In recent years, with the rapid development of small unmanned aerial vehicles, researchers are concerned with quadrotor because it has simple structure and is easy to control. Quadrotor can not only carry out vertical takeoff, landing and autonomous hover function, but also the task efficiently under the uncertain and dangerous environment. However, faced with diverse combat missions, single UAV is more and more difficult to meet the needs of the multi-UAV collaborative control concept come into being. UAV formation control is an important basic research track in the field of multi-UAV collaborative control. Reasonable formation control method can make the UAV formation quickly get the air superiority and finish the combat task more efficiently. On the future battlefield, UAV will play an irreplaceable role.
The research of methods of formation control mainly contains leader-follower method, virtual structure, graph theory and behavior-based[1]. At present, the mainstream is the integration of the above methods. Early formation control mainly uses centralized control method that is characterized by high precision and easy to control, but depends on the calculation capabilities and global communication capabilities of the central control unit. With the increasing number of the formation members, the calculation of the central control unit increases exponentially, therefore this method lacks scalability and flexibility. Later, people propose the distributed control method, in which each UAV only communicates with the adjacent UAV. The surrounding UAV relative position relationship is acquired, compared to the expected formation by using of UAV’s own computing power. The actual position of the UAV is corrected to eliminate the formation error.
In recent years, many scholars aim at problems of the quadrotor formation, and they do a lot of research and design different types of controllers. In literature [2-6], the controller is designed with different degree of simplification to quadrotor model by feedback linearization and small perturbation linearization. However, the quadrotor is a typical underactuated system, a cascade nonholonomic system with complex constraint equations, which has a strong nonlinearity and thus needs a higher requirement for the control system. In literature [7], the quadrotor is also modeled by unit quaternion and the concept of manifold is introduced. The attitude control algorithm is designed under the differential geometry framework and gets a nice effect. Literature [8] combines the nonlinear part of the system identified by neural network online and leader-follower method to realize the formation control of the quadrotor. In literature [9], the kinetics and kinematics models of the quadrotor are described by quaternion and the intermediate control is introduced. The formation is stabilized by setting the appropriate intermediate control for each UAV. In literature [10], the error of the actual position and the expected position is introduced, and the tracking error model is established. Literature [11] also divides quadrotor model into two independent subsystems of position and attitude, and each of them uses backstepping method to design the time-varying feedback in order to make the system stable. The backstepping method is a design method of forward and backward recursion. It makes the system error progressive and stable, and it can also reduce the difficulty of design by designing the Lyapunov function step by step. Based on the above research, the quadrotor mathematical model is established by using the unit quaternion method. Then the geometric center position of the formation is calculated by the consistency algorithm and is used as the expected trajectory. Finally the whole formation is stable by designing Backstepping Control for each quadrotor.
Quaternion is a number such as
Unit quaternary can be seen as a point on the unit sphere
Given a unit quaternion, the relationship between the corresponding attitude matrix and the unit quaternion is:
Where
The error between the actual attitude matrix
Let
The quaternion does not satisfy the exchange law, which is
The quadrotor UAVs are usually divided into Type X and Type +, with four inputs and six outputs, which are typical underactuated systems. The quadrotor carries out the movements, pitch, roll and yaw of the aircraft by controlling the speed of four independent motors and propellers. Quaternion is a simple and effective mathematical tool that describes the rotation and movement of rigid bodies in three-dimensional space, which can effectively avoid the appearance of singularity and have the characteristics of high efficiency etc. The quadrotor UAV is regarded as the rigid body structure. Assuming that the center of gravity is located at the origin of the coordinate system of the body. The motor has no installation error angle, and the motor lift surface is located on the same plane as the aircraft center of mass. Using the quaternion to establish the quadrotor of the dynamics model and kinematics model, the quadrotor is decomposed into the position subsystem Σ1 and the attitude subsystem Σ2, and getting the following model:
In the formula,
Considering the UAV formation has
Regardless of the difference in the quadrotor in the formation, the quadrotor is isomorphic and conforms to the model (6):
Define the position error of the
In the formula,
The work which is done in this paper is to design the virtual control volume
In the formula,
According to the cascade system analysis method, the quadrotor error system model is decomposed into position error subsystem and attitude error subsystem.
The following reference [] uses the BackStepping method to design the position subsystem controller and the attitude subsystem controller separately. First step is to define the error between the system state and the virtual feedback:
In the formula,
Step 1: Find the derivative of x
Define the first Lyapunov function:
Obviously, if
take
In the formula,
The reference signal is negotiated by the formation members through a coherence algorithm. The reference signal needs to meet certain constraints. Assuming that the reference signals
The desired position
It translates the formation problem into a trajectory tracking problem for a given reference signal. In the formula,
The formation of the ith UAV can be guaranteed through the inversion control law.
Regardless of the differences among the quadrotor systems, the system quality is assumed
Setting the formation size
The pilot does uniform circular motion in a fixed height, the movement trajectory is:
The relative positional deviation of the formation is described by:
The simulation time is 9.5s and the simulation step is 0.001s
According to the above parameters, the system model is built with Simulink module in MATLAB, and the control algorithm is verified. Figure 2 is the three-dimensional map of the formation simulation. In the figure, the black curve is the trajectory of the pilot, and the triangle “△” indicates the final position of each quadrotor UAV. At the beginning of the simulation, the pilot does circular motion in the air, followers locates on the different positions on the ground, taking off after receiving the state information of the pilot, the first team members calculate the formation center position through the negotiation of calculation, and as a reference tracking signal, Algorithm (18) eventually converges the five quadrilateral to the intended formation. A variety of formation can be easily designed by setting a different relative position error.
Fig. 3-Fig. 5 shows the change of the position and velocity of the fleet members in the formation and maneuvering process. It can be seen from the figure that the followers quickly move closer to the pilot after takeoff. At t = 1s, the distance from the pilot tends to be the designated position and remains at the same speed as the pilot, and is well tracked by the pilot. The validity of the tactics proposed in this paper is verified.
Fig. 6 shows the error between the formation center trajectory and the pilot’s trajectory calculated by the formation member’s negotiation. It can be seen that the formation center finally converges to the position of the pilot.
Aiming at the control problem of quadrotor formation, the nonlinear dynamic model and kinematic model are described by unit quaternion. The formation control is achieved by tracking the geometric center of the formation. At the beginning of the formation, team members calculate the formation of the geometric center as a reference signal through the wireless network exchange position, speed and other state information. Each UAV is designed with time-varying feedback control law through the Back stepping method which translates the formation problem into a tracking problem for a given reference signal. The simulation results of matlab show that the method is fast and accurate and the convergence speed of the formation system is improved. The simulation results show that the proposed method is fast and accurate. At present, the method has not yet considered the formation problem and the formation problem with the maximum velocity constraint. The subsequent study will consider the problem of formation, maintenance and reconstruction of the formation under the constraint of uniformity and maximum speed.