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

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.


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 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.

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 q 1 , q 2 , q 3 , q 4 T i q T i , q i T , i 1, . . ., n.Here q i e i sin 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 i q i 1 21 .The product of two unit quaternions p i and q i is defined by q i p i q i p i p i q i q i × p i q i p i − q T i p i .

2.1
The conjugate of the unit quaternion q i is defined by q −1 i − q T i , q i T .Attitude kinematics and dynamics of each spacecraft using the unit quaternion are given by 21 qi 1 2 E i q i ω i , i 1, . . ., n, where q i ∈ R 4 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 where I 3 is the 3 × 3 identity matrix, S • denotes a 3 × 3 skew-symmetric matrix, and J i ∈ R 3×3 and τ i ∈ R 3 are, respectively, the inertia tensor and control torque of the ith spacecraft.

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 {v 1 , . . ., 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 i 1 , v i 2 , v i 2 , v i 3 , . ...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 2.4

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 ∈ t 0 − τ, t 0 .Also suppose that the solution x t 0 , φ t through t 0 , φ is continuous in t 0 , φ, t in the domain of definition of the function, where t 0 ∈ R. Definition 3.1 see 23 .The solutions x t 0 , φ of the RFDE 3.1 are uniformly asymptotically stable if i for every κ > 0 and for every t 0 ≥ 0 there exists a δ δ κ independent of t 0 such that for any φ ∈ C δ n,τ the solutions x t 0 , φ of the RFDE 3.1 satisfies for all t ≥ t 0 , ii for every η > 0 and for every t 0 ≥ 0 there exists a T η independent of t 0 and a v 0 > 0 independent of η and t 0 such that for any φ ∈ C n,τ , φ c < v 0 implies that x t t 0 , φ c < η, for all t ≥ t 0 T η .
, then the solutions of 3.1 are uniformly ultimately bounded.

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 timevarying 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 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 −1 j q 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 1 a ij q ij − n j 1 a ij ω 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 as where q d and ω d denote, respectively, the time-varyingly desired attitude and angular velocity of the leader spacecraft, 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 proposed 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 modelindependent control torque τ i is proposed, 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 ωd are bounded and define γ J i J i , γ d sup t≥0 ω d t , and β 1 ωd ω d 2 in this paper.

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 K 1 , . . ., 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 closedloop systems 2.2 with 4.3 .
, 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 and c, ρ ij , D, K, M, B, C, D, and F are defined in the proof.
Proof.Consider the following Lyapunov function candidate 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 > c 2 λ 2 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

5.3
where we have used the fact that ω T i ω i ×J i ω i 0 and Leibniz-Newton formula q j t−T ij t q j t − 0 −T ij ˙ q j t μ dμ 23 .It thus follows that T ij a ij ω T j q j I 3 S q j T q j I 3 S q j ω j

5.4
where q q T 1 , q T 2 , . . ., q T n T , ω ω T 1 , ω T 2 , . . ., ω T n T , K diag K 1 , . . ., K n , M L K, and and F diag γ J 1 , . . ., γ J n .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.

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 where Δq Δq

6.3
Before moving on, we need the following lemma.
r k 1 T k q λ max P 2 /λ min P 1 , 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 P 2 b/λ min P 1 θλ min Q (c, q, P 1, P 2, b, θ will be defined in the proof).
Proof.Consider the following Lyapunov function candidate where c is a positive constant.By using the fact that Δq i − q I 2 ≤ 2 Δq i 2 , we know that x T P 1 x ≤ V ≤ x T P 2 x, where

and J
diag J 1 , . . ., 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 > c 2 λ 2 max J i , for all i.Taking the derivative of V gives Δq t μ dμ,

6.6
where we have used 6.1 , and the facts that

Δq
Δq t μ T Δq t μ dμ, 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 we know that λ min P 1 Δω 2 t θ < qλ max P 2 Δω 2 Δq 2 .Note that this is a property inherited from Lyapunov-Razumikhin uniformly ultimately bounded theorem.Thus, we have that where y Δq 1 , . . ., Δq n , Δω 1 , . . ., Δω n T , Q is defined in Theorem 6.2, and w 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 Therefore, the uniformly ultimate boundedness of x follows from Lemma 3.4.In addition, the ultimate bound is λ max P 2 b/λ min P 1 θλ min Q by following a similar analysis to that in 28 .

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.
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 qd , reference torque τ d and reference inertia satisfy 2.2 with q d 0 0.3921, 0.5502, 0.5287, 0. 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.
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.

Conclusions
In this paper, the cooperative attitude regulation problem in the presence of multiple timevarying 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.

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.

1
a ij ω T i ω 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

Lemma 6 . 1 Theorem 6 . 2 .
27 .For any a, b ∈ R n and any symmetric positive definite matrix Φ ∈ R n×n , one has 2a T b ≤ a T Φ −1 a b T Φb.Using 4.4 for 2.2 , if M σ > 0 in each time interval, D i > 0, for all i, and

Figure 4 :
Figure 4: Rigid body control torques with control law 4.3 .
Definition 3.2 see 23 .The solutions x t 0 , 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 T 0 α > 0 such that |x t 0 , φ t | ≤ β for t ≥ t 0 T 0 α for all t 0 ∈ R, φ ∈ C, and |φ| ≤ α.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 → ∞.