Modeling the Willingness to Interact
in Cooperative Multi-robot Systems
Mirgita Frasheri
1 a
, Lukas Esterle
2 b
and Alessandro Vittorio Papadopoulos
1 c
1
M
¨
alardalen University, V
¨
aster
˚
as, Sweden
2
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-
a
https://orcid.org/0000-0001-7852-4582
b
https://orcid.org/0000-0002-0248-1552
c
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
62
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
Copyright
c
2022 by SCITEPRESS Science and Technology Publications, Lda. All rights reserved
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
1
,a
2
,... ,a
n
}, and a set of targets or objects of inter-
est O = {o
1
,o
2
,. .. ,o
m
} in this problem. Both robots
and objects can freely move within Z, with (noncon-
stant, yet limited) velocities v
i
, where i = 1,...,n, and
v
j
, 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
i
. 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
j
. 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
l
j
at a discrete time-step t.
Every agent i can calculate its willingness w
i
to
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
i
,
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
63
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
a
(t) = v
a
(t)cos(θ
a
(t))
˙y
a
(t) = v
a
(t)sin(θ
a
(t))
˙
θ
a
(t) = ω
a
(t)
(1)
where x
a
(t) and y
a
(t) are the x- and y-coordinate on
the map and define the position p
a
= (x
a
,y
a
) of a robot
a at time t, θ
a
is the orientation of the robot, v
a
is the
forward velocity of the robot, and ω
a
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
a
O,
each of which has a different level of interest l. The
direction d
a
over which the robot moves is thus com-
puted as
d
a
(t) =
iO
a
l
i
(p
a
(t) p
i
(t))
iO
a
l
i
(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
a
and the forward
velocity of the robot are therefore computed as:
θ
a
(t) = d
a
(t), (3)
˜v
a
(t) = kd
a
(t)k (4)
where p [0,2π) is the angle of the vector p =
(p
x
, p
y
) in its reference frame, and it is obtained as
p = atan2(p
y
, p
x
). In order to compute the proper
value of the angular velocity, we can just use a simple
proportional controller with tracking error e
a
normal-
ized between [π,π):
e
a
(t) = θ
a
(t) θ
a
(t) (5)
˜
ω
a
(t) = K
p
atan2(sin(e
a
(t)),cos(e
a
(t))) (6)
Finally, we include saturations on the forward and an-
gular velocities:
v
a
(t) = min( ˜v(t), v
max
) (7)
ω
a
(t) = min(max(
˜
ω
a
(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
64
Inspect
Inspect
& Follow
Evaluate
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:
w
i
(t + 1) = min(max(w
i
(t) + B
>
f(t),1), 0), (9)
f(·) = [ f
1
(·),. .. , f
m
(·)]
>
is an m × 1 vector of the
m factors that affect the willingness, while B =
[β
1
,. .. ,β
m
]
>
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
i
is given by
f
i
(k) = φ
i
(k) φ
i,min
, (10)
where φ(k) represents the current measurement of that
factor (e.g., the current battery level), while φ
min
is
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
a
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
O
) 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
O
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:
β
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
65
weight for an optional factor β
opt
is defined as:
β
opt
(t) =
0, if φ
nec
,φ
nec
(t) φ
nec,min
< 0,
sgn(φ
opt
(t) φ
opt,min
)
m
, 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 κcoverage,
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
No
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:
w
i j
(t) = w
i
(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
66
a
i
d
in
d
tg
1
tg
2
Figure 3: Agent a
i
with visibility range d, indicated with
the red circle, and internal range d
in
, indicated with the blue
circle. The targets tg
1
and tg
2
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
in
around the robot, where d
in
< d
(Figure 3). When a target, e.g., tg
2
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
1
, 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
1
, 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-
1
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
67
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
A
= 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
j
= 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
t
(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
t
(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:
t
(κ)
avg
=
1
N
e
1
|O|
j
t
(κ)
j,e
(14)
t
(κ)
min
=
1
N
e
max
j
t
(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
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68
0 50 100 150 200 250 300
t
(κ)
min
0
20
40
60
80
100
120
t
(κ)
avg
scenario
S0
S1
S2
S3
S4
S5
S6
method
BC-AV
BC-NN
BC-RE
RA-AV
RA-NN
RA-RE
W
(a) Average vs Minimum Time to Coverage with at least one
agent when tasked with κ 3.
0 50 100 150 200 250 300
t
(κ)
min
0
25
50
75
100
125
150
175
t
(κ)
avg
scenario
S0
S1
S2
S3
S4
S5
S6
method
BC-AV
BC-NN
BC-RE
RA-AV
RA-NN
RA-RE
W
(b) Average vs Minimum Time to achieve κ 3.
0 50 100 150 200 250 300
t
(κ)
min
0
25
50
75
100
125
150
175
t
(κ)
avg
scenario
S0
S1
S2
S3
S4
S5
S6
method
BC-AV
BC-NN
BC-RE
RA-AV
RA-NN
RA-RE
W
(c) Average vs Minimum Time to Coverage with at least one
agent when tasked with κ 5.
0 50 100 150 200 250 300
t
(κ)
min
0
50
100
150
200
t
(κ)
avg
scenario
S0
S1
S2
S3
S4
S5
S6
method
BC-AV
BC-NN
BC-RE
RA-AV
RA-NN
RA-RE
W
(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
2
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-
2
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
τ
(κ)
avg
=
1
N
e
1
|O|
j
τ
(κ)
j,e
(16)
α
avg
=
1
N
e
1
|O|
j
α
j,e
(17)
Modeling the Willingness to Interact in Cooperative Multi-robot Systems
69
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
70
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
S0
S1
S2
S3
S4
S5
S6
method
BC-AV
BC-NN
BC-RE
RA-AV
RA-NN
RA-RE
W
(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
S0
S1
S2
S3
S4
S5
S6
method
BC-AV
BC-NN
BC-RE
RA-AV
RA-NN
RA-RE
W
(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
S0
S1
S2
S3
S4
S5
S6
method
BC-AV
BC-NN
BC-RE
RA-AV
RA-NN
RA-RE
W
(c) Active Coverage κ 5
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
S0
S1
S2
S3
S4
S5
S6
method
BC-AV
BC-NN
BC-RE
RA-AV
RA-NN
RA-RE
W
(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 κcoverage 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 κcover 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
71
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.
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