School of Automation,Nanjing University of Science and Technology,Nanjing 210094,China

Improved optimal steering law for SGCMG and adaptive attitude control of fl xible spacecraft

Lu Wang1,Yu Guo2,*,Liping Wu3,and Qingwei Chen4

School of Automation,Nanjing University of Science and Technology,Nanjing 210094,China

The issue of attitude maneuver of a f exible spacecraft is investigated with single gimbaled control moment gyroscopes (SGCMGs)as an actuator.To solve the inertia uncertainty of the system,an adaptive attitude control algorithm is designed by applying a radial basis function(RBF)neural network.An improved steering law for SGCMGs is proposed to achieve the optimal output torque.It enables the SGCMGs not only to avoid singularity, but also to output more precise torque.In addition,global,uniform, ultimate bounded stability of the attitude control system is proved via the Lyapunov technique.Simulation results demonstrate the effectiveness of the new steering law and the algorithm of attitude maneuver of the fl xible spacecraft.

fl xible spacecraft,adaptive,steering law,attitude control.

1.Introduction

In recent years,more and more spacecraft adopt single gimbaledcontrolmomentgyroscopes(SGCMGs)as anactuator.Though the SGCMGs have many advantages,there exists configuratio singularity issue in practical applications.Moreover,there exists a lot of uncertainties due to the change of inertia and various environmental disturbances.Also,the vibration caused by the strong coupling between hub and fl xible appendages influence the attitude control performance seriously.Actually,the attitude control system of fl xible spacecraft is an multiple input multiple output(MIMO)nonlinear system with strong rigid fl xible coupling effect.

Recentlytheuncertaintyissue inspacecraftattitudecontrol system has attracted a great deal of attention.Jiang et al.used a neural network control method to compensate the saturation nonlinearity of actuator and the adaptive controlmethodis also used to estimate the boundof uncertainties[1].Jiao et al.presented an adaptive output feedback controller to suppress the disturbance,but the dynamic of actuator is not considered[2].Jia et al.discussed the attitude maneuverof spacecraft whose actuators are reaction wheels,and designed an adaptive control law[3]. Yoon et al.proposed an adaptive attitude tracking control law for spacecraft in which the actuator dynamics is involved[4].However,in the control law,a lot of uncertain parametersneed to be estimated.Taking accountof the uncertainties of inertia,MacKunis et al.designed a nonlinear adaptive controller,but this controller is only workable for rigid spacecrafts[5].Singla et al.proposed a model reference adaptive controller to cope with the parametric disturbancesas well as attitude measurement errors[6].However,the dynamics of actuator is not involved.To avoid singularityofSGCMGs,Nazarethet al.proposeda Moore-Penrose pseudo inverse steering law with a non-directional null-motion algorithm and an singularity robust inverse steering law for redundant SGCMGs.And the singularity robust inverse steering law allowed torqueerrors to be produced in the vicinity of singular states to provide better steering law performance[7].Wie presented an improved generalized singularity robust inverse steering law to provide singularity avoidance at the cost of certain torque errors which worsens the performance of the control system [8].To minimize the torque error and avoid the singularity,Tekinalp et al.proposed a blended inverse steering law which blends the expected gimbal rates and the required torque in a weighted form,and the system can be driven to a desired gimbal configuratio[9].Yasuyuki et al.presentedanewCMGsteeringlawusingpreferredinitialgimbalanglesandnullmotiontoachieveaccurateattitudecontrol for an agile spacecraft[10].Zhang et al.proposed an installation scheme for SGCMGs to avoid singularity,in which a rotator for the support bracket of each control moment gyrois adoptedandits applicationon reconfiguratioand singular escape is discussed[11].Besides,Kojima did the research in an adaptive-skew pyramid-type CMG(ASCMG)systemanddevelopedanovelsteeringlaw,inwhich the skew angle is one of the control variables for generating torque for a rigid spacecraft[12].It’s known that steering laws are grouped as the dynamic inverse method,offline planning,angular momentum constraining and so on. However,the off-lineplanningnecessitates bruteforcecalculationandis thereforetime-consuming.With the angular momentum constraining method,the angular momentum workspaceofthis methodis muchsmaller than the original maximum workspace,which means that the momentum workspaceofthesystemisnoteffectivelyusedandtheeffi ciency of the actuator is also decreased.Thus the dynamic inverse steering law is more widely used.In addition,it is well known that the attitude in a f exible spacecraft is more likely to vibrate than that in a rigid one because of the coupling effects between hub and fl xible appendages. For this reason,it is necessary to study the availability of thesteeringlawto theSGCMGs in a f exiblespacecraft.To the best of our knowledge,up to now there does not exist such a powerful steering law which can consider singularity avoidance,outputaccuracy,as well as vibrationrestrain for the SGCMGs in fl xible spacecrafts together.

In this paper,we investigate the dynamic friction existed in the dynamics of SGCMGs in a f exible spacecraft, and improve the blended inverse steering law by minimizing a new mixed quadratic cost function,which enables the SGCMGs to escape singularities in a f exible spacecraft as well as to output precise torque nearly without vibration.As for the attitude control system of the fl xible spacecraft,an adaptive attitude control algorithm is given by applying a radial basis function(RBF)neural network to cope with the inertia uncertainty.In addition,the global, uniform,ultimate bounded stability of the attitude control system is proved via the Lyapunov theory.The simulation results show that the stability of attitude maneuver of a fl xible spacecraft is improved.The paper is organized as follows.Section 2 states the mathematical model of a f exible spacecraft and SGCMGs.Section 3 presents an attitude control law based on an RBF neural network and an improved optimal steering law.In Section 4,the results of numerical simulation are presented.Finally,the paper is completed with some concluding comments.

2.Mathematical model

2.1Attitude kinematics

The quaternion kinematics equations of a spacecraft attitude[13]are governed by

If we defin the algebraic quaternionq=q1i+q2j+q3k+q4,then its inverse isq?1=?q1i?q2j?q3k+q4. Withqdas a desired attitude quaternion vector andωdas a desired angular velocity vector,the attitude error quaternion vector can be written aswhere?is multiplication of quaternion.Then we have

In addition,the angular velocity error vector is denoted asand

is a transformational matrix based on quaternion.

2.2Attitude dynamics

Attitude dynamics of a f exible spacecraft[13]is governed by

whereJ∈R3×3is the inertia matrix of the spacecraft which is positive definit and symmetric,andC∈Rn×3is the coupling matrix between hub and fl xible appendages,η∈Rn×1is the modal coordinate vector of the fl xible appendages,wherenis the number of fl xible modes considered,τcmg∈R3×1is the control torque exerted on the hub,andTd∈R3×1is the external disturbance torque,whose magnitude‖Td‖is bounded withTdmax.‖·‖denotes 2-norm of a vector.ζ∈Rn×nandΛ∈Rn×nare the damping ratio matrix and mode frequency matrix,respectively.

Besides,accordingto the coupling relationship between the hub and fl xible appendages in the attitude dynamics of the spacecraft,it can be deduced clearly that the angu-lar acceleration of the hub affects the fl xible modes and deform the appendages,and in return the vibration of the fl xible modes affects the attitude of hub simultaneously.

2.3Dynamics of SGCMG

In this paper a pyramid-type SGCMGs is considered.Dynamics of the SGCMGs including the dynamic friction of the SGCMGs[5]are governed by

whereFdis a coefficien matrix of friction of the SGCMGs,andhcmgis the angular momentum of the SGCMG system.The angle vector of the gimbals isδ= [δ1δ2δ3δ4]T,whereδi(i=1,2,3,4)is theith angle of the gimbals.˙δis a angular velocity vector of the gimbal; ˙hcmgis the derivative of angular momentum with respect to time and it has the following relationship:

wherehais the constant magnitude of angular momentum of each SGCMG;Ais the Jacobian matrix[14],which is described as

whereβis the skew angle of the SGCMG.

3.Attitude control law and improved optimal steering law design

3.1Attitude control law based on RBF neural network

Aiming at reducing the adverse impact of the inertia uncertainty to the stability of the system,we design an adaptive attitude control algorithm with a RBF neural network. To avoid the singularity of SGCMGs and output more precise torque,we improve the blended inverse steering law [9].Moreover,we design a novel attitude control system with initial attitude guidance[15]as depicted in Fig.1,in which,qv0andω0are the initial attitude quaternion and the initial angular velocity,respectively,whileqvrandωrare the desired quaternionand the desired angularvelocity, respectively.

Fig.1 Structure of attitude control system

We defin a vectorras follows:

whereρis a positive number.Furthermore,as is mentioned above,it can be deducedso we have

MultiplyingJon both sides of(9)and substituting(3)and (4)into it,we have

Consideringthe inertia uncertainty,we definwhereJ0is the nominal inertia,andis the inertia uncertainty.Here,we defin a nominal function

along with a functiony(w,r)which contains the uncertainty of inertia as follows:

Moreover,an adaptive attitude control algorithm is designed to estimatey(w,r)by applying the RBF neural network with a middle layer of fve neurons described by

wherew=[w1w2···w5]T∈R5×1is the weight vector;h=[h1h2···h5]Tis the radial basis vector;ci=[ci1ci2ci3]Tis the central vector of theith node;biis a positive constant.

To estimate the weight vector,an adaptive law is designed as

whereΓ1andΓ2are positive numbers andis the estimation of the weight vector.

Based on the error of attitude angular velocity and the attitude quaternion,we design the control law as follows:

wherekv,kn,wM＞0 and they are the gains that can be chosen by designer.The bound ofwMmust satisfystands forF-norm of a vector.

3.2Improved optimal steering law

To design the improved optimal steering law,individual gimbal angular velocityshould be close to the desired gimbal angular velocityMeanwhile,the output torque error is expected to be minimal as far as possible.Tekinalp designed the blended inverse via mixed quadratic cost function[9]described as minwhere theQandPare symmetric positive definit matrices;in whichare the desired gimbal angular velocity and desired torque respectively.

Because of the dynamic friction,the torque producing capacity of SGCMGs deteriorates over time due to the changes in the dynamics such as bearing degradation and increased friction in the gimbals.The aftereffect of SGCMGs’friction accumulation include increasing power consumptiondue to energy dissipation.These all may lead to serious problems which may even cause actual satellite failures.In Tekinalp’s method,an idealized model is utilized.In this paper,we investigate the dynamic friction existed in the SGCMGs in a f exible spacecraft,and improve the blended inverse steering law by minimizing a new mixed quadratic cost function,which enables the SGCMGs to escape singularities in fl xible spacecraft as well as to output smoothly precise torque nearly without error.

The attitude maneuver of the spacecraft is considered in the dynamics of the SGCMGs,with the skew-symmetric matrix of angular velocity of spacecraftω×.These improvements make it convenient for actual application. Moreover,based on the mixed quadratic cost function and optimization method,different values of the diagonal entries of the weighting matricesQandPcan be acquired by a trial and error approach to restrain the errorandTe,which enable the SGCMGs to escape singularities in a f exible spacecraft.According to(6)and(7),we use a mixed quadratic cost function as follows:

whereQ,Pare symmetric positive definit matrices.

To minimizeL,letting?L/?˙δ=0 yields

Therefore,we have

whereΓ3is a positive matrix.This law compensates the friction uncertainty.

Thedesired gimbalangularvelocity isis theith desired gimbal angular velocity,fori=1,2,3,4.The error betweenδiandδdiisΔδdi=arccos(〈hi,hdi〉/‖hi‖2)·sgn(〈hi×hdi,gi〉),?π≤Δδdi≤π.hiis the angular momentum of single SGCMG;hdi=gi×Tc/‖gi×Tc‖is the desired angularmomentumof each SGCMG[16],giis a gimbal axis vector.

whereWi＞0(i=1,2,3,4),ε=ε0sin(ω0t+φ), α1=α10e?μD,α2=α20e?μD,the singularity measurementD=det(AAT),ω0,μ,ε0,α10,α20＞0 and they can be chosen by designer.

3.3Stability proof

Assumingw?∈R5×1is the ideal identificatio weight vector ofy,we haveand the modeling error of neural network is

According to(11)and(16),the equation of closed-loopattitude control system leads to

Denoting the estimation error of the weight vector of the neural network asand the total disturbance to the system asthen(21)can be rewritten as

We assume that the disturbance is bounded[17],then we can have

withc0,c1andc2as known positive constants.

Lemma 1[18] Suppose that a positive define LyapunovfunctioncandidateV(t)satisfieβ0,whereλ＞0 andβ0＞0 are constants,then the system is globally,uniformly,ultimately bounded and stable.

Theorem1Forthe system in(4)and(5),assumingthat the disturbanceis boundedwith the attitude controlsystem of the fl xible spacecraft described by(4),(5)and(11), then the adaptivecontrol law(20)as well as controller(16) are globally,uniformly,ultimately bounded and stable.

ProofConsider the following Lyapunov function candidate:

According to(3),(11)and(16),the derivative ofValong the trajectories of the system is

From the Lyapunov function candidate,we can see thatwhereλ2is the maximum eigenvalue ofJand

Therefore,according to Lemma 1,the attitude control system of fl xible spacecraft is globally uniformly ultimately bounded and stable.

4.Simulation results

4.1Simulation of SGCMGs optimal steering law

To verify numerically the results provedin this section,we considerthe pyramid-typeSGCMGs givenin(6).The constant magnitude of angular momentum of each SGCMGha=1.The true value of SGCMG friction uncertaintyFd=0.1I4and the skew angle of the SGCMG isβ= 53.13°.The initial gimbal angular velocity of SGCMGs iss and the initial gimbal angle of SGCMGs isδ0=[0 0 0 0]T.The desired torque isτu= [1 N·m 0.5 N·m 0.2 N·m]T.The parameters in the optimal steering law areεi=0.1cost,α20=0.01,α10= 0.01,μ=10,Wi=i(i=1,2,3,4).

From Fig.2,we can see that,the estimation of the SGCMG frictionuncertaintycan convergeto its truevalue. In contrast to the optimal steering law,we adopt the blended inverse steering law which did not consider the compensation of the SGCMG friction uncertainty[9].The simulation results are depicted in Figs.3–6,which can also show the effectiveness of the adaptive law.

Fig.2 Estimation of SGCMG friction uncertainty

Fig.3 Output torque of improved steering law

Fig.4 Singularity measurement of improved optimal steering law

Fig.5 Output torque of blended inverse steering law

Fig.6 Singularity measurement of blended inverse steering law

From Figs.3–4,we can see that when the singularity measurementDchanges fast and tends to the vicinity of zero,it means the SGCMGs are nearly singularity, and the small torque error is utilized to avoid singularity. Since then,though in the presence of SGCMG friction uncertainty,the error of output torque is still almost zero. Besides,the singularity measurementDbecomes steady and is away from zero,which verifie that the singularity is avoided.From Figs.5–6,in contrast,we can see that the blended inverse steering law does not perform well in the presence of SGCMG friction uncertainty.The torque error always exists.The system can not output torque as exact as the desired output torque.The singularity measurementDstill changes fast and this is manifested as the continuous appearance of output torque error.Therefore, the results show that the improved optimal steering law can output torque much more smoothly and precisely than the blended inverse steering law though in the presence of SGCMG friction uncertainty,and meanwhile it does well in avoiding singularity.

4.2 Simulation of attitude maneuver control of flexibl spacecraft

To illustrate the results presentedin this paper,we consider the system dynamics in(4)and(5),with the parameters given in[19].

Numerical simulations for a f exible spacecraft with the proposed attitude control scheme have been completed.

The initial gimbal angle of the SGCMGs isδ0= [0 0 0 0]Tand the initial gimbal angular velocity is ˙δ0=[0 0 0 0]T.The parameters of the steering law areα10=0.01,α20=0.01,εi=0.1cost,μ=10,ha=10 andWi=i(i=1,2,3,4),and the skew angle of the SGCMG isβ=53.13°,ω0=[0 0 0]T.The initial attitude angle of the spacecraft is[60°20°15°]Tand the corresponding initial attitude quaternion is calculated as[0.468 6 0.213 4 0.197 4 0.834 2]T.The desired attitude angle vector isωd=[0°0°0°]T.The parame-ters in the attitude controller are set asΓ1=10,Γ2=5,wM=100,kv=15,kz=1,kn=15,ρ=1.Simulation results are shown in Fig.7.

Fig.7 Response of the proposed attitude control system

From Fig.7(a)–(c),we can see that,by guiding the initialattitude,theEulerangleandtheerrorofattitudequaternion converge much more smoothly during the maneuver. Zooming in the figures during 100–300 s,the attitude pointing accuracy is better than 0.005°,while the stability of attitude angular velocity is better than 0.001(°)/s.It means that the spacecraft maneuvers rapidly and steadily with high pointing accuracy and stability.From the modes of fl xible appendages in Fig.7(d),it can be seen that the appendages’vibrations are rapidly suppressed.

For comparison,a PID control method with initial attitude guidance is also used to show the advantage of the attitude control law based on an RBF neural network.The simulation results are shown in Fig.8.It can be seen that compared with the PID control law,the proposed control law has better attitude pointing accuracy and stability of attitude angular velocity.From Fig.7(d)and Fig.8(d),we can see that with the proposed control law the vibration suppression process is much faster,which means that the proposed control law has a much better vibration suppression property.

Fig.8 Response of PID attitude control system

5.Conclusions

In this paper,an adaptive attitude control algorithm is designed by applying an RBF neural network for inertia uncertainty of the system.A new improved optimal steering law considering the dynamic friction and the attitude maneuver of the spacecraft existed in the dynamics of SGCMGs in a f exible spacecraftis proposedforSGCMGs to escape singularities in fl xible spacecraft and output torque with high precision as well.The simulation results show that with the new optimal steering law,the SGCMGs can output precise torque.It can be concluded that,with the proposed attitude control scheme,high-precision,high stability of attitude maneuver of fl xible spacecraft can be achieved.However,the static friction in the SGCMGs dynamics is not considered,which should be further investigated in fl xible spacecraft control system design.

[1]Y.Jiang,Q.L.Hu,G.F.Ma.Intelligentadaptive variable structure attitude maneuvering control for fl xible spacecraft with actuator saturation.Journal ofAstronautics,2009,30(1):188–193.

[2]X.H.Jiao,L.Y.Zhang.Adaptive output feedback control of attitudemaneuver and vibration suppression for fl xible spacecraft.Electric Machines and Control,2011,15(7):94–100.

[3]F.L.Jia,W.Xu,H.N.Li,et al.Design of controller for spacecraft attitude large angle maneuver with three reaction wheels.Flight Dynamics,2010,28(2):78–81.

[4]H.Yoon,P.Tsiotras.Adaptive spacecraft attitudetracking control with actuator uncertainties.Journal of Astronautic Sci-ences,2008,56(2):251–268

[5]W.MacKunis,K.Dupree,N.Fitz-Coy,et al.Adaptive satellite attitudecontrolinthepresence ofinertiaandCMGgimbalfriction uncertainties.The Journal of the Astronautical Sciences, 2008,56(1):121–134.

[6]P.Singla,K.Subbarao,J.L.Junkins.Adaptiveoutputfeedback control for spacecraft rendezvous and docking under measurement uncertainty.Journal of Guidance,Control,and Dynamics,2006,29(4):892–902.

[7]N.S.Bedrossian,J.Paradiso,E.V.Bergmann.Steering law design for redundant single-gimbal control moment gyroscopes.Journal of Guidance,Control,and Dynamics,1990, 13(6):1083–1089.

[8]B.Wie.New singularity escape/avoidance steering logic for control moment gyro systems.Journal of Guidance,Control, and Dynamics,2005,28(5):948–956.

[9]O.Tekinalp,E.Yavuzoglu.A new steering law for redundant control moment gyroscope clusters.Aerospace Science and Technology,2005,9(7):626–634.

[10]Y,Nanamori,M,Takahashi.Steering law of control moment gyros using optimization of initial gimbal angles for satellite attitude control.Journal of System Design and Dynamics, 2011,5(1):30–41.

[11]J.R.Zhang,J.Jin,Z.Z.Liu.An improved installation for control moment gyros and its applications on reconfiguratio and singular escape.Acta Astronautica,2013,85(1):93–99.

[12]H.Kojima.Singularity analysis and steering control laws for adaptive-skew pyramid-type control moment gyros.Acta Astronautica,2013,85(1):120–137.

[13]P.C.Hughes.Spacecraft attitude dynamics.New York: Courier Dover Publications,2012.

[14]R.W.Zhang.Dynamics and control of satellite orbit and attitude.Beijing:Beijing University of Aeronautics and Astronautics Press,1998:278–293.

[15]A.F.Lai,Y.Guo,L.J.Zheng.Active disturbance rejection control for spacecraft attitude maneuver and stability.Control Theory and Application,2012,29(3):401–407.

[16]Z.Y.Sun,G.Jin,K.Xu,etal.Steering law design for SGCMG based on optimal output torque capability.Chinese Journal of Space Science,2012,32(1):113–122.

[17]F.L.Lewis,A.Yesildirek,K.Liu.Multilayer neural-net robot controller with guaranteed tracking performance.IEEE Trans. on Neural Networks,1996,7(2):388–399.

[18]Z.Qu,D.M.Dawson,S.Y.Lim.A new class of robust control laws for tracking of robots.The International Journal of Robotics Research,1994,13(4):355–363.

[19]Q.Hu.Robust adaptive backstepping attitude and vibration control with L2-gain performance for fl xible spacecraft under angular velocity constraint.Journal of Sound and Vibration, 2009(327):285–298.

Biographies

Lu Wangwas born in 1990.He received his B.S. degree in 2012 in Electrical Engineering and Automation from Nanjing University of Science and Technology.He is currently a Ph.D.degree candidate in Control Theory and Control Engineering at Nanjing University of Science and Technology. His research interests include design of singlegimbaled control moment gyroscopes(SGCMGs) steering law,spacecraft attitude control considering actuator and nonlinear control.

E-mail:timwang21@126.com

Yu Guoreceived her B.S.degree and M.S.degree in Automation,both from Huazhong University of Science and Technology in 1984,1987 respectively. She received her Ph.D.degree in Control Science and Engineering from Nanjing University of Science and Technology.In 1987,she joined the faculty of the School of Automation,Nanjing University of Science and Technology,and is currently a professor of automatic control there.Her major research interests are adaptive control,robust control,nonlinear control and multi-object optimization for complicated systems.Now she is interested in fl xible spacecraft attitude control and servo control system with high accuracy.

E-mail:guoyu@njust.edu.cn

Liping Wuwas born in 1990.She received her B.S. degree in 2013 in Automation from Nanjing University of Science and Technology.She is currently a Ph.D.degree candidate in control theory and control engineering at Nanjing University of Science and Technology.Her research interests include spacecraft attitude control and spacecraft orbit control.

E-mail:wulp0724@163.com

Qingwei Chenreceived his M.S.degree and Ph.D. degree in control science and engineering,both from Nanjing University of Science and Technology.In 1988,he joined the faculty of the School of Automation,Nanjing University of Science and Technology, and is now a professor of automatic control.His major research interests are intelligent control,nonlinear control,network control systems,high accuracy servo systems and optimization for complicated systems.

E-mail:cqw1002@sina.com

10.1109/JSEE.2015.00139

Manuscript received December 30,2014.

*Corresponding author.

This work was supported by the National Natural Science Foundation of China(61473152)and the China Scholarship Council and the Educational Innovation Project for Graduate Students of Jiangsu Province (KYLX15 0399).