Elliptical encirclement control capable of reinforcing performances for UAVs around a dynamic target

Fei Zhang, Xingling Shao, Yi Xia, Wendong Zhang

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

Keywords:Elliptical encirclement Reinforced performances Wind perturbations UAVs

ABSTRACT Most researches associated with target encircling control are focused on moving along a circular orbit under an ideal environment free from external disturbances.However, elliptical encirclement with a time-varying observation radius,may permit a more flexible and high-efficacy enclosing solution,whilst the non-orthogonal property between axial and tangential speed components, non-ignorable environmental perturbations, and strict assignment requirements empower elliptical encircling control to be more challenging, and the relevant investigations are still open.Following this line, an appointed-time elliptical encircling control rule capable of reinforcing circumnavigation performances is developed to enable Unmanned Aerial Vehicles (UAVs) to move along a specified elliptical path within a predetermined reaching time.The remarkable merits of the designed strategy are that the relative distance controlling error can be guaranteed to evolve within specified regions with a designer-specified convergence behavior.Meanwhile, wind perturbations can be online counteracted based on an unknown system dynamics estimator (USDE) with only one regulating parameter and high computational efficiency.Lyapunov tool demonstrates that all involved error variables are ultimately limited, and simulations are implemented to confirm the usability of the suggested control algorithm.

1.Introduction

With the rapid increase of autonomous technology, motion control of Unmanned Aerial Vehicles(UAVs)has gained widespread spotlights attributed to its outstanding features with low cost,powerful mobility,and high flexibility[1-3].Classical applications of motion controls range from search and convoy, geography mapping,localization,target tracking,and area surveillance[4-8].Among which, target enclosing serves as an enabling technology for the aforementioned missions, which aims to appoint UAVs to approach a stationary or moving target while maintaining a stable motion along a user-defined orbit, such that multi-dimensional information can be effectively gathered and the possible events of target missing can be remarkably reduced.

In recent years, target circling has attracted tremendous attention,and plenty of technologies have been considered in the fields of simple integrator agents [9,10], ground robots [11-17], UAVs[18-21],and autonomous surface vehicles[22,23].For instance,the contribution of Ref.[9] lies in proposing a positioning scheme and an encircling controller that can force agents to move along a circular path around the drifting target.Then this enclosing scheme is tailored to address the multiple targets in Ref.[10].A bearing-based circling control policy is developed for ground nonholonomic robots in Ref.[11].Later,two cooperative enclosing control algorithms are deployed in Refs.[12,13], respectively, such that multiple vehicles can succeed in circumnavigating the static and dynamic targets with a fixed radius.The target tracking problem by single integrators is addressed by using a cyclic pursuit control protocol,and a circular enclosing formation is established [14].Based on a line-of-sight guidance strategy, a circular motion of vehicles is performed concerning a moving target [22].In Ref.[15], a consensus-based entrapment scheme is designated by inserting a distributed observer to a circle-guided formation control,such that a multicircular circumnavigation around a dynamic target can be established.In Ref.[16], a distance-based cooperative encircling controller is constructed for a cluster of robots to generate a circular formation without global position measurements.The feature of Ref.[17] aims to propose a distributed enclosing control protocol for multiple robots with constrained velocity while the communication among nodes is eliminated based on local measurements.In Ref.[18], enclosing a nonstationary target is investigated for UAVs to carry out the standoff tracking mission by using a modified Lyapunov vector field guidance (LVFG).Moreover, a revised curvature-constrained LVFG is provided in Ref.[19]to steer aircraft to realize surveillance assignments by setting the initial positions of UAVs.In Ref.[20],a planar circumnavigation behavior is fulfilled for UAVs around a static target by utilizing range measurements.The novelty of Ref.[21] aims to synthesize the target tracking and formation cooperation among UAVs and ground targets, such that circumnavigation can be well realized based on a composite system mechanism.In Ref.[23], a cooperative target enclosing control protocol is devised for autonomous surface vehicles to entrap the target over a common circle collaboratively.

After overviewing the existing literature, the widespread shortcoming is that the permanently invariant encirclement radius is frequently constructed based on circular enclosing control protocols[9-23],which may impose a weaker adaptability and a lower circling efficiency under the premise of closely entrapping multiple targets, as illustrated in Fig.1.It can be observed that the circular entrapping cannot yield the optimal solution with respect to energy consumption and observation efficiency.To repair this weakness,recently, by explicitly inferring the non-orthogonal relationships between the axial and tangential unit vectors in terms of bearing angles, an elliptical encirclement control protocol has been developed for agents to enclose multi-targets [24].Nevertheless, only a single integrator is taken into account, which cannot be directly employed for UAVs described by nonholonomic constraints [25].Moreover, it is worth declaring that current circumnavigation strategies [9-24] can only guarantee robots to asymptotically approach the specified circle centered at the target,i.e., the robots will arrive at the predesigned path as time goes to infinity,resulting in a sluggish settling time, which is inadvisable for escorting mission of UAVs subject to critical time constraints in practices.Recently, a finite-time circling controller has been proposed to enable robots to approximate the desired circle in finite time [26].However, the regulation of decaying time completely depends on initial states and controller parameters, which faces the conservative convergence time and unexpected transient behavior under different initialization errors.Thus,it is urgent and considerable to regulate the transient elliptical escorting convergence with a guaranteed user-specified rate.

Prescribed performance control (PPC) presented in Ref.[27],which attempts to convert the original constrained dynamics to the equivalent unconstrained form by constructing transformation functions, such that user-defined performance behavior can be satisfied for all the operational time, i.e., the decaying rate, overshoot as well as steady accuracy of tracking error can be strictly confined within the predetermined region irrespective of tediously parameter tuning, and fruitful outcomes have been studied in numerous fields [28-35].In Ref.[30], the attitude tracking performance of spacecraft formation can be pre-specified based on a PPC without regulating control arguments repeatedly.In Ref.[31],to enhance the trajectory tracking responses, an adaptive control protocol with performance guarantees is constructed for surface vessels.The consensus errors of multiagent systems can be ensured to converge within the preassigned envelopes by using PPC in Ref.[33].A PPC-based formation controller is devised such that the leader-follower system can accomplish the desired configuration within the prescribed bound in Ref.[34].In Ref.[35], an adaptive fuzzy controller is presented for robotic manipulators to enforce tracking errors to decay within prescribed performance boundaries.However, although predetermined tracking zones are commonly enforced in the foregoing works, the tracking errors can only be asymptotically driven to the specific residual sets, i.e., infinite setting time is usually imposed on system behaviors, which is undesirable for motion missions with rigid time specifications.To remedy this defect, an appointed-time prescribed performance control (APPC) technique is devised for spacecrafts in Ref.[36],where a novel performance function formulated by a piecewise and continuous behavior envelope is delicately constructed,resulting in that the convergence of system states to final values can be ensured before a user-designated time.Note that the spacecraft dynamics considered in Ref.[36] is formulated in a simple strict-feedback form, while the non-negligible nonholonomic constraints make the target escorting issue by UAVs more complicated.

Another crucial concern in existing target encircling control is to perform target surrounding under an ideal environment free from external disturbance, while wind perturbation is inevitably encountered, bringing serious performance deterioration during target enclosing.Currently, to handle disturbances online and enhance robustness, function approximators such as neural networks (NNs) [37-44] have been widely studied.For example,Ref.[37] incorporates the NN into the tracking controller to solve unknown nonlinearities for mobile robots.A formation control scheme is constructed for a cluster of mechanical systems in Ref.[38], where an artificial NN is employed to approximate the uncertain system dynamics.In Ref.[41], a modified backstepping controller is designed to stabilize a class of nonlinear systems by using a NN technique.In Ref.[42],a NN is utilized to deal with the lumped disturbances to enhance the robustness of quadrotors.Moreover,an adaptive practical optimal formation tracking control protocol is developed for multiagent systems in Ref.[43], where NNs are adopted to approximate the value function.However, the involved NNs impose a demanding computational burden and exhibit a sluggish convergence, which is unsuitable for implementing rapid target capturing by UAVs with limited computational resources.Recently,based on an invariant manifold criterion and algebraic manipulations, a straightforward unknown system dynamics estimator (USDE) with only one filtering argument is elegantly proposed to counteract lumped disturbances by means of auxiliary filtering operation upon measurable system states [45],permitting an easy in implementation and a low computational burden.Benefiting from aforesaid superiorities, some applications of USDE have been reported.In Ref.[46],the USDE is incorporated into the controller for robotic systems subject to external disturbances and internal uncertainties, exhibiting an improved interference rejection ability.Subsequently, the USDE is constructed to compensate unknown dynamics for servo systems, quarter-car active suspension systems, and half-vehicle active suspension in Refs.[47-49], respectively.Regrettably, although USDEs [45-50]have obtained fruitful results, how to generalize USDE to elliptical escorting scenarios with enhanced performances is still unresolved and challenging.Owing to the fact that the appearance of nonorthogonal speed vectors, nonholonomic constraints, and strict time requirements,as well as disturbance estimation errors,it leads to a more nontrivial design and analysis procedure.Thus, it is urgent and rewarding to develop a novel appointed-time elliptical escorting control scheme for UAVs subject to exogenous disturbances, which stimulates us to provide the related solution.

Fig.1.Encirclement behavior of circular and elliptical schemes around multiple targets.

Enlightened by previous studies, this article focuses on an elliptical target encircling control issue for UAVs in the existence of external interferences.Primary contributions are outlined as follows.

(1) Contrasting to the universal circular target circumnavigation algorithms [9-23] confined to a permanently fixed circling semidiameter, by deriving the non-orthogonal geometrical relationship between axial and tangential vectors, a more realistic elliptical escorting control scheme with a timevarying circling radius is formulated, empowering a more flexible encircling pattern with the reinforced surrounding and observation efficiency.Additionally, the nonholonomic constraints, time constraints, and wind disturbances are explicitly addressed for UAVs,which is more reasonable than the existing encirclement methods considering simple integrators [9,10] or without interferences [9-14,15-21,24].Moreover, compared to prevalent NNs [37-43] exposing complicated parameter regulation and lagging convergence behaviors, by implementing simple filtering upon the measurable system states, the designed USDE only necessitates regulating a filtering coefficient, rendering a less demanding calculation complexity.In addition, unlike previous USDE-based control protocols [45-49] that cannot ensure a prior tracking performance, a robust enclosing controller is proposed without violating performance constraints.

(2) Distinguishing from the current encircling consequences with an asymptotical convergence [9-24], enforcing an infinite decaying time to approximate the designed circular orbit, herein by converting constrained error dynamics into an equivalent unconstrained type, an elliptical enclosing control rule with reinforced behaviors is skillfully established, such that relative range controlling error can be quantitatively devised without violating user-defined regions.In addition, unlike published PPC-based outcomes[28-35]subject to an infinite prescribed time due to the use of an exponential-type performance boundary, by the inclusion of a piecewise and smooth appointed-time behavior envelope, a novel appointed-time elliptical escorting controller is established, rendering that convergence time can be arbitrarily designated by the user-chosen parameter,which can well match the target enclosing missions with various arriving time requirements.Furthermore, the enclosing error profile can be guaranteed within devised envelops free from tedious controller parameter tuning,which is more preferable and ingenious than finite-time tracking [26].Brief comparison between current control strategies and our proposed scheme is summarized in Table 1.To the best knowledge of the authors,the appointedtime elliptical encircling control of uncertain UAVs with performance constraints is considered for the first time.

The remainders of this article are planned as follows: Preliminaries and problem description are reviewed in Section 2.Main results of elliptical escorting control with enhanced performances are displayed in Section 3.Stability analysis is exhibited in Section 4.Section 5 provides simulations.Section 6 summarizes this paper.

2.Problem description and preliminaries

2.1.Problem description

Consider a UAV encompassing a planar dynamic target located at pt=[xt,yt]Talong an elliptical path, and assume that UAV is manipulated at a fixed plane, which can offer a reliable detection mode for target enclosing, thus inspired by Refs.[51,52], a planar nonholonomic UAV kinematics is formulated in the earth coordinates as

where p = [x,y]T, v, ψ, and ω determine the position, liner speed,heading angle, and angular rate of UAVs, respectively.vmdenotes the unmeasurable wind speed and ψmmeans the wind direction.Generally, vehicle velocity and angular rate should be fulfilled vmin≤v ≤vmax, ωmin≤ω ≤ωmaxwhere vminand vmaxdenote the minimum and maximum vehicle speeds, ωminand ωmaxare the minimum and maximum angular rates.Then, consider a dynamic target whose dynamics is expressed as [53]

where pt=[xt,yt]Tis the target position,ut=[vxt,vyt]Tdenotes the target velocity vector.Notice that the UAV coordinate p and target coordinate ptare available by onboard Global Positioning System(GPS)device.The speed v and heading angle ψ can be measured by mounting pitot tubes and magnetic compasses.

2.2.PPC

To restrict controlling errors within the user-defined monotonous decaying region,by revisiting the design principle of PPC[27],the following inequality is always imposed as

To ensure error profiles within a prescribed envelope, aperformance transformation strategy is introduced, such that restrained error e(t) can be converted into an equivalent unrestrained form as

Table 1 Comparison between the existing enclosing control schemes and our method.

Fig.2.Illustration of PPC.

Remark 1.Although the controlling error can be confined based on predesigned convergence specification by a PPC paradigm[28-35], the infinite reaching time is usually imposed on system behaviors with an exponential rate, yielding a possible time delay or even failure in escorting mission since the expected encircling performance cannot be theoretically assured before the missionassigned time, which is undesirable for time-urgent escorting flight missions.

2.3.Control objectives

The design aims to propose an elliptical escorting control protocol for UAVs around a nonstationary target while conforming the prescribed performance constraints despite of exogenous disturbances, as illustrated in Fig.3, such that.

3.Main results

In this section,an appointed-time elliptical encirclement control protocol is devised for UAVs around a dynamic target, and the corresponding framework is illustrated in Fig.4.Firstly, different from the existing encircling control scheme [9-24] exposed to sluggish convergence speed, an APPC is enforced on relative range error such that a user-specified settling time can be assured.Then,an elliptical control rule with reinforcing performances is skillfully devised.Besides,a straightforward USDE is introduced to overcome unknown wind disturbances, such that enclosing precision can be enhanced without triggering a heavy computational burden.

3.1.APPC

To ensure a user-defined reaching time without regulating control arguments repeatedly, a smooth and piecewise performance function [36] is devised as follows:

where T is the prescribed time that can be suitably selected to regulate the convergence time of behavior function.μ∊(0,1) adjusts the decaying slope of Sr(t) during the transient.Sr0,Sr∞represent initial and final states of Sr(t), respectively, which are applied to constrain the transient overshoot and maximum steadystate allowable bound.Clearly, one has Sr0>Sr∞,

Theorem 1.For any specified prescribed time T>0, the designed performance function Sr(t) satisfies monotonous decreasing over 0 ≤t

Proof.As 0 ≤t

Fig.4.Block diagram of the proposed elliptical encircling control protocol.

which implies that ˙Sr(t) satisfies continuous property on ∀t∊[0,∞).Referring to the boundedness of a continuous function defined on a closed interval, we can deduce that ˙Sr(t) is bounded.

Remark 2.Resorting to a special appointed-time boundary behavior function, a user-defined settling time can be ensured before a bounded appointed value, rather than the exponential decaying ability inherent in PPC [28-35].Additionally, a quantitative comparison between PPC and APPC is arranged to verify the convergence performance, as illustrated in Fig.5, where S(0) =Sr(0) = 40, S(∞) = Sr(∞) = 2.Obviously, distinguishing from the prevailing PPC [28-35] exposed to an infinite reaching time, the performance profile of APPC can affirm an appointed fast decaying time by tuning different arguments of T,which offers a more userfriendly and convenient alternative for missions with demanding time limitations.

Similar to PPC,to guarantee relative range controlling error er(t)without violating predetermined boundaries, the following performance constraint is constructed as

3.2.Elliptical escorting controller with reinforced performance design

This subsection will provide the design procedure of the elliptical enclosing controller, which consists of the following four procedures.

Fig.5.Illustration of convergence performances between PPC and APPC.

Step 1.In this step, the design process of elliptical escorting parameters is provided for UAVs.To guide UAVs to track the dynamic target located at ptwhile following the predetermined ellipse path,a generalized elliptical trajectory [24,54], as depicted in Fig.6, is formulated related to a long semi-axis a,a short semi-axis b,and a counterclockwise rotation angle δ as

Then, let the vector [rd(θ)cos θ,rd(θ)sin θ]Tbe the desired range vector between any point on the elliptical path and the target ptformulated in a polar framework, where θ denotes the polar angle from the elliptical horizontal axis to the appointed point.Then,we can get

Fig.6.Motion profile for UAVs around the target along an elliptical path.

Note that rd(φ) corresponds to the predetermined escorting radius,and when the UAV is steered to the desired ellipse,then the velocity direction of UAVs will be parallel to the tangent component along the track.Therefore, we formulate the unit vector η=[cos ψ1,sin ψ1]Tto denote the elliptical tangent velocity component,where ψ1is the angle from the x-axis to the tangential velocity of the ellipse.And in terms of Eqs.(18) and (19), the tangent of ψ1can be derived as

Step 2.In this step, without considering external disturbances,the design aims to drive UAVs to escort the target along an elliptical orbit with prescribed performance constraints.Typically, an elliptical encircling control scheme without APPC can be formulated in terms of the radial and tangential vectors as follows:

Remark 3.As mentioned in Refs.[9-23], the circular enclosing

control laws can be readily devised owing to the orthogonality between axial and tangential velocity vectors.Conversely, with respect to the more realistic elliptical encirclement scenarios, the non-orthogonality relationship between axial and tangential components enables target escorting to be more difficult, and the relevant researches are still rare.Furthermore, by deriving a timevarying circling distance rather than a predesigned invariable radius in the circular encircling schemes [9-23], the designed elliptical surrounding provides a significant promotion concerning observation efficiency and operational flexibility.

Remark 4.Distinguishing from the current encircling controllers[9-24] holding an infinite convergence property, herein the proposed circling control scheme is capable of pursuing a reinforced performance with an appointed convergence, yielding that the reaching time of relative distance error can be arbitrarily specified in accordance with realistic requirements.

Step 3.In this step, to counteract the impact of exogenous disturbances on UAVs,two USDEs are designed as compensation terms to enhance the robustness and enclosing precision.To facilitate the implementation, Eq.(1) can be rewritten as

Remark 5.Assumption 1 serves as a sufficient condition to facilitate the latter decaying analysis of USDE, and it has been widely discussed as an indispensable factor in USDE-based control design[45-49].Additionally,it is worth emphasizing that a specific upper boundary of differentials is not necessary to be acquired,and the boundedness can be readily inferred owing to the fact that wind field energy is usually constrained.Even for the dramatic changing of wind represented by a set jumping signal, one can utilize highorder Taylor expansion polynomials to represent the sudden change with an acceptable level, such that the boundedness of derivatives at the jumping point can be ensured.Therefore, the related assumption is not too strong following engineering practices.

Next, following the classical USDE idea, a series of filtering operations are enforced on measurable terms p and V, which can be expressed as

where κ>0 represents a filtering coefficient.Then,by employing an invariant manifold, a quantitative relationship between filtered signals and external disturbances is explored as below.

Theorem 2.Consider Eqs.(24) and (25), an auxiliary term is constructed as

where ε is limited and decreases exponentially toward a domain around the origin for κ>0, and it has

Then, based on the estimates offered by USDE, an enhanced control version of Eq.(23) with anti-disturbance rejection is tailored as

Remark 6.Unlike the existing enclosing control strategies[9-14,15-21,24,26] that are merely implemented under an ideal environment ignoring ambient disturbances, the USDE with one design argument is elegantly constructed by borrowing simple filter operations to build an invariant manifold, such that the robustness of UAV system can be assured based on the accurate perturbation estimation and compensation with low computational complexity.In addition, referring to Theorem 2, it can be concluded that asymptotic decaying of observation errors can be achieved by setting filtering gain κ, and as κ decreases, the upper bound can be regulated to a sufficiently small size.

Step 4.In this step,the following process is provided to yield the velocity and angular rate of UAVs.According to Eq.(31), one has

where ψddenotes the desired heading angle.Define the heading angle error as eψ= ψ- ψd.To stabilize eψ, the angular rate ω is generated as

where k2is the nonnegative gain.

4.Stability analysis

In this section, the stability and performance analysis of UAV systems are summarized.Firstly,the error dynamics utilized in the subsequent process is displayed.

Resorting Eqs.(37)and(40),the differential of V1is expressed as

Substituting Eq.(43) into (42), it follows that

Thus, the relative range controlling error can be constrained within the user-specified region before an appointed reaching time,implying that the time-critical enclosing missions can be accomplished for UAVs subject to external interferences.

Remark 7.To promote the execution of the developed appointedtime elliptical escorting control protocol,some fundamental tuning regulations of parameters are outlined here:

3) Regarding USDEs, the magnitude of observation error ～Gmis related to the filtering parameter κ,and a minor κ can contribute to a high estimation level in estimation errors, which can effectively enhance the robustness of the whole system.

5.Simulation results

In this section,a series of simulations are implemented based on MATLAB/SIMULINK platform to verify the efficacy of the presented control strategy in the existence of external disturbances, where the sampling time is chosen as 5 ms.Supposing that the nonstationary target path is generated by pt=[0.2t,0.1t+sin(πt/50)]Tm.The physical limitations imposed on speed and angular rate are vmin=1 m/s,vmax=8 m/s,ωmin= -0.8 rad/s, and ωmax= 0.8 rad/s.The exogenous disturbances are represented by Gm=[vmcos(ψm),vmsin(ψm)]Twith vm=cos(0.2t)and ψm=π cos(0.5t +π)/10+π/5.Moreover,in the following,to illustrate clearly,the pentagon represents the initial coordinates of UAVs, and the square denotes its instantaneous position.

Table 2 Parameters for the presented elliptical enclosing control scheme.

Fig.7.Evolutions of UAV's elliptical enclosing motions along dynamic target under presented control scheme Eq.(31)and contrastive control method Eq.(52):(a)Absolute position for UAVs around the target during t∊[0,200]s;(b)Relative position for UAVs around the target during t∊[0,200]s;(c)Episode of the transient approaching path during t∊[0,3]s;(d) Episode of steady-state relative positions at t = 100 s; (e) Episode of steady-state relative positions at t = 150 s.

Fig.8.Enclosing performances of the proposed controller Eq.(31)and the contrastive controller Eq.(52):(a)Relative range controlling error during the transient;(b)Steady-state relative range controlling error.

A.Comparison simulations with Ref.[24].

To demonstrate the superiorities of the proposed protocol in terms of a reinforced enclosing performance, comparison simulations against existing elliptical encircling controller [24] are implemented.The initial conditions of UAVs are set as p(0) =[20,0]Tm,ψ(0) = 3π/4 rad.The associated control arguments are listed as below.

1) Proposed controller:In terms of Theorem 3 and Remark 7,the related parameters of the developed appointed-time elliptical enclosing controller are offered in Table.2

2) Contrastive controller in Ref.[24]:Note that Ref.[24]is mainly concentrated on driving agents to perform elliptical escorting without considering uncertainties and arriving time constraints,in which a simple integrator is involved to describe the agent dynamics, and the related elliptical control protocol is represented as

Fig.9.Observed disturbances by USDE.

Fig.10.Speed and angular rate.

where fu3= [f1u3,f2u3]T.k3denotes a nonnegative number.k4is a design parameter.Then,to promote a relatively fair comparison,we refer to the design processes Eqs.(32)and(33)for UAVs to yield the corresponding speed and angular rate.The configurations of the desired elliptical orbit are consistent with the proposed scheme.Additionally, the associated arguments are finely set as k3= 3,k4=2,such that the magnitude of the error feedback component is kept as the same as our solution.

The comparison profiles between the proposed appointed-time elliptical enclosing controller and the contrastive controller[24]are illustrated in Figs.7-10.The evolutions of enclosing motions for UAVs are described in Fig.7,demonstrating that both schemes can permit UAVs to approximate the predetermined ellipse centered at the target while maintaining an anticipated enclosing.However,it is observed that the contrastive control exhibits a sluggish convergence time and fails in precisely approximating the orbit owing to the deployment of asymptotical decaying criterion, and even for t = 150 s, there still exists an obvious displacement deviation between UAVs and designed ellipse.Oppositely, our

Table 3 Performance comparison between the proposed method and contrastive scheme[24].

alternative is superior in reaching the orbit before the user-selected appointed time rather than in an exponential decaying manner,allowing for the satisfaction of strict time constraints, which is primarily originated from the use of APPC in guaranteeing transient behaviors and the employment of USDE to promptly observe and accommodate the unknown uncertainties,as exhibited from Figs.8 and 9.Clearly,the profile of relative range controlling error is well preserved within the predesigned envelope and can be confined to a small domain around the origin, which is consistent with Theorem 3,while a fairly longer settling time and a greater steady deviation are accompanied with the contrastive method, meaning that it is inferior in obtaining a satisfied enclosing with consideration of wind perturbations and definite time limitations.Responses of speed and angular rate are provided in Fig.10,demonstrating that physical constraints are not violated.Moreover,quantitative evaluations in terms of convergence time, standard deviation of relative distance errors are reflected in Table 3.It can be observed that the proposed control scheme can permit a faster settling time and a higher control precision than existing elliptical enclosing controller [24], which can be attributed to prescribed performance mechanism and accurate cancellation of disturbances endowed by USDE.Thus, the remarkable superiorities of the presented appointed-time elliptical escorting control scheme concerning user-designed decaying time and reinforced robustness are obviously illustrated.

B.Feasibility verification considering different scenarios

Case 1.Enclosing performances under different elliptical orbits and initial states.

To confirm the feasibility of the proposed controller under different initial states and elliptical parameters,a set of simulations are considered, and related configurations are shown in Table 4.Then, following the consistent controller arguments with the previous subsection,the simulation outcomes are presented in Figs.11 and 12, composed of episodes of elliptical circumnavigation behaviors and relative distance errors.It can be found that UAVs can all arrive at the devised elliptical routes before a user-defined time irrespective of different choices in orbit parameters and initial states, while imposed transient and steady specifications are not surpassed for involved cases, which further expounds that the presented escorting controller is able to achieve reinforced enclosing behaviors before a user-assigned reaching time.Meanwhile, the consistent enclosing behaviors and bounded control deviations imply that the closed-loop system can achieve UUB stability.

Case 2.Performance confirmation by setting different envelopes of APPC

In the following, to certify the effectiveness of the presented elliptical enclosing controller in fulfilling performance constraints,simulations with various envelopes are executed, and the corresponding parameter setting is listed in Table 5.It can be obtained from Fig.13 that,by virtue of APPC,the convergence performances of controlling errors,including transient settling times and steadystate precision, can be readily regulated by predesigned behavior functions, such that different target enclosing behaviors can be implemented according to specific mission requirements, which clearly demonstrates the appointed-time decaying ability.Moreover, from Table 6, we can see that the decaying times of relative distance errors are no more than the prescribed reaching times no matter how the envelope changes,while control precision can also be restrained within enforced boundaries,which indicates that the proposed enclosing control scheme can maintain the stability androbustness of whole systems.

Table 4 Different initial states and elliptical orbits for UAVs.

Fig.11.Evolutions of UAV's elliptical enclosing motions along dynamic target under different conditions: (a) Evolution under initial condition 1 during t∊[0,100]s; (b) Evolution under initial condition 2 during t∊[0,100]s;(c)Evolution under initial condition 3 during t∊[0,100]s;(d)Episode of the transient approaching paths during t∊[0,3]s;(e)Episode of steady-state relative positions at t = 50 s; (f) Episode of steady-state relative positions at t = 100 s.

Fig.12.Encircling control performances of the presented controller under different conditions: (a) Relative range controlling errors during the transient; (b) Steady-state relative range controlling errors.

Table 5 Different envelope settings.

Fig.13.Evolutions of controlling errors based on the devised APPC by taking different envelopes:(a)Envelope 1;(b)Envelope 2;(c)Envelope 3;(d)Envelope 4;(e)Envelope 5;(f)Envelope 6.

Table 6 Convergence performances of relative distance errors under different envelopes.

Fig.14.Evolutions of UAV's elliptical enclosing motions along dynamic target under example 1:(a)Absolute position for UAVs around the target during t∊[0,100]s;(b)Episode of the transient approaching path during t∊[0,5]s; (c) Episode of steady-state relative positions at t = 100 s.

Fig.15.Enclosing performances of the proposed controller Eq.(31) under example 1:(a)Relative range controlling error during the transient;(b)Steady-state relative range controlling error.

Fig.16.Observed disturbances by USDE under example 1.(a).^Gmx; (b).^Gmy.

Fig.17.Evolutions of UAV's elliptical enclosing motions along dynamic target under example 2:(a)Absolute position for UAVs around the target during t∊[0,100]s;(b)Episode of the transient approaching path during t∊[0,5]s; (c) Episode of steady-state relative positions at t = 100 s.

Fig.18.Enclosing performances of the proposed controller Eq.(31) under example 2:(a)Relative range controlling error during the transient;(b)Steady-state relative range controlling error.

Fig.19.Observed disturbances by USDE under example 2: (a).^Gmx; (b) ^Gmy.

Case 3.Performance verification considering different maneuvering targets and disturbances

Then,in order to demonstrate the effectiveness and robustness of the proposed elliptical encirclement control scheme,simulations are performed under different exogenous disturbances and maneuvering types of the target, assigned as example 1:Gm=[vmcos(ψm),vmsin(ψm)]Twith vm=3 sin(0.2t) and ψm=π cos(t + π)/4, pt= [0.8t,0.2t+1.5 sin(πt/50)]Tm, example 2:Gm=[vmcos(ψm),vmsin(ψm)]Twith vm=4 sin(0.2t)cos(0.3t) and ψm= sin(0.1t)cos(0.5t), pt=[1.5t+0.5 cos(t/5),0.5t+0.2 sin(t/10]Tm , and example 3: Gm=[vmcos(ψm),vmsin(ψm)]Twith vm=4 sin(0.4t)cos(0.1t) and ψm= sin(0.3t)cos(0.2t), pt=[2+0.3t+0.2 cos(t/5) +exp(cos(2t/5)),3+0.4t*sin (2 ln(t+1))+0.2 cos(7t/20)]Tm.Then, following the consistent controller arguments with the previous subsection, the simulation outcomes are presented in Figs.14-22, composed of episodes of elliptical circumnavigation behaviors,relative distance errors,and observed unknown uncertainties.It can be found that UAVs can arrive at the predetermined elliptical orbit before a user-appointed settling time irrespective of different choices in disturbances and maneuvering types of the target.Meanwhile,the transient and steady-state error responses can be ensured at the imposed specifications for involved cases, and the unknown uncertainties can be promptly observed and accommodated by USDE, which further demonstrates the robustness of the proposed escorting controller.

6.Conclusions

In this work, an appointed-time elliptical encirclement control protocol for UAVs subject to external disturbances is interviewed,yielding that UAVs can be driven to approach and maintain a specified elliptical pattern centered at the dynamic target with reinforcing performances.First,an elliptical enclosing scheme with a time-varying radius is deduced to realize a more flexible and high-efficacy escorting solution.Then, to further enhance the surrounding performance, a novel appointed-time convergence regulation is embedded into the control design,such that UAVs can be arranged on the desired elliptical orbit before a user-chosen reaching time.Moreover, a convenient USDE with only one filtering argument is established to counteract disturbances.Theoretical analysis verifies that the involved errors are convergent.Simulation outcomes are conducted to testify superiority and feasibility of the developed control algorithm.

Fig.20.Evolutions of UAV's elliptical enclosing motions along dynamic target under example 3:(a)Absolute position for UAVs around the target during t∊[0,100]s;(b)Episode of the transient approaching path during t∊[0,5]s; (c) Episode of steady-state relative positions at t = 100 s.

Fig.21.Enclosing performances of the proposed controller Eq.(31) under example 3:(a)Relative range controlling error during the transient;(b)Steady-state relative range controlling error.

Fig.22.Observed disturbances by USDE under example 3: (a) ^Gmx; (b) ^Gmy.

Promising investigation in the future may contain pursuing a coordinated elliptical enclosing based on reinforcement learning to empower UAVs with enhanced autonomy and intelligence, while the three-dimensional elliptical encirclement without prior positions of the target[58] will be considered.

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

The related work is supported by National Natural Science Foundation of China (Grant Nos.61803348, 62173312, 51922009),and Shanxi Province Key Laboratory of Quantum Sensing and Precision Measurement (Grant No.201905D121001).The authors would also like to appreciate the reviewers and the editor for their comments and suggestions that helped to improve the paper significantly.