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Distributed Progressive Formation Control with One-Way Communication for Multi-Agent Systems

Authors:

Abstract

The cooperation of multiple robots towards a common goal requires a certain spatial distribution, or formation configuration, of the agents in order to succeed. Centralized controllers that have information about the absolute or relative positions of all agents, or distributed approaches using communication to share system-wide information between agents, are able to calculate optimal individual paths. However, this reserves important computational resources as the number of agents in the system increases. We address the problem of distributed formation control with minimal communication and minimal computational power required. The algorithm introduced in this paper progressively generates a directed path graph to uniquely assign formation positions to all agents. The benefits of the proposed algorithm compared to previous include the need for one-way communication only, low computational complexity, ability to converge without any a priori assignments under certain geometric conditions and need for limited sensing information only. The algorithm can be deployed to computationally constrained devices, enabling its deployment in robots with simpler hardware architectures. The ability to converge and distributively assign positions, or roles, with only one-way communication makes this algorithm robust during deployment, at which time all agents are equivalent and anonymous. Moreover, we account for limited communication and sensing range, and agents only need to have information about other agents in their vicinity in order to make decisions. Communication is simple and allows for scalability without an impact on performance or convergence latency, with a linear dependence on the number of agents. Agents only need to broadcast their status to other neighboring agents and do not reply any message. Finally, this algorithm enables almost-arbitrary configurations. The main limitation for the choice of formation configuration is that all pairs of points forming an edge in the polygonal line defining the boundary of the convex hull must be within sensing range.
Distributed Progressive Formation Control with One-Way
Communication for Multi-Agent Systems
Jorge Pe˜
na Queralta2, Li Qingqing1,2, Tuan Nguyen Gia2, Hannu Tenhunen3, Z. Zou1and Tomi Westerlund2
1School of Information Science and Technology, Fudan University, China
2Department of Future Technologies, University of Turku, Finland
3Department of Electronics, KTH Royal Institute of Technology
Emails: {jopequ, tunggi, tovewe}@utu.fi, {qingqingli16, zhuo}@fudan.edu.cn, hannu@kth.se
Abstract—The cooperation of multiple robots towards a com-
mon goal requires a certain spatial distribution, or formation
configuration, of the agents in order to succeed. Centralized
controllers that have information about the absolute or relative
positions of all agents, or distributed approaches using commu-
nication to share system-wide information between agents, are
able to calculate optimal individual paths. However, this reserves
important computational resources as the number of agents in
the system increases. We address the problem of distributed
formation control with minimal communication and minimal
computational power required. The algorithm introduced in
this paper progressively generates a directed path graph to
uniquely assign formation positions to all agents. The benefits of
the proposed algorithm compared to previous include the need
for one-way communication only, low computational complexity,
ability to converge without any a priori assignments under certain
geometric conditions and need for limited sensing information
only. The algorithm can be deployed to computationally con-
strained devices, enabling its deployment in robots with simpler
hardware architectures. The ability to converge and distributively
assign positions, or roles, with only one-way communication
makes this algorithm robust during deployment, at which time
all agents are equivalent and anonymous. Moreover, we account
for limited communication and sensing range, and agents only
need to have information about other agents in their vicinity in
order to make decisions. Communication is simple and allows
for scalability without an impact on performance or convergence
latency, with a linear dependence on the number of agents.
Agents only need to broadcast their status to other neighboring
agents and do not reply any message. Finally, this algorithm
enables almost-arbitrary configurations. The main limitation for
the choice of formation configuration is that all pairs of points
forming an edge in the polygonal line defining the boundary of
the convex hull must be within sensing range.
Index Terms—Multi-Agent Systems; Formation Control; Dis-
tributed Control; Progressive Formation Control; Distributed
Role Assignment; Multi-Agent Cooperation;
I. INTRODUCTION
The robotics field often finds its inspiration in nature [1].
This has had a significant impact on fields such as swarm
robotics [2], [3], or multi-agent and multi-robot systems [4],
[5]. With part of the robotics community aiming at developing
robots with capabilities that match or even outperform those
of nature [6], others explore the possibilities of creating
complex behaviour from relatively simple robots [7], [8]. The
coordination of multiple robots can originate in robust robotic
systems with the ability of complex behaviour [9]. We explore
one of the basic elements of multi-robot cooperation and
collaboration: formation control or pattern configuration [10].
Multi-robot and multi-agent systems have been increas-
ingly studied by the research community over the past few
decades [11]–[14]. Formation control algorithms are one of
the key aspects for collaborative operation in multi-agent
systems [15], [16]. Cooperation of multiple agents to perform
a predefined task often involves the definition of a specific
spatial distribution of the agents [17]. Hence the interest in
formation control algorithms for multi-robot and multi-agent
systems as the basis towards advanced collaboration.
A. Background and Related Works
Formation control algorithms can be broadly classified as
a function of the variables that agents actively control when
converging to the desired configuration [14]. These variables
are often an absolute position in a global reference frame [18],
[19]; a relative position, or displacement, with respect to other
agents and measured in a local reference frame [20], [21]; or
the distance or bearing to neighboring agents [22]–[25]. Equiv-
alently to the case of distance-based formation control, other
approaches propose the use of inter-agent bearing instead [25].
Alternatively, both distance and bearing have been used for
non-holonomic formation control problems [26].
A formation control problem typically has two differentiated
parts: (i) the assignment of positions or roles; and (ii) the
definition of control laws that ensures convergence towards
the assigned objectives for all units in the system [27], [28].
The second step can also be considered as path planning
in a multi-robot scenario. In order for a group of agents
to converge to a predefined configuration, positions in the
formation have to be first assigned to individual agents via a
bijective correspondence. This can be done a priori, in the case
that agents are indexed and preassigned to a certain position,
or decided after deploying agents to their initial positions [29]–
[34]. This step is not necessary if all positions in the formation
are equivalent from the point of view of the controlled variable,
as is the case of flocking [35]. In this paper, we propose an
algorithm that allows agents to autonomously self-assign their
objective position after deployment, and use the assignment
of local neighbors to perform autonomous path planning and
converge to the objective configuration. We have focused
on a solution that requires less a priori information given
to agents because of the effect this has on scalability and
deployment flexibility. In terms of scalability, the proposed
algorithm requires a time complexity that is linear with respect
to the number of agents to complete the position assignment
of all agents. Regarding flexibility, problems can arise with a
large number of agents converging towards a given formation
configuration. If positions have been assigned a priori, an
unfavourable initial distribution where agents are far from their
objective position might significantly increase the time needed
for convergence and total travelled distance in the system.
Centralized algorithms that use relative or global positioning
of all agents to calculate the individual optimal paths have been
traditionally used [14]. However, distributed approaches have
gained increasing popularity over the past decades, enabling
operation in more diverse environments. We propose a dis-
tributed formation control algorithm that requires no a priori
position assignments and minimal one-way communication
between agents. Positions are assigned autonomously by the
agents in a progressively through a directed acyclic graph.
Different distributed approaches exist depending on the
communication within the agents, and the a priori informa-
tion given such as predefined position assignments. If no
assignments are made, when no communication is allowed,
only formation configurations where all positions have an
equivalent definition in terms of their neighbors is possible.
This is the case of flocks, where distance between neighbors
is constant, or regular polygons, where the angle between the
positions of the two nearest neighbors is also constant [35],
[36]. When agents can achieve consensus through communica-
tion, then arbitrary formation shapes are possible. This can be
done either by using system-wide communication to achieve
consensus or via local interactions only. In the former case,
auction mechanisms have been used to perform task allocation,
which is equivalent to assign positions in a formation [37],
[38]. Regarding the latter scenario, a very interesting approach
has been proposed by Pinciroli et al., in which positions are
assigned progressively via local interactions between neigh-
boring agents [29], [30]. The authors propose a solution that
is easily scalable, robust, and specifically suited for the natural
scenario where agents are deployed progressively and not at
once. Moreover, only minimal communication is required at
the time of joining the formation, and a given agent only needs
to exchange information with two agents that are already part
of the objective configuration. Compared to their approach,
we propose a method that reduces further the amount of nec-
essary communication. The proposed algorithm only requires
one-way communication by broadcasting status information.
However, this simplification limits the number of possible
configurations. We prove that if the set of initial positions
of deployed agents is a convexly layered set with a constraint
below the agents’ sensing range, then agents converge to the
objective formation. This mainly requires that agents can be
assigned to the vertices of a series of convex polygonal layers,
such that they follow an inclusion relation, and all pairs of
agents forming a polygon edge are within sensing range.
The progressive position assignment inherently introduces
latency in the system when compared to algorithms that
do not require communication between agents [32], [34].
However, it also allows us to ensure the existence of consensus
during the position assignment and enables the convergence
to almost arbitrary configurations. Distributed formation con-
trol algorithms that require, to some extent, communication
between agents and measuring the relative positioning of
neighboring agents often rely on situated communication [33],
[39]. This type of communication refers to data exchanges
using wireless technology and devices capable of estimating
the relative position of a message sender. Two-way situated
communication enables mutual localization of agents. Other
approaches have been proposed for very-large-scale systems
where individual positions are not assigned but instead a
certain number of agents must be located in a particular area
or volume in space. This idea, introduced by Bandyopadhyay
et al., uses probability distributions to define the formation
configuration [31]. Even though agents do not communicate
to achieve consensus regarding their objective positions, they
need to be aware of the global spatial distribution of the
swarm.
Our work has been partially inspired from previous works
from Pinciroli et al. [29], [30]. Nevertheless, the differences
are significant. First, we propose a method that requires
only one-way communication. Second, we also assume that
agents share a common orientation reference. This, which
can be implemented via inexpensive magnetometers or other
kind of inertial sensors or digital compasses, enables us to
define configurations with a predefined orientation. This has
an important impact when agents should move towards a
given direction after or while converging towards the objective
configuration. Furthermore, our algorithm does not require of
a predefined pair of identified agents which are necessary for
assigning the rest of roles in Pinciroli’s work. Instead, we
propose a method that enables autonomous self-assignment of
all roles or positions in the objective pattern.
B. Contribution and Organization
The main contributions of this paper are the following: (i)
the definition of an algorithm that enables distributed and
autonomous position assignment in multi-robot systems with
one-way communication; (ii) the introduction of a control
law that ensures convergence and utilizes the information ac-
quired during the position assignment; and (iii) the simulation
and analysis of multiple scenarios and pattern configurations
showing that our proposed approach is scalable and. A more
realistic simulation has been implemented and analyzed using
the Robot Operating System (ROS), the Gazebo simulator
and the RotorS module for dynamics modelling in another
paper [33]. These tools enable a more advanced modelling of
unmanned aerial vehicle (UAV) dynamics and simulation and
have allowed us to demonstrate the efficiency and applicability
of our algorithm. Nonetheless, in this paper we focus on the
theoretical foundations of our algorithm and on analyzing its
performance and scalability for different objective patterns and
initial distributions of an increasing number of agents.
The rest of this paper is organized as follows. Section
II introduces the formation control problem and notation
used within the paper. Also, a methodology for labeling the
positions in a formation via the creation of a directed path
graph is introduced. In Section III, we present an algorithm
to progressively assign the positions through the creation of
an analogous path graph when agents are deployed. Section
IV, we propose a basic control law that ensures convergence,
with simulation results illustrated in Section V. Section VI
concludes the paper and outlines future work directions.
II. PRO BL EM FO RM UL ATIO N
In this paper, we use the following notation. We use
[N] = {kZ+:kN}to denote the set of the first
Npositive integers. Given a set of vectors x1, . . . , xNRn,
x= [xT
1, . . . , xT
N]RnN denotes the stacking of the vectors.
Given a set of points in Rn, the convex envelope, hull or
closure is the smallest convex set containing all points. We
denote by Conv(q)the convex hull of qand by δConv(q)
its boundary. The algorithms presented in this paper are for-
mulated for two-dimensional formation control, and therefore
we implicitly assume n= 2 and all points belong to R2.
Consider a planar formation configuration defined as a set
of points, represented by a set q= [q1, . . . , qN]R2N. Given
a set of Nagents with positions p(t) = [p1(t), . . . , pN(t)]
R2N, we address the problem of achieving a spatial distribu-
tion equivalent to qwith respect to a translation.
Problem 1 (Formation Objective).Given an objective point
set qand a set of agents represented by their positions p(t),
we consider that the formation has been achieved at a time
t=t0if a permutation σ: [N][N]exists such that
kpi(t0)p0(t0) + qσ(i)qσ(0)k< ε
k˙pi(t0)k< δ (1)
for predefined constants ε, δ > 0that represent the maximum
error allowed for positions and speed. We assume that agents
are able to measure the position of any other agent in line of
sight up to a predefined sensing distance δs.
In order to solve Problem 1, we provide a methodology
for uniquely assigning a position in the formation to each
agent through the definition of a directed path graph. One-
way communication is used to progressively assign positions
in the formation. The definition of the path graph requires the
following two conditions, where the sensing range of agents
is taken into account.
Assumption 1. The set of points can be divided in a set
of Lconvex polygons, or layers, {l1,...,lL}such that the
polygonal areas they delimit {Al1, . . . , AlL}, defined as com-
pact sets in R2, follow a strict inclusion relation Al1
Al2⊃ ·· · ⊃ AlL. Any two consecutive points in a given
layer are separated by a distance smaller than the agents’
maximum sensing distance, klki lk(i+1 mod Lk)k< δs. We
denote by Lkthe number of points if the k-th layer, and
define lk={lk1, . . . , lkLk}in an order such that any pair
of consecutive points (lki, lk(i+1 mod Lk))defines an edge of
the convex polygon.
(b)Assignmentorder
13
14
12
11
112
23 22 21
111
110
32 31
15 19
24
16 18
25
17
(a)Identifiers,layers
Fig. 1. Representation of convex layers in a 2D point set and SDPG
generation. Non straight connections between nodes are merely illustrative.
Assumption 2. For each point lki,1k < L, there exists
1jLk+1 such that klki l(k+1)jk< δsand klki
l(k+1)(j+1 mod Lk+1 )k< δs. Equivalently, we assume that for
each point lki,1< k L, there exists 1jLk1such
that klki l(k1)jk< δsand klki l(k1)(j+1 mod Lk1)k< δs.
This essentially implies that, for each point, there are at least
two points in the previous and next layer within sensing range,
unless the point is in the first or last layer. If there is a single
point in the innermost layer, then at least three other points
must be within sensing range. Intermediate layers cannot have
one or two points only, as any three points form a triangle and
all triangles are convex polygons.
The parameter δs>0is a lower limit of the agents’ sensing
range. This value is different in each application scenario and
should be chosen lower than the real range to ensure agents
are able to sense their neighbors consistently through time.
The first layer l1is the boundary of the convex hull or
convex envelope of the point set l. Therefore, we can easily
calculate the points in any layer using the following relations:
l1=δConv(q),lk=δConv
q\[
1k0<k
lk0
1< k L
(2)
The convex hull of a finite point set in R2, or R3, can be
calculated in O(nlog h)time with Chan’s algorithm, where
his the output size, i.e., the number of points defining the
convex hull [40].
Definition 1 (Convexly layered set).Given a set of points in
the plane, represented by a stacked vector q= [qT
1, . . . , qT
N]
R2N, and a distance δsR, δs>0, we define the pair (q, δs)
as a convexly layered set with constraint δsif Assumption 1
and Assumption 2 hold.
We should note than given any set q, there always exists a
value δslarge enough such that (q, δs)is a convexly layered
set with constraint δs. This can be easily proven from the
definition of lkin Eq. 2, as the layers can be calculated first
and then the minimum sensing range is obtained from the
conditions in Definition 1. Figure 1 (a) shows three convex
layers for a set of 19 points. The first digit of each node’s
identifier references the layer number {1,2,3}and the rest of
digits are unique for each layer, with layer sizes {12,5,2}.
This identifiers have been merely chosen for illustration pur-
poses; in a real application, a sequence of increasing natural
numbers can be used as agents are given a priori information
about the objective configuration. This information can then
include the number of agents in each layer.
We can now define a directed path graph on the point layer
that uniquely assigns identifiers to each point progressively.
The directed path graph is generated as follows. First, a node is
chosen as the graph root. Any edge node can be chosen at this
point. A node is an edge node if it is a point qi= (qix, qiy)that
belongs to the convex hull and there exist constants m, n 0,
defining a line f(x) = mx+n, and constants s1, s2∈ {−1,1},
such that the edge node qibelongs to the line, all other points
in the set belong to only one of the two half-planes defined
by the line, and other points that belong to the line belong to
only one of the two half-lines in which qidivides the line.
Mathematically, this means that
qiy=f(qix)
qjys1f(qjx)j[N], j 6=i
j6=i|qjy=f(qjx) =qjx< s2qix
(3)
Second, a clockwise or counterclockwise assignment direction
is chosen. This direction is used in all layers to define the
order in which identifiers are assigned to points in the set.
Finally, starting at the graph root, the next node in the graph
is one of the two points that share an edge with the root in the
outer layer, chosen accordingly with the assignment direction.
The assignment continues iteratively in the same direction
through the outer layer until all points in the layer have been
assigned an identifier. When the last point in the first layer
has been identified, the assignment continues to inner layers
by choosing the closest point in the next layer and repeating
the same process as in the first layer. This process continues
until all points in the set have been assigned a position.
Definition 2 (SDPG from a convexly layered set).Given a
convexly layered set (q, δs), we define a Spiral Directed Path
Graph (SDPG) following the next steps:
1. Choose an edge node as the graph root, and constants
m, n, s1, s2that uniquely define the point within the set q.
The constants m, s1, s2will be used by agents to self decide
whether they are the root node after deployment. The value
nis not necessary because we consider any configuration
equivalent with respect to a translation. Therefore, any line
from the set of parallel lines defined by mcan be used to
uniquely identify the root, together with constants s1, s2.
2. Choose an assignment direction, clockwise or counter-
clockwise.
3. Identify the nodes following the assignment direction,
starting from the root node, and following the previous in-
dications when all nodes in a layer have been identified.
A path graph is a tree where only two nodes have degree
one, and all other nodes have degree two. Because the SDPG
is a directed graph, the root and terminal nodes have degree
one. All nodes except the root have a parent node, and all
nodes except the terminal have a child node.
To solve Problem 1, we first assign a unique identifier to
the positions in a given formation configuration by generating
an SDPG. Then agents perform a progressive self-assignment
of positions until an equivalent SDPG is generated from the
moment they are deployed. This analog process is defined in
Section III. At this point, agents actively control their position
with respect to the nodes they are connected with in the
SDPG according to the displacement defined in the objective
formation configuration. The desired pattern is achieved when
all agents error are below a predefined threshold. However,
agents are not aware of the global error due to the lack of
communication, and local errors are used instead to estimate
the global system state.
Figure 1 (b) shows the SDPG generated from a given set
of points as the desired formation configuration. The edge
node is given by constants m= 0, n =q1y, defining a line
f(x) = q1y, where q1is the point with identifier 11. The
half-plane where all other points lay is defined by s1= 1.
The value s2is not relevant in this case because there is no
other point in the line, i.e., qjy6=q1yj6= 1. However,
the origin choice might affect the graph generation in the
agent set when assigning positions in the corresponding SDPG.
Therefore, even if not significant, a value for s2∈ {±1}must
be chosen. The assignment direction is counter-clockwise in
this case. We use the term spiral to refer to the directed path
graph because of the decreasing distance to the center of mass
when considering different layers, even though this does not
necessarily happen within a certain layer.
III. PROG RE SS IV E POSITION ASS IG NM EN T ALGORITHM
In this section, we describe a position assignment algorithm
that only requires one-way, minimal communication between
agents. One-way communication is used to enable the assign-
ment of positions in a random spatial distribution of agents.
Suppose a set of Nagents with positions represented by
p(t)=[pT
1(t), . . . , pT
N(t)] R2Nand maximum sensing
range δsis given. An objective formation configuration is
defined by a point set q= [qT
1, . . . , qT
N]R2N. We assume
that the set p(0), together with the agent sensing range δs, is
a convexly layered set {p(0), δs}.
During the position assignment process, agents can have
one of three different states: (1) Assigned: agents that know
their position and identifier; (2) Known layer: agents with
unassigned position but that are aware of their layer; and (3)
Unknown layer: agents with unknown layer.
All agents in the outer layer are in the known layer state
immediately after deployment, and all other agents in the
unknown layer state. Agents continuously transmit their state
through a broadcast signal. Situated communication can be
used to both transmit the signal and measure the position
of other agents [39]. The broadcasted signal is used by
neighboring agents to track the position of the transmitting
agent while, at the same time, make decisions with respect
to their objective position. If an agent is in the assigned
state, then the broadcast signal includes information about its
objective position. We assume that if agent iis able to receive
agent j’s signal at a certain time, then agent jis able to receive
agent’s isignal at the same time.
The progressive position assignment (PPA) then proceeds
as follows: (1) With the same constants m, s1, s2used in the
definition of the SDPG from the objective configuration, each
agent in position pi= (pix, piy)checks if the three conditions
in Equation 3 hold for n=piypixm. with f(x) = mx +n.
If an agent ifulfills all three conditions, with f(x) = m(x
pix) + piy, then the agent is the root node and it changes its
state to assigned.(2) The root agent starts broadcasting signal.
Then, there are two agents next to it in the outer layer which
self-decide whether they are or not the next node in the graph
based on the assignment direction. In the case of a positive
decision, then the next node starts broadcasting signal as well.
After all nodes in the outer layer have a position assigned, then
nodes in the next layer can automatically change their state to
known layer.(3) Finally, each node starts to actively control
its position immediately after it is in the assigned state.
Theorem 1 (Uniqueness of the PPA).Let pR2Nrepresent
the spatial distribution of a set of Nagents with sensing
range δs>0, and capable of one-way communication by
continuously broadcasting a signal over time. If {p, δs}is
a convexly layered set, then the position assignment process
defined by the previous three steps uniquely assigns a position
to each agent.
Proof. Positions are self-assigned in an autonomous way by
agents, based on the positions and states of neighboring agents.
Therefore, in order to prove the theorem, we first need to
demonstrate that a single agent will self-assign itself the role
of graph root, and that at every step the appropriate agent, and
only that agent, will self-assign the next node in the SDPG.
Suppose there are two agents such that, immediately after
deployment, claim that they are the root agent. Let pi, pjbe the
two positions of such agents. Without any loss of generality,
we can assume j > i. With constants m, s1, each of the points
defines a half-plane. If both half-planes are the same, then pi
and pjlay on the same line. If agents iand jare within
sensing range, then s2has different value for each of them
and this is a contradiction. If they are not able to sense each
other, then there exists another agent in the outer layer with
position pk,i < k < j, such that it belongs to the half-plane
defined by the piand pj. If it belongs to the boundary, then the
same problem arises with the sign of s2(and new points can
be equally generated if kis not within sensing range of both
iand j). If it belong to the interior of the half-plane, then
the segment pipjis outside of the convex outer layer, and
this contradicts the definition of convex polygon and breaks
Asumption 1.
If the half-planes are not the same, we can assume without
any loss of generality, and based on the value of s2, that the
half-plane generated by piis contained within the half-plane
generated by pj. This necessarily means that they do not sense
each other. Let kbe now the nearest edge to iin the outer layer,
such that it is between iand j. Then pknecessarily belongs
to the half-plane defined by pibased on Eq. 3. However, then
pipj, which lies completely outside of that same half-plane, is
not within the outer layer and a contradiction arises.
The above implies that all agents in the external layer can
be uniquely identified following the SDPG generation, without
two agents claiming to have the same objective position. Now
we just need to prove that a single agent in the next layer will
claim to have the appropriate objective position. The problem
can be reduced to prove that a single agent will claim to be
the closest to the last agent identified in the outer layer. Let
pibe the position of the last agent assigned to the outer layer,
and assume that two agents in position pj,pkclaim to be the
closest. Both pjand pkknow that they are in the next layer in
the same manner than initially all agents in the outer layer are
aware of that, since all these have been already identified. If pj
and pkare within sensing range, then they can both masure the
other’s distance to pi, and only one will claim to be the closest.
In case of equivalent distance, the decision is made based on
the assignment direction. Therefore, for agents jand kto make
the claim, they cannot be within sensing range of each other.
Based on Assumptions 1 and 2, this means there is at least one
other agent hin the same layer in between. However, because
his farther from i, it is outside the triangular area formed by
agents i, j, k. This implies that the segment pjpklays outside
any convex poligonal area with vertices including pj, pk, ph,
that also leaves pioutside. This contradicts the definition of
convex polygon, and we started assuming that pj, pk, ph.
As previously noted, all agents except the root have a parent
node, and the root can be uniquely identified. This allows us to
propose a control law for each agent based on a basic leader-
follower formation, where the parent is the leader and the
agent itself is the follower. We assume that the parent and
child nodes of any agent, which are within sensing range at
time t= 0, stay within sensing range at any time t0>0. If
a new objective configuration is required, then agents loose
their assignments and the process is restarted.
IV. CON TROL INPUT
We propose a control law that enables agents to converge
to the objective configuration while avoiding inter-agent col-
lisions. A natural solution is for agents to actively control
the displacement with respect to their parent node in the path
graph. However, this has the disadvantage of increasing the
system error with small drifts in the displacement of each
parent-child pair. We propose a methodology for reducing
the system error after individual agent errors are below a
certain threshold, but its analysis is not wihin the scope of
this paper. The main contribution of this paper is on the
progressive assignment algorithm that uniquely generates a
SDPG given an spatial distribution of agents, and not on
the control input that enables convergence. Multiple leader-
follower formation control solutions exist in the literature and
——(a) Objective formation F1 ——(b) Initial positions ——(c) Agent paths ——(d) Agent errors
Fig. 2. Illustration of 13 agents in random initial positions converging towards the formation configuration F1 defined in (a).
——(a) Objective formation F2 ——(b) Initial positions ——(c) Agent paths ——(d) F2 errors
Fig. 3. Illustration of 13 agents in random initial positions converging towards the formation configuration F1 defined in (a).
can be easily adapted to work with our proposed progressive
position assignment algorithm [14], [26], [41], [42].
Let qbe a set of Npoints representing the objective
formation, and p(t)the position of Nagents actively trying
to converge to a spatial configuration equivalent to qwith
respect to a translation ~
dp(t). Without any loss of generality,
we assume ~
dp=q0p0(t)and agents are indexed such
that pi(t)
t→∞ qi+~
dp. Let pi
j(t)be the position of
agent jmeasured by agent iin its local reference frame,
pi
j(t) = pj(t)pi(t). Then, agent i’s objective is fulfilled at
time t0, from the point of view of a leader-follower formation,
if pi
i1(t0) = qi1qi, for i > 1. Let Nibe the set of agents
that agent iis able to sense. Necessarily, this set contains at
least the parent node, pi1∈ Nii > 0.
In order to test the feasibility of this method, we propose a
simple control law based on a single integrator model, ˙pi=ui,
where uiis the control input for agent i. For agent i= 0, a
trivial solution is to have ui= 0 f orallt > 0. For all other
agents, a simple follower equation can be written generically
as ui=ϕ(pi
i1(t)qi1+qi)such that is ensures asymptotical
convergence towards the objective displacement. This function
should minimize a cost function such as
(4)
Ji(t) = γ1kpi
i1(t)qi1+qik2
+γ2X
j<i, j ∈Nikpi
j(t)qj+qik2+γ3k˙pik2
In this paper, we use γ2= 0. However, we introduce this
term as a proposal to reduce the overall system error when
the parent-child displacement is already within a predefined
limit. We introduce the constraint j < i because the smaller
the index, the smaller the agent position error due to less ac-
cumulated drift. Moreover, agents with smaller index converge
faster as they are closer to the static root node from the point
of view of the directed path graph. Therefore, agents can take
as reference any other agents that they can sense, and that
precede them in the SDPG.
For collision avoidance, we use a potential such as [43]:
Vij (t) =
min (0,kpi
j(t)k2R2
kpi
j(t)k2r2)!2
kpi
j(t)k> r
∞ kpi
j(t)k< r
(5)
where R, r represent the warning and danger distance, re-
spectively. These constants are defined in a way such that an
agent actively tries to avoid another agent when the distance
that separates them is smaller than the warning distance, and
it must never be below or equals to the danger distance.
During the simulations we assume that, for any given agent
other than the root, its parent agent is always within sensing
range. However, this is unrealistic as even line of sight could
be lost when another agent passes by through both. To solve
this, since parent agents are also able to measure the position
of their child agent, we propose to reduce the parent speed
closer to 0 as the distance between parent and child increases
towards the sensing range or a predetermined limit.
V. EX PE RI ME NTAL RESULTS
In order to test the feasibility and effectiveness of the pro-
posed algorithm, we have run a set of simulations to analyze
——(a) Objective formation F3 ——(b) Objective formation F4
Fig. 4. Objective distributions F3 anf F4 utlized in the experimental
simulations.
F1 F2 F3 F4
0
200
400
——(c)
Fig. 5. Boxplot illustrating the number of iterations (vertical axis) needed
to converge to different objective patterns. In each simulation, the initial
distribution of agents is either random (red) or has been generated adding
noise to the objective distribution (blue). In the horizontal axis, each of the
objective formation configurations introduced in Figures 2, 3 and 4.
——(a) Collision avoidance in F1 ——(a) Collision avoidance in F2
Fig. 6. Collision avoidance triggers
10 20 30
0
200
400
600
800
——(a) Random initial distributions
10 20 30
0
200
400
600
800
——(b) Noise in initial distribution
Fig. 7. Number of iterations converging to a random configuration
its performance under favorable and unfavorable conditions.
Figures 2 and 3 show two example configurations, F1 and
F2, with 13 agents each. Both configurations can be divided
in a set of three convex layers. In both cases, random initial
distributions have been used. These two examples serve as an
initial illustration of the feasibility of our proposed algorithm.
Figure 6 shows the number of agents that have at least one
neighboring agent closer than the warning distance.
Common to all simulation results presented in this paper are
the following parameters. When distributing agents randomly,
a minimum distance of 15 is kept between agents. The inter-
agent warning distance to avoid collisions is set to 8, and the
danger distance to 4. The maximum speed of agents is 1. The
assignment direction is counterclockwise, and the root node
is chosen with constants m= 0, s1= 1, s2= 1. In order to
avoid that agents lose sight of their parent node, we have also
successfully tested a modification of the collision avoidance
scheme in which roles are assigned a priority based on their
distance to the root node in the graph. When the collision
avoidance potential is activated, only the agent with lower
priority actively avoids the collision. Moreover, if a parent
node notices that its follower is getting to a distance near
the maximum sensing range, it can reduce its speed until the
distance is reduced.
In Figures 5 and 7, we analyze how the objective configura-
tion and the initial distribution of agents affect the performance
of the algorithm in terms of time to convergence. Figure 5 (c)
summarizes the number of iterations needed to converge to the
objective formations F1, F2, F3 and F4 over 800 simulations,
100 for each configuration and type of initial distribution.
In red is shown the results of simulations where the initial
distribution of agents is random over an area similar to the
objective one. Formation F1 clearly requires less time. This is
due to the sparse distribution of agents in space. Formations
F2, F3 and F4 show similar complexity, with F4 requiring
fewer iterations presumably due to the lower number of agents.
The box graphs in blue show the number of iterations needed
to converge in the case in which the initial positions of
agents are calculated by adding a random drift to the objective
positions. In this case, the space sparsity of the formation does
not play such a significant role, as the initial distribution is
similar. Therefore, very similar complexity is shown by the
different formations.
In Figure 7, we show the result of running 280 simulations
for 2 different types of initial distributions and 7 different
number of agents. The objective configuration is a random
distribution of Nagents over an area with side length Lgrid =
502pN/5. In Figure 7 (a), the initial distribution of agents
is also random over an area of the same size. However, in
(b), agents are individually deployed nearby the objective
distribution, in areas of side length 30. This resembles the
idea of adding noise to the objective system. We can see that
the complexity of the system rapidly grows in the case of a
random distribution, while the number of iterations increases
slower when the shapes are more similar. In a real scenario,
a person deploying agents in an objective scenario probably
has an idea of the final spatial distribution that agents will
converge to. Therefore, it is natural to expect that the original
distribution is not totally random and is in some way correlated
to the objective distribution.
VI. CONCLUSION AND FUTURE WO RK
We have presented a distributed progressive formation
control algorithm that enables a wide range of formation
configurations. Any formation configuration is possible if the
sensing range enables the definition of a convexly layered
set. Compared to a previous progressive formation control
algorithm proposed by Pinciroli et al., our approach requires
only one-way communication, and all agents are equivalent
and anonymous upon deployment. Therefore, the proposed
methodology does not require any agent to have a preassigned
role or objective position. Furthermore, we assume that agents
share a common orientation reference. This enables conver-
gence to a formation configuration with respect to a transla-
tion, keeping the desired orientation. Finally, we propose a
control law based on a leader-follower scheme that requires
only the same information utilized during the role assignment
process. The algorithm is lightweight and can be implemented
in resource constrained devices.
Future work will include an implementation of the proposed
algorithm in terrestrial and aerial vehicles, introducing the
dynamics of different agents and adapting control inputs
accordingly. The algorithms introduced in this paper will be
also extended to 3D formation control.
ACKNOWLEDGMENT
This work has been supported by NFSC grant No.
61876039, and the Shanghai Platform for Neuromorphic and
AI Chip (NeuHeilium).
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We address the problem of progressively deploying a set of robots to a formation defined as a point cloud, in a decentralized manner. To achieve this, we present an algorithm that transforms a given point cloud into an acyclic directed graph. This graph is used by the control law to allow a swarm of robots to progressively form the target shape based only on local decisions. This means that free robots (i.e., not yet part of the formation) find their location based on the perceived location of the robots already in the formation. We prove that for a 2D shape it is sufficient for a free robot to compute its distance from two robots in the formation to achieve this objective. We validate our method using physics-based simulations and robotic experiments, showing consistent convergence and minimal formation placement error.
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We tackle the problem of achieving any given shape defined as a point cloud in a distributed manner with a swarm of robots. The contributions of this paper are (i) An algorithm that transforms a point cloud into a acyclic directed graph; (ii) A motion control law that, from the acyclic directed graph, allows a swarm of robots to achieve the target shape in a decentralized manner; and (iii) A theoretical model, which provides sufficient conditions on the convergence of the control law. The key idea of our approach is to achieve the target shape progressively by inducing an ordering among the robots. More precisely, we construct an acyclic directed graph so that any free robot (i.e., not part of the shape) finds its location with respect to the already placed robots. We prove that, for a 2D shape, it is sufficient for a free robot to calculate its location with respect to two already placed robots to achieve this objective. We validate our method through accurate physics-based simulations of non-holonomic robots.
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