Vibration Isolation Platform for Control MomentGyroscopes on Satellites

Yao Zhang1 and Shijie Xu2

Abstract: The vibration isolation platform of control moment gyroscopes (CMGs) on satellites is discussed. First, the theory of single-axisvibration isolation is presented and an isolation scheme for CMGs is put forward and applied to a satellite, which can provide an ultraquietenvironment for optical payloads. Second, an integrated satellite dynamic model including the vibration isolation platform and pyramid con-figuration of the CMGs is built. The validity of this dynamic model is then verified by ADAMS. Third, the influence of the vibration isolationsystem on the attitude control system of the satellite is analyzed in detail and the appropriate parameters of the proportional-integral-derivative(PID) controller are selected. Finally, the attitude stabilization performance with the vibration isolation system is predicted by integratedsimulation. The analysis and simulation results show that the vibration isolation platform can attenuate the disturbances to a certain extent—thus, improving the attitude stability—and when the vibration isolation platform is used, the closed-loop control system is conditionally stable,which means that the gain parameter of the PID controller should be within a range according to the root locus diagram.DOI: 10.1061/(ASCE)AS.1943-5525.0000156. © 2012 American Society of Civil Engineers.

CE Database subject headings: Vibration; Control systems; Satellites.

Author keywords: Control moment gyroscope; Jitter control; Vibration isolation platform; Attitude control.

Introduction

Presently, the control moment gyroscope (CMG) is widely used asthe actuator of attitude control in modern satellites (Heiberg et al.2000; Jin and Xu 2009; Wie 2005). Because of the static and dy-namic imbalances of the rotor and the bearing disturbances, theCMG becomes one of the main vibration sources on satellites. Toachieve attitude control with high orientation precision and stabi-lization, providing an ultraquiet environment for the optical pay-loads, it is necessary to reduce the vibration propagated into thesatellite, which is caused by the CMG.

Previous research has proven that the disturbance frequenciesproduced by the rotor of the CMG is higher than 10 Hz, and thishigh-frequency vibration is directly propagated into the satellitebecause the actuator of the attitude control cannot give a quick re-sponse (Masterson et al. 1999). Vibration isolation is one of thetechniques used to solve this problem, in which the object is isolatedfrom the vibration source by adding a vibration isolator betweenthe two. The D-strut (Wilson and Wolke 1997) manufactured byHoneywell, Inc., was mainly used in isolating the vibration of theflywheel of the Hubble Space Telescope (HST). The effectivenessof the vibration isolator was sufficiently proved in orbit. During thepast decades, many researchers have focused on how to improve theperformance of the vibration isolator element and have designed

some vibration isolators, for instance, the D-Strut, 1.5 Hz D-Strut,Adaptable D-Strut, andHybridD-Strut (Anderson et al. 1991;DavisandWorkman 1992; Davis et al. 1994a, b, 1995). There is abundantresearch regarding vibration isolation systems using these vibrationisolator elements. Davis et al. (1986) tested the efficiency of thevibration isolator in the HST by conducting a hardware simulationon the ground. Pendergast and Schauwecker (1998) designed apassive vibration isolation system for a reaction wheel to meet theimaging performance requirement of the Advanced X-ray Astro-physics Facility. The isolation system aligned in the geometrycommonly known as the Stewart platform arrangement can reducemultidimensional disturbances. Bronowicki (2006) gave a detaileddescription of the vibration isolation system of the James WebbSpace Telescope. The jitter performance using the vibration iso-lation system of spacecraft was predicted in some missions, such asin the Space Interferometry Mission (Basdogan et al. 2000), Ter-restrial Planet Finder Coronagraph (Dewell et al. 2005; LoBoscoet al. 2004), Geostationary Operational Environmental Satellite-N(GOES-N) spacecraft (Miller et al. 2007), and Solar DynamicsObservatory (Liu and Maghami 2008). In these studies, the struc-tures of the spacecraft are all modeled using the finite-element (FE)method, where some use a vibration isolation system consistingof six decoupled second-order low-pass filters to approximatethe effect of a passive fly wheel mount (Gary et al. 1998; Mezaet al. 2005). These accurate FE models are necessary to predictthe jitter performance of the spacecraft. However, the analysismethod using FE models has some limitations. For instance, thespacecraft structure should be well known, the dynamic couplingcannot be obtained, and the FE model is not suitable for attitudecontroller design based on modern control theory.

Despite the abundant research concerning vibration isolators andjitter performance prediction, research on analytical models of in-tegrated satellites including the vibration isolation system and theinfluence of the vibration isolation system on the attitude controlsystem is rare. In this study, an integrated satellite dynamic modelincluding a vibration isolation platform and pyramid configuration

1Ph.D. Student, Dept. of Astronautics, Beijing Univ. of Aeronauticsand Astronautics, Beijing, China 100191 (corresponding author). E-mail:zhangyao7000@yahoo.cn

2Professor, Dept. of Astronautics, Beijing Univ. of Aeronautics andAstronautics, Beijing, China 100191. E-mail: starsjxu@yahoo.com.cn

Note. This manuscript was submitted on January 25, 2011; approved onSeptember 2, 2011; published online on September 5, 2011. Discussionperiod open until March 1, 2013; separate discussions must be submitted forindividual papers. This paper is part of the Journal of Aerospace Engi-neering, Vol. 25, No. 4, October 1, 2012. ©ASCE, ISSN 0893-1321/2012/4-641–652/$25.00.

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CMGs was built using the Newton-Euler method. Furthermore, theinfluence of the vibration isolation system on the attitude controlsystem is analyzed in detail and the parameter design of theproportional-integral-derivative (PID) controller is described.

Theory of Single-Axis Vibration Isolation andDescription of CMG Disturbances

With the purpose of reducing the influence of a vibration sourceon the payload of a satellite, the passive vibration isolationtechnique on a satellite intends to isolate the vibration source(such as the flywheel or CMG) from the satellite by adding a vi-bration isolator. The spring-damp system, which is shown in Fig.1(a), is used with a commonly applied passive vibration isolator(Agrawal 2009).

The transfer function of the passive vibration isolator can bewritten as

GðsÞ ¼ cs þ kMs2 þ cs þ k

ð1Þ

The undamped natural frequency of the transfer function isvn and the damping ratio is j. Substituting s5 jv into Eq. (1) and

letting the frequency ratio g5v=vn, the following equation isobtained to represent the vibration isolation effect:

mF ¼ jfsjjfdj ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 þ 4j2g2

ð12 g2Þ2 þ 4j2g2

sð2Þ

By taking the damping ratio j as 0.1, 0.5, and 1, three curves of theamplitude of the frequency response function are obtained [shown inFig. 1(b)].

It is evident in Fig. 1(b) that the design of a passive isolatorinvolves a trade-off between the resonance amplification and thehigh-frequency attenuation; the ideal isolator should have frequency-dependent damping with high damping below the critical frequencyffiffiffi2

pvn to reduce the amplification peak and low damping above

ffiffiffi2

pvn

to improve the decay rate.For convenience in building the analytical model, the current

study focuses on the disturbances of the CMG caused by the staticand dynamic imbalances of the rotor; i.e., the nonverticality andnoncrossing between the gimbal axis and rotor axis as well as thedeviation of the mass center of the CMG system from the gimbalrotation axis. The rotor static imbalance rwj 5 ½ g j z �T , whichshows the offset of the center of mass of the rotor from its spinaxis. Dynamic imbalance AIwj is caused by the misalignment ofthe rotor’s principle axis and the rotation axis; Afgj means the

Fig. 1. (a) Spring-damp system; (b) frequency response curve

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installation error of the rotor spin axis and rfgj is the vector from themass center of the gimbal to the geometric center of the rotor(Jin 2008). In the following equation AIwj and Afgj are shown:

AIwj ¼

264

1 h 2m

2h 1 0

m 0 1

375;  Afgj ¼

264

1 a 2b

2a 1 x

b 2 x 1

375 ð3Þ

where h and m 5 small variables describing the dynamic imbalanceand a, b, and x 5 small angles caused by the installation error ofthe bearing rotor. In the current study, the errors are boundedwithinmicronmagnitude. From these errors, it is determined that the staticand dynamic imbalances are about 1.36 and 17 g/cm2, respectively.

Control Moment Gyroscope Isolation Scheme

It is common in applications to separately isolate the flywheel ofa satellite. The HST uses viscous fluid dampers called D-Struts toattenuate the axial disturbances of each flywheel (Davis et al. 1986).The Defense Satellite Communications System III spacecraft usesfour damped stainless steel spring isolators supporting a wheelto provide six degrees of freedom (DOF) wheel isolation (Strainand Mittal 1988). The Chandra X-ray Observatory employs ahexapod isolator at each of its six reaction wheels (Pendergast andSchauwecker 1998). Kamesh et al. (2010) designed a flexibleplatform to act as a mount for each fly wheel, and the platform con-sists of four folded continuous beams arranged in three dimensions.

However, CMGs employed on a satellite are in some kind of con-figuration [L. P. Davis and T. T. Hyde, “Moment control unit forspacecraft attitude control,” U.S. Patent No. 6340137.2002-01-22(2002)] (Kim and Agrawal 2006; Peck and Cavender 2003), and thedirection of CMGs’ output torque keeps changing, which makes itcostly to isolate them separately. The advantages of isolating the vi-bration with one platform are that it increases the suspended mass,enabling a lower break frequency; it requires fewer isolators, whichshould lower the cost; and it can save space for other payloads ona satellite. Passive vibration suppression and isolation are recommendedas a first approach. Therefore, it is common to use a six DOF passivevibration isolation platform to interface the cluster of CMGs with thesatellite bus, which not only isolates disturbances but also transmits the

effective torques to the satellite to realize the attitude control. Someoptical imaging satellites have already used this method to isolatethe disturbances caused by theCMGs, such asWorldview I,WorldviewII, and Pleiades-HR (Baudoin et al. 2001). Based on this kind of instal-lationmethod, the integrated satellite dynamicmodelwill be establishedin the current study to testify to the effectiveness of the passive vibrationisolation platform. The CMG isolation scheme is shown in Fig. 2.

In this research, the CMGs are in a pyramid configuration and themass of every CMG is about 19 kg. The passive multileg vibrationisolation platform is a Stewart platform with a cubic configurationand it has six extensible legs (Dasgupta and Mruthyunjaya 1998;Mahboubkhah et al. 2009). Each leg of the platform includes anupper part, lower part, and a spring and damper connecting the twoparts. The legs are connected to the upper platform by sphericaljoints and the base platform is connected by universal joints.

Dynamic Model of the Integrated Satellite with theCMGs and Vibration Isolation Platform

This section presents the dynamic modeling of the integrated sat-ellite with the CMGs and vibration isolation platform using theNewton-Euler method. Because the CMGs are fixed at the upperplatform, the CMGs and the upper platform are taken as the upperplatform system, while the satellite and the base platform are takenas the base platform system. First, the kinematic and dynamicmodels of the each leg are derived. Then, based on the forces andtorques that are given to the upper platform system and the baseplatform system, the dynamic models are established. According tothe kinematic and dynamic model of the base platform system, theattitude information of the satellite can be obtained.

Description of a Leg

In the inertial frame, the kinematic equations of the ith leg can bedescribed as

Si ¼ t þ Aeupi 2 ðd þ AedqiÞ ð4Þ

li ¼ kSik ð5Þ

sui ¼ Si=li ð6Þ

Fig. 2. CMG isolation scheme

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vSi ¼ _t þ Aeuv�p pi 2

�_d þ Aedv

�q qi

� ð7Þ

aSi ¼ €t þ Aeu _v�p pi þ Aeuv

�p v

�p pi2 aqi ð8Þ

aqi ¼ €d þ Aed _v�q qi þ Aedv

�q v

�q qi ð9Þ

where the variables with index i 5 ith leg variables, indicating thatin general the equations are applicable to any leg; Si 5 leg vector;vSi and aSi 5 velocity and acceleration of the leg, respectively; aqi 5acceleration of the base platform connection point of the leg; t 5position vector from the origin of the inertial frame to themass centerof the upper platform; d 5 position vector from the origin of theinertial frame to the mass center of the base platform; pi 5 vectorfrom the mass center of the upper platform to the upper platformconnection point; qi 5 vector from the mass center of the baseplatform to the base platform connection point;vp andvq 5 angularvelocities of the upper platform system and the base platform, re-spectively; sui 5 unit vector along the leg; and li 5 leg length. Thecoordinate systems of the vibration isolation platform are shown inFig. 3, in which fe is the inertial frame; fb is the satellite body frame;fu is the upper platform system frame; fd is the base platform frame;and Aab is the transformation matrix from fb to fa.

Kinematics of a Leg

If the installation of the universal joint is determined, then the unitvector along the fixed axis of the universal joint, ki, is also de-termined, where yi is the unit vector along the moving axis of theuniversal joint and hi is the normal vector of the plane formed byki and yi. The expressions of ki and yi are as follows:

yi ¼ ðk�i suiÞ=kk�i suik ð10Þ

hi ¼ y�i ki ð11Þ

The angular velocity of each leg is

vli ¼ vi þ vsisui ð12Þ

where

vi ¼ s�uivSi=li

vsi ¼ 2 ðvTi hiÞ=ðsTuihiÞ

The angular acceleration of each leg is

«li ¼ «i þ «sisui ð13Þ

where

«i ¼ 1li

�s�uiaSi 2 2l_ivli þ 2l_ivsisui 2 livsiðs�uiviÞ

� ð14Þ

«si ¼�ð«Ti kiÞðkTi suiÞ þ �

vkivyi�ðk�i yiÞTsui��½12 ðkTi suiÞ2� ð15Þ

vki ¼ vTli ki ð16Þ

vyi ¼ vTi yi ð17Þ

The sliding velocity and acceleration between the two parts of theleg are

l_i ¼ sTuivSi ð18Þ

l€i ¼ sTuiaSi 2 sTui½v�li v

�li Si� ¼ sTuiaSi þ liv

Tli vli 2 liv

2si ð19Þ

The accelerations of the gravity centers of the lower and upperparts of the leg are

aui ¼ aqi þ «�li rui þ v�li v

�li rui þ l€isui þ 2l_iv

�li sui ð20Þ

adi ¼ aqi þ «�li rdi þ v�li v

�li rdi ð21Þ

where rui and rdi 5 position vectors of the gravity centers of thelower and upper parts in the inertial frame, respectively.

Dynamic Equations of a Leg

This section presents the leg dynamic equations that are built usingthe Newton-Euler approach. With consideration of the momentsacting on the leg in the inertial frame, Euler’s equation for thewhole leg can be obtained as follows:

ðIui þ IdiÞ«li þ v�li ðIui þ IdiÞvli

¼ S�i Fsi þ Muihi 2 csi�vli 2Aeuvp

�2 cui

�vli 2Aedvq

�2 ðmuirui þ mdirdiÞ�aqi ð22Þ

and Newton’s equation for the whole leg is obtained as follows:

muiaui þ mdiadi ¼ Fsi þ Fui ð23Þ

wheremui and mdi 5masses of the upper and lower parts of the leg,respectively; Iui and Idi 5moments of inertia of the upper and lowerparts of the leg in the inertial frame, respectively; Fsi 5 constraintforce at the spherical joint acting on the upper part; Fui 5 constraintforce at the universal joint acting on the lower part;Mui 5magnitudeof the constraint moment at the universal joint acting about the legaxis; and csi and cui 5 coefficients of viscous friction in the sphericaland universal joints, respectively.

Here, Fsi, Fui, and Mui can be solved from Eqs. (22) and (23) asfollows:

Fsi ¼ Xisui þ Ki ð24Þ

Fui ¼ muiaui þ mdiadi 2Fsi ð25Þ

Mui ¼ ðsTuiCiÞ=ðsTuihiÞ ð26Þ

where

Fig. 3. Coordinate frames of the vibration isolation platform

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Xi ¼ sTuiFsi ¼ muisTuiaui þ kiðli2 l0iÞ þ ci_li

Ci ¼ ðIui þ IdiÞ«li þ v�liðIui þ IdiÞvli þ csi

�vli 2Aeuvp

�þ cui

�vli 2Aedvq

� þ ðmuirui þ mdirdiÞ�aqi

Ki ¼ C�i sui 2Muih�i sui

li

where ki 5 stiffness coefficient of the leg, ci 5 damping coefficientof the leg, and l0i 5 nominal length of the leg.

Dynamic Equations of the Integrated Satellite

Based on the theorem of angular momentum and the theorem of mo-mentum, and according to the disturbance types, the dynamic equa-tions of the upper platform system can be organized and simplified as

mp _vp þ v�p p9 þ _p9 ¼ 2Aue

PNi¼ 1

Fsi þ Fp

ðA1 þ A2Þ _vp ¼ 2PNi¼ 1

p�i AueFsi þ PNi¼ 1

Fpi þ Tp

8>>><>>>:

ð27Þ

Fig. 4. (a) Position response curve of the base platform system obtained by using MATLAB; (b) position response curve of the base platform systemobtained by using ADAMS

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Fpi ¼ csi�Auevli 2vp

� ð28Þ

wheremp 5mass of the upper platform system; vp 5 velocity of themass center of the upper platform system;vp 5 angular velocity ofthe upper platform system; p9 5 momentum of the upper platformsystem that does not contain the disturbances; Fp 5 disturbanceforces caused by the CMGs; Tp 5 combination of the outputtorque provided by the CMGs and disturbance torques that arecaused by the coupling of theCMGs and the satellite;A1 5 coefficientmatrix associated with the disturbances; A2 5 coefficient matrix notrelated to the disturbances; N 5 number of legs; and Fpi 5 momentof viscous friction at the spherical joint.

The dynamics of the base platform system can be described as

mb�_vb þ v�

b vb� ¼ 2Abe

PNi¼ 1

Fui þ Fext

Ib _vb þ v�b Ibvb ¼ 2

PNi¼ 1

ðAbdqi þ rdbÞ�ðAbeFuiÞ

þ PNi¼ 1

Fqi þ Mext2AbePNi¼ 1

Muihi

8>>>>>>>><>>>>>>>>:

ð29Þ

Fqi ¼ cui�Abevli 2vb

� ð30Þ

wheremb and Ib 5mass and moment of inertia of the base platformsystem, respectively; vb 5 velocity of the mass center of the baseplatform; vb 5 angular velocity of the base platform system; Fext

and Text 5 external disturbing force and torque acting on the sat-ellite, respectively; rdb 5 vector from the origin of fb to the originof fd ; and Fqi 5moment of viscous friction at the universal joint.

Thus, the integrated satellite dynamic model with the vibrationisolation platform and pyramid configuration CMGs is a combi-nation of Eqs. (27)–(30).

Validation of the Dynamic Model of the VibrationIsolation Platform

The CMGs are considered to be locked. The validity of the dynamicmodel of the integrated satellite is verified by using the commercialsoftware ADAMS. Obtained from the Stewart platform (DasguptaandMruthyunjaya 1998; Mahboubkhah et al. 2009), the matrices ofthe upper and base platform points are as follows:

p¼24 0:5 2 0:25 2 0:25 0:5 2 0:25 2 0:25

0 0:25ffiffiffi3

p2 0:25

ffiffiffi3

p0 2 0:25

ffiffiffi3

p0:25

ffiffiffi3

p

0 0 0 0 0 0

35m

q¼24 0:25 2 0:5 0:25 0:25 2 0:5 0:25

0:25ffiffiffi3

p0 2 0:25

ffiffiffi3

p2 0:25

ffiffiffi3

p0 0:25

ffiffiffi3

p

0 0 0 0 0 0

35m

The unit vectors along the fixed axis of the universal joints are asfollows:

k ¼

264 2

ffiffiffi3

p=2 0

ffiffiffi3

p=2

ffiffiffi3

p=2 0 2

ffiffiffi3

p=2

0:5 2 1 0:5 0:5 2 1 0:5

0 0 0 0 0 0

375

where ki and ci 5 20,000 N/m and 500 N×s/m of the leg, re-spectively, and csi and cui 5 0.0001 N×m×s/rad. Letting a set ofexternal forces ð½ 10 0 30 �TNÞ act on the upper platform, the

Fig. 5. Control system block diagram

Fig. 6. Root locus diagram in the x-rotation direction

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position response curves of the base platform system can beobtained by using MATLAB and ADAMS, respectively, as shown inFigs. 4(a and b). The simulation results show that the response curvesof the base platform system are the same, which means that thedynamic model of the integrated satellite using the Newton-Eulermethod is valid and could be applied to the latter research.

Analysis of the Influence of the Vibration IsolationPlatform on the Attitude Control System

The additional compliance introduced by the vibration isolationplatform has a major impact on the low-frequency dynamics of thesystem, and its interaction with the attitude control system must be

Fig. 7. Bode plot in the x-rotation direction

Fig. 8. Nyquist plot in the x-rotation direction

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taken into account (de Marneffe et al. 2009). Therefore, the in-fluence of the platform on the attitude control system will beanalyzed in detail.

Because the amplitude of the vibration is small, it is assumedthat the platform configuration does not change. Under the at-titude stabilization control, both the attitude angles and theangular velocities are small. Thus, the transformation matri-ces Aeu and Aeb can be seen as identity matrices and the squareof the angular velocities can be ignored. Because of thesmall influence of the moment of inertia and the mass of theleg on the transfer function of the vibration isolation platform,they can be ignored while analyzing the effect of the vibra-tion isolation platform on the attitude control system. Themoment of inertia and the mass of the upper platform system

are chosen as diagð½ 8:8 8:8 15 �Þ kg×m2 and 82.8 kg (themass of each CMG is 19 kg and the mass of the upper plat-form is 6.8 kg), respectively. The transfer function of thevibration isolation platform of each main channel can beobtained.

The vibration isolation platform can transmit both the forcesin three translational directions and the torques in three rota-tional directions. However, the attitude controller can onlyprovide torques in the three rotational directions. Thus, it isimportant to analyze the influence of the vibration isolationplatform on the attitude control system in the three rotationaldirections.

The transfer functions of the three rotational directions are asfollows. For the x-rotation ðuxÞ channel

Fig. 9. (a) Attitude angular velocities of the satellite without the vibration isolation platform; (b) attitude angular velocities of the satellite with thevibration isolation platform

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GRðsÞ ¼ cs3 þ �0:036c2 þ k

�s2 þ 0:072kcs þ 0:036k2

70:3s4 þ 3:69cs3 þ ð3:69k þ 0:036c2Þs2 þ 0:072kcs þ 0:036k2ð31Þ

For the y-rotation ðuyÞ channel

GPðsÞ ¼ cs3 þ �0:036c2 þ k

�s2 þ 0:072kcs þ 0:036k2

70:3s4 þ 3:69cs3 þ ð3:69k þ 0:036c2Þs2 þ 0:072kcs þ 0:036k2ð32Þ

For the z-rotation ðuzÞ channel

GYðsÞ ¼ cs þ k20s2 þ cs þ k

ð33Þ

where k and c 5 unknown and Eq. (31) is the same as Eq.(32) because of the symmetry of the vibration isolation platform.

A simplified control system block diagram is shown in Fig. 5,which assumed unity dynamics for the sensors and CMGs (Rybaket al. 1973). With the assumption of the decoupling of the inte-grated satellite dynamic model, each channel of the attitude controlsystem can be analyzed individually and the PID controller can bedescribed as

PIDðsÞ ¼ K

�s þ tp

��s þ tq

�s

ð34Þ

where tp and tq 5 known parameters and K 5 gain parameter.The transfer function of the satellite dynamics and kinematics can

be described as

uðsÞ ¼ 1Is2

ð35Þ

Only the x-rotation channel is analyzed in the present paper. Theanalyses of the y-rotation and z-rotation channels are omitted be-cause they are similar to that of the x-rotation channel. The open-looptransfer function of the system is as follows:

GðsÞ ¼ PIDðsÞGRðsÞuðsÞ ¼ K

�s þ tp

��s þ tq

�Ixs3

cs3 þ �0:036c2 þ k

�s2 þ 0:072kcs þ 0:036k2

70:3s4 þ 3:69cs3 þ ð3:69k þ 0:036c2Þs2 þ 0:072kcs þ 0ð36Þ

The satellite parameter Ix and the vibration isolator parameters cand k are chosen as Ix5 1,100 kg×m2, c5 500N×s/m, and k5 20,000N/m. The parameters of the PID controller are chosen as tp5 0.079and tq 5 5 3 1024.

The root locus diagram (shown in Fig. 6), can be obtained by usingMATLAB. Fig. 6 shows that the vibration isolation platform can

introduce three zero points and four pole points to the open-loopsystem compared with the system without the vibration isolationplatform. It can also be seen in Fig. 6 that a pair of conjugate poleswill move toward the right half-plane with the increase of K, whichcan make the closed-loop system unstable. Thus, when the transferfunction of the vibration isolation platform is added, the closed-loop

Fig. 10. Attitude angles of the satellite with the vibration isolation platform in the large attitude maneuver

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system is conditionally stable. Therefore, to make the closed-loopsystem stable, the gain parameter of the PID controller should bewithin the range according to the root locus diagram. In this range,an appropriate gain parameter should be selected. Then, the Bodeplot can be drawn according to the selected gain parameter. Fur-thermore, the stability margins, which include the amplitude marginand the phase margin of the system, can be obtained from the Bodeplot. The gain parameter of the attitude controller can be verified asappropriate if the stability margins are reasonable.

From Fig. 6, the scope of K can be obtained as follows:

0:5 , K , 9;130 ð37Þ

where K 5 2,500. Hence, the Bode and Nyquist plots of the open-loop system shown in Figs. 7 and 8 can be drawn.With Figs. 7 and 8,the stability of the closed-loop system can be analyzed and thestability margins can be obtained. According to the Nyquist stability

criterion, the closed-loop system is stable if Z 5 0, where Z is thenumber of poles of the closed-loop system in the right half-plane;i.e., Z5P2 2N . Here, P is the number of poles of the open-loopsystem in the right half-plane and N is the number of the circles ofNyquist contour anticlockwise encircling the point (21, j0) on theNyquist plot.

Fig. 7 shows that the amplitude margin is 11.2 dB and the phasemargin is 88�. In Fig. 8, the number of the negative crossover is 1:N2 5 1; near the left of the point (21, j0), the number of thepositive crossover is 1: N1 5 1. Thus, N5N1 2N2 5 0 can beobtained. FromEq. (36),P5 0 is obtained. Therefore, Z5 0,whichmeans the closed-loop system is stable and the chosen gain pa-rameter of the attitude controller is appropriate.

Through computation, the vibration isolation platform’s cornerfrequencies of the three rotationalmodes are 2.46, 2.46, and 4.58Hz.The bandwidth of the attitude control system is about 0.37 Hz. Fromthis result, it can be preliminarily obtained that the corner frequency

Fig. 11. (a) Attitude angular velocities of the satellite without the vibration isolation platform in the large attitude maneuver; (b) attitude angularvelocities of the satellite with the vibration isolation platform in the large attitude maneuver

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of the isolator should be about one decade above the attitude controlbandwidth for the stability of the control system.

Simulations and Discussion

The attitude angular velocities of the integrated satellite with andwithout the vibration isolation platform have been simulated for theintegrated satellite dynamic model with the vibration isolation plat-form and pyramid configuration CMGs. A PID controller was usedto realize three-axis stabilization. As elaborated previously, K waschosen as 2,500 and parameters Kp, Ki, and Kd of the PID con-troller were obtained as 200, 0.1, and 2,500, respectively. In thepresent paper, the three initial attitude angles were all chosen as1.5�. The steering law of the CMGs was to distribute the gimbalrates according to the situation of gimbal angles and commandtorque. To avoid possible singularity, a robust pseudo inversesteering law was used. The rotor motor, gimbal motor, attitudemeasurement, and external disturbing torque acting on the satellitewere ignored. The mass and the moment of inertia of the satelliteare as follows (Wie et al. 2002):

mb ¼ 1;000 kg;  Ib ¼24 1;100 220 210

220 900 215

210 215 800

35kg=m2

The simulation results are shown in Figs. 9(a and b), which in-dicate the attitude angular velocities converge well to their equi-libriums and the response time is appropriate. This shows that thesatellite can realize attitude stabilization and the chosen parametersof the PID controller are reasonable. In Fig. 9(a), it is evident that themagnitude of the attitude stability is 3.4 3 1024�/s under the dis-turbances of the CMGs. In Fig. 9(b), it is evident that the magnitudeof the attitude stability is 1.13 1025�/s with the vibration isolationplatform. By comparing Figs. 9(a and b), it is evident that thevibration isolation platform is able to reduce the vibrationamplitude of the attitude angular velocities by 3% of that underthe disturbances of the CMGs. Therefore, the vibration isolation

platform can attenuate the disturbances to a certain extent andcan improve the attitude stability.

When the satellite was required to realize a large angle attitudemaneuver, the attitude angles and the angular velocities were notsmall and the square of the angular velocities was not able to beignored. Therefore, the dynamic model of the integrated satelliteobviously has a nonlinear characteristic. For this problem, it isthought that the analysis method based on the assumption of smallattitude angles can be regarded as a foundation. Then, the chosenparameters of the PID controller are able to be used to simulate thelarge angle attitude maneuver and their validity can be tested by thesimulation results.

The three initial attitude angles were chosen as zero degrees, whilethe target angles were chosen as 60, 30, and 260�. The attitudemaneuver simulation results are shown in Figs. 10–12. In Figs. 10and 11, it is shown that the attitude angles and attitude angularvelocities converge well to their equilibriums and the response timeis appropriate, which indicates that the chosen parameters of the PIDcontroller are reasonable and the satellite is able to realize the largeangle attitude maneuver. The simulation results (see Fig. 12) of thegimbal angles show that the CMGs work normally and the CMGsare able to provide sufficient torque to realize the large angle at-titude maneuver. From the comparison between Figs. 11(a and b),it is obvious that the vibration isolation platform is able to reducethe vibration amplitudes of the attitude angular velocities by 3.3%of those under the disturbances of the CMGs. The vibration iso-lation platform can also improve the attitude stability in the largeangle attitude maneuver.

Conclusions

In this paper, a vibration isolation platform for CMGs on a sat-ellite was studied. The dynamic model of the integrated satellitewith the vibration isolation platform and CMGs was built usingthe Newton-Euler method. This dynamic model has been provedto be valid and was able to be applied in this research. Accordingto the assumption of small attitude angles, the influence of the

Fig. 12. History of the gimbal angles in the large attitude maneuver

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vibration isolation platform on the attitude control system wasanalyzed. It was obtained that the closed-loop control system isconditionally stable because of the application of the vibrationisolation platform for the CMGs. Moreover, the parameter designmethod of the PID controller was described to make the closed-loopsystem sufficiently stable. Selected parameters of the PID controllerwere used to simulate the attitude stabilization and the large angleattitude maneuver, respectively, and their validity was tested by thesimulation results. The simulation results showed that the applica-tion of the vibration isolation platform to CMGs could improve theattitude stability of the satellite to 0.04 arcsec and the parameterdesignmethod of the PID controller is valid according to the analysismethod based on the assumption of small attitude angles.

In the present study, valuable findings for the vibration isolationplatform of CMGs on a satellite were obtained. However, the dy-namic model of the integrated satellite is an ideal one, ignoring theinfluence of flexible appendages, which limits the scope of its ap-plication. In future studies, flexible appendages should be taken intoaccount while building the integrated satellite dynamic model andthe influence of the flexible appendages on the vibration isolationplatform should be analyzed.

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