Event-triggered attitude synchronization of multiple rigid-body systems

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Abstract

In this paper, an attitude synchronization problem of multiple rigid-body systems is investigated by using an event-based approach. The leader and followers are described by unit quaternions. A nonlinear distributed observer with event-triggered observations is proposed to estimate the attitude and angular velocity of the leader without continuous information exchange. The triggering mechanism is intermittent and asynchronous; and a positive lower bound of inter-event triggering times is given to show that Zeno behavior can be excluded in the intermittent communication sequence for any agent. Based on the estimated attitude and angular velocity of the leader, a distributed controller is synthesized for each follower to achieve attitude synchronization with the leader via intermittent communication. Finally, an example is provided to illustrate the effectiveness of the theoretical results.

Introduction

Attitude synchronization of rigid-body systems has received great attention due to its widespread applications in the practice, e.g., spacecrafts [1], [2], quadrotors [3] and underwater vehicles [4], [5]. Attitude synchronization attempts to drive all the rigid-body systems into a common rotational movement. The attitude synchronization has been studied under various conditions using different approaches. For example, Attitude synchronization and tracking of multiple rigid-body systems were treated in [6] by converting each rigid-body system to an Euler–Lagrange system. Leader-following attitude synchronization of multiple spacecrafts was investigated in [7] by using the back stepping design and state feedback. The issue of communication delays in attitude synchronization was studied in [1]. To achieve attitude synchronization with non-accessible leaders, a nonlinear distributed observer-based approach was proposed for static and switching networks, respectively, in [8] and [9]. Another type of distributed observers based on sliding modes has been proposed to achieve attitude synchronization, for example, see [10], [11], where a bounded velocity and a bounded acceleration were assumed in [10] and [12]. However, all the aforementioned results require continuous (static networks) [8], [10], [12] or piecewise continuous (switching networks) [9] sensing and communication within the networks, which are over-idealized due to inevitable communication and energy constraints such as transmission delays, packet dropouts, data rates, and limited battery capacity [11], [13], [14], [15], [16], [17].

A more realistic way is to use the so-called event-triggered strategy. The principle lies in that agents exchange information or execute control actions only when absolutely necessary, which can dramatically reduce the resource consumption. Over the past decade, a large number of event-triggered approaches have been proposed to achieve cooperation and regulation of multi-agent systems with reduced consumption of communication and energy [18], [19], [20], [21]. However, these results required continuous sensing and communication among the followers, which contradicts the original intention of event-triggered control to reduce resource consumption.

In order to remedy this issue, some efforts have been made. [22] and [23] used the open-loop estimator framework proposed in [24] to replace the continuous inter-agent communication and designed an event-triggered mechanism determining when to broadcast the state of an agent to its neighbors’ linear open-loop estimators. [25] proposed a distributed observer utilizing an edge-based triggering strategy, where predictors were embedded in each agent to remove the continuous inter-agent communication. [26] applied an edge-based triggering strategy to design a fully distributed control law to solve the cooperative tracking control problem for linear multi-agent systems subject to a linear leader with a bounded control input. As the leader considered in this paper is a nonlinear system described by unit quaternions, the existing observers and triggering conditions in [22], [23], [24], [25], [26] cannot be applied directly, and developing a nonlinear counterpart is necessary.

Another issue in event-triggered control is finding a lower bound of inter-event times, which is critical to choose the step size for solving underlying ordinary differential equations of the event-triggered nonlinear systems. However, it is difficult to find the lower bound of inter-event times for nonlinear systems. Most existing results such as [19], [27] and [20] cannot drive the steady state tracking error to zero, but just make it bounded.

Motivated by the studies mentioned above, an attitude synchronization problem of multiple heterogeneous and nonlinear rigid-body systems is investigated in this paper by using an event-triggered observer approach. The main contributions are summarized as follows:

  • (1)

    A nonlinear distributed observer is proposed to estimate the attitude and angular velocity of the leader.

  • (2)

    A novel event-triggered condition together with an auxiliary nonlinear observer is designed to enable asynchronous and localized triggering.

  • (3)

    A distributed control law based on the distributed observer is synthesized to achieve attitude synchronization based on the designed observer.

  • (4)

    A lower bound on inter-event times is established for each agent to exclude the existence of Zeno behavior.

  • (5)

    Comparing with [8] and [9], the continuous (static networks) or piecewise continuous (switching networks) sensing and communication assumption has been removed in the attitude synchronization.

The remainder of this paper is organized as follows. In Section 2, the problem of attitude synchronization is formulated and some standard assumptions are introduced. The main results on observer and controller design will be presented in Section 3 followed by a numeric example in Section 4. Finally, the conclusions of this paper are drawn in Section 5.

Notation: is the Euclidean norm of a vector and induced norm of a matrix. denotes the Kronecker product of matrices. For any PRn×n, let λp and λP denote the minimum and maximum eigenvalues of P, respectively. For biRni×p, i=1,,m, col(b1,,bm)[b1TbmT]T. Let M():R3R4×4 be a matrix-valued function, such that y=coly1,y2,y3R3, M(y)=0y3y2y1y30y1y2y2y10y3y1y2y30. Q denotes the set of all quaternions, i.e., Q={q|q=qˆ,q̄R3×R}, where qˆ and q̄ are the vector part and scalar part of q, respectively. Qu denotes the set of all unit quaternions, i.e., Qu={q|qQ,q=1}. For x=col x1,x2,x3R3, Q(x)x,0Q and x×0x3x2x30x1x2x10R3×3.For qi,qjQ, the product of two quaternions is defined as [3] qiqjq̄iqˆj+q̄jqˆi+qˆi×qˆj,q̄iq̄jqˆiTqˆj,and has the quaternion 0ˆ,1 as the identity element with 0ˆcol 0,0,0. For qQ, q=(qˆ,q̄) is the conjugate of q. It is noted that, for qQ, qq1=q1q=0ˆ,1, where q1=qˆ,q̄q2 denotes the inverse of q. If qQu, then q1=q=qˆ,q̄. At last, for qQ, define C(q)q̄2qˆTqˆI3+2qˆqˆT2q̄qˆ×.If, in particular, qQu, then C(q) is the direction cosine matrix which corresponds to q.

Section snippets

Problem formulation and preliminaries

Following [3] and [28], we assume that the attitude q0Qu of the leader is generated by the following exosystem: q̇0=12q0Qw0,where w0R3 is the angular velocity of the leader.

The N followers are described by the rigid-body systems of the following form: for i=1,,N, q̇i=12qiQwi,Jiẇi=wi×Jiwi+ui, where qi=(qˆi,q̄i)Qu is the unit quaternion expression of the attitude frame Bi relative to the underlying inertial frame I; wiR3 is the angular velocity of Bi to I; JiR3×3 is the positive

Event-triggered nonlinear distributed observer

We do not assume that the attitude and angular velocity of the leader is accessible to all the followers. To estimate the state of the leader, we propose the following nonlinear observer: η̇i=12ηiQFξi+μjNi(ηjηi),ξ̇i=Sξi+μjNi(ξjξi), where μ is to be defined later; η0q0 and ξ0v; ξiRm and ηiQ, for i=1,,N, are the estimation of v and q0, respectively. ηiQ and ξiRm are generated by the following auxiliary observer t[tki,tk+1i) with kKi being the event-triggered steps of

Numerical example

In this section, we consider a group of six rigid-body systems described by (3) where Ji=1+iI3, i=1,,6. The communication topology is shown in Fig. 1, which satisfies Assumption 1. By Theorem 2.5.3 of [32], we obtain D=I6 such that Y=H+HT>0 with λy=0.3194.

The leader’s signal is generated by (2) with w0(t)=col(A1sin(t+ψ1),A2cos(2t+ψ2),A3sin(3t+ψ3)), where A1, A2, A3, ψ1, ψ2 and ψ3 are unknown real numbers. Then, w0(t) can be produced from (6) with S=diag(a,2a,3a),F=100000000100000001anda=0110

Conclusions

In this paper, the leader-following attitude synchronization problem of multiple rigid-body systems has been investigated. A nonlinear distributed observer with event-triggering observations has been proposed for each agent to estimate the state of the leader without continuous sensing and communication. All agents have only local interaction with their neighbors and access their neighbors’ signals by using intermittent communication. A positive lower bound on inter-event times for each agent

CRediT authorship contribution statement

Shimin Wang: Main conceptual ideas provider, Responsible for the process of proof outline, Majorly contributes to the original draft preparation. Zhan Shu: Responsible for working out a significant number of the technical details, and the numerical example. Tongwen Chen: Supervisor for the process of writing, reviewing and editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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    This work was supported by University of Alberta (RES0049729), University of Alberta, NSERC, Canada, and an Alberta EDT Major Innovation Fund, Canada .

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