Inﬂuence of Ship-to-Ship Interaction on Formation Control of

Multi-Vessel Systems

Xin Xiong

, Rudy R. Negenborn

and Yusong Pang

Department of Maritime and Transport Technology, Delft University of Technology, Delft, The Netherlands

Keywords:

Formation Control, Multi-Vessel Systems, Ship-to-Ship Interaction.

Abstract:

The formation control of autonomous surface vessels (ASVs) has received increasing attention, with research

focusing on the formation generation and maintenance. However, the majority of existing researches neglect

the effects of ship-to-ship interactions. Considering that in some scenarios, the distance between ships in a

formation is small, it is necessary to conduct relevant research on formation control. Based on existing litera-

ture on ship hydrodynamic effects, this paper proposes a semi-empirical formula to describe the ship-to-ship

interaction forces, with the main factors being the relative distance and velocity between ships. Subsequently,

experiments were designed to independently analyze these two inﬂuencing factors, and control simulations

were executed for a formation consisting of two homogeneous ASV. The simulation outcomes demonstrate

that ship-to-ship interaction forces indeed inﬂuence control performance, with control performance errors di-

rectly correlating with the variations in interaction forces between the two ASV. Among these factors, speed

exerts a greater inﬂuence than distance, rendering it challenging for a conventional PID controller to satisfy

the stringent control requirements.

1 INTRODUCTION

In recent years, with the escalating complexity of

maritime transport tasks and the increased variability

of marine environments, a single Autonomous Sur-

face Vessel (ASV) may prove insufﬁcient to satisfy all

operational demands (Negenborn et al., 2023)(Xiong

et al., 2015). The cooperation of ASVs enables

the execution of more intricate tasks and scenarios,

thereby enhancing the overall efﬁciency of the wa-

terborne transportation system (Chen et al., 2021).

Furthermore, the deployment of ASVs can enhance

the system’s reliability and redundancy, also signif-

icantly fortiﬁes the system’s fault tolerance and op-

erational continuity. Meanwhile, with the increasing

of the number of controlled ASVs, the cooperation

also encounter numerous challenges, particularly in

the realm of cooperative control (Peng et al., 2020b;

Du et al., 2021a; Oh et al., 2015).

In the domain of cooperative control, formation

control represents a pivotal branch that concentrates

on maintaining prescribed geometric conﬁgurations

https://orcid.org/0009-0009-7265-5077

https://orcid.org/0000-0001-9784-1225

https://orcid.org/0000-0001-8094-3436

and relative positioning among ASVs engaged in col-

lective task execution (Liu et al., 2023). In the exist-

ing literature, Various formation control methodolo-

gies have been explored, such as the leader-follower

approach (Park and Yoo, 2018; Wang et al., 2022),

behavioral approach (Tan et al., 2023; Qin et al.,

2017), and virtual structure approach (Mu et al.,

2020; Zhou et al., 2020), along with strategies includ-

ing consensus-based (Peng et al., 2020a; Gu et al.,

2019), relation-based (Liu et al., 2018), and position-

based methods (Liu et al., 2022; Liu et al., 2017).

Meanwhile, there are also some literature that inves-

tigate collision avoidance in formation control (Mon-

dal et al., 2017; He et al., 2019), as well as controller

design under environmental disturbances (Du et al.,

2021b). However, most studies overlook the ship-to-

ship interaction within the formation. In some for-

mation scenarios, ships are close to each other, where

ship-to-ship interaction signiﬁcantly impact ship mo-

tions. Overlooking this hydrodynamic effect can re-

sult in diminished control performance or potentially

lead to control failures (He et al., 2021).

The maneuvering characteristics of a single ASV

are signiﬁcantly inﬂuenced by hydrodynamic effects,

with hydrodynamic coefﬁcients of the ASV undergo-

ing changes in response to variations in speed and en-

Xiong, X., Negenborn, R. and Pang, Y.

Inﬂuence of Ship-to-Ship Interaction on Formation Control of Multi-Vessel Systems.

DOI: 10.5220/0012911700003822

Paper published under CC license (CC BY-NC-ND 4.0)

In Proceedings of the 21st International Conference on Informatics in Control, Automation and Robotics (ICINCO 2024) - Volume 1, pages 25-36

ISBN: 978-989-758-717-7; ISSN: 2184-2809

vironmental conditions (Skjetne et al., 2004). Hy-

drodynamic interaction of multi ships under differ-

ent situations have been analyzed, including ships in

the process of lightering operation (De Decker, 2006;

Sano and Yasukawa, 2019), overtaking and encoun-

ters (Chen et al., 2019; Setiawan and Muin, 2018; Yu

et al., 2019). These ﬁnds indicate that when two ships

are in close proximity, phenomena such as "bow re-

pulsion" and "bow attraction" occur. These are due

to the changes in interaction forces between the ships.

Thus it is essential to further investigate the inﬂuence

of ship-to-ship interactions on formation control.

The methods for investigating ship-ship interac-

tion can generally be categorized into three types:

First, physical model testing, which entails the use

of scaled model ships within towing tanks (Lataire

et al., 2011; Lataire et al., 2009). This method has

high precision and can accurately reﬂect the behavior

of ships, but it is costly, time-consuming, and has is-

sues with scale effects; Second, numerical simulation,

employing Computational Fluid Dynamics (CFD) or

potential ﬂow panel methods to simulate hydrody-

namic interactions under speciﬁc conditions-a method

favored for its high ﬂexibility and cost -effectiveness

(Wn˛ek et al., 2018; Yuan et al., 2016; He et al., 2022);

Third, theoretical analysis and empirical model devel-

opment, involving the derivation of empirical formu-

las from experimental data to model hydrodynamic

effects (Varyani et al., 2002; Vantorre et al., 2002;

Dong et al., 2022). This method enables rapid predic-

tion of changes in hydrodynamic effects, rendering it

well-suited for addressing real-time control. It is also

cost-effective and easy to use; therefore, this paper

employed this method to model ship-ship interaction.

This paper focuses on the inﬂuence of the ship-

ship interaction on formation control. By summa-

rizing and processing formulas and data from exist-

ing literature, a semi-empirical formula suitable for

control system was proposed to describe these varia-

tions, which was applied to formation control. Sub-

sequently, relevant formation control simulations can

be conducted to investigate the impact of the factors

of formulas on formation control.

This paper is organized as follow. In Section 2,

we establish models for the formation system, includ-

ing the kinematic model of the ASV and the ship-

to-ship interaction model. In the Section 3, we de-

ﬁne some parameters required for the simulation ex-

periments and reﬁned the Tito-Neri kinematic model.

Then, we design simulation experiments focusing on

two important factors: speed and distance. The re-

sults of simulation experiments are provided in Sec-

tion 4 to demonstrate the inﬂuence of the ship-to-ship

interaction of control, followed by the analysis and

explanation. Finally, conclusions and future research

directions are given in Section 5.

2 ASVs SYSTEM MODELING

2.1 ASV Motion Model

An ASV is a complex system characterized by large

inertia and time-delay properties. To establish the mo-

tion model of ASV is the foundational task for forma-

tion control. This paper primarily focuses on underac-

tuated ASV as shown in Fig. 1, with the system input

consisting of two degrees of freedom, namely thrust

and torque τ = [τ

]

. The plane motion of a vessel

can be described by the 3-DOF (degree of freedom)

kinematics and kinetics model which is widely called

Fossen model (Fossen, 2011), which is expressed as:

˙x = f (x

)

= M

˙v + M

˙v +C (v (t))v(t) + D(v(t))v (t) = τ (t)

(1)

η(t) = R (ψ (t))v (t), (2)

Where x = [x y ψ u v r]

as the system’s state

variable,

η(t) = (x, y,ψ) is the position vector

in the world frame and the heading. v(t) =

[u(t) v (t) r (t)]

ε R

is the velocity vector in the

Body-ﬁxed frame containing the velocity of surge

u(t) ,sway v(t) , and yaw r (t). The terms M ∈ R

3×3

C ∈ R

3×3

,D ∈ R

3×3

are the Mass (inertia), Coriolis-

Centripetal, and Damping matrix, respectively. u =

′

= [τ

, τ

]

as the system’s control input, RεR

3×3

is the standard rotation matrix, which is deﬁned as:

R(ψ(t)) =





cos(ψ(t)) −sin(ψ(t)) 0

sin(ψ(t)) cos(ψ(t)) 0

0 0 1





(3)

2.2 Model of Ship-to-Ship Interaction

Typically, the initial phase in formation control in-

volves the establishment of the formation conﬁgura-

Figure 1: Three degrees of freedom motion model.

ICINCO 2024 - 21st International Conference on Informatics in Control, Automation and Robotics

tion. For the purpose of facilitating the study of ship-

to-ship interactions, the formations described in this

paper are composed of two homogeneous ASVs, sys-

tematically categorized into three conﬁgurations as

depicted in the Fig. 2, labeled as tandem, parallel and

semi-triangle. The origin of the body-ﬁxed coordinate

system is set at mid-ship, and the ⃗x,⃗y,⃗z axis are point

to the starboard, bow, and downwards, respectively.

In the parallel formation (as shown in Fig. 2a) , the

hull’s bows are aligned, and the transverse separations

is S

without any longitudinal offset.In the Tandem

formation (as shown in Fig. 2b), the longitudinal dis-

tances between the two ships is S

, and the transverse

offset is zero.The layout of the semi-triangle forma-

tion as shown in Fig. 2c. The parameters of the for-

mation conﬁgurations in the subsequent calculations

are presented in Table 1, where L and B represent the

length and breadth of ship, respectively.

There are three main methods for studying hy-

drodynamic effects as mentioned in Section1. Al-

though the accuracy of theoretical analysis and em-

pirical model development is constrained, their ca-

pacity to offer rapid predictions and ease of applica-

tion are critically valuable in real-time control. Con-

sequently, this section will draw upon these methods

and, by integrating formulas and data from the exist-

ing literature, will introduce a more universally appli-

cable semi-empirical formula. Employing Vantorre’s

experimental results (Vantorre et al., 2002), Decker

(De Decker, 2006) attempted to determine the appli-

cability of Vantorre’s results for lightering operations.

The analysis employed regression techniques to focus

on three dimensionless coefﬁcients:C

; C

and C

corresponding to surge, sway, and yaw movements,

respectively. These coefﬁcients are primarily inﬂu-

enced by the ships’ longitudinal and lateral positions

and their speeds, which can be described as follows:

ρBTU

(4)

ρLTU

(5)

ρBLTU

, (6)

where B is the beam(m), T is the draught(m), U is the

speed(m/s), L is the length(m), ρ is the water density.

These formulas reveal that ship-ship interaction are

predominantly governed by the speeds and relative

positions of the two vessels. As the relative distance

diminishes, the interactions begin to signiﬁcantly in-

tensify, a trend that is similarly observed with increas-

ing vessel speeds.Furthermore, the presence of these

interaction forces variably inﬂuences the maneuver-

ability of the vessels.

For formation situations, He (He et al., 2022) Uti-

lizing the Reynolds-Averaged Navier-Stokes (RANS)

equations in conjunction with the K − ω turbulence

models, a series of numerical simulations were con-

ducted. The numerical simulation explores the rela-

tionship between ship resistance and formation con-

ﬁgurations under various formation shapes, where

these conﬁgurations primarily encompass the relative

distance and speed of the ships. Subsequently, a third-

order polynomial regression analysis was employed

to derive the relationship between the resistance coef-

ﬁcient, relative position, and speed of the ships. The

relationship of the follower ship under various forma-

tion shapes is shown as:

Tandem Formation:

′

Leader

= −0.0118 +6.3582 ×Fr

−0.0017 ×S

−1.6286 ×Fr

+ 0.0069 ×S

+ 0.1680 ×Fr

−0.0089 ×S

+ 0.0003 ×Fr ×S

(7)

Parallel Formation:

′

Leader

= −0.0191 +9.0151 ×Fr

+ 0.0011 ×S

−2.2171 ×Fr

−0.0035 ×S

+ 0.2206 ×Fr

+ 0.0044 ×S

−0.0071 ×Fr ×S

(8)

Triangle Formation:

′

Leader

= −0.0093 +6.0171 ×Fr

−0.0001 ×S

−1.5498 ×Fr

+ 0.0091 ×S

+ 0.0005 ×S

+ 0.1604 ×Fr −0.0139 ×S

−0.0029 ×S

+ 0.0022 ×Fr ×S

+ 0.0009 ×Fr ×S

+ 0.0044 ×S

×S

, (9)

where Fr is the Froude Number,which can be ex-

pressed as:

Fr =

√

. (10)

From these formulas, it can be inferred that across

all formation types, ship resistance demonstrates a

fundamentally linear and positive correlation with

speed, indicating that an increase in speed corre-

sponds to a rise in the total resistance coefﬁcient. In

tandem and parallel formation shapes, when the speed

remains constant, the resistance coefﬁcient is inﬂu-

enced by the lateral or longitudinal distances. In the

case of the triangle formation, the relative distance en-

compasses both lateral and longitudinal dimensions,

making its expression overly complex. Therefore, we

usually assume a constant speed in the equations to

ensure that the inﬂuencing factors are limited to two

dimensions.

Owing to the unique characteristics of hydrody-

namics, the follower ship is more susceptible to the

Inﬂuence of Ship-to-Ship Interaction on Formation Control of Multi-Vessel Systems

Figure 2: The geometric conﬁguration of formation.

wake and wave effects generated by the leader. There-

fore, this paper primarily concentrates on the follower

ship and provides a comprehensive summary of the

resistance formulas applicable in various scenarios.

Given that total hull resistance is a function of hull

form, ship speed, and water properties, the coefﬁcient

of total hull resistance similarly depends on these fac-

tors. The coefﬁcient of total hull resistance is deter-

mined by the following equation:

ρSV

, (11)

where R

is the total hull resistance and s is the wet-

ted surface area of the underwater hull, ρ is the water

density.

Based on equations (7) to (11), the three degrees

of freedom ship-to-ship interaction forces for the fol-

lower ship can be expressed as follows:

ρ Sv

(γv + αL + βB + ε), (12)

where τ





represent the interaction

forces and moments of three degrees of freedom re-

spectively. γ = [a

]

is the velocity parameters

in the three degrees of freedom. L =





and B =





. The terms α ∈ R

3×3

, β ∈

3×3

are longitudinal distance weights matrix and

transverse separations weights matrix, respectively,

which is deﬁned as:

α =









,β =









(13)

Table 1: Parameters of the formation conﬁgurations.

Formation shape Relative distance Velocity (m/s)

Single ASV – 0.1–0.4

Tandem formation 0.1–2.0L 0.1–0.4

Parallel Formation 0.1–1.5B 0.1–0.4

Semi-Triangle 0.1–2.0L, 0.1–1.5B 0.1–0.4

3 SIMULATION EXPERIMENT

DESIGN

In this section, we design simulation experiments for

a formation composed of two homogeneous ASVs,

introducing the parameter settings and experimental

setup. Considering that the main inﬂuencing factors

of ship-to-ship interaction are speed and relative dis-

tance, and that formation control also needs to man-

age speed and distance, we have designed two sets of

experiments. These experiments separately consider

the impact of variations in speed and distance on for-

mation control.

3.1 Parameter Setup

Considering that acquiring hydrodynamic parameters

for ships is often an extremely tedious process, this

section applies the "Tito-Neri", which is developed

by Delft University of Technology(TU Delft). "Tito-

Neri" is a 1:30 replica model tugboat (Bruggink et al.,

2018). The hydrodynamic model parameters of "Tito-

Neri" are shown in Table 2.

Table 2: Parameters of the Tito-Neri.

Parameters Value Parameters Value

m 16.9 Y

˙v

-49.2

0.51L Y

˙r

0.0

0.0 N

˙v

0.0

˙u

−1.2 N

˙r

-1.8

It can be observed that the parameters in the mass

(M) and Coriolis-Centripetal (C) matrices are identi-

ﬁed, but the damping matrix (D) is not acquired. The

Tito-Neri replace the whole damping part D(v

)v(t) to

the drag forces vector τ

drag

(t) :

drag(t)





drag/u(ϕ

(t)

, u(t))

drag/v(ϕ

(t)

, v(t))

drag/r(r(t))





, (14)

where τ

drag/u

, τ

drag/v

and τ

drag/r

represent the drag

forces and moment in x , y and z direction. The

polynomial functions in x and y direction are deter-

mined by the velocity in the corresponding direction

and heading angles. The polynomial function in z di-

rection is calculated by the yaw velocity.

Based on the data presented in the report (Brug-

gink et al., 2018),which is shown as Fig. 3. We em-

ployed third-order polynomial regression analysis to

derive regression functions for the x and y directions.

velocities ranging from 0−0.4(m/s) and angles from

ICINCO 2024 - 21st International Conference on Informatics in Control, Automation and Robotics

0 −1(π). The derived functions are as follows:

drag/u

= −0.1849 + 0.5327 ×V (t) + 0.8117 ×ϕ(t)

+ 0.5387 ×v(t)

−0.0784 ×ϕ(t)

+ 6.3668 ×v(t)

−0.7143 ×ϕ(t)

(15)

drag/v

= −0.0348 + 0.1416 ×v(t) + 0.2760 ×ϕ(t)

+ 0.4710 ×v(t)

−0.5299 ×ϕ(t)

+ 0.9844 ×v(t)

+ 0.2930 ×ϕ(t)

−0.1159 ×v(t) ×ϕ(t) (16)

The simulation in the next section will conduct

path following control for the follower ship based on

this motion model. In path following, controlling the

ship’s speed (e.g., maintaining a constant velocity)

enables the prediction of the target position the ship

can attain at a future time. Since the focus of this pa-

per is on investigating the impact of ship-to-ship inter-

action on control performance, a simple and easy-to-

implement PID controller will be used. For a simple

PID control system , the transfer function of PID con-

troller is shown as:

u(t) = k

e(t) + k

e(t)dt + k

(17)

3.2 Simulation Experiment Design

According to (12), the main inﬂuencing factors of

ship-to-ship interaction forces are velocity and rela-

tive distance. Therefore, this section will study their

impact on control performance based on different ge-

ometric conﬁgurations. The simulation experiments

will be divided into velocity experiments and distance

experiments: In the velocity experiments, the initial

distance between the two ASVs is set at the minimum

and is maintained consistently. Then, the speeds of

the two ships are precisely controlled to achieve the

desired values. After completing a set of experiments,

the initial distance is increased until it reaches twice

the ship length, The process of velocity simulation is

as shown in the Fig 4a; Similarly, for the distance ex-

periments, the initial set maintains a constant desired

speed of 0.1 m/s for both ASVs. Then, the distance

between the two ships is gradually reduced from twice

the ship length to 0.1 times the ship length. After

completing this set of experiments, the desired speed

of the two ships is increased until the desired speed

reaches 0.3 m/s, as shown in the Fig 4b. The speciﬁc

simulation parameters are as follows:

• In speed experiments, the two ships are in a tan-

dem formation with an initial relative distance

of SL

. The initial positions and speeds of the

two ships: y

leader

= (1, SL

) , y

f ollower

= (1, 0) ,

v = (u

) = (0,0,0) , Desired Speed v

0.01i(i = 1,2,...) .

• In distance experiment, the initial speed is v

the initial positions of the two ships are: y

leader

(1,2L) , y

f ollower

= (1,0) , and the desired relative

distance is SL

= 2L −0.01iL (i = 1, 2, . . .).

4 SIMULATION RESULTS AND

DISCUSSION

This section begins with simple simulation experi-

ments to verify the feasibility of the follower ship

controller. In this simulation, ship-to-ship interac-

tion forces are ignored, focusing only on the follower

ship’s ability to follow the leader ship. Subsequently,

velocity and distance simulations are conducted sepa-

rately. These simulations focus on the impact of ship-

to-ship interaction forces on control, recording the

follower ship’s relevant data to analyze how changes

in these two factors affect the results. The settings for

the controller test simulations are as follows: The ini-

tial position of the follower ship is located at the (0, 1)

with 0 degree of heading and no speed, the leader ship

is y

leader

= (1, 0) with no speed. The desired distance

is 1m. The simulation results for the follower ship are

presented as ﬁg. 5. From the simulation results, it

shows that when ship-to-ship interaction are not con-

sidered, the follower ship is able to track the leader

ship well and maintain the desired distance.

The velocity simulation experiments and distance

simulation experiments in this section are both based

on the following assumptions:

• Assumption 1: This study concentrates on the

control dynamics of the trailing ship under the in-

ﬂuence of ship-to-ship interactions.Consequently,

it is assumed that the leader ship adheres pre-

cisely to the predetermined trajectory, unaffected

by ship-to-ship interactions.

• Assumption 2: In tandem formations, given that

the centers of the two ships are perfectly aligned,

it is posited that the ship-to-ship interactions are

negligible in the Y and Z directions. So the 12 can

be set as follow:

= [τ

,0,0]

ρSv



×v + α

+ α

SL + ε



v(t) ∈ [0,v

max

]

SL ∈ [0.1L,2L]

(18)

4.1 Velocity Simulation

The velocity experiment is structured into 20 distinct

groups, with the distance between the two ships in-

crementally increasing from 0.1 to 2 times the ship

Inﬂuence of Ship-to-Ship Interaction on Formation Control of Multi-Vessel Systems

(a) x-direction. (b) y-direction.

Figure 3: Corresponding ﬁtting surface of drag forces.

(a) The ﬂowchart of velocity simulation.

(b) The ﬂowchart of distance simulation.

Figure 4: The ﬂowchart of Simulation.

length in each group. It is hypothesized that the in-

teraction between the two ships varies solely based

on the velocity of the follower ship.The simulation

results, which display the velocity variations and ve-

locity errors of the trailing ship across each group, are

illustrated in the below.

The simulation results delineate the performance

at distances of 0.1L, 1L, and 2L, shown as ﬁg. 6

to ﬁg. 8 with additional results presented in the ac-

companying table. The simulation outcomes clearly

demonstrate that the interaction forces between the

ships signiﬁcantly impact the performance of the con-

troller. For the Follower ship, observable ﬂuctuations

in speed and the emergence of a steady-state error are

documented, as illustrated in Fig. 7a. This ﬁnding un-

derscores that a solitary PID controller is inadequate

to effectively counteract the complex effects induced

by the interaction forces between ships. As the initial

relative distance incrementally increases, the average

speed error initially escalates and subsequently sub-

sides. As delineated in Table 3, the error rate peaks

when the relative distance reaches 1.2L. This phe-

nomenon is attributed to the inﬂuence of the initial

relative distance; the equation reveals that at a dis-

tance of 1.2L, the force exerted on the trailing ship

maximizes before beginning to wane. Fig. 7d illus-

trates that the interaction force between the ships is

positively correlated with speed and exhibits signiﬁ-

cant sensitivity to speed variations.

4.2 Distance Simulation

In the distance experiment, ten distinct trial groups

will be conducted. The initial expected speed for the

experiments will incrementally increase from 0.1 m/s

to 0.3 m/s. In each experimental group, the distance

between the two ships will be systematically reduced

to investigate changes in interaction forces and their

ICINCO 2024 - 21st International Conference on Informatics in Control, Automation and Robotics

(a) Path following. (b) Velocity. (c) Relative Distance.

Figure 5: Simulation test results Without considering Ship-to-Ship Interaction.

Table 3: The velocity errors of Velocity Experiments.

Longitudinal distances (SL) Average Velocity errors E

(m/s) Average Error rate D

(%)

0.1L 0.0110 3.84

0.2L 0.0124 4.66

0.3L 0.0165 5.33

0.4L 0.0197 5.95

0.5L 0.0212 6.44

0.6L 0.0224 6.96

0.7L 0.0251 7.74

0.8L 0.0265 8.32

0.9L 0.0271 8.93

1.0L 0.0279 9.95

1.1L 0.0297 10.51

1.2L 0.0345 11.56

1.3L 0.0376 10.01

1.4L 0.0274 8.81

1.5L 0.0245 8.16

1.6L 0.0224 7.05

1.7L 0.0188 5.85

1.8L 0.0156 5.27

1.9L 0.0102 4.58

2.0L 0.0115 3.96

effects on control performance. The average distance

error and average error rate for the experiments are

presented in Table 4. The overall trend indicates that

as the speed increases, the error correspondingly in-

creases. This suggests that the increase in expected

speed results in a heightened maximum interaction

force between the ships, thereby exerting a more sub-

stantial impact on control.

Fig. 9 through 11 display the simulation outcomes

for the follower ship at expected speeds of 0.1, 0.2,

and 0.3 m/s, respectively. From Fig 9b, it is evident

that an increase in expected speed correlates with a

rise in the maximum value of the distance error. Fig

9c illustrates that during the interval between 20 and

40 seconds, there are pronounced speed ﬂuctuations.

During this interval, with the distance between the

two ships ranging from 1.5L to 0.5L, it is indicated

that distance exerts a signiﬁcant inﬂuence on ship

control. Fig. 9d delineates the trend of interaction

forces between the ships relative to distance; however,

overall, the inﬂuence of distance remains compara-

tively minor. In contrast, higher expected speeds re-

sult in increased overall forces exerted on the trailing

ship, demonstrating that speed plays a more substan-

tial role in inﬂuencing interaction forces between the

ships than distance does.

Table 4: Distance Error of Distance experiments.

Velocity (m/s) Average Distance Deviation Rate D

(%) Average Distance Error (m)

0.10 1.177 0.0154

0.12 1.385 0.0181

0.14 1.592 0.0202

0.16 1.774 0.0246

0.18 2.012 0.028

0.20 2.234 0.0305

0.22 2.382 0.0331

0.24 2.557 0.0367

0.26 2.831 0.039

0.28 3.157 0.0414

0.30 3.406 0.0452

5 CONCLUSIONS AND FURTHER

RESEARCH

This paper addresses the frequently overlooked ef-

fects of ship-to-ship interactions in formation con-

trol. In scenarios where ships operate in close prox-

imity, the interaction forces between them play a piv-

otal role. Consequently, this paper ﬁrst proposes a

semi-empirical formula for ship-to-ship interaction,

followed by conducting pertinent formation control

experiments using the Tito-Neri vessel which is de-

veloped by TU Delft.

The semi-empirical formula precisely delineates

the primary factors inﬂuencing ship-to-ship interac-

tion, speciﬁcally the relative positions and velocities

of the two ships. Consequently, this study designs two

experimental sets to systematically explore the effects

of position and velocity, respectively, recording the

simulation outcomes for the follower ship. However,

when accounting for the interaction between the two

ships, the overall control performance of the trail-

ing ship deteriorates signiﬁcantly. Speed ﬂuctuations

not only occur but also precipitate the emergence of

steady-state errors. This is attributed to speed being

the principal factor inﬂuencing the interaction force,

which functions as an external disturbance and signif-

icantly impacts the control of the trailing ship. Simu-

lation results reveal that as speed increases, the inter-

action force between the two ships intensiﬁes, thereby

amplifying its impact on control.

Inﬂuence of Ship-to-Ship Interaction on Formation Control of Multi-Vessel Systems

(a) Velocity. (b) Velocity errors.

Figure 6: Simulation of 0.1L distance.

(a) Velocity. (b) Velocity errors.

Figure 7: Simulation of 1L distance.

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(a) Velocity. (b) Velocity errors.

Figure 8: Simulation of 2L distance.

(a) Distance. (b) Distance errors.

Figure 9: Simulation of 0.1m/s.

Inﬂuence of Ship-to-Ship Interaction on Formation Control of Multi-Vessel Systems

(a) Distance. (b) Distance errors.

Figure 10: Simulation of 0.2m/s.

(a) Distance. (b) Distance errors.

Figure 11: Simulation of 0.3m/s.

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Concerning distance, while the relative distance

exerts a lesser impact on control performance com-

pared to speed, it still markedly inﬂuences the control

performance of the follower ship within the range of 1

to 1.2 times the ship’s length. Within this range, speed

ﬂuctuations are particularly pronounced. Overall, the

interaction force between the two ships is governed by

both speed and distance, demonstrating greater sensi-

tivity to changes in speed. If these variations in force

are not incorporated into the controller design, they

could readily result in degraded control performance

or even failure.

The current study exclusively investigates a sce-

nario involving two ships in tandem and is designed

to visually demonstrate the impact of ship-to-ship in-

teraction forces on control. Controllers are speciﬁ-

cally tailored for the follower ship, with the tracking

path constrained to straight lines. Future research will

broaden the scope of experimental scenarios to en-

compass various formation conﬁgurations, incorpo-

rating the interaction forces among ships within these

formations. Additionally, to verify the impact of ship-

to-ship interaction forces on control, this paper em-

ployed a relatively simple PID controller. This type

of controller has poor robustness and struggles to ac-

count for the inherent limitations of ASV, such as rud-

der angle rate and thruster output. Therefore, it is nec-

essary to design a more suitable controller speciﬁcally

tailored to these conditions to ensure the performance

of the control system.

ACKNOWLEDGMENTS

This research is supported by the "Researchlab Au-

tonomous Shipping (RAS)" of Delft University of

Technology.

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