SHI Zhen，XIE Yaen，DENG Chengchen，ZHAO Kun，HE Yushan，and HAO Yong

1. College of Automation，Harbin Engineering University，Harbin 150001，China;2. College of Aerospace and Civil Engineering，Harbin Engineering University，Harbin 150001，China

Abstract: To solve the problem of attitude synchronization control for spacecraft formation flying (SFF) suffering from external disturbances under a directed communication topology，a sliding mode disturbance observer (SMDO) based on the finite-time control strategy is developed to observe the time-varying external disturbance via estimating the upper bound of its first derivative. Meanwhile，the rotation matrix is employed to describe the attitude of SFF for the purpose of the avoidance of singularity and unwinding phenomenon. As for the attitude synchronization and the tracking control architecture，a sliding mode surface(SMS) is given such that the control objective can be achieved.The effectiveness and the validity of the proposed method are elaborated via theoretical analysis and numerical simulations.

Keywords: coordinated attitude control，disturbance observer，rotation matrix，attitude synchronization control.

1. Introduction

Spacecraft formation flying (SFF) plays an increasingly significant role in space missions，after its emerging in the 1990s with the development of the micro-satellite technology，including but not limited to on-orbit servicing，deep space exploring，earth monitoring and Mars exploration. Owing to its inherent advantages that do not exist in the large single space system，such as low cost，high fault tolerance，flexibility and easy maintenance，SFF has gained extensive attention recently. Being one of the most attractive technologies for SFF，the development of the control system always determines the success of a variety of space missions. However，the strong nonlinear dynamics of the spacecraft and complex space environment bring distinctive problems for the controller design，involving communication delays [1-3]，actuator faults [4-6]，input saturation constraints [7-9]and external disturbances [10-12]. Additionally，modern space missions require that the control system possesses abilities of fast convergence rate and accessible robustness against disturbances，which makes it a tough work to design the controller for SFF.

Considering the adverse effect on SFF caused by the external disturbance，a majority of methods have been proposed for the attitude synchronization and tracking control problem，involving back-stepping control [13-15]，adaptive control [16-18]，sliding mode control [19-21]，event-triggered control [22-24]and disturbance observer based control [10，25，26]. Back-stepping based control strategies possess satisfactory ability for disturbance rejection via combining other methods，such as sliding mode control [13]and adaptive control [15]. Nevertheless，the drawback of “explosion of complexity” caused by repeated derivative on the virtual command must be accounted during space missions. This is because the explosion of complexity means the control of the nonlinear system is more complicated than that of the linear system.In this case，the back-stepping method ensures all signals of the system optionally designed if the provisions are satisfied，and this algorithm can achieve high accuracy of convergence. However，the conventional back-stepping method has limitations for a class of nonlinear systems，namely the complexity explosion problem. To improve the methods proposed in [13]and [15]，the command filter is utilized in the back-stepping procedure in this paper，thereby realizing the attitude containment objective when there exists external disturbance. Compared with the back-stepping based control，the sliding mode control technology possesses satisfactory capability of disturbance attenuation，which brings about fruitful research results [19-21]. In [19，21]，a time-varying sliding mode surface (SMS) was utilized for the purpose of attitude syn-chronization in cases of healthy actuators and failure actuators. However，the mentioned strategies in [13-21]has neglected the limited communication resource among members in the SFF. By utilizing the event-triggered control method [22-24]，communication among neighbors is only required when the predefined event is satisfied，which will effectively release the pressure on communication channels. Disturbance observer，possessing excellent performance for disturbance compensation，has been widely used for SFF [10，25，26]. In [27]，a terminal sliding mode control method was proposed based on the neural-network for the trajectory tracking of the tethered space-tug. In [28]，an integrated robust H infinity control method was proposed based on an output feedback component and a feedforward component for the problem of multi-objective attitude tracking. However，these observers have been designed by assuming that the upper bound of the disturbance always exists，which may be inapplicable for some situations. In this paper，a sliding mode disturbance observer (SMDO) will be constructed without assuming an upper bound for the disturbance.

It must be noted that a variety of attitude controllers have been deduced through the unit quaternion [29]，which will lead to the unwinding problem. From a practical point of view，this issue must be solved to reduce unnecessary energy consumption. Considering the unwinding problem during the design process of the attitude controller，there are emerged various achievements[30-34]. The hybrid control strategy is one of the most effective methods to construct unwinding-free attitude controllers for spacecraft [30，31]. For a single spacecraft，the quaternion-based state-feedback control strategies were proposed to solve the problem of global attitude tracking [30]. In the spacecraft formation flying case，the results in [30]were extended to globally synchronize the attitudes of a network of spacecraft [31]. It must be noted that the presented methods in [30]and [31]were obtained by the application of unit quaternion. As an attitude description method，the rotation matrix can represent the attitude motion uniquely，which will eliminate the unwinding phenomenon properly. Consequently，the attitude coordinated control problem for multiple spacecraft was solved via the rotation matrix with communication delays and the uncertain initial matrix [32]. With the consideration of actuator faults，two rotation matrix-based controllers were proposed by integration with the distributed observer，which will estimate the leader's attitude information for the follower spacecraft [33，34]. Obviously，the controllers in [32-34]only possess asymptotic stability for the attitude synchronization mission. In contrast to the asymptotically stable methods，controllers with finitetime convergence have much more superior qualities in convergence rate，robustness against disturbance and control accuracy [35-38]. This paper is dedicated to developing a rotation matrix-based finite-time controller such that the convergence rate in [32-34]could be improved. In[39]，a new SMDO was developed using the terminal sliding mode technique，and the SMDO was employed to estimate unknown external disturbances and modeling uncertainties in finite time. However，the derivative of the disturbance was assumed to be bounded.

Considering the above discussion and analysis，a coordinated attitude synchronization and tracking controller for SFF will be established in this paper by combination with a novel SMDO. The rotation matrix and the directed communication topology are employed to build the mathematical model of the SFF system. The sliding mode technology will be applied in both the disturbance observer design part and the attitude controller design part.Taking the disadvantages of existing studies into account，the main contributions of this paper are stated as follows.

(i) The finite-time stability will be ensured for the tracking errors under the proposed controller，which will improve the convergence rate for [32-34]. Additionally，the unwinding phenomenon will be eliminated by the utilization of the rotation matrix.

(ii) By resorting to a constructive SMS，the proposed SMDO will estimate the disturbance without assuming an upper bound for it，effectively improving the proposed methods in [10，25，26，39].

The follow-up architecture of this paper is given as follows. The system model and the preliminaries are introduced in Section 2. In Section 3，it shows the design of the disturbance observer and the attitude controller，in which rigorous mathematical derivation and proof are given. In Section 4，the simulation results and the corresponding analysis are included. Finally，the conclusion of this paper is given in Section 5.

2. Mathematical model and preliminaries

2.1 Mathematical model of SFF

For the advantage of anti-unwinding，the rotation matrix will be utilized to structure the mathematical model，which is introduced as

where the subscripticorresponds to theith spacecraftRi∈SO(3)where SO(3) is the rotation group，and ωi∈ R3×1are defined as the rotation matrix of the attitude and the angular velocity in the spacecraft body frame，respectively;Ji∈ R3×3denotes the inertia matrix，which is the inherent parameter of the spacecraft;ui∈ R3×1anddi∈ R3×1are introduced as the control torque and the disturbance torque. A new operation is introduced marked as × in (3)，which rewrites a vector into a unique corresponding skew-symmetric matrix.

The map ∨ ，which is the inverse of × ，means restoring a skew-symmetric matrix to a vector. Some of their properties are shown as follows:

To complete the attitude controller design，the desired attitudeRd，the desired angular velocity ωd，the attitude error，and the angular velocity errorare introduced.Specifically，the relationships among the above variables are given as follows:

With (1)，(2)，and (7)-(9)，the error dynamics are described as

2.2 Algebraic graph theory

The algebraic graph，denoted asG=(ν，ζ，C)，is composed of the node set ν ={ν1，ν2，···，νn}，the edge set ζ ∈n×n，and the weighted adjacency matrixC=[cij]. When applied to the communication topology of SFF，each node is used to represent a single spacecraft，and the edge is regarded as the communication path between two spacecraft. Obviously，all the communications are considered as bidirectional in the undirected graph，whereas the direction of information transmission depends on the arrow direction in the directed graph. Then，the elements in the weighted adjacency matrixCare defined ascij＞0 when(νi，νj)∈ ζ; otherwise，cij=0. It is generally assumed that the node has no connectivity with itself，which meanscii=0. The undirected graph is connected only if paths connect two arbitrary nodes. Likewise，as long as all nodes are fitted together by paths，the directed graph is just weakly connected. To proceed further，information can go from any node to all other nodes in a strongly connected directed graph.

The Laplace matrixLof the graphGis defined as

Obviously，Lis a symmetric matrix in the undirected graph and it is usually not symmetric in the directed graph.

2.3 Notations

The norm of a vector or a matrix is denoted by the notation ‖ ·‖，which follows the algorithm of 2-norm in this paper. For arbitrary ξ ∈ Rn×1，s i gγ(ξ) is introduced to represent the relation s igγ(ξ)=[sign(ξ)|ξ|γ]. As for the vector ξ =[ξ1，ξ2，···，ξn]T，si gγ(ξ) meanssigγ(ξ)=[sign(ξ1)|ξ1|γ，sign(ξ2)|ξ2|γ，···，sign(ξn)|ξn|γ]T.

2.4 Preliminaries

Lemma 1[32]If the Laplace matrixLbelongs to a strongly connected graph，there must be a positive vector η =[η1，η2，···，ηn]Tto satisfy ηTL=0.

Lemma 2[8]The condition of finite-time stabilization for the Lyapunov functionVis shown as

wherekis a positive constant and α satisfies the relation 0＜α＜1.

Assumption 1The communication topology of SFF is a directed graph with strong connection.

Assumption 2The first derivative of the external disturbance has an upper bound，e.g.，the equation

For the spacecraft control problem，the disturbance mainly comes from the interference of the space environment and the spacecraft. In fact，the disturbance does not mutate，that is to say，the first derivative of the external disturbance has an upper bound.

3. Finite-time controller design

In this section，the SMDO will be combined with a finitetime controller to ensure the finite-time convergence for the tracking errors. In order to eliminate the negative impact of the disturbance on system performance，a novel finite-time SMDO is firstly exploited without assuming an upper bound for the disturbance. Subsequently，a finitetime SMS is derived such that the tracking error will converge to the origin within limited time.

3.1 Disturbance observer design

3.2 Attitude controller design

With the construction of the finite-time SMDO in the previous part，the external disturbance will be observed when the upper bound of its first derivative is unavailable. Integrated with this SMDO，a coordinated attitude control architecture is presented in this part to realize attitude tracking and synchronization objective. Initially，a novel SMS is designed for ensuring the finite-time stability of the closed-loop system. Subsequently，the control law is designed by combing the SMDO，which can further endow the attitude errors with finite-time stability and satisfactory disturbance rejection performance. To derive the synchronization controller，the communication topology is assumed be rc.

4. Numerical simulations

Simulation results are shown in this part to illustrate the effectiveness and superiority of the designed method. In the simulation scenario，the SFF consists of one virtual leader spacecraft and four follower spacecraft. The detailed information of the communication topology is given in Fig. 1，where the nodei(i=1，2，3，4) represents theith spacecraft.

Fig. 1 Network topology

Based on the directed communication topology，the weighted adjacency matrix is given as

The inertia matrices of the four following spacecraft are set as

For the purpose of comparison，the simulation results of the existing method in [32]will be given in this part.The attitude and angular velocity tracking errors are presented in Fig. 9 and Fig. 10. By comparing with Fig. 6 and Fig. 7，it reveals that the proposed method in this paper possesses a better ability in the convergence rate. Actually，the controller in [29]was developed based on a linear SMS，which can only ensure asymptotical convergence for the error dynamics. Different from this controller，the control architecture in this paper is constructed through a terminal SMS，thus realizing the tracking task with a faster convergence.

5. Conclusions

In this paper，a robust controller is designed to solve the attitude synchronization and tracking control issue of multiple rigid spacecraft. By resorting to the SMDO and terminal sliding mode technology，the proposed method possesses satisfactory performance in fast convergence and disturbance rejection. Compared with the existing references，the assumption of an upper bound for the external disturbance is released，enlarging the application of SMDO to a certain extent. Additionally，finite-time convergence is ensured for the global tracking error system when the rotation matrix is directly utilized for controller development. Finally，simulation results reveal the effectiveness and advantage of the presented controller.