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Manoeuvring Control of Fully-actuated Marine Vehicles. A

Port-Hamiltonian System Approach to Tracking

Alejandro Donaire, Tristan Perez, and Christopher Renton

Abstract— This paper presents a novel control strategy fortrajectory tracking of marine vehicles manoeuvring at lowspeed. The model of the marine vehicle is formulated as a Port-Hamiltonian system, and the tracking controller is designedusing energy shaping and damping assignment. The controllerguarantees global asymptotic stability and includes integralaction for output variables with relative degree greater thanone.

I. INTRODUCTION

Models of marine vehicles present a significant degree

of uncertainty, and the dynamic response changes with seastate, velocity, wave direction, water depth, and other craft in

close proximity. Since vehicle motion follows physical laws

of energy, passivity-based control designs have been very

successful [1]. A controller designed so that stability depends

only on dissipativity properties can result in closed-loop

stability under large parametric uncertainty—even changes

in model order may be tolerated provided that dissipativity

properties remain unchanged [2].

Port-Hamiltonian systems (PHS) have a particular form

that explicitly incorporates a function of the total energy

stored in the system (energy function) and functions that

describe interconnection and dissipation structure of thesystem. As its name indicates, the input and output variables

of PHS constitute a port. Hamiltonian models have their root

in analytical mechanics, which, in contrast to the vectorial

mechanics of Newton, describes the motion of mechanical

systems in terms of two scalar quantities: potential and

kinetic energy. From a point of view of pure modelling of

mechanical systems, Hamiltonian models are not particularly

superior to Lagrangian and Newtonian models; what is more,

the latter models can be derived from the Hamiltonian model

[3]. Hamiltonian models, however, and in particular port-

Hamiltonian models, readily show energy and dissipativity

properties which enable control design based on physical

consideration. This class of design, known as energy-based

techniques, has attracted the attention of the control commu-

nity in the last two decades [4], [5], [6].

In this paper, we write the classical kinetic model of

marine vehicles, see [7], into a PHS form. We then consider

a control design for trajectory tracking such that the closed-

loop system retains a PHS form. We further show how

A. Donaire, T. Perez, and C. Renton are with the School of Engineeringat The University of Newcastle, Callaghan, NSW-2308, Australia.

T. Perez is also with the Centre for Ships and Ocean Structures (Ce-SOS) at the Norwegian University of Science and Technology, Trondheim,Norway.

integral action can be added and exploit the PHS-form to

provide a procedure for control design that ensures stability.

Finally, we present a numerical case study to illustrate the

performance of the proposed control design. The results in

this paper extend the previous work of the authors related to

position regulation control [8], [9].

II. PORT-HAMILTONIAN SYSTEMS

In this paper, we focus on a class of port-Hamiltonian

systems for which there are no constraints on the state.

This type of PHS are referred to as input-state-output port-

Hamiltonian systems (ISO PHS) [10]1. Dynamic systems

represented as ISO PHS have the following form:

x = [E(x) − F(x)]∂H (x)

∂ x+ G(x) u, (1)

y = GT (x)∂H (x)

∂ x, (2)

where x ∈ Rn is the state vector, H : R

n → R is

known as the Hamiltonian. This function can represent the

total energy stored in the system. Since the states define

the Hamiltonian, they are called energy variables. The pair

u, y ∈ R

m

are the input and output variables. These areconjugate variables; that is, their inner product represents

the power exchanged between the system and the environ-

ment. The matrix E(x) is skew-symmetric. It describes the

power conserving interconnection structure through which

the components of the system exchange energy. The matrix

F(x) ≥ 0 is symmetric and captures dissipative phenomena

in the system. The matrix G(x) weighs the action of the

input on the system and defines the output.

III. MODELS OF MARINE CRAFT

To describe the motion of a marine craft, we consider two

reference frames: Earth-fixed ({n}-frame) and body-fixed ({b}-frame). The position of the craft is given by the relative

position of {b} with respect to {n}. The components of this

vector are North, East and Down positions. The orientation

of the vessel is given by the Euler angles ψ–yaw, θ–

pitch, and φ–roll. The generalised-position vector (position-

orientation) is defined by η [N, E, D, φ, θ, ψ]T . The

velocities are conveniently expressed in terms of body-fixed

coordinates and denoted by the generalised velocity vector

(linear-angular) defined by ν [u, v, w, p, q, r]T .

1This class of models has been originally referred to as port-controlled

Hamiltonian systems (PCHS)–see [4].

© 2011 Engineers Australia

2011 Australian Control Conference 10-11 November 2011, Melbourne, Australia

ISBN 978-0-85825-987-4 32



The linear-velocity vector [u, v, w]T is the time derivative

of the position vector as seen from the frame {n} with

components in {b}. These components are the surge, sway,

and heave velocities respectively. The vector [ p, q, r]T is

the angular velocity of the body with respect to the {n}frame with components in {b}. These components are the

roll, pitch, and yaw rates respectively.

Using Kirchhoff’s equations, one can derive the classicalmodel of marine craft, which includes hydrodynamic (feed-

back) forces due to the motion of the craft [7]:

η = J(η)ν , (3)

Mν + C(ν )ν + D(ν )ν + g(η) = τ , (4)

where M is the total generalised mass matrix due to rigid-

body mass distribution and fluid added mass, C(ν ) is the

total Coriolis-centripetal matrix, D(ν ) is the total damping,

g(η) gives the restoring forces, and τ represents the vector

of forces and moments. We consider the vector of forces and

moments as the sum of the control input vector τ c and the

disturbance vector τ d. For further details on how this model

can be derived, see [1] and [7].

In this paper, we restrict our study to the positioning

problem in the horizontal plane, i.e. we take into account

only the degrees of freedom of surge, sway and yaw, and we

consider the following state vector:

x =

x1

x2

, x1 = M

u

v

r

, x2 =

N

E

ψ

, (5)

with the Hamiltonian H (x) = 12 xT

1 M−1x1 that corresponds

to the total kinetic energy.

The model (3)-(4) in the degrees of freedom of interestcan be written as

x1

x2

=

−C(x1) −RT (x2)

R(x2) 0

−

D(x1) 0

0 0

∂H (x1)∂ x1

0

+

τ c

0

+

τ d

0

,

(6)

y =∂H (x1)

∂ x1= M−1x1, (7)

where C1(x) = C(M−1x1) is skew-symmetric, D(x1) =D(M−1x1) is symmetric and positive semi-definite, and

R(x2) =

cos ψ − sin ψ 0sin ψ cos ψ 0

0 0 1

(8)

is the rotation matrix.

IV. POSITIONING CONTROL RETAINING THE ISO-PHS

FORM

In our previous work, see [8], [9], we designed a position

regulation control system in two steps. First, we exploit the

structure of the system (6) to stabilise the desired equilibrium

point by shaping the energy function. We also assign a new

interconnection structure and dissipation function such that

the closed-loop dynamics retains the ISO-PHS form. Second,

we add integral action for the position error also retaining

the ISO-PHS form.

In the first step, we follow the standard interconnection

and damping assignment passivity-based control method

proposed by [5]. This technique results in a static-feedback

control law which guarantees passivity of the closed loop. In

many applications, the influence of unknown disturbances,parameter uncertainties and measurement noise deteriorate

the performance of the static-feedback controller, even if

the closed loop system is passive [11], [12]. A classical

and widely accepted practice to deal with this problem is

the incorporation of integral action on the controller [11],

[13]. Hence, in the second step, we use a technique proposed

in [14] to add integral action to the variables with relative

degree greater than one whilst preserving the ISO PHS form.

The full control law τ c results from the combination of

the regulation and integral laws, τ r and τ i respectively, as

τ c = τ r+τ i. In this paper, we follow this approach to design

a tracking controller that ensure internal stability, outputregulation and rejection of unknown constant disturbances.

In addition, we prove the input-to-state stability (ISS) of the

control system. This property ensures that the states remain

bounded when the disturbances are bounded, and the states

converge to the equilibrium when the disturbances converge

to zero.

V. TRACKING CONTROL FOR MANOEUVRING

In this section, we consider the asymptotic tracking of

time-varying position references. In addition, the control law

must incorporate integral action to reject constant unknown

disturbances due to ocean currents. This provides an exten-

sion to the position regulation controller discussed in the

previous section.

A. Reference signals and control objective

Given the position reference vector x∗2 and its derivatives

x∗2 and x∗2 provided by the guidance system, we define the

momentum reference vector and its derivative as follows:

x∗1 := MRT (x2)x∗2, (9)

x∗1 := MRT (x2)x∗2 + MRT (x2)x∗2. (10)

The control objective is then to drive the vessel such that it

asymptotically tracks the positions and momenta references,

namely x → x∗.

B. Controller Design

The control objective can be expressed using the momen-

tum and position errors. We define these errors as

x1 := x1 − x∗1, (11)

x2 := x2 − x∗2. (12)

33



Taking the time derivative of the errors and using the model

(6), we can formulate the error dynamics as follows:

˙x1 = − C(x1 + x∗1)M−1(x1 + x∗1) − D(x1 + x∗1)

M−1(x1 + x∗1) + τ c + τ d − x∗1, (13)

˙x2 =R(x2 + x∗2)M−1(x1 + x∗1) − x∗2. (14)

Using the error dynamics, we transform the trajectory track-ing problem into the stabilisation control problem of the error

dynamics around the origin [12].

The first step is to design the regulation control law τ r to

stabilise the error dynamics. With this purpose, we consider

the following desired closed-loop PHS:˙x1˙x2

=

−C(x1 + x∗1) −RT (x2 + x∗2)

R(x2 + x∗2) 0

−

Dd(x1) 0

0 0

∂H d

∂ x(x) +

τ i + τ d

0

,

(15)

with

H d(x) = 12

xT 1 M−1x1 + 1

2xT 2 K2x2, (16)

where K2 is a positive-definite constant, and Dd(x1) is a

positive-definite function that represents the desired dissipa-

tion of the error dynamics.

The desired error dynamics are chosen such that it pre-

serves the PHS form. Therefore, some terms of the original

system are preserved and new terms are introduced using

physical considerations. In the closed-loop system, the form

of the Coriolis terms is preserved since it does not produce

instabilities (but it could deteriorate the performance). Hence,

we try to avoid cancelling nonlinear terms, which otherwise

can jeopardise the system stability—in the case of Coriolisand centripetal terms, the model structure is certain, but the

value of the parameters may not be so. The energy function

is shaped to consider the momentum and position errors

such that the Hamiltonian is a state function. In addition,

a new damping function Dd(x1) is assigned to ensure a

desired dissipation. The PHS form of the error dynamics will

ensure asymptotic stability of the positions to the specified

trajectory.

The regulation control law τ r that tranforms the open loop

dynamics (13)-(14) into the desired PHS (15) is obtained by

matching the state equations of the open- and closed-loop

dynamics. That is,

− C(x1 + x∗1)M−1(x1 + x∗1) − D(x1 + x∗1)M−1

(x1 + x∗1) + τ r + τ i + τ d − x∗1 = −C(x1 + x∗1)∂H d

∂ x1−

RT (x2 + x∗2)∂H d

∂ x2− Dd(x1)

∂H d

∂ x1+ τ i + τ d, (17)

R(x2 + x∗2)M−1(x1 + x∗1) − x∗2 = R(x2 + x∗2)∂H d

∂ x1.

(18)

The second matching equation (18) is satisfied using the

definition (9) for the vector of momentum references x∗1.

The control law that ensures regulation is computed from

the first matching equation (17). This yields

τ r = C(x1 + x∗1)M−1x∗1 + x∗1 − Dd(x1)M−1x1−

RT (x2 + x∗2)K2x2 + D(x1 + x∗1)M−1(x1 + x∗1). (19)

Proposition 1: The closed-loop system obtained by using

the control law (19) in the error dynamics (13)-(14) can be

described as the port-Hamiltonian system (15). Assume that

τ i and τ d are zero, then the closed-loop PHS (15) has an

asymptotically stable equilibrium at the origin of the error

state space.

Proof: Since the control law (19) verifies (17) and that

(18) is satisfied by the reference signal (9), then the error

dynamics of the closed loop is the PHS (15). The stability

of the origin can be proved using standard properties of

port-Hamiltonian systems. The Hamiltonian function (16)

has a minimum at the origin of the error state space. The

Hamiltonian can be used as a Lyapunov candidate function.

Computing the derivative with respect to time of the Hamil-

tonian results in

H d(x) = −∂ T H d

∂ x1Dd(x1)

∂ T H d

∂ x1. (20)

Negative semi-definiteness of H d only ensures stability of

the origin. Asymptotic stability follows using Krasovskii-

LaSalle’s invariance principle [2]. Indeed, since the maxi-

mum invariant set contained in {x|∂ T H d

∂ x1Dd(x1)∂

T H d

∂ x1= 0}

is zero, then, the origin of the error state space is asymp-

totically stable. Given that the Hamiltonian H d is radially

unbounded, the stability property is global.

The regulation controller above is a static nonlinear feed-back controller, and it cannot effectively reject unknown

disturbances. The classical solution to reject constant or

slowly varying disturbances is the addition of integral action,

which depends on the signal to be suppressed, in this case,

the position-tracking error. A second objective, then, is to

design an integral controller via state augmentation whilst

preserving the form of the original PHS. This is not possible

when the integral action depends on the states with relative

degree greater than one [15], [14]. Thus, we need to find a

state transformation that allows us to preserve the PHS form

and ensure integral action of position-tracking error. This is

done using a procedure similar to that proposed in [14] for

the case of set-point regulation. Therefore, in the second step

of the manoeuvring control design, we propose to design a

integral control law τ i such that the closed loop results in

the PHS with dynamicsz1

z2z3

=

−C(z1) −RT (z2 + x∗2) −D3

R(z2 + x∗2) 0 −R(z2 + x∗2)DT

3 RT (z2 + x∗2) 0

−

Dd(z1) 0 0

0 D2 0

0 0 D3

∂H dz

∂ z(z) +

τ d0

0

,

(21)

34



with

H dz(z) =1

2zT 1 M−1z1 +

1

2zT 2 K2z2 +

1

2zT 3 K3z3. (22)

The matrices and functions K2, K3, Dd(z1), D2 and D3 are

to be chosen as part of the controller tuning. The constants

K2 and K3 are positive definite. Dd(z1) is the desired

nonlinear damping. The constants D2 and D3 are positivedefinite and represent the linear damping in the states z2 and

z3 respectively. However, D2 and D3 could be nonlinear

positive-definite functions if nonlinear damping is desired.

To build the target PHS (21), we consider the same

interconnection structure and Hamiltonian form as we did in

the regulation control design, but here we augment the state

vector by adding the state z3 which produces the desired

integral action. We also add to the Hamiltonian function a

term depending on the state z3, and we inject damping in

all the states. The dynamics of z3 are given by

z3 =D3M−1x1 + D3RT (z2 + x∗2)D2+

RT (z2 + x∗2)

K2x2. (23)

The change of variable is chosen such that

z2 = x2. (24)

The change of variable for z1 is obtained by taking the time

derivative of (24) and replacing the derivative of the state by

the corresponding state equations. This gives

z1 = x1 + MRT (x2 + x∗2)D2K2z2 + MK3z3 (25)

The integral control law τ i that renders the regulation loop

as the augmented PHS (21) is obtained by performing the

derivative with respect to time of (25) and replacing thederivative of the state by the corresponding state equations.

By doing so, the integral control law yields

τ i = −

C(z1) + Dd(z1)

M−1z1 +

C(x1 + x∗1)+

Dd(x1) − MRT (x2 + x∗2)D2K2R(x2 + x∗2) − MK3D3

M−1x1 −

MK3

D3RT (x2 + x∗2)D2 + RT (x2 + x∗2)

−

MRT (x2 + x∗2)D2

K2x2 − D3K3z3. (26)

Proposition 2: The error dynamics (13)-(14) in closed

loop with the tracking control law τ c = τ r + τ i obtained

from (19) and (26) can be described as the closed-loopport-Hamiltonian system (21). Assume that there are no

disturbances, i.e., τ d = 0, then closed-loop PHS has a global

asymptotically stable equilibrium point at the origin of the z-

state space, which ensure asymptotic tracking of the desired

trajectory.

Proof: From proposition 1, the regulation feedback

control system can be represented as the PHS (15). Using

the derivative of the change of variable (25) with respect to

time, the change of variables (24) and (25), and the integral

control law (26) in (15) results in the state equation for z1.

The state equation for z2 is obtained by replacing (24) and

(25) in the state equation of x2 in (15). The closed-loop PHS

(21) is completed using (24) and (25) in the dynamics of the

controller states (23).

We can show stability by using the Hamiltonian (22) as a

Lyapunov candidate function. Then,

H dz(z) = −∂ T H dz

∂ z1

Dd(z1)∂H dz

∂ z1

−∂ T H dz

∂ z2

D2∂H dz

∂ z2

−

∂ T H dz

∂ z3D3

∂H dz

∂ z3< 0. (27)

The negative definiteness of H dz ensures asymptotic stability

of the origin. Given that the Hamiltonian H dz is radially

unbounded, the stability property is global. The tracking of

the desired trajectory is ensured noting that convergence

of z to zero implies that x → 0, and then the position

and momentum vectors asymptotically converge to their

reference trajectories.

In the remainder of this section, we discuss the action

of the disturbance vector on the vehicle control systems

described by the PHS (21). We will prove that the closed-loop system is input-state-stable [16]. This means that there

exists β ∈ KL and γ ∈ K∞ such that

||z(t)|| ≤ β (||z(o)||, t) + γ (||τ d||∞). (28)

Proposition 3: Assume that the parameters of the con-

troller Dd, D2 and D3 are chosen so that there exist

constants c1, c2, c3 > 0 that satisfies

z1M−T Dd(z1)M−1z1 < c1||M−1z1||2, (29a)

z2KT

2 D2K2z2 < c2||z2||2, (29b)

z3KT

3 D3K3z3 < c3||z3||2. (29c)

Then, the closed-loop PHS (21) with input τ d is input-state

stable.

Proof: We propose the Hamiltonian H dz in (22) as a

ISS-Lyapunov candidate function (we refer to [16] for further

details on ISS). Then,

H dz(z) ≤1

2c1||τ d||2 −

c1

2||M−1z1||2 − c2||z2||2 − c3||z3||2

≤1

2c1||τ d||2 −

c1λmin(M−1)

2||z1||2 − c2||z2||2 − c3||z3||2

≤1

2c1||τ d||2 − cmin||z||2, (30)

with cmin = min{12c1λmin(M−1), c2, c3}. Since H dz is a

smooth, proper and positive definite function, and satisfies

(30), then H dz is a ISS-Lyapunov function of the system

(21). Therefore, the closed-loop PHS is ISS.

The ISS property of the closed-loop system (21) ensures

i) that the states are bounded for bounded time-varying

disturbances, and ii) that the tracking errors converge to zero

when the disturbance vanishes. Properties i) and ii) follows

from (28). In addition, ISS combined with the integral action

ensure that iii) the system tracks the desired trajectory when

it is under the action of constant unknown disturbances.

Indeed, for each constant input a ISS system has a steady

35



0 50 100

0

5

10

Time [s]

S u r g e p o s i t i o n [ m ]

0 50 100

−0.5

0

0.5

1

1.5

2

Time [s]

S u r g e v e l o c i t y [ m / s ]

demanded

actual

demanded

actual

Fig. 1. Demanded and actual motion variables (surge).

0 50 100−1

0

1

2

3

4

5

Time [s]

S w a y p o s i t i o n [ m ]

0 50 100−0.2

0

0.2

0.4

Time [s]

S w a y v e l o c i t y [ m / s ]

demanded

actual

demanded

actual

Fig. 2. Demanded and actual motion variables (sway).

state [16]. Given the integral action, the steady state of the

PHS (21) ensures that x1 = 0 and x2 = 0 for all constant

unknown disturbances, which proves iii).

V I. CAS E STUDY

In this section, we consider a simulation study based on

a model of an open frame remotely operated underwater

vehicle [8]. The parameters of the vehicle are included inAppendix I.

Figures 1 to 3 show the displacements and velocities in

the degrees of freedom of interest. As can be seen the actual

position and velocities of the vehicle track their reference,

which corroborates the theoretical results. Figures 4 to 6

show the tracking errors corresponding to the positions

and velocities. The control forces are shown in Figure 7.

A few seconds into the simulation, a constant disturbance

representing an ocean current is added into the model. Then

at 40 seconds into the simulation, a reference trajectory

is generated by the guidance system and passed on to the

tracking controller. The reference trajectory is composed of

a change of surge at time=40s, a change of sway at time=60s,

and a change of yaw at time=80s. As we can see from these

figures, the designed controllers perform satisfactorily both

during regulation and tracking.

VII. CONCLUSION

This paper presents a novel control strategy for trajectory

tracking of marine vehicles manoeuvring at low speed.

The model of the marine vehicle is formulated as a Port-

Hamiltonian system. Then the tracking controller is designed

by formulating the error dynamics as a set-point regulation

Port-Hamiltonian control problem. This approach fits into

0 50 100−20

0

20

40

Time [s]

Y a w a

n g l e [ d e g ]

0 50 100

−20

0

20

40

Time [s]

Y a w r

a t e [ d e g / s ]

demanded

actual

demanded

actual

Fig. 3. Demanded and actual motion variables (yaw).

0 50 100−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

Time [s]

S u r g

e p o s i t i o n [ m ]

0 50 100−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

Time [s]

S u r g e

v e l o c i t y [ m / s ]

Fig. 4. Tracking errors in position and velocity (surge).

0 50 100−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

Time [s]

S w a y p o s i t i o n [ m ]

0 50 100−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

Time [s]

S w a y v e l o c i t y

[ m / s ]

Fig. 5. Tracking errors in position and velocity (sway).

0 50 100−12

−10

−8

−6

−4

−2

0

2

Time [s]

Y a w a

n g l e [ d e g ]

0 50 100−20

−15

−10

−5

0

5

10

Time [s]

Y a w r

a t e [ d e g / s ]

Fig. 6. Tracking errors in position and velocity (yaw).

36



0 50 100

0

500

1000

Time [s]

S u r g e c o n t r o l f o r c e [ N ]

0

200

400

600

800

0

200

400

600

S w a y c o n t r o l f o r c e [ N ]

0 50 100Time [s]

0 50 100Time [s]

Y a w c

o n t r o

l f o r c e [ N ]

Fig. 7. Control forces of an underwater vehicle manoeuvring at low speed.

the framework previously used by the authors for the design

of position regulation controllers for marine systems. We

proved the stability of the closed loop system and illustrate

its performance using a case study. Our future work will

focus on extending the results to the case of under-actuatedmarine vehicles.

VIII. ACKNOWLEDGMENTS

The first author gratefully acknowledges The University of

Newcastle for providing financial support under its research

fellowship program and the research grant G1100066.

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APPENDIX I

VEHICLE MODEL PARAMETERS

We consider an open-frame underwater vehicle with a

mass of 140kg. The vehicle has four thrusters in an x-type

configuration, which provides actuation in all the degrees of

freedom of interest. The parameters of the model considered

in this paper are

M =

290 0 0

0 404 500 50 132

,

D =95 + 268|v| 0 00 613 + 164|u| 0

0 0 105

,

C =

0 0 −404v − 50r

0 0 290u

404v + 50r −290u 0

,

and the controller parameters are

K2 =

700 0 0

0 700 00 0 400

,

K3 =0.0034 0 0

0 0.0025 00 0 0.0076

,

D2 =

1 0 0

0 1 00 0 1

, D3 =

400 0 0

0 400 00 0 200

,

Dd =

191 + 100|v| 0 0

0 1228 + 100|u| 00 0 210

.

37

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