This paper investigates the flocking and the coordinative control problems of multiple mobile agents with the rules of avoiding collision. We propose a set of control laws using hysteresis in adding new links and applying new potential function to guarantee that the fragmentation of the network can be avoided, under which all agents approach a common velocity vector, and asymptotically converge to a fixed value of interagent distances and collisions between agents can be avoided throughout the motion. Furthermore, we extend the flocking algorithm to solve the flocking situation of the group with a virtual leader agent. The laws can make all agents asymptotically approach the virtual leader and collisions can be avoided between agents in the motion evolution. Finally, some numerical simulations are showed to illustrate the theoretical results.

In recent years, the problem of coordinated motion of multiple mobile agents, especially the flocking control [

In this work, we consider how to maintain network connectivity by constructing artificial potential function and investigating the flocking algorithm with avoiding collision rules. This algorithm is using hysteresis in adding new links [

The rest of this paper is organized as follows. We define the multiagent flocking problem in Section

In order to improve the understanding of the flocking control problem and facilitate the narrative for the next part, some basic knowledge including the undirected graph

Assume that each agent has the same sensing radius

the initial edges are satisfied with

if the inequality

if

Here,

The above imply that a new edge will be added when the distance between the two agents is less than induction radius

Define the Laplacian of agent

For a given undirected graph

In 1986, Reynolds introduced three heuristic rules that led to creating the first computer animation of flocking [

Flock centering: attempt to stay close to nearby agents.

Velocity matching: attempt to match velocity with nearby agents.

Collision avoidance: avoid collisions with nearby agents.

In our background, the Reynolds model was proposed based on the position and velocity of agents; consider a group of mobile agents moving in a

The control purpose of this work is to make all agents approach a uniform velocity and asymptotically converge to a fixed value of interagent distances, and collisions can be avoided among agents in the motion evolution. Namely, for all

And consider the situation of the group with a virtual leader agent; the control input

We have assumed that multiple mobile agents are moving in

the potential function

if

We can know that no distance between agents will tend to 0 or

For the physical properties of the dynamic agent, we define the total energy of agent system as follows:

For the convenience of description, denote the velocity and position of the center of mass (COM) of all agents as follows [

Considering a group of

Assume that the initial energy

Part one of Theorem

One has

According to the definition of the artificial potential function, it is easy to know

Using a similar analysis method, we can obtain the time derivative of

Based on this fact, the inequation is reasonable in

According to the above analysis, we know that

In what follows, we will give the proof procedure of parts (ii) and (iii).

Let us consider that there are

From (

The inequality

From (

For given conditions

From the control input (

In a stable state, clearly

At the end of Section

The situation of multiagent systems (

We define the total energy of agent system as follows:

It shows that

Similar to the analysis of flocking behavior without the virtual leader, considering a group of

Assume that the initial energy

Part one of Theorem

Assume that

According to the definition of

Referencing the proof of part (i) of Theorem

Then, we give the proof procedure of part (ii). We can get that the positive invariant set is

The following equation is given combined with LaSalle’s invariance principle [

We now prove part (iii) of Theorem

At the end of this section, we will prove part (iv). We can get the following equation from the control protocol (

We can get the solution of the equation

which implies that the initial velocity of the COM will exponentially converge to the desired velocity

In this section, several numerical examples of the proposed control laws are presented to illustrate the rationality of the theoretical analysis.

The simulation is performed with 10 agents moving in a two-dimensional Euclidean space under the control protocol (

Initial states.

Final states.

Paths and final states.

Velocity convergence without virtual leader (

Velocity convergence without virtual leader (

The process of position convergence for

The process of position convergence for

This simulation is performed with a virtual leader and 10 agents moving in a two-dimensional Euclidean space under the control protocol (

Initial states with a virtual leader.

Final states with a virtual leader.

Paths and final states.

Velocity convergence with a virtual leader (

Velocity convergence with a virtual leader (

The process of position convergence (

The process of position convergence (

In this paper, we have investigated the flocking and the coordinative control problems of mobile autonomous agents with preserved network connectivity and proposed the flocking algorithm with avoiding collision rules. This algorithm has proposed using hysteresis in adding new links and applying new potential function method to ensure that the network always stays connected and collisions between agents can be avoided. The extended application of the flocking algorithm with a virtual leader has been investigated. The simulation has proved that the laws can make all agents approach a common velocity vector and asymptotically converge to the fixed value of interagent distances and collisions between any agents can be avoided throughout the motion. The laws also satisfy the situation that there exists a virtual leader in the group of all agents, and it is proved that the value of desired velocity of the group is the same as that of the virtual leader. Future work will pay attention to the situation of how to make the network topology be connected as the initial network is not satisfied.

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

This work was supported in part by the National Natural Science Foundation of China, Grant no. 61473129.