Complexity
Volume 2018 (2018), Article ID 1814653, 9 pages
https://doi.org/10.1155/2018/1814653

Research Article

Adaptive 3D Distance-Based Formation Control of Multiagent Systems with Unknown Leader Velocity and Coplanar Initial Positions

Xuejing Lan,1,2 Wenbiao Xu,³ and Yun-Shan Wei¹

¹School of Mechanical and Electrical Engineering, Guangzhou University, Guangzhou 510006, China
²Center for Intelligent Equipment and Internet-Connected Systems, Guangzhou, China
³Guangdong 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.

Abstract.

This paper considers the distributed 3-dimensional (3D) distance-based formation control of multiagent systems, where the agents are connected based on an acyclic minimally structural persistent (AMSP) graph. A parameter is designed according to the desired formation 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 position measurements with respect to their local coordinate systems. In addition, the proposed formation controller provides a new way for the agent to leave the initial coplanar location. The 3D formation control law is globally asymptotically stable and has been demonstrated 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 applications such as surveillance, exploration, and search and rescue missions. In particular, formation control problem is one of the important aspects and has received significant attention recently with the development of the information communication technique [1–4].

According to the different requirements on the sensing capability and the interaction topology, the existing formation control schemes are categorized into position-based, displacement-based, and distance-based formation control. In the position-based formation control, the desired positions are given with respect to a global coordinate system, and the global position sensing is required [5]. The desired displacements are given and controlled in the displacement-based formation, where the relative positions of the neighboring agents are sensed with respect to a global coordinate system [6]. The distance-based formation is prescribed by the desired interagent distances, where the interagent distances are controlled and the relative positions of the neighboring agents are sensed with respect to their local coordinate systems [7–10]. The orientations of the local coordinate systems need not be aligned with each other. This means that a global sensing is not required 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-based formation control is how to maintain the formation shape, while the agents are tracking a reference trajectory or moving with a reference velocity [11]. A number of research works have considered the movement of formation in the displacement-based control. However, only few results in the distance-based control are available because of its extreme complexity in analysis [12, 13]. Further, in some real-world applications, such as the formation of unmanned flight vehicles and submarines [14, 15], the agents are actually moving in a 3-dimensional (3D) space, which will make the formation control scheme more complicated. Therefore, the study of 3D distance-based formation control is gaining increasing attention [16, 17]. But they have not yet considered the movement of the formation. Then, Zhang et al. presented a 3D formation that can move with a given velocity while achieving the desired formation shape by directly adding a term of formation maneuvering velocity in the formation control law [18]. It is shown from the paper that the agents should know the reference velocity so as to remove the formation control error. In view of this, Kang et al. designed a distance-based formation control law in the leader-follower type with a moving leader [19, 20]. The follower agents in [19, 20] can estimate the velocity of the moving leader adaptively by only measuring the relative positions from their neighboring agents, which promotes the development of distance-based formation control. It is true that all the agents should not be collocated at a common point initially when using the steepest descent control law in the distance-based formation. The initial positions of the agents are usually set to noncollinear and close to the target formation shape [21, 22]. To solve this initial collinear problem, Park and Ahn modified the gradient control law by introducing a rotation matrix into the controller, which can change the descent direction and help the agents escape from the collinear position [23]. In [24], a formation control law was proposed based on two mutually perpendicular vectors, which provided a way for the agents to leave the initially collinear location. Another way to solve this problem is to set the initial velocity of the agent with a different orientation from the line [25]. Although the formation control problem with collinear initial positions of the agents has been solved [23–25], it is still challenging when the initial position of the agents is coplanar in the 3D distance-based formation.

In this paper, we aim to propose an adaptive 3D distance-based formation control law for a multiagent system with four kinds of agents, where the underlying graph of the formation is an acyclic minimally structural persistent (AMSP) graph. Compared with the global leader, first follower, and second follower, it is more difficult to design the controller for the ordinary follower, which follows three agents and has the problem of trapping in a plane. The proposed formation controller of the ordinary follower constructs a vector perpendicular to the plane determined by its three leaders, which provides a new way for the ordinary follower to leave the coplanar location. Moreover, a parameter is designed according to the desired formation shape and is used in the formation controller to solve the problem that there are two formation shapes satisfying the same distance requirements. The unknown moving velocity of the leader is adaptively estimated by the followers requiring only the relative position measurements with respect to their local coordinate systems. The 3D formation control law is globally asymptotically stable and has been demonstrated based on the Lyapunov theorem. The outline of this paper is listed as follows. Background and preliminaries are introduced in Section 2. The procedure of distributed formation control scheme design with the velocity estimator is presented in Section 3. Numerical simulations are completed in Section 4, and we reach a conclusion in Section 5.

2. Background and Preliminaries

2.1. Graph and Formation Structure

The formation problem of a multiagent system is modeled by a directed graph , which consists of a vertex set and a directed edge set . The vertices represent the agents, and the weighted edges represent the interagent distance constraints. The neighboring set of agent is defined as . A directed edge from to means that agent can measure the relative position between agent and . Then, we call agent a “follower” of agent and correspondingly call agent a “leader” of agent . The formation shape can be maintained during any continuous motion, if the underlying graph is rigid and the distance constraints of each agent are satisfied. A formation graph is minimally persistent if it is rigid and constraint consistent with the minimum possible number of edges [26–28].

For a 3D minimally persistent formation, the sum of the degrees 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 DOFs among non-0-DOF agents. Moreover, the concept of structural persistence should be taken into consideration for 3D application [29]. For example, the graph in Figure 1(a) is an acyclic minimally structural persistent (AMSP) graph, while the graph in Figure 1(b) is minimally persistent but not structurally persistent with two free leaders.

(a)

(b)

Figure 1: Examples of graph in a 3D space: (a) an acyclic minimally structural persistent graph; (b) minimally persistent but not structurally persistent graph.

In this paper, the AMSP graph is applied to construct the 3D leader-follower formation structure, which is the most convenient structure to design distributed control schemes. In the AMSP graph, there are one 3-DOF agent called the global leader, one 2-DOF agent called the first follower, one 1-DOF agent called the second follower, and some 0-DOF agents called ordinary followers.

2.2. Problem Statement

In the distance-based formation, the desired formation is prescribed by the desired interagent distances. The desired distance between agent and agent is denoted by and apparently . It is assumed that each agent measures the relative positions of its neighboring agents via an onboard sensor with respect to its local coordinate system . The orientations of the local coordinate systems are not aligned with each other. All the agents move in a 3-dimensional space. Although the control law of each agent is implemented in in practice, it is more convenient to represent the agents with respect to a global coordinate system for stability analysis. In addition, the state in can be transformed into by a suitable coordinate transformation. In this paper, the state of each agent will be represented with respect to . The position and the velocity of agent at time in are denoted by and , respectively. The dynamics of agent is modeled as a single integrator:

where , , is the control input of agent . is used to represent the relative position vector as follows:

In this paper, only the relative position can be measured directly by agent , where agent is a follower of agent . A follower is responsible for maintaining the desired distances from its leaders, while the leader does not perform any action to maintain the distance. Then, the formation control procedure is to design a decentralized formation control law for each follower agent such that

3. Results

In this section, firstly, we design the formation control laws for an AMSP formation with four agents (one global leader agent , one first follower agent , one second follower agent , and one ordinary follower agent ). Then, the proposed formation control laws are extended to an AMSP formation with agents.

3.1. Controller Design for Formation with Four Agents

The global leader does not follow any other agents and determines where the entire formation goes. The control input for the global leader is shown as

where is the designed velocity of the entire formation. We consider the situation that the velocity of the leader is not known to all the followers. And an adaptive method is applied to estimate the velocity of the leader.

The first follower only follows the global leader and maintains the desired distance towards the global leader. The control law with an estimator for the first follower is shown as follows:

where and is the estimation for by the first follower.

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

where , , and is the estimation for by the second follower. In addition, the convergences of the first follower and second follower to the desired formation have been proven in [20].

Assumption 1. In this paper, the desired formation is realizable. All the corresponding desired distances satisfy the triangular inequality constraints. For example, and . Further, the first follower and the second follower have converged to the desired formation by the controllers and estimators. That is, , , , , and are known.

Therefore, in this paper, we focus on designing the controller for the ordinary follower, which follows three agents and 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 ordinary follower that satisfy the same distance requirements in a 3D space, shown in Figure 2. In the sequel, we call the formation in Figure 2(a) as orientation 1 and the formation in Figure 2(b) as orientation 2. To achieve the global convergence of the system, a new formation control law with an adaptive estimator for the ordinary follower is proposed as follows:

where is the estimation for by the ordinary follower. The cross product is a vector perpendicular to the plane determined by and , which provides a way for the ordinary follower to escape from the coplanar position. is the inner product of and when the expected formation is achieved, is the inner product of and when the expected formation is achieved, and is the inner product of and when the expected formation is achieved. and for the two different formations are the same as follows:

(a)

(b)

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

is designed according to the desired formation shape and is used in the formation controller to solve the problem that there are two different formation shapes satisfying the same distance requirements. When the expected formation is as shown in Figure 2(a), is designed by (13). When the expected formation is as shown in Figure 2(b), is designed by (14).

where V is the expected volume of the tetrahedron constructed by the agents , , , and , which can be calculated by the Carley-Menger determinant:

Lemma 3.1. The distance error and velocity estimation error of the ordinary follower are bounded (i.e., , , , and are bounded).

Proof 1. Define the following Lyapunov function:

which is continuously differentiable and satisfies that with equality if and only if , , , and . Based on Assumption 1, it is known that and are not collinear. Then, is perpendicular to the planar defined by and . Thus, the three noncoplanar vectors of , , and can form a base of space. Then, and can be reexpressed as follows:

where , , and are the corresponding components of , while , , and are the corresponding components of . Then,

where

The time derivative of is

The time derivative of is

The time derivative of is

Then, the time derivative of is

which is negative semidefinite. From (19), is continuously differentiable and satisfies that . Therefore, is bounded. From (19), it holds that . In addition, is continuously differentiable and satisfies that from (16). Therefore, is bounded, and hence , , , and are bounded.

Lemma 3.2. The distance errors converge to zero (i.e., , , and as ).

Proof 2. From the fact that is continuously differentiable and bounded in Lemma 3.1, must converge to a constant. Then, applying Barbalat’s lemma gives the condition (i.e., ). Based on Assumption 1, , , and are not zero. Thus, we have .

Lemma 3.3. The velocity estimation error of the ordinary follower converges to zero (i.e., as ).

Proof 3. Define a function as follows:

Then, we have the time derivative of which is

From (9), is obtained because is already verified in Lemma 3.2. Then, is continuously differentiable and bounded. Thus, from Barbalat’s lemma, is obtained (i.e., ). Based on Assumption 1, is not zero. Further, is already proved. To satisfy , should converge to zero (i.e., ). Define a function as follows:

Then, we have the time derivative of which is

From (10), is obtained because is already verified in Lemma 3.2. Then, is continuously differentiable and bounded. Thus, from Barbalat’s lemma, is obtained (i.e., ). Based on Assumption 1, is not zero. Further, is already proved. To satisfy , should converge to zero (i.e., ). Define a function as follows:

Then, we have the time derivative of which is

From (11), is obtained because is already verified in Lemma 3.2. Then, is continuously differentiable and bounded. Thus, from Barbalat’s lemma, is obtained (i.e., ). Based on Assumption 1, is not zero. Further, is already proved. To satisfy , should converge to zero (i.e., ).
In conclusion, from (17) and (18), it is straightforward to obtain that , because ,, and .

Theorem 3.4. The ordinary follower converges to the desired states by using the controller in (7) and estimator in (8) (, , , , and as t → ∞). That is, the system converges to the desired formation.

Proof 4. From Lemma 3.1, it was proven that the distance errors , , and and the velocity estimation error are bounded. Further, the convergence of distance errors , , and to zero is obtained. In addition to the convergence of velocity estimator to , from (7), the velocity of the ordinary follower converges to . As a result, all of the agents converge to the desired formation as follows: , , , , , , , , and .

3.2. Extension to -Agent Case

Then, we extend the proposed formation control law to an AMSP formation with () agents. Based on the AMSP graph, each ordinary follower () has exactly three neighbors, which are denoted by , , and (), respectively. Without loss of generality, , , and denote the expected distance between the corresponding agents, respectively. From the previous analysis, the convergence of four agents to the desired formation was achieved. Therefore, based on the control law designed for agent ((7) and (8)), the control law for agent can be inferred inductively as follows:

where is the estimation for by agent . is the inner product of and when the expected formation is achieved, is the inner product of and when the expected formation is achieved, and is the inner product of and when the expected formation is achieved. and for the two different formations are the same and calculated as follows:

is designed according to the desired formation shape and is used in the formation controller to solve the problem that there are two different formation shapes satisfying the same distance requirements. When the expected formation is as shown in Figure 2(a), is designed by (33). When the expected formation is as shown in Figure 2(b), is designed by (34).

where is the expected volume of the tetrahedron constructed by the agents , , , and , which can be calculated by the Carley-Menger determinant:

Assume that the agents constructed by an AMSP graph converge to the desired formation. Add a new agent to the graph of agents.

Then, the graph of agents is still an AMSP graph, as the added agent follows three agents. The neighbors of the added agent belong to the graph of agents and converge to the desired formation as assumed. Then, the lemmas and theorems in Section 3 can be applied to the added agent by replacing the name of the agent. As a result, the agents converge to the desired formation if the agents converge. Since both the basis and the inductive steps have been performed, by mathematical induction, the agents globally converge to the desired formation under the control law with an adaptive estimator.

4. Simulations

In this section, two simulations are presented to support our theoretical analysis. Firstly, we will show that two different formation shapes that satisfy the same distance requirements are achieved, respectively. Then, the simulation verifies that the ordinary follower can leave the initial coplanar location even when the velocity of the leader agent is in the same initial plane.

4.1. Two Different Formation Shapes

The initial positions of the five agents are given in a plane: , . The velocity of the leader agent is given as . The desired distances between agents are assigned as follows: , , and . Then, the parameters are calculated as follows: , , , and . It is clear that the underlying graph of the desired formation is an AMSP graph, and there exist two different formations that satisfy the same distance requirements, which can be achieved based on the proposed control law as follows.

Case 1. Agent and agent are set to the same Orientation 1. Then, and . The trajectories of agents in a 3D space are illustrated in Figure 3. It shows that agent and agent are on the same side of the plane determined by the agents , , and .

Case 2. Agent is set to Orientation 2, while agent is set to Orientation 1. Then, and . The trajectories of agents in a 3D space are shown in Figure 4. It shows that agent and agent are on the different sides of the plane determined by the agents , , and .

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

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

It is shown from Case 1 and Case 2 that the two different formation shapes satisfying the same distance requirements can be achieved by designing different and . Further, the five agents approach the desired formation and maintain the formation shape while moving. The distance errors of the formation converge to zero quickly as shown with time in Figures 5 and 6.

Figure 5: Distance errors: agent and agent are in the same Orientation 1.

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

4.2. Escape from the Initial Coplanar Position

As shown in Section 4.1, the agents can leave the initial coplanar location for the reason that the velocity of the leader agent is not in the plane. Moreover, based on the proposed formation controller, the ordinary follower can leave the initial coplanar location even when the velocity of the leader agent is also in the plane. Thus, in this section, the velocity of the leader agent is set as . The other simulation conditions are set the same as those of Case 2 in Section 4.1. Then, the trajectories of agents in a 3D space are illustrated in Figure 7. It shows that agent and agent leave the initial coplanar location and approach the desired formation. The distance errors of the formation converge to zero quickly as shown with time in Figure 8.

Figure 7: Trajectories of agents in a 3D space: the velocity of the leader agent is in the initial coplanar plane.

Figure 8: Distance errors: the velocity of the leader agent is in the initial coplanar plane.

5. Conclusion

In this paper, we investigate a decentralized 3D formation control law for a multiagent system. The proposed approach can achieve the different formation shapes that satisfy the same distance requirements, which extends the existing distance-based 3D formation control laws. Although the underlying AMSP graph of the formation is not globally rigid, the multiagent system is still globally asymptotically stable. Moreover, a stable 3D formation motion can be realized, even when the initial positions of the agents are coplanar and the velocity of the leader agent is also in the plane. The performed numerical simulation results show the effectiveness of the formation control strategy.

In the future, we will further design more advanced formation control algorithms with robustness in mind. With the development of the adaptive neural network [30, 31], learning control [32], adaptive observer, and parameter estimation [33–35], the dynamics of the agent can be extended to the scenarios involving unknown nonlinear dynamics and external disturbances to further validate this formation control scheme.

Data Availability

The data used to support the findings of this study are available 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 Science Foundation of China (Grant no. 61473124) and Scientific Research Foundation of Guangzhou University (Grant no. 2700050356).

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