Elsevier

Automatica

Volume 122, December 2020, 109245
Automatica

Brief paper
Event-triggered attitude consensus with absolute and relative attitude measurements

https://doi.org/10.1016/j.automatica.2020.109245Get rights and content

Abstract

In this paper, we consider the event-triggered attitude consensus of multiple rigid-body systems. Two event-triggered attitude consensus protocols are designed under the absolute attitude and relative attitude measurement, respectively. For the first case, the gnomonic projection is utilized to project the attitude to the Euclidean plane almost globally. Then, a distributed attitude consensus protocol based on the projections is proposed under the event-triggered mechanism. By using the proposed protocol and event-triggered condition (ETC), the almost global attitude consensus is achieved on a positively invariant set. Next, in order to remove the requirement of the absolute attitude information, we propose an event-triggered attitude protocol with relative attitude measurements. The Riemannian gradient descent approach is utilized to design the attitude consensus protocol on a geodesically convex set of attitude configuration space. Moreover, to overcome the continuous monitoring in the event-detection, a self-triggered strategy is presented based on the event-triggered protocol only with the relative attitude measurement. Finally, simulation studies are conducted to verify the effectiveness of the proposed protocols.

Introduction

Attitude consensus is a fundamental topic in multiple rigid-body systems, aiming to drive all rigid-body agents to the same attitude (Mayhew et al., 2012, Zhang et al., 2018). It has been widely studied, which is motivated by extensive real applications in the fields like multi-robot systems, vision-based localization and camera networks (Jin et al., 2020, Montijano et al., 2013, Tron et al., 2016).

One of the main difficulties in attitude consensus is that its configuration space is not diffeomorphic to the Euclidean space (Chaturvedi et al., 2011). Due to this fact, most of the existing works study the attitude consensus by using different kinds of attitude parametrizations like Euler angles, Modified Rodrigues Parameters and unit quaternions (Abdessameud and Janabi-Sharifi, 2015, Abdessameud et al., 2012, Cai and Huang, 2016, Meng et al., 2017, Meng et al., 2010). It should be noted that only the rotation matrix can represent attitudes both globally and uniquely (Chaturvedi et al., 2011), which belongs to the compact manifold SO(3), i.e., the set of all rotation matrices in R3×3. However, attitude consensus by using rotation matrices is seldom considered in the literature. The challenge lies in that SO(3) is a Lie group which has the non-Euclidean structure (Chaturvedi et al., 2011). Furthermore, due to the topological constraint on SO(3), the global attitude consensus cannot be achieved on SO(3) by using continuous time-invariant feedback (Markdahl et al., 2018). Therefore, several attempts have been made on the almost global attitude consensus (Markdahl et al., 2018, Thunberg et al., 2016, Thunberg et al., 2018, Tron et al., 2012) in the literature recently.

At present, there are increasing real applications of attitude consensus in the aerospace community, such as small satellite networks (Guo & Zhang, 2019) and fractionated spacecraft (Wang et al., 2020). In these scenarios, each individual system is generally connected with a wireless communication network. The data sampling and control action signals are all transmitted over the wireless channels, which implies that the congestion may dramatically increase when all agents share the common channels. Moreover, the wireless communication bandwidth between spacecraft is quite limited due to the power constraint and far communication distance (Xu et al., 2019). Hence, it is practically demanded to consider the communication and control efficiency in the design of attitude consensus protocols. On the other hand, the event-triggered control has been studied for few decades, which gains much popularity in multi-agent systems recently (Li et al., 2020, Xiao et al., 2018, Yi et al., 2017). In the event-triggered sampling setting, the update of the event-triggered control protocol only happens at some certain discrete instants, when a pre-designed event-triggered condition (ETC) is violated, which makes the computation and communication resources utilization more efficient. In addition, for some applications of attitude consensus, such as deep-space exploration, only relative attitude information can be measured by cameras or optical navigation sensors but the absolute attitude information cannot be precisely obtained (Hadaegh & Smith, 2005). Note that each agent can obtain the relative attitude information by using cameras without sending messages, which means that the communication cost can be reduced further. Hence, designing an event-triggered consensus protocol with only relative attitudes available is also well-motivated.

Recently, some literature studied the event-triggered attitude leader–follower tracking (Guo and Zhang, 2019, Wang et al., 2020) and the event-triggered attitude consensus (Weng and Yue, 2016, Weng et al., 2016, Xu et al., 2019), respectively. In Wang et al. (2020), an event-triggered adaptive controller is proposed to guarantee fault-tolerant capability and reduce the communication burden at the same time. The attitude synchronization problem of multiple spacecraft in Xu et al. (2019) considers the model uncertainties and external disturbances under the event-triggered control scheme. However, due to the nonlinear attitude configuration space, very few papers consider the event-triggered almost global attitude consensus. On the other hand, very recently, the almost global attitude consensus problem is considered by using rotation matrices (Markdahl et al., 2018, Tron et al., 2012) and local attitude representations (Thunberg et al., 2017, Thunberg et al., 2016), respectively. In Tron et al. (2012) and Markdahl et al. (2018), it is ensured that the desired equilibrium points in the consensus set are only asymptotically stable equilibrium points whereas all the other undesired equilibrium points are unstable on the manifolds by introducing the candidate function with the specific constraint conditions. Nevertheless, these conditions may be quite conservative which are hard to satisfy in the event-triggered sampling setting. In the other line of research, the axis-angle vector is utilized to represent the attitude almost globally (Thunberg et al., 2017, Thunberg et al., 2016). To reach the almost global consensus, this local attitude representation set is shown to be positively invariant by using a kind of local Lyapunov functions. However, this approach cannot be utilized to solve the event-triggered almost global attitude consensus. The difficulty lies in analyzing the hybrid nonlinearity caused by the attitude kinematics and event-triggered mechanism. Specifically, the positively invariant property of the local attitude representation set is hard to be guaranteed in the event-triggered sampling setting. Furthermore, the existing event-triggered attitude control scheme is strongly dependent on a global reference frame (Weng and Yue, 2016, Weng et al., 2016, Xu et al., 2019). It should be noted that removing the requirement of absolute attitude information in protocols and ETCs in the event-triggered attitude consensus problem is a difficult task.

Based on the above discussion, this paper considers event-triggered attitude consensus under the absolute and relative attitude measurement, respectively. The advantage of the first method is that the attitude consensus can be achieved almost globally. While in the second method, the initial attitudes should be restricted in the largest geodesically convex set on SO(3). The benefit of the second protocol is that it can be used in the application where only relative attitude information is available. Moreover, we extend the second protocol to the self-triggered case, which can avoid the continuous monitoring in event-triggered mechanism.

The main contributions of this paper can be summarized in the following. (1) By using the gnomonic projection, the attitude on SO(3) can be projected into the Euclidean plane almost globally. An event-triggered attitude consensus protocol using projections is proposed to achieve almost global attitude consensus. (2) An event-triggered attitude consensus protocol using only relative attitude measurements is proposed based on the Riemannian gradient descent approach. Furthermore, based on the geometry properties of SO(3), we extend the result to the self-triggered case in order to avoid the continuous measurements in the event-detection. (3) Both of two event-triggered attitude consensus protocols are fully distributed, which means that the global topology information is not required to be known in protocols and ETCs.

This paper proceeds as follows. Some background knowledge and problem formulation are presented in Section 2 and the main theorem and mathematical proof are shown in Section 3. Section 4 gives the numerical simulation to verify the effectiveness of proposed protocols. The conclusion is drawn in Section 5.

Notations: RN denotes the N-dimension vector space, where N represents a positive integer number. R+ and N+ denote the positive real number and positive integer number, respectively. represents the Euclidean norm for vectors and spectral norm for matrices, respectively. denotes the Kronecker product. The transpose of matrix A is defined as A. tr(A) represents the trace of matrix A. SN={xRN+1:x,x=1} denotes the N-dimension unit sphere in RN+1, where , denotes the standard inner product in Euclidean space. Notation “” means “defined as”.

Section snippets

Graph theory

The communication topology is described as a directed graph denoted as G={N,E}, where N={1,,N} is the node set and EN×N is the edge set. For an undirected graph, if (i,j)E then (j,i)E. For a directed graph, one directed path linking the ith node and the jth node is a series of different edges in the form {(i,l1),(l1,l2),,(lk,j)},lkN,kN+. A directed graph which is called strongly connected means that any two nodes in the graph can be connected with each other through a directed path. The

Main results

In this section, we consider event-triggered attitude consensus problems in two different measurement cases. The attitude consensus protocols and ETCs are designed in two subsections, respectively.

Simulation

In this section, we give numerical simulations to verify the validity of the theoretical results. The simulations run in Matlab by utilizing the Euler method to solve the differential equations. We firstly consider a multi-agent system that consists of five rigid-bodies under the strongly connected graph. The adjacency matrix is given by A1=00.5000000.8000000.6000000.310000.In this example, attitudes are described by the local representations yi=tanΘi2ui. The rotation unit vectors ui of each

Conclusion

In this paper, the event-triggered attitude consensus problem has been investigated for multi-agent systems. Two distributed event-triggered consensus protocols are designed under the absolute attitude and relative attitude measurements, respectively. The first protocol uses the absolute attitude based on the gnomonic projection and the almost global attitude consensus can be achieved in the event-triggered sampling setting. The second protocol only depends on the relative attitude

Acknowledgments

This paper was not presented at any IFAC meeting. This research was supported in part by the National Natural Science Foundation of China under Grant 61988101, Grant 61873294, Grant 61751305, and Grant 61673176, in part by the National Key Research and Development Program of China under Grant 2018YFC0809302, in part by the Key Laboratory of Advanced Perception and Intelligent Control of High-end Equipment, Ministry of Education under Grant GDSC202002, in part by the Fundamental Research Funds

Xin Jin received the B.S. degree from the School of Automation, Guangdong University of Technology, Guangzhou, China, in 2016. He is currently working toward the Ph.D. degree from East China University of Science and Technology, Shanghai, China. His research interests include multi-agent systems, multiple rigid body systems and their applications.

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  • Cited by (0)

    Xin Jin received the B.S. degree from the School of Automation, Guangdong University of Technology, Guangzhou, China, in 2016. He is currently working toward the Ph.D. degree from East China University of Science and Technology, Shanghai, China. His research interests include multi-agent systems, multiple rigid body systems and their applications.

    Yang Shi (SM’09-F’17) received B.Sc. and Ph.D. degrees in mechanical engineering and automatic control from Northwestern Polytechnical University, Xi’an, China, in 1994 and 1998, respectively, and the Ph.D. degree in electrical and computer engineering from the University of Alberta, Edmonton, AB, Canada, in 2005.

    From 2005 to 2009, he was an Assistant Professor and Associate Professor in the Department of Mechanical Engineering, University of Saskatchewan, Saskatoon, SK, Canada. In 2009, he joined the University of Victoria, and now he is a Professor in the Department of Mechanical Engineering, University of Victoria, Victoria, BC, Canada. His current research interests include networked and distributed systems, model predictive control (MPC), cyber–physical systems (CPS), robotics and mechatronics, navigation and control of autonomous systems (AUV and UAV), and energy system applications.

    Dr. Shi received the University of Saskatchewan Student Union Teaching Excellence Award in 2007, and the Faculty of Engineering Teaching Excellence Award in 2012 at the University of Victoria (UVic). He is the recipient of the JSPS Invitation Fellowship (shortterm) in 2013, the UVic Craigdarroch Silver Medal for Excellence in Research in 2015, the 2017 IEEE Transactions on Fuzzy Systems Outstanding Paper Award, the Humboldt Research Fellowship for Experienced Researchers in 2018. He is a member of the IEEE IES Administrative Committee and the IES Fellow Evaluation Committee during 2017–2019; he is the Chair of IEEE IES Technical Committee on Industrial Cyber–Physical Systems. Currently, he is Co-Editor-in-Chief for IEEE Transactions on Industrial Electronics; he also serves as Associate Editor for Automatica, IEEE Transactions on Control Systems Technology, IEEE/ASME Transactions on Mechatronics, IEEE Transactions on Cybernetics, etc. He is General Chair of the 2019 International Symposium on Industrial Electronics (ISIE) and the 2021 International Conference on Industrial Cyber–Physical Systems (ICPS).

    He is a Fellow of IEEE, ASME, Engineering Institute of Canada (EIC), and Canadian Society for Mechanical Engineering (CSME), and a registered Professional Engineer in British Columbia, Canada.

    Yang Tang received the B.S. and Ph.D. degrees in electrical engineering from Donghua University, Shanghai, China, in 2006 and 2010, respectively. From 2008 to 2010, he was a Research Associate with The Hong Kong Polytechnic University, Hong Kong. From 2011 to 2015, he was a Post-Doctoral Researcher with the Humboldt University of Berlin, Berlin, Germany, and with the Potsdam Institute for Climate Impact Research, Potsdam, Germany. Since 2015, he has been a Professor with the East China University of Science and Technology, Shanghai. His current research interests include distributed estimation/control/optimization, cyber–physical systems, hybrid dynamical systems, computer vision, reinforcement learning and their applications.

    Prof. Tang was a recipient of the Alexander von Humboldt Fellowship and the ISI Highly Cited Researchers Award by Clarivate Analytics from 2017. He is a Senior Board Member of Scientific Reports, an Associate Editor of IEEE Transactions on Neural Networks and Learning Systems, IEEE Transactions on Emerging Topics in Computational Intelligence and IEEE Systems Journal, etc.

    Xiaotai Wu received the M.S. degree in applied mathematics from Jiangsu University, Zhenjiang, China, in 2006, and the Ph.D. degree in control theory and engineering from Donghua University, Shanghai, China, in 2012. He is currently a Full Professor with the Department of Mathematics, Anhui Polytechnic University, China. His current research interests include stability and control of stochastic hybrid systems and multi-agent systems.

    The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Abdelhamid Tayebi under the direction of Editor Thomas Parisini.

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