Planar rigid-flexible coupling spacecraft modeling and control considering solar array deployment and joint clearance

Accepted Manuscript Planar rigid-flexible coupling spacecraft modeling and control considering solar array deployment and joint clearance Yuanyuan Li, Zilu Wang, Cong Wang, Wenhu Huang PII:

S0094-5765(17)30896-2

DOI:

10.1016/j.actaastro.2017.10.008

Reference:

AA 6497

To appear in:

Acta Astronautica

Received Date: 30 June 2017 Revised Date:

20 September 2017

Accepted Date: 4 October 2017

Please cite this article as: Y. Li, Z. Wang, C. Wang, W. Huang, Planar rigid-flexible coupling spacecraft modeling and control considering solar array deployment and joint clearance, Acta Astronautica (2017), doi: 10.1016/j.actaastro.2017.10.008. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. 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.

ACCEPTED MANUSCRIPT

Yuanyuan Lia, Zilu Wangb, Cong Wanga* and Wenhu Huanga a

School of Astronautics, Harbin Institute of Technology, Harbin, P.R. China

b

School of Aerospace Engineering, Tsinghua University, Beijing, P.R. China

*Corresponding author email address: [email protected]

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ABSTRACT

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Planar Rigid-flexible coupling spacecraft modeling and control considering solar array deployment and joint clearance

Based on Nodal Coordinate Formulation (NCF) and Absolute Nodal Coordinate Formulation (ANCF), this paper establishes rigid-flexible coupling dynamic model of the spacecraft with large deployable solar arrays and multiple

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clearance joints to analyze and control the satellite attitude under deployment disturbance. Considering torque spring, close cable loop (CCL) configuration and latch mechanisms, a typical spacecraft composed of a rigid main-body described by NCF and two flexible panels described by ANCF is used as a demonstration case. Nonlinear contact force model and modified Coulomb friction model are selected to establish normal contact force and tangential friction model, respectively. Generalized elastic force are derived and all generalized forces are defined in the NCF-ANCF frame. The Newmark- β method is used to solve system equations of motion. The availability and superiority of the proposed model is verified through comparing with numerical co-simulations of Patran and ADAMS software. The numerical results reveal the

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effects of panel flexibility, joint clearance and their coupling on satellite attitude. The effects of clearance number, clearance size and clearance stiffness on satellite attitude are investigated. Furthermore, a proportional-differential (PD) attitude controller of spacecraft is designed to discuss the effect of attitude control on the dynamic responses of the whole system.

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1 Introduction

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Keywords: NCF-ANCF, Solar arrays, Clearance joint, Attitude control

Solar arrays, as vital spacecraft appendage, provides necessary power for the whole system. The success of deployment of solar array is the first task for a spacecraft in space and the failure would be a disaster for a space mission. Moreover, the deployment of large flexible solar arrays disturbs the spacecraft flight and has un-neglected effect on satellite attitude in orbit. Thus, to model the deployable solar array system and to study the effects on satellite attitude have deserved attentions from many researchers, which are the foundations of mechanism design, precision analysis and control system design. Paolo et al. [1] analyzed the attitude behavior of the spacecraft with one flexible panel under the sloshing motion described through a spherical pendulum model. Liu et al. [2] proposed a rigid-flexible-thermal coupling dynamic model for satellite multibody system. Alipour et al. [3] derived a precise compact dynamic model for an active stabilized spacecraft system with flexible members and then developed attitude control algorithm. Birhanu et al. [4] used ADAMS and ANSYS computer programs to simulate and model solar panel deployment and locking process, and the results demonstrated the effects on attitude of satellite. Li et al. [5] presented a modeling method for rigid solar array system considering joint

ACCEPTED MANUSCRIPT stick-slip friction and used the fuzzy adaptive PD control method to design attitude controller. Wallrapp et al. [6] used the multi-body program SIMPACK to simulate the deployment of a solar array three-dimensionally and to check the influence of the flexibility of the solar array on the solar generator motions. However, the dynamic model of rigid-flexible deployable solar arrays how to be established accurately is still a difficult issue, compared with simulated using commercial software or simply regarded as rigid multibody system. Moreover, these researches neglected effects of joint clearance that is inevitable due to manufacturing tolerances, imperfections, wear and material deformation. Besides, the emergence of the contact, collision, friction and impact in clearance joints will cause severe vibration and nonlinear dynamic responses of

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the deployable space structures [7].

Study of rigid-flexible multibody systems has been attracting more and more attentions over the past two decades. Absolute Nodal Coordinate Formulation (ANCF) [8] is an accurate, non-incremental finite element method to accurately deal with the dynamics of flexible multibody systems considering both large rotation and large deformation. ANCF defines elemental coordinates as the absolute displacements and global slopes, which forces the mass matrix of the system

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equations to remain constant and the centrifugal and Coriolis forces identically equal to zero [9-11]. And ANCF has witnessed a large number of theoretical and numerical developments and has applied to various fields [12-15]. To solve the rigid-flexible multibody systems, Tian et al. [16] combined ANCF with Natural Coordinate Formulation (NCF) to build

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dynamic models of rigid-flexible multibody systems. Cheng Liu et al. [17] introduced an alternative method to impose the concentrated moments at the joints and to evaluate the joint reaction moments for modeling a rigid-flexible system via the NCF-ANCF method. Similar to the concept of NCF, Shabana [18] employed the ANCF reference nodes (ANCF-RNs) to describe the rigid body. More applications about the ANCF-based method on the rigid-flexible multibody systems can be found in [19,20]. In view of these advantages of ANCF-based method, this paper established precise dynamic model of rigid-flexible coupling spacecraft system with multiple clearance joints via the NCF-ANCF method. On the other hand, the dynamic response of mechanism with clearance has become one of the key problems that need to be solved in mechanical engineering and aerospace engineering [21]. During the past two decades, many studies on the

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influence of the joint clearance in planar and spatial multibody mechanical systems have been conducted. Each well-known friction models [22-25] and contact force models [26,27] has its own distinctive pros and cons by comparing their fundamental characteristics and their performances. Flores et al. [28-30] compared the kinematic and dynamic responses of planar rigid multi-body systems with dry and lubricated friction. Erkaya et al. [31,32] used numerical and experimental methods to investigate effects of joint clearances on vibration and noise characteristics of mechanism. Tian et

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al. [16,33,34] presented a new computational methodology for modeling and analysis of planar flexible multibody systems with dry and lubricated revolute clearance joints based on ANCF, and proposed ElastoHydroDynamic (EHD) lubricated cylindrical joints to study the coupling dynamics of a geared multibody system [35]. Zhang et al. [36] investigated

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effects of various parameters on the displacement and velocity of jointed beams, such as beam stiffness, clearance and joint stiffness. Salahshoor et al. [37] used multiple scales method to conduct a vibration analysis of a mechanical system with multiple clearances. In addition, spatial revolute joints with radial and axial clearances were modeled to study their influences on kinematics and dynamics of multi-body systems [38-40]. Most studies treated a crank-slider mechanism with multiple clearance joints as an example to investigate the clearance effects [41-46].The dynamic behaviors of a crank-slider mechanism considering effects of joint clearance size, number of clearance joints, different clearance locations, contact stiffness and flexible components were discussed, and some relevant optimization methods to obtain a more stable behavior were proposed. Furthermore, research on solar array system considering clearance joint has been attracting attentions in recent years, but is still insufficient. Li et al. [47] and Zhang et al. [48] studied the dynamics of rigid solar array system with only one clearance joint. Considering panel flexibility and multiple clearance joints, Li et al. [49,50] further studied dynamics of deployable solar array system with multiple clearance joints and given some design suggestions of the whole system by using commercial software ADAMS. Hai-Quan Li et al [51] study the deployment and control of flexible solar array

ACCEPTED MANUSCRIPT system considering joint friction without normal contact force. However, to study the effects of joint clearance on attitude of solar array system with flexible panels, considering both contact force and friction force, is still insufficient. Therefore, aims to study the effects of joints clearance and components flexibility on the attitude of deployable solar array system, including the effects of clearance joints number, clearance size and contact stiffness, this paper establishes the rigid-flexible coupling spacecraft model based on NCF-ANCF. The superiority of this presented model is verified through comparing with the co-simulations of Patran and ADAMS software. Then attitude control of rigid-flexible spacecraft with deployable solar arrays and multiple clearance joints is designed and the effect of controller is discussed.

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The paper is organized as follows: section 2 establishes the model of solar arrays consists of torque springs, close cable loops and lock mechanisms. In section 3, both NCF and ANCF are briefly described. Mass matrix and generalized external force and generalized elastic force are derived under NCF-ANCF framework. Section 4 presents full description of the planar revolute clearance joint for the contact model according to nonlinear contact force model and modified Coulomb friction model. In section 5, using Newmark- β method to solve equations of motion for the planar flexible multibody

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systems with revolute clearance joints is described. Section 6 gives calculation parameters of the simulation model and design parameters of the controller. Section 7 shows the numerical results that verify the validity of proposed model, reveal the effects of flexibility and clearance on spacecraft attitude, discuss the effects of clearance number, clearance 8. Nomenclature

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size and local material on space attitude, and show the control effects. Finally, the main conclusions are drawn in Section

torque of mechanism (N ⋅ m)

K

stiffness ( N ⋅ m/rad )

l

distance between two wheels (m)

ϕ

relative rotation angle of two wheels (m)

φ

angle between the center line of two wheels and the belt (m)

θ

angle (rad)

A

rotation matrix

r

position vector in global coordinate

r

position vector in local coordinate

CP

coordinate transformation matrix

S

shape function

M M

qe

q

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fp

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T

concentrated force

mass matrix (kg)

torque (N ⋅ m)

generalized coordinates of element generalized coordinates of system

ρ

density (kg/m3)

Q

generalized force matrix (m)

k

curvature (m-1)

ε

strain

vt

relative tangent velocity (m/s)

d

eccentricity vector (m)

ACCEPTED MANUSCRIPT normal direction

t

tangential direction

δ

penetration depth (m)

Rb

radius of the bearing (m)

Rj

radius of the journal (m)

Ce

restitution coefficient

Fn

normal contact force (N)

E

Young’s modulus

u

Poisson’s ratio

δ& ( − )

initial impact velocity

Ft

tangential friction force (N)

cf

friction coefficient

cd

dynamic correction coefficient

Φq

Jacobi matrix of constraint equation

λ

Lagrange multiplier column matrix

C

damping matrix

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2 Model for solar array system

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n

A typical spacecraft system adopted in this paper is shown in Fig.1, which consists of a main-body and two solar panels connected by revolute joints. Torsion spring mechanism, CCL mechanism and latch mechanism are the essential and vital devices for the whole system, as shown in Fig.2, which play key roles in solar arrays deployment and normal operation

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process. The preloaded torsion springs located at each revolute joint provide the necessary diving force to deploy the arrays. The CCL configuration is applied to achieve the synchronous deployment of the two panels. Besides, latch mechanisms located at each revolute joint play a role of latching the arrays in the proper position. This paper considers both these two

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revolute joints applied in the system as clearance joints.

Fig. 1. Structure of solar array system.

ACCEPTED MANUSCRIPT F2

r1

F2 φ

ϕ

Tccl

Fcclr

r2

F1

Fcclt

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F1

Fig. 2. The three key mechanisms in solar array system.

2.1 Torsion spring mechanism

The preloaded torsion springs drive solar arrays deploy in a preset speed. The driving torque on the i th ( i =1,2) joint can be represented as[5,33]

i Tdrive = K s (θ pre − θ )

respectively. 2.2 CCL mechanism

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where K s is the torsion stiffness of torsion spring; θ pre and θ are the preload angle and practical angle of the i

(1) th

joint,

Folded arrays are released in orbit under the control of the flexible CCL configuration by synchronizing the deployment

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angles between panels during the deployment process. The CCL mechanism comprises synchronous wheels and pre-tensioned close-loop cables. Fig. 2(b) illustrates the forces and torques of a typical CCL. The tight-side tension F1 and

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slack-side tension F2 can be expressed by

F1 = F0 + (l - l0 )Kccl + ϕ r1Kccl F1 ≥ 0  F2 = F0 + (l - l0 )Kccl − ϕ r1Kccl F2 ≥ 0

(2)

where K c c l is the equivalent elastic stiffness; F 0 is the preload in the belt; l0 is the initial distance between two wheels;

l is the real distance under tension; ϕ is the relative rotation angle of two wheels; r1 and r2 are the radius of the wheels.

When the two arrays are not synchronous, the tight-side tension F1 and slack-side tension F2 will provide the radial force Fcclr , tangential force F c clt and torque Tccl . Then

 Fcclr = ( F1 + F2 ) cos φ   Fcclt = ( F1 + F2 )sin φ  2 Tccl = ( F1 − F2 )r1 = 2ϕ r1 Kccl

(3)

where φ is the angle between the center line of two wheels and the belt. CCL configurations synchronize the deployment angles of each panel by applying a passive control torque proportional to

ACCEPTED MANUSCRIPT the angle difference. Therefore, the torques of passive control (Fig 1.(c)) can be regarded as

Tccl1 = nϕ r12 Kccl = K1 (2θ1 − θ 2 )  2 Tccl2 = ϕ r1 Kccl = K 2 (θ1 − θ 2 )

(4)

where Tccl1 and T c c l 2 are the equivalent synchronous torques generated by the CCL; K 1 and K 2 are the equivalent torsion stiffness of wheels;

n

is the ratio of the wheels; θ1 and θ 2 are the deployment angles.

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2.3 Latch mechanism A schematic diagram of latch mechanism is shown in Fig. 2 (c). The joint connects two bodies A and B separately; the two bodies rotate relatively round the joint D. The pin E can move on the surface of C in the deployment process. The pine slide into the groove F until the deployment angle θi reaches the preset latch angle. Thus, the latch mechanism is activated and

the two bodies are latched. In addition, the latch angle between spacecraft and yoke is 0.5π , the angle between other π

.

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panels is

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A STEP function and a BISTOP function are introduced to present the equivalent moment T lo c k in the latch mechanism as  when the mechanism is unlocked T = 0  lock  (5)   when the mechanism is locked Tlock = STEP(θi , x1 , 0, x2 ,1) ×   BISTOP(θ i , θ&i , x3 , x4 , K bs , e, C , d )

STEP(θi , x1 , h1 , x2 , h 2 ) =

(6)

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if θi < x1 : 0  if x1 ≤ θi ≤ x2 : θi − x1 2 θi − x1  h1 + (h2 − h1 )( x − x ) (3 − 2 × x − x ) 2 1 2 1   if θi > x2 : 1

BISTOP(θi ,θ&i , x3 , x4 , Kbs , e, Cmax , d ) =

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 if θi < x3 :  e max( Kbs ( x3 − θi ) − θ&i step(θi , x3 - d , Cmax , x3 ,0),0)   if θi > x4 : min(−K (θ − x )e − θ& step(θ , x ,0, x - d , C ),0) bs i 4 i i 4 4 max 

(7)

where θi is the angle at the i th joint, θ&i the rotation velocity of the i th latch mechanism; Kbs and C max are the stiffness and damping coefficients of the corresponding latch mechanism, respectively;

e

is an exponent; d is the distance depth. The

STEP function approximates the Heaviside step function with a cubic polynomial increases from h 1 to h 2 . The BISTOP function allows free motion between x3and x 4 at which point the contact function begins to push the joint back towards the expect center angle. The specific details are set referencing literature [5,51]. Therefore, the forces acted on the whole solar array system under these three key mechanisms consist of two pairs of driving torques, a couple of equivalent synchronous torques and two groups of lock torques, as shown in Fig.3.

ACCEPTED MANUSCRIPT Tdrive2

Tdrive1

Tdrive2

TCCL

θ2

Tdrive1 θ1

Tlock1

Tlock2

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TCCL

Fig. 3. Torque analysis model.

3 Rigid-flexible multibody systems formulation

In this work, the spacecraft main-body is described as rigid part by using NCF and solar arrays are described as flexible

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parts by using ANCF.

∂ri

θi ∂x

Y

i

f

rP

P

uf

M ri

j

ri 2

-f rP

rj 2

rp

θk

rj

ri1

rj1

ri1

j

ri 2

ri

X

rj

P

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i

∂rj rj 2

∂x

rj1

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Fig.4. Planar NCF-based and ANCF-based elements.

3.1 Relevant formulations of rigid body described by NCF

For a planar rigid body shown in Fig. 4(a), its motion can be defined by using two points. The 4 global generalized coordinates of the rigid body can be expressed as

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qe = {ri

rj }

T

(8)

where ri and rj are the position vectors of the nodes i and j , respectively. The location of a generic point

where

A

of the element is defined by a position vector rp in the inertial system and rp in the

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moving system, so that

P

rp = ri + A rp = ri + c1 ( r j − ri ) + c 2 n

(9)

is the rotation matrix, c1 and c 2 are the components of the vector rp in the basis formed by the orthogonal

vectors rj − ri and

n,

the vector

n

follows the direction of Y. Then

 xp  1 − c c c −c2   ri  rp =   =  1 2 1    = CPqe  yp   −c2 1− c1 c2 c1  rj 

(10)

where C P is a coordinate transformation matrix. The mass matrix can be established as M = ρ ∫ C T C dv

(11)

v

When a concentrated force f p is applied at a point P of an element, the generalized external forces can be established as (12) Q ex = C Tp f p When a concentrated torque

M

is applied at an arbitrary point on the element, the torque may be replaced by an

ACCEPTED MANUSCRIPT and −f , acting on a plane perpendicular to the direction of M

equivalent pair of forces, f

and end of a unit vector u f . This unit vector is defined by uf =

, applied at the beginning

( r j − ri ) × M

(13)

|| ( r j − ri ) × M ||

and f = uf × M

(14)

Therefore the equivalent generalized force with respect to the natural coordinates can be established as Q ex = ( C Ti f − C Ti + uf f )

3.2 Relevant formulations of flexible body described by ANCF

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(15)

The planar ANCF-based deformable beam element with two nodes is shown in Fig. 4(b). The position vector rp of an arbitrary point

P

on the two dimensional beam element in the global coordinate system is given by

where S is the element shape function and it can be obtained S = [ S1 I 2

S2I2

S3I 2

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a0 + a1 x + a2 y + a3 x3  rp =  = Sqe 3   b0 + b1 x + b2 y + b3 x 

(16)

S4I2 ]

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x x x x x x x x x S1 = 1 − 3( )2 + 2( )3 , S 2 = l[ − 2( ) 2 + ( )3 ] , S3 = 3( ) 2 − 2( )3 , S 4 = l[ −( ) 2 + ( )3 ] l l l l l l l l l

(17)

where I2 is the identity matrix of size two and l denotes the element length.

The 8 global generalized coordinates of the deformable beam element can be expressed as ∂r j 1 ∂r j 2 T ∂r ∂r T qe = [ e1 , e2 , e3 , e4 , e5 , e6 , e7 , e8 ] =[ ri1 , ri 2 , i1 , i 2 , r j1 , r j 2 , , ] ∂x ∂x ∂x ∂x

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where x denotes the nodal coordinates in element local coordinate system. The vector [ rk1, rk 2 ] position coordinates defined in the global coordinate system. The vector

∂ rk ∂x

T

(18)

(k = i, j ) indicates the

is tangent to the beam centerline.

The mass matrix can be established as

M = ρ ∫ ST Sdv

is applied at the first node

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established as

τ

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If the a concentrated torque

(19)

v

i of a beam element, the generalized external torques can be

 −τ e4 Qex = 0 0 fi 2 

τ e3 fi

2

 0 0 0 0 

(20)

here fi = (e3 )2 + (e4 )2 .

According to Euler–Bernoulli beams hypothesis, the virtual work of the finite element can be expressed as l

l

0

0

δ W fe = − ∫ EAδε x 0σ x 0 dx − ∫ EI z

1 δ (| k |2 ) dx 2

(21)

where E is the modulus of elasticity, A is the cross-sectional area, and Iz the second moment of area. ε x 0 and k are respectively strain and curvature along the middle line.

ε x 0 can be expressed based on Green-Lagrangian strain tensor as 1  ∂r ∂r  ε x 0 = ( )T ( ) − 1 2  ∂x ∂x  And the precise curvature expression of the beam element is

(22)

ACCEPTED MANUSCRIPT ∂r ∂ 2 r × 2 k = ∂x 3∂x f here f =

(

(23)

∂r T ∂ r ) ( ). ∂x ∂x

Then,

1 2

ε x 0 = (qeT S'TS' qe − 1)

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k = ( S ' q e × S '' q e ) / f

(24)

3

| k | = ( q S S q e ) / f − ( q eT S 'T S ' ' q e ) 2 / f 2

T e

''T

''

4

The variational expressions can be obtained δε x 0 = δ qeT S 'T S ' q e

6

(25) (26) (27)

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1 δ (| k |2 ) = δ qeTS''TS'' qe / f 4 − 2qeTS''TS'' qeδ qeTS'TS'qe / f 6 − qeTS'' TS''qeδ qeT (S'TS'' + S' ' TS' )qe / f 6 + 3(qeTS'TS''qe )2 δ qeTS'TS' qe / f 8 2 (28)

Thus, the precise generalized elastic force of the beam element can be obtained l

l

0

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Q fe = −∫ EAε x 0S'TS'qe dx − ∫ EI z [S''T S''qe / f 4 − 2qeTS''TS'' qe S'TS' qe / f 6 0

(29)

− q S S qe (S S + S S )qe / f + 3(q S S qe ) S S qe / f ]d x T e

'T

''

'T

''T

''

6

'

T e

'T

''

2

'T

'

8

In order to simplify the calculation, the first term of flexible beam bending is retained to establish the generalized elastic force as

l

Q fe = − ∫ EAε x 0 S 'T S ' q e d x −

1l T T qe S ' S ' qe d x . l ∫0

here f =

f

4

l

EI z ∫ S ''T S ''d x q e

(30)

0

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0

1

4 Model for clearance joint based on NCF-ANCF

In order to bring the contact forces into the equations of motion for a rigid-flexible multibody system with revolute

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clearance joints, it is necessary to develop a mathematical model for revolute clearance joints in rigid-flexible multibody system. Fig.5 shows a revolute clearance joint in the NCF-ANCF-based framework. Fig.5 a shows position vector in the global coordinate system. Based on the NCF and ANCF, node

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indicate the center of the bearing and the center of the journal. The eccentricity vector

i and j respectively

d connects the centers of the

bearing and the journal, it can be calculated as The vectors

n

and

t

d = r j − ri

(31)

represent the normal and tangential directions to the collision surfaces between the bearing and the

journal. Then the unit eccentricity vector can be expressed by n=

d || d ||

Then the penetration depth due to local deformation is evaluated as δ =|| d || − c where

δ

is represented by the distance between

P

(32)

(33)

and Q in Fig. 5a . c is the radial clearance, c = R b − R j . R b and

R j denote the radius of the bearing and the radius of the journal, respectively. When the journal is not in contact with the

bearing, the eccentricity is smaller than the clearance and the penetration has a negative value. When the penetration has a value equal or greater than zero, the contact is established.

ACCEPTED MANUSCRIPT Points P and Q indicate the contact points on body bearing and journal, respectively. Then the position of P and Q can be expressed as rP = ri + nRb ,

rQ = rj + nR j .

(34)

The velocity of the contact points P and Q in the global coordinate system can be obtained by taking derivative of Eq. (34) as

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r&P = r&i + n& Rb , r&Q = r&j + n& R j .

r&j − r&i d& = where n& = . || d || || d ||

(35)

The contact-impact process often involves both relative normal velocity and tangential velocity, thus the relative normal velocity can be expressed as

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vn = [(rP − rQ )T n]n and the relative tangent velocity can be obtained by

vt = (r&P − r&Q ) − vn

(36) (37)

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Based on the NCF and ANCF, the contact-impact forces that applied on the contact point should be transformed into the forces that applied at the nodes, as shown in Fig. 5b. Thus, the torques that act at the bearing node i and journal node j can be evaluated respectively by

M i = Fti × Ri , M j = Ftj × R j .

(38)

When impact occurs, an appropriate contact law is applied and the resultant forces are introduced as generalized forces in the motion equations of the system, consisting of normal contact forces and tangential friction force. This paper selects Lankarani and Nikravesh model [53] to evaluate the normal contact forces, since energy dissipation is considered in

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addition to elastic force. Besides, this model is capable of producing accurate results when compared to other contact force estimation models [41]. It has been used for numerous studies [28-32] and has been validated through experimental studies [27]. The model is given as

Fn = Kδn [1 +

3(1 − Ce2 )δ& ] 4δ& ( − )

(39)

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where Fn is the normal contact force, δ is the indentation depth of the contacting bodies, δ& is the relative impact velocity and δ& ( − ) is the initial impact velocity. When the contact is accurately detected, the contact impact force is introduced. The detection of the instant of contact occurs when the sign of penetration changes between the two discrete moments in time,

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tn and t n +1 ,

δ (q (tn ))δ (q (tn +1 )) < 0

(40)

then δ& ( − ) is determined by δ&(t n +1 ) here. The exponent n =1.5 for metallic surfaces and the restitution coefficient here Ce = 0.9 . The equivalent stiffness K depends on the material properties and the shape of the contacting surfaces, which is given as: 1

 R j Rb  2 K=   3π (σ j + σ b )  R j − Rb  1 − uk2 (k = j, b) σk = π Ek 4

(41)

where Rb and R j are the radii of the bearing and journal, respectively. Ek and u k are the Young’s modulus and Poisson’s ratio associated with each sphere. Generally, Coulomb friction model is used to represent the friction response in impact and contact process. However, the

ACCEPTED MANUSCRIPT definition of the Coulomb’s friction law poses numerical difficulties when the relative tangential velocity is in the vicinity of zero. To prevent the numerical difficulties, in this study, a modified Coulomb’s friction law is used to define the friction effect, which has been used for numerous studies[30,31,40]. The influence of the Lund-Grenoble and the modified Coulomb’s friction models on the system dynamics is comparatively studied [34]. That the moments obtained via these two friction models are very close. The friction model is expressed as,

vt || vt ||

(42)

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Ft = −c f cd Fn

where c f is the friction coefficient, vt is the relative tangential velocity. And dynamic correction coefficient cd is given by

if ||v t || ≤ v0 if v0 ≤ ||v t || ≤ v1 if ||vt || ≥ v1

(43)

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0   || v || −v0 cd =  t  v1 − v0 1 

v0 and v1 are given tolerances for the tangential velocity. The parameters are adopted according to literature [34]. The

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dynamic correction factor cd prevents that the friction force changes direction in the presence of almost null values of the tangential velocity, which would be perceived by the integration algorithm as a dynamic response with high frequency contents, forcing it to reduce the time step size.

Mb

i

d

j rj

P

rP

Ftb

t Fnb

Q

rQ

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ri

t

δ

Mj Fnj

Ftj

n

(b) contact load

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(a) position vector in the global coordinate system

n

Fig.5. Clearance revolute joint model based on NCF-ANCF.

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5 Computational strategy to solve the equations of motion

Based on the NCF-ANCF, the assembly of the elements can be carried out by the traditional finite element method. The element nodal coordinates qe can be easily transformed into the rigid-flexible multibody system generalized coordinates q . The equations of motion for a constrained rigid-flexible multibody system can be expressed in a compact form as

M  T  Φq

Qex &&   ΦTq   q     =   0   λ   - Φq q& q q& - 2Φqt - Φ tt 

(44)

&& is the generalized accelerations vector column matrix, M , C and K are the generalized mass matrix, where q

damping matrix and stiffness matrix, respectively. Φq is the Jacobi matrix of constraint equation. Q ex is the generalized force matrix and λ is the Lagrange multiplier column matrix. In this study, Qe includes the driving torques, equivalent synchronous torques, lock torques and the contact force. After the relative penetration and velocity have been determined, the contact and friction forces can be computed and included in the DAE as externally applied loads. Then, the system dynamics can be determined. Using Eq. (33) and Eq. (40) to detect contact start, if yes, the contact forces are determined

ACCEPTED MANUSCRIPT by Eq.(39) and Eq.(42). If no, contact do not occur and the contact forces are zero. It should be noted that no kinematic constraint is introduced while describing the clearance joint.

q& 0

&& 0 q

γ

λ

β

1 && n +β ⋅ q && n +1 ) q n +1 =q n + ∆t ⋅ q& n + ∆t 2 ⋅ (( -β ) ⋅ q 2

&&n +1 = q

1

(q n +1 −q n ) −

1

q& n − (

1

&&n − 1)q

β ⋅ ∆t 2 β ⋅ ∆t 2β γ 1− γ 1− γ & && n q& n +1 = ( q − q ) − q − ( − 1) ⋅ ∆t ⋅ q β ⋅ ∆t 2 n +1 n β ⋅ ∆t n 2β

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&& n +1 + F (q n +1 ) − Q(q n +1 ) + (Φ qT λ )(q n +1 )   Mq rn +1 =  T  &&n +1 - Φ qq& q (q n +1 )q& n +1 - 2Φ qt (q n +1 ) - Φtt  Φ ( q  q n +1 )q

q n +1 , λ n +1 1

1

1 && n − 1)q 2β

M AN U

&& n +1 = q

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M , q0

β ⋅ ∆t 2

(q n +1 −q n ) −

β ⋅ ∆t

q& n − (

&& n + γ ⋅ ∆t ⋅ q && n +1 q& n +1 = q& n +(1 − γ ) ⋅ ∆t ⋅ q

Time=Time+∆t < Time tol

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Fig.6 The numerical iteration procedure for the Newmark- β method

In this paper, the Newmark- β method is applied to solve the systems of algebraic differential equations. Implicit solution is more advantageous to enhance calculating stability and avoids solving the complex generalized force jacobian matrix. The number of unknowns of equations of motion Eq. (43) involved in the Newmark- β method reduces by half

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compared with that involved in Baumgarte Stabilization Method (BSM). Because the solution procedure of BSM needs to convert the n second-order differential equations of motion into 2n first-order differential equations. Classical dynamic equations of motion for constrained mechanical multibody systems were compared in literature [54].

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Fig.6 shows the algorithm flow chart, where n denotes the n th iteration, ∆t is the time step. γ and β are the algorithm parameters, and A-stability is guaranteed for 2 β ≥ γ ≥ 1 / 2 . Here β = 1 / 4 and γ = 1 / 2 . The algebraic equations can alternatively be solved with qn +1 as primary unknowns, by substituting q&&n +1 and q&n +1 in terms of q n , q& n , q&&n and q&&n +1 .

6 Numerical model and control design

A deployable rigid-flexible coupled solar array system with multiple clearance joints is modeled based on the NCF-ANCF to analyze the spacecraft attitude. The physical parameters of main-body and panels are listed in Table 1. The radius of journal is set as rj =0.01m, and the radius of bearing is set as rb =0.0102m. The materials of journal and bearing are Steel and Brass, respectively. Parameters of torsion springs located at two joints are shown in Table 2. Coefficients of restitution Ce is 0.9 and contact stiffness K in clearance joints is 7.39 × 1010 N/ m

3

2

. The flexible panel is divided

into 6 elements, and the whole multibody system has 60 degrees of freedom based on NCF-ANCF in this paper. To prove the validity of this model, a rigid-flexible coupling solar array model is established under co-simulation of Patran and

ACCEPTED MANUSCRIPT ADAMS softwares. The flexible panel is established under the finite element software Patran, which enabled us to obtain the eigenfrequencies and eigenmodes of the panels. The first thirty-six modes with corresponding natural frequencies were chosen for the simulation of dynamic behavior of the solar array system with clearance. The finite element panel model is set up with 40 beam elements, whose four typical eigenmodes under Patran are shown in Fig.7. The flexible panels are then exported into the ADAMS to connect with rigid spacecraft main-body. Thus, we can obtain the rigid-flexible coupling model in the simulation software. Control strategy should be taken into account to adjust the attitude change of the spacecraft due to the influence of the attitude-adjusting. The PD control can be expressed as F(t) = K p ⋅ (eref − e ) + K d ⋅ (e&ref − e&)

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deployment of solar arrays. In this paper, classical PD control method is selected to design the controller of (45)

where eref and e are desired attitude and real attitude of the spacecraft main-body. e&ref and e& are time differential

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of eref and e ; the PD tuner proportional and differential gains K p and K d are chosen as Kp =1000 and K d =200.

(a) 1st order

(b) 2nd order

(c) 3rd order

(d) 4th order

Fig.7 Four typical eigenmodes of flexible panel in Patran.

Table 1. Physical parameters of the solar deployment system Width

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Length

Thickness

Material Density

Young’s Modulus

3

(m)

(m)

( kg/m )

(Pa)

Main-body

1

1

1

1500

/

Panel 1(2)

1.5

1

0.02

200

280e6

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(m)

Table 2. Parameters of torsion springs Torsional

Preloaded

stiffness

angle(rad)

( N ⋅ m/rad ) Torsion spring 1

0.1

0.75 π

Torsion spring 2

0.075

1.25 π

ACCEPTED MANUSCRIPT 7 Result and discussions 7.1 Model verification The validity of the rigid dynamics model is verified firstly through the comparison with ADAMS software. Fig.8 shows the angular displacements of the two solar panels during the deployment. Obviously, the proposed rigid model based on NCF may achieve the same results as ADAMS software. Thus, the correctness of rigid solar array model based on NCF could be proved. Numerical simulations were performed under the MATLAB R2013b on a desktop computer with 3.3 GHz CPU

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and 8 GB RAM. When the time step is 0.0002s for the total simulation setting 24s, the rigid system costs 15704s to obtain the results.

To verify the rigid-flexible coupling dynamics model based on NCF-ANCF, the angular displacement results of the deployable solar array system obtained by co-simulation of Patran and ADAMS softwares are compared with in Fig.9. From Fig.9 (a), comparing angular displacement results of NCF-ANCF and co-simulation under the same time step

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0.0002s, vibration hysteresis phenomenon of co-simulation result happens with time. Measures of reduce the analysis step and increase finite element number have been taken to improve the results accuracy. Both detailed time step and densified mesh skills are help to get more accurate numerical results. As shown in Fig.9 (a), the co-simulation results under the time

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step 0.00005s and with 60 elements are both closer to the NCF-ANCF results. The similar phenomenon is shown in Fig.9 (b) as well. Moreover, it can be seen that co-simulation results under the same time step 0.001s lost some vibration information of the flexible solar panel2. Thus, the superiority of rigid-flexible coupling solar array model based on NCF-ANCF could be proved, which involves more accurate and complete dynamic response of the rigid-flexible coupling

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1.5 1.2 0.9 0.6

NCF ADAMS

0.3 0.0 0

4

8

12 16 Time (s)

20

24

Angular displacement of panel 2 (rad)

1.8

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Angular displacement of panel 1 (rad)

multibody system comparing with co-simulation using Patran and ADAMS softwares.

3.0 2.4 1.8 1.2

NCF ADAMS

0.6 0.0 0

(a) Angular displacement of panel 1

4

8

12 16 Time (s)

20

24

(b) Angular displacement of panel 2

1.5

1.58

1.2 0.9

1.56

22

0.6

24

NCF-ANCF-0.0002 ADAMS-40ele-0.0002 ADAMS-40ele-0.00005 ADAMS-60ele-0.00005

0.3 0.0 0

4

8

12 Time (s)

16

20

(a) Angular displacement of panel 1

24

Angular displacement of panel 2 (rad)

Angular displacement of panel 1 (rad)

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Fig.8 Angular displacement of the rigid solar array system

3 3.15 3.14

2

3.13 15.0

15.3

15.6

15.9

16.2

NCF-ANCF-0.0002 ADAMS-40ele-0.0002 ADAMS-40ele-0.00005 ADAMS-60ele-0.00005

1

0 0

4

8

12 Time (s)

16

20

(b) Angular displacement of panel 2

Fig.9 Angular displacement of the rigid-flexbible coupling solar array system

24

ACCEPTED MANUSCRIPT 7.2 Effects on spacecraft attitude To study the effects of panel flexibility and joint clearance on the spacecraft attitude, four comparison models (rigid solar arrays with ideal joints, rigid solar arrays with clearance joints, rigid-flexible coupling solar arrays with ideal joints and rigid-flexible coupling solar arrays with clearance joints) are established, as shown in Fig.10. Obviously, flexibility of panels leads to impressive oscillation of spacecraft main-body after latch. Because elastic vibration of large flexible panel aggravates the asynchrony of deployment angles. As shown in Fig.11, amplitude of equivalent synchronous torque

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generated from CCL of rigid-flexible solar arrays model increases drastically, which is proportional to the anabatic angle difference according to Eq. (4). The violent fluctuation of equivalent synchronous torque acted on the main-body has significant effects on the attitude of spacecraft and makes concussion of displacement and angle displacement of spacecraft main-body. In addition, clearance joints intensify yaw of the spacecraft to some extent due to impact forces at imperfect joints, as shown in Fig.10 ((d)-(f)). Moreover, the effects of clearance joint on rigid-flexible multibody model is greater than that on rigid multibody model. Because, considering flexibility of solar panels, elastic vibration of large flexible as shown in Fig.12, which is the torque acted on the main-body.

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panels increases both the concussion degree of the whole system and instantaneous impact frequency at clearance joints, Moreover, the displacement and angle displacement of the model with clearance joints shows hysteresis property

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comparing with the model with ideal joints, especially the rigid-flexible model, as shown in Fig.10 ((d)-(f)). Because not only the friction forces at clearance joints delay the deployment process of the solar arrays; but also the flexible panel as suspension damper coupled with clearance decrease the adjustment sensitivity of the CCL mechanism. To prove it, Fast Fourier Transform (FFT) is implemented to transform the equivalent synchronous torques (in Fig.11) of CCL mechanism of rigid-flexible coupling models. Obviously, the vibration frequency of equivalent synchronous torques of model with clearance joints decreases comparing with the model with ideal joints, as shown in Fig.13.The decrease of vibration frequency of synchronous torque reflected at the whole system is the oscillation hysteresis quality of the spacecraft

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0.494

0.488 4

8

12 16 Time (s)

20

24

Displacement in X direction (m)

(a) Displacement in X direction 0.4900

rigid-ideal rigid-clearance flexible-ideal flexible-clearance

0.4895

0.4890 18

20 22 Time (s)

(d) Displacement in X direction (local amplification)

24

rigid-ideal rigid-clearance flexible-ideal flexible-clearance

0.000

-0.002 0

4

8

12 16 Time (s)

20

0.001

rigid-ideal rigid-clearance flexible-ideal flexible-clearance

0.4 0.3 0.2 0.1 0.0 0

4

8

12 16 Time (s)

20

24

(c) Angular displacement in Z direction

rigid-ideal rigid-clearance flexible-ideal flexible-clearance

0.000

0.5

24

(b) Displacement in Y direction Displacement in Y direction (m)

0.490

0

0.002

EP

0.492

Angular displacement in Z direction (rad)

0.496

0.004

Angular displacement in Z direction (rad)

rigid-ideal rigid-clearance flexible-ideal flexible-clearance

0.498

Displacement in Y direction (m)

0.500

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Displacement in X direction (m)

main-body and deployable solar arrays after latch.

-0.001

0.5

rigid-ideal rigid-clearance flexible-ideal flexible-clearance

0.4

-0.002 20

Time (s)

(e) Displacement in Y direction (local amplification)

24

20

(f)

Time (s)

22

24

Angular displacement in Z direction (local amplification)

ACCEPTED MANUSCRIPT Fig.10. Attitude of spacecraft main-body of different models.

4

150

3

rigid-ideal rigid-clearance flexible-ideal flexible-clearance

100

0 -1

rigid-ideal rigid-clearance flexible-ideal flexible-clearance

-2 -3

50 0 -50 -100 -150

-4 0

5

10

15

12

20

Time (s)

20

24

Fig.12. Torque acted on main-body of different models.

main-body

ideal joint

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Amplitude (Nm)

16 Time (s)

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Fig.11. Equivalent synchronous torque generated from CCL acted on

2.0 1.5 1.0 0.5 0.0 0.0

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1

Torque (Nm)

CCL torque (Nm)

2

0.5

1.0

1.5

3 2 1 0 0.0

clearance joint

0.5 1.0 Frequency (Hz)

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Amplitude (Nm)

Frequency (Hz)

1.5

Fig.13. Frequency spectrums of equivalent synchronous torques of rigid-lexible coupling model with ideal joints and with clearance jounts.

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7.3 Effects of clearance number

Fig.14 shows the results of four comparison models to study effect of clearance number and location on spacecraft attitude. The comparison models are rigid solar arrays with two ideal joints (0.0-0.0mm), rigid solar arrays with 0.2mm clearance

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at Joint 2 (0.0-0.2mm), rigid solar arrays with 0.2mm clearance at Joint 1 (0.2-0.0mm) and rigid solar arrays with both two 0.2mm clearance joints (0.2-0.2mm). Obviously, the yawing degree of the model with two clearance joints is the largest. Clearance number has effect on spacecraft attitude and the model considering all the multiple clearance joints has the largest impact on attitude of spacecraft main-body. Because impact force acted on spacecraft main-body of the model with multiple clearance joints is greater and denser than the impact force of the model with only one clearance-joint, as shown in Fig. 15. Besides, due to complex force condition of solar array system with clearance joints under the control of pre-loaded driving torque, CCL synchronization torque and lock torque, the clearance joint located at different joints has different effects on attitude of spacecraft main-body. Thus, the model with one clearance-joint cannot completely reveal force behavior of the solar array system with clearance joints. Considering all the multiple clearance joints is necessary to study the dynamic response of solar array system accurately.

0.4902 0.4900 0.0-0.0mm 0.0-0.2mm 0.2-0.0mm 0.2-0.2mm

0.4896 21

22

23

24

0.0-0.0mm 0.0-0.2mm 0.2-0.0mm 0.2-0.2mm

-0.0010 -0.0012 -0.0014 -0.0016 -0.0018 -0.0020 21

22

Time (s)

23

0.54 0.53 0.52 0.51 0.50 0.49 0.48 0.47 0.46

0.0-0.0mm 0.0-0.2mm 0.2-0.0mm 0.2-0.2mm

21

24

22

(a) Displacement in X direction

23

24

Time (s)

Time (s)

(b) Displacement in Y direction

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0.4898

-0.0008

Angular displacement in Z direction (rad)

Displacement in Y direction (m)

Displacement in X direction (m)

ACCEPTED MANUSCRIPT

(c) Angular displacement in Z direction

Fig.14. Effects of clearance number on attitude of spacecraft main-body.

75

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0.0-0.2mm 0.2-0.0mm 0.2-0.2mm

45 30 15 0 13

14

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Torque (Nm)

60

15

16 17 Time (s)

18

19

20

Fig.15. Torque acted on main-body of different clearance number models.

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7.4 Effects of clearance size Relative increments of yaw between clearance joints model and ideal joints model are obtained to study effect of clearance size on spacecraft attitude. Fig. 16 shows the relative increments of yaw of solar array system with 0.05mm clearance joints, 0.2 mm clearance joints and 0.5 mm clearance joints. Obviously, with the increase of clearance size, the yaw increases, especially after locked. Because the impact force generated by collision at clearance joint plays important

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role in post-lock phase. Besides, the larger clearance size results in larger equivalent contact stiffness according to Eq.(41) and larger relative impact velocity δ& applied in Eq.(39). Therefore, the larger clearance size leads to the lager impact

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force at clearance joints. Although the decrease of clearance size shortens collision distance, which brings intensive impact frequency. Fig. 17 shows the torque acted on main-body of different clearance size models. The main-body obtains greater

0.05mm 0.2mm 0.5mm

-5

6.0x10

-5

4.0x10

-5

2.0x10

0.0 0

4

8

12 16 Time (s)

20

(a) Relative yaw in X direction

24

-4

1.5x10

0.05mm 0.2mm 0.5mm

-4

1.0x10

-5

5.0x10

0.0 0

4

8

12 16 Time (s)

20

(b) Relative yaw in Y direction

24

Relative yaw in Z direction (rad)

-5

8.0x10

Relative yaw in Y direction (m)

Relative yaw in X direction (m)

torque with the increase of clearance size, which causes the larger yaw of spacecraft attitude. 0.012 0.05mm 0.2mm 0.5mm

0.009 0.006 0.003 0.000 0

4

8

12 16 Time (s)

20

(c) Relative yaw in Z direction

Fig.16. Effects of clearance size on attitude of spacecraft main-body.

24

ACCEPTED MANUSCRIPT 0.05mm 0.2mm 0.5mm

Torque (Nm)

40

20

18.0

18.3

18.6

18.9 19.2 Time (s)

19.5

19.8

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0

20.1

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Fig.17. Torque acted on main-body of different clearance size models.

7.5 Effects of local material

To study effect of local material of clearance joints on spacecraft attitude, relative increments of yaw of comparison

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models are shown in Fig.18. The local material of bearing and journal at clearance joints of these three comparison models are steel-brass, steel-steel and aluminium-aluminium, respectively. The local material determines the equivalent contact stiffness according to Eq.(41). The equivalent contact stiffness of steel-brass, steel-steel and aluminium-aluminium material are 7.39e10 N/ m

3

2

, 1.08e11 N/ m

3

2

and 3.85e10 N/ m

3

2

, respectively. Obviously, the solar array system with

steel material clearance joints has the maximum relative increments of yaw. Because the larger equivalent contact stiffness generates larger impact force at clearance joints, which disturb the attitude of spacecraft main-body. It can been

-4

1.0x10

-5

8.0x10

-5

6.0x10

-5 -5

2.0x10

0.0 0

4

8

12 16 Time (s)

20

24

(a) Relative yaw in X direction

steel-brass steel aluminium

-4

1.5x10

-4

1.0x10

-5

5.0x10

EP

4.0x10

-4

2.0x10

0.0 0

4

8

12 16 Time (s)

20

(b) Relative yaw in Y direction

24

Relative yaw in Z direction (rad)

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steel-brass steel aluminium

Relative yaw in Y direction (m)

-4

1.2x10

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Relative yaw in X direction (m)

seen from Fig. 19, the disturb torque acted on spacecraft main-body of steel material clearance joints is the largest.

0.015 steel-brass steel aluminium

0.010

0.005

0.000 0

4

8

12 16 Time (s)

20

(c) Relative yaw in Z direction

Fig.18. Effects of local material on attitude of spacecraft main-body

24

ACCEPTED MANUSCRIPT steel-brass steel aluminium

Torque (Nm)

20 15 10

0 18.6

18.8 19.0

19.2

19.4 19.6 19.8 Time (s)

20.0

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5

20.2

Fig.19. Torque acted on main-body of different local material models.

without control with PD control

0.492 0.488 0

4

8

12 16 Time (s)

20

24

(a) Displacement in X direction

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0.004 0.002 0.000 -0.002 0

4

8

12 16 Time (s)

20

0.6

Angular displacement in Z direction (rad)

0.496

without control with PD control

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0.500

Displacement in Y direction (m)

Displacement in X direction (m)

7.6 Control analysis for spacecraft attitude

0.4 0.3 0.2 0.1 0.0

0

24

(b) Displacement in Y direction

without control with PD control

0.5

4

8

12 16 Time (s)

20

24

(c) Angular displacement in Z direction

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Fig.20. Attitude of main-body of rigid models with and without control.

To study the control performance and control influence of the whole solar array system, four comparison models (rigid solar arrays with control and without control, and rigid-flexible coupling solar arrays with control and without control) are established. Fig.20 and Fig.21 show the attitude control results of main-body of rigid model and rigid-flexible coupling model, respectively. It can be seen that the controller is excellent through comparing the attitude of main-body with and

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without control. Furthermore, the controller influence the expected time of latch. As shown in Fig.22, controller acted on the main-body delays the latch time due to the effects on deployment angle between main-body and panel 1. Similarly, the controller also influence the equivalent synchronous torques of CCL mechanisms with the difference of deployment angles.

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FFT is applied to obtain the frequency spectrums of equivalent synchronous torques of rigid-flexible coupling model with control and without control, as shown in Fig.23. The vibration frequency of equivalent synchronous torques of model with control decreases. It means that the attitude control not only restrain the yaw of spacecraft, but also restrain the vibration of

0.500 without control with PD control

0.496 0.492 0.488 0

4

8

12 16 Time (s)

20

24

without control with PD control

0.004 0.002 0.000 -0.002 0

4

8

12 16 Time (s)

20

24

0.6 Angular displacement in Z direction (rad)

Displacement in X direction (m)

process.

Displacement in Y direction (m)

spacecraft main-body and solar arrays to some extent. It contributes to smooth and steady of the whole system in post-latch

without control with PD control

0.5 0.4 0.3 0.2 0.1 0.0 0

4

8

12 16 Time (s)

20

24

ACCEPTED MANUSCRIPT (a) Displacement in X direction

(b) Displacement in Y direction

(c) Angular displacement in Z direction

1.6 1.5 1.4

Non-control Control

10.811.211.612.012.4

0

4

8

12 16 Time (s)

20

24

2.0 1.5 1.0 0.5 0.0 0.0

1.5 1.0 0.5 0.0

1.6 1.5

Non-control Control

10.811.211.612.012.4

0

4

8

12 16 Time (s)

20

24

(b) Angular displacement of panel 1of rigid-flexible coupling model

1.5

1.5 1.0 0.5 0.0 0.0

Control

0.5 1.0 Frequency (Hz)

1.5

Fig.23. Frequency spectrums of equivalent synchronous torques of rigid-flexible coupling model with control and without control

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Fig.22. Angular displacement comparison between control and non-control models.

1.0

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1.4

0.5

Frequency (Hz)

Amplitude (Nm)

Angular displacement of panel 1 (rad)

(a) Angular displacement of panel 1of rigid model

Non-control

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1.5 1.0 0.5 0.0

Amplitude (Nm)

Angular displacement of panel 1 (rad)

Fig.21. Attitude of main-body of rigid-flexible coupling models with and without control.

.

8 Conclusions

This paper established the dynamic model of planar rigid-flexible coupling solar arrays considering multiple clearance joints through nodal coordinate method describing rigid body and absolute nodal coordinate method describing flexible body. The effect of the clearance revolute joints and large flexible panels on the spacecraft attitude has been investigated,

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and effect of attitude control on the whole system dynamic responses has been discussed. Firstly, the NCF-ANCF based dynamic model of rigid-flexible multibody system exhibits many good features, such as the constancy of the mass matrix of the derived dynamic equation, and taking the vectors rather than rotational coordinates to describe the rotation and deformation of the rigid-flexible bodies. All the proposed formulations, including generalized elastic force and all kinds of generalized external forces, were defined in the global frame to avoid the

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coordinate transformation. System elastic force was derived according to the Green-Lagrangian stress tensor. Besides, implicit Newmark- β method was used to solve the system equations of motion to assure solution stability. Then, the results obtained by using the co-simulation of Patran-ADAMS models with different element numbers and

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under different time steps were compared with to verify the effectiveness of the proposed rigid-flexible coupling multibody model in this paper. The response result from NCF-ANCF model is more accurate than that from co-simulation software model, which contains more vibration information. The model predicting accurately the dynamic responses of the solar array system, which is the basis of mechanism design, precision analysis and control system design.

In addition, panel flexibility has great effects on the satellite attitude and cause the spacecraft main-body violent oscillation, as a result of the enhanced amplitude of equivalent synchronous torque generated from CCL mechanism. Moreover, clearance intensified the satellite yaw due to the impact at clearance joint and lagged the vibration of spacecraft solar arrays, especially when the panel flexibility is considered. Besides, numerical results indicated that the yaw of the rigid-flexible system model with clearance joints is bigger than that of the stiff system model with clearance joints. Furthermore, clearance number and location have effects on the spacecraft attitude with deployable solar array system. Dynamical response in a mechanism with two clearance joints is not a simple superposition of that in a mechanism with one clearance joint. Therefore, in order to accurately simulate the dynamic behavior of a multibody system, all the joints

ACCEPTED MANUSCRIPT in it should be modeled as clearance joints. Besides, the clearance size and clearance stiffness are the important parameters that affect the dynamic responses of a mechanical system by affecting the impact force at clearance joints. The numerical results showed that a larger size clearance and a greater stiffness clearance caused higher peaks of impact force and worse characteristics of the systems.

Finally, the designed attitude controller showed significant control effect. Consequently, it effected the dynamic responses of the whole system through delaying the expectation latch time and controlling the vibration of the solar arrays. Besides, further research could be done on the spatial rigid-flexible solar array system considering joint clearance

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and wear problems. References

[1] Gasbarri P, Sabatini M, Pisculli A. Dynamic modelling and stability parametric analysis of a flexible spacecraft with fuel slosh. Acta Astronautica. 127 (2016) 141-159. Aerospace Science and Technology. 52 (2016) 102-114.

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[2] Liu J, Pan K. Rigid-flexible-thermal coupling dynamic formulation for satellite and plate multibody system. [3] Khalil Alipour · Payam Zarafshan · Asghar Ebrahimi. Dynamics modeling and attitude control of a flexible space

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1.Dynamic model of solar arrays with multiple clearance joints is established based on NCF-ANCF. 2.The rigid-flexible coupling model is verified by co-simulation of Patran and ADAMS softwares. 3.Effect of joint clearance and panel flexibility on satellite attitude are analyzed. 4.Effects of clearance number, clearance size and local stiffness on satellite attitude are studied. 5.Attitude controller is designed and the effect performance of controller is discussed.