Cooperative Attitude Control of Multiple Rigid Bodies with Multiple Time-Varying Delays and Dynamically Changing Topologies

Abstract

Cooperative attitude regulation and tracking problems are discussed in the presence of multiple time-varying communication delays and dynamically changing topologies. In the case of cooperative attitude regulation, we propose conditions to guarantee the stability of the closed-loop system when there exist multiple time-varying communication delays. In the case of cooperative attitude tracking, the result of uniformly ultimate boundedness of the closed-loop system is obtained when there exist both multiple time-varying communication delays and dynamically changing topologies. Simulation results are presented to validate the effectiveness of these conclusions.

1. Introduction

Cooperative control of multiagent system has been followed with extensive interest in recent years. Compared to single-agent system, greater benefits such as greater efficiency, lower costs, and higher robustness can be realized by cooperation of multiagent system. The basic idea of cooperative control of multiagent system is that each agent in the group uses its local interactions such that the common objectives and tasks can be achieved [1]. One important application toward this direction is distributed cooperative attitude control for multiple rigid bodies. In particular, in the context of deep space interferometry, it is often necessary and significant to maintain relative attitude synchronization precisely during and after maneuvers among a formation of spacecraft [2, 3], where cooperative attitude control may serve as an effective tool.

As a decentralized control strategy, cooperative attitude control demonstrates many superior qualities compared with the traditionally centralized approaches. A good survey on cooperative attitude control can be found in [4]. In particular, a leader-following structure was used in [2, 5], where the follower spacecraft are assumed to have access to the information of the leader spacecraft. The authors in [6] proposed a behavioral strategy to realize attitude synchronization, where the behavior of individual spacecraft attitude tracking and that of formation keeping were considered together in an index function. As an alternative to the behavioral strategy, the virtual structure approach was proposed in [7], where the entire formation is treated as a single rigid body. The communication topology was highlighted in [8–10] and the attitude coordination problems were presented by using relative attitude and relative angular velocity information. Attitude containment control for multiple stationary leaders were considered in [11], where the linearly relative attitude was expressed by using Modified Rodriguez Parameters (MRPs) as attitude representation.

Although lots of benefits can be obtained from cooperative control of multiagent system, the performance of such networked system is often subject to communication failure and communication delays. Plenty results on the influence of delays and dynamically changing topologies have been obtained for multiagent system described by simplified models of motion, such as single-integrator dynamics and double-integrator dynamics. In particular, the authors in [12] presented a consensus algorithm with delays and dynamically changing topologies and used the time-domain and frequency-domain approaches to find the stability conditions. Average consensus was considered in [13], where the delays were assumed nonuniform and the communication topology was assumed jointly-connected. The similar problem was discussed in [14] where the communication topology was extended from an undirected graph to a directed one. A second-order consensus regulation algorithm with nonuniform communication delays was studied in [15] with the focus on a flocking problem of large scale multiagent systems. Both delay-independent and delay-dependent conditions were obtained.

The research on the cooperative attitude control problem in the presence of communication delays and dynamically changing topologies was given in [16, 17], where a synchronization variable was used to contain both attitude and angular velocity information. Motivated by the work of [18], the conditions to guarantee cooperative attitude control with communication delays were obtained. Similar problem was discussed in [19] with an emphasis on multiple networked Lagrangian systems. Delays, limited data rates, and bounded disturbance input were considered together in the control law.

This paper is organized as follows. In Sections 2 and 3, we provide basics for spacecraft attitude dynamics, graph theory, and functional differential equation. In Section 4, cooperative attitude regulation and tracking problems are described and the control torques with communication delays and dynamically changing topologies are proposed. The case of cooperative attitude regulation with multiple time-varying communication delays and fixed topology is discussed in Section 5, while the case of cooperative attitude tracking with multiple time-varying communication delays and dynamically changing topologies is discussed in Section 6. Simulation results are given in Section 7 to validate the theoretical results. Section 8 contains our concluding remarks.

2. Preliminaries

2.1. Notations

ℝ and ℂ are, respectively, the set of real numbers and the set of complex numbers. 𝜆_min(𝐴) and 𝜆_max(𝐴) are, respectively, the minimal eigenvalue and the maximum eigenvalue of the matrix 𝐴. ⊗ denotes the Kronecker product.

Consistent with [20], we denote ℂ_𝑛,𝜏 as the Banach space of continuous vector functions mapping the interval [−𝜏,0] into ℝ^𝑛 with the topology of uniform convergence. ℂ^𝑣_𝑛,𝜏={𝜙∈ℂ_𝑛,𝜏∶‖𝜙‖_𝑐<𝑣}, where 𝑣 is a positive real number.

‖⋅‖ stands for the Euclidean vector norm and ‖𝜙‖_𝑐=sup_{−𝜏≤𝑡≤0}‖𝜙(𝑡)‖ stands for the norm of a function 𝜙∈ℂ_𝑛,𝜏.

𝑄>0 means that the matrix 𝑄 is positive definite.

2.2. Spacecraft Attitude Kinematics and Dynamics

In this paper, the attitude of each spacecraft in a formation is represented by the unit quaternion, given by 𝑞_𝑖=[𝑞₁,𝑞₂,𝑞₃,𝑞₄]^𝑇_𝑖=[𝑞^𝑇_𝑖,𝑞_𝑖]^𝑇,𝑖=1,…,𝑛. Here 𝑞_𝑖=𝑒_𝑖sin(𝜙_𝑖/2), 𝑞_𝑖=cos(𝜙_𝑖/2), where 𝑒_𝑖 and 𝜙_𝑖 are the principle axis and the principle angle of the attitude of the 𝑖th spacecraft and 𝑞^𝑇_𝑖𝑞_𝑖=1 [21]. The product of two unit quaternions 𝑝_𝑖 and 𝑞_𝑖 is defined by 𝑞_𝑖𝑝_𝑖=𝑞_𝑖𝑝_𝑖+𝑝_𝑖𝑞_𝑖+𝑞_𝑖×𝑝_𝑖𝑞_𝑖𝑝_𝑖−𝑞^𝑇_𝑖𝑝_𝑖. (2.1)

The conjugate of the unit quaternion 𝑞_𝑖 is defined by 𝑞_𝑖⁻¹=[−𝑞^𝑇_𝑖,𝑞_𝑖]^𝑇. Attitude kinematics and dynamics of each spacecraft using the unit quaternion are given by [21] 𝑞̇_𝑖=12𝐸_𝑖(𝑞_𝑖)𝜔_𝑖𝐽,𝑖=1,…,𝑛,_𝑖𝜔̇_𝑖=−𝜔_𝑖×(𝐽_𝑖𝜔_𝑖)+𝜏_𝑖,𝑖=1,…,𝑛, (2.2) where 𝑞_𝑖∈ℝ⁴ denotes the rotation from the body frame of the 𝑖th spacecraft to the inertial frame, 𝜔_𝑖 is the angular velocity of the 𝑖th spacecraft with respect to the inertial frame expressed in the body frame of the 𝑖th spacecraft, and 𝐸_𝑖(𝑞_𝑖) is given by 𝐸_𝑖(𝑞_𝑖)=𝑞_𝑖𝐼₃+𝑆(𝑞_𝑖)−𝑞^𝑇_𝑖, (2.3)where 𝐼₃ is the 3×3 identity matrix, 𝑆(⋅) denotes a 3×3 skew-symmetric matrix, and 𝐽_𝑖∈ℝ^3×3 and 𝜏_𝑖∈ℝ³ are, respectively, the inertia tensor and control torque of the 𝑖th spacecraft.

2.3. Graph Theory [22]

The communication topology among spacecraft in the formation is modeled using graph theory. An undirected graph 𝒢 consists of a pair (𝑉,ℰ), where 𝑉={𝑣₁,…,𝑣_𝑛} is a finite nonempty set of nodes and ℰ⊆𝑉×𝑉 is a set of unordered pairs of nodes. An edge (𝑣_𝑖,𝑣_𝑗) denotes that nodes 𝑣_𝑖 and 𝑣_𝑗 can obtain information from each other. In such case, nodes 𝑣_𝑖 and 𝑣_𝑗 are neighbors of each other. All the neighbors of node 𝑣_𝑖 are denoted as 𝑁_𝑖∶={𝑣_𝑗∣(𝑣_𝑗,𝑣_𝑖)∈ℰ}, where we assume that 𝑣_𝑖∉𝑁_𝑖.

An undirected path is a sequence of edges in a undirected graph of the form (𝑣_𝑖1,𝑣_𝑖2),(𝑣_𝑖2,𝑣_𝑖3),…. An undirected graph is connected if there is an undirected path between every pair of distinct nodes. In this paper, the communication topology is assumed to be undirected.

The adjacency matrix 𝒜=[𝑎_𝑖𝑗]∈ℝ^𝑛×𝑛 associated with the undirected graph 𝒢 is defined such that 𝑎_𝑖𝑗 is a positive value if (𝑣_𝑗,𝑣_𝑖)∈ℰ, and 𝑎_𝑖𝑗=0 otherwise. We assume that 𝑎_𝑖𝑗=𝑎_𝑗𝑖, for all 𝑖≠𝑗, since (𝑣_𝑗,𝑣_𝑖)∈ℰ implies (𝑣_𝑖,𝑣_𝑗)∈ℰ in the undirected graph. Also, the Laplacian matrix ℒ=[𝑙_𝑖𝑗]∈ℝ^𝑛×𝑛 associated with 𝒜 is defined as 𝑙_𝑖𝑗=^𝑗∈𝑁_𝑖𝑎_𝑖𝑗,𝑖=𝑗,−𝑎_𝑖𝑗,𝑖≠𝑗. (2.4)

3. Definitions and Lemmas

Suppose 𝑓∶ℝ×ℂ^𝑣_𝑛,𝜏↦ℝ^𝑛 is continuous and consider retarded functional differential equation (RFDE) 𝑥̇(𝑡)=𝑓(𝑡,𝑥_𝑡). (3.1) Let 𝜙=𝑥_𝑡 be defined as 𝑥_𝑡(𝜃)=𝑥(𝑡+𝜃),𝜃∈[−𝜏,0]. Suppose that the initial condition satisfies 𝑥(𝜃)=0, for all 𝜃∈[𝑡₀−𝜏,𝑡₀]. Also suppose that the solution 𝑥(𝑡₀,𝜙)(𝑡) through (𝑡₀,𝜙) is continuous in (𝑡₀,𝜙,𝑡) in the domain of definition of the function, where 𝑡₀∈ℝ.

Definition 3.1 (see [23]). The solutions 𝑥(𝑡₀,𝜙) of the RFDE (3.1) are uniformly asymptotically stable if(i)for every 𝜅>0 and for every 𝑡₀≥0 there exists a 𝛿=𝛿(𝜅) independent of 𝑡₀ such that for any 𝜙∈ℂ^𝛿_𝑛,𝜏 the solutions 𝑥(𝑡₀,𝜙) of the RFDE (3.1) satisfies 𝑥_𝑡(𝑡₀,𝜙)∈ℂ^𝜅_𝑛,𝜏 for all 𝑡≥𝑡₀,(ii)for every 𝜂>0 and for every 𝑡₀≥0 there exists a 𝑇(𝜂) independent of 𝑡₀ and a 𝑣₀>0 independent of 𝜂 and 𝑡₀ such that for any 𝜙∈ℂ_𝑛,𝜏, ‖𝜙‖_𝑐<𝑣₀ implies that ‖𝑥_𝑡(𝑡₀,𝜙)‖_𝑐<𝜂, for all 𝑡≥𝑡₀+𝑇(𝜂).

Definition 3.2 (see [23]). The solutions 𝑥(𝑡₀,𝜙) of the RFDE (3.1) are uniformly ultimately bounded if there is a 𝛽>0 such that for any 𝛼>0, there is a constant 𝑇₀(𝛼)>0 such that |𝑥(𝑡₀,𝜙)(𝑡)|≤𝛽 for 𝑡≥𝑡₀+𝑇₀(𝛼) for all 𝑡₀∈ℝ, 𝜙∈ℂ, and |𝜙|≤𝛼.

Lemma 3.3 (Lyapunov-Krasovskii stability theorem [20]). Consider the RFDE (3.1). Suppose 𝑓∶ℝ×ℂ_𝑛,𝜏↦ℝ^𝑛 takes ℝ× (bounded sets of ℂ_𝑛,𝜏) into bounded sets of ℝ^𝑛, 𝑢(𝑠), 𝑣(𝑠), and 𝑤(𝑠) are continuous, nonnegative and nondecreasing functions with 𝑢(𝑠),𝑣(𝑠)>0 for 𝑠≠0 and 𝑢(0)=𝑣(0)=0. If there exists a continuous function 𝑉∶ℝ×ℂ_𝑛,𝜏↦ℝ such that(i)𝑢(‖𝜙(0)‖)≤𝑉(𝑡,𝜙)≤𝑣(‖𝜙‖_𝑐),(ii)𝑉̇(𝑡,𝜙)≤−𝑤(‖𝜙(0)‖), then the solutions of (3.1) are uniformly stable. In addition, if 𝑤(𝑠)>0 for 𝑠>0, then the solutions of (3.1) are uniformly asymptotically stable.

Lemma 3.4 (Lyapunov-Razumikhin uniformly ultimately bounded theorem [23]). Consider the RFDE (3.1). Suppose 𝑓∶ℝ×ℂ_𝑛,𝜏↦ℝ^𝑛 takes ℝ× (bounded sets of ℂ_𝑛,𝜏) into bounded sets of ℝ^𝑛 and 𝑢,𝑣,𝑤∶ℝ⁺↦ℝ⁺ are continuous nonincreasing functions, 𝑢(𝑠)→∞ as 𝑠→∞. If there is a continuous function 𝑉∶ℝ×ℝ^𝑛↦ℝ, a continuous nondecreasing function 𝑝∶ℝ⁺↦ℝ⁺, 𝑝(𝑠)>𝑠 for 𝑠>0, and a constant 𝐻≥0 such that 𝑢(‖𝑥‖)≤𝑉(𝑥)≤𝑣(‖𝑥‖), 𝑡∈ℝ, 𝑥∈ℝ^𝑛, and 𝑉̇(𝑡,𝜙)≤−𝑤(‖𝜙(0)‖) if ‖𝜙(0)‖≥𝐻, 𝑉(𝑡+𝜃,𝜙(𝜃))<𝑝(𝑉(𝑡,𝜙(0))), 𝜃∈[−𝜏,0], then the solutions of (3.1) are uniformly ultimately bounded.

4. Problem Statement

In this paper, we consider cooperative attitude regulation and tracking problems for multiple rigid bodies in the presence of multiple time-varying delays and dynamically changing topologies. The objectives are to guarantee that each spacecraft tracks the constant or time-varying states of the leader spacecraft while aligning their attitudes within the formation. Cooperative attitude regulation control law with zero delay and fixed topology is proposed in [8] as𝜏_𝑖=−𝐾_𝑖𝑞_𝑖−𝐷_𝑖𝜔_𝑖−^𝑛_𝑗=1𝑎_𝑖𝑗𝑞_𝑖𝑗−^𝑛_𝑗=1𝑎_𝑖𝑗𝜔_𝑖𝑗, (4.1) where 𝐾_𝑖 and 𝐷_𝑖 are nonnegative constants, 𝑎_𝑖𝑗 is the (𝑖,𝑗)th entry of the adjacency matrix 𝒜 associated with the graph 𝒢, 𝑞_𝑖𝑗=𝑞_𝑗⁻¹𝑞_𝑖, 𝑞_𝑖𝑗 is the vector part of 𝑞_𝑖𝑗, 𝜔_𝑖𝑗=𝜔_𝑖−𝐴(𝑞_𝑖𝑗)𝜔_𝑗, and 𝐴(𝑞_𝑖𝑗)=𝐴(𝑞_𝑖)𝐴^𝑇(𝑞_𝑗) denotes the rotation matrix [21]. Here 𝑞_𝑖𝑗 represents the relative attitude between spacecraft 𝑖 and spacecraft 𝑗, and 𝜔_𝑖𝑗 represents the relative angular velocity between spacecraft 𝑖 and spacecraft 𝑗. Note that the existence of the attitude consensus terms −∑^𝑛_𝑗=1𝑎_𝑖𝑗𝑞_𝑖𝑗−∑^𝑛_𝑗=1𝑎_𝑖𝑗𝜔_𝑖𝑗 help to guarantee that the attitude of each follower spacecraft will be close to its neighbors. This is necessary in certain spacecraft mission, such as distributed synthetic-aperture imaging mission [24], where the attitude control system is required to have the ability to ensure relative attitude keeping during the maneuver.

Cooperative attitude tracking control law with zero delay and fixed topology is proposed in [9] as𝜏_𝑖=𝐽_𝑖𝐴(Δ𝑞_𝑖)𝜔̇_𝑑+𝜔^𝑏𝑖_𝑑×𝐽_𝑖𝜔^𝑏𝑖_𝑑−𝐾_𝑖Δ𝑞_𝑖−𝐷_𝑖Δ𝜔_𝑖−^𝑛_𝑗=1𝑎_𝑖𝑗𝑞_𝑖𝑗−^𝑛_𝑗=1𝑎_𝑖𝑗𝜔_𝑖𝑗, (4.2) where 𝑞_𝑑 and 𝜔_𝑑 denote, respectively, the time-varyingly desired attitude and angular velocity of the leader spacecraft, Δ𝑞_𝑖=𝑞_𝑑⁻¹𝑞_𝑖, Δ𝑞_𝑖 is the vector part of Δ𝑞_𝑖, 𝜔^𝑏𝑖_𝑑=𝐴(Δ𝑞_𝑖)𝜔_𝑑, 𝐴(Δ𝑞_𝑖)=𝐴(𝑞_𝑖)𝐴^𝑇(𝑞_𝑑), and Δ𝜔_𝑖=𝜔_𝑖−𝜔^𝑏𝑖_𝑑. Here Δ𝑞_𝑖 denotes the relative attitude between spacecraft 𝑖 and the leader, Δ𝜔_𝑖 denotes the relative angular velocity between spacecraft 𝑖 and the leader. By using (4.1) for (2.2), cooperative attitude regulation, that is, 𝑞_𝑖→𝑞_𝐼 and 𝜔_𝑖→0, is achieved, where 𝑞_𝐼 denotes the identity quaternion [0,0,0,1]^𝑇. By using (4.2) for (2.2), cooperative attitude tracking, that is, 𝑞_𝑖→𝑞_𝑑 and 𝜔_𝑖→𝜔_𝑑 is achieved.

In this paper, we extend cooperative attitude regulation and tracking control laws to the cases where there exist multiple time-varying communication delays and dynamically changing topologies. For the first part, we discuss cooperative attitude regulation problem in the presence of multiple time-varying communication delays and assume that the communication topology is fixed. A model-independent control torque 𝜏_𝑖 is proposed𝜏_𝑖=−𝐾_𝑖𝑞_𝑖−𝐷_𝑖𝜔_𝑖−^𝑛_𝑗=1𝑎_𝑖𝑗[𝑞_𝑖−𝑞_𝑗(𝑡−𝑇_𝑖𝑗(𝑡))]−^𝑛_𝑗=1𝑎_𝑖𝑗[𝜔_𝑖−𝑙_𝑖𝑗𝐴_𝑖𝑗(𝑡−𝑇_𝑖𝑗(𝑡))𝜔_𝑗(𝑡−𝑇_𝑖𝑗(𝑡))], (4.3) where 𝐾_𝑖, 𝐷_𝑖, and 𝑎_𝑖𝑗 are defined after (4.1), 𝑙_𝑖𝑗 is a nonnegative constant, 𝑇_𝑖𝑗=𝑇_𝑗𝑖 denotes multiple time-varying communication delay, and 𝐴_𝑖𝑗(𝑡−𝑇_𝑖𝑗(𝑡))=𝐴(𝑞_𝑖)𝐴^𝑇(𝑞_𝑗(𝑡−𝑇_𝑖𝑗)). For the second part, we discuss cooperative attitude tracking problem in the presence of multiple time-varying communication delays and dynamically changing topologies, where a model-independent control torque 𝜏_𝑖 is proposed, 𝜏_𝑖=−𝐾_𝑖Δ𝑞_𝑖−𝐷_𝑖Δ𝜔_𝑖−^𝑛_𝑗=1𝑎^𝜎_𝑖𝑗[Δ𝑞_𝑖−Δ𝑞_𝑗(𝑡−𝑇_𝑖𝑗(𝑡))], (4.4) where Δ𝑞_𝑖 and Δ𝜔_𝑖 are defined after (4.2). Following the similar definition given in [25], the dynamically changing topology is defined as 𝜎∶[0,+∞)→𝜓_Γ, where the set Γ is a finite collection of undirected graphs with a common node set. Then 𝑎^𝜎_𝑖𝑗 denotes the (𝑖,𝑗)th entry of the adjacency matrix 𝒜_𝜎 associated with the communication topology 𝒢_𝜎. Before moving on, we assume that 𝜔_𝑑 and 𝜔̇_𝑑 are bounded and define 𝛾_𝐽𝑖=‖𝐽_𝑖‖, 𝛾_𝑑=sup_𝑡≥0‖𝜔_𝑑(𝑡)‖, and 𝛽₁=‖𝜔̇_𝑑‖+‖𝜔_𝑑‖² in this paper.

Remark 4.1. Compared with (4.3), (4.4) introduces absolute angular velocity damping, thus avoiding introducing the communication delays of relative angular velocity information between the follower spacecraft.

5. Cooperative Attitude Regulation with Multiple Time-Varying Communication Delays and Fixed Topology

In this section, we propose proper conditions to guarantee that cooperative attitude regulation is achieved by using (4.3) for (2.2). Before moving on, we need the following lemma.

Lemma 5.1. The matrix 𝑀=ℒ+diag(𝐾₁,…,𝐾_𝑛) is symmetric and positive definite if the undirected graph 𝒢 is connected and at least one 𝐾_𝑖>0, where ℒ is the Laplacian matrix of graph 𝒢.

Proof. See the proof of Lemma 4 [25].

Motivated by the works of [15, 16, 18], we provide the following theorem for closed-loop systems (2.2) with (4.3).

Theorem 5.2. Using (4.3) for (2.2), if 𝑀>0, 𝐷_𝑖>0, for all 𝑖, 𝜌_𝑖𝑗<𝑎_𝑖𝑗, 4𝜌_𝑖𝑗(1−𝑇̇_𝑖𝑗)(𝑎_𝑖𝑗−𝜌_𝑖𝑗)>𝑙²_𝑖𝑗𝑎²_𝑖𝑗, when 𝑎_𝑖𝑗≠0, and 𝑊>0, cooperative attitude regulation, that is, 𝑞_𝑖→𝑞_𝐼 and 𝜔_𝑖→0, is achieved, where 𝑊 is given by 𝑊=𝑐𝑀−𝑐²1𝒟−2𝑐²ℬ(𝑀−𝐾−𝑐ℬ−𝑐𝒞)2(𝑀−𝐾−𝑐ℬ−𝑐𝒞)21𝐷−ℬ−2𝑐ℱ, (5.1) and 𝑐, 𝜌_𝑖𝑗, 𝐷, 𝐾, 𝑀, ℬ, 𝒞, 𝒟, and ℱ are defined in the proof.

Proof. Consider the following Lyapunov function candidate 𝑉=^𝑛_𝑖=1(𝐾_𝑖+𝑐𝐷_𝑖)‖𝑞_𝑖−𝑞_𝐼‖²+12^𝑛_𝑖=1𝜔^𝑇_𝑖𝐽_𝑖𝜔_𝑖+^𝑛_𝑖=1𝑐𝑞^𝑇_𝑖𝐽_𝑖𝜔_𝑖+^𝑛𝑛_𝑖=1𝑗=1⁰_{−𝑇𝑖𝑗(𝑡)}𝜌_𝑖𝑗‖𝜔_𝑗(𝑡+𝑠)‖²+1𝑑𝑠2^𝑛𝑛_𝑖=1𝑗=1⁰_{−𝑇𝑖𝑗(𝑡)}⁰_𝜇𝑎_𝑖𝑗‖𝑞_𝑗̇(𝑡+𝑠)‖²𝑑𝑠𝑑𝜇, (5.2) where 𝑐 is a positive constant, 𝜌_𝑖𝑗=0 when 𝑎_𝑖𝑗=0, and 𝜌_𝑖𝑗 is a positive constant when 𝑎_𝑖𝑗≠0. It is easy to verify that 𝑉 is positive definite if 2(𝐾_𝑖+𝑐𝐷_𝑖)𝜆_min(𝐽_𝑖)>𝑐²𝜆²_max(𝐽_𝑖), for all 𝑖 [26]. This implies that the selection of sufficiently small 𝑐 guarantees that 𝑉 is positive definite. Taking the derivative of 𝑉 gives 𝑉̇=^𝑛_𝑖=1(𝐾_𝑖+𝑐𝐷_𝑖)𝑞^𝑇_𝑖𝜔_𝑖−^𝑛_𝑖=1(𝜔_𝑖+𝑐𝑞_𝑖)^𝑇[𝐾_𝑖𝑞_𝑖+^𝑛_𝑗=1𝑎_𝑖𝑗(𝑞_𝑖(𝑡)−𝑞_𝑗−(𝑡))]_𝑛𝑖=1(𝜔_𝑖+𝑐𝑞_𝑖)^𝑇{𝐷_𝑖𝜔_𝑖+^𝑛_𝑗=1𝑎_𝑖𝑗[𝜔_𝑖−𝑙_𝑖𝑗𝐴_𝑖𝑗(𝑡−𝑇_𝑖𝑗(𝑡))𝜔_𝑗(𝑡−𝑇_𝑖𝑗(𝑡))]}+^𝑛_𝑖=1𝑐𝑞^𝑇_𝑖(−𝜔_𝑖×𝐽_𝑖𝜔_𝑖)+12^𝑛_𝑖=1𝑐[(𝑞_𝑖𝐼₃+𝑆(𝑞_𝑖))𝜔_𝑖]^𝑇𝐽_𝑖𝜔_𝑖−^𝑛_𝑖=1(𝜔_𝑖+𝑐𝑞_𝑖)^𝑇𝑛_𝑗=1⁰_{−𝑇𝑖𝑗(𝑡)}𝑎_𝑖𝑗𝑞_𝑗̇+1(𝑡+𝜇)𝑑𝜇2^𝑛𝑛_𝑖=1𝑗=1⁰_{−𝑇𝑖𝑗(𝑡)}𝑎_𝑖𝑗[𝑞^𝑇_𝑗̇(𝑡)𝑞_𝑗̇(𝑡)−𝑞_𝑗̇(𝑡+𝜇)^𝑇𝑞_𝑗̇(𝑡+𝜇)]𝑑𝜇+𝜌^𝑛_𝑖𝑗^𝑛_𝑖=1𝑗=1𝜔_𝑗(𝑡)^𝑇𝜔_𝑗(𝑡)−𝜌^𝑛_𝑖𝑗^𝑛_𝑖=1𝑗=1(1−𝑇̇_𝑖𝑗)𝜔_𝑗(𝑡−𝑇_𝑖𝑗)^𝑇𝜔_𝑗(𝑡−𝑇_𝑖𝑗), (5.3) where we have used the fact that 𝜔^𝑇_𝑖(𝜔_𝑖×𝐽_𝑖𝜔_𝑖)=0 and Leibniz-Newton formula 𝑞_𝑗(𝑡−𝑇_𝑖𝑗(𝑡))=𝑞_𝑗(𝑡)−∫⁰_{−𝑇𝑖𝑗}𝑞_𝑗̇(𝑡+𝜇)𝑑𝜇 [23]. It thus follows that 𝑉̇≤12𝑐^𝑛_𝑖=1𝛾_𝐽𝑖‖𝜔_𝑖‖²−𝑐𝑞^𝑇(𝑀⊗𝐼₃)𝑞−𝜔^𝑇(𝑀⊗𝐼₃)𝑞+𝑞^𝑇(𝐾+𝑐𝐷)⊗𝐼₃𝜔+12^𝑛𝑛_𝑖=1𝑗=1𝑇_𝑖𝑗𝑎_𝑖𝑗‖𝜔_𝑖+𝑐𝑞_𝑖‖²−12^𝑛𝑛_𝑖=1𝑗=1⁰_{−𝑇𝑖𝑗}𝑎_𝑖𝑗‖𝜔_𝑖+𝑐𝑞_𝑖+𝑞_𝑗̇(𝑡+𝜇)‖²+1𝑑𝜇2^𝑛𝑛_𝑖=1𝑗=1𝑇_𝑖𝑗𝑎_𝑖𝑗𝜔^𝑇_𝑗(𝑞_𝑗𝐼₃+𝑆(𝑞_𝑗))^𝑇(𝑞_𝑗𝐼₃+𝑆(𝑞_𝑗))𝜔_𝑗−^𝑛𝑛_𝑖=1𝑗=1(𝑎_𝑖𝑗−𝜌_𝑖𝑗)‖(𝜔_𝑖+𝑐𝑞_𝑖𝑙)−^𝑖𝑗𝑎_𝑖𝑗2(𝑎_𝑖𝑗−𝜌_𝑖𝑗)𝐴_𝑖𝑗(𝑡−𝑇_𝑖𝑗)𝜔_𝑗(𝑡−𝑇_𝑖𝑗)‖²+𝜌^𝑛_𝑖𝑗^𝑛_𝑖=1𝑗=1𝜔^𝑇_𝑗𝜔_𝑗+^𝑛𝑛_𝑖=1𝑗=1(𝑎_𝑖𝑗−𝜌_𝑖𝑗)‖𝜔_𝑖+𝑐𝑞_𝑖‖²−^𝑛𝑛_𝑖=1𝑗=1𝑎_𝑖𝑗(𝜔_𝑖+𝑐𝑞_𝑖)^𝑇𝜔_𝑖−^𝑛𝑛_𝑖=1𝑗=1[𝜌_𝑖𝑗(1−𝑇̇_𝑖𝑗𝑙)−^2𝑖𝑗𝑎²_𝑖𝑗4(𝑎_𝑖𝑗−𝜌_𝑖𝑗)]‖𝜔_𝑖(𝑡−𝑇_𝑖𝑗)‖²−^𝑛_𝑖=1𝐷_𝑖‖𝜔_𝑖‖²−^𝑛_𝑖=1𝑐𝐷_𝑖𝑞^𝑇_𝑖𝜔_𝑖, (5.4) where 𝑞=[𝑞^𝑇₁,𝑞^𝑇₂,…,𝑞^𝑇_𝑛]^𝑇,𝜔=[𝜔^𝑇₁,𝜔^𝑇₂,…,𝜔^𝑇_𝑛]^𝑇,𝐾=diag(𝐾₁,…,𝐾_𝑛),𝑀=ℒ+𝐾, and 𝐷=diag(𝐷₁,…,𝐷_𝑛). Based on the conditions that 𝜌_𝑖𝑗<𝑎_𝑖𝑗 and 4𝜌_𝑖𝑗(1−𝑇̇_𝑖𝑗)(𝑎_𝑖𝑗−𝜌_𝑖𝑗)>𝑙²_𝑖𝑗𝑎²_𝑖𝑗 when 𝑎_𝑖𝑗≠0, we have that 𝑉̇≤12𝑐^𝑛_𝑖=1𝛾_𝐽𝑖‖𝜔_𝑖‖²−𝑐𝑞^𝑇(𝑀⊗𝐼₃)𝑞−𝜔^𝑇(𝑀⊗𝐼₃)𝑞+𝑞^𝑇(𝐾⊗𝐼₃)𝜔−𝜔^𝑇(𝐷⊗𝐼₃1)𝜔+2^𝑛𝑛_𝑖=1𝑗=1𝑇_𝑖𝑗𝑎_𝑖𝑗‖𝜔_𝑖+𝑐𝑞_𝑖‖²+12^𝑛𝑛_𝑖=1𝑗=1𝑇_𝑖𝑗𝑎_𝑖𝑗𝜔^𝑇_𝑖𝜔_𝑖+𝑐^𝑛𝑛_𝑖=1𝑗=1(𝑎_𝑖𝑗−2𝜌_𝑖𝑗)𝜔^𝑇_𝑖𝑞_𝑖+𝑐^2𝑛𝑛_𝑖=1𝑗=1(𝑎_𝑖𝑗−𝜌_𝑖𝑗)𝑞^𝑇_𝑖𝑞_𝑖=−𝑞^𝑇(𝑐𝑀−𝑐²1𝒟−2𝑐²ℬ)⊗𝐼₃𝑞+𝜔^𝑇(−𝑀+𝐾+𝑐ℬ+𝑐𝒞)⊗𝐼₃𝑞−𝜔^𝑇1(𝐷−ℬ−2𝑐ℱ)⊗𝐼₃𝜔, (5.5) where we have used the fact that ∑^𝑛_𝑖=1∑^𝑛_𝑗=1𝑎_𝑖𝑗𝜔^𝑇_𝑗𝜔_𝑗=∑^𝑛_𝑖=1∑^𝑛_𝑗=1𝑎_𝑖𝑗𝜔^𝑇_𝑖𝜔_𝑖 (graph 𝒢 is undirected). Here we also define ℬ=[𝑏_𝑖𝑗],𝒞=[𝑐_𝑖𝑗],𝒟=[𝑑_𝑖𝑗], and ℱ as 𝑛×𝑛 matrices, where 𝑏_𝑖𝑗=𝑇_𝑖𝑗𝑎_𝑖𝑗,𝑐_𝑖𝑗=𝑎_𝑖𝑗−2𝜌_𝑖𝑗,𝑑_𝑖𝑗=𝑎_𝑖𝑗−𝜌_𝑖𝑗, and?ℱ=diag(𝛾_𝐽1,…,𝛾_𝐽𝑛). Based on the conditions that 𝑀 is positive definite and 𝐷 is positive definite (𝐷_𝑖>0, for all 𝑖), for the sufficient small 𝑇_𝑖𝑗 and 𝑐, it is easy to verify that there always exist 𝑀 and 𝐷 to guarantee 𝑊 is positive definite. Then, Lemma 3.3 implies the stability of the closed-loop systems (2.2) with (4.3) from the condition that 𝑊>0. Thus, cooperative attitude regulation, that is, 𝑞_𝑖→𝑞_𝐼, and 𝜔_𝑖→0 is achieved under the conditions provided in Theorem 5.2.

Remark 5.3. It follows that 𝑀 is positive definite from Lemma 5.1 if the undirected graph 𝒢 is connected and at least one 𝐾_𝑖>0. This implies conditions that the undirected graph 𝒢 is connected and at least one 𝐾_𝑖>0 can be used to replace condition that 𝑀>0.

Remark 5.4. Note that the parameters 𝜌_𝑖𝑗 and 𝑐 in the proposed conditions in Theorem 5.2 are independent of control parameters in control torque (4.3).

Remark 5.5. The cooperative attitude regulation problem in the presence of communication delays was also discussed in the work of [16]. In contrast to [16], we do not assume that relative attitude information and relative angular velocity information between different follower spacecraft could be described in a united variable. This may increase the flexibility of the design.

6. Cooperative Attitude Tracking with Multiple Time-Varying Communication Delays and Dynamically Changing Topologies

In this section, the conditions to guarantee cooperative attitude tracking in the presence of multiple time-varying communication delays and dynamically changing topologies are obtained. We first transform the closed-loop systems (2.2) to the error kinematic and dynamic as Δ̇𝑞_𝑖=12𝐸_𝑖(Δ𝑞_𝑖)Δ𝜔_𝑖,𝐽_𝑖Δ̇𝜔_𝑖=−𝜔_𝑖×(𝐽_𝑖𝜔_𝑖)−𝐽_𝑖𝐴(Δ𝑞_𝑖)𝜔̇_𝑑+𝐽_𝑖Δ𝜔_𝑖×𝜔^𝑏𝑖_𝑑+𝜏_𝑖, (6.1) where Δ𝑞_𝑖, Δ𝜔_𝑖, and 𝜔^𝑏𝑖_𝑑 are defined after (4.2). Also define 𝜏=[𝜏^𝑇₁,…,𝜏^𝑇_𝑛]^𝑇. We can then transform (4.4) to the matrix expression 𝜏=−(𝑀_𝜎⊗𝐼₃)Δ𝑞−(𝐷⊗𝐼₃)Δ𝜔−^𝑟_𝑘=1(𝒜^𝜎_𝑘⊗𝐼₃)⁰_{−𝑇𝑘(𝑡)}Δ𝑞(𝑡+𝜇)𝑑𝜇, (6.2)

where Δ𝑞=[Δ𝑞^𝑇₁,Δ𝑞^𝑇₂,…,Δ𝑞^𝑇_𝑛]^𝑇,???𝜔=[Δ𝜔^𝑇₁,Δ𝜔^𝑇₂,…,Δ𝜔^𝑇_𝑛]^𝑇,??K =diag(𝐾₁,…,𝐾_𝑛),??𝑀_𝜎=𝐿_𝜎+𝐾,𝐿_𝜎 is the Laplacian matrix of 𝒢_𝜎,𝐷=diag(𝐷₁,…,𝐷_𝑛),𝑇_𝑘(𝑡)∈{𝑇_𝑖𝑗(𝑡)∶𝑖,𝑗=1,…,𝑛} for 𝑘=1,…,𝑟, and 𝑟=𝑛(𝑛−1)/2,𝒜^𝜎_𝑘=[𝑎_𝑖𝑗𝑘] is a corresponding 𝑛×𝑛 matrix, where 𝑎_𝑖𝑗𝑘=𝑎^𝜎_𝑖𝑗,𝑇_𝑘=𝑇_𝑖𝑗0,𝑇_𝑘≠𝑇_𝑖𝑗. (6.3) Before moving on, we need the following lemma.

Lemma 6.1 ([27]). For any 𝑎,𝑏∈ℝ^𝑛 and any symmetric positive definite matrix Φ∈ℝ^𝑛×n, one has 2𝑎^𝑇𝑏≤𝑎^𝑇Φ⁻¹𝑎+𝑏^𝑇Φ𝑏.

Theorem 6.2. Using (4.4) for (2.2), if 𝑀_𝜎>0 in each time interval, 𝐷_𝑖>0, for all 𝑖, and 𝑄=[^𝑄11𝑄12_𝑄12𝑄22]>0, where 𝑄₁₁=𝑐𝑀_𝜎−(1/2)𝑐²∑^𝑟_𝑘=1𝑇_𝑘𝒜^𝜎_𝑘(𝒜^𝜎_𝑘)^𝑇,𝑄₁₂=[−𝐾+𝑀_𝜎−𝑐∑^𝑟_𝑘=1𝑇_𝑘𝒜^𝜎_𝑘(𝒜^𝜎_𝑘)^𝑇]/2, 𝑄₂₂=𝐷−ℱ_∈−(1/2)∑^𝑟_𝑘=1𝑇_𝑘𝒜^𝜎_𝑘(𝒜^𝜎_𝑘)^𝑇,?and?ℱ_∈=diag((3/2)𝑐𝛾_𝐽1+(1/2)∑^𝑟_𝑘=1𝑇_𝑘𝑞(𝜆_max(𝑃2)/𝜆_min(𝑃1)),…,(3/2)𝑐𝛾_𝐽𝑛+(1/2)∑^𝑟_𝑘=1𝑇_𝑘𝑞(𝜆_max(𝑃2)/𝜆_min(𝑃1))), the error state 𝑥 of the closed-loop system is uniformly ultimately bounded, where 𝑥=[Δ𝑞^𝑇,Δ𝜔^𝑇]^𝑇. In particular, the ultimate bound of 𝑥 is 𝜆_max(𝑃2)𝑏/𝜆_min(𝑃1)𝜃𝜆_min(𝑄) (𝑐, 𝑞, 𝑃1, 𝑃2, 𝑏, 𝜃 will be defined in the proof).

Proof. Consider the following Lyapunov function candidate 𝑉=^𝑛_𝑖=1(𝐾_𝑖+𝑐𝐷_𝑖)‖Δ𝑞_𝑖−𝑞_𝐼‖²+12^𝑛_𝑖=1Δ𝜔^𝑇_𝑖𝐽_𝑖Δ𝜔_𝑖+^𝑛_𝑖=1𝑐Δ𝑞^𝑇_𝑖𝐽_𝑖Δ𝜔_𝑖, (6.4) where 𝑐 is a positive constant. By using the fact that ‖Δ𝑞_𝑖−𝑞_𝐼‖²≤2‖Δ𝑞_𝑖‖², we know that 𝑥^𝑇𝑃₁𝑥≤𝑉≤𝑥^𝑇𝑃₂𝑥, where 𝑃₁=(𝐾+𝑐𝐷)⊗𝐼₃12𝑐𝐽⊗𝐼₃12𝑐𝐽⊗𝐼₃12𝐽⊗𝐼₃,𝑃₂=2(𝐾+𝑐𝐷)⊗𝐼₃12𝑐𝐽⊗𝐼₃12𝑐𝐽⊗𝐼₃12𝐽⊗𝐼₃, (6.5) and 𝐽=diag(𝐽₁,…,𝐽_𝑛). We also know that 𝑉 is positive definite if 𝑐 is chosen properly to ensure 2(𝐾_𝑖+𝑐𝐷_𝑖)𝜆_min(𝐽_𝑖)>𝑐²𝜆²_max(𝐽_𝑖), for all 𝑖. Taking the derivative of 𝑉 gives 𝑉̇≤^𝑛_𝑖=1(𝐾_𝑖+𝑐𝐷_𝑖)Δ𝑞^𝑇_𝑖Δ𝜔_𝑖−^𝑛_𝑖=1(Δ𝜔_𝑖+𝑐Δ𝑞_𝑖)^𝑇[𝐾_𝑖Δ𝑞_𝑖+^𝑛_𝑗=1𝑎^𝜎_𝑖𝑗(Δ𝑞_𝑖(𝑡)−Δ𝑞_𝑗(𝑡))+𝐷_𝑖Δ𝜔_𝑖]+^𝑛_𝑖=1{𝛾_𝐽𝑖𝛽₁‖Δ𝜔_𝑖‖+𝑐𝛾_𝐽𝑖𝛽₁‖Δ𝑞_𝑖‖+𝑐𝛾_𝐽𝑖‖Δ𝜔_𝑖‖²+3𝑐𝛾_𝐽𝑖𝛾_𝐽𝑑‖Δ𝑞_𝑖‖‖Δ𝜔_𝑖‖+12𝑐[(Δ𝑞_𝑖𝐼₃+𝑆(Δ𝑞_𝑖))Δ𝜔_𝑖]^𝑇𝐽_𝑖Δ𝜔_𝑖}+(Δ𝜔+𝑐Δ𝑞)^𝑇𝑟_𝑘=1(𝒜^𝜎_𝑘⊗𝐼₃)⁰_{−𝑇𝑘(𝑡)}Δ𝑞(𝑡+𝜇)𝑑𝜇, (6.6) where we have used (6.1), and the facts that Δ𝜔^𝑇_𝑖(−𝜔_𝑖×(𝐽_𝑖𝜔_𝑖)−𝐽_𝑖𝐴(Δ𝑞_𝑖)𝜔̇_𝑑+𝐽_𝑖Δ𝜔_𝑖×𝜔^𝑏𝑖_𝑑)=Δ𝜔^𝑇_𝑖(−𝐽_𝑖𝐴(Δ𝑞_𝑖)𝜔̇_𝑑−𝜔_𝑑^𝑏𝑖×𝐽_𝑖𝜔_𝑑^𝑏𝑖)≤𝛾_𝐽𝑖𝛽₁‖Δ𝜔_𝑖‖ and Δ𝑞^𝑇_𝑖(−𝜔_𝑖×(𝐽_𝑖𝜔_𝑖)−𝐽_𝑖𝐴(Δ𝑞_𝑖)𝜔̇_𝑑+𝐽_𝑖Δ𝜔_𝑖×𝜔^𝑏𝑖_𝑑)=Δ𝑞^𝑇_𝑖(−𝐽_𝑖𝐴(Δ𝑞_𝑖)𝜔̇_𝑑−𝜔_𝑑^𝑏𝑖×𝐽_𝑖𝜔_𝑑^𝑏𝑖+𝐽_𝑖Δ𝜔_𝑖×𝜔^𝑏𝑖_𝑑−𝜔^𝑏𝑖_𝑑×𝐽_𝑖Δ𝜔_𝑖−Δ𝜔_𝑖×𝐽_𝑖Δ𝜔_𝑖−Δ𝜔_𝑖×𝐽_𝑖𝜔^𝑏𝑖_𝑑)≤𝛾_𝐽𝑖𝛽₁‖Δ𝑞_𝑖‖+𝛾_𝐽𝑖‖Δ𝜔_𝑖‖²+3𝛾_𝐽𝑖𝛾_𝑑‖Δ𝑞_𝑖‖‖Δ𝜔_𝑖‖. We then have that 𝑉̇≤Δ𝑞^𝑇(𝐾+𝑐𝐷)⊗𝐼₃Δ𝜔−𝑐Δ𝑞^𝑇(𝑀_𝜎⊗𝐼₃)Δ𝑞−Δ𝑞^𝑇(𝑀_𝜎⊗𝐼₃)Δ𝜔−Δ𝜔^𝑇(𝐷⊗𝐼₃)Δ𝜔−𝑐Δ𝑞^𝑇(𝐷⊗𝐼₃+)Δ𝜔_𝑛𝑖=1𝛽₁𝛾_𝐽𝑖‖Δ𝜔_𝑖‖+^𝑛_𝑖=1𝑐𝛽₁𝛾_𝐽𝑖‖Δ𝑞_𝑖3‖+2^𝑛_𝑖=1𝑐𝛾_𝐽𝑖‖Δ𝜔_𝑖‖²+^𝑛_𝑖=13𝑐𝛾_𝐽𝑑𝛾_𝐽𝑖‖Δ𝜔_𝑖‖+12^𝑟_𝑘=1⁰_{−𝑇𝑘(𝑡)}(Δ𝜔+𝑐Δ𝑞)^𝑇[𝒜^𝜎_𝑘(𝒜^𝜎_𝑘)^𝑇]⊗𝐼₃+1(Δ𝜔+𝑐Δ𝑞)2^𝑟_𝑘=1⁰_{−𝑇𝑘(𝑡)}Δ𝑞(𝑡+𝜇)^𝑇Δ𝑞(𝑡+𝜇)𝑑𝜇, (6.7) where we have used the fact that ‖Δ𝑞_𝑖‖≤1, for all 𝑖 and Lemma 6.1 to derive the inequality. Take 𝜙(𝑠)=𝑞𝑠 for some constant 𝑞>1. In the case of 𝑉(𝑥(𝑡+𝜃))<𝑞𝑉(𝑥(𝑡)),−sup_𝑘{𝑇_𝑘}≤𝜃≤0, (6.8) we know that 𝜆_min(𝑃₁)‖Δ𝜔‖²(𝑡+𝜃)<𝑞𝜆_max(𝑃₂)(‖Δ𝜔‖²+‖Δ𝑞‖²). (Note that this is a property inherited from Lyapunov-Razumikhin uniformly ultimately bounded theorem.) Thus, we have that 𝑉̇≤Δ𝑞^𝑇(𝐾−𝑀_𝜎)⊗𝐼₃Δ𝜔−𝑐Δ𝑞^𝑇(𝑀_𝜎⊗𝐼₃)Δ𝑞−Δ𝜔^𝑇(𝐷⊗𝐼₃+)Δ𝜔_𝑛𝑖=1𝛽₁𝛾_𝐽𝑖‖Δ𝜔_𝑖‖+^𝑛_𝑖=1𝑐𝛽₁𝛾_𝐽𝑖‖Δ𝑞_𝑖3‖+2^𝑛_𝑖=1𝑐𝛾_𝐽𝑖‖Δ𝜔_𝑖‖²+^𝑛_𝑖=13𝑐𝛾_𝐽𝑑𝛾_𝐽𝑖‖Δ𝜔_𝑖‖+12^𝑟_𝑘=1𝑇_𝑘(Δ𝜔+𝑐Δ𝑞)^𝑇[𝒜^𝜎_𝑘(𝒜^𝜎_𝑘)^𝑇]⊗𝐼₃+1(Δ𝜔+𝑐Δ𝑞)2^𝑟_𝑘=1𝑇_𝑘𝑞𝜆_max(𝑃₂)𝜆_min(𝑃₁)(‖Δ𝜔‖²+‖Δ𝑞‖²)=−𝑥^𝑇(𝑄⊗𝐼₃)𝑥+𝑤^𝑇𝑦, (6.9) where 𝑦=[‖Δ𝑞₁‖,…,‖Δ𝑞_𝑛‖,‖Δ𝜔₁‖,…,‖Δ𝜔_𝑛‖]^𝑇, 𝑄 is defined in Theorem 6.2, and 𝑤=[𝑐𝛽₁𝛾_𝐽1+(1/2)∑^𝑟_𝑘=1𝑇_𝑘𝑞(𝜆_max(𝑃2)/𝜆_min(𝑃1)),…,𝑐𝛽₁𝛾_𝐽𝑛+(1/2)∑^𝑟_𝑘=1𝑇_𝑘𝑞(𝜆_max(𝑃2)/𝜆_min(𝑃1)),(3𝑐𝛾_𝐽𝑑+𝛽₁)𝛾_𝐽1,…,(3𝑐𝛾_𝐽𝑑+𝛽₁)𝛾_𝐽𝑛]^𝑇. Based on the conditions that 𝑀_𝜎>0 in each time interval and 𝐷>0 (𝐷_𝑖>0, for all 𝑖), for the sufficient small 𝑇_𝑖𝑗 and 𝑐, it is easy to verify that there always exist 𝑀_𝜎 and 𝐷 to guarantee 𝑄 is positive definite. Therefore, we have that 𝑉̇≤−𝜆_min(𝑄)‖𝑥‖²+𝑏‖𝑦‖=−𝜆_min(𝑄)‖𝑥‖²+𝑏‖𝑥‖, (6.10) where 𝑏=∑^𝑛_𝑖=1(𝑐𝛽₁𝛾_𝐽𝑖+(1/2)∑^𝑟_𝑘=1𝑇_𝑘𝑞(𝜆_max(𝑃2)/𝜆_min(𝑃1)))²+∑^𝑛_𝑖=1𝛾²_𝐽𝑖(3𝑐𝛾_𝐽𝑑+𝛽₁)². Thus, for 0<𝜃<1, if ‖𝑥‖≥𝑏/𝜃𝜆_min(𝑄), we have that 𝑉̇=−(1−𝜃)𝜆_min(𝑄)‖𝑥‖²−𝜃𝜆_min(𝑄)‖𝑥‖²+𝑏‖𝑥‖≤−(1−𝜃)𝜆_min(𝑄)‖𝑥‖². (6.11) Therefore, the uniformly ultimate boundedness of 𝑥 follows from Lemma 3.4. In addition, the ultimate bound is 𝜆_max(𝑃2)𝑏/𝜆_min(𝑃1)𝜃𝜆_min(𝑄) by following a similar analysis to that in [28].

Remark 6.3. Note that both the case of cooperative regulation and that of cooperative tracking discussed in this paper introduce model-independent control laws. The final errors converge to zero for the cooperative regulation case while the final errors are bounded for the cooperative tracking case. The authors in [26] showed that the final errors will be decreased effectively if the control parameters are chosen large enough for the tracking case. Similar conclusion also holds for our control law.

Remark 6.4. Note that the stability or uniformly ultimate boundedness conditions given in Theorems 5.2 and 6.2 are just sufficient conditions, not the necessary conditions.

Remark 6.5. Note that the bounds of the communication delays to guarantee the stability or uniformly ultimate boundedness of the closed-loop systems are implied in the conditions that 𝑊>0 for the regulation case and 𝑄>0 for the tracking case. Also note that the bounds of the communication delays are related to the control parameters and their relationship is indirect.

Remark 6.6. Reference [17] also discussed the cooperative attitude tracking problem in the presence of communication delays and dynamically changing topologies, where stability result was obtained by using Lyapunov-Krasovskii Theorem. In contrast, here we use Lyapunov-Razumikhin Theorem to derive uniformly ultimate boundedness results of the closed-loop systems.

7. Simulation

In this section, control laws (4.3) and (4.4) are used in simulation to achieve cooperative attitude regulation and tracking among three follower spacecraft. The spacecraft specifications are given in Table 1.

Table 1 

Spacecraft specifications.

For control law (4.3), we choose the control parameters as 𝐾_𝑖=2,𝐷_𝑖=5, and??𝑙_𝑖𝑗=0.3, for all 𝑖,𝑗. 𝑞_𝑖(0) and 𝜔_𝑖(0),𝑖=1,2,3 are generated randomly. For control law (4.4), we choose the control parameters as 𝐾_𝑖=2.2?and?𝐷_𝑖=11. 𝑞_𝑖(0) and 𝜔_𝑖(0),𝑖=1,2,3 are generated randomly. Suppose that the reference attitude 𝑞_𝑑(𝑡), reference angular velocity 𝜔_𝑑(𝑡)=2𝐸⁻¹(𝑞_𝑑)𝑞̇_𝑑, reference torque 𝜏_𝑑 and reference inertia satisfy (2.2) with 𝑞_𝑑(0)=[0.3921,0.5502,0.5287,0.5139]^𝑇 (the corresponding Euler Angles are 𝜓=80deg,𝜙=60deg, and??𝜃=40deg), 𝜔_𝑑(0)=[0.021,−0.012,0.014]^𝑇rad/s, 𝜏_𝑑=[0,0,0]^𝑇Nm, and 𝐽_𝑑=[22.0,0.2,−0.1;0.2,23.1,0.3;−0.1,0.3,21.3]?kg?m². For communication topology, we assume that 𝑎_𝑖𝑗=1 if (𝑗,𝑖)∈ℰ, and 𝑎_𝑖𝑗=0 otherwise. For control law (4.3), the communication topology for follower spacecraft is fixed and determined by 𝒢₁ in Figure 1(a). For control law (4.4), the communication topology for follower spacecraft is switching between 𝒢₁ in Figure 1(a) and 𝒢₂ in Figure 1(b) every one second. The time-varying communication delays are chosen as 𝑇₁₂=𝑇₂₁=0.4|sin(0.2𝑡)|,𝑇₁₃=𝑇₃₁=0.3|cos(0.4𝑡)|, and𝑇₂₃=𝑇₃₂=0.5|sin(0.3𝑡)|.

(a) Communication topology 𝒢1

(b) Communication topology 𝒢2

Figure 1 

Communication topologies.

Figures 2, 3, and 4 show, respectively, the attitudes, angular velocities, and control torques of follower spacecraft 1, 2, and 3 using (4.3) for (2.2). We can see from the figures that if the control parameters are selected properly, all spacecraft can regulate their attitude and angular velocity to zero even if there exists multiple time-varying communication delays.

(a)

(b)

(c)

Figure 2 

Rigid body attitudes with control law (4.3).

(a)

(b)

(c)

Figure 3 

Rigid body angular velocities with control law (4.3).

(a)

(b)

(c)

Figure 4 

Rigid body control torques with control law (4.3).

Figures 5, 6 and 7 show, respectively, the attitudes, angular velocities and control torques of follower spacecraft 1, 2 and 3 using (4.4) for (2.2). We can see from the figures that if the control parameters are selected properly, all spacecraft can track time-varyingly desired attitude and angular velocity even if there exists multiple time-varying communication delays and dynamically changing topologies.

(a)

(b)

(c)

Figure 5 

Rigid body attitudes with control law (4.4).

(a)

(b)

(c)

Figure 6 

Rigid body angular velocities with control law (4.4).

(a)

(b)

(c)

Figure 7 

Rigid body control torques with control law (4.4).

8. Conclusions

In this paper, the cooperative attitude regulation problem in the presence of multiple time-varying communication delays and the cooperative attitude tracking problem in the presence of multiple time-varying communication delays and dynamically changing topologies are discussed. Lyapunov-Krasovskii Theorem and Lyapunov-Razumikhin Theorem are used to derive the conditions to guarantee the stability or uniformly ultimate boundedness of the closed-loop system. Simulation results validate the effectiveness of the theoretical results. Future work will include proposing a more practical design by addressing the sign ambiguity problem for the unit quaternion description and discussing the cooperative attitude regulation problem in the presence of both communication delays and dynamically changing topologies.