37
1 INTRODUCTION
Recently, intensive research has been conducted on the
development of automatic control technologies to
bring maritime autonomous surface ships into
practical service [1]. Among the various ship-handling
tasks, berthing and unberthing manoeuvres in
confined waters, such as ports, pose a significant
challenge to automation because of the high control
accuracy required. One contributing factor is the
change in the manoeuvrability caused by acceleration
and deceleration. The effectiveness of the rudder used
for the heading control depends on the surrounding
inflow velocity. Therefore, the controller must output
commands that account for the variations in
manoeuvrability caused by changes in speed and
propeller thrust. Moreover, the influence of
disturbances, such as wind and current, becomes
relatively more significant when manoeuvrability is
reduced. Consequently, in berthing and unberthing
controls where speed changes are required, controllers
must be designed based on motion models that can
represent such variations and disturbances.
Nonlinear motion models are often used to
represent ship motions during berthing and
unberthing, and several control methods based on such
models have been proposed. Nonlinear models are
capable of expressing speed-dependent nonlinear
variations such as hull resistance and the effectiveness
of actuators, thereby contributing to high control
accuracy during acceleration and deceleration. The
parameters of these nonlinear models have been
identified through captive model tests in tanks, where
scaled ship models were moved under constrained
conditions [2]. However, tank experiments require
large-scale facilities and extensive trials under various
conditions, making them time-consuming and costly.
As an alternative, methods have been proposed to
estimate parameters from motion data acquired during
free-running tests using model or full-scale ships [3–5].
These approaches enable the identification of motion
Experimental Evaluation of Unberthing Manoeuvre
Control Using an Online Estimation Model Under Actual
Sea Conditions
H. Kashiwagi & T. Okazaki
Tokyo University of Marine Science and Technology, Tokyo, Japan
ABSTRACT: Ship manoeuvrability varies with speed and is affected by environmental disturbances, making
unberthing control challenging. To address this, a control method using an online estimation model is proposed,
in which the parameters of a linear motion model are sequentially updated based on the observed ship velocities
and actuator inputs. A linear quadratic regulator was applied for route tracking, and a bias term was introduced
to estimate and compensate for steady external forces, such as wind and tidal currents. The proposed method
was validated through actual sea experiments using a full-scale vessel. The control system achieved accurate
tracking using an unberthing manoeuvre. It also demonstrated effective disturbance compensation and
adaptability to changes in actuator effectiveness. These results confirmed the practicality of the proposed
approach for real-time unberthing operations under actual sea conditions.
http://www.transnav.eu
the International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 20
Number 1
March 2026
DOI: 10.12716/1001.20.01.05
38
models from limited data, even for vessels with
unknown motion characteristics. However, parameter
estimation for nonlinear motion models generally
requires high computational power and extended
convergence times. The mathematical modelling group
model [6] offers another approach in which empirical
formulas derived from extensive experimental data
across various ship types are used to estimate model
parameters from principal particulars [7–9]. These
formulas allow for the design of motion models
without the need for physical experiments. Based on
these formulas, Miyoshi et al. [10] designed a linear
motion model, which was used for route-tracking and
berthing control [11, 12]. This method was applied to
three vessels and its effectiveness was demonstrated
through experiments under actual sea conditions.
Although linear models are relatively easy to design,
they often require parameter tuning to accurately
represent actual ship behaviour, leading to non-
negligible implementation costs for each target ship.
Furthermore, when controlling manoeuvres involving
speed changes using linear models, an additional
model design for various speed ranges is necessary
[12]. To address these challenges, recent studies have
proposed online model estimation methods that
sequentially update the motion model parameters
using input-output data acquired during manoeuvring
[13]. These approaches allow for control without
predefining a motion model by continuously updating
it in real time. Nonetheless, for actual deployment in
ship operations, linear models are preferred because of
their predictability and reliability [11, 12].
To overcome this issue, the authors previously
proposed a model estimation method that updates a
linear model, initially derived from principal
particulars, using input-output data obtained during
control [14]. As the speed of a ship changes gradually,
the associated changes in manoeuvrability are also
gradual. Thus, sequential updates to linear model
parameters can effectively capture speed-dependent
variations in dynamics. The estimated parameters
were those of the sway and yaw motions, which were
controlled by the rudder in a three-degree-of-freedom
linear model. In this model estimation, the time delay
between the actuator inputs and the resulting motions
was considered by shifting the input-output time
series. In addition, to compensate for steady external
disturbances, such as tidal currents, a bias term was
introduced in the sway motion model to estimate the
steady forces acting on the hull. The rudder command
was then adjusted based on the estimated disturbance
effect. The effectiveness of this method was previously
verified using simulations of unberthing manoeuvres,
showing that it can accurately reproduce ship motion
even in the presence of sensor noise and current
disturbances [14]. However, under actual sea
conditions, multiple time-varying disturbances—such
as wind and current—exist, and sensor data may
contain irregular noise, which can affect both model
estimation and control performance. Therefore, an
experimental validation using an actual ship is
necessary to demonstrate its practical effectiveness.
In this study, we implemented a control system
based on the proposed online estimation model on a
full-scale ship and evaluated its performance through
real sea trials. The target ship was Shioji Maru, a 60 m
long, 775-ton training ship comparable in size to a
typical 500-ton coastal cargo vessel. The control was
conducted based on a manoeuvring plan created by
referencing the actual unberthing operations of the
target ship. The experiments were conducted in a large
open-sea area within a bay to ensure the safety of the
ship and surrounding structures, and virtual quays
were defined to reproduce the unberthing scenario.
Although the bank effects cannot be replicated, the
setup allows for the evaluation of responses to wind
and current disturbances. Owing to the operational
limitations of the actuators, the unberthing manoeuvre
is divided into two phases: unberthing and leaving.
The implemented system included online model
estimation and route-tracking control modules
proposed in a previous study [14]. The model
parameters were updated using the time-series data of
the actuator inputs and the measured ship velocities
via the gradient descent method. Based on the
estimated linear model, a state-feedback controller
using a linear quadratic regulator (LQR) algorithm is
applied for route tracking. The model update and
control frequency were set to 2 Hz in accordance with
the system cycle of the ship. Moreover, the actuator
commands from the control system were constrained
according to the operational limits of the target ship to
prevent overloading. The experimental results
demonstrate that the proposed control method can
estimate external disturbances and achieve accurate
unberthing control under actual sea conditions.
The remainder of this paper is organized as follows.
Section 2 describes the unberthing motion model used
in this study. Section 3 presents the online model
estimation method and control system. Section 4
describes the experimental setup and Section 5
presents the experimental results. Finally, Section 6
concludes the study.
2 SHIP MOTION MODEL IN UNBERTHING
MANEUVER
2.1 Manoeuvring Phases
In this study, an unberthing manoeuvre is defined as
an operation in which a ship departs from a stationary
position on a quay and accelerates while following a
planned course line. As shown in Figure 1, the
unberthing manoeuvre is divided into two phases, the
unberthing and leaving phases, based on the actuators
used for attitude control. The unberthing phase refers
to the initial stage of the manoeuvre when the ship
away from the quay at speeds below 3 kn (5.6 km/h). In
this phase, attitude control is achieved using side
thrusters because rudder effectiveness is limited at low
speeds. The leaving phase is defined as the stage at
which the ship’s speed exceeds 3 kn and continues to
accelerate outward from the port. In this phase, control
was performed using the main propeller and rudder,
which become effective at higher speeds. Accordingly,
different actuator configurations are used in the two
phases. The unberthing phase employs the main
propeller, bow thruster (B/T), and stern thruster (S/T),
whereas the leaving phase uses the main propeller and
rudder. In this study, separate motion models,
parameter estimation processes, and controller designs
were developed for each phase.
39
Figure 1. Manoeuvring phases in the unberthing manoeuvre
2.2 Ship motion model
In this study, a linear model was used to describe the
motion of a ship to be able to apply the linear control
theory. Because ship dynamics are inherently
nonlinear, a steady-state condition is assumed, in
which the ship is proceeding straight ahead at a
constant speed US. The linear model represents
variations in the state relative to the steady-state
condition. The model describes the motion of a ship
with three degrees of freedom–surge, sway, and yaw–
using the coordinate system shown in Figure 2. Here,
X and Y represent the longitudinal and lateral positions
[m], respectively, ψ denotes the heading angle [rad], u
denotes the surge speed [m/s], v denotes the sway
speed [m/s], and r denotes the yaw rate [rad/s]. The
actuator variables include the propeller blade angle θP
[rad], the rudder angle δ [rad], B/T blade angle θB [rad],
and the S/T blade angle θS [rad]. Positive values of
these variables indicate the motion of the ship in the
forward or starboard directions. The linear motion
model with three degrees of freedom used in this study
is express:
0 0 0 0 0
0 , 0
00
,
P B S
uu uP
vv vr vδ vB vS
rv rr rδ rB rS
vr θ δ θ θ
ab
a a b b b
a a b b b
u
=+
==
==
x Ax Bu
xu
AB
(1)
where x is the state vector, and u is the control input
vector. A is the system matrix a represents its elements,
B is the input matrix, and b represents its elements.
Each of the element a and b uses subscripts, in which
the first character indicates the left-hand-side state
variable, and the second character indicates the state
variable or control input to which the coefficient is
applied. For example, avr refers to the coefficient
multiplied by r in the equation for
v
.
In this study, to enable discrete-time control with
sampling period TS [s], the continuous-time model in
Equation (1) is discretized. The resulting discrete-time
state-space model for the unberthing manoeuvre is
expressed as follows:
( 1) ( ) ( )
0 0 0 0 0
0 , 0
00
uu uP
vv vr vδ vB vS
rv rr rδ rB rS
t t t
φγ
φ φ γ γ γ
φ φ γ γ γ
+ = +
==
x Φx Γu
ΦΓ
(2)
where Φ is the state transition matrix and φ represents
its elements; Γ is the input matrix and γ represents its
elements. The subscript notation used for φ and γ
follows the same rule as that for a and b in Equation (1).
The matrices Φ and Γ are derived as follows:
0
s
s
T
T
s
e
e ds
=
=
A
A
Φ
ΓB
(3)
In this study, the control sampling period TS is set
to 0.5 s.
3 CONTROL SYSTEM USING AN ONLINE
ESTIMATION MODEL
3.1 Control system overview
In this study, a control system was developed to
perform an unberthing manoeuvre using an online
estimation model. The system integrates model
estimation and control functionalities into a unified
application. It comprises four main components: model
estimation, route-tracking controller design,
disturbance compensation, and output command
process. The computational flow of each control step is
illustrated in Figure 3. Each system component is
described in detail in the following section.
Figure 3. Flow diagram of the control system
3.2 Online model Estimation
The parameters of the linear model presented in
Equation (2) were estimated online for the sway and
yaw motions, which are controlled by the rudder, B/T,
and S/T, and their effectiveness varies with the ship
speed. The estimation was conducted using a batch
gradient descent method, as in our previous study [14],
based on the observed ship velocity and actuator input
data. Because time delays between actuator inputs and
40
the resulting ship responses can affect the accuracy of
the model estimation [15], the time-series input-output
data were shifted according to the delay before being
used in the parameter update process. Additionally,
during unberthing, a ship is exposed to environmental
disturbances such as tidal currents. These disturbances
affect not only the route-tracking performance but also
the model estimation accuracy because such effects are
not included in the linear models [15]. To account for
this, we introduced the bias term cv into the sway
motion model during the leaving phase, focusing on
the steady component of environmental disturbances
affecting sway. The models used to estimate the sway
and yaw motions are given below. As the actuator
configurations differ between the unberthing and
leaving phases, Equation (4) is applied to the former,
and Equation (5) to the latter:
( ) ( 1) ( 1) ( ) ( )
( ) ( 1) ( 1) ( ) ( )
vv vr vB B v vS S v
rv rr rB B r rS S r
vn φ v n φ r n γ θ n τ γ θ n τ
rn φ v n φ r n γ θ n τ γ θ n τ
= − + − + − + −
= − + − + − + −
(4)
( ) ( 1) ( 1) ( )
( ) ( 1) ( 1) ( )
vv vr vδ v v
rv rr rδr
vn φ v n φ r n γ δ n τ c
rn φ v n φ r n γ δ n τ
= − + − + − +
= − + − + −
(5)
Here, v(n), r(n), and δ(n) represent the sway speed,
yaw rate, and rudder angle at time step n, respectively.
The term cv represents the bias force due to steady
disturbance. The delays τv and τr denote the time
delays [s] in the sway and yaw motions, respectively.
Based on operational data from the target ship, τv and
τr were set to 15 s and 10 s, respectively.
In the proposed estimation method, the parameters φ,
γ, cv and in Equations (4) and (5) are updated by
minimizing the loss function L, which represents the
squared error between the observed and predicted
states. Loss function L is defined as follows:
2
1
ˆ
( ) { ( ) ( )}
r
n
t n t
r
L n y t y n
T
=−
=−
(6)
where Tr is the number of input-output data samples
used in the estimation, y is the observed state (e.g., v or
r), and
ˆ
y
is the predicted state from Equations (4) and
(5). The model parameters are iteratively updated
using the following gradient descent method:
( 1) ( )
L
θ n θ n α
θ
+ = −
(7)
where θ denotes the set of parameters to be estimated
(e.g., φ, γ, and cv), and α is the learning rate. In this
study, the parameters were updated 100 times for each
control cycle.
3.3 Route-tracking control
In this study, a state-feedback control approach was
applied to enable a ship to follow a planned unberthing
route based on the estimated linear model. The control
design was obtained using an LQR algorithm, which
minimizes the deviation from the planned route and
the heading error between the ship’s orientation and
route direction, as illustrated in Figure 4., where Xd and
Yd represent the longitudinal and lateral distances [m]
to the target waypoint, respectively; ψd denotes the
heading error [rad] with respect to the desired
trajectory. The control model for the unberthing
manoeuvres formulated separately for the unberthing
and leaving phases, reflecting the differences in
manoeuvring methods and actuator configurations.
The general state-space representation is given by
=+x Ax Bu
(8)
where x is the state vector; u is the control input vector;
A is the system matrix; and B is the input matrix. The
specific models used for control in the unberthing
(lateral and forward motions) and leaving phases are
defined by Equations (9)– (11), respectively:
0 0 0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
,
1 0 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0 0
d d d
P B S
uu uP
vv vr vB vS
rv rr rB rS
u v r X Y ψ
θθθ
ab
a a b b
a a b b
=
=
==
x
u
AB
(9)
0 0 0 0 0 0
0 0 0 0
,
0 0 0 0
0 1 0 0 0 0 0 0
0 0 1 0 0 0 0 0
dd
P B S
uu uP
vv vr vB vS
rv rr rB rS
u v r Y ψ
θθθ
ab
a a b b
a a b b
=
=
==
x
u
AB
(10)
0 0 0 0 0
0 0 0 0
,
0 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0 0
dd
P
uu uP
vv vr vδ
rv rr
t
u v r Y ψ
θδ
ab
a a b
aa
U
=
=
==
x
u
AB
(11)
These continuous-time models were discretized
using Equation (3) and then used for the controller
design. In Equations (9)–(11), the parameter values for
the surge motion were taken from a linearized model
previously reported in [14], whereas the parameters for
the sway and yaw motions were obtained from the
online model estimation.
Because the control model in this study is updated
online, it constitutes a time-varying system. However,
because changes in the ship speed occur gradually, the
model can be regarded as time-invariant over short
intervals. Therefore, a design approach for discrete-
time, time-invariant systems based on the LQR
algorithm was adopted. The control input is computed
using the following state-feedback law:
( ) ( ) ( )n n n=−u F x
(12)
41
where F is the feedback gain matrix that minimizes the
following cost function:
( ) ( ) ( ) ( )
TT
tn
J t t t t
=
= +
x Qx u Ru
(13)
The gain matrix F is calculated based on the solution
P of the discrete-time algebraic Riccati equation:
1
()
T T T T−
= + − +PQΦ PΦ Φ PΓ R Γ PΓ Γ PΦ
(14)
1
()
TT−
=+FRΓ PΓ Γ PΦ
(15)
where Q and R are the weighting matrices for the state
and control inputs, respectively.
Figure 4. Coordinate system of the control system
3.4 Disturbance compensation
In the online estimation model described in Section 3.2,
a bias term was introduced during the leaving phase to
estimate the influence of the external force in real time.
In this study, disturbance compensation was
performed by computing a rudder angle correction
based on the estimated bias term cv. The rudder
correction amount, denoted as Δδ, was calculated
using the following equation:
δψ v
Δδ F Kc=
(16)
where Fδψ is an element of the feedback gain matrix F
obtained in Equation (15) and K is the scaling
coefficient, which was set to 16 in this study. The
corrected rudder angle was then obtained by adding
Δδ to the rudder command computed by the state
feedback control law in Equation (12), and the adjusted
value is used as the final control input.
3.5 Output limiting
In actual ship control, excessive actuator commands
can cause an overload and lead to equipment failure.
This is particularly critical for propulsion systems with
propellers, such as the main propeller, B/T, and S/T, in
which smooth operation is essential. To address this
issue, output limiting was implemented in the control
system to comply with the operational constraints of
the actuators of the actual ship. For the propeller and
rudder, both the upper bounds and rate-of-change
limits were imposed, as shown in Table 1. For the B/T
blade angle, the maximum allowable value was set to
11°, with a rate limit of 2°/s. For the S/T blade angle, the
maximum value was set to 9°, with a rate limit of 2°/s.
Additionally, to prevent excessive control inputs
caused by sudden changes in reference values, a rate
limit of 0.05 kn/s was applied to the target speed.
Table 1. Output limit value
Speed [kn] (km/h)
Propeller
Rudder
Range [°]
Rate [°/s]
Range [°]
< 4.0 (< 7.4)
-4.0–6.0
0.25
± 35
4.0–5.0 (7.4–9.3)
1.2–8.7
± 30
5.0–6.0 (9.3–11.1)
± 25
6.0–7.0 (13.0–14.8)
± 20
7.0–8.0 (13.0–14.8)
± 15
8.0 < (14.8 <)
± 10
4 TARGET SHIP AND EXPERIMENT SCENARIO
4.1 Target ship
Sea trials were conducted using a full-scale ship to
evaluate the actual sea performance of the unberthing
control system based on an online estimation model.
The target ship was the Shioji Maru, a training ship
operated by the Tokyo University of Marine Science
and Technology, as shown in Figure 5. The principal
specifications of the ship are listed in Table 2. Shioji
Maru is a single-shaft ship equipped with a B/T and
two S/Ts. The rudder is a conventional rudder, and the
main propellers, B/T and S/T, are controllable pitch
propellers. During the experiments, various sensors
were used for motion measurements. A GNSS
navigation unit (Japan Radio Co., Ltd.) was used for
the position data, a gyrocompass (YDK Technologies
Co., Ltd.) for the heading and yaw rates, and a fiber-
optic compass (iXblue) for the surge and sway speeds.
To reduce the effect of noise on the measured data, a
low-pass filter was applied to the observed sway
speeds and yaw rates. In addition, to evaluate the
effects of environmental disturbances, wind direction
and speed were measured using an automatic weather
observation system (ANEOS Corporation), and current
direction and velocity were measured using an
ultrasonic multilayer current profiler (Teledyne RD
Instruments).
Figure 5. Shioji Maru
Table 2. Output limit value
Length overall
60.727 m
Length between perpendiculars
54.00 m
Breadth
11.10 m
Draft (average)
3.33 m
Gross tonnage
775 t
ψ
θ
θ
δ
θ
42
4.2 Experiment scenario
The unberthing manoeuvring plan used in the full-
scale sea experiments was created based on actual
unberthing manoeuvre data of the target ship from the
Tsukishima Wharf in the Port of Tokyo, Japan. The
planned manoeuvre is shown in Figure 6. In this plan,
the unberthing manoeuvre is divided into five steps
according to predefined waypoints. As shown in
Figure 6, Step 1 involved a 40 m lateral movement to
the port using B/T and S/T. In Step 2, the ship was
accelerated to 3 kn (5.6 km/h) while maintaining its
heading through attitude control using both thrusters
over a 300 m forward movement. Then, from Steps 3 to
5, the ship performed a course change using a rudder
while further accelerating to 8 kn (14.8 km/h). In this
study, Steps 1 and 2 were designated as the unberthing
phase, and Steps 3–5 were designated as the leaving
phase, and were used to define the experimental
scenario.
Figure 6. Manoeuvring plan
5 ACTUAL SHIP EXPERIMENT
To ensure the safety of both the vessel and nearby
coastal facilities, unberthing experiments were
conducted within a wide area of the bay by defining a
virtual quay. The experiments were carried out using
unberthing and leaving scenarios and took place off the
coast of Hakkeijima in Tokyo Bay, Japan, on January
23, 2025. The initial conditions for each experiment
corresponded to the starting conditions of each phase:
for the unberthing phase, the ship started from the
stationary state (0 kn), whereas for the leaving phase, it
began moving straight ahead at 3 kn (5.6 km/h).
Multiple experiments were conducted under different
environmental conditions to estimate the external
disturbance forces during the leaving phase. In the
following sections, the experiment during the
unberthing phase is referred to as Exp. 1, and those in
the leaving phase are referred to as Exps. 2 and 3.
5.1 Results of unberthing phase
In this study, full-scale experimental verification was
conducted using the Step 1 and Step 2 scenarios
described in Section 4 as the unberthing phase (Exp. 1).
The results of Exp. 1 are shown in Figure 7. The state
variables and control inputs are shown in Figure 7 (a).
The left side of the figure shows, from top to bottom,
the trajectory [m], surge speed u [m/s], sway speed v
[m/s], and yaw rate r [°/s], while the right side shows,
from top to bottom, the cross-track error (defined as Xd
in Step 1 and Yd in Step 2) [m], heading angle ψ [°],
propeller blade angle θP [°], B/T blade angle θB [°], S/T
blade angle θS [°]. Figure 7 (b) shows the wind and tidal
current conditions observed during the experiment. In
this figure, the wind direction indicates the direction
from which the wind is blowing, and the current
direction indicates the direction toward which the
water is flowing. The positions and headings in Figures
7 (a) and (b) are plotted in a coordinate system, where
the initial position is set to the origin and the initial
heading is aligned at 0°. In the trajectory plot in Figure
7 (a), the dashed line represents the planned route, the
solid line represents the actual trajectory, and the ship-
shaped markers indicate the ship heading every 60 s.
The horizontal axes of the other plots represent the
time elapsed from the beginning of the experiment [s].
From the results, the maximum cross-track error was
1.6 m during Step 1 (lateral motion) and 7.1 m during
Step 2 (forward motion). Overload warnings were not
triggered for any of the actuators during the
experiments. The estimated model parameters
obtained from the online estimation model in Exp. 1 are
shown in Figure 7 (c). The right-side graphs show, from
top to bottom, φvv, φrv, γvB, and γrB, while the left-side
graphs show φvr, φrr, γvS, and γrS.
The results show that throughout Exp.1, the cross-
track errors remained below 10 m, which is considered
acceptable from an operational standpoint, given that
vessels typically move at least one ship width away
from the quay during the unberthing manoeuvre. As
the breadth of the target ship is 11.10 m, the observed
deviation is within a tolerable range. However, in Step
2, the ship deviated from the planned route to the
starboard. At approximately 520 s, both thrusters were
temporarily operated in the direction opposite to that
required to reduce the deviation. This was likely due to
the tidal current of approximately 0.4 kn acting from
the port side, with the maximum flow observed at
approximately 440 s. Hence, even though the thrusters
attempted to move the ship to the port, the sway speed
decreased and eventually reversed direction at
approximately 480 s. This reversal caused the sign of
γvS to become negative, as observed around 500 s in
Figure 7 (c). Consequently, the S/T was operated in a
direction opposite to that needed to reduce the cross-
track error, and the B/T was followed to decrease the
yaw motion. Subsequently, the increased sway speed
led to the recovery of the estimated parameter values.
In contrast, the parameters related to the B/T, γvB and
γrB, decreased as the ship’s speed increased. This
reflects the known characteristic that the effectiveness
of the B/T worsens at higher speeds, which indicates
that the model appropriately estimates the actuator
effectiveness. However, at the end of the experiment,
after approximately 600 s, the sign of γvB became
negative. This change is linked to the unintended
thruster behavior observed near 520 s, suggesting that
the model parameter at this point may not be suitable
for use in controller design and requires further
countermeasures.
43
(a) Ship state and control inputs
(b) Disturbance conditions
Estimated parameters
Figure 7. Results of Exp.1
5.2 Results of leaving phase
In this study, Steps 3–5 described in Section 4 were
used as the leaving phase scenarios, and a full-scale
experimental verification was conducted using Exp. 2
and 3. The results are presented in Figures 8 and 9. The
graph layout in Figures 8 and 9 (a) follows the same
format as that in Figure 7 (a), but the actuators used are
rudders instead of thrusters. Figures 8 and 9 (b) show
the wind and tidal current conditions measured during
the experiments, whereas Figures 8 and 9 (c) show the
parameters estimated by the online model. In Figures 8
and 9 (c), the left-side graphs show, from top and
bottom, φvv, φrv, γvδ, and cv, while the left-side graphs
show, φvr, φrr, and γrδ from top to bottom. Table 3
summarizes the maximum cross-track errors observed
at each step.
Based on Figures 8 and 9 (a) and Table 3, both Exp.
2 and 3 showed successful route-tracking when the
ship was accelerating. The cross-track errors remained
within 5 m throughout Steps 3–5, indicating accurate
control performance during the leaving phase.
Furthermore, from Figures 8 and 9 (c), the rudder-
related parameters γvδ and γvδ increased as the ship
accelerated, which reflects increasing rudder
effectiveness at higher speeds. This suggests that the
model estimation appropriately captured the rudder
dynamics. However, the bias term cv in the model
estimation was estimated in the port-side (negative)
direction, starting at approximately 200 s after the
heading change in both experiments. Although direct
observation of the external forces acting on a ship
under actual sea conditions is not possible, the
measured wind and current conditions as well as the
observed sway speeds indicate that the ship likely
experienced a port-side disturbance. Looking at the
cross-track errors in Figures 8 and 9, the ship moved to
starboard in both experiments, which was opposite to
the direction of the sway speed. This behavior suggests
that the disturbance compensation based on the bias
term cv is effective in counteracting external forces.
Table 3. Maximum deviation of each step
Step 3
Step 4
Step 5
Exp. 2
1.4 m
-4.1 m
-3.5 m
Exp. 3
3.1 m
-4.0 m
-3.9 m
44
(a) Ship state and control inputs
(b) Disturbance conditions
(c) Estimated parameters
Figure 8. Results of Exp. 2
(a) Ship state and control inputs
(b) Disturbance conditions
(c) Estimated parameters
Figure 9. Results of Exp. 3
45
6 CONCLUSION
In this study, sea experiments were conducted using a
full-scale vessel to evaluate the actual sea performance
of an unberthing control system based on an online
estimation model that updates the motion model
parameters from the time-series data of ship velocity
and control inputs. The experimental scenarios were
designed based on actual unberthing operations and
were divided into two phases: unberthing and leaving.
To ensure safety, experiments were conducted over a
wide area of the bay using a virtual quay setup. The
experimental results demonstrate that the proposed
system achieves accurate route tracking in both the
unberthing and leaving phases. Furthermore, in the
leaving phase, the disturbance compensation based on
the estimated bias term successfully identified the
direction of the disturbance and generated control
inputs to compensate for its effect. The results indicate
that the proposed control system can accurately
perform unberthing manoeuvres under actual sea
conditions. However, during the unberthing phase,
unstable control behaviour was observed in the
segments where the accuracy of the model estimation
deteriorated. This suggests the need for further
improvement which could involve the development of
reliability assessment methods that can prevent low-
accuracy model estimates from being used in the
controller design.
ACKNOWLEDGEMENT
The authors express their gratitude to the crew of Shioji Maru
for their cooperation in conducting the ship experiments in
this study. This study was supported by JST SPRING under
Grant JPMJSP2147.
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