J.R. Marx et al. 
Table 5 
MPC parameter settings for vessel DENEB. 
Symbol Description Valne 
Sampling time 1 
Prediction horizon RO 
Control horizon } 
Control update interval 
Unit 
Samples 
Samples 
Samples 
+ 
20000 0 d 
20000 0 
3 0 15625000 
0.4 0 o 1 
016 0 | 
7 0 = 0.007% 
I 
State weighting matrix 
SI 
R 
Input weighting matrix 
SI 
Icean Engineering 343 (2026) 123388 
Table 6 
MPC parameter settings for vessel BELA. 
Symbol Description Val 
Sampling time 0? 
D?rediction horizon 75 
Zontrol horizon 8 
Zontrol update interval 10 
19 
Unit 
Ss 
Samples 
Samples 
Samples 
State weighting matrix 
0 
20 0 |] 
I 156.25 
u.16 0 (0 
) 0.4 0 ! 
x N 0.4087 
SI 
Input weighting matrix 
5] 
Fable 7 
MPC parameter settings for vessel MESSIN. 
Symbol Description Value 
Sampling time 0.7 
Prediction horizon 7 
Zontrol horizon 
Control update interval 
Unit 
Ss 
Samples 
Samples 
Samples 
) 
State weighting matrix 
"40 0 0 
) 40 0 
9 0 3125 
1.2344 0 0 | 
0 0.0010 0 
a a:1212 
SI 
Input weighting matrix 
SI 
While the initial maneuver planning trajectories are used for the first 
‚teration, the optimized trajectories are used as a basis for each subse- 
quent iteration. The algorithm works with the respective motion models 
of the vehicles involved as constraints. Inputs to each model are the spe- 
cific force vectors rt. The state vector x is defined as output of the model 
with x = (x, y, P,u,v,r)! , where x, y,F represent the position and the 
neading angle of the vehicle as well as u, v,r are the velocities in the 
‘hree DoFs. A head-on situation must take place with each vehicle evad- 
ng to starboard. The defined distances of the ships to each other and 
:©O the port facilities are achieved by circles that are drawn around the 
respective object and are thus taken into account in the optimization. 
The second problem OCP 2 is dedicated to the crossing situation. The 
zreen ship must allow the other two ships to pass before entering the 
[airway. Optimally, this behavior is achieved by reducing the speed just 
enough so that the green vehicle does not have to stop. Actual stopping 
's very energy-intensive for watercraft, as they have long stopping dis- 
:ances depending on their current speed, which can only be achieved by 
angaging reverse gear. From a nautical perspective, it is preferable that 
‘he potential danger spot is only reached when the risk of collision no 
‚onger exists. For the optimization, this means that a time must be cal- 
zulated when the trajectory tracking is started. In the second optimiza- 
tion cycle, the resulting trajectories of OCP 1 are involved as obstacles, 
marked as gray lines in the lower, right box in Fig. 8. The integration of 
SOLREG rules into optimal control formulations for head-on and cross- 
ing encounters, including suitable cost functions and geometric safety 
zones, has been presented in detail by Eisenblätter et al. (2025). 
{t should be noted that the successive decomposition into two ship 
scenarios (OCP 1, OCP 2,) may lead to a loss of global optimality, since 
the solution of a higher-priority sub problem can restrict the feasible set 
of subsequent problems. This trade-off between computational tractabil- 
ty and optimality is inherent in the proposed approach. Future work 
could address this by considering simultaneous multi-ship optimization 
or by integrating global and local optimization layers. 
An important aspect for the practical deployment of IPOPT is its com- 
autational complexity, especially when considering embedded hard- 
ware. Since optimization algorithms can be demanding in terms of com- 
utation time, this may restrict their use in applications with fast re- 
;ponse requirements or on resource-constrained platforms (Jerez et al., 
2014). In IPOPT, a major part of the effort arises from solving linear 
systems in each iteration (Wächter and Biegler, 2006), which can be 
challenging on embedded processors with limited capability. Therefore, 
while IPOPT is well suited for prototyping and for large-scale problems 
on desktop-class CPUs, its applicability to real-time embedded imple- 
mentations is more limited, unless problem sizes are small or dedicated 
salver frameworks are used (Zanelli et al.. 2017). 
3. Results and discussion 
rhis section is dedicated with the test environments for the simula- 
don and the real-world experiments as well as the results of these tests.