Research ArticleAdaptive 3D Distance-Based Formation Control ofMultiagent Systems with Unknown Leader Velocity andCoplanar Initial Positions

Xuejing Lan ,1,2 Wenbiao Xu ,3 and Yun-Shan Wei 1

1School of Mechanical and Electrical Engineering, Guangzhou University, Guangzhou 510006, China2Center for Intelligent Equipment and Internet-Connected Systems, Guangzhou, China3Guangdong Institute of Metrology, Guangzhou 510405, China

Correspondence should be addressed to Yun-Shan Wei; weiys@gzhu.edu.cn

Received 11 May 2018; Accepted 25 July 2018; Published 6 September 2018

Academic Editor: Andy Annamalai

Copyright © 2018 Xuejing Lan et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper considers the distributed 3-dimensional (3D) distance-based formation control of multiagent systems, where the agentsare connected based on an acyclic minimally structural persistent (AMSP) graph. A parameter is designed according to the desiredformation shape and is used to solve the problem that there are two formation shapes satisfying the same distance requirements.The unknown moving velocity of the leader agent is estimated adaptively by the followers requiring only the relative positionmeasurements with respect to their local coordinate systems. In addition, the proposed formation controller provides a new wayfor the agent to leave the initial coplanar location. The 3D formation control law is globally asymptotically stable and has beendemonstrated based on the Lyapunov theorem. Finally, two numerical simulations are presented to support the theoretical analysis.

1. Introduction

Cooperative control of a multiagent system has many appli-cations such as surveillance, exploration, and search andrescue missions. In particular, formation control problem isone of the important aspects and has received significantattention recently with the development of the informationcommunication technique [1–4].

According to the different requirements on the sensingcapability and the interaction topology, the existing forma-tion control schemes are categorized into position-based,displacement-based, and distance-based formation control.In the position-based formation control, the desired positionsare given with respect to a global coordinate system, and theglobal position sensing is required [5]. The desired displace-ments are given and controlled in the displacement-based for-mation, where the relative positions of the neighboring agentsare sensed with respect to a global coordinate system [6]. Thedistance-based formation is prescribed by the desired intera-gent distances, where the interagent distances are controlledand the relative positions of the neighboring agents are sensed

with respect to their local coordinate systems [7–10]. The ori-entations of the local coordinate systems need not be alignedwith each other. This means that a global sensing is notrequired in the distance-based formation control. Moreover,with the application of the rigid and persistent graph theory,only a part of the interagent distances needs to be controlled.Thus, the distance-based formation control has better cost-effectiveness than the other two approaches.

One of the hotspot problems in the distance-basedformation control is how to maintain the formation shape,while the agents are tracking a reference trajectory or mov-ing with a reference velocity [11]. A number of researchworks have considered the movement of formation in thedisplacement-based control. However, only few results inthe distance-based control are available because of itsextreme complexity in analysis [12, 13]. Further, in somereal-world applications, such as the formation of unmannedflight vehicles and submarines [14, 15], the agents are actu-ally moving in a 3-dimensional (3D) space, which will makethe formation control scheme more complicated. Therefore,the study of 3D distance-based formation control is gaining

HindawiComplexityVolume 2018, Article ID 1814653, 9 pageshttps://doi.org/10.1155/2018/1814653

increasing attention [16, 17]. But they have not yet consid-ered the movement of the formation. Then, Zhang et al. pre-sented a 3D formation that can move with a given velocitywhile achieving the desired formation shape by directly add-ing a term of formation maneuvering velocity in the forma-tion control law [18]. It is shown from the paper that theagents should know the reference velocity so as to removethe formation control error. In view of this, Kang et al.designed a distance-based formation control law in theleader-follower type with a moving leader [19, 20]. The fol-lower agents in [19, 20] can estimate the velocity of the mov-ing leader adaptively by only measuring the relative positionsfrom their neighboring agents, which promotes the develop-ment of distance-based formation control. It is true that allthe agents should not be collocated at a common point ini-tially when using the steepest descent control law in thedistance-based formation. The initial positions of the agentsare usually set to noncollinear and close to the target for-mation shape [21, 22]. To solve this initial collinear prob-lem, Park and Ahn modified the gradient control law byintroducing a rotation matrix into the controller, whichcan change the descent direction and help the agentsescape from the collinear position [23]. In [24], a formationcontrol law was proposed based on two mutually perpendic-ular vectors, which provided a way for the agents to leave theinitially collinear location. Another way to solve this prob-lem is to set the initial velocity of the agent with a differ-ent orientation from the line [25]. Although the formationcontrol problem with collinear initial positions of theagents has been solved [23–25], it is still challenging whenthe initial position of the agents is coplanar in the 3Ddistance-based formation.

In this paper, we aim to propose an adaptive 3Ddistance-based formation control law for a multiagent sys-tem with four kinds of agents, where the underlying graphof the formation is an acyclic minimally structural persis-tent (AMSP) graph. Compared with the global leader, firstfollower, and second follower, it is more difficult to designthe controller for the ordinary follower, which follows threeagents and has the problem of trapping in a plane. Theproposed formation controller of the ordinary followerconstructs a vector perpendicular to the plane determinedby its three leaders, which provides a new way for the ordi-nary follower to leave the coplanar location. Moreover, aparameter is designed according to the desired formationshape and is used in the formation controller to solve theproblem that there are two formation shapes satisfyingthe same distance requirements. The unknown movingvelocity of the leader is adaptively estimated by the fol-lowers requiring only the relative position measurementswith respect to their local coordinate systems. The 3D for-mation control law is globally asymptotically stable and hasbeen demonstrated based on the Lyapunov theorem. Theoutline of this paper is listed as follows. Background andpreliminaries are introduced in Section 2. The procedureof distributed formation control scheme design with thevelocity estimator is presented in Section 3. Numerical sim-ulations are completed in Section 4, and we reach a conclu-sion in Section 5.

2. Background and Preliminaries

2.1. Graph and Formation Structure. The formation problemof a multiagent system is modeled by a directed graph G ,which consists of a vertex set V = 1, 2,⋯, n and a directededge set ℰ = i, j : i, j ∈V , i ≠ j . The vertices represent theagents, and the weighted edges represent the interagent dis-tance constraints. The neighboring set of agent i is definedasN i = j ∈V ∣ i, j ∈ℰ . A directed edge from i to jmeansthat agent i can measure the relative position between agent iand j. Then, we call agent i a “follower” of agent j and corre-spondingly call agent j a “leader” of agent i. The formationshape can be maintained during any continuous motion, ifthe underlying graph is rigid and the distance constraints ofeach agent are satisfied. A formation graph is minimally per-sistent if it is rigid and constraint consistent with the mini-mum possible number of edges [26–28].

For a 3D minimally persistent formation, the sum of thedegrees of freedoms (DOFs) of agents is always six [29].Then, there exist various structures of the formation graph,according to the different distributions of these 6 DOFsamong non-0-DOF agents. Moreover, the concept of struc-tural persistence should be taken into consideration for 3Dapplication [29]. For example, the graph in Figure 1(a) is anacyclic minimally structural persistent (AMSP) graph, whilethe graph in Figure 1(b) is minimally persistent but not struc-turally persistent with two free leaders.

In this paper, the AMSP graph is applied to construct the3D leader-follower formation structure, which is the mostconvenient structure to design distributed control schemes.In the AMSP graph, there are one 3-DOF agent called theglobal leader, one 2-DOF agent called the first follower, one1-DOF agent called the second follower, and some 0-DOFagents called ordinary followers.

2.2. Problem Statement. In the distance-based formation, thedesired formation is prescribed by the desired interagent dis-tances. The desired distance between agent i and agent j isdenoted by dij > 0 and apparently dij = dji. It is assumed thateach agent imeasures the relative positions of its neighboringagents via an onboard sensor with respect to its local coordi-nate system i∑. The orientations of the local coordinate sys-tems are not aligned with each other. All the agents move in a3-dimensional space. Although the control law of each agentis implemented in i∑ in practice, it is more convenient to

5

2

1

43

(a)

45

1

2

3

(b)

Figure 1: Examples of graph in a 3D space: (a) an acyclic minimallystructural persistent graph; (b) minimally persistent but notstructurally persistent graph.

2 Complexity

represent the agents with respect to a global coordinate sys-tem g∑ for stability analysis. In addition, the state in i∑ canbe transformed into g∑ by a suitable coordinate transforma-tion. In this paper, the state of each agent will be representedwith respect to g∑. The position and the velocity of agenti at time t in g∑ are denoted by pi t = Xi, Yi, Zi

T ∈ R3 and

vi t = VXi , VY

i , VZi

T ∈ R3, respectively. The dynamics ofagent i is modeled as a single integrator:

pi t = ui t , 1

where ui ∈ R3, i = 2,… , n, is the control input of agent i.z ij ∈ R3 is used to represent the relative position vectoras follows:

z ij = pi − pj 2

In this paper, only the relative position z ij can bemeasured directly by agent j, where agent j is a followerof agent i. A follower is responsible for maintaining thedesired distances from its leaders, while the leader doesnot perform any action to maintain the distance. Then,the formation control procedure is to design a decentralizedformation control law for each follower agent such that

limt→∞

z ij = dij 3

3. Results

In this section, firstly, we design the formation controllaws for an AMSP formation with four agents (one globalleader agent ν1, one first follower agent ν2, one second fol-lower agent ν3, and one ordinary follower agent ν4). Then,the proposed formation control laws are extended to anAMSP formation with n n ≥ 4 agents.

3.1. Controller Design for Formation with Four Agents. Theglobal leader does not follow any other agents and deter-mines where the entire formation goes. The control inputfor the global leader is shown as

u1 = v, 4

where v is the designed velocity of the entire formation. Weconsider the situation that the velocity of the leader is notknown to all the followers. And an adaptive method isapplied to estimate the velocity of the leader.

The first follower only follows the global leader andmaintains the desired distance towards the global leader.The control law with an estimator for the first follower isshown as follows:

u2 = v̂ 2 + e12 z 12,v̂2 = z 12e12, 5

where e12 = z 122 − d212 and v̂2 is the estimation for v by the

first follower.

The second follower follows the global leader and the firstfollower. The control law with an estimator for the secondfollower is shown as follows:

u3 = v̂ 3 + z 13e13 + z 23e13,v̂ = z 13e13 + z 23e23, 6

where e13 = z 132 − d213, e23 = z 23

2 − d223, and v̂3 is theestimation for v by the second follower. In addition, the con-vergences of the first follower and second follower to thedesired formation have been proven in [20].

Assumption 1. In this paper, the desired formation is real-izable. All the corresponding desired distances satisfy thetriangular inequality constraints. For example, d12 < d13 +d23, d23 < d12 + d13 and d13 < d12 + d23. Further, the firstfollower and the second follower have converged to thedesired formation by the controllers and estimators. Thatis, z 12 → d12, z 13 → d13, z 23 → d23, p2 → v, andp3 → v are known.

Therefore, in this paper, we focus on designing the con-troller for the ordinary follower, which follows three agentsand measures the relative positions of the three neighbors.It should be noted that the formation is not globally rigid,as the agents are connected based on an AMSP graph.Obviously, there exist two different formations for the ordi-nary follower that satisfy the same distance requirements ina 3D space, shown in Figure 2. In the sequel, we call theformation in Figure 2(a) as orientation 1 and the formationin Figure 2(b) as orientation 2. To achieve the global con-vergence of the system, a new formation control law withan adaptive estimator for the ordinary follower is proposedas follows:

u4 = e14 ⋅ z 12 × z 13 + e24 ⋅ z 12 + e34 ⋅ z 13 + v̂ 4, 7

v̂4 = e14 ⋅ z 12 × z 13 + e24 ⋅ z 12 + e34 ⋅ z 13, 8

e14 = z 12 × z 13 ⋅ z 14 − E14, 9

e24 = z 24 ⋅ z 12 − E24, 10

e34 = z 34 ⋅ z 13 − E34, 11

1 3

2

4

d14

d13

d23d12

d24

d34

(a)

4

1

2

3

d14 d24d34

d23d12

d13

(b)

Figure 2: Two different formations for the ordinary follower thatsatisfy the same distance requirements: (a) orientation 1; (b)orientation 2.

3Complexity

where v̂4 is the estimation for v by the ordinary follower.The cross product z 12 × z 13 is a vector perpendicularto the plane determined by z 12 and z 13, which providesa way for the ordinary follower to escape from the coplanarposition. E14 is the inner product of z 12 × z 13 and z 14when the expected formation is achieved, E24 is the innerproduct of z 24 and z 12 when the expected formation isachieved, and E34 is the inner product of z 34 and z 13when the expected formation is achieved. E24 and E34 forthe two different formations are the same as follows:

E24 =d224 − d212 − d214

2,

E34 =d234 − d213 − d214

2

12

E14 is designed according to the desired formationshape and is used in the formation controller to solve theproblem that there are two different formation shapes satis-fying the same distance requirements. When the expectedformation is as shown in Figure 2(a), E14 is designed by(13). When the expected formation is as shown inFigure 2(b), E14 is designed by (14).

E14 = 6 ⋅ V , 13

E14 = −6 ⋅ V , 14

where V is the expected volume of the tetrahedron con-structed by the agents v1, v2, v3, and ν4, which can be cal-culated by the Carley-Menger determinant:

V =1288

0 1 1 1 1

1 0 d212 d213 d214

1 d212 0 d223 d224

1 d213 d223 0 d234

1 d214 d224 d234 0

15

Lemma 3.1. The distance error and velocity estimation errorof the ordinary follower are bounded (i.e., e14, e24, e34, andv1 − v̂4 are bounded).

Proof 1. Define the following Lyapunov function:

Vp = e214 + e224 + e234 + v − v̂42, 16

which is continuously differentiable and satisfies that Vp ≥ 0with equality if and only if e14 = 0, e24 = 0, e34 = 0, and v1− v̂4 = 0. Based on Assumption 1, it is known that z 12and z 13 are not collinear. Then, z 12 × z 13 is perpendicularto the planar defined by z 12 and z 13. Thus, the three non-coplanar vectors of z 12 × z 13, z 12, and z 13 can form abase of R3 space. Then, v and v̂4 can be reexpressed as follows:

v = α ⋅ z 12 × z 13 + β ⋅ z 12 + χ ⋅ z 13, 17

v̂ 4 = α ⋅ z 12 × z 13 + β ⋅ z 12 + χ ⋅ z 13, 18

where α, β, and χ are the corresponding components of

v, while α, β, and χ are the corresponding components ofv̂4. Then,

Vp = e214 + e224 + e234 + α − α ⋅ z 12 × z 13

+ β − β ⋅ z 12 + χ − χ ⋅ z 132

≤ e214 + e224 + e234 + α − α ⋅ z 12 × z 132

+ β − β ⋅ z 122+ χ − χ ⋅ z 13

2

= V1 + V2 + V3 ≜ VΘ,

19

where

V1 = e214 + α − α ⋅ z 12 × z 132,

V2 = e224 + β − β ⋅ z 122,

V3 = e234 + χ − χ ⋅ z 132

20

The time derivative of V1 is

V1 = 2e14e14 − 2e14 α − α ⋅ z 12 × z 132

= 2e14 α ⋅ z 12 × z 132 − e14

⋅ z 12 × z 132 − z 12 × z 13 ⋅ v̂

− 2e14 α − α ⋅ z 12 × z 132

= 2e14 α ⋅ z 12 × z 132 − e14

⋅ z 12 × z 132 − z 12 × z 13

⋅ α ⋅ z 12 × z 13 + β ⋅ z 12 + χ ⋅ z 13

− 2e14 α − α ⋅ z 12 × z 132

= 2e14 α − e14 − α ⋅ z 12 × z 132

− 2e14 α − α ⋅ z 12 × z 132

= −2e214 ⋅ z 12 × z 132 ≤ 0

21

The time derivative of V2 is

V2 = 2e24e24 − 2e24 β − β ⋅ z 122

= 2e24 z 12 ⋅ β ⋅ z 12 − e24 ⋅ z 12 − β ⋅ z 12

− 2e24 β − β ⋅ z 122

= 2e24 β − e24 − β ⋅ z 122

− 2e24 β − β ⋅ z 122

= −2e224 ⋅ z 122 ≤ 0

22

4 Complexity

The time derivative of V3 is

V3 = 2e34e34 − 2e34 χ − χ ⋅ z 132

= 2e34 z 13 ⋅ χ ⋅ z 13 − e34 ⋅ z 13 − χ ⋅ z 13

− 2e34 χ − χ ⋅ z 132

= 2e34 χ − e34 − χ ⋅ z 132

− 2e34 χ − χ ⋅ z 132

= −2e234 ⋅ z 132 ≤ 0

23

Then, the time derivative of VΘ is

VΘ =V1 +V2 + V3 ≤ 0, 24

which is negative semidefinite. From (19), VΘ is continu-ously differentiable and satisfies that VΘ ≥ 0. Therefore, VΘt ≤VΘ 0 is bounded. From (19), it holds that Vp t ≤VΘ t ≤VΘ 0 . In addition, Vp is continuously differentiableand satisfies that Vp ≥ 0 from (16). Therefore, Vp is bounded,and hence e14, e24, e34, and v − v4 are bounded.

Lemma 3.2. The distance errors converge to zero (i.e., e14 → 0,e24 → 0, and e34 → 0 as t→∞).

Proof 2. From the fact that VΘ t is continuously differen-tiable and bounded in Lemma 3.1, t

0VΘdt must convergeto a constant. Then, applying Barbalat’s lemma gives thecondition lim

t→∞VΘ = 0 (i.e., lim

t→∞V1 +V2 + V3 = 0). Based on

Assumption 1, z 12 × z 13, z 12, and z 13 are not zero. Thus,we have lim

t→∞e14 = lim

t→∞e24 = lim

t→∞e34 = 0.

Lemma 3.3. The velocity estimation error of the ordinaryfollower converges to zero (i.e., v̂4 → v as t→∞).

Proof 3. Define a function Q1 as follows:

Q1 = z 12 × z 13 ⋅ z 14 25

Then, we have the time derivative of Q1 which is

Q1 = α ⋅ z 12 × z 132 − e14

⋅ z 12 × z 132 − z 12 × z 13 ⋅ v̂

= α ⋅ z 12 × z 132 − e14 ⋅ z 12 × z 13

2

− z 12 × z 13 ⋅ α ⋅ z 12 × z 13

+ β ⋅ z 12 + χ · z 13

= α − α − e14 ⋅ z 12 × z 132

26

From (9), Q1 = z 12 × z 13 · z 14 → E14 is obtainedbecause e14 → 0 is already verified in Lemma 3.2. Then, Q1is continuously differentiable and bounded. Thus, from Bar-

balat’s lemma, Q1 → 0 is obtained (i.e., α − α − e14 ·z 12 × z 13

2 → 0). Based on Assumption 1, z 12 × z 13 isnot zero. Further, e14 → 0 is already proved. To satisfy Q1→ 0, α − α should converge to zero (i.e., α→ α). Define afunction Q2 as follows:

Q2 = z 24 ⋅ z 12 27

Then, we have the time derivative of Q2 which is

Q2 = z 12 ⋅ β ⋅ z 12 − e24 ⋅ z 12 − β ⋅ z 12

= β − β − e24 ⋅ z 122

28

From (10), Q2 = z 24 · z 12 → E24 is obtained becausee24 → 0 is already verified in Lemma 3.2. Then, Q2 is contin-uously differentiable and bounded. Thus, from Barbalat’s

lemma, Q2 → 0 is obtained (i.e., β − β − e24 z 122 → 0).

Based on Assumption 1, z 12 is not zero. Further, e24 → 0 is

already proved. To satisfy Q2 → 0, β − β should converge to

zero (i.e., β→ β). Define a function Q3 as follows:

Q3 = z 34 · z 13 29

Then, we have the time derivative of Q3 which is

Q3 = z 13 ⋅ χ · z 13 − e34 ⋅ z13 − χ ⋅ z 13

= χ − χ − e34 ⋅ z 132 30

From (11), Q3 = z 34 · z 13 → E34 is obtained becausee24 → 0 is already verified in Lemma 3.2. Then, Q2 is contin-uously differentiable and bounded. Thus, from Barbalat’s

lemma, Q3 → 0 is obtained (i.e., χ − χ − e34 · z 132 → 0).

Based on Assumption 1, z 13 is not zero. Further, e34 → 0 isalready proved. To satisfy Q3 → 0, χ − χ should convergeto zero (i.e., χ→ χ).

In conclusion, from (17) and (18), it is straightforward to

obtain that v̂4 → v, because α→ α,β→ β, and χ→ χ.

Theorem 3.4. The ordinary follower converges to the desiredstates by using the controller in (7) and estimator in (8)(i e , e14 → 0, e24 → 0, e34 → 0, v̂4 → v, and p4 → v ast→∞). That is, the system converges to the desired formation.

Proof 4. From Lemma 3.1, it was proven that the distanceerrors e14, e24, and e34 and the velocity estimation error v1− v̂4 are bounded. Further, the convergence of distanceerrors e14, e24, and e34 to zero is obtained. In addition to theconvergence of velocity estimator v̂4 to v, from (7), the veloc-ity of the ordinary follower p4 converges to v. As a result, all

5Complexity

of the agents converge to the desired formation as follows:p2 → v, p3 → v, p4 → v, z 12 → d12, z 13 → d13, z 23→ d23, z 14 → d14, z 24 → d24, and z 34 → d34.

3.2. Extension to n-Agent Case. Then, we extend the pro-posed formation control law to an AMSP formation withn (n ≥ 4) agents. Based on the AMSP graph, each ordinaryfollower i (4 ≤ i ≤ n) has exactly three neighbors, which aredenoted by j, k, and w (1 ≤ j, k,w ≤ i − 1), respectively.Without loss of generality, dji, dki, and dwi denote theexpected distance between the corresponding agents,respectively. From the previous analysis, the convergenceof four agents to the desired formation was achieved.Therefore, based on the control law designed for agent ν4((7) and (8)), the control law for agent νi can be inferredinductively as follows:

ui = eji ⋅ z jk × z jw + eki ⋅ z jk + ewi ⋅ z jw + v̂i,

v̂i = eji ⋅ z jk × z jw + eki ⋅ z jk + ewi ⋅ z jw,

eji = z jk × z jw ⋅ z ji − Eji,

eki = z ki ⋅ z jk − Eki,

ewi = z wi ⋅ z jw − Ewi,

31

where v̂i is the estimation for v by agent νi. Eji is the inner

product of z jk × z jw and z ji when the expected forma-

tion is achieved, Eki is the inner product of z ki and z jk whenthe expected formation is achieved, and Ewi is the inner prod-uct of z wi and z jw when the expected formation is achieved.Eki and Ewi for the two different formations are the same andcalculated as follows:

Eki =d2ki − d2jk − d2ji

s,

Ewi =d2wi − d2jw − d2ji

2

32

Eji is designed according to the desired formation shapeand is used in the formation controller to solve the problemthat there are two different formation shapes satisfying thesame distance requirements. When the expected formationis as shown in Figure 2(a), Eji is designed by (33). Whenthe expected formation is as shown in Figure 2(b), Eji isdesigned by (34).

Eji = 6 ⋅ Vi, 33

Eji = −6 ⋅ Vi, 34

where Vi is the expected volume of the tetrahedron con-structed by the agents νj, νk, νw, and νi, which can becalculated by the Carley-Menger determinant:

Vi =1288

0 1 1 1 1

1 0 d2jk d2jw d2ji

1 d2jk 0 d2kw d2ki

1 d2jw d2kw 0 d2wi

1 d2ji d2ki d2wi 0

35

Assume that the n − 1 agents constructed by an AMSPgraph converge to the desired formation. Add a new agentto the graph of n − 1 agents.

Then, the graph of n agents is still an AMSP graph,as the added agent follows three agents. The neighborsof the added agent belong to the graph of n − 1 agents andconverge to the desired formation as assumed. Then, thelemmas and theorems in Section 3 can be applied to theadded agent by replacing the name of the agent. As a result,the n agents converge to the desired formation if the n − 1agents converge. Since both the basis and the inductive stepshave been performed, by mathematical induction, the agentsglobally converge to the desired formation under the controllaw with an adaptive estimator.

4. Simulations

In this section, two simulations are presented to support ourtheoretical analysis. Firstly, we will show that two differentformation shapes that satisfy the same distance require-ments are achieved, respectively. Then, the simulation ver-ifies that the ordinary follower can leave the initial coplanarlocation even when the velocity of the leader agent is in thesame initial plane.

4.1. Two Different Formation Shapes. The initial positions ofthe five agents are given in a plane: pi 0 2 cos 2π/5 i , 2 sin2π/5 i , 0 T, i = 1, 2, 3, 4, 5. The velocity of the leader agent

is given as v = −1 5, −1, 0 5 T. The desired distances betweenagents are assigned as follows: d12 = d13 = d23 = 2 23, d14 =d24 = d34 = 2 25, and d15 = d25 = d35 = 2 2. Then, theparameters are calculated as follows: E24 = E34 = −6, V4 = 43, E25 = E35 = −6, and V5 = 2 3 . It is clear that the

underlying graph of the desired formation is an AMSPgraph, and there exist two different formations that satisfythe same distance requirements, which can be achievedbased on the proposed control law as follows.

Case 1.Agent ν4 and agent ν5 are set to the same Orientation1. Then, E14 = 6 ·V4 = 24 3 and E15 = 6 · V5 = 12 3 . Thetrajectories of agents in a 3D space are illustrated inFigure 3. It shows that agent ν4 and agent ν5 are on the sameside of the plane determined by the agents ν1, ν2, and ν3.

Case 2. Agent ν4 is set to Orientation 2, while agent ν5 is setto Orientation 1. Then, E14 = −6 · V4 = −24 3 and E15 = 6 ·V5 = 12 3 . The trajectories of agents in a 3D space areshown in Figure 4. It shows that agent ν4 and agent ν5 are

6 Complexity

on the different sides of the plane determined by the agentsν1, ν2, and ν3.

It is shown from Case 1 and Case 2 that the two differentformation shapes satisfying the same distance requirementscan be achieved by designing different E14 and E15. Further,the five agents approach the desired formation and maintainthe formation shape while moving. The distance errors of theformation converge to zero quickly as shown with time inFigures 5 and 6.

4.2. Escape from the Initial Coplanar Position. As shown inSection 4.1, the agents can leave the initial coplanar locationfor the reason that the velocity of the leader agent is not in theplane. Moreover, based on the proposed formation control-ler, the ordinary follower can leave the initial coplanar loca-tion even when the velocity of the leader agent is also in theplane. Thus, in this section, the velocity of the leader agent

is set as v = −1 5, −1, 0 T. The other simulation conditionsare set the same as those of Case 2 in Section 4.1. Then, thetrajectories of agents in a 3D space are illustrated inFigure 7. It shows that agent ν4 and agent ν5 leave the initialcoplanar location and approach the desired formation. Thedistance errors of the formation converge to zero quickly asshown with time in Figure 8.

5. Conclusion

In this paper, we investigate a decentralized 3D formationcontrol law for a multiagent system. The proposed approachcan achieve the different formation shapes that satisfy thesame distance requirements, which extends the existing

6

4

2

0

z-ax

is

x-axis

y-axis

−2

−4

−6−20

−15−10

−10 −8 −6 −4 −2 0 2−5

05

Agent 1Agent 2Agent 3

Agent 4Agent 5

Figure 3: Trajectories of agents in a 3D space: agent ν4 and agent ν5are in the same Orientation 1.

x-axis

6

45

21

3

0

z-ax

is

−1−2−3

y-axis

−20−15

−10

−10 −8 −6 −4 −2 0 2−5

05

Agent 1Agent 2Agent 3

Agent 4Agent 5

Figure 4: Trajectories of agents in a 3D space: agent ν4 is inOrientation 2, while agent ν5 is in Orientation 1.

‖z14‖2 − d2

14‖z24‖

2 − d224

‖z34‖2 − d2

34

‖z15‖2 − d2

15‖z25‖

2 − d225

‖z35‖2 − d2

35

0−15

−10

−5

0

5

Dist

ance

erro

r

10

15

20

25

30

1 2 3 4 5Time (s)

6 7 8 9 10

Figure 5: Distance errors: agent ν4 and agent ν5 are in the sameOrientation 1.

−15

−10

−5

0

5

Dist

ance

erro

r

10

15

20

25

30

‖z14‖2 − d2

14‖z24‖

2 − d224

‖z34‖2 − d2

34

‖z15‖2 − d2

15‖z25‖

2 − d225

‖z35‖2 − d2

35

0 1 2 3 4 5Time (s)

6 7 8 9 10

Figure 6: Distance errors: agent ν4 is in Orientation 2, while agentν5 is in Orientation 1.

7Complexity

distance-based 3D formation control laws. Although theunderlying AMSP graph of the formation is not globallyrigid, the multiagent system is still globally asymptoticallystable. Moreover, a stable 3D formation motion can be real-ized, even when the initial positions of the agents are copla-nar and the velocity of the leader agent is also in the plane.The performed numerical simulation results show the effec-tiveness of the formation control strategy.

In the future, we will further design more advanced for-mation control algorithms with robustness in mind. Withthe development of the adaptive neural network [30, 31],learning control [32], adaptive observer, and parameterestimation [33–35], the dynamics of the agent can beextended to the scenarios involving unknown nonlineardynamics and external disturbances to further validate thisformation control scheme.

Data Availability

The data used to support the findings of this study are avail-able from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (Grant no. 61473124) and ScientificResearch Foundation of Guangzhou University (Grant no.2700050356).

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