# Consensus of Multiagent System in the Sense of Curvature and Torsion

Authors: Jinping Mou
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

In this paper, the consensus problem is investigated for a distributed multiagent system (MAS), where the consensus is characterized by curvature function and torsion function. According to the Frenet-Serret formulas, a distributed consensus protocol is designed for the tangent, normal, and binormal unit vectors of trajectory of each agent, and then it gets a closed-loop system. Based on the Lyapunov function, several sufficient conditions for consensus are derived for the closed-loop system. Also the consensus problem of multiagent system on a surface is studied. Finally, the numerical examples show the reliability of the proposed methods.


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## Body

## 1. Introduction

In the last decades, curvature and torsion are taken as the efficient tools to solve some problems in different communities, such as fluid dynamics [1, 2] and mathematics physics [3]. In recent years, the further research on curvature and torsion has been investigated in [4–8], and the results provide more ways to solve the consensus problems. However, as the efficient tools, curvature and torsion have not attracted much attention in the control field of multiagent system (MAS).Consensus of MAS is the basic requirement for some applications, which requires the state of each agent converging to an interesting value. For the past decade, consensus problem of MAS has attracted much attention due to its presence in formation, environmental monitoring, and unmanned aerial vehicles control [9]. Therefore, many results on consensus problems of MAS have been obtained (see, e.g., consensus of the first order MAS [10–19]; consensus of the second order MAS [20, 21]; consensus of the fraction order MAS [22]; and the consensus of the scale free networks [23]).Notice that, in the above results, consensus refers to the fact that the state of each node converges to a same value, or the errors among all agents are limited for the target tracking system. In [14, 15], the consensus results show that all trajectories of nodes converge to a curve. In [17], the tracking errors among all trajectories are limited in the permitted range. However, in some real applications, we require that the distances among all agents keep nonzero constants; for example, we need all unmanned aerial vehicles to keep the same track with nonzero distances, and we call it the same trajectory except for the different locations here (STEDL). But the vast majority of results require the zero error among all agents, and the target tracking system with limited errors also cannot remain the nonzero distances, so the listed results are not suitable for solving the STEDL problems.The motivation of this paper is that a trajectory of the moving node is decided by its curvature function and torsion function entirely. That is to say, to solve STEDL problem, each agent should have the same curvature function and torsion function, respectively. However, for the different initial locations, the different agents may have the different curvature and torsion functions. Hence, the main challenge of solving STEDL is how to let the different curvature and torsion functions converge to a curvature function and a torsion function asymptotically.Generally, most of us believe that, without considering the curvature function and the torsion function, under the traditional average consensus protocols, the connected networks can achieve the average consensus (see, e.g., [14–21], and the references therein). However, under the traditional consensus protocols, all trajectories of agents converge to a curve with zero distance or converge to a line if we apply the protocol to the tangent, normal, and binormal vectors of each agent directly without considering the curvature function and the torsion function, and the results are not what we expected sometimes. For example, we want a group of unmanned aerial vehicles having the same trajectory, and the track may be a curve. In this paper, curvature function and torsion function will be considered when we design the consensus protocol, and the results show that the trajectories of all agents converge to a curve except for the different locations.The aim of this paper is to describe and analyze the STEDL problem. Our approach begins with some conceptions about the consensus in the sense of the curvature and the torsion, and we call itCT consensus here. The conception of CT consensus extends the traditional consensus that is discussed in [18]; when we ignore curvature function and the torsion function, the discussed model is the same as that studied in [18]. Then the consensus protocol about curvature and torsion is proposed, we get a closed-loop system, and the stability of the system is discussed. For updating of all trajectories can be expressed by the unit vectors of tangent, normal, and binormal, hence, to discuss CT consensus, the system is described by unit vectors of the tangent, normal, binormal and the Frenet-Serret formulas.As two special cases, we study STEDL problems on a plane and on a spherical surface. For the STEDL problem on a plane, since the torsion of a curve on a plane is zero, the consensus problem of MAS boils down to the consensus of curvature, and, for the spherical surface, the consensus of MAS comes down to the relation between torsion and curvature [24, 25]. Finally, the general STEDL problem of MAS on a surface is investigated.The remainder of this paper is organized as follows. In Section2, it introduces some concepts, and the main notations are shown in Table 1. Section 3 proposes the STEDL problem. In Section 4, several consensus criteria are proposed for MAS. Section 5 provides some numerical examples to illustrate the reliability of the proposed methods. In Section 6, we get several conclusions.Table 1
General notations.
Notations Meaning Z + the positive integer set R n the n- dimensional Euclidean space C 3 the 3- times continuously differentiable I n the n times n unit matrix s s is the length of arc on trajectory vi 0 zero vector α i ′ ( t ) the first derivative ofα(t) α ˙ i ( s ) the first derivative ofα(s) τ ^ max ⁡ { τ i ( t ) ∣ 1 ≤ i ≤ n }, κ ^ max ⁡ { κ i ( t ) ∣ 1 ≤ i ≤ n }, τ ˇ min ⁡ { τ i ( t ) ∣ 1 ≤ i ≤ n }, κ ˇ min ⁡ { κ i ( t ) ∣ 1 ≤ i ≤ n } d i ∑ j ∈ N i ‍ a i j κ i 0 ( t ) 1 d i  ∑ j ∈ N i ‍ κ j ( t ), τ i 0 ( t ) 1 d i  ∑ j ∈ N i ‍ τ j ( t ), α ( α 1 T ( t ) , α 2 T ( t ) , … , α n T ( t ) ) T, β ( β 1 T ( t ) , β 2 T ( t ) , … , β n T ( t ) ) T, γ ( γ 1 T ( t ) , γ 2 T ( t ) , … , γ n T ( t ) ) T, K d i a g  κ 1 t , κ 2 t , … , κ n t ⊗ I 3, J d i a g  τ 1 t , τ 2 t , … , τ n t ⊗ I 3 .


## 2. Preliminaries

LetV={v1,v2,…,vi,…,vn} be the agent set, where i∈Z+, 1≤i≤n. Let A=(aij)n×n be a weighted symmetric adjacent matrix, where aij is the weight of the edge (i,j), aij takes value in 1 or 0, and aij=aji. aij=1 shows that there exists information flow between nodes vj and vi, or else aij=0. E={(i,j)∣aij=1,i≠j} refers to the edge set, where (i,j) denotes an edge between vi and vj. Ni denotes the neighbor set of vi, where Ni={vj∣(vi,vj)∈E}. Hence, the topology of the multiagent system (MAS) can be denoted by an undirected weighted graph G=(V,E,A).∀ v i ∈ V, let (1)rit=xit,yit,zitT∈R3be the state of agent vi, where t∈R, and xi(t),yi(t),zi(t) are the C3 functions, so the state of MAS can be denoted by (2)rt=r1tT,r2tT,…,ritT,…,rntTT.According to the theory of the differential geometry, the trajectory ofvi is decided entirely by the curvature function and the torsion function. For convenience, let κi(t) and τi(t) be the curvature function and torsion function of vi, respectively. Therefore, ∀vi and vj, if (3)κit=κjt,τit=τjt,then vi and vj have the same trajectory except for the difference of their locations. Consequently, based on the curvature function and torsion function, a consensus definition is proposed in the following.Definition 1.
∀ v i, vj∈V, if (4)limt→∞κit-κjt=0,limt→∞τit-τjt=0,then MAS is the consensus in the sense of curvature and torsion. For short, we call it CT consensus here.Remark 2.
In Definition1, CT consensus is different from the consensus mentioned in the recent literatures (e.g., see [18–23]). In these literatures, the consensus requires that, for every two vectors ri(t) and ri(t), they satisfy (5)limt→∞rit-rjt=0,and all trajectories of them converge to interest curve; the state error between every two agents converges to zero vector, and (6)limt→∞κit-κjt=0,limt→∞τit-τjt=0.However, the results cannot ensure that every two agents keep the nonzero distance. Similarly, for the second-order MAS [21], based on the different initial values, consensus of the velocity and acceleration leads that trajectories of all agents to converge to a curve; however, every two agents cannot keep the nonzero distance.Definition 3 (see [26]).
The tangent unit vector is denoted byαi(t), the normal unit vector is denoted by βi(t), and the binormal unit vector is denoted by γi(t), where (7)αit=ri′tri′t,γit=ri′t×ri′′tri′t×ri′′t,βit=γit×αit.Remark 4.
In Definition1, CT consensus is decided by the curvature function and the torsion function. On the other hand, the curvature function and torsion function are influenced by the updating of the tangent vector αi(t), normal vector βi(t), and binormal vector γi(t) of each agent vi, which satisfy the following Frenet-Serret formulas:(8)αi′t=κitβit,βi′t=-κitαit+τitγit,γi′t=-τitβit.
According to (8), if(9)limt→∞αit-αjt=0,limt→∞βit-βjt=0,limt→∞γit-γjt=0, then we have (10)limt→∞κit-κjt=0,limt→∞τit-τjt=0.Hence, we get the following result.Proposition 5.
∀ v i , v j ∈ V, if (9) holds, then MAS achieves the CT consensus.


## 3. Problem Statement

In this section, the dynamics of tangent vector, normal vector, and binormal vector are displayed for each agent firstly and then the dynamics of tangent vectors, normal vectors, and binormal vectors are described for MAS.∀ v i ∈ V, the dynamics of αi(t),βi(t),γi(t) are described by(11)αi′t=κitβit+ui1t,βi′t=-κitαit+τitγit+ui2t,γi′t=-τitβit+ui3t, where ui1(t), ui2(t), and ui3(t) are the controls of agent vi, and they are defined by(12)ui1t=∑j∈Niαjt-αit-1di∑j∈Niκitβit,ui2t=∑j∈Niβjt-βit-1di∑j∈Niτitγit+1di∑j∈Niκitαit,ui3t=∑j∈Niγjt-γit+1di∑j∈Niτitβit,From (11) and (12), we obtain the dynamics of vectors α, β, and γ below:(13)α′=-Lα+I-A^Kβ,β′=-I-A^Kα-Lβ+I-A^Tγ,γ′=-I-A^Tβ-Lγ, where L is the Laplacian matrix: (14)
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.For convenience, denoteX=(αT,βT,γT)T; then (13) can be represented as(15)X′=AX,where (16)A=-LI-A^K0-I-A^K-LI-A^J0-I-A^J-L.In the following, we are going to discuss theCT consensus of system (15).


## 4. The Main Results

In this section, the stability of the error system of system (15) is studied firstly. Then the consensus problem is investigated for system (15) in R3 space, where the ranges of the curvature functions and torsion functions are gotten, and the criteria of CT consensus are proposed. Finally, the CT consensus problem is studied for system (15) in some surfaces.Denotee(t)=X(t)-X0(t), where X0(t)=(α0(t),β0(t),γ0(t)),(17)α0t=1n∑i=1nαi,β0t=1n∑i=1nβi,γ0t=1n∑i=1nγi,and it obtains that(18)et=I-1n×nnXt,and the error system(19)e′t=I-1n×nnAtXt.Notice that(1n×n/n)L=0, and (20)∑j∈Niαjt-αit=∑j∈Niαjt-α0t-αit+α0t=∑j∈Niejt-eit;hence, LX(t)=Le(t), and the error system (19) can be shown by(21)e′t=-Let+0QK0-QK0QJ0-QJ0XtIn order to study the stability of (19), we should discuss the relation between the matrices (I-1n×n/n)A(t) and A(t)(I-1n×n/n). Denote(22)I-1n×nnAt-AtI-1n×nn(23)=At1n×nn-1n×nnAt=Bt,where(24)Bt=0B120B210B230B320,B12=-B21=K1n×nn-1n×nnK+1n×nnA^K-A^K1n×nn=bijn×n,B23=-B32=J1n×nn-1n×nnJ+1n×nnA^J-A^J1n×nn=cijn×n,bij=κit-κjtn+1n∑v≠iavidvκit-∑v≠iaivdiκvt,cij=τit-τjtn+1n∑v≠iavidvτit-∑v≠iaivdiτvt,and then it gets that(25)B12β=∑i=1nb1iβitT,∑i=1nb2iβitT,…,∑i=1nbliβitT,…∑i=1nbniβitTT,B21α=-∑i=1nb1iαitT,∑i=1nb2iαitT,…,∑i=1nbliαitT,…∑i=1nbniαitTT,B23γ=∑i=1nc1iγitT,∑i=1nc2iγitT,…,∑i=1ncliγitT,…∑i=1ncniγitTT,B32β=-∑i=1nc1iβitT,∑i=1nc2iβitT,…,∑i=1ncliβitT,…∑i=1ncniβitTT,where (26)∑i=1nbliβit=1n∑j=1nκit-κjt+aijκitdj-aijκjtdiβjt,∑i=1nbliαit=1n∑j=1nκit-κjt+aijκitdj-aijκjtdiαjt,∑i=1nbliγit=1n∑j=1nτit-τjt+aijτitdj-aijτjtdiγjt,∑i=1nbliβit=1n∑j=1nτit-τjt+aijτitdj-aijτjtdiβjt,To study the stability, we need the following.Assumption 6.
Suppose thatαi(t), βi(t), γi(t) satisfy (27)2κ^t-κˇtβ0t≤fte2t,2κ^t-κˇtα0t≤fte1t,τ^t-τˇt+τ^tdm-τˇdMγ0t≤fte3t,τ^t-τˇt+τ^tdm-τˇdMβ0t≤fte2t,where (28)ft=max⁡2κ^-κˇ,τ^-τˇ+τ^dm-τˇdM.
Then, we discuss the stability of system (15).
Let the Lyapunov candidate function be presented as(29)Vt=12etTet;it follows that(30)V′t=12e′tTet+12etTe′t=12X′tTI-1n×nnet+12etI-1n×nnX′t=12XtTAtTI-1n×nnet+12etI-1n×nnAtXt.Based on (22), it gets(31)V′t=12XtTI-1n×nnAtT+BtTet+12etI-1n×nnAt+BtXt=etTAtT+A(t2et+12XtTBtTet+12etTBtTXt=-e1TtLe1t-e2TtLe2t-e3TtLe3t+12[βtTB12e1t+e1tTB12βt+αtTB21e2t+e2tTB21αt+γtTB23e3t+e3tTB23γt+βtTB32e3t+e3tTB32βtTheorem 7.
IfG is connected, and if f(t)<λ/2, then system (19) is the asymptotically stable, where λ is the second largest eigenvalue of L.Proof.
Following inequality (31), it holds that (32)V′t≤2λVt+12[βTB12e1t+e1tTB12β+αTB21e2t+e2tTB21α+γTB23e3t+e3tTB23γ+βTB32e3t+e3tTB32β,since it holds that(33)B12β≤∑l=1n∑i=1nbliβit≤1n∑l=1n∑j=1nκit-κjt+aijκitdj-aijκjdiβjt≤2κ^t-κˇtβ⋆t≤fte2t;similarly, we have (34)αtTB21≤fte1t,γtTB23≤fte3t,βtTB32≤fte2t;hence, (35)V′t≤-2λVt+ft[4e2tTe1t+2e3tTe3t+2e2tTe3t≤-2λVt+4fte1tTe1t+e2tTe2t+e3tTe3t=-2λVt+4ftVt<0.System (19) is asymptotically stable.Theorem 8.
Under (11), if G is connected, then system (15) achieves consensus.Proof.
From the proof of Theorem7, if t→∞, then V(t)→0. Hence, ∀vi,vj∈V(36)limt→∞⁡αit-αjt=0,limt→∞⁡βit-βjt=0,limt→∞⁡γit-γjt=0,so system (15) achieves the consensus.Remark 9.
Theorem8 shows the sufficient conditions of system (15) about the consensus. However, it does not provide the values about κi(t) and τi(t); the following results show the values.Corollary 10.
If system (15) achieves the consensus, then κi(t) and τi(t) satisfy the following equations:(37)limt→∞κit=1di∑j∈Niκjt+1n∑i=1nκit-1di∑j∈Niκjt,(38)limt→∞τit=1di∑j∈Niτjt+1n∑i=1nτit-1di∑j∈Niτjt,Proof.
from (11) and (12), we have(39)αi′t=κit-1di∑j∈Niκitβit,βi′t=-κit-1di∑j∈Niκitαit+τit-1di∑j∈Niτitγit,γi′t=-τit-1di∑j∈Niτitβit,On the one hand, following the result of Theorem 7, it is obtained that limt→∞e(t)=0 and ∀i∈V, (40)limt→∞αit-α0t=0,limt→∞βit-β0t=0,limt→∞γit-γ0t=0;then (41)limt→∞αit-αjt=0,limt→∞βit-βjt=0,and (42)limt→∞αi′tβit-αj′tβjt=0. On the other hand, (43)αi′tβi=κit-1di∑j∈Niκit,αj′tβj=κjt-1dj∑k∈Njκjt,and, hence, (44)limt→∞κit-1di∑j∈Niκit=limt→∞κjt-1dj∑k∈Njκjt. Similarly, we have (45)limt→∞τit-1di∑j∈Niτit=limt→∞τjt-1dj∑k∈Njτjt.
Denote(46)κt=limt→∞κit-1di∑j∈Niκit,(47)τt=limt→∞τit-1di∑j∈Niτit,so(48)κt=κ1t-1d1∑j∈N1κ1t+δ1t,κt=κ2t-1d2∑j∈Niκ2t+δ2t,…κt=κnt-1dn∑j∈Nnκnt+δnt,where limt→∞δit=0,i=1,2,…,n, and we have(49)limt→∞κt=1n∑i=1nκit-1di∑j∈Niκjt.Substitute (49) into (46); it holds that (50)limt→∞κit=1di∑j∈Niκjt+1n∑i=1nκit-1di∑j∈Niκjt; similarly, we get (38).Following Theorem7, we obtain the following result.Remark 11.
If the consensus protocol is chosen as(51)αi′t=ui1t=∑j∈Niαjt-αit,βi′t=ui2t=∑j∈Niβjt-βit,γi′t=ui3t=∑j∈Niγjt-γit,the method is widely used in many literatures (see, e.g., [13, 23] et al.).
If we apply (51) to vectors αi(t),βi(t), and γi(t) directly, then we get that all trajectories of agents become some parallel lines finally (see the simulation results of numerical Example 2). This is because αi(t),βi(t), and γi(t) satisfy (51); on the other hand, αi(t),βi(t), and γi(t) satisfy (8). Thereby, ∀i∈V, κi(t)→0, τi(t)→0 while t→∞. Thus, the protocol (51) is a special case of protocol (11).Definition 12.
STEDL problem on a curved surface refers to all agents moving on a curved surface and, under protocol (51), system (15) achieves CT consensus.In the following, STEDL problems on a curved surface are investigated. First, as the special cases, STEDL problems on a plane and on a spherical surface are studied.Corollary 13.
IfG is connected, and all agents are moving in a plane, then, under protocol (11), system (15) achieves the consensus if(52)2κ^t-κˇt≤λ2.Proof.
If all agents are moving in a plane, then∀vi∈V, τi(t)=0. From Assumption 6 and (6), we have κi(t)=0 and (52).Corollary 14.
Under the protocol (12), if G is connected, all agents are moving on a spherical surface, and system (15) achieves the consensus; then the following inequality holds:(53)2A1cos⁡∫τisds+A2sin⁡∫τisds-1-κˇis≤λ2,where s is the length of arc, and A1 and A2 are the constants.Proof.
According to literatures [27, 28], the following holds:(54)A1cos⁡∫τisds+A2sin⁡∫τisdsκis=1.Hence, from (54), we obtain that for each agent vi, if its τi(s) satisfies that (55)2A1cos⁡∫τisds+A2sin⁡∫τisds-1≤κˇis+λ2. This completes the proof.Remark 15.
System (15) achieves the consensus if and only if ∀vi,vj∈V, (56)lims→s0αis-αjs=0,lims→s0βis-βjs=0,lims→s0γis-γjs=0,
Since(57)ri′s=αis,βis,γis,rj′s=αjs,βjs,γjs,And, hence, (58)lims→s0ri′s-rj′s=0if and only if(59)lims→s0ris-rjs=cij,where cij is a constant vector.
According to Remark15, it is not difficult to get the following equivalent statement. ∀vi,vj∈V, (60)lims→s0ris-rjs=cijif and only if (9) hold. So we get another equivalent statement of Definition 1.Proposition 16.
∀ v i , v j ∈ V, if (61)lims→s0ris-rjs=cij,then MAS (15) achieves the consensus.In the following, CT-consensus problem of MAS on curved surfaces is studied. To solve the problem, we should discuss the relations among the curvature function, torsion function, surface, and curves.Letr=r(li,l^i)∈R3 be the C2 function, where (li,l^i)∈H⊂R2, and H is an open region. Suppose ri(s)=r(li(s),l^i(s)) is the trajectory of agent vi on the surface r(li,l^i), where s is the length of arc of vi, li(s),l^i(s) satisfy(62)l˙is=ai,l^˙is=bi,and ai,bi are the constants. Hence, (63)r˙is=rl1l˙is+rl2sl^˙i=airl1+birl2, where (64)rl1=∂ri∂li,rl2=∂ri∂l^i.Then we obtain that(65)αi=r˙is,(66)βi=α˙isα˙is.Since(67)α˙i=ai∂rl1∂dl1dl1ds+ai∂rl1∂l2dl2ds+bi∂rl2∂dl1dl1ds+bi∂rl2∂l2dl2ds=ai2∂rl1∂l1+aibi∂rl2∂l1+aibi∂rl1∂l2+bi2∂rl2∂l2, denote (68)∂rl1∂l1=r11′′,∂rl1∂l2=r12′′,∂rl2∂l1=r21′′,∂rl2∂l2=r22′′;we get(69)α˙i=ai2r11′′+2aibir12′′+bi2r22′′.Notice thatαi˙=κiβi; the following holds:(70)κi=βi·α˙i=α˙i.On the other hand, sinceγ˙i(s)=-τi(s)βi(s), the following hold:(71)τis=-βiγ˙is=γis·β˙isand(72)τis≤γis·β˙is≤β˙is.Following inequalities (70) and (72), we have the following corollary.Corollary 17.
IfG is connected, and all agents are moving on a surface r=r(l1,l2), and l1,l2 satisfy (62), then, under protocol (11), system (15) achieves CT consensus if for each agent vi,(73)max⁡2α˙is,2+2dmβ˙is≤λ2.Proof.
From Assumption6, (70), and inequality (72), it holds that κ^≤max⁡{α˙j∣vj∈V}, |τ^|≤{max⁡β˙j(s)∣vj∈V}, and τˇ≤max⁡β˙j(s)∣vj∈V; then (74)2κ^-κˇ≤2κ^≤λ2,τ^-τˇ+τ^dm-τˇdM≤2+2dmmaxβ˙js∣vj∈V<λ2;this completes the proof.In the next section, several numerical examples show reliability of the proposed methods.


## 5. Numerical Examples

Comparing with the traditional consensus protocol, the methods proposed in this paper get the better effect.Example 1.
Consider a connected MAS (15) with three agents; the topology of (15) is shown as Figure 1, where V={v1,v2,v3}, (75)Xt=r1t,r2t,r3tT,r10=17,0,0T,r20=50,0,0T,r30=20,0,0T,
and sthe curvature functions and torsion functions of the system are listed as follows.(76)κ1t=112+113t,τ1t=-112+113t,κ2t=113+114t,τ2t=-113+114t,κ3t=122+125t,τ3t=-115+117t.and (77)A=010101010.L=1-10-12-10-11.According to Corollary 13, under the control of protocol (11), MAS (15) achieves the consensus. The simulation result is shown as Figure 2, where three agents begin from x-axis, and their trajectories become the same shape; namely, all agents achieve the consensus.Figure 1
The topology of MAS (15).Figure 2
All agents of MAS (15) achieve CT consensus.Example 2.
Following Example1, if we ignore the curvature function and torsion function and apply the general consensus protocol to the tangent, normal, and binormal unit vectors of each agent, then we get (51). Under protocol (51), we get another simulation result that is shown as Figure 3, where the trajectory of each agent converges to straight line, and all trajectories are parallel lines finally.Figure 3
All agents of MAS (15) achieve the special consensus.Example 3.
Following Example1, if we ignore the curvature function and torsion function and apply the general consensus protocol of literatures [18, 19] to ri(t) directly, namely,(78)ri′t=∑j∈Nirjt-rit, then we get(79)r′t=-L⊗I3rt, where (80)rt=r1tT,r2tT,r3tTT.The simulation result is shown as Figure 4, where all agents cannot keep nonzero distance.Figure 4
All agents of MAS (15) achieve the traditional consensus.The above simulation results show that, under protocol (11), all agents achieve the consensus, where, among all agents, they keep the nonzero constant distances, and all trajectories are the curves with the same shape; if we ignore the curvature function and torsion function, the trajectories of all agents are parallel lines asymptotically; if we apply the general consensus protocol to ri(t)  vi∈V, all agents achieve the consensus with zero distance. In fact, in some applications, we would prefer to choose the result of Example 1.


## 6. Conclusion

In this paper, we have discussed the STEDL problem which is the same trajectory except for the different locations. We have offered the concept of consensus firstly; then a kind of protocol of the consensus has been proposed; finally, several consensus criteria have been obtained. However, there are several problems worth considering. One is that the criteria of the consensus are too strong, there might exist others ways to solve STEDL problems, and some weaker conditions could be found. Another is that if there are some disturbances among the agents, the STEDL problem is very difficult to investigate, and the related consensus problems will be considered by me in the future.



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*Source: 8084209-2019-02-28.xml*