Inspired from the collective behavior of biological entities for the group motion coordination, this paper analyzes the formation control of mobile robots in discrete time where each robot can sense only the position of certain team members and the group behavior is achieved through the local interactions of robots. The main contribution is an original formal proof about the global convergence to the formation pattern represented by an arbitrary Formation Graph using attractive potential functions. The analysis is addressed for the case of omnidirectional robots with numerical simulations.

In the last years, the control community has a special interest in the study of the coordination of multiple mobile robots [

According to [

This paper is related to a motion coordination problem, named Formation control, where the robots converge to some static formation patterns [

The formation control strategies based in biological systems can be classified in two schemes. The first proposes reactive schemes that includes the majority of the

This FG-based scheme can provide specific postures of robots in the formation and the facility to analyze eventual changes of formation and leader roles, intermittent and delayed communication, and other dynamic behaviors [

On the other hand, the discrete-time formation control has been primarily studied by some consensus algorithms [

The rest of the paper is organized as follows. Section

Denote by

Let

According to [

A Formation Graph

If

Example of Formation Graph.

The matrix of a FG that captures the topological properties of the graph is called the

The Laplacian matrix of a Formation Graph

For a connected FG, the Laplacian matrix has a single zero eigenvalue and the corresponding eigenvector is the vector

Finally, a FG is said to be

For completeness of the paper, we introduce the next matrix operation [

Let

The Kronecker product allows a more compact notation for systems’ equations.

For system (

The functions

Consider the system (

The proof requires some preliminary lemmas.

Let

If the matrix

The closed-loop system, formed by (

The matrix

Note that

Substituting

Using Lemma

Note that the convergence of the formation errors is translated into the stability of the equilibrium point

Thus, the solution of

Figure

Formation control of a mixed FG.

Formation Graph

Trajectories of the robots

Error coordinates

To compare the previous simulation, Figure

Formation control of an undirected cyclic pursuit FG.

Formation Graph

Trajectories of the robots

Error coordinates

Note that the performance of the control actions improves as the sampling period

This paper deals the case of discrete-time formation control for the case of omnidirectional robots. The main contribution is a formal proof about the global convergence of the robots to the desired formation pattern, showing the stability of multiple equilibrium points when the formation errors converge to zero. The approach is based on the coordination of biological entities were the motion coordination is defined by the local communication between robots and the available information is the position of some robots measures by local sensors. In further researches, the collision avoidance, the flocking behavior with group path-following, and the extension for the case of nonlinear models, like unicycle-type robots, will be addressed.

The authors acknowledge financial support from UIA, Mexico, and the closely academic collaboration with the Mechatronics Section from CINVESTAV-IPN, Mexico.