Lingming CHEN, Chunjing LI, Ynning GUO, Gungfu MA,Bolong ZHU

a Department of Control Science and Engineering, Harbin Institute of Technology, Harbin 150001, China

b School of Electrical Engineering and Automation, Qilu University of Technology (Shandong Academy of Sciences), Jinan 250353,China

KEYWORDS

Abstract This paper investigates the problem of Spacecraft Formation-Containment Flying Control(SFCFC)when the desired translational velocity is time-varying.In SFCFC problem,there are multiple leader spacecraft and multiple follower spacecraft and SFCFC can be divided into leader spacecraft’s formation control and follower spacecraft’s containment control.First,under the condition that only a part of leader spacecraft can have access to the desired time-varying translational velocity, a velocity estimator is designed for each leader spacecraft. Secondly, based on the estimated translational velocity,a distributed formation control algorithm is designed for leader spacecraft to achieve the desired formation and move with the desired translational velocity simultaneously. Then, to ensure all follower spacecraft converge to the convex hull formed by the leader spacecraft, a distributed containment control algorithm is designed for follower spacecraft. Moreover, to reduce the dependence of the designed control algorithms on the graph information and increase system robustness, the control gains are changing adaptively and the parametric uncertainties are handled, respectively. Finally, simulation results are provided to illustrate the effectiveness of the theoretical results.

1. Introduction

Recently, the distributed coordination of Spacecraft Formation Flying(SFF)has drawn much attention for its wide applications, such as Earth observation, distributed aperture radar and deep-space exploration.1To increase flexibility,adaptivity,and robustness, the spacecraft in SFF system usually coordinate with a distributed manner,in which each spacecraft communicates the state information with its neighbors.2Note that the spacecraft formation flying system can be regarded as a special kind of Multi-Agent Systems(MAS)by regarding each spacecraft as an agent with orbital dynamics.Therefore,in the following, we review the related work in a more general way,more specifically, from the control perspective of multi-agent systems.

1.1. Related works

(1) Fundamental coordinated control problemsRecently,the fundamental coordinated control problems which include the leaderless consensus,3,4leader-follower tracking,5containment,6,7and formation,8,9have been widely investigated. Note that these research3–9mainly focused on the linear multi-agent systems. However, many physical systems are inherently nonlinear in reality.Euler–Lagrange(EL) system, one kind of second-order nonlinear systems,can be used to describe the dynamics of many mechanical systems, including robotic manipulators, orbital dynamics of spacecraft formation flying system, and flying dynamics of unmaned aerial vehicles, to name a few.10,11Recent results on distributed coordination of networked nonlinear Euler–Lagrange systems have concentrated on the consensus,12leader–follower tracking with one leader,13containment with multiple leaders,14and formation.15The spacecraft formation flying control problem was investigated in Ref. 1,16.

(2) Formation-containment control problem

DefinitionFormation-containment system consists of multiple leaders and multiple followers, in which followers can receive information from leaders,but leaders do not receive information from followers. In formation-containment problem,leaders are responsible for achieving a desired formation, which means that the coordinated formation control algorithm needs to be proposed for leaders to form a desired static or time-varying configuration. Followers are responsible for achieving containment,in which the containment control algorithm needs to be designed such that all followers can converge to the convex hull spanned by leaders.17,18Since the leaders’motion has an effect on followers’containment,the formation-containment control problem is not the simple combination of formation control problem of leaders and containment control problem of followers,which poses a great challenge for the formationcontainment control algorithm design for leaders and followers simultaneously.19The main advantage of the formation-containment framework is that it can be applied in many complex missions, including deep-space exploration of multiple spacecraft,reconnaissance of coordinated robots, and formation flying of multiple unmanned aerial vehicles.One of these examples is that:a group of spacecraft is assigned to explore the deep space,but only some of them are equipped with advanced sensors to detect the hazardous environment. One effective way to reduce the system costs and guarantee the motion safety simultaneously is that these spacecraft with advanced sensors can be designed as leader spacecraft and the other spacecraft can be designed as follower spacecraft. Thus, in the exploration process, leader spacecraft need to form a safe configuration firstly and then follower spacecraft can converge into the convex hull formed by leader spacecraft, which conforms to the definition of formation-containment problem.

Literature reviewDimarogonas et al. 17 addressed the formation-containment problem for multi-agent systems under undirected communication graph.For high-order linear swarm systems, Dong et al. 20 designed formationcontainment control algorithms under directed graph. The control algorithms were designed in Ref. 21,22 for linear formation-containment system with communication delays and output feedback.With more consideration on communication constraints, Zheng et al. 23, Wang et al. 24, Han et al. 25 proposed control algorithms for linear formationcontainment system with only sampled position data,intermittent communication, and switching topology, respectively. The above results mainly concentrated on the linear formation-containment systems. For nonlinear Euler-Lagrange systems, our recent work studied the formationcontainment problem without using relative velocity information26and with the performance of collision avoidance.27It is worthwhile to point out that in Ref.18,20–22,24,25,the desired formation trajectory should be pre-specified and known for each leader, which is difficult to be obtained in practical missions.In Ref.23,the desired formation configuration consists of a constant translational moving velocity and relative positions among agents.Note that the constant translational velocity should be known for all the agents in Ref.23.Our recent work28,27considered a static desired configuration and dynamic configuration with constant translational velocity, respectively. However, the formationcontainment system cannot maneuver with a time-varying velocity in Ref. 26–28, which is usually hard to meet the requirements of the complicated task and environment.

1.2. Motivation of this paper

In practice, the formation configuration is usually required to be dynamic to adapt the changeable environment. For example, multiple unmanned aerial vehicles are assigned to track a moving target on ground, and multiple formation flying spacecraft aim at completing the observation or measurement of an asteroid. In such cases, the formation configuration is usually time-varying to satisfy the requirements from the tasks and environment.29Therefore, it is necessary to study more general formation-containment control problem, especially when the translational velocity of the formation configuration is time-varying.

Motivated by the fact that in the previous study the dynamics of agents are linear and the desired formation trajectories are required to be pre-specified for each leader on the formation-containment problem, this paper aims at designing distributed formation-containment control algorithm with time-varying translational velocity for the spacecraft formation flying system, in which the motion dynamics of each spacecraft are nonlinear and the desired formation trajectories are not required to be pre-specified for each spacecraft.

1.3. Contribution of this paper

In comparison to the existing results, this research mainly has the following three advantages.

(1) The desired configuration of the formation-containment system is time-varying.The time-varying configuration consists of a whole time-varying translational velocity and constant relative positions among spacecraft.The time-varying translational velocity is only available to a portion of leader spacecraft and the relative positions are distributed among leader spacecraft. Compared with the static desired configuration26,28and the desired configuration with constant translational velocity,23,27the configuration with timevarying translational velocity is more general.

(2) The proposed formation-containment control algorithm is fully distributed and continuous. Compared with Ref.18,20–22,24, the formation-containment control algorithm does not require the graph information and the knowledge of the upper bound of the time-varying translational velocity.

(3) The parametric uncertainties are considered for all spacecraft’s nonlinear motion dynamics. Compared with the linear multi-agent systems,17,18,20–25the nonlinear SFCFC system is more difficult for the control algorithm design but more practical than linear MAS.

1.4. Notations

2. Problem formulation and preliminaries

In this section,we first introduce the spacecraft relative motion dynamics and basic graph theory. Then, the research problem is formally formulated.

2.1. Spacecraft relative motion dynamics

We consider a group of spacecraft labeled from 1 toNformation flying in a Low-Earth orbit.The relative motion dynamics of the spacecraft in the Local-Vertical and Local-Horizontal(LVLH) rotating frame can be described as30

wherei=1，2，...，N，qi=[xi，yi，zi]T，τi=[τix，τiy，τiz]T,

Then, the following three properties are satisfied31:

(P1)Ciis skew symmetric. Then,one hasxTCix=0 for all vectorsx∈R3.

2.2. Graph theory

Lemma 1(32,33).LetGMbeadirectedgraphandLMbethe associatedLaplacianmatrix.Then,thefollowingstatement holds:

Definition 134. Let C be a set in a real vector space S ?Rn.The set C is convex if, for anyxandyin C, the point（1-t）x+ty∈C for anyt∈[0，1].The convex hull for a set of pointsX=｛x1，x2...，xn｝in S is the minimal convex set which contains all points inX. Then, Co（X） is used to denote the convex hull ofX. In particular,

Definition 235. LetZn?Rn×nbe the set of all square matrices of dimensionnwith non-positive off-diagonal entries.A matrixA∈Rn×nis a nonsingularM-matrix ifA∈Znand all eigenvalues ofAhave positive real parts.

Lemma 236.AmatrixA∈ZnissaidtobeanonsingularMmatrixifandonlyifA-1existsandeachentryofA-1is nonnegative.

2.3. Problem description

Suppose that the spacecraft formation-containment system consists ofNleader spacecraft andMfollower spacecraft, whose motion dynamics are described by Eq. (1). Let subscriptsE=｛1，2，...，N｝ andF=｛N+1，N+2，...，N+M｝ denote the leader spacecraft set and follower spacecraft set, respectively. The communication topology of the formationcontainment system is denoted asGA,in which follower spacecraft can receive information from leader spacecraft but not vice versa. According to the definition of formation-containment problem, the Laplacian matrixLA∈R（N+M）×（N+M）ofGAcan be written as

whereLE∈RN×Ndescribes the communication among leader spacecraft,L1∈RM×Mdescribes the communication among follower spacecraft,andL2∈RM×Ndescribes the communication from leader spacecraft to follower spacecraft.

In this study,we consider that the desired time-varying formation configuration consists of a time-varying translational velocity and constant relative positions. In the following, we give the definitions of formation, containment, and formation-containment.

Definition 3.The desired formation with a time-varying translational velocityvr（t）∈R3and constant relative positions δik∈R3is said to be achieved by the leader spacecraft(?i，k∈E) if

Definition 5.Spacecraft formation flying system achieves SFCFC if the desired formation with a time-varying translational velocity is achieved by all leader spacecraft （?i∈E）,and all the follower spacecraft（?j∈F）converge to the convex hull spanned by the leader spacecraft, which means that Eq.(3), Eq. (4), and Eq. (5) hold simultaneously fori∈E，j∈F.

In this study, we aim to solve the SFCFC problem by designing distributed formation control algorithm for leader spacecraft （?i∈E） and distributed containment control algorithm for follower spacecraft （?j∈F）.

3. Main results

In this section, the SFCFC algorithm is proposed for the spacecraft formation flying system with parametric uncertainties. First, we design the distributed formation control algorithm for leader spacecraft. Then, we design the distributed containment control algorithm for follower spacecraft. To handle the constraint that the desired time-varying translational velocity is only known for a part of leader spacecraft and not known for all follower spacecraft, the distributed velocity estimators are designed first for leader spacecraft and then follower spacecraft.

3.1. Distributed formation control for leader spacecraft

In this subsection,a distributed formation control algorithm is proposed for each leader spacecraft to achieve Eqs.(3)and(4).The information interaction topologyGLamong leader spacecraft and the desired translational velocityvrare assumed to satisfy the following assumptions.

Assumption 1.The directed communication topologyGLamong the leader spacecraft is strongly connected.

Assumption 2.The desired translational velocityvr（t） and its time-derivatives are bounded.

Because only a part of leader spacecraft can obtain the desired translational velocityvr, we first design a distributed velocity estimator such that each leader spacecraft can obtain the desired translational velocity information.

Proof.Define the relative velocity estimation error

Then, Eq. (6) can be written as

Consider the Lyapunov function candidate as

Taking the time-derivate of Eq. (9) gives

According to Ref. 37,38, we can get that?T1＞0，?t＞T1，ei（t）=0，?i∈E. Write Eq. (7) as the compact form

where

Then, we define the auxiliary variables

where β is a positive constant.

According to Property (P3) of the spacecraft relative motion dynamics, we have

Then, we propose the following distributed formation control algorithm

where ηi，εiare positive constants, ^miis the estimation ofmibecause the mass of each spacecraft cannot be precisely known due to the uncertainty and Eq. (16) is used to estimatemi.

Now,we aim to analyze the stability of the formation flying system under control algorithms Eq. (18)-Eq. (21) whent≥T1. According to Ref.40, if the directed communication graph among leader spacecraft is strongly connected, there always exist δiand δksuch that δik=δi-δk, for all aik＞0.Substituting Eq. (19) into Eq. (1), we can get

We construct the following Lyapunov function candi date

where we have used Property (P1) of the spacecraft relative motion dynamics and adaptive law Eq. (20). Using the adaptive law Eq. (21), we can get

On the other hand,

Note that

We write Eq. (18) as the compact form

Substituting Eq. (30) into Eq. (27), we obtain

Note that

which implies that the conditionxTζ=0 in Lemma 1 is satisfied. From Lemma 1, we have

One also has

where κ1is a positive constant.

Combining Eq. (25) and Eq. (26), Eq. (31), Eq. (33),Eq. (34), we obtain

wherekmin=miniki.

Thus, we can choose

Remark 1.Note that the conditions Eqs.(36)and(37)are only used in the proof and are not required in the practical realization.It is unnecessary for the formation control algorithm Eq.(19)to use the information of κ1，kminin Eqs. (36),(37), which is beneficial from the adaptive updating law Eq. (17). Therefore, the proposed formation control algorithm is distributed.

3.2. Distributed containment control for follower spacecraft

In this subsection,we design a distributed containment control algorithm under the directed communication graphGAsuch that all follower spacecraft can converge to the convex hull formed by all leader spacecraft. First, we give the following assumption about the graphGA.

Assumption 3.For each of theMfollower spacecraft, there exists at least one leader spacecraft that has a directed path to the follower spacecraft.

Then, the following lemma holds.

Proof.By following the same steps in Lemma 3, one can also get the finite-time stability of the estimator Eq. (39). □

Then, based on the estimated translational velocity, we define the auxiliary variables

where ρ is a positive constant. According to Property (P3) of the spacecraft relative motion dynamics, we have

We design the containment control algorithm as

Substituting Eq.(47)into Eq.(1),the closed-loop system of thejth follower spacecraft can be written as

Taking the derivative ofV2along Eq. (49) yields

Using Property (P1) of the spacecraft relative motion dynamics and the adaptive law Eq. (48), we obtain

Note that Eq. (46) can be written in a vector form

Remark 3.In Ref. 18,20,22, the desired time-varying formation consists of time-varying flying trajectories for all formation agents.Then,all agents can form the desired time-varying formation by only using their own desired flying trajectories. In this study, we consider that the time-varying formation has a common time-varying translational velocity which is partially known by leader spacecraft, and constant relative positions which is distributed for connected leader spacecraft.

Remark 4.One of the advantages of the spacecraft formationcontainment flying system with time-varying translational velocity is the capability of obstacle avoidance. Fig.1 shows how the formation-containment system works to pass through the three obstacles (A，B，C). This formation-containment system is composed of 6 leader spacecraft (labeled from 1 to 6)and 4 follower spacecraft (labeled from 7 to 10). Suppose that the first and sixth leader spacecraft are equipped with the advanced sensors which can be used to detect the positions of obstacles (the dashed blue lines represent the detection,the dashed black line represents the planned flying velocity).The first and sixth leader spacecraft can plan a feasible moving velocityvr（t）when they detect that there are obstacles ahead of them. Then, leader spacecraft 2 to 5 will use the distributed finite-time estimator to estimate this moving velocityvr（t）.

Fig.1 Obstacle avoidance of the formation-containment system.

At the same time,all leader spacecraft will keep the desired relative positions δij. For follower spacecraft, they do not have advanced sensors. Thus, they can use the communication information from leader spacecraft to estimate the moving velocityvr（t）. At the same time, to guarantee the motion safety, all follower spacecraft are required to be inside of the convex hull spanned by all leader spacecraft.

4. Illustrative example

In this section, a simulation example is given to illustrate the effectiveness of the designed SFCFC algorithm.

We consider a group of 10 spacecraft with 6 leader spacecraft (E=｛1，2，...，6｝) and 4 follower spacecraft(F=｛7，8，...，10｝). In this simulation example, the flying reference of the formation-containment system follows a nearcircular orbit with the initial orbit elements

The directed communication topologyGAamong 10 spacecraft is described in Fig.2.

The Laplacian matrix of the directed graphGAis

where

Fig.2 Communication graph among the 10 spacecraft.

The desired time-varying formation consists of a common time-varying translational velocityvr（t） and distributed relative positions δik（i，k=1，2，...，6） among connected leader spacecraft. In this example,vr=[0，1，50 sin（0.0081t）]Tm/s.Only the first and sixth leader spacecraft know the desired velocityvr, i.e.,b1=1，b6=1，bi=0，i=2，3，...，5. The relative positions among connected leader spacecraft are

δ15=[50，150，0]Tm, δ21=[50，-50，0]Tm, δ23=[-50，150，0]Tm, δ36=[100，-100，0]Tm, δ54=[50，-50，0]Tm,δ56=[50，-150，0]Tm, δ62=[-150，50，0]Tm.

The initial states of the 10 spacecraft are

The simulation parameters in control algorithms Eq. (15),Eq. (43) are chosen as: εi=1，γ1=γ2=0.5，ηk=10，β=0.01，kj=4，ρ=0.002，i=1，2，...，6，j=7，8，...，10，k=1，2，...，10.

The simulation results are given as follows.

Fig.3 Trajectories of the 10 spacecraft.

Fig.4 Positions of the 10 spacecraft.

Fig.3 gives the flying trajectories of the 10 formationcontainment spacecraft. Fig.4 gives the positions of the 10 formation-containment spacecraft. Fig.5 gives the containment errors of the 4 follower spacecraft.Fig.6 shows the control forces of the 10 formation-containment spacecraft.

We can see from Fig.3 that the formation-containment of the 10 spacecraft is reached under the SFCFC algorithm Eqs.(15), (43), in which all leader spacecraft form the desired formation shape and move with the desired time-varying velocity finally and all follower spacecraft move into the convex hull spanned by the leader spacecraft. According to Fig.4, we can get that all spacecraft keep the desired relative positions inX-axis,move with a constant velocity and keep relative positions simultaneously inY-axis, and oscillate with sinusoidal form inZ-axis, which corresponds to the desired timevarying configuration. From Fig.5, we can get that the containment errors of the four follower spacecraft are asymptotically stable within 600 s. Fig.6 shows that the magnitude of control forces of leader spacecraft and follower spacecraft are within 30 N,and the control forces inZ-axis oscillate with sinusoidal form. All of these illustrate the effectiveness of the proposed SFCFC algorithm.

Fig.5 Containment errors of the 4 follower spacecraft.

Fig.6 Control forces of the 10 spacecraft.

5. Conclusion

In this study,the spacecraft formation-containment flying control problem has been investigated under a directed graph.First, we have designed a distributed formation control algorithm such that all leader spacecraft can converge to a desired formation with time-varying translational velocity. Then, we have proposed the distributed containment control algorithm such that all follower spacecraft can converge into the convex hull formed by the leader spacecraft. To handle the uncertainties and make the control algorithm fully distributed,the adaptive updating laws have been designed. Future work on the SFCFC problem will take the collision avoidance among spacecraft into consideration.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Nos. 61876050, 61673135, 61603114).