Dynamic Modeling for a Flexible Spacecraft With

Lun Liu School of Astronautics, Harbin Institute of Technology, P.O. Box 137, Harbin 150001, China e-mail: [email protected]

Dengqing Cao1 School of Astronautics, Harbin Institute of Technology, P.O. Box 137, Harbin 150001, China e-mail: [email protected]

Dynamic Modeling for a Flexible Spacecraft With Solar Arrays Composed of Honeycomb Panels and Its Proportional–Derivative Control With Input Shaper A high-precision dynamic model of a flexible spacecraft installed with solar arrays, which are composed of honeycomb panels, is established based on the nonconstrained modes of flexible appendages (solar arrays), and an effective cooperative controller is designed for attitude maneuver and vibration suppression by integrating the proportional–derivative (PD) control and input shaping (IS) technique. The governing motion equations of the system and the corresponding boundary conditions are derived by using Hamiltonian Principle. Solving the linearized form of those equations with associated boundaries, the nonconstrained modes of solar arrays are obtained for deriving the discretized dynamic model. Applying this discretized model and combining the IS technique with the PD controller, a hybrid control scheme is designed to achieve the attitude maneuver of the spacecraft and vibration suppression of its flexible solar arrays. The numerical results reveal that the nonconstrained modes of the system are significantly influenced by the spacecraft flexibility and honeycomb panel parameters. Meanwhile, the differences between the nonconstrained modes and the constrained ones are growing as the spacecraft flexibility increases. Compared with the pure PD controller, the one integrating the PD control and IS technique performs much better, because it is more effective for suppressing the oscillation of attitude angular velocity and the vibration of solar array during the attitude maneuver, and reducing the residual vibration after the maneuver process. [DOI: 10.1115/1.4033020] Keywords: flexible spacecraft, honeycomb panel, dynamic modeling, input shaping technique, vibration control

1

Introduction

Large-span solar arrays are critical appendages for most of modern spacecraft employed for communications, remote sensing or numerous other applications. They can provide sufficient energy to achieve various functions of the spacecraft and ensure that the spacecraft life is long enough. The solar arrays consist of solar cells and support components such as back board or membrane-boom structures. For the former case, the back boards of solar arrays are made from honeycomb panels [1,2] to reduce the launch mass and save the launch cost. As a result, those spacecraft are extremely flexible and have low-frequency vibration modes, which are coupled with the spacecraft attitude motion and might be excited by orbital operations such as attitude maneuver and quick tracking. Hence, it is of great importance to establish accurate rigid-flexible coupling dynamic model and design efficient controller used to achieve vibration suppression of flexible spacecraft during attitude maneuver. Because the spacecraft consists of a rigid central hub and largespan solar arrays composed of honeycomb panels, it is a typical rigid–flexible coupling dynamic system, and often simplified as a hub-beam system (see Fig. 1) [3–10]. The traditional model of this 1 Corresponding author. Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received October 22, 2015; final manuscript received February 26, 2016; published online May 25, 2016. Assoc. Editor: Ming Xin.

dynamic system is referred as the zeroth-order approximate coupling (ZOAC) model [11,12], which assumes that the axial and transverse deflections at any point in the beam are not coupled. However, this model cannot describe the dynamic characteristics of the system at a higher rotating speed since the dynamic stiffening phenomenon is not taken into account in this model [13]. Then the so-called first-order approximate coupling (FOAC) model is developed by considering a second-order coupling term of axial displacement caused by transverse displacement of the flexible beam [14–16]. By considering the higher-order coupling terms of deformation in the longitudinal and transverse deflections, Deng et al. [17] and Liu et al. [18] developed a higher-order approximate coupling (HOAC) dynamic model. The numerical results obtained from this model demonstrate that the ZOAC and FOAC models may not be correct when the rotating speed of the hub is large. It should be noted that the FOAC and HOAC models are generally used to study the hub-beam systems with high rotating speed, such as the blades of helicopter rotor or turbine machine rotor. For flexible spacecraft, the rotating speed of the rigid central hub, i.e., the attitude angular velocity of spacecraft, is so small that its ZOAC models are sufficiently precise for establishing the dynamic equations of the system. Therefore, ZOAC models are widely used to design the control system of flexible spacecraft in the published literature [3–10]. For the ZOAC dynamic models of flexible spacecraft, the solar panels are simplified as isotropic Euler–Bernoulli beams with small deformation. However, the main structures of the solar

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Fig. 1 Model of spacecraft with solar arrays: (a) sketch of spacecraft and (b) geometric relations

arrays in practical applications are honeycomb panels due to their lightweight, high specific bending stiffness and strength [19]. Applying the classical plate theory (CPT), Mindlin’s first-order shear deformation plate theory (FSDPT), and Reddy’s third-order shear deformation plate theory (TSDPT), Yu and Cleghorn [20] investigated the free flexural vibration of symmetric rectangular honeycomb panels with simply supported boundaries. Their numerical results indicate that the CPT is accurate only for thin plates with small thickness-to-length ratio, and the FSDPT and TSDPT can be used to study much thicker honeycomb panels. Using the TSDPT, Li and Jin [21] analyzed the effects of honeycomb core thickness to panel thickness ratio on flexural vibration of honeycomb panels. Guillaumie [22] proposed an analytical formula of eigenmodes for rectangular honeycomb panel with composite faces and investigated its vibroacoustic bending properties from the wavenumber modulus to the mechanical impedance. Using the homotopy analysis method, Li et al. studied the geometrically nonlinear free vibrations of honeycomb panels with simply supported boundaries [23], as well as the geometrically nonlinear forced vibrations of honeycomb panels with clamped supported boundaries [24]. Luo et al. [25,26] derived the governing equations of honeycomb panel with Hamilton’s Principle and the TSDPT, and proposed a novel multilayer piezoelectric actuator to control the vibration of honeycomb panels. Based on the studies of honeycomb panels in the above literature, some researchers further investigated the honeycomb panels applied in aerospace engineering by considering the specific circumstances, such as the variation of the thermal environment and the rigid-flexible coupling effects of spacecraft. Li et al. [2] developed a thermal analysis model of solar array with honeycomb panels subjected to space heat flux, and then studied the thermally induced vibration of the solar array in low earth orbit [27]. Considering the coupling effects between the attitude motion of rigid hub and the vibration of flexible solar array which is modeled by honeycomb panel, Johnston and Thornton [28] derived the governing equations of a flexible spacecraft using a generalized form of Lagrange’s equations for hybrid coordinate systems, and analyzed the thermally induced vibration of the spacecraft. Although the dynamic model in Ref. [28] can be used to study the dynamic characteristics and design controller of attitude maneuver and vibration suppression for flexible spacecraft with solar arrays (a rigid-flexible coupling system), it should be pointed out that this model may result in an unacceptable error for a flexible spacecraft. The main reason lies that the assumed modes, such as beam functions of cantilever beam [10,28,29] or other admissible basis functions [3,4,9,30,31] satisfying the geometric and physical boundary conditions, are used to discretize the displacement of flexible solar arrays. The boundary for those modes is assumed to be clamped-free, i.e., the mode shapes are constrained ones. However, the boundary of solar arrays cannot be assumed to be clamped-free for a flexible spacecraft. So their elastic modes 081008-2 / Vol. 138, AUGUST 2016

should be nonconstrained and are influenced by the rigid central hub. This paper is focused on establishing a high-precision dynamic model for a flexible spacecraft installed with solar arrays composed of honeycomb panels, and designing effective controllers for attitude maneuver and vibration suppression of the spacecraft. First, the honeycomb core is modeled as a solid layer, where the equivalent material properties of the layer are calculated by using Gibson’s cellular material theory [32]. And then, the honeycomb panel is considered as a three layered laminate structure. In this work, the flexible spacecraft is simplified as a rigid-flexible coupling hub-beam system with tip masses, and the governing motion equations of the system, as well as corresponding boundary conditions, are derived by using Hamiltonian Principle. Employing the solving procedure proposed by Cao et al. [33] and Song et al. [34], the nonconstrained modes of solar arrays are obtained from the linearized model of the system with associated boundaries. Then those modes are used to discretize the nonlinear governing equations. Subsequently, based on the discretized dynamic model, a hybrid control strategy combining the IS technique [35–37] and PD controller is designed to accomplish the attitude maneuver of a spacecraft and vibration suppression of its flexible solar arrays. Using numerical simulation, the influence of spacecraft flexibility and the honeycomb panel parameters on nonconstrained modes of the system is studied. Moreover, the performance of PD control with IS technique is assessed by comparing with the pure PD controller. The rest of the paper is as follows: Sec. 2 presents the modeling process of a flexible spacecraft installed with solar arrays. In Sec. 3, the hybrid control scheme is briefly described which integrates the PD control and IS technique. The model validation and analysis of numerical results are shown in Sec. 4. Finally, the paper is concluded in Sec. 5.

2

System Modeling

2.1 Model Description. Figure 1 shows a typical model of a flexible spacecraft, which consists of a cubic rigid hub with half of side length r0 and moment of inertia 2JR , two solar arrays with length L and width b, and two tip masses with mass mt . The solar arrays are installed symmetrically on the two sides of the cubic rigid hub, and the tip masses are used to model components tuning the frequencies of the spacecraft or other concentrated masses such as spreader bars [6]. o XYZ and o xyz are coordinate systems defined as the inertial frame (global coordinate system) and the body frame rotating with the rigid hub (local coordinate system), respectively. Similar to the previous reports [4,6,9,10], the maneuver of the spacecraft considered in this study is limited in the x z plane and its angular displacement is denoted by h. It is assumed that the solar arrays vibrate antisymmetrically in the Transactions of the ASME


Fig. 2 panel

Solar array model: (a) sectional view of solar array and (b) schematic of honeycomb

x z plane since the symmetrical deflection of solar arrays does not influence the attitude motion of the hub. In this research, each solar array is modeled by a beam fixed on the rigid central hub because its length is much larger than its width. Point P is the initial position of an arbitrary point on the solar array, and symbol “P*” represents the position of point P with deformation. uðx; tÞ and wðx; tÞ are displacements of point P in the x and z directions, respectively. 2s, which depends on time t, is a control torque implemented on the hub. As illustrated in Fig. 2, the solar array consists of a back board made from honeycomb panel and solar cells which are installed on the back board and covered by glass fiber sheets. In this research, only honeycomb panels are considered because they are main structures of solar arrays. The honeycomb panel is composed of honeycomb core with height 2h c and face sheet with height hf , where the subscripts c and f represent the honeycomb core and face sheet, respectively. The total height of honeycomb panel is 2h. The cell of honeycomb core is regular hexagon. lc and dc are length and thickness of honeycomb wall. The honeycomb core and face sheet are both made of aluminum. Based on Gibson’s cellular material theory [32], the equivalent material properties of honeycomb core, including elastic modulus Ec , equivalent mass density qc , and shear modulus Gc , can be expressed as 3 " 2 # 4 dc dc 2 dc G0 d c 13 Ec ¼ pffiffiffi E0 ; qc ¼ pffiffiffi q0 ; Gc ¼ pffiffiffi lc 3 lc 3 lc 3 lc

The position vector of point P on the solar array in o xyz and its inertial velocity vector in o XYZ are denoted by r and v, respectively. As shown in Fig. 1(b), they can be expressed as

2.2 Governing Equations. According to the conclusions of Yu and Cleghorn [20], the CPT can be used to model the solar arrays because they are thin plates with small thickness-to-length ratio. The displacement field of the structure is uð x; tÞ ¼ z

@wð x; tÞ @x

(3)

where i, j, and k are the unit vectors of o xyz in x, y, and z directions, respectively. (•) denotes the time derivative. Substituting Eq. (2) into Eq. (3), the total kinetic energy of the system can be obtained as follows: 1 1 2 T ¼ 2JR h_ þ 2 mt vð LÞ vð LÞ þ 2 2 2 # ð " ð hc ð hc ðh 1 L qb v vdz þ qf b v vdz þ qf b v vdz dx 2 0 c hc hc h (4) where qf ¼ q0 is the mass density of the face sheet. The strain–displacement relation is given as e¼

2 @u 1 @w @u @w þ þ ; c¼ @x 2 @x @z @x

(5)

where e and c are the normal and shear strains, respectively. The constitutive equations for the honeycomb panel are

(1) where E0 , q0 , and G0 are the elastic modulus, mass density, and shear modulus of aluminum, respectively. Then the honeycomb panel can be considered as a three layered laminate structure, as shown in Fig. 3.

_ r v ¼ r_ þ hj

r ¼ ðr0 þ x þ uÞi þ wk;

rc ¼ Ec ec ; sc ¼ Gc cc ; rf ¼ Ef ef ; sf ¼ Gf cf

(6)

where rc , sc , and rf , sf are normal and shear stresses for honeycomb core and face sheet, respectively. Ef ¼ E0 and Gf ¼ G0 are the elastic and shear modulus of face sheet. Then, the total strain energy of the spacecraft is expressed as ð " ð hc ðh 1 L b U ¼2 ðrc ec þ sc cc Þdz þ b ðrf ef þ sf cf Þdz 2 0 hc hc ð hc þb (7) ðrf ef þ sf cf Þdz dx h

(2)

The virtual work done by the external control torque is given by W ¼ 2sh

(8)

Using the Hamiltonian Principle which is expressed as ð t2 t1

dðT UÞdt þ

ð t2

dWdt ¼ 0

(9)

t1

Fig. 3 Coordinates in the thickness direction for the honeycomb core and two face sheets

the governing equations of motion for the spacecraft are derived as follows:

Journal of Dynamic Systems, Measurement, and Control

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" ð L

ðJR þ Js þ Jt Þ€h þ PðtÞ þ q

2

w dx þ I

ðL

0

# 2

PðtÞ ¼ P0 x2 pðtÞ

0

" ð # ðL L 0 0 _ þ 2I w w_ dx þ 2mt wðLÞwðLÞ _ h_ ¼ s þ q 2w wdx 0

Substituting Eq. (18) into Eq. (14), PðtÞ is rewritten as

ðw Þ dx þ mt w ðLÞ €h 0 2

(19)

where

0

(10)

P0 ¼ q

ðL

ðx þ r0 ÞuðxÞdx þ mt ðL þ r0 ÞuðLÞ

(20)

0 2 € Iw € 00 h_ ðqw Iw00 Þ 3Hðw0 Þ2 w00 ¼ 0 Dw0000 þ q½€hðx þ r0 Þ þ w (11)

and the corresponding boundary conditions are also obtained as follows: 8 _2 0 > € 0 ðLÞ þ Dw000 ðLÞ H½w0 ðLÞ3 > < I h w ðLÞ Iw 2 € þ mt ½ðL þ r0 Þ€h wðLÞ þ wðLÞh_ ¼ 0 > > : 00 w ðLÞ ¼ 0; w0 ð0Þ ¼ 0; wð0Þ ¼ 0

For free vibration analysis, s ¼ 0. Solving the expression of €h from Eq. (15) and then substituting it with Eqs. (18) and (19) into Eq. (16), a fourth-order differential equation related to uðxÞ is obtained u0000 ð xÞ k4 uð xÞ ¼ k4

(12)

P0 ð x þ r0 Þ J

where J ¼ JR þ Js þ Jt , k4 ¼ x2 q=D. The solution of Eq. (21) is expressed as follows: uðxÞ ¼ C1 R1 ðxÞ þ C2 R2 ðxÞ þ C3 R3 ðxÞ þ C4 R4 ðxÞ

All parameters in Eqs. (10)–(12) are given by

0

D ¼ Ec b

hc ð hc

z2 dz þ 2Ef b

hc

hc ðh

(13)

z2 dz

hc

H ¼ Ec bhc þ Ef bhf

€ þq PðtÞ ¼ mt ðL þ r0 ÞwðLÞ

ðL

€ þ r0 Þ dx wðx

(14)

0

Where Js and Jt represent the moment of inertia for each solar array and tip mass, respectively. Equation (10) describes the motion of rigid central hub, and Eq. (11) the motion of solar arrays. There are two kinds of inertial moment for solar arrays in expression (13), i.e., Js and I. The former’s value is much larger than that of the latter, which results in the omission of terms related to I in Eqs. (10)–(12). Then, the following simplified linearized dynamic equations (Eqs. (15) and (16)) with associated boundary conditions (Eq. (17)) can be obtained from Eqs. (10)–(12):

(

ðJR þ Js þ Jt Þ€h þ PðtÞ ¼ s

(15)

€ ¼0 Dw0000 þ q½€hðx þ r0 Þ þ w

(16)

€ Dw000 ðLÞ þ mt ½ðL þ r0 Þ€h wðLÞ ¼0 w00 ðLÞ ¼ 0; w0 ð0Þ ¼ 0; wð0Þ ¼ 0

(17)

2.3 Discretization of Governing Equations. The homogeneous equations of Eqs. (15) and (16) can be solved by using the method called “separation of variables” displayed in Refs. [33] and [34]. wðx; tÞ is selected as wðx; tÞ ¼ uðxÞpðtÞ;

pðtÞ ¼ sinðxt þ aÞ

081008-4 / Vol. 138, AUGUST 2016

(22)

where C1 C4 are undetermined constants. The expressions of Ri ðxÞ ði ¼ 1–4Þ are listed in the Appendix. The constants C1 C4 should be determined using the boundary conditions given in Eq. (17). Substituting expression (22) into Eq. (17), the eigenvalue equation for the linearized system of the flexible spacecraft is obtained, and its matrix form is expressed as follows:

q ¼ 2hc bqc þ 2hf bqf ðL Js ¼ q ðx þ r0 Þ2 dx Jt ¼ mt ðL þ r0 Þ2 ð hc ðh I ¼ qc b z2 dz þ 2qf b z2 dz

(21)

(18)

FX ¼ 0 ; X ¼ ½C1 ; C2 ; C3 ; C4 T

(23)

where F is a 4 4 matrix whose elements are given in the Appendix. For a nontrivial solution of Eq. (23), the determinant of F must be the zero, i.e., jFj ¼ 0. By solving this equation, the natural frequencies xi ði ¼ 1; 2; …Þ of the linearized system for a flexible spacecraft can be obtained, and then the unknown constants C1 C4 are determined by using Eq. (23), and the corresponding nonconstrained modes ui ðxÞ ði ¼ 1; 2; …Þ of the system are obtained by using expression (22). To obtain a finite dimensional dynamic model from Eqs. (10) and (11), the elastic deformation wðx; tÞ of solar arrays is represented as the linear combination of mode shape uðxÞ multiplied by time-dependent generalized coordinate pðtÞ, i.e., wðx; tÞ ¼ uðxÞpðtÞ

(24)

where uðxÞ is a 1 Nt row vector and pðtÞ is a Nt 1 column vector. Using the nonconstrained mode uðxÞ solved from Eqs. (22) and (23) as the element of uðxÞ, the expressions of uðxÞ and pðtÞ are given by uðxÞ ¼ ½u1 ðxÞ; u2 ðxÞ; … ; uNt ðxÞ; pðtÞ ¼ ½p1 ðtÞ; p2 ðtÞ; … ; pNt ðtÞT

(25)

Substituting expression (24) into Eqs. (10) and (11), the discretized rigid-flexible coupling dynamic equations for flexible spacecraft are yielded "

#" # " #" # €h 0 0 J þ pT Mp U h_ þ T 0 C p_ U M p€ # " #" # " _ T Mp_ h 0 0 s 2hp (26) ¼ þ 2 0 Kl þ Kn ðpÞ h_ M p 0

where the elements of M (mass matrix), Kl (linear stiffness matrix), Kn (nonlinear stiffness matrix), and U (rigid-flexible coupling terms) are given in the Appendix. C ¼ jM M þ jK Kl is the Transactions of the ASME


proportional viscous damping matrix. jM and jK are proportional constants.

3

To combining the PD control with IS technique, hd in Eq. (32), a single value, is replaced by an impulse sequence hdIS which are obtained by

Control Strategies Design

hdIS ¼ hd Amult

3.1 State Space Model. The linearized model of flexible spacecraft is obtained from Eq. (26), and expressed as follows: " # " # €h h s J U 0 0 0 0 h_ ¼ (27) þ þ 0 UT M p€ 0 C p_ 0 Kl p It can be rewritten in a compact form as ^ Y_ þ KY ^ ^Y € þC ^ ¼ Ss M

(28)

where " # " # h J U ^ ¼ Y¼ ; M ; p UT M " # " # 1 0 0 ^ ^ ; S¼ K¼ 0 0 Kl

" ^ ¼ C

# 0 0 ; 0 C (29)

Then the state space model is given by Z_ ¼ AZ þ Bs

(30)

where

Y Z¼ _ ; Y

0 A¼ ^ ^ 1 K M

I 1 ^ ; ^ C M

B¼

0

1

^ ^ S M (31)

3.2 PD Control With IS. IS is a technique of convolving an input shaper, which is a sequence of impulses constituted by appropriate amplitudes and time locations, with the desired command so that the residual vibration of the system is less than that with original unshaped command [6,35–37]. This technique is developed based on linear system theory, and the key to designing an input shaper is the precise knowledge of natural frequency and damping ratio of the system. The IS is an open-loop controller so that the state variables of the flexible spacecraft cannot be fed back in this controller. To achieve better control performance, it can be integrated with closed-loop control strategy such as PD control schemes [6,37]. The expression for a PD feedback controller is given by _ s ¼ Kp ðhd hÞ þ Kd ðh_ d hÞ

(32)

where Kp and Kd are the PD gains. hd represents the desired attitude angle of the spacecraft and h_ d is the desired angular velocity.

(33)

where Amult is a multimode input shaper, and given by Amult ¼ A1s A2s Ais Ans

(34)

where Ais ði ¼ 1; 2; …; nÞ is the impulse sequence of the ith mode of the system, and “” is the convolution operator. The amplitude Aj and time location tj of zero-vibration (ZV) shaper for Ais ði ¼ 1; 2; …; nÞ are expressed as follows [36,37]: 2 3 1 K Aj 6 7 ¼ 4 1 þ K 1 þ K 5; j ¼ 1; 2 (35) tj T1 T2 pffiffiffiffiffiffiffiffiffiffiffiffiffi where K ¼ expðfp= 1 f2 Þ, Tj ¼ ðj 1Þp=xd ðj ¼ 1; 2; :::Þ. f and xd are the ith damping ratio and damped frequency of the system, which are given by qffiffiffiffiffiffiffiffiffiffiffiffiffi (36) ksys ¼ fxd 6 ixd 1 f2 where ksys is the eigenvalue of the system matrix A in Eq. (30). Symbol “ i ” is the imaginary unit. Actually, ZV shaper may be sensitive to the errors in f and xd . To improve the robustness of the IS method, a zero-vibration-derivative (ZVD) IS shown in Eq. (37) is designed by forcing the derivation of the residual vibration (with respect to xd ) to equal zero 2 3 1 2K K2 Aj 6 7 ¼ 4 ð1 þ K Þ2 ð1 þ K Þ2 ð1 þ K Þ2 5; j ¼ 1; 2; 3 (37) tj T1 T2 T3 Similarly, the zero-vibration-derivative–derivative (ZVDD) shaper can be obtained as follows [6,37]: 2 3 " # 1 3K 3K 2 K3 6 7 Aj ð1 þ K Þ3 ð1 þ K Þ3 ð1 þ K Þ3 ð1 þ K Þ3 7 ¼6 4 5; j ¼ 1; 2; 3; 4 tj T1 T2 T3 T4 (38)

4

Simulation Results

4.1 Model Validation. Before the main topic of this research, verifications for the methodology, formulations, and the MATLAB codes are conducted. The natural frequencies of a hub-beam system with tip masses are calculated, and corresponding mode shapes are plotted. The geometric and material parameters of the system are taken as: r0 ¼ 1 m, JR ¼ 100 kg m2 , mt ¼ 0:3 kg,

Table 1 Comparison for natural frequencies of a hub-beam system with tip masses x (rad/s) 1st L ðmÞ 10 12 14 16 18 20

2nd

3rd

Ref. [38]

Present model

Ref. [38]

Present model

Ref. [38]

Present model

2.15 1.74 1.48 1.29 1.15 1.03

2.147 1.745 1.482 1.295 1.150 1.032

10.13 7.23 5.47 4.31 3.51 2.94

10.133 7.234 5.466 4.309 3.512 2.942

28.92 20.31 15.08 11.67 9.31 7.61

28.913 20.310 15.081 11.664 9.307 7.613


AUGUST 2016, Vol. 138 / 081008-5


Fig. 4 The first three mode shapes of a hub-beam system with tip masses obtained from different models (L 5 20 m): (a) and (b) non-normalized and normalized mode shapes

Table 2 Geometric and material parameters of the flexible spacecraft Components

Parameters

Values

Solar array

Length L ðmÞ Width b ðmÞ Height of honeycomb panel 2h ðmÞ Core-to-thickness ratio hc =h Height of face sheet hf ðmÞ Length of honeycomb lc ðmÞ Thickness of honeycomb dc ðmÞ Elastic modulus of aluminum E0 ðPaÞ Shear modulus of aluminum G0 ðPaÞ Mass density of aluminum q0 ðkg m3 Þ Damping coefficients jM , jK

8.0 0.5 0.02 0.7 h hc 1.833 103 0.0254 103 6.89 1010 2.6 1010 2.8 103 0.008, 0.00025

The hub and tip mass

Size of the hub r0 ðmÞ Inertial moment of the hub 2JR ðkg m2 Þ Mass of the tip mass mt ðkgÞ

1.0 200.0 0.3

Af ¼ 7:2986 105 m2 , L ¼ 10, 12, 14, 16, 18, 20 m, If ¼ 8:2189 109 m4 , Ef ¼ 6:8952 1010 Pa, q ¼ 2766:7 kg m3 . As shown in Table 1, the first three natural frequencies of nonconstrained modes for the system calculated by using the eigenvalue equation (23) agree well with those in Ref. [38] which are obtained from a finite element model. Figure 4 shows the first three mode shapes of the system (L ¼ 20 m) given by the model proposed in this paper (Eqs. (22) and (23)) and the finite element method (FEM) model in Ref. [38]. Figure 4(a) illustrates the nonnormalized mode shapes and significant differences of relative amplitude between the two kinds of mode shapes are observed. In fact, the amplitude of the mode shapes is the ratio of absolute vibration amplitude for each point on the flexible beam, so one cannot directly compare different kind of mode shapes as shown in Fig. 4(a). In order to compare those mode shapes, they are normalized and presented in Fig. 4(b). It can be seen that the mode shapes given by the present model and the FEM model are 081008-6 / Vol. 138, AUGUST 2016

entirely similar. Based on the comparisons on frequencies and mode shapes with results of FEM model, the validity of the present dynamic model is demonstrated. 4.2 The Influence of the Core-to-Thickness Ratios and the Spacecraft Flexibility on the Frequencies. The geometric and material parameters of the flexible spacecraft studied in the following analyses are given in Table 2. When the parameters of solar arrays remain, the flexibility of spacecraft is denoted by the moment of inertia for rigid central hub 2JR , i.e., small JR represents large spacecraft flexibility. Using the eigenvalue equation (23), the first four nonconstrained frequencies of the system are calculated for various JR and core-to-thickness ratio hc =h. The results are depicted in Figs. 5(a)–5(d), where the dashed lines are frequencies of corresponding cantilever honeycomb panel (constrained frequencies). It is observed from those figures that the natural frequencies increase with the core-to-thickness ratio hc =h and then sharply Transactions of the ASME


Fig. 5 Variation of the nonconstrained frequencies of the flexible spacecraft with respect to hc =h for various JR . The dashed lines are frequencies of corresponding cantilever honeycomb panel (constrained frequencies). —3—, JR 5 1 kg m2 ; —䊊—, JR 5 10 kg m2 ; —䉮—, JR 5 102 kg m2 ; —$—, JR 5 103 kg m2 ; —䉫—, JR 5 104 kg m2 : (a)–(d), the first to the fourth frequencies.

Fig. 6 The normalized mode shapes of the first four nonconstrained and constrained modes of solar arrays (JR 5 100 kg m2 , hc =h 5 0:7): (a)–(d), the first to the fourth mode shapes

drop when hc =h exceeds a particular value where the frequency is maximum, which is similar with the conclusion in Refs. [20] and [21]. The dots in Figs. 5(a)–5(d) represent the maximum frequencies for each curve, which illustrates that, for each order frequency, the value of hc =h where the frequency is maximum decreases first and then increases as JR grows. Therefore, for a flexible spacecraft with a particular inertial moment, it is necessary to choose a suitable value of core-to-thickness ratio hc =h of honeycomb panels to make sure that the frequencies of the system are relatively higher. On the other hand, Figs. 5(a)–5(d) reveal that the differences between the nonconstrained frequencies and the constrained ones are growing when JR decreases, especially for low–order frequencies. Figs. 6(a)–6(d) show the normalized mode shapes of the first four nonconstrained and constrained modes of solar arrays (JR ¼ 100 kg m2 , hc =h ¼ 0:7). As displayed in those figures, significant differences also exist between nonconstrained and

constrained mode shapes. The differences can be explained as follows: the effects of central rigid hub on the modes of solar arrays are considered in Eqs. (22) and (23) which are employed to calculate the nonconstrained modes; however, the constrained modes do not take into account those effects. Then one can conclude that constrained modes, i.e., assumed modes such as the modes of beam with clamped-free boundary, can only be used to derive the discretized dynamic models of the flexible spacecraft whose flexibility is small. For other spacecraft which have large flexibility, the discretized dynamic models based on the constrained modes in Refs. [3,4,9,10], and [28–31] may lead to an unacceptable error. In this case, nonconstrained modes, which can reflect the effect of the rigid central body on the elastic modes of the flexible appendages, should be used to discretize the system’s governing equations of motion. According to the discussions on frequencies and mode shapes for nonconstrained and constrained modes of solar arrays


AUGUST 2016, Vol. 138 / 081008-7


Fig. 7

The structure of pure PD control

composed of honeycomb panels, it can be concluded that the nonconstrained modes are more accurate than the constrained ones. As a result, the discretized model based on nonconstrained modes is more accurate than that based on the constrained ones. 4.3 Attitude Maneuver and Vibration Suppression. In this section, numerical simulations are conducted and presented to demonstrate the effectiveness of the control schemes designed in this paper. The parameters of the flexible spacecraft are listed in Table 2. In this work, the first two nonconstrained modes are chosen to design the input shaper, while the first four nonconstrained modes are used to calculate the dynamic responses of the system. The input shaper is the convolution of the four impulses ZVDD shaper for the first mode and the three impulses ZVD shaper for the second mode. In this simulation, the flexible spacecraft is commanded to a rest-to-rest maneuver, and the attitude angle hðtÞ varies from the initial state to the desired angle hd ¼ 1 rad. The desired attitude angular velocity h_ d equals zero. The PD gains of PD controller are taken as Kp ¼ 16:5 and Kd ¼ 350, respectively. For comparison, two control strategies for attitude maneuver and vibration suppression are simulated: (1) using pure PD controller and (2) using PD þ IS controller. Those two control schemes are discussed as follows. 4.3.1 Pure PD Control. The block diagram of pure PD con_ trol is illustrated in Fig. 7. The corresponding responses of h, h, wðL; tÞ, and sðtÞ are shown in Figs. 8(a)–8(d). In this case, the desired attitude angle of the flexible spacecraft can be accurately achieved within 70 s, and no overshoot occurs during attitude maneuver. However, at the beginning of attitude maneuver, the _ wðL; tÞ, and sðtÞ oscillate intensely. The relatively curves for h, large amplitude vibration of solar array is observed and the maximum amplitude of wðL; tÞ reaches up to 0.016 m. In addition, the maximum control torque is about 18 N m. 4.3.2 PD þ IS. In order to further improve the performance of the pure PD controller, a hybrid control scheme for control of spacecraft attitude motion and vibration suppression of solar arrays is presented here by integrating the PD control with IS. The block diagram of this hybrid control strategy is illustrated in Fig. 9. The parameters of the flexible spacecraft and PD controller

Fig. 9 The structure of PD control with IS

remain the same for a fair comparison. Figure 10 displays the simulation results of employing the PD controller with IS. The dashed line in Fig. 10(a) is the impulse sequence hdIS obtained by multiplying the desired attitude angle hd and the multimode IS Amult . It is observed from this figure that the spacecraft accurately reaches the demanded attitude angle with a settling time about 70 s without overshoot. The orders of magnitude for amplitude of residual vibrations of h_ and wðL; tÞ are both 105 and the same as those of the PD control case. However, compared with the responses of PD control case illustrated in Fig. 8, the oscillation of angular velocity h_ and the tip vibration of solar array wðL; tÞ excited by rapid maneuver at the beginning are suppressed effectively for _ wðL; tÞ, and sðtÞ PD þ IS control case. Therefore, the curves of h, shown in Figs. 10(b)–10(d) are much smoother than those of the PD control case. Moreover, the maximum amplitude of wðL; tÞ is less than 0.01 m and much smaller than that displayed in Fig. 8, as well as the control torque sðtÞ with less than 15 N m. All the comparisons demonstrate the validity of IS technique in the respect of active vibration suppression. From the analyses above, it is concluded that the PD control with IS (PD þ IS control case) is a hybrid control scheme with high-effectiveness for attitude maneuver and vibration suppression. It can not only accurately accomplish the attitude maneuver but also effectively suppress the oscillation of attitude angular velocity and the vibration of solar arrays, as well as the corresponding residual vibration. The algorithm of PD þ IS control is very simple and thus conducive to real-time control of spacecraft. On the one hand, a full-state measurement in practical applications may neither be possible nor feasible. Hence, the controllers requiring full-state feedback, such as linear quadratic regulator (LQR) controller, may be limited to this factor. For PD control with input shaper, the full-state feedback is not necessary, and only attitude angle and angular velocity are needed, i.e., only angle and angular velocity sensors installed on the central rigid hub of the spacecraft are needed for the measurement system. What is more, the sequence of impulses of the input shaper is a modified command signal that can move the system at the maximum rate possible without exciting vibrations. So, for the hybrid control scheme (PD þ IS), only the control torque calculated by Eq. (32) is needed to achieve the attitude maneuver of the spacecraft and vibration suppression of its flexible solar arrays. The

Fig. 8 Time responses of the flexible spacecraft for using the pure PD control (JR 5 100 kg m2 )

081008-8 / Vol. 138, AUGUST 2016

Transactions of the ASME


Fig. 10 Time responses of the flexible spacecraft for using the PD 1 IS control (JR 5 100 kg m2 )



control torque is implemented on the central rigid hub, and can be generated by flywheels or on-off jet thrusters. In addition, the PD control with IS can also ensure the achievement of attitude maneuver and vibration suppression for spacecraft with large or small flexibility (JR ¼ 10 kg m2 or JR ¼ 1000 kg m2 ) as shown

in Figs. 11 and 12. It should be pointed out that, even though an over shoot phenomenon occurs in Fig. 12(a) when the PD gains of PD controller are taken as Kp ¼ 16:5 and Kd ¼ 350, it can be eliminated easily by adjusting the values of Kp and Kd . The investigations above demonstrate that the hybrid control strategy designed in this


AUGUST 2016, Vol. 138 / 081008-9


research by integrating the PD control with IS provides effective theoretical basis for the attitude control of flexible spacecraft installed with solar arrays composed of honeycomb panels.

where ci ðkÞ ði ¼ 1–4Þ are given as following: c1 ðkÞ ¼

ðL

qðx þ r0 Þ cosh kxdx þ mt ðL þ r0 Þ cosh kL

0

5

Conclusions

c2 ðkÞ ¼

In this paper, the high-precision dynamic model of a flexible spacecraft installed with solar arrays composed of honeycomb panels has been established based on the nonconstrained modes of solar arrays, and then a hybrid control scheme has been designed by integrating the PD controller with IS to effectively achieve the attitude maneuver of spacecraft and vibration suppression of its solar arrays. The influence of spacecraft flexibility and the honeycomb panel parameters on nonconstrained modes of the system has been investigated. Also, the performance of PD control with IS are assessed by comparing with the pure PD controller. Some main conclusions are summarized as follows: (1) The dynamic model established in this work is valid, which can be used to obtain the nonconstrained modes of the system. The discretized dynamic model based on those nonconstrained modes is more accurate than that based on constrained modes. Moreover, the present model is closer to the real engineering than others, because the solar arrays in this model are composed of honeycomb panels rather than isotropic plates/beams. (2) The spacecraft flexibility and honeycomb panel parameters significantly influence the nonconstrained modes of the system. Therefore, for a flexible spacecraft with a particular flexibility, it is necessary to choose the optimal geometric parameters of honeycomb panels, used as the back boards of solar arrays, to make sure that the frequencies of the system are relatively larger. In addition, the differences between the nonconstrained modes and the constrained ones are growing as the spacecraft flexibility increases. Hence, to obtain a high-precision discretized dynamic model of a flexible spacecraft with large flexibility, the corresponding nonconstrained modes which can reflect the effect of spacecraft flexibility should be used. (3) The IS technique can effectively suppress the oscillation of attitude angular velocity and the vibration of solar array during the attitude maneuver, and reduce the residual vibration after maneuver. The PD control with IS only requires attitude angle and angular velocity measurement. This hybrid control scheme is very simple, and can ensure the achievement of attitude maneuver and vibration suppression for spacecraft with different flexibility.

qðx þ r0 Þ sinh kxdx þ mt ðL þ r0 Þ sinh kL

0

c3 ðkÞ ¼

ðL

c4 ðkÞ ¼

ðL

qðx þ r0 Þ sin kxdx þ mt ðL þ r0 Þ sin kL

0

F in Eq. (23) is a 4 4 matrix. The expressions of the elements of this matrix are given as follows: F11 ¼ Dk3 sinh kL "ð

mt ð L þ r0 Þx2 J

#

L

qð x þ r0 ÞR1 ð xÞdx þ mt ð L þ r0 ÞR1 ð LÞ þ mt x2 R1 ð LÞ

0


F12 ¼ Dk3 cosh kL "ð

#

L


0

F13 ¼ Dk3 sin kL "ð


#

L


0

F14 ¼ Dk3 cos kL "ð


L

#


0

F21 ¼ k2 cosh kL; F22 ¼ k2 sinh kL F23 ¼ k2 cos kL; F24 ¼ k2 sin kL F31 ¼

c1 ðkÞ c2 ðkÞ c3 ðkÞ c4 ðkÞ ; F32 ¼ k þ ; F33 ¼ ; F34 ¼ k þ JR JR JR JR r0 c1 ðkÞ r0 c2 ðkÞ ; F42 ¼ JR JR r0 c3 ðkÞ r0 c4 ðkÞ ¼1þ ; F44 ¼ JR JR

F41 ¼ 1 þ

This research has been supported by the National Natural Science Foundation of China (Grant No. 11472089). The authors are grateful to the reviewers for their helpful comments, and also to Dr. Yang Yang for his support on preparing the paper.

F43

Expressions of M, Kl , Kn , and U in discretized dynamic equations (Eq. (26)) are given by

Appendix The expressions for Ri ðxÞ ði ¼ 1; 2; 3; 4Þ in Eq. (22) are x þ r0 c1 ðkÞ; JR

R2 ð xÞ ¼ sinhkx þ

x þ r0 c2 ðkÞ JR

M¼q Kl ¼ D

ðL 0 ðL

½uðxÞT uðxÞdx þ I

x þ r0 c3 ðkÞ; JR

R4 ð xÞ ¼ sin kx þ

081008-10 / Vol. 138, AUGUST 2016

x þ r0 c4 ðkÞ JR

ðL

½u0 ðxÞT u0 ðxÞdx þ mt ½uðLÞT uðLÞ

0

½u0 ðxÞT u0 ðxÞdx

0

Kn ¼ H R3 ð xÞ ¼ cos kx þ

qðx þ r0 Þ cos kxdx þ mt ðL þ r0 Þ cos kL

0

Acknowledgment

R1 ð xÞ ¼ coshkx þ

ðL

U¼q

ðL

½u0 ðxÞT u0 ðxÞppT ½u0 ðxÞT u0 ðxÞdx

0 ðL

ðx þ r0 ÞuðxÞdx þ mt ðL þ r0 ÞuðLÞ

0

Transactions of the ASME


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References
