Weibin CHEN,Yangyang CHEN?,Ya ZHANG

1School of Automation,Southeast University,Nanjing 210096,China

2Key Laboratory of Measurement and Control of Complex Systems of Engineering,Ministry of Education,Southeast University,Nanjing 210096,China

3Robot Department,Jiangsu Automation Research Institute,Lianyungang 222061,China

Abstract:This paper presents applications of the continuous feedback method to achieve path-following and a formation moving along the desired orbits within a finite time.It is assumed that the topology for the virtual leader and followers is directed.An additional condition of the so-called barrier function is designed to make all agents move within a limited area.A novel continuous finite-time path-following control law is first designed based on the barrier function and backstepping.Then a novel continuous finite-time formation algorithm is designed by regarding the path-following errors as disturbances.The settling-time properties of the resulting system are studied in detail and simulations are presented to validate the proposed strategies.

Key words:Finite-time;Coordinated path-following;Multi-agent systems;Barrier function

1 Introduction

Currently,the theory of the formation control problem has emerged as a hot topic and attracted great attention from researchers.To achieve better measurements of biological variables across a range of spatial and temporal scales in the applications in oceanic and planetary explorations(Bertozzi et al.,2005;Fiorelli et al.,2006),unmanned systems are required to simultaneously follow a set of given orbits with a desired formation,which is a special formation control problem called the coordinated pathfollowing control problem.

In the area of coordinated path-following control,many scholars focused on the asymptotic stability of the resulting multi-agent systems.In Cao et al.(2009),a discrete-time consensus-based algorithm was developed to force each follower to track a leader with the desired dynamics,which is also called the consensus tracking control problem.The continuoustime consensus tracking control laws were given in cases of time-invariant formation in Cao and Ren(2012),the time-varying formation in Yu et al.(2018),and the containment motion in Zhang FX and Chen(2022).In Ghabcheloo(2007),a coordinated path-following control law was designed by parameterizing the desired trajectories while synchronizing the orbital parameters.This idea was used in the case of uncertain dynamics in Peng et al.(2013).Noting the geometry of the orbit,a novel geometry extension method was proposed and then integrated into the consensus of the generalized arc-lengths(thesmooth functions)to achieve the coordinated pathfollowing task in Zhang FM and Leonard(2007)and Chen and Tian(2015).The geometry extension method was also used to solve the asymptotic coordinated path-following problem with time-varying flows in Chen et al.(2021a,2021b).However,the coordinated path-following control problem within a finite settling time is still unsolved.

Recently,finite-time control laws in multi-agent systems concentrate on the consensus(or consensus tracking)problems.In Xiao et al.(2009),a finitetime consensus tracking law was designed for a structure that consists of one leader and bidirectional connected followers based on the sliding-mode method.The sliding-mode method was used in the case of directed topologies in Cao et al.(2010)and Wang L and Xiao(2010),in the case of uncertainties in Khoo et al.(2009),and in under-actuated systems in Li TS et al.(2018).The finite-time properties of a sliding-mode-based consensus tracking system can be analyzed using the degree of homogeneity;details can be found in Guan et al.(2012)and Dou et al.(2019).Note that the above control laws are nonsmooth and thus sometimes cannot be directly used in actual continuous systems(Qian and Lin,2001).There is a trend toward designing a continuous finitetime controller for the coordinated control problem.In Li SH et al.(2011),a continuous finite-time consensus law was designed for second-order multi-agent systems under one leader and bidirectional connected followers.A similar idea was designed in Du et al.(2013)using dynamic output feedback.In Huang et al.(2015),an adaptive finite-time consensus algorithm was designed for uncertain nonlinear mechanical systems.The continuous finite-time consensus method was developed to deal with high-order nonholonomic mobile robots with bidirectional topologies in Du et al.(2017)and surface vehicles under the assumption that all followers can access to the leader in Wang N and Li(2020).Note that the objectives of the coordinated path-following control problem include path-following and formation,which are different from those of the consensus problem.It is essential to give a finite-time method to the coordinated path-following problem.

This paper gives a continuous solution to the finite-time control problem of coordinated pathfollowing under directed topologies.To solve the trajectory restriction problem,we present a new barrier function definition that is integrated into backstepping to design a novel continuous finite-time path-following control input projected on the normal vector on the orbit.Another continuous finitetime formation control input projected on the tangential vector on the orbit is designed by regarding the path-following errors as disturbances.Note that the proposed method in this paper is different from our previous adaptive method in Chen et al.(2021b)concerning two conditions:(1)directed networked second-order agents are under consideration and the first-order systems are replaced with bidirectional topologies;(2)a continuous finite-time design method is used to replace the adaptive methods.

2 Preliminaries and problem formulation

2.1 Graph theory and barrier functions

The network topology of the coordinated pathfollowing system can be described by a digraphG={V,E},where nodesV={V0,V1,···,Vn}are associated with a virtual leader labeledV0andnvehicles labeledV1,V2,···,Vn,andE?V×Vis a set of network links.A directed path from nodeVito nodeVjis a sequence of edges(Vi,Vi1),in the network topology with distinct nodesVik,k=1,2,...,l.A digraph is called a directed tree if there exists a node,called the root,that has directed paths to all the other nodes in the digraph.Let,fori,j=0,1,...,n,aii=0 andaij=1 if(Vi,Vj)∈E,andaij=0 otherwise.In addition,define the Laplacian matrixandlij=-aij,for anyi/=j,i,j=0,1,...,n.

Assumption 1The digraph consisting of a virtual leader andnvehicles contains a directed spanning tree with rootV0.

For the considered coordinated path-following system,the Laplacian matrixLcan be written as

wherel0=[l10,l20,···,ln0]T∈Rn×1andL1∈Rn×n.Suppose that Assumption 1 holds.L1is a nonsingular M-matrix and all eigenvalues ofL1have positive real parts(Zhang Y and Tian,2009).ρL1denotes the smallest eigenvalue ofL1.

To keep each agent’s trajectory staying in a restricted area when applying the geometry extension method,a new definition of the barrier functionΨiis given:

Definition 1AC2functionΨi:(-εi,εi)→R is a barrier function with barrier 2εi＞0 if the following conditions hold:

Remark 1The barrier function in Definition 1 is different from those in traditional definitions,because condition(C4)is added and used to yield the finite-time convergence of the resulting system with the state constraintΩi=

wherec1andc2are positive constants.In this case,one can select parameterc2from 0.3 to 0.8 to yield condition(C4),as shown in Fig.1.

Fig.1 Sketches of Ψi,?Ψi,and cΨλi with respect to different c2 values:(a)c2=0.3;(b)c2=0.8(cΨ=0.7,εi=2,and c1=0.2)

2.2 Some lemmas

Lemma 1(Chen and Tian,2015) Consider any simple,closed,and regular orbit satisfying the following conditions:

Identifying the orbit by mapCi0,there exists a constantεi＞0 such thatCi(·)(·,·)is a diffeomorphism on[0,2π)×(-εi,εi).Moreover,there exists an open setΩi?R2,which is a tubular neighborhood of the orbit,and a smooth functionλi:Ωi→(-εi,εi),which is called the orbit function(its value is called the orbit value),such that the following conditions hold:

(C9)λi(pi)=c,for all pointspion the orbit identified byCicwithc∈(-εi,εi).

Cicis a level line of the orbit functionλi(pi),and the orbit value associated with orbitCi0is zero.

Definition 2(Chen et al.,2021a)The generalized arc-lengthsξiareC1functions of the arc-lengthssi

Lemma 2(Chen et al.,2021a)TheC1invertible mappingsξi:R→R define a change of coordinates,which allows formulation of the coordinated pathfollowing problem with state variablessiinto the consensus problem described by

with state variablesξi(si).To form the desired formation,the desired arc-lengths*iof theithfollower is determined by the arc-length of the leaders0vias*i=gsis0(s0),wheregsis0:R→R is an invertible mapping explicitly defined by the desired formation.By Eq.(2),ξ0andξiare such that

Note thats0=(s*i)yieldsξi=ξ0,which properly definesξifor any givenξ0,sincegsis0is known.Here,“?”is the symbol for composition of functions;that is,ξiis the composition ofξ0and

2.3 Problem formulation

In this study,we first consider the cases of static virtual leader and dynamic virtual leader.In a fixed inertial reference frame,the model of the static virtual leader is the first-order dynamics such that ˙p0=0,wherep0=[px0,py0]T∈R2is its position.Fori=1,2,···,n,the dynamic equation for theithfollower satisfying the second-order dynamics is given by

wherepi=[pxi,pyi]T∈R2andvi=[vxi,vyi]T∈R2denote the position and velocity,respectively.ui=[uxi,uyi]T∈R2denotes the control input.

Suppose that the desired orbit associated with each agent is a simple,closed,and regular curve with nonzero curvature.According to Lemma 1,this curve can be extended geometrically to a set of level curves,which can be defined by a smooth function(the orbit function)λi:Ωi→(-εi,εi);the desired orbit can be defined byλi(pi)=0,whereΩi?R2is an open set andpi∈R2.

The path-following error can be described by the value of the orbit function and the path-following task is achieved if

with a finite timeT＞0 andpi(t)∈Ωi,for allt≥0,where

Let the arc-lengths be given by

whereφ*iis the parameter associated with the starting point of the arc ofsi.The generalized arclengthsξi:R→R ofsiare used to describe the formation along the curves.?ξi/?siis a constant and and satisfiescξ≤|?ξi/?si|≤cˉξwith two positive constantscξandcˉξ.From Lemma 2,the task of achieving coordinate formation in finite timeTcan be described as follows:

The finite-time coordinated path-following control problem is as follows:Fori=1,2,···,n,consider system(3)and the initial positionpi(0)∈Ωi.Suppose that Assumption 1 holds.Design a finitetime coordinated path-following controluisuch that the closed-loop system satisfies Eqs.(4)and(7).

Remark 2The discontinuous laws based on sgn(·)(Khoo et al.,2009;Xiao et al.,2009;Cao et al.,2010;Wang L and Xiao,2010;Guan et al.,2012;Li TS et al.,2018;Dou et al.,2019)might cause signal chattering in the closed-loop system.In practice,it is difficult to accomplish these discontinuous laws.

Remark 3This paper is devoted to designing a continuous finite-time control law for directed networking second-order agents for the coordinated path-following problem.However,Chen et al.(2021b)dealt with the adaptive design for first-order agents with unknown time-varying parameters and bidirectional topologies.

3 Main results

In this study,we first consider the cases of static virtual leader and dynamic virtual leader.The design precedence is as follows:(1)decouple the whole system as a path-following subsystem and a formation subsystem;(2)regardvNias a virtual controller?vNiand designuNiby backstepping to achieve finite-time path-following along the given orbits(Theorem 1);(3)regardvTias a virtual controller?vTiand designuTiby backstepping to achieve finite-time formation along the given orbits(Theorem 2);(4)according to Theorems 1 and 2,give Theorem 3 to show the finite-time convergence of the coordinated path-following control system.Then a corollary is given to show the case of the dynamic virtual leader.

Section 3.1 gives the open-loop system(i.e.,the error equations of the coordinated path-following control system),which is used to design the pathfollowing control law in Section 3.2 and the formation control law in Section 3.3.

3.1 Open-loop system

By differentiatingλi,the path-following dynamics of agentiis obtained as follows:

wherevNi=NTi videnotes the velocity projected on vectorNiwhich is normal to the level orbit of the current position of agentiandNi=.Differentiating both sides ofvNiyields

whereuNi=NTi uidenotes the control input projected on the normal vectorNi,ΔNi=,and

LetTidenote the vector which is tangent to the level orbit of the current position of agentiandTi=RTNi=[R1,R2]TNi,whereR1=[0,1]TandR2=[-1,0]T.Then the dynamics ofξiis given by

wherevTi=TTi viandΔξi=The proof of Eq.(10)is provided in the supplementary materials.Differentiating both sides ofvTiyields

whereuTi=TTi uidenotes the control input projected on the tangent vectorNiandΔTi=

Letdenote the formation errors.The dynamics of?iis described by

As a result,the equations of the formation tracking control system are given by Eqs.(8),(9),(11),and(12).

3.2 Path-following controller design

Let us first consider the path-following subsystem consisting of Eqs.(8)and(9)and let the virtual controlbe

where 1≤α=,p1andp2are positive odd integers,and the control gaink1will be selected later.Consider the path-following candidate Lyapunov function as

whereThe first term on the righthand side of Eq.(14)contributes to achieving the path-following objective,i.e.,Eq.(4).The second term contributes to guaranteeing the convergence of the differences.Differentiating both sides of Eq.(14)along the trajectories of Eqs.(8),(9),and(13)yields

where

The proof of inequality(15)is provided in the supplementary materials.

whereφ1=2-1-1α(1+α)/α,φ2=1+α,cφ1=

The proof of inequality(16)is provided in the supplementary materials.

On the setfor somecP＞0,one haswith someandcΨ2＞0.Exploiting inequalities(16)and(17),we conclude that

which yields

where

The proof of inequality(18)is provided in the supplementary materials.

Suppose thatVP(t)/=0.Substituting Eq.(19)into inequality(18)yields

whereBy Eq.(19),the closed-loop equation associated with the path-following subsystem for theithfollower is

Remark 4It is obvious that the closed-loop system(22)for path-following is not homogeneous.Therefore,the finite-time stability analysis methods given in Guan et al.(2012)and Dou et al.(2019)cannot be applied in this study.

Note that 0＜(1+)/2＜1.To apply Theorem 4.2 in Bhat and Bernstein(2000)(which is provided in the supplementary materials),we will show thatgPhas a lower bound.From condition(C4),we have

withλi0=λi(0),which yields

whereβP3=maxAs a result,we have

whereβP4is positive and bounded.According to Theorem 4.2 in Bhat and Bernstein(2000),we establish the following theorem:

Theorem 1Suppose that the initial positions of vehicles are such thatpi(0)∈Ωi.Assume moreover that Assumption 1 holds.Then the path-following objective(Eq.(4))can be achieved by the finite-time controluNigiven in Eq.(19),fori=1,2,···,n.

ProofFrom inequality(21),we conclude that the functionVPis bounded all the time,which implies that the objective(Eq.(5))is satisfied according to conditions(C1)and(C2).From inequalities(21)and(24),we conclude thatλi=0 and thatˉvNi(i=1,2,···,n)are the finite-time stable equilibria of the closed-loop path-following subsystem(22).

3.3 Coordinated formation controller design

In the following,we will consider the formation subsystem consisting of Eqs.(11)and(12).Let the virtual controlbe

wherek3is a positive control gain and will be selected later.Consider the coordinated formation candidate Lyapunov function as

whereIn Eq.(26),the first term on the right-hand side contributes to achieving the formation objective,i.e.,Eq.(7),and the second term contributes to guaranteeing the convergence of the differences.Differentiating both sides of Eq.(26)along thetrajectories of Eqs.(11),(12),and(25)yields

where

The proof of Eq.(27)is provided in the supplementary materials.

Note that

where

andγ4=max?i,j{aij}.From Lemmas A.1 and A.2 in Qian and Lin(2001),we have

whereSubstituting inequality(32)into inequality(31)yields

where

where

Due to the fact that

from Eq.(25),inequality(32),and Eq.(35),one has The proof of inequality(37)is provided in the supplementary materials.

From Lemmas A.1 and A.2 in Qian and Lin(2001),we have

withwhich yields

wherecvT2=cg1+cg2cF2+ncg3+ncg3cF2+ncg4cF2andcσ2=cg2+ncg4.

From inequalities(34)and(39),we have

Substituting the above inequality andinequality(33)into Eq.(27)yields

which makes the choices such that

where the control gaink4will be set later.As a result,the closed-loop formation subsystem for theithfollower is

in which control gains are chosen as follows:

whereβF1is an arbitrary positive constant.As a result,we have

which yieldsSuppose thatVF(t)/=0.Inequality(44)can be rewritten as

whereandgF3=gF1+gF22.Due to 0＜＜1,βF2has a lower bound.gF3approaches zero as limt→T λi(t)=0 and limt→T(t)=0,as proved in Theorem 1.We give the following result directly:

Theorem 2Suppose that the initial positions of vehicles are such thatpi(0)∈Ωi.Assume moreover that Assumption 1 holds.Then the formation objective(Eq.(7))can be achieved by the finite-time controluTigiven in Eq.(41),fori=1,2,···,n.

ProofThe proof follows the same argument as the proof of Theorem 5.3 in Bhat and Bernstein(2000)(which is provided in the supplementary materials).Hence,it is omitted.

Theorems 1 and 2 yield the following result:

Theorem 3Suppose that the initial positions of vehicles are such thatpi(0)∈Ωi.Assume moreover that Assumption 1 holds.Fori=1,2,···,n,the finite-time coordinated path-following control problem is solved by the coordinated path-following control:

whereuNianduTiare as given in Eqs.(19)and(41),respectively.

Remark 5Different from the consensus problem studied in Li SH et al.(2011),this study addresses the coordinated path-following control problem,which includes two subproblems,i.e.,pathfollowing and formation control.Moreover,the digraph in this paper,assumed to consist of a virtual leader andnvehicles,contains a directed spanning tree with rootV0,while in Li SH et al.(2011),each follower was required to access to the leader’s states.

Now,let us consider a special case:the virtual leader has a velocity=η0along the responding orbit and its velocity accesses to each follower,whereη0and˙η0are bounded signals.In this case,the open-loop equations of the path-following subsystem are Eqs.(8)and(9),which are the same as those of the static case.Let=ξi-ξ0.The time derivative of

which is similar to Eq.(12).As a result,the expressions ofandare the same as Eqs.(13)and(19)in the static case,respectively.The expressions ofandare also the same as Eqs.(25)and(41),respectively,in the static case by replacing?iandΔTiwithand,respectively.

We now give the following corollary directly:

Corollary 1Consider that a virtual leader has a velocityη0along the responding orbit and that the access to each follower has been considered.Suppose that the initial vehicle positions are such thatpi(0)∈Ωi.Fori=1,2,···,n,the finite-time coordinated path-following control problem is solved by the coordinated path-following control law(47),whereuNianduTiare as given in Eqs.(19)and(41),respectively.

4 Simulations

In this section,we first apply the proposed control laws in Theorem 3 to coordinate the vehicles moving along the elliptic orbits with a triangle pattern in case 1,and then use the control algorithm proposed in Corollary 1 to achieve the in-line formation in case 2.The selected trajectories of the agents are concentric ellipses with a different semi-major axis and semi-minor axis,that is,Cl0:whereel=1+0.5l,a=3,b=2,andl=0,1,2,3,4.

1.Case 1:static virtual leader

The topology for the virtual leader and followers is shown in Fig.2.The parameters are selected ask1=k3=2.7,k2=k4=34,andα=.The initial generalized arc-length of the virtual leader isξ0(0)=0.The motion of the agents is illustrated in Fig.3,where“?”,“□”,“★”,and“+”denote the agents’positions att=0,1,2,and 7 s,respectively.In this figure,one can see that the four followers converge to the given orbits and achieve the desired formation.The path-following errorsλiand formation errorsξi-ξ0are plotted in Figs.4 and 5,respectively.The above figures show that path-following and formation tracking are achieved.

Fig.2 A directed topology

2.Case 2:dynamic virtual leader

We use the coordinated path-following control algorithm given in Corollary 1 to achieve the in-line pattern.The parameters are selected ask1=k2=k3=k4=10 andα=.The motion of the agents,the path-following errorsλi,and the formation errorsξi-ξ0are illustrated in Figs.6,7,and 8,respectively.

Fig.4 Path-following errors in the static virtual leader case

Fig.5 Formation errors in the static virtual leader case

Fig.6 Motion of the agents in the dynamic virtual leader case

Fig.7 Path-following errors in the dynamic virtual leader case

5 Conclusions

Fig.8 Formation errors in the dynamic virtual leader case

A continuous feedback method to solve the finite-time coordinated path-following control problem is presented,where the topology for the virtual leader and followers is directed.Because the movable ranges of the agents are restricted,a novel barrier function is given.A finite-time coordinated pathfollowing control law in the static virtual leader case is designed first.Then the control law is obtained in the dynamic virtual leader case,where its velocity can be accessed by each follower.Conditions on the control gains to guarantee that the path-following errors and the formation errors converge to zeros in finite time are presented.In ongoing work,the experiments involving finite-time coordinated pathfollowing problems will be considered.

Contributors

Weibin CHEN designed the research.Yangyang CHEN drafted the paper.Ya ZHANG helped organize the paper.Weibin CHEN and Yangyang CHEN revised and finalized the paper.

Compliance with ethics guidelines

Weibin CHEN,Yangyang CHEN,and Ya ZHANG declare that they have no conflict of interest.

List of electronic supplementary materials

Theorem 4.2 in Bhat and Bernstein(2000)

Theorem 5.3 in Bhat and Bernstein(2000)

Proof of Eq.(10)

Proof of inequality(15)

Proof of inequality(16)

Proof of inequality(18)

Proof of Eq.(27)

Proof of inequality(37)