Robust attitude coordinated control for spacecraft formation with communication delays

Jin ZHANG,Qinglei HU,Dnwei WANG,Wenbo XIE

aDepartment of Control Science and Engineering,Harbin Institute of Technology,Harbin 150001,China

bSchool of Automation Science and Electrical Engineering,Beihang University,Beijing 100083,China

cSchool of Electrical and Electronic Engineering,Nanyang Technological University,Singapore 639798,Singapore

dCollege of Automation,Harbin University of Science and Technology,Harbin 150080,China

Robust attitude coordinated control for spacecraft formation with communication delays

Jian ZHANGa,Qinglei HUb,*,Danwei WANGc,Wenbo XIEd

aDepartment of Control Science and Engineering,Harbin Institute of Technology,Harbin 150001,China

bSchool of Automation Science and Electrical Engineering,Beihang University,Beijing 100083,China

cSchool of Electrical and Electronic Engineering,Nanyang Technological University,Singapore 639798,Singapore

dCollege of Automation,Harbin University of Science and Technology,Harbin 150080,China

*Corresponding author.

E-mail address:huql_buaa@buaa.edu.cn(Q.HU).

Peer review under responsibility of Editorial Committee of CJA.

Production and hosting by Elsevier

http://dx.doi.org/10.1016/j.cja.2017.01.014

1000-9361©2017 Chinese Society of Aeronautics and Astronautics.Production and hosting by Elsevier Ltd.

This is an open access article under the CC BY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).

In this paper,attitude coordinated tracking control algorithms for multiple spacecraft formation are investigated with consideration of parametric uncertainties,external disturbances,communication delays and actuator saturation.Initially,a sliding mode delay-dependent attitude coordinated controller is proposed under bounded external disturbances.However,neither inertia uncertainty nor actuator constraint has been taken into account.Then,a robust saturated delaydependent attitude coordinated control law is further derived,where uncertainties and external disturbances are handled by Chebyshev neural networks(CNN).In addition,command filter technique is introduced to facilitate the backstepping design procedure,through which actuator saturation problem is solved.Thus the spacecraft in the formation are able to track the reference attitude trajectory even in the presence of time-varying communication delays.Rigorous analysis is presented by using Lyapunov-Krasovskii approach to demonstrate the stability of the closed-loop system under both control algorithms.Finally,the numerical examples are carried out to illustrate the efficiency of the theoretical results.

©2017 Chinese Society of Aeronautics and Astronautics.Production and hosting by Elsevier Ltd.This is an open access article under the CC BY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).

Actuator saturation;

Attitude control;

Communication delays;

Neural networks;

Spacecraft formation

1.Introduction

Spacecraft formation flying has become an attractive topic during the last decade,and extensive effort has been put on this novel technique theoretically and practically.From the existing research literatures,a large number of advantages of the decentralized control algorithms have been shown,such as better system reliability and higher robustness.The control architectures for formation coordination can be categorized into four types,namely,the leader-follower structure,1the behavioral based approach,2the virtual structure3and graph-theoretical based technique.4

Practically speaking,the spacecraft formation subjects to various uncertainties such as the unknown disturbance and the time-varying inertia moment.Yang et al.5accomplished accuracy estimation of the external disturbances by using an extended observer.Considering parametric uncertainties and disturbances,Zhang et al.6proposed a sliding mode and adaptive coordinated tracking controller for multiple rigid spacecraft.As for a group of flexible spacecraft,Du and Li7employed backstepping control approach to solve the attitude synchronization problem.Neural networks(NNs)show effective performance to approximate the continuous functions over any arbitrary accuracy,which makes it be a favorable technique in nonlinear system controller design.Based on Chebyshev neutral network,Zou et al.8proposed a robust control law for the spacecraft in the presence of unknown inertia moment and external disturbances.Fazlyab et al.9designed an adaptive fault-tolerant controller for a rigid spacecraft by three-layered neutral network approximation technique.Zhao and Jia10developed a finite-time attitude synchronization control algorithm for multiple spacecraft formation by utilizing neural networks and modified fast terminal sliding mode.

Apart from the issues stated above,input saturation caused by the limitation of actuator is another challenge for spacecraft attitude maneuver.11–20Su and Zheng13proposed globally asymptotic saturated control laws for spacecraft attitude stabilization by modifying traditional proportional-derivative controller.Boskovic et al.14constructed a time-varying sliding mode control scheme to achieve attitude stabilization for spacecraft under input saturation and uncertainties.Bustan et al.15applied the similar approach to design fault-tolerant attitude tracking control law for a rigid spacecraft.Loria and Nijeijer16adopted hyperbolic tangent function to derive bounded outputfeedback for Euler-Lagrange system.Homogenous system theory combined with hyperbolic tangent function was used to obtain the finite-time coordinated control algorithms for a group of spacecraft formation in Refs.18,19.Zou and Kumar20investigated attitude coordinated control problem for multiple spacecraft,where neural network was employed and an auxiliary signal was used to compensate the exceeding value of control torque.

Moreover,time-delay is unavoidable whenever information transfers between neighboring spacecraft due to communication bandwidth limitation and obstructions in space.The ignorance of such phenomenon may degrade control performance and bring unpredictable effect to system stability.Under such circumstance,the need for developing control algorithms with consideration of the communication delays arises.Li and Liu21derived the adaptive sliding mode control laws for spacecraft formation with uncertainties and constant non-uniform delays.Zhou et al.22proposed a velocity-free decentralized attitude synchronization algorithm with time-varying delays.On basis of the previous research works,coordinated control schemes for spacecraft formation under directed communication topology were carried out.Regardless of external disturbance and parametric uncertainties,Li et al.23designed control law by using backstepping technique and finite-time method.The uncertain factors were also neglected in Ref.24.For the purpose of handling system uncertainties and unknown disturbances,adaptive attitude synchronization control laws were investigated in the presence of communication delays.25–27Especially,Du and Li27proposed attitude synchronized control law for multiple flexible spacecraft,where finite-time control technique was also involved.More recently,input saturation has been considered in the time-delay systems as well.Xu et al.28proposed a robust controller for uncertain discrete time-delay system with bounded external disturbances and input saturation.Abdessameud and Tayebi29studied synchronization algorithm for Euler-Lagrange systems with communication delays,where the upper bound of control input was related to control gains and the number of formation members.It is worth emphasizing that extending these results to spacecraft formation attitude coordinated problem is not straightforward owing to the inherent coupled nonlinear dynamics of spacecraft.

Inspired by the previous research works,we concentrate on developing attitude coordinated tracking control laws for spacecraft formation with communication delay,external disturbance,inertia uncertainties and even actuator constraints.Although attitude coordinated control problem for spacecraft formation has been widely studied,control strategy taking these above factors into consideration simultaneously can hardly be found,especially when the communication topology is directed.First,a sliding mode attitude coordinated controller is presented for a formation subjected to environmental disturbance,where communication delays are considered explicitly.Then,CNN approximation approach is applied to enhance system robustness to uncertainties and disturbances.Backstepping technique with command filter has been adopted to develop the robust saturated control law.Additionally,the information exchange topology between neighboring spacecraft is supposed to be directed,which not only leads to practical significance but also poses more challenges on controller design.

The organization of this paper is as follows.In Section 2,mathematical model of spacecraft formation attitude system and some preliminaries on graph theory are presented.The main work is stated in Section 3,where two novel attitude coordinated control laws are derived and sufficient conditions for system stability are obtained.Then,the effectiveness of the theoretical results is verified by numerical examples in the following section.Finally,the conclusion and remarks are made in Section 5.

2.Preliminaries

2.1.Spacecraft formation attitude kinematics and dynamics

Unit quaternion is introduced for attitude kinematics of theith spacecraft30

The dynamic model of theith spacecraft is governed by

As attitude coordinated tracking problem is addressed in this article,the reference trajectory for the spacecraft forma-tion is described bylar velocity ωd∈R3. The attitude tracking errorand angular velocity tracking error ωei∈ R3are introduced,which are defined asand ωei= ωi-Reiωd.Rei=represents the rotation matrix from the body frame to the reference frame.LetQ（qei）which is denoted asQeifor brevity and satisfiesand we obtain

Assumption 1.The external disturbancediis assumed to be bounded,i.e.‖di‖ ≤dMwithdMbeing a positive constant.

Assumption 2.The desired angular velocity ωdand itsfirsttime derivative˙ωdare supposed to be bounded.

2.2.Graph theory

We assume that the information exchange of the spacecraft formation isdescribed by a fixed and directed graphG（A）≅ ｛V，E，A｝.Each spacecraft is denoted as a node in the graph,andV= ｛vi｝ （i=1，2，...，n）denotes the set of nodes.E⊆V×Vpresents the set of the edges andA= ［aij］∈ Rn×nis the adjacency matrix of the graph with non-negative adjacency elementsaij.The directed edge（vi，vj）∈Eexits if there is an available information channel from theith spacecraft to thejth spacecraft,and thus it hasaij＞0 correspondingly.Otherwiseaij=0,if nodeiand nodejare not connected.Besides,it is supposed thataii=0 for alli=1，2，...，n.The Laplacian matrix of the graphGis definedi≠j.

Assumption 3.Throughout this paper,the communication graphGof the formation is supposed to be fixed and directed,and contains a spanning tree.Moreover,the reference information can be reached by at least one spacecraft in the formation.

Lemma 1.31If the directed graph G has a spanning tree,then0is a simple eigenvalue of L*with INas its right eigenvector,and all the other N-1eigenvalues have positive real parts.

3.Main results

As the main results of this paper,two novel distributed attitude tracking control algorithms are proposed for the spacecraft formation with communication delays.First,a sliding mode control law is proposed on the condition that the inertia matrices can be measured accurately and there is no constraint on the actuator torque.Then,improvements are made to accomplish the control objective of developing a bounded control law with high robustness.Neural network is involved to handle the system parametric uncertainties and the external disturbances.Backstepping technique with command filter is adopted to solve input saturation.

3.1.Delay-dependent attitude coordinated controller design

In this subsection,the inertia matrixJiis supposed to be precisely known asa prior.The sliding mode manifoldsei= ωei+ εqeiis employed with ε being a positive constant.For theith spacecraft,a sliding mode delay-dependent attitude coordinated control algorithm is proposed:

whereTdenotes time-varying communication delays between the neighboring spacecraft,which always satisfies 1-＞0;L=L*⊗I3×3and λmax（LLT）denotes the maximum eigenvalue of matrixLLT;the robust control term-ρsgn（sei）is introduced to restrain external disturbance with the positive constant ρ satisfying ρ ＞dM;the control gainsk1,k2andk3are positive constants satisfying 2（k2+ εk1）λmin（J）≥where λmin（J）and λmax（J）denote the minimum and maximum eigenvalue of matrixJandJ=diag（J1，J2，...，Jn）.

In order to address the stability of systems with time-delays,the following lemmas are given.

Lemma 2.32For a continuous function（x）and its derivative（x）defined on［a，b］,we have

Lemma 3.32Foranysymmetric positive definite matrix M∈ Rn×n,constants r1and r2satisfy r1＜r2,and the vector functionυ:［r1，r2］→ Rnsuch that

Theorem 1.Considering spacecraft formation governed by Eqs.(1)–(3),if the sliding mode attitude coordinated controller Eq.(7)is adopted and the system parameters are selected to satisfy the following inequalities,

Then,the closed-loop system achieves asymptotically stable,i.e.the attitude error and angular velocity error of each spacecraft converge to zero asymptotically.

Proof.Substituting controller Eq.(7)into Eq.(6)and rewriting the closed-loop system into vector form lead to

Due to the fact that 1+qei0∈ （0，2）,it can be obtained that

Obviously,V1is proved to be positive and definite if 2（k2+ εk1）λmin（J）≥（J）holds.Then,taking the time derivative ofV1gives

By using Young’s inequality,33the following inequalities can be verified:

If ｜λmax（LLT）｜I3n×3n-LLTis positive definite and Eqs.(8)–(10)are satisfied,it can be concluded that˙V1≤0.Hence,it causes thatV1is bounded,which implies ωe∈ L2∩L∞,qe∈ L2∩L∞, ωe（t-T）∈ L∞andqe0∈ L∞.Recalling the closed-loop system,we get˙ωe∈L∞.Additionally,it can be proved that˙qe∈L∞from the kinematic model Eq.(5).By using the Barbalat’s lemma,34it follows that limt→∞qe=0 and limt→∞ωe=0,i.e.the attitude and angular velocity of each spacecraft converge to the reference trajectory asymptotically.□

Remark 1.Given Lemma 1 and the fact that the communication topology contains a spanning tree,λmax（LLT）is a positive constant.Hence,it leads to ｜λmax（LLT）｜I3n×3n-LLT＞ 0,which guarantees the above analysis.

Remark 2.In Theorem 1,the inertia tensorJiis assumed to be known precisely and the limitation of actuator torque has not been considered.However,such assumptions can barely be satisfied in practical cases.Thus,a robust saturated controller is further derived in the following section.

3.2.Robust saturated delay-dependent controller design

In the preceding section,the sliding mode delay-dependent control law is proposed without inertia uncertainties and input saturation.Although the robustness against external disturbance and communication delays is ensured,it is inevitable to bring large gain as the disturbances and uncertainties increase.Nevertheless,the control torque ought to be bounded because of the physical limitation of actuator onboard.There-fore,the necessity of developing control law with input saturation is obvious.

Before the control algorithms are derived,the following assumption is made.

Assumption 4.The inertia of each spacecraft is divided intoJi=J0i+ ΔJi,whereJ0idenotes the nominal part and ΔJithe uncertain part.The norm ofJ0isatisfiesandbeing positive constants.

Then,the kinematic and dynamic model of the spacecraft formation are reformed:

Thus the signaluiwill be constraint to an allowable scope¯u0before acting on individual spacecraft.Remark 3.In the view ofand Assumption 4,it is easy to verify thatg0（qei）=

Remark 4.Note thatFeiwill exit whenQeiis invertible,which implies that det（Qei）qei0（t）≠ 0 fort≥ 0.Thus,the initial state and control strategy should be chosen to guaranteeqei0（t）≠ 0 for all time.

From Eq.(20),it can be seen that the nonlinear termDi（·）is important for system analysis.Given the remarkable performance of neural network for approximating uncertain dynamics and nonlinear functions,the Chebyshev neural network(CNN)is employed to facilitate control law design.Consequently,the unknown function is approximated by

whereXi=the vectorXi; θi（Xi）= ［1，N1（Xi，1），...，Np（Xi，1），...，N1（Xi，13），...，Np（Xi，13）］T∈ R13p+1is the CNN basis function withpbeing the order of Chebyshev polynomials;∈ R3×14p+1denotes the optimal approximation weight matrix of the output layer.The Chebyshev polynomials can be obtained by using the twoterm recursive formula given by

Assumption 5.The optimal function approximation is bounded so that tr（W*TW*）≤is a positive constant.

It is clear that as the states of the closed-loop system are driven to the reference trajectory,qeiandveiconverge to zero,and the uncertain termDi（·）will converge to the reference nonlinear functionDr（·）=Assumptions 1–5 show that the CNN approximation error denoted as φiis bounded such that｜φi｜≤is a positive constant.

To facilitate the backstepping command filter technique,an auxiliary system is introduced:

where ϑiis the virtual control input;ξ1iand ξ2iare applied to obtain the saturated control law.

We denote Δui=u0i-uias the extra part of control signal,which should be compensated in the design procedure.For this purpose,ξ1iand ξ2iare defined in the form of command filter:

where η1and η2are positive constants to be selected.With the filter Eqs.(25)and(26),Δuiis involved in system analysis so that the saturated control law can be verified explicitly.

The saturated attitude coordinated tracking control algorithm with the adaptive tuning law for estimating the optimal weight matrix is proposed:

whereh1,h2,h3,γ and μ are positive parameters;is used to estimate the matrixdenote the weight matrix estimation e

Theorem 2.Considering spacecraft formation governed by Eqs.(1)–(3),if control law defined as Eqs.(27)and(28)is adopted and system parameters are selected to satisfy inequalities(29)–(31),

Then,the trajectory of the closed-loop system can be guaranteed to be uniformly ultimately bounded(UUB).

Proof.In order to derive the virtual control signal,the Lyapunov candidate function is selected:

Let the virtual control be ϑi=-η1qei,and thus it leads to

In view of Eq.(35),taking the control signaluidesigned as Eq.(27)into system stability analysis directly is possible and reasonable.□

Then,to show the stability ofz2iandi,a Lyapunov function is constructed as

and adaptive law Eq.(28)into Eq.(36)results in

Due to the fact that

Hence,Eq.(37)further leads to

In addition,ξ1iand ξ2ishould be taken into account to guarantee stability of the closed-loop system.Thus,we choose a Lyapunov functionVcas

where ξ2= ［ξ21，ξ22，...，ξ2n］Tis defined.Taking the time derivative ofVcalong the command filter(25)and(26),we have

whereg0（qe）=diag（g0（qe1），g0（qe2），...，g0（qen））, Δu=［Δu1，Δu2，...，Δun］T.

Moreover,considering the communication delays,we adopt the following Lyapunov-Krasovskii candidate:

To this end,the Lyapunov function for the closed-loop system is constructed as

Based on the analysis above,the time derivative ofV2is easy to get:

where ς1,ς2,ς3,ς4and Θ are defined as the following:

Remark 5.In fact,the magnitude ofudecreases as the states of each spacecraft converge to the reference trajectory,which ensures that the final value of Δuis zero,i.e.limt→∞Δu=0.Then,it can be deduced from Eq.(26)that ξ2is asymptotically stable,and it also implies that ξ1can achieve asymptotical stability from Eq.(25).Thus,system Eqs.(23)and(24)reduce toz1i=qeiandz2i=vei-ϑiafter certain time,which is accordant with the conclusion of Theorem 2.

Remark 6.It should be mentioned that to construct the Lyapunov candidate functionVd,the communication delayTis used in Eq.(43).However,the appearance of unknownTis acceptable,which would not bring any trouble to control law design,because the exact value ofTis related to the selection of control parameters rather than the structure of controller Eq.(27).

Remark 7.From the proof procedure,it is noticeable that the distributed control law defined as Eqs.(27)and(28)is free from the restriction of communication topology.Hence,the attitude coordinated tracking can be achieved only if Eqs.(29)–(31)are satisfied.Thus the condition on communication within the spacecraft formation is further relaxed.

4.Simulation and analysis

A formation of four spacecraft under a fixed directed communication topology is employed in the numerical work.Additionally,the performance of control algorithm in Ref.35has been examined as well,which is presented as below:

Thus the improvement of the proposed control law can be verified.To ensure the fair comparison,the system states and communication topology for the proposed control laws and controller Eq.(47)are selected to be the same.

The initial attitudes,angular velocities and inertia matrices for each spacecraft are presented in Table 1.A virtual dynamic leader ξrprovides reference trajectory for the multiple spacecraft formation,which rotates with the angular velocity of ωd= ［0.1cos（0.01t），-0.1sin（0.01t），-0.1cos（0.01t）］Trad/s,and the initial attitude is chosen asd（0）=［0.8062，-0.1，0.5，-0.3］T.The communication topology is described in Fig.1,where ‘SC1”, ‘SC2”, ‘SC3” and ‘SC4”represent the four spacecrafts in the formation.The communication time-delay between the neighboring spacecraft is supposed to beT=0.15sin（0.02t）s.Without loss of generality,the external disturbance is selected as

The response curves of closed-loop system by using controller Eq.(7)are carried out in Figs.2–4.The control gains are selected as ε=0.5,ρ =0.05,k1=3,k2=5,k3=1.Figs.2 and 3 illustrate the absolute attitude errors and angular veloc-ity errors of each spacecraft.And it can be observed that the attitude tracking error converges to zero with the precision of｜qeiw｜＜ 6 × 10-3（w=1，2，3）in less than 20 s.

Table 1 System parameters of spacecraft formation.

The performance of angular velocity tracking during the transient phase and the final accuracy are acceptable.The time history of the control torques under controller Eq.(7)is depicted in Fig.4.As there is no restriction effect on the actuators,the magnitude generated by the control law reaches｜uiw｜max=4 N ·m approximately.

Then,simulated results under controller Eq.(47)are shown in Figs.5–7 with the control gains chosen asci=1,=3.5,Fig.5,it is observed that the attitude tracking errors drop to the attraction of｜qeiw｜＜ 2 × 10-2（w=1，2，3）within 40 s.

The corresponding angular velocity errors are given in Fig.6,and the response curves of control torque are described in Fig.7.Compared with the performance under control law Eq.(7),the final precision of attitude tracking has been reduced whereas the control torque remains the same level.Hence,the conclusion can be made that the proposed controller Eq.(7)guarantees better robustness and higher accuracy.

System performance under the robust saturated delay

-Acknowledgements

dependent controller(denoted as RSDDC)is presented in Figs.8–10.The control parameters are chosen as η1=5,η2=1,h1=1,h2=0.5,h3=0.1,γ=0.1,μ =0.05.The attitude tracking errors are presented in Fig.8,from which it can be seen that the vector parts ofqeidrop to a small region around zero,i.e.｜qeiw｜＜ 7 × 10-3（w=1，2，3）at 20 s.

The angular velocity tracking errors are recorded in Fig.9,correspondingly.Fig.10 presents the actual control torque under the RSDDC with the maximum allowance of0=2 N·m.It shows that the control signals calculated by RSDDC have exceeded the given upper bound at the beginning.However,the actual control torque can be constrained within ±2 N·m during the whole simulating period.Then,control torques decrease into the allowable scope as the state errors illuminate,which reduce to a small region of zero finally.Although the control torque of the RSDDC has been cut down considerably compared with that in Fig.4,system performance has not been degraded.

In order to compare system performance,the allowable bound of control torque generated by control law Eq.(47)should be set as same as that under RSDDC,i.e.｜uiw｜max=2 N ·m.The control parameters are chosen as the former case,and thus the work in the following Figs.11–13 is carried out.The attitude tracking errors are shown in Fig.11,which imply that the vector parts of quaternion converge to a close neighborhood around the equilibrium,i.e.｜qeiw｜＜ 2 × 10-2（w=1，2，3）at 40 s.The angular velocity errors are shown in Fig.12,and the response curves of control torque with constraint are described in Fig.13.It is observed that the control toque is restricted within ±2 N·m during the simulation period.However,the attitude tracking accuracy is apparently lower than that of RSDDC.Thus the superiority of the proposed robust saturated delay-dependent controller has been illustrated.

The effectiveness of the two attitude coordinated control laws for spacecraft formation with communication delays are verified by summarizing both cases above.Moreover,better robustness and faster response speed can be achieved by using the robust saturated control algorithm.

5.Conclusions

(1)The attitude coordinated tracking problem for the spacecraft formation is studied in this article,where inertia uncertainties,external disturbances,input saturation and communication time-delay are taken into accounted simultaneously.

(2)With the Lyapunov-Krasovskii approach,two attitude coordinated control laws are developed under the fixed directed topology,thus communication delays are explicitly addressed and overcome.Moreover,the delay-dependent control schemes reduce the conservatism of the concerned system.

(3)In addition,the time-delay between different formation members are supposed to be symmetrical,which limits the feasibility of the proposed algorithms.Hence,asymmetric communication delays and control strategies under switching topology will be further investigated in our future research work.

This study was co-supported by the National Natural Science Foundation of China(Nos.61633003 and 61522301),Heilongjiang Province Science Foundation for Youths(Nos.QC2012C024 and QC2015064),and the Research Fund for Doctoral Program of Higher Education of China(No.20132302110028).

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24 May 2016;revised 26 September 2016;accepted 31 October 2016 Available online 15 February 2017