Distributed formation building algorithms for groups of wheeled mobile robots

doi:10.1016/j.robot.2015.08.006

Robotics and Autonomous Systems

Volume 75, Part B, January 2016, Pages 463-474

https://doi.org/10.1016/j.robot.2015.08.006 Get rights and content

Highlights

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We present a method for decentralized flocking and global formation building for a network of unicycle-like robots.
•
These robots are described by the standard kinematic equations with hard constraints on the robots linear and angular velocity.
•
We prove the convergence of the proposed algorithm with probability 1.
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The effectiveness of the proposed control algorithm is illustrated via computer simulations and experiments with real robots.

Abstract

The paper presents a method for decentralized flocking and global formation building for a network of unicycle-like robots described by the standard kinematics equations with hard constraints on the robots linear and angular velocities. We propose decentralized motion coordination control algorithms for the robots so that they collectively move in a desired geometric pattern from any initial position. There are no predefined leaders in the group and only local information is required for the control. The effectiveness of the proposed control algorithms is illustrated via computer simulations and experiments with real robots.

Introduction

The study of decentralized control laws for groups of mobile autonomous robots has emerged as a challenging new research area in recent years (see, e.g., [1], [2], [3], [4], [5], [6], [7], [8], [9], [10] and references therein). Broadly speaking, this problem falls within the domain of decentralized control, but the unique aspect of it is that groups of mobile robots are dynamically decoupled, meaning that the motion of one robot does not directly affect that of the others. Researchers in this new emerging area are finding much inspiration from biology, where the problem of animal aggregation is central in both ecological and evolutionary theory. Animal aggregations, such as schools of fish, flocks of birds, groups of bees, or swarms of social bacteria, are believed to use simple, local motion coordination rules at the individual level that result in remarkable and complex intelligent behaviour at the group level; see e.g., [11], [12]. Such intelligent behaviour is expected from very large scale robotic systems. The term ‘very large scale robotic system’ was introduced in [13] for a system consisting of autonomous robots numbering from hundreds to tens of thousands or even more. Because of decreasing costs of robots, interest in very-large-scale robotic systems is growing rapidly. In such systems, robots should exhibit some forms of cooperative behaviour.

In 1995, Vicsek et al. proposed a simple, but interesting discrete-time model of a system consisting of several autonomous agents, e.g., particles, moving in the plane [14]. The motion of each agent is updated using a local rule based on its own state and the state of its neighbours. This model can be viewed as a special case of a computer model proposed in [15] for the computer animation industry and mimicking animal aggregation. Simulation results in [14] show that in Vicsek’s model, all agents might eventually move in the same direction, despite the absence of centralized coordination. In [5], a modification of the Vicsek’s model is introduced and considered. Here, the heading of each robot is updated as the average headings of its neighbours. Compare this with [14], where the heading is given as the heading of the average velocity vector of its neighbours. The modification in [5] results in simpler mathematical analysis and allows linear tools to be applied. The first results on mathematical analysis of this model were given in [5]. The main results of [5] are sufficient conditions for coordination of the system of agents that are given in terms of a family of graphs characterizing all possible neighbour relationships among agents. Some further results have been obtained in other papers; see e.g., [4], [7]. There is a number of other recent results which show that for certain local rules the flocking problem will be solved, i.e. all vehicles will eventually move in the same direction despite the absence of centralized coordination.

A more difficult problem is to design a decentralized control law which guarantees that all vehicles will eventually move not only in the same direction but also in a desired geometric configuration. In this paper, we address this more difficult problem.

It should be pointed out that many papers in this area consider simplest first- or second-order linear models for the motion of each robot; see, e.g., [16], [17], [18]. Therefore, the obtained results are heavily based on tools and methods from linear system theory, such as stochastic matrices or graph Laplacians. It is known, that there are examples of unrealistic physically embodied behaviour that would be possible under such simplified models. In particular, the robot motion in such linear models does not satisfy the standard hard constraint on either robot speed or angular velocity. Furthermore, it can be shown that the models proposed in [7] and many other papers will result in arbitrarily large robot angular velocity and arbitrarily small robot turning radius, which is impossible on actual wheeled robots. This paper considers the much more difficult problem to design a similar decentralized control law for a multirobot system, in which the motion of each robot is described by a nonlinear model with the standard hard constraints on the vehicle angular acceleration and linear velocity. Such models can describe the kinematics of unmanned aerial vehicles (UAVs) or wheeled mobile robots; see, e.g., [19], [20]. In this situation, linear system approaches of [5] and many other papers are not applicable. In [9], an algorithm of flocking for a group of wheeled robots was proposed, however, the much more difficult problem of formation building was not considered. We present a constructive and easily implementable decentralized control law for a group of autonomous wheeled robots such that the vehicles will eventually move in the same direction with the same speed. Furthermore, we give a mathematically rigorous analysis of the proposed decentralized robot navigation scheme. In our problem, there are no leaders assigned a priori, and the robots have to coordinate with each other in the group relying on some global consensus in order to achieve and maintain a desired pattern. Furthermore, we consider the problem of formation building with anonymous robots. In this problem statement, each robot does not know a priori its position in the desired configuration, and the robots should reach a consensus on their positions. We propose a randomized decentralized navigation algorithm and prove its convergence with probability 1.

The effectiveness of the proposed control algorithms is verified via computer simulations and experiments with real robots.

Potential applications of our formation control algorithms for a group of wheeled mobile robots are for sweep coverage [21], [10] in operations like mine sweeping [22], boarder patrolling [23], environmental monitoring of disposal sites on the deep ocean floor [24], and sea floor surveying for hydrocarbon exploration [25].

The reminder of the paper is organized as follows. Section 2 describes the networked multi-robot system under consideration. Section 3 presents our formation building algorithm and its mathematical analysis. Section 4 is devoted to the formation building problem for anonymous robots. Computer simulations of the proposed formation building algorithms are given in Section 5. Section 6 presents the results of experiments with real wheeled robots. Finally, brief conclusions are given in Section 7.

Section snippets

Multi-robot system

The system under consideration consists of $n$ autonomous robots labelled 1 through $n$ . All these robots are moving in a plane with continuous time dynamics. Let $(x_{i} (t), y_{i} (t))$ be the Cartesian coordinates of the vehicle $i$ . Also, let $θ_{i} (t)$ be the orientation of this vehicle with respect to the $x$ -axis, that is $θ_{i} (t)$ is measured from the $x$ -axis in the counterclockwise direction, it takes values in the interval $(- π, π]$ (so the $x$ -axis corresponds to the orientation $θ = 0$ ).

Furthermore, let $v_{i} (t)$ be the

Formation building

We propose the following rules for updating the consensus variables ${\tilde{θ}}_{i} (k), {\tilde{x}}_{i} (k), {\tilde{y}}_{i} (k)$ and ${\tilde{v}}_{i} (k)$ : ${\tilde{θ}}_{i} (k + 1) = \frac{{\tilde{θ}}_{i} (k) + \sum_{j \in N_{i} (k)} {\tilde{θ}}_{j} (k)}{1 + | N_{i} (k) |};$ ${\tilde{x}}_{i} (k + 1) = \frac{x_{i} (k) + {\tilde{x}}_{i} (k) + \sum_{j \in N_{i} (k)} (x_{j} (k) + {\tilde{x}}_{j} (k))}{1 + | N_{i} (k) |} - x_{i} (k + 1);$ ${\tilde{y}}_{i} (k + 1) = \frac{y_{i} (k) + {\tilde{y}}_{i} (k) + \sum_{j \in N_{i} (k)} (y_{j} (k) + {\tilde{y}}_{j} (k))}{1 + | N_{i} (k) |} - y_{i} (k + 1);$ ${\tilde{v}}_{i} (k + 1) = \frac{{\tilde{v}}_{i} (k) + \sum_{j \in N_{i} (k)} {\tilde{v}}_{j} (k)}{1 + | N_{i} (k) |} .$

The algorithm (3.5) can be summarized as follows. The mobile robots use the consensus variables to achieve a consensus on the heading, speed and mass centre of the formation.

Lemma 3.1

Suppose that

Formation building with anonymous robots

In this section, we consider the problem of formation building with anonymous robots. The term “anonymous” means that each robot does not know a priori its position in the configuration $C = {X_{1}, X_{2}, \dots, X_{n}, Y_{1}, Y_{2}, \dots, Y_{n}}$ , and the robots should reach a consensus on their positions in the formation in the process of formation building.

Definition 4.1

A navigation law is said to be globally stabilizing with anonymous robots and the configuration $C = {X_{1}, X_{2}, \dots, X_{n}, Y_{1}, Y_{2}, \dots, Y_{n}}$ if for any initial conditions $(x_{i} (0), y_{i} (0), θ_{i} (0))$ ,

Computer simulations

In this section, some computer simulation results are presented. Fig. 3, Fig. 4, Fig. 5, Fig. 6 show the simulation results for the algorithms proposed in Section 3. Here we consider five robots whose motions are governed by (2.1), (2.2), (2.3) with the decentralized law (3.5), (3.15) employed. The shape of the undirected graph $G (k)$ may change from time to time. It is shown that with any initial conditions and various configurations $C$ , global stability is achieved.

In Fig. 7, Fig. 8, Fig. 9,

Experiments with real robots

In this section, we present our experiments with real robots guided by the algorithm proposed in Section 3. In these experiments, the proposed algorithm is implemented on three Pioneer 3-DX (P3) mobile robots. The necessary information for the computation of the algorithm (positions, orientations and the velocities) can be acquired by the encoders that are attached to the wheels of the robots.

At the beginning of each experiment, the robots are placed in random positions with random

Conclusions

We have considered the problems of decentralized flocking and global formation building for a group of wheeled robots described by the standard kinematics equations with hard constraints on the robots linear and angular velocities. There are no predefined leaders in the group and only local information is available for the control. A distributed motion coordination control algorithms for the robots so that they collectively move with the same speed in a desired geometric pattern from any

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Finally, it is shown that the quantization error of the control torque is bounded. Different from the previous formation control works [5–17,20,21], this analysis is required to deal with the quantized state feedback and communication problems in the distributed formation tracker design. Furthermore, for the proof of Lemma 3, the local adaptive laws (37)–(41) using quantized state feedback information are presented to compensate for unknown wheel slippage and system nonlinearities and their boundedness is proved.
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The formation control problem of multiple autonomous mobile robots has been studied because of its significant applications such as transporting large objects, surveillance, and exploring hazardous areas (see Cebollada et al. (2021), Liu and Jiang (2013), Ou et al. (2014), Ren and Sorensen (2008) and reference therein). Distributed control methodologies based on the graph theory (Han & Zheng, 2021; Zhan et al., 2021) have been developed for multiple nonholonomic mobile robots under several forms of networks where the formation problems have been addressed at the robot kinematic level (Liang et al., 2020; Lin et al., 2021; Lu et al., 2020; Savkin et al., 2016; Sharma et al., 2021; Yamchi & Esfanjani, 2017; Yang et al., 2020) and dynamic level (Park & Yoo, 2021; Peng et al., 2016; Wang et al., 2014; Yoo & Kim, 2015, 2017; Yoo & Park, 2018, 2019, 2020). In these methodologies, each robot can communicate with only their neighbor robots connected by networks and its controller is designed using the local state information.
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Among them, the distributed formation control problem of nonholonomic mobile robots has been widely investigated owing to nonholonomic constraints. Most research results tackle the distributed formation problem under the unlimited communication ranges of the robots where the controllers at the kinematic and dynamic levels were designed in Cao, Jiang, and Yue (2014), Liu and Jiang (2013), Liu and Jiang (2014), Montijano, Cristofalo, Schwager and Sagues (2016), Peng, Wen, Rahmani, and Yu (2015), Savkin, Wang, Baranzadeh, Xi, and Nguyen (2016), Yamchi and Esfanjani (2017), Yu and Liu (2016) and in Peng, Wen, Yang and Rahmani (2016), Peng, Yang, Wen, Rahmani and Yu (2016), Wang, Huang, Wen, and Fan (2014), Yoo and Kim (2015) and Yoo and Kim (2017), respectively. Contrary to them, much interest seems not to be gathered into the distributed formation problem in the presence of limited communication ranges of the robots.
This paper proposes a unified error transformation strategy for the distributed formation tracking of networked uncertain nonholonomic mobile robots with different communication ranges. To guarantee the connectivity preservation and collision avoidance of all robots without employing any potential functions, a novel nonlinear formation error function using the relative distance and angles between robots is introduced. Then, the approximation-based local adaptive tracking design methodology using the dynamic surface design technique is recursively established to achieve (i) the connectivity preservation and collision prevention of initially connected robots, (ii) the collision avoidance of initially non-connected robots, and (iii) the desired formation tracking of robots in a fully distributed manner. The stability of the total closed-loop control system is analyzed in the Lyapunov sense.
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A multi-robot tractor system for conducting agriculture field work was developed in order to reduce total work time and to improve work efficiency. The robot tractors can form a spatial pattern, I-pattern, V-pattern or W-pattern, during the work process. The safety zones of each robot were defined as a circle and a rectangle. The robots can coordinate to turn to the next lands without collision or deadlock. The efficiency of the system depends on the number of robots, the spatial pattern, the setting distance between each robot, and the field length. Three simulations were carried out to determine the usefulness of the system. The simulation results showed that the efficiency range of three robots using the I-pattern is from 83.2% to 89.8% at a field length of 100 m. The efficiency range of seven robots using the W-pattern is from 59.4% to 65.8% at a field length of 100 m. However, the minimum efficiency of seven robots using the W-pattern is 84.9% at a field length of 500 m. The efficiency would be higher than 85% if the field length was larger than 500 m. Thus, the newly developed multi-robot tractor system is more effective in a large field.
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Andrey V. Savkin was born in 1965 in Norilsk, USSR. He received the M.S. and Ph.D. degrees from the Leningrad State University, USSR in 1987 and 1991, respectively. From 1987 to 1992, he was with the Television Research Institute, Leningrad, USSR From 1992 to 1994, he held a postdoctoral position in the Department of Electrical Engineering, Australian Defence Force Academy, Canberra. From 1994 to 1996, he was a Research Fellow with the Department of Electrical and Electronic Engineering and the Cooperative Research Center for Sensor Signal and Information Processing at the University of Melbourne, Australia. In 1996–2000, he was a Senior Lecturer, and then an Associate Professor with the Department of Electrical and Electronic Engineering at the University of Western Australia, Perth. Since 2000, he has been a Professor with the School of Electrical Engineering and Telecommunications at the University of New South Wales, Sydney, Australia. His current research interests include robust control and state estimation, hybrid dynamical systems, guidance, navigation and control of mobile robots, applications of control and signal processing in biomedical engineering and medicine. He has authored/co-authored five research monographs and numerous journal and conference papers on these topics. He has served as an associate editor for several international journals.

Chao Wang was born in 1987 in China. He received his Ph.D. degree in 2014 from University of New South Wales (UNSW). He is currently continuing his research as a postdoctoral Researcher in UNSW. His research interests include control system analysis, nonholonomic system, and navigation of wheeled mobile robot in cluttered dynamic environment.

Ahmad Baranzadeh received the B.Sc. and M.Sc. degrees in Electrical Engineering in 1994 and 1998, respectively, both from The University of Tehran, Iran. He has been a faculty staff with the Department of Electrical and Computer Engineering, The University of Hormozgan, Iran since 1999. Currently, he is a Ph.D. student at the School of Electrical Engineering and Telecommunications, The University of New South Wales (UNSW), Sydney, Australia. His current research interests include control of mobile robots, networked control systems and nonlinear control.

Zhiyu Xi received the B.Eng. degree in Control Science and Engineering from Harbin Institute of Technology, China in 2004. She then received M.Eng. and Ph.D. degrees in Automatic Control from The University of New South Wales, Australia in 2007 and 2011 respectively from the School of Electrical Engineering & Telecommunications, University of New South Wales, Australia. She is now an Associate Lecturer in the School of Electrical Engineering & Telecommunications, University of New South Wales.

Her research interest includes: sliding mode control, fuzzy control, network control systems, hybrid systems, model predictive control and control applications.

Hung T. Nguyen is a Professor of Electrical Engineering at the University of Technology, Sydney (UTS). He is Dean of the Faculty of Engineering and Information Technology and Director of the Centre for Health Technologies. He received his Ph.D. in 1980 from the University of Newcastle, Australia. His research interests include biomedical engineering, advanced control and artificial intelligence. He has developed biomedical devices for diabetes, disability, and cardiovascular diseases. He is a senior member of the Institute of Electrical and Electronic Engineers; and a Fellow of the Institution of Engineers, Australia, the British Computer Society and the Australian Computer Society.

^☆: This work was supported by the Australian Research Council.

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Distributed formation building algorithms for groups of wheeled mobile robots☆

Highlights

Abstract

Introduction

Section snippets

Multi-robot system

Formation building

Formation building with anonymous robots

Computer simulations

Experiments with real robots

Conclusions

Robot. Auton. Syst.

Automatica

Robot. Auton. Syst.

J. Theoret. Biol.

Robot. Auton. Syst.

Robot. Auton. Syst.

Robot. Auton. Syst.

Automatica

Robot. Auton. Syst.

Robot. Auton. Syst.

Decentralized control of cooperative robotic vehicles: theory and application

IEEE Trans. Robot. Autom.

A general algorithm for robot formations using local sensing and minimal communication

IEEE Trans. Robot. Autom.

Coordination of groups of mobile autonomous agents using nearest neighbor rules

IEEE Trans. Automat. Control

Coordinated collective motion of groups of autonomous mobile robots: analysis of Vicsek’s model

IEEE Trans. Automat. Control

Coordinated collective motion of groups of autonomous mobile robots with directed interconnected topology

J. Intell. Robot. Syst.

Decentralized navigation of groups of wheeled mobile robots with limited communication

IEEE Trans. Robot.

Decentralized control for mobile robotic sensor network self-deployment: Barrier and sweep coverage problems

Robotica

Social networks and models for collective motion in animals

Behav. Ecol. Sociobiol.

Novel type of phase transitions in a system of self-driven particles

Phys. Rev. Lett.