/.R. Marx et al. 
Decoupling & 
Prioritization 
3-Vehicles- 2-Vehiles- 
Scenarios 
INPUT 
nitial U) 
| ) 
Parametrizatioh 
Scenarios 
+ Prio 
OCP - Optimal 
Control Problems 
SA 
N 
A a 
Ucre T 
r [vehide Model] x 
Yillr 
Ocean Engineering 343 (2026) 123388 
OUTPUT 
Energy-optimal 
& collision-free 
Trajectorv 
nn 
Fig. 8. Successive steps of the developed optimization method. 
most important extension of the RRT algorithm is RRT*, where a defined 
cost value is optimized by restructuring the edges of a tree (Karaman and 
Frazzoli, 2010). For example, each node in the tree can be assigned a 
cost value, which is the sum of the edge lengths starting from the root 
node. This makes it possible to search for optimal connections in the 
:ree that have a minimum cost value. The authors applied the RRT* 
[or real-time path planning for maritime, autonomous surface vehicles 
‚Damerius and Jeinsch, 2022). 
The potential field method offers another option for motion planning 
Gu et al., 2019; Lin et al., 2020). The target point is assigned to a sink 
in a potential field, while obstacles are assigned high potentials. The 
minimum cost for a path is achieved by continuously taking steps in 
the direction of the steepest gradient. The decisive disadvantage of this 
method is that a minimum found may not correspond to the target, but 
cO a local minimum (Likhachev and Ferguson, 2009). 
Trajectory planning can also be formulated as an optimization prob- 
lem (Lewis et al., 2012), as applied in this contribution. If xg is the initial 
state at time fg and x, is the final state at time f-, then the manipulated 
variable u(f) is sought which minimizes a given cost function / under a 
series of constraints. 
min J(f,,x(f), u(f)) 
Ur + 
subject to x = f(x(P), u(f)) 
Urin < u(f) < Urax 
X(f9) = X9 
x(f£) =X+ 
hix()) <0 
The dynamic motion behavior is taken into account by the differential 
zonstraint x = f(x(f), u(f)). The constraint u; < u(f) < Upmax Ensures that 
he input variables lie within a permissible range. At the same time, 
ihe initial and final states are given by the two constraints x(f9) = xg 
and x(f/) = x. Obstacles have to be defined by a non-linear constraint 
h(x(f)) < 0 must be described. A non-linear optimization problem arises, 
which must be solved with appropriate solvers, for which an overview 
is given in Betts (1998) and Rao (2010). 
4.1. Basic optimization method 
Like the control system, the trajectory optimization is based on dy- 
21amic motion models of the three vehicles. The equation of motion in 
‘hree degrees of freedom (DoF) is representing Newton’s second law as 
‚eferenced in Fossen (2011) with 
Mvyv+Cv)v=DvV)vVv+rT (21) 
where M is the inertia matrix, v is the velocity vector, D and C are the 
damping and the Coriolis and centripedal matrices, and r is the vector 
of virtual forces. Generally, it is calculated from the sum of the actuators 
thrust 7,., and additional external forces 7,,, such as wind or current for 
a motion model to 
r=(X, Y, NY = Tot 4 Toxı (22) 
where X and Y are the forces in longitudinal and lateral direction and N 
's the yaw moment. In this approach, the external forces are neglected or 
't is assumed that the controller is robust against external disturbances. 
Trajectory optimization is handled as a nonlinear optimal control 
problem (OCP), which is solved with an interior-point optimizer (IPOPT, 
Wächter and Biegler (2006)). This method is an efficient tool for nonlin- 
car programming in case of large-scale optimization problems. In order 
:©0 optimize cooperatively the trajectories of the three vehicles, the con- 
straints involved are very complex, the specific dynamic motion model 
of each vessel, the limitations of the manipulated variables, the distances 
‘o each other and to static objects and the expressions for the corre- 
sponding COLREG rule. To minimize the energy required to maneuver 
all three vehicles, the objective function / is minimized 
* @ 
A = Zz / wiApT Ap; + wtf en, dt} 
10 
. T 
with: Ap; = (Xgeri — Xinit,i> Yacıi — Yınita) 
subject t0: Tin < T; £ Tmax: 
where n represents the number of vehicles in the optimization problem, 
Ap; denotes the position deviation from the initial trajectory (X; Yınir) 
of the i-th vehicle, and 7; represents the control inputs of the i-th vehicle. 
The weights w, and w, are penalizing the deviation from the initial 
trajectory and the control effort and thus the total energy consumption. 
4.2. Successive optimization procedure 
The successive application of the IPOPT method is schematically pre- 
sented in Fig. 8. The initial trajectories from the maneuver plans serve 
as an input into the optimization. At the beginning of each maneuver 
aptimization, the basic situation is analyzed once. The aim of the analy- 
sis is to decompose the multi-vehicle maneuver into several two-vehicle 
naneuvers. The analysis not only assesses the collision risk, but also de- 
:‚ermines which evasive maneuver should be used in accordance with the 
COLREGs, whether it is a head-on, crossing or overtaking situation. The 
wo-vehicle scenarios are prioritized for processing during optimization 
according to the risk. In the example scenario, the head-on situation has 
‘he higher priority with OCP 1 and is therefore optimized first. In con- 
rast to the analysis step, the optimizations are executed periodically, 
once every 60 s. In this study, the maneuver optimization is applied to a 
aredefined scenario. Therefore, no unforeseen obstacles appear during 
»xecution and the 60 s cycle time is sufficient to update the trajectories 
(Or the given situation.