J.R. Marx et al.
Ocean Engineering 343 (2026) 123388
5.1. Simulation tests
namics, and actuation characteristics. The sampling time T, is chosen
to match the dynamic response of each vessel. The smaller and lighter
;hips MESSIN and BELA exhibits fast maneuvering behavior and thus re-
]uires a finer temporal resolution to capture control-relevant dynamics.
in contrast, the larger vessel DENEB can be controlled adequately with
A coarser discretization due to it’s higher inertia and slower response
characteristics. The prediction horizon H, and control horizon H, are
:uned accordingly to provide sufficient foresight while keeping com-
utational demands manageable. The input update interval /;„ defines
ı10w often the controller is allowed to modify the control input within
‘he prediction horizon. This interval is set higher for the DENEB to re-
luce actuator workload and imitate natural steering behavior. Smaller,
nore agile vessels instead benefit from more frequent updates to some
extend.
The weighting matrices Q and R reflect the relative importance of
accurate trajectory tracking and control efficiency. To ensure compara-
>ility across states and inputs with different physical units (e.g., linear
and angular positions in Q; forces and torques in R), the matrices are
scaled accordingly. This normalization guarantees that each component
:ontributes meaningfully to the optimization cost, regardless of its mag-
ıitude or physical unit. Simulations mainly were used to tune the Q and
R matrices. Initially, each diagonal element was scaled by an appropri-
ate denominator to normalize the impact of different physical units. For
‘he Q matrix, the initial diagonal was set to [1/0.5?, 1/0.5?, 1/0.0174?]
such that a position error of 0.5 m is weighted equally to a heading er-
:or of 1° (0.0174 rad). For the R matrix, illustrated here for the vessel
DENEB, the diagonal was initialized with [1/1? 1/1? 1/100°] implying
‘hat an effort of 1 N is considered equivalent to a moment of 100 Nm.
As the cost function is quadratic, all denominators are squared accord-
.ngly.
To tune Q against R, multiple simulation trials were conducted.
ıf large deviations from the reference trajectory occurred, the weights
n the Q matrix were increased. Conversely, if the tracking behavior
showed osecillations or instability, the Q values were reduced to allow
;moother control actions. Simultaneously, input weights are adjusted
:o favour or disfavour specific force directions or yaw moment. In gen-
aral, lateral forces and frequent changes in heading are considered un-
desirable. In contrast, adjustments to the forward velocity, primarily
governed by the surge force in X direction, can typically be made in a
nore energy-efficient and cost-effective manner. These considerations
are taken into account in the selection of the parameter values for the
R matrix and to fine-tune the controller.
While the relative weighting within the Q matrix remains constant
across all vessels, the entries of R, as well as the overall balance be-
5.1.1. Environment of simulation tests
The simulation environment served multiple purposes, for the de-
velopment of basic control and optimization functionalities, but also of
che test procedures, quantification of disturbance influences or suitable
visualization tools for the responsible skippers and the captain as the
aighest supervisor. In its final configuration, which can be seen in Fig. 9,
't differs from the equipment that was installed on board the Deneb only
ın that motion models were used instead of the real vehicles.
On the far right of the picture, the boxes can be seen which are
mounted on BELA or MESSIN and which mainly contain an industrial
PC (IPC) for control and the router for the network communication. The
dynamic motion models run on the PCs to the left. The rack in the cen-
cer of the picture contains the IPC for the control system for the DENEB.
The switch for the connection to the ship’s network, ie. to the actuators,
sensors and bridge, is also located here. The monitor to the right shows
a human machine interface (HMI) for online parameterization of the
:ontrol systems. The devices on the left-hand side of Fig. 9 are used to
»perate the overall system and to visualize current operating modes or
:ommands. This includes the RCD and a musical instrument digital in-
:erface (MIDI) board, which are primarily used for manual control and
visualization of the commands to the DENEB in automatic mode, but
also for implementing the entire cooperative scenario, as described in
Section 5.2.1.
The visualization was tested in the simulation to optimize the set-
iungs for the engineers and the captain alike, so that they can jointly
evaluate the maneuvering in the back of the DENEB bridge during the
ceal-world tests. The visualization interface could already be seen in
Fig. 4. In addition, the effects of wind influences were investigated by
applying heading dependend wind forces to the motion models.
5.1.2. Results of simulation tests
The simulation tests were primarily used to examine and parame-
cerize the algorithms developed for optimization and control without
endangering people or materials. The decisive indicator for suitable con-
croller parameters were the safety aspects mentioned above, which were
defined in Section 2.3. The parameters for each MPC are summarized in
che Tables 5-7. They contain the final values for the sampling time T,,
che number of samples for prediction horizon H,, the control horizon
H, and the control update interval /,„, as well as the state weighting
matrix Q and input weighting matrix R.
While the overall structure of the controller remains consistent, the
specific values differ significantly due to variations in vessel size, dyv-
PCs DENEB
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