This paper addresses the formation control problem without collisions for multiagent systems. A general solution is proposed for the case of any number of agents moving on a plane subject to communication graph composed of cyclic paths. The control law is designed attending separately the convergence to the desired formation and the noncollision problems. First, a normalized version of the directed cyclic pursuit algorithm is proposed. After this, the algorithm is generalized to a more general class of topologies, including all the balanced formation graphs. Once the finite-time convergence problem is solved we focus on the noncollision complementary requirement adding a repulsive vector field to the previous control law. The repulsive vector fields display an unstable focus structure suitably scaled and centered at the position of the rest of agents in a certain radius. The proposed control law ensures that the agents reach the desired geometric pattern in finite time and that they stay at a distance greater than or equal to some prescribed lower bound for all times. Moreover, the closed-loop system does not exhibit undesired equilibria. Numerical simulations and real-time experiments illustrate the good performance of the proposed solution.

During the last years, formation control in multiagent systems has received much attention due to the wide range of applications in which they can be used as exploration, rescue tasks, toxic residues cleaning, and so forth, [

Initially, the proposed solutions to the noncollision problem were designed as the sum of attractive and repulsive vector fields, in most cases obtained as the negative gradient of potential functions. Attractive potential functions are centred, for each agent, at its desired position, while the repulsive potential functions are centred at the positions of the rest of agents or even at obstacles’ positions, [

Related works can be found in [

In a recent work, a new strategy for designing the repulsive vector fields has been proposed [

In this paper we study the noncollision problem in formation control using discontinuous vector fields for an arbitrary number of agents. In one hand we undertake the design of attractive vector fields based on the well known cyclic pursuit algorithm but, unlike the results reported in the literature [

This paper is organized as follows. We start in Section

As we mentioned before, in this section we state some useful definitions [

A formation graph

For a directed communication graph,

There exists a path between the vertices

If there is a path between any two vertices of the formation graph, then the graph is called connected. If a formation graph is connected and the vector

The Laplacian matrix associated with a formation graph

Consider the dynamical system

Since

Consider a group of

Agent

The initial conditions of all agents satisfy

The goal is to design control laws

the agents reach a desired formation; that is,

there are no collisions among agents; moreover, at all times robots remain at some distance greater than or equal to a predefined distance

The control design is presented in two parts, one of each attending a different objective according to the control goal. We start proposing a control law based on normalized attractive vector fields to ensure finite time convergence of the agents to the desired formation.

The control law to reach the desired formation pattern is designed based on attractive vector fields proportional to the position error; that is,

Cyclic pursuit formation graphs.

Consider system (

Consider the error coordinates

Then, the stability of the system is simplified to the analysis of the matrix

In the last proof, the closed formation condition was useful in order to reduce the original system. In the same way, we can state a corollary to show the stability for the case of an undirected cyclic pursuit formation graph.

Consider the closed-loop system (

The closed-loop system in terms of the Laplacian matrix and using initially (

To cover a more general class of communication graphs, let us consider as an example the graphic shown in Figure

Four agents formation graphs with two cycles.

Consider a group of

For this case, knowing that the stability of the closed-loop system can be analysed using the system without normalization

Corollary

Once we have shown the finite time convergence for agents, we attend the noncollision problem by designing a proper complementary control law based on repulsive vector fields focusing on the distance among agents. For this purpose it is useful to define the relative distance variables

Relative distance between

The first step to design the repulsive vector fields is done regarding the most simple case of a scenario when only two robots are in risk of collision. Then, the situation is geometrically generalized to the case of a robot rounded by a group of possible colliding robots. The vector fields are proposed in such a way that for robot

Consider the system (

Assume first that there are

Even though the control laws proposed in this paper are not intended to produce a sliding mode motion, in case that the attractive vector fields outside the surface point to the surface

In order to generalize the problem under discussion, it is insightful to consider now the situation of having three different robots

Agent

Consider the system (

In this case, due to the possible collision among the different agents, the discontinuous surface consists of two different components, each one being related with a pair of agents; that is,

At this point, we would add more agents to analyse a more complex collision problem. Geometrically, the most general case occurs when a robot

Agent

Consider the system (

For simplicity, and without loss of generality, let us denote by

If we take the proposed condition

A numerical simulation was carried out in order to illustrate the performance of the proposed algorithm. The simulated system consists of nine mobile robots

Desired formation, where the agents have been located according to the desired geometric pattern.

Results are shown in Figures

Distances among the nine agents.

Agents distribution on the plane at different times.

For real-time experiments, we used unicycle-type robots as agents. For this reason, the control strategies previously developed are modified for the case of this type of mobile robots. The kinematic model for each robot

Kinematic model of the unicycle-type mobile robot.

The kinematics of point

When a robot

The real-time experiments were carried out over an experimental setup composed of the following elements.

Four differential-drive mobile robots, model AmigoBot manufactured by MobileRobots Inc. (Figure

Note that in (

A positioning system: The position and orientation of each robot is measured through a vision system composed of

One Intel core i3-based computer: The software Motive calculates the position of the centroid of the geometric figure formed by the markers and its orientation. The control law is calculated in Visual C++ using Aria libraries which are also used to communicate with the robots. The protocol VRPN is used to share information between Motive and Visual C++. Finally, the velocities of each wheel are sent through Wi-Fi to the robots.

AmigoBot robots.

Cameras Flex 13.

The experiment consists of four agents

Desired formation.

Trajectory in the plane of the four agents.

Figure

Position errors of the four agents.

Position error of

Position error of

Position error of

Position error of

In Figure

Distances among the four agents.

Distances in the experiment

Distances in the simulation

Finally, Figure

Control inputs.

Velocities of

Velocities of

Velocities of

Velocities of

A solution to the general noncollision problem in formation control has been proposed. This solution is based on the combination of attractive and repulsive vector fields. The attractive forces are designed proportionally to the error of each robot. The repulsive vector fields are designed as unstable focus centred at the position of the other robots. Besides, the attractive field was normalized to ensure the agents move at constant velocity when no danger of collision exists. As a by-product, finite time convergence is ensured. We analysed geometrically all the possible cases of multiple collisions and we proved the proposed control law is suitable in all situations, ensuring that the agents reach the desired geometric pattern in finite time and that they stay at a distance greater than a predefined bound. As a further research, the analysis can be extended not only to a general class of formation graph possessing a spanning tree but also to nonholonomic robots. Moreover, the differences that can be seen between the real-time experiment and the numerical simulation, due to unmodeled dynamics, motivate us to extend our work considering second-order agents.

The authors declare that there is no conflict of interests regarding the publication of this paper.

J. F. Flores-Resendiz, J. González-Sierra, and J. Santiaguillo-Salinas acknowledge partial financial support from CONACyT, Mexico, in the form of Grants no. 202955, 219257, and 243226, respectively.