PhD Thesis - Federico Gaggiotti

Università Politecnica delle Marche

Dipartimento di MeccanicaDottorato di Ricerca in Sistemi Artificiali Intelligenti

nell’Ingegneria dell’Informazione e nell’Ingegneria Industriale

Deep Water Pipe Laying: fromMooring-Based Station Keeping to

Dynamic Positioning

Ph.D. Dissertation of:Federico Gaggiotti

Advisor:Prof. Massimo Callegari

Supervisor of the Ph.D Program:Prof. Sauro Longhi

VIII Edition - New Series


Dipartimento di MeccanicaDottorato di Ricerca in Sistemi Artificiali Intelligenti

nell’Ingegneria dell’Informazione e nell’Ingegneria Industriale

Deep Water Pipe Laying: fromMooring-Based Station Keeping to

Dynamic Positioning

Ph.D. Dissertation of:Federico Gaggiotti

Advisor:Prof. Massimo Callegari

Supervisor of the Ph.D Program:Prof. Sauro Longhi

VIII Edition - New Series


Dipartimento di MeccanicaVia Brecce Bianche – 60131 – Ancona, Italy

A C K N O W L E D G E M E N T S

This project was developed with the Department of Mechanicsof the Polytechnic University of Marche, Ancona, Italy and SaipemEnergy Services, Fano, Italy. I wish to thank Prof. Massimo Callegarifor the opportunity to participate in such an important experience,Roberto Bruschi and Luigino Vitali from the offshore departmentin Fano for the opportunity to develop a project in contact withindustry, and for the continuous technical support.

Many people contributed to the successful completion of thiswork: Paolo Monti and his group in Milan, Michele Rattin in partic-ular for his work on the tensioner model and inverse filtering of firstorder motion, Stefano Meggio for the precious contribution to thesimulator that made possible to test the new Wave Feed-Forwardapproach proposed.

I also wish to thank the other people I worked with, in Saipemand at the University, who shared their experiences and made theefforts spent at work fun and less wearing.

i

C O N T E N T S

introduction 1

I basics 3

1 pipelaying technology 5

1.1 Main pipelaying technologies 5

1.1.1 Onshore welded pipelines 6

1.1.2 Offshore welded pipelines 7

1.2 Pipelaying in deep water 9

1.3 Pipe structural model 10

1.3.1 Pipe stiffness 11

1.4 Tensioner 12

2 environment 21

2.1 Basic potential theory 22

2.1.1 Kinematic Boundary Conditions 23

2.1.2 Dynamic Boundary Conditions 23

2.2 Regular wave theory 24

2.2.1 Particle trajectories 29

2.3 Sea state modeling 33

2.3.1 Wave spectra 33

2.3.2 Pierson-Moskowitz spectrum 37

2.3.3 JONSWAP spectrum 38

2.3.4 Torsethaugen spectrum 40

2.4 From spectra to time series 41

2.4.1 Sampling at constant frequency intervals 43

2.4.2 Random frequency sampling 43

2.4.3 Constant energy sampling 43

2.5 Statistics on water elevation time histories 44

2.6 Directional wave distribution 46

2.7 Wave spectra approximation 48

2.8 Software for sea state modeling 51

2.9 Wind spectra 56

3 seakeeping 59

3.1 Ship motion and degrees of freedom 59

3.1.1 First and second order motion 60

3.2 First order motion by means of RAOs 62

3.2.1 Statistical approach to first order motion cal-culation 63

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3.3 Drift forces 66

3.3.1 Newman’s approximation 67

3.3.2 Simplified method for drift force calculation 68

3.3.3 Frequency domain analysis of drift forces 69

3.4 Software for ship dynamics modeling 70

3.4.1 First order motion 70

3.4.2 Second order forces and motion 71

II from moorings to dynamic positioning 79

4 moorings 81

4.1 Mooring-based pipelaying 81

4.2 Static catenary solution 82

4.2.1 Mooring stiffness 84

4.3 Spread mooring systems 86

5 dp basics 89

5.1 DP architecture 89

5.2 Observer 91

5.2.1 Kalman filter in dynamic positioning 91

5.3 Controller 95

5.4 Thrust allocation logic 99

5.4.1 Pseudo-inverse algorithm, fixed-azimuths 102

5.4.2 Nonlinear programming 105

5.4.3 Simulation 106

5.5 Simplified performance assessment 110

5.5.1 Filtering delay 113

5.6 Acceleration feedback 116

6 measurement systems 123

6.1 Absolute position 123

6.1.1 GPS 125

6.1.2 DGPS 129

6.2 Underwater positioning 132

6.2.1 USBL systems 132

6.2.2 LBL systems 133

6.3 Vessel accelerations 134

6.4 Sea state 135

6.4.1 Wave radar 135

6.4.2 Ship motions-based estimate 136

7 wave feed-forward 137

7.1 Literature 137

7.2 Real-time estimate 139

7.2.1 Performance in Positioning 142

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7.3 Data Analysis 147

7.3.1 Motivation for predictive approach 152

7.4 Neural networks 152

7.4.1 Neuron model 154

7.4.2 Basic network architectures 155

7.5 ANNs for drift force prediction 158

7.5.1 Feed-forward input time-delay back-propagationnetwork 159

7.5.2 Recurrent input time-delay network 162

7.5.3 Pattern recognition network 164

7.5.4 Multiple input recurrent time-delay network 165

7.5.5 Network architecture selection 168

7.6 Integration of neural networks into DP algorithm 168

7.7 Future developments 170

conclusions 173

bibliography 175

index 183

v

L I S T O F F I G U R E S

Figure 1.1 Surface tow method 6

Figure 1.2 Mid-depth tow method 7

Figure 1.3 Mid-depth tow method 8

Figure 1.4 S- and J-lay modes 9

Figure 1.5 Static pipelay analysis, deformed shape in 20 mwater depth 12

Figure 1.6 Pipe displacement-tension variation relation-ship 13

Figure 1.7 Picture of a tensioning device manufacturedby RE.MAC.UT. 14

Figure 1.8 Tensioner transfer function, input is pipe ten-sion, output is pipe payout 16

Figure 1.9 The system resonant period diverges to infi-nite 17

Figure 1.10 The system resonant period converges to a fi-nite value 18

Figure 2.1 Profile of quantities describing a wave over awave length 28

Figure 2.2 Different methods of profile extension over themean free surface 30

Figure 2.3 Particle trajectories in shallow water 31

Figure 2.4 Particle trajectories in deep water 32

Figure 2.5 Irregular sea with spreading as superpositionof regular waves 34

Figure 2.6 Representation of a sea state as a wave spec-trum 35

Figure 2.7 Location of the WAVESCAN buoy in Mahón,Menorca (ES) 36

Figure 2.8 Extract from recorded significant wave heightand wind speed signals in july 2009, Mahónbuoy 37

Figure 2.9 Relation between wind speed and significantwave height 37

Figure 2.10 Relation between wind speed and significantwave height 38

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Figure 2.11 Relation between wind speed and significantwave height, with the wind time history de-layed of 3 h 39

Figure 2.12 Pierson-Moskowitz spectrum for Hs = 3.5 m,Tp = 10 s 40

Figure 2.13 JONSWAP spectra for Hs = 3.5 m, Tp = 10 sand some γ values, compared with a Pierson-Moskowitz spectrum with the same parame-ters 41

Figure 2.14 Torsethaugen spectra for Hs = 3.5 m and vari-ous peak periods 42

Figure 2.15 Pierson-Moskowitz spectrum for Hs = 3.5 m,Tp = 10 s, sampled at constant frequency in-tervals 44

Figure 2.16 Pierson-Moskowitz spectrum for Hs = 3.5 m,Tp = 10 s, sampled at random frequencies in-side constant frequency intervals 45

Figure 2.17 Pierson-Moskowitz spectrum for Hs = 3.5 m,Tp = 10 s, sampled at constant energy inter-vals 46

Figure 2.18 Water elevation time history, generated by aconstant energy sampling of a JONSWAP spec-trum, Hs = 3.5 m, Tp = 10 s, γ = 3.3, 200 har-monic components 48

Figure 2.19 Directionality functions plotted for different nvalues 49

Figure 2.20 Pierson-Moskowitz spectrum for Hs = 3.5 m,Tp = 10 s, with spreading factor n = 2, maindirection East 50

Figure 2.21 Pierson-Moskowitz spectrum for Hs = 3.5 m,Tp = 10 s, with spreading factor n = 8, maindirection East 51

Figure 2.22 JONSWAP spectrum for Hs = 3.5 m, Tp = 10 s,γ = 3.3 values, compared with linear approxi-mation fitted by least-squares algorithm 52

Figure 2.23 SIMULINK model for wave time history gen-eration and water elevation measurement at ageneric location 53

vii

Figure 2.24 Water elevation time history, generated by aconstant energy sampling of a JONSWAP spec-trum, Hs = 3.5 m, Tp = 10 s, γ = 3.3, 100 har-monic components, with the SIMULINK modelrepresented in Figure 2.23 56

Figure 2.25 Harris wind spectrum for 12 m s−1 mean windspeed 57

Figure 3.1 Reference system and degrees of freedom 60

Figure 3.2 Components of the motion of a vessel subjectto waves 61

Figure 3.3 Response Amplitude Operator as a black box 62

Figure 3.4 Example of RAO functions for a semisubmer-sible (Ultramarine, Inc., 2008) 64

Figure 3.5 Roll response spectrum of a vessel subject toa sea state represented by a JONSWAP spec-trum with Hs = 3.5 m, Tp = 10 s and γ =3.3 65

Figure 3.6 Hsu and Blenkarn (1970) simplified methodfor drift force calculation 68

Figure 3.7 Drift force spectral densities for different seastates 70

Figure 3.8 SIMULINK model for first order motion calcu-lation 71

Figure 3.9 Sub-system of the MSS Motion RAO block, mod-ified for zero speed 72

Figure 3.10 Simulated first order motion for a tanker, Hs =3.5 m, Tp = 20 s, γ = 3.3, quartering sea 73

Figure 3.11 SIMULINK model for second order force cal-culation 74

Figure 3.12 SIMULINK model for second order force cal-culation, Newman algorithm 76

Figure 3.13 SIMULINK model for second order forces, ac-cording to Newman formulation 77

Figure 3.14 Simulated second order forces for the case ofa tanker, Hs = 3.5 m, Tp = 10 s, γ = 3.3, beamsea 78

Figure 4.1 Catenary line reference scheme for analyticalmodel 85

Figure 4.2 Equilibrium of an infinitesimal length of cate-nary mooring line 85

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Figure 4.3 Example mooring line configuration 86

Figure 4.4 Mooring system 87

Figure 5.1 DP system overview 90

Figure 5.2 Steady-state Kalman filter example of transferfunction, input is total motion and output issecond order motion 96

Figure 5.3 Wind speed time history, raw and filtered with50 s time constant low-pass filter 98

Figure 5.4 Drag and lift coefficients as a function of theangle of attack for a typical rudder 101

Figure 5.5 SIMULINK model for pseudo-inverse thrustallocation 108

Figure 5.6 Global generalized forces on vessel, demandedand obtained 111

Figure 5.7 Demanded and obtained thrust for each singlepropeller 112

Figure 5.8 Demanded and obtained azimuth for each az-imuthal thruster 113

Figure 5.9 Polar plot of extreme surge response 114

Figure 5.10 Polar plot of extreme sway response 115

Figure 5.11 Polar plot of extreme surge response with 10 sdelay, comparative results 116

Figure 5.12 Polar plot of extreme sway response with 10 sdelay, comparative results 117

Figure 5.13 Low-pass mass frequency-domain characteris-tic 119

Figure 5.14 Force to velocity transfer function of a mass-damper system without and with accelerationfeedback 121

Figure 6.1 Positioning error due to wrong geodetic refer-ence system 125

Figure 6.2 Dilution of Precision: low DOP and high DOPsituations 128

Figure 6.3 Dilution of Precision map for September 16th,2009, made by GPS Operations Center 128

Figure 6.4 SIMULINK model for GPS simulation 129

Figure 6.5 Simulated GPS trace for a stationary receiver,apparent motion due to noise 130

ix

Figure 6.6 Errors in x and y directions during a 10 minlong simulation, obtained autocorrelations com-pared with the theoretical one 131

Figure 6.7 Differential GPS scheme 131

Figure 6.8 USBL transceiver 132

Figure 6.9 USBL transceiver 133

Figure 6.10 LBL operational scheme 134

Figure 7.1 Approximation of the inverse heave RAO bymeans of parametric filter 141

Figure 7.2 SIMULINK model for real-time drift force es-timation 143

Figure 7.3 Simulated and estimated water elevation timehistory 144

Figure 7.4 Simulated and estimated sway drift force timehistory 144

Figure 7.5 Performance comparison between simulationswith and without wave feed-forward 145

Figure 7.6 Sway offset and velocity compared with lat-eral thrust force 148

Figure 7.7 Cross-correlations between sway, sway veloc-ity and lateral thrust signals. 149

Figure 7.8 Sway offset and velocity compared with lat-eral thrust force from thrusters feedback, withWFF (0.5 gain) 150

Figure 7.9 Cross-correlations between sway, sway veloc-ity and lateral thrust signals, with WFF (0.5gain) 151

Figure 7.10 Neuron model 155

Figure 7.11 Examples of activation functions 156

Figure 7.12 Single-layer feed-forward network 157

Figure 7.13 Feed-forward network with an hidden layerand two outputs 157

Figure 7.14 Recurrent network with an hidden layer 158

Figure 7.15 Tapped delay line for prediction use 159

Figure 7.16 Feed-forward input time-delay backpropaga-tion network 160

Figure 7.17 Network performance and training time (fac-tor with respect to single-pass training) as afunction of the number of passes 161

x

Figure 7.18 Example of prediction performance with single-pass incremental training, compared with a 10

passes training 161

Figure 7.19 Recurrent input time-delay network 162

Figure 7.20 Example of prediction performance with 10

passes incremental training, 10 s advance pre-diction 163

Figure 7.21 Pattern recognition network 164

Figure 7.22 Pattern recognition network results 165

Figure 7.23 Multiple input recurrent time-delay networkwith three inputs and a single output 166

Figure 7.24 Example of prediction performance with 10

passes incremental training, 20 s advance pre-diction 167

Figure 7.25 Performance comparison between real-time andneural network wave feed-forward 169

Figure 7.26 Simulated and estimated sway drift force timehistory, by means of neural networks 171

L I S T O F TA B L E S

Table 1.1 Pipe data 12

Table 1.2 Iterative procedure does not converge to a fi-nite resonant period 18

Table 1.3 Iterative procedure converges to a finite reso-nant period, decoupling not effective 19

Table 2.1 Analytical expressions from linear wave the-ory, general equations and simplified for deepwater 27

Table 2.2 Particle movement in deepwater, with respectto wave amplitude 31

Table 2.3 Most probable largest wave heights for differ-ent numbers of observed waves 47

xi

Table 4.1 Axial cable tension at upper end as a func-tion of water depth, for a horizontal pull of300 kN 87

Table 6.1 WGS84 and ED50 coordinate systems 124

Table 7.1 Performance improvement achieved with wavefeed-forward, Hs = 3.0 m 146

Table 7.2 Performance improvement achieved with wavefeed-forward, Hs = 3.5 m 146

Table 7.3 Motion-actuation lags for maximum correla-tion during simulated beam sea without wavefeed-forward, with regard to sway degree offreedom 149

Table 7.4 Motion-actuation lags for maximum correla-tion during simulated beam sea with wave feed-forward at 0.5 gain, with regard to sway de-gree of freedom 151

L I S T O F L I S T I N G S

Listing 2.1 Example of initialization code for wave spec-trum parameters 53

Listing 2.2 JONSWAP spectrum, algorithm for samplingat constant energy intervals 53

Listing 5.1 MATLAB code for pseudo-inverse thrust allo-cation 107

N O M E N C L AT U R E

β Fraction of system mass added as acceleration feedback

xii

∆t Sampling interval

ε Small constant number

λ Wave spectrum shape parameter, see Section 2.7

Ω Weight matrix for the change of azimuth angles

ωi Resonant circular frequency in i-th degree of freedom

ωp Wave peak circular frequency

Pj Mean quadratic transfer for j-th degree of freedom

φ Velocity potential function

ψp Low-frequency heading

ψw Wave-frequency heading

ρ Fluid density

ρ Weight of manoeuvrability requirement in quadratic pro-gramming thrust allocation

ρb Range error due to GPS receiver clock error

ρi GPS receiver range from the satellite i

σU Wind speed standard deviation

τ Filter-induced delay

ϕ(·) Neuron activation function

0 Null matrix/vector

α Array of azimuth angles

α0 Fixed azimuth angle vector

α0 Vector of azimuth angles at previous time step

ηp Low-frequency position vector

ηw Wave-frequency position vector

λ Vector of Lagrange multipliers

ν Low-frequency velocity vector

xiii

τ Thrust vector

τenv Generalized force vector exerted by all environmental ac-tions

τext Vector of generalized external forces

ξ See Eq. 5.1

A State matrix

a Array of polynomial coefficients of A(s)

B Damping matrix

b Array of polynomial coefficients of B(s)

BDP Linear damping introduced by DP

c Current velocity vector

c Wave number vector

CDP Stiffness introduced by DP

D Input matrix

epos Position error

evel Velocity error

f Thrust vector

f e Extended thrust vector

H Output matrix

I Identity matrix

KD Derivative gain matrix

K I Integral gain matrix

KP Proportional gain matrix

Kw Wave spectrum gain matrix, see Section 5.2.1

M Mass matrix

MA Hydrodynamic added mass

xiv

MAF Acceleration gain

Q Slack weight matrix

ri Position of the satellite i

ru Position of the GPS user

s Slack vector

T Thrust matrix

Te Extended thrust matrix

u Measurement noise vector

v Fluid velocity field

W Thrusters weight matrix

x Vessel state vector, see Eq. 5.1

ζ Water elevation

ζi Fraction of critical damping in i-th degree of freedom

ζa Wave amplitude

a Logistic function slope parameter

a Semi-major axis of ellipsoid

A(s) Denominator of parametric filter

Ai Vessel projected area along direction i

b Semi-minor axis of ellipsoid

B(s) Numerator of parametric filter

bk Bias of the k-th neuron

C Cost function for thrust allocation

c Speed of light

CD(θ) Drag coefficient

C f Cost function for parametric filter optimization

CL(θ) Lift coefficient

xv

f Ellipsoid flattening

f Frequency

F(w)i Aerodynamic force on ship along degree of freedom i

Ft Thrust force

f ∗t Propeller rotational velocity (fraction of maximum rpm)

g Gravity acceleration

H Wave height

h Water depth

Hs Significant wave height

k Wave number, see Table 2.1

KI Integral gain

KP Proportional gain

Kt Proportionality constant between actuation signal and thrust

Keq Equivalent stiffness

Kpipe Pipe stiffness

L Length of catenary line

L Wave length

LU Integral length scale

M Mass

Mn Spectral moment of order n

Nt Number of available thrusters

p Pressure

Pi Power consumption weight coefficient for the i-th thruster

r Low-frequency heading change rate

T Period

tb Time lag due to GPS receiver clock error

xvi

Tf Acceleration feedback cut-off period

TI Integral time constant

Ti Natural period for i-th degree of freedom

Tn Natural period

Tp Wave peak period

tu GPS user reception time

Tz Zero up-crossing period

tsi GPS signal sending time by the satellite i

u Low-frequency longitudinal velocity

uc Slope of catenary line

U10 10 min mean wind speed

v Low-frequency transversal velocity

vk Induced local field

vn GPS error at time increment n

v19.4 Wind speed at 19.4 m height

wc Linear weight of catenary line

wk Weight of frequency wk in parametric filter optimization

wn Gaussian white noise at time increment n

wk,j Synaptic weight of the j-th input of the k-th neuron

xj j-th input of neural network

xp Low-frequency longitudinal position

xw Wave-frequency longitudinal position

yk Output of the k-th neuron

yp Low-frequency transversal position

yw Wave-frequency transversal position

s Complex Laplace variable

xvii

- Dove ce li ha, gli occhi, il mare?- . . .- Perché ce l’ha, vero?- Sì.- E dove cavolo sono?- Le navi.- Le navi cosa?- Le navi sono gli occhi del mare.

(A. Baricco, Oceano Mare)

I N T R O D U C T I O N

Trunklines projects as strategic carriers of hydrocarbons acrossthe sea are currently moving to the depths, and to locations wherevery rough weather conditions insist over significant periods of theyear. Competition factors in the offshore construction market tendto maximise production. S-lay technology is brought to a new lightthanks to its productivity, with the requirement of optimization fordeep water and harsh environment.

Dynamic positioning opposed to mooring-based positioning ishighly desirable, unique solution for deep water projects. Mooring-based positioning is characterised by operative concerns due to com-plexity of move-up and area interested by spread mooring systems.The Dynamic Positioning capability is therefore a strict requirementfor production maximisation. Positioning performance and safetywith respect to weather conditions must be optimised, rising newneed during design phase. The Dynamic Positioning control sys-tem, pipe lay equipment, and operational procedures are now partof an integrated design.

The simulation tools developed for the design of next generationpipelayers shall be able to integrate most of the aspects involved.Further, a flexible simulation environment shall be designed withsufficient modularity to allow for substitution of components basedon the scope of the analysis. Detailed models are needed for com-ponents on which the study is focused, while simplified models arevaluable tools for sensitivity analyses and to speed up calculationwhen insight view is not needed. Most interesting is the possibilityto use the simulation model as a test environment for new featuresstudies. When a proper modular model is available, substitutingcomponents with improved versions offers an immediate way toevaluate the performance improvement, if present.

In this work the components needed to build a complete sim-ulation environment are presented, and the obtained tool is usedto test a proposal for enhancement of Dynamic Positioning perfor-mance based on prediction of incoming wave groups by means ofvessel motion measurements and neural networks.

Part I contains a basic introduction of pipelaying technology, en-vironment modeling and seakeeping problems connected to posi-

1

2 List of Listings

tioning. Part II deals with positioning systems: moorings are brieflypresented before a detailed description of the Dynamic Positioningcomponents and models is given. Further, measurement systemsfor current and future applications are presented and a new controlstrategy is proposed.

A summary of the content of each chapter follows. Chapter 1

presents the main pipelaying technologies, with the differences andfields of application. A specific device designed to hold the pipeon the barge while laying is described in detail with regard to itsdynamic response, due to the implications in Dynamic Positioningoperation. Chapter 2 presents the models for description and sim-ulation of the environmental conditions. Chapter 3 discusses theloads induced on the ship by the environment, which constitute theperturbing loads counteracted by the Dynamic Positioning system.Chapter 4 gives an brief introduction on the mooring systems forpositioning, particularly in pipelay applications. Chapter 5 describesthe architecture an principles of operations of the Dynamic Position-ing systems, which guarantee the correct positioning of the vesselby proper actuation of the thrusters, without any constraint as themooring lines. In Chapter 6 the sensors on which the Dynamic Po-sitioning systems currently rely are described, and some other arepresented for future applications like the Wave Feed-Forward whichis discussed in detail and validated through simulation in Chapter 7.Simulation models are described all along the mentioned chaptersto give a pretty complete picture of the developed components.

Part I

B A S I C S

1P I P E L AY I N G T E C H N O L O G Y

Contents1.1 Main pipelaying technologies 5

1.1.1 Onshore welded pipelines 6

1.1.2 Offshore welded pipelines 7

1.2 Pipelaying in deep water 91.3 Pipe structural model 10

1.3.1 Pipe stiffness 11

1.4 Tensioner 12

This chapter contains an introduction to the problem of offshorepipe laying and the technologies connected to the problem itself. Inparticular, an historical introduction shows the various laying tech-niques, and the vessels developed on purpose. Then an overviewis given on the currently applied methods for positioning (moor-ing lines vs. dynamic positioning) and on other devices interestingfor automation and control problems (i.e. the tensioner, the stingerrollers).

1.1 main pipelaying technologies

Offshore pipeline projects gained a crucial importance as strate-gic carriers of hydrocarbons across seas world-wide. The pipelayingtechnologies can be divided in two groups:

onshore welded pipelines The pipe segments are assembledonshore in strings of suitable length and then carried withdifferent methods in place for installation.

offshore welded pipelines The pipelaying barge carries shortpipe segments, which are then assembled during laying oper-ations. This method is the only one suited for practical instal-lation of very long pipelines.

5

6 pipelaying technology

1.1.1 Onshore welded pipelines

This group includes towing and reeling methods: both imply thewelding and control of the joints onshore, while the transportationmethod is different.

The towing methods envisage the pipe strings being transportedTowing

to the offshore location of installation by means of one or more tugboats, while being kept in a suitable configuration by a combinationof pull cables, floaters and ballast weights.

In Figure 1.1 the surface-tow method is depicted. The pipe stringis kept in tension between two tug boats to guarantee to be able tocontrol its configuration against lateral bending, while the floaterscontrol the pipe shape along the vertical plane. This way of opera-tion forces the pipe to withstand the actions from waves, if present,and therefore cannot be used in situations where significant roughweather is expected.

tug boat tug boatfloaters

pipe

seabed

Figure 1.1: Surface tow method

The mid-depth tow uses floaters or weights to make the pipe stringnearly neutrally buoyant, such that the configuration can be con-trolled by the pull applied by the tug boats connected at the twoends, see Figure 1.2. The objective is to keep the pipe string suffi-ciently in depth not to be excited by the action of waves, shouldthey be present somewhere during the route.

The bottom tow method is adopted when the pipe has sufficientspecific gravity1 to guarantee stability on the seabed, see Figure 1.3a.A tow head is applied on a single end and the string is pulled whiledragging the seabed. The off-bottom method is slightly different,requiring two tug boats to guarantee alignment, see Figure 1.3b.

Tow methods are only practical for short lines, often in remoteareas where transit costs of conventional pipelayers could be overlyexpensive.

The reeling technology is based on onshore bars assembling asReeling

1 The specific gravity of an object is defined as the ratio between the weight submergedin water and the buoyancy

1.1 main pipelaying technologies 7

tug boat tug boat

trim chains

pipe

seabed

Figure 1.2: Mid-depth tow method

well. The strings obtained away from the installation location arewrapped around a reel which is then mounted on the vessel deck,with horizontal or vertical rotational axis. The laying operations arevery rapid because there is not any welding and controlling cycle,but the method is limited to small diameter pipelines: the bendingradius is the parameter which determines the stress envisaged dur-ing reeling and straightening, and if of course limited by the reelsize to be fitted on-board.

1.1.2 Offshore welded pipelines

This group of technologies is employed in most strategic trunk-lines, being able to lay linepipe along very long routes (currentlyeven longer than 1000 km), with increasingly high diameters.

Submarine pipelines are assembled then launched and laid onthe seabed by dedicated mono-hull (keel or flat-bottom) or semi-submersible vessels. On these vessels there is a firing line alongwhich line pipe joints are brought, aligned and coupled, weldedand controlled, then held and carefully released moving up the ves-sel for launching in the depths. The integrity of the pipeline duringinstallation, see Figure 1.4, is a crucial task due to the intended oper-ational targets, as far as the safe-life and leak-less carrying capacityof the pipeline over the design lifetime is concerned. Pipe layingtechnology (Bruschi, 2002) for such strategic life-lines (20 in to 48 indiameter, 20 to 40 mm wall thickness) includes:

s-lay The pipe is assembled along a horizontal welding or firingline, it leaves the lay vessel as supported by a curved launch-ing ramp that moves the configuration from horizontal to in-clined to the depths, it crosses the water column to reach the


tug boat tug boat

tow headpipe

seabed

(a)

tug boat tug boat

trim chainspipe

seabed

(b)

Figure 1.3: Mid-depth tow method

seabed in a S shape, it is borne and kept within pipe curvatureallowance limits by a lay pull applied by the tensioner placedat the end of the firing line, Figure 1.4a.

j-lay The pipe is assembled in a vertical tower, it leaves the lay ves-sel vertically towards the depths, it crosses the water columnand smoothly bends to reach the sea-bed due to a weak hori-zontal pull applied by the lay vessel, particularly by propellers,it is borne by the tensioning device placed on the tower afterthe welding station, usually on tracks, sometimes assisted byactive clamps, Figure 1.4b.

In both pipe-lay modes, the horizontal lay pull is applied by moor-ing lines or/and propellers, which guarantee station keeping andregular pipe pay-out at the end of each welding cycle (Anselmi andBruschi, 1993). This work is focused on the configuration withoutDP pipelaying

mooring lines, classified as dynamically positioned pipe-lay vessels.In the early stage of the Oil & Gas industry, DP was mainly adoptedfor drilling ships (the earliest in the ’70ies) and vessels assisting off-shore operations i.e. diving or ROV survey of the seabed or subseainfrastructure. Offshore projects are currently moving to the depths,particularly strategic trunklines (Bianchi and Bruschi, 2008). DP willbe unique solution for station keeping and move-up of pipe lay

1.2 pipelaying in deep water 9

vessels, and in a few circumstances even in shallow waters acrossbrown fields or in presence of features that advice against mooringspread (unexploded weapons).

(a) (b)

Figure 1.4: S- and J-lay modes

1.2 pipelaying in deep water

The recent market evolution lead to larger diameters and deeperinstallations for the new strategic trunklines. The pipe wall thick-ness approaches the manufacturing limits, being also tightly linkedto material specifications. The characteristics of the pipes and theinstallation depth imply a huge weight of the suspended span to beheld by the tensioning machine onboard the vessel and by the pro-pellers system. For these reasons, the availability of holding capacityis a factor in the optimization process to be carried out in projectdesign, together with conditions determining flow properties, aspressure, diameter and number of lines. The design approach tendsnowadays to integrate specific studies on barge capability, such thatsuitable analytic tools need to be developed (De Hartog et al., 2009).

Due to its productivity, the S-lay technology is being pushed S-lay indeepwaterto achieve safe installation in deeper water, meaning that longer

stingers with complex structures and machines are being developed.In this renewed scenario, operation in DP is required, due to poorperformance of long mooring lines and simplified operative proce-dures.

The requirements on a DP lay vessel performance can be sum-marised as follows:


a. DP station keeping and pipeline pay-out control capacity;

b. Pipeline response to significant slow-drift displacements, tak-ing into account the configuration at the seabed and over thestinger, as well as reactions on rollers, in particular at thestinger exit;

c. First order vessel response, stress/strain state induced on thepipe span, and loads on equipment.

Particular attention must be paid to environmental loads espe-cially in ship-shaped vessels, where beam seas are the main concern,due to the large area exposed to the action of waves. The problemis relevant due to the limited weather vaning capability in S-lay,Weather vaning

where the heading is mainly constrained by the pipeline route. Theweather vaning is tolerated for significant angles in J-lay, and thebenefits are such that the design of suitable procedures is underevaluation also for S-lay. Should the environmental conditions, com-bined with the allowable vessel heading with respect to weather,become critical, the abandon procedure must be initiated: an aban-don head is fitted on the upper pipe end, hooked with a cable andlaid down with a winch controlling the pay-out. The same head isthen pulled on-board when the weather conditions allow to recoverthe pipe and continue operation.

1.3 pipe structural model

Modeling a pipe lay span, particularly in deep water, despite ofthe quite simple geometry, is not a trivial task due to a number ofnon-linearities that be included.

The most evident non-linearity regards the large displacementsthat the slender pipeline undergoes from the straight undeformedshape to the S or J shape assumed during laying. The contacts withthe rollers on the stinger and the seabed are monolateral constraints,inherently non-linear. Further, the constitutive law of the pipe mate-rial is often assumed to be elastic non-linear or, even worse for calcu-lation complexity, elasto-plastic. For concrete-coated pipelines, alsostress intensification factors at the uncoated field joints must be con-sidered (Bruschi et al., 1995). The problems connected to these non-linear effects are solved by iteration, such that general-purpose finiteelement codes make the calculation very time-consuming. Specifi-cally designed codes (Bruschi et al., 1994) are able to guarantee con-

1.3 pipe structural model 11

vergence in a limited number of iterations thanks to proper initial-ization and design oriented at the particular problem.

Analytical methods have been developed that deal with the prob-lem by using different formulations in different segments of the layspan. See Lenci and Callegari (2005) where the catenary solution isadopted from the stinger exit point up to the sagbend region wherebending becomes significant and the beam formulation is used.

For the dynamic positioning problem, the main concern is thepipe stiffness that, if not correctly decoupled from low-frequencybarge motion, can lead to instability.

1.3.1 Pipe stiffness

An important characteristic relevant to the pipelaying activity totake into account in a specific DP system design is the pipe stiffness,as it behaves like a mooring line for the vessel. The stiffness param-eter is associated with the whole laying span, rather than the beamsection, as it is strongly influenced by the deformed shape. As canbe easily noted, the span stiffness varies very much with the waterdepth.

From a practical point of view, the stiffness can be calculated by fi- Computation

nite element models, changing the tensioner pull from an optimizedvalue and measuring the displacement occurring at the barge. Staticanalyses are enough for the purpose, while non-linear bending-curvature relationships are needed for accurate results. The pipestiffness Kpipe is easily computed as the ratio between the tensionvariation ∆T and the barge displacement ∆x. A non-linear relation-ship is expected, that can be linearised around a particular settingidentified by the stress/strain allowance limits.

Consider e.g. a 36 in diameter pipe, laid in 20 m water depth, seeTable 1.1. The static shape under the action of the laypull is plottedin Figure 1.5, as per finite element analysis. The distance from thetensioner to a node constrained on the seabed can be measured bythe same model, and its variation logged when the laypull at ten-sioner is varied around the setpoint. The displacement-force vari-ation plot in Figure 1.6 is easily produced, with the least squareslinear approximation that indicates the pipe stiffness with regard tothe vessel surge degree of freedom, around the configuration result-ing from the laypull in Table 1.1.

As indicated in Figure 1.6 the coefficient of the least squares linearapproximation is the stiffness that the pipe introduces with respect


to the surge degree of freedom. In the present example the stiffnessis 1907 kN m−1. This value will be used in Section 1.4 to assess thetensioner response and in can be used to show by time simulationand frequency domain techniques the problem of DP instability in-duced by coupling with the pipe stiffness.

Table 1.1: Pipe data

Parameter Value Unit

Outside diameter 36 inWall thickness 25.2 mm

Concrete coating thickness 75 mmWeight in air 12 465 N/m

Submerged weight 3356 N/mAxial force at tensioner 1000 kN

Longitudinal coordinate (m)

xVe

rtic

alco

ordi

nate

(m)

z

seabed

−200 −100

−10

10

Figure 1.5: Static pipelay analysis, deformed shape in 20 m water depth

1.4 tensioner

The tensioner is a machine designed to link the pipe being laidwith the vessel, applying the correct tension and providing suitedhandling during move-up. These machines, see Figure 1.7 are based

1.4 tensioner 13

Displacement (m)

∆x

Tens

ion

vari

atio

n(k

N)

∆T

−0.020−0.015−0.010−0.005 0.0200.0150.0100.005

−100

−200

−300

100

200

300

∆T = 1907∆x

Figure 1.6: Pipe displacement-tension variation relationship

on a couple of tracks, fitted with suitable friction pads, pushed oneagainst the other by an actuator acting along a vertical axis2. Thetracks rotation is powered by electrical motors, controlled in ten-sion or current during different phases of laying operations. Often apipelaying vessel is equipped with more than one tensioner, sharingthe total load. This practice, besides the ability to apply higher laypull without any damage to the pipe coating, provides redundancyon such a critical machinery.

The control characteristics of a tensioning machine in a dynami- Dynamicresponsecally positioned vessel, especially in shallow waters, must be able to

decouple the ship from the pipe, which would act as a stiff mooringrising the problem of instability due to resonance.

The tension control loop behaves according to a PI law, with theintegral term much predominant over the proportional one. Theinput is the pipe tension, and the output is the track speed, withthe following transfer function:

h(s) = KP

(1 +

1TIs

)(1.1)

2 RE.MAC.UT. S.r.l., Rivoli, Turin, Italy, http://www.remacut.com

http://www.remacut.com


Figure 1.7: Picture of a tensioning device manufactured by RE.MAC.UT.

1.4 tensioner 15

where TI = KP/KI is the integral time constant, defined as the ra-tio of the proportional and integral gains. The transfer function be-tween pipe tension and pipe payout is immediately determined byan integration:

g(s) = KP

(TIs + 1

TIs2

)(1.2)

The Bode plot of the transfer function in Eq. 1.1 is displayed inFigure 1.8, for typical values of the gains.

Being the proportional term small, the transfer function of Eq. 1.1is simplified to:

h(s) =KPTIs

(1.3)

The transfer function where the output is the pipe payout coordinateinstead of speed is easily determined multiplying by 1/s:

g(s) =KP

TIs2 (1.4)

The tensioner acts as a spring between the pipe and the vessel.Looking globally at the system, we notice therefore that the vesselis linked to the ground by two springs in series: one is the tensioner,the other one is the pipe itsel, see e.g. the pipe analysed at Sec-tion 1.3.1. The tensioner transfer function should behave in a waysuch that the dynamic positioning system does not interfere withthe pipe stiffness, to avoid resonance.

Consider as an example the linearised pipe stiffness determined Example

for the scenario described by Table 1.1 and Figure 1.6, a proportionalgain set to 1× 10−5, and an integral time constant of 0.8 s. Coupledwith the vessel mass, an equivalent resonant period is calculated as:

Tn = 2π

√M

Kpipe= 2π

√80 000 t

1907 kN m−1 = 40.7 s (1.5)

The inverse of the tensioner stiffness at that period is calculated asthe magnitude of its transfer function:∣∣∣∣g( 2π

40.7rad/s

)∣∣∣∣ = 5.284× 10−4 m kN−1 (1.6)


Frequency (Hz)

f

Mag

nitu

de(m

/kN

)

|h(s)|

10−2 10−1

10−4

10−3

10−2

10−1

Frequency (Hz)

f

Phas

e(°

)

∠h(s)

10−2 10−1

−180

−90

90

Figure 1.8: Tensioner transfer function, input is pipe tension, output is pipepayout

1.4 tensioner 17

such that the overall stiffness of the series of springs is:

Keq =1

11907 kN m−1 + 5.284× 10−4 m kN−1 = 949.891 kN m−1 (1.7)

The system natural period is then updated to

Tn = 2π

√MKeq

= 2π

√80 000 t

949.891 kN m−1 = 57.7 s (1.8)

The tensioner stiffness must then be evaluated at the new resonantperiod to obtain a new equivalent stiffness, in an iterative proce-dure. It is possible to verify that, if the proportional gain KP of thetensioner is sufficiently high, the iterative procedure does not con-verge to a finite period, thus ensuring that the resonance is avoided,see Figure 1.9 and Table 1.2. If the KP gain is lowered by e.g. onethird, the tensioner is less reactive, and may not able to decoupleanymore the vessel from the dynamics of the pipe being laid, seeFigure 1.10 and Table 1.3.

Iteration number

i

Res

onan

tPe

riod

(s)

Tn

10 20 30 40 50 60 70 80 90

100

200

300

400

Figure 1.9: The system resonant period diverges to infinite

The decoupling action exerted by the tensioner is fundamentalfor the stability of the dynamic positioning control system. It can beshown using the simplified system model described in Section 5.2,that when a stiffness due to the pipe being laid with a blocked ten-sioner is activated, the system behaviour changes completely. Thestate-space model proposed in following chapters of this work make


Table 1.2: Iterative procedure does not converge to a finite resonant period

Iteration Natural Period Tn (s)∣∣∣g( 2π

Tn

)∣∣∣ (m/kN) Keq (kN/m)

1 40.7 5.284× 10−4 949.8912 57.7 1.057× 10−3 632.4603 70.7 1.585× 10−3 474.0444 81.6 2.114× 10−3 379.0915 91.3 2.642× 10−3 315.8296 100.0 3.170× 10−3 270.6627 108.0 3.699× 10−3 236.7978 115.5 4.227× 10−3 210.4649 122.5 4.755× 10−3 189.40110 129.1 5.284× 10−3 172.171. . . . . . . . . . . .50 288.8 2.642× 10−2 37.115

Iteration number

i

Res

onan

tPe

riod

(s)

Tn

10 20 30 40 50 60 70 80 90

20

40

60

80

Figure 1.10: The system resonant period converges to a finite value

1.4 tensioner 19

Table 1.3: Iterative procedure converges to a finite resonant period, decou-pling not effective

Iteration Natural Period Tn (s)∣∣∣g( 2π

Tn

)∣∣∣ (m/kN) Keq (kN/m)

1 40.7 3.522× 10−4 1140.7332 52.6 5.871× 10−4 899.7143 59.2 7.436× 10−4 788.6304 63.3 8.480× 10−4 728.6545 65.8 9.176× 10−4 693.4936 67.5 9.640× 10−4 671.8797 68.6 9.949× 10−4 658.2028 69.3 1.016× 10−3 649.3909 69.7 1.029× 10−3 643.64510 70.0 1.038× 10−3 639.871. . . . . . . . . . . .50 70.7 1.057× 10−3 632.454

evident the presence of system roots with positive real part, whenthe pipe is stiffer than the dynamic positioning system.

2E N V I R O N M E N T

Contents2.1 Basic potential theory 22

2.1.1 Kinematic Boundary Conditions 23

2.1.2 Dynamic Boundary Conditions 23

2.2 Regular wave theory 242.2.1 Particle trajectories 29

2.3 Sea state modeling 332.3.1 Wave spectra 33

2.3.2 Pierson-Moskowitz spectrum 37

2.3.3 JONSWAP spectrum 38

2.3.4 Torsethaugen spectrum 40

2.4 From spectra to time series 412.4.1 Sampling at constant frequency intervals 43

2.4.2 Random frequency sampling 43

2.4.3 Constant energy sampling 43

2.5 Statistics on water elevation time histories 442.6 Directional wave distribution 462.7 Wave spectra approximation 482.8 Software for sea state modeling 512.9 Wind spectra 56

This chapter provides information regarding the modeling tech-niques to represent the sea states which cause the perturbing loadsto counteract in Dynamic Positioning studies and seakeeping anal-yses. After a basic introduction to wave theory, the most popularwave spectra are presented, together with suitable techniques to ob-tain wave elevation time histories for time-domain simulation. Windspectra and simulation techniques are also briefly mentioned. Thekey role of this part of the complete system model is evidenced asit is the basis for a correct simulation of the effects exerted by theenvironment on the barge and for advanced prediction techniquesimplementation.

21

22 environment

2.1 basic potential theory

When dealing with sea waves modeling, the water can be as-sumed as inviscid and incompressible, therefore irrotational flow con-ditions hold, and the potential function can be used to represent thefluid velocity field. Representing with v = v(x, y, z, t) the vectorialvelocity field and with φ = φ(x, y, z, t) the scalar potential function,the following relation holds:

v = ∇φ =

∂φ

∂x∂φ

∂y∂φ

∂z

(2.1)

For an irrotational flow the vorticity is everywhere equal to zero:

∇× v = 0 (2.2)

and the incompressibility implies that

∇ · v = 0 (2.3)

which combined with Eq. 2.1 states that the potential satisfies theLaplace equation

Laplace equation:

∇2φ =∂2φ

∂x2 +∂2φ

∂y2 +∂2φ

∂z2 = 0 (2.4)

Bernoulli’s equation describes the field of pressure p, being in thefollowing expression z the vertical direction, positive upward. Whenthe only force applied on the fluid is gravitational:

p + ρgz + ρ∂φ

∂t+

12

ρv · v = C (2.5)

The time dependency of C can be taken into account in potentialfunction φ, such that C becomes a constant related to the pressure atthe free surface (atmospheric pressure). The free surface is supposedto be located at z = 0.

The velocity field can be determined with appropriate boundaryconditions. In particular, for gravity waves, the boundary conditionsare

a. Kinematic Boundary Conditions

b. Dynamic Boundary Conditions

2.1 basic potential theory 23

2.1.1 Kinematic Boundary Conditions

These boundary conditions, as the name suggests, are related tothe movement of the fluid particle.

At the seabed, taken as flat, horizontal and located at z = −h,where h is the mean water depth, a non-penetration condition isformulated: Non-penetration

condition∂φ

∂z= 0 on z = −h (2.6)

To formulate adequate boundary conditions at the free surface,recall the meaning of the substantial derivative of F = F(x, y, z, t):

DFDt

=∂F∂t

+ v · ∇F (2.7)

In this case the function F can be defined as:

F(x, y, z, t) = z− ζ(x, y, t) (2.8)

where ζ(x, y, t) describes the wave amplitude in a defined location,in a defined time instant. Under the hypothesis that a water particleon the surface remains on the surface during the wave cycle, thefollowing condition holds:

DFDt

= 0 on z = ζ(x, y, t) (2.9)

Using the definition of substantial derivative given by Eq. 2.7 it ispossible to write

∂ζ

∂t+

∂φ

∂x∂ζ

∂x+

∂φ

∂y∂ζ

∂y− ∂φ

∂z= 0 on z = ζ(x, y, t) (2.10)

2.1.2 Dynamic Boundary Conditions

The dynamic boundary condition is valid at the free surface, andregards the pressure equilibrium as stated by Eq. 2.5. If the constant Pressure

equilibriumC is equal to the ambient pressure p0, then the boundary conditionis expressed by:

gζ +∂φ

∂t+

12

[(∂φ

∂x

)2+(

∂φ

∂y

)2+(

∂φ

∂z

)2]

= 0 on z = ζ(x, y, t)

(2.11)

24 environment

2.2 regular wave theory

It can be seen that the boundary conditions expressed by Eqs. 2.10

and 2.11 are non-linear since they depend on the knowledge of thefree surface geometric description, which is not known until theproblem is solved. In most cases the conditions can be linearised,Linearising

boundaryconditions at thefree surface

assuming that the velocity potential is proportional to the wave am-plitude, which is a valid hypothesis when the wave amplitude issmall compared with the wavelength and the body dimension. Thefree surface conditions are moved to the mean free surface level(z = 0).

The following Eqs. 2.12 and 2.13 express the linearised kinematicand dynamic boundary conditions, respectively:

∂ζ

∂t=

∂φ

∂zon z = 0 (2.12)

gz +∂φ

∂t= 0 on z = 0 (2.13)

Once determined the velocity potential, the water surface elevationcan be determined by Eq. 2.13, while deriving the same equationwith respect to time and combining with Eq. 2.12 it is found:

∂2φ

∂t2 + g∂φ

∂z= 0 on z = 0 (2.14)

that, if the potential is assumed to be a harmonic function withcircular frequency ω, can be written as

−ω2φ + g∂φ

∂z= 0 on z = 0 (2.15)

With these linearised boundary conditions, the Airy linear wavetheory can be developed for regular waves (Airy, 1841). The hypoth-Airy linear wave

theory esis that wave is repeating periodically equal to itself leads to thefollowing guess for the velocity potential:

φ = φ0(z) sin(kx−ωt) (2.16)

where k = 2πL is said wave number (L is the unknown wave length),

and ω = 2πT is the circular frequency (the wave period T is known).

We assume that the wave is propagating along the x direction, suchthat Eq. 2.4 can be rewritten as

∂2φ

∂x2 +∂2φ

∂z2 = 0 (2.17)

2.2 regular wave theory 25

Substituting the velocity potential expression as per Eq. 2.16:(−k2φ0(z) +

∂2φ0(z)∂z2

)sin(kx−ωt) = 0 (2.18)

and, being sin(kx−ωt) in general not equal to zero

−k2φ0(z) +∂2φ0(z)

∂z2 = 0 (2.19)

The solution is a function like:

φ0(z) = Aekz + Be−kz (2.20)

with A and B to be determined with the boundary conditions.Recall the non-penetration condition on the seabed (Eq. 2.6), that

becomes for this specific φ expression Boundaryconditions at theseabed

Ake−kz = Bkekz =1D

(2.21)

such that

φ0(z) = D(

ek(z+h) + e−k(z+h))

= D cosh [k(z + h)] (2.22)

and

φ = D cosh [k(z + h)] sin(kx−ωt) (2.23)

On the free surface, it is possible to write, as a consequence of Boundaryconditions at thefree surface

Eqs. 2.13 and 2.16:

ζ(t) =H2

cos(kx−ωt) (2.24)

with H the wave height. Then, substituting Eqs. 2.24 and 2.23 inEq. 2.13, the following is determined:

−ωD cosh [k(z + h)] cos(kx−ωt) + gH2

cos(kx−ωt) = 0 (2.25)

and finally, considering that this relation holds on the free surface(z = 0)

D =gHT4π

(1

cosh(kh)

)(2.26)

In the previous equation the substitution T = 2πω has been made.

26 environment

The final solution for velocity potential is therefore:Solution forvelocity potentialφ

φ =gHT4π

cosh [k(z + h)]cosh(kh)

sin(kx−ωt) (2.27)

The relation between k and T is still to be determined, and therequired equation can be determined substituting the velocity po-tential expression in Eq. 2.14:

tanh [k(ζ + h)] =ω2

kg(2.28)

that, with the hypothesis that ζ + h ≈ h becomes

tanh(

2πhL

)=

2πLgT2 (2.29)

The previous Eq. 2.29 is solved by numerical iteration, or by anapproximation valid in deep water: if 2πh L, then

tanh(

2πhL

)= 1 (2.30)

and therefore

L0 =gT2

2π(2.31)

In Table 2.1 the equations to determine the most interesting quanti-ties characterising the wave are summarised, according to the lineartheory as developed in the previous pages. Note that the substitu-tion ζa = H/2 has been made, since it is more straightforward towork with wave amplitudes rather than with wave heights whendealing with sine waves. In Figure 2.1 the profiles of the some in-teresting quantities are plotted to make clear the phasing linkingthem.

The described linear wave theory was developed by Airy, pub-lished in 1841. Higher order wave theories include the cnoidal one,Higher order

wave theories the Stokes’ second, third and fifth order theories (Stokes et al., 1880).These are not described in this document, but some notes are herereported about how to extend the equations derived from linearwave theory outside of the mean water plane. In fact, the hypoth-esis on which the linear wave theory is founded is that the freesurface remains on the mean water plane, which is valid only when


Tabl

e2

.1:A

naly

tica

lexp

ress

ions

from

linea

rw

ave

theo

ry,g

ener

aleq

uati

ons

and

sim

plifi

edfo

rde

epw

ater

Ge

ne

ra

lEq

ua

tio

ns

De

ep

Wa

te

r

Velo

city

pote

ntia

lφ

=gζ

aω

cosh

[k(z

+h)

]co

sh(k

h)si

n(kx−

ωt)

φ=

gζa

ωekz

sin(

kx−

ωt)

Rel

atio

nbe

twee

nk

and

ωω

2 g=

kta

nh(k

h)ω

2 g=

k

Rel

atio

nbe

twee

nλ

and

Tλ

=g 2π

T2

tanh

2π λh

λ=

g 2πT

2

Wat

erel

evat

ion

ζ=

ζa

cos(

kx−

ωt)

ζ=

ζa

cos(

kx−

ωt)

Dyn

amic

pres

sure

p D=

ρgζ

aco

sh[k

(z+

h)]

cosh

(kh)

cos(

kx−

ωt)

p D=

ρgζ

aekz

cos(

kx−

ωt)

Hor

izon

talp

arti

cle

disp

lace

men

ts x

=−

ζa

cosh

[k(z

+h)

]si

nh(k

h)si

n(kx−

ωt)

s x=−

ζae

kzsi

n(kx−

ωt)

Vert

ical

part

icle

disp

lace

men

ts z

=ζ

asi

nh[k

(z+

h)]

sinh

(kh)

cos(

kx−

ωt)

s z=

ζae

kzco

s(kx−

ωt)

Hor

izon

talp

arti

cle

velo

city

v x=

ωζ

aco

sh[k

(z+

h)]

sinh

(kh)

cos(

kx−

ωt)

v x=

ωζ

aekz

cos(

kx−

ωt)

Vert

ical

part

icle

velo

city

v z=

ωζ

asi

nh[k

(z+

h)]

sinh

(kh)

sin(

kx−

ωt)

v z=

ωζ

aekz

sin(

kx−

ωt)

Hor

izon

talp

arti

cle

acce

lera

tion

a x=

ω2 ζ

aco

sh[k

(z+

h)]

sinh

(kh)

sin(

kx−

ωt)

a x=

ω2 ζ

aekz

sin(

kx−

ωt)

Vert

ical

part

icle

acce

lera

tion

a z=−

ω2 ζ

asi

nh[k

(z+

h)]

sinh

(kh)

cos(

kx−

ωt)

a z=−

ω2 ζ

aekz

cos(

kx−

ωt)

28 environment

Length along propagation direction

λ

λ

λ

λ

λ

λ

λ

λ

Potential

Dynamic pressure

x-displacement

z-displacement

x-velocity

z-velocity

x-acceleration

z-acceleration

Figure 2.1: Profile of quantities describing a wave over a wave length


the wave height is much smaller than the wave length. The equa-tions listed in Table 2.1 are therefore not suited to determine whathappens over z = 0.

The first effect to take into account is that waves, especially thehigher ones, tend to extend over the water surface more than halfthe nominal wave height: this is accounted for in non-linear wavetheories, but not in the linear one. It is commonly assumed that awave will extend up to z = 0.6÷ 0.7H. Next, the profile has to be Profile extension

extended over the z = 0 plane up to the maximum height reachedby water. Different strategies can be used, the most common being(Journée and Massie, 2001):

extrapolation The same equations from linear wave theory aresimply extended for z > 0. This leads to very conservativeestimation of drag and hydrodynamic inertia forces, linked towave velocity and acceleration.

constant extension This simple method simply extends thesame values found at the mean free surface for all the heightsabove. This yields a constant profile over the z = 0.

wheeler profile stretching The water motion is calculatedat the vertical coordinate z′, with −h ≤ z′ ≤ 0, in place of areal vertical coordinate z, with −h ≤ z ≤ ζ (Wheeler, 1969).The quantities calculated for z′ are simply moved to anotherheight z. Wheeler proposes the following relation between thetwo coordinates:

z′ = qz + h(q− 1) with q =h

h + ζ(2.32)

The results obtained with the different methods are qualitativelyplotted in Figure 2.2.

2.2.1 Particle trajectories

The equations for the horizontal and vertical displacements in Ta-ble 2.1 can be combined exploiting the well-known relation sin2 α + Particle

trajectory inshallow water

cos2 α = 1, to find the trajectory described by a particle placed at(xp, zp):

(sx − xp)2(ζa

cosh[k(zp + h)

]sinh(kh)

)2 +(sz − zp)2(

ζasinh

[k(zp + h)

]sinh(kh)

)2 = 1 (2.33)

30 environment

Profile

Vert

ical

Coo

rdin

ate

(m)

z

−60

−50

−40

−30

−20

−10

0

10Constant Ext.

ExtrapolatedWheeler

Figure 2.2: Different methods of profile extension over the mean free sur-face


which describes an ellipsoid. Moving to deep water Eq. 2.33 is fur- Particletrajectory in deepwater

ther simplified:

(sx − xp)2 + (sz − zp)2 =(

ζaekzp)2

(2.34)

then the trajectory is a circle. The different particle trajectories canbe seen in Figures 2.3 and 2.4 for shallow and deep water, respec-tively. Note that in shallow water, the particles still have and hori-zontal movement at the seabed, while in deep water the radius ofthe circle decreases rapidly at increasing distances from the free sur-face. In particular, as shown in Table 2.2, the radius of the circularparticle trajectory becomes negligible at a depth equal to the wave-length.

seabed

Figure 2.3: Particle trajectories in shallow water

Table 2.2: Particle movement in deepwater, with respect to wave amplitude

Depth/λ Particle movement/ζa

0.00 1.000−0.25 0.208−0.50 0.043−0.75 0.009−1.00 0.002

32 environment

Figure 2.4: Particle trajectories in deep water

2.3 sea state modeling 33

2.3 sea state modeling

The waves that appear on the surface of a sea under the actionof winds can be modelled as being originated by the superpositionof regular waves. A sea state is generally described by its spectrum,which can be derived from measurements on a specific region, orfrom some spectrum shape functions. These shape functions arethemselves derived from measurements, but can be used to describea wide variety of sea states rather than the single sea state observedin a specific region at a specific time.

A brief discussion on the most widely used spectra follows, but itcan be anticipated that in this work the JONSWAP one will be usedwhen not specified. The following spectra are defined for a fullydeveloped sea, which is a sea where no more energy is transferred bythe wind from the atmosphere to the water. This condition is notlikely to be achieved in seas delimited by narrow borders, but isrealistic for oceans.

Wave spectra are generally single-peaked, as this is the condition Single anddouble-peakedspectra

most frequently observed. However, after the wind has stopped, adecaying sea is formed, with a shift towards lower frequencies. Thisis called swell and sometimes can interact with the waves developedby another storm, generating therefore a low-frequency peak. Alsotidal waves can generate a similar effect, and that’s why also double-peaked spectral forms have been developed.

After the different spectra have been presented, also the tech-niques to develop a water elevation time history are explained, to-gether with the rules that govern the propagation over water sur-face.

2.3.1 Wave spectra

A sea state can be considered to be stationary for a duration up to3–6 hours, and under this condition a state can be characterized by a Sea state

characterisingparameters

limited amount of parameters. The most common are the significantwave height Hs, the peak period Tp, and the zero up-crossing period Tz.

significant wave height It is defined as the average height ofthe highest 1

3 waves, and it is a measure of the total energyrepresented by the spectrum

peak period It is the wave period at which the wave spectra showsits maximum value

34 environment

Regular waves

Irregular waves

Figure 2.5: Irregular sea with spreading as superposition of regular waves

zero up-crossing period The mean time interval between twoconsecutive instants when the sea surface moves upward acrossthe mean sea level.

Although wave spectra are experimentally determined by dedi-cated measurement campaigns, analytical formulations are widelyused to approximate those data and provide a straightforward toolfor analysis and simulation.

The link between the characterising parameters and the analyti-Spectral moments

cal wave spectra formulations are the spectral moments. The spectralmoment of order n is given by:

Mn = (2π)n∫ ∞

0f nS( f ) df =

∫ ∞

0ωnS(ω) dω (2.35)

The significant wave height Hs is therefore practically calculatedas

Hs = 4√

M0 (2.36)


while the zero up-crossing period Tz is:

Tz = 2π

√M0

M2(2.37)

Other relations are collected in DNV-RP-C205, together with theanalytical wave spectra equations which will be shown in the fol-lowing sections.

The idea behind the spectral representation of a sea state is that Irregular seastate as regularwaves’superposition

every water elevation time history can be synthetically representedby the sum of a number of sinusoidal time histories superimposed.This is also the key for the process of time history realisation froma spectrum, operation widely adopted in simulation, see Figure 2.6.

t

ω

ζ(t)

S(ω)

sampled wavespectrum

irregular water elevationtime history

each componenthas its phase

Figure 2.6: Representation of a sea state as a wave spectrum

As waves are supposed to be wind-generated, there are relation-ships expected between significant wave height and wind speed.For example, the following equation is valid for a sea described by Wind-wave

relationshipa Pierson-Moskowitz spectrum, see Section 2.3.2 (Fossen, 2002):

Hs =2.06g2 v2

19.4 (2.38)

where v19.4 is the wind speed at 19.4 m.

36 environment

As an example, the data from an instrumented buoy1 placed nearthe isle of Menorca, Spain, are taken into account (see Figure 2.7 forbuoy position). In Figure 2.8 the significant wave height and windspeed recordings are represented, for a couple of days in july 2009.The relation between the two signals is quite clear, but the repre-sentation in Figure 2.9, in which a quadratic fitting was performed,shows quite a high uncertainty. Looking at Figure 2.10 it is possi-ble to recognise that the Hs signal is delayed with respect to windof about 3 h. In fact, if the fitting is repeated shifting of three hoursone signal with respect to the other, the fitting is much more success-ful, see Figure 2.11. This effect is due to the time needed to completethe energy transfer from wind to the water surface, and is not takeninto account by the simple relation stated in Eq. 2.38.

MahónWAVESCAN buoy

Menorca

Mallorca

Figure 2.7: Location of the WAVESCAN buoy in Mahón, Menorca (ES)

1 See http://www.puertos.es/externo/clima/Rayo/indexeng.html for the list ofbuoys and download recorded data for several locations near Spanish coasts. Thesite is managed by Puertos del Estado and Instituto Español de Oceanografia

http://www.puertos.es/externo/clima/Rayo/indexeng.html


Date

Sign

ifica

ntW

ave

Hei

ght(m

)

Win

dSp

eed

(m/

s)

0:00 12:00 0:00 12:00 0:00 12:00

26-7-2009 27-7-2009 26-7-2009

0.5

1.0

1.5

3.0

6.0

9.0

Figure 2.8: Extract from recorded significant wave height and wind speedsignals in july 2009, Mahón buoy

2.3.2 Pierson-Moskowitz spectrum

The Pierson-Moskowitz spectrum (Pierson and Moskowitz, 1964)was developed from experimental analyses carried out in the NorthAtlantic Ocean. It is described by the following formulation:

SPM(ω) =516

H2s ω4

pω−5 exp

(−5

4

(ω

ωp

)−4)

(2.39)

with ωp = 2π/Tp the peak circular frequency.

Sign

ifica

ntW

ave

Hei

ght(m

)

Wind Speed (m/s)

1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

0.5

1.0

1.5 R2 = 0.523

Figure 2.9: Relation between wind speed and significant wave height

38 environment

Cro

ss-C

orre

lati

on

Lag (h)

−12 −9 −6 −3 0 3 6 9 12

−0.25

0.00

0.25

0.50

0.75

1.00

max. cross-correlation

Figure 2.10: Relation between wind speed and significant wave height

No parameters other than the significant wave height and thepeak period are needed to define this spectrum. This means thatLimitations

there is no way to differentiate sea states for their peakedness.Eq. 2.38 is supposed to be valid under the hypothesis that waves

can be considered a Gaussian random process and that the spec-trum is narrow-banded.

An example of Pierson-Moskowitz spectrum is represented in Fig-ure 2.12.

2.3.3 JONSWAP spectrum

This spectrum was derived as a result of measurement campaignsduring the Joint North Sea Wave Project (Hasselmann and Olbers,1973). It was proved to give a good description also for sea states dif-ferent from the original experimental condition. The spectral formis defined by:

SJ(ω) = AγSPM(ω)γexp

[−0.5

(ω−ωp

σωp

)2]

(2.40)

with

σ =

σa ω ≤ ωpσb ω > ωp

(2.41)


Sign

ifica

ntW

ave

Hei

ght(m

)

Wind Speed (m/s)

1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

0.5

1.0

1.5 R2 = 0.780

Figure 2.11: Relation between wind speed and significant wave height, withthe wind time history delayed of 3 h

This is clearly an expression similar to the Pierson-Moskowitz one,but with γ multiplying factor which amplifies the function mainlynear the peak value. The Aγ factor is a normalising factor that comesout from a numerical fitting procedure to get a constant spectrumintegral as γ varies:

Aγ = 1− 0.287 ln(γ) (2.42)

The constant spectrum integral is needed to get a valid link be-tween significant wave height and zero-order spectral moment (seeEq. 2.36).

The following assumption for σ is often adopted (see DNV-RP-C205):

σ =

0.07 ω ≤ ωp0.09 ω > ωp

(2.43)

If the γ value is not available from direct measurements, Det NorskeVeritas suggests the following: Choice of γ

γ =

5Tp√Hs

< 3.6

exp(

5.75− 1.15Tp√Hs

)3.6 ≤

Tp√Hs

< 5

1Tp√Hs≥ 5

(2.44)

It can be verified that the total energy associated to the sea state Sea state energycontent

40 environment

Circular Frequency (rad/s)

ω

Wav

eSp

ectr

um(m

2 s)

SPM(ω)

0.5 1.0 1.5 2.0 2.5

0.33

0.67

1.00

1.33

1.67

Figure 2.12: Pierson-Moskowitz spectrum for Hs = 3.5 m, Tp = 10 s

depends only on significant wave height:

EJ =H2

s16

(2.45)

See Figure 2.13 for the effect of γ on the spectrum shape.

2.3.4 Torsethaugen spectrum

This spectrum was developed from experimental data taken in theNorth Sea (Torsethaugen and Haver, 2004). Its main feature is thatit is double-peaked, therefore suited to represent sea states where asuperposition of two main wave frequencies takes place. This con-dition happens when wind-generated waves develop in a locationOrigin of

double-peaked seastates

reached by waves generated somewhere else, or decaying after astorm. When the wind is giving no more energy to the sea, thewaves tend to group in low frequency waves, and travel away fromthe location where the storm was generated. This kind of waves isreferred to as swell. This condition is quite common in the North Sea,and makes difficult to achieve optimal wave filtering performances.

2.4 from spectra to time series 41


ω

Wav

eSp

ectr

um(m

2 s)S(ω)

0.25 0.5 0.75 1.0 1.25 1.5 1.75 2.0

1.0

2.0

3.0

4.0

5.0JONSWAP γ = 5.0

JONSWAP γ = 3.3

JONSWAP γ = 2.0

Pierson-Moskowitz

Figure 2.13: JONSWAP spectra for Hs = 3.5 m, Tp = 10 s and some γ values,compared with a Pierson-Moskowitz spectrum with the sameparameters

The equations are reported in DNV-RP-C205, and implemented inMarine Systems Simulator (Fossen and Perez, 2004) for convenientuse in a simulation environment. In Figure 2.14 three spectra withthe same Hs but different peak periods are represented.

2.4 from spectra to time series

Spectral information is often not enough for the detailed analysescarried out for offshore operations. When time-domain simulationis required, the wave time history is usually the driving signal ofthe problem, and as such must be simulated carefully.

The aim of the time-domain simulation of the seaway is not toreproduce a particular water elevation recording that originated thespectrum, but to produce a water elevation time history with thesame frequency content and statistical properties. The procedure isgraphically represented in Figure 2.6, corresponding to the math-

42 environment


ω

Wav

eSp

ectr

um(m

2 s)

ST(ω)

0.33 0.67 1.00 1.33 1.67 2.00 2.33 2.67

0.33

0.67

1.00

1.33

1.67

2.00

Tp = 10 s

Tp = 6 s

Tp = 5 s

Figure 2.14: Torsethaugen spectra for Hs = 3.5 m and various peak periods

ematical description of the water elevation as a sum of sinusoids:

ζ(t) =N

∑i=1

ζa,i cos (ωit + ϕi) (2.46)

While the amplitude ζa,i for each circular frequency ωi is deter-mined by the spectrum to be represented, the phase ϕi can be ran-domly chosen for each harmonic component. Indeed, the choice ofRandom phases

the phases is the degree of freedom that makes possible to produceand infinite number of time series with the same spectral character-istics.

Even though the wave spectra are analytically described, in calcu-Sampling

lations one must choose a finite number of harmonics to be summedup in order to describe the irregular seaway. The way the samplingis carried out is not a trivial matter, as it has consequences on the fea-tures of the resulting water elevation time history. Three commonlyadopted methods are presented in the following sections.

2.4 from spectra to time series 43

2.4.1 Sampling at constant frequency intervals

The easiest sampling method is to simply use constant frequencyintervals, as shown in Figure 2.15. This is straightforward but raisesthe problem of the record length. In fact, it can be shown that thegenerated time history is periodic, with period

Tl =2π

∆ω(2.47)

After this time the wave time history repeats itself (Faltinsen, 1993).When using this sampling method, it must be always verified that The problem of

periodic timehistories

the simulation time is shorter than Tl . When a considerable simu-lation time is needed, many samples have to be taken, significantlyslowing down the calculation. As an example, take the spectrumrepresented in Figure 2.15, which can be conveniently representedwith samples between ωmin = 0.1 rad s−1 and ωmax = 2.5 rad s−1. Ifa simulation 1 h long is required, 1376 samples are required. This isa considerable number as the water elevation, velocity and accelera-tion are determined treating the sea state as a superposition of thisnumber of regular waves. Moreover, to correctly represent the statis-tical parameters of a sea state, longer simulations are often carriedout.

2.4.2 Random frequency sampling

To avoid the problem of the periodic wave elevation time history,a slightly more complex technique can be used. One can choose todivide the ω range into constant intervals, and then sample the spec-trum at frequencies randomly chosen between ωi− ∆ω

2 and ωi + ∆ω2 .

The process, described in Figure 2.16, is effective but still simple toimplement, and makes possible to reduce significantly the numberof harmonics involved in calculations.

2.4.3 Constant energy sampling

The last presented technique is another way of producing a non-periodic water elevation time history. Here each sample is in themiddle of two frequencies that bound a constant energy interval.

44 environment


ω

Wav

eSp

ectr

um(m

2 s)

SPM(ω)

0.5 1.0 1.5 2.0 2.5

0.33

0.67

1.00

1.33

1.67

Analytical spectrum

Sampled spectrum

Figure 2.15: Pierson-Moskowitz spectrum for Hs = 3.5 m, Tp = 10 s, sam-pled at constant frequency intervals

When the total energy is known (see Eq. 2.45), the energy of each ofthe N slices is

dE =H2

s /16N

(2.48)

This sampling method has the advantage, besides producing non-periodic output signals with a limited number of sinusoids, of repre-Physical

significance ofconstant energysampling

senting with more detail the most physically important harmonics,the ones near the peak period.

2.5 statistics on water elevation time histories

Water elevation time histories generated for simulation purposeswith the methods described in Section 2.4 have to be validatedagainst intended characteristic parameters. In fact, two main fac-tors could have detrimental effects on the representativeness of theproduced signal. First, the random phases are responsible for theEnsuring water

elevation timeseries quality

variety of time histories that can be produced starting from a sin-gle spectrum. The choice of sets of phases giving rise to particularcombinations must be avoided. Second, the produced time history

2.5 statistics on water elevation time histories 45


ω

Wav

eSp

ectr

um(m

2 s)

SPM(ω)

0.5 1.0 1.5 2.0 2.5

0.33

0.67

1.00

1.33

1.67

Analytical spectrum

Sampled spectrum

Figure 2.16: Pierson-Moskowitz spectrum for Hs = 3.5 m, Tp = 10 s, sam-pled at random frequencies inside constant frequency intervals

must be long enough to contain a number of waves that fits thenominal wave height distribution. In particular, in simulation is of-ten important to verify that the highest wave is encountered, sinceit is expected to produce the worst loading conditions.

By experimental data analysis it has been found that the Rayleighdistribution fits well the distribution of wave heights. Then the an-alytical expression of the Rayleigh distribution can be used to cal- Most probable

largest waveheight

culate expected values for the maximum wave height. It is shown(Bhattacharyya, 1978) that the most probable largest wave height amongN observed waves (MPLH) is equal to:

MPLH = 2√

2 ln N√

M0 =√

2 ln NHs

2(2.49)

If the distribution is different from the Rayleigh one, a correctionfactor must be applied, see Bhattacharyya (1978). In Table 2.3 the ra-tio between the MPLH and the significant wave height is reportedfor different numbers of observed waves. Note that, since the du-ration of a storm is conventionally of 3 h, it contains around 1000waves, such that the largest wave height is often taken as 1.86Hs.

46 environment


ω

Wav

eSp

ectr

um(m

2 s)

SPM(ω)

0.5 1.0 1.5 2.0 2.5

0.33

0.67

1.00

1.33

1.67

Analytical spectrum

Sampled spectrum

Figure 2.17: Pierson-Moskowitz spectrum for Hs = 3.5 m, Tp = 10 s, sam-pled at constant energy intervals

A short extract from a water elevation time history generated ar-tificially is shown in Figure 2.18, where a single wave is highlighted.The mean zero up-crossing period and the significant wave heightcan then be measured from time series data. In particular, the Hs isimmediately verified with Eq. 2.36 and taking into account that:√

M0 = std ζ(t) (2.50)

where std indicates the standard deviation of the water elevationsamples.

2.6 directional wave distribution

The wave spectra shown in previous sections are usually assumedto be referred to a mono-directional wave propagation. This is oftennot the case, such that directional energy spreading must be takeninto account.

A more general description of a wave spectrum includes the prop-agation direction as a variable:

S(ω, θ) = S(ω)D(ω, θ) (2.51)

2.6 directional wave distribution 47

Table 2.3: Most probable largest wave heights for different numbers of ob-served waves

Number of waves N MPLH/Hs ratio

10 1.07100 1.521000 1.86

10 000 2.15100 000 2.40

In practice the following simplification is often used:

S(ω, θ) = S(ω)D(θ) (2.52)

This means that the frequency effect and the direction effect are splitin two different functions, such that the same spectral functions canbe used for mono-directional or spread seas. In addition, DNV-RP-C205 mentions the possibility to apply, for two-peaked seas, twodirectionality functions, one for each spectrum.

To ensure that the total energy content described by the spectrum Directionalityfunctionrequirements

is not modified by the directionality function, the following equa-tion must be fulfilled:∫ θmax

θmin

D(θ) dθ = 1 (2.53)

where θmin and θmax are the extremes of the direction interval inwhich wave energy is found.

The most common directionality function is the following:

D(θ) =Γ(1 + n

2)

√π Γ(

12 + n

2

) cosn(θ − θp) (2.54)

where Γ is the Gamma function and −π2 ≤ θ− θp ≤ π

2 . The functionis plotted for different n factors in Figure 2.19.

In Figures 2.20 and 2.21 the same Pierson-Moskowitz spectrum ispresented in a polar plot with different spreading factors. Eq. 2.54

is used with n = 2 and n = 8, typical values for wind sea andswell, respectively. Note that the lower n factor distributes the waveenergy over a greater angular interval, lowering therefore the spec-trum peak value. The energy spreading level has important conse-quences on the loads produced on vessels and offshore structures.

48 environment

Time (s)

t

Wat

erEl

evat

ion

(m)

ζ(t)

20 40 60 80 100 120

1

2

−1

−2

wave period

wav

ehe

ight

Figure 2.18: Water elevation time history, generated by a constant energysampling of a JONSWAP spectrum, Hs = 3.5 m, Tp = 10 s,γ = 3.3, 200 harmonic components

2.7 wave spectra approximation

In Dynamic Positioning studies, the wave-frequency motion istreated as noise, as it acts as a disturbance superimposed on thesecond-order motion to be counteracted, see Section 3.1.1. For thisreason, a rough linear approximation of the wave kinematics can beoften used when simulating closed-loop systems (Fossen, 2002).

Consider the linear approximation y(s) of the spectral densityS(ω) as the result of a suitably filtered white noise:

y(s) = h(s)w(s) (2.55)

Being w(s) a zero-mean Gaussian noise, it satisfies the followingrelation regarding its power spectral density (Orfanidis, 2007):

Pωω(ω) = 1 (2.56)

The filter h(s) that makes y(s) approximate the wave spectrum hasto be determined, and this is accomplished making Pyy(ω) approxi-Wave spectrum

as a linear filter mate S(ω), considering the following relation:

Pyy(ω) = |h(j ω)|2Pωω(ω) = |h(|ω)|2 (2.57)

2.7 wave spectra approximation 49

Angle wrt Main Direction (rad)

θ − θp

0

D(θ − θp)

−π

4−π

2π

4π

2

0.2

0.4

0.6

0.8

1.0

n = 2

n = 4

n = 8

Figure 2.19: Directionality functions plotted for different n values

A second-order linear approximation has been proposed by Balchenet al. (1976), and later modified by Sælid et al. (1983). The followingexpression is used for the filter:

h(s) =Kws

s2 + 2λωps + ω2p

(2.58)

The gain Kw is defined as

Kw = 2λωpσ (2.59)

The σ constant is proportional to the sea state energy and λ is aspectrum shape parameter. Then, substituting s = j ω and Eq. 2.59

into Eq. 2.58, the following is obtained:

h(j ω) =j 2λωpσω(

ω2p −ω2

)+ j 2λωpω

(2.60)

and

|h(j ω)| =2λωpσω√(

ω2p −ω2

)2+ 4

(λωpω

)2(2.61)

50 environment

EW

S

N

0.20.4

0.60.8

1.01.2

ω (rad/s)

SPM(ω) (m2s)

0.20.40.60.81.0

Figure 2.20: Pierson-Moskowitz spectrum for Hs = 3.5 m, Tp = 10 s, withspreading factor n = 2, main direction East

From Eq. 2.57 we have

Pyy(ω) =4(λωpσ

)2ω2(

ω2p −ω2

)2+ 4

(λωpω

)2(2.62)

The constants σ and λ have to be determined. The first helpfulconstraint is that Pyy(ω) and S(ω) both have the maximum value atω = ωp, such that

σ2 = max S(ω) (2.63)

The λ constant is instead determined considering that the integralof the two power spectral densities must be equal.

Another technique could rely on a numerical optimization rou-tine based on approximation of Pyy(ω) to S(ω) in a least-squaressense. Popular non-linear algorithms able to solve this problem areimplemented in many scientific libraries, see The MathWorks, Inc.(2008a) and Press et al. (1992). An example of result from this kind

2.8 software for sea state modeling 51

EW

S

N

0.20.4

0.60.8

1.01.2

ω (rad/s)

SPM(ω) (m2s)

0.51.01.52.0

Figure 2.21: Pierson-Moskowitz spectrum for Hs = 3.5 m, Tp = 10 s, withspreading factor n = 8, main direction East

of approach is presented in Figure 2.22, where a JONSWAP spec-trum has been fitted by a least-squares curve fit function from theMATLAB Optimization Toolbox (The MathWorks, Inc., 2008c).

2.8 software for sea state modeling

The concepts developed in this chapter are of crucial importancefor simulation and dynamic positioning studies in general. The ac-tions exerted by the environment are the perturbations that must becounteracted by the thrusters, and incorrect modelling could pro-duce misleading evaluations of the system’s performance.

The main simulation tool on which the studies proposed in thiswork rely is built upon Mathworks SIMULINK, with the use ofsome blocks from the Marine Systems Simulator (Fossen and Perez,2004), which provides convenient means for modeling the most im-portant parts of the overall system.

52 environment


ω

Wav

eSp

ectr

um(m

2 s)

S(ω)

0.25 0.5 0.75 1.0 1.25 1.5 1.75 2.0

1.0

2.0

3.0

4.0

JONSWAP

Fitted linear approximation

Figure 2.22: JONSWAP spectrum for Hs = 3.5 m, Tp = 10 s, γ = 3.3 values,compared with linear approximation fitted by least-squares al-gorithm

In particular, a very simple model is here presented, as it will beused in most of the more complex models presented over this work.

The block Waves is part of the Marine Systems Simulator and pro-vides a way to produce spectral data for a number of wave spectra.Option are present to take into account wave spreading, visualisethe sea state over a defined area, set the number of required har-monic components. Another option is to use an external functionto calculate the harmonic components, providing greater control onthe calculation procedure. In this example the following initializa-tion function (Listing 2.1) is used, which in turn references the func-tion in Listing 2.2 to sample the JONSWAP spectrum at constantenergy intervals, and as such the waves are generated with the am-plitude selected by a user function, instead of using the facilitiesprovided by the toolbox. Note the seed parameter which providesa way to reproduce the same simulation by picking a particular setof phases, or to generate a new set of random phases each time thefunction is called.


Waves

Waves

wave directionWave velocitycalc

[x y z ] NED

Waves

Zeta

V_w NED

To Workspace

simout

ScopeLocation

(0 0 0)

Figure 2.23: SIMULINK model for wave time history generation and waterelevation measurement at a generic location

Listing 2.1: Example of initialization code for wave spectrum parameters1 Hs = 3.52 Tp = 103 gamma = 3.34 nsamples = 1005 seed = 193466 dir = 07

8 [ omega, deltaE, s, phase, domega ] = jonspec( Hs, Tp, gamma, seed, ...9 nsamples, 1 );

10 etaamp = ones(nsamples, 1) * sqrt( 2*deltaE );11 k = omega.^2/9.806;12 dir = ones(nsamples, 1) * dir;

The block Wave velocitycalc calculates the water elevation andvelocity at a particular location, in this case identified by the coordi-nates provided in Location constant block.

The simple example here described implements a water elevationmeasurement, sent to a Scope block for rapid visualisation, and aTo Workspace block for subsequent processing. The water elevationtime history of which an extract is shown in Figure 2.24 is the resultof this simple model simulation.

Listing 2.2: JONSWAP spectrum, algorithm for sampling at constant energyintervals

1 function [ omec, deltaE, s, phase, domega, Einterval, ome ] = ...2 jonspec( Hs, Tp, gamma, seed , nfr, displayplots )3 %JONSPEC Sample a JONSWAP spectrum at constant energy intervals, generate4 %random phases.5 % INPUT:6 % Hs Significant wave height (m)7 % Tp Peak period (s)8 % seed Seed for random phases generation (if 0, then initialize9 % number generator with clock)

10 % nfr Number of spectrum samples11 % displayplots 1-display 0-don’t display12 % OUTPUT:

54 environment

13 % omec Central frequency of each interval (rad/s)14 % s Value of the Jonswap spectrum for each harmonic15 % component, all equal (m^2 s)16 % phase Random phase generated for each harmonic component (rad)17 % domega Frequency span associated with each harmonic component18 % (rad/s)19 % Einterval Energy associated with each interval (m^2)20 % ome Frequencies delimiting intervals (rad/s)21

22 omegap = 2*pi/Tp;23

24 % Select minimum and maximum omega25 omemin = 0.2 * omegap;26 omemax = 5 * omegap;27

28 % Total energy29 E = Hs^2/16;30 % Energy for each component31 deltaE = Hs^2/16/nfr;32

33 % Initialize variables34 ome = zeros( nfr, 1 );35 s = zeros( nfr, 1 );36 phase = zeros( nfr, 1 );37

38 % Plot the spectrum with a fixed sampling39 if displayplots40 fig = figure;41 subplot(2,1,1)42 omega = linspace( omemin, omemax, 1000);43 plot( omega, jonswap( omega ), ’LineWidth ’, 2 );44 grid on; hold on;45 title( ’Jonswap Spectrum ’);46 xlabel( ’\omega (rad/s ) ’);47 ylabel( ’S(\omega) (m 2 s ) ’);48 xlim([ omemin omemax ]);49 end50

51 % Look for circular frequencies delimiting constant energy intervals52 ome1 = omemin;53 ome = zeros( nfr-1, 1 );54 for k = 1:nfr-155 ome2 = fminbnd( @E_Err , ome1 , ome1+omegap , ...56 optimset( ’Display ’ , ’ notify ’ , ’TolX ’ , 1e-6 ) );57 if ome2 == ome1+omegap58 disp( ’Warning! Maximum omega reached ! ’);59 end60 ome(k) = ome2;61 ome1 = ome2;62 end63

64 % Plot samples over spectrum65 if displayplots66 plot( ome, jonswap( ome ), ’ ro ’ );67 plot( omemin, jonswap( omemin ), ’go ’ );68 plot( omemax, jonswap( omemax ), ’go ’ );69 end70

71 % Append omemin and omemax to ome72 ome = [ omemin; ome; omemax ];73


74 % Check energy for each interval75 Einterval = zeros( nfr, 1 );76 omec = zeros( nfr, 1 );77 domega = zeros( nfr, 1 );78 for k = 1:nfr79 Einterval(k) = quad( @jonswap, ome(k), ome(k+1) );80 omec(k) = 0.5 * (ome(k+1)+ome(k));81 if omec(k) > omemax82 disp( ’Warning! Maximum frequency exceeded! ’);83 end84 domega(k) = ome(k+1)-ome(k);85 if domega(k) < 086 disp( ’Warning! Frequency array are not in ascending order ! ’);87 end88 end89 if displayplots90 subplot(2,1,2)91 plot(omec, Einterval, ’−ob ’);92 grid on93 title( ’Energy distribution ’);94 xlabel( ’Central \omega (rad/s ) ’);95 ylabel( ’Energy (m 2) ’);96 xlim([ omemin omemax ]);97 end98

99 % Values of spectrum for each central frequency (constant)100 s = jonswap(omec);101

102 % Random phases103 if seed == 0104 rand( ’ twister ’, sum(100*clock));105 phase = 2*pi * rand( nfr, 1 );106 else107 rand( ’ twister ’,seed);108 phase = 2*pi * rand( nfr, 1 );109 end110 % Plot phases111 if displayplots112 figure113 plot( phase, ’−om’ );114 title( ’Randomly generated phases ’);115 xlabel( ’Harmonic n. ’);116 ylabel( ’Phase (rad) ’);117 grid on118 axis tight119 end120

121 %%%%%% Functions122

123 function S_J = jonswap( omega )124 % Calculate the value of the JONSWAP spectrum125 S_PM = 5/16 * Hs^2 * omegap^4 * omega.^-5 .* ...126 exp( -5/4*(omega./omegap).^-4 );127 Agamma = 1-0.287*log(gamma);128 sigma = zeros( size(omega) );129 for i = 1:length(omega)130 if omega(i) <= omegap131 sigma(i) = 0.07;132 else133 sigma(i) = 0.09;134 end

56 environment

135 end136 S_J = Agamma * S_PM .* gamma.^( exp( -0.5 * ...137 ((omega-omegap)./(sigma*omegap)).^2) );138 end139

140 function E_Err = E_Err( ome )141 % E_ERR Calculate error between desired and actual energy step142 E_Err = abs( deltaE - quad( @jonswap, ome1, ome ) );143 end144

145 %%%%%%146

147 end

Time (s)

t

Wat

erEl

evat

ion

(m)

ζ(t)

20 40 60 80 100 120 140 160 180

1

2

−1

−2

Figure 2.24: Water elevation time history, generated by a constant energysampling of a JONSWAP spectrum, Hs = 3.5 m, Tp = 10 s,γ = 3.3, 100 harmonic components, with the SIMULINK modelrepresented in Figure 2.23

2.9 wind spectra

Similarly to the sea surface, the wind speed distribution can bedescribed by means of spectra as well. The most accurate procedureis to derive site-specific spectra for the particular location under in-vestigation, or to determine some synthetic parameters to be usedin analytical spectra formulations.

2.9 wind spectra 57

The Harris spectrum is a popular one, reported by DNV-RP-C205. Harris spectrum

It is described by the following equation:

SU( f ) = σ2U

4 LUU10[

1 + 70.8(

f LUU10

)2] 5

6(2.64)

where U10 is the 10 min mean wind speed and σU is the windspeed standard deviation (can be determined as a function of U10and terrain roughness, see DNV-RP-C205). The integral length scaleLU can be determined by the following relation, corresponding tothe Eurocode 1 specification, (Eurocode 1):

LU = 300( z

300

)0.46+0.074 ln g0(2.65)

where z is the vertical coordinate and g0 is the soil roughness pa-rameter (see DNV-RP-C205 for roughness parameters on differentterrain types).

In Figure 2.25 the Harris spectrum is plotted for a 12 m s−1 meanwind speed.

Frequency (Hz)

f

Win

dSp

ectr

um(m

2 /s)

SU( f )

0.1 0.2 0.3 0.4 0.5

5

10

15

20

25

Figure 2.25: Harris wind spectrum for 12 m s−1 mean wind speed

3S E A K E E P I N G

Contents3.1 Ship motion and degrees of freedom 59

3.1.1 First and second order motion 60

3.2 First order motion by means of RAOs 623.2.1 Statistical approach to first order motion

calculation 63

3.3 Drift forces 663.3.1 Newman’s approximation 67

3.3.2 Simplified method for drift force calcu-lation 68

3.3.3 Frequency domain analysis of drift forces 69

3.4 Software for ship dynamics modeling 703.4.1 First order motion 70

3.4.2 Second order forces and motion 71

This chapter deals with the basic notions of seakeeping, definedas the ability of a marine vehicle to operate under the actions ex-erted by the environment. The theories used in the seakeeping stud-ies are presented, with the mathematical tools adopted to modelthe complex systems involved. The distinction between first-orderand second-order motion is stressed, as the two contributions needdifferent modeling and remedial actions. Also efficient methods forsimulations are explained, with particular attention to techniquessuited for Dynamic Positioning studies.

3.1 ship motion and degrees of freedom

The kinematics of any floating body is described by six degreesof freedom, three linear displacements and three rotations (Bertram,2000), which for the case of a ship are referred to as surge-sway-heave and roll-pitch-yaw, respectively, see Fig. 3.1.

With regard to these degrees of freedom, the system is differentlymodelled, depending on the presence of the hydrostatic stiffness:

59

60 seakeeping

sway

surge

heave

roll

pitch

yaw

Figure 3.1: Reference system and degrees of freedom

surge , sway, yaw The system can be viewed as a simple mass-damper system, all the positions for each degree of freedomare equilibrium positions, as there is no hydrostatic stiffness.These degrees of freedom define the pose (position and head-ing) of the vehicle on the water plane.

heave , roll , pitch The system can be viewed as a mass-spring-damper system, where the stiffness is determined by the hy-drostatic characteristics of the floating body.

3.1.1 First and second order motion

If the motion for example of a moored vessel is observed withregard to the generic motion direction x, the typical signal wouldresemble the one schematically represented by the blue plot in Fig-ure 3.2: high frequency oscillations are superimposed on a low fre-quency motion. These two components of motion are named firstorder and second order motions, respectively, for some reasons thatwill be explained in the following paragraphs.

Each contribution to the total vessel motion correspond to a forc-ing term, by which is caused. In this sense, it is also common tospeak about first order force and second order force. This terms areboth caused by the incident waves, but the first is caused by thefirst term in the series expansion of the potential flow equation,while the other is caused by the second term of the same equation.

3.1 ship motion and degrees of freedom 61

Time (s)

t

Dis

plac

emen

t(m

)x(t)

50 100 150 200

5

10

15

20

Slow Drift Motion

Total Motion

Wave-Frequency Motion

Figure 3.2: Components of the motion of a vessel subject to waves

This should clarify the naming conventions commonly used. How-ever, an useful name interpretation considers the fact that first andsecond order also refer to the order of magnitude of the forces in-volved. It is clear that if a forcing action causes a big mass to moveat the wave periods (5–15 s), it is much more intense than a forcethat makes the same mass move with a period of 200 s.

An important observation is that while first order forces are oscil-lating with zero mean, second order ones are not. Precisely, secondorder forces are always pushing in the waves direction of propaga-tion: the force oscillation are around a mean value, but they neverinvert. These forces would therefore make the ship drift away fromits initial position without any possibility of coming back and that’sthe reason why these are often called drift forces. Note that in Fig-ure 3.2 the slow drift motion is represented as sinusoidal because ofthe presence of the moorings, that bring back the vessel in the timeperiods when the applied forces are lowering.

The distinction between the two motion (and force) componentsis of fundamental importance for the study of the Dynamic Position-ing problem, because the thrusters must be controlled in such a wayto counteract only the second order motion. In fact it is not possiblefor the machinery to follow loading variations like the ones appliedat the waves periods, and however there wouldn’t be any point indoing this because they cause zero mean oscillations around the setpoint.

62 seakeeping

With regard to linearity, it is possible to verify that the first orderloading is linearly proportional to the wave height, while the secondorder forces are proportional to the square of the wave height. Forthis reason, the first order motion is simply calculated by means ofResponse Amplitude Operators (RAOs), while the second order onesneed more detailed modeling to obtain a comparable degree of ac-curacy.

3.2 first order motion by means of raos

As stated in Section 3.1.1, the wave frequency motion is propor-tional to the wave height, such that the system can be modeled, withregard to this motion component, as a traditional linear dynamicsystem. It is very useful to adopt the Response Amplitude Operatorsas tools to study this system, while keeping in mind that this is aconcept completely general for linear systems, even if it will be usedmostly as a black box that transforms a wave signal into a barge mo-tion signal.

RAOWaves timeseries

First ordermotion

black box

Figure 3.3: Response Amplitude Operator as a black box

If the excitation is small enough, any dynamic system can be mod-elled as a linear system, fact that provides two useful properties:

scaling The amplitude of the input signal and the amplitude ofthe output signal are linearly dependent, such that e.g. doubleinput amplitude means double output amplitude.

superposition The effect of the harmonics that constitute the in-put signal can be superimposed to reconstruct the output sig-nal.

The two properties together provide a simple method to calculatethe response of a system for which a RAO is known, for any in-put signal. Note that the RAOs are complex functions, usually ex-pressed in amplitude/phase notation. In Figure 3.4 a set of RAO forall the six degrees of freedom is reported as an example (this setrefers to a semisubmersible present in the MOSES software library,

3.2 first order motion by means of raos 63

see Ultramarine, Inc. (2008)). Note that the phase plots look quiteconfused because the phase is expressed as an angle in the [−π; π]interval, such that the lines jump from one side to the other whencrossing the extremes values.

The following example illustrates the simple process used in thiskind of calculations. If a system is subject to a loading function

i(t) = I sin(ωt) (3.1)

and the RAO corresponding to ω is represented by amplitude Aand phase φ, then the response is:

o(t) = IA sin(ωt + φ) (3.2)

If now the input signal is formed by the superposition of two si-nusoidal waves at ω1 and ω2 circular frequencies, the response iseasily determined reading the RAO function at those frequencies:

i(t) = I1 sin(ω1t) + I2 sin(ω2t) (3.3)

o(t) = I1 A1 sin(ω1t + φ1) + I2 A2 sin(ω2t + φ2) (3.4)

Phases different from zero in the input sinusoids have been omitted,but their presence would not change the procedure.

From the previous example is immediate to recognise that theextension to a number of harmonic components adds no other dif-ficulties, making it possible to calculate the response from a waveinput signal built as explained in Section 2.4.

3.2.1 Statistical approach to first order motion calculation

In offshore engineering problems, the input signal to a dynamicsystem is often treated as a random signal, exploiting therefore thestatistical properties derived from time histories studies.

It is possible to derive a response spectrum for each degree offreedom once the input signal spectrum and the RAOs are known.The relation, when the RAO is considered as a linear filter, is fromthe classical linear theory:

So(ω) = [RAO(ω)]2 Si(ω) (3.5)

where So and Si are the system’s input and output spectra.

64 seakeeping

Beam Sea Quartering Sea Following Sea

T (s)

T (s)

Surg

ePh

.(°)

Surg

eA

mp.

(m/m

)

10 20 30

0.33

0.67

180

−180

T (s)

T (s)

Sway

Ph.(

°)Sw

ayA

mp.

(m/m

)

10 20 30

0.33

0.67

180

−180

T (s)

T (s)

Hea

vePh

.(°)

Hea

veA

mp.

(m/m

)

10 20 30

0.4

0.8

180

−180

T (s)

T (s)

Rol

lPh.

(°)

Rol

lAm

p.(°

/m)

10 20 30

0.33

0.67

180

−180

T (s)

T (s)

Pitc

hPh

.(°)

Pitc

hA

mp.

(°/m

)

10 20 30

0.25

0.50

180

−180

T (s)

T (s)

Yaw

Ph.(

°)Ya

wA

mp.

(°/m

)

10 20 30

0.04

0.08

180

−180

Figure 3.4: Example of RAO functions for a semisubmersible (Ultramarine,Inc., 2008)

3.2 first order motion by means of raos 65

As represented in Figure 3.5, the RAO acts as a classical linearfilter, able to amplify the harmonic components near the resonancepeak, while suppressing harmonic components present in the inputsignal. The single-peaked roll RAO is typical of ship-shaped vehi-cles.


ω

S i(ω

),R

AO

(ω),

S o(ω

)

0.25 0.5 0.75 1 1.25 1.5

1.0

2.0

3.0

4.0

5.0

JONSWAPspectrum (m2s)

Roll RAO(deg/m)

Roll responsespectrum (deg2s)

Figure 3.5: Roll response spectrum of a vessel subject to a sea state repre-sented by a JONSWAP spectrum with Hs = 3.5 m, Tp = 10 s andγ = 3.3

Once the response spectrum has been obtained, the standard statis-tic techniques can be applied to determine the extreme responsevalues, which are the most important data for design activities. Oneimportant aspect is the distribution of maxima: it can be shown(Barltrop and Adams, 1991) that if a random time history is repre-sented by a narrow-banded, Gaussian distribution of instantaneousvalues, the peaks will be distributed according to a Rayleigh dis-tribution, such that all the statistical properties analysed for wavesextend to the system’s response. Note that, being this theory basedon the linear wave theory, non-linear interactions between wave har-monic components are neglected, leading to a slight over-estimateof maxima, as a consequence of the real sea state distributions be-

66 seakeeping

ing not exactly Gaussian and Rayleigh. It is possible, under certaincircumstances, to apply adequate correction factors to take into ac-count these effects not considered by the linear wave theory (Bhat-tacharyya, 1978).

3.3 drift forces

These forces are as already explained much smaller than first or-der ones, but still important because the mean is different from zero,making the floating body drift away under the action of waves. Thesmaller order of magnitude implies difficulties in experiments andin calculation.

Sometimes in literature the drift forces are divided in two contri-butions: mean force and slowly-varying forces. However, the physicalorigin of the two is the same, and as such they can be treated as asingle force contribution.

When dealing with drift forces, one is interested in the effecton the degrees of freedom that do not show hydrostatic stiffness,namely surge, sway and yaw. If the pressure integration over thewetted surface of the floating body is carried out with regard tothese degrees of freedom, it is possible to recognise the three con-tributions from hydrostatic pressure, first order pressure, secondorder pressure. The development of the mathematical procedure isdetailed in Journée and Massie (2001), where it is possible to verifythat the second order term of two forces and the yaw moment areproportional to the square of the wave height.

As an example, we focus on the surge degree of freedom, identi-fied on the following equations by the subscript index 1. The firstand second order contributions are instead identified by the super-script (1) and (2), respectively. The second order surge force as aresult of the direct integration (Journée and Massie, 2001) can bewritten as:

F(2)1 (t) = −

∮wl

12

ρg(

ζ(1)r (t, l)

)2n1dl (3.6)

where n1 is the 1-direction component of the unit vector normal tothe hull surface, ζ

(1)r (t, l) is the first order relative water elevation as

a function of time and of the coordinate over the water line l. Thislast function is defined as the difference between the water elevation

3.3 drift forces 67

and the first order vertical position of the hull at the water line.When the first order water elevation is expressed as (see Eq. 2.46)

ζ(t)(1) =N

∑i=1

ζ(1)a,i cos (ωit + ϕi) (3.7)

then ζ(1)r (t, l) is calculated as:

ζ(1)r (t, l) =

N

∑i=1

ζ(1)a,i ζ

(1)′r,i (l) cos (ωit + ϕi + ϕr,i(l)) (3.8)

Substituting Eq. 3.8 in Eq. 3.6, the result is:

F(2)1 (t) =

N

∑i=1

N

∑j=1

ζ(1)a,i ζ

(1)a,j Pij cos

[(ωi −ωj

)t +(

ϕi − ϕj)]

+N

∑i=1

N

∑j=1

ζ(1)a,i ζ

(1)a,j Qij cos

[(ωi −ωj

)t +(

ϕi − ϕj)] (3.9)

where all the higher frequency terms have been omitted (sum offrequencies). In Pij and Qij all the time independent terms are col-lected, and these are the terms which characterise the vessel fromthe second order forces point of view. In Journée and Massie (2001)the analytical expression is developed, however these terms, for realvessel shapes, are determined by numerical calculations by meansof software like e.g. MOSES (Ultramarine, Inc., 2008).

3.3.1 Newman’s approximation

In 1974, Newman proposed an approximation which is still widelyused to calculate efficiently the second order forces applied on a ves-sel subject to the action of waves. The idea is to approximate Pij andQij with Pii, Pjj, Qii and Qjj. This implies the following assumptions:

Pij = Pji =12(

Pii + Pjj)

(3.10)

Qij = Qji = 0 (3.11)

This is motivated by the fact that usually we are interested in theterms where ωi is near ωj because when ωi −ωj is too large, we are

68 seakeeping

going to higher frequency terms, away from the range of interest formooring and dynamic positioning analyses.

A further approximation, proposed by Newman (1974), too, is thefollowing:

F(2)1 (t) = 2

(N

∑i=1

ζ(1)a,i (Pii)

12 cos (ωit + ϕi)

)2

(3.12)

This allows for the calculation of N terms instead of N2 terms, en-suring faster program execution.

3.3.2 Simplified method for drift force calculation

Another method that allows for fast drift force evaluation waspresented by Hsu and Blenkarn (1970). The method was proposedas an intuitive way of understanding the rules governing the phe-nomenon of slow oscillations, and is schematically represented inFigure 3.6.

Time

t

Wat

erEl

evat

ion,

Dri

ftFo

rce

drift force

Figure 3.6: Hsu and Blenkarn (1970) simplified method for drift force calcu-lation

3.3 drift forces 69

Each half-wave is identified between subsequent zero-crossings,then the period of the single wave can be determined as the dou-ble of the time distance between the two crossings, and the ampli-tude is the maximum water elevation in the same time period. Thisinformation is used to calculate one point of the slow drift forcetime history, as the QTF for the found wave period, multiplied bythe square of the wave amplitude. The force is always pushing inthe waves’ direction, regardless of the half-wave being a crest or atrough.

This method is very simple and provides immediately a roughestimate that can be useful in simplified simulations.

3.3.3 Frequency domain analysis of drift forces

The slow drift force on a floating body can be conveniently writ-ten in frequency domain, to allow for spectral calculations and pre-liminary mooring and DP performance assessment. According toPinkster (1980) the spectral density of the drift force can be ex-pressed by:

S(2)F (µ) = 8

∫ ∞

0S(ω)S(ω + µ)

F(2)i(ω + µ

2)

ζ2a

2

dω (3.13)

where F(2)i is the mean second order force for a wave with the given

circular frequency.The drift force spectral density is represented in Figure 3.7 for dif-

ferent sea states. All the considered wave spectra have a peak periodTp = 8 s. It is visible that the peak of the drift force happens at zerofrequency (mean force), and the force value is negligible at wavefrequency, especially when the presence of first order wave forcesis considered. These are in fact, as already pointed out, much moreintense, making the second order contribution at wave frequencycompletely negligible.

The spectral behavior of the drift forces is of practical impor-tance because it can be used for a preliminary estimation of DPsystem performance. A rough yet relatively reliable estimate canbe deduced with a simple mass-spring-damper model, as shown inSection 5.5.

70 seakeeping


µ

Dri

ftFo

rce

Spec

tral

Den

sity

(kN

s2 )

S(2)F (µ)

0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

2×106

4×106

6×106

8×106

waves peak frequency

Hs = 2.5 mHs = 3.0 mHs = 3.5 m

Figure 3.7: Drift force spectral densities for different sea states

3.4 software for ship dynamics modeling

In this section the calculation methods put together to model thedynamics of a ship under the action of waves are presented. TheSIMULINK models are obviously closely related to the ones for thegeneration of waves, described in Section 2.8. The following sectionsdescribe separately the models developed for first and second ordership dynamics.

3.4.1 First order motion

A simple SIMULINK model able to calculate the first order mo-tion of a barge is shown in Figure 3.8. The Waves generator blockcontains the scheme shown in Figure 2.23, which is the origin ofthe wave time history in each of the models presented in this work.The 1st order block takes the Wave out signal containing all theharmonic components of the sea state, with their direction, ampli-tude, phase, circular frequency and wave number, and calculates theresponse for each of them, by means of the following equation:

ηWF,j(t) =N

∑i=1

ζa,i Aj(ωi, ψi) cos(kix−ωit− ϕi + Φj(ωi, ψi)) (3.14)

3.4 software for ship dynamics modeling 71

In previous equation, ηWF,j(t) is the wave frequency motion of thebarge centre of gravity with regard to the j-th degree of freedom,Aj(ωi, ψi) and Φj(ωi, ψi) are the amplitude and phase of the re-sponse operator with regard to the same degree of freedom for ωicircular frequency and ψi incoming direction.

Waves generator

Wave Out

Wave Direction

WaterElev

V_w NED

Scope 1

Scope

LF motion

[0 0 0 0 0 0 ]’

1st order

etaLF

Wave In

etaWF

nuWF

nuWFdot

Figure 3.8: SIMULINK model for first order motion calculation

The scheme in Figure 3.9 is one of the sub-systems of the MSSblock Motion RAO, which has been modified into a new block namedMotion RAO (U=0) to take into account the fact that in our simula-tions the ship travel speed is always negligible. This modificationsimplifies a bit the use of the block and the program input, withoutany significant loss of detail. Note that the 1st order block takes asinput signal also a 6-component constant array, to state that the ves-sel has no low frequency motion. This input is used when the vesselis also subject to forces causing slow or non-oscillatory motion thatcan not be modelled by means of Response Amplitude Operators.

In this model, and everywhere this is possible, the Time Series

data structure, defined in MATLAB, is used, to have access to theproperties and methods specifically designed for easy processingand plotting of signals in time domain (The MathWorks, Inc., 2008a).In Figure 3.10 the result of a short simulation for a tanker is shown,for a quartering sea scenario.

3.4.2 Second order forces and motion

While for wave-frequency motion a direct derivation of motionis performed thanks to the definition of RAOs, for second order

72 seakeeping

nu_

dot r

ate

4

nu_

dot W

F

3

nu W

F

2

eta

_h

WF

1S

elec

tor5

U Idx1

1

Y

Sel

ecto

r4

U Idx1

1

Y

Sel

ecto

r3

U Idx1

1

Y

Sel

ecto

r2

U Idx1

1

Y

RA

O ta

bles

psi_

r

omeg

a

U

amp

phas

e

Pro

duct

7P

rodu

ct6

Pro

duct

5

Pro

duct

4

Pro

duct

3

Pro

duct

2

Pro

duct

1

Mem

ory

4

Mem

ory

3

Mem

ory

2

Mem

ory

1

Mat

hF

unct

ion

1

|u|2

Mat

hF

unct

ion

|u|2

Gai

n1

−1Gai

n

−1F

or It

erat

or

For

Itera

tor

N1

: N

Con

stan

t

0

1

cos

sin

sin

cos

Zet

a_

a

5

Pha

se_

tot

4

Om

ega

3

Psi

_r

2

num

ber

of

itera

tions

1

psi_

r

omeg

a

phas

e

i=1:

N

zeta

_a

Figu

re3

.9:S

ub-s

yste

mof

the

MSS

Motion

RAO

bloc

k,m

odifi

edfo

rze

rosp

eed


Time (s)

t

Wat

erEl

evat

ion

(m)

ζ(t)

25 50 75 100 125 150 175 200

−1.0

1.0

2.0

Water Elevation

Time (s)

t

Surg

e,Sw

ay,H

eave

(m)

η1,2,3(t)

25 50 75 100 125 150 175 200

−1.0

1.0

SurgeSwayHeave

Time (s)

t

Rol

l,Pi

tch,

Yaw

(deg

)

η4,5,6(t)

25 50 75 100 125 150 175 200

−1.0

1.0

RollPitchYaw

Figure 3.10: Simulated first order motion for a tanker, Hs = 3.5 m, Tp = 20 s,γ = 3.3, quartering sea

74 seakeeping

dynamics the drift forces are first calculated and then applied to amodel of the ship. This different approach is obviously because ofnon-linearity, which doesn’t allow the use of the classical definitionof transfer function.

The simplified Newman formulation is used in this model (seeEq. 3.12), to speed up the simulation. Higher detail is most of theSimplified drift

force calculation times not required for DP system simulation. The overall model isnow represented by the scheme in Figure 3.11, where the additionof the 2nd order block is evident. This block contains a first subsys-tem (Figure 3.12) where data is prepared to be fed to the real force-calculating block (Newman1974). The algorithm implemented by thisblock is represented in Figure 3.13. As highlighted by the diagram,the Q coefficients of the QTFs are not used by this simplified theory.For the purpose of being consistent in the dimensions of the signalsets exchanged by different blocks, also the force related to heave,roll and pitch have been calculated and sent to the output port. Ofcourse this terms will be completely negligible with respect to thefirst order forces and the hydrostatic stiffness forces.

Waves generator

Wave Out

Wave Direction

WaterElev

V_w NED

Scope 2

Scope 1

Scope

LF motion

[0 0 0 0 0 0 ]’

2nd order

eta LF

Wave In

Drift Force

1st order

etaLF

Wave In

etaWF

nuWF

nuWFdot

Figure 3.11: SIMULINK model for second order force calculation

In Figure 3.14 the results of a simulation with the model depictedExample

in Figure 3.11 are plotted. It is evident that the sway force is muchmore significant that the surge force, as it is related to a linear di-mension which is in general an order of magnitude higher for aship-shaped hull. This consideration is important to understand the


problem of dynamic positioning and the efforts that are being spentto gain performance on the control side.

Since in dynamic positioning studies we are mainly interested indisplacements from a defined set-point, the forces calculated with From force to

motionthe method shown above (or any other sufficiently accurate) mustbe applied to a ship model. With regard to low-frequency motion onthe water plane, the ship can be conveniently represented as a pointmass with three degrees of freedom, appropriately damped. Theonly complication is due to the hydrodynamics involved in bargemotion: the mass to account for is not barely the ship mass, with allits equipment and cargo, but also the added hydrodynamic mass.This additional mass can be intuitively viewed as the mass of thewater which follows the ship motion, and participates to the over-all vessel inertia. This contribution is obviously not the same foreach degree of freedom, and the same holds for the damping, whichis also an hydrodynamic effect caused by drag forces. Then, 3-dofmass and damping matrices are needed to build a simple but reli-able ship model for low-frequency dynamics. The added mass andhydrodynamic damping are an ancillary result from the RAO cal-culation programs, as they are important terms for wave-frequencydynamics. The values calculated by potential theory for the lowestfrequency of interest can be used, as well as the asymptotic values.The following system of three equations is here represented as a Low-frequency

ship modelsingle vectorial equation:

Mν + Bν = F(2) (3.15)

where the M mass matrix includes both the vessel mass and thehydrodynamic added mass. The linear damping matrix B has beenused here, despite the fact that the damping is for vessels providedby drag forces, which are known to depend on the fluid velocitysquared. The reason for that is twofold: first, to obtain a linearmodel with the resulting simplification, second, potential theory isinherently linear, such that the damping matrices produced by com-mon seakeeping software packages refer to a linear model too. Thevelocity vector ν is the low-frequency motion, therefore determinedbefore the first order motion obtained by RAOs is added.

76 seakeeping

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Figure 3.14: Simulated second order forces for the case of a tanker, Hs =3.5 m, Tp = 10 s, γ = 3.3, beam sea

Part II

F R O M M O O R I N G S T O D Y N A M I CP O S I T I O N I N G

4M O O R I N G S

Contents4.1 Mooring-based pipelaying 814.2 Static catenary solution 82

4.2.1 Mooring stiffness 84

4.3 Spread mooring systems 86

The present chapter deals with the station keeping alternative toDP, making use of mooring lines. First the application of mooringsto pipelaying is discussed, with the limitations and problems con-nected to the complexity of the system and the need for move-upalong the pipe route. Then the analytical model of catenary cableis presented. Some general information on the design of mooringsystems composed by several lines is presented in the last part ofthis chapter.

4.1 mooring-based pipelaying

Different types of moorings are used in offshore industry for dif-ferent applications, with a variety of materials and principles ofoperation. The following classification (Journée and Massie, 2001)includes the most common: Tupes of

mooringstaut lines The lines are in general made of synthetic fibres, with

a very light submerged weight. The restraining action is basedon the stretching of the lines themselves.

tension legs This scheme is used in particular in tension leg plat-form, whose buoyancy exceeds the submerged weight, suchthat the static equilibrium position is maintained by the verti-cal downward force applied by the mooring. When the float-ing object is displaced horizontally, the mooring lines moveaway from vertical direction and a restoring action is put inplace.

catenary lines The weight of the lines, made of steel chainsor rope, or a combination of both, is significantly influencing

81

82 moorings

the shape assumed by the suspended length. The stiffness themoored object is subject to is mainly associated with liftingand lowering of the lines. A significant length of mooring lineis usually lying on the seabed, such that little vertical resis-tance is required by the anchors attached at the end.

Pipelaying assisted by a mooring system is an operation charac-Practicalconcerns terised by a number of practical concerns. The positioning require-

ments are constrained by the pipe stress/strain state. An additionalproblem specific for pipelaying is the move-up: the barge must movealong the route as the pipe is being laid. The ability to follow thedesign corridor within the given tolerances must be ensured. Thisimplies the need for anchors that must be moved by assisting bargesto make the vessel follow the route. The anchors employed for thistask are therefore characterised by small vertical resistance, and thehorizontal holding capacity is supplied also by a significant lengthof mooring line lying on the seabed.

Several lines attached fore and aft, port and starboard, and spreadaround the hull. The area interested by the presence of the lines, sus-pended or lying on the seabed, is extending over a circle that easilyexceeds a km in radius. The extension of the area clearly rises prac-tical problems, particularly in case of the presence of other deviceson the seabed or structures like platforms, or unexploded weapons.

4.2 static catenary solution

The analytical static solution of the catenary mooring line is herederived to analyse the working principle of the mooring systemsadopted in pipelaying. The nomenclature for the mathematical ex-pressions is in Figures 4.1 and 4.2a.

Suppose that the shape of a given length L, linear weight wc ofline suspended between point A and B shall be determined. Theslope uc is:

uc =dzdx

(4.1)

The infinitesimal length of line ds corresponding to dx and dz is:

ds =√

dx2 + dz2 =√

1 + u2c dx (4.2)

4.2 static catenary solution 83

Said c the ratio between the horizontal force component HB andthe linear weight, the variation of slope is determined by (see Fig-ure 4.2b):

duc = d(

dzdx

)=

wcdsHB

=dsc

(4.3)

Recalling Eq. 4.2:

duc =√

1 + u2c

cdx (4.4)

Rearranging:

dx = cduc√1 + u2

c(4.5)

The previous equation can be integrated:∫dx = c

∫ duc√1 + u2

c(4.6)

x = c ln(

uc +√

1 + u2c

)+ C1 =

csinh uc

+ C1 (4.7)

uc = sinh( x

c− C1

)(4.8)

The condition that horizontal slope holds for x = 0 is imposed, thenC1 = 0 and

uc = sinh( x

c

)(4.9)

The deflected shape is determined by another integral:

dz = sinh( x

c

)dx (4.10)

∫dz =

∫sinh

( xc

)dx (4.11)

z = c cosh( x

c

)+ C2 (4.12)

84 moorings

Choosing C2 = 0, z = c holds for x = 0, as depicted in Figure 4.1.The curvilinear abscissa s as a function of the horizontal coordinatex is calculated:

s =∫ x

0

√1 + u2

c dx =∫ x

0

√1 + sinh2

( xc

)dx = c sinh

( xc

)(4.13)

Combining Eqs. 4.12 and 4.13 the following is obtained:

z2 − s2 = c2 (4.14)

Previous equation makes available the relation to obtain by numer-ical solution the value of c given the length L of the line and therelative coordinates of the points A and B.√

L2 + h2 = c√

2 sinh(

k2c

)(4.15)

The coordinates of the point A with respect to the x-z frame are:

xA = c ln

(√L + h

2(L− h)

)− k

2(4.16)

zA = c cosh( xA

c

)(4.17)

The forces acting on the section of the line are:

H = cwc (4.18)

V = swc (4.19)

T = zwc (4.20)

4.2.1 Mooring stiffness

The equations derived in previous section completely describe thegeometric configuration assumed and the forces exerted by the lineunder static conditions. The stiffness of the mooring with respectto horizontal displacement of the moored object is an important

4.2 static catenary solution 85

x

z

B

A

TB

HB

VB

dxdz

ds

c

h

k

φ

Figure 4.1: Catenary line reference scheme for analytical model

T + dT

T

φ wc ds

(a)

T + dT

T

wc ds

φ

(b)

Figure 4.2: Equilibrium of an infinitesimal length of catenary mooring line

86 moorings

concern, because it is directly related to the excursions experiencedunder variable force conditions (Faltinsen, 1993).

An example is used to show the effects of the main geometri-cal parameters on the resulting stiffness. A linear weight of wc =981 N m−1 (100 kg m−1) is assumed for a mooring line that is askedto exert a horizontal pull H = 300 kN in 30 m water depth. The an-chor is assumed to be able to withstand only horizontal pull, suchthat a length of line lying on the seabed must be ensured.

The stiffness of the mooring is calculated by determining via an-Calculationmethod alytical formulae the increase or decrease in projected length of

the suspended cable as a result of horizontal pull variation. In thepresent example, the horizontal force has been increased to 350 kNand decreased to 250 kN, with the projected span length evidencedin Figure 4.3. The stiffness is determined as ratio between the hori-zontal force variation and projected span length:

Km =∆H∆k

= 4.38 kN m−1 (4.21)

The maximum axial force is T = 329.43 kN. In shallow water,the force is applied to the moored object almost horizontally, withno significant tension increase due to suspended span weight. Asthe depth increases, the vertical component of the sectional force atthe surface grows rapidly, see Table 4.1, up to critical values. Thestiffness trend is opposite: the decrease is significant in deep water.

seabed

k = 134.4 mk = 145.3 mk = 122.5 m

30 m

30m

Figure 4.3: Example mooring line configuration

4.3 spread mooring systems

To guarantee station keeping and correct heading defined by thepipeline route, at least three mooring lines acting along intersecting

4.3 spread mooring systems 87

Table 4.1: Axial cable tension at upper end as a function of water depth, fora horizontal pull of 300 kN

Depth (m) Axial force T (kN) Stiffness (kN/m)

30 329.43 4.3850 349.05 3.38100 398.10 2.36200 496.20 1.64300 594.30 1.33

directions must be provided. In terms of analysis, the tools achievedfor a single line can easily be extended to systems of multiple moor-ings.

x

y

(xi , yi) θi

Figure 4.4: Mooring system

With reference to nomenclature as indicated in Figure 4.4, thefollowing mean forces are balanced by the mooring system:

F(M)1 =

n

∑i=1

H cos θi (4.22)

F(M)2 =

n

∑i=1

H sin θi (4.23)

F(M)6 =

n

∑i=1

H (xi sin θi − yi cos θi) (4.24)

88 moorings

where n is the number of lines connected.The stiffness matrix too is determined by superposition of the

effects of the single moorings:

C(M)11 =

n

∑i=1

ki cos2 θi (4.25)

C(M)22 =

n

∑i=1

ki sin2 θi (4.26)

C(M)66 =

n

∑i=1

ki (xi sin θi − yi cos θi)2 (4.27)

C(M)26 = C(M)

62 =n

∑i=1

ki (xi sin θi − yi cos θi) sin θi (4.28)

The coupled terms C(M)16 = C(M)

61 and C(M)12 = C(M)

21 are equal tozero in the usual case of symmetry about the longitudinal plane(Faltinsen, 1993).

In design of mooring for operation, the many degrees of freedom(number of lines, tension, anchor positions, length of lines) must beset for the specific scenario, in terms of weather conditions, pipe laytension, water depth. The problem is quite complex and subject con-straints that may be imposed by the seabed configuration or by thepresence of structures or devices like platforms, existing submarinepipelines or cables, or other dangerous features (e.g. unexplodedweapons). Further, the move-up procedure must be planned to opti-mise productivity, one of the most critical competition factors.

The complexity of the problem in design and most importantlyin operation is one of the reasons for which dynamic positioning isconsidered more desirable. Further, the gradual loss of performancein increasingly deep waters makes impossible to lay the pipelinesfor the next generation of strategic projects.

5D P B A S I C S

Contents5.1 DP architecture 895.2 Observer 91

5.2.1 Kalman filter in dynamic positioning 91

5.3 Controller 955.4 Thrust allocation logic 99

5.4.1 Pseudo-inverse algorithm, fixed-azimuths 102

5.4.2 Nonlinear programming 105

5.4.3 Simulation 106

5.5 Simplified performance assessment 1105.5.1 Filtering delay 113

5.6 Acceleration feedback 116

This chapter regards the standard architecture of a Dynamic Po-sitioning System, as it is currently applied in pipelayers and manyother vessels. The general scheme will be presented, then an insighton each component will be given to find out critical aspects andmotivate the development of the approach proposed in this work.

5.1 dp architecture

A DP installed on a pipe-lay vessel operates in a way to controlthe effects of the actions that the environment exerts on the vessel it-self, notably the offset from a setpoint suitable for operation. Layingoperations are performed in increasingly severe conditions, and theDP system is asked to counteract effects of various environmentalforces acting on the pipe-lay vessel and to control the pipe tensionas well. This ensures that the pipe is kept in a configuration notexceeding allowable stress/strain conditions.

Dynamic positioning only deals with three of the six degrees offreedom of a floating body, i.e. surge, sway and yaw, which definethe vessel pose1 on the water plane.

1 The term pose, inherited from robotics, defines the set of coordinates that completelydefine the placement of a body. For a rigid body lying on a plane, this set is composedof two translational and one rotational coordinates.

89

90 dp basics

In some applications the DP task is simplified by the fact thatthe vessel in operation can be conveniently headed towards the di-rection of impacting sea state, thus minimizing the forces derivingfrom the interaction with the hull. On the contrary, for S-pipelayactivity, the control of the heading is very important to ensure pipeintegrity. The control of pipe-lay setpoint and heading is performedby a suitable configuration of azimuthal, tunnel, longitudinal pro-pellers, which counteract environmental loads from the action ofwaves, current, wind, and by the pipe being laid.

The control action is based on ship pose as measured by a seriesof sensors, often redundant to minimise effects from measurementnoise. The logical entities that constitute the system are depicted inFigure 5.1 (Bruschi and Gaggiotti, 2008).

Ship

Environmentalloads

Observer

+

Measurementnoise

Controller

Set-point

Thrust Allocation Logic

state

measuredheading/-position

predicted state

force/momentto be applied

to the ship

thrust/angle com-mand to eachthruster

Figure 5.1: DP system overview

The ship is subject to environmental loads and action applied bythrusters counteracts second order forces. Ship motions are com-posed by first and second order components, giving a time his-tory record that evidences wave-frequency motion superimposed toslowly-varying motion (see Figure 3.2). As the thrusters are asked toreact only to slow displacements, the DP system needs some meansfor separating the two contributions, thus a state observer is em-ployed. This separation is reflected in mathematical models usedfor DP studies and design (see Section 5.2). The observer makes useof Kalman filtering techniques to provide the best estimate of po-sition and velocity. Once the component of ship motion useful for

5.2 observer 91

control purposes has been extracted, the error with respect to a de-fined setpoint is passed to the controller, which decides the systemof forces to be applied by the thrusters (Section 5.3). The overallforce/moment that the controller has decided to apply to keep theship in position must then be split among the available thrusters(Thrust Allocation, Section 5.4). Then the control action is appliedto the ship and fed to the observer for an effective state estimation.

5.2 observer

As anticipated, the purpose of the observer is to filter out anynoise and wave-frequency motion from the position and velocitysignals acquired by the sensors the vessel is equipped with. Nowa-days, the de facto standard is the Kalman filter technology (Kalman,1960), since the first introduction by Balchen et al. (1976).

5.2.1 Kalman filter in dynamic positioning

The Kalman filter is a recursive estimator, because the current esti-mated state depends only on the previous estimate and the currentmeasurement. Further, the Kalman filter is model-based, because itrelies on a model of the system to estimate the state of the real sys-tem.

Vessel model for the Kalman filter

The vessel is conveniently modelled in state-space notation, ac-counting for wave-frequency and low-frequency separation. The state2

is defined as follows:

x =

ξ

ηwηpν

(5.1)

where ηw =[

xw yw ψw]T is the wave-frequency position vec-

tor, ηp =[

xp yp ψp]T is the low-frequency position vector, and

ν =[

u v r]T is the low-frequency velocity vector.

2 The state of a system is defined as the smallest set of variables required to describethe system’s dynamic behavior, or the set of variables that at some initial time t0,together with the input variables completely determine the system’s behavior fort > t0 (Burns, 2001).

92 dp basics

A state-space representation can then be obtained writing theequations for two models describing the vessel. For the wave-frequencyWave-frequency

model behavior, we can use the following formulation:

ξ = ηw (5.2)

ηw + 2λωpηw + ω2pξ = Kww (5.3)

The λ and ωp parameters in previous expression are the ones corre-sponding to the linear approximation of the wave spectrum, whileKw is a scaling factor for the driving white noise w (see Section 2.7).

For the low-frequency behavior, we can assume that the ship is aLow-frequencymodel mass-damper system, described by the equations:

ηp = ν (5.4)

Mν + Bν = τ (5.5)

with M the vessel mass matrix (included hydrodynamic added mass),B the damping matrix, and τ =

[τx τy τψ

]T the thrust vector.The M and B matrices depend on the hydrodynamic properties ofthe vessel, and can be determined with the same techniques usedto obtain the RAOs. While this matrices are frequency-dependent,being this part of the model specific for low-frequency dynamics,the values obtained for a sufficiently long period can be assumed.

Note that the vessel model is an highly simplified vision of thereal system. All the effects not modelled fall in the noise definition,Kalman filter

modifications and therefore a properly working Kalman filter is able to accountfor them. In general, the Kalman filter is only able to deal withlinear models (note for example that the damping term has been as-sumed to be linear while it is well known that it is not), but suitablemodifications are available. For example the extended Kalman filter isable to overcome this limitation by linearising the model around thecurrent system state, while the unscented Kalman filter can give evenbetter results when the system is highly non-linear.

It can be seen that the model here formulated is exactly the sameas the one used in the SIMULINK model described in Section 3.4.2.In fact, if the Kalman filter is instructed exactly with the mass anddamping matrices used in the model, the simulation results can be

5.2 observer 93

somewhat optimistic, due to the perfect correspondence betweenthe two models, which can never be achieved in real systems. Thisproblem can be dealt with instructing deliberately the Kalman filterwith slightly wrong vessel parameters. This situation is much morelikely to happen in operation, since while the observer is instructedonly one time, the vessel characteristics change repeatedly duringits life: for example different loading conditions, presence of thecargo, operating loads, can impact on the hydrodynamic properties.In practice the danger to over-estimate the observer performance isquite low, since the most important disturbance comes from first or-der motion and measurement noise, rather than from un-modelledbehavior.

The two models together are represented by the following couple Complete systemmodelof equations:

x = Ax + Dτ + Ew (5.6)

y = Hx + u (5.7)

The Eq. 5.7 is the measurement equation in which the measurementnoise u is introduced. The following matrices were used in previousequations:

A =

0 I 0 0−ω2

p I −2λλ0 I 0 00 0 0 I0 0 0 −M−1B

(5.8)

D =

000

M−1

(5.9)

E =

0 0 0

Kw 0 00 0 00 I 0

(5.10)

H =[

0 I I 0]

(5.11)

94 dp basics

In the previous equations, the matrices 0 and I indicate 3-by-3 nulland identity matrices, respectively.

When the system model is discretised in time, the model formu-lation becomes as follows:

xk = Akxk−1 + Dkτk + Ekwk (5.12)

yk = Hkxk + uk (5.13)

Algorithm

As stated, the Kalman filter is recursive. In practice that doesmean that no history record is needed, but only the previous es-timated state and the current measurement. Using the notation xn|mto indicate the state estimated at time step m for the system at timestep n, the phases of the filter algorithm are presented. Being thisfilter recursive, n − m = 1. Another important variable is the er-ror covariance matrix Pn|m, which is a measure of how accurate thestate estimation is, compared with the measurement-corrected stateestimate. The process noise is assumed to have Qn covariance, giventhat is distributed according to a zero-mean Gaussian distribution.Similarly, the measurement noise is normally distributed with Rncovariance.

In the following sections, as the system characteristic matrices areassumed to be time-invariant, the index is dropped.

The filter works essentially in two distinct phases named hereafterKalman filterphases predict and update.

predict In this step, the model is used to project the system stateforward in time (from time step k− 1 to time step k):

xk|k−1 = Axk−1|k−1 + Dτk−1 + Ewk−1 (5.14)

Also the estimate covariance is projected ahead of one timestep:

Pk|k−1 = APk−1|k−1 AT + Qk−1 (5.15)

update Once a new (noisy) measurement is available, the esti-mated state is updated according to that. First, an innovationis calculated:

yk = yk − Hxk|k−1 (5.16)

5.3 controller 95

The covariance of the innovation is determined by:

Sk = HPk|k−1HT + R (5.17)

Then the optimal Kalman gain is calculated. It is a parameterwhich decides the weight corresponding to a measure of howmuch the measurement is trusted. It is decided at each itera-tion on the basis of the steps described.

Kk = Pk|k−1HTS−1k (5.18)

Now the state estimate and estimate covariance can be up-dated, including the information from the newly-acquired mea-surement:

xk|k = xk|k−1 + Kkyk (5.19)

Pk|k = (I − Kk H) Pk|k−1 (5.20)

Transfer functionIt can be shown (Fossen, 2002) that a wave filter based on a steady-

state Kalman filter (the gains are constant) has a transfer functionsimilar to the one plotted in Figure 5.2. Note that the notch in themiddle is tuned to correspond to the wave peak frequency, thatmust be suppressed. High-frequency terms are also faded to leaveout measurement noise. The transfer function looks similar to a com-bination of a notch filter and a low-pass filter. Indeed, before theKalman filter-based observers became popular, the combinations ofthe two named filters was employed. In 1978, Grimble wrote a pa-per to underline the similarity, and to encourage engineers to trustthe new observers featuring Kalman filters.

5.3 controller

The kind of controllers used in general dynamic positioning ap-plications is a very simple and standard one: most of the times theassumption of having a PID controller is right.

The general equation includes three terms proportional to the po-sition error

epos = ηp − ηp,re f (5.21)

96 dp basics

Circular frequency (rad/s)

ω

Mag

nitu

de(d

B)

|h(s)|

10−1 1 101

−10

0

−10

Circular frequency (rad/s)

ω

Phas

e(°

)

∠h(s)

10−1 1 101

−90

−45

45

Figure 5.2: Steady-state Kalman filter example of transfer function, input istotal motion and output is second order motion

5.3 controller 97

and velocity error

evel = ν− νre f (5.22)

Note that since the thrusters must be employed only to react tothe low-frequency motion, ηp and ν are the only components ofthe system state to be taken into account when computing errorsthat must be counteracted. In case station keeping is required, thevelocity error evel is equal to the low-frequency velocity ν.

The control action is expressed by the thrust vector τ containing Control law

the three components of the generalized force required to be put inplace by the actuators. The following expression clarifies the controllaw:

τ = −M(

KPepos + K I

∫ t

0epos dτ + KDν

)(5.23)

It is easy to recognise that the KP represents a stiffness for the Stiffness anddampingsystem, while KD is associated with a damping. As such, they can

be easily linked to the desired resonant period and fraction of criticaldamping for each degree of freedom.

KP =

ω21 0 0

0 ω22 0

0 0 ω26

(5.24)

KD = 2

ζ1ω1 0 00 ζ2ω2 00 0 ζ6ω6

(5.25)

The resonant circular frequency for i-th degree of freedom are in-dicated by ωi = 2π/Ti, while ζi is the fraction of critical dampingrelative to the i-th degree of freedom. The integral term weightedby K I ensures zero error for constant system input and takes careof all the constant or slowly varying disturbances.

An interesting enhancement of the described control law is the Windfeed-forwardwind feed-forward which is a quite standard way of managing the

drag forces developing at the portion of the ship exposed to wind.Since the wind speed is quite efficiently measured by anemometers(usually more than one mounted on a craft), and a good approxima-tion for the aerodynamic coefficients is achievable by model tests,the applied force is easily calculated:

F(w)i (θ) =

12

ρAiCD,i(θ)v2w for i = 1, 2, 6 (5.26)

98 dp basics

Since the anemometers catch also very rapid variations of the windspeed (wind gusts) the wind speed signal is properly low-pass fil-tered before force computation, with a time constant of, say, 50 s.The force is then added directly to the τ vector. In Figure 5.3 awind time history corresponding to a 12 m s−1 mean speed is plot-ted (Harris spectrum, see Section 2.9), being the smooth curve thefiltered counterpart (50 s time constant) of the gusty one.

Time (s)

t

Win

dSp

eed

(m/

s)

vw(t)

50 100 150 200 250 300 350 400 450

9

10

11

12

13

14

15

Raw wind speedFiltered wind speed

Figure 5.3: Wind speed time history, raw and filtered with 50 s time con-stant low-pass filter

The advantage in doing so is evident: the propulsion system re-acts to the perturbing action before a displacement from set point ismeasured. The wind perturbation is therefore compensated in open-loop, being the system stability ensured by the closed-loop controlaction. The forces from wind are usually quite small compared tothe ones from waves, such that errors in wind measurement and inforce calculation are compensated.

5.4 thrust allocation logic 99

The strategy used for wind force counteraction and its wide-spread Control logicenhancementsand successful application suggest to use the same logic to counter-

act actions from current and, most importantly, from waves. Theidea of a wave feed-forward has in fact been proposed multiple timesin literature, with different strategies, even with encouraging results,but did not break through commercial applications up to now, seee.g. (Pinkster, 1978). The proposal of an innovative strategy for wavefeed-forward is the main contribution of this work, and will be ex-posed in Chapter 7.

Another interesting enhancement of the basic control strategy isthe acceleration feedback, which is still a promising research topicbased on inertial measurements, described in Section 5.6.

The move-up is generally performed moving the setpoint alongthe desired route. A trapezoidal velocity profile is adopted for mo-tion law generation.

5.4 thrust allocation logic

The thrust allocation problem must be solved once the overallgeneralized force vector τ to be applied to guarantee station keep-ing has been determined by the controller. The τ vector has a sizeof three, with the elements corresponding to surge force, sway forceand yaw moment required to be put in place. In general, the num-ber of controlled actuators (propellers) parameters is such that theproblem is under-constrained.

In fact, consider the thruster types commonly adopted in pipelay- Thruster types

ers are the following:

longitudinal thrusters These thrusters are able to provide athrust force along the surge direction. Because of the relativelyshort beam dimension of ship shaped hulls, the yaw momentthat longitudinal thrusters can provide is limited. A substan-tial quote of the total power is dedicated to these thrusters, asthey are used for navigation rather than during station keep-ing.

tunnel thrusters Sway-direction force is provided by thesethrusters, whose name is due to the fact that they are gener-ally enclosed in a circular transversal tunnel made through thehull. The thrust is in general limited because of the small sizedue to their location inside the hull, but they can give a sig-nificant contribution to guarantee that the correct heading is

100 dp basics

maintained, thanks to the long moment arm available. Thesethrusters are indeed often used to improve manoeuvrability,also for vessels not operating in DP.

azimuthal thrusters As the name suggests, thrusters of thiskind have an additional degree of freedom: the propeller isable to rotate around a vertical axis, such that the thrust can bedirected in any direction lying on the horizontal plane. Theycan be positioned anywhere in the vessel, but to maintain de-cent efficiency they need to extend away form the hull surface,to ensure that the effect of the wall is not significant. In veryshallow waters this can be a limitation. These thrusters areheavily used in dynamic positioning applications thanks totheir intrinsic flexibility.

Some particular devices can be used to gain partial directional con-Control surfaces

trol on mono-directional thrusters. It is common to use rudders po-sitioned downstream the main propellers and exploit the lift forceto obtain lateral thrust. There is obviously a drawback due to thefact that also drag occurs, therefore impacting on the efficiency. Seefor example the typical curves for the drag and lift coefficients of arudder, represented in Figure 5.4 (Bertram, 2000; Tupper and Raw-son, 2001). At an angle of about 35° the lift coefficient falls suddenlydue to the flow separation. At the same angle of attack the dragcoefficient rises significantly, therefore the available rudder angle isoften limited under this threshold.

The controlled parameter in thrusters is in general the rotationalThrust vector

velocity of the propeller. Very often it is assumed that the relationwith the obtained thrust is linear:

Ft = Kt ft (5.27)

but in other cases a quadratic relation is more exact:

Ft = Kt f ∗t | f ∗t | (5.28)

In previous equations f ∗t is the propeller rotational velocity, oftenexpressed as a fraction of the maximum rpm for the thruster, Kt is aconstant of proportionality for that particular thruster. The subscriptidentifies the thruster among the set of available ones for the vehicle.


Attack angle (deg)

θ

Dra

gan

dLi

ftC

oeffi

cien

tsCD , CL

10 20 30 40 50 60 70 80 90

0.2

0.4

0.6

0.8

1.0

CD(θ)CL(θ)

Figure 5.4: Drag and lift coefficients as a function of the angle of attack fora typical rudder

A single thruster exerts on the vessel an action which can be de-scribed by a generalized force vector τt:

τt =

Ft,xFt,y

‖r× Ft‖

=

Ft,xFt,y

rxFt,y − ryFt,x

(5.29)

where r =[

rx ry 0]T is the vector pointing from the refer-

ence point (usually the ship COG) to the thruster, where the Ft =[Ft,x Ft,y 0

]T force is applied. Several thrusters are acting ondynamically positioned vessel, such that the overall thrust vector isdetermined summing up all of the contributions:

τ =Nt

∑t=1

τt (5.30)

A convenient representation of the overall force/moment vectorτ can be written in matrix notation:

τ = T(α)K f ∗ (5.31)

where K and f ∗ are the matrix and vector, respectively, containingall the relevant Kt and f ∗t . In particular, K = diag K1, K2, . . . , KNt .

102 dp basics

From now on, the term K f ∗ will be reduced to a thrust vector f ,such that Eq. 5.31 is rewritten:

τ = T(α) f (5.32)

In thrust allocation, the problem is reversed, as τ is known andThe thrustallocationproblem

the proper set of ft for each thruster must be determined. Amongthe available solutions, and efficient thrust allocation algorithmshould choose one which is optimum with respect to some definedcriteria. The most common algorithm is based on the pseudo-inversematrix, while more sophisticated methods rely on quadratic program-ming. The most simple solution is instead to aggregate thrustersinto groups (equivalent thrusters), such that the degrees of freedomare reduced and the problem has a unique solution.

5.4.1 Pseudo-inverse algorithm, fixed-azimuths

The following algorithm solves the thrust allocation problem forfixed-azimuth thrusters, condition that implies

α = α0 = constant (5.33)

T(α) = T(α0) = constant (5.34)

A weight matrix W can be used to express the condition that ispreferable to use some actuators with respect to some others, e.g.control surfaces are less expensive in terms of power than azimuthalthrusters that must be rotated.

Under the conditions expressed by Eqs. 5.33 and 5.34 the costfunction C to be minimised is:

C = minf

f TW f subject to τ − T f = 0 (5.35)

with W positive definite.The solution of the optimization problem formulated in Eq. 5.35

can be obtained by the Lagrange multipliers method (Fossen, 1994).Define the Lagrangian function as:

L( f , λ) = f TW f + λT (τ − T f ) (5.36)


where λ ∈ RNt is a vector of Lagrange multipliers. DifferentiatingEq. 5.36:

∂L∂ f

= 2W f − TTλ (5.37)

f =12

W−1TTλ (5.38)

Then, if we assume that TW−1TT is not singular, it is possible toexpress the Lagrange multipliers as follows:

τ = T f =12

TW−1TTλ (5.39)

λ = 2(

TW−1TT)−1

τ (5.40)

Substituting Eq. 5.40 into Eq. 5.38 yields:

f = W−1TT(

TW−1TT)−1

τ = T†τ (5.41)

The matrix T† = W−1TT(TW−1TT)−1 is the generalized inverse, alsocalled pseudo-inverse that in the particular case where W = I reducesto the Moore-Penrose pseudo-inverse (Penrose and Todd, 1955).

The algorithm does not guarantee that it is possible for the thrustersto put in place the actions required: if the required force for athruster exceeds the available one, it will be saturated afterwards.This means that the algorithm is not able to recognise situationswhere one or more thrusters are not putting in place the force theyhave been asked for, and to transfer some load to other thrustersthat could be not used at their full power e.g. because of their higherweight factor.

Another limitation is due to the fact that thrusters dynamics isnot taken into account. The load variation for unit time is of courselimited, also to minimise wear on the machinery, and this limitationis not part of the algorithm, but a filter that acts downstream.

Pseudo-inverse algorithm extended

The limitation on which the algorithm proposed in previous Sec-tion 5.4.1 is founded is for sure impacting too much on the per-formance of the dynamically positioned vessel, but offers the pos-sibility to be extended to make it applicable with good results inpractice.

104 dp basics

The extension is based on the fact that an azimuthal thruster, orExtended thrust

any other kind of steerable thruster can be viewed as the composi-tion of two thrusters, one acting longitudinally, and the other actingtransversally. An extended thrust vector f e and an extended thrust ma-trix Te can be used to write the problem is a similar manner asbefore.

As an example of extended thrust vector, consider the following,relative to a vessel with a longitudinal thruster, an azimuthal one,and a tunnel one:

f e =

f1

f2,xf2,yf3

(5.42)

The difference between the directions in which the thrusters operateis visible when one looks at the extended thrust matrix:

Te =

1 1 0 00 0 1 1−r1,y −r2,y r2,x r3,x

(5.43)

The solution of the problem is carried out exactly as stated byEq. 5.41, such that the components of the f e are easily determined.One more passage is needed to obtain the real vector f that rep-resents the command input, together with the α vector: the thrustfrom steerable propellers that was split artificially along the two di-rections must be reassembled taking into account the real capabilityof the thrusters. With reference to the example above, f and α (not avector because this imaginary craft has only one azimuthal thruster)are obtained as:

f =

f1√f 22,x + f 2

2,y

f3

(5.44)

α = arctanf2,y

f2,x(5.45)

In a calculation program, as needs to be implemented on board, it isadvisable to use the atan2 function which is present in all the mostpopular programming languages, and which is able to discern the


right direction taking into account the x and y components insteadof their ratio alone.

It is evident that the procedure described could lead to demandforces beyond the thrusters capability, and in this case the demandis saturated. The same holds for the rate of turn of the azimuthalunits, which is limited and not taken into account in the solution.

5.4.2 Nonlinear programming

To overcome the most significant disadvantages of the pseudo-inverse solution, a different formulation of the cost function hasbeen proposed. Johansen et al. (2004) suggested a function to be min-imised that, in combination with suitable constraint equations takesinto account power consumption, exceedance of available thrust, ex-ceedance of available azimuth angle change rate, and singularitiesof the thrust configuration.

The new optimization problem can be formulated as:

CNL = minf ,α,s

Nt

∑i=1

Pi | fi|32 + sTQs + (α− α0)

T Ω (α− α0)

+ρ

ε + det(

T(α)W−1TT(α))

subject to

T(α) f = τ + s

fi,min ≤ fi ≤ fi,max for i = 1, 2, . . . , Nt

αi,min ≤ αi ≤ αi,max for i = 1, 2, . . . , Nt

∆αi,min ≤ (αi − α0) ≤ ∆αi,max for i = 1, 2, . . . , Nt (5.46)

The first term represents the power consumption, and Pi is a weightcoefficient for the i-th thruster. The following sTQs is a term relatedto the slack s, where the slack is the part of generalized force whichis not provided to the vessel because of saturation, with a properweight matrix Q. The weights in the matrix Q must be large enoughto make the optimal solution guarantee zero slack. Two constraintequations are used to take into account the force saturation andthe angular sectors available for the azimuthal thrusters to rotate.Another term in the cost function takes into account the penaltyassociated with the change of azimuth angles with respect to the

106 dp basics

configuration at the previous time step α0, weighted by the Ω ma-trix. The last contribution to the CNL function makes it possible toavoid singular configurations, in which no generalized force is ob-tained even for f 6= 0. Consider Eq. 5.41, which ensures a solutionif det

(T(α)W−1TT(α)

)6= 0. The ε constant is a small number put

to avoid division by 0 when the pseudo-inverse solution becomessingular, while the constant ρ weights the importance of the ma-noeuvrability with respect to the other requirements.

The problem expressed in Eq. 5.46 is non-linear and in generalProblem solution

non-convex, such that the solution implies significant computationaleffort (Gill et al., 1982). Possible strategies include the reduction toa quadratic programming problem, or the reduction to a linear pro-gramming problem.

Johansen et al. (2004) proposed the following reduction to a quadraticprogramming problem, based on the assumptions that the powerconsumption can be expressed as a quadratic function of f and thatthe avoidance of the singularities can be obtained by a penalty termlinearised around the azimuth configuration at the previous timestep α0. The problem is then convex and it is possible to use stan-dard numerical routines for solution (see Johansen et al. (2004) forthe details about the function to be minimised, and

Another possible strategy is to approximate the nonlinear prob-lem with a linear one (Webster and Sousa, 1999), still taking intoaccount maximum thruster capability and azimuth change rate.

Consider that the use of any solution strategy involving iterativeAlternativestrategies computations must be adapted to the real-time context where the

thrust allocation algorithm has to operate. This means that, if the al-gorithm does not converge to an acceptable solution, or if the avail-able computational time is exceeded, an alternative strategy mustbe invoked to comply with the robustness requirements for marineoperation.

5.4.3 Simulation

The extended pseudo-inverse thrust allocation algorithm has beenimplemented in a simple SIMULINK model to be included in arbi-trarily complex systems. The scheme in Figure 5.5 implements thethrust allocation algorithm: most of the procedure is executed in-side the pseudoinverseTA block which embeds the MATLAB codein Listing 5.1. In this application an hypothetic vessel with a tun-nel, three azimuthal thrusters and one longitudinal main propeller


is used to demonstrate the functioning of such a logic, where thedynamic limits and saturation do not affect the thrust demand.

Listing 5.1: MATLAB code for pseudo-inverse thrust allocation1 function [ X_thrust, Y_thrust, N_thrust, fe, f_dem, alfa_dem, f, alfa, rpm ]

= pseudoinverseTA(X_dem, Y_dem, N_dem, alfa0, rpm0)2

3 % Get current time step4 dt = 1.0;5

6 % Required generalized force7 tau_dem = [ X_dem; Y_dem; N_dem ];8

9 % Thrust capacity10 tcap = [ 200 % tunnel11 600 % azimuthal12 600 % azimuthal13 600 % azimuthal14 1200 ]; % main propeller15

16 % Number of thrusters17 n_t = length(tcap);18

19 % Location of thrusters20 tloc.All = [ 100, 0, 021 50, 0, 022 -20, -15, 023 -20, 15, 024 -100, 0, 0 ];25 tlocx = tloc.All(:,1);26 tlocy = tloc.All(:,2);27

28 % Extended thrust matrix29 Te = [ 0 1 0 1 0 1 0 130 1 0 1 0 1 0 1 031 tlocx(1), -tlocy(2), tlocx(2), -tlocy(3), tlocx(3), -tlocy(4), ...32 tlocx(4), -tlocy(5) ];33

34 % Weight matrix35 W = eye(size(Te,2));36

37 % Risetimes38 risetimes = [ 10 % tunnel39 15 % azimuthal40 15 % azimuthal41 15 % azimuthal42 20 ]; % main propeller43

44 % Maximum thrust variation45 maxrpmvar = zeros(n_t,1);46 for i = 1:n_t47 maxrpmvar(i) = dt/risetimes(i);48 end49

50 % Azimuth variation in rpm51 azivar = [ 2 % azimuthal52 2 % azimuthal53 2 ]; % azimuthal54

55 % Number of azimuthal thrusters

108 dp basics

Thr

ustF

orce

1

zero0

Uni

t Del

ay4

z1U

nit D

elay

3

z1U

nit D

elay

2

z1

Uni

t Del

ay1

z1

Uni

t Del

ay

z1

Ste

p2

Ste

p1

Ste

p

Sco

pe7

Sco

pe6

Sco

pe5

Sco

pe4

Sco

pe3

Sco

pe2

Sco

pe1

Sco

pe

Em

bedd

edM

AT

LAB

Fun

ctio

n

X_de

m

Y_d

em

N_d

em

alfa

0

rpm

0

X_th

rust

Y_t

hrus

t

N_t

hrus

t fe

f_de

m

alfa

_dem

f

alfa

rpm

pseu

doin

vers

eTA

rpm

alfa

X_th

rust

Y_t

hrus

t

N_t

hrus

t

fe f_de

m

alfa

_dem

f

thru

stfo

rce

glob

dem

and

Figu

re5

.5:S

IMU

LIN

Km

odel

for

pseu

do-i

nver

seth

rust

allo

cati

on


56 n_a = length(azivar);57

58 % Maximum azimuth variation (rad)59 maxalfavar = zeros(n_a,1);60 for i = 1:n_a61 maxalfavar(i) = 2*pi/60*azivar(i)*dt;62 end63

64 %%%%% SOLUTION65

66 fe = inv(W)*Te’*inv(Te*inv(W)*Te’)*tau_dem;67

68 f_dem = [ fe(1)69 sqrt(fe(2)^2+fe(3)^2)70 sqrt(fe(4)^2+fe(5)^2)71 sqrt(fe(6)^2+fe(7)^2)72 fe(8) ];73

74 alfa_dem = [ atan2( fe(3), fe(2) )75 atan2( fe(5), fe(4) )76 atan2( fe(7), fe(6) ) ];77

78 %%%%% Verify if solution is allowable79

80 f = zeros( n_t, 1 );81 alfa = alfa_dem;82

83 % Verify maximum capacity84 for i = 1:n_t85 if f_dem(i)>tcap(i)86 f(i) = tcap(i);87 else88 f(i) = f_dem(i);89 end90 end91

92 % Find rpm93 rpm = zeros( n_t, 1 );94 for i = 1:n_t95 rpm(i) = sign(f(i))*sqrt(abs(f(i))/tcap(i));96 end97

98 % Verify maximum rpm variation99 for i = 1:n_t

100 if abs(rpm(i)-rpm0(i))>maxrpmvar(i)101 if rpm(i)-rpm0(i)>=0102 rpm(i) = rpm0(i)+maxrpmvar(i);103 else104 rpm(i) = rpm0(i)-maxrpmvar(i);105 end106 end107 end108

109 % Verify maximum alfa variation110 for i = 1:n_a111 if abs(alfa(i)-alfa0(i))>maxalfavar(i)112 if alfa(i)-alfa0(i)>=0113 alfa(i) = alfa0(i)+maxalfavar(i);114 else115 alfa(i) = alfa0(i)-maxalfavar(i);116 end

110 dp basics

117 end118 end119

120 %%%%% Obtained Thrust121

122 T = [ 0 cos(alfa(1)) cos(alfa(2)) cos(alfa(3)) 1123 1 sin(alfa(1)) sin(alfa(2)) sin(alfa(3)) 0124 tlocx(1) (sin(alfa(1))*tlocx(2)-cos(alfa(1))*tlocy(2)) ...125 (sin(alfa(2))*tlocx(3)-cos(alfa(2))*tlocy(3)) ...126 (sin(alfa(3))*tlocx(4)-cos(alfa(3))*tlocy(4)) (-tlocy(5)) ];127

128 for i = 1:n_t129 f(i) = sign(rpm(i))*rpm(i)^2*tcap(i);130 end131

132 tau = T*f;133

134 X_thrust = tau(1);135 Y_thrust = tau(2);136 N_thrust = tau(3);137

138 end

In Figure 5.6 the example thrust demand time history is plotted. Alongitudinal thrust, transversal thrust, and yaw moment are askedto the propulsion system at steps delayed by 20 s. In the lower plot,the generalized force put in place by the system is plotted. Duringthe transients the required thrust is not obtained, but in this case thesystem capability is sufficient to guarantee convergence in a shorttime. Figure 5.7 shows the thrust asked to each propeller, while thelower plot displays the thrusters feedback, where the delay due tothe response time is visible. The azimuthal thrusters are asked tosteer the propeller axis as a consequence of the required thrust, asshown in Figure 5.8, where the slightly delayed response takes intoaccount the limit rotation speed.

5.5 simplified performance assessment

A complete DP system including the components presented inthis chapter is quite a complex system to be studied. Preliminaryanalyses based on the complete model require significant modelingefforts and assumptions on parameters that may be unknown in thefirst phases of a project. This leads to the necessity of a simple pro-cedure which is also extremely useful to understand the relevanceof the factors involved in the problem.

An intuitive way of modeling a DP vessel in station keeping isthrough the well-known damped oscillating system, with the matrix

5.5 simplified performance assessment 111

Time (s)

t10 20 30 40 50 60 70 80 90 100

Time (s)

t10 20 30 40 50 60 70 80 90 100

Thr

ust

dem

and

(kN

/kN

m)

τdem

1000

2000

3000

4000

5000

Obt

aine

dth

rust

(kN

/kN

m)

τ

−1000

1000

2000

3000

4000

5000

τdem,1

τdem,2

τdem,6

τ1

τ2

τ6

Figure 5.6: Global generalized forces on vessel, demanded and obtained

notation which encloses the three equations for surge, sway andyaw: (

M + MA)

ηp +(

B + BDP)

ηp + CDPηp = τenv (5.47)

In previous Eq. 5.47 the vessel mass and hydrodynamic mass areseparately indicated. The BDP and CDP matrices are used to accountfor the presence of the DP system, which can be preliminarily mod-elled as a damper and a stiffness.

Let the environmental forces be described by the spectral densityin Eq. 3.13. This assumption can be considered acceptable in thisphase since the main perturbing load that influences the DP perfor-mance is the one from surface waves. Treating the input process as a

112 dp basics

Time (s)

t10 20 30 40 50 60 70 80 90 100

Time (s)

t10 20 30 40 50 60 70 80 90 100

Thr

ust

dem

and

(kN

)

f dem

100

200

300

400

500

Obt

aine

dth

rust

(kN

)

f

100

200

300

400

500

fdem,1

fdem,2

fdem,3

fdem,4

fdem,5

f1

f2

f3

f4

f5

Figure 5.7: Demanded and obtained thrust for each single propeller

random time signal, the squared standard deviation of the responseis derived for each degree of freedom (Faltinsen, 1993), e.g. for thesurge:

σ21 =

∫ ∞

0

S(2)F (µ)dµ[

CDP11 −

(M11 + MA

11)

µ2]2 +

(B11 + BDP

11)2

µ2(5.48)

The extreme displacement value is generally 3 to 4 times σi. Therough estimate obtained with this very simplified model can be sur-prisingly accurate, considering the limited amount of data requiredfor calculation, and very useful to determine immediately the feasi-bility of prescribed objectives given general design information.

As a result, producing radial plots like in Figures 5.9 and 5.10 andis possible in the early phases of the project, to be used as a guide


Time (s)

t10 20 30 40 50 60 70 80 90 100

Time (s)

t10 20 30 40 50 60 70 80 90 100

Azi

mut

han

gle

dem

and

(rad

)αdem

π4

π2

Obt

aine

dth

rust

(kN

)

α

π4

π2

αdem,1

αdem,2

αdem,3

α1

α2

α3

Figure 5.8: Demanded and obtained azimuth for each azimuthal thruster

to understand the main concerns to be faced in detail. The plot inFigure 5.9 shows the expected extreme surge motion for a 80 000 tvessel (plus hydrodynamic added mass, quantified by Journée andMassie (2001) as 5–8% of the total), with a stiffness correspondingto a resonant period of 200 s, and 90 % of the critical damping dueto DP, 15 % due to hydrodynamics (Faltinsen, 1993). A typical setof QTFs for a tanker has been used in this sample calculation. InFigure 5.10 the same calculation is repeated for the sway degree offreedom, with an added mass of 40 000 t.

5.5.1 Filtering delay

The need for a wave filter to avoid the thrusters trying to react tothe zero-mean first order motion has been explained. The effect ofhaving such a filter upstream the controller is of course beneficial interms of machinery wear and power generation, but the drawback

114 dp basics

90°

0°

Extr

eme

Off

set

(m)

180°

135°

45°

0.5

1.0

1.5

2.0

3 σ1

4 σ1

Figure 5.9: Polar plot of extreme surge response

of the time delay introduced by the filter itself can be evidenced bysimplified modeling.

Assume that the control action is put in place, due to the filter-induced delay, τ seconds after the displacement happened. Eq. 5.47

is rewritten as:(M + MA

)ηp(t)+ Bηp(t)+ BDPηp(t− τ)+ CDPηp(t− τ) = τenv(t)

(5.49)

where the time dependency is explicit. The τ delay is assumed tobe small with respect to the motion period, such that linearisationis allowed:(

M + MA)

ηp(t) + Bηp(t) + BDP(

ηp(t)− ηp(t)τ)

+

CDP(

ηp(t)− ηp(t)τ)

= τenv(t) (5.50)


90°

0°

Extr

eme

Off

set

(m)

180°

135°

45°

5

10

15

20

3 σ2

4 σ2

Figure 5.10: Polar plot of extreme sway response

Rearranging previous equation:(M + MA − BDPτ

)ηp(t) +

(B + BDP − CDPτ

)ηp(t)+

CDPηp(t) = τenv(t) (5.51)

It is noticed that the effect of the delay is to decrease the mass andthe damping introduced by the DP system, with detrimental effectson performance. The extent of the degradation can be evaluatedwith the same method explained in Section 5.5, with the correctionin mass and damping as in Eq. 5.51. It is shown in Figures 5.11 and5.12 that the difference with the no delay dynamics can be quitesignificant even for a time delay of 10 s, which is not an overconser-vative assumption.

116 dp basics

90°

0°

Extr

eme

Off

set

(m)

180°

135°

45°

0.5

1.0

1.5

2.0

3 σ1

3 σ1 with delay4 σ1

4 σ1 with delay

Figure 5.11: Polar plot of extreme surge response with 10 s delay, compara-tive results

5.6 acceleration feedback

PID-like controllers applied to a second order system lead, as onecan easily notice, to oscillatory motion, since control action is onlyput in place as a result of positioning errors. The derivative term iseffective in reducing the amplitude of oscillation, but with the gain(and therefore the effectiveness) being limited by the presence ofunfiltered disturbances at wave frequency or higher.

Lindegaard (2003) proposed a DP application based on accelera-tion measurements, on the basis of which an additional feedbackterm to control action is computed. The active use of measured ac-celerations is equivalent to a manipulation of the system’s mass:the idea is to make the body apparently heavier than it is, at therange of frequencies of the disturbances to reject. Measured accel-erations are indeed subject to two possible uses: De Jager (1994)

5.6 acceleration feedback 117

90°

0°

Extr

eme

Off

set

(m)

180°

135°

45°

0.5

1.0

1.5

2.0

3 σ2

3 σ2 with delay4 σ2

4 σ2 with delay

Figure 5.12: Polar plot of extreme sway response with 10 s delay, compara-tive results

named direct the approach in which the accelerations are employedto modify the control law, and indirect the use of accelerations toimproved the system observation. The combined use of the two isin general not beneficial, as for De Jager’s robotic application, but isprobably the way to go for the dynamic positioning problem, sincethe accelerations measured on a vessel will be mostly characterisedby wave-frequency content, to be filtered out by a suitable observer(Lindegaard, 2003).

The analogy with the mass-damper system is here used to presentthe effect of the acceleration feedback in ship dynamics. Including thenewly added acceleration feedback term, Eq. 5.47 is rewritten as:(

M + MA + MAF)

ηp +(

B + BDP)

ηp + CDPηp = τenv (5.52)

where MAF is the acceleration gain, which is also the apparent massintroduced by acceleration feedback. Instead of a virtual mass con-

118 dp basics

stant for all the frequencies, behaving like a structural mass, the ac-celeration feedback term can be shaped to obtain improved response.This technique is named inertia shaping and has interesting applica-tions also in compensation of non-symmetric mass effects for highspeed marine crafts, manoeuvring and transit conditions of ships(Fossen et al., 2002). A proper response characteristic for the acceler-ation feedback inertia shall add mass only at slow-drift frequencies(for a single degree of freedom):

MAF(s) =βM

1 + Tf s(5.53)

where the mass added to the system is expressed as a fraction β ofthe physical mass M. The acceleration feedback mass behaves likea low-pass filter with cut-off period Tf , see Figure 5.13 where anexample frequency characteristic is plotted for a single degree offreedom.

It is possible to derive, for a vessel with the mass modified as perEq. 5.53, the transfer functions from force to velocity, and evaluatethe effect of the acceleration feedback. The vessel can be modelledas a simple mass-damper system, and enclose the DP and environ-mental forces in a single τext vector of generalized external forces:

(M + MA

)ηp + Bηp = τext (5.54)

where the equation holds for each single degree of freedom. Thetransfer function from force to velocity is then written:

ηp

τext(s) =

1B (Tms + 1)

(5.55)

with the substitution

Tm =B

M + MA (5.56)

Then the acceleration feedback-induced mass in Eq. 5.53 is added:

ηp

τext(s) =

1

B(

1 + Tms + βTms1+Tf s

) (5.57)

The following two characteristic frequencies can be identified asdelimiting a frequency interval in which the acceleration feedbackmass influences the system response:

ωa =1

Tm

11 + β

(5.58)


1Tf

Frequency (Hz)

f

Mag

nitu

de(k

g)

|(M + MAF)(s)|

10−4 10−3 10−2 10−1

M

M + MAF

Frequency (Hz)

f

Phas

e(°

)

∠(M + MAF)(s)

10−4 10−3 10−2 10−1

−45

−90

Figure 5.13: Low-pass mass frequency-domain characteristic

120 dp basics

ωe =1 + β

1 + Tf(5.59)

In Figure 5.14 the transfer function from force to velocity is plot-ted for a system without and with acceleration feedback. The com-parison evidences a reduced response in the frequency range be-tween ωa and ωe.


ωa2π

ωe2π

Frequency (Hz)

f

Mag

nitu

de(m

/s/N

)

| ηpτext

(s)|

10−4 10−3 10−2 10−1

10−7

10−6

10−5

Frequency (Hz)

f

Phas

e(°

)

∠ηp

τext(s)

10−4 10−3 10−2 10−1

−45

−90

Figure 5.14: Force to velocity transfer function of a mass-damper systemwithout and with acceleration feedback

6M E A S U R E M E N T S Y S T E M S

Contents6.1 Absolute position 123

6.1.1 GPS 125

6.1.2 DGPS 129

6.2 Underwater positioning 1326.2.1 USBL systems 132

6.2.2 LBL systems 133

6.3 Vessel accelerations 1346.4 Sea state 135

6.4.1 Wave radar 135

6.4.2 Ship motions-based estimate 136

In this chapter the measurement systems used in Dynamic Po-sitioning applications are presented, with their principles of oper-ation and modeling techniques. Moreover, innovative sensors andprocessing techniques like inertial platforms and wave radars aredescribed, being the basis for the DP improvement technologies pro-posed in this work.

6.1 absolute position

When referring to the absolute position of a vessel, the positionwith respect to the earth is meant. This idea is indeed not simple toexpress univocally, since the shape of the earth itself is a complextopic, studied by the discipline named geodesy.

The reference shape for the earth is called geoid, which is an highly The geoid and theellipsoidsirregular shape defined by the idealised water surface in absence of

currents, wave, tides, and continued under the continents. A geo-metrical tool to approximate the geoid is an ellipsoid, described byits semi-major axis a and the flattening factor, f = (a− b)/a, whereb is the semi-minor axis. The ellipsoid to approximate the geoid isnot unique: one can be chosen offers better approximation in a re-gion, while another fits better for another location. When expressingthe position of a point with the usual latitude and longitude angles,

123

124 measurement systems

explicit reference to the chosen ellipsoid shall be made, to avoidpositioning errors that can, in some operations, be critical.

Since the aim of an absolute positioning system is to be able toWGS84

identify the location of the vehicle in whatsoever point on the sur-face of the earth, a suited ellipsoid must be chosen to fit reasonablywell everywhere on the geoid. The standard ellipsoid in GPS sys-tems is identified by the code WGS84 (U.S. National Imagery andMapping Agency, 2000). Since the continents are drifting, using afixed and unique ellipsoid means that the geographic features onthe earth will be represented by coordinates which are slowly vary-ing. Beyond accuracy, this is another reason why other continent-specific ellipsoids have been developed.

The conversion between coordinates expressed with respect toCoordinateconversion different ellipsoids is simple once a few parameters are known, but

despite that the process must be treated with care because errorscan lead to unexpected operational problems. As an example, con-sider the ED50 ellipsoid, used in western Europe before the WGS84

was formulated, and still widely used in European cartography. SeeTable 6.1 for the characteristic parameters of the two systems. If aparticular point is expressed in ED50 coordinates, e.g. the entranceof the marina in the city of Ancona, Italy, and then erroneously fedto a GPS, the referenced point would be about 134 m away from theintended location, see Figure 6.1. It is evident that such a position-ing error could prevent the successful completion of some offshoreoperations, making the contractor incur in considerable additionalcosts.

Table 6.1: WGS84 and ED50 coordinate systems

Parameter ED50 WGS84

Spheroid International 1924 WGS84

Semi-major Axis 6 378 388.0000 m 6 378 137.0000 mSemi-minor Axis 6 356 911.9461 m 6 356 752.3142 m

Inverse Flattening (1/ f ) 297 298.257

The most popular absolute positioning system is the Global Po-GPS andcompetitors sitioning System, which is easily found also in many pieces of con-

sumer electronics, due to the low cost of the receivers. Other com-petitors are under development or already active, but no other sys-

6.1 absolute position 125

ED50 Coordinates

WGS84 Coordinates

Figure 6.1: Positioning error due to wrong geodetic reference system

tem has reached such an extended coverage and number of re-ceivers. Beside GLONASS1 and COMPASS, Russian and Chinesesystems, respectively, the European Union and the European SpaceAgency are developing the GALILEO system, which is claimed tobe operational in 2013

2. The European system is accounted to de-liver much better accuracy than GPS also for open services to whichanyone will have free access, and like GPS will make available anencrypted Commercial Service signal with accuracy of less than 1 mor less than 10 cm if complemented by ground-fixed stations.

6.1.1 GPS

The basic principle on which the GPS system operates is the range Operationalprinciplemeasurement from a satellite in a known position to a receiver. The

measurement is carried out indirectly, since a time measurement isindeed performed. The satellite continuously emits a signal carryingthe information about the time instant in which the signal itself wasemitted. When the signal is received, the range from the satellite ican be computed with the following simple equation:

ρi = c(tu − tsi) (6.1)

1 See the official website http://www.glonass-ianc.rsa.ru/pls/htmldb/f?p=202:1:

82029356773286546172 See the official website http://ec.europa.eu/transport/galileo/index_en.htm

http://www.glonass-ianc.rsa.ru/pls/htmldb/f?p=202:1:8202935677328654617

http://www.glonass-ianc.rsa.ru/pls/htmldb/f?p=202:1:8202935677328654617

http://ec.europa.eu/transport/galileo/index_en.htm


where c is the speed of light, tu is the user reception time, tsi is thesignal sending time by the satellite i.

A distance from a satellite in a known position identifies a spher-Three-satellitespositioning ical surface on which the receiver lies. Two distances identify two

spheres that intersect in a circle. The intersection of this circle witha third sphere determines two possible points. A fourth range fromanother satellite seems to be necessary, but it is not since one of thetwo points will lie away from the earth surface. Given that the threesatellites positions are expressed by the following vectors:

ri =

xiyizi

i = 1, 2, 3 (6.2)

the following system of equations must be solved to determine theposition of the receiver:

ρ1 =√

(x1 − xu)2 + (y1 − yu)2 + (z1 − zu)2

ρ2 =√

(x2 − xu)2 + (y2 − yu)2 + (z2 − zu)2

ρ3 =√

(x3 − xu)2 + (y3 − yu)2 + (z3 − zu)2

(6.3)

where the receiver position ru =[

xu yu zu]T is the unknown.

The theoretical process stated by the previous Eq. 6.1 is indeedDisturbances

affected by a number of errors: both the clocks have unknown er-rors, the presence of the troposphere and the ionosphere introducesdelays, the receiver is affected by noise, and the relativistic effectsmust be corrected. Some errors can be corrected or neglected, butnot the receiver clock error. This latter error has the effect of addingFour-satellites

positioning and unknown lag which is systematically added to the time intervalmeasurement, such that Eq. 6.1 is modified to:

ρi = c(tu − tsi + tb) (6.4)

A range error ρb = ctb is then added to each of the ranges measuredfrom the available satellites. Then and equation is added to the sys-


tem in Eq. 6.3, with the result that a fourth satellite is needed tocompensate for user clock bias:

ρ1 =√

(x1 − xu)2 + (y1 − yu)2 + (z1 − zu)2 + ρb

ρ2 =√

(x2 − xu)2 + (y2 − yu)2 + (z2 − zu)2 + ρb

ρ3 =√

(x3 − xu)2 + (y3 − yu)2 + (z3 − zu)2 + ρb

ρ4 =√

(x4 − xu)2 + (y4 − yu)2 + (z4 − zu)2 + ρb

(6.5)

The range error is an additional unknown needed to compute theintersection between the spheres with the radius corrected for thereceiver clock error. The system of equations is solved numericallyby the receiver.

Other variations in positioning accuracy are caused by the partic- Dilution ofPrecisionular position of the satellites with respect to the user: since the orbits

on which the satellites are moving are known in advance, Dilutionof Precision (DOP) maps can be produced, see e.g. Figure 6.3, wherea DOP map for a particular day is shown3. The DOP can be inter-preted as the reciprocal of the volume delimited by the 4 satellitesand the receiver, see Figure 6.2, such that the degradation of accu-racy is much more important in the situation depicted in Figure 6.2bthan in the one depicted in Figure 6.2a.

The position measured by a GPS receiver, due to all the issues GPS sensormodelingnamed before, is affected by an error, which is important to model

for simulation purposes. It has been determined by specific studies(Rankin, 1994) that most of the errors affecting the range measure-ments can be modelled as a Gauss-Markov process: Gauss-Markov

process

vn+1 = e−β ∆tvn + wn (6.6)

This means that the error vn+1 at step n + 1 is made up by a Gaus-sian white noise wn and an influence from the error at the previoustime increment vn. The sampling interval is indicated with ∆t. Theβ coefficient defines how long the process memory is; a typical value,used in the simulations here presented, is 1/β = 60 s. The autocorre-lation of such a noise signal is an exponentially decreasing function,see Rankin (1994).

The SIMULINK model in Figure 6.4 was developed considering Simulation

valid the hypothesis that the total error in horizontal position has

3 See http://gps.afspc.af.mil/gpsoc/

http://gps.afspc.af.mil/gpsoc/


(a) (b)

Figure 6.2: Dilution of Precision: low DOP and high DOP situations

Figure 6.3: Dilution of Precision map for September 16th, 2009, made byGPS Operations Center


the same characteristics of the errors affecting the range measure-ments, described by Rankin.

e^(−beta *Dt)

−K−

XY Graph

Scope

Memory

Barge Position

0

Band −LimitedWhite Noise 1

Band −LimitedWhite Noise Add2

Add1

AddXnoise

Ynoise

Measured Position

Figure 6.4: SIMULINK model for GPS simulation

The results of a simulation are plotted in Figure 6.5, where astandard deviation of 0.5 m has been set for both the Band-Limited

White Noise blocks. In Figure 6.6 the same results are plotted sep-arately for x and y directions, together with the autocorrelations(scaled to 1), to be compared with the theoretical one.

6.1.2 DGPS

The Differential GPS (DGPS) uses a technique to enhance the po-sitioning accuracy, with the support of ground-fixed stations. Thepositions of these reference stations are known with a high degreeof accuracy. When the position of a reference station is determinedby GPS measurement, the errors made in range measurement arecalculated, and then broadcasted to the mobile GPS user via radioor satellite communication (Figure 6.7).

The DGPS operational technique permits to sensibly improve the Accuracyimprovementpositioning accuracy. For example, the NDGPS4 (Nationwide Differ-

ential GPS) system, operated by the United States Department ofTransportation, claims to be able to reach a 1 m accuracy near thereference stations, 3 m at the edge of the coverage area (up to 402 kmaway).

4 See http://www.tfhrc.gov/its/ndgps/

http://www.tfhrc.gov/its/ndgps/


W E

S

N

2.0

4.0

6.0

8.0

Error (m)

t = 0 st = 60 st = 120 s

Figure 6.5: Simulated GPS trace for a stationary receiver, apparent motiondue to noise


Erro

r(m

)v(t)

Aut

ocor

rela

tion

R(τ)

t(s)

τ

100 200 300 400 500 600

100 200 300 400 500 600

−5.0

5.0

0.5

1.0

x-noisey-noise

RxxRyyTheoric

Figure 6.6: Errors in x and y directions during a 10 min long simulation,obtained autocorrelations compared with the theoretical one

Geostationarysatellite

Reference Station

Figure 6.7: Differential GPS scheme


6.2 underwater positioning

Underwater positioning systems rely on acoustics to measure rangesand angles that identify the position of a target with respect toa transceiver. The main technologies here presented differ for thelength of the base line, as explained next.

6.2.1 USBL systems

The acronym stands for Ultra Short Base Line, since the distance be-tween different transceivers is actually only some centimetres long,see Figure 6.8.

Receiver 1Receiver 2

Receiver 3

Trasmitter

Longitudinal axis

Transversal axis

Figure 6.8: USBL transceiver

The transceiver is mounted in general on the vessel, such thatthe power supply and signal logging do not need any umbilical.The signal transmitted is sent back to the receivers by the seabed-mounted unit, and received with a small delay related to the angleof view, see Figure 6.9. The distance is calculated as the productbetween the speed of sound and the travel time. The angles aredetermined from the relative delay between the reception instant atthe three receivers.

Crucial for measurement accuracy, especially in deep water, isthe availability of accurate values of the speed of sound profile. Thespeed of sound is a function of temperature, salinity (conductivity)

6.2 underwater positioning 133

Transceiver

Receiver 2 Receiver 1

Figure 6.9: USBL transceiver

and pressure (depth); a CTD5 probe is indeed employed for speedof sound profile calibration before operation.

6.2.2 LBL systems

This system represents a variation of the previously describedUSBL, in that the baseline is expanded and moved to the seabed(LBL stands for Long Base Line). An array of transponders must bedeployed such that the vessel to be positioned operates in the rangeof at least three of them. The position computation is in fact car-ried out by intersection of the three spheres of known radius. Eachradius is determined by response time measurement and speed ofsound profile as before. Due to errors, the spheres are never inter-secting in a single point. The most likely position is in general de-termined by least squares solution.

The problem of the placement of beacons on the seabed must besolved taking care of the problem of the dilution of precision, similarlyto what happens for GPS. Since the measured position is relativeto the seabed-mounted beacons, the array of transponder must becalibrated after deployment.

5 Acronym of conductivity, temperature, depth


beacon 1

beacon 2

beacon 3

measured position

Figure 6.10: LBL operational scheme

6.3 vessel accelerations

Inertial measurements are gaining popularity as the cost of theunits decreases (Fossen, 2002). The technology is well-establishedas derived from aerospace applications, and very common also inunderwater ROV applications.

Applications of inertial units include integration with state ob-servers for improved response, and research applications as the ac-celeration feedback briefly described in Section 5.6 of this document(Lindegaard, 2003).

An Inertial Measurement Unit is composed by the sensors (in gen-eral 3-axis accelerometers and gyros are employed), the filtering andtemperature compensation. An IMU is completed with software in-tegrating the motion equations to obtain an Inertial Navigation Sys-tem able to supply the user with position and velocity signals. Dueto integration, the errors tend to build up a drift that must be com-pensated by an absolute positioning system, typically a GPS sensor.In absence of a suitable integration for drift correction, an high-passfilter must be applied to catch only the oscillatory part of the mo-tion.

The error in oscillatory wave-frequency displacement measure-ment can be as low as some centimetres, and a hundredth of degree

6.4 sea state 135

for rotational motion. The drift in absence of GPS or other referencefor 1 min can be lower than 2 m.

6.4 sea state

The knowledge of sea state parameters can be useful during off-shore operations to recognise sufficiently in advance dangerous con-ditions developing. Such an alarm can be important to maximiseworking time still being able to identify the moment in which safetyoperations must be started (e.g. abandonment for subsequent recov-ery in pipelaying). In the following sections a specifically designedclass of sensors and a strategy relying on ship motion (inertial) mea-surements are described.

6.4.1 Wave radar

The wave radar technology is highly appealing thanks to the factthat a standard X-band radar present on most vessels for offshoreoperations is the only needed sensor. A processing unit need to beconnected to the analog radar signal for processing and informationextraction. Once the images have been digitised, a temporal and spa-tial distribution of backscatter or sea clutter is available for processing.While this information is regarded as noise for navigation purposes,it is linked to the sea state (Borge et al., 2000).

When the wind is blowing at least at about 3 m s−1, small rip-ples are produced on the water surface, modulated by longer waveslike wind waves and swell. The radar is able to capture the clutterfrom ripples, and an almost real-time calculation can detect the seastate parameters as the significant wave height and the peak period.Further, the directional distribution of both wind sea and swell isdetermined.

Given the dispersion relation between wave numbers and fre- Surface current

quencies, also the surface current c can be estimated. See the fol-lowing modified version of Eq. 2.28:

ω =√

kg tanh(kd) + k · c (6.7)

Looking at the most intense component characterised by a knownω and k, previous Eq. 6.7 gives the current velocity.


The wave radars have been proved to be reliable and accurate inmaking available sea state parameters in almost real-time, see alsoReichert and Lund (2007), Reichert et al. (2006), Borge et al. (2004).

The applications cited up this point only regard average proper-ties over the surface scanned by the radar device, implying that onlyvery slow variations of the general seaway conditions can be de-tected. For control purposes, the instantaneous distribution of wavefield surrounding the vessel is much more interesting, up to the ex-tent that it could be functional for a wave feed-forward control logic,see Chapter 7. Several publications (e.g. Hessner and Reichert (2007)and Rosenthal et al. (2003)) deal with processing techniques able todeliver a map of the punctual sea surface elevation over the scannedsurface, with the scope of spotting the maximum single wave, inter-esting as extreme event.

Specific results on image processing techniques suited for spot-Advancedprocessingtechniques

ting and tracing the trajectory of high wave groups seems to bemissing in literature. Such a procedure could deliver a very reliableprediction of forces impacting on the ship where the wave radar ismounted, for convenient use in DP control algorithm.

6.4.2 Ship motions-based estimate

Based on growing affordability and quality of inertial sensors formarine applications, a number of techniques exploiting their func-tionality have been developed. Particularly interesting are the meth-ods to infer the directional wave spectrum from the ship response.Parametric and non-parametric methods have been developed andtested by several authors, see e.g. (Nielsen, 2006; Tannuri et al., 2003),with satisfactory results both in simulation and in basin tests.

If sufficiently accurate procedures can be implemented for on-board real-time use, these could avoid the use of a wave radar orprovide useful information when the wave radar is not working dueto e.g. presence of structures or supply vessels disturbing the pro-cessing of radar images. On the other hand, it must be noted that themethods rely on the knowledge of the RAOs, and the performanceis heavily influenced by the accuracy of the available operators.

7WAV E F E E D - F O RWA R D

Contents7.1 Literature 1377.2 Real-time estimate 139

7.2.1 Performance in Positioning 142

7.3 Data Analysis 1477.3.1 Motivation for predictive approach 152

7.4 Neural networks 1527.4.1 Neuron model 154

7.4.2 Basic network architectures 155

7.5 ANNs for drift force prediction 1587.5.1 Feed-forward input time-delay back-propagation

network 159

7.5.2 Recurrent input time-delay network 162

7.5.3 Pattern recognition network 164

7.5.4 Multiple input recurrent time-delay net-work 165

7.5.5 Network architecture selection 168

7.6 Integration of neural networks into DP algorithm 1687.7 Future developments 170

In this chapter the wave feed-forward (WFF) technique is presented.After a review of the strategies proposed in literature, a new ap-proach is proposed, based on a combination of ship movement in-ertial measurements and other independent measurements of theincoming waves, like the wave radar. Preliminary simulations arecarried out with theoretically-calculated forcing inputs as well asin-field measured data from specific campaigns.

7.1 literature

The idea of wave feed-forward has been proposed during theearly times of the dynamic positioning era, with the wind feed-forward having already shown its beneficial capabilities (Sjouke andLagers, 1971), see Section 5.3. Pinkster (1978) proposed to estimatethe instantaneous value of the drift forces by integration along the

137

138 wave feed-forward

static water line, as in Eq. 3.6. This requires in principle a continuousmeasurement of the water elevation all around the hull, but a goodapproximation can be obtained with a finite number of sensors. Thehigher frequencies must be removed by filtering, possibly trying toavoid adding filter delay to the time needed for calculation. Sincethe slow drift and the higher frequency contribution are well sepa-rated, being the latter terms characterised by about twice the wavefrequency (omitted terms in Eq. 3.9), the task for the filter is not toodemanding. The method was experimented at the Netherlands ShipModel Basin1, with 8 wave probes mounted on a model, 4 on eitherside, with promising results.

The same dutch institute (MARIN) carried out studies (Aalbersand Nienhuis, 1987) to include the real-time wave direction infor-mation when determining the optimum heading set-point for thepresent wind and wave conditions. Further, known the wave di-rection, a drift force estimate is computed depending on the shiphydrodynamic properties, in a wave feed-forward logic intendedto maintain the requested heading. The wave direction has beendetermined by means of only two probes, one on each side: theweather side experiences greater water surface relative motion thanleeward side. The experiments were carried out on three differentbarge types, with the most beneficial effects on the larger one, whichis the most affected by wave drift forces. Wind forces are more im-portant for small barges, such that wind feed-forward dominatesthe response.

The technique was then enhanced (Aalbers et al., 2001), includingmore probes around the hull, and combining more accurate driftforce and wave direction estimates. The force estimate is also fed tothe Kalman filter observer, such that the signal is exploited to for-mulate a better low frequency position and velocity estimate. Newmodel tests were executed, showing significant reduction of the lowfrequency deviation from set-point.

The method explained above was used to carry out simulationsof offshore operations in a safety point of view (Aalbers et al., 2004).The Captain’s Decision Mimic algorithm decides wether an opera-tion comprising a number of tasks can be completed in environmen-tal scenarios selected by Monte Carlo algorithm. The operability, de-fined as the maximum allowable environmental condition in whichthe operation can be successfully completed, results enhanced.

1 Now Maritime Research Institute Netherlands (MARIN), see http://www.marin.nl/.

http://www.marin.nl/

7.2 real-time estimate 139

7.2 real-time estimate

All the methods proposed in literature are characterised by thefact that drift forces are estimated in real-time, such that the de-lay between the perturbation and the action put in place by thethrusters is reduced, but still dependent on computation time andmachinery responsiveness.

Before showing an innovative approach based on prediction, asimple method to analyse the effectiveness of real-time estimatestrategies is here shown, for simulation and comparison.

The directional wave spectrum is supposed to be known withina reasonable degree of accuracy, from measurements (Section 6.4.1)or by estimate based on ship motions (see methods presented andreferenced in Section 6.4.2). When the motion response amplitudeoperators are known, the response to the particular directional dis-tribution can be determined:

RAOj(ω) =∫ 2π

0RAOj(ω, θ)

√D(ω, θ)dθ (7.1)

with the notation introduced in Chapter 2. The RAOj(ω) operator isthe complex valued transfer function between the wave height andthe corresponding barge motion for the j-th degree of freedom, withthe particular directional distribution characterising the sea state.

To obtain the expected wave elevation measuring barge motion, InverseapproximateRAO filter

the response amplitude operator must be used inversely. Precisely,a parametric filter can be found to approximate the RAO in thefrequency range where most of the energy is concentrated. The filteranalytical expression is written as:

H(s) =B(s)A(s)

=b1sn + b2sn−1 + · · ·+ bn+1

a1sm + a2sm−1 + · · ·+ am+1(7.2)

with the coefficients of the polynomials B(s) and A(s) to be deter-mined fitting the inverse RAOj(ω) operator. If the coefficients of thepolynomials B(s) and A(s) are grouped into the arrays:

b =[

b1 b2 · · · bn+1]T (7.3)

a =[

a1 a2 · · · an+1]T (7.4)


the problem may be formulated as follows:

C f = minb,a ∑k

wk

∣∣∣∣∣ 1RAOj(j ωk)

− B(j ωk)A(j ωk)

∣∣∣∣∣2

(7.5)

The weights k can be used to direct the best fit range around thewave peak frequency. The obtained filter transfer function can beapplied in real-time to the measured barge motion, making avail-able an estimate of the instantaneous wave elevation.

Drift forces can be estimated roughly under a couple of prettyForce estimation

strong hypotheses: the quadratic transfer functions can be simpli-fied using only the elements on the diagonal (see Section 3.3.1),and are constant in the frequency range around the wave peak fre-quency. The second hypothesis is reasonable when the spectrum issufficiently narrow banded.

F(2)j (t) = 2Pjζ(t)2 (7.6)

The mean quadratic transfer function Pj is a mean over the fre-quency range but also taking into account the directional distribu-tion of the waves:

Pj =∫ 2π

0

∫ ω2

ω1

Pj(ω, θ)S(ω, θ) dω dθ (7.7)

where Pj(ω, θ) is the real part of the QTF for the j-th degree of free-dom, see Section 3.3, and S(ω, θ) is the directional wave spectrum,see Section 2.6.

As an example, the sway drift force time history could be esti-mated assuming that heave motion is measured. The hydrodynamicdata for the vessel mentioned at Section 3.2, with the RAOs repre-sented in Figure 3.4. In a sea state characterised by a period of 7 s(0.90 rad s−1), the inverse of the heave response amplitude operatorcan be approximated by a second order filter in the frequency rangefrom 0.80 to 1.1 rad s−1:

B(s)A(s)

=b1s2 + b2s + b3

a1s2 + a2s + a3(7.8)

The following values are found for the arrays of parameters:

b =[−4.00 4.48 −2.40

]T (7.9)


a =[

1.00 0.08 1.29]T (7.10)

In Figure 7.1 the inverse heave RAO is compared with its discrete-time approximation (0.5 s sample interval). As shown, the local ap-proximation is obtained by using a weight different from zero onlyfor the desired range of harmonic components. The calculated trans-fer function fits well the target one for the selected range around thewave peak period.

Wei

ght

wk

Mag

nitu

de(m

/m)

|·|

Phas

e(r

ad)

∠·


ω


ω


ω

0.7 0.8 0.9 1.0 1.1 1.2

0.7 0.8 0.9 1.0 1.1 1.2

0.7 0.8 0.9 1.0 1.1 1.2

1

40

80

π2

− π2

Inverse RAOParametric filter

Inverse RAOParametric filter

Figure 7.1: Approximation of the inverse heave RAO by means of paramet-ric filter


The model in Figure 7.2 passes the simulated measure of theheave signal through the approximated filter, then estimates thesway drift force with the formulation expressed by Eq. 7.6.

It can be noticed that the wave elevation signal is quite well cap-tured by filtering the simulated heave motion, see Figure 7.3. Theaccuracy of the sway drift force calculation (results plotted in Fig-ure 7.4) suffers from the delay which is inherited from the water el-evation estimate and from the approximation of the QTF by meansof the mean value around the wave peak period. It can be noticedthat the accuracy is quite good, but it should be pointed out that,despite the assumptions made, the theory on which the simulationand the approximation are based is exactly the same, and the QTFis assumed to be exactly known. As a consequence, the results canbe considered somewhat optimistic.

The delay between simulated force and corresponding estimateis only due to filtering, but in real world one should foresee anadditional delay due to measurement signal acquisition and the de-scribed processing to be performed in real time.

7.2.1 Performance in Positioning

The method explained above can be used in a closed-loop simu-lation to preliminarily assess the capability to improve the perfor-mance in terms of deviation from a defined set-point. A completemodel integrating the units described along the previous chapters isevidences the behaviour of the dynamically positioned vessel underthe action of a beam sea state, which is critical for a ship-shapedhull.

In particular, a 3 m significant wave height sea state with a peakperiod of 8 s is generated, and the performance of the DP system isevaluated simply comparing the sway offset time history when thewave feed-forward is switched on or off.

The results plotted in Figure 7.5 show that the sway offset is sig-nificantly reduced. The feed-forward term is fed to the controllerwith a gain that halves the contribution, to account for estimationerrors and leave the rest to the PID controller. The gain value hashere been determined by rough trial and error approach, but it isa crucial topic for performance and stability, that will be discussedlater on.

While Figure 7.5 shows a 20 min extract, some parameters canbe computed for the complete 1 h long simulation to evaluate the


eta

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[0 0

0 0

0 0

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Wav

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Wav

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re7

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odel

for

real

-tim

edr

ift

forc

ees

tim

atio

n


Time (s)

t

25 50 75 100 125 150 175 200

Wat

erel

evat

ion

(m)

ζ(t)

1

−1

Simulated water elevationEstimated water elevation

Figure 7.3: Simulated and estimated water elevation time history

Time (s)

t25 50 75 100 125 150 175 200

Dri

ftFo

rce

(m)

F(2)2 (t)

1000

2000

3000

4000Simulated sway drift forceEstimated sway drift force

Figure 7.4: Simulated and estimated sway drift force time history


Time (s)

t40 80 120 160 200 240 280 320 360 400

Sway

offs

et(m

)η2

−4

−8

4

8

No WFFWFF gain 0.5

Time (s)

t440 480 520 560 600 640 680 720 760 800

Sway

offs

et(m

)

η2

−4

−8

4

8

No WFFWFF gain 0.5

Time (s)

t840 880 920 960 1000 1040 1080 1120 1160 1200

Sway

offs

et(m

)

η2

−4

−8

4

8

No WFFWFF gain 0.5

Figure 7.5: Performance comparison between simulations with and withoutwave feed-forward


improvement. Table 7.1 shows that the improvement is sensible interms of offset from set-point. In Table 7.2, where the significantwave height is higher, also the beneficial effect on the time overwhich the thrusters are not able to guarantee the thrust demand isevidenced.

Table 7.1: Performance improvement achieved with wave feed-forward,Hs = 3.0 m

std(η2) max(η2) min(η2) Saturation Time %

No WFF 3.47 11.05 −12.83 2.94WFF gain 0.5 1.86 7.24 −7.82 2.53

Table 7.2: Performance improvement achieved with wave feed-forward,Hs = 3.5 m

std(η2) max(η2) min(η2) Saturation Time %

No WFF 4.90 10.89 −19.95 15.83WFF gain 0.5 3.04 13.14 −12.17 10.89

In conclusion, the approach here presented and tested by meansof simulation results highly promising. Besides the main advantageof reducing deviation from set-point, also thruster saturation is di-minished. The second aspect, though not the first objective of thestudy, can be critical for the problem of wear and tear on machinery.In severe environments where the availability of full propulsion isneeded for operation, an out of order thruster can lower the allow-able sea state enough to enforce suspension. Even if the sea state towithstand is not any dangerous, authorities ask marine contractorsto operate in DP3 mode in certain sensible areas, which means fullredundancy must be achieved. Again, failures could stop operationdue to having made the propulsion system insufficient with regardto safety requirements, if not directly unable to keep position.

7.3 data analysis 147

7.3 data analysis

The analysis of data derived from simulation or on-board logstaken by the DP software evidences the role of delays in perfor-mance degradation, and insight into the improvements shown inSection 7.2.1. The following pages analyse the sway motion andthrust in a beam sea situation, as it is the most critical for DP per-formance of a ship-shaped vessel.

Figure 7.6 shows the time histories of sway offset, sway velocityand lateral thrust during a 1 h long simulation in beam sea, sig-nificant wave height Hs = 3.5 m, without any wave feed-forwardstrategy included in control algorithm. The delays between sway ve-locity and sway offset, sway thrust and sway offset, sway thrust andsway velocities are displayed by means of cross-correlation plots inFigure 7.7. It is evident from the upper plot that as expected thevelocity anticipates the offset. The plot in the middle shows that thethrust is applied indeed slightly before displacement actually hap-pens, due to the derivative control action. The lower plot evidencesthat the thrust is put in place significantly later the velocity buildsup.

The same discussion is developed for the simulated signals whenthe wave feed-forward with 0.5 gain is switched on. Figure 7.8shows the sway-direction displacement and velocity and the lat-eral thrust as a consequence of the same sea of the previous sim-ulation (the same random seed is used in waves generation). Thecross-correlation plots in Figure 7.9 evidence how the delays havechanged switching on the wave feed-forward based on real-timeestimation. The dashed lines mark the maximum-correlation lagsof the previous example. In particular, it is noted that the thrustdelay with respect to velocity is reduced. Further, the correlationbetween second order motion signals and the thrust is, as expected,lower due to the introduction of the wave drift estimated as a feed-forward. Table 7.3 and Table 7.4 compare the lags for maximumcorrelation between the signals analysed in this section.

The above observations clearly state that the delays that exist be-tween the in-plane (second order) motion and the counteraction bythe propulsion system is still significant. Further, the analysis of in-field measurement logs evidence that due to un-modelled systems,e.g. the power generators, some delays could be affected by the mod-els presented along this work and by particular operating conditions


Time (s)

600 1200 1800 2400 3000

Sway

offs

et(m

)

η2

−4

−8

4

8

Sway

velo

city

(m/s

)

η2

−0.25

−0.50

0.25

0.50

Time (s)

t

600 1200 1800 2400 3000 3600

Late

ralT

hrus

t(m

)

τ2

−2000

−4000

2000

4000

Figure 7.6: Sway offset and velocity compared with lateral thrust force


Lag (s)10 20 30−10−20−30

Rxy

−1

1

Sway Velocity - Sway

Lag (s)10 20 30−10−20−30

Rxy

−1

1

Sway Thrust - Sway

Lag (s)10 20 30−10−20−30

Rxy

−1

1

Sway Thrust - Sway Velocity

Figure 7.7: Cross-correlations between sway, sway velocity and lateralthrust signals.

Table 7.3: Motion-actuation lags for maximum correlation during simulatedbeam sea without wave feed-forward, with regard to sway degreeof freedom

Thrust Displacement Velocity

Thrust - −12 38Displacement 12 - 57

Velocity −38 −57 -


Time (s)

600 1200 1800 2400 3000

Sway

offs

et(m

)

η2

−4

−8

4

8

Sway

velo

city

(m/s

)

η2

−0.25

−0.50

0.25

0.50

Time (s)

t

600 1200 1800 2400 3000 3600

Late

ralT

hrus

t(m

)

τ2

−2000

−4000

2000

4000

Figure 7.8: Sway offset and velocity compared with lateral thrust force fromthrusters feedback, with WFF (0.5 gain)


Lag (s)10 20 30−10−20−30

Rxy

−1

1

Sway Velocity - Sway

Lag (s)10 20 30−10−20−30

Rxy

−1

1

Sway Thrust - Sway

Lag (s)10 20 30−10−20−30

Rxy

−1

1

Sway Thrust - Sway Velocity

Figure 7.9: Cross-correlations between sway, sway velocity and lateralthrust signals, with WFF (0.5 gain)

Table 7.4: Motion-actuation lags for maximum correlation during simulatedbeam sea with wave feed-forward at 0.5 gain, with regard to swaydegree of freedom

Thrust Displacement Velocity

Thrust - −10 32Displacement 10 - 48

Velocity −32 −48 -


which may arise (e.g. failure in one or a group of generators mayincrease response time).

7.3.1 Motivation for predictive approach

An innovative approach to wave feed-forward is here proposed,with the aim to fill the gap that real-time estimation methods stillshow in simplified simulations carried out to preliminarily assessthe possibility to extract useful information from first order motion.The main concern in severe environments is the duration of peri-ods of time in which high drift forces are applied due to high wavegroups. During such events the limited thrust capacity is insuffi-cient to cope with the environmental forces. On the other side, it isnot practical to try to deal with the problem by increasing thrustcapacity, because the operational drawbacks linked to storage andworking areas reduction, and power generation requirements wouldevidence the choice being far from optimal. The well-establishedpractice in dynamic positioning design is to recover from periodsin which the thrust capacity is exceeded by exploiting the relativelylow forces applied by the waves following a high group. Rather, itis intuitively expected that shall be possible to enhance operabilityif the thrusters could be powered to higher regimes before the wavegroups impact the hull, preventing motion with long recovery peri-ods associated with the vessel inertia.

The scope of strategy here proposed is to operate a very shortArtificial NeuralNetworks term forecast on wave groups, exploiting the regularity of the wave

pattern. Neural networks are a powerful tool for signal prediction,not yet applied to the field of dynamic positioning. The followingSection 7.4 gives an introduction to the Artificial Neural Networks(ANNs) and proposes different architectures to be applied to thepresent problem. Finally, in Section 7.6 the integration of a suitedcombination of the presented neural networks designs into the stan-dard DP algorithm is shown, with performance comparison.

7.4 neural networks

A definition of neural networks is the following (Haykin, 1998):

A neural network is a massively parallel distributed pro-cessor made up of simple processing units, which hasa natural propensity for storing experiential knowledge

7.4 neural networks 153

and making it available for use. It resembles the brain intwo respects:

a. Knowledge is acquired by the network from its en-vironment through a learning process.

b. Interneuron connection strengths, known as synap-tic weights, are used to store the acquired knowl-edge.

See also Aleksander and Morton (1995) for an alternative definition.Haykin also lists a number of properties and capabilities that Properties and

capabilitiesmake the use of neural networks appealing in multiple fields ofinterest. In particular, ANNs are nonlinear, thus able to catch thebehavior of inherently nonlinear systems, and adaptive, such thatthey can update the internal synaptic weights to the changes inthe surrounding environment. The latter ability is very importantwhen the process to be identified is subject to changes dependingon weather conditions and geographical location as in offshore ap-plications. Another important characteristic of the neural networkswhen it comes to task-critical role, is to be able, with some specificarchitecture, to give information about the confidence of the classifi-cation or prediction. In the particular application here studied, theconfidence evaluation is an important parameter to tune the wavefeed-forward weight. By the definition given before, the most im-portant aspect of neural networks is immediately understood: theyare able to deduce information on a system or a process without anyexternally-induced knowledge as a model or equations. This charac-teristic makes ANNs useful when a sufficiently reliable model is notavailable or process formalisation through equations is too complex.

The black-box nature of the neural network does not mean it isuncontrollable. Notably, invariances are purposely built in networksin several situations, e.g. in text recognition from images, there mustbe an invariance with respect to rotations (a character must be recog-nised as the same character if it is rotated). Other means are avail-able also to build in the network some prior information about theprocess, as the possibility to restrict some connections or constrainweights. The user’s ability is to model a network with all the avail-able knowledge built in but still able to adapt to the process drivenby data.

The fundamental concern in neural networks is training. This isthe way the ANNs acquire knowledge from environment, as pertheir definition. The training algorithm solves in general an opti-


mization problem, since it is mathematically aimed at finding theset of parameters (synaptic weights) that best fit the training data.Many training algorithms have been developed, often linked to theparticular network architecture. Some of them will be named orbriefly presented when used in the applications useful for the presentwork.

As the definition states, neural networks are made up of simpleprocessing units. The details on the mathematical expression of thesesimple units will be presented in next Section 7.4.1.

7.4.1 Neuron model

The brief description of neural networks give in previous para-graphs did not specify any detail on the computational structure onwhich ANNs are based, since it is not necessary when the scope isto describe their characteristics and field of application.

See Figure 7.10 for a schematic representation of a simple neuron,identified by the k index. The inputs x1, x2 . . . xm are multiplied bythe synaptic weights wk,1, wk,2 . . . wk,m and summed up. The bias bkis added as well. The sum vk (induced local field) is the argument ofthe function ϕ(vk), which finally builds the neuron output yk. Theoutput of the neuron is therefore calculated by the following Eq. 7.11

yk = ϕ

(m

∑j=0

wk,jxj + bk

)(7.11)

Note that the bias can be regarded as a weight applied to an ad-ditional constant input with unitary value. For this reason, in thefollowing sections the bias will not be explicitly represented, to sim-plify the graphical network schemes.

The function ϕ(vk), said activation function, has the aim of limitingActivationfunction the range of the neuron output. Different activation function can be

used for the scope, the most common being the threshold function,the piecewise linear function, and the sigmoid function (see Fig-ure 7.11). In particular a very common sigmoid function massivelyused in ANNs is the logistic function:

ϕ(v) =1

1 + exp(−av)(7.12)

with a the slope parameter.


wk,1

wk,2

...

wk,m

x1

x2

...

xm

bk

∑ ϕ(vk) ykvk

Figure 7.10: Neuron model

7.4.2 Basic network architectures

A single neuron like the one described at previous Section 7.4.1 isable to model only very simple input-output relations. The power ofANNs, similarly to the human brain, develops as the number of neu-rons grows. The basic architecture is the single-layer feed-forward net-work, as in Figure 7.12. In the depicted example three source nodesare connected to two neurons. The output of those neurons is theoutput of the network, therefore in this case the number of neuronsof the only layer must be equal to the number of outputs.

The limitation is removed by addition of a layer, as in Figure 7.13

where a multi-layer feed-forward network is shown. The hidden layercan be composed by an arbitrary number of neurons, while the out-put layer has a number of neurons equal to the number of networkoutputs.

The data flow can be redirected such that output from a layer orfrom the network itself returns (with unit or greater delay) as an in-put for a particular layer, see Figure 7.14. The use of feedback loopsis mainly useful when the network output from the previous stepscarries important information to determine the next output. Thisis particularly applied e.g. in time prediction of system response,when the input to the network is indeed delayed and a lag existswith the desired output, as will be shown in applications in the fol-lowing section Section 7.5. A widely used tool for such operationsis the tapped delay line, see Figure 7.15. The input signal connected to


Induced local field

v−2 −1 0 1 2

Act

ivat

ion

func

tion

ϕ(v)

0.33

0.66

1

(a) Threshold function

Induced local field

v−2 −1 0 1 2

Act

ivat

ion

func

tion

ϕ(v)

0.33

0.66

1

(b) Piecewise linear function

Induced local field

v−2 −1 0 1 2

Act

ivat

ion

func

tion

ϕ(v)

0.33

0.66

1

a = 0.5a = 1.0a = 0.5

(c) Sigmoid (logistic) function

Figure 7.11: Examples of activation functions


Input layer Output layer

Figure 7.12: Single-layer feed-forward network

Input layer Hidden layer Output layer

Figure 7.13: Feed-forward network with an hidden layer and two outputs


the single node at the left is passed through a number of unit delayoperators, such that the array of network inputs is composed by theinstantaneous value of the input signal and the values at the previ-ous time steps. If the first connections are cut, a delayed feed of theinput signal to the network is simulated, allowing for prediction inadvance of an arbitrary number of samples.

z−1

Unit Delay

Input layer Hidden layer Output layer

Figure 7.14: Recurrent network with an hidden layer

7.5 anns for drift force prediction

The neural networks have been proved to be an effective tool inwave forecast applications, see (Deo and Naidu, 1998; Makarynskyy,2004). In this section some neural networks applications are shownfor the particular problem tackled in this work. The networks here

7.5 anns for drift force prediction 159

z−1

z−1

...

z−1

Figure 7.15: Tapped delay line for prediction use

presented are mainly focused on prediction, but also some strategieshave been developed to obtain a measure of reliability in systemidentification. The simulations presented here were developed inMATLAB environment, supported by the Neural Networks Toolbox(The MathWorks, Inc., 2008b), which provides functions for networkdesign and training.

7.5.1 Feed-forward input time-delay back-propagation network

The network architecture is represented by the simple scheme inFigure 7.16. The single input node depicted is indeed a tapped delayline, where the composition of an array with a delay from 5 to 30samples takes place. With a sample period of 1 s the output of thenetwork will be a 5 s prediction of the input signal. The input signalis therefore also the target signal to use in training.

The back-propagation term refers to the training algorithm, seeHaykin (1998). In particular, since it is highly desirable to have anetwork learning to recognise the environment during operation,the adapt learning function is used wherever possible. This learn-ing strategy continues to update the network weights as new databecome available, such that the network is always in line with thechanging conditions.


z−5 : z−30

...

Input tappeddelay line

Sigmoidhidden

layer (10neurons)

Piecewiselinearoutputlayer

Figure 7.16: Feed-forward input time-delay backpropagation network

An important parameter of the back-propagation algorithm is thenumber of passes that is the number of times the same data is used fortraining. The higher the number of passes the higher the accuracy(up to saturation), but the computation time is increased as well,see Figure 7.17. The computation time is plotted as increase factorwith respect to the single-pass training. A factor influencing the ef-ficiency of training is the initialization of weights. Here the Nguyenand Widrow (1990) algorithm has been selected; the initializationis performed with the aim of distributing evenly the active regionof each neuron with respect to the input space. The algorithm con-tains a degree of randomness, such that repeating the initializationresults in slightly different performance, at least in the first phase ofincremental training.

In Figure 7.18 a network like the one depicted in Figure 7.16 isused to predict the envelope of the heave time history for a bargeunder the action of a 3.5 m significant wave height, peak period 10 s.The envelope has been calculated as the absolute value of the nu-merical Hilbert transform. The time history sampling period is 1 s,and the tapped delay line prepares the input array composed bysamples with a delay from 5 to 30. The prediction is thus performed5 s in advance. The plot shows the fitting performance after 3000 sof incremental training. The 10 passes fitting achieves better perfor-mance as visible, with delay significantly reduced.


Number of passes

1 2 3 4 5 6 7 8 9 10

Mea

nsq

uare

der

ror

0.05

0.07

0.09

0.11

0.13

Com

puta

tion

Tim

eM

ulti

plie

r

2

4

6

8

10

Figure 7.17: Network performance and training time (factor with respect tosingle-pass training) as a function of the number of passes

Time (s)

t3000 3050 3100 3150 3200 3250 3300 3350 3400 3450

Hea

veen

velo

pe(m

)

η3,env

0.25

0.5

0.75

1.00

1.25

TargetSingle pass10 passes

Figure 7.18: Example of prediction performance with single-pass incremen-tal training, compared with a 10 passes training


7.5.2 Recurrent input time-delay network

The network described at the previous section completely ignoresthe measurements between up to the minimum delay of the tappeddelay line. This is the main source of inaccuracy, that can be partiallyrecovered by means of recursion. The net in Figure 7.19 uses thepredictions made at the last time steps to fill the gap between thecurrent time and the first available measurement. Besides the inputtapped delay line, the 30 neurons in the hidden layer are fed byanother delay line acting on previous network outputs.

z−10 : z−50

...

z−1 : z−9

Input tappeddelay line

Sigmoidhidden

layer (30neurons)

Piecewiselinearoutputlayer

Figure 7.19: Recurrent input time-delay network

As evidenced by Figure 7.20 the prediction after 3000 s of incre-mental training is quite good, the higher incoming waves groups arealmost correctly predicted. Note that the minimum external inputdelay is here 10 s, equal to the wave period. The improvement hasbeen made possible without significant increase of computationalcost.


Time (s)

t

3000 3050 3100 3150 3200 3250 3300 3350 3400 3450

Hea

ve(m

)

η3,env

−0.25

−0.50

−0.75

−1.00

0.25

0.50

0.75

1.00

TargetPrediction

Figure 7.20: Example of prediction performance with 10 passes incrementaltraining, 10 s advance prediction


7.5.3 Pattern recognition network

Pattern recognition networks are useful tools generally appliedto classification problems, where the most suited category must befound given the properties of an object. Even though this is notstrictly the case, a particular network property can be exploited toallow for reliability estimation.

The network structure is depicted in Figure 7.21. The scheme isvery similar to the one in Figure 7.16, with a modified number ofoutputs. Each output corresponds to a class, each class is associatedto a level of the signal to be predicted, in this case the heave motionenvelope. The target signals to train the network have been shapedsuch that only the one corresponding to the level is equal to unity,while the others are zero. The network outputs will be ranging from0 to 1. This way, not only an estimate of the predicted signal level ismade available, but a confidence as well, see Figure 7.22. The value ofeach output is represented by the grey level, with white correspond-ing to zero and black corresponding to unity. The maximum confi-dence is indicated by a single output being equal to unity, whereassimilar grey levels indicate low confidence.

z−5 : z−30

...

Figure 7.21: Pattern recognition network

The output of the network carries twice the information in it, withimportant implications on the reliability of the wave feed-forwardsystems. In particular, it is proposed the use of the obtained addi-tional information in the global control architecture as a means totune the weight of the feed-forward term with respect to the stan-dard feedback-based control action.


Time (s)

t3000 3050 3100 3150 3200 3250 3300 3350 3400 3450

Hea

veEn

velo

peLe

vel

η3,lev

1

2

3

4

Figure 7.22: Pattern recognition network results

7.5.4 Multiple input recurrent time-delay network

A more complex neural network architecture is here proposed,based on multiple inputs. In particular, the 3-d.o.f. first order motionis used, to benefit from different phasing. The roll motion is e.g. ingeneral in advance with respect to heave, because tends to be in-phase with the wave slope for long wave periods, rather than withwave elevation as it is for the heave. Each of the three signals is fedto the network through a tapped delay line, see Figure 7.23. Further,a feedback loop is implemented to keep the heave signal updatedwith the estimates filling the gap with measurements.

The resulting network is trained as before with an incremental al-gorithm. The complexity of the architecture and the increased num-ber of neurons make training more expensive in computational time.The feasibility of real-time incremental training with the requirednumber of passes must be checked for the final implementation.

Figure 7.24 shows the performance of the described network whilepredicting the heave motion with 20 s in advance. The fitting is rea-sonably good, with the network being able to catch the periods oftime when the motion envelope is increasing or decreasing. Notethat the delay of 20 s is quite an extreme requirement, since the typ-ical response delay of the DP system is smaller (see Section 7.3).


z−1 : z−19

...

Heavez−20 : z−60

Rollz−20 : z−60 ...

Pitchz−20 : z−60 ...

Figure 7.23: Multiple input recurrent time-delay network with three inputsand a single output


Time (s)

t

3000 3050 3100 3150 3200 3250 3300 3350 3400 3450

Hea

ve(m

)

η3,env

−0.25

−0.50

−0.75

0.25

0.50

0.75

1.00

TargetPrediction

Figure 7.24: Example of prediction performance with 10 passes incrementaltraining, 20 s advance prediction


7.5.5 Network architecture selection

Some of the network architectures designed in this work havebeen described in previous sections, and a few conclusions can bededuced about the selection of the most correct strategy to follow.The best neural network is not univocally defined. A tradeoff ob-viously exist between prediction time horizon and computationalresources needed for training. Not secondary is the aspect of thereliability of the prediction, which is, even with a properly trainednetwork, decreasing when the input delay increases.

For the reasons explained above, the final controller design shallinclude a number of networks working in parallel and cooperatingto make available the best possible wave feed-forward signal, alongwith the right weight to assume in the total demand. The followingsection presents a possible implementation that tries to integrate thefindings of the previous analyses into a working DP system model.

7.6 integration of neural networks into dp algorithm

A neural network has been designed to predict the first orderheave motion under the action of waves. The output of the networkis used in the same way as in Figure 7.2, to obtain a wave elevationsignal through the approximated inverse RAO filter. The wave driftforce is then estimated as explained.

The network designed for this application is similar to the mul-tiple input recurrent time-delay network described in Section 7.5.4,with 10 s prediction time (20 samples with a simulation time stepof 0.5 s). The performance with regard to sway positioning underbeam sea conditions is plotted in Figure 7.25. The optimal weight ofOptimal weight

the feed-forward term (determined by trial and error) is a bit higherthan in the real-time case. The comparison plot evidences that theneural network guarantees the same performance of the real time-estimate, but the prediction time of 10 s makes the computation andbuild up of the required thrust fully feasible with no additional de-lay. The performance obtained is therefore very near to what thereal-time estimation method permits only in principle.

Figure 7.26 shows the applied sway drift force compared withthe prediction. A number of wave groups is predicted with suffi-cient accuracy, some others are predicted in time but not correctlyestimated in amplitude, a small number is missed. In general, theestimation quality can be found to be quite good, particularly with

7.6 integration of neural networks into dp algorithm 169

Time (s)

t40 80 120 160 200 240 280 320 360 400

Sway

offs

et(m

)η2

−4

−8

4

8

No WFFWFF gain 0.5ANN WFF gain 0.6

Time (s)

t440 480 520 560 600 640 680 720 760 800

Sway

offs

et(m

)

η2

−4

−8

4

8


Time (s)

t840 880 920 960 1000 1040 1080 1120 1160 1200

Sway

offs

et(m

)

η2

−4

−8

4

8


Figure 7.25: Performance comparison between real-time and neural net-work wave feed-forward


respect to the fact that the peaks are often anticipated in time, if notcorrectly guessed in amplitude. Despite of the imperfect superposi-tion of the two signals, the DP behaviour is significantly improved,with potentially great impact on operability.

7.7 future developments

The wave feed-forward strategy proposed in previous sections ofthis document demonstrated to be highly promising in simulation.Still a number of practical concerns must be solved before a robustapplication ready for field operations can be deployed.

First of all, the proposed evaluation of prediction reliability mustbe integrated with the overall architecture, to avoid the need ofproper weight selection by the operator. The network architecturecan be optimised based on logged data from sensors during testcampaigns or normal operation, to find the configuration whichguarantees the best tradeoff between accuracy and prediction time.Finally, the system must be tested from FMEA point of view (Phillips,1997) to ensure robustness and allow for certification by marine au-thorities.

Despite the fact that much work has to be done to get the sys-tem up and running, the results are already good enough to justifyinvestment in further research and engineering.

Significant improvements could result from adoption of innova-tive measurement systems like wave radar, with additional process-ing developed on-purpose, see Section 6.4.1. In particular, suitabletechniques to spot the high wave groups, follow their trajectory andpropagate their motion in the unobserved area near the hull aredesirable.

7.7 future developments 171

Time (s)

t40 80 120 160 200 240 280 320 360 400

Dri

ftfo

rce

(kN

)η2

4000

8000

12000

16000

20000

24000

Simulated sway drift forceEstimated sway drift force

Time (s)

t440 480 520 560 600 640 680 720 760 800

Dri

ftfo

rce

(kN

)

η2

4000

8000

12000

16000


Time (s)

t840 880 920 960 1000 1040 1080 1120 1160 1200

Dri

ftfo

rce

(kN

)

η2

4000

8000


Figure 7.26: Simulated and estimated sway drift force time history, bymeans of neural networks

C O N C L U S I O N S

This work has been prepared at the conclusion of a Ph.D. courseduring which the topics linked to positioning of vessels for offshoreconstruction, in particular pipe lay vessels, have been examined. Thescope of the study is justified by the new challenges to be tackledby the offshore industry, regarding the maximisation of operabilityin harsh environment, and safety.

Dynamic Positioning is a requirement for future lay vessels sup-posed to be able to complete the next generation strategic trunklinesfor hydrocarbon transportation. Solutions based on moorings be-came less appealing in today’s offshore construction market, whilethrust capacity on dynamically positioned vessel has reached satura-tion due to the request of working and storage space. These factorsjustify the efforts spent in control technology, with the aim of max-imising the performance without excessive demand on machinery.

The concurrent design of the ship and its propulsion and con-trol systems is nowadays a requirement reflected also in the designtools needed by industry. Simulation routines relative to differentengineering areas need now to be collected under a common frame-work, to allow for analysis of the different aspects of the same prob-lem. This need has been addressed in the present dissertation bydeveloping a number of models, thought to be integrated in a modu-lar environment. This approach guarantees sufficient completenessand expandability of the framework, both in the direction of moredetailed analysis on specific components, and in the direction ofsimplified aggregated models which have demonstrated to be veryuseful to understand the influence of the great number of factorsinvolved.

Artificial intelligence techniques have shown their power in multi-ple field of engineering. In particular, the typical application for Ar-tificial Neural Networks, is where modeling by mathematical equa-tions is not possible due to problem complexity. The Dynamic Posi-tioning problem appears to be efficiently approached by such tools.The Wave Feed-Forward control technique, on which many effortshave been spent since the early times of Dynamic Positioning, hasbeen showed to be successfully implemented in simulation with Ar-tificial Neural Networks estimation of drift forces.

173


The activities described shall continue on two main aspects: Neu-ral Networks optimization and integration, and innovative sensorsfor wave groups prediction. The first task regards the architectureand cooperation of different Neural Networks working in parallel toprovide both a drift force prediction based on vessel first order mo-tion and a reliability information to weight the Wave Feed-Forwardcontribution with respect to the position and velocity feedback term.The second task regards the development of suitable processingtechniques to exploit the capabilities of innovative sensors, namedwave radars, and achieve effective tracking of high wave groups.

In conclusion, the present work aimed at the development of sim-ulation tools and control techniques to push Dynamic Positioningsystems beyond the limits of currently operating vessels, ready forthe challenges of the next strategic projects.

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I N D E X

Aabsolute position, 123

acceleration feedback, 99, 117

acoustics, 132

adaptive, 153

added mass, 75

Airy linear wave theory, 24

autocorrelation, 129

azimuthal thrusters, 100

Bbase line, 132

Bernoulli’s equation, 22

bottom tow, 6

boundary conditions, 22, 23

Ccnoidal wave theory, 26

conclusions, 173

concrete-coated, 10

Constant energy sampling, 43

constant extension, 29

contact, 10

convex, 106

cost function, 102, 105

critical damping, 113

Ddeep water, 26, 31

delay, 113, 139

Dilution of Precision, 127

direct acceleration approach,117

Directional wave distribution,46

directionality functions, 47

double-peaked, 40

drag, 100

drift forces, 71

Dynamic Boundary Condi-tions, 22

Dynamic boundary condi-tions, 23

EED50, 124

elasto-plastic, 10

ellipsoid, 31, 123, 124

extended Kalman filter, 92

extended thrust matrix, 104

extended thrust vector, 104

extrapolation, 29

Ffilter, 48, 113

fully developed sea, 33

GGamma function, 47

Gauss-Markov process, 127

geodesy, 123

geoid, 123

GPS, 125

GPS sensor modeling, 127

gravity waves, 22

Hharmonic function, 24

Harris spectrum, 57

Hilbert transform, 160

I

183

184 index

incompressibility, 22

incompressible, 22

indirect acceleration ap-proach, 117

introduction, 1

inviscid, 22

irrotational, 22

JJONSWAP, 33, 38, 52

KKalman filter, 91

Kalman filter phases, 94

Kinematic Boundary Condi-tions, 22

kinematic boundary condi-tions, 23

LLagrange multipliers, 102

Laplace equation, 22

latitude, 123

least squares, 11

least-squares, 50

lift, 100

linear theory, 26

linear wave theory, 26

linearised boundary condi-tions, 24

longitude, 123

longitudinal thrusters, 99

low-frequency model, 92

Mmanoeuvrability, 106

MARIN, 138

measurement equation, 93

mid-depth tow, 6

model-based, 91

monolateral constraints, 10

Moore-Penrose pseudo-inverse, 103

moorings, 81

most probable largest waveheight, 45

Nnarrow-banded, 38

non-convex, 106

non-linear wave theories, 29

non-linearity, 74

non-penetration, 23

nonlinear programming, 105

Oobserver, 91

optimal Kalman gain, 95

oscillating system, 110

PParticle trajectories, 29

pay-out, 10

peak period, 33

periodic time history, 43

Pierson-Moskowitz spectrum,37

potential function, 22

power consumption, 105

power spectral density, 48

predict, 94

prediction, 139

pressure, 22

productivity, 9

pseudo-inverse, 102

pseudo-inverse matrix, 102

QQTF, 74

QTFs, 113

quadratic programming, 106

Rradial plots, 112

Random frequency sampling,43

index 185

Random phases, 42

range, 125

Rayleigh distribution, 45

real-time, 139

receiver clock error, 126

recursion, 162

recursive, 91, 94

reeling, 6

Regular wave theory, 24

resonance, 13, 15

rudders, 100

SSampling, 42

saturation, 105

sea state, 33

Sea state modeling, 33

seakeeping, 21

second order, 71, 75

seed, 52

semi-submersible, 7

shallow water, 31

significant wave height, 33

single-peaked spectra, 33

singularities, 105

Software for sea state model-ing, 51

specific gravity, 6

spectral moments, 34

spectrum, 33

spreading, 46

state-space, 91

stationary, 33

Statistics, 44

stiffness, 11

Stokes’ wave theory, 26

substantial derivative, 23

surface-tow, 6

swell, 33, 40

synaptic weights, 153

synthetic fibres, 81

Ttensioner, 8, 12

thrust allocation, 99

thrusters, 99

time history realisation, 35

time series, 41

time-domain simulation, 21

Torsethaugen, 40

towing, 6

transceiver, 132

transfer function, 74

trunklines, 8

tug boats, 6

tunnel thrusters, 100

Uunderwater positioning, 132

unscented Kalman filter, 92

update, 94

Vvelocity potential, 26

virtual mass, 117

vorticity, 22

Wwave feed-forward, 137

wave filtering, 40

wave groups, 136

wave length, 24

wave number, 24

wave spectra, 21, 33

Wave spectra approximation,48

wave spreading, 52

wave theory, 21

wave-frequency model, 92

Weather vaning, 10

weight matrix, 102

WGS84, 124

Wheeler profile stretching, 29

white noise, 129

186 index

wind feed-forward, 97

wind spectra, 56

wind speed, 35

Zzero up-crossing period, 33

PhD Thesis - Federico Gaggiotti

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