Modeling and Flocking Consensus Analysis for Large-Scale UAV Swarms

Abstract

Recently, distributed coordination control of the unmanned aerial vehicle (UAV) swarms has been a particularly active topic in intelligent system field. In this paper, through understanding the emergent mechanism of the complex system, further research on the flocking and the dynamic characteristic of UAV swarms will be given. Firstly, this paper analyzes the current researches and existent problems of UAV swarms. Afterwards, by the theory of stochastic process and supplemented variables, a differential-integral model is established, converting the system model into Volterra integral equation. The existence and uniqueness of the solution of the system are discussed. Then the flocking control law is given based on artificial potential with system consensus. At last, we analyze the stability of the proposed flocking control algorithm based on the Lyapunov approach and prove that the system in a limited time can converge to the consensus direction of the velocity. Simulation results are provided to verify the conclusion.

1. Introduction

UAV is an advanced system with high autonomy for intelligent combat [1]. In the future, UAVs will be used for complex tasks, such as surveillance, reconnaissance, and precision strike missions. Many organizations have foreseen that in the near future, swarms of UAVs will replace single ones for more complicated missions in more uncertain and possibly hostile environments [2]. Therefore, many researchers are studying groups of cooperative UAVs.

(A) Related Work on the UAV Swarms Problem. The new challenges imposed by UAV swarms have attracted many researchers. New control mechanisms, application domains, simulation models, and simulation tools have been developed to tackle issues in different aspects of the swarm. Currently, a completely new topic is opening up in the area of UAV swarms performing different missions cooperatively. [3], the method of evolutionary pinning control is applied to UAV swarms successfully. Path planning and routing are investigated in [4, 5], using multiobjective evolutionary algorithm. The path planning problem in three-dimensional environment without any obstacles is addressed in [6, 7] and with only static obstacles in [8]. [9], cooperative searching problem is discussed for the purpose of detecting moving and evading targets in a hazardous environment. A similar cooperative searching problem is also discussed in [10, 11]. Reference [12] investigates the automatic target recognition (ATR) problem in UAV control and proposes a distributed strategy for UAV swarms. Task allocation problem is discussed in [13–18] using different methodologies. Some applications of using a UAV swarm to search and destroy targets could be found in [19–22]. [23], a swarm simulator for target searching is implemented with Java. Garcia introduces a multi-UAVs simulator implemented with X-Plane—a commercial flight simulator [24]. Russell et al. present a parallel swarm simulation environment which utilizes an existing parallel emulation and simulation tool called SPEEDS [25]. MASON [26] is a general purpose multiagents simulation library utilized in our previous work, along with MATLAB based UAV simulator [27].

(B) Related Work on the Consensus Flocking Problem. Most research on formation of agent swarms uses distributed techniques by Reynolds’ seminal work on the mobility of flocks [28], which prescribes three fundamental operations for each robot to realize distributed flocking—separation, alignment, and cohesion [29, 30]. One of the earliest attempts to realize flocking through a set of basic behaviors including safe wandering, aggregation, dispersion, and homing to implement flocking is by Mataric [31]. Kelley and Keating realize flocking with robots based on leader-following behavior [32]. Hayes and Dormiani-Tabatabaei [33] propose a flocking algorithm based on two behaviors: collision avoidance and velocity-matching flock centering. Holland et al. [34] propose a flocking algorithm for UAV similar to Reynolds’. A host is used as an intermediate station for receiving each UAV’s range, bearing, and velocity and sending them to other UAVs to simulate the sensing process of one UAV for perceiving range, bearing, and heading of its neighbors. Ferrante et al. [35] introduce a new communication strategy called the information aware communication for alignment behavior. Recently, Stranieri et al. [36] perform flocking with a swarm of behaviorally heterogeneous mobile robots.

In this paper, we consider models for flocking swarms. Firstly, a mathematical model of cooperative system is established by using Markov stochastic process and calculus analysis. Then, the control law for UAV swarm is established based on artificial potential field. At last, we analyze the stability of the proposed flocking control algorithm based on the Lyapunov approach and prove the conclusion that the system in a limited time can converge to the consensus direction of the velocity. Simulation results are provided to verify the conclusion.

2. The Model of the UAVs Swarms

2.1. Differential Integral Model

Let 𝐶(𝑡) denote the state of the UAV swarms at time 𝑡;𝐶(𝑡)=0 identifies the state that UAV swarms are stable at time 𝑡. The state of UAV 𝑖 at time 𝑡 is denoted by 𝑐_𝑖(𝑡)=(𝑝_𝑖(𝑡),𝑜_𝑖(𝑡)), in which the first element 𝑝_𝑖(𝑡)=(𝑥_𝑖(𝑡),𝑦_𝑖(𝑡),𝑧_𝑖(𝑡)) is the UAV’s position in the environment at time 𝑡, and the second element 𝑜_𝑖(𝑡) is the UAV’s orientation. The UAV’s dynamics is subject to its physical curvature radius constraints, leading to the fact that it can only change its orientation by at most one step, which is described as go straight, go up, go down, turn left, turn upper left, turn lower left, turn right, turn upper right, and turn lower right.

In order to obtain Markov random process, the new state of process is derived by supplement of variable [37, 38], which is described as follows: 𝑃_𝑖(𝑡)(𝐶(𝑡))𝑃=𝑃=𝑖,𝑖=0,1,2,…,𝑁,_𝑗𝑘(𝑥,𝑡)=𝑃𝐶(𝑡)=𝑗,𝑥<𝑦_𝑗,𝑘(𝑡)<𝑥+𝑑𝑥,𝑗=𝑁+1,…,𝑀,𝑘=0,1,2,…,𝑁, (1) where 𝑦_𝑖(𝑡) is the dwell time after state 𝑖. So it is easy to verify that {𝑈(𝑡),0≤𝑡≤𝑇₀}=(𝐶(𝑡),𝑦(𝑡)) is a broad Markov random process.

The probability of state transition after Δ𝑡 can be obtained using total probability theorem: 𝑃₀(𝑡+Δ𝑡)=^𝑀_𝑖=0𝑃(𝐶(𝑡)=𝑖,𝐶(𝑡+Δ𝑡)=𝑖)=𝑃₀(𝑡)1−^𝑁_𝑖=1𝜆_0𝑖Δ𝑡+𝑜(Δ𝑡)+^𝑁_𝑖=1𝜆_𝑖0𝑃_𝑘(𝑡)+Δ𝑡_{𝑀𝑗=𝑁+1𝑇0}𝑃_𝑗(𝑥,𝑡)𝑟_𝑗(𝑥)Δ𝑡𝑑𝑥+𝑜(Δ𝑡). (2) According to (2) we can get the all probability: 𝑃_𝑘(𝑡+Δ𝑡)=𝑃_𝑘(𝑡)1−^𝑁_{𝑖=1,𝑖≠𝑘}𝜆_𝑘𝑖()+Δ𝑡+𝑜Δ𝑡_{𝑁𝑖=1,𝑖≠𝑘}𝜆_𝑖𝑘𝑃_𝑘(𝑡)(),𝑃Δ𝑡+𝑜Δ𝑡𝑘=1,2,…,𝑁,_𝑗(𝑥,𝑡)=𝑃𝐶(𝑡)=𝑗;thetimeofself-organised𝐶_system𝑥,𝐶()is𝑡+Δ𝑡=𝑗=𝑃_𝑗(𝑥,𝑡)1−𝑟_𝑗(𝑥)Δ𝑡+𝑜(Δ𝑡),𝑗=𝑁+1,…,𝑀, (3) where 𝜆_𝑖𝑗 is the average sustained rate of each state and 𝑟_𝑗(𝑥) is the average repair rate at state 𝑗. Similarly, the expression of state transition rate for 𝑃_𝑗(𝑥+Δ𝑥,𝑡+Δ𝑡) can be derivated.

Differentiate the expression for state transition probability to derive its limit. Then the mathematical model can be described using integral-differential equations as follows: 𝑑𝑃₀(𝑡)+𝑑𝑡^𝑁_{𝑖=0,𝑖≠𝑘}𝜆_0𝑘𝑃₀(𝑡)=^𝑁_{𝑖=0,𝑖≠𝑘}𝜆_𝑖0𝑃_𝑖(𝑡)+^𝑀_𝑗=𝑁+1^𝑇₀𝑟_𝑗(𝑥)𝑝_𝑗(𝑥,𝑡)𝑑𝑥𝑑𝑃_𝑘(𝑡)+𝑑𝑡^𝑁_{𝑖=0,𝑖≠𝑘}𝜆_𝑘𝑖𝑃_𝑘(𝑡)=^𝑁_{𝑖=0,𝑖≠𝑘}𝜆_𝑖𝑘𝑃_𝑖(𝑡),𝑘=1,2,…,𝑁,𝜕𝑝_𝑗(𝑥,𝑡)+𝜕𝑥𝜕𝑝_𝑗(𝑥,𝑡)𝜕𝑡+𝑟_𝑗(𝑥)𝑝_𝑗(𝑥,𝑡)=0,𝑗=𝑁+1,…,𝑀. (4) The boundary and initial conditions are 𝑃₀(0)=1,𝑃₁(0)=0,…,𝑃_𝑖(0)=0,…,𝑃_𝑁(0)𝑃=0,_𝑗(0,𝑡)=^𝑁_𝑖=0𝜆_𝑖𝑗𝑃_𝑖(𝑡),𝑗=𝑁+1,…,𝑀. (5)

Theorem 1. The reliability of coordination system has uniqueness and nonnegative solution on 𝐶[0,𝑇].

Proof. According to the initial conditions we can get the analytic solution of the partial differential equation [39, 40].
Set 𝑎₀=^𝑁_𝑖=1𝜆_0𝑖,…,𝑎_𝑗=^𝑁_𝑖=1𝜆_𝑗𝑖,…,𝑎_𝑁=^𝑁_𝑖=1𝜆_𝑁𝑖,𝑃_𝑗(𝑥,𝑡)=𝑃_𝑖(0,𝑡−𝑥)exp−^𝑥₀𝑟_𝑖𝜇𝑑𝜇,𝑖=𝑁+1,…,𝑀𝑑𝑃₀(𝑡)𝑑𝑡=−𝑎₀𝑃₀(𝑡)+^𝑁_{𝑖=0,𝑖≠𝑘}𝜆_𝑖0𝑃_𝑖(𝑡)+^𝑀_𝑗=𝑁+1^𝑇₀𝑃_𝑗(0,𝑡−𝑥)exp−^𝑥₀𝑟_𝑖𝜇𝑑𝜇×𝑟_𝑖(𝑥)𝑃𝑑𝑥,₀(𝑡)=exp−𝑎₀𝑡+^𝑁_𝑖=1𝜆_𝑖0^𝑇0₀𝑃_𝑖(𝑠)exp−𝑎₀(𝑡−𝑠)𝑑𝑠+^𝑇0₀^𝑀_𝑖=𝑁+1^𝑠₀𝑃_𝑖(0,𝜏)exp−₀^𝑠−𝜏𝑟_𝑖𝜇𝑑𝜇×𝑟_𝑖(𝑠−𝜏)𝑑𝜏×exp−𝑎₀(𝑡−𝑠)𝑑𝑠=exp−𝑎₀𝑡+^𝑁_𝑖=1𝜆_𝑖0^𝑇0₀𝑃_𝑖(𝑠)exp−𝑎₀()+𝑡−𝑠×𝑑𝑠_{𝑀𝑖=𝑁+1𝑇00}𝑃_𝑖()×0,𝜏𝑑𝜏_{𝑇00−𝜏}exp−𝑎₀𝑇₀−𝜏+𝑎₀𝑣−^𝑣₀𝑟_𝑖𝜇𝑑𝜇𝑟_𝑖(𝑣)𝑑𝑣=exp−𝑎₀𝑡+^𝑁_𝑖=1𝜆_𝑖0^𝑇0₀𝑃_𝑖(𝑠)𝑘₀+𝑑𝑠_{𝑀𝑖=𝑁+1𝑇00}𝑃_𝑖(0,𝜏)𝐾_𝑖(𝑡−𝜏)𝑑𝜏, (6) where 𝑘₀=exp−𝑎₀()𝑘𝑇−𝑠_𝑖(𝑇−𝜏)=₀^𝑇−𝜏𝑘₀∗exp𝑎₀𝑣−^𝑣₀𝑟_𝑖𝜇𝑑𝜇𝑟_𝑖(𝑣)𝑃𝑑𝑣,𝑖=𝑁+1,…,𝑀,_𝑗(𝑡)=^𝑁_{𝑖=0,𝑖≠𝑗}^𝑇₀exp−𝑎_𝑗(𝑇−𝑠)𝜆_𝑖𝑗𝑃_𝑖(𝑠)=𝑑𝑠_{𝑁𝑖=0,𝑖≠𝑗𝑇0}𝑘_𝑗𝜆_𝑖𝑗𝑃_𝑖(𝑠)𝑑𝑠,𝑗=1,2,…,𝑁, (7) where 𝑘_𝑗=exp(−𝑎_𝑗(𝑇−𝑠)),𝑗=1,2,…,𝑁.
So we can get the following equation: 𝑃_𝑗(0,𝑡)=^𝑁_𝑖=0𝜆^𝑁_{𝑖𝑗𝑙=0,𝑙≠𝑗}^𝑇₀𝑘_𝑗𝜆_𝑙𝑗𝑃_𝑙(𝑠)𝑑𝑠=^𝑇₀𝑘_𝑗^𝑁_𝑖=0𝜆_𝑖𝑗^𝑁_{𝑙=0,𝑙≠𝑗}𝜆_𝑙𝑗𝑃_𝑙(𝑠)𝑑𝑠,𝑗=𝑁+1,…,𝑀. (8) Assuming 𝑃(𝑡)=𝑃₀(𝑡),𝑃₁(𝑡),…,𝑃_𝑁(𝑡),𝑃_𝑁+1(0,𝑡),…,𝑃_𝑀(),𝑓(𝑡)=𝑓0,𝑡₀(𝑡),𝑓₁(𝑡),…,𝑓_𝑁(𝑡),𝑓_𝑁+1(0,𝑡),…,𝑓_𝑀()=0,𝑡exp−𝑎₀𝑡,0,…,0,…,0, (9)
then, the solution of the system can be converted into vectors format as follows:

𝑃(𝑡)=𝑓(𝑡)+^𝑇₀𝑘()𝑃(𝑠)𝑘()=𝑇−𝑠𝑑𝑠,𝑇−𝜏^{0𝜆10𝑘0⋯𝜆𝑁0𝑘0𝑘𝑁+1()𝑇−𝑠⋯𝑘𝑀()𝜆𝑇−𝑠01𝑘10⋯𝜆𝑁1𝑘1𝜆0⋯0⋮⋮⋯⋮⋮⋯⋮0𝑁𝑘𝑁𝜆2𝑁𝑘𝑁⋯00⋯0}_{𝑁𝑙=0,𝑙≠𝑗𝜆𝑙𝑗𝜆0𝑁+1𝑁𝑙=0,𝑙≠𝑗𝜆𝑙𝑗𝜆1𝑁+1⋯𝑁𝑙=0,𝑙≠𝑗𝜆𝑙𝑗𝜆𝑁𝑁+10⋯0⋮⋮⋯⋮⋮⋯⋮𝑁𝑙=0,𝑙≠𝑗𝜆𝑙𝑗𝜆0𝑀𝑁𝑙=0,𝑙≠𝑗𝜆𝑙𝑗𝜆1𝑀⋯𝑁𝑙=0,𝑙≠𝑗𝜆𝑙𝑗𝜆𝑁𝑀0⋯0}. (10)

Any component of 𝐹(𝑡) and 𝐺(𝑡−ℎ) vector is nonnegative. The functions 𝐹_𝑗(𝑡) and 𝐺_𝑗(𝑡−ℎ),𝑗=1,2,…,𝑀are limitary on the domain 0<𝑇<+∞. The solution of integral equation is unique and nonnegative on 𝐶[0,𝑇]. So the reliability of coordination system has unique and nonnegative solution on 𝐶[0,𝑇].

2.2. Probabilistic Analysis Based on State Transformation

The behavior evolution of the UAV swarm system is a limited Markov decision process. Suppose that the probability distribution of the system state is𝑃(𝑁,𝑡)at time 𝑡. Then at time 𝑡+𝜏, the probability distribution is 𝑃(𝑁,𝑡+𝜏). According to the relationship of the probability density at different time, the marginal probability density 𝑃(𝑁,𝑡+𝜏) is 𝑃(𝑁,𝑡+𝜏)=∫𝑃(𝜂,𝑡)𝑃(𝑁,𝑡+𝜏∣𝜂,𝑡)𝑑𝜂.

And the time derivative of the 𝑃(𝑁,𝑡+𝜏) is 𝑑𝑃(𝑁,𝑡)𝑑𝑡=lim_𝜏→0𝑃(𝑁,𝑡+𝜏)−𝑃(𝑁,𝑡)𝜏. (11) Define 𝑊_𝑡(𝜂,𝑁) as the transition probability density from state 𝜂 to state 𝑁 in unit time during time interval [𝑡,𝑡+𝜏]. So the transition probability from state 𝜂 to state 𝑁 during time interval [𝑡,𝑡+𝜏] is 𝜏𝑊_𝑡(𝜂,𝑁). Then the probability by which the transition does not happen is 1−𝜏𝑊_𝑡𝜂,𝑁𝛿𝑁−𝜂𝑑𝑛, (12) where 𝛿(𝑁−𝜂)=𝑃(𝑁,𝑡+𝜏∣𝜂,𝑡), when 𝜏=0. Thus, 𝑑𝑃(𝑁,𝑡)=𝑊𝑃𝑃(𝑑𝑡𝜂,𝑛𝜂,𝑡−𝑊𝑛,𝜂𝑛,𝑡)𝑑𝜂. (13) Equation (13) describes the evolution of the system states over time, which is the primary equation model of the UAV swarms behavior.

3. Flocking Control of UAV Swarms

3.1. Flocking Control Law

In this section, first we design a distributed flocking control law. Assuming that each UAV senses its own position and velocity and is able to obtain its neighbors’ position and velocity, the UAV swarms form flocking behaviour model structure control law as follows: 𝑈_𝑖=Uniformity(𝑣)^𝑁_{𝑗=1,𝑗≠𝑖}𝑓𝑝_𝑖,𝑗𝑝_𝑖−𝑝_𝑗+𝛽𝑝_𝑖−𝑝_goal+^𝑁_{𝑗=1,𝑗≠𝑖}𝑎_𝑖,𝑗𝑝_𝑖,𝑗𝑣_𝑗−𝑣_𝑖, (14) where Uniformity(𝑣)=(𝛼∑_𝑖≠𝑗‖𝑣_𝑖−𝑣_𝑗‖₂). Note that alignment at a common velocity is equivalent to Uniformity(𝑣)=0.𝑃_𝑖,𝑗 is the distance between the individual 𝑖 and 𝑗.  𝑈_𝑖𝑗 is potential function and satisfies the condition [29, 30]:(i)𝑈_𝑖𝑅(‖𝑝_𝑖−𝑝_𝑗‖)→∞,‖𝑝_𝑖−𝑝_𝑗‖<𝑅_min,(ii)𝑅_min≤‖𝑝_𝑖−𝑝_𝑗‖≤𝑅_max,∃!min𝑈_𝑖𝑅.

3.2. Stability Analysis

Consider the following positive semidefinite function: 1𝐸=2^𝑁_𝑖=1^𝑁_{𝑗=1,𝑗≠𝑖}𝑈_𝑖,𝑗+𝐾_𝑖𝑈_𝑖,goal+𝑣^𝑇_𝑖𝑣_𝑖. (15) In order to facilitate writing, we simplify the certification process variable substitution as follows: 𝑈_𝑖=^𝑁_{𝑗=1,𝑗≠𝑖}𝑈_𝑖,𝑗+𝐾_𝑖𝑈_𝑖,goal,𝐸̇=12^𝑁_𝑖=1𝑈̇‌_𝑖−^𝑁_𝑖=1𝑣^𝑇_𝑖𝑈_𝑖=12^𝑁_𝑖=1𝑈̇‌_𝑖+^𝑁_𝑖=1𝑣^𝑇_𝑖−^𝑁_{𝑗=1,𝑗≠𝑖}∇_𝑝𝑖𝑈_𝑖𝑗𝑝_𝑖,𝑗−𝐾_𝑖∇_𝑝𝑖𝑈_𝑖𝐴𝑝_𝑖,𝑗−^𝑁_{𝑗=1,𝑗≠𝑖}𝛼_𝑖,𝑗𝑣_𝑖−𝑣_𝑗=^𝑁_𝑖=1𝑣^𝑇_𝑖∇_𝑝𝑖𝑈_𝑖+^𝑁_𝑖=1𝑣^𝑇_𝑖−^𝑁_{𝑗=1,𝑗≠𝑖}∇_𝑝𝑖𝑈_𝑖𝑗𝑝_𝑖,𝑗−𝐾_𝑖∇_𝑝𝑖𝑈_𝑖,goal𝑝_𝑖,goal−^𝑁_{𝑗=1,𝑗≠𝑖}𝛼_𝑖,𝑗𝑣_𝑖−𝑣_𝑗=^𝑁_𝑖=1𝑣^𝑇_𝑖∇_𝑝𝑖^𝑁_{𝑗=1,𝑗≠𝑖}𝑈_𝑖,𝑗+𝐾_𝑖𝑈_𝑖,goal+^𝑁_𝑖=1𝑣^𝑇_𝑖−^𝑁_{𝑗=1,𝑗≠𝑖}∇_𝑝𝑖𝑈_𝑖𝑗−𝐾_𝑖∇_𝑝𝑖𝑈_𝑖,goal−^𝑁_{𝑗=1,𝑗≠𝑖}𝛼_𝑖,𝑗𝑣_𝑖−𝑣_𝑗=^𝑁_𝑖=1𝑣^𝑇_𝑖−^𝑁_{𝑗=1,𝑗≠𝑖}𝛼_𝑖,𝑗𝑣_𝑖−𝑣_𝑗=−𝑣^𝑇𝐿_𝐶⊗𝐼₃𝑣, (16) where 𝐿_𝑐 is UAV swarms system satisfying the Laplacian matrix of the communication conditions. Therefore, the quadratic form is explicitly described as follows: 𝐸̇=−𝑣^𝑇_𝑥𝐿_𝑐𝑣_𝑥−𝑣^𝑇_𝑦𝐿_𝑐𝑣_𝑦−𝑣^𝑇_𝑧𝐿_𝑐𝑣_𝑧≤0. (17) Consider the following collections: {𝑣_𝑖,𝑃_𝑖,𝑗∣𝐸≤𝐶} is a closed set. The following is to verify that it is a compact set, and there is a clear conclusion that 𝑃_𝑖,𝑗≤𝐶. Similarly 𝑣^𝑇_𝑖𝑣_𝑖≤𝐶,‖𝑣_𝑖‖≤𝐶^1/2, and according to the definition of the potential field we obtain ‖𝑃_𝑖,𝑗‖≤𝑈⁻¹_𝑖,𝑗(𝐶(𝑁−1)). According to the LaSalle invariance principle, the system will converge to the largest invariant set in the area and meet 𝐸̇=0. According to 𝐸̇=0, when the system enters the steady state, the speed of each individual is equal, and all individuals move to the target position 𝑃_goal, making the overall potential energy minimum.

Theorem 2. Consider the UAV swarms consisting of 𝑁 UAVs. The position of individual 𝑖 is 𝑝_𝑖. All individuals in the swarms will eventually build up to the spherical region: 𝑝_𝑖−𝑝_𝑐≤2𝜆_max(𝐺)(𝑁−1)𝑙𝐶𝜆_min𝐴^𝑇.𝐺+𝐺𝐴 (18)

Proof. Consider 𝑃_𝑐̇=𝑉_𝑐𝑉̇_𝑐=1𝑁^𝑁_𝑖=1𝑣̇_𝑖=1𝑁^𝑁_𝑖=1−^𝑁_{𝑗=1,𝑗≠𝑖}∇_𝑝𝑖𝑈_𝑖𝑗𝑝_𝑖−𝑝_𝑗−𝛽_𝑖∇_𝑝𝑖𝑈_𝑖𝐴𝑝_𝑖−𝑝_𝑐−^𝑁_{𝑗=1,𝑗≠𝑖}𝛼_𝑖,𝑗𝑣_𝑖−𝑣_𝑗=0, (19) where 𝑃_𝑐(𝑡)=(1)/𝑁∑^𝑁_𝑖=1𝑝_𝑖(𝑡),𝑉_𝑐=(1)/𝑁∑^𝑁_𝑖=1𝛼_𝑖,𝑗𝑣_𝑖.
By making the variable replacement 𝜀_𝑖𝑝=𝑃_𝑖−𝑃_𝑐,𝜀_𝑖𝑝=𝑣_𝑖−𝑣_𝑐, we get 𝜀̇_𝑖𝑝=𝜀_𝑖𝑣,𝜀̇_𝑖𝑝=−^𝑁_{𝑗=1,𝑗≠𝑖}∇_𝑝𝑖𝑈_𝑖𝑗𝑝_𝑖−𝑝_𝑗−𝛽_𝑖∇_𝑝𝑖𝑈_𝑖𝐴𝑝_𝑖−𝑝_𝑐^𝑁_{𝑗=1,𝑗≠𝑖}𝛼_𝑖,𝑗𝑣_𝑖−𝑣_𝑗=−𝛽_𝑖𝜀_𝑖𝑝−𝑁^𝑁_{𝑗=1,𝑗≠𝑖}𝛼_𝑖,𝑗𝜀_𝑖𝑣−^𝑁_{𝑗=1,𝑗≠𝑖}∇_𝑝𝑖𝑈_𝑖𝑗𝑝_𝑖−𝑝_𝑗. (20) Then 𝜁_𝑖̇=𝜀̇_𝑖𝑝𝜀̇_𝑖𝑣=01−𝛽_𝑖−𝑁^𝑁_{𝑗=1,𝑗≠𝑖}𝛼_𝑖,𝑗𝜁_𝑖−_{𝑁𝑗=1,𝑗≠𝑖}0∇_𝑝𝑖𝑈_𝑖𝑗𝑝_𝑖−𝑝_𝑗,𝐴=01−𝛽_𝑖−𝑁^𝑁_{𝑗=1,𝑗≠𝑖}𝛼_𝑖,𝑗,0𝐵=^𝑁_{𝑗=1,𝑗≠𝑖}∇_𝑝𝑖𝑈_𝑖𝑗𝑝_𝑖−𝑝_𝑗. (21) Since 𝐾_𝑖,𝑁>0, 𝜆(𝐴)=−𝑁∑^𝑁_{𝑗=1,𝑗≠𝑖}𝛼_𝑖,𝑗2±𝑁∑^𝑁_{𝑗=1,𝑗≠𝑖}𝛼_𝑖,𝑗²−4𝑘_𝑖2,Re(𝜆(𝐴))<0. (22) The largest and the smallest eigenvalues of symmetric positive definite matrix 𝑆 are 𝜆_max(𝑆) and 𝜆_min(𝑆), respectively. The symmetric positive definite matrix𝑆with appropriate dimensions satisfies the following conclusion [41, 42]. 𝜆_min(𝑆)𝜁_𝑖²≤𝜁^𝑇_𝑖𝑆𝜁_𝑖≤𝜆_max(𝑆)𝜁_𝑖². (23)
Finally, select Lyapunov function 𝐸_𝑖=𝜁^𝑇_𝑖𝐺𝜁_𝑖,𝐺^𝑇=𝐺,𝜆(𝐺)>0. (24)
Time derivative can be obtained: 𝐸_𝑖̇=𝜁^𝑇_𝑖̇𝐺𝜁_𝑖+𝜁^𝑇_𝑖𝐺𝜁_𝑖̇=𝐴𝜁_𝑖−𝐵^𝑇𝐺𝜁_𝑖+𝜁^𝑇_𝑖𝐺𝐴𝜁_𝑖−𝐵=−𝜁^𝑇_𝑖𝐴^𝑇𝐺+𝐺𝐴𝜁_𝑖−2𝜁^𝑇_𝑖𝑈𝐺𝐵,_𝑖𝑗𝑝_𝑖−𝑝_𝑗𝑝_𝑖−𝑝_𝑗≤𝑙𝐶. (25) Therefore, according to the above formula we obtain 𝐸_𝑖̇≤−𝜆_min𝐴^𝑇𝐺+𝐺𝐴𝜁_𝑖²+2𝜆_max(𝐺)𝜁_𝑖(𝑁−1)𝑙𝐶. (26) When 𝜁_𝑖>2𝜆_max(𝐺)(𝑁−1)𝑙𝐶𝜆_min𝐴^𝑇,𝐺+𝐺𝐴 (27)𝐸̇_𝑖<0. The system continues to move closer to the population centre. Therefore, eventually the system stabilizes at a known system of 𝜁_𝑖≤2𝜆_max(𝐺)(𝑁−1)𝑙𝐶𝜆_min𝐴^𝑇.𝐺+𝐺𝐴 (28)

4. Simulation of System Flocking Formation Behavior

According to the UAV’s physical characteristics, this paper will discretize the time with high frequency. Thus, a UAV 𝑖 makes its path decision 𝑃_𝑖(𝑡+1) at time-step 𝑡 and will execute an action as the following equation: 𝑃_𝑖(𝑡+1)=𝑃_𝑖(𝑡)+𝑣Δ𝑡. (29)

The movement of the individual is not only controlled by itself but also affected by the state of other individuals. Therefore, the individual direction of movement at a certain time is not only relative to its direction one moment before, but also relative to the directions of its surrounding individuals’ movements. The influence of all the individuals to the individual 𝑖 can be described as the following equation: 1∑1/𝑝^𝑛_{𝑖𝑗𝑗=1}𝜃_𝑗(𝑡)𝑝_𝑖𝑗. (30)

Then, the speed direction of the UAV 𝑖 at time (𝑡+1) can be modified as the following equation: 𝜃_𝑖(𝑡+1)=𝛼𝜃_𝑖(𝑡)1+𝛽∑1/𝑝^𝑛_{𝑖𝑗𝑗=1}𝜃_𝑗(𝑡)𝑝_𝑖𝑗𝑦+𝛾arctan⁰−𝑦₁𝑥₀−𝑥₁. (31)

We consider the swarms of 100 UAVs with six degrees of freedom. The weights of the cost function are set to 𝛼=0.3,𝛽=0.5,𝛾=0.2, 𝑉=[200,0,0] m/s and 𝑚=25kg. Direction is the rand variable from −2 * pi to 2 * pi. The position of the UAVs is the rand variable. The factors of the influence on the flight are wind and airstream. The results for the case of 100 UAVs are shown in Figures 1 and 2.

Figure 1 

Velocities with respect to time.

Figure 2 

Trajectories with respect to time.

From Figure 1, at 𝑡=19s, the velocities of the swarms achieve consensus at 𝜃=3.2.

Figure 2 describes the trajectories with respect to time. The UAV swarms system will eventually be able to form a stable distance between each individual and the same velocity vectors. Collision between individuals is thus avoided.

Figures 3, 4, and 5 show the Pitch, Roll, and Attack with respect to time. From the simulation results, we can conclude that the UAVs based on the method successfully fly after the adjustment at the initial stage.

Figure 3 

Angle of the Pitch with respect to time.

Figure 4 

Angle of the Roll with respect to time.

Figure 5 

Angle of the Attack with respect to time.

Figure 6 shows the Sideslip with respect to time. Through the analysis of the Sideslip Angle, we can find that the Angle of the Sideslip is less than 0.5 degrees and tends to zero, to ensure the turning flight control.

Figure 6 

Angle of the Sideslip with respect to time.

5. Conclusion

This paper analyzed current researches and existent problems of UAV swarms. Afterwards, by the theory of stochastic process and supplemented variables, a differential-integral model was established. The existence and uniqueness of the solution of the system were discussed. The flocking control law is given based on artificial potential with system consensus. At last, we analyzed the stability of the proposed flocking control algorithm based on the Lyapunov approach and proved the conclusion that the system in 28 s can converge to the consensus direction of the velocity. And we performed simulation tests to verify the conclusion.

Acknowledgment

This paper is supported by The National Defense Pre-Research Foundation of China (Grant no. B222011XXXX).