Nonlinear maneuvering theory and path-following control

Thor I. Fossen1,2

1Department of Engineering Cybernetics, Norwegian University of Science and Technology, Trondheim2Centre for Ships and Ocean Structures, Norwegian University of Science and Technology, Trondheim

ABSTRACT: Speed and path-following control systems for accurate maneuvering of ships can be designedusing line-of-sight (LOS) guidance principles. The basis for the control system is a nonlinear maneuveringmodel that describes the coupled motions in surge, sway and yaw. In order to follow a predefined path, the shipmust be equipped with minimum two actuators that produce a surge force and a yaw moment. Moreover, onlytwo controls are needed to control an underactuated ship in three degrees of freedom (DOF). Path following isachieved by a geometric assignment based on a LOS projection algorithm for minimization of the cross-trackerror to the path. The LOS algorithm computes heading angle commands to the autopilot system. The desiredspeed along the path is controlled independently of the heading autopilot. The path-following controller ismodel based and a nonlinear hydrodynamic model is used to derive the motion control system. A ship casestudy using a scale model illustrates the path-following capabilities.

1 INTRODUCTION

In this chapter maneuvering theory is used to derivethe equations of motion for a ship. Models for wind,waves and ocean currents are included to obtain a re-alistic model for a ship in a seaway. The maneuveringmodel is the foundation for the design of a speed andpath-following control system.

1.1 Maneuvering theory

The study of a ship moving at constant positive speedU in calm water within the framework of maneuver-ing theory is based on the assumption that the ma-neuvering (hydrodynamic) coefficients are frequencyindependent (no wave excitation). This chapter de-scribes a practical approach which can be used to de-rive a nonlinear maneuvering model for simulationand control of ships in a seaway. Emphasis is placedon using first principles to the derive the maneuver-ing coefficients and thus avoid a nonphysical repre-sentation based on curve fitting of a large number ofhydrodynamic coefficients. The maneuvering modelwill in its simplest representation be linear while thenonlinear representations is derived using the square-law surge resistance model and the cross-flow dragprinciple to represent dissipative forces. This is com-bined with models describing the potential coeffi-cients, kinematics and rigid-body kinetics to obtaina complete model.

1.2 Path-following control

A trajectory describes the motion of a moving objectthrough space as a function of time. The object mightbe a craft, projectile or a satellite, for example. A tra-jectory can be described mathematically either by thegeometry of the path, or as the position of the objectover time. Path following is the task of following apredefined path independent of time–that is, there areno temporal constraints. This means that no restric-tions are placed on the temporal propagation alongthe path. Spatial constraints, however, can be added torepresent obstacles and other positional constraints.

A frequently used method for path-following con-trol is LOS guidance. A LOS vector from the ship tothe next waypoint or a point on the path between twowaypoints can be used for both course and headingcontrol. If the ship is equipped with a heading autopi-lot, the angle between the LOS vector and the pre-described path can be used as setpoint for the head-ing autopilot. This will force the ship to track thepath. The advantages of the LOS path-following con-troller to linear methods, minimizing the deviation toa straight-line, is that it copies the behavior of thehelmsman. This is done by introducing a lookaheaddistance that defines the desired heading angle. It isstraightforward to implement LOS guidance systemssince an existing autopilot system can be used to trackthe computed heading commands. If the LOS guid-

xb

yb

zb

CO CG

CB

Figure 1: The body-fixed coordinate system {b} =(xb, yb, zb) is located on the centerline in the point CO andits rotating about the North-East-Down coordinate system{n} = (xn, yn, zn).

ance law is linearized, it can be shown that LOS guid-ance is equivalent to minimizing the lateral deviationlocally. This is the standard solution for straight-linepath following while LOS guidance, mimicking thebehavior of the helmsman, can be used to track curvedpaths.

2 MANEUVERING MODELSManeuvering theory assumes that the ship is movingin restricted calm water–that is, in sheltered watersor in a harbor. Hence, the maneuvering model is de-rived for a ship moving at positive speed U under azero-frequency wave excitation assumption such thatadded mass and damping can be represented by us-ing hydrodynamic derivatives (constant parameters).Added mass and potential damping are also referredto as potential coefficients.

A linear mass-damper system describing a floatingvessel under proportional-derivative (PD) control canbe formulated in the time-domain by applying the ap-proach of Cummins (1962). Consider,

[m+A(∞)]ξ + [B(∞) +Kd]ξ

+∫ t

0K(t− τ)ξ(τ)dτ +Kpξ = Λ cos(ωt) (1)

whereKp > 0 andKd > 0 are the PD controller gains,m is the mass, A(ω) and B(ω) are the frequency-dependent potential coefficients, ξ is the position andK(t) is a time-varying retardation function given bythe integral:

K(t) =2

π

∫ ∞0

[B(ω)−B(∞)]ω (2)

Under forcing Λ cos(ωt), the potential coefficientsA(ω) and B(ω) will depend on the frequency ω. Thismodel can be used to describe a ship in surge, sway

and yaw where the only restoring forces Kpξ are dueto the control system.

The frequency-domain representation of (1) is (seeNewman, 1977; Faltinsen 1990):(

−ω2[m+A(ω)]− jω[B(ω) +Kd] +Kp

)ξ(jω)

= Λ cos(jω) (3)

where ξ(jω) is a complex variable:

ξ(jω) = ξ cos(jω + ε) ⇒ ξ(jω) = ξ exp(jε) (4)

This approach is common in seakeeping analysis andit follows the well celebrated methods of Cummins(1962) and Ogilvie (1964).

Naval architects often write the seakeeping model(1) as a pseudo-differential equation:

[m+A(ω)]ξ + [B(ω) +Kd]ξ +Kpξ = Λ cos(ωt)(5)

mixing time and frequency. Unfortunately this isdeeply rooted in the literature of hydrodynamics eventhough it is not correct to mix time and frequency inone single equation. Consequently, it is recommendedto use use the time- and frequency-domain represen-tations (1) and (3).

The integral:

µ =∫ t

0K(t− τ)ξ(τ)dτ (6)

is recognized as the fluid-memory effects. The retar-dation functions (2) can be fitted to ordinary differ-ential equations (ODEs) but more than 100 ODEsare needed to describe the fluid-memory effects of aship in a seaway (see Perez and Fossen, 2007, 2008a,2008b). However, identification of the fluid-memorytransfer functions in the frequency domain can re-duce the number of parameters significantly (Perezand Fossen, 2011). At higher speeds, in a maneuver-ing situation, fluid-memory effects will be dominatedby nonlinear lift and drag forces as well as other vis-cous effects. This motivates a different approach forship maneuvering at higher speed.

In maneuvering theory, it is common to remove thefrequency-dependent matrices under a zero-frequencyassumption. However, the zero-frequency assumptionis only valid for surge, sway and yaw since the naturalperiod of a PD controlled ship will be in the range of100–150 s. For 150 s the natural frequency is:

ωn =2π

T≈ 0.04 rad/s (7)

which clearly gives support for the zero-frequency as-sumption. The natural frequencies in heave, roll andpitch are much higher so it is not straightforward to

remove the frequency dependencies in these modes.For instance, a ship with roll period of 10 s will havea natural frequency of 0.63 rad/s which clearly vio-lates the zero frequency assumption. This means thatthe potential coefficients should be evaluated at a fre-quency of 0.63 rad/s in roll if a pure rolling motionis considered. As a consequence of this, it is commonto formulate the ship maneuvering model as a cou-pled surge-sway-yaw model and thus neglect heave,roll and pitch motions (Fossen 1994, 2011):

m[u− vr− xgr2 − ygr

]= X (8)

m[v + ur− ygr2 + xgr

]= Y (9)

Iz r+m [xg(v + ur)− yg(u− vr)] = N (10)

whereX,Y andN denote the external forces and mo-ment while (xg, yg, zg) describes the location of thecenter of gravity (CG).

2.1 Kinematics and rigid-body kineticsThe rigid-body kinetics (8)–(10) can be expressed invector form according to (Fossen 1994, 2011):

η = J(ν)ν (11)

MRBν +CRB(ν)ν = τRB (12)where MRB =M> > 0 is the rigid-body inertia ma-trix, CRB(ν) = −C>RB(ν) is a matrix of rigid-bodyCoriolis and centripetal forces and τRB is a vectorof generalized forces. The Coriolis and centripetalterm is due to the rotation of the body-fixed referenceframe around the inertial reference frame.

Since the horizontal motion of a ship is by the mo-tion components in surge, sway and yaw, the statevectors are chosen as:

η = [N,E,ψ]> (13)

ν = [u, v, r]> (14)

where (N,E) represents the North-East positions, ψis the heading angle, (u, v) are the linear velocities insurge and sway, respectively and r is the yaw rate.This implies that the dynamics associated with themotion in heave, roll and pitch are neglected–that is,w = p = q = 0. For the horizontal motion of a vesselthe kinematic equations of motion can be describedby the principal rotation about the z-axis:

J(η) :=R(ψ) =

cos (ψ) − sin (ψ) 0sin (ψ) cos (ψ) 0

0 0 1

(15)

It is also common to assume that the vessel has ho-mogeneous mass distribution and xz-plane symmetrysuch that:

Ixy = Iyz = 0 (16)

+

yb

zb

North

East

Vc

gc

bc

yxb

Figure 2: Ocean current speed Vc, direction βc and angle ofattack γc relative to the bow.

Let the {b}-frame coordinate origin be set in the cen-ter line of the ship at the point CO such that yg = 0(See Figure 1). Under the previously stated assump-tions, the matrices associated with the rigid-body ki-netics become (Fossen 1994, 2011):

MRB =

m 0 00 m mxg0 mxg Iz

(17)

CRB(ν) =

0 0 −m(xgr+ v)0 0 mu

m(xgr+ v) −mu 0

(18)

An alternative skew-symmetric representation ofCRB(ν) avoiding the use of the linear velocities (u, v)is:

CRB(ν) =

0 −mr −mxgrmr 0 0mxgr 0 0

(19)

Notice that surge is decoupled from sway and yaw inMRB due to symmetry considerations of the systeminertia matrix.

The generalized force acting on the ship is:τRB = τhyd + τwind + τwave + τcontrol (20)

where the subscripts stand for:• hyd: hydrodynamic added mass, potential damp-

ing due to wave radiation and viscous dampingincluding the effect of ocean currents

• wind: wind forces represented by wind currentcoefficients

• wave: wave forces (1st- and 2nd-order theory)

• control: control and propulsion forces

2.2 Hydrodynamic forcesThe generalized hydrodynamic force can be written:

τhyd = −MAνr −CA(νr)νr −D(νr)νr (21)

where MA is the hydrodynamic added mass matrix.Since the body-fixed frame {b} is rotating about theNorth-East-Down inertial reference frame {n}, hy-drodynamic added mass contributes with a Coriolisand centripetal matrix CA(ν) = −CA(ν)>.

For conventional ships, it is common to decouplethe surge motion from the sway-yaw motions. Hydro-dynamic added mass is approximated under a zero-frequency assumption such that:

MA =

−Xu 0 00 −Yv −Yr0 −Yr −Nr

(22)

≈

A11(0) 0 00 A22(0) A26(0)0 A62(0) A66(0)

(23)

The corresponding hydrodynamic added mass Corio-lis and centripetal matrix is given by:

CA(ν) =

0 0 Yvv + Yrr0 0 −Xuu

−Yvv− Yrr Xuu 0

(24)

Care must be taken when representing the dissipativeforcesD(νr)νr in (21) to avoid double counting. Forinstance, the potential term Yvvr due to the productCA(νr)νr can sometimes be included in the dampingcoefficient Xvrvr. Hence, it is important to know thephysical interpretation of each of the coefficients. Asa consequence of this, the damping matrix D(νr) isdefined to include the following effects (excludes thepotential terms in CA(νr)):

D(νr)νr :=Dνr + d(νr) (25)

where Dνr represents linear potential and viscousdamping and d(νr) is a vector of nonlinear dampingforces.

The generalized hydrodynamic force τhyd can berepresented by linear or nonlinear model techniques:

Linearized models: In the linear 6 DOF case therewill be a total of 36 mass and 36 dampingelements proportional to velocity and acceler-ation, respectively. In addition to this, therewill be restoring forces, propulsion forces andenvironmental forces. If the generalized forceτhyd is written in component form using theSNAME (1950) notation, the linear added mass

and damping forces become:

X1 = −Xuu−Xvv−Xww−Xpp−Xqq−Xrr

−Xuu−Xvv−Xww−Xpp−Xq q−Xrr

...

N1 = −Nuu−Nvv−Nww−Npp−Nqq−Nrr

−Nuu−Nvv−Nww−Npp−Nq q−Nrr

where Xu,Xv, ...,Nr are the linear damping co-efficients and Xu,Xv, ...,Nr represent hydrody-namic added mass.

Nonlinear models: Application of nonlinear theoryimplies that many elements must be included inaddition to the 36 linear elements. This is usuallydone by one of the following methods:

– Truncated Taylor-series expansions usingodd terms (1st- and 3rd-order) which arefitted to experimental data, for instance(Abkowitz 1964):

X1 = Xuu+Xuu+Xuuuu3

+ Xvv +Xvv +Xvvvv3 + · · · (26)

...

N1 = Nuu+Nuu+Nuuuu3

+ Nvv +Nvv +Nvvvv3 + · · · (27)

In this approach added mass is assumedto be linear and damping is modeled bya 3rd-order odd function. Alternatively,2nd-order modulus terms can be used(Fedyaevsky and Sobolev 1963)::

X1 = Xuu+Xuu+X|u|u|u|u

+ Xvv +Xvv +X|v|v|v|v + · · · (28)

...

N1 = Nuu+Nuu+N|u|u|u|u

+ Nvv +Nvv +N|v|v|v|v + · · · (29)

This is motivated by the square-law damp-ing terms in fluid dynamics and aerody-namics. When applying Taylor-series ex-pansions in model-based control design,

the system (12) becomes relatively com-plicated due to the large number of hy-drodynamic coefficients on the right-handside needed to represent the hydrody-namic forces. This approach is quite com-mon when deriving maneuvering modelsand many of the coefficients are diffi-cult to determine with sufficient accuracysince the model can be overparametrized.Taylor-series expansions are frequentlyused in commercial planar motion mecha-nism (PMM) tests where the purpose is toderive the maneuvering coefficients experi-mentally.

– First principles where hydrodynamic effectssuch as lift and drag are modeled usingwell established models. This results inphysically sound Lagrangian models thatpreserve energy properties. Models basedon first principles usually require a muchsmaller number of parameters than mod-els based on 3rd-order Taylor-series expan-sions.

2.3 Surge resistanceIn surge, the viscous damping for ships may be mod-eled as (Lewis 1989):

X = −1

2ρS(1 + k)Cf (ur)|ur|ur (30)

where ρ is the density of water and S is the wettedsurface of the hull. The from factor k is used for vis-cous correction. For ships in transit k is typical 0.1whereas this value is much higher in DP, typicallyk = 0.25 (Hoerner 1965). The friction coefficient ismodeled as:

Cf (ur) = CF +CR (31)

CF =0.075

(log10Rn − 2)2(32)

where CF is the flat plate friction from the ITTC(1957) line and CR represents residual friction dueto hull roughness, pressure resistance, wave makingresistance and wave breaking resistance.

The relative surge velocity:

ur = u− uc= u− Vc cos(βc − ψ) (33)

is expressed in {b} (see Figure 2).The resistance coefficient CF depends on the

Reynolds number:

Rn =urLppν

(34)

where ν = 1 · 10−6 m/s2 is the kinematic viscosity at20oC. For small values of log10(Rn− 2) in the expres-sion for CF , a minimum value of Rn should be usedin order to avoid that CF blows up at low speed. Forships, a typically value is Rn,min = 106 which lim-its the friction coefficient to CF,max < 0.05 at lowerspeeds (see Figure 3). The damping model in surgecan also be written as:

X = X|u|u|ur|ur (35)

X|u|u| = −1

2ρS(1 + k)Cf < 0 (36)

For low-speed maneuvering, this formula gives to lit-tle damping compared to the quadratic drag formula:

X|u|u| =1

2ρAxCx(γc) (37)

where CX(γc) > 0 is the current coefficient as a func-tion of angle of attack γc (see Figure 2) and Ax is thefrontal project area. The current coefficients are usu-ally found from experiments using a model ship upto 1.0 m/s currents. The resistance and current coeffi-cients in (36) and (37) are related by:

CX(γc) =S

AxCf (ur) (38)

One way to obtain sufficient damping at low speed isto modify the resistance curve Cf (ur) according to:

Cnewf (ur) = Cf (u

maxr )

+(AxSCX(γr)−C(umax

r ))

exp(−αu2r) (39)

where α > 0 (typically 1.0) is a user specifiedshaping parameter. The maximum friction coefficientCf (u

maxr ) is computed for maximum relative veloc-

ity umaxr . The modified resistance curve Cnew

f (ur) isplotted together with Cf (ur) in Figure 3. Notice thatthe resistance is increased at lower velocities to matchthe experimental current coefficient CX . The secondplot shows the current coefficient CX at zero speedtogether with Cnew

X = 1/2ρAxCnewf . The effect of the

current coefficient vanishes at higher speeds and theITTC resistance line is captured thanks to the expo-nentially decaying weight exp(−αu2

r).

2.4 Cross-flow drag principleFor relative current angles |βc − ψ| � 0, where βcis the current direction, the cross-flow drag princi-ple may be applied to calculate the nonlinear dampingforce in sway and the yaw moment (Faltinsen 1990):

Y = −1

2ρ∫ Lpp

2

−Lpp2

T (x)C2Dd (x)|vt(x)|vt(x)dx (40)

0 1 2 3 4 5 6 7 80

0.005

0.01

u (m/s)

0 1 2 3 4 5 6 7 80

0.05

0.1

0.15

0.2

u (m/s)

0 1 2 3 4 5 6 7 80

50

100

150

u (m/s)

Modified resistance coefficient Cfnew

ITTC resistance coefficient Cf

Current coefficient Cfnew = (Ax/S)*Cf

new

Current coefficient Cx

Modified drag forceITTC drag force

Cf

Cfnew

Cfnew

X(kN)

Figure 3: Modified resistance curve Cnewf (ur) and ITTC friction curve Cf (ur) as a function of ur when CR = 0. The

friction curve is modified at low speed to match the experimental current coefficient CX such that Cnewf (0) = (Ax/S)CX .

The example uses CX = 0.16 and α = 1.0 (tunable parameter).

N = −1

2ρ∫ Lpp

2

−Lpp2

T (x)C2Dd (x)x|vt(x)|vt(x)dx(41)

where C2Dd (x) is the 2-D drag coefficient, T (x) is the

draft, andvt(x) = vr + xr (42)

In order to include the effect of ocean currents, therelative sway velocity

vr = v− vc= v− Vc sin(βc − ψ) (43)

is used where Vc and β are the current speed and direc-tion in {n}. This is a strip theory approach where eachhull section contributes to the integral. Drag coeffi-cients for different hull forms may be found in Hooft(1994). A constant 2-D current coefficient can alsobe estimated using Hoerner’s curve (Hoerner 1965)as shown in Figure 4.

The 2-D drag coefficients C2Dd can be computed

as a function of beam B and length T using Ho-erner’s curve. A 3-D representation of (40)–(41) elim-inating the integrals can be found by curve fittingformulas (40) and (41) to 2nd-order modulus terms

(Fedyaevsky and Sobolev 1963):

Y = Y|v|v|vr|vr + Y|v|r|vr|r+ Yv|r|vr|r|+ Y|r|r|r|r(44)

N = N|v|v|vr|vr +N|v|r|vr|r+Nv|r|vr|r|+N|r|r|r|r(45)

where Y|v|v,Y|v|r,Yv|r|,Y|r|r|r|r|,N|v|v,N|v|r,Nv|r|, andN|r|r are maneuvering coefficients defined accordingto the SNAME notation.

2.5 Nonlinear maneuvering model based on surgeresistance and cross-flow drag

It is possible to use the surge resistance and cross-flowdrag models in Sections 2.3–2.4 to derive a nonlinearmaneuvering model that include viscous effects. Con-sider the rigid-body kinetics:

MRBν +CRB(ν)ν = τRB (46)

whereτRB = τhyd + τwind + τwave + τ (47)

If (19) is used to represent the Coriolis and centripetalforces instead of (18), the following property is satis-fied:

MRBν +CRB(ν)ν ≡MRBνr +CRB(νr)νr (48)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50.5

0.7

0.9

1.1

1.3

1.5

1.7

1.9

2

Cd2D

B/2T

Figure 4: 2-D cross-flow drag coefficientC2Dd (x) as a func-

tion of B/2T (Hoerner 1965).

Hence, combining (46)–(48) with the kinematic ex-pression (12) and the hydrodynamic forces (21), thefollowing model is obtained:

η =R(ψ)ν (49)

Mνr +N(νr)νr = τ + τwind + τwave (50)where the matrixN (νr) is defined according to:

N(νr) := CRB(νr) +CA(νr) +D(νr) (51)

Notice that added mass Coriolis and centripetal termsand hydrodynamic damping terms are collected intothe same matrix in order to avoid double counting.This is indeed convenient since it is difficult to distin-guish terms in CA(νr) with similar terms in D(νr).Hence, only the sum of these terms are used in themodel in order to avoid overparametrization.

The nonlinear matrix N (νr) in (51) can also berepresented by:

N (νr)νr = [CRB(νr) +CA(νr) +D]νr + d(νr)(52)

whereD is a constant linear damping matrix:

D =

−Xu 0 00 −Yv −Yr0 −Nr −Nr

(53)

of potential and viscous skin friction damping terms.In this formulation, the nonlinear vector field d(νr)can be modeled by using formulas for surge resistanceand cross-flow drag:

d(νr) =

12ρS(1 + k)Cnew

f (ur)|ur|ur12ρ∫ Lpp/2−Lpp/2

T (x)C2Dd (x)|vt(x)|vt(x)dx

12ρ∫ Lpp/2−Lpp/2

T (x)C2Dd (x)x|vt(x)|vt(x)dx

(54)

The linear damping termD in (52) is important forlow-speed maneuvering and stationkeeping (dynamicpositioning) while the nonlinear term d(νr) domi-nates at higher speeds. Linear damping is necessaryin order for the velocity to converge exponentially tozero.

2.6 Nonlinear maneuvering model based on2nd-order modulus functions

The idea of using 2nd-order modulus functions to de-scribe the nonlinear terms in N(νr) dates back to(Fedyaevsky and Sobolev 1963). A simplified formof Norrbin’s nonlinear model (Norrbin 1970) whichretains the most important terms for steering andpropulsion loss assignment has been proposed byBlanke (1981).. This model corresponds to fitting thecross-flow drag integrals (40) and (41) to 2nd-ordermodulus functions according to1:

N (νr)νr = CRB(νr)νr +CA(νr)νr +D(νr)νr(55)

which can be expanded as:

N (νr)νr =

−Y vvrr+ Y rr2

Xuurr(Y v−X u)urvr+Y rurr

+

−X |u|u |ur|ur−Y |v|v|vr|vr−Y |v|r|vr|r− Y v|r|vr|r| − Y |r|r|r|r|−N |v|v|vr|vr−N |v|r|vr|r−Nv|r|vr|r| −N |r|r|r|r|

From this expression it is seen that:

CA(ν) =

0 0 Yvvr + Yrr0 0 −Xuur

−Yvvr − Yrr Xuur 0

(56)

D(νr) =

−X |u|u |ur|0 −Y |v|v |vr|−Y |r|v |r|0 −N |v|v |vr|−N |r|v |r|

0−Y |v|r |vr|−Y |r|r |r|−N |v|r |vr|−N |r|r |r|

(57)

For large ships |r| r and |r|v are small. For this caseBlanke (1981) suggests to simplify (57) according to:

Dn(νr) =

−X |u|u |ur| 0 00 −Y |v|v |vr| −Y |v|r |vr|0 −N |v|v |vr| −N |v|r |vr|

(58)

1The CA-terms can also be denoted as Xvrvrr, Xrrr2,

Yururr, Nuvurvr, and Nururr. If these terms are experimen-tally obtained, viscous effects will be included in addition to thepotential coefficients Yv,Xu and Yr.

2.7 Nonlinear maneuvering model based on oddfunctions

So far, we have discussed nonlinear maneuveringmodels based on first principles such as surge resis-tance (2.3) and cross-flow drag (section 2.4) and itsapproximations to 2nd-order modulus functions (Sec-tion 2.6).

In many cases a more pragmatic approach is usedfor curve fitting of experimental data (Clarke 2003).This is typically done by using Taylor series of first-and third-order terms (Abkowitz 1964) to describe thenonlinear terms inN(νr).

Consider the nonlinear rigid-body kinetics:

MRBν +CRB(ν)ν = τRB (59)

withτRB = [X(x), Y (x),N(x)]> (60)

andx = [u, v, r, u, v, r, δ]> (61)

where δ is the rudder angle. Based on these equations,Abkowitz (1964)proposed a 3rd-order truncated Tay-lor series expansion of the functions X(x), Y (x) andN(x) at:

xo = [U,0,0,0,0,0,0]> (62)

This gives:

X(x) ≈ X(x0)+n∑i=1

(∂X(x)

∂xi

∣∣∣∣∣x0

∆xi

+1

2

∂2X(x)

(∂xi)2

∣∣∣∣∣x0

∆x2i +

1

6

∂3X(x)

(∂xi)3

∣∣∣∣∣x0

∆x3i

)

Y (x) ≈ Y (x0)+n∑i=1

(∂Y (x)

∂xi

∣∣∣∣∣x0

∆xi

+1

2

∂2Y (x)

(∂xi)2

∣∣∣∣∣x0

∆x2i +

1

6

∂3Y (x)

(∂xi)3

∣∣∣∣∣x0

∆x3i

)

N(x) ≈ Z(x0)+n∑i=1

(∂N(x)

∂xi

∣∣∣∣∣x0

∆xi

+1

2

∂2N(x)

(∂xi)2

∣∣∣∣∣x0

∆x2i +

1

6

∂3N(x)

(∂xi)3

∣∣∣∣∣x0

∆x3i

)

where ∆x = x− x0 = [∆x1,∆x2, ...∆xn]>. Unfor-tunately, a 3rd-order Taylor series expansion resultsin a large number of terms. By applying some phys-ical insight, the complexity of these expressions canbe reduced.

Abkowitz (1964) makes the following assump-tions:

1. Most ship maneuvers can be described with a3rd-order truncated Taylor expansion about thesteady state condition u = u0.

2. Only 1st-order acceleration terms are consid-ered.

3. Standard starboard-port symmetry simplifica-tions except terms describing the constant forceand moment arising from single-screw pro-pellers.

4. The coupling between the acceleration and ve-locity terms is negligible.

Simulations of standard ship maneuvers show thatthese assumptions are quite good. Applying these as-sumptions to the expressions X(x), Y (x) and N(x)yields:

X = X∗ +Xuu+Xu∆u+Xuu∆u2 +Xuuu∆u3

+ Xvvv2 +Xrrr

2 +Xδδδ2 +Xrvδrvδ+Xrδrδ

+ Xvδvδ+Xvvuv2∆u+Xrrur

2∆u+Xδδuδ2∆u

+ Xrvurvu+Xrδurδ∆u+Xvδuvδ∆u

Y = Y ∗ + Yu∆u+ Yuu∆u2 + Yrr+ Yvv+ Yrr+ Yvv

+ Yδδ + Yrrrr3 + Yvvvv

3 + Yδδδδ3 + Yrrδr

2δ

+ Yδδrδ2r+ Yrrvr

2v+ Yvvrv2r+ Yδδvδ

2v+ Yvvδv2δ

+ Yδvrδvr+ Yvuv∆u+ Yvuuv∆u2 + Yrur∆u

+ Yruur∆u2 + Yδuδ∆u+ Yδuuδ∆u

2

N = N∗ +Nu∆u+Nuu∆u2 +Nrr+Nvv+Nrr

+ Nvv+Nδδ +Nrrrr3 +Nvvvv

3 +Nδδδδ3

+ Nrrδr2δ +Nδδrδ

2r+Nrrvr2v+Nvvrv

2r

+ Nδδvδ2v+Nvvδv

2δ +Nδvrδvr+Nvuv∆u

+ Nvuuv∆u2 +Nrur∆u+Nruur∆u2 +Nδuδ∆u

+ Nδuuδ∆u2 (63)

The hydrodynamic derivatives are defined using thenotation:

F ∗ := F (x0), Fxi :=∂F (x)

∂xi

∣∣∣∣∣x0

Fxixj :=1

2

∂2F (x)

∂xi∂xj

∣∣∣∣∣x0

, Fxixjxk :=1

6

∂3F (x)

∂xi∂xj∂xk

∣∣∣∣∣x0

where F ∈ {X,Y,N}.

2.8 PMM modelsThe hydrodynamic coefficients can be experimentallydetermined by using a planar-motion-mechanism(PMM) system which is a device for experimentallydetermining the hydrodynamic derivatives required inthe equations of motion. This includes coefficientsusually classified into the three categories of static-stability, rotary-stability and acceleration derivatives.

The PMM device is capable of oscillating a shipor submarine model while it is being towed in a test-ing tank. The forces are measured on the scale modeland fitted to odd functions based on Taylor-series ex-pansions. The resulting model is usually referred to asthe PMM-model and this model is scaled up to a fullscale ship by using Froude number similarity. Thisensures that the ratio between the inertial and grav-itational forces is kept constant.

2.9 Linearized maneuvering modelFor marine craft moving at constant or at least slowlyvarying forward speed:

U =√u2 + v2 ≈ u (64)

the nonlinear maneuvering models can be decoupledin a forward speed (surge) model and a sway-yaw sub-system for maneuvering.

• Forward speed model (surge subsystem).Starboard-port symmetry implies that surge isdecoupled from sway and yaw. Hence, the surgeequation can be written as:

(m−Xu)u−Xuu−X|u|u |u|u = τ1 (65)

where τ1 is the sum of control forces in surge.

• Linearized maneuvering model (sway-yawsubsystem). The linearized maneuvering modelknown as the potential theory representation canbe written; see Fossen (1994, 2011) and Clarkeand Horn (1996):

Mν +N(uo)ν = bδ (66)

where ν = [v, r]> and δ is the rudder angle isbased on the assumptions that the cruise speed:

u = uo ≈ constant (67)

and that vr and r are small. In this model, theocean current forces are treated as a linear termN(uo)[vc,0]>. The potential theory representa-tion is obtained by extracting the 2nd and 3rdrows in CRB(ν) and CA(ν) with u = uo, result-ing in:

C(ν)ν ≈[

(m−Xu)uor(m− Yv)uov + (mxg − Yr)uor

−(m−Xu)uov

]

=

[0 (m−X u)uo

(X u−Y v)uo (mxg−Y r)uo

] [vr

]

Linear damping takes the following form:

D =[ −Yv −Yr−Nv −Nr

](68)

Assuming that the ship is controlled by a singlerudder:

τ = bδ

=[ −Y δ

−N δ

]δ (69)

and that N (uo) contains the linearized termsfrom C(ν) and the linear damper D, finallygives:

M =[m− Y v mxg−Y r

mxg−Y r Iz−N r

](70)

N (uo) =[ −Y v

(X u−Y v)uo−N v

(m−X u)uo−Y r

(mxg−Y r)uo−N r

](71)

b =[ −Y δ

−N δ

](72)

Comment: Davidson and Schiff (1946) assumed thatthe hydrodynamic forces τRB are linear in δ, ν and ν(linear strip theory) such that:

τRB = −[YδNδ

]︸ ︷︷ ︸

b

δ +[Yv YrNv Nr

]︸ ︷︷ ︸

MA

ν +[Yv YrNv Nr

]︸ ︷︷ ︸

D

νr

(73)This gives:

N(uo) =[ −Y v muo−Y r

−N v mxguo−N r

](74)

Notice that the Munk moment (Xu − Yv)uov is miss-ing in the yaw equation–that is, element N21 in (74)when compared to (71). This is a destabilizing mo-ment known from aerodynamics which tries to turn

the vessel (Faltinsen 1990 [pp. 188–189]). Also noticethat the less important terms Xuuor and Yruor areremoved from N when compared to (71). All miss-ing terms terms are due to theCA(ν) matrix which isomitted in the linear expression (73). Consequently, itis recommended to use (71) which includes the termsfrom the CA(νr) matrix.

3 SPEED AND PATH-FOLLOWING CONTROLFor fully actuated vessels, trajectory tracking in surge,sway and yaw (3 DOF) is easily achieved since inde-pendent control forces and moments are simultane-ously available in all degrees of freedom. For slowspeed, this is referred to as dynamic positioning (DP)where the vessel is controlled by means of tunnelthrusters, azimuths and main propellers. Conventionalcraft, on the other hand, are usually equipped withone or two main propellers for forward speed controland rudders for turning control. The minimum con-figuration for waypoint tracking control is one mainpropeller and a single rudder. This means that onlytwo controls are available, thus rendering the ship un-deractuated for the task of 3 DOF trajectory-trackingcontrol.

Conventional waypoint guidance systems are usu-ally designed by reducing the output space from3 DOF position and heading to 2 DOF heading andsurge (Healey and Marco 1992). In its simplest formthis involves the use of a classical autopilot systemwhere the commanded yaw angle ψd is generatedsuch that the cross-track error is minimized. A path-following control system is usually designed such thatthe ship moves forward with reference speed ud at thesame time as the cross-track error to the path is min-imized. As a result, ψd and ud are tracked using onlytwo controls.

This section presents a maneuvering controller in-volving a LOS guidance system and a nonlinear feed-back tracking controller (Fossen et al. 2003). The de-sired output is reduced from (xd, yd, ψd) to ψd and udusing a LOS projection algorithm. The heading track-ing task ψ(t) → ψd(t) is then achieved using onlyone control (normally the rudder), while tracking ofthe speed assignment ud is performed by the remain-ing control (the main propeller). Since we are dealingwith segments of straight lines, the LOS projection al-gorithm will guarantee that the task of path followingis satisfied.

First, a LOS guidance procedure is derived. This in-cludes a projection algorithm and a waypoint switch-ing algorithm. To avoid large bumps in ψd whenswitching, and to provide the necessary derivativesof ψd to the controller, the commanded LOS head-ing is fed through a reference model. Second, a non-linear 2 DOF tracking controller is derived using thebackstepping technique. Three stabilizing functions

α = [α1, α2, α3]> are defined where α1 and α3 arespecified to satisfy the tracking objectives in the con-trolled surge and yaw modes. The stabilizing functionα2 in the uncontrolled sway mode is left as a free de-sign variable. By assigning dynamics to α2, the result-ing controller becomes a dynamic feedback controllerso that α2(t)→ v(t) (sway velocity) during path fol-lowing. This idea was first proposed by Fossen et al.(2003) and it adds to the extensive theory of backstep-ping. The presented design technique results in a ro-bust controller for underactuated ships since integralaction can be implemented for both path-followingand speed control.

3.1 Problem StatementThe problem statement is stated as a maneuveringproblem with the following two objectives (Skjetneet al. 2004):

LOS geometric task: Force the vessel positionp = [x, y]> to converge to a desired path byforcing the course angle χ to converge to:

χd = atan2 (ylos − y,xlos − x) (75)

where the LOS position plos = [xlos, ylos]> is the

point along the path which the vessel should bepointed at and atan2(y,x) is the four-quadrantarctangent of the real parts of the elements x andy, satisfying:

−π ≤ atan2(y,x) ≤ π (76)

Notice that ψ = χ− β and ψd = χd − β implythat ψ = χ when designing the controller.

Dynamic task: Force the speed u to converge to adesired speed assignment ud such that:

limt→∞

[u(t)− ud(t)] = 0 (77)

where ud is the desired speed composed alongthe body-fixed x-axis.

A conventional trajectory-tracking control systemis usually implemented using a standard PID autopilotin series with a LOS algorithm. Hence, a state-of-the-art autopilot system can be modified to take the LOSreference angle as input (see Figure 5). This adds flex-ibility since the default commercial autopilot systemof the ship can be used together with the LOS guid-ance system. The speed can be adjusted manually bythe captain or automatically using the speed profileof the path. A model-based nonlinear controller thatsolves the control objective is derived next.

North-Eastpositions

way-points

controlsystem

controlallocation

observer andwave filter

windfeedforward

windforces

LOSalgorithm

autopilot

Yaw rateand angle

Figure 5: Existing heading autopilot augmented with a LOS path-following guidance system for generation of desiredheading angles ψd.

3.2 Model for Speed and Path-Following ControllerConsider a ship equipped with one main propeller anda rudder. The control forces and moment in 3 DOFcan be written:

τ =

(1− t)TYδ δNδ δ

(78)

where t is the thrust deduction number, T is the thrustand δ is the rudder angle. Consider the 3 DOF nonlin-ear maneuvering model (50) in the following form:

Mν +N (ν)ν =

τ1

Yδ δτ3

(79)

where τ1 and τ3 are the control force and moment insurge and yaw while sway is left uncontrolled. No-tice that rudder motion affects the sway equation butit will not be used to actively control sway. The con-troller computes τ1 and τ3 which can be allocated tothrust and rudder angle commands using (78). More-over,

T =τ1

1− t(80)

δ =τ3

Nδ

(81)

This can easily be modified to cover other actuatorconfigurations using control allocation methods (Fos-sen, 2011).

The matricesM andN are taken as:

M =

m11 0 00 m22 m23

0 m32 m33

=

m−X u 0 00 m− Y v mxg−Y r

0 mxg−N v Iz−N r

N (ν) =

n11 0 00 n22 n23

0 n32 n33

=

−Xu 0 00 −Y v mu− Y r

0 −N v mxgu−N r

+

−X |u|u |u| 0 00 −Y |v|v |v| −Y |v|r |v|0 −N |v|v |v| −N |v|r |v|

3.3 Backstepping designThe design is based on the model (79) where M =M> > 0 and M = 0. Define the error signals z1 ∈[0,2π] and z2 ∈ <3 according to:

z1 := χ− χd = ψ− ψd (82)

z2 := [z2,1, z2,2, z2,3]> = ν −α (83)

where χd and its derivatives are provided by properfiltering of the LOS angle, ud ∈ L∞ is the desiredspeed and α = [α1, α2, α3]> ∈ <3 is a vector of stabi-lizing functions to be specified later. Next, let

h = [0,0,1]> (84)

such that:

z1 = r− rd = h>ν − rd= α3 +h>z2 − rd (85)

where rd = ψd and

Mz2 = Mν −Mα

= τ −Nν −Mα (86)

Consider the control Lyapunov function:

V =1

2z2

1 +1

2z>2 Mz2, M =M> > 0 (87)

Differentiating V along the trajectories of z1 and z2,yields:

V = z1z1 + z>2 Mz2

= z1(α3 +h>z2 − rd)

+ z>2 (τ −Nν −Mα) (88)

Choosing the virtual control α3 as:

α3 := −cz1 + rd (89)

while α1 and α2 are yet to be defined, gives:

V = −cz21 + z1h

>z2 + z>2 (τ −Nν −Mα)

= −cz21 + z>2 (hz1 + τ −Nν −Mα) (90)

Suppose we can assign:

τ =

τ1

Yδ δτ3

:=Mα+Nν −Kz2 −hz1 (91)

whereK = diag{k1, k2, k3} > 0. This results in:

V = −cz21 − z>2 Kz2 < 0, ∀z1 6= 0,z2 6= 0 (92)

and by standard Lyapunov arguments, this guaranteesthat (z1,z2) is bounded and converges to zero.

3.4 Resulting Speed and Path-Following ControllerNotice from (91) that we can only prescribe values forτ1 and τ3–that is,

τ1 = m11α1 + n11u− k1(u− α1) (93)

τ3 = m32α2 +m33α3

+ n32v + n33r− k3(r− α3)− z1 (94)

Choosing α1 = ud solves the dynamic task since theclosed-loop surge dynamics become:

m11 (u− ud) + k1 (u− ud) = 0 (95)

which clearly is globally exponentially stable (GES).

Sway and yaw are coupled so this analysis is morecomplicated. First we notice that the sway dynamicsgiven by (91) results in a dynamic equality constraint:

m22α2 +m23α3 + n22v+ n23r− k2(v−α2) =YδNδ

τ3

(96)affected by the control input τ3. Substituting (94) into(96), gives:(m22 −

YδNδ

m32

)α2 +

(m23 −

YδNδ

m33

)α3

+(n22 −

YδNδ

n32

)v +

(n23 −

YδNδ

n33

)r

−k2(v− α2) +YδNδ

(k3(r− α3) + z1) = 0 (97)

Application of α3 = c2z1 − cz2,3 + rd, α3 = −cz1 +rd, v = α2 + z2,2 and r = α3 + z2,3 then gives:

mα2 + nα2 = γ(z1,z2, rd, rd) (98)

where

m = m22 −YδNδ

m32 (99)

n = n22 −YδNδ

n32 (100)

and

γ(z1,z2, rd, rd) =

−(m23 −

YδNδ

m33

)(c2z1 − cz2,3 + rd

)

−(n22 −

YδNδ

n32

)z2,2

−(n23 −

YδNδ

n33

)(−cz1 + rd + z2,3)

+ k2z2,2 −YδNδ

(k3z2,3 + z1) (101)

The variable α2 becomes a dynamic state of the con-troller according to (97) and (98). Furthermore,m22 >(Yδ/Nδ)m32 and n22 > (Yδ/Nδ)n32 imply that (98)is a stable differential equation driven by the con-verging error signals (z1,z2) and the bounded ref-erence signals (rd, rd) within the expression of γ(·).According to (92) z2,2(t) → 0, and it follows that|α2(t)− v(t)| → 0 as t→∞.

The LOS maneuvering problem for the 3 DOF un-deractuated craft (79) is implemented using the con-trol laws:

τ1 = m11ud + n11u− k1(u− ud) (102)

τ3 = m32α2 +m33α3

+ n32v + n33r− k3(r− α3)− z1 (103)

where k1 > 0, k3 > 0, z1 := ψ−ψd, z2 := [u−ud, v−α2, r− α3]>, and

α3 = −cz1 + rd, c > 0 (104)

α3 = −c(r− rd) + rd. (105)

The reference signals ud, ud, ψd, rd and rd are pro-vided by the LOS guidance system. Notice that thesmooth reference signal ψd ∈ L∞ must be differenti-ated twice to produce rd and rd, while ud ∈ L∞ mustbe differentiated once to give ud. This is most eas-ily achieved by using reference models representedby low-pass filters.

3.5 Closed-loop stability analysisThe equilibrium point (z1,z2) = (0,0) of the closed-loop system:[

z1

z2

]=

[−c h>

−M−1h −M−1K

] [z1

z2

](106)

mα2 + nα2 = γ(z1, z2, rd, rd) (107)is uniformly globally asymptotically stable (UGAS)if m > 0 and n > 0, while α2 ∈ L∞ satisfies:

limt→∞|α2(t)− v(t)| = 0 (108)

It follows directly from the Lyapunov arguments(87) and (92) that the equilibrium (z1,z2) = (0,0)of the z-subsystem is UGAS. The unforced α2-subsystem (γ = 0) is clearly exponentially stable.Since (z1,z2) ∈ L∞ and (rd, rd) ∈ L∞, then γ ∈ L∞.This implies that the α2-subsystem is input-to-statestable (ISS) from γ to α2. This is seen by applying forinstance:

V =1

2mα2

2 (109)

which differentiated along the solutions of α2 gives:

V ≤ −1

2nα2

2, ∀ |α2| ≥2

n|γ(z1,z2, rd, rd)| (110)

By standard comparison functions, it is then possibleto show that for all |α2| ≥ 2

n|γ(z1,z2, rd, rd)| then:

|α2(t)| ≤ |α2(0)| e−n2t (111)

Hence, α2 converges to the bounded set:{α2 : |α2| ≤

2

n|γ(z1,z2, rd, rd)|

}(112)

Since z2,2(t)→ 0 as t→∞.

Figure 6: CyberShip 2 in action at the MCLab.

4 EXAMPLE: EXPERIMENT PERFORMEDWITH THE CS2 MODEL SHIP

The proposed path-following controller and guidancesystem were tested out at the Marine CyberneticsLaboratory (MCLab). MCLab is an experimental lab-oratory for testing of scale models of ships, rigs, un-derwater vehicles and propulsion systems. The soft-ware is developed by using rapid prototyping tech-niques and automatic code generation under Mat-lab/Simulink.

In the experiment,CyberShip 2 (CS2) was used. Itis a 1:70 scale model of an offshore supply vessel witha mass of 15 kg and a length of 1.255 m. The maxi-mum surge force is approximately 2.0 N, while themaximum yaw moment is about 1.5 Nm. The MCLabtank is L × B × D = 40 m × 6.5 m × 1.5 m.

Figure 6 shows CS2. Three spheres can be seenmounted on the ship, ensuring that its position andorientation can be computed using infrared cameras.Due to a temporary bad calibration, the camera mea-surements vanished when the ship assumed certainyaw angles and regions of the tank. This affected theresults of the experiment and also limited the avail-able space for maneuvering. Nevertheless, good re-sults were obtained. The cameras operate at 10 Hz.The desired path consists of a total of eight waypoints:

wpt1= (0.372,−0.181) wpt5= (6.872,−0.681)wpt2= (−0.628,1.320) wpt6= (8.372,−0.181)wpt3= (0.372,2.820) wpt7= (9.372,1.320)wpt4= (1.872,3.320) wpt8= (8.372,2.820)

representing an S-shape. CS2 was performing the ma-neuver with a constant surge speed of 0.1 m/s. By as-suming equal Froude numbers, this corresponds to asurge speed of 0.85 m/s for the full scale supply ship.A higher speed was not attempted because the conse-quence of vanishing position measurements at higher

−6 −4 −2 0 2 4 6−2

−1

0

1

2

3

4

5

6

7

8

9Measured and desired XY−position

East [m]

Nort

h [m

]

Measured pathDesired path

Figure 7: xy-plot of the measured and desired geometrical path during the experiment.

speed is quite severe. The controller used:

M =

25.8 0 00 33.8 1.01150 1.0115 2.76

N (ν) =

2 0 00 7 0.10 0.1 0.5

c = 0.75, k1 = 25, k2 = 10, k3 = 2.5

In addition, a reference model consisting of three first-order low-pass filters in cascade delivered continuousvalues of ψd, rd, and rd. The ship’s initial states were:

(x0, y0, ψ0) = (−0.69 m, −1.25 m, 1.78 rad)

(u0, v0, r0) = (0.1 m/s, 0 m/s, 0 rad/s)

Both the ship enclosing circle and the radius of accep-tance for all waypoints was set to one ship length.

Figure 7 shows an xy-plot of the CS2’s position to-gether with the desired geometrical path consisting ofstraight line segments. The ship is seen to follow thepath very well. To illustrate the effect of the position-ing reference system dropping out from time to time,Figure 8 is included. It shows the actual heading an-gle of CS2 alongside the desired LOS angle. The dis-continuities in the actual heading angle is due to the

camera measurements dropping out. When the mea-surements return, the heading angle of the ship is seento converge nicely to the desired angle.

5 CONCLUSIONSThis section has presented nonlinear models forship maneuvering and path following. A LOS path-following controller for marine craft based on Fossenet al. (2003) has been designed using a nonlinear ma-neuvering model. The performance of the controllerhas been demonstrated through experiments with amodel ship scale 1:70.

The experiments indicate that the path-followingcontroller is capable of tracking 2-D curved pathswith feasible curvatures and at the same time con-trol the speed along the path using only two controls(rudder and main propeller). Hence, the LOS guid-ance algorithm, path-following controller and speedcontroller effectively solve the maneuvering problem(geometric and dynamic tasks) as defined by Skjetneet al. (2004).

0 20 40 60 80 100 120 140 160

−40

−20

0

20

40

60

80

100

120

140

160

Measured and desired heading angle

Time [s]

He

ad

ing

[d

eg

]

Measured headingDesired heading

Figure 8: The actual yaw angle of the ship tracks the desired LOS angle well.

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