positioning heading kalman

8/2/2019 Positioning Heading Kalman

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32 IEEE CONTROL SYSTEMS MAGAZINE DECEMBER 2009 1066-033X/09/$26.002009IEEE

PHOTO BY D.S. BERNSTEIN

Kalman Filtering

for Positioning andHeading Control ofShips and Offshore Rigs

THOR I. FOSSEN and TRISTAN PEREZ

ESTIMATING THE EFFECTSOF WAVES, WIND,AND CURRENT

Authorized licensed use limited to: ELETTRONICA E INFORMATICA PADOVA. Downloaded on January 7, 2010 at 03:46 from IEEE Xplore. Restrictions apply.


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DECEMBER 2009 IEEE CONTROL SYSTEMS MAGAZINE 33

Digital Object Identifier 10.1109/MCS.2009.934408

Ships and offshore rigs perform missions that requiretight motion control. During the past 30 years, therehas been an increasing demand for higher accuracyand reliability of ship motion-control systems to suchan extent that it is difficult to conceive of operations

in which motion-control systems are not an essential part, or

even an enabling factor. Modern marine vessels are equippedwith sophisticated motion-control systems that have differ-ent objectives depending on the particular operations beingperformed. Some of these control objectives include positionand heading regulation, path following, trajectory tracking,and wave-induced motion reduction.

The stochastic nature of environmentally induced forceshas led to extensive use of the Kalman filter for estimatingship-motion-related quantities and for filtering disturbances,

both of which are of paramount importance for implement-ing marine motion-control systems. In this article, we intro-duce the elements of a ship motion-control system, describe

the models used for position-regulation and course-keepingcontrol, and discuss how the Kalman filter is used for wavefiltering, that is, removing the oscillatory wave-inducedmotion from velocity and positioning measurements.

SHIP MOTION CONTROL AND WAVE FILTERING

Ships and offshore rigs are exposed to wind, waves, and cur-rent forces. The requirements of the various operations per-formed by the vessel, together with the characteristics of theenvironmental forces, define the modes of operation and theobjectives of the motion-control system. Figure 1 illustratesthe components of a ship motion-control system [1], [2]. Theguidance system generates a smooth and feasible desired ref-erence trajectory described in terms of position, velocity, andacceleration. The trajectory is generated by algorithms thatuse the ships actual and desired position and a mathematicalmodel of the ship together with information regarding mis-sions, operator decisions, weather, and fleet operations. Thecontrol system processes motion-related signals to infer thestate of the ship, to filter disturbances, and to generate anappropriate command for the actuators so as to reduce thedifference between the actual and desired ship trajectories.The controller may have multiple modes of operation

depending on the type of mission performed, for example,course keeping, positioning, and roll- and pitch-motiondamping. For some ships and particular operations, thedesired control action can be delivered in several ways due tooveractuation, which provides increased reliability to actua-tor faults. Thus, multiple combinations of actuator demandscan yield the same control action. In these cases, the controlsystem must also solve a control allocation problem based onan optimization criteria [2]. The navigation system, whichprovides reliable measurements of position and heading, col-lects information from the various sensors, such as GPS,speed log, compass, gyros, radar, and accelerometers. The

navigation system also performs signal quality checking andtransforms the measurements to a common reference frameused by the control and guidance systems [1][3].

Environmental forces due to waves, wind, and current areconsidered disturbances to the motion-control system. Theseforces, which can be described in stochastic terms, are concep-

tually separated into wave- and low-frequency components.Waves produce a pressure change on the hull surface,

which in turn induces forces. These pressure-induced forceshave an oscillatory component that depends linearly on thewave elevation. Hence, these forces have the same fre-quency as that of the waves and are therefore referred to aswave-frequency forces. Wave forces also have a componentthat depends nonlinearly on the wave elevation [4], [5].Nonlinear wave forces are due to the quadratic dependenceof the pressure on the fluid-particle velocity induced by thepassing of the waves. If, for example, two sinusoidal inci-dent waves have different frequencies, then their quadratic

relationship gives pressure forces with frequencies at boththe sum and difference of the incident wave frequencies aswell as zero-frequency or mean forces. Hence, the nonlinearwave forces present both lower and higher frequencies thanthe wave frequencies. The mean wave forces cause thevessel to drift. The forces with a frequency content at thedifference of the wave frequencies can have low-frequencycontent, which can lead to resonance in the horizontalmotion of vessels with mooring lines or under positioningcontrol [5]. The high-frequency, wave-pressure-induced

Motion-Control System

Observer(Wave Filter)

Controller Allocation

ActuatorCommands

Actuators

ControlForces

Desired Positionand Velocity

Measured Position

and VelocityMotion

NavigationSystem

GuidanceSystem

Information

Task Operator

Weather

Fleet

Desired Forces

FIGURE 1 The basic components of a modern ship motion-control

system can be grouped into three subsystems. The navigationsystem processes information from the motion sensors to provide

reliable signals related to the actual trajectory of the vessel. Theguidance system provides desired feasible trajectories based on

information related to the vessel state and mission in progress.The control system processes the information coming from the

navigation and guidance system to produce forces that correct

deviations of the vessel trajectory from the desired vessel trajec-tory. (Picture of courtesy of Austal Ships, Australia.)



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34 IEEE CONTROL SYSTEMS MAGAZINE DECEMBER 2009

forces, which have frequency content at the sum of the wavefrequencies, are normally too high to be considered in shipmotion control, but these forces can contribute to structuralvibration in the hull. For further details about wave loadsand their effects on ship motion, see [4] and [5].

Like waves, wind and current induce forces due to pres-sure variation on the vessel structure. Wind forces have amean component and a random fluctuating component dueto gusts. In ship motion control, only the mean wind forcesare compensated since the frequency of gusts is often outsidethe bandwidth of the vessel response [2]. Current-induced

forces affect vessels requiring positioning control and vesselsat mooring. These forces have low-frequency content. Thevariations of these forces can be induced by changes in thespeed and direction of the current relative to the vessel.

Wind, current, and nonlinear wave forces that are ofinterest to ship motion control are referred to as low-frequencyforces. Figure 2 shows a typical example of observed motionin marine surface vessels. The motion components derivedfrom the inducing forces are thus referred to as the low-frequency motion and wave-frequency motion. Due to thecharacteristics of ship motion depicted in Figure 2, theobjective of a ship motion-control system may be to controlonly the low-frequency motion, only the wave-frequencymotion, or the total motion, that is, the sum of the low- andwave-frequency motions [1][3].

In low-to-medium sea states, the frequency of oscillationsof the linear wave forces do not normally affect the operationalperformance of the vessel. Hence, controlling only low-frequency motion avoids correcting the motion for every singlewave, which can result in unacceptable operational conditionsfor the propulsion system due to power consumption andpotential wear of the actuators. Operations that require thecontrol of only the low-frequency motion include dynamicpositioning, heading autopilots, and thruster-assisted positionmooring. Dynamic positioning refers to the use of the propul-sion system to regulate the horizontal position and heading of

the vessel. In thruster-assisted position mooring, the propul-sion system is used to reduce the mean loading on the mooringlines. Additional operations that require the control of only thelow-frequency motion include slow maneuvers that arise, forexample, from following underwater remotely operated vehi-cles. Operations that require the control of only the wave-frequency motions include heave compensation for deployingloads at the sea floor [6] as well as ride control of passengervessels, where reducing roll and pitch motion helps avoidmotion sickness [3].

The control of only low-frequency motion is achieved byappropriate filtering of the wave-frequency componentsfrom the position and heading measurements and estimatedvelocities. This filtering is performed before the signals arepassed on to the controller as indicated in Figure 1.

Early course-keeping autopilots used a proportional con-troller with a deadband nonlinearity. The deadband providedan effect similar to wave filtering since it delivered a null con-trol action until the control signals were large enough to beoutside the deadband. The amount of deadband in the auto-pilot could be changed, and this setting was called weathersince the size of the deadband was selected by the operator

based on weather conditions [7], [8]. Other systems used

lowpass and notch filters, which introduced significant phaselag, and thus performance degradation when a high gaincontrol is required. An alternative to traditional filtering con-sists of using a wave-induced motion model and an observerto separate the wave motion from the low-frequency motion.For this approach, which requires a spectral factorizationmodel of the disturbances, the Kalman filter is the preferredobserver [9]. The Kalman filter can use measurements frommultiple sensors with different levels of accuracy to produceestimates of the ship velocities, which are not measured inmost ship-positioning applications.

In the remainder of this article, we focus on two opera-

tions where only low-frequency motion control is required,dynamic positioning and course keeping, and we present

0 20 40 60 80 100 120 140 160 180 2002

0

2

4

6

8

10

12

14

Time (s)

HeadingAngle()

LF Heading (Filtered)

Total Heading (LF+WF)

FIGURE 2 Example of motion of a marine vessel. The total motioncan be thought of as the superposition of a low-frequency (LF) and

a zero-mean oscillatory wave-frequency (WF) component. This

plot shows the total motion and low-frequency components of theheading angle of a vessel during a course change.

The requirements of the various operations performed by the vessel define

the modes of operation and the objectives of the motion-control system.



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the north-east positions (n, e) of the vessel relative to thelocal geographical frame and the yaw or heading angle c relative to the north. The components of the velocity vectorn5 3u, v, r 4^ are the surge and sway velocities ( u, v) andthe yaw rate r. These variables are depicted in Figure 4. The

body-fixed velocities u and v are the time derivatives ofthe position of the origin of the body-fixed frame relative tothe origin of the local geographical frame expressed in the

body-fixed frame. Similarly, the yaw rate r is a componentof the angular velocity of the body-fixed frame with respectto the local geographical frame expressed in the body-fixedframe. For ship-positioning and heading control, the trans-lational motion is assumed to be confined to the horizontalplane, and thus the angular velocity has only one component,

namely, r, which the rotation rate about the axis perpen-dicular to the horizontal plane.

A model for vessel dynamics can be expressed as

h? 5 R(c )n, (1)(MRB1MA)n

#1CRB(n )n1d(Vrc, gc) 5tcontrol1twind 1 twaves.

(2)

The kinematic transformation (1) relates the body-fixedvelocities to the time derivative of the positions in the localgeographical frame. This transformation is given by therotation matrix

R (c ) 5 cos c 2sin c 0sin c cos c 00 0 1

, R21(c ) 5 R^(c ) . (3)For further details about kinematic models used in marinecontrol systems and transformations, see [10].

The terms on the right-hand side of (2) represent the vec-tors of forces due to control, wind, and waves, respectively.These forces are directed along the body-fixed directionsxb and yb, and the moment is taken about the vertical axiszb, so that t 5 3X, Y, N4T, where X is the surge force, Y isthe sway force, and N is the yaw moment. Table 1 summa-rizes this notation and indicates the reference frames inwhich the variables are expressed.

The positive-definite, rigid-body mass matrix MRB and

the skew-symmetric Coriolis-centripetal matrix CRB(n ) are given by

HydroacousticPositioning

SystemTautWire

SurfaceReferenceSystem

Satellite NavigationSystem(DGPS/GLONASS)

HydroacousticPositioning

SystemTaut

SurfaceReferenceSystem

Satellite NavigationSystem(DGPS/GLONASS)

FIGURE 3 Examples of sensors for vessel positioning. For reliability,some vessels have multiple sensors. The surface position reference

systems include laser-doppler sensors and radio navigation systems.

As a satellite system, GPS and differential GPS are the most common.For vessels operating at fixed locations, such as oil rigs, it is common

to use hydroacoustic sensors and taut wires. The latter consist of aweight tied to a wire, where deviations of the desired position of the

ship are determined from the length of the wire deployed.

v (Sway) q (Pitch)

p (Roll)

u (Surge)

r (Yaw)

w (Heave)xb

zb

yb

FIGURE 4 Motion variables for a marine vessel based on the

SNAME convention [32]. The linear velocities in surge, sway, and

heave are expressed in a body-fixed frame. Similarly, the threecomponents of the angular velocity vector are expressed in the

same body-fixed frame. These angular velocity components canbe related to the time-derivatives of the roll, pitch, and yaw Euler

angles using a kinematic transformation.

Variable Name Frame Units

n North position Earth fixed me East position Earth fixed mc Heading or yaw angle radu Surge speed Body fixed m/sv Sway speed Body fixed m/sr Yaw rate Body fixed rad/sX Surge force Body fixed NY Sway force Body fixed NN Yaw moment Body fixed N-mh5 3n, e,c 4T Generalized positionn5

3u, v, r

4T Generalized velocity

t5 3X, Y, N4T Generalized force

TABLE 1 Summary of ship motion variables used for

dynamic positioning and maneuvering applications.



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MRB 5 m 0 00 m mxg0 mxg Iz

, CRB(n ) 5 0 0 2m(xgr 1 v)0 0 mu

m(xgr 1 v) 2mu 0 , (4)

where xg denotes the longitudinal position of the center ofgravity of the vessel relative to the body-fixed frame. TheCoriolis-centripetal terms appear as a consequence of express-

ing the equations of motion in body-fixed coordinates. Thisformulation simplifies the model since, when expressed inthe body-fixed frame, the moments of inertia become timeinvariant and the direction of the actuator forces are simplerto describe. When a vessel operates under positioning con-trol, the velocities are small. Hence, the Coriolis-centripetalterms CRB(n )n in (2) can be ignored for control design.

When a vessel moves in the water, the changes in pres-sure on the hull are proportional to the velocities and accel-erations of the vessel relative to the fluid. The coefficientsused to express the pressure-induced forces proportionalto the accelerations are called added-mass coefficients.These forces reflect the change in momentum in the fluiddue to the vessel accelerations. The positive-definite hydro-dynamic added mass matrix MA is represented by

MA 5 2Xu# 0 00 2Yv# 2Yr#0 2Yr# 2Nr#

, (5)where the added-mass coefficients Xu# , Yv# , Yr# , Yr# , and Nr# depend on the hull shape. Despite the terminology, notethat only Xu# and Yv# have mass units.

The term d(Vrc,

gc)

on the left-hand side of (2) repre-sents the current and damping forces. These nonconser-vative forces, which reflect the transfer of energy fromthe vessel to the fluid, depend on the speed and direc-tion of the current relative to the vessel is given by

Vrc 5 "urc2 1 vrc2 5 "(u 2 uc) 2 1 (v 2 vc) 2, (6) grc 5 2atan2(vrc, urc) , (7)

where uc and vc are the components of the current velocityin the vessel body-fixed frame. Thus, the angle of the cur-rent grc is defined relative to the bow of the vessel.

It is common practice to write the current forces insurge, sway, and yaw as functions of nondimensional

current coefficients CXc (grc) , CYc (grc) , CNc ( grc) , [1], [2],that is,

d(Vrc, grc) 512

rV2rc AFcCXc(grc)ALcCYc( grc)ALcLoaCNC (grc)

, (8)where r is the water density, AFc and ALc are frontal andlateral projected areas of the submerged part of the hull,and Loa is the length of the ship. Typical current coeffi-cients for a dynamically positioned vessel are shown in

Figure 5. These coefficients are determined experimen-tally based on scale models or using computational fluiddynamic models [2].

Unless the vessel under consideration is the subject ofan extensive hydrodynamic analysis and scale-model test-ing, the current coefficients CXc (grc) , CYc (grc) , CNc (grc) in(8) are difficult to estimate with accuracy. In such cases, it iscommon practice to simplify the model (8) in terms of alinear damping term and a bias term [33] of the form

d(Vrc, grc) < Dn2 R^(c )b, (9)

0 50 100 150 200 250 300 350 4000.2

0

0.2

Relative Current Angle ()

0 50 100 150 200 250 300 350 400Relative Current Angle ()

0 50 100 150 200 250 300 350 400Relative Current Angle ()

CX

1

0

1

CY

0.5

0

0.5

CN

FIGURE 5 Current coefficients of an offshore supply vessel deter-mined using computational fluid dynamics. The current coeffi-

cients are normalized nondimensional forces in surge CX, swayCY, and yaw CN. The normalization is based on current speed,

water density, and the projected area of the vessel below the water

line. These coefficients can also be determined experimentallyfrom scale model tests.

The objective of a ship motion-control system may be to control only the

low-frequency motion, only the wave-frequency motion, or the total motion.



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where

D 5 D^5 D11 0 00 D22 D230 D32 D33

, b 5 b1b2b3

. (10)The linear damping term also models the transfer of energyfrom the vessel to the fluid due to the waves that are gener-ated as a consequence of the vessel motion.

Using (9), the simplified vessel model (1)(2) becomes

h? 5 R(c )n, (11)Mn? 1 Dn5 R^(c )b 1 tcontrol 1 twind 1 twaves, (12)

b?

5 0, (13)

where M5MRB 1MA in (12). The bias term is constant inEarth-fixed coordinates, under the assumption of constantor slowly varying currents. This assumption is appropriateconsidering the tide periods relative to the ship dynamicresponse characteristics. The bias term must therefore berotated to be included into the equation of motion (12). Thisrotation captures the effect that the current forces changewith the heading of the vessel.

The bias is estimated by the observer shown in Figure 1,which allows the resulting force term in (12) to be compen-sated by the motion controller. This approach thus achievesintegral action for positioning control since the bias repre-sents the low-frequency disturbances [2].

Vessel Parallel Model Formulation

The model (11)(13) is nonlinear due to the kinematictransformations. This model can be linearized dynami-cally by introducing the vessel parallel coordinates [2],

which are defined in a reference frame fixed to the vesselwith axes parallel to the Earth-fixed reference frame. Thevessel parallel coordinateshp are thus defined by using thetransformation

hp J R^(c )h, (14)

where hp is the position-attitude vector expressed in bodycoordinates. For dynamic positioning applications, rotationabout the z-axis is often slow. Therefore, r# < 0 andR#

(c ) < 0 are good approximations. Consequently, (11)(13) can be written in terms of hp, and the resulting model,

which is referred to as the vessel parallel reference model, isgiven by

h?p 5 n, (15)Mn? 1 Dn5 bp 1 tcontrol 1 twind 1 twaves, (16)

b?

p 5 0, (17)

where bp J R^(c )b. Since (15)(17) is linear, a linearobserver can be used to estimate the velocity n and biasbp using the force vectors tcontrol, twind, and twaves alongwith hp, which is known from the measurement of h through (14).

Wind Force Models

The wind forces and moments can be represented in a sim-ilar way to the current forces and moments by means ofnondimensional force coefficients [2], namely,

twind 512

raV2rw AFwCXw (grw)ALwCYw(grw)

ALwLoaCNw (grw) , (18)

where ra is the air density, AFw and ALw are frontal and lat-eral projected wind areas, and Loa is the vessels overalllength. The wind speed Vrw and direction grw relative tothe vessel are given by

Vrw 5 "u2rw 1 v2rw, (19) grw 5 2atan2(vrw, urw) , (20)

with

urw 5 u 2 Vwcosbw, (21)

vrw 5 v 2 Vwsin bw. (22)

Here Vw and bw are the wind speed and direction in Earth-fixed coordinates. Typical wind coefficients CXw (grw) ,CYw (grw) , and CNw (grw) for an offshore vessel are shownin Figure 6.

The wind coefficients can be obtained either from com-putational fluid dynamics, model tests, or by scaling coef-ficients of similar vessels. For control design purposes,however, wind speed and direction measurements areoften used for approximate feedforward compensation,and the errors associated with this compensation are mod-eled as a bias in (13) and (17). That is, the bias accounts for

the simplified current forces as well as the wind forces. Iffeedforward compensation is not used, then a sluggish

3

uw

3

vw

Most of the vessels operating in the offshore industry worldwide use

Kalman filters for velocity estimation and wave filtering.



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0 50 100 150 200 250 300 350 400Relative Wind Angle ()



CX

CY

CN

0.5

0

0.5

1

1

0

1

0.2

0

0.2

FIGURE 6 Wind coefficients of an offshore supply vessel deter-

mined using computational fluid dynamics. The wind coefficientsare normalized forces in surge CX, sway CY, and yaw CN. The

normalization is done in terms of wind speed, air density, and theprojected area of the vessel above the waterline. These coeffi-

cients are determined experimentally from scale model tests for

multiple wind directions measured by the wind angle grw relativeto the bow of the ship.

response to changes in the direction of the wind relative tothe vessel may result [2].

Models for Wave Loads and Wave-Induced Motion

As discussed above, wave forces are usually modeled asthe sum of a linear and a nonlinear component, namely,

twaves 5 tlinwaves 1 t

nlinwaves. (23)

The linear oscillatory component has the same frequencyas the wave elevation. The nonlinear component has bothlower and higher frequency than the wave elevation. Thelinear and low-frequency nonlinear wave forces are rele-vant to ship motion control.

The low-frequency nonlinear wave forces are treated asan input disturbance and modeled by a bias term (17). Thatis, the bias term represents a combination of nonlinearwave and current. The linear wave forces, however, areusually transformed into an equivalent output disturbance.With this point of view, the measured position vector can

be represented as

htot 5h 1hw, (24)

where htot is the total position, hw is wave-induced posi-tion due to linear wave forces, and h represents the low-frequency position due to control, wind, current, andnonlinear wave forces.

The sea surface elevation is typically modeled as therealization of a stationary Gaussian random process [34].Stationarity can be considered for short periods between 20min and 3 h depending on weather conditions, while

Gaussianity depends on the wave height and depth [35].The deeper the ocean at the location considered, the higherthe waves for which Gaussianity can be assumed [34], [35].Under these conditions, the wave elevation is completelycharacterized by the wave spectrum Szz (v ) , which changeswith the sea state, namely, the dominant amplitudes andwave periods.

Under a linear response assumption, the wave-inducedmotion spectrum can be represented as

Shwhw (v ) 5|H(jv )F(jv )|2 Szz (v ) , (25)

where F(jv ) represents the frequency response functionfrom the wave elevation to the pressure-induced forces on

the hull andH(jv ) represents the frequency response fromforce to motion [4], [5]. The frequency response functionF(jv ) , which is known as the force response amplitudeoperator (RAO) in naval architecture, depends on the shapeof the hull and the angle from which the waves approachthe vessel. The frequency response H(jv ) , known as theforce-to-motion RAO, depends on the shape of the hull andthe mass distribution. These frequency response functions,which can be computed using hydrodynamic computer

codes, are the basis of ship motion evaluation [3].The spectrum of the wave-induced motion can be

approximated using spectral factorization. That is, (25) can be approximated by the spectrum of the signal obtainedusing a linear filter driven by Gaussian noise w(t ) with aflat spectrum Sww over the frequencies of interest. Then,

Shwhw (v ) < |G(jv )|2Sww. (26)

A second-order, noise-driven filter is usually appropriatefor modeling wave-induced motion [1][3], namely,

To describe the motion of a ship, two reference frames are used,

a local geographical Earth-fixed frame and a body-fixed frame,

which is attached to the vessel.



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G(s) 5v 20 s

s2 1 2zv0 s 1 v 20. (27)

The parameters of the shaping filter (27) as well as theintensity Sww of the noise depend on the sea state, the

vessel shape, and the angle from which the waves approachthe vessel.

A filter transfer function of the form (27) is consideredfor each degree of freedom. The combination of thesetransfer functions can be transformed into a single state-space model

j?

5Awj1 Eww, (28)hw 5 Cwj, (29)

where the matrices Aw, Bw, and Cw depend on the chosenstate-space realization adopted. Since a model for dynamic

positioning has three degrees of freedom (surge, sway, andyaw), a second-order noise filter approximation yields astate vector j with six components. The state vector j mayor may not have a physical interpretation depending on theparticular state-space realization used.

Linear Model for Dynamic Positioning Observer Design

By combining the models of the vessel response togetherwith the disturbance models discussed above, we obtain amodel structure of the form

j#

5Awj1 Eww1, (30)h?p 5 n, (31)

Mn? 1 Dn5 bp 1 tcontrol 1 twind 1 w2, (32)b?

p 5 w3, (33)y 5hp 1 Cwj1 v. (34)

The variables wi, i 5 1, 2, 3, are Gaussian noise vectors,which represent model uncertainty. Note that no uncertaintyis associated with the kinematic equation (31) since this rela-tion describes known geometrical aspects of motion. Themeasurement positions (34) are the sum of the wave motionCwj and the motion hp due to wind forces, current, and con-

trol. The measurement vector also contain some noise v. Thecontrol forces usually have two components

t 5 2 t wind 1 Buu, (35)

where t wind is an estimate of the wind forces implemented by using feedforward compensation and Buu representsactuator forces. The wind feedforward term, which is pro-portional to the square of the measured wind velocity,depends on the vessels projected area in the direction ofthe wind [2]. The vector u is the command to the actuators,which are assumed to have much faster dynamic response

than the vessel; thus the coefficient Bu represents the map-ping from the actuator command to the force generated by

the actuator. For example, if the command to a propeller isthe rotation speed, then the corresponding coefficient in Bu maps the speed to the generated thrust.

The resulting model for a dynamic positioning observerdesign is the 15th-order state-model

x?

5Ax 1 Bu 1 Ew, (36) y 5Hx 1 v, (37)

wherex 5 3j^, h^p , b^p , n^ 4^ [ R15 is the state vector, u [ Rp(p $ 3) is the control vector, w 5 3w^1 , w^2 , w^3 4^ [ R9 rep-resents the process noise vector, and

A 5 Aw 0633 0633 06330336 I333 0333 03330336 0333 0333 0333

0336 2M21D M21 0333

, B 5 063303330333

M21Bu

, (38)H5 3Cw I333 0333 0333 4, E 5 Ew0333I333

M21 . (39)

KALMAN FILTER DESIGN FOR

SHIP DYNAMIC POSITIONING

The model given in (36) forms the basis of a Kalman filterdesign. To implement the observer on a computer, the modelneeds to be discretized. The discretization has the form

x(k 1 1) 5Fx(k) 1 Du(k ) 1Gw(k) , (40) y(k) 5Hx(k ) 1 v(k) , (41)

where

F5 exp(Ah ) , (42) D 5A21 (F2 I)B, (43) G5A21 (F2 I)E, (44)

h is the sampling time, and the equivalent discrete-timenoises w(k) and v(k) are Gaussian and white with zeromean. Appropriate sampling times can be determined from

step responses in the various degrees of freedom of the vessel.As a rule of thumb, the sampling period is chosen to be in therange of 2040 samples within the rise time of the force tovelocity response of the fastest degree of freedom. For largeoffshore vessels and rigs, the sampling time is normally inthe range of 100500 ms. Small vessels with faster dynamicsmay require sampling times as low as 50 ms.

To implement a Kalman filter, the parameters of themodel (40), (41) as well as the covariance of the state mea-surement noises in the model are necessary. The mass anddamping parameters of the model can be initially estimatedfrom hydrodynamic computations. Then, an update of the

parameter estimates can be based on data obtained fromtests performed in calm water [27]. The structure of the



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model matrices (38), (39) determines the structure of thematrices of the discrete-time model (40), (41). Then, forexample, standard grey-box, state-space prediction-errormethods can be used to estimate the parameters [12].

The measurement noise is usually associated with sen-sors. After correcting the sensor bias, the covariance s2vi of the

measurement noise of the sensor i can be estimated by thesample covariance from a data record taken while the vesselis at port with no motion. Since the noise of different sensorsis often uncorrelated, the covariance matrix of the vector ofmeasurement noise is chosen to be diagonal, that is,

R 5 diag(s2v1, s2v2,c, s2vp) .

The estimation of the covariance Q of the state noise w in(40) is more complex since it depends on the sea state, theheading of the vessel relative to the environmental distur-

bances, and how uncertain the model (40), (41) is [3]. This

covariance matrix is chosen to be block diagonal, that is,

Q 5 diag(Q1, Q2, Q3 ) . (45)

The matrix Q1 [ R333 is the covariance of the noise w1,which drives the noise filter representing linear wave-in-duced motion, Q2 [ R333 is the covariance of the noise w2,which represents the uncertainty in the equation of motion,and Q3 [ R333 is the covariance of the noise w3, whichrepresents the uncertainty in the bias term that models therest of the environmental forces, as indicated in (30) (33).

The matrices Q2 and Q3 are chosen to be diagonal. Theentries of the matrix Q2 are taken as a fraction of the vari-ance of the position measurement noises. The entries of Q3 are high values. These choices provide a filter with anappropriate balance of the uncertainty in various parts ofthe model. The covariance Q1 is estimated together with theparameters of the wave-frequency motion model (27) fromdata measured before and during the operation of thevessel. The parameters are re-estimated after significantchanges in heading or at regular time intervals of 20 min,which is the time period for which the sea state can be con-sidered to be stationary [3], [34], [35]. Since the vessel is in a

positioning control mode, the total motion measured can berecorded and detrended to obtain an estimate of the wave-induced motion vector h pw(k ) , or, equivalently, a first-orderhighpass filter can be used [14]. These data can then be usedto estimate the parameters of the wave-induced motionmodel, for which it is convenient to consider the directlyparameterized innovations form [12]

j (k 1 1) 5Aw(u)j (k) 1 Lw(u)e (k) , (46)hpw(k ) 5 Cw(u)j (k ) 1 e(k ) , (47)

where u is the vector of parameters to be estimated, and

e(k) is the vector of innovations. The parameter estimationcan then be formulated as

u 5 arg minu

detaN

k51`(k, u)`(k, u)T (48)

with

e(k, u) 5h pw(k) 2 Cw(u)j (k) , (49)

j

(k 1 1) 5Aw(u)j

(k) 1 Lw(u)e (k, u) , (50)

where hpw(k) is replaced by the estimate h pw(k) obtainedfrom detrending the measured data. Equations (48)(50)comprise a standard prediction error estimation problemwhose solution is related to the maximum likelihood esti-mate of the parameter vector u [11], [12].

Once the parameters of the mode are estimated, thecovariance Q

Pof the innovations can also be estimated

from the sample covarianace of the predictions errors.Then, the Kalman filter can be implemented with the inno-vation wave-frequency model, and thus we can choseQ1 5 Q P. This choice entails no loss of information.

An alternative to the procedure described above consistsof fixing the damping of the filters (27) to a value in the range0.01 to 0.1 as suggested in [13] and estimate only the naturalfrequency and noise covariance [13], [14]. This estimationapproach is summarized in [1], where recursive least squaresis used for parameter estimation. A related approach, also

based on recursive least squares, is given in [3].From the procedures discussed above, all the parame-

ters necessary to implement the Kalman filter for a dynam-ically positioned vessel are obtained. The Kalman filter can

be formulated as follows [16]. The state estimate or correc-tion is given by

x (k) 5x (k) 1K(k ) 3y(k) 2Hx(k ) 4, (51)while the state prediction is given by

x(k 1 1) 5Fx (k) 1Du(k ) . (52)

The Kalman gain K(k) is computed as

K(k) 5 P(k)HT(HP(k)HT1 R) 21, (53)

where P(k

)satisfies the recursion

P(k) 5 (I2K(k)H) 21P(k) , P(k 1 1) 5FP(k)FT1 Q. (54)

The matrices P(k) and P(k) are the conditional covari-ances of the estimation errors x (k) 2x (k) and x(k ) 2x(k ) ,respectively.

The algorithm (51)(54) is a standard Kalman filter.The recursive form (54) may not be the best choice forreal-time implementation since numerical issues may beassociated with matrix inversion and covariance matrix

propagation. These errors could result in covariancematrices that are not positive semidefinite. To address



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these issues, various implementations of the equations areused to propagate and update the estimates and the cova-riance. These implementations, known as square root fil-tering, are algebraically equivalent to (54), but exhibitimproved numerical precision and stability. For detailssee [16] and [17].

The filter (51)(54) gives the state estimate

x (k) 5 3j (k )^, h p(k)^, b p (k)^, n (k)^ 4^. (55)The components h p(k ) and n (k) , which are the low-fre-quency motion components, are used by the controller, andin this way, the wave filtering is achieved. That is, theKalman filter uses a model of the wave-frequency motionthat facilitates estimation of the low- and wave- frequencycomponents of the position and heading measurements.The bias estimate provided by the Kalman filter can beused in the controller to implement integral action unless

the controller has an integrator; see [2] for further detailson dynamic positioning control design.

Example of Kalman Filter Performance

for a Vessel Under Positioning Control

Figure 7 shows simulation data of a Kalman wave filter ona 15-m fishing vessel under positioning control. In this sim-

ulation, the wave filter is switched on 500 s after the vesselis under positioning control. Then, at 600 s the vessel posi-tion is changed 10 m forward. Figure 7(a) shows the mea-sured and wave-filtered surge position. Figure 7(b) showsthe measured and wave-filtered surge velocity. Figure 7(c)shows the force generated by the controller.

During the first 600 s, while the wave filter is switchedoff, the wave-indued motion produces significant controlaction. Once the wave-filter is switched on, the controlaction at wave-frequencies is reduced.

Observer Based on Nonlinear Models

A more advanced implementation of the observer resultsby using the nonlinear vessel model (1), (2), which includesthe quadratic damping forces and current loads. This modelcan give better accuracy, but the resulting observer is com-putationally more intensive, and the parameters of the non-linear model are more difficult to estimate. The filtering

problem for this case can be solved using the extendedKalman filter. An alternative to the extended Kalman filteris to use a passivation design and Lyapunov theory. Thisapproach is demonstrated in [33].

KALMAN FILTER DESIGN FOR SHIP

COURSE-KEEPING AUTOPILOTS

Wave filtering for positioning systems must be imple-mented in three degrees of freedom, surge, sway, and yaw.When designing ship autopilots for automatic heading con-trol, it is only necessary to consider a model and performwave filtering for the yaw degree of freedom [1], [2], [8],[15]. In this sect ion, we discuss the type of models that arenormally used to implement observers for autopilot wavefiltering and how the Kalman filter is used. For a discus-sion on the development of ship autopilots, see HistoricalAspects of Ship Motion-Control Systems Related to Autopi-lots and Dynamic Positioning.

The Nomoto Model for Ship Heading Response

Autopilots are used to correct deviations from a desiredequilibrium heading, and thus a linear model is sufficientfor control design [15]. The response in yaw rate due to a

small deviation in the angle of a control actuator, such asa rudder or the steering nozzle of a water jet, can bederived from (2) by isolating the yaw motion, which isgiven by

(Iz 2 Nr# )r#

2 Nrr 5 Ndd, (56)

where Iz is the moment of inertia in yaw, Nr# , Nr, and Nd arehydrodynamic coefficients, r is the yaw rate, and d is theactuator angle. This model, which is known as the first-orderNomoto model [38], can be written as the transfer function

r(s)d(s)

5 K1 1 Ts. (57)

300 350 400 450 500 550 600 650 700 750 8005

05

10

15

SurgePos

ition(m)

Time (s)

(a)

300 350 400 450 500 550 600 650 700 750 800Time (s)

(b)

300 350 400 450 500 550 600 650 700 750 800Time (s)

(c)

1

0.5

0

0.5

1

SurgeVelocity(m/s)

1

0.5

0

0.5

1 104

SurgeControlForce

(N)

Measured

Filtered

DemandActual

Measured Filtered

FIGURE 7 Simulated data of Kalman-based-wave filter perfor-

mance for a 15-m fishing vessel under positioning control. Thewave filter is switched on at 500 s, and the vessel is moved for-

ward by 10 m at 600 s. (a) shows the measured and wave-filtered

surge position. (b) shows the measured and wave-filtered surgevelocity. (c) shows the surge force generated by the controller.



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The time constant and low-frequency gain are given by

T5Iz 2 Nr#

2Nr, (58)

K 5 2Nd

Nr, (59)

which can be estimated from trials in calm water.Using the motion superposition assumption, as in the case

of positioning control design, we model low-frequency envi-

ronmental disturbances with a bias moment term in the equa-tion of motion. Then, the state-space model can be written as

c#

5 r, (60)

r#

5 21T

r 11m

tN 1 b, (61)

b#

5 0, (62)

where m 5 Iz 2 Nr# and

tN 5 mK

Td 5 Ndd (63)

denotes the control yaw moment.

Autopilot Design

For autopilot control design, it is common to design a pro-portional-integral-derivative controller with feedforwardfrom wind and a smooth time-varying reference signalcd (t) according to

tN(s ) 5 2t wind 1 mar# d 2 1Trdb 2m

aKpc|

1 Kdr|1 Ki3

t

0

c|(t )dt

b, (64)

where tN is the controller yaw moment and tFF is a feedfor-ward term using the reference signal rd 5 c

#

d. The headingand yaw rate errors are denoted by c|5 c 2 cd and r| 5 r 2 rd,respectively. The control gains Kp, Kd, and Ki must be chosensuch that the third-order linear error dynamics

r|#

1 a 1T

1 Kdb r| 1 Kpc| 1 Ki3 t0

c|(t )dt 5 0 (65)

is asymtotically stable. Notice that the control law (64)depends on wind feedforward, where

t wind 512

raV2rwALwLoaCNw (grw) . (66)

The uncertainty in (66) has a low-frequency content, which iscompensated by the integral action of the controller [2]. Therudder command is computed from the control input tN as

d 51

NdtN. (67)

To implement the control law, both c and r are needed.Most ships have only compass measurements c, and thusthe turning rate r must be estimated. In addition, it is neces-sary to perform wave filtering such that the oscillatorywave-induced motions are avoided in the feedback loop.

Models for Course Autopilots Observer Design

As in the case of positioning, we consider the first-order,wave-induced motion as an output disturbance. Hence themeasured yaw angle can be decomposed into

ctot 5 c 1 cw, (68)

where c is the response due to the control action and low-frequency disturbance and where cw represents the first-order wave-induced motion.

To estimate of c and r, and thus wave filtering, themodel used for a Kalman filter design is given by

j#

w 5 cw, (69)

c#

w 5 2v20jw 2 2zv0cw 1 w1, (70)

c#

5 r, (71)

r

#

5 2

1Tr 1

1m (twind 1 tN) 1 b 1 w2, (72)

b#

5 w3, (73)

where v0 and z are the parameters of the filter (27) used torepresent the wave-induced yaw motion, b is the bias term,and w1, w2, and w3 are Gaussian white noises. The mea-surement equation is

y 5 c 1 cw 1 v, (74)

where v represents zero-mean Gaussian white measure-ment noise introduced by the sensor. Notice that neither

the yaw rate r nor the wave states jw and cw are measured.The resulting state-space model is

5

tFF

8

tPID

When a vessel moves in the water, the changes in pressure on the

hull are proportional to the velocities and accelerations

of the vessel relative to the fluid.



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x? 5Ax 1 Bu 1 Ew, (75)y 5Hx 1 v, (76)

where

x 5

3jw, cw, c, r, b

4^, (77)

u 5 twind 1 tN, (78)w 5 3w1, w2, w3 4^, (79)

and

A 5 E 0 1 0 0 02v20 22zv0 0 0 00 0 0 1 00 0 0 2T21 10 0 0 0 0

U , b 5 E 000m21

0

U , (80)E 5 E

0 0 0

1 0 00 0 00 1 00 0 1

U , h^5 30, 1, 1, 0, 0 4. (81)

To implement a Kalman filter for a course autopilot, themodel (75) can be discretized as in the case of dynamic posi-tioning, and the parameters T and m of the response fromthe control forces can be estimated from tests performed incalm water. Then, the parameters v0 and z of the first-order,wave-induced model and the covariance of the driving

noise w1 can be estimated from data collected during theoperation of the vessel [3]. The procedure for the parameterestimation is similar to that discussed for dynamic position-ing. The only significant difference is that the parameters ofthe first-order, wave-induced model must be re-estimatednot only if significant heading changes occur but also if sig-nificant speed changes occur. Changes in the speed of thevessel produce a Doppler effect, and the frequencies of thewaves seen from the vessel change with both the speed anddirection of the waves relative to the vessel [3].

Example Kalman Filter Design

for Course-Keeping AutopilotTo illustrate the performance of Kalman-filter-based wave fil-tering, we consider the case of an autopilot application takenfrom theMarine Systems Simulator (MSS) [40]. This simulationpackage implemented in Matlab-Simulink provides models ofvessels and a library of Simulink blocks for heading autopi-lot control system design and blocks for a Kalman-filter-basedwave filter from heading-only measurements.

The vessel considered is a 160-m mariner class vesselwith a nominal service speed of 15 kn, or 7.7 m/s. Theparameters of a complete and validated nonlinear model ofthe form (1)(2) are given in [1]. From the step tests per-formed on the nonlinear model, a first-order Nomoto model

1 0 ,0 0

0 ,10

0

0

0

FIGURE 9 Performance of a heading autopilot for a mariner cargoship with a wave filter. (a) shows the time series of the desired

heading cd and the actual vessel heading c. (b) shows the rudderangle d. Because only the low-frequency estimates of the heading

angle and rate produced by the Kalman filter are passed to the

controller, the motion of the rudder does not respond to the wavemotion, and thus wave filtering is achieved.

0 50 100 150 200 25010

0

10

20

30

40

Time (s)

(a)

0 50 100 150 200 250Time (s)

(b)

Heading()

5

0

5

10

15

RudderAngle

()

VesselDesired

0 50 100 150 200 2500

10203040

Time (s)

(a)

0 50 100 150 200 250Time (s)

(b)

0 50 100 150 200 250Time (s)

(c)

Heading()

0.20

0.20.40.6

HeadingRate(/s)

42

024

Heading()

LowFrequency Motion KF Estimate

WaveFrequency Motion KF Estimate

LowFrequency Motion KF Estimate

FIGURE 8 Performance of a Kalman filter for heading autopilot fora mariner cargo ship. (a) shows the true low-frequency heading c

and its Kalman filter estimate c . (b) shows the true low-frequencyheading rate r and its Kalman filter estimate r . (c) shows the

wave-frequency component of the heading cw and its Kalmanfilter estimate c w. The Kalman filter, which uses measurements of

the sum of low- and wave-frequency heading, estimates these

states and the rate using a model of the vessel and a model of thewave-induced motion.



14/15


(57) is identified with the parameters K 5 0.185 s21 andT5 107.3 s. Based on the time constant, a sampling periodof 0.5 s is chosen for the implementation of the Kalmanfilter. The standard deviation of the noise of the compasssensor is 0.5. From a record of heading motion while therudder is kept constant, the parameters of the first-order,wave-induced motion model are estimated, namely, z 50.1, v0 5 1.2 rad/s, and the standard deviation of the noisedriving the filter is sw1 5"300 rad/s.

Using the above data, a Kalman filter is designed. Fig-ures 8 and 9 show the performance of the Kalman filter.

Figure 8(a) and (b) shows the true low-frequency headingangle and rate together with the Kalman filter estimates.Figure 8(c) shows the first-order, wave-induced headingangle component and its estimate. Figure 9 shows the per-formance of the control loop. Figure 9(a) shows the desiredand the actual heading angle of the controlled vessel. Figure9(b) depicts the rudder angle. In this figure, we can appreci-ate the effect of the wave filtering since the rudder angle hasno motion at the wave frequency.

CONCLUSIONS

Ships and oil rigs perform operations that require accuratepositioning and heading control. Vessel motion is affected

by environmental forces, which can be separated into low-and wave-frequency forces. The low-frequency forces arecaused by wind, current, and wave forces that depend non-linearly on the wave elevation. The wave-frequency forcesare caused by oscillatory pressure changes that depend lin-early on the wave elevation.

The objective of a positioning and heading control systemis regulation while compensating only for the low-frequencyenvironmental forces. This mode of operation results inenergy savings and prevents actuator wear. For example, in

a 24-h positioning offshore operation, the propulsion systemmust not compensate for every single wave but only for thelow-frequency disturbances. The implementation of suchcontrol systems often requires estimates of velocities frommeasurements of position and heading, while also filteringthe oscillatory components of motion due to the waves. Thistype of filtering, known as wave filtering, is a key aspect inthe motion control of marine surface vessels.

An effective way to address both velocity estimationand wave filtering is by augmenting the vessel model witha model of the oscillatory wave-induced motion and usinga Kalman filter to estimate the states of the various parts of

the model. That is, the model has state variables associatedwith the low-frequency motion and other state variables

associated with wave-frequency motion. Once these statesare estimated by the Kalman filter, only the low-frequencystates are used as feedback signals in the controller.

In this article, we have described the main componentsof a ship motion-control system and two particular motion-control problems that require wave filtering, namely,dynamic positioning and heading autopilot. Then, we dis-cussed the models commonly used for vessel response andshowed how these models are used for Kalman filterdesign. We also briefly discussed parameter and noisecovariance estimation, which are used for filter tuning. To

illustrate the performance, a case study based on numericalsimulations for a ship autopilot was considered.

The material discussed in this article conforms tomodern commercially available ship motion-controlsystems. Most of the vessels operating in the offshoreindustry worldwide use Kalman filters for velocity esti-mation and wave filtering. Thus, the article provides anup-to-date tutorial and overview of Kalman-filter-basedwave filtering.

AUTHOR INFORMATION

Thor I. Fossen received the M.Sc. degree in naval architecturein 1987 from the Norwegian University of Science and Tech-nology (NTNU) and the Ph.D. in control systems from NTNUin 1991. In 19891990, he was a Fulbright Scholar in aerody-namics and flight control at the Department of Aeronauticsand Astronautics at the University of Washington, Seattle. In1993, he was appointed professor of guidance and control atNTNU. He is the author of Guidance and Control of Ocean Ve-hicles andMarine Control Systems and coauthor of New Direc-tions in Nonlinear Observer Design. He has contributed to thedevelopment of several industrial autopilot and dynamicpositioning systems. He is a founder of Marine Cybernetics,

which offers services for hardware-in-the-loop testing of ma-rine control systems. He has also been involved in the designof the SeaLaunch trim and heel correction systems. His workon weather optimal positioning control for marine vessels re-ceived the Automatica Prize Paper Award in 2002.

Tristan Perez ([email protected]) com-pleted his electronics engineering degree at the NationalUniversity of Rosario, Argentina, in 1999 and the Ph.D. incontrol engineering at the University of Newcastle, Austra-lia, in 2003. In 2002, he worked as a naval architect at ADI-Limited Australia, where he evaluated ship motion perfor-mance in waves. In 20032004 he was a research fellow at the

Mechatronics Research Centre, University of Wales, UnitedKingdom, where he worked on control and fault detection

The objective of a positioning and heading control system is regulation while

compensating only for the low-frequency environmental forces.



15/15


of thrusters for underwater vehicles. From 2004 to 2007, hewas a senior researcher at the Centre for Ships and OceanStructures (CeSOS), Trondheim, Norway, and a lecturer inmarine control systems at the Norwegian University of Sci-ence and Technology (NTNU). Since October 2007, he has

been with the Centre for Complex Dynamic Systems and

Control at the University of Newcastle, Australia, where heleads a research program on performance optimization ofmarine control systems. In 2008, he was appointed adjunctassociate professor of ship dynamics at NTNU, Norway.He is the author of Ship Motion Control, a member of theIFAC Technical Committee on Marine Control Systems,and a member of the Royal Institution of Naval Architects,United Kingdom. He provides consultancy services for themarine industry and has collaborated on the developmentof commercial ship autopilots, positioning systems, andride control. He can be contacted at the Centre for ComplexDynamic Systems and Control (CDSC), The University of

Newcastle, Callaghan, NSW 2308, Australia.

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positioning heading kalman

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