Modeling the Willingness to Interact

in Cooperative Multi-robot Systems

Mirgita Frasheri

1 a

, Lukas Esterle

2 b

and Alessandro Vittorio Papadopoulos

1 c

alardalen University, V

aster

as, Sweden

Aarhus University, DIGIT, Aarhus, Denmark

Keywords:

Robot Collaboration, κ-Coverage Problem, Adaptive Autonomy.

Abstract:

When multiple robots are required to collaborate in order to accomplish a speciﬁc task, they need to be coor-

dinated in order to operate efﬁciently. To allow for scalability and robustness, we propose a novel distributed

approach performed by autonomous robots based on their willingness to interact with each other. This will-

ingness, based on their individual state, is used to inform a decision process of whether or not to interact with

other robots within the environment. We study this new mechanism to form coalitions in the on-line multi-

object κ-coverage problem, and compare it with six other methods from the literature. We investigate the

trade-off between the number of robots available and the number of potential targets in the environment. We

show that the proposed method is able to provide comparable performance to the best method in the case of

static targets, and to achieve a higher level of coverage with respect to the other methods in the case of mobile

targets.

1 INTRODUCTION

Robots collaborating with each other in order to

tackle a task perform faster and more efﬁciently

than their individually operating counterpart. Some

tasks even require collaboration of multiple robots

and cannot be accomplished by individual robots at

all. However, such collaboration requires coordina-

tion of the individual robots, and formation of coali-

tions between them. Numerous coalition formation

approaches have been proposed which either rely on

central components (Garc

ıa et al., 2018; Burgard

et al., 2002; Shehory and Kraus, 1998) or focus on

a single task to be accomplished (Qureshi and Ter-

zopoulos, 2007) in order to achieve meaningful inter-

action and collaboration. These approaches require

dissipation of information about available coalitions

as well as negotiations about participation of each po-

tential coalition member (Shehory and Kraus, 1998;

Ye et al., 2013; Qureshi and Terzopoulos, 2007).

While coalitions are usually formed around single

tasks, the use of multiple teams has been shown to

be beneﬁcial when pursuing goals that require multi-

https://orcid.org/0000-0001-7852-4582

https://orcid.org/0000-0002-0248-1552

https://orcid.org/0000-0002-1364-8127

ple tasks to be accomplished concurrently (Theraulaz

et al., 1998; Esterle, 2018). Moreover, when con-

sidering autonomously operating robots that aim to

achieve multiple tasks, the individuals have to make

decisions on when and how to form coalitions, and to

what end the coalition is formed.

In this work, we are interested in the ability of au-

tonomously operating robots to interact and collab-

orate in order to provision varying sets of tasks ef-

ﬁciently, without a central component involved. We

propose an approach where each robot makes indi-

vidual decisions about whether or not to provision a

speciﬁc task, employing local information about its

own status, e.g., its battery level, its ability, and its

interest (i.e., expected performance value the robot

contributes to the collective) in performing such task.

More speciﬁcally, we propose a novel distributed

coalition formation and study this approach in the

online multi-object κ-coverage problem (Esterle and

Lewis, 2017; Esterle and Lewis, 2019), which is re-

lated to the cooperative multi-robot observation of

multiple moving targets (CMOMMT) problem pro-

posed by Parker and Emmons (Parker and Emmons,

1997), and consists of a varying number of tasks re-

quired to be tackled concurrently.

First, the robots need to discover initially un-

known moving objects in the environment. They do

Frasheri, M., Esterle, L. and Papadopoulos, A.

Modeling the Willingness to Interact in Cooperative Multi-robot Systems.

DOI: 10.5220/0008951900620072

In Proceedings of the 12th International Conference on Agents and Artiﬁcial Intelligence (ICAART 2020) - Volume 1, pages 62-72

ISBN: 978-989-758-395-7; ISSN: 2184-433X

not possess any a priori information about the number

or location of these objects. Furthermore, objects may

be mobile, requiring robots to change their own loca-

tion respectively in order to continuously provision

them. Second, each object needs to be provisioned

with at least κ robots concurrently, i.e., κ robots hav-

ing the object within their sensing/actuating region at

the same time. Here, detecting new targets is consid-

ered the ﬁrst task, however, every newly discovered

target generates a new task for the collective of cover-

ing this known target. This generates a trade-off be-

tween detecting new objects and covering known ob-

jects with κ robots when the collective tries to maxi-

mize the duration and number of targets covered by

κ robots. However, a robot not only needs to de-

cide between provisioning a speciﬁc target or explor-

ing the area to discover new targets, but also which

of the different known targets it wants to provision.

In order to achieve an efﬁcient outcome in this trade-

off, the robots are required to form new coalitions for

each individual target. According to the taxonomy

of Robin and Lacroix (Robin and Lacroix, 2016), the

on-line multi-objective κ-coverage problem is hunt-

ing mobile search, monitoring multiple targets, with

different viewpoints.

In this paper, we present a novel distributed coali-

tion formation algorithm considering several tasks. At

its core, we propose to introduce a willingness to in-

teract to each individual robot as the main driver for

the coalition formation. The willingness is depen-

dent on the state of the robot, such as local conditions

like battery level, and current level of activity. Uti-

lizing this willingness, robots can make decisions on

whether or not to interact and provision a speciﬁc ob-

ject which eventually leads to forming coalitions with

other robots. This approach is evaluated over several

scenarios of increasing number of targets, consider-

ing both static and mobile targets separately. The per-

formance is assessed through different metrics, e.g.,

the average number of agents covering one target, the

average coverage time with at least κ agents. Fur-

thermore, the proposed approach is compared against

six other methods presented in the literature. The

proposed approach shows performance that is either

comparable with the best of the methods it is com-

pared with – in the case of static objects – while it

exhibits a higher coverage in the case of moving tar-

gets.

The remainder of this paper is structured as fol-

lows. Section 2 gives a formal deﬁnition of the on-

line multi-object κ-coverage problem and Section 3

covers the behaviour of the agents and targets, their

interaction as well as our novel coalition formation

algorithm. Section 4 gives an overview of the experi-

mental setup, the performed experiments, and the ob-

tained results. Section 5 discusses the generalization

of the proposed approach, while Section 6 concludes

the paper and outlines future work.

2 PROBLEM FORMULATION

In the online multi-object κ-coverage problem, we

assume a discrete 2D area Z with a given width

and height w and h, respectively, without any obsta-

cles. We also consider a set of active robots A =

,... ,a

}, and a set of targets or objects of inter-

est O = {o

,. .. ,o

} in this problem. Both robots

and objects can freely move within Z, with (noncon-

stant, yet limited) velocities v

, where i = 1,...,n, and

, where j = 1,. .. ,m; in their motion the robots will

always remain in Z. It is assumed that any robot can

move faster than the objects, and that the number of

robots and targets is constant, i.e., targets cannot ap-

pear or disappear. Each robot is controlled by an in-

ternal, autonomous software agent. We refer to both

as a

. Each robot has a visibility range, with radius

r. An object can be perceived by any robot, only if

it is located within its visibility range. At this point,

the robot will determine the number of already provi-

sioning robots for this object. Therefore, it will either

initiate a new coalition, in the case of no robots fol-

lowing the target, or join the existing coalition, in the

case that less than κ agents are following the target.

All objects are associated with a prescribed constant

interest level l

. Levels of interest are not necessarily

the same between objects, and deﬁne the utility u

i j

(t)

of a robot i for following an object j with interest level

at a discrete time-step t.

Every agent i can calculate its willingness w

interact with others (as detailed in Section 3.3) at

each time-step. This can occur in different situations,

e.g., (i) when a robot i ﬁrst detects an object j en-

tering or leaving its sensing area an object j is en-

tering the sensing area of the robot i and (ii) when

robot i receives an invitation to provision an object

j from another robot. Robots are assumed to com-

municate with one another via broadcast, as imple-

mented in ROS (Quigley et al., 2009). Thus, the will-

ingness to interact shapes the cooperative behavior of

an agent and its respective robot in relation to the oth-

ers. Robots are able to change and keep track of their

own state and behavior, as well as the state and be-

havior of other robots. Speciﬁcally, robot’s n state is

composed of the following variables: battery level b

range d, location `

x,y

, and velocity v

a,i

. Without loss

of generality, we assume that the level of interest for

the targets is robot-independent, i.e., there is a shared

Modeling the Willingness to Interact in Cooperative Multi-robot Systems

knowledge among the agents on the level of interest

of different targets.

The online multi-object κ-assignment problem is

solved by having at least κ robots covering any target

in the set. Consequently, two tasks should be achieved

concurrently: (i) maximizing the number of provi-

sioned objects, and (ii) provisioning the targets with

at least κ robots. This paper addresses the following

questions:

1. What is the average time for which at least κ

robots can cover all targets moving around in an

environment when using the proposed coalition

formation algorithm?

2. What is the average number of agents that can

cover a target with the proposed coalition forma-

tion algorithm?

3. Is the motion of the targets affecting the obtained

performance?

4. How does the deﬁned value for κ affect the per-

formance of the robot cooperation?

5. How does the proposed method compare with

other state-of-the-art techniques for the κ-

coverage problem?

We address these questions using two experimental

setups with varying number of either mobile or static

(immobile) targets, according to metrics that analyze

the obtainable performance in terms of time to cover

targets with at least κ robots, and the average num-

ber of robots that cover the targets. Furthermore, we

compare our results with six other methods previously

proposed in the literature (Esterle and Lewis, 2017).

3 AGENT MODEL

In this section, we describe how a robot operates, how

the agent, embodied in a robot, updates the willing-

ness to interact, and how these agents form decisions

to cooperate through the proposed interaction proto-

cols. In the following we are using the terms robot

and agent interchangeably.

3.1 Robot Kinematics

Every robot a ∈ A follows a simple unicycle kinematic

model











˙x

(t) = v

(t)cos(θ

(t))

˙y

(t) = v

(t)sin(θ

(t))

(t) = ω

(t)

(1)

where x

(t) and y

(t) are the x- and y-coordinate on

the map and deﬁne the position p

= (x

) of a robot

a at time t, θ

is the orientation of the robot, v

is the

forward velocity of the robot, and ω

is its angular

velocity. We assume that the robot can localize itself

within the map, and that it can detect the obstacles

within its visibility range.

A robot a ∈ A follows a set of objects O

⊆ O,

each of which has a different level of interest l. The

direction d

over which the robot moves is thus com-

puted as

(t) =

∑

i∈O

(t) − p

(t))

∑

i∈O

(2)

making the robot to move towards all the followed

objects, weighted by their respective interest. In this

way, the robot will prioritize targets with higher level

of interest. The target orientation θ

◦

and the forward

velocity of the robot are therefore computed as:

◦

(t) = ∠d

(t), (3)

˜v

(t) = kd

(t)k (4)

where ∠p ∈ [0,2π) is the angle of the vector p =

, p

) in its reference frame, and it is obtained as

∠p = atan2(p

, p

). In order to compute the proper

value of the angular velocity, we can just use a simple

proportional controller with tracking error e

normal-

ized between [−π,π):

(t) = θ

◦

(t) − θ

(t) (5)

(t) = K

atan2(sin(e

(t)),cos(e

(t))) (6)

Finally, we include saturations on the forward and an-

gular velocities:

(t) = min( ˜v(t), v

max

) (7)

(t) = min(max(

(t),−ω

max

),ω

max

) (8)

3.2 Agent Behavior

Software agents, embodied in physical robots, oper-

ate autonomously and their behavior can be described

as a state machine composed of four states: inspect,

evaluate, inspect & follow, and evaluate & follow.

Figure 1 shows the state machine that describes the

behavior structure of an agent. At run-time, any agent

starts its operation in the state inspect, in which it

moves in Z according to a given pattern. In case a

new target is spotted, or a request is received, an agent

switches from inspect to evaluate. In the evaluate

state, an agent decides how it wants to interact with

the spotted target or the request for help, based on

its current state. The proposed interaction protocol is

described in detail in Section 3.4. The result of the

interaction is a coalition of agents that will start fol-

lowing the spotted target. If the agent is not part of

the coalition after the interaction, it will switch back

ICAART 2020 - 12th International Conference on Agents and Artiﬁcial Intelligence

Inspect

& Follow

Evaluate

& Follow

New object

or New request

Not part of

the coalition

Lost

all targets

New object

or New request

Updated coalitions

Part of the

coalition

Figure 1: Agent operation state machine.

to the inspect state, looking for other targets in the

environment. Otherwise, if the agent is part of the

coalition, then it will switch to the inspect & follow

state. In this new state, the agent follows the target,

while it simultaneously inspects for new ones. In case

an agent loses track of all the targets it is following,

then it switches to inspect. In case an agent has neg-

ative willingness, spots a new target, detects that a

target is going outside its visibility region, or it gets

a request for help, then it switches to the evaluate &

follow state. In this state, the agent either generates

a help request, or it responds to a help request. In

both cases, the agent decides if it will be part of a new

coalition, or if it is going to drop a target. Once the

interaction is complete, an agent switches back to the

inspect & follow state with an updated set of targets

to follow.

Note that an agent can be part of more than one

coalition, i.e., can follow several targets simultane-

ously according to their level of interest (as per Eq. 2),

but a target is only followed by a single coalition.

Also, notice that transitions between states are con-

sidered to be instantaneous.

When an agent is following a set of targets, its mo-

tion is described by the dynamic model (1), and by

control strategy deﬁned in (3)–(8). The interest level

of a target affects the motion of the agent according

to (2), i.e., the agent’s direction is mostly affected by

the level of interest of the targets.

3.3 Willingness to Interact

The willingness to interact w shapes the cooperative

behavior of an agent, i.e., when an agent should ask

for help and when it should give help. This parameter

does not refer to a particular task that should be com-

pleted, but rather reﬂects the general disposition of

any agent to cooperate with the others. The willing-

ness to interact w takes values in [−1,1]. When w ≥ 0

the agent is willing to provide support to other agents

that have requested help. When w < 0, the agent will

raise requests for help, and for w = −1 it cannot con-

tinue with the execution of a task on its own. The

value of the willingness is updated by each individual

agent i based on several individual factors, at discrete

time instants t, according to the dynamics:

(t + 1) = min(max(w

(t) + B

f(t),−1), 0), (9)

f(·) = [ f

(·),. .. , f

(·)]

is an m × 1 vector of the

m factors that affect the willingness, while B =

[β

,. .. ,β

]

is an m × 1 vector that contains the

weights of the corresponding factors on the calcula-

tion of the willingness.

The calculation of a factor f

is given by

(k) = φ

(k) − φ

i,min

, (10)

where φ(k) represents the current measurement of that

factor (e.g., the current battery level), while φ

min

a minimal threshold considered acceptable (e.g., the

minimal battery level to perform a task). The terms φ

and φ

min

take values in [0, 1], where 0 is the minimum

value of the measured quantity, and 1 its maximum.

In this work, we consider two factors that affect

the willingness to interact. These are the battery level

b, and the number of objects in O

currently provi-

sioned by a. Other factors can be included in the cal-

culation of the willingness, without loss of generality

of the proposed approach.

Factors can be divided into two categories: nec-

essary and optional. The battery level is a necessary

factor, since a robot with a battery level lower than a

certain threshold may not be able to reach the moving

target, or to complete an assigned task. Therefore, an

agent with a low battery level should try to receive

help from the other agents. On the other hand, the

number of targets (n

) an agent is tracking is con-

sidered as optional, since an agent can follow several

targets, but this makes its task more difﬁcult, e.g., if

1/n

goes below a certain threshold – the agent is

following too many targets – then the agent decreases

its willingness to give help and consequently increase

its willingness to ask for help. The effect of different

factors is deﬁned by their corresponding weights. The

weight for a necessary factor β

nec

is deﬁned as:

nec

(t) =

(

1/m, φ

nec

(t) − φ

nec,min

> 0,

−(1 + w(t)), otherwise,

(11)

where m is the number of all the factors, whereas the

Modeling the Willingness to Interact in Cooperative Multi-robot Systems

weight for an optional factor β

opt

is deﬁned as:

opt

(t) =







0, if ∃φ

nec

,φ

nec

(t) − φ

nec,min

< 0,

sgn(φ

opt

(t) − φ

opt,min

)

, otherwise.

(12)

This ensures that necessary factors have the highest

impact on the willingness to interact. As an exam-

ple, in the case the battery level is below a thresh-

old, then the agent should ask for help, irrespective

of other factors (w = −1). Thus, the weights of other

factors should be set to zero. While we provided an

example for factors approaching a minimum, factors

approaching a maximum can also be applicable. In

such a case the calculation for factors and weights has

to be adapted accordingly. More examples on factors

that can affect the willingness can be found in (Frash-

eri et al., 2018).

3.4 Interaction Protocol

The interaction protocol deﬁnes how agents create

coalitions for any given target and elect the corre-

sponding leaders for these coalitions. The proposed

protocol mostly complies with the SCR design pat-

tern (Casadei et al., 2019), however differently from

SCR an agent can belong to different coalitions, hence

it can have more than one leader. An agent can trigger

a help request in case it spots a new target, or it wants

to extend an existing coalition to reach κ−coverage,

or it perceives that targets in its visibility range are

moving away from itself, and if it is necessary to

ask for help (e.g., battery level is under the accepted

minimum). Furthermore, agents can decide to inter-

act with one another when they receive help requests

from others. The interaction protocol is illustrated in

Figure 2.

When an agent spots a new target, it broadcasts an

information request to other agents together with its

willingness and respective utility for provisioning the

target. The agent waits for a speciﬁed time ∆t to re-

ceive a response from other robots. We assume that

agents can identify commonly observed objects and

assign common labels. In case a coalition exists al-

ready for the given target, the corresponding leader

will reply whether or not further agents are needed

to reach the κ-coverage. If no help is needed, then

the agent continues its previous activities. If help

is needed, then the agent will receive an assignment

from the leader of the coalition, based on the previ-

ously sent willingness and utility. In case the agent

does not receive a response within time ∆t to its ini-

tial information request, it assumes no other agent is

following the target. Subsequently, the process for

creating a coalition and electing a leader responsi-

Request Information

Wait ∆t

Check

Response

Deﬁne own w

Broadcast help

request

Collect answers

Order by w

target

Select κ agents

& leader

Notify κ agents

Got assigned

Ignore target

response

Help is

needed

Help is

not needed

Figure 2: Activity diagram of the agent’s behavior when a

new target is spotted.

ble for following the target is triggered. Initially, the

agent calculates its own willingness to help in the fu-

ture coalition, and the utility from following the tar-

get. The mechanism follows the logic of a fast bully

algorithm (Lee and Choi, 2002), well known in dis-

tributed systems. A request for help to follow the ob-

ject is broadcast to all other agents. Other agents send

their willingness to help, i.e, the willingness to enter

the coalition, and their utility for following the spe-

ciﬁc target. After the responses are collected, agents

with a negative willingness w < 0, are not considered

further. Positive willingness of an agent i to interact

is combined with its utility u

i j

to form the willingness

to interact to provision a speciﬁc object j at time t:

i j

(t) = w

(t) + u

i j

(t). (13)

Utilities are deﬁned by each agent for the individual

target and can generally vary between the different

agents as well as the different targets. Examples for

this could be the size, speed, or direction of movement

of the object. In our experiments, we consider dif-

ferent interest levels that are agent-independent, i.e.,

the agents share the same interest for the same tar-

gets. The received values w

i j

(t) are ordered, and the

κ agents with highest w

i j

(t) are selected for the coali-

tion. The agent with the highest w

i j

(t) is elected the

leader. The outcome is propagated to the other agents.

ICAART 2020 - 12th International Conference on Agents and Artiﬁcial Intelligence

Figure 3: Agent a

with visibility range d, indicated with

the red circle, and internal range d

, indicated with the blue

circle. The targets tg

and tg

are indicated with crosses,

and they are moving towards and away from the agent, re-

spectively.

The initiating agent does not necessarily need to be

part of the coalition.

Furthermore, every agent keeps track of whether

the targets in its visibility range are moving away

from the robot. We introduce another internal thresh-

old with radius d

around the robot, where d

< d

(Figure 3). When a target, e.g., tg

in Figure 3, moves

out of the internal range, yet remains within the vis-

ibility range, then a request for help is triggered. If

a target, e.g., tg

, is moving towards the agent while

being within the internal and visibility range, no re-

quest is issued. In case the willingness of an agent

becomes negative, help requests are generated. At the

same time, an agent will consider dropping its targets

one by one. If the willingness remains negative or

becomes −1, eventually all targets will be dropped.

A help request means that either an agent is look-

ing for a replacement for itself, or it is looking for an

additional agent that can enter the coalition. This is

illustrated in Figure 4. If an agent needs to leave a

coalition, we distinguish between leading agents and

ordinary members of the coalition. If a leader agent

needs to replace itself, then the leader election needs

to be repeated. The process can include other agents

not yet in the coalition, if κ-coverage is not achieved

at that point in time. On the other hand, if a com-

mon agent needs to drop a target, then it ﬁrst notiﬁes

its leader. Leaders are also responsible for triggering

continuously the extension of a coalition in order to

maintain κ-coverage, following a monotonically in-

creasing period.

Is leader?

Is there

a coalition?

Notify leader

Start full

leader election

Start partial

leader election

Drop target

Yes No

NoYes

Figure 4: Activity diagram of the agent’s behavior when it

needs to replace itself in a coalition.

4 SIMULATION SETUP

The behavior of the agents was evaluated with com-

puter simulations

, based on the robot operating sys-

tem (ROS) (Quigley et al., 2009; Hellmund et al.,

2016) to model the agents kinematics, behavior, and

interaction.

The method utilizing the willingness to interact,

as described in this paper, was compared to six other

methods that were previously proposed in the liter-

ature for solving the multi-object κ-coverage prob-

lem (Esterle and Lewis, 2017). Each of the six meth-

ods is a combination of one communication model

and one response model. Two communication mod-

els are considered, broadcast BC and random RA. In

the broadcast model an agent broadcasts help request

to everyone, whereas in the random model it sends a

help request to κ random agents. As for the response

models, three are considered: (i) newest-nearest NN,

(ii) available AV , and (iii) received calls RE. In the

newest-nearest model an agent will answer to the re-

quest that is newest, and if there are multiple request

at the same time, it will respond to the one that is

nearest. In the available model the agent answers to

requests according to the newest-nearest strategy only

if it is not engaged in following other target(s). In re-

ceived calls, an agent will answer to requests for ob-

The code for running the simulations is publicly avail-

able at https://gitagent@bitbucket.org/gitagent/gitagent 2.

git

Modeling the Willingness to Interact in Cooperative Multi-robot Systems

jects with the least coverage, only if it is not follow-

ing other targets. The six methods chosen for com-

parison are: BC-NN, BC-AV , BC-RE, RA-NN, RA-

AV, and RA-RE. Such selection is due to previous

results (Esterle and Lewis, 2017), where the broad-

cast and random communication models were evalu-

ated better with respect to the rest, and the response

models were reported to have a signiﬁcant impact on

the κ-coverage.

In all of our simulations, we consider a total num-

ber of n

= 10 robots starting from the same initial

position (0,0), with a random direction, and v

i,max

= 2

units per time-step. If an agent hits any boundary in Z,

it will bounce back at a 90

◦

angle, i.e., we are consid-

ering a limited area surrounded by walls. The objects

to be covered are distributed uniformly in the map Z

of size 100m×100m. We consider 7 different scenar-

ios for our experiments with an increasing number of

objects. The number of objects distributed in the envi-

ronment are 1, 4, 7, 13, 16 and 19 for the correspond-

ing scenarios S0 to S6. For each simulation the in-

terest level of any target was randomly sampled from

a set of levels L = {0.3,0.6,0.9}. We performed 20

experiments for each scenario, with each experiment

having a duration of T

sim

= 300 discrete time steps

and a speciﬁed seed. The latter impacts the initial lo-

cation of the targets, the initial direction for agents

and mobile targets, as well as the level of interest of

targets for each experiment corresponding to a sce-

nario. Given these settings, we analyzed the behavior

of our agents to achieve κ-coverage where κ ≥ 3, and

κ ≥ 5.

4.1 Results for Static Targets

In the ﬁrst set of experiments, we consider only tar-

gets that remain in their initial location (v

= 0). Once

the targets are covered, they remain covered for the

rest of the simulation, as such, the time for reaching

the desired coverage is considered one of the perfor-

mance indicators for evaluation.

For every scenario, we run N different experi-

ments. For every experiment e = 1,...,N, we com-

pute for every target j the time to reach 1-coverage

(1)

j,e

, and the time to reach κ-coverage t

(κ)

j,e

. Based on

this information we can calculate: (i) the average time

to get one target to be covered by at least by κ agents

(κ)

avg

, and (ii) the average minimum time to get all the

targets covered by at least κ agents t

(κ)

min

. These two

metrics give an indication of a minimum coverage,

and a complete coverage, and the respective timing

properties. They are formally deﬁned as:

(κ)

avg

∑

|O|

∑

(κ)

j,e

(14)

(κ)

min

∑

max

(κ)

j,e

(15)

In particular, in these experiments we study (i) the av-

erage time to get one target to be covered by at least

by 1 agent, t

(1)

avg

, (ii) the average time to get one target

to be covered by at least by κ agent, t

(κ)

avg

, (iii) the aver-

age minimum time to get all the targets covered by at

least 1 agent, t

(1)

min

, and (iv) the average minimum time

to get all the targets covered by at least κ agents, t

(κ)

min

In all the metrics, the lower, the better.

Results for κ ≥ 3 as well as κ ≥ 5 are shown in

Figure 5, where t

(κ)

min

is given on the x-axis, and t

(κ)

avg

is given on the y-axis. In both metrics, the lowest

value, the better. In the graph we also indicate the cor-

responding Pareto frontier to highlight the best per-

forming methods. We also compare this directly to

the cases for κ ≥ 1 (only a single agent covers the tar-

get), however, the agents still aim to cover all targets

with κ ∈ {3,5} and therefore might cluster at speciﬁc

objects even when reporting results for κ ≥ 1.

It can be observed that on average there are no

differences between the utilized methods for scenario

S0 for κ ≥ 1, as shown in Figure 5. This is due to

the fact that there is only one static target in the envi-

ronment, which will be discovered at the exact same

time irrespective of the method for an experiment ini-

tiated with the same seed. There could be a shift with

a couple of time-steps in the discovery times, in case

there is an occasional failure in the ROS service calls

or broadcast used by the agent when handling targets

that appear in the visibility range. Nevertheless, for

κ ≥ 5, Figure 5d, the minimum times are not neces-

sarily the same, e.g., the result for method RA-AV as

compared to the six other methods. With the increase

of number of targets in each scenario, the average and

minimum times to coverage also increase. For each

scenario S1–S6, there is a difference on average be-

tween the different methods. Mostly, the proposed

method, indicated with the ‘W’ in the legend, is ei-

ther the best on average or at least on the Pareto fron-

tier, for scenarios S3 in Figure 5b; S2 and S5 in Fig-

ure 5c; S2, S5, and S6 in Figure 5d; and for scenarios

S2 and S3 in Figure 5a; S2 in Figure 5b; S4 and S6

in Figure 5c; and S4 in Figure 5d, respectively. Sim-

ilar performance is displayed by the BC-NN method,

which is the best performing method among the ones

considered in this study.

When only one target is involved, the average

minimum time to coverage is lowest. In all cases, the

agents will move in Z and eventually ﬁnd and cover

ICAART 2020 - 12th International Conference on Agents and Artiﬁcial Intelligence

0 50 100 150 200 250 300

(κ)

min

100

120

(κ)

avg

scenario

method

BC-AV

BC-NN

BC-RE

RA-AV

RA-NN

RA-RE

(a) Average vs Minimum Time to Coverage with at least one

agent when tasked with κ ≥ 3.

0 50 100 150 200 250 300

(κ)

min

100

125

150

175

(κ)

avg

scenario

method

BC-AV

BC-NN

BC-RE

RA-AV

RA-NN

RA-RE

(b) Average vs Minimum Time to achieve κ ≥ 3.

0 50 100 150 200 250 300

(κ)

min

100

125

150

175

(κ)

avg

scenario

method

BC-AV

BC-NN

BC-RE

RA-AV

RA-NN

RA-RE

agent when tasked with κ ≥ 5.

0 50 100 150 200 250 300

(κ)

min

100

150

200

(κ)

avg

scenario

method

BC-AV

BC-NN

BC-RE

RA-AV

RA-NN

RA-RE

(d) Average vs Minimum Time to achieve κ ≥ 5.

Figure 5: Average time vs minimum time to cover all stationary targets (i.e., not moving) with 1 (left) or κ agents (right). The

top row show results where agents are tasked to cover targets with κ ≥ 3 while the lower row shows results where agents are

tasked to cover targets with κ ≥ 5.

the targets. However, when increasing the number

of targets (S1–S6), it can happen that agents gather

on the ﬁrst targets found, leaving remaining targets

undiscovered for the rest of the simulation. As such,

all metrics are affected, and the t

(κ)

j,e

is saturated to the

duration of the simulation T

sim

−1

for the targets that

were not discovered. In Figure 5, the points are accu-

mulated at the t

(κ)

min

−1, which means that in those sce-

narios there were undiscovered targets for the whole

duration of the simulation.

4.2 Results for Dynamic Targets

In our second set of experiments, targets move within

the map Z by randomly changing direction, with ve-

In the ﬁnal time-step the multi-agent system shuts

down, hence this time-step is not considered when dealing

with the results.

locity v

t,max

= 1.5 m per time-step. In both cases,

agents move with a higher velocity v

a,max

= 2 m per

time-step. Nevertheless, we still use the same sets of

scenarios. As for the performance, for a single ex-

periment e = 1, .. ., N, we consider the average time

for which a target j is covered with at least κ agents

over the simulation, τ

(κ)

j,e

, and the average amount of

agents that cover the target j over the simulation α

j,e

Based on these two quantities we compute the fol-

lowing metrics: (i) the average time for which at least

κ agents cover the targets, τ

(κ)

avg

, and (ii) the average

amount of agents that cover the targets α

avg

. These

quantities are computed as

(κ)

avg

∑

|O|

∑

(κ)

j,e

(16)

avg

∑

|O|

∑

j,e

(17)

Modeling the Willingness to Interact in Cooperative Multi-robot Systems

While in our ﬁrst set of experiments, featuring a set of

static targets, agents will cover a target for the whole

duration of the simulation once they joined a coali-

tion, in the dynamic case, such assumption cannot be

made, because agents can lose targets as all objects

are moving in Z. Furthermore, these metrics are cal-

culated twice for active and passive coverage, i.e., by

(i) considering the targets that agents are actively fol-

lowing by adjusting their own motion and being part

of a coalition, and (ii) considering targets that are not

being actively followed, but are within the visibility

range of agents, without necessarily being part of a

coalition.

Results are shown in Figure 6, where the average

time of coverage is given along the x-axis, and the av-

erage number of agents is given on the y-axis. It is

possible to observe that for κ ≥ 3 the method with the

willingness is overall on the Pareto frontier, with an

exception for scenario S3, shown in Figure 6a. Re-

garding κ ≥ 5, the method with the willingness, in-

dicated with W in the legends of Figure 6, is on the

Pareto frontier for scenarios S0–S4, and the best on

average for S5–S6, Figure 6c. The same is observed

for passive following in Figure 6d. Furthermore, our

approach tends toward maximizing the number of

agents covering a target, thus it lies on the left side

of the Pareto frontier. Similarly to the results in the

static case, the performance of the BC-NN strategy is

comparable to the method with the willingness.

We can observe that for both κ ≥ 3 and κ ≥ 5

the average coverage time is highest when the num-

ber of targets is lower, in S0 and S1, falls for S2–6

when the number of targets to be covered increases.

We speculate that an increase in the number of tar-

gets, whilst the size of the area is unchanged, might

increase the average coverage time as agents can join

multiple coalitions. However, this remains subject to

further research. The impact of the chosen values for

κ can be observed as well in Figure 6, by inspecting

the average coverage times, which are lower for κ ≥ 5

than κ ≥ 3. Taking into account what is being covered

passively increases the average number of agents that

cover a target.

Note that, the averages are taken over all time-

steps of the simulation including the time to discover

the objects in the ﬁrst place. As such the lack of

coverage before the discovery naturally penalizes the

shown results.

In our current approach, the agents are not aim-

ing to exceed the desired coverage. Nevertheless, this

can happen due to race-conditions in the coalition for-

mation process. Furthermore, this can also take place

when an agent detects that a target is moving away.

In this case it will try to ﬁnd another agent that can

join the coalition. At the same time, the target is

not dropped by the former agent until it actually goes

out of its visibility range while the new agent already

joined the coalition and might have the target within

its visible range.

5 GENERALIZATION OF THE

APPROACH

In this paper, a collaborative approach based on the

willingness to interact has been tailored to solve the

κ-coverage problem for a multi-robot system. The de-

scribed framework, composed of the agent behaviour,

willingness to interact, and interaction protocol can

be applied in other problems as well. Regarding the

agent behaviour, the state machine presented in Sec-

tion 3.2 can be generalized by considering the fol-

lowing abstract states: idle, interact, idle & execute,

and interact & execute adapted from (Frasheri et al.,

2018). The latter can be specialized depending on the

behaviours that the robots should have for solving dif-

ferent problems, e.g., moving by randomly changing

direction and inspecting the space for new targets can

be used to instantiate the idle state into the inspect

state as done in this paper for solving the κ-coverage

problem. Whereas the execute state can be instanti-

ated into either the inspect & follow or evaluate &

follow, by adding the target following behaviour to

the agents.

The willingness to interact formalism can be eas-

ily adopted to account for additional relevant factors

in a given application domain. The framework al-

lows for the factors to be grouped into two categories,

necessary and optional, as well as giving a speciﬁc

weight to each factor. In this paper we have consid-

ered the battery level and the number of targets an

agent is already following, which correspond to the

necessary and optional factors respectively. Weights

are determined in a simple way, i.e., if no necessary

factor is under the the minimum threshold, then fac-

tors are weighted the same, otherwise the necessary

factors will override the optional ones, thus determin-

ing the ﬁnal value of the willingness.

Finally, the interaction protocol is independent of

the application and problem to be solved, apart for the

κ parameter which can be adjusted depending on the

size of the coalitions that the robots should be able to

form, and the triggers that agents use to initiate the in-

teraction. In the current application domain agents are

tasked with discovering and tracking targets in their

environment. Therefore, the triggers for executing the

interaction protocol are application dependent such as

(i) spotting a new target in the visibility range, (ii) de-

ICAART 2020 - 12th International Conference on Agents and Artiﬁcial Intelligence

0 50 100 150 200 250

Average Coverage Time κ ≥ 3

2.5

3.0

3.5

4.0

4.5

Average Agents per Target κ ≥ 3

scenario

method

BC-AV

BC-NN

BC-RE

RA-AV

RA-NN

RA-RE

(a) Active Coverage κ ≥ 3

0 50 100 150 200 250

Average Coverage Time κ ≥ 3

2.5

3.0

3.5

4.0

4.5

Average Agents per Target κ ≥ 3

scenario

method

BC-AV

BC-NN

BC-RE

RA-AV

RA-NN

RA-RE

(b) Passive Coverage κ ≥ 3

0 50 100 150 200 250

Average Coverage Time κ ≥ 5

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

6.5

Average Agents per Target κ ≥ 5

scenario

method

BC-AV

BC-NN

BC-RE

RA-AV

RA-NN

RA-RE

0 50 100 150 200 250

Average Coverage Time κ ≥ 5

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

6.5

Average Agents per Target κ ≥ 5

scenario

method

BC-AV

BC-NN

BC-RE

RA-AV

RA-NN

RA-RE

(d) Passive Coverage κ ≥ 5

Figure 6: Average number of agents covering targets vs average coverage time of all targets of the entire duration of the

simulation. On the left we show active coverage where we only consider the agents actively following a target (left) while the

right includes passive coverage - agents having multiple targets in their visibility range while following another target. The

top row show results where agents are tasked to cover targets with κ ≥ 3 while the lower row shows results where agents are

tasked to cover targets with κ ≥ 5.

tecting that a target is moving away and might soon be

outside of the visibility range, and (iii) extending an

existing coalition in order to reach κ-coverage. The

fourth trigger captures the moment when an agent de-

cides that it needs to ask for help, which is based on

the willingness to interact. This trigger is not applica-

tion dependent.

6 CONCLUSION AND FUTURE

WORK

This paper presented a novel, distributed, agent-

centric coalition formation approach, based on the

willingness to interact for adaptive cooperative behav-

ior. We showed that we can use this novel approach

to solve the κ−coverage problem for a set of targets.

The performance of this approach is measured along

two different sets of metrics for two different cases,

(i) with static targets, and (ii) with mobile targets, and

compared with six methods previously proposed in

the literature. In the former case, the average time

to get one target covered with κ agents, and the aver-

age minimum time to κ−cover all objects are consid-

ered. In the latter case, the average coverage time and

average number of agents per target are considered.

Results show that our approach either performs com-

parably good in the case of static targets with respect

to the BC-NN method (the best performing among the

ones considered in the paper), and that it performs

better than the other methods in terms of achieving

a higher level of coverage when it comes to moving

targets.

There are three main lines of inquiry for future

work. First, it is of interest to compare further

the performance of our approach with those meth-

Modeling the Willingness to Interact in Cooperative Multi-robot Systems

ods that reached similar performance like the BC-NN

method. Such investigation can include the explo-

ration of other experimental settings that might bet-

ter highlight possible trade-offs for the utilization of

the BC-NN method or the one based on the willing-

ness proposed in this paper. Furthermore, issues re-

lated to how the studied models scale up in terms

of, e.g., bandwidth capacity and latency, can also be

considered in the analysis. Second, security aspects

can be introduced, by considering the trustworthi-

ness of agents. Such information can be included in

the calculation of the willingness to interact, in or-

der to facilitate the cooperation between agents that

are more trustworthy, e.g., open systems where new

agents may be introduced or removed, similarly to

recent approaches (Castell

o Ferrer, 2019; Calvaresi

et al., 2018). Third, some assumptions made in this

paper can be relaxed, e.g., targets can appear and dis-

appear at random times, or leave the area deﬁned by

the map, in order to adapt the current approach for

solving a more general κ-coverage problem.

ACKNOWLEDGEMENTS

This work was supported by the DPAC research pro-

ﬁle funded by KKS (20150022), the FIESTA project

funded by KKS, and the UNICORN project funded

by VINNOVA.

REFERENCES

Burgard, W., Moors, M., and Schneider, F. (2002). Col-

laborative exploration of unknown environments with

teams of mobile robots. In Beetz, M., Hertzberg, J.,

Ghallab, M., and Pollack, M. E., editors, Advances in

Plan-Based Control of Robotic Agents, pages 52–70.

Calvaresi, D., Dubovitskaya, A., Calbimonte, J. P., Taveter,

K., and Schumacher, M. (2018). Multi-agent sys-

tems and blockchain: Results from a systematic lit-

erature review. In Demazeau, Y., An, B., Bajo, J., and

Fern

andez-Caballero, A., editors, Advances in Practi-

cal Applications of Agents, Multi-Agent Systems, and

Complexity: The PAAMS Collection, pages 110–126,

Cham. Springer International Publishing.

Casadei, R., Pianini, D., Viroli, M., and Natali, A. (2019).

Self-organising coordination regions: A pattern for

edge computing. In Riis Nielson, H. and Tuosto, E.,

editors, Coordination Models and Languages, pages

182–199, Cham. Springer International Publishing.

Castell

o Ferrer, E. (2019). The blockchain: A new frame-

work for robotic swarm systems. In Arai, K., Bha-

tia, R., and Kapoor, S., editors, Proceedings of the

Future Technologies Conference (FTC) 2018, pages

1037–1058, Cham. Springer International Publishing.

Esterle, L. (2018). Goal-aware team afﬁliation in collectives

of autonomous robots. In Proc. of the Int. Conf. on

Self-Adaptive and Self-Organizing Systems (SASO),

pages 90–99.

Esterle, L. and Lewis, P. R. (2017). Online multi-object k-

coverage with mobile smart cameras. In Proc. of the

Int. Conf. on Distributed Smart Cameras, pages 1–6.

ACM.

Esterle, L. and Lewis, P. R. (2019). Distributed autonomy

and trade-offs in online multiobject k-coverage. Com-

putational Intelligence.

Frasheri, M., C

ukl

u, B., Ekstr

om, M., and Papadopou-

los, A. V. (2018). Adaptive autonomy in a search and

rescue scenario. In Proc. of the Int. Conf. on Self-

Adaptive and Self-Organizing Systems, pages 150–

155.

Garc

ıa, S., Menghi, C., Pelliccione, P., Berger, T., and

Wohlrab, R. (2018). An architecture for decentralized,

collaborative, and autonomous robots. In Proc. of the

Int. Conf. on Software Architecture, pages 75–7509.

Hellmund, A., Wirges, S., S¸. Tas¸, O., Bandera, C., and

Salscheider, N. O. (2016). Robot operating system:

A modular software framework for automated driving.

In Proc. of the Int. Conf. on Intelligent Transportation

Systems, pages 1564–1570.

Lee, S.-H. and Choi, H. (2002). The fast bully algorithm:

For electing a coordinator process in distributed sys-

tems. In Revised Papers from the International Con-

ference on Information Networking, Wireless Com-

munications Technologies and Network Applications-

Part II, ICOIN ’02, pages 609–622, Berlin, Heidel-

berg. Springer-Verlag.

Parker, L. E. and Emmons, B. A. (1997). Cooperative multi-

robot observation of multiple moving targets. In Proc.

of the Int. Conf. on Robotics and Automation, vol-

ume 3, pages 2082–2089 vol.3.

Quigley, M., Conley, K., Gerkey, B., Faust, J., Foote, T.,

Leibs, J., Wheeler, R., and Ng, A. Y. (2009). ROS: An

open-source robot operating system. In ICRA work-

shop on open source software, volume 3, page 5.

Qureshi, F. and Terzopoulos, D. (2007). Distributed coali-

tion formation in visual sensor networks: A virtual

vision approach. In Aspnes, J., Scheideler, C., Arora,

A., and Madden, S., editors, Proc. of the Int. Conf.

on Distributed Computing in Sensor Systems, pages

1–20.

Robin, C. and Lacroix, S. (2016). Multi-robot target detec-

tion and tracking: taxonomy and survey. Autonomous

Robots, 40(4):729–760.

Shehory, O. and Kraus, S. (1998). Methods for task allo-

cation via agent coalition formation. Artiﬁcial Intelli-

gence, 101(1):165 – 200.

Theraulaz, G., Bonabeau, E., and Deneubourg, J.-L. (1998).

Response threshold reinforcements and division of

labour in insect societies. Proc. of the Royal Society

B: Biological Sciences, 265(1393):327–332.

Ye, D., Zhang, M., and Sutanto, D. (2013). Self-

adaptation-based dynamic coalition formation in a

distributed agent network: A mechanism and a brief

survey. IEEE Trans. on Parallel & Distributed Sys-

tems, 24(5):1042–1051.

ICAART 2020 - 12th International Conference on Agents and Artiﬁcial Intelligence