where R n () and H(r p) are de ned y (6) and (5) respectively. Correspondingly, the generalized force vector p with point of attack p is transformed to the origin O y: = H (r p) p : (45) For details, see Fossen (). These tools can e used as input to an interacting system, e.g. to nd the velocity and accelerations of the tip of a crane. The force generated y an interacting system, e.g. the force on the crane tip from a suspended load, can then e transformed ack to the origin of the - frame, and e included in the equations of motion. 5

6 5. First-order wave forces The rst-order wave excitation forces in the h- frame W = [ W ; ; 6 W ] are calculated from the force transfer functions T i (! j ; j ) (for DOF i), which are outputs from a hydrodynamic analysis program and taulated as functions of the wave frequency! j and the wave heading relative to the vessel rj = j. The total excitation force is the sum over N harmonic wave components, i.e. for i = ::6: i W = NX j T i j cos(!j t + j + arg(tj i )): (46) j= 5.3 Wave drift forces The wave drift loads are a signi cant part of the total excitation forces, and may divided into a mean and a slowly varying component. For long-crested waves, the determination of the drift forces can e done y means of a set of approximations given in Newman (974) and Faltinsen (99). A hydrodynamic program like WAMIT (4) can determine the quadratic transfer functions Tjj i (for wave component j and DOF i) from rst-order potential theory, with Tjj i taulated as functions of! j and rj. For i = ::6, the total wave drift force i W is then given y: i W = NX NX j= k= j k T i jk cos (! k! j ) t + ' k ' j ; (47) where ' j is a phase angle and Tjk i = Tkj i = T i jj + Tkk i. Note that the mean wave drift force given y j = k is valid for oth long- and shortcrested waves, whereas the slowly-varying drift force for j 6= k only is valid for long-crested waves. The resulting wave drift load vector in the -frame W is found from (45) using the vector from O to the point of attack of the wave drift forces r W : W = H (r W )[ W ; ; 6 W ] : (48) 5.4 Nonlinear damping/current forces The linear drag component from the current is included in the equations of motion (39) y replacing the vessel velocity in the damping term D with the relative velocity r : r = v c: (49) Quadratic damping is not as easily included, ut several options exist. In surge, the viscous damping f c may e modelled y e.g. (Lewis, 989): f c = w Su r( + k)(c f + C f ); (5) :75 C f = (log R n ) ; (5) where w is the density of water, S is the wetted surface of the hull, u r = u U c cos( c ) is the relative velocity in surge, k is the form factor giving a viscous correction, C f is the at plate friction from the ITTC 957 line, C f represents friction due to hull roughness, and R n is the Reynolds numer. For relative current angles cr = j c j the cross ow principle (Faltinsen, 99) may e applied to calculate the nonlinear damping force in sway f c and moment in yaw f c6 : f c = Z w T (x)c D (x)vr x (x)jvr x (x)jdx; (5) L pp f c6 = Z w xt (x)c D (x)vr x (x)jvr x (x)jdx:(53) L pp Here, T (x) is the draft, C D (x) is the dimensional drag coe cient, and v x r (x) = v r + rx is the relative cross- ow velocity at x. v r = v U c sin( c ) is the relative velocity in sway, and r is the angular velocity in yaw. Drag coe cients for di erent hull forms may e found in e.g. Hooft (994). From (45) the resulting nonlinear damping term in the -frame d( r ) ecomes: d( r ) = H (r c)[ f c f c f c6 ] ; (54) where r c is the vector from O to the point of attack of the drag forces c. A typical value could e r c = [; ; T=]. If availale, quadratic drag coe cients from experiments or a CFD analysis may replace the ITTC drag and cross- ow formulations. The relative generalized velocity r should replace in all hydrodynamic computations. 5.5 Wind forces The wind angle relative to the vessel is de ned as wr = w : The wind drag coe cients C w, C w, and C w6 in surge, sway, and yaw are taulated as functions of wr, giving the wind forces f w, f w, and f w6 : f wi = :5 a C wi ( wr )A wi U w ju w j ; (55) where i f; ; 6g, a is the density of air, A w and A w are the transverse and lateral projected areas of the hull aove the waterline and the superstructure, and A w6 = A w L oa, where L oa is the overall length of the vessel. Wind drag coe cient tales may e found in e.g. Blendermann (986). The resulting wind load vector in the -frame wind is found in a similar manner as in (54) using the vector from O to the point of attack of the wind forces r wind. 6 THE RESULTING MODEL Comining the nonlinear state-space model (39) and the load models from Section 5, the resulting 6

7 Tale : The S-75 container ship main particulars. Length etween perpendiculars L pp 75 m Beam B 5.4 m Draught T 9.5 m Displaced volume r 44 m 3 Block coe cient C B.57 LCG relative to midships -.48 m 8 CONCLUSIONS Figure : Block diagram of a DP operation as modelled in MSS, with focus on the vessel dynamics. vessel model implemented in the MSS is: _ = J(); M _ = D r d( r ) J > (C r + D r J r ) g() + W + W + wind + P + I ; _ = A r + B r J r ; () = : (56) Figure shows a lock diagram of a DP operation as given y (56), focusing on the vessel dynamics with the rigid-ody equations of motion, and forces from nonlinear damping, radiation, rstorder wave excitation, wave drift, wind, propulsion and external loads. 7 SIMULATION RESULTS Simulations were performed with the S-75 container ship, with main particulars given in Tale. ShipX (VERES) (Fathi, 4) was used to compute the frequency dependent added mass and damping, as well as the rst-order exciting wave force transfer functions. For details on the frequency dependent coe cients and computation of the retardation functions, consult Fossen and Smogeli (4). The simulation shows a DP operation with mean wave direction 45 degrees o the ow. The sea state was generated from the JONSWAP wave spectrum with signi cant wave height H s = 4 m, wave peak frequency! p = :6 rad/s, spectrum peakedness factor = 3:3, and a directional spreading function giving short-crested waves. At time t = s a crane load of kn is applied at the point r p = [; 5; ]. Figure 3 shows time series of the 6DOF generalized rstorder exciting wave forces W and the radiation forces R, and Figure 4 shows time series of the generalized position vector. Even though the crane load is applied as a step, the force superposition principle leads to physically natural vessel motions: the main e ect is a new mean roll angle. A nonlinear state-space model for surface vessels performing station-keeping and low-speed manoeuvring in a seaway has een presented. The model representation using force superposition rather than motion superposition was a key aspect, since it allowed a modular description of complex marine operations. This enaled the interaction of various physical systems like vessels, cranes, risers, and motion control systems. The presented models have een implemented in the Marine Systems Simulator (MSS) developed at NTNU. 9 ACKNOWLEDGMENT This work has een carried out at the Centre for Ships and Ocean Structures (CESOS) at NTNU, sponsored y the Norwegian Research Council. REFERENCES Bailey, P. A., W. G. Price and P. Temarel (998). A Uni ed Mathematical Model Descriing the Maneuvering of A Ship Travelling in a Seaway. Trans. RINA 4, Bishop, R. E. D. and W. G. Price (98). On the Use of Equilirium Axes and Body Axes in the Dynamics of a Rigid Ship. Journal of Mechanical Engineering Science 3(5), Blendermann, W. (986). Die Windkräfte am Schi : Bericht Nr Institut für Schi au der Universität Hamurg. In German. Cummins, W. E. (96). The Impulse Response Function and Ship Motions. International Symposium on Ship Theory, Hamurg, Germany. Det Norske Veritas (). Classi cation Notes No. 3.5: Environmental Conditions and Environmental Loads. DNV, Norway. Faltinsen, O. M. (99). Sea Loads on Ships and O shore Structures. Camridge Ocean Technology Series, Vol. 6. Camridge University Press. Fathi, D. (4). ShipX Vessel Responses (VERES) Users Manual. Marintek AS, Trondheim, Norway. Fossen, T. I. (). Marine Control Systems: Guidance, Navigation and Control of Ships, Rigs and Underwater Vehicles. Marine Cyernetics AS. Trondheim, Norway. 7

8 τ 6 [MNm] ψ [deg] τ 5 [MNm] θ [deg] τ 4 [MNm] φ [deg] τ 3 [MN] z [m] τ [MN] y [m] τ [MN] x [m] time [s] Figure 3: Time series of the 6DOF generalized rstorder wave exciting forces W (dotted) and radiation forces R (solid). 5 5 time [s] Figure 4: Time series of the generalized position vector = [x; y; z; ; ; ]. Fossen, T. I. and Ø. N. Smogeli (4). Nonlinear Time-Domain Strip Theory Formulation for Low-Speed Maneuvering and Station- Keeping. Modelling Identi cation and Control (MIC) 5(4),. Hooft, J. P. (994). The Cross-Flow Drag on a Manoeuvring Ship. Ocean Engineering (3), Journée, J. M. J. and L. J. M. Adegeest (3). Theoretical Manual of Strip Theory Program "SEAWAY for Windows". Amarcon. Delft University of Technology. DUT-SHL Report 37. Kristiansen, E. and O. Egeland (3). Frequency- Dependent Added Mass in Models for Controller Design for Wave Motion Damping. In: IFAC Conference on Maneuvering and Control of Marine Systems (MCMC 3). Girona, Spain. Lewis, Edward V., Ed.) (989). Principles of Naval Architecture. nd ed.. Society of Naval Architects and Marine Engineers (SNAME). MSS (4). Marine Systems Simulator. Norwegian University of Science and Technology, Trondheim, Norway, < Newman, J. N. (974). Second-Order, Slowlyvarying Forces on Vessels in Irregular Waves. In: Proc. Int. Symp. Dynamics of Marine Vehicles and Structures in Waves. London. pp NORSOK (999). NORSOK STANDARD N-3: Actions and Action E ects. < Ochi, M. K. (998). Ocean Waves, The stochastic Approach. Camridge Ocean Technology Series, Vol. 6. Camridge University Press. Ogilvie, T. (964). Recent Progress towards the Understanding and Prediction of Ship Motions. In: 5th Symposium on Naval Hydrodynamics. Sørensen, A. J., E. Pedersen and Ø. N. Smogeli (3). Simulation-Based Design and Testing of Dynamically Positioned Marine Vessels. In: Proc. Int. Conference on Marine Simulation And Ship Maneuveraility (MARSIM 3). Kanazava, Japan. WAMIT (4). WAMIT User Manual. WAMIT, Inc., MA, USA, < 8