Mathematical Problems in Engineering

Volume 2018, Article ID 6291082, 11 pages

https://doi.org/10.1155/2018/6291082

## Collision Avoidance Mechanism for Symmetric Circular Formations of Unitary Mass Autonomous Vehicles at Constant Speed

^{1}National Institute for Space Research, 1758 Av. dos Astronautas, 12227-010 Sao Jose dos Campos, Brazil^{2}Federal University of Sao Paulo, 1201 Cesare Mansueto Giulio Lattes, 12247-014 Sao Jose dos Campos, Brazil

Correspondence should be addressed to Vander L. S. Freitas; moc.liamg@pmocrednav

Received 11 September 2017; Revised 31 March 2018; Accepted 3 May 2018; Published 7 June 2018

Academic Editor: Oleg V. Gendelman

Copyright © 2018 Vander L. S. Freitas and Elbert E. N. Macau. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Collective motion is a promising field that studies how local interactions lead groups of individuals to global behaviors. Biologists try to understand how those subjects interplay in nature, and engineers are concerned with the application of interaction strategies to mobile vehicles, satellites, robots, etc. There are several models in literature that employ strategies observed in groups of beings in nature. The aim is not to literally mimic them but to extract suitable strategies for the chosen application. These models, constituted of multiple mobile agents, can be used in tasks such as data collection, surveillance and monitoring. One approach is to use phase-coupled oscillators to design the mobile agents, in which each member is an oscillator and they are coupled according to an interconnection network. This design usually does not keep track and handle the possible collisions within the group, and real applications obviously must manage these situations to prevent the equipment from crashing. This paper introduces a collision avoidance mechanism to a model of particles with phase-coupled oscillators dynamics for symmetric circular formations.

#### 1. Introduction

Phase-coupled oscillators may be employed to model several systems of interacting individuals, such as neurons in a brain, coupled pendulums, groups of living beings [1] and applications of collective motion [2, 3]. One of the most prominent paradigms of the field of phase-coupled oscillators is the Kuramoto model [4], in which the oscillators’ interactions are mediated via sinusoidal coupling. This model opened possibilities for the study of synchronization [5–9].

Biologists try to identify local interaction rules that lead groups of agents to ordered collective motion such as flocks of birds [10, 11], schools of fish [12], and several other living beings [13]. Some models consider such knowledge, as in the case of the well-known* boids* [14] which interact with each other according to three rules: attraction, repulsion and alignment.

A well-known model for collective motion was proposed by Vicsek et al. [15] in which, depending on the density of the particles and the noise amplitude, the particles converge to an ordered motion. This motion is also known as* flocking* and is characterized by particles moving in parallel to the same direction.

Sepulchre et al. [2, 16] developed a model to lead particles with coupled-oscillator dynamics to synchronized and balanced states, showing parallel and circular formations with symmetric patterns. Jain and Ghose [3] stabilized their models to formations whose centroid converges to a desired spatial coordinate. Besides, the radii of the circular formations may vary for each agent, with the same (or not) center of rotation, and is divided into synchronization subgroups.

Circular formations are found in groups of fishes and bacteria [13]. Engineering applications of this kind of motion are related to tasks that require the agents to visit places periodically, like in surveillance, data collection, transport, etc.

In this work we improve the control for symmetric circular formations proposed in Sepulchre et al. [2, 16] by adding a collision avoidance term, based on the balanced states of Kuramoto model. The purpose is to guarantee that the agents do not collide with the neighbors in their vicinity, so that the model can be applied to more general scenarios such as the ones founded in autonomous vehicles and satellite constellations.

Moreover, we solve the inverse problem of optimization to find the optimal control parameters, using the Multiobjective Generalized Extremal Optimization (M-GEO) algorithm [17–19]. The aim is to establish a combination of parameters that leads the group to the symmetric circular formations and guarantees that the agents are not going to crash.

The application of such model in real applications may require the usage of a data assimilation technique [20–22]. Its role is to incorporate the sensor readings into the model to correct the discrepancies between its state and the real state of the device. Still, the model may also be deployed along with a control strategy [23] whose role is to apply changes in the machine state, considering its inertia, limited speed, maximum angular velocity, etc.

We consider here that the agents communicate according to three network schemes: (1) all-to-all, (2) a ring-like topology, and (3) a dynamic network. The first case was used in the optimization procedure and deals with agents that are able to communicate with neighbors at any distance. The ring-like topology is the circulant network topology with the fewer possible number of connections. It is applicable when the agents’ hardware is limited to the point of not being able to process all the data coming from the other agents. In this topology, each agent receives data from only two neighbors. Lastly, the agents communicate through a dynamic network whose connections are created or removed according to the Euclidean distance between agents. When a neighbor approaches the sensory region (radius) of an agent, one adds a connection between them. On the other hand, when the neighbor leaves this radius, the connection is broken.

#### 2. Particles with Coupled-Oscillator Dynamics

Our underlying system consists of identical individuals with unitary mass and velocity. They maneuver at constant speed according to steering controls. Each particle position is given by , the direction of the velocity vector by , and phase for . The phase represents the heading angle of the particle, pointing out the direction of motion.

Let be the vector of particles’ positions, the velocity vector, and the maneuver control (feedback control).

The particle model is the following:

When the particles move in a straight line towards their initial phase (heading angle) . Still, when , i.e., if not coupled with others, particle moves in circular trajectories centered at with radius , as shown in the following:

The center of mass of the group is

When for all pairs and , particles are synchronized. On the other hand, if the phases are spread in such a way that they cancel each other with opposite values, they are said to be balanced. These two extreme states are measured with the Kuramoto order parameter [24], given by the following equation:with , and . The order parameter corresponds to the velocity of the particles center of mass, since . When , they are in a balanced state with the center of mass in a steady position, and stands for the synchronized state, in which the center of mass moves at unitary velocity.

The order parameter represents the centroid of the first harmonic of the particle headings and is used to define the potential (5) [2]:whose maximum and minimum are characterized by synchronized and balanced phases, respectively. This potential is a direct candidate to be a Lyapunov function, because it is strictly positive, and it is zero only in the equilibrium point, that is, when the phases are balanced, and therefore . It reaches its unique minimum (balancing) when , and its unique maximum (synchronization) when all phases are identical. All other critical points of are saddle points [2]. Its gradient, with relation to phase , is described by the following:

This inner product is defined by , for , and is the conjugate of the complex number [2]. Thus,with a gain.

Sepulchre et al. [2] showed that all solutions of this system are asymptotically stable for the described case, with all-to-all coupling.

##### 2.1. Circular Formations

Circular formations are achieved with a composed potential which contains an orientation control and a spacing control. Their aim is to stabilize both the relative orientations of the velocity vectors and the positions relative to the center of mass of the group, respectively [25].

Let the rotation centroids of particles be represented by the vector .

*Definition 1 (circular formations). *A circular formation is a relative equilibrium regime in which all particles travel around the same centroid, i.e., for every pair and .

Consider the following potential:so that is the Laplacian matrix, withand being the set of neighbors of in the interconnection network. For an all-to-all coupling, .

The control for the -th particle associated with this potential is shown in the following [26]:in which is the -th row of the Laplacian matrix. It is able to stabilize the particles rotation centers to the same position.

The circular formations can be used in systems of multiple agents to perform tasks like area monitoring or data collection, since the particles visit the same places periodically. Hereafter, we show how to form clusters of particles inside the circular formation.

Consider the following rewritten order parameter [2]:representing the -th harmonic of the particle phases, with . corresponds to the synchronization of the -th phase harmonic, and is its antisynchronized state. The centroid of the phase harmonic is [25].

The synchronized state is characterized by . Note that implies for . The splay state is given by , ; i.e., the phases are evenly distributed around the unit circle.

Now consider a natural generalization of the potential (5):

Theorem 2. *Let . The potential reaches its minimum when (balancing modulo ) and its unique maximum when the phase difference between any two phases is an integer multiple of (synchronization modulo ). All other critical points of are saddle points of (proof in [2]).*

The idea is to use linear combinations of this potential to achieve arrangements of particles inside the formation. For this purpose, consider a positive integer, divisor of . A pattern is a symmetric arrangement of phases divided into clusters. corresponds to the synchronized state, in which all particles have the same phase, and the pattern relates to the splay state, when the phases are uniformly distributed around the unitary circle (balanced state).

An arrangement of particles, with clusters is a pattern . For example, if there are 12 particles in the system and they form two clusters, of 6 particles each, this arrangement represents a pattern.

Consider the potential formed with the linear combinations of :

An arrangement is formed if and only if the set of phases leads to its global minimum, with , for and (proof in [2]).

The resulting control to achieve symmetric circular formations with all-to-all interaction is the composition of controls (10) and (16), as follows:for is the natural frequency, a gain, is the set of particles’ centers, and is the projection matrix -th row, defined as ,and potential gradient

Figure 1 shows simulations with particles with random initial conditions and control (14) for all-to-all interaction.