Adaptive super-twisting distributed formation control of multi-quadrotor under external disturbance

In this paper, the adaptive super-twisting distributed formation control of multi-quadrotor in the presence of external disturbances and uncertainties is studied. First, the quadrotor formation system is separated into a position subsystem and an attitude subsystem which are represented by unit-quaternions. And then a composite adaptive super-twisting control method is proposed for the position subsystem and attitude subsystem respectively. For the position subsystem, an adaptive multivariable super-twisting controller is designed such that the positions of formation converge to the desired formation configuration and generate the desired attitude. And the adaptive fast super-twisting controller is designed for the attitude subsystem to track the desired attitude in finite time. Based on Lyapunov-based stability analysis, finite time convergence stability of the whole closed-loop system is proved. Finally, a numerical simulation result is provided to illustrate the effectiveness of the proposed formation control scheme.


I. INTRODUCTION
In recent years, the quadrotor control problem has received much interest from many researchers. The main reason is the diversity of quadrotor applications, such as geographic survey, agricultural plant protection, military reconnaissance and so on [1]- [5]. However, due to its characteristics including that nonlinear, strong coupling multi-variable, underactuated and so on, it puts forward high requirements for its control ability. The quality of control performance greatly affects the safety and stability of quadrotor.
Compared with a single quadrotor, the quadrotor formation may provide a better performance for the difficult tasks [6]- [10]. Furthermore, system stability and reliability can also be improved through the exchange of information among multiple quadrotors. Meanwhile, formation control of quadrotor formation is more challenging. And the main issue of formation control is how to achieve and maintain the desired configuration while tracking a desired trajectory. Hence, it is practical significance to study the formation control problem for quadrotors [11].
For achieving the formation control, the typical control strategies can be categorized as leader-follower [12]- [14], behavior-based [15] and virtual [16], [17]. These strategies all lack their own drawbacks [18]. However, for the reason of simplicity and flexibility of leader-follower strategy, this strategy is the most widely employed and most efficient approach for quadrotor formation control. Meanwhile, to avoid the formation falling caused by the failure of the leader, leader-follower strategy is integrated with distributed formation control approach, which is more suitable for a large number of quadrotors to maintain the formation configures. And a great deal of literature has been presented for the formation control problem by this strategy.
For the leader-follower formation problem of quadrotors, Wesam Jasim [19] developed a suboptimal H-infinity controller. However, this paper is only expected to perform a formation flight in the x-y plane and use a PID control law to control the inner-loop, which may lead to a poor control per-formance. Hao Liu [20] proposed a robust formation control based on backstepping approach for a group of quadrotors. The convergence performance is not discussed in this paper. A consensus formation control algorithm is also proposed and utilized in [21], the inner-loop modeled by quaternion can be stable in finite time, but the outer-loop is not. Haibo Du [22] presented a finite-time formation control algorithm in the basis of backstepping approach and the finite time convergence characteristic is proved in outer-loop and innerloop respectively, but the finite time convergence of closed system is not discussed. In addition to the above methods, the optimal control method is also used in the formation control problem of quadrotor. In [23], a formation control algorithm was proposed based on optimal control method, which aims to achieve the desired formation behaviors of multiple quadrotors in asymptotic stability with not finite time stability. Although the above methods improve the control performance to a certain extent, the further improvement performance is urgently needed, especially the convergence rate.
Over the years, sliding mode control has has been applied into many nonlinear robust control systems because of its insensitivity to matched disturbances and finite time convergence [24]- [28]. However, many sliding mode controllers have the problem of chattering, and then the second order sliding mode control is developed. It can hide the switching terms within the derivatives of the controllers themselves. In addition, the super-twisting algorithm has been immensely popular since its inception [29]- [32]. Recently, there has been a lot of interest in adaptive versions of the super-twisting algorithm [33], [34]. The adaptive law can change the control gain. When a second order sliding surface is established, the adaptive law stops and the gain remains constant. One advantage of this adaptive method is that the boundary information of the disturbance can be unknown. However, for the characteristics of quadrotor formation system mentioned above, there are so many problems that need to be considered in the application of this algorithm, such as external interference, communication between quadrotors, the selection of adaptive parameters and so on. These problems increase the difficulty of applying this method to multi-quadrotors. And this paper considers a multivariable adaptive super-twisting sliding mode control method for the quadrotor formation system.
Motivated on above analyses, to improve control performance, this study focuses on the whole closed-loop stability control algorithm for quadrotor formation modeled with unknown bounded external disturbances and uncertainties. For the characteristics of inner loop and outer loop of quaternion modeled, the different adaptive super twisting control algorithms are designed to improve the convergence rate and accuracy respectively. The finite time stability analysis of whole closed-loop quadrotor formation is proved by a novel Lyapunov function. The main contributions of this paper are summarized as follows: 1) Based on the quaternion model, an super-twisting dis-tributed formation control is designed, which ensures the stability of quadrotor formation under the condition of unknown upper bound of external disturbance. And the stability of closed-loop finite time convergence is proved.
2) For the typical second-order dynamic model, the proposed control law has the advantages of fast convergence speed and high convergence accuracy, which satisfies the speed requirements of quadrotor in fast maneuverability.
The remainder of this paper is organized as follows. In Section II, notations and graph theory, some useful lemmas and problem formulation are given. In Section III, the control laws and adaptive laws are proposed, and the stability of the whole closed-loop system is proven via Lyapunovbased analysis. In Section IV, simulation results are given to demonstrate the effectiveness of the control laws proposed above. Finally, the conclusions are given in Section V.

A. GRAPH THEORY
For the formation with n quadrotors, the interaction among the n quadrotors is modelled by weight undirected graph G which is (V, E, F ). In undirected graph G, V = {v j , j = 1, ..., n} is the set of nodes, E ⊆ V × V is the set of edges. F is the weighted adjacency matrix of the graph G. For K, we have a ii = 0 and,a ji = 1 if there is an edge between agent j and agent i while a ji = 0 otherwise. The set of neighbors of a ji . The degree matrix of undirected graph G is D = diag{d 1 , d 2 , . . . d n } and the Laplacian matrix of graph G is K = D − F . The path from v j to v i is a series of different nodes, starting with v j and ending with v i , so that the continuous node are adjacent. The physical structure of the quadrotor is shown in Figure 1 [35].
Assumption 1: The communication topology for n agents is connected at the initial time.

B. USEFUL LEMMAS
Lemma 1. [36]: Consider the following nonlinear systeṁ exists an open neighborhood Θ of the origin, a C 1 positivedefinite function V :Θ → Ω and real number α > 0, such thaṫ V + αV κ is negative semidefinite on Θ. Then the origin is a finite-time stable equilibrium of nonlinear system. Lemma 2. [37]: Suppose that there exists a continuous differential positive definite function V (t), and α > 0, β > 0, 0 < r < 1 are real numbers. If V (t) satisfies the differential inequality Eq.(1). Then, V (t) will converge to the equilibrium point in a finite time t f , where t f can be obtained by Eq.(2)V + αV + βV r ≤ 0 (1) FIGURE 1: Physical structure of a quadrotor The model differential equations of the jth quadrotor quadrotor are expressed as where m j is the mass of the jth quadrotor and g is the gravitational acceleration. p j and v j are the position and linear velocity expressed in the inertial frame. T j is the magnitude of the control thrust and T j ≥ 0. Γ j = (Γ xj , Γ yj , Γ sj ) T ∈ R 3 represents the control torque applied on the system expressed in the body frame F . d U j and d Γj are composed of the aerodynamic disturbances and parameter uncertainties. I f j = diag(I xxj , I yyj , I zzj ) is the symmetric positive definite constant inertia matrix of the jth quadrotor.
Assumption 2: For the formulation control of quadrotors, the communication topology graph is undirected.
Assumption 3: The disturbance d U j and d Γj are bounded and we denote |d U j | ≤ D U j and |d Γj | ≤ D Γj , where D U j and D Γj are unknown upper bound.

D. PROBLEM FORMULATION
Assuming that p d = [p dx , p dy , p dz ] T and v d = [v dx , v dy , v dz ] T denote the desired position and velocity of the quadrotor, respectively, and the position and velocity tracking error of the jth quadrotor is defined as where ∆ ji = [∆ xji , ∆ yji , ∆ zji ] T represents the relative distance between the ith quadrotor and jth quadrotor, which determines the desired formation pattern. For the leader quadrotor, the desired trajectory ∆ ji = ∆ 10 = [0, 0, 0] T , , q d2 , q d3 ] T denote the desired orientation expressed in the desired frame, and ω d denotes the desired angular velocity expressed in the desired frame.Q = [q T ,η] T andq = [q 1 ,q 2 ,q 3 ] T denote the relative orientation error from the body frame to the desired frame, andQ is defined bỹ where the inverse of the unit-quaternion is defined by Let ω d denote the desired angular velocity and the angular velocity error from the body frame to the desired frameω is described bỹ

E. CONTROL OBJECTIVES
In this paper, the main objective is to design a formation control law for quadrotor formation to achieve the desired formation pattern and track the desired formation trajectory under parametric uncertainties and external disturbances.

III. FORMATION CONTROL DESIGN
To facilitate the design of formation controller separately, the quadrotor model is divided into the outer-loop subsystem and the inner-loop subsystem. From Eq. (7), it's clear that by designing T j cannot maintain the stability of jth quadrotor(because it is under-actuated), so, another control input U j = T j R T (Q j )e 3 is designed, which aims to solve underactuation of the position system. For the system (7) and (8), the outer-loop subsystem and inner-loop subsystem can be written as Outer-loop subsystem: Inner-loop subsystem: The position formation control for out-loop subsystem and the attitude tracking control for the inner-loop subsystem will be designed respectively.

A. POSITION FORMATION CONTROL DESIGN
The error system can be written in a matrix form as Submitting Eq.(9) to Eq.(15) and it yieldṡ where The virtual displacement controller U j is designed as following where υ(x) = x ||x|| x ||x||+ε0 is the ratio of x to the norm of x , ε 0 > 0 is a very small constant to avoid singularities. b 1 > 0 defined aṡ 2b 1 e T pj e pj + e pj e T vj e vj Let U j = [u j1 , u j2 , u j3 ] T , so in order to get a unique set of solutions, let q dj3 = 0, and then we get So, we can extract the thrust force T j And the jth η dj exception is Then, using the first two equations of Eq. (21), extract q dj1 and q dj2 as The required angular velocity ω dj can be obtained as Theorem 1. Consider the outer-loop subsystem (7) with the control strategy (19), the tracking error converges to the region around zero within a finite time.
Proof. Choose the following Lyapunov function candidate: where s = δ(e pj ), e T vj T , let γ < 2b is a positive definite symmetric matrix, x = 0 is the ratio of x inverse to the square root of the norm of x. Consider the Lyapunov function has a number of variables, for the sake of brevity, it is proved in two parts below. where where The time derivative of Eqs.(29)-(33) are given as follows: Substituting Eq. (16) to Eq. (29), together with Eq.(29)-(33) yieldṡ Let χ= e pj Submitting it to Eq.(40),then we havė Moving forward a single step we can obtaiṅ where By analyzing the Eq.(43) with Eq. (44), there is so it follows that 1 (e pj , e vj ) where ε 1 is a constant greater than 0.
Note that It follows thaṫ Thus the outer-loop proof is completed. Remark 1: The parameter ε 1 affects the convergence rate of position error. In order to accelerate the convergence rate, it should be given a smaller value. The parameter ε 0 is to avoid singularity in the system, it should be as small as possible in its range of values. The parameter b 1 determines the bandwidth of the sliding mode dynamics. A larger b 1 leads to a larger bandwidth indicating a faster response rate and higher tracking accuracy but it will cause larger measurement noises.

B. ATTITUDE CONTROLLER DESIGN
Consider the Inner-loop subsystem (14) and Assumption1-3, let the virtual attitude controller Γ j be designed as below. where where ξ 1j = e qj e qj 1 2 sgn(e qj ) where e qj is the error ofq j andω j , σ is a constant greater than 0.
Take its first time derivativė Submitting Eq.(59) and Eq.(61) to the Eq.(67),we can havė Theorem 2. Consider the inner-loop subsystem (8) with the control strategy (59), the attitude error converges to the region around zero within a finite time.
Proof. Choose the following Lyapunov function candidate: where Q 1 = Q 11 Q 12 Q 21 λ is a symmetric matrix.
in which This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication.
IfV 20 is negative definite, then it needs Q 1 is positive definite, only needing to det(Q 1 ) > 0. Then it satisfies that where It's obvious that , then we can get that the matrix Q 1 is positive defined.
Taking into account a well-known inequality λ min (Q 1 ) ς j 2 ≤ ς T j Qς j with Eq.(72), there iṡ where N 1 = Substituting (76) to (71), there iṡ The Eq.(77) can be rewritten aṡ If e qj = 0, there isḃ q1j = k γ1 2 and So there is τ = 0, substitute it to (78), then there iṡ Thus the inner-loop proof is completed. Remark 2: The parameter k in Eq.(64) will affect the convergence of attitude error of quadrotor. A larger k leads to faster convergence but at the cost of introducing more measurement noises. The parameter λ also affects the convergence of attitude error of quadrotor and it is suggested to be in the range of 1 < λ < 3.

C. CONVERGENCE ANALYSIS OF THE CLOSE-LOOP SYSTEM
In section III.A and III.B, the stability analysis of the outerloop and inner-loop have been proved. In this section, we will prove the stability analysis and convergence characteristic for the closed-loop of quadrotor formation.
Theorem 3. For the whole closed-loop system given by outer-loop subsystem (13) controller by (19) with adaptive law (20) and inner-loop subsystem (14) controller by (59) with adaptive law (64) under the upper bounded disturbances and uncertainties. The whole closed-loop system is inputstate stable in finite convergence time. To this end, the control objectives can be established. Proof. Consider the Lyapunov function candidate of whole closed-loop system as follows: where V outer and V inner are the Lyapunov function of outerloop system and inner-loop system. As V outer and V inner are positive definite, the whole closed-loop system Lyapunov function is positive definite. Differentiating the Lyapunov function V whole along the trajectory of the system, we can derive thaṫ For the convenience of the proof, divide the space [V outer , V inner ] T ∈ +2 into two different areas, O and R, area R further divided into two areas R 1 and R 2 , which are defined as follows: When [V outer , V inner ] is in the area O, Eq.(84) can be rewritten as: From Theorem 1, the space can be satisfied to be stable in finite time as shown in Eq.(89) and Eq.(90). And it is hard to determine the convergence characteristic in the area , because the outer-loop and inner-loop Lyapunov function cannot be easily replaced to satisfy the finite time convergence theorem. However, it does not affect the proof.
As shown in theorem 2, the independent inner-loop system can be stable within finite time. The inner-loop Lyapunov function also will convergence to zero. Hence, the space T ∈ +2 will not stay in R 2 forever, but cross from R 2 to R 1 within the finite time. Once the space enter R 1 . Therefore, the finite time stability of whole closed-loop system can be guaranteed. Thus the proofs are completed.

IV. SIMULATION
To prove the effectiveness of the proposed control laws and adaptive laws for the formation modeled on quaternion, several simulations are carried out in this section. In this section, we suppose a quadrotor formation including five quadrotors. Quadrotor communicates with other quadrotor within the range of communication, and the communication graph is presented in Figure 2. Five quadrotors have different initial position and attitude. The simulation will verify that quadrotors can form and maintain the formation with high performance by using the proposed control method. For further prove the superiority of proposed algorithm, it is compared with the finite time convergence control algorithm in [20]with quaternion-based system.

FIGURE 2: The communication graph
Assuming that the leader quadrotor flies along the predetermined trajectory: the reference trajectory of the leader used in the simulation can be expressed as: T To verify the validity of the adaptive law equation (19) and (59) Table 1, and the parameters of the controllers and proof process are shown in Table 2.

C. SIMULATION RESULTS
In this part, we compare the proposed algorithm with finite time control(FTC) algorithm [20], and the purpose of this comparison is to explain which method can form and maintain the formation configuration with higher precision and faster convergence speed. The simulation results are provided in Figure 3- Figure  16. Figure 3 shows the x-coordination, y-coordination and z-coordination trajectories of five quadrotors with proposed algorithm. Figure 4 and Figure 5 show the control forces and torques of the ASTDFC. And Figures 6-15 show the comparison results between the two algorithms. From Figure  16, the curve changes of the adaptive law and disturbance are clearly presented. Figure 3 shows the control strategy designed in this paper can form and maintain formation quickly in the x y z plane. From Figure 4 and Figure 5, it can be observed that the control forces and torques can converge in short time. Figures  6-9 illustrate that the method proposed in this paper converge to 0 within 2s, which is faster than that by finite time convergence method. Moreover, the errors convergences to 0 with higher convergence precision and faster convergence rate by the proposed algorithm. From Figures 10-15, it can be observed that convergence of ASTDFC is faster than FTC. In Figure 16, it's clear that both the adaptive b 1 and b1 2 are larger than the disturbance d U j , and finite-time stability can be established.

D. SIMULATION II
In this section, we test the the convergence performance of quadrotor formation under irregular disturbances. The Real white Gaussian noises are considered as the irregular disturbances. We design a new trajectory for the quadrotor formation, and the other parameters are the same as in Simulation I. The reference linear velocity is v d = sin(0.1t) 0.5 cos(0.1t) 1 The simulation results are shown in Figure 17 - Figure 24. Figures 17-18 show the noise added to the position subsystem and attitude subsystem. Figure 19 is the specific trajectory show the quaternion errors of each quadrotor. As shown in Figure 19, it can be seen that the quadrotor formation can maintain unity at about 2.5s. From , it can be seen that the quaternion errors of each quadrotor can converge before 3s. From these figures, it can be seen that the proposed method has an improved tracking performance of the closed-loop control system. Distributed formation control problem for multi-quadrotors described by unit-quaternions has been studied in this paper. Based on super-twisting control and finite time control method, considering the characteristic of the position subsystem and attitude subsystem, the different control methods are developed and adopted to improve the convergence rate and accuracy of the system respectively. Then with the theoretical analysis, the stability of whole closed-loop system is proved and quadrotor formation can converge to the desired formation configures in finite time. Finally, the effectiveness of the proposed method has been illustrated with numerous simulations. In the future work, the actuator deadzones, unmodeled