Multi-Agent Collision Avoidance System Based on Centralization and Decentralization Control for UAV Applications

A collision avoidance method for multi-agent systems based on the centralization and decentralization effects for cooperative control is presented in this article. In this context, a matrix called Anti-Laplacian is proposed to control the agents when the UAVs are on a conflicting route. The matrix adjustment occurs through proportionality relations with the Laplacian Matrix from an interaction graph between the agents. The adjustment method aims a balance between centering and decentering to avoid collisions. Controlled quadcopters follow the trajectory indicated by virtual agents that act as guides for the real ones. The tests are performed via simulation for the most critical cases, with protocols involving flock centralization. As a virtual agent, a first-order model is used in the simulation, the method efficiency is observed by varying the number of agents involved. The proposed method presents allows different forms to control the collision avoidance parameter, such as deterministic and machine learning methods.


I. INTRODUCTION
Multi-agent [1] systems represent a group's behavior, given the individual characteristics of the elements that it. For the purpose of this study, they are the agents. From the computational and engineering points of view, cooperative dynamic systems aims a common goal, be it a task or a position. Therefore, these protocols obey set of rules, called Reynold's rules [2]. These are divided into three categories: collision avoidance between an agent and its neighbors, speed matching (maintaining a relative speed to its neighbors), and centralization (uniting all agents in the entire goal).
Based on the first rule, the development of an collision avoidance system addresses the weaknesses of the protocols, given the characteristics of centralization, formation, and leadership.
The associate editor coordinating the review of this manuscript and approving it for publication was P. Venkata Krishna .
Several recent works, such as [3], [4], [5], and [6], have addressed the problem of the formation of homogeneous multi-agent systems through collisions in free convergence by traditional methods, such as trajectory planning and the balance between formation and centralization, to avoid the problem. In another approach, the authors [7] proposes a collision avoidance system for a second-order nonlinear agent that is surrounded by moving targets. An estimator is used to determine the center of each target. An adaptive controller is designed for the agent to keep a distance from other elements in a collaborative form. The strategy is based on an error function with a parameter indicating the offset that must be kept between an agent and the target.
Another example of developing collision avoidance systems is reported in [8]. In this appoach, the navigation problem for a distributed formation of agents with second-order dynamics, speed limitations, and control input is addressed. The controller considers both collision avoidance and VOLUME 11, 2023 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ connectivity limitations. It uses a minimum cost spanning tree based on a barrier function to prevent the loss of communication and avoid collisions. These barrier functions change the controller, and resolve the problem. Furthermore, it proposes that this controller fits graph switching; however does not guarantee collision avoidance in any initial condition. It uses a centralized control with the swarm spacial formation and a quadratic barrier function that limits the action of this control. There are different strategies to avoid collisions, such as [9], which uses reinforcement learning. In [10] addresses the issue of training based on distance and field potential. It shows that the proposal for collision avoidance of this research is relevant through the different strategies presented. It can be applied to a four-engine [11].
Aiming to emphasize the importance of algorithms that approach cooperative behavior, the authors of the reference [12] present the development of swarm intelligence algorithms to solve the optimization associated with multiagent problems. Application trends of this methodology are reported in reference [13].
This research presents a collision avoidance system based on flock decentralization be defining an anti-Laplacian matrix. Critical regions are defined relative to the distances between two agents. Base on this distances an collision avoidance factor is defined and applied to the protocol that acts in response to an agent's presence in the other's critical region. The tests take place through quadcopter simulations guided by a first-order virtual agent and the protocols. The worst-case protocol evaluates the developed system, with only a centralization effect.
In the next sections, the theoretical bases and development of the collision avoidance method are presented. The Section II presents the multi-agent concepts, agent models, protocols, and schemes used. Section III presents the model used for quadricopters. Section IV presents the developmental collision avoidance method. Results and dicursions are presented in Section V. Finally, the conclusions are presented in Section VI.

II. MULTI-AGENT SYSTEMS
According to [2], multi-agent systems are based on the science of networks with a conceptual paradigm. It includes interconnected networks ranging from social networks and sensor networks to multi-agent coordination and control. The elements that make up this network must act collaboratively in these systems to achieve a global objective. Flocks, swarms, and schools are examples of collaboration, according to [1] and [2], which highlight the application of multi-agent theory as cooperative control.
In this context, uncrewed vehicles act collaboratively for more efficiency, cost reduction, and time savings. In addition, the range of action in a given task is increased, making the execution more precise.
Based on these examples, an agent is the element that receives information from the environment and responds to stimuli through actions. These can be determined according to the agent's characteristics, resulting in changes in the social context. The individual state change influences the entire chain to which the element is connected, directly or indirectly. It is about information and how it is arranged to add to the agent's perception, resulting in an interference in the environment, as presented in Fig. 1.
Representation of an agent's perception of the environment through sensors [14].
For a matrix representation of this concepts, such as information flow between agents, the Graph theory is used and presented in the next section [1].

A. GRAPHS THEORY
Graph theory consists of abstract structures representing the interdependence of a set of elements. It uses multi-agent modeling. This relationship is expressed through arcs or edges that interconnect several nodes (or vertices). A graph does the makeup of N nodes (agents) and edges (representing information flow). It is defined by the pair G = (V , E), with V = [v 1 , . . . , v N ] representing the set of vertices and E representing the set of edges or arcs. The elements of E are made of pairs of (v i , v j ), which represent the flow of information between v j and v i .
Another parameter inherent to the graph is the strength of the connection associated with an edge, defined through a ij . A graph G can be represented in a matrix form by an adjacent or connectivity matrix A = {a ij }, whose edges have weights a ij > 0 for (v j , v i ) ∈ E and a ij = 0 otherwise. If the agent does not have the feedback information, a ii = 0. Therefore, a i,j is defined by where w is the incidence value. Once the weights that represent the influence of information are defined, the degree of entry is determined. It is the sum of the weights corresponding to the information flow that reaches the agent, that is, the line sum of the matrix A, which is given by Given these properties, the set of agents arranged in a graph with a connectivity matrix A has a Laplacian matrix representing the multi-agent network's behavior. Thus, using the definition presented in [1], where D = diag(d i ), the Laplacian matrix L is given by (3)

B. MODEL FOR AGENTS
Multi-agent systems need laws regulating their behavior based on their individual dynamics. Therefore, individualized behavior ranges from simple dynamics to non-linear ones. As shown in [1], there are various representative models for agents. Those based on Newtonian models, such as displacement and velocity are examples. As a result, the dynamic equation for the agent i in continuous time is given byẋ i = v i andv i = u i , with the acceleration u i ∈ R, velocity v i ∈ R, and position x i ∈ R. As this study focused on discrete time, the reference discretized model adopted is given by where n is a step in period T . In this article, the model presented in Eq. (4), with a control protocol as entry, is used as a guide for the UAV, indicating how the quadcopter should move in space until reaching a goal. This schematic diagram is presented in Fig. 2, where a virtual agent and multi-agent protocol serve as a reference for the quadcopter displacement. This strategy is used to evaluate the performance of the collision avoidance system. It is inserted in the control protocol, as presented in the following sections.

C. CONTROL PROTOCOLS
The protocols are the laws that control agents' actions, that is, their state transition, based on Reynold's rules. To complement it, concepts such as centralization, spacial formation, and leadership in a group of agents are inserted [15].
Leadership in multi-agent systems is needed when the group needs to move towards a predefined goal. In this situation, only some elements receive information from the leader, whereas the others are based only on the information from their neighbors.
The formation is a function of the desired distribution for these agents and can take different forms, including geometric forms. This training distributes around the agents' movement center defined by the leader/reference state.
Centralization aims to make agents act on a common goal, such as following the leader's actions. This is represented by the connectivity between the agent and its neighbors around the established purpose. Then, consider an agent i in a graph, with N agents and c as the coupling factor. The protocol that promotes centralization is given by By including the leader, the protocol makes all agents centralized in the leader state, which is represented by x o ∈ R as follows: (6) where g o is the proportionality constant with respect to the leader state.
The global matrix form of Eq.

III. QUADCOPTER MODELING
This work uses a fully controlled four-engine model similar to that of [16]. The simulated model is based on the dynamic equations used in [16], [17], and [18]. The model is divided into body rotation dynamics, through the angles φ (roll angle), θ (pitch angle), and ψ (yaw angle), and translation in Cartesian axes, as show in Fig. 3. The quadcopter dynamics depends on the moments of inertia in the x-,y−, and z-axes and moments generated by the actuators, resulting in the angular accelerations given bÿ Here, l is the distance between the axes of rotation in opposite propellers. I x , I y , I z and J are the moments of inertia VOLUME 11, 2023 of the body and propeller, respectively. The moments U 1 , U 2 , and U 3 depend on the motor speeds, with as a net speed, be defined later.
The dynamics for the attitude angles φ, θ, and ψ is influenced by the moments generated by the actions of the engines. The thrust U , which is responsible for altitude (z − axis), U 1 for rotation around x − axis (φ), U 2 for rotation around y − axis (θ ), and U 3 around z − axis (ψ), are given by where b is the thrust or lift factor and d is the drag. 1 , 2 , 3 , and 4 represent, the speeds of engines 1, 2, 3 and 4, respectively. Thus, the non-linear model of the four-engine for the Cartesian coordinates (x, y, z), which is a function of thrust and rotation angles, is given bÿ The dynamic model used for the motors (actuator) is given by˙ where k m , r, and η are the engine torque constant, engine gearbox reduction, and reduction system efficiency, respectively. Including the propeler parameters in place of τ d , yields:

IV. COLLISION AVOIDANCE SYSTEM
The development of an collision avoidance system aims to address the weaknesses of the protocols, given the characteristics of centralization, formation, and leadership. Thus, a proposed system takes advantage of the centralization, and decentralization effects of the protocols [4], [19].

A. CENTRALIZATION AND DECENTRALIZATION
The centralization of the flock is the fundamental principle of cooperative control. It aims to bring all agent i ∈ [1, N ] to a common point or objective and is Decentralization, in contrast, is performed by a similar formulation, with the opposite sign, inspired by [1], and is given as with f ij providing the weakening effect of centering on the communication graph. The balancing equation between these effects is given by Replacing the Eq. (13) and (14) in (15) results in Using the entry-degree d i and Laplacian matrix in Eq. (3), the anti-Laplacian matrix represented by L is given by where Considering the simplest case in Eq. (16) where x i ∈ R, the individual form for u i is obtained by means of In its generalized form it is represented by Taking into account, equations (17) and (3), the cooperative protocol is given by Equation (20) creates a balance between centralization and decentralization. Through the difference L − L, we have

B. COLLISION AVOIDANCE MATRIX ADJUSTMENT
In the cooperative control theory the agents centralization is linked to the eigenvalues position of L on a complex plane [20]. For discrete time systems the centering occurs when the eigenvalue is inside of unit circle. In this context, the relationship between the eigenvalues of L and L acts on the centralize/decentralize effect on the protocol. It is enough to relate the two matrices by a proportionality factor α, that is given To better understand this relationship, consider M as the Jordan transformation matrix, such that J L = M −1 LM . Thus, 7034 VOLUME 11, 2023 L = αL results in an equivalent matrix J L = αJ L = αM −1 LM . Consequently, the singular values of L become proportional to those of L, weighted by α, resulting in L i = α L i for a ith agent. Therefore, the resulting singular value by the protocols, is given by Given the stability criterion for discrete-time implies that i must be inside the unit circle for a convergence effect. Thus, for convergence and divergence, the centralisation and decentralisation regions are given by In terms of the difference between L and L, Eq.
Considering the relationship between the Laplacian and Anti- resulting in absolute values for α that comprise the respective ranges: As a consequence of these regions, the ranges in which α can vary and cause the centralisation/decentralisation effects are: • Decentralisation range: • Centralisation range: The variational ranges of α, suggest that the system is convergent for all α when L i = 0 ( I L indicates the eigenvalue of the corresponding agent i of the matrix L). This means that for a spanning tree graph where 1 = 0 (agent 1), α 1 can take any value such that the convergence effect is maintained, unlike for agents with i ̸ = 0. Also, there is no need to use the entire range of available values of α. Therefore, considering only the positive range, we have A single α parameter for all agents causes an undesired interference involving elements, not on a collision course. To prevent this, each agent has its own α i , specifying the adjustment of the collision avoidance parameter. As a reference to the two approaching agents, i and j, the collision avoidance will adjust for only one of them, in this case, the element j, causing α j to act.
Considering a vector vec(α) = [α 1 , α 2 , . . . , α N ], ∀i ∈ [1, N ], we define the diagonal collision avoidance matrix, given by This matrix, multiplied by the Jordan form of L represented by J L , presents the individualized result for each agent, as shown in the expression Then, from the single parameter formulation (α − 1) L → ( − I N ) L, the global form of the protocol, given by Once the collision avoidance system is defined, it remains to be determined how this parameter will change. For this, four evaluation zones are determined.

D. EVALUATION ZONES
The zones are divided into contact, critical, moderate, and free zones. The α j changes depend on a each zone may be null, convergent, and divergent. Therefore, considering the agents i and j, we have: • Contact zone (CZ): an agent j directly interferes with agent i, that is, direct contact is observed; • Critical zone (CTZ): it is a high-risk zone where the collision is imminent, and an α j divergent act to avoid reaching the CZ case.
• Moderate zone (MZ): it is a zone where the approach is observed with a moderate alert level, with α j in the converging region and attenuating the approach between the agents.
• Free zone (FZ): in this zone, the circulation of j around i is free, that is, the moderator parameter is null without influencing the protocol. VOLUME 11, 2023 Adopting circular or spherical zones, the radius limiting each region from the agent's center were determined. These radii are depicted in Fig. 4, where r c is the radius that characterizes the collision. r ct is the limiting radius of the critical area (CTZ) where the divergence should be predominant. r m limits the moderate region (MZ). After defining the radii that limit the regions, is necessary to relate them to the cases presented in Fig. 5. It shows the different situations for an agent j is in relation to an agent i and the zones, be it critical, moderate or collision. As show in Fig. 5, consider two agents, i and j. Agent i is centered on (x i , y i , z i ) and j on (x j , y j , z j ), with d ij being the distance between them. If both have different characteristic r c , the CZ is given by CZ : d ij ≤ r i c + r j c . If these agents have equal radii, For CTZ, the upper limiting distance is d max = r i c + r j c + r ct − r i c = r j c + r ct . The lower limiting distance is given by the CZ, as shown in Fig. 5. For equal and different r c radii, respectively, CTZ : r i c + r j c < d ij ≤ r j c + r ct (i and j different), and CTZ : 2r c < d ij ≤ r c + r ct (i and j are identical).
For MZ, the calculation is analogous. With CTZ as the lower limiting region, MZ : r j c + r ct < d ij ≤ r j c + r m (different), and MZ : r c + r ct < d ij ≤ r c + r m (equal).
Finally, FZ is everything that is not understood by the other cases. That is, FZ : d ij > r j c + r m (different), and FZ : d ij > r c + r m (equal).
After defining the critical zones and limiting radii, it is necessary to determine how the collision avoidance parameter α varies with distance d ij . This adjustment can occur deterministically or via computational intelligence, such as Fuzzy Logic.

E. EXPONENTIAL ADJUSTMENT OF THE COLLISION AVOIDANCE PARAMETER
For any evaluation zone (CZ, CTZ, and MZ) one particular equation or value is defined, considering the limits of centralisation and decentralisation presented in Eq. (27).

1) ADJUSTMENT FOR CZ
Considering the form α(d ij ) = a · b −d ij , the parameters a and b are determined according to the boundary regions of each critical zones. If an agent j is in the FZ of i, α j must be null. Then, we have

2) ADJUSTMENT FOR MZ
For the MZ, α must act in the convergent range. It increases as d ij decreases until the threshold of the critical zone, as shown in Fig. 6.

The final result for MZ is given by
(32)

3) ADJUSTMENT FOR CTZ
As there is no upper limit for CTZ, the parameter p > 1 is adopted, indicating how often the upper limit is greater than the lower one. This bound is associated with the threshold d ij of transition to the contact region, as shown in Fig. 7. Considering the limits shown in Fig. 7, with α ct max = It results in the solution b = p and the final result for the CTZ region is given by The final equation for α is a composite of the expressions for each zone, resulting in the formulation: Here, d ij is measured and classified within the critical zones, according to the flowchart in Fig. 8. The distance d ij is given by the Euclidean norm relative to the agent states is Once the collision avoidance method is defined, the protocol's final equation (Eq. (37)) presents the effects of centralization, leadership, formation, and corresponding speed. Then, the collision avoidance effect is applied to the centralization part of the protocol, where the most critical activities related to collisions are found. As seen in Eq. (30), it includes the leadership. According to Eq. (6), results of a protocol with collision avoidance are given by

V. RESULTS AND DISCUSSIONS
In this section, the results are presented with the virtual agents used as target to real agents. Therefore, in front of a simulation platform, tests are carried out with and without the developed method.
To evaluate the collision avoidance system, simulations are performed with agents. The objective is to verify the performance and effect of increasing the number of agents on the presented method. The results are obtained via simulation performed in Python 3.10, according to the decision process shown in the flowchart in Fig. 8, checking the distance between agents (Eq. (36)), and adjusting the collision avoidance matrix = diag{α j }, ∀j ∈ [1, N ]. α j is given by Eq. (35), where N is the number of agents.
The tests are performed according to the diagram shown in Fig. 2, with the real agent as specified in Section III and using a protocol given in Eq. (37). The results are presented for cases with and without collision avoidance system. It is worth mentioning that the singular value j of each agent are determined according to the graph structure adopted.
In an iteration interval T , the existence of a collision is checked and counted whenever d i,j ≤ r c . Whenever a collision condition is identified the coordinates are stored to be displayed on a graph as a function of time. For better understanding, consider three agents. The collision verification is performed through the possible combinations of agents, excluding repetitions. If agent 1 collides with agent 2, it is equivalent to agent 2 colliding with 1. At this moment, a counter increments the number of collisions.
As the objective is to verify the existence of collisions and time of occurance, the agents involved were not graphically differentiated, only coordinates as a function of time are displayed. For a better understanding of the agent's behavior, the error graphs on the respective Cartesian axes are presented. They complement the analysis with the 3D images of the simulation. The lines indicate the trajectory. In addition, the protocol with only centralization is the most critical case possible. Thus it is considered the ideal formulation to evaluate the proposed method.
The error is presented as a function of the difference between the leader's and agent's states, taking into account the centralization when x o (n) − x(n) → 0, for x coordinate. Similar process is followed for y and z coordinates.
The DC motor used for simulation is model 16G88 211E1 Brush DC whose data can be found on the manufacturer's website Portescap [22].
The PID internal controllers design method is given by [21]. The controllers tuning is performed by poles allocation, the proportional (k p ), integral (k i ), and derivative (k d ) gains are • Controller for DC motors: k p = 0.45 e k i = 4.5; • Controller for Euler angles: k p = 0.4, k d = − 0.1 e k i = 0.1; • Control in z: k p = 0.4, k d = −0.9 e k i = 0; • Control in x and y: k p = 0.31, k d = 0.5 e k i = 0.049.

A. THREE AGENTS CASE WITHOUT COLLISION AVOIDANCE
In the first case, 3 drones are considered, with their predefined internal controllers (Section III)), The trajectory of 3 virtual agents controlled by multi-agent protocols is taken as reference, according to the method presented in Section II-B and given by Eq. (4). Results without collision avoidance are presented by way of comparison, using the protocol given by Eq. (7). The connections (graph) and the corresponding adjacency matrix of agents are in Fig. 9. The coupling factor used is c = 0.5 and indicates the strength of centralization. The adopted coupling factor referent to the leader is g o = 0.8, as show in Eq. (7). The simulation use a sampling time of T = 5 ms, as well as verifying and accounting collisions.  Fig. 11, which marks the coordinates where the impacts occurred. In this figure, two curves stand out, indicating the occurrence of two simultaneous collisions at T = 14 s. Figure 12 shows the image taken from the simulation in which the overlapping drones stand out when they reach the final position, that is, in a collision, a consequence of the centralization protocol.
Although the protocol performed its function (Fig. 10), this cooperative law provides an undesired situation by causing a financial loss. Given this emphasis, the next step is to present identical simulation context including the developed system.
Observing the Figs. 10 and 12, the drone D0 is the first to hit the leader, followed later by D1 (the second fastest) and D2 which executes a longer path. Compared to Fig. 11, it indicates that the first collision occurred between D0 and D1 (at approximately 11 s) and continued colliding until the end test. D2 collided after 14 s until all drones were in steady state collision.

B. THREE AGENTS CASE WITH COLLISION AVOIDANCE
For the case of three drones with collision avoidance system, the parameters considered are r c = 0.3 m, r ct = 0.9 m,   where zero collisions were counted. As the collision figure, such as the Fig. 11, did not show any results, it is omitted.
Analyzing the Figs. 13 and 14 shows how the collision avoidance parameters act as the agents try to centralize (approximately 6 s of simulation). At this point, an approximation between agents (entry in the r m region) causes each α i to balance the system between the forces of centralization and decentralization. As a consequence, the agents got very close in y and x axes, while keeping a relative distance in z axes, as observed in both Figs 13 and 15. As noted, the parameter α of D0 does not work, as its corresponding singular value in L is null. Therefore, decentralization is not applicable. For other agents, D1 and D2 enter each other's moderate risk zones between 6 and 7 s, causing their corresponding α to act. As the approximation continued, these parameters increased until equilibrium and no collisions between agents.  Result of the guide system: collision avoidance, centralization, and leadership for three agents -Factor α j for each agent.

C. FIVE AGENTS CASE WITHOUT COLLISION AVOIDANCE
As mentioned initially, to assess the impact of increasing the number of agents, the number of elements is increased to five VOLUME 11, 2023 as shown in Fig. 16. It shows the graph and corresponding adjacency matrix. It is worth mentioning that the parameters c and g o , the reference and protocols, are equal to that in the case with three agents. In this case, the critical radii values do change to better adapt to the problem. As there was an increase in the number of agents, the new values are r ct = 1.5 m and r m = 2.5 m, with r c at 0.3 m. These values have been changed in relation to the previous case. This was to improve performance given the increase in the agents number. In the simulation of approximately 23 s, the system reached the objective of centralizing the agents in the reference point (5 m, 5 m, 5 m), resulting in a final average error of 0 m and accounting for 2586 collision iterations, on average, that is, given an observation period of 5ms, this amount of direct contact between agents was accounted.
As observed in Figs. 17-19, if there is no collision avoidance, all agents are superimposed at the centering point, entering a collision situation as show in Fig. 19. The first collisions are detected at approximately 8s, as shown in Fig. 18. As in the 3-agent case, collisions occur when the system tends to a steady state, as shown in Fig. 17.

D. FIVE AGENTS CASE WITH COLLISION AVOIDANCE
Including the collision avoidance effect, the results show that this new system did not interfere with agents' behavior during their displacements, acting only when necessary, that is, when they entered the respective risk zones. As shown in . The performance of the collision avoidance system can be observed, maintaining the spacing between the agents, mainly on the z axis. The α i stabilization   occurs around 15s, when the balance between centralization/decentralization is reached for all agents, resulting in zero collisions.
In addition to these investigations, a test for six agents was carried out, with a noticeable reduction in collisions. However, they could not be completely avoided. The speed of advance of the agents impacts the system's ability to avoid collisions, especially when the density of agents increases, as in the case above. Therefore, considering the worst case, that is, without the formation and corresponding speed that   help to avoid the occurrence of unwanted results, the system showed an excellent performance of the proposed collision avoidance system, which acts through the collision avoidance matrix, composed of the factors α i .
The advantage of the presented method is the collision avoidance matrix construction, allowing to change the dynamic characteristics of the collision avoidance effect through the α j reaction curves that is given by Eq. (35). In addition, it allows an easy application of other methodologies, such as machine learning that can act on the α j value, depending on local factors, dimensions and number of agents. As an example, there are the optimization techniques presented in [12].
The scalability of the proposed method depends on factors such as the number of agents, the complexity of the collision avoidance matrix and the adjustment of the parameters of this matrix. In experimental terms, it depends on the hardware capability that considers real-time constraints and processing effort. Furthermore, a purely centralizing protocol was evaluated, considered the worst case in question. In this case, increasing the number of agents increases the dimension of the collision avoidance matrix and parameter, decreasing the efficiency of the method. Increasing the number of agents generates a natural increase in the complexity of the collision avoidance matrix and the complexity of adjusting the α j parameter. For this, intelligent techniques as in [12] can help improve performance as N increases.

VI. CONCLUSION
Given the collision problem in multi-agent systems, this article presented the development of a collision avoidance system based on the divergence effect of cooperation protocols. Critical zones were determined, and collision avoidance matrix was defined, creating a proportionality relationship concerning the Laplacian Matrix. It resulted in an individual α parameter that varies exponentially as an agent invades the critical regions of another agent.
To evaluate the collision avoidance system, a multi-agent guide system was considered for a set of quadcopters. The critical radii were adjusted. The protocol was applied considering the worst case, that is, involving only the centralization of the flock around the leader. The number of agents and consequently the graph that represents the interactions between them were varied. Collisions have been avoided, so far, for up to 5 agents and with a significant reduction to 6 and without change in the dynamic behavior of the system. Therefore, the proposed method presents satisfactory results, allowing several other ways of controlling the collision avoidance parameter, such as intelligent methods based on machine learning.