Formation Analysis for a Fleet of Drones: A Mathematical Framework

Emiliano Traversi

, Michal Barcis

, Lorenzo Bellone

, Agata Barcis

, Dina Ahmim-Bonaldi

Eliseo Ferrante

and Enrico Natalizio

IDO Department, ESSEC Business School, Cergy-Pontoise, France

Technology Innovation Institute, Abu Dhabi, U.A.E.

New York University Abu Dhabi, Abu Dhabi, U.A.E.

Keywords:

Robot and Multi-Robot Systems, Task Planning and Execution, Formation Study.

Abstract:

We consider a dynamic coverage scenario, where a group of agents (e.g., Unmanned Aerial Vehicles (UAVs))

is exploring an environment in search of a moving target (e.g., survivors on a lifeboat). We assume UAVs are

capable to achieve, maintain, and move in formation (e.g., to maintain connectivity). This paper addresses

the question “Which formation maximizes the chance of ﬁnding the target?”. We propose a mathematical

framework to answer this question. The proposed framework is generic and can be easily applied to various

formations and missions. We show how the framework can identify which formation will result in better

performance in the type of missions we consider. We analyze how different factors, namely the target speed

relative to the group, affect the performance of the formations. We validate the framework against simulations

of the considered scenarios. The supplementary video material including the real-world implementation is

available at https://youtu.be/ mYmTnAJi-I?si=dSmVVNZOjj5NbSG1.

1 INTRODUCTION

Groups of agents or robots can be advantageous over

single agents in several applications. Search missions

(looking for survivors or faults or oil spills, etc.) con-

stitute important applications and are also the sub-

ject of active research (Ivi

c et al., 2020). Deploy-

ing groups of searching agents can increase the ef-

ﬁcacy and efﬁciency in such missions. An important

decision is how to divide the search among agents.

A possible choice is to divide the area into partitions

and assign agents to independently explore them (Liu

et al., 2023; Mazdin et al., 2020; Kovacina et al.,

2002). In other situations, it may be advantageous to

deploy search parties consisting of multiple agents to

explore a common large area. For example, searching

in groups can be advantageous when diverse capabil-

ities are available within each search party (Dorigo

et al., 2013). For autonomous (unmanned) search-

ing agents, the need to travel in search parties is mo-

tivated by limited access to long-range communica-

tion devices: For instance, only some robots may

be equipped with satellite modems allowing commu-

nication with the mission control station, while all

robots are equipped with shorter range transceivers.

Agents employed in group search must travel in

close proximity, but the exact relative positions of the

agents vary. The speciﬁc choice of the set of relative

positions of the agents can be referred to as a forma-

tion. Agents operating in formations present advan-

tages: For instance, birds ﬂy in speciﬁc formations

to save energy (Bajec and Heppner, 2009), which in-

spired the development of systems that can reduce

fuel consumption in aircrafts (Afonso et al., 2023;

Antczak et al., 2022; Hartjes et al., 2019). Formation

ﬂights can also be used in the decentralized handling

of control tasks for surveillance to optimize a system

with respect to cost and weight constraints (Anderson

et al., 2008). Formation control – the study on how to

achieve and maintain a formation – is an active ﬁeld of

research (Justh and Krishnaprasad, 2002; Wang et al.,

2007; Paul et al., 2008; Chao et al., 2012). The focus

of our work is not on formation control. We instead

assume a reliable method for formation control is al-

ready available.

The main hypothesis in this work is that some for-

mations are better than others in group search mis-

sions, depending on the mission proﬁle. We propose

a mathematical model that can analyze how differ-

ent choices of formation inﬂuence the target detec-

tion chance. We consider the dynamic coverage (Pa-

patheodorou and Tzes, 2018; Atınc¸ et al., 2020), or

sweeping coverage (Cheng and Savkin, 2009; Zhai

and Hong, 2012; Zhai and Hong, 2013; Saska et al.,

2013) problem. To the best of our knowledge, we fo-

cus on the ﬁrst time the effect of speciﬁc formation

Traversi, E., Barcis, M., Bellone, L., Barcis, A., Ahmim-Bonaldi, D., Ferrante, E. and Natalizio, E.

Formation Analysis for a Fleet of Drones: A Mathematical Framework.

DOI: 10.5220/0013189400003890

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

In Proceedings of the 17th International Conference on Agents and Artiﬁcial Intelligence (ICAART 2025) - Volume 1, pages 471-480

ISBN: 978-989-758-737-5; ISSN: 2184-433X

471

for intercepting a moving target.

The pursuer-evader task is related to the one con-

sidered in this paper (Hespanha et al., 1999). In (Yu

et al., 2019; de Souza et al., 2021), multiple pursuers

have been used to maximize their chances. In (Wei

and Yanq, 2018), pursuers are used to lure the evader

into their paths, while others simultaneously establish

an encirclement to capture the target. In the above

work, the pursuers’ remain close to the evader, rather

than ﬁnding it in an unknown environment.

In contrast to the above literature, this paper

presents a different approach to dynamic coverage,

focusing on the analysis of the use of different for-

mations to search for a moving target with an un-

known trajectory. The main contribution of this work

is a mathematical framework for this analysis (Sec-

tion 3). We validate the correctness of the framework

in a simulated mission with Unmanned Aerial Vehi-

cles (UAVs), and show a proof of concept with real

UAVs (Section 4) where the framework can be use-

ful. We conclude the paper in (Section 5).

2 PROBLEM STATEMENT

We consider a search mission in which a group of

robots in formation has to ﬁnd a moving target. We

introduce a mathematical framework that models the

effect of the formation on the search task. To sim-

plify the mathematical treatment, we consider arbi-

trary linear search trajectories followed by the group

of agents. We assume these are planned in a way to

cover the parts of the environment where the target

may appear. Each sweeping trajectory is modeled as

a straight corridor depicted vertically: the agents tra-

verse it from bottom to top. Let n

be the number

of agents. Each agent is modeled as a rectangular

area with size ℓ

× ℓ

that represents the agent ﬁeld

of view. If agents correspond to UAVs, the rectan-

gular area represents the ﬁeld of view of the camera

equipped on the drone. The agents move within the

corridor without changing their relative positions and

at a constant speed of v

. We assume a reference sys-

tem that is ﬁxed and centered on the group (in other

words, the group does not move with respect to the

considered reference frame). Figure 1 shows an illus-

trative example.

We introduce: The maximum and minimum entry

points of the target, x

min

and x

max

; The starting and

ending points of agent exploration, x

min

and x

max

The agent group has to detect the presence of a

moving target within the corridor. The mission be-

gins with the agents group positioned at the point x

min

(lower dotted horizontal lines in Figure 1), and it tra-

min

max

min

(a) Start of the exploration.

min

max

min

(b) Target’s appearance.

min

max

min

max

min

(d) Target is intercepted.

Figure 1: Example of a successful exploration.

verses the corridor at a constant speed v

up to the

point x

max

(upper dotted horizontal lines in Figure 1).

We assume that at any moment during the exploration,

a target (represented as a red dot in Figure 1) enters

the corridor from one of its edges at a randomly gen-

erated point within the interval [x

min

max

] (see the

yellow left border in Figure 1) and with a random

orientation (distributed in the range [0,π]), constant

speed v

, and following a straight trajectory. We indi-

cate with P

(x) the probability that the target shows

up at a given position x ∈ [x

min

max

During the exploration, if the target enters any of

the agent’s rectangular area, it is considered inter-

cepted. If instead the target exits the other side of the

corridor without entering any rectangle, it is consid-

ered missed. Furthermore, exploration stops when all

agents have crossed the maximum exploration point,

max

. If the target has not been intercepted by that

time, it is considered missed.

3 MODEL DEFINITION

In this section, we present the framework enables to

compute the probability for a given group formation

to intercept a target (hereby referred to as the inter-

ception probability) in a mission with a customizable

proﬁle.

ICAART 2025 - 17th International Conference on Agents and Artiﬁcial Intelligence

472

3.1 Framework Overview

Let x represent the distance between the target’s entry

point and the group of agents

. Deﬁne S as the set of

all subsegments of [0,π], and S as the set of all sub-

sets of S (i.e., S is the power set of S). Key to our

analysis is representing the set of interceptable direc-

tions S

⊆ S , that is the set of maximal (with respect

to inclusion) segments that correspond to values of θ

that lead to interception.

Figure 2: Example of S

for a speciﬁc formation.

Figure 2 provides an example of S

for a speciﬁc

formation and distance x between the formation and

the target entry point (in red). Here, S

contains the

two segments in purple.

Let P

(θ) be the probability that, for an entry point

x, the direction of the target is equal to θ and let P

the probability that the target is intercepted if it enters

at a distance x from the formation of agents. We can

compute P

as:

∑

[θ

,θ

]∈S

(θ)dθ (1)

In this equation, only entrance angles that belong to

and that yield to interceptions are integrated. Con-

sider the case where all entry angles are equally prob-

able (P

(θ) =

, ∀θ ∈ [0,π]). In this case, P

(denoted

as P

) is:

∑

[θ

,θ

]∈S

· dθ =

∑

[θ

,θ

]∈S

(θ

− θ

) .

(2)

In the rest of the paper we will focus only on this case.

3.2 Modeling Formations

The type of formation selected impacts signiﬁcantly

the complexity of the description of the set S

. In this

In this paper we will denote by x a position on the ver-

tical axis. Depending on the context, it may indicate the

target entry point or the position of the formation.

work, we focus on formations that allow the set S

have the following properties:

• Symmetry. S

is symmetric if its value does not

depend on the entry side of the target but only on

the distance x. This property allows us to assume,

without loss of generality, that the target enters

from only one of the two sides.

• Scalar-representability. S

is scalar-representable

if it consists of one single segment of the form

[0,θ

]. Loosely speaking, this holds when forma-

tions do not have “holes” in the range of angles

θ that lead to interception. This property allows

us to see S

as a function S

: R → [0, π] that for

each entry point x returns the maximal entry angle

that leads to an interception. The entry point x

is to be understood as an absolute value, so the

above holds regardless of whether the target en-

ters in front or behind the formation.

Working with a set of interceptable directions that

is symmetric and scalar-representable allows us to

have a more tractable mathematical model. Having

said this, the framework is general enough to be ex-

tendable also to the cases where “holes” are present.

Figure 3: Formations considered.

Among the formations that meet the above proper-

ties, we study the three of them: line, arrow, and vee

formations (see Fig. 3). We conclude by noting that

the choice of the formation is not the only factor that

can affect whether the set S

is scalar representable or

not, as it will become clear in the next section. Thus,

conditions on the formation are necessary but not suf-

ﬁcient for having a scalar-representable set of inter-

ceptable directions.

3.3 Calculating P

Except when explicitly stated, in the rest of this doc-

ument we assume a set of interceptable directions S

that is symmetric and scalar-representable. To make

the analysis easier to tackle, we represent S

as a com-

position of two functions, f

(θ) and f

(x):

= [0, f

( f

(x))] (3)

The function f

(x) : R → [0,π] (where the “S”

stands for static) calculates the maximal entry angle at

which we have an intersection between the target and

the group of agents in a given formation, for a given

snapshot of the mission whereby the agents do not

Formation Analysis for a Fleet of Drones: A Mathematical Framework

473

move (i.e., v

= 0). This function takes into account

the contribution of the speciﬁc formation but does not

consider the magnitude of the speed of the target v

since agents are considered static (v

= 0). The func-

tion f

(x) : [0, π] → [0,π] (where the “D” stands for

dynamic) models how the speed of the target v

and

the speed of the agents v

modify the maximal entry

angle calculated with f

(x).

Eq. (3) allows to simplify the computation of P

(see Eq. (2)):

∑

[θ

,θ

]∈S

(θ

− θ

) =

( f

(x))

. (4)

Eq. (4) shows that, if all target entry angles are equally

probable, it is sufﬁcient to determine the maximum

entry angle for the given entry point and divide it by

π to obtain the probability of interception.

In the following, we detail how to calculate f

(θ)

and f

(x) for speciﬁc scenarios, as a guideline for ap-

plying the framework to any scenario with symmetric

and scalar-representable S

3.3.1 Calculation of f

(x)

We formalize the value of f

(x) as a function of the

entry point of the target. According to the formation

used, the deﬁnition will differ and incorporate the pa-

rameters ℓ

, ℓ

, and the size of the group n

. For sim-

plicity, we assume that we only consider an odd num-

ber of agents n

and we introduce k that identiﬁes the

“‘depth” of the formation k = (n

− 1)/2.

ℓ

Figure 4: How to compute f

(x) for a line formation.

Line-Formation. The function f

(x) applied to a

line formation is denoted as f

(x). Figure 4 shows

how to calculate f

(x) by applying the deﬁnition of

sine to the right triangle with vertices in the target en-

try point and in two extreme points of the formation:

(x) = arcsin





(2k + 1)ℓ

(2k + 1)

ℓ

+ x





(5)

If the reference system is deﬁned appropriately,

Eq. (5) can also be used for computing f

(x) when

the target enters from the back of a line formation, in

front of a vee-formation or from the back of an arrow-

formation (see Fig. 5 for some examples).

ℓ

(a) Target enter-

ing in the back of

a line formation.

ℓ

(b) Target enter-

ing in the front of

a vee formation.

Figure 5: When to apply f

(x) to other conﬁgurations.

Vee and Arrow Formations. In this paragraph, we

describe f

for a target entering in the front of an ar-

row formation and in the back of a vee formation, us-

ing the same equation thanks to symmetry. We denote

it as f

in both cases.

The more complex nature of the arrow and vee for-

mations makes it necessary to split the analysis into

three cases, depending on where the target hits the

formation when it has a direction corresponding to its

maximal entry angle: (1) The nearest side of the ar-

row (if 0 ≤ x ≤ kℓ

, see Figure 6a); (2) The “head” of

the arrow (if kℓ

≤ x ≤ (2k + 1)ℓ

, see Figure 6b); (3)

The farthest side of the arrow (if x ≥ (2k + 1)ℓ

, see

Figure 6c):

(x) =











+ arcsin





kℓ

− x

(kℓ

− x)

+ k

ℓ





0 ≤ x ≤ kℓ

arcsin





(k + 1)ℓ

(k + 1)

ℓ

+ (x − kℓ

)





kℓ

≤ x ≤ (2k + 1)ℓ

arcsin





(2k + 1)ℓ

(2k + 1)

ℓ

+ x





(2k + 1)ℓ

≤ x

(6)

ℓ

θ −

ℓ

kℓ

− x

kℓ

(a) Target touching the

nearest side of the arrow.

ℓ

x − kℓ

(k + 1)ℓ

(b) Target touching

the head of the arrow.

ℓ

the farthest side of the

arrow.

Figure 6: f

(x) for different target entry points.

3.3.2 Calculation of f

(x)

does not depend on the formation because it mod-

els how the maximum entry angle obtained in the

static case is altered when the agents have a veloc-

ity greater than zero. Instead, f

depends on the entry

point of the target. In the following, we indicate with

(θ) and f

(θ) the values of f

when the target en-

ters from the front or from the back.

If the agents are in motion, it becomes necessary

to employ a formula that indicates how the speed of

the target is perceived by the moving agents (recall

ICAART 2025 - 17th International Conference on Agents and Artiﬁcial Intelligence

474

= 1

(θ)

π − f

(θ)

(a) Computation of f

(θ)

if the target enters in front

of the agents.

= 1

(θ)

θ − f

(θ)

π − f

(θ)

(b) Computation of f

(θ)

if the target enters behind

the agents

Figure 7: Computation of f

(θ) and f

(θ).

the reference system is ﬁxed with the agent group).

We observe that, when the target enters in front of the

agents, moving agents increase the chances of inter-

cepting the target, which corresponds to an increased

maximum entry angle in the dynamic case. Similarly,

when the target enters from the rear of the forma-

tion, the maximum entry angle also increases in the

dynamic case.

The formulas for f

(θ) do not depend on the ex-

act values of the two velocities v

and v

but only on

their ratio. For this reason, we suppose to have v

= 1

w.l.o.g.

Target Entering in Front of the Agents. Figure 7a

shows the quantities used to compute f

(θ). We re-

mind that the input of the function is the maximal en-

try angle in the static case (deﬁned as θ in the ﬁgure)

and that v

and v

are given as input. We can therefore

use the law of sines to obtain the following relation:

sin



(θ) − θ



sin(θ)

(7)

and therefore f

(θ) = θ+arcsin(v

sin(θ)). The func-

tion arcsin is deﬁned only in the interval [−1, 1],

therefore we have the following constraint on the val-

ues of θ: −1 ≤ v

sin(θ) ≤ 1. If v

≤ 1 then the in-

equalities are always satisﬁed. On the other hand, if

> 1 we have that θ ≤ arcsin





, that has also

a practical explanation: if the value of v

is high

enough, regardless of the direction of the target and

of the value of θ, it will always be intercepted. We

obtain:

(θ) =











θ + arcsin(v

sin(θ)) if v

≤ 1

or θ ≤ arcsin





π otherwise

(8)

that takes into account the values of v

Target Entering Behind the Agents. In Fig. 7b we

show how to compute f

(θ). The law of sines leads

to the following relation:

sin(θ − f

(θ))

sin(π − θ)

(9)

and therefore f

(θ) = θ − arcsin(v

sin(π − θ)). The

function arcsin is deﬁned only in the interval [−1,1],

therefore we have that the only valid values for θ are

the ones satisfying the following constraints: −1 ≤

sin(π − θ) ≤ 1. If v

≤ 1 then the inequalities are

always satisﬁed. On the other hand, if v

> 1 we have

that θ ≥ π − arcsin





. Similarly to the previous

case, this requirement implies that if the value of v

too high, regardless of the direction of the target and

on the value of θ, it will never be intercepted. We

obtain:

(θ) =







θ − arcsin(v

sin(π − θ)) if v

≤ 1

or θ ≥ π − arcsin





0 otherwise

(10)

that takes into account the values of v

3.4 Modeling Missions

Once the formation and speeds of the agent ﬂeet have

been determined, the next step is to include the spe-

ciﬁc context of the mission within the model. This in-

volves specifying the following parameters: the max-

imum and minimum entry points of the target (x

min

and x

max

), the starting and ending points of agent ex-

ploration (x

min

and x

max

), the probability (P

(θ)) that

the angle associated with the direction of the target is

equal to θ, the speed of the target (v

) and the prob-

ability P

(x) that the target enters the corridor when

the ﬂeet is at a given point x.

The choice of mission can signiﬁcantly impact the

complexity of calculating the corresponding intercep-

tion probability. Therefore, in this paper we con-

sider missions that are meaningful from an applica-

tion standpoint and are also easily manageable from a

formal perspective. Speciﬁcally, this involves search-

ing for missions that possess the following property:

Formation Analysis for a Fleet of Drones: A Mathematical Framework

475

• Scalar Representability. Mission choice can

change the way the set S

is calculated. A mis-

sion is considered scalar-representable if it leads

to a set of interceptable directions that is scalar-

representable.

This is an example of another type of condition that

is necessary to have a scalar-representable set S

, on

top of those associated with the formation and pre-

sented in Section 3.2. In this work we consider scalar-

representable missions, that is cases where x

max

+∞, in other words where the ﬂeet of agents can ex-

plore inﬁnitely the corridor until either the target is

intercepted or reaches the other side of the corridor.

This assumptions allows to consider a target as missed

only if its trajectory does not intercept an agent, re-

gardless of where such an agent is intercepted.

To utilize the framework for comparing the ad-

vantages of one formation over another in a speciﬁc

mission, an additional mathematical step is required.

Speciﬁcally, we must calculate the integral of the tar-

get interception probabilities between the maximum

and minimum entry points of the target (and possibly

also between the starting and ending points of agent

exploration). This calculation will subsequently yield

the ﬁnal interception probability for the given forma-

tion within the given mission context.

In Section 4, we will elaborate on these steps for

validation purposes.

4 APPLICATION OF THE

FRAMEWORK

In this section we ﬁrst introduce I

(resp. I

), the

quantity that represents the integral of the punctual

probability of intercepting the target entering in front

(resp. in the back) of the drones up to a maximum

distance that will be used in the mission considered.

Intuitively, this value takes into account the fact that,

while the drones are exploring, the distance where the

target might appear in front (back) changes. By inte-

grating various distances, I

allows us to argue about

the general performance of the mission.

Let I

F,L

max

) represent the integral of the proba-

bility of intercepting the target entering up to a maxi-

mum distance of x

max

, in front of the drones ﬂying in

the line formation:

F,L

max

) =

max



(x)



dx . (11)

Similarly, we can deﬁne the quantities I

B,L

max

F,AV

max

) and I

B,AV

max

) where F stands for front

and B stands for back of the formation, while L stands

for line formation and AV stands for either arrow or

vee formation.

4.1 Mission Description

We select two scalar-representable missions, called

entry-time aware and entry-time agnostic. They differ

on the assumptions on the probability P

(x) that the

target enters the corridor when the ﬂeet is at a given

point x.

Entry-Time Aware Missions. The basic idea of

this mission is that the ﬂeet is alerted to the entry

of a target and consequently already begins to move.

This assumption translates in practice to the assump-

tion that the target enters the corridor at instant t = 0

at a uniformly distributed random distance within the

range [0,x

], where x

is the length of the corridor.

The main variable is the position of the ﬂeet when

the target arrives. We therefore deﬁne as P

awe

(x)

the probability of intercepting the target in an entry-

time aware environment when the ﬂeet is at position

x ∈ [0,x

]. When the formation is at a distance x from

the beginning of the corridor, the probability of inter-

cepting the target for a line formation P

awe

(x) is equal

to:

awe

(x) =



F,L

− ℓ

− x) + 1 · ℓ

+ I

B,L

(x)



(12)

The computation of P

awe

(x) consists of three com-

ponents, the ﬁrst computes the contribution associated

to the case where the target enters in front of the for-

mation, the middle one takes into account the fact that

the target can enter “touching” the formation and the

third component takes into account the contribution

due to the target entering in the back of the formation.

Similarly, for the arrow and vee formations we

have:

awe

(x) =



F,AV

− ℓ

− x) + 1 · ℓ

+ I

B,L

(x)



(13)

awe

(x) =



F,L

− ℓ

− x) + 1 · ℓ

+ I

B,AV

(x)



(14)

Entry-Time Agnostic Missions. For these mis-

sions we assume that we do not know the exact time

of entrance of the target, we only assume that it will

enter while the ﬂeet of drones is moving. We assume

that the drones start ﬂying at t = 0 at the bottom of the

corridor and ﬂy through the corridor up to its end. The

target can enter at any time during the movement. Let

ICAART 2025 - 17th International Conference on Agents and Artiﬁcial Intelligence

476

agn

be the probability of interception for an entry-

time agnostic mission of a line formation. The for-

mula for computing P

agn

reduces to the integral over

the whole corridor of the quantity P

awe

, where P

(x)

is the probability that the target enters at distance x.

For this mission, we assume that P

(x) is uniform and

therefore it can be considered constant, leading to the

following calculation:

agn

−ℓ

awe

(x)dx =

−ℓ

F,L

− ℓ

− x) + 1 · ℓ

+ I

B,L

(x))dx

and for the other two formations we have:

agn

−ℓ

awe

(x)dx =

−ℓ

F,AV

− ℓ

− x) + 1 · ℓ

+ I

B,L

(x))dx

agn

−ℓ

awe

(x)dx =

−ℓ

F,L

− ℓ

− x) + 1 · ℓ

+ I

B,AV

(x))dx .

4.2 Simulation-Based Validation

To assess the validity of the mathematical framework

presented in Section 3, we implemented a lightweight

simulator using Python. It allows us to deﬁne mul-

tiple moving entities and control them through ve-

locity and yaw rate input. We used it to simulate a

ﬂeet of aerial agents and the moving target that are

present in the two missions presented in Section 4.1.

Each aerial agent is equipped with a detection sensor,

which provides the capability of detecting a target if

it falls within a certain rectangular area simulating the

sensor’s ﬁeld of view. This area corresponds to the vi-

sual range of a real-world camera-equipped drone.

With the agents in formation, the ﬂeet starts to

move in a direction within a ﬁxed space interval. Dur-

ing the mission, the target can appear at different mo-

ments, according to the analyzed mission. A target

will be considered detected if it falls within the visual

range of at least one aerial agent during the mission.

In order to carry out a fair comparison between

simulation and mathematical framework, we ran sev-

eral missions in different conditions. In detail, each

experiment is parametrized with the following vari-

ables: i) ﬂeet velocity, ii) target’s starting point,

iii) target’s heading, and iv) target’s appearance time.

While the model considers these parameters as con-

tinuous variables, in simulation we uniformly sam-

ple each of them in 150 instances for the entry-time

aware, and 24 instances for the entry-time agnostic

mission. The introduction of an additional variable

in the entry-time agnostic scenario signiﬁcantly in-

creases the computation time. Hence, the choice to

reduce the sampling rate for this case allowed us to ﬁx

the simulation time to a given maximum (i.e., around

4 hours).

The validation of the mathematical framework is

carried out by analyzing the same scenario both in

simulation and using the framework. For each of

them, we compute the probability of successful target

detection at different ratios between target and ﬂeet

velocity. Eventually, we expect to get similar results

using both simulation and the mathematical frame-

work. The parameters of the missions considered are

reported in Table 1. The simulations were run on a

workstation with an Intel(R) Core(TM) i9-10980XE

processor running at 3.00GHz.

Table 1: Parameters of the considered scenarios used in both

mathematical framework and simulation.

Parameter Value

Corridor length 1km

Detection range 90m × 67.5m

Formation width 450m

Arrow/Vee formation height 202.5m

Maximum ﬂeet velocity 10

Target velocity 5

The results from the validation are presented in

Fig. 8. Each subﬁgure displays the probability of

successful target detection as a function of the ra-

tio between target and ﬂeet velocities. Qualitatively,

the same trend in performance for both mathematical

framework and simulation stands out. The discrep-

ancy between the plot obtained with the mathematical

framework and the simulation performance never ex-

ceeds 1.2ppt for the entry-time aware and 2.3 ppt for

the entry-time agnostic missions, which is expected

due to a slight discrepancy between the theoretical as-

sumptions and the simulator. Such similar results be-

tween our framework and the simulation demonstrate

that the mathematical model is able to properly assess

the performance of different formations in both of the

presented scenarios. At the same time, it allows for

almost immediate computation of results, in contrast

to the simulation used for validation, which takes sig-

niﬁcant time to complete.

4.3 Extended Results with the

Framework

We used the framework to obtain additional insights

into the proposed scenarios. To do so, we use the

Formation Analysis for a Fleet of Drones: A Mathematical Framework

477

(a) Mathematical framework results, entry-time aware

mission

(b) Simulation results, entry-time aware mission

mission

(d) Simulation results, entry-time agnostic mission

Figure 8: Comparisons of mathematical framework vs simulation results for entry-time aware and agnostic missions.

mathematical framework to calculate the aforemen-

tioned probabilities in various scenarios.

2 4 6 8 10 12 14 16 18 20

Corridor Length (km)

Improvement (%)

arrow, v

= 1

vee, v

= 1

arrow, v

= 1.1

vee, v

= 1.1

arrow, v

= 1.25

vee, v

= 1.25

Figure 9: Results of the experiment showing how the per-

formance depends on the length of the corridor. The vertical

axis shows the relative improvement of detection chance wrt

to the line formation.

To support the reproducibility of our research, we

publish the source code of the mathematical frame-

work and of this study in [redacted in initial submis-

sion to maintain anonymity]. We start by analyzing

the results presented in Fig. 8. First of all, for both

scenarios, the vee formation results in the best per-

formance, being slightly better than the arrow for-

mation. The line formation performing signiﬁcantly

worse. Furthermore, in both cases, the formations

achieve the same performance starting from a velocity

ratio of around 1.25, which conﬁrms the intuition that

for fast-moving search parties, the formation choice

is less important. For the velocity ratio above 1.5,

they converge to the 50 % of successful detections.

This observation matches the intuition that, with high

speed, the 50% of the targets that appear in front of

the formation will all be detected before they can es-

cape from the ﬂeet detection range. Furthermore, the

results of the entry-time aware mission present a per-

haps counter-intuitive ﬁnding: if the searching drones

cannot move as fast as the target, it might be better to

slow them down even further. This can be explained

by the fact that moving slower gives more chances for

the targets appearing behind the formation to catch

up with the formation and the formation will not be

able to catch up to many targets appearing in front of

ICAART 2025 - 17th International Conference on Agents and Artiﬁcial Intelligence

478

it. This behavior does not appear in the entry-time

agnostic mission, where a target entrance time is in-

cluded.

In Fig. 9 we show how the performance in the

entry-time agnostic mission changes depending on

the corridor length. We present it as a relative im-

provement wrt to the line formation to emphasize the

beneﬁts of choosing the correct formation. We show

the plots for multiple values of v

to show how this

plot changes in relation to the ﬁndings previously pre-

sented in Fig. 8. The obtained results are very intu-

itive: the longer the corridor, the lower the beneﬁt of

choosing the better formation.

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

100

200

300

400

500

600

700

800

900

Group position when target appears (m)

Arrow

Vee

Figure 10: Results of the experiment showing which for-

mation outperforms the other according to different velocity

ratios and positions of the group when the target appears.

Finally, to explicitly address the question pre-

sented as the motivation of this research, we present

Fig. 10. It presents the difference in performance be-

tween the vee and arrow formation and could serve as

an easy way to choose which one to prefer, depend-

ing on the scenario at hand. The line formation is

not presented, as it is never performing better than the

aforementioned two.

4.4 Robotic Proof of Concept

Our proof of concept involves deploying a ﬂeet of

six UAVs (speciﬁcally, Holybro PX4 Vision) in each

of the formations introduced above. The UAVs are

equipped with Arducam 12 MP, IMX477 camera with

an 80

◦

horizontal ﬁeld of view, ﬂying at an altitude of

50 meters with an exploration depth (corridor length)

of 1 kilometer. This setup was successfully tested

over a maritime environment to detect a target in the

sea and in a deserted area to detect a car using a

YOLO algorithm (Redmon et al., 2016), demonstrat-

ing that our model is applicable in realistic scenarios.

5 CONCLUSIONS

The mathematical framework presented in this paper

offers a convenient and efﬁcient means of analyzing

the formation ﬂights of a ﬂeet of drones. It allows

for the estimation of the probability of target detec-

tion during a search mission. Speciﬁcally, it serves

as a valuable tool for selecting the most effective

formation in a given context. The proposed scenar-

ios demonstrate the practical value of this framework

in real-world applications such as search and rescue,

border surveillance, and environmental monitoring.

Although our initial experiments yielded promising

results, we are planning further in-ﬁeld experiments

to gain insights and quantitatively validate the math-

ematical framework under realistic conditions and

across various scenarios.

REFERENCES

Afonso, F., Sohst, M., Diogo, C. M. A., Rodrigues, S. S.,

Ferreira, A., Ribeiro, I., Marques, R., Rego, F.

F. C., Sohouli, A., Portugal-Pereira, J., Policarpo, H.,

Soares, B., Ferreira, B., Fernandes, E. C., Lau, F., and

Suleman, A. (2023). Strategies towards a more sus-

tainable aviation: A systematic review. Progress in

Aerospace Sciences, 137:100878.

Anderson, B. D., Fidan, B., Yu, C., and Van der Walle, D.

(2008). UAV Formation Control: Theory and Appli-

cation. In Recent advances in learning and control,

volume 371 of Lecture Notes in Control and Informa-

tion Sciences, pages 15–33. Springer edition.

Antczak, A., Lasek, M., and Sibilski, K. (2022). Efﬁcient

Positioning of Two Long-Range Passenger Aircraft

in Formation Flight. Transactions on Aerospace Re-

search, Nr 3 (268).

Atınc¸, G. M., Stipanovi

c, D. M., and Voulgaris, P. G.

(2020). A swarm-based approach to dynamic cov-

erage control of multi-agent systems. Automatica,

112:108637.

Bajec, I. L. and Heppner, F. H. (2009). Organized ﬂight in

birds. Animal Behaviour, 78(4):777–789.

Chao, Z., Zhou, S.-L., Ming, L., and Zhang, W.-G. (2012).

UAV Formation Flight Based on Nonlinear Model

Predictive Control. Mathematical Problems in Engi-

neering, 2012:e261367. Publisher: Hindawi.

Cheng, T. M. and Savkin, A. V. (2009). Decentralized co-

ordinated control of a vehicle network for deployment

in sweep coverage. In 2009 IEEE International Con-

ference on Control and Automation, pages 275–279.

ISSN: 1948-3457.

de Souza, C., Newbury, R., Cosgun, A., Castillo, P., Vi-

dolov, B., and Kuli

c, D. (2021). Decentralized Multi-

Agent Pursuit Using Deep Reinforcement Learning.

IEEE Robotics and Automation Letters, 6(3):4552–

4559. Conference Name: IEEE Robotics and Au-

tomation Letters.

Formation Analysis for a Fleet of Drones: A Mathematical Framework

479

Dorigo, M., Floreano, D., Gambardella, L. M., Mondada,

F., Nolﬁ, S., Baaboura, T., Birattari, M., Bonani, M.,

Brambilla, M., Brutschy, A., Burnier, D., Campo,

A., Christensen, A. L., Decugniere, A., Di Caro,

G., Ducatelle, F., Ferrante, E., Forster, A., Gonzales,

J. M., Guzzi, J., Longchamp, V., Magnenat, S., Math-

ews, N., Montes de Oca, M., O’Grady, R., Pinciroli,

C., Pini, G., Retornaz, P., Roberts, J., Sperati, V., Stir-

ling, T., Stranieri, A., Stutzle, T., Trianni, V., Tuci,

E., Turgut, A. E., and Vaussard, F. (2013). Swar-

manoid: A Novel Concept for the Study of Hetero-

geneous Robotic Swarms. IEEE Robotics & Automa-

tion Magazine, 20(4):60–71. Conference Name: IEEE

Robotics & Automation Magazine.

Hartjes, S., Visser, H. G., and van Hellenberg Hubar,

M. E. G. (2019). Trajectory Optimization of Ex-

tended Formation Flights for Commercial Aviation.

Aerospace, 6(9):100. Number: 9 Publisher: Multi-

disciplinary Digital Publishing Institute.

Hespanha, J., Kim, H. J., and Sastry, S. (1999). Multiple-

agent probabilistic pursuit-evasion games. In Pro-

ceedings of the 38th IEEE Conference on Decision

and Control (Cat. No.99CH36304), volume 3, pages

2432–2437 vol.3. ISSN: 0191-2216.

Ivi

c, S., Crnkovi

c, B., Arbabi, H., Loire, S., Clary, P., and

Mezi

c, I. (2020). Search strategy in a complex and

dynamic environment: the MH370 case. Scientiﬁc Re-

ports, 10:19640.

Justh, E. W. and Krishnaprasad, P. S. (2002). A Simple

Control Law for UAV Formation Flying.

Kovacina, M., Palmer, D., Yang, G., and Vaidyanathan, R.

(2002). Multi-agent control algorithms for chemical

cloud detection and mapping using unmanned air ve-

hicles. In IEEE/RSJ International Conference on In-

telligent Robots and Systems, volume 3, pages 2782–

2788 vol.3.

Liu, H. Y., Chen, J., Huang, K. H., Cheng, G. Q., and Wang,

R. (2023). UAV swarm collaborative coverage con-

trol using GV division and planning algorithm. The

Aeronautical Journal, 127(1309):446–465. Publisher:

Cambridge University Press.

Mazdin, P., Barci

s, M., Hellwagner, H., and Rinner, B.

(2020). Distributed Task Assignment in Multi-Robot

Systems based on Information Utility. In 2020 IEEE

16th International Conference on Automation Science

and Engineering (CASE), pages 734–740. ISSN:

2161-8089.

Papatheodorou, S. and Tzes, A. (2018). Theoretical and

Experimental Collaborative Area Coverage Schemes

Using Mobile Agents.

Paul, T., Krogstad, T. R., and Gravdahl, J. T. (2008). Mod-

elling of UAV formation ﬂight using 3D potential

ﬁeld. Simulation Modelling Practice and Theory,

16(9):1453–1462.

Redmon, J., Divvala, S., Girshick, R., and Farhadi, A.

(2016). You Only Look Once: Uniﬁed, Real-Time

Object Detection. arXiv:1506.02640 [cs].

Saska, M., Von

asek, V., and P

reu

cil, L. (2013). Trajectory

Planning and Control for Airport Snow Sweeping by

Autonomous Formations of Ploughs. Journal of Intel-

ligent & Robotic Systems, 72(2):239–261.

Wang, X., Yadav, V., and Balakrishnan, S. N. (2007).

Cooperative UAV Formation Flying With Obsta-

cle/Collision Avoidance. IEEE Transactions on Con-

trol Systems Technology, 15(4):672–679. Conference

Name: IEEE Transactions on Control Systems Tech-

nology.

Wei, X. and Yanq, J. (2018). Optimal Strategies for Mul-

tiple Unmanned Aerial Vehicles in a Pursuit/Evasion

Differential Game. Journal Guidance, Control and

Dynamics, 41.

Yu, Y., Li, Z., Wang, X., and Shen, L. (2019). Bearing-

only circumnavigation control of the multi-agent sys-

tem around a moving target. IET Control Theory &

Applications, 13(17):2747–2757.

Zhai, C. and Hong, Y. (2012). Decentralized sweep cover-

age algorithm for uncertain region of multi-agent sys-

tems. In 2012 American Control Conference (ACC),

pages 4522–4527. ISSN: 2378-5861.

Zhai, C. and Hong, Y. (2013). Decentralized sweep cover-

age algorithm for multi-agent systems with workload

uncertainties. Automatica, 49(7):2154–2159.

ICAART 2025 - 17th International Conference on Agents and Artiﬁcial Intelligence

480