Formation control of UAVs based on artificial potential field

The problem of UAVs’ formation control in the process of motion is investigated in this paper. A formation control method based on artificial potential field of UAVs is proposed, established on the collision avoidance, aggregation and speed matching rules of UAVs. First establish the UAVs’ kinetic model in accordance to the motion rules, then design the formation control algorithm based on artificial potential field function, which is used to control the formation during the movement of UAVs. Finally, the results of simulation experiment show that the proposed formation control method in this paper is effective and has the advantages of easy realization, good real-time performance and excellent robustness.

path planning in the planning space, and achieved the swarms' evasion to the fixed threat. In [7], Liu Huan proposed a state estimation algorithm and a distributed leader-follower formation control method based on back-stepping, accomplished the simulation experiment of the formation flight of multiple UAVs. In 2014, Keio University, Kuriki invested the cooperative formation control problem of UAVs, a cooperative formation control strategy with anti-collision ability was proposed, which realized the cooperative formation flight and collision avoidance of the quadrotor UAVs [8] . In 2015, A distributed formation controller based on the behavior rules was proposed by Shin, the controller is designed by the adjacent UAVs' status information, which can improve the robustness of UAVs formation [9].
According to the problem of UAVs' formation control, a formation control method based on artificial potential field of UAVs is proposed, established on the collision avoidance, aggregation and speed matching rules of UAVs, and simulation experiment results show that the method is easy to be realized and suitable for the formation control of UAV swarms.

Kinetic Model of UAVs
The kinetic model of UAVs is established on rules, which mainly include aggregation/disaggregation rules, formation transformation rule and so on. These rules constitute the rule library of UAVs' motion. To avoid the distraction or collision of UAVs, three basic principles should be obeyed for each UAV, that is collision avoidance, speed matching and aggregation, which is described by the R-A model in Figure 1.
Establish three concentric circles with UAV i as the center of the circles, the outermost circle of Figure 1 is the perception range of UAV i, the inner region near to UAV i is called repulsive region, which will provide repulsive force to the UAV i, and UAV i should try to adjust the distance from other UAVs in this region to avoid collision, this region is expressed as The outermost grey region far from UAV i is called attractive region, other UAVs in this region will provide attractive force to the UAV i, and UAV i should try to pull closer the distance from other UAVs in this region to avoid dispersion, this region is expressed as The middle blue region between repulsive region and attractive region is called unanimous region, which mean that the distance between UAV i and other UAVs in this region is suitable, the repulsive force and attractive force between them is balanced, this region is expressed as From the R-A model, we know that the velocity of UAVs will be influenced by three kinds of virtual forces, assume that in the repulsive region, unanimous region and attractive region respectively, which can be expressed as follows: In (2), ij k represent the control coefficient of repulsive force between UAV i and UAV j, which is much smaller than the r d .

Formation Control Algorithm of UAVs
It can be inferred from (1) that, in order to control the UAVs' position to form the expected formation, we should first design the function configuration of i v .
Set a virtual leader in the swarms, then there would exist two kinds of forces acting on the UAV i, one is from the virtual leader, the other one is from the other UAVs in the UAV swarms, and the strength and direction of force is determined by the distance between UAV i and other UAVs or the virtual leader. According to the aggregation and collision avoidance rules of the UAV swarms, establish the motion equation of UAV i as follow: In (3)  UAVs (collision avoidance, speed matching and aggregation), which is expressed as follow: In (4) and (5), i a   ¡ represent the velocity parameter, which determines the modulus of the virtual force produced by the artificial potential field. And i    ¡ is called equilibrium distance, which mean that when the distance between UAV i and virtual leader is equal to i  , then the repulsive force and attractive force between the virtual leader and the UAV i are equal. Under the control of the artificial potential field, when the UAV swarms are in motion, two kinds of forces will have an effect on the swarms. One is the attractive/repulsive forces between UAV i and other UAVs, which mainly decide the formation style of swarms; the other one is the attractive/repulsive force from the virtual leader, which provide a constraint on the formation style and play a role in guiding the flight direction of UAVs, and the UAV swarms will form a specific formation according to the value of a , i  and ij  , which is shown in Figure 2

Simulation Results
To verify the feasibility and correctness of the algorithm, the UAVs' formation control algorithm is simulated and analysed. The objective of the experiment is to realize that, when the task command is reached, UAVs can form the default formation quickly under the control of artificial potential field, then following the virtual leader, move as a swarm in the direction of destination with stable formation, reach the target position with default formation finally.
In the simulation experiment, the UAV swarms consist of 6 UAVs and one virtual leader, the default formation of UAVs is shown in Figure 3, the red circle represent the virtual leader, six green blocks represent six UAVs, and they have formed an equilateral triangle formation. Firstly, the velocity parameter a in formula (4) and (5) is set as 5, the location of the selected virtual leader is set as (35, 35,30), as well as the destination. Running the simulation program, the distribution of 6 UAVs in three-dimensional space is shown in Figure 4. It can be seen from Figure 4 to Figure 7 that six UAVs are in a random position at t = 0, when t=1.6s, the UAVs have clustered into swarms and move toward the virtual leader, form the default formation at t=3.8s, finally, the six UAVs have form the default formation with the virtual leader. It can be seen from the simulation results that six UAVs started to gather when the simulation started and would quickly assembled from a dispersed state into the default formation. When moving toward the destination, the formation of UAVs can always be maintained. Finally, the UAVs could reach the target position with default formation, which is shown in Figure 7. The simulation results show that the objective of the simulation experiment have been achieved. The track of UAVs formation flight in three-dimensional space and x-y plane are shown in Figure 8 and Figure 9 respectively, the red circle represents the virtual leader, and six paths with different color represent the flight path of six UAVs. It can be seen from the Figure 8 and Figure 9 that the motion of UAVs is irregular at the beginning, when they cluster into a swarm, they would flight toward the virtual leader neatly, in a consistent direction. The track diagrams of UAVs illustrate that under the control of proposed formation control algorithm, UAVs would form a swarm quickly, and maintain the default formation stably during the motion, reached the destination with default formation finally. The simulation results show that the proposed formation control method is effective, and the objective of experiment has been realized.

Conclusion
According to the problem of UAVs' formation control in motion, a formation control method of UAVs based on artificial potential field is proposed, established on the collision avoidance, aggregation and speed matching rules. In the simulation experiment, UAV swarms could quickly assemble from a dispersed state into the default formation under the control of proposed method, and move toward the destination with a stable formation, reach the target position with default formation finally. The simulation results show that the proposed formation control method in this paper is effective and has the advantages of easy realization, good real-time performance and excellent control effect. Future work should focus on the motion control of UAVs in complicated environment with multiple obstacles, as well as formation transformation problems during the UAVs movement.