105
1 INTRODUCTION
The reduction of energy consumption and greenhouse
gas emissions has emerged as a critical focus in the
sustainable development of marine transportation.
Weather routing, a methodology that optimizes ship
trajectories by harnessing weather-induced forces—
such as wind, waves, and ocean currents—to minimize
propulsion demand, has garnered considerable
attention in maritime path planning research. Notably,
ship energy consumption is affected by ship motion
states, weather characteristics, which are both spatio-
temporal variables. Thus, the development of an
advanced weather routing framework, integrating
weather effects on ship motion states is imperative for
enhancing energy efficiency and operational
sustainability.
Weather routing is commonly formulated as a path-
planning problem under constant-velocity
assumptions, typically addressed using graph-search
methods such as the isochrone method [1], [2], A*
algorithm [3], or probabilistic roadmap approach [4]. In
these studies, energy consumption between adjacent
nodes is estimated according to the ship constant
velocity and weather conditions, and serving as
directed edge weight for path searching. Furthermore,
various studies introduce the time or velocity
dimensions into grid map developments, while the
improved path with velocity profile can be directly
searched in 3D configuration space [5]. Meanwhile, to
trade off various demands (navigation time, energy
consumption, etc.) proposed by ship owner, numerous
heuristic algorithms have been introduced and
employed in this field, such as genetic [6] and ant
colony algorithms [7]. Despite these methodological
A Ship Weather Routing Optimization Method
Incorporating Ship Seakeeping Model
Z. Han
1
, Y. Zhou
2
, J. Zhang
1
& D. Wu
1
1
Wuhan University of Technology, Wuhan, China
2
Beibu Gulf Port Co., Ltd., Nanning, China
ABSTRACT: Weather routing problem represents one of the most significant advancements in ship energy
conservation and emission reduction. Weather routing problems involve a numerous spatio-temporal ship
motion states and weather characteristics, leading to challenges in consumption estimation and computational
efficiency. To address this, an improved weather-state lattice-based method is first proposed, taking into account
the ship's seakeeping model, weather characteristics, and consumption optimal control. The optimal energy
conservation trajectory considering ship course under discrete motion states and weather characteristics is
determined by optimal control and defined as motion primitives. Furthermore, the graph search method
combined with the motion primitives is applied to generate the optimal energy trajectories. The results indicate
that the framework can generate lower-consumption trajectories by leveraging weather-induced forces.
Moreover, incorporating the seakeeping model into the weather routing framework not only enhances the
accuracy of energy consumption estimation but also significantly improves the manoeuvrability of the generated
trajectories.
http://www.transnav.eu
the International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 19
Number 1
March 2025
DOI: 10.12716/1001.19.01.13
106
advancements, several limitations persist due to
simplifying assumptions in these studies. For instance,
most existing methods neglect the ship manoeuvring
model under varying weather conditions and only
account for weather-induce forces in the surge
direction, while disregarding their influence on sway
and yaw. It leads to the resulting paths cannot be
directly applied as reference trajectories, requiring
additional smoothing to adapt to actual ship
navigation.
To address this limitation, the weather-induced
effects on ship motion have attracted significant
attention, particularly their influence on other degrees
of freedom. For instance, weather-induced lateral sway
forces can significantly increase navigational drift,
altering effective travel distance. Similarly, weather-
induced yaw moments may restrict planned course
alterations, compromising path feasibility.
Traditionally, ship manoeuvring models primarily
served controller design for path tracking, treating
environmental forces as disturbances to be eliminated
[8]. With advancements in wind, wave, and ocean
current simulation, the influence of these weather
characteristics on ship motion are further analysed and
incorporated into ship manoeuvring model, referred to
the seakeeping model [9]. Leveraging this model, the
weather characters are considered as constraints, and
the optimal control has been introduced to improve the
trajectory performance [10]. However, these studies are
typically applied in the static environment. In other
words, the force and moments induced by weather
characters are not a spatio-temporal variables, which is
inconsistent with the transoceanic and long-term ship
navigation.
To address this, the improved lattice-based path
planning method considering discrete weather
condition is proposed and utilized to plan the optimal
trajectory. The motion primitives incorporate ship
seakeeping model and discrete weather characteristics
through optimal control, with energy consumption
explicitly minimized in the objective function.
Crucially, these motion primitives capture weather-
induced force variations resulting from course
alterations. The resulting navigation graph,
constructed from these motion primitives, is then
solved using graph search methods to determine the
optimal path between origin and destination.
The paper is organized as follows: Section 2 details
the proposed methodology. Section 3 presents
numerical validation through comparative case
studies. Conclusions are drawn in Section 4.
2 METHODOLOGY
Section 2.1 presents a 3-DOF ship seakeeping model
that calculates weather-induced forces and moments
acting on ship motion. This model serves as critical
constraints for energy optimal control in Section 2.2.
The energy-optimal motion primitives under discrete
weather conditions are derived by optimal control and
used in graph searching process to plan the minimum-
energy trajectories in Section 2.3.
2.1 3-DOF ship seakeeping manoeuvring model and
energy optimal control
The ship motion considering weather-induced force
and moment is typically described by the 3-DOF
seakeeping model as shown in Figure. 1. The ship's
position and posture are typically described by the
pose vector
=[N,E,
]
T
in North-East-Down (NED)
coordinate system, where N and E denote the ship's
position in the north and east directions, and
represents the yaw angle. The time derivative of the
pose vector is determined by the velocity vector
v =[u,v,r]
T
in body-fixed coordinates, where u, v, and r
correspond to the surge, sway linear velocities, and
yaw angular velocity, respectively and the rotation
matrix J(left
), as expressed in Eq. (1) and Eq. (2):
( )
,
=η J υ
(1)
with
( )
( ) ( )
( ) ( )
0
0.
0 0 1
cos sin
sin cos
−
=
J
(2)
The weather effects on the ship motion are
formulated in the body-fixed coordinate. Generally,
the irrotational ocean current is marked as voc, which
combines with the ship's relative velocities vr to
determine the ships velocity over ground, as shown in
Eq. (3). Furthermore, the wind-induced
wind = [fu
wind
,fv
wind
,fr
wind
]
T
, wave-induce
wave = [fu
wave
,fv
wave
,fr
wave
]
T
and thrust force and rudder
force moment
= [fu,fv,fr]
T
determine the time derivative
of relative velocity vr, as indicated in Eq. (4),
=−
rc
υ υ υ
(3)
( ) ( )
++ + = +
r r r r r wind wave
Mυ C υ υ D υ υ τ τ τ
(4)
where, M includes both the rigid body and added mass
terms, C represents the Coriolis-centripetal matrix, D is
the hydrodynamic damping matrix, vc signifies the
ocean current velocity, which combines with the
relative velocity vector to constitute the speed over
ground.
The total ship resistance comprises the calm water
resistance, wind resistance, and wave add resistance.
While previous studies focus on the added force solely
on surge direction, this study considers the weather-
induced force and moments in 3-DOF (surge, sway,
and yaw) direction. This comprehensive treatment
captures the complete spectrum of environmental
interactions affecting ship performance.
107
Figure 1. The seakeeping model in NED and body-fixed
coordinates.
The ocean current force and moments is considered
int hydrodynamic damping D matrix, as indicated in
Eq. (5),
( )
00
0,
0
r
uu
rr
v v r v v r r r
rr
v v r v v v r r
Xu
Y v Y r Y v Y r
N v N r N v N r
= + +
++
r
Dv
(5)
where X|u|u, Y|v|v, Y|r|v, N|v|v and N|r|v represent the
hydrodynamic coefficients. The calm water resistance
in the surge direction is determined by X|u|u in line with
other weather routine research.
The wind force and moments acting on the ship
seakeeping model are computed by ship velocity u and
v, wind velocity Vw and angle of attack
w, as shown in
Eq. (6) and Eq. (7),
( )
( )
( )
22
arctan ,
cos
sin
w w w
w w w
ru w
ru w
ru ru ru
ww
ru ru ru
uV
vV
u u u
v v v
vu
V u v
=−
=−
=−
=−
=−
= − −
=+
(6)
( )
( )
( )
2
1
2
x ru fw
a ru y ru lw
n ru lw oa
CA
V C A
C A L
=
wind
τ
(7)
where, uru and vru denote the surge and sway relative
velocity,
ru denote the course between the ship velocity
and wind velocity,
a denote the air density, Afw and Alw
represent the frontal and lateral projected areas, Loa
represent the length of ship, Cx, Cy and Cn represent the
wind parameter depends on
ru.
The wave-induced forces and moments are
functions of both the wave characteristics and the
vessel's kinematic state. The sea state is modelled using
the JONSWAP spectral density function S(w|Hs,TP),
parameterized by significant wave height Hs, peak
spectral period TP, and mean wave direction
wave,
which accurately represents wave energy distribution
for fully-developed open ocean conditions. The wave
encounter frequency
e is derived by ship velocity Vs ,
wave frequency
, the angle between the wave
direction and the ship's course 𝜇, while g represents
the gravitational acceleration. in Eq. (8), thereby
characterizing the effective wave energy spectrum
acting on the ship.
2
cos
s
e
V
g
=−
(8)
The hydrodynamic response of a vessel to wave
excitation is commonly characterized by the Response
Amplitude Operator (RAO), denoted as Caw. The wave
energy density within a given frequency band can be
derived from the wave spectrum S
(
e), adjusted for
the encounter frequency 𝜔
𝑒
, yielding the wave
amplitude component
0 according to Eq. (9):
( )
0
2
ee
S
=
(9)
Through spectral integration across all frequency
components, the resultant mean wave-induced forces
and moments are obtained as expressed in Eq. (10):
( )
0
2
aw aw
R C S d
=
(10)
2.2 Motion primitives determined by energy optimal
control
This section presents a quantitative evaluation of
energy consumption for different path, and illustrates
the application of optimal control to determine the
most energy-efficient trajectory.
As illustrated in Figure 2, while all trajectories share
identical origin, destination, and weather conditions,
substantial energy consumption discrepancies emerge
due to variations in initial and terminal course.
Moreover, even with same origin and destination
courses, course adjustments along the path generate
different force profiles and corresponding energy
consumption.
Figure 2. Weather-induced force in different trajectories.
Weather routing for energy conservation can be
fundamentally formulated as an energy optimization
108
problem subject to weather-induced force and moment
constraints. In the Section 2.1, the 3-DOF seakeeping
model is introduced to describe the weather
characteristic influence on ship motion, while is then
transformed into equality constraints in the energy
optimal control problem. The path planning algorithm
enforces specified initial and end motion states as
boundary conditions for trajectory optimization. Note
that, the navigation time tf is also regarded as the
variable, which also be optimized by optimal control:
( ) ( ) ( ) ( ) ( ) ( )
0 , 0 , 0 , 0 , 0 , 0 , , , ,0,0 ,
init init init r
N E u v r N E u
=
(11)
( ) ( ) ( ) ( ) ( ) ( )
, , , , , , , , ,0,0
f f f f f f end end end r
N t E t t u t v t r t N E u
= ，
(12)
where Ninit, Einit,
init, Nend, Eend, and
end represent the
initial and end states, respectively, and
r
u
denotes the
constant surge velocity under the chosen propulsion
force.
In addition to this, the control inputs are bounds as
Eq. (13) - Eq. (14),
( )
min max
u u u
f f t f
(13)
( )
min max
r r r
f f t f
(14)
In this problem, the optimization objective is the
ship energy consumption, which is a combination of
navigation time tf, control input
(t) and velocity
ur(t).This study employs a constant relative velocity
assumption, thereby reducing the objective function as
shown in Eq. (15)
( )
( ) ( )
( )
1
0
,,
arg dt,
Seakeeping maneuverability constraints Eq.(1) - Eq.(10),
s.t. Control input constraints Eq.(13) - Eq.(14),
Initial and terminal constraints Eq.(11) and Eq.(12).
f
fu
x t t t
J t mint f t=
τ
(15)
The optimal control problem, which necessitates
consideration of the ship's motion state and weather
conditions, presents a large configuration space that is
challenging to solve directly. Thus, the optimal energy
trajectory is derived under time-invariant
environmental conditions. Moreover, the trajectory
optimization problem for long-term navigation under
spatio-temporal weather conditions is addressed
through graph search method as shown in Section 2.3.
2.3 State-lattice planner considering weather
characteristic
The graph search-based methods, such as the A*
algorithm, serve as fundamental path planning
techniques. Given the significant impact of weather-
induced forces and moments on the ship sway and yaw
DOF, the path between adjacent nodes is no longer
regarded as a straight path. Consequently, the state
lattice-based motion-planning approach, employed as
a path search method accounting for ship seakeeping
model and energy consumption optimal control, is
integrated with weather condition in this study.
However, the computational burden of real-time
optimal control between successive nodes remains
prohibitive for practical applications in dynamic
weather conditions. More critically, the combinatorial
explosion of possible wind-wave-current interactions
makes complete enumeration of energy-optimal
solutions for all potential weather scenarios
computationally intractable. the weather characters
including ocean current velocity, wind velocity, wind
angle of attack, significant wave height, wave period
and wave direction are discretized. Directional
parameters follow the standard 16-point compass
division (22.5° resolution), maintaining compatibility
with conventional navigational practices (north-east,
north-west, etc). Weather condition intensity
parameters are discretized according to the Beaufort
scale and Douglas sea state classifications, as detailed
in Tables 1 and 2, with threshold values adapted for
regional conditions in the target navigation area.
However, ship manoeuvrability becomes severely
compromised under sea state code force 5 conditions,
which generally exceed operational safety thresholds
for commercial shipping. Consequently, this study
explicitly excludes such hazardous conditions, treating
all sea states beyond 5 as navigationally infeasible.
Table 1. Beaufort number table
Beaufort
Number
Wind Description
Wind Speed
(m/s)
Average Wind
Speed (m/s)
0
Calm
0–0.51
0.255
1
Light Air
1.03–1.54
1.285
2
Light Breeze
2.06–3.60
2.83
3
Gentle Breeze
4.12–5.66
4.89
4
Moderate Breeze
6.17–8.23
7.20
5
Fresh Breeze
8.75–10.80
9.78
6
Strong Breeze
11.32–13.89
12.605
7
Near Gale
14.40–16.97
15.685
8
Gale
17.49–20.58
19.035
9
Strong Gale
21.09–24.69
22.89
10
Storm
25.20–28.80
27.00
11
Violent Storm
29.32–33.40
31.36
12
Hurricane-force
> 32.7
32.7
Table 2. Sea state code table
Sea State Code
Description
Wave Height (m)
0
Calm (like a mirror)
0
1
Calm (rippled)
0–0.1
2
Smooth (small wavelets)
0.1–0.5
3
Slight
0.5–1.25
4
Moderate
1.25–2.5
5
Rough
2.5–4.0
6
Very rough
4.0–6.0
7
High
6.0–9.0
8
Very high
9.0–14.0
9
Phenomenal
Over 14.
Furthermore, the ship's motion states require
discretization to facilitate graph construction. As
weather data typically exists in gridded format, the
ship motion state N and E positional coordinates are
discretized correspondingly. For comprehensive
analysis of course-dependent energy expenditure, the
yaw angle
is quantized in
increments. To maintain
computational tractability, velocity states are
simplified to a constant velocity
r
u
, with zero lateral
and rotational velocity components, thereby bounding
the search space.
After determining the initial, end motion states and
weather character combinations, the motion primitives
consist of representative trajectories under various
weather condition is obtained by optimal control as
shown in Figure 3. The four circles in Figure 3 represent
the level and direction of wave and ocean current.
Note that, the trajectories with course information are
109
used to connect the adjacent nodes, which accounts for
the ship's seakeeping model. Furthermore, due to
weather difference, the minimum energy consumption
trajectory derived from optimal control also varies.
Compared with related approach that only considers
the force in the ship surge direction, this method
further improves the trajectory performance with the
help of weather to affect the lateral force and torque of
the ship.
Figure 3. Motion primitives in different weather condition.
An example of the motion primitives with an initial
course of 0 are presented, along with their fuel
consumption and navigation time as shown in Figure
4. Due to the difference in weather conditions, the
trajectories from the same initial state to the target
position differ, and the trajectories are not symmetric.
Both energy consumption and navigation time can be
simultaneously reduced when navigating with
favourable waves and currents. Therefore, a graph
search algorithm must be employed to consider both
the distance and energy-saving performance in order
to generate the optimal path.
Figure 4. Propulsion difference in different weather
conditions.
In line with A*, the cost function is divided into
accumulated cost G and heuristic cost H. The path cost
G is determined as the energy accumulation of
multiple motion primitives, and the heuristic cost H is
determined as the energy consumption estimation
between nodes and destination without wind, wave
and ocean current. It is noteworthy that, as the voyage
duration to each node remains preserved, the varied
weather conditions at individual nodes will undergo
continuous temporal updates during the search
process.
3 CASE STUDY
In this section, weather routine problem with different
weather characteristics is analysed to evaluate the
performance and efficiency of the algorithms. A
container ship seakeeping model s175 with estimated
RAOs in 3-DOF is used. To reduce the difficulty of the
optimal control, only the quadratic terms of the ship's
higher-order hydrodynamic effects in Damper matrix
are retained. The estimated energy by this algorithm
caused by course change is compared with other
weather routing problems. Next, the advantage of the
approach will be demonstrated caused by considering
the seakeeping model. Lastly, the computational
efficiency is assessed. All evaluations are conducted on
a PC with an Intel i7 CPU@1.8GHz and 16GB of RAM.
In this study the proposed algorithm is validated in
the North Atlantic within the longitude range of 74W
to 5W and the latitude range of 45N to 55N. The
weather data is obtained from ERA5 datasets with a
time interval of 3h. Since the wind and current data in
ERA5 have a resolution of 0.5° and the wave data has a
resolution of 1°, all data will be interpolated to a
uniform resolution of 1°. In this section, the great circle
routing and conventional grid-based weather routing
method A* is compared. The trajectories planned by
three methods as shown in Figure 5, Figure 6 and
Figure 7. In contrast to the influence of wind and
current on energy consumption, wave impact exhibits
a more substantial and immediate effect on ship power.
Consequently, a comparative analysis is conducted
between the shaft power output and wave-induced
power, as shown in Figure 8– Figure 10.
The great circle route represents the shortest
distance between the starting and destination points in
the coordinate system, thus offering the minimum
distance and shortest voyage time. However, as shown
in Figure 6, the vessel's heading during 15 - 23 hours
presents a bow-sea condition, resulting in significantly
increased wave resistance. The vessel's shaft power
approaches 10000 KW, exceeding the main engine's
available power range. Furthermore, navigation under
such sea conditions proves exceptionally challenging
to maneuver, rendering this route unsafe.
Both the weather-conditioned A* algorithm and the
weather state lattice approach generated navigational
routes that strategically avoided regions with high
wave conditions. Moreover, both algorithms
optimized trajectories favoring following-sea
navigation, leveraging wave action to provide
additional propulsive energy and thereby reducing
main engine power demand. While both conventional
A* and weather state lattice algorithms produce paths
with similar trajectory patterns, their energy
consumption differs due to two primary factors. First,
the A* algorithm employs fixed-path connections
between adjacent nodes, whereas the state-lattice
method dynamically adjusts vessel navigation
between waypoints. Within each grid cell, it optimizes
course variations to minimize wave resistance, as
illustrated in Section 2.2 and Section 3.3. The second
contributing factor stems from the seakeeping model's
comprehensive approach. Rather than solely
accounting for surge direction, it incorporates the
lateral displacement effects induced by waves and
110
wind, thereby facilitating additional energy savings
through optimized course corrections.
Figure 5. The trajectories planned by great circle, A* and
proposed method in wave map.
Figure 6. The trajectories planned by great circle, A* and
proposed method in wind map.
Figure 7. The trajectories planned by great circle, A* and
proposed method in ocean current velocity map.
Figure 8. The shaft power and wave power along with the
great circle routine.
Figure 9. The shaft power and wave power along with the
path planned by A*.
Figure 10. The shaft power and wave power along with the
path planned by weather-state latticed.
Table 3. Trajectory method comparison.
Method
Navigation time
(hour)
Energy consumption
(kw*h)
Great Circle
121.89
577190.07
A* with weather condition
140.49
557656.45
State lattice
145.70
461831.49
Furthermore, conventional A* and great circle
methods inherently lack vessel maneuverability
constraints, frequently generating kinematically
infeasible course alterations (e.g., abrupt 90° or 180°
turns) that necessitate trajectory smoothing before
being executable by the ship's control system. In
contrast, our proposed state-lattice framework
intrinsically incorporates the seakeeping dynamics,
yielding navigation paths that are both optimized for
energy efficiency and inherently compliant with the
ship's maneuvering characteristics, thus requiring no
additional preprocessing for execution.
4 CONCLUSIONS
This study presents an innovative method that
integrates seakeeping model and optimal control for
weather routine. The core of the methodology lies in a
weather state-lattice approach that precomputes
energy-optimal motion primitives across discretized
environmental conditions while incorporating 3-DOF
seakeeping model and optimal control. A graph search
algorithm then connects these motion primitives,
accounting for spatiotemporal weather transitions to
enable global energy optimal between departure and
destination. Validated through simulations in North
Atlantic confirm the method's effectiveness in energy
reduction and trajectory performance. The proposed
method enables ship-specific weather routing
optimization incorporating its seakeeping model.
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