Adaptive control for attitude coordination of leader-following rigid spacecraft systems with inertia parameter uncertainties

Xiaokui YUE,Xianghong XUE,*,Haowei WEN,Jianping YUAN

aNational Key Laboratory of Aerospace Flight Dynamics,Northwestern Polytechnical University,Xi'an 710072,China

bSchool of Astronautics,Northwestern Polytechnical University,Xi'an 710072,China

KEYWORDS Adaptive control;Attitude coordination;Leader-following consensus;Non-certainty equivalence;Spacecraft formation flying

Abstract This paper studies the leader-following attitude coordination problems of multiple spacecraft in the presence of inertia parameter uncertainties.To achieve attitude coordination in the situation that even the leader's attitude is only applicable to a part of the following spacecraft,a nonlinear attitude observer is proposed to obtain an accurate estimation of the leader's attitude and angular velocity for all the followers.In addition,a distributed control scheme based on noncertainty equivalence principle is presented for multiple spacecraft'attitude synchronization.With a dynamic scaling,attitude consensus can be achieved asymptotically without any information of the bounds of the uncertain inertia parameters.Furthermore,once the estimations of inertia parameters reach their ideal values,the estimation process will stop and the ideal value of inertia parameter will be held.This is a special advantage of parameter estimation method based on non-certainty equivalence.Numerical simulations are presented to demonstrate that the proposed non-certainty equivalence-based method requires smaller control toque and converges faster compared with the certainty equivalence-based method.

1.Introduction

Spacecraft formation flying has drawn extensive attention in the last decade due to its higher flexibility and robustness,greater efficiency and lower fuel consumption.1-4Attitude coordination has been expected to be a necessary and important technology for many space missions,such as the CanX-4&5 primarily supported by Canadian Space and the SULFRO mission concept led by Chinese Academy of Sciences.5Compared with the leaderless case studied in Ref.6,the leader-following attitude coordination control is more challengeable since it not only ensures the consensus of the followers'attitude,but also enables the followers'attitude and angular velocity to be in line with those of the leader.7,8If there is only one follower,the leader-following problem degenerates into the attitude tracking problem of spacecraft.9,10

According to how the followers access the leader's states,the methods for tackling leader-following problem can be classified into decentralized and distributed methods.The decentralized control algorithms assume that the leader's state is accessible to every follower in the group.11-16A passivitybased controller for a group of spacecraft without inertial frame information was developed in Ref.13.Meng et al.designed two distributed containment control laws for attitude consensus of spacecraft,including both static and dynamic leaders.14The result indicates that the attitude of the followers will converge to the convex hull formed by all leaders.A nonsingular fast terminal sliding mode controller was proposed to counteract the impact of external disturbances and actuator failures in Ref.15Attitude coordination with communication delays,caused by the distance between satellites,was taken into consideration in Ref.16,17

In practice,it is more realistic that only a part of the followers can access the leader's attitude.Several distributed control laws have been presented to solve this problem in Ref.18-22Ren firstly studied distributed attitude synchronization of the leader-following problem with attitudes represented by modifi ed Rodriguez parameters.18Cai and Huang proposed a distributed nonlinear estimation method of leader's state for followers under the assumption that the communication network of the spacecraft systems is undirected and connected.19This work was extended to the case without the measurements of angular velocity by employing an angular velocity auxiliary system in Ref.20Zou et al.also studied the attitude coordination of the leader-following spacecraft systems without angular velocity measurement under undirected communication graph and proposed a finite-time observer to obtain the unmeasurable angular velocity of the leader.21Du extended attitude consensus of leader-following spacecraft systems to the situation with both rigid and flexible spacecraft.22However,all the above literatures did not consider attitude coordination with uncertain inertia parameters.

Uncertainties of the spacecraft,caused by fuel consumption,appendage deployment and onboard payload transformation, are challenging to spacecraft attitude synchronization.Attitude tracking for single spacecraft with unknown inertia parameter has been investigated in Ref.9,10,23-25However,the development from attitude tracking problem to leader-following problem is nontrivial,especially when the leader's information cannot be directly accessed by all the followers.Several authors have studied the leaderfollowing attitudecoordination problem associated with parameter uncertainties.26-30Wu et al.studied the attitude synchronization and tracking problem of spacecraft with parameter uncertainties.26,27In these papers,the uncertainties were assumed to be a known nominal part and an uncertain part.A robust nonlinear controller for both relative position tracking and attitude synchronization with external disturbance and model uncertainty was presented in Ref.28However,initial values of the inertia parameters were regarded as known constants.Recently,Cai and Huang proposed a distributed control algorithm based on Certainty-Equivalence(CE)for multiple spacecraft systems in the presence of inertia parameter uncertainties.31The authors later extended their research to the condition with both uncertainties and external disturbances.32Although the asymptotic convergence of the tracking errors is easy to achieve in the existing literatures,the parameter estimation process is directly affected by tracking error dynamics,which could result in bad transient performance.

The main aim of this paper is to provide a highperformance solution for the problem of attitude coordination for leader-following spacecraft in the presence of inertia parameter uncertainties.Unlike the CE-based approaches in attitude consensus with uncertainties,a distributed control law based on dynamic-scaling-based non-certainty equivalence is proposed.By setting an upper bound for dynamic scaling factor,the proposed method can not only make the convergence rate of attitude coordination faster,but also avoid the bad transient performance compared with CE-based method.In addition,the asymptotic convergence of tracking error under all the possible initial conditions is guaranteed without considering the boundary information of system uncertainties.Moreover,the distributed observer proposed in Ref.20is modified and applied to overcome the difficulty that the leader's attitude and angular velocity are only accessible to a part of the followers.This distributed observer guarantees that the attitude estimation made by the followers will converge to the leader's attitude,and then all the followers can track the estimation to achieve attitude consensus with the leader.

The following of this paper is organized as follows.The attitude dynamics of spacecraft and problem formulation are brie fl y described in Section 2.Section 3 introduces a nonlinear distributed observer.The controller is given and its stability analysis is made in Section 4.Simulations are presented in Section 5 to illustrate our results.Conclusions are given in Section 6.

2.Preliminaries and problem formulation

2.1.Algebraic graph theory

In this subsection,algebraic graph theory is brie fl y introduced to describe the communication networks among the leaderfollowing spacecraft system.A directed graph(or digraph)is represented as,in whichrepresents a vertex set,and E（ G）?V（ G）×V（ G）represents an edge set of ordered pair of vertexes.The edge set consists of elements of the formif vertex vjcan get some information from vertex vi. An adjacency matrixof digraph G is de fined as: ifaij＞0,otherwise aij=0.With the adjacency matrix A（ G）,the Laplacian matrixdigraph G is given as:if

In the leader-following spacecraft system,v1，v2，...vNrepresent the N followers and G represents the communicating relations between all following spacecraft.The leader is denoted as v0and the edges between the leader and all followers are denoted as Elf.Then,the corresponding communication network s of leader-following spacecraft system can be represented asisthevertex set,is the edge set andis the adjacency matrix ofˉG.The adjacency matrix between the leader and all following spacecraft is denoted bylower can access the state of the leader,otherwise ai0=0.Since there is no edge from the follower to the leader,a0i=0 for all i=1，2，...，N.If not otherwise specified,matrix L（ G）andare abbreviated as L and B in the following sections respectively.

Assumption 1.The communication digraphˉG of leaderfollowing spacecraft system contains a spanning tree,in which the leader spacecraft(v0inˉG)is the root.Remark 1.Assumption 1 is the weakest condition to guarantee the accessibility of the leader's attitude information by all the followers.

2.2.Problem formulation of attitude synchronization

Consider a system containing N rigid spacecraft with the following attitude dynamic equations:

where Ji∈ R3×3is the symmetric and positive definite inertia matrix of the i-th spacecraft. A quaternion q=［ˉq，^q］T∈R×R3is denoted as q∈Q,and a unit quaternion is denoted as q∈Quif it satisfies the constraint=1.qi∈Quis a unit quaternion describing the orientation of the body frame Biof the i-th spacecraft with respect to the inertial frame;denotes the attitude velocity of the body frame Biof the i-th spacecraft relative to the inertial frame I;ui∈R3denotes the control torque of the i-th spacecraft.The variable quantities ωi，Ji，uiare all expressed in Bi.The operator q（·）converts a vector x ∈ R3to a quaternionThe notation ω×forrepresents the following skew symmetric matrixThe operator⊙ represents the quaternion product: for qi，qj∈ Q,

The reference attitude q0of the leader is generated according to the following system:

In view of Eqs.(1)and(2),the attitude tracking errors between the ith spacecraft and the leader are given as whererepresent the unit quaternion attitude tracking error and direction cosinematrix between frameBiand B0,respectively;I3∈R3×3denotes the identity matrix;~ωirepresents the angular velocity tracking error between ωiand ω0.From the Definition above,the attitude tracking error dynamics are obtained as follows:

The purpose of this paper is to solve the following problem associated with parameter uncertainties.

Problem 1.Given systems(1)and(2),under Assumption 1,design a distributed control law such that,for all ωi（0）∈ R3and all

Assumption 2.The leader's angular velocity ω0（t）is generated the same as19

where ω0∈ R3，S ∈ R3×3,and system(5)is assumed to be marginally stable.

Assumption 3.The leader's angular velocity ω0（t）satisfies the persistent excitation inequality29

for positive constants t0，T0and ?.

Remark 2.Assumption 2 implies that the leader's angular velocity ω0is bounded.Additionally,according to system(2),q0is bounded too.The model of similar form has been widely used in Ref.19,20,30-32However,these literatures all assume that the matrix S is known to all followers to make the follower be able to generate the same angular acceleration as the leader.To remedy the unknown of matrix S,an estimation of S is proposed based on Immersion&Invariance(I&I)theory with Assumption 3.It is noted that persistent excitation is widely used to ensure the convergence of parameter.

2.3.Lemmas

To prove the main result in this paper,we need the following lemmas:

Lemma 1.The matrix H=L+B is positive stable if and only if digraphˉG contains a spanning tree with the leader spacecraft as the root.33

Lemma 2.Let x1∈ Qu，x2∈ Q，x3=x1⊙x2and x4=x2⊙x1.Then xT2x2=xT3x3=xT4x4.20

3.Distributed observer

Since the attitude of the leader is not available to all the following spacecraft,a distributed observer is proposed to estimate the leader's attitude for all other spacecraft.To estimate the angular velocity and angular acceleration of the leader,we de fine a dynamic compensator similar as19

where

and ηi∈Q,ξi∈R3,i=1，2，...，N represent the leader's attitude and angular velocity estimated by the i-th follower,respectively;η0=q0,ξ0=ω0,ν1， ν2are positive real numbers,anddenotes the estimation of S by the i-th spacecraft.Letdenotes the vector of all ones,and IN∈RN×Ndenotes the identity matrix.In view of these Definitions,Eqs.(5)and(7)can be rewritten in a compact form as

For i=1，2，...，N,let ζi=ηi-q0,?i=ξi-ω0denote the attitude and angular velocity error between the leader and the followers' estimation,andand then,the time derivative of ? is given as

Since S is marginally stable, letdenotes the elements of matrix S,and then it can be easily veri fi ed thatwhere

Let γSbe assigned as

and thus,the dynamics of the estimation error~θSis further reduced to

Next,we de fine the estimation βSas

where

and Q1（ξi），Q1（ξi），Q1（ξi） ∈R6are the column vectors of QT（ξi）.

Lemma 3.Considering systems(2)and(5),and the estimations given in Eqs.(7),(11),(12)and(14)with Assumptions 1-3,if ν1， ν2＞0,q0（0） ∈Qu,ω0（0） ∈R3,ηi（0） ∈Q,ξi（0） ∈R3,then ηi（t）and ξi（t）exist,are bounded for all t＞0,and satisfyandexponentially for all i=1，2...，N.

Remark 3.Assumption 1 is the premise of the establishment of Lemma 3 and it ensures that all the followers can access the leader's state.Then,the followers can utilize the distributed observer given in Eq.(7)to estimate the leader's attitude,angular velocity and angular acceleration.Lemma 3 indicates that the estimation ηiand ξiwill converge to the leader's attitude q0and angular velocity ω0,respectively.In addition,Remark 1 shows that q0and ω0are bounded and then ηiand ξiare also bounded.

Since the leader's state might not be accessible to all the followers,the error Eqs.(3)and(4)cannot be employed directly.Instead,for each follower,the error signal is de fined as20,31

where k1i＞0 denotes a control gain,ei∈Q is the attitude error,is the angular velocity error, anddenotes the corresponding rotation matrix.Differentiating Eq.(15)leads to the following error system:

where pi=C（ ei）ξi，pdi=C（ ei）^Siξiand

Remark 4.Eq.(4)represents the real errors between the followers'attitude and that of the leader;however,Eq.(16)represents the errors between the followers'attitude and the estimated attitude of the leader by the i-th follower.Lemma 2 ensures that the estimated attitude of the leader by the followers is convergent to the leader's attitude.Therefore,the convergence of system(16)guarantees the convergence of system(4).In addition,Lemma 3 indicates→0 exponentially ast→∞,and then by Remark 3.2 in Ref.20,ε1i， ε2i， ε3i→0 exponentially as t→ ∞.

4.Dynamic-scaling-based non-certainty equivalent controller

4.1.Distributed controller design

To deal with the unknown inertia matrix

a model parameter vector is de fined asThen it can be easily veri fi ed that

where

For the system de fined in Eq.(16),the corresponding parameter estimation^θiof θiis given by

where βiand γiare two functions to be specified,and Φidenotes the measurable or obtainable signals that are independent of ωi.Therefore,the parameter estimate error becomes

Consequently,the dynamics of the parameter estimation error can be obtained by substituting Eq.(1)into the time derivative of Eq.(22)to yield the following:

where denotes the j-th element of Φi.

Consider a regression matrix Wi∈R3×6，i=1，2，...，N

where δi， φi， αi，i=1，2，...，N are positive constants;k2i∈R denotes a dynamic control gain to be determined.Hence,the time derivative of~θican be written as

Let γiand the control input uibe assigned as

Substitute Eqs.(26)and(27)into Eq.(25),and the time derivative of the parameter estimate error is simplified as

As in Ref.34,an‘‘a(chǎn)pproximate”solution βican be generated as

where ρiis a constant real number,Wi（is the regression matrix in Eq. (24) whereis replaced byare the column vectors ofand the estimated angular velocity^ωiis de fined as

where ψi∈R is a dynamic observer gain to be determined.Then partial derivative?βi/?ωican be obtained as

From the illustration in Appendix,the following inequality holds between Δiand the observation error^ωi-ωi

The details of ΔiandˉΔiare given in Appendix A.2.

Substitute Eq.(31)into Eq.(28),and the time derivative of the parameter estimate error becomes

By applying Eqs.(24)and(27),Eq.(16)can be further derived as

Remark 5.In Eq.(34),the dynamics ofis directly related to,which is different to the parameter estimation scheme given in Ref.20,31based on CE principle.Therefore,≡0 holds when=0,i.e.,once the estimationreaches the value of θi,the estimation process will stop and the ideal values of inertia parameter will be held.

4.2.Dynamic scaling

Since the inertial matrix Jiis unknown,the minimum eigenvalue of the matrix cannot be used in controller.To overcome this difficulty,a scaled estimate error is de fined as

where λi＞0 is the minimum eigenvalue of Ji.The scaling factor is de fined by the following differential equations:

From Eq.(37),it can be obtained that ri（t） ≥1 holds for all t≥0.By differentiating Eq.(36)and using Eqs.(34)and(37),it follows that

Consider a Lyapunov function

With Eq.(38),its derivative with respect to time is obtained as

where

Since Jiis positive definite,λminis positive too.It can be concluded that˙Vz≤0,and then,and ziare bounded.

4.3.Stability analysis

Theorem 1.Given systems(1),(2)and(5)with unknown inertia matrix Ji,using the control law(27),the parameter estimation in Eqs.(7),(11),(12),(14),(25),(26)and(29)with the dynamic gains k2iand ψiare de fined as

where k3i，k4i，i=1，2，...，N are positive constants.Then,the parameter estimation error~θ is bounded,^ωi（t）-ωi（t）will converge to zero,and Problem 1 is solvable.

Proof.Utilizing Eqs.(1),(24)and(30),the time derivative of-ωi（t）can be obtained as

Now,consider a Lyapunov-like function

Since ri（0） ≥1 due to Eq.(37),V1is positive definite.Using Eq.(33),Eq.(43)and the inequality

the time derivative of V1is obtained as

Taking the time derivative of Eq.(48)yields

Finally,to show the convergence of tracking errors,we consider the following Lyapunov function:

Taking the derivative of V3in Eq.(50)and using Eqs.(35),(42)and(46),we can obtain a s t→∞.With Eq.(35),one hasand=0.By Remark 3.3 in Ref.20,we obtain that=0 and=0.

Since ωi,and^e are bounded,Wiand Δiare bounded with Eqs.(24)and(32).By Eq.(37),is bounded and then by Eqs.(38)and(44),andare bounded too.Thusis bounded and using Barbalat'slemma again onehas=0 and hence=0.Therefore,-ωi=0 according to Eq.(44).The proof is completed.

Remark 6.Although the external disturbance is not considered in Eq.(1),σ-modification can be employed to ensure the attitude tracking and parameter estimation errors against bounded external disturbance.Considering a bounded external disturbance di（t）∈ R3,is added to the right-hand side of Eq.(1),where dmaxis a positive constant.Add a leakage term-ρiσi（βi+ γi）to the right-hand side of Eq.(18),where σiis a positive constant,and then the dynamics of~θ is given by

Consider the same Lyapunov function Vzin Eq.(39)with the same dynamic scaling mechanism in Eq.(36),the timederivative of Vzbecomes

where

Via Eq.(16),it is obtained that˙ˉωiand˙eiare bounded,and then Eq.(50)ensures that¨V3is bounded too.According to Barbalat's lemma,˙V2and hence˙ˉωiand˙^e will converge to zero

where

Letusde finethescaling factorwith thefollowing derivative:

where k5iis a positive constant.Then˙Vzbecomes

Eq.(55)ensures ri≥ri（0） ≥1.Hence,the following inequalities are established by utilizing exp（ x） ＞x:

Let σi＞k5i/λi,and then the boundedness of zican be obtained according to the second term on the right-hand side of Eq.(58).Following the same design process in the proof of Theorem 1,if we chose the constant k3isatisfying k3i＞1/2,the observation error^ωi-ωiwill be ultimately attracted to the set:

Similarly,if φi＞ αi/2 and δi＞1/2,the attitude tracking errorsandwill also be attracted to the set:

Therefore,in view of Eqs.(55),(59)and(60),Δiand riare bounded.In addition,the size of S1iand S2ican be made arbitrarily small by increasing the constants k3i,φiand δi.Moreover,if the external disturbance is slowly time-varying,it can be handled as unknown constant parameters,which will lead to the asymptotic convergence of^ωi-ωi,^eiandˉωi.

5.Simulation

Consider a leader-following spacecraft system,the followers'attitude dynamics are described in Eq.(1)with the following inertia parameters:

The leader's angular velocity ω0is generated by Eq.(5)with the following matrix:

The simulation is conducted with the following initial conditions: ω0（0）=［2，1，1］T,ωi（0）=［0，0，0］T,andThe initial attitudes of spacecraft are given asIn addition,the control gains are de fined as ν1=20,ν2=20,k1i=10,k3i=2,αi=5,δi=0.5,φi=10,ρi=0.1,k4i=10-6and ri（0）=1.The results based on CE in Ref.31are selected as the control group,in which the control gain is given as k1=10 and k2=10.

The communication graphˉG is shown in Fig.1 and the corresponding adjacency matrix is

Fig.1 Communication digraphˉG.

Fig.2 Norm of angular acceleration errors||^Siξi（t）-Sω0（t）||2.

Fig.3 Norm of attitude tracking errors

Fig.4 Norm of angular velocity tracking errors

Simulation results of four followers are presented in Figs.2-6.To simplify the figures,the norm of errors is used to replace the errors in these figures.Fig.2 shows that the angular velocity error decays to zero,which indicates that the estimation angular acceleration^Siξi（t）ultimately coincides with the practical angular acceleration Sω0（t）.The attitude and angular velocity tracking errors of the four followers relative to the leader are compared in Figs.3 and 4,respectively.The results show that the errors of the attitude and angular velocity all converge to zero under both control schemes,while the convergence rate of the proposed method is obviously faster than that of the CE-based method for all the spacecraft.Fig.5 shows the comparison of parameter estimation errors,which indicates that the proposed method converges faster than the CE-based method.The profiles of the control torque are given in Fig.6.It can be observed in Fig.6 that the maximum torque required by the CE-based algorithm is about three times that of the proposed controller.Since tracking of the same timevarying reference,the control torque of steady-state for both methods are almost identical and remain time-varying.

6.Conclusion

Fig.5 Norm estimation errors

Fig.6 Norm of control torque

In this paper,a dynamic-scaling-based non-certainty equivalent adaptive control scheme has been presented to address theleader-following attitude coordination problem with unknown inertia parameters.The non-certainty equivalent principle and dynamic scaling skill are used both in the distributed estimator and the distributed control law.The main advantage of the proposed method is that it does not need the boundary of inertia parameter matrix.In addition,the followers can track the attitude of the leader without any information of the leader by the proposed distributed estimation.The numerical simulations have illustrated the effectiveness of the proposed distributed control scheme.Compared with the existing schemes based on CE,our method has better convergence accuracy and faster convergence rate.Furthermore,with an upper bound of dynamic-scaling factor,the control process requires a smaller control torque to ensure the convergence of attitude and angular velocity.

Acknowledgements

This paper was supported by the National Natural Science Foundation of China(Nos.11402200,11502203).

Appendix A

A.1 Proof of Lemma 1

Substituting Eq.(14)to Eq.(13),we obtain

To eliminate the second items in Eq.(61),the dynamic scaling skill is used again.Consider a scaled parameter estimation errorand then its derivative is given as

De fine a Lyapunov function

and then its time derivative is obtained as

where the Young inequality has been employed.Let

Replacing Eq.(65)in Eq.(64)yields

Since ω0is a persistent excitation signal,the matrixˉQ（ ωS）is row full rank.Therefore,there exist some positive constants αSsuch thatand then Eq.(66)can be rewritten as

Therefore,ˉQ（ ωS）zS→0 exponentially as t→∞.Similar as the construction of Lyapunov-like function Eq.(45)of Theorem 1 and the following deduction,we have rS,zSand ? which are all bounded.Therefore,

Theorem 2.5.3 in Ref.35guarantees the existence of a positive definite diagonal matrix C=diag｛ c1，c2，...，cN｝such thatis positive definite. Letand be the smallest eigenvalue ofLet

where P is a positive definite matrix such that

By direct calculating,we have

where

In the derivation of Eq. (71), the identity‖q0⊙q（ ?i）‖=‖q（ ?i）‖implied by Lemma 2 is employed.Substituting(71)into(70)gives

for some αU＞0.Hence,it can be obtained that||?||,||ζ||andconverge to zero exponentially. There-exponentially.

A.2 Details of ΔiandˉΔi

In this part,the detailed form of ΔiandˉΔiwill be given to complete the controller derivation.For notational convenience,we de fine

By applying Eqs.(14),(16)and(18),Δican be obtained as

The details are given below:

Since all elements of Δiin Eqs.(73)(77)can be written aswhere Θ ∈R3,Δican be given in a compact form as

where denotes the corresponding matrix.Then one has

with

It can be concluded for the Definition of 2-norm that

Thereby the detailed forms of ΔiandˉΔiare given.