Nonlinear Observer-Based Ship Control and Disturbance Compensation

Transcription

1 Preprints, 10th IFAC Conference on Control Applications in Marine Systems September 13-16, Trondheim, Norway Nonlinear Observer-Based Ship Control and Disturbance Compensation Harald Aschemann Stefan Wirtensohn Johannes Reuter Chair of Mechatronics, University of Rostock, D Rostock, Germany (Tel: +49 (0) ; Fax: +49 (0) ; Institute for System Dynamics, HTWG Konstanz, D Konstanz, Germany, (Tel: +49 (0) ; {stwirten, Abstract: In this paper, a gain-scheduled nonlinear control structure is proposed for a surface vessel, which takes advantage of extended linearisation techniques. Thereby, an accurate tracking of desired trajectories can be guaranteed that contributes to a safe and reliable water transport. The PI state feedback control is extended by a feedforward control based on an inverse system model. To achieve an accurate trajectory tracking, however, an observer-based disturbance compensation is necessary: external disturbances by cross currents or wind forces in lateral direction and wave-induced measurement disturbances are estimated by a nonlinear observer and used for a compensation. The efficiency and the achieved tracking performance are shown by simulation results using a validated model of the ship Korona at the HTWG Konstanz, Germany. Here, both tracking behaviour and rejection of disturbance forces in lateral direction are considered. Keywords: trajectory tracking, disturbance rejection, gain scheduling, extended linearisation, ship control 1. INTRODUCTION Despite the nonlinear character of the corresponding mathematical models for the ship dynamics, cf. Fossen [1994], PI and PID output feedback have been often used for ship control. Nevertheless, nonlinear approaches have been proposed in the last decade that fully address the system dynamics and allow for a superior tracking accuracy. Here, nonlinear designs based on differential flatness (Sira-Ramirez [2005]), feedback linearisation (Han and Zhao [2003]), backstepping techniques (Pettersen and Nijmeijer [1998]), and approaches for the exponential stabilisation using Lyapunov functions as in Han and Zhao [2003], Reyhanoglu [1997] have to be cited. Moreover, neural network-based control techniques have been investigated, where classical state space or output feedback controllers are either replaced by or combined with neural controllers, cf. Grigoryev et al. [2010]. Recently, an integral line-of-sight controller has been proposed by Caharija et al. [2015]. In this paper, a model-based strategy for a combined feedforward and feedback control is proposed that makes use of extended linearisation techniques. It extends earlier work, cf. Aschemann and Rauh [2010], by a refined ship model that explicitly addresses nonlinear damping effects. In Fossen [2000], a passivity-based observer for wave filtering has been developed. This concept is altered in this paper by using a reduced-order observer approach that counteracts both wave-induced measurement disturbances and a lateral disturbance force. This paper is structured as follows: In Sect. 2, the mathematical modelling of the surface vessel Korona from the HTWG Konstanz, Germany, which is shown in Fig. 1, is discussed in detail. Sect. 3 presents extended linearisation techniques for the design of a gain-scheduling feedforward and feedback Fig. 1. Solar surface vessel Korona (HTWG Konstanz, Germany) control approach. Sect. 4 focuses on the design of a reducedorder disturbance observer that estimates both wave-induced measurement disturbances as well as an unknown lateral disturbance force. These estimates are used for a subsequent disturbance compensation that is important for an accurate trajectory tracking. In Sect. 5, numerical simulation results are shown to demonstrate the applicability of the selected approaches to the control of the ship Korona. Finally, this paper is concluded with an outlook on future work in Sect MATHEMATICAL MODELLING OF SHIP MOTIONS 2.1 Kinematic Modelling Usually, a local earth-fixed coordinate system (x I,y I ) is employed to specify the motion task. This local coordinate system is connected to a segment describing the geometric path the ship should follow. The overall geometric path is given as a sequence of straight-line segments, where the corresponding Copyright 2016 IFAC 297

2 inertial coordinate system is valid only for a segment and switching is necessary with consistent initial conditions when changing the relevant segment. With the ship-fixed coordinates (x, y) and the corresponding velocities (u, v) in this coordinate system, the kinematics of a ship can be described by a three DOF manoeuvring model ẋ I = u cos(ψ) v sin(ψ), (1) ẏ I = u sin(ψ) + v cos(ψ), and (2) ψ = ω. (3) The corresponding angular velocity of the ship is given by ω, see Fig. 2. In (1) (3), the variable ψ represents the orientation of the ship with respect to the local earth-fixed coordinate system. y I R y v u 0 x I Fig. 2. Definition of earth-fixed and body-fixed coordinate systems for the description of ship motions. 2.2 Dynamic Modelling In the sequel, the mathematical modelling of the solar surface vessel Korona is presented briefly, see Fig. 1. The Korona, which is 7.2 m long and 2.5 m wide, serves as an experimental platform for ship control experiments at the Institute of System Dynamics at the HTWG Konstanz, Germany. In addition to several environmental sensors it is equipped with a differential GPS and automotive grade MEMS gyroscopes for each axis. The data streams are fused by using an Unscented Kalman Filter. The underacted propulsion system consists of a fixed propeller and a single rudder, which both can be controlled by a rapid prototyping control unit. To derive a set of ordinary differential equations for the (angular) velocities u, v, and ω in surge, sway, and yaw, respectively, it is necessary to balance the forces and torques acting on the ship, see Fossen [1994], Reyhanoglu [1997]. In the equations of motion u = v ω d 11 u d 11q u u + 1 f u, (4) m 11 m 11 m 11 m 11 v = m 11 u ω d 22 v d 22q v v f v, (5) ω = m 11 u v d 33 ω d 33q ω ω + 1 τ ω, (6) m 33 m 33 m 33 m 33 roll, pitch and heave motions are not considered due to the assumption that they have only little influence on the manoeuvre dynamics of the ship for the considered operational scenarios. The parameters m ii > 0, i = {1,2,3}, represent the ship mass m with additive hydrodynamic masses m xx and m yy in x- and y- direction according to m 11 = m + m xx and = m + m yy. The mass moment of inertia with respect to the centre of gravity x (COG) is given by m 33 = I z +I hz. This corresponds to a diagonal mass matrix and accounts for the most important terms. The parameters d ii > 0 as well as d iiq > 0 describe damping effects covering the whole operational range of the Korona. Since the Korona is not actuated in the sway direction, only the force f u and the torque τ ω can be provided by the actuators. The lateral force f v is unknown and represents environmental disturbances as well as non-modelled effects. The unknown model parameters have been determined based on real test data. Different maneuvers with varying excitations were carried out to reflect the dynamic of the Korona over a wide range. For more information concerning the identification and the validation of the ship model, see Wirtensohn et al. [2013]. The identified model parameters are stated in Table B Wave-Induced Motion To model the influence of the wave-induced motion on the position and heading measurements, cf. Fossen and Perez [2009], second-order models driven by a white noise for each degree of freedom are used ξ i = 2ζ ω 0 ξi ω 2 0 ξ i + Ω i, (7) with the dominating wave frequency ω 0, wave damping ζ and the noise input Ω i. Here, the index i {ψ,v,u} denotes the three DOFs. The corresponding numerical values can be found in Table B.2. Each measured state variable is, hence, superimposed with a wave disturbance η wi = ξ i,i {ψ,v,u}. 2.4 Propulsion The propulsion system of the Korona consists of a fixed propeller and a single rudder. In this subsection, the modelling of the rudder and propeller force is discussed briefly. The thrust force F p = c 1 ρd 4 pn n c 2 ρd 3 pu a n (8) can be mathematically described as a function of the rotational speed of the propeller n and the speed of advance u a, cf. Blanke [1981]. Here, D p is the propeller diameter, and c 1 and c 2 are determined from an open water diagram of the propeller. The modelling of the rudder force is based on the work of Gronarz [1997], where the force acting in perpendicular direction on the rudder area is given by F R = 1 2 C FNρ(A R V 2 R sin(δ er ) + A P V 2 P sin(δ ep )), (9) with the geometric constant C FN. The whole rudder area can be separated in the area A R which is not affected by the propeller and the area A P which is affected by the propeller. The inflow of the corresponding areas can be described with the inflow velocities V R and V P and the angles of the fluid inflow δ er and δ ep. Using (8) and (9), the resulting actuator force and torque become f u = F p F R sin(δ R ), (10) τ ω = F R cos(δ R )x R, (11) with the rudder angle δ R and the distance x R from the center of rudder to the origin of the body-fixed frame. The corresponding relations and parameters of this model can be found in the appendix in Table B.2. It has to be mentioned that the nonlinear mappings f u to δ R and τ ω to r = ω - representing inverse actuator characteristics - have been calculated numerically and stored in look-up tables. 298

3 3. CONTROL DESIGN USING EXTENDED LINEARISATION 3.1 Cascaded Feedback Control Structure For trajectory control, a cascaded approach using partial feedback linearisation is chosen: the ship velocity u(t) in longitudinal direction is controlled by an underlying fast control loop using the propeller force f u (t). Similarly, the angular velocity ω(t) = ψ(t) is controlled in a subsidiary nonlinear control loop employing the rudder torque τ ω (t). The inverse dynamics of the two control laws can be stated as f u = vω + d 11 u + d 11q u u + m 11 υ 1 and, (12) τ ω = ( m 11 )uv + d 33 ω + d 33q ω ω + m 33 υ 2, (13) respectively. Thereby, all nonlinear couplings and the damping terms are compensated for. The resulting longitudinal dynamics becomes a single integrator, which can be asymptotically stabilised by an underlying PI controller t υ 1 = u d + k Pu (u d u) + k Iu 0 (u d u)dτ, (14) with an appropriately chosen proportial gain k Pu, integral gain k Iu, and the desired ship velocity u d as well as the desired acceleration u d in longitudinal direction. In the same manner, the yaw dynamics is asymptotically stabilised by t υ 2 = ω d + k Pω (ω d ω) + k Iω 0 (ω d ω)dτ, (15) with the proportial gain k Pω, the integral gain k Iω, and the desired trajectories for the yaw motions ω d as well as ω d. As a consequence, these controlled kinematic variables can be considered as kinematic control inputs in a superordinate control structure. Note that the integral control parts counteract both disturbances and model uncertainties in the corresponding error dynamics. The remaining nonlinear subsystem describes the dynamics of the ship in y I -direction, where the desired path of the ship is assumed to be oriented in x I -direction. The nonlinear dynamic model can then be stated by the following quasi-linear state equations [ẏi ] 0 cos(ψ) u si(ψ) [ ] yi 0 v = 0 d 22+d 22q v 0 v + m 11 u u y + e f v, ψ ψ 1 (16) with the function si(ψ) = sin(ψ)/ψ, where si(0) = 1. Furthermore, the disturbance input vector e [ T e = 0 m ], (17) the lateral disturbance force f v due to cross currents or wind, and the kinematic control input u y = ω are introduced. It becomes obvious that the system matrix depends on the state variables ψ and the sway velocity v as well as the longitudinal velocity u. The system represents a fully controllable quasilinear system, which can be easily shown by computing the controllability matrix Q ys = [ b y A y b y A 2 yb y ]. (18) Checking Kalman s rank condition point-wise shows clearly that rank(q ys ) = n = 3 holds within the entire workspace. 3.2 Eigenvalue Assignment Using Extended Linearization Without the disturbance input f v, which is considered separately at the disturbance observer design, the quasi-linear dynamical system to be stabilised is given by ẋ y (t) = A y (ψ,v,u)x y (t) + b y (u)u y (t), y y (t) = c T (19) y x y (t), leading to state- and parameter-dependent matrices or vectors A y = A y (ψ,v,u), b y = b y (u), and c T y. The design approach involves the symbolic computation of a state-dependent gain vector k T y by a comparison of the desired characteristic polynomial specifying the eigenvalues s = s Ri,i {1,2,3} of the closed-loop system p R (s) = (s + s R1 )(s + s R2 )(s + s R3 ), (20) with the actual characteristic equation p(s) = det ( si 3 A yr ( ψ,u,k T y )). (21) Here, I 3 is the 3 3 identity matrix and A yr the system matrix A yr ( ψ,u,k T y ) := Ay (ψ,u) b y (u)k T y (ψ,u) (22) of the closed-loop. As the system is fully controllable for u 0 and si(ψ) 0, a unique solution for k T y = [k R1 k R2 k R3 ] (23) exists in the single-input single-output case such that the actual and desired characteristic equations of the closed-loop system have identical roots. For example, the first control gain depends on the angle ψ and the velocity u and becomes k R1 = s R1s R2 s R3 u si(ψ) d 22. (24) The possible singularities are not critical and can be avoided by model-based trajectory planning. The stability of the timevarying system can be shown in a multi-model approach based on a joint quadratic Lyapunov function V (x y ) = x T y P x y. (25) Here, P = P T > 0 is the symmetric and positive definite solution of a set of Lyapunov inequalities corresponding to the multimodel set. A sufficient condition for asymptotic stability is the negative definiteness of all matrices A T yrip + PA yri < 0, (26) where A yri = A y b y k T y represents matrix i of the multimodel set. Additionally, thorough simulation studies have been performed to assess both the closed-loop stability and the tracking performance. 3.3 Feedforward Control Design Using Extended Linearisation For feedforward control design, the lateral position of the ship y I (t) with respect to the reference direction is considered as the controlled variable. Thus, the output equation becomes y(t) = y I (t) = [ ] x y (t) = c T y x y (t). (27) The control transfer function can be derived as Y (s) U y,ff (s) = ( ct y si Ay + b y k T ) 1 y by = (b 0 + b 1 s). (28) N (s) Obviously, the numerator of the control transfer function contains a first-order polynomial in s, leading to one transfer zero. This shows that the considered output represents a non-flat output variable that makes feedforward control design more difficult. 299

4 The calculation of the feedforward control law u y,ff involves a modification of the numerator of the control transfer function by introducing a polynomial ansatz for the feedforward control action in the Laplace domain according to U y,ff (s) = [ k V 0 + k V 1 s + k V 2 s 2 + k V 3 s 3] Y d (s). (29) For its realisation the desired trajectory y Kd (t) as well as the first three time derivatives are available from a trajectory planning module. The feedforward gains can be computed from a comparison of the corresponding coefficients in the numerator as well as the denominator polynomials of Y (s) Y d (s) = (b 0 + b 1 s) [ kv k V 3 s 3] N (s) = b V 0 (k V j ) + b V 1 (k V j ) s b V 4 (k V j ) s 4 a 0 + a 1 s + a 2 s 2 + s 3, (30) which leads to a i = b Vi (k V j ),i {0,...,3}. (31) This leads to parameter-dependent feedforward gains k V j = k V j (ψ,u). The first two feedforward gains, e.g., are given by k V 0 = k R1, k V 1 = k R3 u si(ψ). (32) It is obvious that related to the higher numerator degree in the modified control transfer function a remaining dynamics must be accepted. Though perfect tracking could not be achieved due to the transfer zero of the open-loop system, this easily implementable feedforward control contributes significantly to an improved tracking behaviour. Besides the application mentioned above, the model-based feedforward control law can be employed offline to investigate the feasibility of desired state and output trajectories with respect to constraints on the control inputs. If such constraints are violated, a time scaling of the desired trajectory could be applied to comply with given constraints on the angular velocity u y = ω. 4. OBSERVER-BASED COMPENSATION OF EXTERNAL DISTURBANCES Disturbance behaviour and tracking accuracy in view of cross currents or wind forces in lateral direction can be improved significantly by introducing a compensating control action provided by a disturbance observer. Note that the resulting disturbance force in longitudinal direction as well as the resulting torque are counteracted by the fast underlying control loops according to (14) and (15). Aiming at a fast observation and a low implementation effort, a reduced-order observer design is proposed. The observer design for the sway direction is based on the state-space description of the lateral dynamics of the ship (16), where the variable f v takes into account the resulting disturbance force in lateral direction. As disturbance model for this slowly varying disturbance force f v, a single integrator ḟ v = 0 is employed. For the wave-induced signal η wv present in the measurement v m = v + η wv, a secondorder oscillator η wv = 2ζ ω 0 η wv ω 2 0 η wv (33) is employed as disturbance model. Given the disturbed sway velocity measurement y m = v m, the reduced-order disturbance observer results in the following state-space representation ẏ m = f m (y m,x e, f v,u y ), ẋ e = A e x e, ḟ v = 0, (34) where ẏ m = v m = f m ( ) is the state equation for the disturbed sway velocity measurement v m. Moreover, x e = [η wv, η wv ] T is the vector of the oscillator states, f v represents the disturbance force and u y = ω denotes the scalar control input. The estimated quantities ˆx e and the disturbance force ˆf v follow from [ ] ˆxe ˆτ = = hy ˆf m + ẑ, (35) v where h = [h 1,h 2,h 3 ] T represents the observer gain vector. The ansatz for the state equations for the observer state vector z is chosen as ẑ = Φ(y m, ˆτ,u y ) (36) The vector of nonlinear functions Φ is determined in such a way that the steady-state observer error τ = τ ˆτ converges to zero. Thus, Φ results from the demand for a vanishing steady-state estimation error according to τ = 0 = τ h ẏ m Φ(y m,τ 0,u y ). (37) Considering the disturbance models ẋ e = A e x e and ḟ v = 0, (37) can be solved for Φ and becomes with f m ( ) = f m (y m,x e, f v,u y ) η wv h 1 f m ( ) Φ(y m,τ, u y ) = ω0 2η wv 2ζ v ω 0v η wv h 2 f m ( ). (38) h 3 f m ( ) Asymptotical stability of the linearised error dynamics is achieved by placing all eigenvalues of the Jacobian in the left complex half-plane according to ( det si Φ(y ) 3 m,τ, u y )! = τ (s + s Bi ). (39) With negative eigenvalues s = s Bi < 0, i {1,2,3}, the observer gains follow directly from Eq. (39). A disturbance compensation is performed by introducing the estimated force ˆf v into the static compensation law u DC = cos(ψ) k R3 u si(ψ)k R2 ˆf v. (40) u si(ψ) d 22 + d 22q v Moreover, the disturbed sway velocity measurement can be corrected by removing the wave-induced measurement disturbance according to ˆv = v m ˆη wv. i=1 5. SIMULATION RESULTS Simulations have been performed to assess both the tracking as well as the disturbance behaviour. Here, the parameters of the validated Korona ship model according to Tab. B.1 have been used. The initial states have been chosen as y I (0) = 0.1 m, v(0) = 0 ms 1, ψ(0) = 0.01 rad, and x I (0) = 0 m. The control structure has been assigned a fixed eigenvalue specification, where all system eigenvalues have been chosen as s Ri = 0.4. The goal of tracking control is to minimize the deviations between the actual and desired positions y I (t) and y Id (t). The non-zero initial conditions at the beginning, cf. Fig. 5, lead to a transient phase, where the desired value y Id = 0 m is reached after approx. 5 s. The ship moves in the body-fixed x-direction, see Fig. 2, where the longitudinal velocity u(t) changes during this motion according to Fig. 3. In the time interval 5 s to 30 s, a desired lateral displacement of the ship in the positive y I -direction by 10 m in form of a smooth polynomial function is specified. From 60 s till 90 s, a lateral motion with a polynomial shaped displacement of 10 m in negative y I - direction is performed. The achieved tracking performance is emphasized by Fig. 5, where the desired and the actual lateral 300

5 Fig. 3. Changing longitudinal velocity u. Fig. 6. Comparison of the disturbance force f v (t) and the corresponding estimated disturbance force ˆf v (t). Fig. 4. Ship motion in the inertial coordinate system (x I,y I ). displacements can be compared to each other. Additionally, the benefits of the disturbance compensation are shown. At 60 s, an additional cross current disturbance is built up in the simulation and acts on the ship in lateral direction according to Fig. 6. The disturbance compensation leads to a maximum tracking Fig. 7. State variables during the maneuver. Fig. 8. Comparison of the measured and the estimated lateral velocity: v m and ˆv. Fig. 5. Comparison of the desired values y Id (t) and the actual values y I (t) with and without disturbance compensation (DC) for the lateral ship position. error of 0.36 m and gurantees a vanishing steady-state tracking error in spite of the persistent lateral force that is acting on the ship after 90 s, see Fig. 6. Without the disturbance compensation, a steady-state control error of 0.58 m occurs due to the persistent disturbance force f v, whereas the maximum tracking error increases up to 0.84 m. The estimated disturbance force ˆf v matches the actual disturbance force f v with small deviations as depicted in Fig. 6. The time history of the state variables is depicted in Fig. 7. It can be noticed that both the lateral velocity v and the heading angle ψ are non-zero to account for the persistent disturbance force f v after 90 s. The estimated lateral velocity ˆv derived from the disturbance observer allows for a smooth reconstruction of v by means of a correction of the wave-induced measurement disturbance η wv, see Fig. 8. In Fig. 9, the variations in the control gains are depicted. Thereby, a constant eigenvalue configuration in the s-plane is maintained. Fig. 9. Gain-scheduled adaptation of the control gains with fixed eigenvalues. 6. CONCLUSIONS In this paper, a cascaded nonlinear control approach is proposed for the surface vessel Korona. The longitudinal velocity as well as the angular velocity are controlled in fast control loops, respectively, and can be considered as kinematic inputs of the superordinate control structure. In these underlying control loops, partial input-output linearisation contributes to a compensation 301

6 of nonlinear coupling terms, whereas disturbances and parameter uncertainties are taken into account by an integral control action in the corresponding stabilising PI controllers. For a gainscheduled control of the lateral ship displacement in a local earth-fixed coordinate system, an eigenvalue assignment is proposed that makes use of extended linearisation techniques. The feedback control part is extended by appropriate feedforward control to improve the tracking properties concerning desired trajectories. A disturbance observer is employed to estimate both the resulting disturbance force in lateral direction as well as a wave-induced measurement disturbance. This estimated force can be used to correct the angular velocity as control input of the lateral ship dynamics and contributes significantly to an accurate trajectory tracking. In the near future, the presented control strategy will be investigated experimentally at the ship Korona and compared to existing ones. REFERENCES H. Aschemann and A. Rauh. Nonlinear control and disturbance compensation for underactuated ships using extended linearisation techniques. In Proc. of 8th IFAC Conference on Control Applications in Marine Systems, CAMS 2010, Rostock-Warnemuende, Germany, M. Blanke. Ship Propulsion Losses Related to Automated Steering and Prime Mover Control. PhD thesis, Technical University of Denmark, Lyngby, Denmark, W. Caharija, K.Y. Pettersen, M. Bibuli, P. Calado, E. Zereik, J. Braga, J.T. Gravdahl, A.J. Sørensen, M. Milovanović, and G. Bruzzone. Integral line-of-sight guidance and control of underactuated marine vehicles. Transactions on Control Systems Technology, T. I. Fossen and T. Perez. Kalman filtering for positioning and heading control of ships and offshore rigs. Control Systems, IEEE, 29(6):32 46, T.I. Fossen. Guidance and Control of Ocean Vehicles. John Wiley & Sons, Chichester, UK, T.I. Fossen. Nonlinear passive control and observer design for ships. In Modeling, Identification and Control, volume 21, pages , doi: /mic V. Grigoryev, A. Rauh, H. Aschemann, and M. Paschen. Development of a Neural Network-Based Controller for Ships. In Proc. of the 1st Joint International Conference on Multibody System Dynamics, Lappeenranta, Finland, ISBN A. Gronarz. Rechnerische Simulation der Schiffsbewegung beim Manövrieren unter besonderer Berücksichtigung der Abhängigkeit von der Wassertiefe (in German). PhD thesis, University of Duisburg-Essen, Germany, B. Han and G.L. Zhao. Course-Keeping Control of Underactuated Hovercraft. Journal of Marine Science and Application, 3(1):24 27, K.Y. Pettersen and H. Nijmeijer. Global Practical Stabilization and Tracking for an Underactuated Ship. A Combined Averageing and Backstepping Approach. In Proc. of the IFAC Conf. on Systems Structure and Control, pages 59 64, Nantes, France, M. Reyhanoglu. Exponential Stabilization of an Underactuated Autonomous Surface Vessel. Automatica, 33(12): , H. Sira-Ramirez. On the Control of the Underactuated Ship: A Trajectory Planning Approach. Automatica, 41(1):87 95, S. Wirtensohn, J. Reuter, M. Blaich, M. Schuster, and O. Hamburger. Modelling and identification of a twin hull-based autonomous surface craft. In 18th International Conference on Methods and Models in Automation and Robotics (MMAR), 2013, pages IEEE, Appendix A. RUDDER MODEL The relations used in the rudder model (9) are given by C FN = 6.13a RAR a RAR , with a RAR = b2 R and A R = b R c R. A R b R is the average rudder height, whereas width is denoted by c R. The whole rudder area A R is separated in two areas due to the inflow of the propeller A R = A Prop + A Frei = D p a R + (b D p e p )a R, with propeller diameter D p and ( C s + 1 1) e p = 1 + k PR ( C s + 1 1), with k PR = d PR /D p, the water density ρ and C s = 8F p ρu 2 P πd2 p. The inflow velocity u P = (1 w H )u of the Propeller depends on the the hull damping w H. Considering the motion of the vessel Korona, the inflow velocities for the two areas are given by v R = (v ωx R ) 2 + u 2 RP, v P = (v ωx R ) 2 + u 2 RP, and the inflow angles become with δ er = δ R atan( v ωx R u RP ), δ ep = δ R atan( v ωx R u RP ), u RP = u P + ((k RP 0.5)sign(u P ) + 0.5)[(sign(u P ) u 2 P + sign(u P) 8F p ρπd 2 u P )]. p Appendix B. PARAMETERS OF THE KORONA SHIP MODEL Parameter Value Unit Parameter Value Unit m kg 2375 kg m kgm 2 N d 11 4 m/s d N m/s d Nm rad/s d 11q 60 N (m/s) 2 d 22q 836 N (m/s) 2 d 33q Nm (rad/s) 2 Table B.1. Model parameters of the Korona. Parameter Value Unit Parameter Value Unit c c D p 0.36 m w H b R 0.38 m c R 0.21 m ω rad ζ, x R 2.86 m Table B.2. Propulsion Parameters and parameters for wave-induced motions of the Korona. 302