UAV Swarm Formation Control Based on Disturbance Observer and Backstepping Controller

The formation control problem was researched for a fixed-wing unmanned aerial vehicle (UAV) swarm. The kinematic model was built for a UAV considering the lumped disturbances. The disturbance observers were designed to estimate the lumped disturbances in finite time. The desired UAV swarm formation was represented by a virtual structure. The backstepping controller was designed for every UAV to complete the formation maintaining a task. The sigmoid tracking differentiator (STD) was added to the backstepping controller, in order to settle the problem of “explosion of complexity”. The numerical simulation was executed to show the formation maintaining procedure of the UAV swarm. The simulation results demonstrate that the disturbance observer has good disturbance rejection capability and the backstepping controller based on the STD has good formation maintaining performance.


Introduction
With the fast development of navigation technology, control technology, sensor, image processing, microprocessor, etc., the application of unmanned aerial vehicles (UAV) extends widely in both the military field and the civil field.A single UAV has limited energy, so it cannot complete a complex extensive task.Compared with a single UAV, the UAV swarm has many advantages, such as high efficiency, low energy-consuming, fault-tolerant, extensibility, etc. [1].It possesses extensive application prospects that the UAV swarm completes cooperatively a complex extensive task.
The swarm formation control problem is very important in the UAV swarm system, and it is a research focus that attracts many scholars in recent decades.The primary swarm control methods include leader-follower methods, consensus-based methods, behavior-based methods, virtual structure methods, etc. [2].The research achievements on the former three methods are plentiful, whereas those on the virtual structure (VS) method are not enough.The VS method is superior to other methods in swarm formation control because the VS expresses directly the swarm formation.Every node in the VS represents the desired position of a UAV in the swarm.Shao et al. [3] designed a nonlinear feedback controller for UAV formation based on the VS, which keeps the stable flight formation.Askari et al. [4] applied the VS to express the UAV formation and designed the controller based on the physics and inverse dynamics.Kownacki [5] introduced the flock and separation behaviors into the VS, and in the flocking process, the UAVs could avoid collisions among the swarm.Zhou et al. [6] applied the VS method to realize formation-keeping, transformation, and obstacle avoidance.The swarm controller was designed by combining the artificial potential field with the VS, and the UAVs tracked the formation reference points to form a desired formation [7].A nonlinear robust control scheme was designed for the formation control of multiple UAVs based on the VS, and it improved the control precision and stability [8].The virtual formation guidance points were constructed to finish the cooperative formation control work [9].
In the former research on the VS method, the effects of model uncertainties and external disturbances were rarely researched.In this paper, we design the disturbance observers to estimate them.In addition, we develop a backstepping controller based on the sigmoid tracking differentiator (STD) to design the UAV swarm formation controller.The rest of this paper is arranged in the following.Section 2 builds the model of the UAV swarm.Section 3 designs the UAV swarm formation controller by combining the disturbance observer and the backstepping controller based on the STD.In Section 4, the numerical simulation is executed to illustrate the formation maintaining process.The conclusions are given in Section 5.

Model of UAV Swarm
The UAV swarm consists of several homogeneous fixed-wing UAVs.For the sake of simplicity, these UAVs are numbered by 1, 2, , N  .The communication topology of the UAV swarm is described by an undirected graph.In order to reduce the UAV's communication burden, every UAV communicates only with some neighbor UAVs, which situate in its communication range.The kinematic model of the i th ( 1, 2, , i N   ) UAV is expressed as follows, considering one-order autopilots [10].
cos cos cos sin sin ( ) ( ) where T ( , , ) According to Dong et al.'s work [1] and combining the model uncertainties and external disturbances, we transform the kinematic model (1) into the following two-order model, where i v is the velocity vector; i u is the acceleration vector, which is the controlled input variable of the i th UAV; i w and i υ represent the lumped disturbances, which include the model uncertainties and external disturbances and so forth.
As to the lumped disturbances, the following assumptions are taken.Assumption 1: i w and i υ are uniformly bounded, but their upper bounds are unknown.Assumption 2: i w and i υ vary slowly, and their derivatives are bounded too.

UAV Swarm Formation Controller Design
The UAV swarm formation controller design is actually to design a controller for every UAV.Because all UAVs are the same, their controllers are also the same.The controller of the i th ( 1, 2, , i N   ) UAV is designed by combining the disturbance observer with the backstepping controller.In order to settle the problem of "explosion of complexity" in the backstepping design process, we adopt the sigmoid tracking differentiator (STD) [11] to give the derivative of the virtual control variable.

Disturbance Observer Design
The lumped disturbances i w and i υ are estimated by the adaptive finite-time disturbance observers, which are developed by Dong et al. [1].For the i th UAV, the disturbance observer is designed as, The gains  are updated by the following adaptive laws,   where

Backstepping Controller Based on STD
The VS represents the desired UAV swarm formation, and the i th node in the VS represents the desired position of the i th UAV.The position vector of the i th node is defined as vs i p .The displacement vector between the i th node and the j th node is defined as p .The design procedures of the backstepping controller are presented in the following section.
Step 1: The formation track error of the i th UAV is defined as,  p e p p p p p ( 7 ) where 1i k and 2i k are weights, 1 2 communicates with the j th UAV, then 1 ij a  ; otherwise 0 ij a  .The following formula can be gotten by taking the derivative of Formula ( 7) and adding the disturbance observer, where vs p  is the desired relative velocity between the i th node and the j th one.We take i v as the virtual control variable and design the control law as, where 1 0 i   is a control gain.i v is called the desired velocity.
Step 2: The tracking error between the actual velocity and the desired one is defined as, To settle the problem of "explosion of complexity" in the backstepping design process, we adopt the STD to produce the derivative of i v .The STD is designed as follows.
     [11], the tracking estimation 0 ς converges to i v , and the differential estimation 1 ς converges to the derivative of i v .
Using the disturbance observer and the STD, the derivative of Formula (13) yields, The backstepping controller of the i th UAV is designed as follows, where 2 0 i   is a control gain.It can be proved by the Lyapunov stability theory that the tracking errors i p e and i v e converge to zero asymptotically.In other words, the UAV swarm formation is maintained well by the designed backstepping controller.

Numerical Simulation
The numerical simulation program is built by MATLAB R2014a software to validate the effect of the disturbance observer and the backstepping controller based on the STD.The UAV swarm consists of ten homogeneous fixed-wing UAVs.The VS is set as the equilateral triangle structure in a plane at a height of 1000 m, which is shown in Figure 1.In Figure 1, ten nodes are ten solid dots with numbers from 1 to 10, and the first node lies in the head.The solid lines between two neighbour nodes express the communication lines, and vs v expresses the velocity vector.All ten nodes move with the same velocity to maintain a stable structure.
In the simulation, the mass of every fixed-wing UAV is 550 kg, its wingspan is 8 m, and its communication distance is 50 m.The time constants in Formula (1)  .The desired swarm flight trajectory is set as a sine curve at a height of 1000 m.The desired swarm flight velocity is 50 m/s.After repeated simulation adjustments, we choose the observer parameters as 1.1 1.8 0.2  Taking the 5 th UAV as an example, the varying processes of the distances between it and its neighbor UAVs are shown in Figure 3, where d 5j (j=2, 3, 4, 6, 8, 9) denotes the distance between the 5 th UAV and the j th UAV, and the amplified figure shows that the distances converge approximately their desired values 30 m at 11.3 s.In addition, the UAV swarm formation in the plane at 11.3 s is depicted in Figure 4. Figures 3 and 4 show that the UAV swarm forms a stable equilateral triangle at 11.3 s and keeps always the desired formation after 11.3 s.
The simulation results validate that the disturbance observer possesses good disturbance rejection capability and the backstepping controller has good formation maintaining performance.

Conclusions
In order to complete the task of the UAV swarm formation maintenance, we develop a backstepping controller based on the STD and the disturbance observer.The UAV swarm consists of many homogeneous fixed-wing UAVs, and its desired formation is represented by the virtual structure.The
a UAV is built considering the lumped disturbances.The disturbance observers are designed to estimate the lumped disturbances.Furthermore, the swarm formation controller is designed by the backstepping controller based on the STD.The numerical simulation is carried out to illustrate the formation maintaining process of the UAV swarm.The simulation results validate that the disturbance observer possesses good disturbance rejection capability and the backstepping controller based on the STD has good formation maintaining performance.In future work, we will study the obstacle avoidance problem in the UAV swarm flight process.

Figure 3 .
Figure 3. Distances between the 5 th UAV and its neighbor UAVs.

Figure 4 .
Figure 4. UAV swarm formation in the plane at 11.3 s.
According to the research results in Dong et al.'s work [1], ˆi