IDENTIFICATION OF DYNAMICALLY POSITIONEDSHIPSThor I. Fossen�, Svein I. Sagatun��and Asgeir J. S�rensen����The Norwegian Institute of Technology, Department of Engineering Cybernetics, N-7034 Trond-heim, NORWAY (E-mail:tif@itk.unit.no).��ABB Industri AS, Hasleveien 50, P.O.Box 6540 Rodel�kka, N-0501 Oslo, NORWAY (E-mail:sis@ttsinit.no)���ABB Industri AS, Hasleveien 50, P.O.Box 6540 Rodel�kka, N-0501 Oslo, NORWAY (E-mail:Asgeir.Sorensen@noina.abb.telemax.no)Abstract. Todays model-based dynamic positioning (DP) systems require that the ship and thrusterdynamics are known with some accuracy in order to use linear quadratic optimal control theory.However, it is di�cult to identify the mathematical model of a dynamically positioned (DP) shipsince the ship is not persistently excited under DP. In addition the ship parameter estimation problemis nonlinear and multivariable with only position and thruster state measurements available forparameter estimation. The process and measurement noise most also be modelled in order to avoidparameter drift due to environmental disturbances and sensor failure. This article discusses an o�-line parallel extended Kalman �lter (EKF) algorithm utilizing two measurement series in parallel toestimate the parameters in the DP ship model. Full-scale experiments with a supply vessel are usedto demonstrate the convergence and robustness of the proposed parameter estimator.Key Words. Dynamic positioning, identi�cation, full-scale sea trials, Kalman �ltering, self-tuningcontrol, marine systems.1. INTRODUCTIONModern dynamic positioning (DP) systems arebased on model-based feedback control. The stateestimator and control law are designed by apply-ing a low-frequency (LF) mathematical model ofthe ship motions caused by currents, wind and2nd-order wave loads, and a high-frequency (HF)model of the 1st-order ship motions caused by 1st-order wave disturbances; see Fossen (1994).Model-based control systems utilizing stochasticoptimal control theory and Kalman �ltering tech-niques was �rst employed with the DP problemby Balchen et al. (1976). Later extensions andmodi�cations of this work have been reported byBalchen et al. (1980a), Balchen et al. (1980b),Grimble et al. (1980a), Grimble et al. (1980b),Fung and Grimble (1983) and S�lid et al. (1983).In order to achieve good performance of the con-trol system it is necessary to have a su�cient de-tailed mathematical model of the ship. ABB In-dustri AS in Oslo has marketed a new self-tuningmodel-based DP system based on the results pre-sented in this article whereas the state estima-tor and control system design are discussed inSagatun et al. (1994) and S�rensen et al. (1995),respectively.

2. SHIP AND THRUSTER MODELSThis section describes the mathematical model ofthe thrusters and the LF motion of the ship.2.1. Thruster ModelMost DP ships use thrusters and main propellersto maintain their position and heading. Thethrust force of a pitch-controlled thruster can beapproximated by:F (n; p) = K(n) jp� p0j (p� p0) (1)where the force coe�cient K(n) is assumed to beconstant for constant propeller revolution n, P isthe \traveled distance per revolution", D is thepropeller diameter and:p = P=D (2)is the pitch ratio. p0 is pitch ratio o�-set de�nedsuch that p = p0 yields zero thrust, that is:F (n; p0) = 0 (3)Thrust Forces and Moment. The thrust forcesand moment vector � 2 IR3 (surge, sway and yaw)IFAC Workshop on Control Applications in Marine Systems (CAMS'95) 1

Fig. 1. Picture showing the supply vessel which was used during the sea trials in the North Sea (L = 76:2 m).

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Fig. 2. Experimentally measured thrust (asterisks) and thruster model approx. Eq. (1) versus p = P=D. Leftplot: F (122; p) = 370 pjpj and F (160; p) = 655 pjpj. Right plot: F (236; p) = 137 pjpj. Propeller revolutionis in rpm.for the supply vessel in Figure 1 can be written:� = T K u (4)where u 2 IRr is a control variable de�ned as:u = [jp1 � p10j(p1 � p10); jp2 � p20j(p2 � p20);:::; jpr � pr0j(pr � pr0)]T (5)where pi0 (i = 1:::r) are the pitch ratio o�-sets forthruster no. i and r is the maximum number ofthrusters.Thrust Force Coe�cient Matrix. The thrust forcecoe�cient matrixK is a diagonal matrix of thrustforce coe�cients de�ned as:K = diagfK1(n1); K2(n2); :::; Kr(nr)g (6)where ni (i = 1::::r) is the propeller revolution ofpropeller number i. The thrust forces Ki(ni)uiare distributed to the surge, sway and yaw modesby a 3� r thruster con�guration matrix T .

Thruster Con�guration Matrix. Consider the shipin Figure 1 which is equipped with two main pro-pellers, three tunnel thrusters and one azimuththruster which can be rotated to an arbitrary an-gle �. The control variables are assigned accord-ing to: u1 = port main propelleru2 = starboard main propelleru3 = aft tunnel thruster Iu4 = aft tunnel thruster IIu5 = bow tunnel thrusteru6 = bow azimuth thrusterThe following thruster con�guration matrix is ob-tained:T = " 1 1 0 0 0 cos�0 0 1 1 1 sin�l1 �l2 �l3 �l4 l5 l6 sin� # (7)where li (i = 1:::6) are the moment arms in yaw.It is also seen that l2 = l1 (symmetrical location of2 IFAC Workshop on Control Applications in Marine Systems (CAMS'95)

main propellers). The thrust demands are de�nedsuch that positive thrust force/moment results inpositive motion according to the vessel parallelaxis system de�ned such that positive x-directionis forwards, positive y-direction is starboard andpositive z-direction is downwards. The origin islocated in the center of buoyancy.2.2. LF Ship DynamicsThe LF ship model in surge, sway and yaw can bedescribed by (Fossen 1994):M _� +C(�)� +D (� � �c) = � +w (8)where � = [u; v; r]T denotes the LF velocity vec-tor, �c = [uc; vc; rc]T is a vector of current veloc-ities, � is a vector of control forces and momentsand w = [w1; w2; w3]T is a vector of zero-meanGaussian white noise processes describing unmod-elled dynamics and disturbances. Notice that rcdoes not represent a physical current velocity, butcan be interpreted as the e�ect of currents in yaw.The current states are useful in the parameter es-timator since they represent slowly-varying non-zero bias terms.The inertia matrix including hydrodynamic addedmass terms is assumed to be positive de�niteM =MT > 0 for a dynamically positioned ship,whereas D > 0 is a strictly positive matrix repre-senting linear hydrodynamic damping. Nonlineardamping is assumed to be negligible for station-keeping of ships whereas the assumption of star-board and port symmetry implies thatM and Dcan be written:M = " m�X _u 0 00 m� Y _v mxG � Y _r0 mxG � Y _r Iz �N _r #(9)D = " �Xu 0 00 �Yv �Yr0 �Nv �Nr # (10)The Coriolis and centrifugal matrix C(�) is in-cluded in the model to improve the convergenceof the parameter estimator. Moreover this ma-trix may be signi�cant for a ship operating atsome speed whereas C(�) = 0 for a ship at rest.It should be noted that inclusion of C(�) in themodel will not increase the number of parametersto be estimated sine C(�) only is a function ofthe elements mij of the inertia matrix; see The-orem 2.2. on page 27 in Fossen (1994). In factM = fmijg yields:C(�) =" 0 0 �m22v �m23r0 0 m11um22v +m23r �m11u 0 # (11)where the non-zero elements mij = �mji are de-

�ned according to (9) such that:m11 = m�X _u m23 = mxG � Y _rm22 = m� Y _v m33 = Iz �N _r (12)2.3. KinematicsThe kinematic equation of motion for a ship is:_� = J(�)� (13)where � = [x; y; ]T and J(�) is a rotation matrixde�ned as:J(�) = " cos � sin 0sin cos 00 0 1 # (14)3. OFF-LINE PARAMETER ESTIMATORThe o�-line parameter estimator is based on thestate augmented extended Kalman �lter (EKF).3.1. State Augmented Extended Kalman FilterConsider the following nonlinear system:x(k + 1) = f(x(k);u(k);�(k)) +w1(k) (15)�(k + 1) = �(k) + �(k) (16)where x 2 IRn is the state vector, u 2 IRr is theinput vector, � 2 IRp is the unknown parametervector to be estimated and w1;� 2 IRn are zero-mean Gaussian white noise processes. This modelcan be expressed in augmented state-space formas: �(k + 1) = F(�(k);u(k)) +w(k) (17)where � = [xT ;�T ]T is the augmented state vec-tor, w = [wT1 ;�T ]T and:F(�(k);u(k)) = � f(x(k);u(k);�(k))�(k) � (18)Furthermore, it is assumed that the measurementequation can be written:z(k) =H(�(k)) + v(k) (19)where z 2 IRm and m is the number of sensors.The discrete-time extended Kalman �lter algo-rithm in Table 1 can then be applied to estimate� = [xT ;�T ]T in (17) by means of the measure-ment (19). For details on the implementation is-sues see Gelb et al. (1988).3.2. O�-Line EKF for Parallel ProcessingIn order to improve the convergence and perfor-mance of the parameter estimator the same quan-IFAC Workshop on Control Applications in Marine Systems (CAMS'95) 3

Table 1 Summary of discrete-time extended Kalman �lter (EKF).System model �(k + 1) = F(�(k);u(k)) + � w(k); w(k) � N(0;Q(k))Measurement z(k) =H(�(k)) + v(k); v(k) � N(0;R(k))Initial conditions �(0) = �0; P (0) = P 0State estimate propagation �(k + 1) = F(�(k);u(t))Error covariance propagation P (k + 1) = �(k) P (k)�T (k) + � (k)Q(k) � T (k)Gain matrix K(k) = P (k)HT (k) hH(k)P (k)HT (k) +R(k)i�1State estimate update �(k) = �(k) +K(k) [z(k)�H(�(k))]Error covariance update P (k) = [I �K(k)H(k)]P (k) [I �K(k)H(k)]T+K(k)R(k)KT (k)De�nitions �(k) = @F (�)@�(k) ����(k)=�(k) H(k) = @H(�)@�(k) ����(k)=�(k)tity can be measured N � 2 times for di�er-ent excitation sequences. Moreover, let the inputui 2 IRr correspond to the state vector xi 2 IRnand measurement vector zi 2 IRm for (i = 1:::N).Under the assumption of constant parameters, theparameter vector � 2 IRp will be the same for allthese subsystems. This can be expressed mathe-matically as:x1(k + 1) = f(x1(k);u1(k);�(k)) +w1(k)x2(k + 1) = f(x2(k);u2(k);�(k)) +w2(k)...xN (k + 1) = f(xN (k);uN (k);�(k)) +wN (k)�(k + 1) = �(k) + �(k) (20)with measurements:z1(k) = h1(x1(k);�(k)) + v1(k)z2(k) = h2(x2(k);�(k)) + v2(k)... (21)zN (k) = hN (xN (k);�(k)) + vN (k)Hence, this system can be written in augmentedstate-space form according to:�(k + 1) = F(�(k);u(k)) +w(k) (22)z(k) = H(�(k)) + v(k) (23)where u = [uT1 ; :::;uTN ]T , z = [zT1 ; :::; zTN ]T , � =[xT1 ; :::;xTN ;�T ]T and:F(�(k);u(k)) = 2666664 f(x1(k);u1(k);�(k))f(x2(k);u2(k);�(k))...f(xN (k);uN (k);�(k))�(k)3777775(24)H(�(k)) = 26664 h(x1(k);�(k))h(x2(k);�(k))...h(xN (k);�(k)) 37775 (25)

It is observed that dim x = Nn+ p, dim u = Nrand dim z = Nm. It is then evident that moreinformation about the system is obtained by usingmultiple measurement sequences. Increased infor-mation improves parameter identi�ability and re-duces the possibility for parameter drift. How-ever it should be noted that parallel processingimplies that the parameters estimation must beperformed o�-line.For ship applications signi�cant performance im-provement is reported already for N = 2; seeAbkowitz (1975), Abkowitz (1980) and Hwang(1980).EKF

SHIPEq.3 y2Dy1

Kp1p2 u1u2MFig. 3. Parallel con�guration of EKF for N = 2.4. IDENTIFICATION OF A SUPPLY VESSELFull-scale experiments with the supply vessel inFigure 1 will be used to demonstrate the conver-gence of the proposed parameter estimation algo-rithm.4.1. System Identi�cation ModelThe system identi�cation model is based on themathematical model presented in Section 2. As-suming no environmental disturbances a dynam-ically positioned ship can be described by thefollowing non-dimensional model (Bis-system) insurge, sway and yaw (see page 178 in Fossen4 IFAC Workshop on Control Applications in Marine Systems (CAMS'95)

1994):M 00 _� 00 +C00(� 00)� 00 +D00 � 00 = T 00 K 00u00 (26)whereM 00 = " 1�X00_u 0 00 1� Y 00_v x00G � Y 00_r0 x00G � Y 00_r k2z �N 00_r # (27)D00 = " �X00u 0 00 �Y 00v �Y 00r0 �N 00v �N 00r # (28)The thruster con�guration matrix was computedto be:T 00 = " 1:0000 1:0000 00 0 1:00000:0472 �0:0472 �0:41080 0 01:0000 1:0000 1:0000�0:3858 0:4554 0:3373 # (29)whereasK 00 = diagfK 001 ; K 002 ; K 003 ; K 004 ; K 005 ; K 006 g isthe unknown matrix to be estimated. In additionto this it will be assumed that D00 is unknown.An apriori estimate of M 00 is calculated by ap-plying semi-empirical methods. For more detailsabout the computation of the added mass deriva-tives X 00_u ; Y 00_v ; N 00_r and Y 00_r , see Faltinsen (1990).The inertia matrix M 00 was computed to be:M 00 = " 1:1274 0 00 1:8902 �0:07440 �0:0744 0:1278 # (30)Hence K 00 and D00 are the only remaining un-known matrices in the DP model (26).4.2. Sea TrialsIn order to improve the convergence of the pa-rameter estimator it is proposed to use several o�-line measurement series generated by a number ofcarefully prede�ned maneuvers. For instance, itis advantageous to decouple the surge mode fromthe sway and yaw modes in order to improve theconvergence of the parameter estimator. This ismotivated by the block diagonal structure of M 00and D00.Decoupled Ship Maneuvers. The following threedecoupled ship maneuvers are proposed:(1) uncoupled surge: the ship is only allowedto move in surge (constant heading) by meansof the main propellers u1 and u2. The head-ing is controlled by means of one of the bowthrusters. At least two maneuvers should beperformed; see Figure 5.(2) coupled sway and yaw: the ship shouldperform two coupled maneuvers in sway andyaw by means of the three tunnel thruster

u3; u4 and u5. Two maneuvers should be per-formed; see Figure 6.(3) azimuth test: the last test involves runningthe azimuth thruster u6 alone. Two measure-ments series are required; see Figure 7.4.3. Implementation IssuesThis implies that 6 at least sea trials must be per-formed forN = 2. The �rst two sea trials are usedto identify the parameters K 001 = K 002 and X 00u inthe decoupled surge equation:(1�X 00_u) _u00 �X 00uu00 = K 001 u001 +K 002u002 (31)_x00 = u00 (32)where X 00_u is computed by using strip theory(Faltinsen 1990). The parameter vector corre-sponding to this system is denoted as �001 =[K 001 ; X 00u ]T . The estimated parameter vector �001in surge is frozen and used as input for the sec-ond system identi�cation scheme (SI2), that is thecoupled sway and yaw identi�cation. Similarly,the output from the second parameter estimationscheme �002 is frozen and used as input for thelast parameter estimation scheme (SI3), that is�003 . The last scheme is used to estimate only oneparameter, that is �003 = K 006 whereas the secondparameter estimation scheme is used to estimatethe coupling terms in sway and yaw; see Figure4. In the last to parameter estimation schemesSI2 SI3SI1 y3y2y1�0 �1 �2u2 u3u1 �3Fig. 4. Decoupled parameter estimation in terms ofthree system identi�cation schemes SI1{SI3.the ship is commanded to change heading duringthe maneuvers which implies that the nonlinearkinematic equation:_�00 = J 00(� 00) �00 (33)where �00 = [x00; y00; 00]T should be used togetherwith the dynamic model (26). Hence the unknownparameter vector corresponding to sea trials 2 is�002 = [Y 00v ; Y 00r ; N 00v ; N 00r ;K 003 ;K 005 ]T . In this examplethe tunnel thrusters at the stern are of same type(K 004 = K 003 ). It is convenient to rewrite (26) and(33) in terms of the vessel momentum:h00 =M 00 � 00 (34)IFAC Workshop on Control Applications in Marine Systems (CAMS'95) 5

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Fig. 8. Parameter estimates �001 {�009 versus time for the supply vessel.and a momentum bias term bh which yields thefollowing model:_h00 + Ch(h00)h00 =A00h(�00) h00 + T 00 K00(�00)u00 + bh +wh (35)_�00 = J 00(�00)M 00�1 h00 (36)_bh = wb (37)_�00 = w� (38)Here wh;wb and w� are zero-mean Gaussianwhite noise processes, bh is a slowly-varying pa-rameter representing unmodelled dynamics andenvironmental disturbances, �00 is the parametervector to be estimated. The new matrices in themodel are de�ned according to:Ah = �DM (39)Ch(h) = C(M�1h) (40)Since M is assumed to be known with su�cientaccuracy the only unknown quantities in (35){(38) are Ah and K. The main motivation forusing the momentum equation instead of the stan-dard dynamic equations of motion is improvedperformance of the state and parameter estima-tor. Moreover estimation of the states h =M�;�and bh together with the parameter vector � iseasier to perform than estimation of �;�; bh and�. Hence, the resulting model can be written:x1(k + 1) = f (x1(k);u1(k);�(k)) +w1(k) (41)x2(k + 1) = f (x2(k);u2(k);�(k)) +w2(k) (42)�(k + 1) = �(k) +w�(k) (43)where xi = [hTi ;�Ti ; bThi]T , wi = [whi;wTbi]T(i = 1; 2) and with obvious de�nition of f . Ifposition (x; y) and heading ( ) are measured (23)

becomes:z1(k) = H1 x1(k) + v1(k) (44)z2(k) = H2 x2(k) + v2(k) (45)4.4. Experimental ResultsSix maneuvers with the supply vessel were used toestimate the DP model. The unknown parametervector �00 = [�001 ; :::; �009 ]T is organized accordingto: A00h = " �001 0 00 �002 �0030 �004 �005 # (46)K 00 = diagf�006 ; �006 ; �007 ; �007 ; �008 ; �009g (47)The parameter estimates for the o�-line parallelcon�guration of the EKF algorithm is shown inFigure 8 whereas the steady-state numerical val-ues are given below.Identi�ed Momentum Equation:.A00h = " �0:0318 0 00 �0:0602 0:06180 �0:0075 �0:2454 # (48)K 00 = 10�3diagf9:3; 9:3; 2:0; 2:0; 2:8; 2:6g (49)Identi�ed State-Space Model:. The estimatedmodel (35) can be related to the DP control modelby assuming that Ch(h) = 0. Hence:_� 00 = A00�00 +B� 00 (50)IFAC Workshop on Control Applications in Marine Systems (CAMS'95) 7

whereA00 =M 00�1A00hM 00 andB00 =M 00�1. Thenumerical values are:A00 = " �0:0318 0 00 �0:0628 �0:00300 �0:0045 �0:2428 # (51)B00 = " 0:0082 0:0082 00:0001 �0:0001 0:00080:0035 �0:0035 �0:00590 0 00:0008 0:0020 0:0017�0:0055 0:0113 0:0079 # (52)For more details about the DP control system de-sign, see S�rensen et al. (1995).5. CONCLUSIONSIn this paper a new approach for identi�cation ofdynamically positioned ships has been proposed.Three di�erent ship maneuvers were used in a de-coupled identi�cation scheme based on an o�-lineparallel con�guration of the extended Kalman �l-ter algorithm. Simulation studies showed thatthe proposed parameter estimation scheme wasremarkable accurate for ship models that werecoupled in surge, sway and yaw. The parameterestimation algorithm has been implemented andtested on a supply vessel. The estimated model ofthe supply vessel has been implemented and usedfor model-based DP control system design. Theestimated values of this ship showed good agree-ment with experimental results from model tests.6. REFERENCESAbkowitz, M. A. (1975). System Identi�cationTechniques for Ship Maneuvering Trials. In:Proceeding of Symposium on Control Theoryand Navy Applications. Monterey, CA. pp. 337{393.Abkowitz, M. A. (1980). Measurement of Hydro-dynamic Characteristics from Ship ManeuvringTrials by System Identi�cation. In: Transac-tions on SNAME, 88:283{318.Balchen, J. G., N. A. Jenssen and S. S�lid (1976).Dynamic Positioning Using Kalman Filteringand Optimal Control Theory. In: IFAC/IFIPSymposium on Automation in O�shore OilField Operation. Holland, Amsterdam. pp. 183{186.Balchen, J. G., N. A. Jenssen and S. S�lid(1980a). Dynamic Positioning of Floating Ves-sels Based on Kalman Filtering and Opti-mal Control.. In: Proceedings of the 19thIEEE Conference on Decision and Control.New York, NY. pp. 852{864.

Balchen, J. G., N. A. Jenssen, E. Mathisen andS. S�lid (1980b). Dynamic Positioning Sys-tem Based on Kalman Filtering and OptimalControl.. Modeling, Identi�cation and ControlMIC-1(3), 135{163.Faltinsen, O. M. (1990). Sea Loads on Shipsand O�shore Structures. Cambridge UniversityPress.Fossen, T. I. (1994). Guidance and Control ofOcean Vehicles. John Wiley and Sons Ltd.Fung, P. T-K. and M. J. Grimble (1983). DynamicShip Positioning Using a Self Tuning KalmanFilter. IEEE Transactions on Automatic Con-trol AC-28(3), 339{349.Gelb, A., J. F. Kasper, Jr., R. A. Nash, Jr., C. F.Price and A. A. Sutherland, Jr. (1988). AppliedOptimal Estimation. MIT Press. Boston, Mas-sachusetts.Grimble, M. J., R. J. Patton and D. A. Wise(1980a). The Design of Dynamic Position-ing Control Systems Using Stochastic OptimalControl Theory. Optimal Control Applicationsand Methods 1, 167{202.Grimble, M. J., R. J. Patton and D. A. Wise(1980b). Use of Kalman Filtering Techniquesin Dynamic Ship Positioning Systems. In: IEEProceedings Vol. 127, Pt. D, No. 3. pp. 93{102.Hwang, Wei-Yuan (1980). Application of SystemIdenti�cation to Ship Maneuvering. Master'sthesis. Massachusetts Institute of Technology.S�lid, S., N. A. Jenssen and J. G. Balchen (1983).Design and Analysis of a Dynamic PositioningSystem Based on Kalman Filtering and Opti-mal Control. IEEE Transaction on AutomaticControl AC-28(3), 331{339.Sagatun, S. I., A. S�rensen and T. I. Fossen(1994). Dynamic Positioning of Marine Ves-sels. Submitted to the IEEE Transactions onControl Systems Technology.S�rensen, A., S. I. Sagatun and T. I. Fossen(1995). The Design of a Dynamic PositioningSystem Using Model Based Control. In: Sub-mitted to the IFAC Workshop on Control Appli-cations in Marine Systems (CAMS'95). Trond-heim, Norway.

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