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Integrated power and vibration control of gyroelastic body with variable-speed control moment gyros
Journal Pre-proof Integrated Power and Vibration Control of Gyroelastic Body with Variable-Speed Control Moment Gyros Chuandong Guo, Quan Hu, Yao Zhang, Jun Zhang PII:
S0094-5765(19)31460-2
DOI:
https://doi.org/10.1016/j.actaastro.2019.12.027
Reference:
AA 7810
To appear in:
Acta Astronautica
Received Date: 20 September 2019 Revised Date:
12 December 2019
Accepted Date: 20 December 2019
Please cite this article as: C. Guo, Q. Hu, Y. Zhang, J. Zhang, Integrated Power and Vibration Control of Gyroelastic Body with Variable-Speed Control Moment Gyros, Acta Astronautica, https:// doi.org/10.1016/j.actaastro.2019.12.027. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier Ltd on behalf of IAA.
Integrated Power and Vibration Control of Gyroelastic Body with Variable-Speed Control Moment Gyros Chuandong Guo1, Quan Hu2, Yao Zhang3, Jun Zhang4
Beijing Institute of Technology, 100081, Beijing, People’s Republic of China The gyroelastic body refers to a flexible structure with distributed angular momentum exchange devices (AMEDs), such as the momentum wheels (MWs) or the control moment gyros (CMGs). The amplitude or the direction of the angular momentum of the rotors in the AMEDs can be changed to produce torques for vibration suppression of the flexible structure. In order to take full advantage of the AMEDs on the flexible structure, the AMEDs are also used for energy storage in this study, thus, an integrated power and vibration control system (IPVCS) is obtained. In this paper, the single-gimbal variable speeds control momentum gyros (VSCMGs) are chosen as the distributed AMEDs mounting on a constrained flexible structure. It can work in a “CMG & MW” mode and a “VSCMG” mode for simultaneous vibration suppression and energy storage. First, the dynamics of a constrained gyroelastic body is established, whereas the power equation of the rotors is deduced. Second, the feedback control laws are designed when the VSCMGs work in the CMG & MW mode and the VSCMG mode, respectively. Finally, numerical examples are presented for a flexible
1
Ph.D student, Beijing Institute of Technology, School of Aerospace Engineering, 100081 Beijing, People's
Republic of China; [email protected]. 2
Assistant Professor, Beijing Institute of Technology, School of Aerospace Engineering, 100081 Beijing,
People's Republic of China; [email protected]; Corresponding Author. 3
Associate Professor, Beijing Institute of Technology, School of Aerospace Engineering, 100081 Beijing,
People's Republic of China; [email protected]. 4
Researcher, National Laboratory of Space Intelligent Control, Beijing Institute of Control Engineering, 100094,
Beijing, People’s Republic of China; [email protected]. 1
truss with distributed VSCMGs to verify the effectiveness of the proposed feedback control strategies. Key words: Gyroelastic Body; Vibration Suppression; Power Control; Single-Gimbal VSCMGs.
I. Introduction Angular momentum exchange devices (AMEDs), such as momentum wheels (MWs) and control moment gyros (CMGs), contains a rotor rotating at a high speed. It can produce control torques by changing the amplitude or the direction of the angular momentum of the rotor. Due to its superiorities, such as no consumption of propellent, precise and continuous torque capabilities, the AMEDs have been extensively used as the actuator for attitude control for large space station[1–6] and agile satellite[7–16]. Recently, it has also been considered for vibration suppression of large space structures. This concept was first proposed by D’Eleuterio and Hughes[17,18]. They assumed that there existed a continuous distribution of AMEDs on the structure. The resultant system was named as “gyroelastic body”. The equations of motion of the gyroelastic body were established[17,18] and the control strategies for vibration suppression were designed[19,20]. In recent years, Hu et al. assumed that a discretized distribution of CMGs was mounted on the structure. They formulated the dynamic model of the gyroelastic body by taking the dynamics of the CMGs into account. The attitude control and vibration suppression of the gyroelastic body were considered[2–5,21]. However, there is no denying the fact that, mounting AMEDs on a flexible structure would increase the system mass and inertia, which conflicts with the design objective of the flexible spacecraft—light weighted. One solution is making the AMEDs have a smaller size and lower mass[22,23]. On the other hand, we can make full use of the AMEDs to make them have more functionalities so that the mass of other subsystems can be reduced. In this work, the rotors in the AMEDs on a gyroelastic body was also used for energy storage, so that the AMEDs could simultaneously produce control torque and store/drain the primary source energy (e.g., from the solar panel). This idea is inspired by the integrated power and attitude control system (IPACS) using MWs or variable speeds CMGs (VSCMGs)[23–30]. In an IPACS, the AMEDs can not only in traditional satellite pointing control roles but also as dynamos to store the energy and drain it, passing secondary power to the satellite’s subsystems during the eclipse. Similarly, if the rotors in the AMEDs on a gyroelastic body are adopted for simultaneous vibration suppression and energy storage, we would haven the integrated power and vibration control system (IPVCS, Figure 2
1). The IPVCS could achieve following control targets: first, when the structure is vibrating, the IPVCS could actively suppress the vibration; second, when the structure is not vibrating (including the situation that the amplitude of the vibration is very small), the IPVCS can absorb or release energy while keeping the structure well-damped; third, the rotors in the AMEDs can absorb excess energy when the energy supply to other subsystems exceeds the demand, while the rotors can release energy as a supplement when the energy supply to other subsystems is insufficient. Therefore, the energy storage system and the vibration suppression system are integrated into one system, so that the weight of the energy storage system can be reduced. It makes the weight of the distributed AMEDs have less influence on the mass and inertia of the overall system. The VSCMGs combines the advantages of torque amplification of a CMGs and the singularity avoidance and power tracking of the variable speed wheels, so we use the single-gimbal VSCMGs as the actuators in this work. It believed that the double gimbal VSCMGs would have much more complexity than other AMEDs, so the following “VSCMGs” refers to the single gimbaled ones. State Reference
Other Subsystems
qb = 0 Pd
Steering Laws
βc P
Sensors
Power Manegment
Solar Array Power System
Legend Physical Motion Logical Connection Electrical Connection
δ, Ω System Dynamics Figure 1. Sketch of IPVCS
The VSCMGs can work as an MW, a CMG, or a VSCMG for attitude control. Analogously, it can work in different modes when mounting it on a flexible structure. The first is CMG mode, in which the modal forces are only produced by the gimbal rotation. The second is MW mode, in which the modal forces are only provided by the rotor acceleration. The third is VSCMG mode, in which the modal forces are generated by the gimbal rotation and 3
rotor acceleration at the same time. The modal forces for vibration suppression are induced by the coupling of the output torque of the VSCMGs and structural vibration. It should be noted that the modal forces produced by the MW mode are much smaller than CMG mode and VSCMG mode. Therefore, the MW mode is used to keep the structure well-damped, while CMG mode and VSCMG mode are used for vibration suppression in this work. It is also noted that the rotors can be used to absorb or release energy only when the VSCMGs work in MW mode or VSCMG mode. As a result, in order to simultaneously achieve vibration suppression and energy storage, two control strategies can be devised. One is the combined “CMG & MW” mode. As shown in Figure 2, when the structure is vibrating, CMG mode can be used for vibration suppression; when the structure is not vibrating or the amplitude of the vibration is very small, the MW mode can be used to absorb or release energy while keeping the structure well damped. Another is the “VSCMG” mode (Figure 3), which means the VSCMGs are always working as a VSCMG. The commands for the gimbal and the wheel are calculated at the same time. MW Mode Power Requirement Pd
No
Legend Physical Motion Logical Connection Electrical Connection
Rotor Acceleration Law
Store/Drain Energy
Output Modal Forces Ω
Vibrating? Yes
δ
Gimbal Steering Law
Output Modal Forces CMG Mode
βc
Sensors Flexible Structure
Figure 2. Sketch of the combined “CMG & MW” mode
To demonstrate the basic idea and design procedure of the IPVCS, a cantilever gyroelastic body with VSCMGs is considered in the formulations. The dynamics of the system is first derived, in which the dynamic characteristics of the VSCMGs, as well as the interactions between the VSCMGs and the flexible structure, are revealed. Then, the two control strategies for the IPVCS are developed. For the CMG & MW mode, when the structure is vibrating, the 4
VSCMGs work in CMG mode using a Lyapunov feedback control law; when the structure is not vibrating, the VSCMGs work in MW mode using a feedback control law based on the projection matrix method. For the VSCMG mode, a feedback control law is designed to guarantee the modal forces are only produced by the gimbal rotation, whereas the rotors are only used to absorb or release energy. The remainder of this paper is organized as follows: Section Section
introduces the dynamics of the system. The CMG & MW mode and VSCMG mode are detailed in and
, respectively, followed by the conclusions in Section Rotor Acceleration Law
Power Requirement
Ω
.
Store/Drain Energy
Pd
Gimbal Steering Law
Controller
δ
Output Modal Forces
Legend Physical Motion Logical Connection Electrical Connection
βc
Sensors Flexible Structure
Figure 3. Sketch of the VSCMG mode
II. Integrated Power and Vibration Control Modal of the Constrained Gyroelastic Body A. System Description The constrained gyroelastic body consists of a constrained flexible structure B with a set of distributed VSCMGs in an arbitrary configuration (Figure 4). fb is the body frame of the structure. f gi and f ri are the gimbal-fixed frame and the rotor-fixed frame, respectively. Their origins coincide at the node Qgi , where is the
r r r r r r mass center of the ith VSCMG. The triaxial unit vectors of f gi and f ri are xgi , ygi , zgi , and xri , yri , zri , r
r
respectively, where xgi signifies the rotation direction of the gimbal; ygi represents the direction of the rotor
r r r r r angular momentum; zgi = xgi × ygi ; xri indicates the spinning position of the rotor; yri points to the same r r r r direction with ygi ; z ri = xri × yri .
5
xb fb ob
rm,b zb
yb
l gi
xgi δ i
um dm Ωi
δi Qgi
u gi
xri Ωi
y gi yri
B
f ri
f gi
zri z gi
Figure 4. Sketch of a constrained flexible body with distributed VSCMGs
B. Equations of Motion A detailed dynamics model should be first established for the gyroelastic body, based on which the steering law of the VSCMGs can be designed for IPVCS. In the following developments, the dynamic model derived by Hu et al. in Ref.[3] is briefly introduced. In their formulations, the dynamic characteristics of the VSCMGs, as well as the interactions between the VSCMGs and the flexible structure B , are taken into consideration. The assumed mode method is applied to describe the elastic motion of the structure under the assumption that the elastic motion of the
r r flexible structure is small. So the translational deformation u m and elastic rotation angle β m of an elementary mass dm in the flexible structure relative to fb can be expanded by a set of selected modes. Since the structure has a cantilevered boundary condition, only the vibration equation of the gyroelastic body is derived from Eq. (7) in Ref.[3]. By eliminating the higher-order nonlinear terms, and retaining the interactions between the VSCMGs and the flexible structure, the vibration equation is written as
& & & & Ea q& b + ( Db + G )qb + Λb qb = BΩ Ω + Bδ δ , where qb ∈ ¡
m×1
modes; Ea ∈ ¡
is the modal coordinates of the flexible structure B ; m is the number of the selected structural
m× m
VSCMGs; Db ∈ ¡
(1)
is a symmetric matrix denoting the modal mass matrix of the flexible structure effected by m× m
and Λb ∈ ¡
matrix, respectively; G ∈ ¡
m× m
m× m
are diagonal matrices, which represent the damping matrix and stiffness
is an antisymmetric matrix, which denotes the gyro coupling coefficient matrix;
BΩ Ω& signifies the generalized modal force caused by the rotor acceleration, BΩ ∈ ¡
m× n
is the rotor torque
coefficient matrix, Ω = [Ω1 ,K , Ω n ]T , where Ω i is the rotor angular rate of the ith VSCMG, n is the number of the VSCMGs; Bδ δ& signifies the generalized modal force caused by the gimbal rotation, Bδ is the gyro torque &T & coefficient matrix, δ&= [δ& 1 ,K , δ n ] , where δ i is the gimbal angular rate of the ith VSCMG.
6
The variables E a , G , BΩ , and Bδ are given by the following equations,
Ea = Eb + ∑ [(mgi + mri )TgiT Tgi + RgiT Ab, gi ( J gi + J ri ) Agi ,b Rgi ] , i
G = ∑ J riy Ωi cby,1gi ( Rgi2T Rgi3 − Rgi3T Rgi2 ) + cby,2gi ( Rgi3T R1gi − R1giT Rgi3 ) + cby,3gi ( R1giT Rgi2 − Rgi2T R1gi ) , i
T BΩ = −[ RgT1cby, g1 J ry1 ,K , Rgn cby, gn J rny ] ,
T z Bδ = −[ RgT1cbz, g1 J ry1Ω1 ,K , Rgn cb , gn J rny Ω n ] ,
Eb ∈ ¡
where
m× m
denotes the modal mass matrix of the flexible structure without VSCMGs, and
Eb = ∫ TmT Tm dm , where Tm ∈ ¡
3× m
b
is the translational modal vector of dm ; mgi and J gi indicate the mass and
the inertia matrix of the gimbal of the ith VSCMG, respectively, and J gi = diag[ J gix , J giy , J giz ] ; mri and J ri indicate the mass and the inertia matrix of the rotor of the ith VSCMG, respectively, and J ri = diag[ J rix , J riy , J riz ] ,
J rix = J riz ; Tgi ∈ ¡
3×m
and Rgi ∈ ¡
3×m
are the translational and rotational modal vector of node Qgi , respectively;
Ab, gi signifies the rotational transformation matrix from f gi to fb , and Ab, gi =[cbx, gi ,cby, gi ,cbz, gi ] , where cbx, gi , cby, gi , and cbz, gi are the column submatrices of Ab, gi ; Agi ,b = AbT, gi ; cby,1gi , cby,2gi , and cby,3gi are the elements of cby, gi , and
cby, gi = [cby,1gi , cby,2gi , cby,3gi ]T ; R1gi , Rgi2 , and Rgi3 are the row submatrices of Rgi . It is easy to verify the antisymmetry of G , viz. G T = −G .
C. Rotor Energy Storage In the IPVCS, the rotors in the VSCMGs are used to absorb or release energy. The actual power of the IPVCS is given by Eq. (10) in Ref.[25],
P = ΩT J ry Ω&,
(2)
where J ry = diag[ J ry1 ,… J rny ] .
D. Sensor Modal The measurement of the IPVCS is provided by a set of angular velocity sensors. They are collocated with the VSCMGs. The angular velocities of the mounting sites induced by the vibration can be used for control synthesis. The direction of each angular velocity sensor is in the direction of the torque produced by the co-located VSCMG 7
working in CMG mode at the initial state. The measurements are written as
β&c = [ β&c1 ,K , β&cn ]T = Rs q&b ,
(3)
where β&ci is the angular velocity measured by the ith sensor. It should be calculated by β&ci = −cbz,0giT Rgi q&b , where T z0 T cb , gn ] . the superscript “ 0 ” in cbz,0gi indicate that cbz,0gi is equal to the initial value of cbz, gi ; Rs = −[ RgT1cbz,0g1 ,K , Rgn
Observing Eqs. (1) and (2), several remarks can be made. Firstly, when the VSCMGs work in CMG mode, the control goal is vibration suppression, so the necessary condition to get the solution for the command of the gimbals is n ≥ m . Secondly, when the VSCMGs work in MW mode, the control goal is absorbing or releasing energy while keeping the structure well damped, so the necessary condition to get the solution for the command of the rotors is n ≥ m + 1 . Thirdly, when the VSCMGs work in VSCMG mode, the control goal is integrated power and vibration
control, so the necessary condition to get the solution for the command of the gimbals and the rotors is 2 n ≥ m + 1 . As a conclusion, the necessary condition to get the solution for the command of the VSCMGs is n ≥ m + 1 . Only in this manner, an IPVCS can be realized.
III. “CMG & MW” Mode In this section, the combined “CMG & MW” mode of the VSCMGs will be explored. As shown in Figure 2, when the structure is vibrating, the rotor angular rates are fixed, then the VSCMGs work in CMG mode. A feedback control law will be designed for vibration suppression based on the Lyapunov approach. When the vibration amplitude of the structure is small, the gimbals will be locked, the VSCMGs work in MW mode. Based on the projection matrix method, a Lyapunov feedback control law will be applied to absorb or release energy while keeping the structure well damped.
A. CMG Mode for Vibration Suppression When the structure is vibrating, the VSCMGs work in CMG mode. By assuming that the gimbal angles are restricted to a small range, the rotational transformation matrix Ab, gi can be regarded as a time-invariant matrix, so that E a , G , BΩ , and Bδ are time-invariant parameter matrices, too. The generalized modal force BΩ Ω& can be dropped from Eq. (1) because the rotor angular rates are constant in the CMG mode. So Eq. (1) can be simplified as & & & E a q& b + ( Db + G )qb + Λb qb = Bδ δ , 8
(4)
Noting that the VSCMGs will convert to MW mode after the vibration amplitude of the structure becomes very small, the gimbal angles are desired to tend to the mounting position, which also keeps the system satisfying the assumption that the gimbal travels are restricted to a small range. Consider the Lyapunov function[2,31]
1 1 1 V1 = q&bT Ea q&b + qbT Λb qb + δT Qδ ≥ 0 , 2 2 2 where Q ∈ ¡
n× n
(5)
is a positive definite matrix; the first two items represent the vibrational energy of the system; the
last item indicates that the gimbal angles are desired to tend to the initial state. We have
1 1 T T T & & & & V& ( Ea q& q&b ( Ea q& 1= b + Λb qb ) qb + b + Λb qb ) + δ Qδ 2 2 1 1 = ( Bδ δ&− ( Db + G )q&b )T q&b + q&bT ( Bδ δ&− ( Db + G )q&b ) + δT Qδ& 2 2 = −q&bT Db q&b + q&bT Bδ δ&+ δT Qδ&, where the antisymmetric property of G is utilized from the second step to the third step. It should be noted that when the VSCMGs work in CMG mode, Bδ is a time-invariant matrix; when the VSCMGs work in MW mode, the gimbal of the VSCMGs will be locked. As a result, in this section, Bδ can be simplified as T z0 Bδ = −[ RgT1cbz,0g1 J ry1Ω1 ,K , Rgn cb , gn J rny Ω n ] .
Comparing Rs and Bδ , we have
Bδ = RsT hd , where hd = diag[ J ry1Ω1 ,…, J rny Ω n ] . Here we have
q&bT Bδ = q&bT RsT hd = β&cT hd . Noting that the measurements are β&c , the feedback steering law of the gimbals can be chosen as
δ&= −kd β&c − kδ δ ,
(6)
where k d = diag [k d 1 ,K , k dn ] , and k di > 0 ; kδ = diag[kδ 1 ,K , kδ n ] , and kδ i > 0 . Q can be defined as hd Q , where Q = diag[kδ 1 kd 1 ,K , kδ n kdn ] . Thus, T & & &T & &T V& 1 = −qb Db qb − βc hd ( k d βc + kδ δ ) − δ hd Q ( k d βc + kδ δ)
= −q&bT Db q&b − β&cT hd kd β&c − β&cT hd kδ δ − δT hd Qkd β&c − δT hd Qkδ δ 9
= −q&bT Db q&b − β&cT hd kd β&c − 2 β&cT hd kδ δ − δT hd Qkδ δ
= −q&bT Db q&b − (∑ J riy Ωi kdi β&ci2 + 2∑ J riy Ωi kδ i β&ciδ i + ∑ J riy Ωi kδ2iδ i2 kdi ) i
i
= −q&bT Db q&b − ∑ J riy Ω i ( kdi β&ci + kδ i
i
kdi δ i ) 2 .
i
In the above derivation, in the first step, q&bT Bδ is replaced by β&cT hd ; δ& is replaced by − kd β&c − kδ δ ; and Q
is replaced by hd Q . From the second step to the third step, Qkd = kδ is utilized. Obviously, the proposed feedback steering law of the gimbals guarantees the Lyapunov stability of the vibration system.
According
to
LaSalle’s
{(q&b , β&ci , δ i ) | q&b = 0, kdi β&ci + kδ i
theorem,
the
system
convergent
to
the
invariant
set
kdi δ i = 0} . When q&b = 0 , β&ci = 0 . Hence it is easy to get that δ i = 0 . Thus,
the system is asymptotically stable, and the feedback steering law of the gimbals can keep the gimbal angles to tend to the mounting position while suppressing the vibration of the structure.
B. MW Mode for Energy Control and Damping Maintenance When the structure is not vibrating or the vibration amplitude of the flexible structure is very small, the VSCMGs work in MW mode. In this situation, the variables E a and BΩ are regarded as time-invariant parameter matrices, while G and Bδ are regarded as time-variant parameter matrices. So Eq. (1) can be simplified as & & E a q& b + ( Db + G ) qb + Λb qb = T
(7)
where T is the generalized modal force caused by the rotor acceleration. T is acted on the flexible structure directly, the expression equation of which is T = BΩ Ω&
(8)
Clearly, the acceleration law of the rotors must be solved to satisfy Eqs. (2) and (8) at the same time. The control logic is shown in Figure 5. Based on the projection matrix method, Ω& can be decomposed into two orthogonal parts, the one is applied to keep the structure well damped and the other is used to absorb or release energy[32]. In this way, the acceleration law of the rotors can be chosen as Ω&= BΩT ( BΩ BΩT ) −1 Tc + S1u1 ,
(9)
where Tc is the desired generalized force, which is needed to be designed; S1 = I n − BΩT ( BΩ BΩT )−1 BΩ , where I n 10
is an n-dimensional identity matrix; u1 is an n×1 column matrix needing to be solved. Clearly, we have S1T = S1 , and S1 S1 = S1 . Pd
Power Requirement
u1
Tc
Rotor Acceleration Law
Legend Physical Motion Logical Connection Electrical Connection
Store/Drain Energy
Output Modal Forces
Ω
βc
T
Sensors Flexible Structure Figure 5. Sketch of the MW mode
Then, submitting Eq. (9) into Eq. (2), the solution of u1 is
u1 = S1 J ry Ω ( ΩT J ry S1 J ry Ω )−1 Pf ,
(10)
where Pf is the power excluding vibration suppression power, viz. Pf = Pd − ΩT J ry BΩT ( BΩ BΩT ) −1 Tc . Pd is the power requirement for the VSCMGs. Now the final task to solve the acceleration law of the rotors is to find the appropriate Tc , the function of which is keeping the structure well damped. Similar to Eq. (5), considering the Lyapunov function
1 1 V2 = q&bT Ea q&b + qbT Λb qb ≥ 0 , 2 2 we have 1 1 T T & & & V& ( E a q& q&b ( E a q& 2= b + Λb qb ) qb + b + Λb qb ) 2 2 =
1 1 (T − ( Db + G )q&b )T q&b + q&bT (T − ( Db + G )q&b ) 2 2
= −q&bT Db q&b + q&bT T . The antisymmetric property of G is utilized from the second step to the third step, too. Choosing the feedback control law of the desired generalized modal force as 11
Tc = − RsT kΩ β&c ,
(11)
where kΩ = diag[kΩ1 ,K , kΩn ] , kΩi > 0 . Substituting Eqs. (9) and (11) back into Eq. (8) leads to
T = BΩ ( BΩT ( BΩ BΩT ) −1 Tc + S1u1 ) = BΩ (− BΩT ( BΩ BΩT )−1 RsT kΩ β&c + S1u1 ) = − BΩ BΩT ( BΩ BΩT ) −1 RsT kΩ β&c + BΩ S1u1 = − RsT kΩ β&c + BΩ ( I n − BΩT ( BΩ BΩT ) −1 BΩ )u1 = − RsT kΩ β&c + ( BΩ I n − BΩ BΩT ( BΩ BΩT ) −1 BΩ )u1 = − RsT kΩ β&c + ( BΩ − BΩ ) u1 = −RsT kΩ β&c .
(12)
In the first step, Eq. (9) is substituted into Eq. (8). From the second step to the third step, Eq. (11) is substituted into the equation. From the fifth step to the sixth step, S1 is replaced by I n − BΩT ( BΩ BΩT )−1 BΩ . In this way, &T & &T T & V& 2 = −qb Db qb − qb Rs kΩ βc
= −q&bT Db q&b − q&bT RsT kΩ Rs q&b ≤ 0 . In the above derivation, from the first step to the second step, β&c = Rs q&b is utilized. Therefore, the proposed feedback control law based on the projection matrix method guarantees the asymptotical stability of the vibration system. According to LaSalle’s theorem, the system convergent to the invariant set {(q&b ) | q&b = 0} . As a conclusion, when the structure is stable or the vibration of the structure is small, the VSCMGs can work in MW mode. Owing to the projection matrix method, the control law of Ω& can ensure the power requirement while guaranteeing the asymptotical stability of the system.
C. Illustrative Example In this section, the effectiveness of the proposed feedback control law is verified by a numerical simulation of a flexible aluminum alloy truss with 8 VSCMGs. The length of the truss is 10m , whereas the length and width of the cross-section are 50mm and 30mm , respectively. Based on the assumed mode method, selecting the first four structural modes to describe the vibration of the truss, the corresponding frequencies of the truss are 1.824Hz , 3.018Hz , 11.030Hz , and 15.697Hz , respectively. The damping coefficient of the truss is 0.005. 12
The mounting locations of the VSCMGs are shown in Figure 6. Relative to fb , the mounting vectors are [0, 6.5, 0]m , [0,7,0]m , [0, 7.5, 0]m , [0,8, 0]m , [0,8.5, 0]m , [0,9, 0]m , [0,9.5,0]m , and [0,10, 0]m , respectively. The initial of the rotors and the gimbals are J gi = diag[3,5,3] × 10−5 kg gm 2 and J ri = diag[3,16,3] × 10−5 kg gm 2 , respectively, while each VSCMG weights 1kg . The mounting matrices of the VSCMGs are shown in Table 1. The gimbal angular rate of the VSCMGs can work in the range of ±0.001 : 1rad s . xb
yb
ob yb
c1
zb
c5
c6
c2
c3
c7
c8
xb
c4
Figure 6. Sketch of the flexible truss with 8 VSCMGs
As a numerical simulation, a simple power requirement is considered by assuming the truss is mounted on a satellite, the orbital period of which is 5750s . The sunlight exposure time is 3750s , and the charging power Pd = 5W . The eclipse time is 2000s , and the discharging power Pd = −5W at the first 1750s of the eclipse
time; the discharging power Pd = −40W at the last 250s . Table 1 Mounting matrices of the VSCMGs
SGVSCMG number
Mounting matrix Ab0, gi
c1 , c2
[0 1 0;0 0 1;1 0 0]
c3 , c4
[0 -1 0;0 0 -1;1 0 0]
c5 , c6
[0 0 -1;0 1 0;1 0 0]
c7 , c8
[0 0 1;0 -1 0;1 0 0]
The control task is suppressing the vibration of the truss based on the CMG mode, and keeping the structure well-damped when utilizing the rotors to absorb or release energy based on the MW mode. The initial states of the system are qb = [1,1, 0, 0]T and q&b = [0,0, 0, 0]T . All the initial rotor angular rates are equal to 30000rpm , and all the initial gimbals angles are equal to zero. At 3000s per orbital period, a perturbation of 57.5s , the value of which is 0.5 rad s , is applied to gimbal angular rates. The feedback control parameters for each VSCMG are chosen the same as k di = 1000 , kδ i = 0.01 , kΩi = 100 ,
13
i = 1,K ,8 . When the elastic deformation of the truss tip is less than 0.1mm , we consider the vibration amplitude of the flexible truss is very small, then the VSCMGs will convert to the MW mode. Complete dynamics of the system is considered in the simulation, which will test the robustness of the proposed control law. After simulating for 12000s , the results show that the proposed feedback control law can achieve the control task under disturbance. Figure 7 depicts the time-varying process of the modal coordinates. Figures 8 and 9 show that the VSCMGs work in CMG mode for the first 532s , and the gimbal angles are traveling in a small range from beginning to the end, which confirms the previous assumptions in this section. After 532s , the VSCMGs convert to the MW mode, and the gimbal angles are nearly equal to zero. Figure 10 indicates that when the VSCMGs work in MW mode, the rotor angular rates vary with the power requirement. The rotors spin down when discharging power while accelerating when charging power. However, the angular rates of each rotor are quite different, and at the end of each orbital period, rotor angular rates are much smaller than the initial. Figure 11 shows that the power requirement is met in MW mode. The elastic deformation of the truss tip is shown in Figure 12. Figures 7 and 12 indicate that the elastic deformation of the truss is decreased rapidly in CMG mode, and the truss is well damped in MW mode. 1
1 0 -1 0
0.5 0
10-6
500 10-4
-0.5 20 2 10 0 -1 0 -2 2000 7000 12000 500 550 -1.5 0 2000 4000 6000 8000 10000 12000 time s Figure 7. Modal coordinates, CMG & MW mode
10 5 0 -7 0
0.03 0 -0.03 2000 4000 6000 8000 10000 12000 13 0 -13 0 550
0.2 0 -0.2 500 2000
4000
6000 time s
550
8000 10000 12000
Figure 8. Gimbal angles, CMG & MW mode 14
57.3
1 0 -1 2000
0
-57.3
0
7000
57.3 0 -57.3 0 2000
1.3 0 -1.3 12000 500
550
110 220 330 440 550 4000 6000 8000 10000 12000 time s
rotor angular rates, rpm
Figure 9. Gimbal angular rates, CMG & MW mode
Figure 10. Rotor angular rates, CMG & MW mode
10 0 -10 6 4 -20 2 0 531 -30 -40
0
Pd P 532 2000
533 4000
6000 time s
8000 10000 12000
Figure 11. Power, CMG & MW mode
1000
500
0 0
1000 500 0 0
2000
0.2 0.1 0 550 500 0.01 0.005 0 2000 4000
6000 time s
550
7000 8000
12000 10000
12000
Figure 12. Elastic deformation of the truss tip, CMG & MW mode 15
Therefore, the proposed feedback control law can exert active modal force to the structure for vibration suppression in CMG mode, and absorb or release energy while keeping the structure well-damped in MW mode, even though there is a perturbation in each orbital period. However, it should be noted that when the VSCMGs work in CMG mode, the power requirement cannot be met. As a result, another control strategy is needed to meet the power requirement while suppressing the vibration. When the VSCMGs work in VSCMG mode, this goal can be achieved.
IV. VSCMG Mode A. Steering Law of the Gimbals and the Rotors Based on the above control strategy, when the VSCMGs work in VSCMG mode, the new feedback steering law of the gimbals and the rotors should combine the characteristics of CMG mode and MW mode. As shown in Figure 3, the steering law expects the rotors do not exert modal force on the structure, while the modal force is only produced by the gimbal rotation. In this way, according to Eqs. (9) and (10), the steering law of the rotors can be chosen as Ω&= S1 J ry Ω ( ΩT J ry S1 J ry Ω ) −1 Pd .
(13)
At the same time, with the assumption that the gimbal angles are restricted to a small range, E a and BΩ are time-invariant parameter matrices. However, G , hd and Bδ are time-variant parameter matrices. Therefore, if the solution of δ& is chosen the same as Eq. (6), it is hard to prove the stability of the feedback steering law. Considering the simplest feedback steering law of the gimbals, δ&= −kd β&c .
In order to verify the stability of the feedback control law, choosing the Lyapunov function
1 1 V3 = q&bT Ea q&b + qbT Λb qb ≥ 0 , 2 2 we have 1 1 T T & & & V& ( Ea q& q&b ( E a q& 3= b + Λb qb ) qb + b + Λb qb ) 2 2 =
1 1 ( Bδ δ&− ( Db + G )q&b )T q&b + q&bT ( Bδ δ&− ( Db + G )q&b ) 2 2
16
(14)
= −q&bT Db q&b + q&bT Bδ δ& = −q&bT Db q&b − q&bT RsT hd kd β&c = −q&bT Db q&b − q&bT RsT hd kd Rs q&b ≤ 0 . In the above derivation, the antisymmetric property of G is utilized from the second step to the third step, too. From the third step to the fourth step, Bδ = RsT hd is substituted into the equation. And from the fourth step to the fifth step, β&c = Rs q&b is also utilized. Similarly, the proposed feedback steering law of the gimbals guarantees the asymptotical stability of the vibration system, and the system convergent to the invariant set {(q&b ) | q&b = 0} .
B. Illustrative Example In this section, all the simulation parameters are the same as Section
.C. After simulating for 12000s , the
results show that the proposed feedback steering law of the gimbals and the rotors can meet the power requirement and suppress the vibration of the truss at the same time. Figure 13 shows that the proposed feedback steering law in this section has similar vibration suppression ability as that proposed in Section
. Figure 14 and Figure 15 show
that the gimbal angles are traveling in a small range from beginning to the end. However, comparing with Figure 8 and Figure 9, the gimbal angles don’t converge to zero, because the gimbal angles are not fed back to the proposed control law. Figure 16 depicts the time-varying process of the rotor angular rates varying with the power requirement. In contrast to the CMG & MW mode, the rotor angular rates at the end of each orbital period are almost equal to the initial values. However, the angular rates of each rotor are still quite different. Figure 17 shows that the power requirement is met throughout the simulation process, which can’t be achieved in CMG &MW mode. Figure 13 and Figure 18 indicate that the elastic deformation of the truss is decreased rapidly and the truss is well damped in VSCMG mode. 1
1 0 -1 0
0.5 0
2 0 -2 -4 2000
-0.5 -1 -1.5
0
500
10-5
10-4
7000
2000
2 0 -2 12000 500
4000
6000 time s
550
8000 10000 12000
Figure 13. Modal coordinates, VSCMG mode 17
14
0.5 0 -0.5 500
7
550
0 13 0 -13 0 2000 4000
-7 -14
0
550 6000 8000 10000 12000 time s
Figure 14. Gimbal angles, VSCMG mode
57.3
0
-57.3
0
0.2 0 -0.2 2000 57.3 0 -57.3 0 2000
7000
0.1 0 -0.1 12000 650
700
350 4000
6000 time s
700 8000 10000 12000
rotor angular rates, rpm
Figure 15. Gimbal angular rates, VSCMG mode
Figure 16. Rotor angular rates, VSCMG mode
10 0 -10
Pd
-20
P
-30 -40
0
2000
4000
6000 time s
8000 10000 12000
Figure 17. Power, VSCMG mode 18
1000 1000 500 0 0
500
0 0
2000
500
4000
0.2 0.1 0 500 0.02 0.01 0 2000 6000 time s
550
7000 8000
12000 10000
12000
Figure 18. Elastic deformation of the truss tip, VSCMG mode
Therefore, when the VSCMGs work in VSCMG mode, the proposed feedback steering law can meet the power requirement and suppress the vibration of the truss at the same time, which can’t be achieved in CMG & MW mode.
V. Conclusions An integrated power and vibration control system (IPVCS) was proposed in this paper. VSCMGs were distributed on flexible structures for vibration suppression and energy storage. Based on the different working modes of the VSCMGs, two control strategies for the IPVCS were explored. One is the CMG & MW mode. When the structure is vibrating, the VSCMGs work in CMG mode. A simple feedback control law utilizing the angular velocity sensor measurements and gimbal angles as feedback values is designed. When the structure is stable or the amplitude of the vibration is very small, the VSCMGs work in MW mode. A feedback control law based on the projection matrix method, which only utilizes the angular velocity sensor measurements as feedback values, is designed. The other is VSCMG mode, in which the modal force is only produced by the rotation of the gimbals, while the rotors are only used to absorb or release energy. The control law of the gimbal angular rates only utilizes the angular velocity sensor measurements. Based on the projection matrix method, the control law guarantees the rotors not exerting modal force on the structure. The simulation results show that both the two proposed control strategies can achieve the control tasks. When the VSCMGs work is CMG & MW mode, the vibration can be suppressed in CMG mode. In MW mode, the flexible structure is well-damped while the power requirement can be met. When the VSCMGs work is VSCMG mode, comparing with CMG & MW mode, the proposed feedback control law can meet the power requirement and suppress the vibration at the same time.
19
Acknowledgments This work is supported by the National Natural Science Foundation of China (11872011). The authors would like to thank the anonymous reviewers for the valuable comments and suggestions to improve the paper.
References [1] S. Jia, Y. Jia, S. Xu, Q. Hu, Maneuver and Active Vibration Suppression of Free-Flying Space Robot, IEEE Trans. Aerosp. Electron. Syst. 54 (2018) 1115–1134. [2] Q. Hu, Y. Jia, S. Xu, Dynamics and vibration suppression of space structures with control moment gyroscopes, Acta Astronaut. 96 (2014) 232–245. [3] Q. Hu, Y. Jia, S. Xu, Recursive Dynamics Algorithm for Multibody Systems with Variable-Speed Control Moment Gyroscopes, J. Guid. Control. Dyn. 36 (2013) 1388–1398. [4] Q. Hu, J. Li, J. Zhang, Passivity control with practically finite-time convergence for large space structures, Acta Astronaut. 131 (2017) 152–158. [5] Q. Hu, C. Guo, J. Zhang, Singularity and steering logic for control moment gyros on flexible space structures, Acta Astronaut. 137 (2017) 261–273. [6] Y. Jia, A.K. Misra, Robust trajectory tracking control of a dual-arm space robot actuated by control moment gyroscopes, Acta Astronaut. 137 (2017) 287–301. [7] F. Zhang, L. Jin, S. Xu, Fault Tolerant Attitude Control for spacecraft with SGCMGs under actuator partial failure and actuator saturation, Acta Astronaut. 132 (2017) 303–311. [8] Y. Zhang, Y. Zang, M. Li, Y. Wang, W. Li, Active-passive integrated vibration control for control moment gyros and its application to satellites, J. Sound Vib. 394 (2017) 1–14. [9] Y. Zhang, C. Sheng, Z. Guo, Y. Wang, W. Li, Dynamic characteristics and performance evaluation for the part strut failure of the vibration isolation platform on satellites, Acta Astronaut. 133 (2017) 403–415. [10] Y. Zhang, M. Li, Z. Song, J. Shan, X. Guan, L. Tang, Design and analysis of a moment control unit for agile satellite with high attitude stability requirement, Acta Astronaut. 122 (2016) 90–105. [11] S. Keshtkar, J.A. Moreno, H. Kojima, K. Uchiyama, M. Nohmi, K. Takaya, Spherical gyroscopic moment stabilizer for attitude control of microsatellites, Acta Astronaut. 143 (2018) 9–15. [12] X. Cao, B. Wu, Robust spacecraft attitude tracking control using hybrid actuators with uncertainties, Acta Astronaut. 136 (2017) 1–8. [13] Y.H. Wu, F. Han, B. Hua, Z.M. Chen, Null motion strategy for spacecraft large angle agile maneuvering using hybrid actuators, Acta Astronaut. 140 (2017) 459–468. [14] J. Zhang, Z. Guo, Y. Zhang, Dynamic characteristics of vibration isolation platforms considering the joints of the struts, Acta Astronaut. 126 (2016) 120–137. [15] S. Kawajiri, S. Matunaga, A low-complexity attitude control method for large-angle agile maneuvers of a spacecraft with control moment gyros, Acta Astronaut. 139 (2017) 486–493. [16] J. Tayebi, A.A. Nikkhah, J. Roshanian, LQR/LQG attitude stabilization of an agile microsatellite with CMG, Aircr. Eng. Aerosp. Technol. 89 (2017) 290–296. [17] G.M.T. D’Eleuterio, P.C. Hughes, Dynamics of Gyroelastic Spacecraft, J. Guid. Control Dyn. 10 (1987) 401– 405. [18] G.M.T. D’Eleuterio, P.C. Hughes, Dynamics of Gyroeiastic Continua, J. Appl. Mech. 51 (1984) 415–422. [19] C.J. Damaren, G.M.T. D’eleuterio, Optimal control of large space structures using distributed gyricity, J. Guid. Control. Dyn. 12 (1989) 723–731. [20] C.J. Damaren, G.M.T. D’eleuteriot, Controllability and observability of gyroelastic vehicles, J. Guid. Control. Dyn. 14 (1991) 886–894. [21] Q. Hu, Y. Jia, S. Xu, Adaptive Suppression of Linear Structural Vibration Using Control Moment Gyroscopes, J. Guid. Control. Dyn. 37 (2014) 990–996. [22] B.J. Kawak, Development of a low-cost, low micro-vibration CMG for small agile satellite applications, Acta Astronaut. 131 (2017) 113–122. [23] D.J. Richie, V.J. Lappas, G. Prassinos, A practical small satellite variable-speed control moment gyroscope for combined energy storage and attitude control, Acta Astronaut. 65 (2009) 1745–1764. [24] D.J. Richie, V.J. Lappas, P.L. Palmer, Sizing/Optimization of a Small Satellite Energy Storage and Attitude Control System, J. Spacecr. Rockets. 44 (2007) 940–952. 20
[25] D.J. Richie, V.J. Lappas, B. Wie, Practical steering law for small satellite energy storage and attitude control, J. Guid. Control. Dyn. 32 (2009) 1898–1911. [26] R.I. Corporation, Integrated Power / Attitude Control System ( IPACS ), J. Spacecr. Rockets. 12 (1975) 485– 491. [27] H. Zhao, F. Liu, Y. Yao, Optimization design steering law for VSCMGs with the function of attitude control and energy storage, Aerosp. Sci. Technol. 65 (2017) 9–17. [28] Y. Yao, F. Liu, H. Zhao, Constrained steering law for VSCMGs with the function of attitude control and energy storage, Aerosp. Sci. Technol. 58 (2016) 341–350. [29] D. Kim, W. MacKunis, N. Fitz-Coy, W.E. Dixon, Precision integrated power and attitude control system (IPACS) in the presence of dynamic uncertainty, J. Astronaut. Sci. 58 (2011) 99–120. [30] Z. Chu, J. Cui, Integrated Power and Attitude Control Design of Satellites Based on a Fuzzy Adaptive Disturbance Observer Using Variable-Speed Control Moment Gyros, Int. J. Aerosp. Eng. 2016 (2016) 1–13. [31] J. Shi, C.J. Damaren, Control Law for Active Structural Damping Using a Control Moment Gyro, J. Guid. Control Dyn. 28 (2005) 550–553. [32] P. Tsiotras, H. Shen, C. Hall, Satellite Attitude Control and Power Tracking with Energy/Momentum Wheels, J. Guid. Control. Dyn. 24 (2001) 23–34.
21
Highlights: An integrated power and vibration control system (IPVCS) is proposed. VSCMGs are distributed on the structure for vibration suppression and energy storage. The control laws of the “CMG & MW” mode and the “VSCMG” mode are designed. The IPVCS can absorb or release energy while keeping the structure well-damped.
Declaration of Interest Statement We declare that we have no financial and personal relationships with other people or organizations.
S0094-5765(19)31460-2
DOI:
https://doi.org/10.1016/j.actaastro.2019.12.027
Reference:
AA 7810
To appear in:
Acta Astronautica
Received Date: 20 September 2019 Revised Date:
12 December 2019
Accepted Date: 20 December 2019
Please cite this article as: C. Guo, Q. Hu, Y. Zhang, J. Zhang, Integrated Power and Vibration Control of Gyroelastic Body with Variable-Speed Control Moment Gyros, Acta Astronautica, https:// doi.org/10.1016/j.actaastro.2019.12.027. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier Ltd on behalf of IAA.
Integrated Power and Vibration Control of Gyroelastic Body with Variable-Speed Control Moment Gyros Chuandong Guo1, Quan Hu2, Yao Zhang3, Jun Zhang4
Beijing Institute of Technology, 100081, Beijing, People’s Republic of China The gyroelastic body refers to a flexible structure with distributed angular momentum exchange devices (AMEDs), such as the momentum wheels (MWs) or the control moment gyros (CMGs). The amplitude or the direction of the angular momentum of the rotors in the AMEDs can be changed to produce torques for vibration suppression of the flexible structure. In order to take full advantage of the AMEDs on the flexible structure, the AMEDs are also used for energy storage in this study, thus, an integrated power and vibration control system (IPVCS) is obtained. In this paper, the single-gimbal variable speeds control momentum gyros (VSCMGs) are chosen as the distributed AMEDs mounting on a constrained flexible structure. It can work in a “CMG & MW” mode and a “VSCMG” mode for simultaneous vibration suppression and energy storage. First, the dynamics of a constrained gyroelastic body is established, whereas the power equation of the rotors is deduced. Second, the feedback control laws are designed when the VSCMGs work in the CMG & MW mode and the VSCMG mode, respectively. Finally, numerical examples are presented for a flexible
1
Ph.D student, Beijing Institute of Technology, School of Aerospace Engineering, 100081 Beijing, People's
Republic of China; [email protected]. 2
Assistant Professor, Beijing Institute of Technology, School of Aerospace Engineering, 100081 Beijing,
People's Republic of China; [email protected]; Corresponding Author. 3
Associate Professor, Beijing Institute of Technology, School of Aerospace Engineering, 100081 Beijing,
People's Republic of China; [email protected]. 4
Researcher, National Laboratory of Space Intelligent Control, Beijing Institute of Control Engineering, 100094,
Beijing, People’s Republic of China; [email protected]. 1
truss with distributed VSCMGs to verify the effectiveness of the proposed feedback control strategies. Key words: Gyroelastic Body; Vibration Suppression; Power Control; Single-Gimbal VSCMGs.
I. Introduction Angular momentum exchange devices (AMEDs), such as momentum wheels (MWs) and control moment gyros (CMGs), contains a rotor rotating at a high speed. It can produce control torques by changing the amplitude or the direction of the angular momentum of the rotor. Due to its superiorities, such as no consumption of propellent, precise and continuous torque capabilities, the AMEDs have been extensively used as the actuator for attitude control for large space station[1–6] and agile satellite[7–16]. Recently, it has also been considered for vibration suppression of large space structures. This concept was first proposed by D’Eleuterio and Hughes[17,18]. They assumed that there existed a continuous distribution of AMEDs on the structure. The resultant system was named as “gyroelastic body”. The equations of motion of the gyroelastic body were established[17,18] and the control strategies for vibration suppression were designed[19,20]. In recent years, Hu et al. assumed that a discretized distribution of CMGs was mounted on the structure. They formulated the dynamic model of the gyroelastic body by taking the dynamics of the CMGs into account. The attitude control and vibration suppression of the gyroelastic body were considered[2–5,21]. However, there is no denying the fact that, mounting AMEDs on a flexible structure would increase the system mass and inertia, which conflicts with the design objective of the flexible spacecraft—light weighted. One solution is making the AMEDs have a smaller size and lower mass[22,23]. On the other hand, we can make full use of the AMEDs to make them have more functionalities so that the mass of other subsystems can be reduced. In this work, the rotors in the AMEDs on a gyroelastic body was also used for energy storage, so that the AMEDs could simultaneously produce control torque and store/drain the primary source energy (e.g., from the solar panel). This idea is inspired by the integrated power and attitude control system (IPACS) using MWs or variable speeds CMGs (VSCMGs)[23–30]. In an IPACS, the AMEDs can not only in traditional satellite pointing control roles but also as dynamos to store the energy and drain it, passing secondary power to the satellite’s subsystems during the eclipse. Similarly, if the rotors in the AMEDs on a gyroelastic body are adopted for simultaneous vibration suppression and energy storage, we would haven the integrated power and vibration control system (IPVCS, Figure 2
1). The IPVCS could achieve following control targets: first, when the structure is vibrating, the IPVCS could actively suppress the vibration; second, when the structure is not vibrating (including the situation that the amplitude of the vibration is very small), the IPVCS can absorb or release energy while keeping the structure well-damped; third, the rotors in the AMEDs can absorb excess energy when the energy supply to other subsystems exceeds the demand, while the rotors can release energy as a supplement when the energy supply to other subsystems is insufficient. Therefore, the energy storage system and the vibration suppression system are integrated into one system, so that the weight of the energy storage system can be reduced. It makes the weight of the distributed AMEDs have less influence on the mass and inertia of the overall system. The VSCMGs combines the advantages of torque amplification of a CMGs and the singularity avoidance and power tracking of the variable speed wheels, so we use the single-gimbal VSCMGs as the actuators in this work. It believed that the double gimbal VSCMGs would have much more complexity than other AMEDs, so the following “VSCMGs” refers to the single gimbaled ones. State Reference
Other Subsystems
qb = 0 Pd
Steering Laws
βc P
Sensors
Power Manegment
Solar Array Power System
Legend Physical Motion Logical Connection Electrical Connection
δ, Ω System Dynamics Figure 1. Sketch of IPVCS
The VSCMGs can work as an MW, a CMG, or a VSCMG for attitude control. Analogously, it can work in different modes when mounting it on a flexible structure. The first is CMG mode, in which the modal forces are only produced by the gimbal rotation. The second is MW mode, in which the modal forces are only provided by the rotor acceleration. The third is VSCMG mode, in which the modal forces are generated by the gimbal rotation and 3
rotor acceleration at the same time. The modal forces for vibration suppression are induced by the coupling of the output torque of the VSCMGs and structural vibration. It should be noted that the modal forces produced by the MW mode are much smaller than CMG mode and VSCMG mode. Therefore, the MW mode is used to keep the structure well-damped, while CMG mode and VSCMG mode are used for vibration suppression in this work. It is also noted that the rotors can be used to absorb or release energy only when the VSCMGs work in MW mode or VSCMG mode. As a result, in order to simultaneously achieve vibration suppression and energy storage, two control strategies can be devised. One is the combined “CMG & MW” mode. As shown in Figure 2, when the structure is vibrating, CMG mode can be used for vibration suppression; when the structure is not vibrating or the amplitude of the vibration is very small, the MW mode can be used to absorb or release energy while keeping the structure well damped. Another is the “VSCMG” mode (Figure 3), which means the VSCMGs are always working as a VSCMG. The commands for the gimbal and the wheel are calculated at the same time. MW Mode Power Requirement Pd
No
Legend Physical Motion Logical Connection Electrical Connection
Rotor Acceleration Law
Store/Drain Energy
Output Modal Forces Ω
Vibrating? Yes
δ
Gimbal Steering Law
Output Modal Forces CMG Mode
βc
Sensors Flexible Structure
Figure 2. Sketch of the combined “CMG & MW” mode
To demonstrate the basic idea and design procedure of the IPVCS, a cantilever gyroelastic body with VSCMGs is considered in the formulations. The dynamics of the system is first derived, in which the dynamic characteristics of the VSCMGs, as well as the interactions between the VSCMGs and the flexible structure, are revealed. Then, the two control strategies for the IPVCS are developed. For the CMG & MW mode, when the structure is vibrating, the 4
VSCMGs work in CMG mode using a Lyapunov feedback control law; when the structure is not vibrating, the VSCMGs work in MW mode using a feedback control law based on the projection matrix method. For the VSCMG mode, a feedback control law is designed to guarantee the modal forces are only produced by the gimbal rotation, whereas the rotors are only used to absorb or release energy. The remainder of this paper is organized as follows: Section Section
introduces the dynamics of the system. The CMG & MW mode and VSCMG mode are detailed in and
, respectively, followed by the conclusions in Section Rotor Acceleration Law
Power Requirement
Ω
.
Store/Drain Energy
Pd
Gimbal Steering Law
Controller
δ
Output Modal Forces
Legend Physical Motion Logical Connection Electrical Connection
βc
Sensors Flexible Structure
Figure 3. Sketch of the VSCMG mode
II. Integrated Power and Vibration Control Modal of the Constrained Gyroelastic Body A. System Description The constrained gyroelastic body consists of a constrained flexible structure B with a set of distributed VSCMGs in an arbitrary configuration (Figure 4). fb is the body frame of the structure. f gi and f ri are the gimbal-fixed frame and the rotor-fixed frame, respectively. Their origins coincide at the node Qgi , where is the
r r r r r r mass center of the ith VSCMG. The triaxial unit vectors of f gi and f ri are xgi , ygi , zgi , and xri , yri , zri , r
r
respectively, where xgi signifies the rotation direction of the gimbal; ygi represents the direction of the rotor
r r r r r angular momentum; zgi = xgi × ygi ; xri indicates the spinning position of the rotor; yri points to the same r r r r direction with ygi ; z ri = xri × yri .
5
xb fb ob
rm,b zb
yb
l gi
xgi δ i
um dm Ωi
δi Qgi
u gi
xri Ωi
y gi yri
B
f ri
f gi
zri z gi
Figure 4. Sketch of a constrained flexible body with distributed VSCMGs
B. Equations of Motion A detailed dynamics model should be first established for the gyroelastic body, based on which the steering law of the VSCMGs can be designed for IPVCS. In the following developments, the dynamic model derived by Hu et al. in Ref.[3] is briefly introduced. In their formulations, the dynamic characteristics of the VSCMGs, as well as the interactions between the VSCMGs and the flexible structure B , are taken into consideration. The assumed mode method is applied to describe the elastic motion of the structure under the assumption that the elastic motion of the
r r flexible structure is small. So the translational deformation u m and elastic rotation angle β m of an elementary mass dm in the flexible structure relative to fb can be expanded by a set of selected modes. Since the structure has a cantilevered boundary condition, only the vibration equation of the gyroelastic body is derived from Eq. (7) in Ref.[3]. By eliminating the higher-order nonlinear terms, and retaining the interactions between the VSCMGs and the flexible structure, the vibration equation is written as
& & & & Ea q& b + ( Db + G )qb + Λb qb = BΩ Ω + Bδ δ , where qb ∈ ¡
m×1
modes; Ea ∈ ¡
is the modal coordinates of the flexible structure B ; m is the number of the selected structural
m× m
VSCMGs; Db ∈ ¡
(1)
is a symmetric matrix denoting the modal mass matrix of the flexible structure effected by m× m
and Λb ∈ ¡
matrix, respectively; G ∈ ¡
m× m
m× m
are diagonal matrices, which represent the damping matrix and stiffness
is an antisymmetric matrix, which denotes the gyro coupling coefficient matrix;
BΩ Ω& signifies the generalized modal force caused by the rotor acceleration, BΩ ∈ ¡
m× n
is the rotor torque
coefficient matrix, Ω = [Ω1 ,K , Ω n ]T , where Ω i is the rotor angular rate of the ith VSCMG, n is the number of the VSCMGs; Bδ δ& signifies the generalized modal force caused by the gimbal rotation, Bδ is the gyro torque &T & coefficient matrix, δ&= [δ& 1 ,K , δ n ] , where δ i is the gimbal angular rate of the ith VSCMG.
6
The variables E a , G , BΩ , and Bδ are given by the following equations,
Ea = Eb + ∑ [(mgi + mri )TgiT Tgi + RgiT Ab, gi ( J gi + J ri ) Agi ,b Rgi ] , i
G = ∑ J riy Ωi cby,1gi ( Rgi2T Rgi3 − Rgi3T Rgi2 ) + cby,2gi ( Rgi3T R1gi − R1giT Rgi3 ) + cby,3gi ( R1giT Rgi2 − Rgi2T R1gi ) , i
T BΩ = −[ RgT1cby, g1 J ry1 ,K , Rgn cby, gn J rny ] ,
T z Bδ = −[ RgT1cbz, g1 J ry1Ω1 ,K , Rgn cb , gn J rny Ω n ] ,
Eb ∈ ¡
where
m× m
denotes the modal mass matrix of the flexible structure without VSCMGs, and
Eb = ∫ TmT Tm dm , where Tm ∈ ¡
3× m
b
is the translational modal vector of dm ; mgi and J gi indicate the mass and
the inertia matrix of the gimbal of the ith VSCMG, respectively, and J gi = diag[ J gix , J giy , J giz ] ; mri and J ri indicate the mass and the inertia matrix of the rotor of the ith VSCMG, respectively, and J ri = diag[ J rix , J riy , J riz ] ,
J rix = J riz ; Tgi ∈ ¡
3×m
and Rgi ∈ ¡
3×m
are the translational and rotational modal vector of node Qgi , respectively;
Ab, gi signifies the rotational transformation matrix from f gi to fb , and Ab, gi =[cbx, gi ,cby, gi ,cbz, gi ] , where cbx, gi , cby, gi , and cbz, gi are the column submatrices of Ab, gi ; Agi ,b = AbT, gi ; cby,1gi , cby,2gi , and cby,3gi are the elements of cby, gi , and
cby, gi = [cby,1gi , cby,2gi , cby,3gi ]T ; R1gi , Rgi2 , and Rgi3 are the row submatrices of Rgi . It is easy to verify the antisymmetry of G , viz. G T = −G .
C. Rotor Energy Storage In the IPVCS, the rotors in the VSCMGs are used to absorb or release energy. The actual power of the IPVCS is given by Eq. (10) in Ref.[25],
P = ΩT J ry Ω&,
(2)
where J ry = diag[ J ry1 ,… J rny ] .
D. Sensor Modal The measurement of the IPVCS is provided by a set of angular velocity sensors. They are collocated with the VSCMGs. The angular velocities of the mounting sites induced by the vibration can be used for control synthesis. The direction of each angular velocity sensor is in the direction of the torque produced by the co-located VSCMG 7
working in CMG mode at the initial state. The measurements are written as
β&c = [ β&c1 ,K , β&cn ]T = Rs q&b ,
(3)
where β&ci is the angular velocity measured by the ith sensor. It should be calculated by β&ci = −cbz,0giT Rgi q&b , where T z0 T cb , gn ] . the superscript “ 0 ” in cbz,0gi indicate that cbz,0gi is equal to the initial value of cbz, gi ; Rs = −[ RgT1cbz,0g1 ,K , Rgn
Observing Eqs. (1) and (2), several remarks can be made. Firstly, when the VSCMGs work in CMG mode, the control goal is vibration suppression, so the necessary condition to get the solution for the command of the gimbals is n ≥ m . Secondly, when the VSCMGs work in MW mode, the control goal is absorbing or releasing energy while keeping the structure well damped, so the necessary condition to get the solution for the command of the rotors is n ≥ m + 1 . Thirdly, when the VSCMGs work in VSCMG mode, the control goal is integrated power and vibration
control, so the necessary condition to get the solution for the command of the gimbals and the rotors is 2 n ≥ m + 1 . As a conclusion, the necessary condition to get the solution for the command of the VSCMGs is n ≥ m + 1 . Only in this manner, an IPVCS can be realized.
III. “CMG & MW” Mode In this section, the combined “CMG & MW” mode of the VSCMGs will be explored. As shown in Figure 2, when the structure is vibrating, the rotor angular rates are fixed, then the VSCMGs work in CMG mode. A feedback control law will be designed for vibration suppression based on the Lyapunov approach. When the vibration amplitude of the structure is small, the gimbals will be locked, the VSCMGs work in MW mode. Based on the projection matrix method, a Lyapunov feedback control law will be applied to absorb or release energy while keeping the structure well damped.
A. CMG Mode for Vibration Suppression When the structure is vibrating, the VSCMGs work in CMG mode. By assuming that the gimbal angles are restricted to a small range, the rotational transformation matrix Ab, gi can be regarded as a time-invariant matrix, so that E a , G , BΩ , and Bδ are time-invariant parameter matrices, too. The generalized modal force BΩ Ω& can be dropped from Eq. (1) because the rotor angular rates are constant in the CMG mode. So Eq. (1) can be simplified as & & & E a q& b + ( Db + G )qb + Λb qb = Bδ δ , 8
(4)
Noting that the VSCMGs will convert to MW mode after the vibration amplitude of the structure becomes very small, the gimbal angles are desired to tend to the mounting position, which also keeps the system satisfying the assumption that the gimbal travels are restricted to a small range. Consider the Lyapunov function[2,31]
1 1 1 V1 = q&bT Ea q&b + qbT Λb qb + δT Qδ ≥ 0 , 2 2 2 where Q ∈ ¡
n× n
(5)
is a positive definite matrix; the first two items represent the vibrational energy of the system; the
last item indicates that the gimbal angles are desired to tend to the initial state. We have
1 1 T T T & & & & V& ( Ea q& q&b ( Ea q& 1= b + Λb qb ) qb + b + Λb qb ) + δ Qδ 2 2 1 1 = ( Bδ δ&− ( Db + G )q&b )T q&b + q&bT ( Bδ δ&− ( Db + G )q&b ) + δT Qδ& 2 2 = −q&bT Db q&b + q&bT Bδ δ&+ δT Qδ&, where the antisymmetric property of G is utilized from the second step to the third step. It should be noted that when the VSCMGs work in CMG mode, Bδ is a time-invariant matrix; when the VSCMGs work in MW mode, the gimbal of the VSCMGs will be locked. As a result, in this section, Bδ can be simplified as T z0 Bδ = −[ RgT1cbz,0g1 J ry1Ω1 ,K , Rgn cb , gn J rny Ω n ] .
Comparing Rs and Bδ , we have
Bδ = RsT hd , where hd = diag[ J ry1Ω1 ,…, J rny Ω n ] . Here we have
q&bT Bδ = q&bT RsT hd = β&cT hd . Noting that the measurements are β&c , the feedback steering law of the gimbals can be chosen as
δ&= −kd β&c − kδ δ ,
(6)
where k d = diag [k d 1 ,K , k dn ] , and k di > 0 ; kδ = diag[kδ 1 ,K , kδ n ] , and kδ i > 0 . Q can be defined as hd Q , where Q = diag[kδ 1 kd 1 ,K , kδ n kdn ] . Thus, T & & &T & &T V& 1 = −qb Db qb − βc hd ( k d βc + kδ δ ) − δ hd Q ( k d βc + kδ δ)
= −q&bT Db q&b − β&cT hd kd β&c − β&cT hd kδ δ − δT hd Qkd β&c − δT hd Qkδ δ 9
= −q&bT Db q&b − β&cT hd kd β&c − 2 β&cT hd kδ δ − δT hd Qkδ δ
= −q&bT Db q&b − (∑ J riy Ωi kdi β&ci2 + 2∑ J riy Ωi kδ i β&ciδ i + ∑ J riy Ωi kδ2iδ i2 kdi ) i
i
= −q&bT Db q&b − ∑ J riy Ω i ( kdi β&ci + kδ i
i
kdi δ i ) 2 .
i
In the above derivation, in the first step, q&bT Bδ is replaced by β&cT hd ; δ& is replaced by − kd β&c − kδ δ ; and Q
is replaced by hd Q . From the second step to the third step, Qkd = kδ is utilized. Obviously, the proposed feedback steering law of the gimbals guarantees the Lyapunov stability of the vibration system.
According
to
LaSalle’s
{(q&b , β&ci , δ i ) | q&b = 0, kdi β&ci + kδ i
theorem,
the
system
convergent
to
the
invariant
set
kdi δ i = 0} . When q&b = 0 , β&ci = 0 . Hence it is easy to get that δ i = 0 . Thus,
the system is asymptotically stable, and the feedback steering law of the gimbals can keep the gimbal angles to tend to the mounting position while suppressing the vibration of the structure.
B. MW Mode for Energy Control and Damping Maintenance When the structure is not vibrating or the vibration amplitude of the flexible structure is very small, the VSCMGs work in MW mode. In this situation, the variables E a and BΩ are regarded as time-invariant parameter matrices, while G and Bδ are regarded as time-variant parameter matrices. So Eq. (1) can be simplified as & & E a q& b + ( Db + G ) qb + Λb qb = T
(7)
where T is the generalized modal force caused by the rotor acceleration. T is acted on the flexible structure directly, the expression equation of which is T = BΩ Ω&
(8)
Clearly, the acceleration law of the rotors must be solved to satisfy Eqs. (2) and (8) at the same time. The control logic is shown in Figure 5. Based on the projection matrix method, Ω& can be decomposed into two orthogonal parts, the one is applied to keep the structure well damped and the other is used to absorb or release energy[32]. In this way, the acceleration law of the rotors can be chosen as Ω&= BΩT ( BΩ BΩT ) −1 Tc + S1u1 ,
(9)
where Tc is the desired generalized force, which is needed to be designed; S1 = I n − BΩT ( BΩ BΩT )−1 BΩ , where I n 10
is an n-dimensional identity matrix; u1 is an n×1 column matrix needing to be solved. Clearly, we have S1T = S1 , and S1 S1 = S1 . Pd
Power Requirement
u1
Tc
Rotor Acceleration Law
Legend Physical Motion Logical Connection Electrical Connection
Store/Drain Energy
Output Modal Forces
Ω
βc
T
Sensors Flexible Structure Figure 5. Sketch of the MW mode
Then, submitting Eq. (9) into Eq. (2), the solution of u1 is
u1 = S1 J ry Ω ( ΩT J ry S1 J ry Ω )−1 Pf ,
(10)
where Pf is the power excluding vibration suppression power, viz. Pf = Pd − ΩT J ry BΩT ( BΩ BΩT ) −1 Tc . Pd is the power requirement for the VSCMGs. Now the final task to solve the acceleration law of the rotors is to find the appropriate Tc , the function of which is keeping the structure well damped. Similar to Eq. (5), considering the Lyapunov function
1 1 V2 = q&bT Ea q&b + qbT Λb qb ≥ 0 , 2 2 we have 1 1 T T & & & V& ( E a q& q&b ( E a q& 2= b + Λb qb ) qb + b + Λb qb ) 2 2 =
1 1 (T − ( Db + G )q&b )T q&b + q&bT (T − ( Db + G )q&b ) 2 2
= −q&bT Db q&b + q&bT T . The antisymmetric property of G is utilized from the second step to the third step, too. Choosing the feedback control law of the desired generalized modal force as 11
Tc = − RsT kΩ β&c ,
(11)
where kΩ = diag[kΩ1 ,K , kΩn ] , kΩi > 0 . Substituting Eqs. (9) and (11) back into Eq. (8) leads to
T = BΩ ( BΩT ( BΩ BΩT ) −1 Tc + S1u1 ) = BΩ (− BΩT ( BΩ BΩT )−1 RsT kΩ β&c + S1u1 ) = − BΩ BΩT ( BΩ BΩT ) −1 RsT kΩ β&c + BΩ S1u1 = − RsT kΩ β&c + BΩ ( I n − BΩT ( BΩ BΩT ) −1 BΩ )u1 = − RsT kΩ β&c + ( BΩ I n − BΩ BΩT ( BΩ BΩT ) −1 BΩ )u1 = − RsT kΩ β&c + ( BΩ − BΩ ) u1 = −RsT kΩ β&c .
(12)
In the first step, Eq. (9) is substituted into Eq. (8). From the second step to the third step, Eq. (11) is substituted into the equation. From the fifth step to the sixth step, S1 is replaced by I n − BΩT ( BΩ BΩT )−1 BΩ . In this way, &T & &T T & V& 2 = −qb Db qb − qb Rs kΩ βc
= −q&bT Db q&b − q&bT RsT kΩ Rs q&b ≤ 0 . In the above derivation, from the first step to the second step, β&c = Rs q&b is utilized. Therefore, the proposed feedback control law based on the projection matrix method guarantees the asymptotical stability of the vibration system. According to LaSalle’s theorem, the system convergent to the invariant set {(q&b ) | q&b = 0} . As a conclusion, when the structure is stable or the vibration of the structure is small, the VSCMGs can work in MW mode. Owing to the projection matrix method, the control law of Ω& can ensure the power requirement while guaranteeing the asymptotical stability of the system.
C. Illustrative Example In this section, the effectiveness of the proposed feedback control law is verified by a numerical simulation of a flexible aluminum alloy truss with 8 VSCMGs. The length of the truss is 10m , whereas the length and width of the cross-section are 50mm and 30mm , respectively. Based on the assumed mode method, selecting the first four structural modes to describe the vibration of the truss, the corresponding frequencies of the truss are 1.824Hz , 3.018Hz , 11.030Hz , and 15.697Hz , respectively. The damping coefficient of the truss is 0.005. 12
The mounting locations of the VSCMGs are shown in Figure 6. Relative to fb , the mounting vectors are [0, 6.5, 0]m , [0,7,0]m , [0, 7.5, 0]m , [0,8, 0]m , [0,8.5, 0]m , [0,9, 0]m , [0,9.5,0]m , and [0,10, 0]m , respectively. The initial of the rotors and the gimbals are J gi = diag[3,5,3] × 10−5 kg gm 2 and J ri = diag[3,16,3] × 10−5 kg gm 2 , respectively, while each VSCMG weights 1kg . The mounting matrices of the VSCMGs are shown in Table 1. The gimbal angular rate of the VSCMGs can work in the range of ±0.001 : 1rad s . xb
yb
ob yb
c1
zb
c5
c6
c2
c3
c7
c8
xb
c4
Figure 6. Sketch of the flexible truss with 8 VSCMGs
As a numerical simulation, a simple power requirement is considered by assuming the truss is mounted on a satellite, the orbital period of which is 5750s . The sunlight exposure time is 3750s , and the charging power Pd = 5W . The eclipse time is 2000s , and the discharging power Pd = −5W at the first 1750s of the eclipse
time; the discharging power Pd = −40W at the last 250s . Table 1 Mounting matrices of the VSCMGs
SGVSCMG number
Mounting matrix Ab0, gi
c1 , c2
[0 1 0;0 0 1;1 0 0]
c3 , c4
[0 -1 0;0 0 -1;1 0 0]
c5 , c6
[0 0 -1;0 1 0;1 0 0]
c7 , c8
[0 0 1;0 -1 0;1 0 0]
The control task is suppressing the vibration of the truss based on the CMG mode, and keeping the structure well-damped when utilizing the rotors to absorb or release energy based on the MW mode. The initial states of the system are qb = [1,1, 0, 0]T and q&b = [0,0, 0, 0]T . All the initial rotor angular rates are equal to 30000rpm , and all the initial gimbals angles are equal to zero. At 3000s per orbital period, a perturbation of 57.5s , the value of which is 0.5 rad s , is applied to gimbal angular rates. The feedback control parameters for each VSCMG are chosen the same as k di = 1000 , kδ i = 0.01 , kΩi = 100 ,
13
i = 1,K ,8 . When the elastic deformation of the truss tip is less than 0.1mm , we consider the vibration amplitude of the flexible truss is very small, then the VSCMGs will convert to the MW mode. Complete dynamics of the system is considered in the simulation, which will test the robustness of the proposed control law. After simulating for 12000s , the results show that the proposed feedback control law can achieve the control task under disturbance. Figure 7 depicts the time-varying process of the modal coordinates. Figures 8 and 9 show that the VSCMGs work in CMG mode for the first 532s , and the gimbal angles are traveling in a small range from beginning to the end, which confirms the previous assumptions in this section. After 532s , the VSCMGs convert to the MW mode, and the gimbal angles are nearly equal to zero. Figure 10 indicates that when the VSCMGs work in MW mode, the rotor angular rates vary with the power requirement. The rotors spin down when discharging power while accelerating when charging power. However, the angular rates of each rotor are quite different, and at the end of each orbital period, rotor angular rates are much smaller than the initial. Figure 11 shows that the power requirement is met in MW mode. The elastic deformation of the truss tip is shown in Figure 12. Figures 7 and 12 indicate that the elastic deformation of the truss is decreased rapidly in CMG mode, and the truss is well damped in MW mode. 1
1 0 -1 0
0.5 0
10-6
500 10-4
-0.5 20 2 10 0 -1 0 -2 2000 7000 12000 500 550 -1.5 0 2000 4000 6000 8000 10000 12000 time s Figure 7. Modal coordinates, CMG & MW mode
10 5 0 -7 0
0.03 0 -0.03 2000 4000 6000 8000 10000 12000 13 0 -13 0 550
0.2 0 -0.2 500 2000
4000
6000 time s
550
8000 10000 12000
Figure 8. Gimbal angles, CMG & MW mode 14
57.3
1 0 -1 2000
0
-57.3
0
7000
57.3 0 -57.3 0 2000
1.3 0 -1.3 12000 500
550
110 220 330 440 550 4000 6000 8000 10000 12000 time s
rotor angular rates, rpm
Figure 9. Gimbal angular rates, CMG & MW mode
Figure 10. Rotor angular rates, CMG & MW mode
10 0 -10 6 4 -20 2 0 531 -30 -40
0
Pd P 532 2000
533 4000
6000 time s
8000 10000 12000
Figure 11. Power, CMG & MW mode
1000
500
0 0
1000 500 0 0
2000
0.2 0.1 0 550 500 0.01 0.005 0 2000 4000
6000 time s
550
7000 8000
12000 10000
12000
Figure 12. Elastic deformation of the truss tip, CMG & MW mode 15
Therefore, the proposed feedback control law can exert active modal force to the structure for vibration suppression in CMG mode, and absorb or release energy while keeping the structure well-damped in MW mode, even though there is a perturbation in each orbital period. However, it should be noted that when the VSCMGs work in CMG mode, the power requirement cannot be met. As a result, another control strategy is needed to meet the power requirement while suppressing the vibration. When the VSCMGs work in VSCMG mode, this goal can be achieved.
IV. VSCMG Mode A. Steering Law of the Gimbals and the Rotors Based on the above control strategy, when the VSCMGs work in VSCMG mode, the new feedback steering law of the gimbals and the rotors should combine the characteristics of CMG mode and MW mode. As shown in Figure 3, the steering law expects the rotors do not exert modal force on the structure, while the modal force is only produced by the gimbal rotation. In this way, according to Eqs. (9) and (10), the steering law of the rotors can be chosen as Ω&= S1 J ry Ω ( ΩT J ry S1 J ry Ω ) −1 Pd .
(13)
At the same time, with the assumption that the gimbal angles are restricted to a small range, E a and BΩ are time-invariant parameter matrices. However, G , hd and Bδ are time-variant parameter matrices. Therefore, if the solution of δ& is chosen the same as Eq. (6), it is hard to prove the stability of the feedback steering law. Considering the simplest feedback steering law of the gimbals, δ&= −kd β&c .
In order to verify the stability of the feedback control law, choosing the Lyapunov function
1 1 V3 = q&bT Ea q&b + qbT Λb qb ≥ 0 , 2 2 we have 1 1 T T & & & V& ( Ea q& q&b ( E a q& 3= b + Λb qb ) qb + b + Λb qb ) 2 2 =
1 1 ( Bδ δ&− ( Db + G )q&b )T q&b + q&bT ( Bδ δ&− ( Db + G )q&b ) 2 2
16
(14)
= −q&bT Db q&b + q&bT Bδ δ& = −q&bT Db q&b − q&bT RsT hd kd β&c = −q&bT Db q&b − q&bT RsT hd kd Rs q&b ≤ 0 . In the above derivation, the antisymmetric property of G is utilized from the second step to the third step, too. From the third step to the fourth step, Bδ = RsT hd is substituted into the equation. And from the fourth step to the fifth step, β&c = Rs q&b is also utilized. Similarly, the proposed feedback steering law of the gimbals guarantees the asymptotical stability of the vibration system, and the system convergent to the invariant set {(q&b ) | q&b = 0} .
B. Illustrative Example In this section, all the simulation parameters are the same as Section
.C. After simulating for 12000s , the
results show that the proposed feedback steering law of the gimbals and the rotors can meet the power requirement and suppress the vibration of the truss at the same time. Figure 13 shows that the proposed feedback steering law in this section has similar vibration suppression ability as that proposed in Section
. Figure 14 and Figure 15 show
that the gimbal angles are traveling in a small range from beginning to the end. However, comparing with Figure 8 and Figure 9, the gimbal angles don’t converge to zero, because the gimbal angles are not fed back to the proposed control law. Figure 16 depicts the time-varying process of the rotor angular rates varying with the power requirement. In contrast to the CMG & MW mode, the rotor angular rates at the end of each orbital period are almost equal to the initial values. However, the angular rates of each rotor are still quite different. Figure 17 shows that the power requirement is met throughout the simulation process, which can’t be achieved in CMG &MW mode. Figure 13 and Figure 18 indicate that the elastic deformation of the truss is decreased rapidly and the truss is well damped in VSCMG mode. 1
1 0 -1 0
0.5 0
2 0 -2 -4 2000
-0.5 -1 -1.5
0
500
10-5
10-4
7000
2000
2 0 -2 12000 500
4000
6000 time s
550
8000 10000 12000
Figure 13. Modal coordinates, VSCMG mode 17
14
0.5 0 -0.5 500
7
550
0 13 0 -13 0 2000 4000
-7 -14
0
550 6000 8000 10000 12000 time s
Figure 14. Gimbal angles, VSCMG mode
57.3
0
-57.3
0
0.2 0 -0.2 2000 57.3 0 -57.3 0 2000
7000
0.1 0 -0.1 12000 650
700
350 4000
6000 time s
700 8000 10000 12000
rotor angular rates, rpm
Figure 15. Gimbal angular rates, VSCMG mode
Figure 16. Rotor angular rates, VSCMG mode
10 0 -10
Pd
-20
P
-30 -40
0
2000
4000
6000 time s
8000 10000 12000
Figure 17. Power, VSCMG mode 18
1000 1000 500 0 0
500
0 0
2000
500
4000
0.2 0.1 0 500 0.02 0.01 0 2000 6000 time s
550
7000 8000
12000 10000
12000
Figure 18. Elastic deformation of the truss tip, VSCMG mode
Therefore, when the VSCMGs work in VSCMG mode, the proposed feedback steering law can meet the power requirement and suppress the vibration of the truss at the same time, which can’t be achieved in CMG & MW mode.
V. Conclusions An integrated power and vibration control system (IPVCS) was proposed in this paper. VSCMGs were distributed on flexible structures for vibration suppression and energy storage. Based on the different working modes of the VSCMGs, two control strategies for the IPVCS were explored. One is the CMG & MW mode. When the structure is vibrating, the VSCMGs work in CMG mode. A simple feedback control law utilizing the angular velocity sensor measurements and gimbal angles as feedback values is designed. When the structure is stable or the amplitude of the vibration is very small, the VSCMGs work in MW mode. A feedback control law based on the projection matrix method, which only utilizes the angular velocity sensor measurements as feedback values, is designed. The other is VSCMG mode, in which the modal force is only produced by the rotation of the gimbals, while the rotors are only used to absorb or release energy. The control law of the gimbal angular rates only utilizes the angular velocity sensor measurements. Based on the projection matrix method, the control law guarantees the rotors not exerting modal force on the structure. The simulation results show that both the two proposed control strategies can achieve the control tasks. When the VSCMGs work is CMG & MW mode, the vibration can be suppressed in CMG mode. In MW mode, the flexible structure is well-damped while the power requirement can be met. When the VSCMGs work is VSCMG mode, comparing with CMG & MW mode, the proposed feedback control law can meet the power requirement and suppress the vibration at the same time.
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Acknowledgments This work is supported by the National Natural Science Foundation of China (11872011). The authors would like to thank the anonymous reviewers for the valuable comments and suggestions to improve the paper.
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Highlights: An integrated power and vibration control system (IPVCS) is proposed. VSCMGs are distributed on the structure for vibration suppression and energy storage. The control laws of the “CMG & MW” mode and the “VSCMG” mode are designed. The IPVCS can absorb or release energy while keeping the structure well-damped.
Declaration of Interest Statement We declare that we have no financial and personal relationships with other people or organizations.