# A Fixed-Time Hierarchical Formation Control Strategy for Multiquadrotors

Authors: Miaojie Huang; Qiang Lu
Journal: Journal of Robotics
 (2021)

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## Abstract

This paper deals with the problem of multiquadrotor collaborative control by developing and analyzing a new type of fixed-time formation control algorithm. The 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. Thus, 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.


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## Body

## 1. 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 [1–6]. 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. Therefore, 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. Therefore, researchers have made improvements on the basis of finite-time 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. The 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 fixed-time 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. Therefore, for a highly complex multiquadrotor system, the main motivation of this paper is to establish a formation control algorithm.The main contribution of this paper is to decouple the originally highly complex quadrotors formation control model and simplify it into two-layer structure. The upper layer designs a coordinating control layer based on a fixed-time method to calculate and get virtual state information of each quadrotor. The 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.


## 2. Preliminaries

### 2.1. Communication Scheme

The communication topology structure among multiple quadrotors is denoted by using an undirected graphGn=V,E,A, where it includes the adjacency matrix A=aij∈ℝn×n, i,j∈B, where aij is the communication weight between the ith quadrotor and the jth quadrotor, when the aircraft i links to the aircraft j; then aij>0. The set of nodes V=v1,v2,…,vn, E is an edge set eij∈E, where eij is the communication link between the ith quadrotor and the jth quadrotor. LGn=lij∈ℝn×n is a Laplacian matrix of Gn. Let Gn+1=Gn∪v0 be an extended graph, where v0 denotes a virtual leader. If the other quadrotors (follows) can get messages from the leader, ai0>0i=1,…,n; otherwise, ai0=0, let M=diaga10,…,an0 , which is true communication matrix.


### 2.2. 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 systemEXe,Ye,Ze and the other is the body coordinate system BXb,Yb,Zb. In particular, the body coordinate system can be transformed by the transformation rotation formula to a geographic coordinate system as follows:(1)R=cψicθisϕisθicψi−cϕisψicϕisθicψi+sϕisψicθisψisϕisθisψi+cϕicψicϕisθisψi−sϕicψi−sθisϕicθicϕicθi,where sθi and cθi denote sin θi and cos θi (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. The coordinates of each quadrotor are expressed as(2)xi,yi,zi,ϕi,θi,ψiT∈ℝ6,i∈1,2,…,n,where ri=xi,yi,ziT∈ℝ3 and Φi=ϕi,θi,ψiT∈ℝ3 represent the Euler angles to denote the attitude of the quadrotor, which include the roll angle ϕi (rotation Euler angles around x-axis), the pitch angle θi (rotation Euler angles around y-axis), and the yaw angle ψi (rotation Euler angles around z-axis) [21].


#### 2.2.1. 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:(3)x¨i=Timicosϕisinθicosψi+sinψisinϕi−kx,imix˙i,y¨i=Timicosϕisinθicosψi+sinψisinϕi−kx,imix˙i,z¨i=Timicosϕicosθi−g−kz,imiz˙i,i∈Γ,where mi is the mass of the quadrotor; kx,i, ky,i, and kz,i are the aerodynamic drag coefficients; and g represents the gravitational acceleration in geographic coordinate system.


#### 2.2.2. Attitude Dynamics Model

From [22], the dynamics model of each quadrotor’s attitude based on Euler angle is given as follows:(4)ϕ¨i=Iy,i−Iz,iθ˙iψ˙i+τϕ,i−kϕ,iϕ˙iIx,i,θ¨i=Iz,i−Ix,iϕ˙iψ˙i+τθ,i−kθ,iθ˙iIy,i,ψ¨i=Ix,i−Iy,iθ˙iϕ˙i+τψ,i−kψ,iψ˙iIz,i,i∈Γ,where kϕ,i, kθ,i, and kψ,i represent the air resistance coefficients and Ix,i, Iy,i, and Iz,i denote the moments of inertia, which are all constant.


### 2.3. Lemmas

In this section, in order to provide theoretical support for subsequent stability proof, we will introduce some lemmas about fixed-time method.Lemma 1.
For a double integral system as shown in equation (2), if the controller meets the following conditions, it is called fixed time stability:(5)limt⟶Txit−xjt+vit−vjt=0,xit−xjt+vit−vjt=0,∀t>T.Lemma 2.
(see [15]). If there exists a continuous radially unbounded function F:ℝn⟶ℝ+∪0 meanwhile 1)Fx=0⇒x∈M; 2) any solution xt of (1) suffice the inequality D∗Fxt≤−αFpxt+βFqxtk for α,β,p,q,k>0; pk<1,qk>1 so M⊆Rn is globally fixed-time attractive for (1) and Tx0≤1/αk1−pk+1/βkqk−1,∀x0∈ℝn.Lemma 3.
Whenpi∈R,i∈Γ, and 0<b≤1, mathematical relationship is presented as follows:(6)∑i=1npib≤∑i=1npib≤n1−b∑i=1npib.


## 2.1. Communication Scheme

The communication topology structure among multiple quadrotors is denoted by using an undirected graphGn=V,E,A, where it includes the adjacency matrix A=aij∈ℝn×n, i,j∈B, where aij is the communication weight between the ith quadrotor and the jth quadrotor, when the aircraft i links to the aircraft j; then aij>0. The set of nodes V=v1,v2,…,vn, E is an edge set eij∈E, where eij is the communication link between the ith quadrotor and the jth quadrotor. LGn=lij∈ℝn×n is a Laplacian matrix of Gn. Let Gn+1=Gn∪v0 be an extended graph, where v0 denotes a virtual leader. If the other quadrotors (follows) can get messages from the leader, ai0>0i=1,…,n; otherwise, ai0=0, let M=diaga10,…,an0 , which is true communication matrix.


## 2.2. 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 systemEXe,Ye,Ze and the other is the body coordinate system BXb,Yb,Zb. In particular, the body coordinate system can be transformed by the transformation rotation formula to a geographic coordinate system as follows:(1)R=cψicθisϕisθicψi−cϕisψicϕisθicψi+sϕisψicθisψisϕisθisψi+cϕicψicϕisθisψi−sϕicψi−sθisϕicθicϕicθi,where sθi and cθi denote sin θi and cos θi (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. The coordinates of each quadrotor are expressed as(2)xi,yi,zi,ϕi,θi,ψiT∈ℝ6,i∈1,2,…,n,where ri=xi,yi,ziT∈ℝ3 and Φi=ϕi,θi,ψiT∈ℝ3 represent the Euler angles to denote the attitude of the quadrotor, which include the roll angle ϕi (rotation Euler angles around x-axis), the pitch angle θi (rotation Euler angles around y-axis), and the yaw angle ψi (rotation Euler angles around z-axis) [21].


### 2.2.1. 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:(3)x¨i=Timicosϕisinθicosψi+sinψisinϕi−kx,imix˙i,y¨i=Timicosϕisinθicosψi+sinψisinϕi−kx,imix˙i,z¨i=Timicosϕicosθi−g−kz,imiz˙i,i∈Γ,where mi is the mass of the quadrotor; kx,i, ky,i, and kz,i are the aerodynamic drag coefficients; and g represents the gravitational acceleration in geographic coordinate system.


### 2.2.2. Attitude Dynamics Model

From [22], the dynamics model of each quadrotor’s attitude based on Euler angle is given as follows:(4)ϕ¨i=Iy,i−Iz,iθ˙iψ˙i+τϕ,i−kϕ,iϕ˙iIx,i,θ¨i=Iz,i−Ix,iϕ˙iψ˙i+τθ,i−kθ,iθ˙iIy,i,ψ¨i=Ix,i−Iy,iθ˙iϕ˙i+τψ,i−kψ,iψ˙iIz,i,i∈Γ,where kϕ,i, kθ,i, and kψ,i represent the air resistance coefficients and Ix,i, Iy,i, and Iz,i denote the moments of inertia, which are all constant.


## 2.2.1. 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:(3)x¨i=Timicosϕisinθicosψi+sinψisinϕi−kx,imix˙i,y¨i=Timicosϕisinθicosψi+sinψisinϕi−kx,imix˙i,z¨i=Timicosϕicosθi−g−kz,imiz˙i,i∈Γ,where mi is the mass of the quadrotor; kx,i, ky,i, and kz,i are the aerodynamic drag coefficients; and g represents the gravitational acceleration in geographic coordinate system.


## 2.2.2. Attitude Dynamics Model

From [22], the dynamics model of each quadrotor’s attitude based on Euler angle is given as follows:(4)ϕ¨i=Iy,i−Iz,iθ˙iψ˙i+τϕ,i−kϕ,iϕ˙iIx,i,θ¨i=Iz,i−Ix,iϕ˙iψ˙i+τθ,i−kθ,iθ˙iIy,i,ψ¨i=Ix,i−Iy,iθ˙iϕ˙i+τψ,i−kψ,iψ˙iIz,i,i∈Γ,where kϕ,i, kθ,i, and kψ,i represent the air resistance coefficients and Ix,i, Iy,i, and Iz,i denote the moments of inertia, which are all constant.


## 2.3. Lemmas

In this section, in order to provide theoretical support for subsequent stability proof, we will introduce some lemmas about fixed-time method.Lemma 1.
For a double integral system as shown in equation (2), if the controller meets the following conditions, it is called fixed time stability:(5)limt⟶Txit−xjt+vit−vjt=0,xit−xjt+vit−vjt=0,∀t>T.Lemma 2.
(see [15]). If there exists a continuous radially unbounded function F:ℝn⟶ℝ+∪0 meanwhile 1)Fx=0⇒x∈M; 2) any solution xt of (1) suffice the inequality D∗Fxt≤−αFpxt+βFqxtk for α,β,p,q,k>0; pk<1,qk>1 so M⊆Rn is globally fixed-time attractive for (1) and Tx0≤1/αk1−pk+1/βkqk−1,∀x0∈ℝn.Lemma 3.
Whenpi∈R,i∈Γ, and 0<b≤1, mathematical relationship is presented as follows:(6)∑i=1npib≤∑i=1npib≤n1−b∑i=1npib.


## 3. Hierarchical Formation Control Architecture

This section will introduce the core layered structure, shown in Figure1; 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. The 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 positionr0=x0,y0,z0T∈ℝ3 and the velocity v0=vx,0,vy,0,vz,0T∈ℝ3. Moreover, the setting of the leader trajectory can be set at will; only make sure that the following is satisfied: (1) v0 is differentiable; (2) the acceleration v0^ is bounded.(2)
For the tracking control level, the technology of PID control method is used. The 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, thrustTi and torque τi are obtained as outputs.Figure 1
The diagram of hierarchical formation control.


### 3.1. 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. The controller is set as follows:(7)v^˙it=−λ1∑j=0naijv^jt−v^it1−a/b−λ2∑j=0naijv^jt−v^it1+a/b,(8)r^˙it=v^it−λ3∑j=0naijr^it−r^jt1−a/b−λ4∑j=0naijr^it−r^jt1+a/b,where λi>0i=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 aij=1; otherwise aij=0. After the calculation of the coordinated controller, the virtual velocity of the quadrotors reaches a fixed-time consistency, i.e., r^i⟶r0 and v^i⟶v0.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 formation pattern, a reference configuration vectorσij was introduced. According to the descriptions above, the controller (7) is rewritten:(9)r^˙it=v^it−λ1∑j=0naijr^it−r^jt−σij1−a/b−λ2∑j=0naijr^it−r^jt−σij1+a/b,where σij=σi−σj 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.


### 3.2. 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.,ri⟶r^i, vi⟶v^i, a position controller and an attitude controller are designed. The 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.


#### 3.2.1. Position Controller

Using the classic PID control, the position controlleruip is presented:(10)uip=v^˙i+kpet+ki∫etdt+kddetdt,i∈Γ,where et=ri^−ri and kp,ki,kd>0 are PID parameters. According to the position dynamical model in (3), the position controller is presented:(11)ux,ip=Timicosϕidcosψidsinθid+sinϕidsinψid−kx,imix˙i,uy,ip=Timicosϕidsinψidsinθid−cosψidsinϕid−ky,imiy˙i,uz,ip=Timicosϕidcosθid−kz,imiz˙i−g,i∈Γ,e1=r^i−ri,e2=v^i−vi,and then, the position formula uip is designed as follows:(12)uip=v^˙i+k1e1+k2∫e1dt+k3e2,i∈B,where k1>0, k2>0, and k3>0.


#### 3.2.2. Attitude Controller

The desired attitudes, rollϕid and pitch θid, of each quadrotor are obtained by the virtual inputs uip=ux,ip,uy,ip,uz,ip. In addition, considering the desired yaw angle ψid is a free variable, for more convenient analysis, it can be set as ψid=0; hence,(13)Tid=miui,xp2+ui,yp2+ui,zp+g2,ϕid=arcsin−miuy,ipTid,θid=arctanux,ipuz,ip+g,i∈Γ.Similarly, the attitude controlleruia=ux,ia,uy,ia,uz,iaT∈ℝ3 is designed; that is,(14)uia=Φ¨i+k4e2+k5∫e2dt+k6e˙2,i∈B,where Φid=ϕid,θid,ψidT is the desired attitude angles, and e2=Φid−Φi,e˙3=Φ˙id−Φ˙i. Next, uia in the attitude dynamic algorithm is(15)ux,ia=1Ix,iIy,i−Iz,iθ˙iψ˙i+τϕ,idIx,i−kϕ,iIx,iϕ˙i,uy,ia=1Iy,iIz,i−Ix,iϕ˙iψ˙i+τθ,idIy,i−kθ,iIy,iθ˙i,uz,ia=1Iz,iIx,i−Iy,iθ˙iϕ˙i+τψ,idIz,i−kψ,iIz,iψ˙i,i∈Γ.According to the results above, the desired torqueτid=τϕ,id,τθ,id,τψ,idT and the velocity of each quadrotor can be obtained.


## 3.1. 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. The controller is set as follows:(7)v^˙it=−λ1∑j=0naijv^jt−v^it1−a/b−λ2∑j=0naijv^jt−v^it1+a/b,(8)r^˙it=v^it−λ3∑j=0naijr^it−r^jt1−a/b−λ4∑j=0naijr^it−r^jt1+a/b,where λi>0i=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 aij=1; otherwise aij=0. After the calculation of the coordinated controller, the virtual velocity of the quadrotors reaches a fixed-time consistency, i.e., r^i⟶r0 and v^i⟶v0.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 formation pattern, a reference configuration vectorσij was introduced. According to the descriptions above, the controller (7) is rewritten:(9)r^˙it=v^it−λ1∑j=0naijr^it−r^jt−σij1−a/b−λ2∑j=0naijr^it−r^jt−σij1+a/b,where σij=σi−σj 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.


## 3.2. 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.,ri⟶r^i, vi⟶v^i, a position controller and an attitude controller are designed. The 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.


### 3.2.1. Position Controller

Using the classic PID control, the position controlleruip is presented:(10)uip=v^˙i+kpet+ki∫etdt+kddetdt,i∈Γ,where et=ri^−ri and kp,ki,kd>0 are PID parameters. According to the position dynamical model in (3), the position controller is presented:(11)ux,ip=Timicosϕidcosψidsinθid+sinϕidsinψid−kx,imix˙i,uy,ip=Timicosϕidsinψidsinθid−cosψidsinϕid−ky,imiy˙i,uz,ip=Timicosϕidcosθid−kz,imiz˙i−g,i∈Γ,e1=r^i−ri,e2=v^i−vi,and then, the position formula uip is designed as follows:(12)uip=v^˙i+k1e1+k2∫e1dt+k3e2,i∈B,where k1>0, k2>0, and k3>0.


### 3.2.2. Attitude Controller

The desired attitudes, rollϕid and pitch θid, of each quadrotor are obtained by the virtual inputs uip=ux,ip,uy,ip,uz,ip. In addition, considering the desired yaw angle ψid is a free variable, for more convenient analysis, it can be set as ψid=0; hence,(13)Tid=miui,xp2+ui,yp2+ui,zp+g2,ϕid=arcsin−miuy,ipTid,θid=arctanux,ipuz,ip+g,i∈Γ.Similarly, the attitude controlleruia=ux,ia,uy,ia,uz,iaT∈ℝ3 is designed; that is,(14)uia=Φ¨i+k4e2+k5∫e2dt+k6e˙2,i∈B,where Φid=ϕid,θid,ψidT is the desired attitude angles, and e2=Φid−Φi,e˙3=Φ˙id−Φ˙i. Next, uia in the attitude dynamic algorithm is(15)ux,ia=1Ix,iIy,i−Iz,iθ˙iψ˙i+τϕ,idIx,i−kϕ,iIx,iϕ˙i,uy,ia=1Iy,iIz,i−Ix,iϕ˙iψ˙i+τθ,idIy,i−kθ,iIy,iθ˙i,uz,ia=1Iz,iIx,i−Iy,iθ˙iϕ˙i+τψ,idIz,i−kψ,iIz,iψ˙i,i∈Γ.According to the results above, the desired torqueτid=τϕ,id,τθ,id,τψ,idT and the velocity of each quadrotor can be obtained.


## 3.2.1. Position Controller

Using the classic PID control, the position controlleruip is presented:(10)uip=v^˙i+kpet+ki∫etdt+kddetdt,i∈Γ,where et=ri^−ri and kp,ki,kd>0 are PID parameters. According to the position dynamical model in (3), the position controller is presented:(11)ux,ip=Timicosϕidcosψidsinθid+sinϕidsinψid−kx,imix˙i,uy,ip=Timicosϕidsinψidsinθid−cosψidsinϕid−ky,imiy˙i,uz,ip=Timicosϕidcosθid−kz,imiz˙i−g,i∈Γ,e1=r^i−ri,e2=v^i−vi,and then, the position formula uip is designed as follows:(12)uip=v^˙i+k1e1+k2∫e1dt+k3e2,i∈B,where k1>0, k2>0, and k3>0.


## 3.2.2. Attitude Controller

The desired attitudes, rollϕid and pitch θid, of each quadrotor are obtained by the virtual inputs uip=ux,ip,uy,ip,uz,ip. In addition, considering the desired yaw angle ψid is a free variable, for more convenient analysis, it can be set as ψid=0; hence,(13)Tid=miui,xp2+ui,yp2+ui,zp+g2,ϕid=arcsin−miuy,ipTid,θid=arctanux,ipuz,ip+g,i∈Γ.Similarly, the attitude controlleruia=ux,ia,uy,ia,uz,iaT∈ℝ3 is designed; that is,(14)uia=Φ¨i+k4e2+k5∫e2dt+k6e˙2,i∈B,where Φid=ϕid,θid,ψidT is the desired attitude angles, and e2=Φid−Φi,e˙3=Φ˙id−Φ˙i. Next, uia in the attitude dynamic algorithm is(15)ux,ia=1Ix,iIy,i−Iz,iθ˙iψ˙i+τϕ,idIx,i−kϕ,iIx,iϕ˙i,uy,ia=1Iy,iIz,i−Ix,iϕ˙iψ˙i+τθ,idIy,i−kθ,iIy,iθ˙i,uz,ia=1Iz,iIx,i−Iy,iθ˙iϕ˙i+τψ,idIz,i−kψ,iIz,iψ˙i,i∈Γ.According to the results above, the desired torqueτid=τϕ,id,τθ,id,τψ,idT and the velocity of each quadrotor can be obtained.


## 4. Theoretical Proof

The stability proof of the coordinating control algorithm is given.Theorem 1.
Suppose that the undirected communication topologyGn=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(16)T≤Tmax1=b2−a/baλ1λ∗Lp1−a/2b+b2a/bn−7a/6baλ2λ∗Lq1+a/2b.Proof.
Define(17)v^¯i=v^i−v^0,ei=v^¯i−1n∑j=1nv^¯j.
Set a Lyapunov constructor as follows:(18)V=12∑i=1neiTei,where V≥0; its time derivative can be calculated by(19)V˙=∑i=1neiTe˙i=−λ1∑i=1n∑j=1naijeiTv^¯i−v^¯j1−a/b−λ2∑i=1n∑j=1naijeiTv^¯i−v^¯j1+a/b.
For the first term inV˙,(20)−λ1∑i=1n∑j=1naijeiTv^¯i−v^¯j1−a/b=−λ1∑i=1n∑j=1naijeiTei−ej1−a/b=−λ12∑i=1n∑j=1naijeiTei−ej1−a/b−λ12∑j=1n∑i=1najiejTej−ei1−a/b=−λ12∑i=1n∑j=1naijeiTei−ej1−a/b+λ12∑i=1n∑j=1naijejTei−ej1−a/b=−λ12∑i=1n∑j=1naijei−ej2−a/b2−a/b.
Then, in a similar way for the second term, it can be obtained that(21)−λ2∑i=1n∑j=1naijeiTv^¯i−v^¯j1+a/b=−λ22∑i=1n∑j=1naijei−ej2+a/b2+a/b.
Therefore, it can be obtained that(22)V˙=−λ12∑i=1n∑j=1naijei−ej2−a/b2−a/b−λ22∑i=1n∑j=1naijei−ej2+a/b2+a/b≤−λ12∑i=1n∑j=1naijei−ej22−a/b−λ22∑i=1n∑j=1naijei−ej22+a/b≤−λ12∑i=1n∑j=1naijei−ej22−a/b−λ22∑i=1n∑j=1naijn−a/2bei−ej22+a/b=−λ12∑i=1n∑j=1naij2b/2b−aei−ej222b−a/2b−λ22n−a/2b∑i=1n∑j=1naij2b/2b+aei−ej222b+a/2b≤−λ12∑i=1n∑j=1naij2b/2b−aei−ej222b−a/2b−λ22n−7a/6b∑i=1n∑j=1naij2b/2b+aei−ej222b+a/2b.
Then, due to [16], we have(23)∑i,j=1naij2b/2b−aei−ej22=2eTLpe≥4λ∗LpV,∑i,j=1naij2b/2b+aei−ej22=2eTLqe≥4λ∗LqV,where e=e1,e2,…,enT,Lp and Lq are the Laplacians of the graphs GA2b/2b−a and GA2b/2b+a, respectively, and λ∗Lp And λ∗Lq are the second smallest eigenvalues of  Lp and Lq, respectively.
Therefore, we get(24)V˙≤−λ124λ∗LpV2b−a/2b−λ22n−7a/6b4λ∗LqV2b+a/2b=−λ124λ∗Lp2b−a/2bV2b−a/2b−λ22n−7a/6b4λ∗Lq2b+a/2bV2b+a/2b.
Invoking Lemma2, the virtual velocity consensus is achieved in a fixed time with a setting time T. Furthermore, v^i⟶v^0 when T≤Tmax1; 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 topologyGn=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(25)T≤Tmax2=b2−a/baλ3λ∗Lp1−a/2b+b2a/bn−7a/6baλ4λ∗Lq1+a/2b.Proof.
Basically, it is the same as the proof of Theorem1, so we only present the main part to avoid redundancy. Based on the virtual velocity consensus results, it can be obtained that v^i=v^j=v^0=r^˙0 when T≤Tmax1. Define(26)r^¯i=r^i−r^0,v^¯i=v^i−v^0.
Hence, protocol (8) can be changed as follows:(27)r^¯˙it=−λ3∑j=0naijr^¯it−r^¯jt1−a/b−λ4∑j=0naijr^¯it−r^¯jt1+a/b.
Similarly, set a Lyapunov constructor(28)V=12∑i=1neiTei.
Then, one finally obtains(29)V˙≤−λ324λ∗Lp2b−a/2bV2b−a/2b−λ42n−7a/6b4λ∗Lq2b+a/2bV2b+a/2b.
Thus, similar to the proof of Theorem1, multiple quadrotors reach the fixed-time consensus within Tmax2, which means the formation shape of multiquadrotors can be obtained in fixed time.


## 5. Simulation Results

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


### 5.1. The Feasibility of the Algorithm

The 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. The 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 Figure2. Therefore, the weight of the edge is, respectively, presented as follows: a21=a31=a41=a51=1, for each quadrotor, the moments of inertia are chosen: Ix,i=0.05,Iy,i=0.05,Iz,i=0.1, the air resistance coefficients are chosen as kx,i=ky,i=kz,i=0.02, and the torque coefficients kϕ,i=kθ,i=kψ,i=0.1. In addition, the parameters of control algorithm are presented in Table 1.Figure 2
Communication topology and the desired formation shape.Table 1
Parameters of control algorithms.
ParameterValueλ13.0λ20.6λ33.5λ40.7a5.00b49.0k16k25k38k4500k5300k630Remark 1.
Due to the control saturation constraints of the real quadrotors and the formation control problem, the quadrotor cannot quickly fly, parametersλii=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:(30)r0,x0,r0,y0,r0,z0,v0,x0,v0,y0,vz,00T=0.3,1.5,2.55,2.78,0.80,0.90T.
The initial acceleration for the leader is(31)a0t=0,1,1T−v0t,0≤t<3,0.9,0,0.6T−v0t,3≤t<12,0.6,0.6,0.6T−v0t,12≤t≤15.
Then, through curve fitting, the leader’s trajectory curve can be approximated by the following function:(32)fx,y=5.798−0.2985sin0.7926πxy+10.56e−3.413y2.
Next, one will choose the initial conditions of four quadrotors as follows:(33)r1,x0,r1,y0,r1,z0,v1,x0,v1,y0,v1,z0T=3.20,2.10,2.50,0.55,0.56,0.47T,r2,x0,r2,y0,r2,z0,v2,x0,v2,y0,v2,z0T=3.30,5.10,1.78,0.25,0.77,0.95T,r3,x0,r3,y0,r3,z0,v3,x0,v3,y0,v3,z0T=1.70,0.50,1.77,0.88,0.69,0.27T,r4,x0,r4,y0,r4,z0,v4,x0,v4,y0,v4,z0T=2.70,3.00,1.85,0.65,0.79,0.28T.
Based on the proposed control architecture, simulation outcomes are shown in Figure3. The 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. The 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.Figure 3
Simulation results. (a) The trajectory of all quadrotors in 3D space. (b) The adjacent distance between every two quadrotors. (c) The virtual velocity of all quadrotors. (d) The real velocity of all quadrotors. (e) The position curves of the leader and the quadrotors. (f) Response of attitude angles of all quadrotors.
(a)(b)(c)(d)(e)(f)


### 5.2. The 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.The comparative controller is a finite-time consistency control protocol; refer to [23]; the controller is selected as follows:(34)v^˙it=γ1sigβ1∑j=0naijv^jt−v^it,r^˙it=v^it−γ2sigβ2∑j=0naijr^it−r^jt−σij,where γ1>0, γ2>0, β1∈0,1, and β2∈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(35)r0,x0,r0,y0,r0,z0,v0,x0,v0,y0,vz,00T=0.2,1.6,2.0,0.50,0.50,0.50T.Next, four sets of initial positions are given for the four follower quadrotors; in order to conform to the reality, the position coordinates in theZ-axis direction are set to ri,z0=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:(36)r1,x0,r1,y0,r1,z0T=15.0,16.3,2.0T,r2,x0,r2,y0,r2,z0T=17.5,16.2,2.0T,r3,x0,r3,y0,r3,z0T=16.9,18.2,2.0T,r4,x0,r4,y0,r4,z0T=19.3,17.6,2.0T.In addition, the initial state of this simulation takes the position as an example, so the initial velocities of the four quadrotors are set asvi,x0,vi,y0,vi,z0T=0.5,0.5,0.5T.Based on the above analysis and data, the simulation results are shown in Figures4–6. 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).Figure 4
The adjacent distance between every two quadrotors under the control algorithm (34). (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).
(a)(b)(c)(d)Figure 5
The 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).
(a)(b)(c)(d)Figure 6
The flight trajectory of all quadrotors in the range of (10, 20). (a) Under the control algorithm (34). (b) Under the control algorithm (9).
(a)(b)Table 2
Comparison results with two control algorithms.
t0,2t3,5t5,10t10,20Control algorithm (34)3.396.499.85>15Control algorithm (9)2.532.512.542.79Performance improvement (%)25.461.374.2>81


## 5.1. The Feasibility of the Algorithm

The 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. The 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 Figure2. Therefore, the weight of the edge is, respectively, presented as follows: a21=a31=a41=a51=1, for each quadrotor, the moments of inertia are chosen: Ix,i=0.05,Iy,i=0.05,Iz,i=0.1, the air resistance coefficients are chosen as kx,i=ky,i=kz,i=0.02, and the torque coefficients kϕ,i=kθ,i=kψ,i=0.1. In addition, the parameters of control algorithm are presented in Table 1.Figure 2
Communication topology and the desired formation shape.Table 1
Parameters of control algorithms.
ParameterValueλ13.0λ20.6λ33.5λ40.7a5.00b49.0k16k25k38k4500k5300k630Remark 1.
Due to the control saturation constraints of the real quadrotors and the formation control problem, the quadrotor cannot quickly fly, parametersλii=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:(30)r0,x0,r0,y0,r0,z0,v0,x0,v0,y0,vz,00T=0.3,1.5,2.55,2.78,0.80,0.90T.
The initial acceleration for the leader is(31)a0t=0,1,1T−v0t,0≤t<3,0.9,0,0.6T−v0t,3≤t<12,0.6,0.6,0.6T−v0t,12≤t≤15.
Then, through curve fitting, the leader’s trajectory curve can be approximated by the following function:(32)fx,y=5.798−0.2985sin0.7926πxy+10.56e−3.413y2.
Next, one will choose the initial conditions of four quadrotors as follows:(33)r1,x0,r1,y0,r1,z0,v1,x0,v1,y0,v1,z0T=3.20,2.10,2.50,0.55,0.56,0.47T,r2,x0,r2,y0,r2,z0,v2,x0,v2,y0,v2,z0T=3.30,5.10,1.78,0.25,0.77,0.95T,r3,x0,r3,y0,r3,z0,v3,x0,v3,y0,v3,z0T=1.70,0.50,1.77,0.88,0.69,0.27T,r4,x0,r4,y0,r4,z0,v4,x0,v4,y0,v4,z0T=2.70,3.00,1.85,0.65,0.79,0.28T.
Based on the proposed control architecture, simulation outcomes are shown in Figure3. The 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. The 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.Figure 3
Simulation results. (a) The trajectory of all quadrotors in 3D space. (b) The adjacent distance between every two quadrotors. (c) The virtual velocity of all quadrotors. (d) The real velocity of all quadrotors. (e) The position curves of the leader and the quadrotors. (f) Response of attitude angles of all quadrotors.
(a)(b)(c)(d)(e)(f)


## 5.2. The 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.The comparative controller is a finite-time consistency control protocol; refer to [23]; the controller is selected as follows:(34)v^˙it=γ1sigβ1∑j=0naijv^jt−v^it,r^˙it=v^it−γ2sigβ2∑j=0naijr^it−r^jt−σij,where γ1>0, γ2>0, β1∈0,1, and β2∈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(35)r0,x0,r0,y0,r0,z0,v0,x0,v0,y0,vz,00T=0.2,1.6,2.0,0.50,0.50,0.50T.Next, four sets of initial positions are given for the four follower quadrotors; in order to conform to the reality, the position coordinates in theZ-axis direction are set to ri,z0=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:(36)r1,x0,r1,y0,r1,z0T=15.0,16.3,2.0T,r2,x0,r2,y0,r2,z0T=17.5,16.2,2.0T,r3,x0,r3,y0,r3,z0T=16.9,18.2,2.0T,r4,x0,r4,y0,r4,z0T=19.3,17.6,2.0T.In addition, the initial state of this simulation takes the position as an example, so the initial velocities of the four quadrotors are set asvi,x0,vi,y0,vi,z0T=0.5,0.5,0.5T.Based on the above analysis and data, the simulation results are shown in Figures4–6. 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).Figure 4
The adjacent distance between every two quadrotors under the control algorithm (34). (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).
(a)(b)(c)(d)Figure 5
The 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).
(a)(b)(c)(d)Figure 6
The flight trajectory of all quadrotors in the range of (10, 20). (a) Under the control algorithm (34). (b) Under the control algorithm (9).
(a)(b)Table 2
Comparison results with two control algorithms.
t0,2t3,5t5,10t10,20Control algorithm (34)3.396.499.85>15Control algorithm (9)2.532.512.542.79Performance improvement (%)25.461.374.2>81


## 6. 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 fixed-time 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. Third, 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.



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*Source: 9979713-2021-05-06.xml*