Multi-objective region reaching control for a swarm of robots

doi:10.1016/j.automatica.2019.01.017

Automatica

Volume 103, May 2019, Pages 81-87

https://doi.org/10.1016/j.automatica.2019.01.017 Get rights and content

Abstract

This paper is concerned with the multi-objective region reaching control for a swarm of robots which are formulated by Lagrangian dynamics. Two distributed multi-objective region reaching control protocols are proposed for the networked robotic systems under directed acyclic topology, and a unifying methodology is presented to perform the convergence analysis for the robotic systems with static and moving target regions. The control strategy is developed by using the potential energy function approach, and the specified shapes of the various desired regions are constructed by selecting appropriate objective functions. In this control strategy, a network of a large number of robots evolves into multiple groups, and the robots in each group only require communicating with their neighbors. Thus, the proposed control strategy is effective for multi-objective region reaching control for a swarm of robots in practical applications. Finally, simulation examples are given to show the validity of the theoretical results.,

Introduction

A significant progress has been made over the past decade in the field of the cooperative control of networked multi-robot systems which are formulated by Lagrangian dynamics (Belta and Kumar, 2004, Brambilla et al., 2013, Cheah et al., 2009, Consolini et al., 2008, Gazi, 2005). The research was originated from two elementary reasons. Firstly, the networked multi-robot systems have been widely regarded as one of the representative models in investigating the coordination problems for multi-agent systems. Secondly, the networked multi-robot systems can accomplish complex tasks more effectively than a combination of individually operating robots. In general, an essential task for decentralized cooperative control is to design an appropriate protocol for the agents to accomplish global tasks cooperatively with local information only (Cao et al., 2013, Chung and Slotine, 2009). Many control strategies, including leader-following method (Meng et al., 2013, Tanner et al., 2004), virtual structure approach (Lewis and Tan, 1997, Ren and Beard, 2004) and potential field technique (Balch and Arkin, 1998, Mastellone et al., 2011, Su et al., 2010, Wang, 2015), have recently been proposed for the networked multi-robot systems. Simultaneously, numerous control protocols (or algorithms) have been developed for different types of coordination problems from various perspectives, such as consensus, formations, rendezvous, swarming and flocking (see refs. Lewis and Tan, 1997, Liu et al., 2015, Mei et al., 2012, Meng et al., 2013, Su et al., 2010 and the relevant references therein).

As an important issue in behavior-based control for a large group of robots, region reaching control has recently received significant interests due to its advantages over the general set-point control in practical applications (Cheah & Wang, 2005). In fact, a large number of experiments and observations from human implementing tasks have demonstrated that the desirable targets are actually the regions of specific shapes instead of the points. In most robotic applications, the desired position for the end effector is commonly regulated as a region rather than a point in task space (Sun & Cheah, 2007), where there exists certain flexibility in realizing the control tasks. Multi-objective region control scheme can well characterize collective behavior as the result of self-organization of swarm intelligence for natural and artificial systems, for example aggregation, flocking, clustering and foraging. In particular, all the agents of the swarm evolve into multiple groups by using simple rules and local communication among agents, and then eventually achieve global deployment of multiple target regions (Brambilla et al., 2013). Furthermore, from the swarm engineering perspective, multi-objective region reaching control is especially suitable for characterizing the practical structure and nature of large-scale modern complex and integrated production process in real-world applications such as automotive and manufacturing applications, where a swarm of robots usually accomplishes a cooperative task of multiple subtasks specified in different target regions. Under the cooperative control scheme, a large group of robots was divided into multiple groups with the robots in different groups completing the corresponding designated subtasks (Brambilla et al., 2013, Gazi, 2005). Obviously, the designed global cooperative task is in essence a multi-objective region reaching control scheme.

The concept of region reaching control was firstly introduced in (Cheah & Wang, 2005) for single-robot, and was later extended for robotic manipulator and underwater vehicle-manipulator systems (UVMS) (Cheah et al., 2007, Sun and Cheah, 2007). A region-based shape controller for a swarm of robots by using the concept of region reaching control was presented in Cheah et al. (2009). Under the designed control scheme, all robots move as a group with a common velocity inside a desired region while keeping a minimum distance among them. However, the designed algorithm requires all followers receiving the information of the desired region. Therefore, this algorithm may not be suitable for controlling a large group of robots as the design of control scheme becomes more complicated with an increase of group size. In addition, the proposed algorithm does not consider the communication topologies among robots, and therefore the reliability, robustness and effectiveness of control performance cannot be fully guaranteed. In practical applications, simple yet effective distributed algorithms are highly desirable for multi-objective region reaching control for networked multi-robot systems having a large number of robots.

Being inspired in part by the above-mentioned works, the main objective of this technical brief is to advance the concept of region reaching control to multi-objective region reaching control for networked multi-robot systems. A simple yet efficient framework will be proposed to solve the multi-objective region reaching control problem of networked multi-robot systems, and two distributed algorithms will be developed for the cases of static and moving target regions. The desired multi-regions can be specified as various shapes and formations appropriate to the practical applications. In comparison with the existing works, the main contributions of this technical brief lie in three aspects. (i) The proposed algorithm for multi-objective region reaching control is fully distributed compared with the work (Cheah et al., 2009), where all followers needed to receive the information of the desired regions. Thus the designed algorithm is particularly applicable to control a large number of robots consistently with the intelligence paradigm of swarm robots. In addition, the developed control strategy can be regarded as an important generalization of pinning-like set point tracking scheme based on the typical PID control strategy (Chen, Liu, & Lu, 2007), and thus it would be easily implemented in practice. (ii) A key feature of the proposed control strategy is that the directed acyclic topology is introduced to formulate the communication interaction among agents. This is different from the existing studies on the region reaching control of robotic systems (Cheah and Wang, 2005, Sun and Cheah, 2007), where the network structure was not considered. In fact, it is now widely accepted that the communication interaction among agents can play an important role in the multi-group distributed cooperative control for networked multi-robot systems. (iii) A novel framework is proposed for developing a unified methodology for multi-objective region reaching control, which distinguishes from the existing research (Liu et al., 2015, Qin and Yu, 2013), where the analysis framework for group consensus problem is not applicable to the multi-objective region reaching control of networked robotic systems. Consequently, the developed methodology could be extended to perform the convergence analysis of coordination problems for more complex networked robot systems.

The rest of this brief is organized as follows. Section 2 provides the mathematical preliminaries. Section 3 describes the problem formulations. Section 4 gives the main results. Examples and simulation results are presented in Section 5 to validate the correctness of the proposed multi-objective region reaching control algorithms. Finally, conclusion is drawn in Section 6.

Section snippets

Graph theory

Throughout this paper, a weighted directed graph $G = (V, E, A)$ is used to represent the interaction among robots in a network of $n$ robots, where the node set is $V = {1, 2, \dots, n}$ , the edge set is $E \in V \times V$ , and a weighted adjacency matrix $A = [a_{i j}] \in R^{n \times n}$ is given by $a_{i j} = 0$ if $(j, i) \notin E$ , and $a_{i j} \neq 0$ , otherwise. $(j, i) \in E$ indicates that robot $i$ can obtain the information from robot $j$ , but not vice versa. Here, $a_{i i} = 0$ for $i \in V$ implies there is no edge between a node and itself. A directed spanning tree in a directed graph

Networked robotic systems

A network consisting of a swarm of robots is considered in this section. Specifically, the dynamics of the $i$ th robot is determined by Lagrangian dynamics (Cheah et al., 2009): $M_{i} (q_{i}) {\ddot{q}}_{i} + C_{i} (q_{i}, {\dot{q}}_{i}) {\dot{q}}_{i} + D_{i} (q_{i}) {\dot{q}}_{i} + g_{i} (q_{i}) = τ_{i},$ where $i = 1, 2, \dots, n$ , $q_{i} \in R^{p}$ is the vector of the generalized coordinates, $M_{i} (q_{i}) \in R^{p \times p}$ is a positive definite inertia matrix, $C_{i} (q_{i}, {\dot{q}}_{i}) \in R^{p \times p}$ is the Coriolis and centrifugal matrix, $D_{i} (q_{i}) {\dot{q}}_{i} \in R^{p}$ represents the damping force in which $D_{i} (q_{i}) \in R^{p \times p}$ is a positive definite matrix, $g_{i} (q_{i}) \in R^{p}$

Static region reaching control protocols

With the above preparation, the distributed static region reaching control protocols for the networked robotic systems will be first formulated.

A reference velocity ${\dot{q}}_{r i} \in R^{p}$ is first introduced as: ${\dot{q}}_{r i} = - \sum_{j \in N_{i}} a_{i j} (q_{i} - q_{j}) - α_{i} q_{i} - α_{i} \int_{0}^{t} Δ ξ_{\hat{i}} d τ, i \in V_{\hat{i}},$ where $α_{i} \geq 0$ can be viewed as the feedback control gain, and $N_{i}$ is a set of neighbors around agent $i$ . $α_{i} > 0$ means that agent $i$ of group $V_{\hat{i}}$ can obtain the information of the desired objective region $Ω_{\hat{i}}$ , and $α_{i} = 0$ otherwise. An $α_{i} > 0$ is assumed to exist for

Numerical simulations

This section presents some simulation results to show the effectiveness of the proposed control algorithms. The equation of motion of each robot is modeled as (Slotine & Li, 1991), $[\begin{matrix} M_{11} & M_{12} \\ M_{21} & M_{22} \end{matrix}] [\begin{matrix} {\ddot{q}}_{i x} \\ {\ddot{q}}_{i y} \end{matrix}] + [\begin{matrix} N_{11} & N_{12} \\ N_{21} & N_{22} \end{matrix}] [\begin{matrix} {\dot{q}}_{i x} \\ {\dot{q}}_{i y} \end{matrix}] = [\begin{matrix} τ_{i x} \\ τ_{i y} \end{matrix}],$ where $M_{11} = a_{1} + 2 a_{3} c o s (q_{i y}) + 2 a_{4} s i n (q_{i y})$ , $M_{12} = M_{21} = a_{2} + a_{3} c o s (q_{i y}) + a_{4} s i n (q_{i y})$ , $M_{22} = a_{2}$ , $N_{11} = - b {\dot{q}}_{i y}$ , $N_{12} = - b ({\dot{q}}_{i x} + {\dot{q}}_{i y})$ , $N_{21} = b {\dot{q}}_{i x}$ , $N_{22} = 0$ , $b = a_{3} s i n (q_{i y}) - a_{4} c o s (q_{i y})$ , $a_{1} = 7, a_{2} = a_{3} = a_{4} = 2$ . We set $α_{1} = 1, α_{3} = 1, α_{6} = 1, α_{i} = 0, i = 2, 4, 5, 7, 8, 9$ , and the directed acyclic network is shown in Fig. 1.

Conclusion

In this technical brief, we have studied the multi-objective region reaching control problem of the networked robotic systems under directed topology graph with acyclic partition. Two distributed multi-objective region reaching control protocols of the networked robotic systems have been proposed for the cases of static and moving target regions. Furthermore, some simple yet generic criteria for solving multi-objective region reaching control problems have been derived by considering the

Acknowledgments

We would like to thank the editor and the reviewers for their valuable suggestions to improve our work.

Zhonghua Miao received his Ph.D. degree in mechatronic engineering from Shanghai Jiao-Tong University, China, in 2010. He is currently a Full Professor and Doctoral Advisor in the School of Mechatronic Engineering and Automation at Shanghai University. His research interests include intelligent robotics, measurement and control, and fault diagnosis, and agricultural machinery equipment.

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Cited by (0)

Jinwei Yu received his Ph.D. degree in dynamics and control from Shanghai Institute of Applied Mathematics and Mechanics at Shanghai University, China, in 2018. He is currently a Lecturer at the School of Mathematics, Taiyuan University of Technology. His research interests include dynamics and control of complex nonlinear systems, neural networks and cooperative control of multi-robot systems.

Jinchen Ji received his Ph.D. in mechanical engineering and is currently an Associate Professor of Mechanical Engineering at University of Technology Sydney, Australia. His research interests include the dynamics and control of complex nonlinear systems and multi-agent systems, nonlinear dynamics and control of mechanical systems, vehicle system dynamics and wind turbine dynamics.

Jin Zhou received his Ph.D. degree in operations research and cybernetics from Shanghai University, China, in 2003. He is currently a Full Professor and Doctoral Advisor of Shanghai Institute of Applied Mathematics and Mechanics at Shanghai University. His research interests include complex systems and complex networks, cooperative control in networked multi-agent systems, and differential equations and system controls.

^☆: This work is supported by the National Science Foundation of China (Nos. 11672169 and 51875331). The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Michael M. Zavlanos under the direction of Editor Christos G. Cassandras.

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Brief paperMulti-objective region reaching control for a swarm of robots☆

Abstract

Introduction

Section snippets

Graph theory

Networked robotic systems

Static region reaching control protocols

Numerical simulations

Conclusion

Acknowledgments

Automatica

Automatica

Automatica

Automatica

Automatica

Systems & Control Letters

Automatica

Behavior-based formation control for multirobot teams

IEEE Transaction Robotics and Automation

Abstraction and control for groups of robots

IEEE Transactions on Robotics

Swarm robotics: a review from the swarm engineering perspective

Swarm Intelligence

An overview of recent progress in the study of distributed multi-agent coordination

IEEE Transactions on Industrial Informatics

Region reaching control of robots: theory and experiments

Brief paper
Multi-objective region reaching control for a swarm of robots☆