A Fixed-Time Hierarchical Formation Control Strategy for Multiquadrotors

*is paper deals with the problem of multiquadrotor collaborative control by developing and analyzing a new type of fixed-time formation control algorithm. *e control strategy proposes a hierarchical control framework, which consists of two layers: a coordinating control layer and a tracking control layer. On the coordinating control layer, according to the fixed-time consistency theory, the virtual position and virtual velocity of each quadrotor are calculated and acquired to form a virtual formation, and the virtual velocity reaches consistency. On the tracking control layer, the real position and the real velocity track the virtual position and the virtual velocity, respectively. *us, multiquadrotor can achieve the required formation shape and velocity consensus. Finally, the comparative simulations are carried out to illustrate the feasibility and superiority of the proposed fixed-time hierarchical formation control method for multiquadrotor collaborative control.


Introduction
Multiquadrotor systems have been widely used in various fields in the last decade, for example, agricultural plant protection, expressing, aerial photogrammetry, and emergency relief [1][2][3][4][5][6]. Compared with a single quadrotor, the advantages of multiquadrotor collaborative work mainly come from the two aspects, information fusion and resource complementation. In the process of executing tasks by multiple quadrotors, even if a single quadrotor fails, it will not cause the failure of the entire mission, which can greatly improve the reliability and fault tolerance of the system, so that it can satisfy the requirements of task time, space, and optimization. However, from the cybernetic perspective, the control of multiquadrotors will be more challenging and difficult, because not only the flight status of the single quadrotor but also the cooperative control between multiple quadrotors is considered. erefore, how to solve the convergence problem of multiple quadrotors and how to increase the convergence rate are the hot topics for current research in cooperative control of multiquadrotor system.
Up to now, researchers have achieved rich research results in the field of cooperative control of multiagents and have put forward many theories of formation control, which include leader-follower methods, virtual structure methods, and behavior-based methods and all of these are widely used in the field of multiquadrotor formation control. By adopting the lead-following formation architecture, the formation control methods which combined state estimation and backstepping method were given in [7]. In [8], a behavior-based decentralized method for robust formation flight control technology for multiple UAVs is proposed. A behavior-based formation flight control method is designed in [9]; it solves the thorny problem that traditional formation flying control based on virtual architecture has a heavier burden on data communication; this theory can enhance the obstacle avoidance and risk avoidance capabilities of UAVs in unknown environments and can provide ideas for aircraft in low-altitude coordinated operations. Furthermore, in order to overcome the difficulties that most of the results of collaborative formation control are asymptotically stable, the finite-time consensus method is proposed and widely used in the field of multiagents [10,11]; the strategy of finite-time control introduced in [12] can ensure that the closed-loop system improves the response characteristics of the system and it is finite-time convergence. Moreover, an optimized new finite-time control strategy given in [13] can provide better disturbance rejection performance. In [14], this paper introduced that the finite-time consensus algorithm was used to construct the position dynamics model and attitude dynamics model, respectively, so that multiquadrotors can achieve the desired formation shape. However, the finite-time consistency algorithm has certain flaws, and its convergence time is also affected by the initial state information.
erefore, researchers have made improvements on the basis of finitetime stability; the fixed-time consistency theory has emerged as the times require.
Compared with the finite-time control theory, fixed-time consistency has better convergence and robustness. In [15], Polyakov proposed the concept of fixed-time consistency on the basis of finite-time stability. For uncertain linear objects, two new nonlinear control algorithms are proposed, which allowed the system trajectory to be adjusted independently under initial conditions to ensure the convergence time. e convergence time will not be affected by the system's initial state. In [16], an edge-based distributed optimization method for multirobot system is proposed, which guaranteed reaching consistency at a fixed time. In [17], aiming at the leader-follower multiagent consistency problem, a fixedtime consistency convergence protocol is proposed to ensure that multiagents converge quickly within a set time. In [18], a fixed-time fault-tolerant controller is designed to solve the problem of spacecraft rendezvous with thruster failure and external disturbance.
As far as the authors know, due to the strong nonlinear coupling and high dimension of multiquadrotor system, there are relatively few references for formation control of multiple quadrotors. For some complex control systems, new types of control methods are often needed. In [19], aiming at the problem that it is difficult to obtain velocity signal in electrohydraulic servomechanisms, a practical adaptive tracking controller is designed and the new kind of controller that can track the unknown parameter estimates only depends on the actual position and desired trajectory. In [20], for the complex hydraulic systems, a new time-delay model is proposed to approximate the valve dynamics and this model will not increase the order of the system. erefore, for a highly complex multiquadrotor system, the main motivation of this paper is to establish a formation control algorithm. e main contribution of this paper is to decouple the originally highly complex quadrotors formation control model and simplify it into two-layer structure. e upper layer designs a coordinating control layer based on a fixed-time method to calculate and get virtual state information of each quadrotor. e lower layer is the PID-based tracking control algorithm, which is used to obtain the real state information to track the virtual position and velocity. On the basis of the given architecture, the multiquadrotors will reach the desired formation and velocity consensus; moreover, the convergence efficiency is significantly improved.

Communication Scheme.
e communication topology structure among multiple quadrotors is denoted by using an undirected graph G n � V, E, A { }, where it includes the adjacency matrix A � [a ij ] ∈ R n×n , i, j ∈ B, where a ij is the communication weight between the ith quadrotor and the jth quadrotor, when the aircraft i links to the aircraft j; then a ij > 0. e set of nodes V � v 1 , v 2 , . . . , v n , E is an edge set (e ij ∈ E), where e ij is the communication link between the ith quadrotor and the jth quadrotor. L G n � [l ij ] ∈ R n×n is a Laplacian matrix of G n . Let G n+1 � G n ∪ v 0 be an extended graph, where v 0 denotes a virtual leader. If the other quadrotors (follows) can get messages from the leader, a i0 > 0(i � 1, . . . , n); otherwise, a i0 � 0, let M � diag a 10 , . . . , a n0 , which is true communication matrix.

Dynamics Model.
In order to describe the position, attitude, velocity, and other information of the multiquadrotors in the spatial range, two coordinate systems were established, where one is geographic coordinate system E(X e , Y e , Z e ) and the other is the body coordinate system B(X b , Y b , Z b ). In particular, the body coordinate system can be transformed by the transformation rotation formula to a geographic coordinate system as follows: where s θ i and c θ i denote sin θ i and cos θ i (i ∈ Γ � 1, 2, . . . , n { }), respectively. Furthermore, it is usually required to use six degree-of-freedom variables to describe the operation of the quadrotor in three-dimensional space. e coordinates of each quadrotor are expressed as where represent the Euler angles to denote the attitude of the quadrotor, which include the roll angle ϕ i (rotation Euler angles around x-axis), the pitch angle θ i (rotation Euler angles around y-axis), and the yaw angle ψ i (rotation Euler angles around z-axis) [21].

Position Dynamics Model.
Due to the fact that the aerodynamic effects cannot be neglected, the position dynamical model for each quadrotor is presented as follows: where m i is the mass of the quadrotor; k x,i , k y,i , and k z,i are the aerodynamic drag coefficients; and g represents the gravitational acceleration in geographic coordinate system.

Attitude Dynamics Model.
From [22], the dynamics model of each quadrotor's attitude based on Euler angle is given as follows: where k ϕ,i , k θ,i , and k ψ,i represent the air resistance coefficients and I x,i , I y,i , and I z,i denote the moments of inertia, which are all constant.

Lemmas.
In this section, in order to provide theoretical support for subsequent stability proof, we will introduce some lemmas about fixed-time method. (2), if the controller meets the following conditions, it is called fixed time stability:

Lemma 1. For a double integral system as shown in equation
Lemma 2 (see [15]). If there exists a continuous radially unbounded function for (1) and

Hierarchical Formation Control Architecture
is section will introduce the core layered structure, shown in Figure 1; it will be divided into two parts to illustrate: (1) For the coordinated control level, due to the fact that this layer of collaborative control algorithm is separated from the control of a single quadrotor, it belongs to model-free control. e virtual leader's state information can be used as input data, and the output data can be used as virtual state information, which are acquired by fixed-time consensus method. Next, we define the leader's position r 0 � [x 0 , y 0 , z 0 ] T ∈ R 3 and the ve- Moreover, the setting of the leader trajectory can be set at will; only make sure that the following is satisfied: (1) v 0 is differentiable; (2) the acceleration v 0 is bounded. (2) For the tracking control level, the technology of PID control method is used. e virtual state information of each quadrotor can be used as input data of the tracking control module, which is composed of an attitude controller and a position controller. Each quadrotor tracks the virtual state which is received from the coordination module. Finally, thrust T i and torque τ i are obtained as outputs.

Coordinating Control.
For the cooperative control part, its advantage is that it is separated from the control part of a single quadrotor; that is to say, the design of the coordinated controller is not involved in dynamics model. e controller is set as follows: where λ i > 0(i � 1, 2, 3, 4 { }) and a is a positive even number, while b is a positive odd number, satisfying 0 < a < b. Furthermore, when information exchanges between the ith and jth quadrotor, then a ij � 1; otherwise a ij � 0. After the calculation of the coordinated controller, the virtual velocity of the quadrotors reaches a fixed-time consistency, i.e., Although a consensus on the virtual location has been reached, the required formation shape has not yet been accomplished. In order for the quadrotors to reach a desired Journal of Robotics formation pattern, a reference configuration vector σ ij was introduced. According to the descriptions above, the controller (7) is rewritten: where σ ij � σ i − σ j is formation reference configuration vector, the proposed coordinating control algorithm without the traditional symbol function sign (·); that is to say, equation (8) is a smooth and continuous consensus protocol, thereby effectively preventing the occurrence of tremor.

Tracking Control.
In this section, in order to make the real position and real velocity converge to the virtual positon and virtual velocity, i.e., r i ⟶ r i , v i ⟶ v i , a position controller and an attitude controller are designed. e tracking control algorithm is based on the PID theory, and its main advantage is that it can be implemented faster and better, and it provides a basis for actual flight experiments in future research.

Position Controller.
Using the classic PID control, the position controller u p i is presented: where e(t) � r i − r i and k p , k i , k d > 0 are PID parameters. According to the position dynamical model in (3), the position controller is presented: and then, the position formula u p i is designed as follows: where k 1 > 0, k 2 > 0, and k 3 > 0.

Attitude
Similarly, the attitude controller u a i � [u a x,i , u a y,i , u a z,i ] T ∈ R 3 is designed; that is, where T is the desired attitude angles, and According to the results above, the desired torque i ] T and the velocity of each quadrotor can be obtained.

Theoretical Proof
e stability proof of the coordinating control algorithm is given.

Theorem 1. Suppose that the undirected communication topology G n � V, E, A
{ } for the quadrotors connected; then, the coordinating control protocol (7) can achieve consensus in fixed-time, and the settling time T satisfies Proof. Define Set a Lyapunov constructor as follows: where V ≥ 0; its time derivative can be calculated by en, in a similar way for the second term, it can be obtained that erefore, it can be obtained that en, due to [16], we have n i,j�1 where e � [e 1 , e 2 , . . . , e n ] T , L p and L q are the Laplacians of the graphs G(A (2b/2b− a) ) and G(A (2b/2b+a) ), respectively, and λ * (L p ) And λ * (L q ) are the second smallest eigenvalues of L p and L q , respectively. erefore, we get Invoking Lemma 2, the virtual velocity consensus is achieved in a fixed time with a setting time T. Furthermore, v i ⟶ v 0 when T ≤ T 1 max ; this completes the proof. Next, the position consensus is considered, and the related theorem and proof are as follows.

Theorem 2. Suppose that the undirected communication topology G n � V, E, A
{ } for the quadrotors connected; then, the coordinating control protocol (8) can achieve consensus in fixed-time, and the settling time T satisfies Proof. Basically, it is the same as the proof of eorem 1, so we only present the main part to avoid redundancy. Based on the virtual velocity consensus results, it can be obtained Hence, protocol (8) can be changed as follows: Similarly, set a Lyapunov constructor  Journal of Robotics 7   Journal of Robotics en, one finally obtains us, similar to the proof of eorem 1, multiple quadrotors reach the fixed-time consensus within T 2 max , which means the formation shape of multiquadrotors can be obtained in fixed time.

Simulation Results
In this section, simulation experiments are used to illustrate the feasibility and superiority of the proposed fixed-time consistency control algorithm. e following content will be divided into two parts for verification of the feasibility and superiority of the control algorithm.

e Feasibility of the Algorithm.
e communication topology of a five-node multiquadrotor system includes four followers and a virtual leader; the four quadrotors form a square as the desired formation shape. e leader points to quadrotor 1, which means that data can be transmitted in a directional manner and the communication method of other quadrotors is presented by the direction of the arrow as presented in Figure 2. erefore, the weight of the edge is, respectively, presented as follows: a 21 � a 31 � a 41 � a 51 � 1, for each quadrotor, the moments of inertia are chosen: I x,i � 0.05, I y,i � 0.05, I z,i � 0.1, the air resistance coefficients are chosen as k x,i � k y,i � k z,i � 0.02, and the torque  coefficients k ϕ,i � k θ,i � k ψ,i � 0.1. In addition, the parameters of control algorithm are presented in Table 1.

Remark 1.
Due to the control saturation constraints of the real quadrotors and the formation control problem, the quadrotor cannot quickly fly, parameters λ i (i � 1, 2, 3, 4 { }) cannot be set too large values, and one can usually set the parameters within (0, 5]. In order to keep the better performance of fixed-time convergence and disturbance rejection, the parameters a and b are usually set to a/b ∈ (0, 1).
In simulations, the leader's initial state information is set: e initial acceleration for the leader is en, through curve fitting, the leader's trajectory curve can be approximated by the following function: f(x, y) � 5.798 − 0.2985 sin(0.7926π(xy)) + 10.56e − (3.413y) 2 . (32) Next, one will choose the initial conditions of four quadrotors as follows:   Figure 5: e adjacent distance between every two quadrotors under the control algorithm (9). (a) In the range of (0, 2). (b) In the range of (3,5). (c) In the range of (5, 10). (d) In the range of (10,20).  2) t (3,5) t (5,10) t (10,20) Control algorithm ( Based on the proposed control architecture, simulation outcomes are shown in Figure 3. e response of position trajectories of all quadrotor in 3D space is shown in Figure 3(a), and the red curve denotes the trajectory of the virtual leader; it can be seen that the desired formation has been formed within 3 seconds; then the four quadrotors followed the leader to fly in the desired square formation. Figure 3(b) shows the adjacent distance between two quadrotors converges to 1.0 m; as can be seen from the figure, the convergence of the four quadrotor is extremely fast. Figures 3(c) and 3(d) represent the virtual velocity curve and real velocity curve of each quadrotor. In the hierarchical formation control framework, the virtual velocity can be obtained by the coordinating control algorithm and making the virtual velocity reach a fixed-time consistency; from Figure 3(c), one can see that the virtual velocity of quadrotors reaches consistency within 2 seconds. Meanwhile, the real velocity can track the virtual velocity by the tracking control algorithm; therefore, the real velocity eventually reaches consistency as shown in Figure 3(d). Furthermore, at the 3rd and 12th seconds, the acceleration of the leader changes, and the velocity of the quadrotor quickly reaches consensus. Figure 3(e) shows the position of all quadrotors which well reflects the position consensus. e attitude angles of each quadrotor aircraft are given in Figure 3(f ), from which one can see that the sudden change of the leader's trajectory will slightly affect the pitch angle and roll angle but overall it maintains a good convergence.

e Superiority of the Algorithm.
In order to verify the superiority of the proposed fixed-time hierarchical control algorithm, that is, the convergence time of the quadrotor system is not affected by its initial state, the comparative simulations were conducted. e comparative controller is a finite-time consistency control protocol; refer to [23]; the controller is selected as follows: where c 1 > 0, c 2 > 0, β 1 ∈ (0, 1), and β 2 ∈ (0, 1). During simulations, the parameters of the controller (34) are all optimal values. In the simulations, the initial position of the leader quadrotor remains unchanged, which is set Next, four sets of initial positions are given for the four follower quadrotors; in order to conform to the reality, the position coordinates in the Z-axis direction are set to r i,z (0) � 2. Meanwhile, the initial positions of the quadrotors in the X-axis and Y-axis directions are in the four ranges (0, 2), (3,5), (5,10), and (10, 20), respectively. As a special case, the initial positions in the range of (10, 20) are given below: In addition, the initial state of this simulation takes the position as an example, so the initial velocities of the four quadrotors are set as Based on the above analysis and data, the simulation results are shown in Figures 4-6. Figure 4 shows the response curves of the distance between the quadrotors in the three-dimensional space under the control algorithm (34), from which it can be seen that all the quadrotors converge to the desired formation shape and move along the desired trajectory within 15 seconds in Figures 4(a)-4(c), except for Figure 4(d). Meanwhile, in the range of (0, 2), (3,5), (5,10), the convergence time of the four quadrotors is 3.39 s, 6.49 s, and 9.85 s, respectively. It can be seen that, under the control algorithm (34), as the distance between the follower quadrotors and the leader increases, the convergence time also increases. However, as can be seen from Figure 5, under the control algorithm (9), the quadrotors reach the convergence faster, and the convergence time is not affected by the initial position. Specifically, the comparison results are shown in Table 2. It is obvious from Table 2 that, compared to the control algorithm (34), the control algorithm (9) converges faster under the same conditions and, no matter how the initial position changes, the convergence time remains basically the same. Furthermore, Figure 6 shows the flight trajectory of all quadrotors in the range of (10,20), and the flight trajectory well shows the superiority of the control algorithm (9).

Conclusion
A hierarchical formation control structure which can solve the problem of multiquadrotor control has been developed through the previous discussion. First, this new kind of formation control structure is composed of a coordinating control part and a tracking control part. Based on the fixedtime consensus methods and PID methods, respectively, a coordinating control algorithm and a tracking control algorithm are given. Second, we have proved the convergence of the coordinating control algorithm in detail and illustrated the feasibility of the algorithm. ird, the proposed control algorithm has a good reference value in practical applications and can be used in actual flight. Finally, the aforementioned simulation results have elaborated the feasibility and superiority of hierarchical formation control structure.

Data Availability
e simulation data used to support the findings of this study have not been made available because the data also form part of an ongoing study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.