Elsevier

Automatica

Volume 125, March 2021, 109388
Automatica

Attitude synchronization and rigid formation of multiple rigid bodies over proximity networks

https://doi.org/10.1016/j.automatica.2020.109388Get rights and content

Abstract

In this paper, the combination of the attitude synchronization and rigid formation problem of multiple moving rigid bodies are considered. The moving rigid bodies communicate with each other via a distance-dependent interaction network, yielding the dynamical neighbor relations and the coupling relationship between positions and attitudes of all rigid bodies. The finite-time control and potential function techniques are combined in the design of distributed control laws for the torques and forces of rigid bodies. We construct the admissible set on the initial states, in which the theoretical results on the attitude synchronization, rigidity maintenance and collision avoidance are established simultaneously without relying on the dynamical connectivity of the neighbor graphs. Furthermore, by transforming the stability of rigid formations into the stability of the parameterized systems, we show that the local asymptotic stability of rigid formations can be achieved. A simulation example is given to illustrate the theoretical results.

Introduction

Over the last two decades, coordination control problems of multiple rigid bodies (or multi-agent systems) have attracted much attention due to their potential applications in practical systems such as underwater vehicles, robotics, UAVs and satellites (Diehl et al., 1999, Krieger et al., 2010, Leonard et al., 2007, Liu et al., 2015, Liu et al., 2017, Shiroma et al., 2005). The agents can cooperate to finish some complicated tasks such as ocean sampling, rescue, surveillance and space missions. The theoretical challenges also arise from the distributed control design and the performance analysis of the whole system.

The states of the rigid bodies are composed of two parts: attitude and position, and correspondingly the motion of rigid bodies in three dimensional space is generally composed of rotation and translation. The attitude of rigid bodies can be globally and uniquely represented by rotation matrix, and there are also some parameterizations, such as rotation vector, Rodrigues parameters, MRPs, and unit quaternion. Generally, these parameterizations can only locally represent the attitude, and are not one-to-one, see Thunberg, Goncalves, and Hu (2016) for details. For some practical scenarios, both the attitudes and the positions of rigid bodies are required to reach the desired states to cooperatively finish the pre-defined task. For example, the rigid bodies are required to cooperatively move a rigid object to a desired destination. To finish such a task, the attitudes of rigid bodies are required to reach expected ones, while the positions need to maintain a specified formation.

In the investigation of the attitude coordination of rigid bodies, the attitude synchronization problem, meaning that all rigid bodies tend to the same attitude, is commonly considered. Thunberg, Song, Montijano, Hong, and Hu (2014) used the rotation vector representation to study the attitude synchronization problem under fixed and switching communication topologies; Cai and Huang (2016) employed unit quaternion representation to develop a distributed observer-based method to investigate the attitude synchronization of leader-following systems, then the results were extended by Cai and Huang (2017) to the case where model uncertainty and external disturbance are existent; Abdessameud, Tayebi, and Polushin (2012) used the unit quaternion representation to address the attitude synchronization problem for the case where the angular velocity is unmeasurable. Note that the above works adopted the parameterizations to represent the attitude of rigid bodies, which are often local and are not one-to-one.

There are some work which use the rotation matrix to represent the attitude of rigid bodies, and the theoretical analysis is generally complicated. For example, Sarlette, Sepulchre, and Leonard (2009) and Sarlette, Sepulchre, and Leonard (2007) studied the attitude synchronization problem of rigid bodies over given fixed and switching communication topologies; Guo, Song, and Li (2016) investigated the finite-time attitude synchronization problem of rigid bodies; Weng, Yue, Xie, and Xue (2016) considered the attitude synchronization problem with event-triggered control mechanism. As far as we know, almost all existing results on the attitude synchronization of rigid bodies rely on the dynamical connectivity assumptions of communication graphs.

For the formation control problem, Oh, Park, and Ahn (2015) classified the current study into three categories according to the types of sensed and controlled variables: position-, displacement-, and distance-based. The position-based formation is developed based on the absolute information of positions with respect to a global coordinate system, which might be costly because the agents are required to carry some advanced sensing equipments such as GPS receivers to measure the absolute positions of the agents (Oh et al., 2015). The displacement-based formation have been studied for single- and double-integrator models, unicycles and general linear agents under undirected or directed interaction graphs, see Oh et al. (2015) and the references therein. For the distanced-based formation control problem, it is usually difficult to carry out theoretical analysis, for which only local convergence results can be obtained. For example, Dörfler and Francis (2009) and Oh and Ahn (2011) showed the local stability of formations for single-integrator agents with the target formation being minimally and infinitesimally rigid, respectively. Oh and Ahn (2014) relaxed the formation condition to be rigid, and further considered the rigid formation problem for double-integrator models. Cai and De Queiroz (2014) investigated the adaptive distance-based formation problem for Euler–Lagrange models with parametric uncertainty. Furthermore, Verginis, Nikou, and Dimarogonas (2019) considered the distance and orientation formation control problem of rigid bodies for fixed tree-graph structures, which, to the best of our knowledge, is the first attempt to this topic.

In this paper, both the attitude synchronization and the formation control problem of multiple moving rigid bodies are considered, that is, the inter-distances of rigid bodies are required to maintain a particular rigid formation while their postures tend to a desired attitude. The moving rigid bodies communicate via a distance-dependent interaction network. Two rigid bodies can communicate with each other if and only if their distance is less than a predefined length, which is a natural way to describe the communication between agents. Note that from the dynamics of rigid bodies, we see that, the attitudes affect the positions, while the positions in turn affect the behaviors of the attitudes via the distance-dependent communication graphs. The coupled relations between attitude and position increase the difficulties for the theoretical analysis, and there are very few results relating with this topic. In order to guarantee the rigid bodies maintaining a predefined rigid formation with the same attitudes, the finite-time control technique is used in the design of control torques such that the angular velocity can track the desired one, while the potential function method is employed to maintain the rigidity of the distance-induced neighbor graph. Some convergence results on the pose synchronization, collision avoidance and the stability of rigidity formation are established, in which the essential problem is the analysis of the coupling relationship between positions and attitudes of all rigid bodies. The main contributions of this paper can be summarized as follows:

(1) We design the control laws for the force and the torque for rigid bodies, and show that if the initial states are chosen from an admissible set, then the rigid bodies can reach attitude synchronization and rigidity maintenance while avoiding collisions with other rigid bodies without relying on the dynamical properties of the neighbor graphs. The Barbalat’s lemma and some essential properties of the rotation matrices are used to deal with the coupled relationship between positions and attitudes.

(2) By transforming the stability of the rigid formation of the dynamical system of rigid bodies into the stability of the parameterized systems, we construct the attractive zone for stability of the rigid formation, and then establish the local stability of the rigid formation of rigid bodies.

(3) Compared with some recent work on attitude synchronization and formation control problems (Verginis et al., 2019) where the network is required to be fixed, the network topology in the current paper is determined by the distance between rigid bodies and may change over time when the rigid bodies are in motion, which makes the theoretical analysis more challenging.

The rest of this paper is organized as follows. In Section 2, we present the problem formulation for the combined attitude synchronization and rigid formation of moving rigid bodies. In Section 3, we design the control laws for the force and the toque, and establish the theorem for the attitude synchronization, collision avoidance and rigidity maintenance of the system with multiple rigid bodies. Furthermore, we analyze the local stability of the rigid formation. In Section 4, we give a simulation result to verify the theoretical results in this paper. Concluding remarks are presented in Section 5.

Section snippets

Problem formulation

In this paper, we study the attitude synchronization and the rigid formation problem of n moving rigid bodies. In order to describe the motion of rigid bodies, the inertial coordinate frame Σw and the frame Σi fixed on the rigid body i(i=1,,n) are introduced. Both Σw and Σi are right-handed Cartesian coordinate frames. The motion of rigid bodies is generally composed of translation and rotation. The position and rotation matrix of the rigid body i in the inertial coordinate frame Σw are,

Control for the torque

In our previous work (Deng, Wang, Liu, & Hu, 2018), we designed control laws for the kinematic model of rigid bodies to achieve attitude synchronization and avoid collisions with nearby rigid bodies while the neighbor graphs keep connected. We denote the angular velocity designed by Deng et al. (2018) as a desired angular velocity, ωid(t)=jNi(t)aij(p(t))sk(Rij(t)),where sk(R)=12(RRT) and the notation “” denotes the inverse operator to “” defined via (2). For the dynamical model (1)

Simulation results

In this section, we will illustrate our results by a simulation example. Consider a system composed of six (i.e. n=6) rigid bodies whose dynamics are described by (1).

For the rigid formation problem, we adopt the control laws according to (7), (14), with two action functions ϕα and ϕβ being taken as follows, ϕα(z)=0,z0,sσ;20[z2(sσ+h1σ)z+sσh1σ],zsσ,h1σ;0,zh1σ,. ϕβ(z)=0,z0,h2σ;20[z2+(h2σ+rσ)zh2σrσ],zh2σ,rσ;0,zrσ,.where s=1,h1=3,h2=7,r=9. The parameters

Concluding remarks

In this paper, we proposed the distributed control law of the force and torque for the dynamical model of multiple moving rigid bodies. The interactions between rigid bodies are defined via their distances, yielding a sequence of state-depended dynamical neighbor graphs. The theoretical results on the attitude synchronization and the rigid formations are established simultaneously, without relying on the dynamical properties of neighbor graphs. Furthermore, the local stability of the rigid

Juan Deng received the B.S. degree from the School of Mathematics and Physics, University of Science and Technology Beijing in 2013, and the Ph.D. degree in systems theory from Academy of Mathematics and Systems Science (AMSS), Chinese Academy of Sciences (CAS), Beijing, China, in 2018. She is currently a lecturer of the College of Liberal Arts and Sciences, National University of Defense Technology. Her research interests include multi-agent systems and distributed control.

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  • Juan Deng received the B.S. degree from the School of Mathematics and Physics, University of Science and Technology Beijing in 2013, and the Ph.D. degree in systems theory from Academy of Mathematics and Systems Science (AMSS), Chinese Academy of Sciences (CAS), Beijing, China, in 2018. She is currently a lecturer of the College of Liberal Arts and Sciences, National University of Defense Technology. Her research interests include multi-agent systems and distributed control.

    Lin Wang received the B.S. and M.S. degrees from the School of Mathematical Sciences, Shandong Normal University, Jinan, China in 2003 and 2006, respectively, and the Ph.D. degree in operations research and control theory from the Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China in 2009. She is currently a Professor with the Department of Automation, Shanghai Jiao Tong University, Shanghai, China. Her current research interests include multi-agent systems, adaptive complex networks, and coordination of multiple manipulators.

    Zhixin Liu received the B.S. degree in Mathematics from Shandong University, Jinan, China, in 2002, and the Ph.D. degree in control theory from Academy of Mathematics and Systems Science (AMSS), Chinese Academy of Sciences (CAS), Beijing, China, in 2007. She is currently a professor of AMSS, CAS, and the director of the Key Laboratory of Systems and Control, CAS. She had visiting positions at KTH Royal Institute of Technology, University of New South Wales, Canberra and University of Maryland, College Park. Her current research interests are complex systems and multi-agent systems.

    This work was supported by the National Key R&D Program of China under Grant 2018YFA0703800 and the National Natural Science Foundation of China under grants 11688101 and 61873167. The material in this paper was partially presented at the 13th World Congress on Intelligent Control and Automation, July 4–8, 2018, Changsha China. This paper was recommended for publication in revised form by Associate Editor Wei Ren under the direction of Editor Christos G. Cassandras.

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