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

Authors: Ziyang Meng; Zheng You; Guanhua Li; Chunshi Fan
Journal: Mathematical Problems in Engineering
 (2010)

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## 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.


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## Body

## 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 Sections2 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. λmin(A) and λmax(A) are, respectively, the minimal eigenvalue and the maximum eigenvalue of the matrix A. ⊗ denotes the Kronecker product.Consistent with [20], we denote ℂn,τ as the Banach space of continuous vector functions mapping the interval [-τ,0] into ℝn with the topology of uniform convergence. ℂn,τv={ϕ∈ℂn,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∈ℂn,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 byqi=[q1,q2,q3,q4]iT=[q̂iT,q¯i]T,i=1,…,n. Here q̂i=eisin(ϕi/2), q¯i=cos(ϕi/2), where ei and ϕi are the principle axis and the principle angle of the attitude of the ith spacecraft and qiTqi=1 [21]. The product of two unit quaternions pi and qi is defined by (2.1)qipi=[q¯ip̂i+p¯iq̂i+q̂i×p̂iq¯ip¯i-q̂iTp̂i].The conjugate of the unit quaternionqi is defined by qi-1=[-q̂iT,q¯i]T. Attitude kinematics and dynamics of each spacecraft using the unit quaternion are given by [21] (2.2)q̇i=12Ei(qi)ωi,i=1,…,n,Jiω̇i=-ωi×(Jiωi)+τi,i=1,…,n,
where qi∈ℝ4 denotes the rotation from the body frame of the ith spacecraft to the inertial frame, ωi is the angular velocity of the ith spacecraft with respect to the inertial frame expressed in the body frame of the ith spacecraft, and Ei(qi) is given by (2.3)Ei(qi)=[q¯iI3+S(q̂i)-q̂iT],where I3 is the 3×3 identity matrix, S(·) denotes a 3×3 skew-symmetric matrix, and Ji∈ℝ3×3 and τi∈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 graph𝒢 consists of a pair (V,ℰ), where V={v1,…,vn} is a finite nonempty set of nodes and ℰ⊆V×V is a set of unordered pairs of nodes. An edge (vi,vj) denotes that nodes vi and vj can obtain information from each other. In such case, nodes vi and vj are neighbors of each other. All the neighbors of node vi are denoted as Ni:={vj∣(vj,vi)∈ℰ}, where we assume that vi∉Ni.An undirected path is a sequence of edges in a undirected graph of the form(vi1,vi2),(vi2,vi3),…. 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 matrix𝒜=[aij]∈ℝn×n associated with the undirected graph 𝒢 is defined such that aij is a positive value if (vj,vi)∈ℰ, and aij=0 otherwise. We assume that aij=aji, for all  i≠j, since (vj,vi)∈ℰ implies (vi,vj)∈ℰ in the undirected graph. Also, the Laplacian matrix ℒ=[lij]∈ℝn×n associated with 𝒜 is defined as
(2.4)lij={∑j∈Niaij,i=j,-aij,i≠j.


## 2.1. Notations

ℝ and ℂ are, respectively, the set of real numbers and the set of complex numbers. λmin(A) and λmax(A) are, respectively, the minimal eigenvalue and the maximum eigenvalue of the matrix A. ⊗ denotes the Kronecker product.Consistent with [20], we denote ℂn,τ as the Banach space of continuous vector functions mapping the interval [-τ,0] into ℝn with the topology of uniform convergence. ℂn,τv={ϕ∈ℂn,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∈ℂn,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 byqi=[q1,q2,q3,q4]iT=[q̂iT,q¯i]T,i=1,…,n. Here q̂i=eisin(ϕi/2), q¯i=cos(ϕi/2), where ei and ϕi are the principle axis and the principle angle of the attitude of the ith spacecraft and qiTqi=1 [21]. The product of two unit quaternions pi and qi is defined by (2.1)qipi=[q¯ip̂i+p¯iq̂i+q̂i×p̂iq¯ip¯i-q̂iTp̂i].The conjugate of the unit quaternionqi is defined by qi-1=[-q̂iT,q¯i]T. Attitude kinematics and dynamics of each spacecraft using the unit quaternion are given by [21] (2.2)q̇i=12Ei(qi)ωi,i=1,…,n,Jiω̇i=-ωi×(Jiωi)+τi,i=1,…,n,
where qi∈ℝ4 denotes the rotation from the body frame of the ith spacecraft to the inertial frame, ωi is the angular velocity of the ith spacecraft with respect to the inertial frame expressed in the body frame of the ith spacecraft, and Ei(qi) is given by (2.3)Ei(qi)=[q¯iI3+S(q̂i)-q̂iT],where I3 is the 3×3 identity matrix, S(·) denotes a 3×3 skew-symmetric matrix, and Ji∈ℝ3×3 and τi∈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 graph𝒢 consists of a pair (V,ℰ), where V={v1,…,vn} is a finite nonempty set of nodes and ℰ⊆V×V is a set of unordered pairs of nodes. An edge (vi,vj) denotes that nodes vi and vj can obtain information from each other. In such case, nodes vi and vj are neighbors of each other. All the neighbors of node vi are denoted as Ni:={vj∣(vj,vi)∈ℰ}, where we assume that vi∉Ni.An undirected path is a sequence of edges in a undirected graph of the form(vi1,vi2),(vi2,vi3),…. 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 matrix𝒜=[aij]∈ℝn×n associated with the undirected graph 𝒢 is defined such that aij is a positive value if (vj,vi)∈ℰ, and aij=0 otherwise. We assume that aij=aji, for all  i≠j, since (vj,vi)∈ℰ implies (vi,vj)∈ℰ in the undirected graph. Also, the Laplacian matrix ℒ=[lij]∈ℝn×n associated with 𝒜 is defined as
(2.4)lij={∑j∈Niaij,i=j,-aij,i≠j.


## 3. Definitions and Lemmas

Supposef:ℝ×ℂn,τv↦ℝn is continuous and consider retarded functional differential equation (RFDE) (3.1)ẋ(t)=f(t,xt).
Let ϕ=xt be defined as xt(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 solutionsx(t0,ϕ) of the RFDE (3.1) are uniformly asymptotically stable if(i)
for everyκ>0 and for every t0≥0 there exists a δ=δ(κ) independent of t0 such that for any ϕ∈ℂn,τδ the solutions x(t0,ϕ) of the RFDE (3.1) satisfies xt(t0,ϕ)∈ℂn,τκ for all t≥t0,(ii)
for everyη>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∈ℂn,τ, ∥ϕ∥c<v0 implies that ∥xt(t0,ϕ)∥c<η, for all t≥t0+T(η).Definition 3.2 (see [23]).
The solutionsx(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×ℂn,τ↦ℝn takes ℝ× (bounded sets of ℂn,τ) into bounded sets of ℝn, 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×ℂn,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×ℂn,τ↦ℝn takes ℝ× (bounded sets of ℂn,τ) into bounded sets of ℝn and u,v,w:ℝ+↦ℝ+ are continuous nonincreasing functions, u(s)→∞ as s→∞. If there is a continuous function V:ℝ×ℝn↦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∈ℝn, 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] as(4.1)τi=-Kiq̂i-Diωi-∑j=1naijq̂ij-∑j=1naijωij,
where Ki and Di are nonnegative constants, aij is the (i,j)th entry of the adjacency matrix 𝒜 associated with the graph 𝒢, qij=qj-1qi, q̂ij is the vector part of qij, ωij=ωi-A(qij)ωj, and A(qij)=A(qi)AT(qj) denotes the rotation matrix [21]. Here qij represents the relative attitude between spacecraft i and spacecraft j, and ωij represents the relative angular velocity between spacecraft i and spacecraft j. Note that the existence of the attitude consensus terms -∑j=1naijq̂ij-∑j=1naijωij 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] as(4.2)τi=JiA(Δqi)ω̇d+ωdbi×Jiωdbi-KiΔq̂i-DiΔωi-∑j=1naijq̂ij-∑j=1naijωij,
where qd and ωd denote, respectively, the time-varyingly desired attitude and angular velocity of the leader spacecraft, Δqi=qd-1qi, Δq̂i is the vector part of Δqi, ωdbi=A(Δqi)ωd, A(Δqi)=A(qi)AT(qd), and Δωi=ωi-ωdbi. Here Δqi denotes the relative attitude between spacecraft i and the leader, Δωi denotes the relative angular velocity between spacecraft i and the leader. By using (4.1) for (2.2), cooperative attitude regulation, that is, qi→qI and ωi→0, is achieved, where qI denotes the identity quaternion [0,0,0,1]T. By using (4.2) for (2.2), cooperative attitude tracking, that is, qi→qd and ωi→ωd 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 torqueτi is proposed(4.3)τi=-Kiq̂i-Diωi-∑j=1naij[q̂i-q̂j(t-Tij(t))]-∑j=1naij[ωi-lijAij(t-Tij(t))ωj(t-Tij(t))],
where Ki, Di, and aij are defined after (4.1), lij is a nonnegative constant, Tij=Tji denotes multiple time-varying communication delay, and Aij(t-Tij(t))=A(qi)AT(qj(t-Tij)). 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 τi is proposed,
(4.4)τi=-KiΔq̂i-DiΔωi-∑j=1naijσ[Δq̂i-Δq̂j(t-Tij(t))],
where Δq̂i and Δωi are defined after (4.2). Following the similar definition given in [25], the dynamically changing topology is defined as σ:[0,+∞)→ψΓ, where the set Γ is a finite collection of undirected graphs with a common node set. Then aijσ denotes the (i,j)th entry of the adjacency matrix 𝒜σ associated with the communication topology 𝒢σ. Before moving on, we assume that ωd and ω̇d are bounded and define γJi=∥Ji∥, γd=supt≥0∥ωd(t)∥, and β1=∥ω̇d∥+∥ωd∥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 matrixM=ℒ+diag(K1,…,Kn) is symmetric and positive definite if the undirected graph 𝒢 is connected and at least one Ki>0, where ℒ is the Laplacian matrix of graph 𝒢.Proof.
See the proof of Lemma4 [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, Di>0, for all i, ρij<aij, 4ρij(1-Ṫij)(aij-ρij)>lij2aij2, when aij≠0, and W>0, cooperative attitude regulation, that is, qi→qI and ωi→0, is achieved, where W is given by
(5.1)W=[cM-c2𝒟-12c2ℬ(M-K-cℬ-c𝒞)2(M-K-cℬ-c𝒞)2D-ℬ-12cℱ],
and c, ρij, D, K, M, ℬ, 𝒞, 𝒟, and ℱ are defined in the proof.Proof.
Consider the following Lyapunov function candidate(5.2)V=∑i=1n(Ki+cDi)∥qi-qI∥2+12∑i=1nωiTJiωi+∑i=1ncq̂iTJiωi+∑i=1n‍∑j=1n∫-Tij(t)0ρij∥ωj(t+s)∥2ds+12∑i=1n‍∑j=1n∫-Tij(t)0∫μ0aij∥q̂j̇(t+s)∥2dsdμ,
where c is a positive constant, ρij=0 when aij=0, and ρij is a positive constant when aij≠0. It is easy to verify that V is positive definite if 2(Ki+cDi)λmin(Ji)>c2λmax2(Ji), 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
(5.3)V̇=∑i=1n(Ki+cDi)q̂iTωi-∑i=1n(ωi+cq̂i)T[Kiq̂i+∑j=1naij(q̂i(t)-q̂j(t))]-∑i=1n(ωi+cq̂i)T{Diωi+∑j=1naij[ωi-lijAij(t-Tij(t))ωj(t-Tij(t))]}+∑i=1ncq̂iT(-ωi×Jiωi)+12∑i=1nc[(q¯iI3+S(q̂i))ωi]TJiωi-∑i=1n(ωi+cq̂i)T∑j=1n∫-Tij(t)0aijq̂j̇(t+μ)dμ+12∑i=1n‍∑j=1n∫-Tij(t)0aij[q̂jṪ(t)q̂j̇(t)-q̂j̇(t+μ)Tq̂j̇(t+μ)]dμ+ρij∑i=1n‍∑j=1nωj(t)Tωj(t)-ρij∑i=1n∑j=1n(1-Ṫij)ωj(t-Tij)Tωj(t-Tij),
where we have used the fact that ωiT(ωi×Jiωi)=0 and Leibniz-Newton formula q̂j(t-Tij(t))=q̂j(t)-∫-Tij0q̂j̇(t+μ)dμ [23]. It thus follows that
(5.4)V̇≤12c∑i=1nγJi∥ωi∥2-cq̂T(M⊗I3)q̂-ωT(M⊗I3)q̂+q̂T(K+cD)⊗I3ω+12∑i=1n‍∑j=1nTijaij∥ωi+cq̂i∥2-12∑i=1n‍∑j=1n∫-Tij0aij∥ωi+cq̂i+q̂j̇(t+μ)∥2dμ+12∑i=1n∑j=1nTijaijωjT(q¯jI3+S(q̂j))T(q¯jI3+S(q̂j))ωj-∑i=1n∑j=1n(aij-ρij)∥(ωi+cq̂i)-lijaij2(aij-ρij)Aij(t-Tij)ωj(t-Tij)∥2+ρij∑i=1n‍∑j=1nωjTωj+∑i=1n‍∑j=1n(aij-ρij)∥ωi+cq̂i∥2-∑i=1n∑j=1naij(ωi+cq̂i)Tωi-∑i=1n‍∑j=1n[ρij(1-Ṫij)-lij2aij24(aij-ρij)]∥ωi(t-Tij)∥2-∑i=1nDi∥ωi∥2-∑i=1ncDiq̂iTωi,
where q̂=[q̂1T,q̂2T,…,q̂nT]T,ω=[ω1T,ω2T,…,ωnT]T,K=diag(K1,…,Kn),M=ℒ+K, and D=diag(D1,…,Dn). Based on the conditions that ρij<aij and 4ρij(1-Ṫij)(aij-ρij)>lij2aij2 when aij≠0, we have that
(5.5)V̇≤12c∑i=1nγJi∥ωi∥2-cq̂T(M⊗I3)q̂-ωT(M⊗I3)q̂+q̂T(K⊗I3)ω-ωT(D⊗I3)ω+12∑i=1n‍∑j=1nTijaij∥ωi+cq̂i∥2+12∑i=1n‍∑j=1nTijaijωiTωi+c∑i=1n∑j=1n(aij-2ρij)ωiTq̂i+c2∑i=1n‍∑j=1n(aij-ρij)q̂iTq̂i=-q̂T(cM-c2𝒟-12c2ℬ)⊗I3q̂+ωT(-M+K+cℬ+c𝒞)⊗I3q̂-ωT(D-ℬ-12cℱ)⊗I3ω,
where we have used the fact that ∑i=1n∑j=1naijωjTωj=∑i=1n∑j=1naijωiTωi (graph 𝒢 is undirected). Here we also define ℬ=[bij],𝒞=[cij],𝒟=[dij], and ℱ as n×n matrices, where bij=Tijaij,cij=aij-2ρij,dij=aij-ρij, and ℱ=diag(γJ1,…,γJn). Based on the conditions that M is positive definite and D is positive definite (Di>0, for all i), for the sufficient small Tij 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, qi→qI, and ωi→0 is achieved under the conditions provided in Theorem 5.2.Remark 5.3.
It follows thatM is positive definite from Lemma 5.1 if the undirected graph 𝒢 is connected and at least one Ki>0. This implies conditions that the undirected graph 𝒢 is connected and at least one Ki>0 can be used to replace condition that M>0.Remark 5.4.
Note that the parametersρij 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 (6.1)Δ̇qi=12Ei(Δqi)Δωi,JiΔ̇ωi=-ωi×(Jiωi)-JiA(Δqi)ω̇d+JiΔωi×ωdbi+τi,
where Δqi, Δωi, and ωdbi are defined after (4.2). Also define τ=[τ1T,…,τnT]T. We can then transform (4.4) to the matrix expression (6.2)τ=-(Mσ⊗I3)Δq̂-(D⊗I3)Δω-∑k=1r(𝒜kσ⊗I3)∫-Tk(t)0Δq̂(t+μ)dμ,whereΔq̂=[Δq̂1T,Δq̂2T,…,Δq̂nT]T,  Δω=[Δω1T,Δω2T,…,ΔωnT]T,  K=diag(K1,…,Kn),  Mσ=Lσ+K,Lσ is the Laplacian matrix of 𝒢σ,D=diag(D1,…,Dn),Tk(t)∈{Tij(t):i,j=1,…,n} for k=1,…,r, and r=n(n-1)/2,𝒜kσ=[aijk] is a corresponding n×n matrix, where
(6.3)aijk={aijσ,Tk=Tij0,Tk≠Tij.
Before moving on, we need the following lemma.Lemma 6.1 ([27]).
For anya,b∈ℝn and any symmetric positive definite matrix Φ∈ℝn×n, one has 2aTb≤aTΦ-1a+bTΦb.Theorem 6.2.
Using (4.4) for (2.2), if Mσ>0 in each time interval, Di>0, for all i, and Q=[Q11Q12Q12Q22]>0, where Q11=cMσ-(1/2)c2∑k=1rTk𝒜kσ(𝒜kσ)T,Q12=[-K+Mσ-c∑k=1rTk𝒜kσ(𝒜kσ)T]/2, Q22=D-ℱ∈-(1/2)∑k=1rTk𝒜kσ(𝒜kσ)T, and ℱ∈=diag((3/2)cγJ1+(1/2)∑k=1rTkq(λmax(P2)/λmin(P1)),…,(3/2)cγJn+(1/2)∑k=1rTkq(λmax(P2)/λmin(P1))), the error state x of the closed-loop system is uniformly ultimately bounded, where x=[Δq̂T,ΔωT]T. In particular, the ultimate bound of x is λmax(P2)b/λmin(P1)θλmin(Q) (c, q, P1, P2, b, θ will be defined in the proof).Proof.
Consider the following Lyapunov function candidate(6.4)V=∑i=1n(Ki+cDi)∥Δqi-qI∥2+12∑i=1nΔωiTJiΔωi+∑i=1ncΔq̂iTJiΔωi,
where c is a positive constant. By using the fact that ∥Δqi-qI∥2≤2∥Δq̂i∥2, we know that xTP1x≤V≤xTP2x, where
(6.5)P1=[(K+cD)⊗I312cJ⊗I312cJ⊗I312J⊗I3],P2=[2(K+cD)⊗I312cJ⊗I312cJ⊗I312J⊗I3],
and J=diag(J1,…,Jn). We also know that V is positive definite if c is chosen properly to ensure 2(Ki+cDi)λmin(Ji)>c2λmax2(Ji), for all i. Taking the derivative of V gives
(6.6)V̇≤∑i=1n(Ki+cDi)Δq̂iTΔωi-∑i=1n(Δωi+cΔq̂i)T[KiΔq̂i+∑j=1naijσ(Δq̂i(t)-Δq̂j(t))+DiΔωi]+∑i=1n{12γJiβ1∥Δωi∥+cγJiβ1∥Δq̂i∥+cγJi∥Δωi∥2+3cγJiγJd∥Δq̂i∥∥Δωi∥+12c[(Δq¯iI3+S(Δq̂i))Δωi]TJiΔωi}+(Δω+cΔq̂)T∑k=1r(𝒜kσ⊗I3)∫-Tk(t)0Δq̂(t+μ)dμ,
where we have used (6.1), and the facts that ΔωiT(-ωi×(Jiωi)-JiA(Δqi)ω̇d+JiΔωi×ωdbi)=ΔωiT(-JiA(Δqi)ω̇d-ωdbi×Jiωdbi)≤γJiβ1∥Δωi∥ and Δq̂iT(-ωi×(Jiωi)-JiA(Δqi)ω̇d+JiΔωi×ωdbi)=Δq̂iT(-JiA(Δqi)ω̇d-ωdbi×Jiωdbi+JiΔωi×ωdbi-ωdbi×JiΔωi-Δωi×JiΔωi-Δωi×Jiωdbi)≤γJiβ1∥Δq̂i∥+γJi∥Δωi∥2+3γJiγd∥Δq̂i∥∥Δωi∥. We then have that
(6.7)V̇≤Δq̂T(K+cD)⊗I3Δω-cΔq̂T(Mσ⊗I3)Δq̂-Δq̂T(Mσ⊗I3)Δω-ΔωT(D⊗I3)Δω-cΔq̂T(D⊗I3)Δω+∑i=1nβ1γJi∥Δωi∥+∑i=1ncβ1γJi∥Δq̂i∥+32∑i=1ncγJi∥Δωi∥2+∑i=1n3cγJdγJi∥Δωi∥+12∑k=1r∫-Tk(t)0(Δω+cΔq̂)T[𝒜kσ(𝒜kσ)T]⊗I3(Δω+cΔq̂)+12∑k=1r∫-Tk(t)0Δq̂(t+μ)TΔq̂(t+μ)dμ,
where we have used the fact that ∥Δq̂i∥≤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
(6.8)V(x(t+θ))<qV(x(t)),-supk{Tk}≤θ≤0,
we know that λmin(P1)∥Δω∥2(t+θ)<qλmax(P2)(∥Δω∥2+∥Δq̂∥2). (Note that this is a property inherited from Lyapunov-Razumikhin uniformly ultimately bounded theorem.) Thus, we have that
(6.9)V̇≤Δq̂T(K-Mσ)⊗I3Δω-cΔq̂T(Mσ⊗I3)Δq̂-ΔωT(D⊗I3)Δω+∑i=1nβ1γJi∥Δωi∥+∑i=1ncβ1γJi∥Δq̂i∥+32∑i=1ncγJi∥Δωi∥2+∑i=1n3cγJdγJi∥Δωi∥+12∑k=1rTk(Δω+cΔq̂)T[𝒜kσ(𝒜kσ)T]⊗I3(Δω+cΔq̂)+12∑k=1rTkqλmax(P2)λmin(P1)(∥Δω∥2+∥Δq̂∥2)=-xT(Q⊗I3)x+wTy,
where y=[∥Δq̂1∥,…,∥Δq̂n∥,∥Δω1∥,…,∥Δωn∥]T, Q is defined in Theorem 6.2, and w=[cβ1γJ1+(1/2)∑k=1rTkq(λmax(P2)/λmin(P1)),…,cβ1γJn+(1/2)∑k=1rTkq(λmax(P2)/λmin(P1)),(3cγJd+β1)γJ1,…,(3cγJd+β1)γJn]T. Based on the conditions that Mσ>0 in each time interval and D>0 (Di>0, for all i), for the sufficient small Tij and c, it is easy to verify that there always exist Mσ and D to guarantee Q is positive definite. Therefore, we have that
(6.10)V̇≤-λmin(Q)∥x∥2+b∥y∥=-λmin(Q)∥x∥2+b∥x∥,
where b=∑i=1n(cβ1γJi+(1/2)∑k=1rTkq(λmax(P2)/λmin(P1)))2+∑i=1nγJi2(3cγJd+β1)2. Thus, for 0<θ<1, if ∥x∥≥b/θλmin(Q), we have that
(6.11)V̇=-(1-θ)λmin(Q)∥x∥2-θλmin(Q)∥x∥2+b∥x∥≤-(1-θ)λmin(Q)∥x∥2.
Therefore, the uniformly ultimate boundedness of x follows from Lemma 3.4. In addition, the ultimate bound is λmax(P2)b/λmin(P1)θλmin(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 Theorems5.2 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 thatW>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.
J1[23.0 0.1 0.1; 0.1 22.2 0.1; 0.1 0.1 23.2] kg·m2J2[22.5 0.1 -0.3; 0.1 22.1 0.1; -0.3 0.1 24.1] kg·m2J3[24.0-0.2 0.1; 0 21.2 0.1; 0.1 -0.1 22.2] kg·m2For control law (4.3), we choose the control parameters as Ki=2,Di=5, and  lij=0.3, for all i,j. qi(0) and ωi(0),i=1,2,3 are generated randomly. For control law (4.4), we choose the control parameters as Ki=2.2 and Di=11. qi(0) and ωi(0),i=1,2,3 are generated randomly. Suppose that the reference attitude qd(t), reference angular velocity ωd(t)=2E-1(qd)q̇d, reference torque τd and reference inertia satisfy (2.2) with qd(0)=[0.3921,0.5502,0.5287,0.5139]T (the corresponding Euler Angles are ψ=80deg,ϕ=60deg, and  θ=40deg), ωd(0)=[0.021,-0.012,0.014]Trad/s, τd=[0,0,0]TNm, and Jd=[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 aij=1 if (j,i)∈ℰ, and aij=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)|.Communication topologies.
(a)
Communication topology𝒢1(b)
Communication topology𝒢2Figures2, 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.Rigid body attitudes with control law (4.3).
(a)(b)(c)Rigid body angular velocities with control law (4.3).
(a)(b)(c)Rigid body control torques with control law (4.3).
(a)(b)(c)Figures5, 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.Rigid body attitudes with control law (4.4).
(a)(b)(c)Rigid body angular velocities with control law (4.4).
(a)(b)(c)Rigid body control torques with control law (4.4).
(a)(b)(c)


## 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.



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*Source: 621594-2010-08-01.xml*