Observer-Based Event-Triggered Circle Formation Control for First- and Second-Order Multiagent Systems

Xu, Peng; Xie, Guangming; Tao, Jin; Xu, Minyi; Zhou, Quan

doi:https://doi.org/10.1155/2020/4715315

Complexity

On this page

Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2020 | Article ID 4715315 | https://doi.org/10.1155/2020/4715315

Observer-Based Event-Triggered Circle Formation Control for First- and Second-Order Multiagent Systems

Peng Xu,¹Guangming Xie,^1,2,3,4Jin Tao ,^2,5Minyi Xu ,^1,4and Quan Zhou⁵

Academic Editor: Átila Bueno

Received25 Sept 2019

Accepted09 Jan 2020

Published24 Mar 2020

Abstract

This paper proposes an observer-based event-triggered algorithm to solve circle formation control problems for both first- and second-order multiagent systems, where the communication topology is modeled by a spanning tree-based directed graph with limited resources. In particular, the observation-based event-triggering mechanism is used to reduce the update frequency of the controller, and the triggering time depends on the norm of the state function and the trigger threshold of measurement errors. The analysis shows that sufficient conditions are established for achieving the desired circle formation, while there exists at least one agent for which the next interevent interval is strictly positive. Numerical simulations of both first- and second-order multiagent systems are also given to demonstrate the effectiveness of the proposed control laws.

1. Introduction

In recent years, many research efforts have been devoted to controlling of multiagent systems (MASs) due to both its practical potentials in a variety of applications [1–3] and theoretical challenges of physical constraints [4–6]. As a significant problem in cooperative control for MASs, formation control, aiming to guide multiple agents to form and maintain predetermined geometries, has attracted considerable interests for its extensive applications in different areas [7–10]. The main focus has been devoted to the design of a distributed formation control framework, especially concerning the robustness against both external disturbances and internal uncertainties [11, 12], as well as the increased number of agents. Moreover, for MASs subjected to aperiodic sampling and communication delays, the problem of cluster formation control was addressed in [10]. Therefore, most existing results on formation control mainly rely on the ideal hypothesis [13–15], e.g., each agent is modeled as having unlimited communication capabilities, unlimited power, and unlimited processing capabilities, which allows arbitrary information to exchange pattern. However, as far as we know, few studies dealt with the limited capacity of communication and the power constraints of agents.

In order to save energy and bandwidth, event-triggered control methods have been presented in [16–20]. One of the most distinct characters of event-triggered control is that control actions only update when specific events occur, which lead to ease the trade-offs among actuator effort, communication, and computation. Moreover, according to the triggered methods, event-triggered control can be mainly divided into state-dependent triggering and time-dependent triggering. A simple state event-triggered schedule based on the feedback control was studied in [16]. The results lead to a guaranteed performance with a fixed sampling rate requirements concerning the optimizing schedules and sampling rates. In [17], under conditions of decreasing thresholds of the measurement errors exponentially, a time-dependent triggered method was designed to guarantee all agents asymptotic converge to a ball centered at the average consensus. Different from most of the existing fixed threshold parameters, the threshold parameter in the improved event-triggered condition is dynamically adjusted by a dynamic rule in [18]. In [21], by proposing a pull-based event-triggered control strategy, a circle formation control problem for first-order MASs with directed topologies was studied. Further, Wen et al. [22] combined event-triggered protocols to solve circle formation problems of first-order MASs. Also, Wen et al. and Xu et al. [23, 24] have investigated a combination algorithm based on quantized communication technology, where the problem of MASs with a limitation of communication was addressed. Given the above reviews, it is noteworthy to mention that most of the existing results on event-triggered control are to prevent the case of Zeno behavior [25, 26], so that within a finite time interval, an infinite number of samplings generate. Typically, the event trigger interval having a strictly positive lower bound is a sufficient condition to exclude Zeno behavior [27].

Different from previous studies, especially [23], which paid attention to the effect of qualitative communication for event-triggered control, the main objective of this paper is to provide an observer-based event-triggered method to solve circle formation problems for both first- and second-order MASs through a set of directed graphs. In our studies, each agent observes the distance from the counterclockwise direction to its nearest neighbor and the counterpart from the clockwise direction through communication, which is similar to Pioneer 3-DX [28]. In comparison to the literature, we have three main contributions: (i) different from [22] concerning the first-order model, combining with a distributed asynchronous event-triggered control algorithm, a more concise form of the event-triggered condition is designed to solve circle formation problems for both first- and second-order dynamics MASs; (ii) different from taking a complex-coordinate system transformation method in [29], the proposed strategy allows for a reduction of the number of control actions without significantly degrading performance using the simple-coordinate system transformation; and (iii) the resulting asynchronous model achieves the desired equilibrium points asymptotically while at least one agent with a positive next event interval exists, i.e., no trajectory generates in a finite time interval.

The remainder of this paper is organized as follows. Preliminary definitions and problem formulation are given in Section 2. In Section 3, a distributed circle formation control law for first-order MASs and the rigorous analysis of its performance are presented. Section 4 uses the event-triggered rule to address a distributed circle formation problem for second-order MASs. Section 5 discusses the simulation results, and Section 6 concludes the paper.

2. Preliminaries and Problem Formulation

2.1. Preliminaries

Let and denote a set of real numbers and a real matrix, respectively. For a finite set , let denote the number of its elements. For a vector or a matrix A, let , , and stand for its Euclidean norm, -norm, and transpose, respectively. Let and be the N dimension column vectors with all entries being 1 and 0, respectively. Let a matrix represent a diagonal matrix whose diagonal entries are .

Let be a directed graph, in which is a set of nodes, stands for a set of edges, and denotes a weighted adjacency matrix. In the directed graph , for all , and for edge , it starts from node j and ends up with node i. It is known that agent i can perceive state information from agent j. Therefore, agent j is called agent i’s in-neighbor. In addition, is applied to describe the in-neighbor set of agent i. Particularly, edge links with the element of a weighted adjacency matrix , if and only if ; otherwise, . We use to denote the in-degree of i-th agent and define as matrix of , where . Subsequently, we can list the eigenvalues of in a descending order as , where is the spectral radius of .

We use the two lemmas listed below to facilitate analysis in this paper.

Lemma 1. For any given and , the following two properties exist:

Lemma 2 (see [30]). Given a directed graph , which is composed of a spanning tree, a vector satisfies and , where ξ denotes the left eigenvector corresponding to zero eigenvalues of the matrix . Furthermore, is semipositive definite, where . After taking square root of each elements of , we get ; consequently, .

Lemma 3 (see [29]). The linear matrix inequalitiesare equivalent to either one of the conditions listed as below:

2.2. Problem Formulation

Given an MAS consisting of N () mobile agents, each agent is initially on a specific circle, and no pair of agents occupies the same position at the same time, as shown in Figure 1. For simplicity, we mark the agents counterclockwise and measure the position of the agent at an angle of . To be specific, the initial positions of all agents are set to satisfy the condition shown as

Here, each agent has only two neighbors, that is, in front of or behind it. Let represent the two neighbors of the mobile agent i, where

The dynamics of agent i are described aswhere and stand for the scalar state and the control input of agent i, respectively.

Based on counterclockwise at time t, are presented as the angular distance from agent i to agent . Along with (5) and (6), it yields towhere , and always holds.

Then, define a vector to determine a desired circle formation, where stands for the desired angular distance between agent i and agent . If d satisfies and , the desired circle formation is for MASs.

3. Circle Formation Control for First-Order MASs

In this section, we first give the definition of the circle formation problem for first-order MASs as follows.

Definition 1 (circle formation problem for first-order MASs). Given an admissible circle formation characterized by d, a distributed control law can be designed, so that under any initial condition (4), the solution to system (7) converges to the equilibrium point . That is, .
A way-point control protocol based on sampled date was designed in [8]:It has been proved that the continuous update control law (9) can move all agents to their equilibrium point but can waste communication bandwidth and unnecessary transmission energy. In order to solve this issue, an observer-based event-triggered circle formation control method for first-order MASs is proposed. It is noteworthy to mention that control actions of each agent only update at the event-triggered sampling instants, where continuous communication between neighboring agents is maintained. Here, we use an increasing sequence to represent the event instants of agent i, such that the state of agent i at the kth event instant is described as . Note that each agent has its own event sequence since all agents are triggered asynchronously.
According to event-triggered strategies, we design the distributed circle formation control law for agent i aswhere represents the last event instant of agent and denotes the observer angular distance.
From (10), agent i’s controller only updates at its own event sequence . For simplicity, let ; consequently, the control law (10) can be represented aswhere , .
Replacing (2) and (5) into (7), the closed-loop form of agent i is rearranged by using asDefine ; then, a compact form of system dynamics iswhere , , andAccording to the control law (5) and MAS (1), the circle formation control for first-order MASs is solvable by Theorem 1.

Theorem 1. For any admissible circle formation characterized by d, considering system (7) and the designed control law (5) on the digraph , the circle formation problem can be solved when the event-triggered condition is designed aswhere is the i^th elements of , is the same diagonal matrix as in Lemma 2, and is the ith diagonal element of matrix . Moreover, in system (7), there exists at least one agent that the next interevent interval is strictly positive under the event-triggered condition (15).

Proof. A function candidate is taken into considerationwhere is the same diagonal matrix as Lemma 2, such that is semipositive definite.
Accordingly, and if and only if the circle formation problem can be solved. The derivative of the function (16) along trajectories of the system is derived asAfter enforcing the event condition (15), we get . Thus, (17) is rewritten asBecause , we can get and if and only if the circle formation problem can be solved.
Next, we explain the achievement of the circle formation in detail. According to the Lemma 2, we getCombining (10) and (15), all conditions result in , where remains constant. In addition, , always satisfies and . Together with , we obtain . More exactly, , which indicates that the designed circle formation can be solved in first-order MASs.
For agent i, the period , increasing from 0 to , is regarded as the event interval between and . Define ; therefore, agent m stands for the maximum norm of among all the agents, which impliesFrom (20), the time attaining is longer than . That is, , where represents positive interval lower bound and τ is the time increasing from 0 to . Thereby, the time derivative of isUsing β to replace , it yields to . Here, , where is the solution of , . According towe know that there exists the interval t which satisfies , such that the event interval between instants and is lower bounded. By solving the difference equation (22), it yields toCalculated from (23), we obtain , where τ is the time from 0 to , which derives to agent m’ lower bound of interval between two event instants:From , we conclude that in system (7), there exists at least one agent , the next interevent interval of which is strictly positive under event-triggered condition (15).

4. Circle Formation Control for Second-Order MASs

Given a second-order MAS with N agentswhere is the angular state of agent i, is its angular velocity state, and denotes its control input.

Similar to transformation of (8), we obtainwhere and always holds.

The definition of the circle formation problem for second-order is described as follows.

Definition 2. (circle formation problem for second-order MASs). Considering an admissible circle formation, which is characterized by d, we can design a distributed control law , such that the solution to system (25) converges to the equilibrium points under any initial condition (4). Namely, and are satisfied.
Let and stand for the observer angular velocity distance of agent i and , respectively. From [8], the control law for agent i can be designed aswhere , and the positive control gain φ is determined in sequel.
For simplicity, we haveSubstituting (8), (26), and (27) into (25), the closed-loop system is written aswhere , , and is the same matrix as in (14).
Define ; we then have the coordinate transformation asConsequently, system (29) is rearranged asGiving and , system (31) can be rewritten asBefore designing any event-triggered conditions, the function candidate is given aswhere r, , and ν are positive constants.

Theorem 2. The function (33) satisfies system (25) when the conditionholds simultaneously, where is considered as the maximum element of vector ξ in Lemma 2 and o denotes a positive constant.

Proof. When the function equals to zero, the circle formation problem can be solved. In this case, the circle formation cannot be achieved. According to (33), it yields toFrom and Lemma 2, we can observe that the desired circle formation is achieved by , and semipositive definite matrix .
We define a positive constant o asSince can be taken as positive (semipositive) definite with a single zero eigenvalue, there exists a unitary matrix such that , where S denotes diagonal matrix with . In addition, is the eigenvalue of associated with eigenvector . Note that assuming , the corresponding eigenvector . By defining , we getFrom (37), we observe that if and only if , and , which indicates that or . Without loss of generality, can be used for further analysis. In that case, we have , which violates the assumptions . Thus, when the circle formation is not achieved. From o, we conclude that . Therefore, we haveAlso, according to Lemma 3, is semipositive definite when andFrom (39), we draw a conclusion that . The function (33) is semipositive definite and if the circle formation problem is solvable, it equals to 0. Therefore, this candidate function (33) satisfies the second-order MAS (25).

Theorem 3. For any admissible circle formation characterized by d, taking into account system (25) and the designed control law (27) over the digraph , the circle formation problem can be solved when the event-triggered condition is designed aswhere , k, , , and ν stand for positive constants and and are the i-th element of vector and , respectively.

Let and the conditionshold simultaneously; there exists at least one agent for which the next interevent interval is strictly positive under the event-triggered condition (40) in system (25).

Proof. By combining (33) and (36), we haveTaking time derivative of function (33) along all trajectories of system (29), we getThen, (43) can be classified into two parts. The former part of (43) is computed byBy and condition (41) and (44) impliesAccording to Lemma 1, the latter part of (43) is rearranged intoCombining (45) and (46), (43) can be written asIn view of the event-triggered function (40) satisfying , we conclude thatFrom (48), we summarize that the function (33) is negative unless the circle formation for the second-order MAS is achievable. Moreover, we have , that is and . Thus, we get , , . Note that and . Then, , , always satisfies and . We conclude that , . To be more precise, .
The result shows that the circle formation can be achieved by all mobile agents. Furthermore, we illustrate the interevent times are positive lower bounded.
Using event-triggered condition (40), we obtain that the interval is the time that increases from 0 to .
Here, we use to represent the interval. Then, is longer or equal to the time increasing from 0 to , where .
Let denote the time from 0 to. From the analysis, the -th event of agent i occurs after the time . Similar to the definition of m, we define , which follows
Referring to the first-order method, the time derivative of iswhere hasand isCombining (50) and (51), (49) is rearranged intowhere .
Additionally, definefor which, . Here, satisfies the bound , where is the solution of , . Using to denote the time that grows from 0 to , we get .
Consequently, is longer than that satisfieswhere .
Thus, . From , we conclude that in system (25), there exists at least one agent that the next interevent interval is strictly positive under event-triggered condition (40). The proof is complete.

5. Simulation Examples

Considering an MAS (7) consisting of six agents, the desired distances between two adjacent agents are set to , and the initial values of the MAS are randomly generated to satisfy (8). In each case, we apply the same initial settings, i.e., same angular positions and same admissible circle formations. Additionally, the unique normalized positive left eigenvector of with respect to eigenvalue 0 is . In order to show advantages of the event-triggered control strategy, it is worth pointing out that the event detection of all those simulations is implemented in a sampled-data fashion, which is proved to be effective, and there is at least one trigger interval with a positive bound. Therefore, the sampling periods h in real-time control is chosen as .

5.1. Simulation of the First-Order MAS

By the permitted range , we set to ensure condition (15) holds in real-time control. Simulation results are shown in Figures 2 and 3. Figure 2(a) shows the evolution difference between the event-triggered angular distance and the expected counterpart, while Figure 2(b) reveals the event sequence of each agent. Figure 3 illustrates the fluttering of the measurement error .

(a)

(b)

We can see from the simulation results that the desired circle formation can be asymptotically achieved by the proposed control law (10) in distributed MASs. We also calculate the average interevent time of all mobile agents from Figure 2, and the result indicates that our method can reduce the amount of control update for first-order MASs.

5.2. Simulation of the Second-Order MAS

We choose coefficients , , , , , , and to ensure condition (40) holds in real-time control. Simulation results are shown in Figures 4 and 5. Figure 4(a) shows the evolution difference between the event-triggered angular distance and the expected counterpart, while Figure 4(b) reveals event sequence of each agent. Figure 5 shows the fluttering of the measurement error , .

(a)

(b)

(a)

(b)

We observe from the simulation results that the desired circle formation can also be asymptotically achieved under the proposed control law (27). Similarly, we calculate the average interevent time of all mobile agents from Figure 4. We conclude that our proposed method has the advantages of reducing the amount of control update for the second-order MASs.

To sum up, compared with traditional time-planning controls in the same simulation environment, the event-triggered control-based formation control strategies proposed in this paper show the effectiveness for both first- and second-order MASs. For the second-order MAS, the average interevent time of all mobile agents is less than the first-order MAS case due to its more complicated modeling. However, from an application point of view, the proposed control strategy for second-order MASs is much more accurate than first-order MASs.

6. Conclusion

This paper investigated the circle formation control problem for both first- and second-order MASs under unbalanced directed networks with limited resource constraints. We first designed the observer-based event-triggered algorithm to reduce dependence on resources, in which, when the value of the event-triggered condition exceeds zero, the agent’s controller will update its states simultaneously. Moreover, it is a fundamental and practical aspect to observe neighbor information on a regular or better basis continuously. Then, we proved that if there is a spanning tree in the underlying graph, the MASs can achieve the desired circle formation by the proposed control laws, and there is at least one agent whose the next interevent interval is strictly positive. At last, we gave numerical simulation examples to illustrate that the proposed event-triggered circle formation control strategies are effective for both first- and second-order MASs. For future work, we will extend our research to more practical operations, e.g., considering the effect of time delays in communication networks, input saturation constraints, and weak links.

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 there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This study was supported in part by the Academy of Finland (grant no. 315660), the National Natural Science Foundation of China (grant nos. 51575005, 61503008, 51879022, 91648120, and 61633002), the Fundamental Research Funds for the Central Universities (grant nos. 3132019037 and 3132019197), and the Beijing Natural Science Foundation (no. 4192026).

References

A. Macwan, J. Vilela, G. Nejat, and B. Benhabib, “Multi-robot deployment for wilderness search and rescue,” International Journal of Robotics and Automation, vol. 31, no. 1, 2016.
View at: Publisher Site | Google Scholar
K. Kim, H. Kim, and H. Myung, “Bio-inspired robot swarm control algorithm for dynamic environment monitoring,” Advances in Robotics Research, vol. 2, no. 1, pp. 1–11, 2018.
View at: Publisher Site | Google Scholar
A.-Q. Hoang and M.-T. Pham, “Swarm intelligence-based approach for macroscopic scale odor source localization using multi-robot system,” in Advances in Information and Communication Technology, pp. 593–602, Springer, Berlin, Germany, 2016.
View at: Publisher Site | Google Scholar
T. Yang, Z. Meng, D. V. Dimarogonas, and K. H. Johansson, “Global consensus for discrete-time multi-agent systems with input saturation constraints,” Automatica, vol. 50, no. 2, pp. 499–506, 2014.
View at: Publisher Site | Google Scholar
G. Nava-Antonio, G. Fernández-Anaya, E. Hernández-Martínez, J.-J. Flores-Godoy, and E. D. Ferreira-Vázquez, “Consensus of multiagent systems described by various noninteger derivatives,” Complexity, vol. 2019, Article ID 3297410, 14 pages, 2019.
View at: Publisher Site | Google Scholar
B. Yang, X. Wang, J.-a. Fang, and Y. Xu, “The impact of coupling function on finite-time synchronization dynamics of multi-weighted complex networks with switching topology,” Complexity, vol. 2019, Article ID 7276152, 15 pages, 2019.
View at: Publisher Site | Google Scholar
M. Villani, L. Sani, R. Pecori et al., “An iterative information-theoretic approach to the detection of structures in complex systems,” Complexity, vol. 2018, Article ID 3687839, 15 pages, 2018.
View at: Publisher Site | Google Scholar
C. Wang, G. Xie, and M. Cao, “Forming circle formations of anonymous mobile agents with order preservation,” IEEE Transactions on Automatic Control, vol. 58, no. 12, pp. 3248–3254, 2013.
View at: Publisher Site | Google Scholar
K.-K. Oh, M.-C. Park, and H.-S. Ahn, “A survey of multi-agent formation control,” Automatica, vol. 53, pp. 424–440, 2015.
View at: Publisher Site | Google Scholar
X. Ge, Q. Han, and X. Zhang, “Achieving cluster formation of multi-agent systems under aperiodic sampling and communication delays,” IEEE Transactions on Industrial Electronics, vol. 65, no. 4, pp. 3417–3426, 2017.
View at: Publisher Site | Google Scholar
Z. Bu, Z. Wu, J. Cao, and Y. Jiang, “Local community mining on distributed and dynamic networks from a multiagent perspective,” IEEE Transactions on Cybernetics, vol. 46, no. 4, pp. 986–999, 2017.
View at: Publisher Site | Google Scholar
D. Tian, J. Zhou, and Z. Sheng, “An adaptive fusion strategy for distributed information estimation over cooperative multi-agent networks,” IEEE Transactions on Information Theory, vol. 63, no. 5, pp. 3076–3091, 2017.
View at: Publisher Site | Google Scholar
G. Wen, H.-T. Zhang, W. Yu, Z. Zuo, and Y. Zhao, “Coordination tracking of multi-agent dynamical systems with general linear node dynamics,” International Journal of Robust and Nonlinear Control, vol. 27, no. 9, pp. 1526–1546, 2017.
View at: Publisher Site | Google Scholar
R. Olfati-Saber and R. M. Murray, “Consensus problems in networks of agents with switching topology and time-delays,” IEEE Transactions on Automatic Control, vol. 49, no. 9, pp. 1520–1533, 2004.
View at: Publisher Site | Google Scholar
W. Xie, B. Ma, T. Fernando, and H. C. Iu, “A new formation control of multiple underactuated surface vessels,” International Journal of Control, vol. 91, no. 10, pp. 1–12, 2017.
View at: Publisher Site | Google Scholar
P. Tabuada, “Event-triggered real-time scheduling of stabilizing control tasks,” IEEE Transactions on Automatic Control, vol. 52, no. 9, pp. 1680–1685, 2007.
View at: Publisher Site | Google Scholar
G. S. Seyboth, D. V. Dimarogonas, and K. H. Johansson, “Event-based broadcasting for multi-agent average consensus,” Automatica, vol. 49, no. 1, pp. 245–252, 2013.
View at: Publisher Site | Google Scholar
X. Ge and Q.-L. Han, “Distributed formation control of networked multi-agent systems using a dynamic event-triggered communication mechanism,” IEEE Transactions on Industrial Electronics, vol. 64, no. 10, pp. 8118–8127, 2017.
View at: Publisher Site | Google Scholar
Y. Liu and X. Hou, “Event-triggered consensus control for leader-following multiagent systems using output feedback,” Complexity, vol. 2018, Article ID 6342683, 9 pages, 2018.
View at: Publisher Site | Google Scholar
D. V. Dimarogonas, E. Frazzoli, and K. H. Johansson, “Distributed event-triggered control for multi-agent systems,” IEEE Transactions on Automatic Control, vol. 57, no. 5, pp. 1291–1297, 2011.
View at: Google Scholar
P. Xu, H. Zhao, G. Xie, J. Tao, and M. Xu, “Pull-based distributed event-triggered circle formation control for multi-agent systems with directed topologies,” Applied Sciences, vol. 9, no. 23, p. 4995, 2019.
View at: Publisher Site | Google Scholar
J. Wen, C. Wang, and G. Xie, “Asynchronous distributed event-triggered circle formation of multi-agent systems,” Neurocomputing, vol. 295, pp. 118–126, 2018.
View at: Publisher Site | Google Scholar
J. Wen, P. Xu, C. Wang, G. Xie, and Y. Gao, “Distributed event-triggered circle formation control for multi-agent systems with limited communication bandwidth,” Neurocomputing, vol. 358, pp. 211–221, 2019.
View at: Publisher Site | Google Scholar
P. Xu, J. Wen, C. Wang, and G. Xie, “Distributed circle formation control over directed networks with communication constraints,” IFAC-PapersOnLine, vol. 52, no. 3, pp. 108–113, 2019.
View at: Publisher Site | Google Scholar
K. H. Johansson, M. Egerstedt, J. Lygeros, and S. Sastry, “On the regularization of zeno hybrid automata,” Systems & Control Letters, vol. 38, no. 3, pp. 141–150, 1999.
View at: Publisher Site | Google Scholar
M. Heymann, F. Lin, G. Meyer, and S. Resmerita, “Control synthesis for a class of hybrid systems subject to configuration-based safety constraints,” Hybrid and Real-Time Systems, vol. 50, no. 3, pp. 376–390, 2005.
View at: Publisher Site | Google Scholar
C. Nowzari, Distributed Event-Triggered Coordination for Average Consensus on Weight-Balanced Digraphs, Pergamon Press, Oxford, UK, 2016.
S. Zaman, W. Slany, and G. Steinbauer, “Ros-based mapping, localization and autonomous navigation using a pioneer 3-dx robot and their relevant issues,” in Proceedings of the Saudi International Electronics, Communications and Photonics Conference (SIECPC), pp. 1–5, Riyadh, Saudi Arabia, April 2011.
View at: Publisher Site | Google Scholar
M. Yu, H. Wang, G. Xie, and K. Jin, “Event-triggered circle formation control for second-order-agent system,” Neurocomputing, vol. 275, no. 31, pp. 462–469, 2017.
View at: Publisher Site | Google Scholar
W. Ren and R. W. Beard, “Consensus seeking in multiagent systems under dynamically changing interaction topologies,” IEEE Transactions on Automatic Control, vol. 50, no. 5, pp. 655–661, 2005.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2020 Peng Xu 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.

PDF Download Citation

Download other formats

Order printed copies