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Acta Astronautica

Acta Astronautica 96 (2014) 232–245

0094-57http://d

n Corrfax: þ8

E-mjia_ying

journal homepage: www.elsevier.com/locate/actaastro

Dynamics and vibration suppression of space structures withcontrol moment gyroscopes

Quan Hu, Yinghong Jia n, Shijie XuSchool of Astronautics, Beihang University, Xueyuan Road No. 37, Haidian District, Beijing 100191, China

a r t i c l e i n f o

Article history:Received 25 March 2013Received in revised form6 August 2013Accepted 25 November 2013Available online 3 December 2013

Keywords:Vibration suppressionLarge space structuresControl moment gyroscopesGyroelastic bodyKane's equation

65/$ - see front matter & 2013 IAA. Publishex.doi.org/10.1016/j.actaastro.2013.11.032

esponding author. Tel.: þ86 10 82339750;6 10 82338798.ail addresses: huquan2690@hotmail.com (Q.h@163.com (Y. Jia), starsjxu@163.com (S. Xu

a b s t r a c t

This paper presents a new and effective approach for vibration suppression of large spacestructures. Collocated pairs of control moment gyroscope (CMG) and angular rate sensorare adopted as actuators/sensors. The equations of motion of a flexible structure with a setof arbitrarily distributed CMGs are developed. The detailed dynamics of the CMGs andtheir interactions between the flexibilities of the structure are incorporated in theformulation. Then, the equations of motion are linearized to describe the small-scalemotion of the system. The optimal placement problem of the actuators/sensors on theflexible structures is solved from the perspective of system controllability and observa-bility. The controller for the vibration suppression is synthesized using the angular rates ofthe locations where the CMGs are mounted and the gimbal angles of the CMGs. Thestability of the controller is proved by the Lyapunov theorem. Numerical examples of abeam structure and a plate structure validate the efficacy of the proposed method.

& 2013 IAA. Published by Elsevier Ltd. All rights reserved.

1. Introduction

Large space structures have been widely used in engi-neering applications such as flexible manipulators, solararrays, and space trusses. Since the structural flexibilitiesmay influence the stability and performance of thesystem, extensive research on active vibration control offlexible structures has been conducted. Various types ofactuators have been considered for vibration suppression.Casella et al. [1] and Sowmianarayanan and Pradeept [2]employed thrusters on large flexible structures. Matunagaet al. [3] adopted multiple proof-mass actuators anddesigned a direct acceleration feedback control law forvibration suppression. Megneto-rheogical fluid damper [4]and electro-rheological fluid damper [5] are two similarand novel actuators for vibration suppression. Piezoelec-tric materials have been widely used in vibration control

d by Elsevier Ltd. All rights

Hu),).

for space structures in recent years [6–10]. Hybrid vibra-tion absorbers were also designed by combining severaltypes of the actuators mentioned above together [11,12].In this study, we investigate another new way for vibrationsuppression of the constrained flexible structures: usingcontrol moment gyroscopes (CMGs) as actuators.

The CMGs have been serving as the attitude controlactuators or the energy storage devices for large spaceplatforms and agile satellites for decades. Owing to thesuperior properties compared to other actuators – they areefficient, clean, and capable of generating large and precisetorques – using CMGs for vibration suppression began todraw much attention. The concept of mounting CMGs onflexible structures was first proposed by D'Eleuterio andHughes [13,14]. They assumed that a continuous distri-bution of stored angular momentum in the form of flywheels (FWs) or CMGs existed on a flexible body. Such asystem was referred to as gyroelastic body. The equationsof motion were first established, and then modal analysiswas performed. It is found that the gyroelastic bodyexhibits many interesting features: the natural frequencyof the structure can be shifted; the mode shapes can be

reserved.

Q. Hu et al. / Acta Astronautica 96 (2014) 232–245 233

coupled; and controllable damping can be added to thesystem. These basic properties make embedding CMGs onflexible structures a feasible approach for vibration reduc-tion. Damaren and D'Eleuterio [15] developed an optimalcontrol method for the shape control of large spacestructures possessing a distribution of CMGs. The condi-tions for controllability and observability are then studiedto provide guidelines for the design of gyroelastic struc-tures [16].

All the theoretical studies mentioned above are basedon the following assumptions: the angular momentumdevices on the structures are continuously distributed, andthe detailed dynamics of the FWs or CMGs are neglected.However, in engineering applications, the embeddedangular momentum devices must be pointwise and thedynamics of the actuators should be considered. Peck andCavender [17,18] mounted a set of FWs with constantvalue for the angular momentum on the momentumcontrol system and line-of-sight (MCS/LOS) testbed. Theyvalidated several important features of gyroelastic bodyand found that adaptive structure tuning could beachieved. Yang et al. [19] installed a scissored pair of CMGson a flexible space truss. A combined feedforward andfeedback control law was designed for the CMGs tosimultaneously achieve maneuvers and vibration controlof the structure. Shi and Damaren [20] installed a singlegimbal CMG at the end of a cantilevered beam to activelydamp the vibration. Comparing the theoretical develop-ments and the engineering applications of the gyroelasticbody, it can be found that the previous gyroelastic theorycannot be strongly connected to the engineering practice.

Another problem that still remains unsolved for thegyroelastic body is the optimal placement of the actuators.It should be first considered while designing a flexiblestructure with FWs or CMGs; however, only two relatedstudies are found in the open literature. Yang et al. [19]developed an optimization approach to find the optimalplacement for the scissored pair of CMGs along the flexibletruss where the interaction between the CMGs and thetruss is minimal. Chee [21] found the optimal distributionof double gimbaled CMGs (DGCMGs) for vibration sup-pression of elastic bodies. He assumed a series of DGCMGswas placed uniformly across a flexible structure. Theamount of angular momentum stored by the spinningwheels was optimally allocated. It can be seen that Yang'sstudy is oriented towards a specific problem, whereasChee focuses on the optimization of the distribution ofthe amount of angular momentum, not the placement ofthe CMGs.

gi

Fig. 1. Gyroelastic body: (a) gyroelas

The contribution of the present paper is a general-purpose methodology for vibration suppression of con-strained large space structures using discretely distributedCMGs. The equations of motion of a gyroelastic bodywhich fully consider the detailed dynamics of actuatorsare first developed. All the mass and inertia properties andtheir interactions with the structure flexibilities are incor-porated in the formulation. The mathematical modelobtained is much more capable of describing the gyroe-lastic structures that are seen in engineering practices.Then, the equations of motion are linearized, based onwhich the modal analysis is performed. The optimalplacement of actuators/sensors is determined by maximiz-ing the indexes indicating system controllability andobservability. A similar control law with the one in Ref.[20] for single CMGs is adopted. We proved that thiscontrol strategy is also applicable to the case that a set ofCMGs is mounted on a structure. The angular velocities ofthe locations where the CMGs are mounted and the gimbalangles are fed back to determine the gimbal rates. Sincemost space structures, such as flexible manipulator arms,space trusses, and solar arrays, can be modeled as beam orplate structure, we demonstrate the efficacy of the pro-posed method by numerical simulations of a gyroelasticbeam and a gyroelastic plate.

In the sequel, we derive the equations of motion of aconstrained gyroelastic body with distributed CMGs inSection 2, followed by the modal analysis in Section 3.In Section 4, controllability and observability of the systemare investigated to define criterion for optimizing theplacement of CMGs and the angular rate sensors. Anangular rate feedback control law is designed in Section5 to actively suppress the vibration. Numerical examples ofgyroelastic beam and plate structures are given in Section6. Conclusion is made in Section 7.

2. Equations of motion

As indicated in Fig. 1, a truss (it can be viewed as abeam structure) and a plate are equipped with CMGs forvibration reduction. In order to establish a generaldynamic model for such systems, an arbitrary flexiblestructure with n discretely distributed CMGs is considered(Fig. 2). ℱℬ is the body-fixed frame. ℱgi and ℱri

are gimbal-fixed frame and rotor-fixed frame of the ith(i¼1,…,n) CMG, respectively. The origin of ℱℬ is on thefixed boundary of the structure, whereas the origins of ℱgi

and ℱri are both located at the mass center of the CMG.It is assumed that the center of mass of the gimbal and

gi

tic beam, (b) gyroelastic plate.

mT

giT gi

ii

i

gix

giz

i

gi

riz

rix

riy

ri

giy

m rirm gir

gidm

ridm

Fig. 2. Sketches of a constrained gyroelastic body and reference frames of a CMG.

Fig. 3. Topology of the flexible structure and the ith CMG.

Q. Hu et al. / Acta Astronautica 96 (2014) 232–245234

rotor coincide with the center of mass of the overall CMGto simplify the formulation. xgi, ygi, and zgi are theconstituent vectors of ℱgi. xgi is the unit vector along thegimbal rate, ygi is the unit vector along the rotor momen-tum, and zgi ¼ xgi � ygi. The gimbal of the CMG is mountedon ℬ by a revolute joint. Therefore, the relative motionbetween ℱℬ and ℱgi consists of two parts: the rigidrotation of the gimbal and the elastic motion caused bythe structural flexibility. ℱri is used to characterize themotion of the rotor with respect to the gimbal. yri is in thesame direction with ygi. The rotor rotates around yri; there-fore, the frame ℱri rotates with respect to the frame ℱgi.

The elastic motion of the structure is assumed to besmall in the following formulation, so that the assumedmode method is applicable. The elastic translational dis-placement of an arbitrary element mass dm will bediscretized in the customary manner as follows:

dm ¼ ∑k

i ¼ 1Tm;iτi9Tmτ ð1Þ

where Tm;i can be the basis function or the modes dataobtained from the finite-element method (FEM) todescribe the modal shape of the ith mode. k is the numberof selected modes to approximate the actual structure.Tm9 ½Tm;1; :::; Tm;k� is referred to as the translational modalmatrix of the elementary mass of the structure. _τ is themodal coordinates of the flexible structure. Similarly, theelastic rotational displacement of dm is discretized as

βm ¼ Rmτ ð2Þ

where Rm is the rotational modal matrix of dm.Kane's method [22] is adopted to establish the equa-

tions of motion of ℬ. The system ℬ is treated as amultibody system (Fig. 3) whose number of degrees offreedom is 2nþk: n is the number of CMGs on ℬ, while kis the number of the selected modes for the flexiblestructure of ℬ.

The generalized speeds for ℬ are chosen as

u1 ¼ _τ;u2i ¼ _δi;u2iþ1 ¼Ωi; ði¼ 1;…nÞ ð3Þ

where _δi is the gimbal rate of the ith CMG and Ωi is thespin angular speed of the rotor with respect to the gimbal.The velocities of the generic mass elements in the flexible

structure, the gimbal, and the rotor can be expressed as

vm;ℬ ¼ℱTℬTm _τ ð4Þ

vm;gi ¼ℱTℬðTgi�Aℬ;gi ~rm;giAgi;ℬRgiÞ_τ�ℱT

gi ~rm;giUx _δi ð5Þ

vm;ri ¼ℱTℬðTgi�Aℬ;ri ~rm;riAri;ℬRgiÞ_τ

�ℱTri ~rm;riAri;giUx _δi�ℱT

ri ~rm;riUyΩi ð6Þ

where ℱℬ, ℱgi, and ℱri are the vectrices [23] of thecorresponding reference frames. The vectrices are adoptedhere to indicate the reference frames in which the followedvectors are expressed. They can also be used to definethe rotational transformation matrix from ℱy to ℱx:Ax;y9ℱxℱ

Ty . Tgi and Rgi have similar definitions with Tm

and Rm, respectively. They are the translational and rotationalmodal matrix of the node located at the mass center of theCMG. rm;gi is the position vector of an elementary mass of thegimbal from the origin of ℱgi. rm;ri is the counterpart for therotor. A symbol with a tilde signifies the cross-product matrixassociated with the 3� 1 column matrix. Ux and Uy aredefined by Ux ¼ ½1 0 0�T and Uy ¼ ½0 1 0�T , respectively.

Kane's equation for ℬ is

FIsþFAs ¼ 0; ðs¼ 1;…2nþ1Þ ð7Þwhere FIs and FAs are the generalized inertial force andgeneralized active force corresponding to the sth general-ized speed, respectively. FIs are calculated by

FIs ¼ �Zℬ

∂vm;ℬ

∂usUdvm;ℬ

dtdm� ∑

n

i ¼ 1

Zgi

∂vm;gi

∂usUdvm;gi

dtdm

� ∑n

i ¼ 1

Zri

∂vm;ri

∂usUdvm;ri

dtdm; ðs¼ 1;…2nþ1Þ ð8Þ

where the dot products are denoted by “U”. The partialderivatives lead to the partial velocities of the elementary

Q. Hu et al. / Acta Astronautica 96 (2014) 232–245 235

mass, whereas the total derivatives conducted in theinertial frame result in the inertial accelerations. Weassume that there are no external active forces or torquesexerted on ℬ; therefore, the generalized active forces areonly determined by the nominal structural elasticity anddamping,

FA ¼ �ðζ_τþΛτÞ ð9Þwhere Λ is the diagonal array of the square of the circularfrequencies ωi of the flexible structure; ζ is the dampingmatrix with 2ζiωi on the main diagonal; and ζi is dampingcoefficient of the ith mode.

Substituting Eqs. (4)–(6) into Eq. (8), then using Eqs. (8)and (9) in Eq. (7), we obtain the vibration equation ofmotion of a constrained flexible body with n CMGs:

M€τþζ_τþΛτ¼ Fgyros ð10Þwhere M is the inertial matrix incorporating the inertialparameters of the CMGs and Fgyros is the active modalforces produced by the CMGs. The details of M and Fgyrosare discussed as follows:

M¼ZℬTTmTmdmþ ∑

n

i ¼ 1ðma

giTTgiTgiþRT

gi_JagiRgiÞ ð11Þ

where magi is the sum of the mass of the gimbal and rotor

of the ith CMG;_Jagi ¼Aℬ;giJ

agiAgi;ℬ, Jagi is the sum of the

moments of inertia of the gimbal and rotor of the ith CMG.

Jagi ¼ Jgiþ Jri ¼ d½Jxgi Jygi Jzgi� þd½Jxri Jyri Jzri�9d½Jaxgi Jaygi Jazgi �ð12Þ

where the operator “d[x]” returns the square matrix withthe element of the vector x on the main diagonal. Fgyros iscalculated by

Fgyros ¼ ∑n

i ¼ 1RTgiTgyros;i ð13Þ

where Tgyros;i is the torque produced by momentumexchange between the structure and the ith CMG. It canbe divided into two parts: the rigid part and the elasticpart:

Tgyros;i ¼ Trgyros;iþTe

gyros;i ð14Þwhere

Trgyros;i ¼ cziJ

yriΩi

_δi ð15Þ

Tegyros;i ¼ ~_βgicxiJ

axgi_δiþ ~_βgicyiJ

yriΩiþ½cziðJaygi � Jazgi ÞcTyi

þcyiðJaygi � Jazgi ÞcTzi� _βgi _δi ð16Þ

where _βgi ¼ Rgi _τ is the elastic rotational speed of the nodeat the mass center of the ith CMG; the vectors cxi, cyi, andczi are defined by

Aℬ;gi9 ½cxi cyi czi� ð17ÞThe physical meaning of each term in Eqs. (15) and (16) isquite obvious. cziJ

yriΩi

_δi is the gyroscope torque exertedon the structure by the ith CMG. It has the same form asthe one produced by a CMG mounted on a rigid body.The direction of the torque is indicated by czi, while themagnitude is calculated by JyriΩi

_δi.~_βgicxiJ

axgi_δi is the interac-

tion between the structure flexibility and the motion ofthe overall CMG in the xgi direction.

~_βgicyiJyriΩi presents the

interaction between the structure and the motion of therotor in yri. The third term on the right-hand side of Eq. (16) is the cross-interaction between theflexibility of the structure and the motion of the CMGalong ygi and zgi. In fact, when we develop the equations ofmotion of a free-flying rigid spacecraft equipped with acluster of CMGs, a similar term will appear if the inertia ofthe gimbal is not neglected [24].

3. Modal analysis

Eq. (10) describes the small-scale elastic motion of thestructures with a cluster of CMGs in arbitrary configura-tion. Because of the complexity of the motion of the CMGs,it is still of high nonlinearity. Noting the fact that the spinrate of the rotor is much higher than a gimbal, Eq. (10) canbe reduced to

E€τþ ∑n

i ¼ 1RTgið ~_βgicyiJyriΩiþcziJ

yriΩi

_δiÞþζ_τþΛτ¼ 0 ð18Þ

where the terms ~_βgicxiJaxgi_δi and ½cziðJaygi � Jazgi ÞcTyiþ

cyiðJaygi � Jazgi ÞcTzi� _βgi _δi are dropped. Then, if the motion of the

CMGs is also restricted to be small, the equations ofmotion of a gyroelastic body can be further simplified toa linear time-invariant form. Modal analysis can be per-formed on such a mathematical model to obtain morefundamental properties of the gyroelastic body, which canguide the system design. Therefore, the following assump-tions are made: the motion of the gimbals is restricted tobe small; besides, the reference frame ℱgi coincide eitherwith ℱℬ or with the one formed by rotating ℱℬ by π=2along yb in the initial status. In this case, the governingequations of ℬ can be linearized as follows by substi-

tuting the expansion Rgi9 ½R1Tgi ;R

2Tgi ;R

3Tgi �T and _βgi ¼ ½_β1gi;

_β2gi; _β

3gi�T ¼ Rgi _τb into Eq. (18):

E€τþðGþζÞ_τþΛτ¼ Bδ_δ ð19Þ

where

E¼ZℬTTmTmdmþ ∑

n

i ¼ 1ðma

giTTgiTgiþRT

giJagiRgiÞ

G¼ ∑n

i ¼ 1G0i ¼ ∑

n

i ¼ 1ð�R1T

gi R3giþR3T

gi R1giÞhi

Bδ ¼ ½B1; :::;Bn� ð20ÞBi differs with the orientation of ℱgi. If ℱgi coincides with

ℱℬ, then Bi ¼ R3Tgi ; otherwise, if ℱgi coincides with the

frame transformed by rotating ℱℬ by π=2 along yb,then Bi ¼ R1T

gi . hi ¼ JyriΩi is the angular momentum of thespinning rotor.

In the following formulations for the modal analysisand the optimization of the actuators, the structuraldamping term will not be considered. It can greatlysimplify the formulation and still reveal the fundamentaldynamic characteristic of the system. Omitting the damp-ing matrix from Eq. (19) leads to the equations of motionof the undamped gyroelastic body. Then, the formulationsfor modal analysis of gyroelastic body in Ref. [13] and [14]are directly applied to the mathematical model given by

Q. Hu et al. / Acta Astronautica 96 (2014) 232–245236

Eq. [19], since Eq. [19] can also be rewritten in a first-orderform as follows,

A _qþSq¼ c ð21Þwhere

A¼ E 00 Λ

� �; S¼ G Λ

�Λ 0

� �; q¼ _τ

τ

� �; c¼ B_δ

0

!

ð22ÞIt is obvious that A is positive definite and S is skewsymmetric.

Setting the control input of Eq. (21) equal to zero leadsto the free-vibration equation of ℬ. The eigenvalue λα andeigenvector qα with respect to the αth ðα¼ 1; ::;2kÞ modesatisfy

λαAqαþSqα ¼ 0 ð23ÞNoting the symmetry of A and the skew symmetry of S,it can be proved that �λHα also satisfies Eq. (23) andλαþλHα ¼ 0. The superscript H denotes conjugate transpose.Therefore, all the eigenvalues satisfy λα ¼ jΩα, ðα¼ 71;:::; 7kÞ, where Ωα ¼ �Ω�α. The eigenvectors qα are allcomplex matrices, and they can be written as

qα ¼ fαþ jkα ð24Þwhere fα and kα are real matrices. Substituting Eq. (24)into Eq. (23) and separating the real part and imaginarypart, we have

�ΩαAkαþSfα ¼ 0 ð25Þ

ΩαAfαþSkα ¼ 0 ð26ÞSubstituting the linear combination defined by k′α ¼aαkαþbαfα and f ′α ¼ aαfα�bαkα for kα and fα in Eqs. (25)and (26), respectively, it is found that the equalitiesstill hold. So it is always possible to choose appropriateaα and bα to make

k�α ¼ fα; kα ¼ f �α ð27ÞTherefore, one of the complete set of kα and fα is sufficientto describe the modal characteristic of ℬ.

Eliminating kα from Eqs. (25) and (26) yields

ðΩ2αAþSA�1SÞfα ¼ 0 ð28Þ

Substituting the subscript α for β in Eq. (28), and supposingthat αaβ, we can obtain

ðΩ2α�Ω2

β ÞfTβAfα ¼ 0 ð29ÞWhen Ω2

αaΩ2β , we have

fTβAfα ¼ 0 ð30ÞIf Ω2

α ¼Ω2β , using the skew symmetry of S in Eq. (25), we

have

ΩαfTαAf �α ¼ fTαSfα ¼ 0 ð31Þ

The first orthogonality condition is finally written as

fTαAfβ ¼ 0; αaβ ð32ÞBy analogy with the treatment in Ref. [13], followingcondition is adopted when α¼ β,

fTαAfα ¼ 2Ω2α ð33Þ

Another orthogonality condition related with S can bederived directly by pre-multiplying Eq. (26) with kTβ andnoting Eqs. (27) and (32):

kTβSkα ¼ �ΩαkTβAfα ¼ 0; ðΩαaΩβÞ ð34Þ

When Ωα ¼Ωβ , substituting Eq. (33) into Eq. (34) leads to

fTαSkα ¼ �2Ω3α ð35Þ

With these developments in hand, the normality conditioncan be summarized as

fTαAfβ ¼ kTαAkβ ¼ 2Ω2αδαβ

fTαSkβ ¼ �2Ω3αδαβ

8<: ð36Þ

The reason for the treatment of setting fTαAfα ¼ 2Ω2α is that the

normality condition in Eq. (36) becomes μTαΛμβ ¼Ω2

αδαβ whenthere is no embedded angular momentum in ℬ, that is, thegyroelastic body ℬ is reduced to a traditional structure [13].The proof is given in Appendix A. Therefore, it makes thegyroelastic theory compatible with those of traditionalstructures.

Noting the fact that the eigenvalues of the system givenby Eq. (21) are complex conjugate pairs, so the super-position of the responses corresponding to Ωα and Ω�α is

q¼ fα cos Ωαt�kα sin Ωαt ð37Þ

Considering q¼ ½_τTb ; τTb �T and Eq. (27), it is natural to definefα and kα in the following form:

fα ¼�Ωαυα

μα

" #; kα ¼

Ωαμα

υα

" #ð38Þ

where the real matrices υα and μα satisfy μα ¼ υ�α,μ�α ¼ υα. The matrix pairs ðμα;υαÞ are referred to as thegyroelastic modes [13]. Because of the orthogonality prop-erties of the eigenvectors fα and kα, the displacementvector q can be expanded in terms of fα as follows:

q¼ ∑k

α ¼ �kfαηαðtÞ ð39Þ

4. Optimization of the actuators/sensors

Co-located CMGs and angular rate sensors are adoptedas actuators/sensors for a gyroelastic body. In this section,an optimization index is derived from the system controll-ability and observability matrices. First, the equations ofmotion in state-space form should be derived.

Substituting Eq. (39) into Eq. (21) and using theorthogonality conditions given by Eq. (36), the linearizedequations of motion of the constrained gyroelastic bodyare rewritten as

_X¼ AXþBu ð40Þ

Q. Hu et al. / Acta Astronautica 96 (2014) 232–245 237

where

X¼

η�k

⋮η�1

η1⋮ηk

26666666664

37777777775; A¼

Ω�k

0 ⋰Ω�1

Ω1

⋰ 0Ωk

26666666664

37777777775;

B¼ 12A�1_Bδ;

_Bδ ¼

μTk

⋮μT1

μT�1

⋮μT

�k

26666666664

37777777775Bδ; u¼ _δ ð41Þ

The output of the sensor is the inertial angular velocity ofthe point where the sensor is mounted,

Y¼ Rg _τ ð42ÞNoting the fact that Eq. (42) can be transformed toY¼ ½Rg ;0�½_τTb ; τTb �T , and using Eqs. (38) and (39), the obser-vation equation is written as

Y¼ CX ð43Þ

C¼ ½ΩkRgμk; :::;Ω1Rgμ1;Ω�1Rgμ�1; :::;Ω�kRgμ�k� ð44ÞEqs. (40) and (43) constitute the state-space model for aconstrained gyroelastic body. Observing B¼A�1_Bδ=2 inEq. (41), the controllability matrix of the system is

Q c ¼ ½A�1_Bδ;_Bδ;A

_Bδ;…;A2k�2_Bδ�=2 ð45Þ

Substituting the expression of A into Eq. (45) and follow-ing the similar procedure in Ref. [16], the rank of thecontrollability matrix is equivalent to the rank of Q ′cQ ′c ¼Q ′c1Q ′c2 ð46Þwhere

Q ′c1 ¼

q1c

q2c

⋱qkc

266664

377775; Q ′c2 ¼

I λ1I ⋯ λk�11 I

I λ2I ⋯ λk�12 I

⋮ ⋮ ⋱ ⋮I λkI ⋯ λk�1

k I

266664

377775

qic ¼

μT� iBδ=Ωi μT

i Bδ

μTi Bδ=Ω� i μT

� iBδ

" #; ði¼ 1; :::; kÞ; λi ¼ �Ω2

i ð47Þ

The determinant of Q ′c2 is nonzero for distinct λi. There-fore, the rank of Q ′c is determined by rank ðQ ′c1Þ which isthe same as

rankðQ ′c1Þ ¼ ∑k

i ¼ 1rankðqi

cÞ ð48Þ

If rankðqicÞ ¼ 2 holds for all qi

c, Q ′c1 is of full rank. Wedefine the modal controllability performance index as thesum of singular values of qi

c,

γc ¼ ∑k

i ¼ 1ðs1ðqi

cÞþs2ðqicÞÞ ð49Þ

where s1ðqicÞ and s2ðqi

cÞ are the two singular values of qic.

γc represents the ability of the actuators to control thesystem. Analogously, the modal observability performance

index is defined as

γo ¼ ∑k

i ¼ 1ðs1ðqi

oÞþs2ðqioÞÞ ð50Þ

where

qi0 ¼

�Ω� iRgμi �ΩiRgμ� i

�Ω2i Rgμ� i �Ω2

� iRgμi

" #; ði¼ 1; :::; kÞ ð51Þ

Now, the algorithm for optimal actuators/sensors place-ments can be summarized as follows:

(1).

Select a set of candidate locations for the collocatedCMGs and angular rate sensors on the flexible structure.

(2).

Assume one pair of CMG and sensor is mounted at thesth candidate locations. The resultant system is denotedas ℬs.

(3).

Establish the equations of motion of ℬs and thenperform modal analysis to obtain the dynamic modelin state-space form.

(4).

Calculate the γc and γo for ℬs. Denote them as γsc andγso, respectively.

(5).

Select a set of locations with the largest γsc � γso tomount CMGs and angular rate sensors. System con-trollability and observability should be guaranteed.

It can be seen that the proposed optimization approachjust requires calculating the optimization indexes for thecandidate locations, and do not require calling for anynonlinear optimization programming.

5. Vibration suppression controller design

Now, let us move on to the problem of vibrationsuppression of the gyroelastic body. The angular velocityfeedback control in Ref. [20] is adopted and extended tothe case where multiple CMGs and sensors are mountedon the structure. For sake of keeping the gimbal anglessmall for long-time operation, it is desirable that thegimbal angles asymptotically go to zero reference aftersuppressing the vibration. The control law is designed asfollows:

_δ¼ �kd_βc�kgδ ð52Þ

_βc ¼ ½_βc1; :::; _βcn�T ; δ¼ ½δ1; :::; δn�T ; kd ¼ d½kd1; :::; kdn�;kg ¼ d½kg1; :::; kgn� ð53Þ

where _βci is the angular rate along the output torque axisof the ith CMG. δi is the gimbal angle of the ith CMG. kd

and kg are positive coefficient matrices. If the followingdefinitions are made:

Q ¼ d½kg1=kd1; :::; kgn=kdn�; Mi ¼ kgi=ffiffiffiffiffiffikdi

p; Ni ¼

ffiffiffiffiffiffikdi

pM¼ d½M1; :::;Mn�; N¼ d½N1; :::;Nn� ð54Þwe have

Qkd ¼ kg ; Qkg ¼M2; MN¼ kg ; N2 ¼ kd ð55ÞWe define the Lyapunov function as

L¼ 12_τTE_τþ 1

2τTΛτþ 1

2δTQδ ð56Þ

b

bx by

bzgi

bxby

Fig. 4. Gyroelastic beam.

Fig. 5. The first four gyroelastic modes.

Fig. 6. The first four normal modes of the beam.

Q. Hu et al. / Acta Astronautica 96 (2014) 232–245238

0 5 10 15 20 25-1.5

-1

-0.5

0

0.5

1

1.5

time/T0 , non-dim

Tip

elas

tic tr

ansl

atio

n , m

tnode21 - xb

tnode21 - zb

22 24-5

0

5x 10-4

0 0.2 0.4 0.6 0.8

-1

0

1

Fig. 8. Time history of the translation at the tip (one CMG).

0 5 10 15 20 25-5

0

5

time/T0 , non-dim

Mod

al c

oord

inat

es ,

non-

dim

1

2

3

4

0 0.1 0.2 0.3

-2024

22 24

-202

x 10-4

Fig. 9. The first four modal coordinates (one CMG).

Q. Hu et al. / Acta Astronautica 96 (2014) 232–245 239

where Q is positive definite. Taking the derivative of theLyapunov function leads to

_L¼ _τTBδ_δ� _τTζ_τþδTQ _δ

¼ � _τTζ_τ� _τTBδðkd_βcþkgδÞ�δTQ ðkd

_βcþkgδÞ¼ � _τTζ_τ� _τTBδN2BT

δ _τ� _τTBδkgδ�δTkgBTδ _τ�δTM2δ

¼ � _τTζ_τ�δTM2δ�2δTMNBTδ _τ� _τTBδN2BT

δ _τ

¼ � _τTζ_τ� ∑n

i ¼ 1ðMiδiþNi

_βciÞ2r0 ð57Þ

Therefore, the system has Lyapunov stability. Using LaSal-le's theorem, the system response will asymptoticallyapproach the invariant set fx¼ ½_τT ; δT �T j_τ¼ 0; δ¼ 0g. Sothe system is globally asymptotically stable.

6. Numerical examples

In this section, all the theories developed above areapplied to a beam structure and a plate structure. Theirgyroelastic modes are demonstrated. Then, the optimiza-tion is conducted to determine the optimal placements forthe CMGs and angular rate sensors. Finally, the efficacy ofthe proposed control law is validated.

6.1. Gyroelastic beam

As shown in Fig. 4, it is the finite-element model of aconstrained beam, which is discretized as 20 beam ele-ments. All the nodes are chosen as candidate locations forthe CMGs and sensors. The fundamental frequency of thebeam structure is f0¼0.025 Hz (T0¼1/f0). We mount fourCMGs at nodes 6, 11, 16, and 21. The first four gyroelasticmodes are given in Fig. 5, whereas the first four normalmodes of the beam structure are shown in Fig. 6. It can beseen that the embedded angular momentum can make themotion in different directions coupled. So it is possible touse only one actuator to control the motion in alldirections.

All the nodes in Fig. 4 are chosen as candidate locationsfor the actuators/sensors. Fig. 6 gives the distribution ofγc � γo. Two different cases are considered in the optimi-zation problem. In case 1, it is supposed that the gimbal-fixed reference frame ℱgi of the CMG coincides with thebody-fixed frame ℱℬ in the initial status. In case 2, ℱgi isassumed to coincide with the frame transformed by

0 5 10 15 20 250

1000

2000

3000

4000

5000

6000

7000

Nodes Number

Opt

imiz

atio

n In

dex

c× o case 1

Fig. 7. Optimization index γc � γ

rotating ℱℬ by π=2 along yb when at the initial position.Since the natural frequencies for the modes in the xb–ybplane is smaller than the one in the yb–zb plane, it requiresless energy to control the motion in the xb–yb plane thanthe motion in the yb–zb plane. Therefore, the optimizationindexes in case one are larger than the indexes obtained incase 2, as shown in Fig. 7. Moreover, it can be seen that

0 5 10 15 20 250

500

1000

1500

2000

Nodes Number

Opt

imiz

atio

n In

dex

c× o case 2

o of the gyroelastic beam.

Q. Hu et al. / Acta Astronautica 96 (2014) 232–245240

locating the CMG and sensor on the node near the edge ofthe beam gives much more controllability and observa-bility to the system. This can be interpreted by observingthe gyroelastic modes in Fig. 5. The slopes at the edges are

0 5 10 15 20 25-1.5

-1

-0.5

0

0.5

1

1.5

time/T0 , non-dim

Gim

bal a

ngle

rate

s ,ra

d/s

d 1/dt

0 0.5 1

-1

0

1

22 24-10

-505

x 10-5

Fig. 11. Time history of the gimbal rate (one CMG).

0 5 10 15 20 25-10

0

10

20

30

40

time/T0 , non-dim

Gim

bal a

ngle

s ,ra

d

1

Fig. 10. Time history of the gimbal angle (one CMG).

0 5 10 15 20 25-1

-0.5

0

0.5

1

1.5

time/T0 , non-dim

Tip

elas

tic tr

ansl

atio

n , m

tnode21 - xb

tnode21 - zb

22 24-2

0

2x 10-7

0 0.5 1-0.5

00.5

1

Fig. 12. Time history of the translation at the tip (two CMGs).

usually bigger in the mode shapes; so the CMGs andsensors can better affect and measure the motion ofthe modes.

0 5 10 15 20 25-1

0

1

2

3

4

5

time/T0 , non-dim

Gim

bal a

ngle

s ,ra

d

20

21

Fig. 14. Time history of the gimbal angles (two CMGs).

0 5 10 15 20 25-4

-2

0

2

4

6

time/T0 , non-dim

Mod

al c

oord

inat

es ,

non-

dim

1

2

3

4

23 24 25

-2

0

2x 10-7

0.2 0.4

0

2

4

Fig. 13. The first four modal coordinates (two CMGs).

0 5 10 15 20 25-1.5

-1

-0.5

0

0.5

1

1.5

time/T0 , non-dim

Gim

bal a

ngle

rate

s ,ra

d/s

d 20/dt

d 21/dt

0 0.2 0.4 0.6-1

0

1

Fig. 15. Time history of the gimbal rates (two CMGs).

Fig. 16. Gyroelastic plate and the candidate locations for CMGs.

Fig. 17. The first gyroelastic mode of the plate.

Fig. 18. The second gyroelastic mode of the plate.

Fig. 19. The third gyroelastic mode of the plate.

Q. Hu et al. / Acta Astronautica 96 (2014) 232–245 241

Q. Hu et al. / Acta Astronautica 96 (2014) 232–245242

We choose Node 20 as the location for the CMG andangular rate sensor according to the optimization results.The first twenty gyroelastic modes are incorporated in thedynamic model. We impose an initial value for the modalcoordinates, and then the control law in Section 5 is usedto suppress the induced vibration. The initial value of thefirst four modal coordinates are set to [5,5,–1,–1]T, respec-tively. The others are set to zero. The maximum gimbalrate is set to be 1.05 rad/s. The time history of the tip

Fig. 20. The fourth gyroelas

Fig. 21. The first four norma

deflection is shown in Fig. 8. Fig. 9 gives the response ofthe first four modal coordinates. The time history of thegimbal angles and rates are shown in Figs. 10 and 11,respectively. It demonstrates that the proposed angularrates feedback control law can effectively suppress thevibration and make the gimbal angle asymptotically go tozero reference. Furthermore, if we choose Nodes 20 and 21for the actuators/sensors, similar numerical results areshown in Figs. 12–15. Because of the additional control

tic mode of the plate.

l modes of the plate.

0 5 10 15 20-1

-0.5

0

0.5

1

time/T0 , non-dim

Tip

elas

tic tr

ansl

atio

n , m

tnode261 - zb

tnode273 - zb

16 18 20-5

0

5x 10-10

0 0.2 0.4 0.6

-0.5

0

0.5

Fig. 23. Response of the deflection at node 261 and 273.

0 5 10 15 20-10

-5

0

5

10

time/T0 , non-dim

Mod

al c

oord

inat

es ,

non-

dim

1

2

3

4

17 18 19

-1

0

1x 10-8

0 0.2 0.4-10

0

10

Fig. 24. Response of the modal coordinates τ1–τ4.

Q. Hu et al. / Acta Astronautica 96 (2014) 232–245 243

input, the settling time is much reduced. Besides, thecontrol precision is improved.

6.2. Gyroelastic plate

A constrained plate structure with multiple CMGs isshown in Fig. 16. The fundamental frequency of the plate isf0¼0.0359 Hz (T0¼1/f0). We select 273 nodes as thecandidate locations for the CMGs and angular rate sensors.All the candidate nodes are mounted with CMGs whilecalculating the gyroelastic modes for this structure.Figs. 17–20 show the first four gyroelastic modes of theplate. The gyroelastic modes are reduced to normal modeswhen the angular momentums for the CMGs are setto zero (Fig. 21). We can see that the mode shapes andthe natural frequencies are changed when there existembedded angular momentum devices on the structure.

The optimal placements for the actuators/sensors arealso found by using the approach developed in Section 4.The two cases for the gyroelastic beam are also consideredhere. Fig. 22 gives the distribution of the optimizationindexes. Clearly, the nodes near the edges and cornersproduce larger indexes, which means the actuators/sen-sors located there lead to better controllability and obser-vability. This result relate well with the fact that the slopesat these nodes are usually bigger in the gyroelastic modes.

The nodes numbered 248, 260–263, and 271–273 arechosen to mount CMGs and angular rate sensors. The firsttwenty gyroelastic modes are incorporated in the dynamicmodel. Similar to the beam example, an initial deformationis imposed to the plate. The initial value of the first fourmodal coordinates are set to [10,10,–10,–10]T, respectively.The others are set to zero. The deflections at nodes 261and 273 are shown in Fig. 23. Figs. 24 and 25 give theresponse of the first eight modal coordinates. The timehistories of the gimbal angles and gimbal rates for theCMGs at nodes 261 and 273 are shown in Figs. 26 and 27,respectively.

We can see that the proposed control law is alsocapable of suppressing the vibration of the plate whilekeeping the gimbal angles small.

Finally, we provide a brief tradeoff discussion of CMG incomparison with RW and thruster. CMGs and RWs areboth momentum exchange devices. They both have beenconsidered for vibration suppressing of flexible structures[17–20]. CMG has high torque capabilities. It is a torque

05

10

-5

0

5-200

0

200

400

600

xy

z

c× o case 1

Fig. 22. Optimization index γc �

amplification device because small gimbal torque inputcan produce large control output. Therefore, it is moresuitable for active vibration control of a flexible structure[19,20]. However, it will increase the complexity of thesystem since the CMG has one more rotating shaft than theRW. Compared with the thrusters, CMG does not consumeany propellant, so it is more appropriate for long-termuse. Moreover, when suppressing the structure vibrationthrough thrusters, the velocities or accelerations of the

05

10

-5

0

5-200

0

200

400

600

xy

z

c× o case 2

γo of the gyroelastic plate.

0 5 10 15 20-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

time/T0 , non-dim

Mod

al c

oord

inat

es ,

non-

dim

5

6

7

8

0 0.2 0.4 0.6

-0.4-0.2

00.20.4

16 18 20

-10123

x 10-10

Fig. 25. Response of the modal coordinates τ5–τ8.

0 5 10 15 20-4

-3

-2

-1

0

1

2

time/T0 , non-dim

Gim

bal a

ngle

s ,ra

d

261

273

Fig. 26. Time history of the gimbal angle.

0 5 10 15 20-1

-0.5

0

0.5

1

time/T0 , non-dim

Gim

bal a

ngle

rate

s ,ra

d/s

d 261/dt

d 273/dt

0 0.2 0.4 0.6

-1

0

1

Fig. 27. Time history of the gimbal rate.

Q. Hu et al. / Acta Astronautica 96 (2014) 232–245244

location of the thrusters should be measured [2,25].However, as shown in the formulation in Section 5, onlythe angular velocities of the mount sites of the CMGs andtheir gimbal angles are needed when adopting the CMGs

as actuators. It is much easier to measure angular informa-tion in practice. In conclusion, using CMGs as actuatorsprovides an attractive option for active vibration suppres-sion of flexible structures.

7. Conclusion

In this paper, a novel approach for vibration suppres-sion of constrained space structures is proposed. Controlmoment gyroscopes (CMGs) are directly mounted on thestructures as actuators, while angular rate sensors collo-cated with the CMGs are used as sensors. The detailedequations of motion incorporating the dynamics of theCMGs are derived by Kane's equation. Then the mathema-tical model is linearized to a state-space form, based onwhich the modal characteristics of the system are inves-tigated. The optimal placement of the actuators/sensorsis solved from the controllability and observability per-spective. An optimization index is defined to find theoptimal locations on which the actuators/sensors shouldbe mounted to maximize the controllability and observa-bility of the structural modes. An angular rates feedbackcontrol law is developed to actively suppress the vibration.Numerical examples of a beam structure and a platestructure are given. It is found that the mode shapes canbe changed by the embedded angular momentum. Theoptimization results indicate that the actuators/sensorsshould be located near the edges and corners of the beamand plate. The proposed control law can efficiently reducethe vibration and make the gimbal angles asymptoticallygo to zero reference.

It should be noted that all the formulations are devel-oped aiming at an arbitrary gyroelastic body. So theproposed approach can be applied to space structuresmore complicated than a beam or a plate.

Acknowledgments

This paper reports the work carried out in the projectsupported by the National Natural Science Foundation ofChina (11272027).

Appendix A

Since a traditional flexible structure can be viewed as aspecial case of the gyroelastic body with zero embeddedangular momentum, the normality condition of the gyro-elastic body should be reduced to the one for the tradi-tional flexible body when setting the angular momentumterm to zero [13]. In order to keep the theory of gyroelasticbody consistent with the theory of structure dynamics, weset the normality condition as Eq. (36). The proof is givenas follows:

Substituting Eq. (38) into Eq. (36) results in

ΩβυTαEυβþΩαμT

αEμβ�υTαGμβ ¼ 2Ωαδαβ ðA:1Þ

When there is no stored angular momentum in thestructure, i.e. G¼ 0. Then, substitute Eqs. (24) and (38) in

Q. Hu et al. / Acta Astronautica 96 (2014) 232–245 245

Eq. (23)

�Ω2αEμαþΛμαþ jð�Ω2

αEυαþΛυαÞ ¼ 0 ðA:2ÞTherefore, we may as well assume μα ¼ υα and �Ω2

αEυαþΛυα ¼ 0 when there is no embedded angular momentumon the structure. Then Eq. (A.1) reduces to

μTαEμβ ¼ δαβ ðA:3Þ

Using Eq. (A.3) in fTαAfβ ¼ 2Ω2αδαβ leads to

μTαΛμβ ¼Ω2

αδαβ ðA:4ÞIt is evident that Eqs. (A.3) and (A.4) are the orthogon-

ality conditions for traditional flexible structures. So, theorthogonality conditions for gyroelastic bodies remainconsistent with the ones for traditional flexible structures.

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