Distributed Multicircular Circumnavigation Control for UAVs with Desired Angular Spacing

Shixiong Li , Xingling Sho ,*, Wendong Zhng , Qingzhen Zhng

a National Key Laboratory for Electronic Measurement Technology,School of Instrument and Electronics,North University of China,Taiyuan 030051,China

b School of Automation Science and Electrical Engineering, Beihang University, Beijing 100086, China

Keywords: Angular spacing Distributed observer Multicircular circumnavigation Moving target UAVs

ABSTRACT This paper addresses a multicircular circumnavigation control for UAVs with desired angular spacing around a nonstationary target.By defining a coordinated error relative to neighboring angular spacing,under the premise that target information is perfectly accessible by all nodes, a centralized circular enclosing control strategy is derived for multiple UAVs connected by an undirected graph to allow for formation behaviors concerning the moving target.Besides, to avoid the requirement of target’s states being accessible for each UAV, fixed-time distributed observers are introduced to acquire the state estimates in a fixed-time sense, and the upper boundary of settling time can be determined offline irrespective of initial properties,greatly releasing the burdensome communication traffic.Then,with the aid of fixed-time distributed observers, a distributed circular circumnavigation controller is derived to force all UAVs to collaboratively evolve along the preset circles while keeping a desired angular spacing.It is inferred from Lyapunov stability that all errors are demonstrated to be convergent.Simulations are offered to verify the utility of proposed protocol.

1.Introduction

During the past few decades,motion control of unmanned aerial vehicles (UAVs) has acquired tremendous spotlights [1-4], and it has been widely practiced in diversified engineering applications ranging from surveillance and tracking missions [5,6], goods delivery [7,8] to search and localization [9,10].Moreover, efficient operation of a cluster of UAVs demonstrates great efficiency, good robustness and high flexibility than deploying a single UAV.And a variety of coordinated control strategies have been researched based on graph theory, such as coordinated trajectory tracking[11,12], cooperative path following [13,14], and collaborative enclosing control [15,16].Among which, the objective of collaborative enclosing control is to drive multiple agents to orbit around interested targets periodically and maintain a desired formation.

Early advancements relative to target enclosing are reported for single integrators [17-19], double integrators [20].Note that nonholonomic constraints are ubiquitous in diversified unmanned systems,thus considerable attentions are dedicated to solve target encircling issues for UAVs [21,22], nonholonomic vehicles [23,24]and mobile robots [25-27].In Ref.[21], an encircling control scheme is designed for UAVs to constitute a circular trajectory around targets.A control strategy based on range measurement is proposed in Ref.[22] to enable all UAVs to maintain a circular motion around a target.In Ref.[23], a distributed localization and circumnavigation algorithm is interviewed for nonholonomic vehicles to disperse evenly along the common circle.In Ref.[24],enclosing a static target is achieved by devising a distributed coordinated control to hold an evenly spaced distribution on a circular curve.A cooperative target enclosing control with respect to static targets is presented in Ref.[25], in which all robots move around a fixed target at a constant speed.In Ref.[26],a distributed control protocol is designated for robots to form a static circular formation around a target with an evenly spaced formation.A distributed containment-maneuvering problem is investigated for multi-agents by using a modular design method in Ref.[27].However, the aforementioned literatures are limited to stationary targets,i.e.,the ratios of linear velocity to angular velocity are time invariant, which cannot be applied to nonstationary targets.In addition, there are relatively fewer reports available for the problem of cooperative moving target enclosing.Aiming at this problem, moving target enclosing control protocols for a group of nonholonomic agents are presented in Refs.[28-32].In Ref.[28],with the aid of the relative positions between vehicles and targets,a circular encircling control law is established for vehicles to configure a desired formation pattern.A moving target formation comprised by nonholonomic vehicles via local measurements is studied in Refs.[29,30].In Ref.[31], by introducing an adaptive distributed observer to reconstruct the position of target, a surrounding control method for multi-agents is considered with the aid of algebraic graph theory.A cooperative cyclic pursuit solution is discussed for both single and double integrators to seek evenly spaced circular formation, rendezvous to a point as well as evenly spaced logarithmic spirals respectively [32].

Notice that the aforementioned protocols with respect to static targets [22-27] and moving targets [28-31] are only effective in traveling along a common circular path centered at the interested targets with an evenly spaced phase distribution.Additionally,the availability of Ref.[32]is closely related to the ring network,where each agent can only exchange information and pursue states with respect to leading neighbors, which is a restrictive scenario for coordinated formation design.However, for real-world aerial assignments such as obtaining the maximum sensor coverage of a target and preventing the moving target from escaping [33], it always imposes control design of UAVs with a flexible angular spacing regulation ability.To this end, a distributed formation control protocol for vehicles is involved in Ref.[34], wherein the global convergence to any desired spaced formation is retained,but all agents involved are assumed to enclose targets along a shared circle.Note that it is more rewarding for a union of UAVs to evolve along multiple circles to boost the execution efficiency and cluster safety.In particular,for the scenarios of cooperative convoying,it is preferred to deploy multiple UAVs along different circular orbits concerning high-value targets, such that the targets can be protected from all-aspect threatens from possible attackers.However,there lack sufficient studies concerning multicircular circumnavigation with a desired spacing, and it should be highlighted that existing enclosing outcomes[28-30] assume that the information of the target can be globally available for each agent, triggering a heavy communication burden,which is not suitable for widespread cases with security concerns and limited transmission bandwidths.

To obviate the demand of global information being distributed to each agent, finite-time distributed observers [35,36] and fixedtime distributed observers [37-39] have attracted much attentions.In Ref.[35],a finite-time distributed estimator is proposed to estimate orientation for a group of agents without information of a global coordinate frame.By introducing a finite-time distributed observer,a formation tracking problem for quadrotors is studied in Ref.[36], wherein the relative state errors between target and quadrotors can be estimated.However, the convergence times of these researches are always associated with initial conditions,and the information on initial conditions is inaccurate or unknown in some cases, which may cause certain uncertainties in accurately predicting settling times and further limit the application range.Fortunately,a fixed-time distributed observer is considered whose settling time is upper bounded by a constant irrelevant with the initial conditions.This nice characteristic stimulates its potential practices in multi-agent systems[37-39].For instance,a variety of fixed-time control protocols are investigated in Refs.[37,38] for a single integrator to ensure that the upper limit of decaying time is independent of initial states.Subsequently,a fixed-time consensus is established for mobile robots with nonholonomic dynamics[39].However,how to generalize the fixed-time distributed observer to achieve multicircular enclosing assignments with any desired angular spacing around moving targets is still open and challenging owing to the strong coupling between circumnavigation and observation subsystems.

Encouraged by previous statements, a distributed multicircular circumnavigation control is presented for UAVs with a desired angular spacing,in which the desired radii around a moving target are not identical.The main contributions are listed as below.

· Distinguishing from the previous enclosing alternatives[21-31]that enforce agents to implement circumnavigation along a common circle with an evenly distributed phase, herein the proposed method can control a group of UAVs to circle around a moving target while maintaining a desired spaced formation over several circles.Different from the encircling control schemes for single integrators [17-19] and double integrators[20], herein a practical circumnavigation control rule for UAVs with nonholonomic constraints is designated to force UAVs to reach predetermined formation while maintaining a desired spacing.Unlike the cooperative cyclic pursuit solution[32]that seeks three special enclosing patterns subject to a ring communication, a general multicircular circumnavigation with arbitrarily controllable angular spacing is pursued, while onpath collaborative motions are manipulated through an undirected graph rather than a ring interaction, which can remove the stringent constraints of communicating with leading neighbors.

· Compared to the existing encircling strategies [28-30], where the information of target is available by all agents, by introducing a fixed-time distributed observer into the cooperative encircling scheme, the suggested method can avoid the requirement of global information being accessible for each UAV and greatly save the limited communication usages.Unlike the prevalent finite-time distributed estimators [35,36], whose settling time depends explicitly on initial conditions of agents,causing certain uncertainties in accurately predicting settling times, herein coordinated circular enclosing behaviors with a uniform convergent speed can be allowed even with different choices of initial conditions by introducing fixed-time distributed observers, and more attempts are made in the stability analysis of resultant enclosing systems comprising of fixed-time distributed observers and circular enclosing controller.

The structure of this paper is organized as follows.Section 2 introduces problem formulation.The main results are elaborated in Section 3.Simulations results are provided in Section 4.Conclusion is drawn in Section 5.

2.Preliminaries and problem statement

2.1.Problem statement

In this article,a multicircle circumnavigation problem consisting of a group ofNUAVs and a moving target is considered with a predetermined formation, as shown in Fig.1.The position ofi-th UAV is denoted as pi=［xi，yi］Tin the earth-fixed referenceX-Y.Revisiting [21], the kinematics ofi-th UAV is formulated as

where ψiis the heading angle, vi＞0 andrirepresent the linear velocity and angular rate, respectively.The UAV states involving velocity viand angular rateriare bounded and measureable for feedback, i.e., 0 ＜vi≤vmaxandrmin≤ri≤rmax, where vmaxis the upper bound of velocity,rminandrmaxare the lower and upper bounds of angular rate respectively.

Fig.1.Geometrical illustration of cooperative circular enclosing.

The kinematics of the moving target pT=［xT，yT］Tis given as

where vTis the target velocity and its derivativeare assumed to be bounded and there must exist the upper bound ofowing to the maneuvering constraints., i.e.,≤ζT, ζTis a positive constant.

2.2.Graph theory

For the considered multi-agent systems,assume that each UAV is a node and the information exchange of UAVs is denoted by an undirected graph G =｛V ，E，A ｝,in which V =｛1，…N｝denotes the finite node set of G, E ?V ×V is the set of edges and A =［aij］N×N∈RN×Nis the adjacent matrix.aijis the link coefficient betweeni-th andj-th UAV,whereinaij=aji=1 implies that there is a connection betweeni-th andj-th UAV in G, otherwise,aij= 0.Besides, D =diag｛d1，…dN｝ denotes an in-degree matrix, whereand L =D -A is the Laplace matrix.Another graph Gsconsisting ofNUAVs and a target (labeled by 0) is defined, which describes communication topology connection between UAVs and target.The connection weight between the target and UAVs can be depicted bybi0,i= 1，2，…，N.We define matrix B = diag｛b10，···bN0｝, ifi-thUAV is connected to the target, i.e.,bi0=1 denotes thati-th UAV can directly receive the data of the target,otherwise,bi0= 0.

3.Main results

In this section, under an undirected topology, a distributed multicircular circumnavigation control is presented for UAVs with a desired angular spacing, and the corresponding control architecture is depicted in Fig.2.

3.1.Centralized circular enclosing control design

In this section,a centralized circular enclosing control approach for multiple UAVs is proposed.The ultimate control objective is to force all UAVs to evolve along a predetermined circular trajectory and maintain a desired neighboring angular spacingunder the precondition that target information can be available for all UAVs,rendering

where ρiis the actual range between target and UAVs, ρdiis the expected circle radius, ?ijis the actual angular spacing between adjacent UAVs.

3.1.1.CircularenclosingcontrolforaUAV

To achieve circular encircling for a UAV around a moving target,a geometrical description between a UAV and a target is shown in Fig.3.Then, using the available actual range ρi（t） between target andith UAV by radar sensors,one has

Define φi（t）as the unit vector on this line passing throughi-th UAV to target, it yields

Fig.2.Control architecture of proposed strategy.

Fig.3.Geometrical description between a UAV and a target.

Then a circular enclosing controller can be defined as

where ui=［uxi，uyi］Tis velocity vector applied to UAVs defined in the earth-fixed reference.κ denotes a nonnegative gain and A = ［0，-1;1，0］.ρdi? is the tangential speed to be adjusted.Then,we have the follow assumption.

Assumption 1.The tangential speed of thei-th UAV is greater than the velocity of target and there exists a nonnegative scalar Ξ such that |ρdi?|- ‖vT‖≥Ξ ＞0.

Assumption 1 provides the premise of the surrounding strategy.When the tangential velocity of thei-th UAV is greater than the target speed all the time,thei-th UAV has the ability to encircle the target.

Remark 1.In order to explain the control effect of controller Eq.(6)on UAVs more clearly,the evolutions ofith UAV under controller Eq.(6)are described in Fig.4.As ρi≠ρdiin Fig.4(a),the radial force κ（ρi（t）-ρdi）φi（t） controls theith UAV to move towards the target.As ρi=ρdiin Fig.4(b), under the action of tangential force ρdi?Aφi（t）,theith UAV does not move towards or leave away from the target but it just stably rotates around the target in a counterclockwise manner, wherein the tangential speed ρdi? ofith UAV should be selected according to Assumption 1.Additionally, note that if ρdi? ＜0, thei-th UAV would circle around the target in an opposite direction.

3.1.2.CentralizedcircularenclosingcontrolformultipleUAVs

Note that previous enclosing researches[21-31]enforce agents to implement circumnavigation along a common circle with a uniform phase distribution, which is inflexible in angular spacing regulation.Herein, based on a graph theory, a centralized circular encircling control method is developed for multiple UAVs to manipulate each UAV to approximate the prescribed circles while holding an anticipated formation pattern with a tunable angular spacing between successive UAVs.First, referring to the angular relation derived in a polar coordinate, the adjacent angular separation is computed as

where ?iand ?jare the angles between the line of sight and horizontal axis.Since the polar angle is monotonic,it can be concluded that a polar angle corresponds to a unique position on the circle,as drawn in Fig.5.Therefore, it is trivial to reckon the spatial geometric of adjacent UAVs based on the sign of ?ij,i.e.,?ij＞0 indicates that for anticlockwise rotation,j-th UAV is moving in front ofi-th UAV, and vice versa.

Fig.4.Evolutions of i-th UAV under controller Eq.(6).

Fig.5.Angular spacing relationship between adjacent UAVs in polar coordinates.

For multiple UAVs connected by an undirected graph, the coordinated angular spacing errore?i（t） is constructed as

Then, by introducing a tangent compensation item resulting from coordinated angular spacing errors, a centralized circular enclosing controller is expressed as

Remark 2.Instead of Eq.(6)that aims at forcing UAVs to perform target enclosing without coordination, herein according to consensus theory, by the incorporation of coordinated angular spacing errore?i（t）defined in the tangential direction,the modified controller Eq.(9)is devised to enable multiple UAVs to preserve an anticipated formation with various enclosing radii and tunable separation angles, which can regulate coordinated on-path behaviors of UAVs effectively.

Besides,the velocity and heading angle are respectively derived as

To stabilize Eq.(11), the angular rate signalriis expressed as

wherekψi＞0 is a control gain.

To move on, the following technical lemmas are introduced to facilitate stability analysis regarding the established cooperative rule.

Lemma 1.[40]:Consider a nonlinear system=f（x，y，t）+Δ,=g（x，y，t）, which is exponentially stable atx=0，y=0 if: 1) there exists a bounded and positive constant ? satisfying Δ ≤?‖x‖‖y‖and 2)systems ˙x=f（x，y，t）and ˙y=g（x，y，t）are exponentially stable atx=0 andy= 0.

Lemma 2.[41]: Based on graph theory, the Kuramoto-coupled oscillator modelsin（φij） can fulfill states synchronization,i.e.,φij=0,where φi∈（-π，π］,λ ＞0 and φij=φj-φi.

Theorem 1.Given multiple UAVs Eq.(1) with an undirected topology, the proposed centralized circular encircling controller Eq.(9) can accomplish target enclosing with a predefined configuration driven by Eqs.(10)and(12),i.e.,control objective Eq.(3)can be fulfilled.

Proof:The entire proof contains two steps,wherein Step 1 is to illuminate that all UAVs approximate the desired multicircular paths.In Step 2, the convergence of coordinated angular spacing errors along the specified circle is assured according to the conclusions from Step 1.

Step 1.Define the encircling controlling error aseρi=ρi-ρdi, to confirm that all UAVs can approximate the desired circle, by invoking Eq.(9), note that= 0, taking the derivative ofeρigives

Establish a Lyapunov function, its time derivative is obtained as

Consequently,eρiis ultimately upper bounded and exponentially converges to zero, indicating that all UAVs can eventually converge to a desired circular trajectory with specified different radii.

Step 2.All UAVs can complete the desired angular spacing along the prescribed circular orbit.It is clear that ?i= arctan（（yi-yT）/（xi-xT））, the time differential of ?iis specified as

where ATA=I2and A =［0， -1;1，0］.By invoking Eq.(8),Eq.(15)can be rewritten as

where -eρi（?+e?i）/ρiis a perturbation term.Clearly, revisiting Step 1, one has that the perturbation term is bounded and convergent.

Remark 3.Different from the previous circular enclosing investigations [21-27] for stationary targets, the control design and stability analysis cannot be directly applied to the case of moving target.Herein moving target circular formation is involved,providing an enhanced adaptiveness during enclosing missions.Distinguishing from the circular encircling [28-31] requiring agents to preserver an desired angular spacing configuration around moving targets, by defining coordinated errors relative to neighboring angular spacing, a centralized circular enclosing control law with adjustable angular spacing is derived.Furthermore,unlike the formation control strategy [34], where the vehicles are forced to move around targets along a common circle with arbitrary spacing, a multicircular circumnavigation control for UAVs is proposed, which expands the scope of practical application.

3.2.Distributed circular circumnavigation control design

Considering that the previous control protocol Eq.(9)is devised based on a centralized-type communication, i.e., each UAV can receive the information of target, which will bring a heavy communication burden.To circumvent this limitation, fixed-time distributed observers are proposed for each UAV to estimate the target’s states with a uniform settling time,where only a portion of UAVs can obtain the information of target.Thus,we are concerned with a distributed circular circumnavigate scene using fixed-time obverses, such that

Lemma 3.[42]: If Gshas a spanning tree with target as the root node, then the matrix H =L +B is symmetric and positive definite.

Lemma 4.[43]: Consider a continuous positive definite function denoted byV（x（t）） fulfilling

where α，β，p，qandk＞0 are positive constants satisfyingpk＜1，qk＞1.Then, the equilibrium point of system Eq.(19) can be asserted as fixed time stable with the settling time upper bounded by

Lemma 5.[44]:For any nonnegative real numbersxT1，xT2，…，xTN,one has

3.2.1.Fixed-timedistributedobserversdesign

To avoid the requirement of target’s states being accessible for each UAV, it is reasonable to estimate the states of target by the local information interaction among UAVs.Thus, following the design of [44], a fixed-time distributed observer is introduced to obtain the accurate estimations of pTand vT,respectively,described as

Remark 4.To further illustrate the practicability of fixed-time distributed observer, adjusting guidelines of fundamental arguments can be summarized as follows: The parametersm＞1 and 0 ＜n＜1 play a decisive role in convergence speed and accuracy.To be specific, ifis satisfied, i.e.,when the tracking deviation is far away from the origin, the exponential-order termplays a dominant role in accelerating the transient decaying rate,whileis satisfied, i.e., when the tracking error is approximating to the origin, then the fractional-order termaccounts for suppressing the steady tracking deviation to a small set.α1and β1correspond to the feedback coefficients of related terms,one can gradually increase the values until fast and chattering-free error profiles are obtained.

Theorem 2.For the undirected graph, the devised fixed-time distributed observer Eq.(23) can estimate the target’s states pT，vTwithin a settling time, which is given by

The augmented velocity estimation error vector is defined as

Select the following Lyapunov function as

Based on Eq.(28), the differential ofV2is derived as

Then,according to Lemma 5 and recalling,the following estimation holds

Besides,whent＞T0,i.e.,approaches to the origin,the dynamics ofbecomes

Establish a Lyapunov function,where, the proof is similar to the convergence analysis of velocity estimation error dynamics,thus it is omitted here.Then,converges to pTwithin a settling time 2T0.Therefore, it can be inferred that the target’s states pTand vTcan be recovered via fixed-time distributed observer Eq.(23) within a fixed time.

Remark 5.Different from the finite-time distributed observers[35,36], the convergence times of these researches are always associated with initial conditions, and the information on initial conditions is inaccurate or unknown in some cases, presenting great difficulties in accurate estimation of settling time.Note that the settling time of the fixed-time distributed observer Eq.(23) is associated with the parametersm，n，α1and β1free from initial conditions, indicating that the upper bound of convergence time can be predefined by adjusting the parameters of the observer.

3.2.2.Distributedcircularcircumnavigationcontrolformultiple UAVs

In what follows, with the assistance of fixed-time distributed observer Eq.(23) and circular enclosing controller Eq.(9), a distributed circular circumnavigation controller is modified as

Remark 6.To obviate the demand of global information being distributed to each UAV,fixed-time distributed observer Eq.(23)is introduced to replace target states with corresponding estimates,rendering the stability analysis of the system to be more complicated compared to previous centralized control policy due to the appearance of estimation errors.

Theorem 3.For multiple UAVs Eq.(1) and a moving target pT, ifAssumption 1preserves to be true, then the control objective Eq.(18) can be performed by fusion of a distributed circular circumnavigation controller Eq.(34) and fixed-time distributed observer Eq.(23).

Proof:The entire proof contains two steps,wherein Step 1 is to illuminate that all UAVs approximate the desired circular path with the aid of fixed-time distributed observers.At Step 2, the convergence of coordinated angular spacing error along the specified circle is assured according to the conclusions from Step 1.

Step 1:Define the relative distance estimation error asρi.Then, according to Eq.(34), the time derivative of the relative range error eρiis given as

Based on Eq.(33), it can be verified thatandare exponentially convergent to zero.Utilizing the result, ~?iexponentially decays to zero ast＞T0.

Select a Lyapunov function as.By virtue of Eq.(35), the derivative ofV4is deduced as

Applying the triangle inequality results in,and resorting to Young’s inequality, it yields

Step 2:With the help of the fixed-time distributed observers,all UAVs can achieve the desired angular spacing along the specified circle.Then, the time differential ofis specified as

4.Simulation results

In this section, the relevant simulation and comparison results are executed to reveal the superiority and utility for the presented control algorithm,which are performed under MATLAB/SIMULINK environment configured with a calculation period being 1 ms.

Fig.6.Communication topology of G s.

Table 1Initial conditions of UAVs.

Table 2Initial states of fixed-time distributed observers.

To substantiate the merit of suggested controller,three cases are considered.For each case, a group of eight UAVs are considered to form a multicircular formation around a moving target with a desired angular spacing.The communication topology among the target and eight UAVs is depicted by Gsshown in Fig.6, where labeled 0 represents the target.The initial states of UAVs are described in Table 1,the upper and lower bounds of UAV’s velocity and angular velocity are chosen as vmax= 100 m/s，ωmin= -5 rad， ωmax= 5 rad.The moving target is given as pT=［2t，15（sin（t/20）+0.1t）］Tm and the desired radii are set as ρd1= 50 m，ρd2= 50 m，ρd3= 50 m，ρd4= 150 m，ρd5= 150 m，ρd6= 100 m， ρd7= 100 m， ρd8= 100 m.The desired angular spacing are set as-7π/5 rad and the desired angular rate of all UAVs is set as ? =1/3（rad/s）.The arguments in the distributed circular circumnavigation controller are specified as κ=3，κ?1=κ?2=κ?3=κ?4=κ?5=κ?6=κ?7=κ?8=3 andkψ1=kψ2=kψ3=kψ4=kψ5=kψ6=kψ7=kψ8= 5, respectively.The initial states of fixed-time distributed observers are elaborated in Table 2, and the gains for fixed-time distributed observers are chosen as β1= α1= 10,m=5/3，n= 3/5，λ1= 0.1.

Case 1.Simulation case of centralized enclosing control algorithm.

Fig.7.Trajectories of UAVs under controller Eq.(9).

Fig.8.Evolutions of UAVs under controller Eq.(9)at different time episodes:(a)Positions of each UAV at t =0 s;(b)Positions of each UAV at t =1 s;(c)Positions of each UAV at t =3 s; (d) Positions of each UAV at t =5 s.

Fig.9.(a) Relative range controlling errors of UAVs; (b) Coordinated angular spacing errors of UAVs.

Fig.10.(a) Linear velocities of UAVs; (b) Angular rates of UAVs.

Fig.11.Trajectories of eight UAVs under controller Eq.(34).

To confirm the viability for proposed control scheme Eq.(9),simulation outcomes are elaborated in Figs.7-10.The trajectories of eight UAVs and target duringt∈［0，150］are depicted in Fig.7,it is illustrated that all UAVs can circumnavigate a moving target in preset circular paths.Moreover,evolutions of each UAV and target att=｛0，1，3，5｝s are depicted in Fig.8, revealing that under the proposed strategy,all UAVs locating at various initial positions can eventually approximate and maintain along the circular trajectories with different radii while retaining a desired formation pattern in terms of specified neighboring angular spacing.Fig.9 demonstrates the relative range errorseρiand coordinated angular spacing errorse?i,it shows that both errors asymptotically converge to the origin,illustrating that coordinated multicircular enclosing assignments can be executed with nice performances,further demonstrating the correctness of Theorem 1.The linear velocity and angular rate profiles of all UAVs are shown in Fig.10 respectively, which are bounded and smooth.

Case 2.Simulation case of distributed enclosing control algorithm

Case 1 depicts the simulation results of eight UAVs under controller Eq.(9).This centralized enclosing control algorithm requires that each UAVs can receive the information of target,which will bring a heavy communication burden.To circumvent this limitation,by introducing fixed-time distributed observer Eq.(23),a distributed circular circumnavigation controller Eq.(34) is designed and the simulation results are shown in Case 2.Figs.11-15 plot simulation outcomes using the controller Eq.(34).With the help of fixed-time observer,the trajectories of eight UAVs and target duringt∈［0，150］are shown in Fig.11,it can be obviously observed that the controller Eq.(34) with the fixed-time distributed observers can enforce all UAVs to evolve along the predefined circular trajectories without the demand on regarding target information as global variable.Besides, evolutions of each UAV and target att=｛0，1，3，5｝s are depicted in Fig.12, indicating that without incurring great communication load among targets and UAVs,all UAVs can still successfully reach the predetermined orbit and maintain the desired angular spacing, such nice performance can be attributed to the fixed-time convergence ability by the suggested observer, as plotted in Fig.13, it is clear that profiles of estimation errors can reach an agreement on settling time.And the convergence times are obviously no more than the upper boundary computed byaccording to Theorem 2, demonstrating that arbitrarily fast convergence rate can be prespecified by practitioners independent of initial states.Simultaneously, as reflected by Fig.14, relative range errorseρiand coordinated angular spacing errorse?iare ultimately constrained within a small compact set around the origin,convincingly revealing that the imposed control goals are well implemented.Fig.15 collects the bounded linear velocity and angular rate responses of all UAVs respectively.

Case 3.Contrastive case with finite-time distributed observer[45].

To further verify the advantages of suggested control method,a comparative strategy[45]adopting finite-time distributed observer to control UAVs is introduced, and the finite-time distributed observer in Ref.[45] is defined as

Fig.12.Evolutions of UAVs under controller Eq.(34)at different time episodes:(a)Positions of each UAV at t =0 s;(b)Positions of each UAV at t =1 s;(c)Positions of each UAV at t =3 s; (d) Positions of each UAV at t =5 s.

Fig.13.(a) Estimated position errors of UAVs by observer Eq.(23); (b) Estimated velocity errors of UAVs by observer Eq.(23).

Simulations are performed with different initial states under Tables 2 and 3.To ensure a fair contrastive study,the gains in finitetime observer are determined as β1= 10, η2= 0.6, η3=0.6 to permit a similar magnitude level with our solution.The contrastive outcomes between considered methods are illustrated in Figs.16 and 17.It can be obviously discerned that both observers can accurately estimate the velocity and position data of the target.However,when the initial states gradually increase to great values,the settling times of finite-time distributed observers Eq.(41)exhibit severe sluggish, implying that the settling times are closely related to initial conditions for finite-time alternatives,thus a degraded controlling performance with a weak convergence rate is retained,as reflected by Fig.17.Oppositely,Fig.16 illustrates that no matter how the initial state changes, the convergence time of fixed-time distributed observers will not be affected at all,showing a uniform convergence rate despite of different choices of initial conditions.Thus,the proposed schemes is superior in coordinating multiple UAVs to flight along multicircular paths with rigid time limitations.

Fig.14.(a) Relative range controlling errors of UAVs; (b) Coordinated angular spacing errors of UAVs.

Fig.15.(a) Linear velocities of UAVs under controller Eq.(34); (b) Angular rates of UAVs under controller Eq.(34).

Table 3Initial states of fixed-time distributed observers with increased values.

Case 4.Contrastive case with Lyapunov guidance vector field(LGVF) [46].

To further demonstrate the advantages of the proposed method,we take three UAVs to execute multicircular enclosing mission,and a representative stand-off tracking solution,i.e.,LGVF[46]is carried out to force UAVs to track a moving target,and the associated LGVF is clarified as

Following Eqs.(42) and (43), we have

Fig.16.Estimation and controlling errors using fixed-time distributed observers Eq.(23): (a) Convergence of ‖~pTi‖ under Table 2; (b) Convergence of ‖~vTi‖ under Table 2; (c)Convergence of eρi under Table 2; (d) Convergence of ‖~pTi‖ under Table 3; (e) Convergence of ‖~vTi‖ under Table 3; (f) Convergence of eρi under Table 3.

then, the velocities and the angular rate signal are expressed as

wherek1,k2andk3are positive parameters.?1，?2，?3are the phase angles, and ?dis desired separation angle.v1，v2，v3are the velocities of three UAVs respectively.

To ensure a fair comparison, the parameters of considered algorithms are finely adjusted to ensure an optimal encircling effect.Figs.18 and 19 show target encircling results using LGVF and involved method, where UAVs can both successfully reach the predetermined orbit and maintain the desired angular spacing,but the contrastive alternative exhibits a sluggish convergence rate.To fully evaluate the superiority of the proposed method, tracking accuracy and convergence time of enclosing errors are used as indicators for comparison, and the accumulated enclosing error is described as

whereTρiis the settling time andTis the total time of simulation.Quantitative metrics are demonstrated in Table 4, it can be discerned that the proposed method enables a shorter convergence time and higher tracking accuracy compared with LGVF [46].

5.Conclusions

In this article, a cooperative multicircular circumnavigation control for UAVs with desired angular spacing is investigated.Two cases of cooperative circumnavigation solutions, i.e., centralized and distributed architectures, are respectively provided to realize accurately approximate the prescribed circles with an expected formation configuration regulated by neighboring angular spacing.In addition, a distributed circular circumnavigation controller is presented for UAVs by introducing a fixed-time distributed observer to obviate the demand of global information being distributed to each UAV, leading to a more complicated controller synthesis and stability analysis.Rigid stability analysis is conducted to show errors convergence to a small compact set around origin.The functionality of explored algorithm is verified via simulations.

Fig.17.Estimation and controlling errors using finite-time distributed observer Eq.(41): (a) Convergence of ‖~pTi‖ under Table 2; (b) Convergence of ‖~vTi‖ under Table 2; (c)Convergence of eρi under Table 2; (d) Convergence of ‖~pTi‖ under Table 3.(e) Convergence of ‖~vTi‖ under Table 3; (f) Convergence of eρi under Table 3.

Fig.18.(a) Target encircling results using involved method; (b) Target encircling results using LGVF [46].

Consider that uncertainties and resource-aware are ubiquitous in various unmanned intelligent systems, thus, in the future, it is rewarding to empower the involved policy with the property of disturbance attenuation [47] and energy saving.

Fig.19.(a) Convergence of eρi using proposed method; (b) Convergence of eρi using LGVF [46].

Table 4Quantitative comparison with LGVF [46].

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This research was supported in part by the National Natural Science Foundation of China under Grant Nos.62173312,61922037,61873115, and 61803348; in part by the National Major Scientific Instruments Development Project under Grant 61927807; in part by the State Key Laboratory of Deep Buried Target Damage under Grant No.DXMBJJ2019-02; in part by the Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi under Grant 2020L0266; in part by the Shanxi Province Science Foundation for Youths under Grant No.201701D221123;in part by the Youth Academic North University of China under Grant No.QX201803;in part by the Program for the Innovative Talents of Higher Education Institutions of Shanxi;in part by the Shanxi“1331 Project” Key Subjects Construction under Grant 1331KSC and in part by the Supported by Shanxi Province Science Foundation for Excellent Youths.