J.R. Marx et al. 
to public traffic and also has a low current speed. Depending on the 
wind direction, there are areas sheltered from wind near the outer piers, 
which are lined with tall trees. The scenario was designed using the 
SAMMON planning software from ISSIMS GmbH based on the dynamic 
notion model of research vessel DENEB, the characteristics of the other 
:wo vehicles and all nautical relevant aspects as safety distances to the 
ort structures and adjusted speeds. To create defined framework con- 
ditions, the maneuver plans start and end with a speed of zero, so that 
‘he vehicles accelerate until they encounter each other and then slow 
down again. DENEB and BELA reach a maximum speed of 4 kn, MESSIN 
according to her propulsion performance of 3kn. The straight section is 
approx. 1000 m. The figure shows the time of 350s after the start of the 
maneuver, whereby several collisions would inevitably occur without 
avasive maneuvers. Following the evasion rules, this is a crossing and a 
ıead-on situation. The evasive maneuvers should be as close as possible 
cO the initial maneuver plans, but maintain the safety distances between 
‘he vessels and to the port structures. For the automatic tests, DENEB 
and BELA can also start from the opposite side, saving time. 
3. Controller approaches 
The increasing need for autonomy, safety, and efficiency in maritime 
operations has made advanced control strategies a cornerstone of mod- 
ern vessel automation. Among these strategies, Model Predictive Control 
‘'MPC) has emerged as a particularly powerful and versatile method due 
:o its inherent ability to prediet future states, incorporate constraints, 
and deal with multivariable, nonlinear systems. 
MPC is especially relevant for inland navigation and nearshore oper- 
ations where confined spaces, hydrodynamic complexity, and environ- 
mental disturbances pose significant challenges. It has been successfully 
applied to dynamic positioning (DP) (Veksler et al., 2016), trajectory 
tracking (Zheng et al., 2014), and energy-optimal docking (Homburger 
et al., 2024b), leveraging its predictive capacity to ensure precision even 
ın uncertain or highly dynamic conditions. 
Various forms of MPC are used in the maritime domain. Classical 
‚inear MPC (LMPC) is computationally efficient but limited in its appli- 
cability to nonlinear systems e.g. simplified motion models. Nonlinear 
MPC, as detailed by Camacho and Alba (2013), or more recently by 
Homburger et al. (2024a), allows accurate modelling but often exceeds 
real-time constraints in onboard applications. 
Modern developments extend MPC with learning components to 
overcome modelling limitations. Iterative Learning Control (ILC)-based 
MPC (Homburger et al., 2024a) or data-driven approaches such as those 
Jased on Deep Reinforcement Learning (DRL) (Zare et al., 2021) have 
shown potential in simulation and experimental trials, although their 
sample inefficiency and need for large datasets remain a challenge. 
Stochastic sampling-based methods like Model Predictive Path Inte- 
gral Control (MPPI), introduced in Theodorou and Todorov (2012), offer 
another promising alternative by approximating the solution to optimal 
:ontrol problems via information-theoretic principles. These approaches 
nave shown excellent performance in docking scenarios involving strong 
zurrents and measurement noise (Homburger et al., 2022). 
In contrast, classical techniques such as PID control, sliding mode 
zontrol (SMC) (Liu et al., 2018), adaptive control (Sgrensen, 2005), and 
teedback linearization (Lutz and Meurer, 2021) are still in use, particu- 
‚arly where system complexity is low or hardware resources are limited. 
However, they generally lack the flexibility and constraint-handling ca- 
pability of MPC. As shown in comparative studies (Wirtensohn et al., 
2021), MPC-based methods tend to outperform traditional approaches 
under realistic conditions. 
Foundational contributions such as those by Fossen (2011, 2002) 
nave laid the groundwork for modern marine control theory, providing 
standard 3DOF vessel models and control-oriented hydrodynamic for- 
nulations. Building on these, current MPC applications address issues 
like underactuation, actuator constraints, and environmental prediction, 
as detailed in reviews such as (Hewing et al., 2020). 
Ocean Engineering 343 (2026) 123388 
In summary, MPC is not only a state-of-the-art control technique in 
naritime automation, but it also continues to evolve rapidly through hy- 
öridization with learning, stochastic, and optimization-based methods. 
[ts adaptability to both theoretical development and real-world applica- 
ion ensures its central role in future maritime control systems. 
Especially in the context of trajectory tracking Model Predictive Con- 
crol is well suited due to several advantages. 
» Prediction capability is enabled through a model-based forecast of 
future system behavior across a prediction horizon. 
Constraint handling allows explicit inclusion of actuator and safety 
limits in the optimization problem. 
Offset-free tracking ensures accurate convergence to the reference 
aven in the presence of constant disturbances and modeling errors. 
Modular adaptability to different vessels is possible by re- 
identifying model parameters. 
Support for nonlinear behavior is realized through nonlinear opti- 
mization. 
'hese properties make MPC ideal for spatially and temporally accu- 
‘ate trajectory tracking, particularly in the presence of external distur- 
jances and system uncertainties, as supported in Marx et al. (2024) and 
Zheng et al (2014). 
A purely nonlinear MPC approach is capable of representing the non- 
inear behavior directly. However, it often comes with a high computa- 
:ional burden and requires complex solver infrastructures that may be 
nfeasible for real-time implementation on embedded systems. Linear 
MIPC, on the other hand, assumes time-invariant linear dynamics, which 
imits its performance for vessels with complex dynamics. 
For this work, a viable middle ground is the use of linearized MPC, 
where the system model is locally linearized at each sampling instant. 
This allows capturing essential nonlinear effects while preserving con- 
vexity of the optimization problem. It enables efficient QP-based so- 
‚utions with strong tracking and disturbance rejection performance, as 
demonstrated in Marx et al. (2023) and Dhar and Bhasin (2018). 
3.1. Motion models 
The base for conventional MPC is the availability of accurate model. 
To capture vessel dynamics sufficiently, a 3-DOF model in surge (uw), 
sway (v), and yaw rate (r) is used. The state vector is defined as 
v = [u v r]'. The dynamic behavior of the vessel is identified experimen- 
:ally using test data from real-world trials and encoded in the follow- 
ıng structure, where the nonlinear dynamic model is split into free and 
forced response terms 
v=f(v,t)= g(v) + Ber 
—— 
free forced 
( vr u u? 
g(v)= F.lAv+Clur|+ D|v* |+ E|v? 
{ UD DS zz 
X 
Be=F-B. [| 
N 
where r is the input vector representing surge force X, sway force Y and 
yaw moment N. The matrices A through F are 3 x 3 and vessel-specific. 
'dentified parameter values differ for each vessel and are summarized in 
‘he tables below. These matrices account for linear coupling, quadratic 
and cubic damping, and parameter scaling (Tables 2-4). 
Vessel trajectories are defined in earth-fixed coordinates, consisting 
of the position vector n = [x, y, v1’, where x,, y, are the Cartesian po- 
sitions and w is the heading angle. The transformation from body-fixed 
to earth-fixed velocity is performed using the rotation matrix Row) as 
[ cosy  -—-siny 0O0' 
sin cos vr 0 
0 f 
‚71No full text available for this image