Distributed Attitude Consensus for Multiple Rigid Spacecraft under Jointly Connected Switching Topologies

We study the distributed leader-following attitude consensus problem formultiple rigid spacecraft with a single leader under jointly connected switching topologies. Two cases are considered, where the first case is with a static leader and the second case is with a dynamic leader. By constructing an auxiliary vector and a distributed observer for each follower spacecraft, the controllers are designed to drive all the attitudes of the follower spacecraft to the leader’s, respectively, for both of the two cases, though there are some time intervals in which the communication topology is not connected. The whole system is proved to be stable by using common Lyapunov function method. Finally, the theoretical result is illustrated by numerical simulations.


Introduction
In recent years, spacecraft formation flying (SFF) has become a new technology that plays an important role in the future space missions, such as Earth Observing (EO) [1,2], Orbit Express (OE), Terrestrial Planet Finder (TPF) [3], Space Telescope Assembly (STA), Stellar Imager (SI), and Synthetic-Aperture Imaging (SAI) [4].This interest is motivated by the advantages gained by replacing a traditionally large and expensive spacecraft by a cluster of microspacecraft to accomplish a common task in a coordinated manner [5].One of the most important control goals for SFF is the attitude consensus or alignment, where every spacecraft updates its own orientation using the orientations of its local neighbors.As a result, the orientations of all spacecrafts approach a common value.For example, in interferometry applications, it is often essential to control different spacecraft to maintain the same or relative attitudes during and after formation manoeuvres.Since the angular velocity of the body cannot be integrated to obtain the attitude of the body directly because of the nonlinear dynamics [6], attitude coordination control problem becomes a particularly interesting problem for the researchers.Reference [7] solves the synchronized multiple spacecraft rotations control problem with a passivity-based damping method, while [8] focuses on the condition that the angular velocity is unknown.Other interesting problems include attitude consensus with time delay [9], with input constraints [10], with attitude constraints [11], and with multiple leaders [12].Certainly, it will be more challenging if only a subset of group agents have access to the virtual leader [13].
As we all know, the biggest difference between multiagent systems and single-agent system lies in the communication network.Therefore, the characteristics of networks decide the performance of the whole system to a great extent.Communication outage, new member's joining or quitting, radio silence, or recovery will change the communication topology (termed as switching topology), which makes it more difficult to design the control laws.Based on relative attitude information and Modified Rodriguez Parameters (MRPs), [14] considers cooperative attitude tracking problem and gives a control law in the presence of a dynamic communication topology.Reference [15] extends this to the condition that there exist both multiple time-varying communication delays and dynamically changing topologies, and the result of uniformly ultimate boundedness of the closed-loop system is obtained.Considering more complicating elements, [16] presents controllers that can render 2 Journal of Control Science and Engineering a spacecraft formation consistent to a given trajectory globally with dynamic information exchange graph and nonuniform time-varying delays while coping with the parameter uncertainties and unexpected disturbances.In [17], a 6-DOF dynamics model of the spacecraft formation flying is established in Euler-Lagrange form, and a control algorithm based on consensus theory is proposed in the presence of dynamic communication topology.Furthermore, almostglobal attitude synchronization is achieved in [18] based on switching joint connection; however auxiliary variables are introduced which make the controllers complicated.In [19], by utilizing Lyapunov direct method and choosing a common Lyapunov function properly, the robustness of the designed position and attitude coordinated controllers to communication delays, switching topologies, parameter uncertainties, and external disturbances is guaranteed.
Note that all the above literatures only consider the uniformly connected topologies.However, the jointly connected case is more challenging because there will be isolated agents during some time intervals, which makes the controller designing more challenging and more difficult.It is worth noting that [20] addresses the attitude synchronization problem of multiple rigid body agents in SO(3) with directed and jointly strongly connected interconnection topologies.Using the axis-angle representation of the orientation, a distributed controller is proposed based on relative orientations between the agents without a global reference coordinate frame.And from the viewpoint of interior metrics, [21] provides a leaderless consensus protocol for strongly convex geodesic balls and applies it to the consensus problem of rotation attitudes under switching and directed communication topologies.Although the topologies are jointly connected, there is no leader in the system, and the control schemes are not used in the leaderfollowing problem and especially not used in the dynamic tracking problem.
In this paper, we focus on the leader-following attitude consensus problem under jointly connected topologies, where the attitude of the leader is only available to a subset of the followers.The difficulty lies in how to construct an effective controller that can drive all the attitudes of the followers not only to a same constant value but also to the attitude of the dynamic leader.We use MRPs to represent the spacecraft attitude for nonredundancy.By constructing a useful auxiliary vector for each spacecraft, we design a distributed controller to each follower to guarantee that the attitude errors between the followers and the static leader converge to zero.By associating a distributed observer for each follower, the controller is designed such that the attitude of the followers will converge to the dynamic leader.
The remainder of this paper is organized as follows.In Section 2, we present the dynamics of rigid body attitude, basic knowledge of graph theory, and the statement of leader-following attitude consensus problem under jointly connected topologies.The details about the construction of auxiliary vectors and distributed observers as well as derivation of the controller are presented in Section 3. In Section 4, we show simulation results for five spacecrafts using the control laws proposed in Section 3 and conclusion follows in Section 5.

Problem Statement and Background Information
where   ∈ R 3 denotes the Modified Rodrigues Parameters that represent the attitude of the th spacecraft.Here   is defined by   = ê tan(  /4), where ê is the Euler axis and   is the Euler angle [22].
denotes the angular velocity of the th spacecraft;   and   are, respectively, the inertial matrix and the external input torque of the th spacecraft. ×  is the skew-symmetric matrix with the form The matrix (  ) is given by which has the following properties [23,24]: Remark 1.We hasten to point out that the use of the MRPs simplifies the analysis and the ensuing formulas, since there is no additional equality constraint to worry about.Another advantage of the MRPs is the fact that they can parameterize eigenaxis rotations up to 360 degrees.In contrast, other three-dimensional parameterizations are limited to eigenaxis rotations of less than 180 degrees.One can refer to [22,25] for more details.The stability results obtained in this paper mean the stability of the corresponding kinematic parameters.That is, the stability is guaranteed for all initial attitudes except for the singular point Φ  = ±360 ∘ , where Φ  is the principle angle of the attitude of the th rigid body.

Graph Theory.
Graphs can be conveniently used to represent the information flow between agents.Let G = {V, E, A} be an undirected graph or directed graph (digraph) of order  with the set of nodes Given a piecewise constant switching signal (), we can define a nonnegative switching matrix A () = [  ()], ,  = 0, 1, . . ., , where, for  = 1, . . ., ,  0 () > 0 if and only if the control input   can access the information of the leader at time instant , and all other elements of A () are arbitrary nonnegative numbers satisfying   () = 0 for any  ≥ 0,  = 0, 1, . . ., .Let G () = (V, E () ) be a dynamic digraph of A () .Then, the node set V = 0, 1, . . .,  with 0 corresponding to the leader system and the integer ,  = 1, . . ., , corresponding to the th subsystem of the follower system, and E () ⊆ V × V and (, ) ∈ E () if and only if   () > 0 at time instant .

Problem Statement.
In this paper, we consider the leaderfollowing attitude synchronization control problem under jointly connected graph with two cases, that is, regulation case and dynamic tracking case.
As for the regulation case, the leader is set to the desired attitude with no angular velocity.The control objective is to drive the attitudes of the followers to the static leader with a dynamic network topology G () ; that is,       () −  0     → 0,       ()     → 0,  = 1, . . ., .
As for the dynamic tracking case, the leader is set to the desired attitude with nonzero angular velocity.The control objective is to drive the attitudes and the angular velocities of the followers to the dynamic leader with a dynamic network topology G () ; that is,       () −  0 ()     → 0,     σ  () − σ 0 ()     → 0,  = 1, . . ., .
Assumption 2. The communication among the followers is bidirectional, and there exists a subsequence {  } of { :  = 0, 1, . ..} with   +1 −    <  for some positive  such that the union graph ⋃  +1 −1 =  G  (  ) contains a spanning tree with the node 0 as the root.

Main Results
In this section, we deal with the leader-following attitude consensus problem with jointly connected topologies.Firstly, we associate each agent with the following auxiliary variable vector: where  is a positive constant.According to (9), we get  0 = σ 0 +  0 for the leader.

Regulation Case.
In order to reflect the isolated agents for jointly connected topologies at some instants, we denote   as the set of the connected agents except for the leader, and   as the set of the isolated agents at time instant , respectively.Obviously, we have   ∪   =  and   ∩   = Ø.The control law for the th follower is designed as where  > 0 is a positive constant,   = − Ġ(  )  −  σ  , and   is defined as in ( 5).
If we define then we get that Combining ( 1) with ( 9)-( 12), the dynamics of the connected spacecraft can be written as where the last equation has used (12), and the dynamics of the isolated spacecraft can be written as which can be synthesized as Remark 5.For jointly connected topologies, there may exist connected spacecraft and isolated spacecraft at the same time in some time intervals.It is possible that some agents are not connected to the leader; however they are connected to each other.In this case, the dynamics of these agents can still be converted to (13).
From the above, we get the following result.
Theorem 6.Under Assumption 2, the leader-following consensus of system ( 1) is achieved by choosing the control protocols as (10).

Dynamic Tracking Case.
The states of the leader are assumed to be dynamic, which can be generated by where We associate a compensator for each follower as follows: with where  is a positive constant.
where  () = L () + diag{ 10 (), . . .,  0 ()}.By Lemma 2 of [29], under Assumptions 2 and 7, the origin of the system (24) is exponentially stable; that is, for all  = 1, . . ., , we have lim Also we get that lim lim For this reason, we call (23) a distributed observer of the leader state.Moreover, define where  is a positive constant; we get that Then we associate another auxiliary signal for each follower as based upon which, the control input can be chosen as Theorem 10.Under Assumptions 2 and 7, the leader-following dynamic consensus of system ( 1) is achieved with the control protocol (31).
Proof.Define a Lyapunov function candidate as With (30), taking the derivative of  gives where we have used (23).According to ( 1) and ( 31), V can be written as Though the network is switching,  is positive in   for all   ; that is,  is lower bounded for all  ≥ 0, and therefore   is bounded.Next we will prove lim →∞   = 0.
Remark 11.For uniformly connected switching topologies, each agent keeps connected all the time such that there are no isolated agents, but this does not hold for jointly connected switching topologies.In order to prevent the divergence of the isolated agents, we design different control laws for the connected and isolated agents, respectively.References [20,21] also study the attitude consensus control problems in the presence of jointly connected switching topologies.However, there is no leader in the system, and the control schemes are not used in the leader-following problem.Meanwhile, the problem in this paper is more difficult as only a subset of followers can access the state information of the leader.

Numerical Example
In this section, we present two numerical examples to illustrate the effectiveness of our protocols ( 9) and (31).Consider the attitude consensus problem for five spacecrafts with one single leader and four followers.The dynamic equation of each spacecraft is described as ].
For the first example, we set the initial attitude and the angular velocity of the leader to be, respectively,  0 = [2, 5, 6]  and  0 = [0, 0, 0]  .The initial attitudes of the three followers are set to be, respectively,   4 shows the auxiliary vector errors.We see that the regulation control goal is achieved under the jointly connected topologies.
which can be generated by a leader system of the form (22) with     shown in Figure 5, and the angular velocities are shown in Figure 6.Besides, Figure 7 shows the auxiliary vectors.We see that the dynamic tracking control goal is achieved under the jointly connected topologies.

Conclusion
In this paper, the leader-following attitude consensus problem with a single leader under jointly connected topologies is studied for two different cases, that is, the regulation case with a static leader and the dynamic tracking case with a dynamic leader.Auxiliary vectors are constructed, based upon which the control protocols are designed for each follower spacecraft.Numerical simulation results are presented to illustrate the effectiveness of the controllers.In the future, we will study the problem under jointly connected topologies coupled with time delays.
and  ∈ R 3×3 .It can be seen that V = V can generate a large class of reference signals, such as step functions of arbitrary magnitudes, ramp functions of arbitrary slopes, or sinusoidal functions of arbitrary amplitudes and initial phases, and therefore  0 can represent various reference signals by linearly combining the components of V.