J.R. Marx et al. 
Ocean Engineering 343 (2026) 123388 
ro 
50 
zZ 50 
5 40 
Lı 
a 
A 30 
20 
GO 
EL A A A a 
—-40 -30 -20 -10 0 10 20 30 40 
Yaw Moment N [Nm] 
Fig. 7. Admissible surge force-yaw moment region for MESSIN. The asymmetric shape reflects actuator limitations, including a required minimum thrust to avoid 
‚dle state. 
via decentralized thrust, which inherently produces a sway component. 
Instead of redesigning the entire MPC and changing the model equa- 
dons to a two-input (X and N force) system, a coupling constraint of 
che form 
Yeah N (18) 
is applied to the predictive control formulation (15) to keep the 
physical relations consistent when switching to underactuated XN- 
allocation. The gain hy represents the effective ratio through which 
yaw moments induce lateral forces. The value of hy is vessel-specific 
and was determined through system identification and geometric anal- 
ysis. The following values are used: 
> MESSIN: hyy = —0.5943 
» BELA: hyy = —0.25 
» DENEB: hyy = —0.045 
This constraint ensures that the controller no longer optimizes over the 
zway component independently, but instead respects the physical cou- 
oling imposed by the reduced actuation configuration. 
4) Low-rate Actuation Updates. As a final measure to reduce wear on 
che actuators and improve commanding smoothness, the controller is 
designed to update the control inputs only at every ith time step rather 
chan at every sampling time. This intentional reduction of the effective 
actuation rate is particularly useful for marine applications, where fre- 
quent changes in actuator commands can lead to unnecessary energy 
consumption, mechanical load, or reduced comfort. 
To implement this behavior, the change in commanded forces Art, = 
Tx — Tx_1 is constrained. Specifically, the following condition is enforced 
over the control horizon: 
Ar =0, VkeE{kosko+i,ko +24... } (19) 
where ; € N denotes the actuation update interval and kg the start index 
jf the horizon. As a result, the optimization can be run less frequently, 
only every i-th control step, without compromising of the prediction 
accuracy, since the sampling time for state updates and predictions re- 
nains unchanged. This approach effectively decouples the optimization 
‚.nterval from the system’s sampling rate, allowing computational effort 
and actuator workload to be reduced simultaneously. 
4. Trajectory optimization for collision avoidance 
Path and motion planning are classified as problems for which no 
:ime-efficient algorithm can be found so that they could be used in prac- 
ice (LaValle, 2006). Therefore, numerous methods have been developed 
‘hat can solve the problem for special problem classes in limited com- 
uting time by making compromises regarding the solution. The most 
mportant method classes are briefly described below. 
Grid-based methods decompose the configuration space into a grid 
f typically equally sized cells. There are free cells and cells occupied by 
>bstacles. Search algorithms from graph theory are used to find an op- 
:imal path from the starting point to the destination. One disadvantage 
öf this method class is that the grid resolution has a decisive influence 
ın performance. The best-known algorithms are A* (Hart et al., 1968) 
and D* (Stentz, 2003). 
Geometric methods create a geometric model for the free space, usu- 
ally with the help of polygons, which in turn is decomposed into cells. 
A graph is created from the neighboring cells, which is systematically 
searched. Examples of this cell decomposition method can be found in 
„ingelbach (2004), Gonzalez et al. (2017) and Sleumer and Tschichold- 
Gürmann (1999). 
Sampling-based methods are another important class of methods for 
solving planning problems. One advantage of this class is that no ex- 
plicit modeling of the configuration space is necessary. Instead, the 
space is explored by a large number of random samples. A black-box 
unction is sufficient to determine whether a sampled configuration is 
collision-free or not. However, this method cannot be used to deter- 
nine whether a solution exists or not, which is why it is referred to 
as probabilistic completeness (LaValle, 2006). The best-known meth- 
nds of this class include probabilistic roadmap (PRM) (Kavraki et al., 
1996a) and rapidly-exploring random tree (RRT) (LaValle, 1998). The 
PRM method is particularly advantageous when many combinations of 
start and target points have to be examined and the configuration space 
does not change (Kavraki et al., 1996b). In contrast to PRM, RRT builds 
a tree based on the starting point by sampling random configurations 
and adding them to the existing tree. RRT has proven its flexibility and 
simple implementation in practice and is considered state-of-the-art for 
;olving planning problems (Chen et al., 2018: Noreen et al., 2016). The