Communication-free and Index-free Distributed Formation Control Algorithm for Multi-robot Systems

Abstract

Pattern formation algorithms for swarms of robots can find applications in many fields from surveillance and monitoring to rescue missions in post-disaster scenarios. Complex formation configurations can be of interest to be the central element in an exhibition or maximize surface coverage for surveillance of a specific area. Existing algorithms that enable complex configurations usually require a centralized control, a communication protocol among the swarm in order to achieve consensus, or predefined instructions for individual agents. Nonetheless, trivial shapes such as flocks can be accomplished with low sensing and interaction requirements. We propose a pattern formation algorithm that enables a variety of shape configurations with a distributed, communication-free and index-free implementation with collision avoidance. Our algorithm is based on a formation definition that does not require indexing of the agents. We show the potential of the algorithm by simulating the formation of non-trivial shapes such as a wedge and a T-shaped configuration. We compare the performance of the algorithm for single and double integrator models for the dynamics of the agents. Finally, we run a preliminary test of our algorithm by implementing it with a group of small cars equipped with a Lidar for sensing and orientation calculation.

Full text

Available online at www.sciencedirect.com
Procedia Computer Science 00 (2018) 000–000
www.elsevier.com/locate/procedia
The 10th International Conference on Ambient Systems, Networks and Technologies (ANT)
April 29 - May 2, 2019, Leuven, Belgium
Communication-free and Index-free Distributed Formation Control
Algorithm for Multi-robot Systems
J. Pe˜
na Queraltaa,, C. Mccorda, T. N. Giaa, H. Tenhunenb, T. Westerlunda
aDepartment of Future Technologies, University of Turku, Turku, Finland
bDeptarment of Electronics, KTH Royal Institute of Technology Stockholm, Sweden
Abstract
Pattern formation algorithms for swarms of robots can find applications in many fields from surveillance and monitoring to rescue
missions in post-disaster scenarios. Complex formation configurations can be of interest to be the central element in an exhibition
or maximize surface coverage for surveillance of a specific area. Existing algorithms that enable complex configurations usually
require a centralized control, a communication protocol among the swarm in order to achieve consensus, or predefined instructions
for individual agents. Nonetheless, trivial shapes such as flocks can be accomplished with low sensing and interaction requirements.
We propose a pattern formation algorithm that enables a variety of shape configurations with a distributed, communication-free and
index-free implementation with collision avoidance. Our algorithm is based on a formation definition that does not require indexing
of the agents. We show the potential of the algorithm by simulating the formation of non-trivial shapes such as a wedge and a T-
shaped configuration. We compare the performance of the algorithm for single and double integrator models for the dynamics of
the agents. Finally, we run a preliminary test of our algorithm by implementing it with a group of small cars equipped with a Lidar
for sensing and orientation calculation.
c
2018 The Authors. Published by Elsevier B.V.
This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/)
Peer-review under responsibility of the Conference Program Chairs.
Keywords: Formation Control; Pattern Configuration; Coordination; Index-Free Control; Communication-Free; Swarm Robotics; Multi-Agent
Systems; Multi-Robot Systems;
1. Introduction
Pattern configuration and formation control algorithms for swarms of robots have been extensively studied for
the past decades [15,9,1,2,8,4,6,7,5]. We can classify pattern formation algorithms among those that require
agent indexing or those in which agents are anonymous. Most of the work to date in formation control algorithms
for multi-agent systems lies in the former category [3,7,14]. Nonetheless, in a swarm of robots where all units are
indistinguishable, a more natural approach is to assume a homogeneous set of agents without identity.
E-mail address: {jopequ, cassandra.l.mccord, tunggi, tovewe}@utu.fi, hannu@kth.se
1877-0509 c
2018 The Authors. Published by Elsevier B.V.
This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/)
Peer-review under responsibility of the Conference Program Chairs.

2J. Pe ˜na Queralta et al. /Procedia Computer Science 00 (2018) 000–000
-
—–(a) Pattern
π/43π/4
5π/47π/4
-(b) —ψ0|τ=0(θ), —ψ0|τ>0(θ)
π/43π/4
5π/47π/4
-(c) —ψ1|τ=0(θ), —ψ1|τ>0(θ)
π/43π/4
5π/47π/4
-(d) —ψ3|τ=0(θ), —ψ3|τ>0(θ)
Fig. 1. Illustration of the spherical indicator distribution. Figure (a) shows a wedge configuration of 7 agents. Figures (b), (c) and (d) show the
spherical indicator distribution of the first, second and fourth (top) agents, if indexed from left to right, for τ=0 and τ > 0.
Many efforts have been devoted to the study and replication of collective motion in nature, such as birds in a wedge
formation or flocks of fishes. In these cases, agents are anonymous and homogeneous, and therefore interchangeable.
Algorithms that model this behavior may use indexing for formulation but are not affected by an indexing permutation
as all agents are considered equal. To the extent of our knowledge, algorithms that rely on anonymous definitions
and do not require communication can only be applied in formations of trivial patterns such as flocks or regular
polygons, where all positions are equivalent. In the case of flocks, the algorithms are usually based on distance-only
measurements [5], while bearing-only measurements can be used to achieve regular polygon configurations [9,1,2].
A limitation of these algorithms is that the set of possible formations is restricted to those configurations in which
all agent positions are equal. More complex configurations are possible with index-free control if communication is
used throughout the system to ensure consensus [15]. Index-free refers to the fact that agents are anonymous and that
a permutation of the indexes does not affect the calculation of the control inputs. However, wireless communications
between robots used in these algorithms can cause a large overhead of latency and energy consumption. When the
number of agents increases, this becomes a serious issue. Some research has been focused on generalizing index-free
control by designing algorithms that are invariable to indexing permutations [16,11,10]. However, these often use an
equivalence relation, or are significantly affected by the non-scalable size of the permutation space with respect to the
number of agents in the configuration[13].
Distance-only measurements or bearing-only measurements also appear in the literature within algorithms that are
able to achieve more complex formation configurations where the positions are not equivalent. For example, bearing-
only measurements are used in [12] to allow non-trivial pattern formation. Alternatively, distance-only measurements
are equally employed by [6,14].
In this paper, we propose a new approach for defining a position within a formation based on the definition of
a spatial distribution that takes into account the position of several agents at once. This approach helps to achieve
complex formation configurations without agent indexing and without communication, given a favorable initial spatial
distribution of the agents. The proposed approach is demonstrated with both simulations for two different pattern
configurations and actual experiments with real robots. The algorithm is simulated using single and double integrator
dynamics models in order to perform an initial analysis of its potential. The performance between both dynamics
models is compared.
The paper is organized as follows: Section 2 introduces the definitions and key features in which the proposed al-
gorithm relies. Section 3 develops the main formulas for different control inputs that enable an actual implementation
of the algorithm. Section 4 covers numerical analysis that demonstrates the effectiveness of the proposed methods.
Section 5 covers a preliminary experiment run with real robots. Section 6 concludes the work and discusses the future
research direction.
2. Formation Definition
Formation control algorithms often define formation positions in a way that are directly related to the variables
sensed by agents. Some widely used formation control algorithms are position-based, distance-based and bearing-

J. Pe ˜na Queralta et al. /Procedia Computer Science 00 (2018) 000–000 3
based algorithms [7]. Position-based formation control algorithms define the positions in the formation with a set of
points in space, which are assigned to individual agents. Distance-based formation algorithms (e.g. algorithms for
achieving flocking formation) define a position in the formation by distances which are calculated by the agent and
its neighbors. Similarly, bearing-based formation control algorithms define a position by the set of bearings, given a
common orientation for the measurement. In this paper, the proposed formation control algorithm is a distributed and
index-free pattern formation algorithm considering the position of neighbor agents simultaneously.
Comparing to index-based formation control algorithms, an index-free algorithm has many advantages. For in-
stance, it is challenging or even impossible for index-based robots to form an accurate formation when one of the
robots cannot get into the predefined position due to a hardware failure or other errors. This issue of an inaccurate
formation can be avoided with an appropriate index-free formation control algorithm as agents are interchangeable.
Furthermore, the proposed algorithm is communication-free, which helps to avoid the overhead of large energy con-
sumption and latency caused by wireless communicating between agents.
2.1. Spherical distribution
Our objective is to design an algorithm that requires, at most, position measurements, and provide a definition for a
formation configuration that is independent of the agent indexing while allowing almost arbitrary patterns. Therefore,
we introduce a new definition for each of the positions in the formation.
Definition 2.1 (Spherical Indicator Distribution).Given a set of N agents with positions (xi,yi,zi)∈R3∀i=1,...,N ,
let Nibe the set of agent i’s neighbors. Their positions are measured within agent i’s local reference frame and
represented by (xi
j,yi
j,zi
j)≡(ri
j, θ j
i, ϕ j
i)∀j∈Niin Cartesian coordinates or spherical coordinates, respectively. Then
we define agent i’s spherical indicator distribution by
ψi(θ, ϕ)=αX
j∈Ni
w(ri
j)(θ, ϕ)∗δ(θ−θi
j, ϕ −ϕi
j) (1)
where δ(θ, ϕ)is the Dirac delta function defined in in the subset [0, π)×[0,2π)⊂R2,α > 0is a constant and w(r)(θ, ϕ)
is a function of the type w(r)(θ, ϕ)=exp −(θ, ϕ)TQ(r)(θ, ϕ)for a positive definite matrix Q(r)0.
The definition of the spherical distribution in (1) is inspired by an agent’s view of its environment. The Dirac
delta marks the two-dimensional bearing of each of the neighbors while Q(r) shapes the space around those points
as a function of the distance. We have defined an agent’s spherical distribution over (θ, φ) only and not over ras
well because the bearing coordinates are defined over a compact subset of R2. Thus, they can be represented by two-
dimensional matrices if we consider a discretization of R2. From a computational point of view, this means a finite
and fixed memory allocation. In particular, for our algorithm formulation, we define Q(r) in Definition 2.1 as
Q(r)=β r20
0r2!+ τ0
0τ!(2)
for constants β, τ ∈R, β, τ > 0. The value of βis the same for all agents and governs the variable width of w(r)
along (θ, ϕ), while τadds a constant minimum width. We can think of 1/(βr2) and 1/τ as variances of two convoluted
Gaussian distributions. We extend on the significance of τlater in this document. Then we can expand an agent’s
spherical distribution (1) into (3). illustrated in Figure 1,
ψi(θ, ϕ)=αX
j∈Ni
exp −βri
j
2+τ


θ−θi
j, ϕ −ϕi
j



2(3)
2.2. Autonomous Position Assignment
The first step in a formation control problem is the position assignment. In displacement-based control this is pre-
defined beforehand, whereas in anonymous distance or bearing-based control this step is skipped due to all positions
being equivalent, for instance. In our algorithm, agents perform autonomous self-position assignment as described by
Definition 2.2.

4J. Pe ˜na Queralta et al. /Procedia Computer Science 00 (2018) 000–000
Definition 2.2 (Position objective).Given a desired position ψ∗
jand an agent in a position ψi, we define the cost of
achieving the position j for agent i as
Dψi(ψ∗
j)=Z2π
0Zπ
0
||ψi(θ, ϕ)−ψ∗
j(θ, ϕ)||2dθdϕ(4)
and, therefore, an agent i decides its objective position by minimizing
ψob j
i=ψ∗
j:j=argmin
j=1,...,N
Dψi(ψ∗
j),Dob j
ψi
.
.=Dψi(ψob j
i) (5)
from where we define the cost of achieving the objective position
In order to achieve the objective position, we give the following problem formulation. Given a formation shape
defined by a set of Npositions {xi}i=1,...,N, and their corresponding spherical indicator distributions {ψi(θ, ϕ)}, then
Definition 2.3 (Formation objective).Given a pattern configuration defined by a set of N positions, from which we
can calculate their individual spherical distributions, the desired formation shape of a group of agents is represented
by the set Eψ={ψ∗
1, . . . , ψ∗
N}of desired spherical indicator distributions. Therefore, the formation objective is to have
a permutation σ∈SNsuch that Dψi(ψ∗
σ(i))< δ for a constant δ∈R,δ > 0. The value of δis the minimum error
allowed to assume a successful convergence to the desired position. The set {ψi(θ, ϕ)}represents the agents’ positions
and {ψ∗
j(θ, ϕ)}the desired positions.
The definition for the position objective in (2.2) does not ensure, in its current form, that there exists a permutation
σthat maps the set of agents into the set of formation positions. This paper presents a preliminary exploration of
the potential of the position definitions introduced in this section. The results included in this document involving
simulations and tests are obtained under the assumption that the initial distribution of agents in space ensures the
existence of σ.
After the autonomous self-position assignment is made by each of the agents, the next step is to minimize some
cost function in order to arrive to the convergence conditions given in Definition 2.3. When minimizing the cost of
achieving the objective position introduced in (5), however, a problem arises from the nature of the definition we are
using for ψi(θ, ϕ) in (3). If the value of τis too small, as happens in the example illustrated in Figure 1, then the
minimization problem can be non-convex. This might happen because ψi(θ, ϕ) is a function formed up by individual
peaks with almost-zero intervals in between. Local minima exist when some of the peaks in an agent’s spherical
distribution and its objective position’s spherical distribution are overlapped but others are not. Therefore, depending
on the desired pattern configuration, the value of τmust be adjusted to prevent this from happening. We have developed
a graphical interface for simulation purposes where we can adjust the parameters of the algorithm in real-time and
during a simulation to see how this affects the position definitions as well as the algorithm performance.
2.3. Collision avoidance
So far, our algorithm does not take into account collision avoidance; for example, in the limits rj→0 and rj→ ∞,
the contribution of an agent jto ψiis reduced to a constant over θor a Dirac delta function in θj, respectively. In the
limit rj→0, this might lead to collisions between agents in the case of distributions in which the angular positions
of neighbors are densely distributed across all θ. In that case, ψiwould be an almost-constant function, and the error
incurred when inter-agent is reduced would be negligible.
In order to introduce collision avoidance to our algorithm, we consider the following modification of the spherical
indicator distribution for a position. Suppose an agent is considered to move safely when its distance with respect
to all other agents is bigger than a threshold rs. An agent is considered to be in collision danger if its distance with
respect to a neighbor is below a threshold rd. Then we can modify (1) into
ψi(θ, ϕ)=X
j∈Ni
αjexp −βri
j
2+τ


θ−θi
j, ϕ −ϕi
j



2, αj=








αi f rs<rj
rj/(rj−rd)i f rd<rj≤rs
∞i f rd≤rj
(6)
where we have introduced the constant αinside the sum and now varies for each of the agents, which always ensures
that rj>rdand therefore agents are safe from collisions.

J. Pe ˜na Queralta et al. /Procedia Computer Science 00 (2018) 000–000 5
3. Control Inputs
In this section, we introduce the control laws that define our algorithm for both a single and double integrator
dynamics model for the agents. For simplicity, we reduce the problem from here on to the two dimensional case.
Therefore, an agent’s spherical distribution is given now in polar variables ψi(θ).
First, we suppose that the dynamics of each of the agents in the swarm follow a single-integrator model given by
pi[k+1] =pi[k]+Tsui[k] where pi[k],ui[k]∈R2denote the position and velocity, respectively, of agent i, at the time
step k, and Tsis the sampling period. We use radial and angular control inputs ui[k]≡(uir[k],uiθ[k]) that represent
the speed and direction of movement, respectively. The polar input is well defined because all agents share a global
orientation. Now we can calculate ri
j[k+1 and θi
j[k+1] based on the control input for agent i, using basic trigonometry,
and define a cost function.
ri
j[k+1]2=ri
j[k]2+T2
suir[k]2−2Tsri
j[k]uir[k] cos θi
j[k]−uiθ(7)
θi
j[k+1] =arccos 







ri
j[k] cos θi
j[k]−Tsuir[k] cos uiθ[k]
ri
j[k+1] 






(8)
Definition 3.1 (Cost function).Given an agent with dynamics described by a single integrator, its position represented
by ψi(θ), and its position objective by ψob j
i(θ), then we define its cost function at a time step k by
Ji[k]=Z2π
0ψi−ψob j
i2dθ+γ||ui||2=Dob j
ψi[k]+γuir[k] (9)
where γ > 0controls the weight of the closed loop control.
From Definition 3.1 we propose a control law that minimizes (9). First, we use numerical methods to calculate the
angular control input, obtaining ψihk+1|uiθ[k]=e
θifrom (7) and (8). Regarding the control input uir, we propose a
control law based on the instantaneous position error.
uiθ[k]=e
θ:e
θ=argmin
e
θ∈[0,2π)Z2π
0ψihk+1|uiθ[k]=e
θi−ψob j
i[k]2dθ(10)
uir[k]=ν
Dob j
ψi
10δ+Dob j
ψi
(11)
where δis the maximum error allowed for each position to assume that the system has converged to its formation
objective, as defined in the problem formulation in Definition 2.3. The factor 10 introduced in the equation is motivated
by a preferred faster convergence of u2
irto 0, with respect to the position error term of the cost function (9).
If we suppose that the dynamics of each of the agents follow a discrete double-integrator model, then the position
of a agent at the next time step is given by pi[k+1] =pi[k]+Tsqi[k],qi[k+1] =qi[k]+Tsui[k], where pi[k], qi[k] and
ui[k] are agent i’s position, velocity and acceleration (control input), respectively. Then, we adapt the cost function
(9) to take into account the agent’s velocity as Ji[k]=Dob j
ψi[k]+γruir[k]+γθuiθ[k]. Now uirand uiθrepresent the
radial and angular acceleration, respectively. We take into account the radial acceleration to avoid rotations. In this
case, instead of defining the control inputs at each time step by a closed formula, we adapt the desired angular speed
from (10) into
qiθ[k]=e
θ:e
θ=argmin
e
θ∈qiθ−uimax
θ,qiθ+uimax
θZ2π
0ψihk+1|uiθ[k]=e
θi−ψob j
i[k]2dθ(12)
where we have only changed the minimization interval by adding the maximum allowed radial acceleration and taking
into account that the operations uiθ−uimax
θand uiθ+uimax
θare modulus 2π. Equivalently, we calculate the ideal radial
speed using (11) and then limit the radial acceleration. Then the control inputs are simply uiθ=qiθ[k]−qiθ[k−1] and
uir=qir[k]−qir[k−1].

6J. Pe ˜na Queralta et al. /Procedia Computer Science 00 (2018) 000–000
50 100
0
50
100 p0[k]
p1[k]
p2[k]
p3[k]
p4[k]
p5[k]
p6[k]
——(a) Single integrator: movement
0 20 40 60
0.0
0.2
0.4
0.6
0.8
1.0 q0[k]
q1[k]
q2[k]
q3[k]
q4[k]
q5[k]
q6[k]
——(b) Single integrator: speeds
0 20 40 60
0
200
400
Dobj
ψ0)[k]
Dobj
ψ1)[k]
Dobj
ψ2)[k]
Dobj
ψ3)[k]
Dobj
ψ4)[k]
Dobj
ψ5)[k]
Dobj
ψi6)[k]
——(c) Single integrator: individual errors
050 100
0
50
100
p0[k]
p1[k]
p2[k]
p3[k]
p4[k]
——(d) Double integrator: movement
0 20 40 60 80
0.0
0.2
0.4
0.6
0.8
1.0 q0[k]
q1[k]
q2[k]
q3[k]
q4[k]
——(e) Double integrator: speeds
0 20 40 60 80
100
101
102PDobj
ψi)[k]
——(f) Double integrator: system error
Fig. 2. A wedge configuration of 7 agents is achieved in (a), where the movement of the agents is displayed with increasing opacity over time.
Graphs in (b) and (c) show the evolution of each of the agents speed and position error. A single integrator model is used for the agents dynamics.
In the second row, a T-shaped configuration of 5 agents is achieved, using double integrator models. Graphs in (e) and (f) now show the speed of
the agents and the system error over time, respectively.
4. Numerical Analysis
In this section, we show the computational simulations results that we have obtained after implementing our al-
gorithm for non-trivial T-shaped and wedge formation configurations. The algorithm has been implemented using
Python. The implementation models the robots as single points in space.
Figure 2shows the result of two simulations. The wedge formation is achieved using a single integrator model for
the dynamics of the agents, while the T-shaped formation is achieved with a double integrator dynamics. We have
tested both single and double integrator models for those two and other formation configurations, including a square,
a square with a fifth agent in the center, and a diamond. We have found little difference in the performance of the
algorithm under the single and double integrator models. The main difference can be seen in Figure 2, graphs (b) and
(e), where we can see that there is an acceleration limit.
We set the maximum speed to ν=1, and the maximum position error allowed to assume convergence is set to δ=1.
This means that when the position error of an agent is Dob j
ψi=1, then its radial speed is qir=ν·Dob j
ψi/(10+Dob j
ψi)≈0.11.
We can see that this effectively happens in Figure 2, (e) and (f), around iteration 50, in which the total error of the
system is approximately PiDob j
ψi≈5 and the speeds are qir≈0.1.
We also partially test the role of the neighborhood set, Ni, in the convergence of the algorithm. In the wedge
configuration simulation, we assume that agents are able to sense the position of all other agents. In the case of the
T-shaped configuration, we introduce a sensing radius, and define each position in the formation based only on the
visible positions. Throughout the simulation, we impose that Agents 0, 3 and 4 are not able to see each other, and the
same for Agents 0 and 2. Agent 1 is the only one able to sense the position of all other agents. The definitions that we
have given so far do not take into account the possibility that the number of neighbor positions that is used to define a
given position might differ from the number of agents that an agent trying to reach that same position is able to sense.
This is an important aspect of the algorithm under study and more results will be reported in future work.

J. Pe ˜na Queralta et al. /Procedia Computer Science 00 (2018) 000–000 7
a) Position 0 (b) Position 2 (c) Position 3 (d) Car with lidar
(e) Lab environment
(f) Lidar output (g) Initial Car Vision (h) Final Car Vision
Fig. 3. Experiment with the MonsterTruck cars. Figures (a) to (c) show the definition of 3 positions in a square configuration (they are all equivalent
with respect to a rotation). Figure (d) shows the car used in the experiment and (e) illustrates the testing environment, where the white column
is used as an orientation reference. Figure (f) shows the lidar output, with one point for each degree, for a car placed near the center of the room
detecting three other cars and the reference column. Figures (g) and (h) show the view of the car, ψ(θ), at iterations it =0 and it =5, respectively.
All the figures, except (d) and (e), are taken from a web application that runs in each of the cars and is used for monitoring.
5. Testing in a lab environment
We have made an initial test of the proposed algorithm with real robots. In order to analyze the effectiveness of our
algorithm in a real scenario, we use a 1:10 Elektro-Monstertruck ”NEW1” BL model, a radio controlled car where we
have replaced the radio receiver by a Raspberry Pi. The motion of the car is controlled via two servo motors, one for
the turning direction and one for speed control. The turning direction is limited to the interval (−0.35rad,0.35rad), of
about 40 degrees, 20 in each direction. This is the value of the maximum radial acceleration uimax
θ. While each iteration
of the algorithm needs a relatively low execution time, reading and processing the lidar data takes more computing
resources. Moreover, we run a web server on the Raspberry Pi in order to have real time monitoring and visualization
of the lidar data and agent view, as well as calculated control inputs. As a result, instead of keeping the car moving
in a continuous way, we move the car in steps. The car speed given by the control input is translated into the distance
that is moved in each step. This can be improved with a more powerful computing platform.
To obtain the position of other cars we use a RPLiDAR A1M8 lidar with a range of 12m that offers a 360-degree
view of the car environment. In order to calculate the car orientation, we use a column in the testing environment
as a reference. We calculate the instantaneous car orientation supposing that the two walls closer to the column
represent north and east directions. In a real scenario, potentially unknown, agents would be equipped with digital
magnetometers or other sensors in order to ensure a shared global orientation. However, the settings we use are
enough for the purpose of the experiment.
The results of our test are shown in Figure 3. We use 4 cars and choose a square pattern configuration as the
formation objective. The first four graphs, (a) to (d) show the definition of the four positions in the square. The lab
environment is illustrated in (e), with the white column used as an orientation reference. The column is not used for
localization but only for aligning the orientation of local refence frames. We follow the movement of a car placed near
the center of the room. The initial position of the car and the other three cars can be inferred from the initial lidar
view shown in (f), where the car being monitored is in the center of the graph. The four cars are initially placed in a
nearly-triangular configuration where one of the cars is near the center of the other two.
Due to the non-negligible size of the cars compared to the room space, and the fact that they are not small robots
that can turn around while keeping their position, the formation can only be achieved to some extent, and the maxi-

8J. Pe ˜na Queralta et al. /Procedia Computer Science 00 (2018) 000–000
mum required error is achieved in just 5 iterations. Graphs (g) and (h) show the initial and final agent view after those
5 iterations, where the car is trying to achieve the position defined in (b). This experiment proves that convergence to
the desired configuration is possible in a real scenario without communication or predefined assignments. The angular
distribution of the four position definitions makes the role assignment simpler in this case, but more complex config-
urations have been simulated. Future work will include extensive testing in larger environments and more complex
formation configurations. We will also take into account more realistic models for the agent dynamics.
6. Conclusion and Future Work
In this paper, we propose an advanced formation control algorithm that enables almost arbitrary shapes to be formed
up without the need of communication between the agents and without identifying each of the agents with a unique
label. This algorithm is based on an index-free definition for a position within a formation that requires distance and
bearing measurements to express the position of a neighbor agent in spherical coordinates. Moreover, the definitions
can be adapted to avoid collision between agents.
The algorithm introduced in this paper consists of a set of preliminary ideas and is in an early stage of develop-
ment. Therefore, further development is required for a robust formulation and implementation. We have assumed the
existence of a permutation that ensures a surjective mapping to the set of desired positions. We base this assumption
in the use of an initial distribution that naturally produces such surjection. Moreover, the role of the neighborhood set
in the convergence has not been studied in depth. We are continuing to study those and other aspects involving the
algorithm convergence and results will be shown in future work.
To the extent of our knowledge, other distributed algorithms that are able to achieve wedge configurations, or a
T-shaped formation, require either communication among the agents to achieve consensus or agent labeling and a
predefined assignment of the positions in the formation to specific agents. The algorithm presented in this paper is
inspired by the anonymous nature of a homogeneous system. It only requires measurement of the position of other
agents, and a shared global orientation. The former can be obtained through multiple solutions such as lidars or
cameras together with computer vision, while the latter can be easily achieved by using a digital compass or similar
sensors.
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