/."...-..-.MAXIMUM' LlkËLIHOOD'.PARAMETER.-.
'IDENTIFICATION OF. FLEXIBLE SPACECRAFT
MAXIMUM LIKELIHOOD PARAMETER
IDENTIFICATION OF FLEXIBLE SPACECRAFT
M A X I M U M LIKELIHOOD PARAMETER
IDENTIFICATION OF FLEXIBLE
SPACECRAFT
Proefschrift
ter verkrijging van de graad van doctor aan de Technische Universiteit Delft, op gezag van de Rector Magnificus, prof.dr. J.M. Dirken, in het openbaar te verdedigen ten overstaan van een
commissie aangewezen door het College van Dekanen op dinsdag 31 maart 1987 te 14.00 uur
door
Ql PING CHU
geboren te Beijing, China
vliegtuigbouwkundig ingenieur
^'£GHN/SQNN
TR diss ^
1531
Samenstelling van de promotiecommissie Rector Magnificus
prof.dr.ir. Ö.H. Gerlach prof.dr.ir. O.H. Bosgra prof.dr. J. Arbocz prof.ir. B.P.Th. Veltman prof.ir. ,K.F. Wakker dr.ir. J.A. Mulder
STELLINGEN
i
Iliff en Maine verwaarlozen ten onrechte de correlatie tussen procesruis en meetruis in hun algorithme voor de berekening van maximum likelihood
schattingen van stabiliteits- en besturingsafgeleiden van vliegtuigen in die gevallen waarin versnellingsmeters deel uitmaken van het
intrumentatiesysteem.
Maine, R. and Iliff, K. "Formulation and Implementation of a Practical Algorithm for Parameter Estimation with Process and Measurement Noises", SIAM J. Appl. Math. Vol. M1 , No. 3, december 1981.
ii
In de Gauss-Newton methode voor het optimaliseren van de likelihood functie, wordt de Hessiaan van de Newton-Raphson methode vervangen door de positief-semi definiete informatiematrix, met als resultaat een aanzienlijke
besparing in rekentijd. Echter zou het beschikbaar zijn van de Hessiaan toch nuttig kunnen zijn in gevallen waarin de likelihood functie niet convex.
Mulder, J.A. "Design and Evaluation of Dynamic Flight Test Manoeuvres", proefschrift, Technische Universiteit Delft, October 1986.
iii
Parameter identificatie van eindige elementen modellen van dynamische responsies kunnen niet alleen toegepast worden op flexibele satellieten in een baan om de aarde doch ook op opgevouwen zonnepanelen gedurende
gesimuleerde lanceringen.
-proefschrift kunhen worden geïmplementeerd op personal computers, waardoor parameter identificatie thuis mogelijk is geworden.
Praktische rekenprogramma's voor parameter identificatie dienen interactief
te zijn.
VI
In veel praktische toepassingen van systeemparameter identificatie kan het systèemmodel significante niet-lineariteiten bevatten. Het zou daarom de moeite waard zijn om de in dit proefschrift ontwikkelde algorithmen uit te breiden tot het niet-lineaire geval. De likelihood functie zou dan met behulp van een 'extended Kalman' filter gegenereerd moeten worden.
vu
Het is betreurenswaardig dat ik nimmer de Chinese Muur bezocht, hoewel ik in Beijing ben geboren en bijna negenentwintig jaar in die streek heb gewoond.
31 maart 1987 Q.P- Chu
-PROPOSITIONS
i
I l i f f and Maine neglect e r r o n e o u s l y the c o r r e l a t i o n between p r o c e s s and m e a s u r e m e n t n o i s e i n t h e i r a l g o r i t h m for the c a l c u l a t i o n of maximum likelihood estimates of s t a b i l i t y - and c o n t r o l d e r i v a t i v e s of a i r c r a f t in those cases where accelerometers are part of the instrumentation system.
Maine, R. and I l i f f , K. " F o r m u l a t i o n and Implementation of a P r a c t i c a l Algorithm for Parameter E s t i m a t i o n with Process and Measurement n o i s e s " , SIAM J . Appl. Math. Vol. 41, No. 3, December, 1981.
i i
In the Gauss-Newton method for the optimization of the likelihood function, the Hessian matrix of the Newton-Raphson method is replaced by the positive-semi definite information matrix. The result is a considerable saving in computing time. However, knowledge of the Hessian matrix might still be use ful in cases where the likelihood function is not convex.
Mulder, J.A. "Design and Evaluation of Dynamic Flight Test Manoevers", Ph.D. thesis, Delft University of Technology, October 1986.
iii
Parameter i d e n t i f i c a t i o n of f i n i t e element s t r u c t u r a l models of dynamic r e s p o n s e s may be applied not only to flexible s a t e l l i t e s in o r b i t but also to folded solar arrays during simulated launch c o n d i t i o n s .
t h e s i s may be implemented on p e r s o n a l c o m p u t e r s , making parameter iden t i f i c a t i o n possible a t home.
v
P r a c t i c a l computer programs for p a r a m e t e r i d e n t i f i c a t i o n s h o u l d be i n t e r a c t i v e .
vi
In many practical applications of system parameter identification the system model may contain significant nonlinearities. It would be worthwhile, there fore, to extend the algorithms developed in the present thesis to the nonlinear case. The likelihood function should then be generated using an extended Kalman filter.
vii
It is a pity that I have never been to the China Wall, although I was born in Beijing and lived in that area for almost twenty-nine years.
To my wife Jian Ying To my grand mother To my parents
ABSTRACT
CHAPTER O LIST OF SYMBOLS AND COORDINATE REFERENCE
FRAME DEFINITIONS 1 0.1 List of Symbols 1 0.2 Coordinate Reference Frame Definitions 11
CHAPTER 1 A GENERAL INTRODUCTION 16 1.1 The Field of Investigation 16
1.2 Problem Statement 20 1.3 Organization of the Thesis 22
References 23
CHAPTER 2 MATHEMATICAL MODELLING OF FLEXIBLE SPACECRAFT DYNAMICS 27
2.1 I n t r o d u c t i o n 27 2.2 The K i n e t i c Energy of t h e F l e x i b l e S p a c e c r a f t 29
2 . 2 . 1 The k i n e t i c energy of the r i g i d main body 29 2 . 2 . 2 The k i n e t i c energy of the f l e x i b l e
s u b s t r u c t u r e s 35 2 . 2 . 3 The t o t a l k i n e t i c energy of t h e s p a c e c r a f t 51
2 . 3 The P o t e n t i a l Energy of the F l e x i b l e S p a c e c r a f t 52 2 . 3 . 1 The p o t e n t i a l energy of t h e undeformed s p a c e c r a f t 52
2 . 3 . 2 The p o t e n t i a l energy due t o the d e f o r m a t i o n s
of t h e f l e x i b l e s u b s t r u c t u r e s 54 2 . 3 . 3 The t o t a l p o t e n t i a l energy of the s p a c e c r a f t 58
2 . 4 A General Set of Linear D i f f e r e n t i a l E q u a t i o n s
of Motion of F l e x i b l e S p a c e c r a f t Dynamics 58 2.5 A P a r t i c u l a r Dynamical Model of a S p a c e c r a f t w i t h
Two Symmetrical Solar P a n e l s 62 2.5.1 Model idealization 62 2.5.2 Element and nodal numbering algorithm 64
i i
2.6 Parameters t o be I d e n t i f i e d i n the Dynamical Model 80 2 . 6 . 1 P a r a m e t e r s t o be i d e n t i f i e d i n t h e r i g i d main body 81 2 . 6 . 2 P a r a m e t e r s t o be i d e n t i f i e d i n t h e f l e x i b l e s u b s t r u c t u r e s 81 2 . 6 . 3 Parameters t o be i d e n t i f i e d i n t h e t o t a l dynamical model 82 References 84
CHAPTER 3 MAXIMUM LIKELIHOOD ESTIMATION OF FLEXIBLE
SPACECRAFT PARAMETERS 87
3-1 I n t r o d u c t i o n 87 3.2 The Mathematical Model t o be Used f o r
Maximum L i k e l i h o o d Parameter I d e n t i f i c a t i o n 89 3.2.1 I n s t u m e n t a t i o n systems and o u t p u t e q u a t i o n of f l e x i b l e s p a c e c r a f t 89 3 . 2 . 2 Order r e d u c t i o n of t h e f l e x i b l e s p a c e c r a f t dynamical model 91 3 . 2 . 3 M o d e l l i n g e r r o r c h a r a c t e r i s t i c s of t h e reduced o r d e r model 106 3 . 2 . 4 D i s c r e t i z a t i o n of t h e system model 112 3 . 2 . 5 P a r a m e t e r s t o be i d e n t i f i e d i n t h e t o t a l system model 115 3 . 3 Maximum L i k e l i h o o d Parameter I d e n t i f i c a t i o n
of Linear Systems with C o r r e l a t e d P r o c e s s
and Measurement Noises 116 3.4 O p t i m i z a t i o n Methods for ML Parameter I d e n t i f i c a t i o n 122
3.5 I d e n t i f i a b i l i t y of the F l e x i b l e S p a c e c r a f t P a r a m e t e r s 141 3.6 P r o p e r t i e s of t h e Maximum Likelihood Parameter E s t i m a t o r 1 43 3.7 S e n s i t i v i t y M a t r i x C a l c u l a t i o n s f o r t h e Reduced Order Model 144 References 154
CHAPTER 4 SIMULATION EXPERIMENTS AND RESULTS OF ML PARAMETER
IDENTIFICATION OF FLEXIBLE SPACECRAFT 160
4.2 A Summary of the Computer Programs for ML Estimation of the Parameters in Linear Sytems with Correlated
Process and Measurement Noises 162 4.3 Evaluation of Computer Program in a Simple
Simulation Test Case 167 4.3-1 Test example without correlated process
and measurement noises 167 4.3.2 Test example with correlated process
and measurement noises 177 4.4 Simulation of Flexible Spacecraft Motions and
Measurement Data Generation 183 4.5 The Selection of a Priori Values of the Parameters
in the Flexible Spacecraft Model 197 4.6 Simulation Results of the Estimation of the
Parameter in the Flexible Spacecraft Model 201 4.6.1 Simulation results of the estimation
of the parameters using the model with
the white process noises 201 4.6.2 Simulation results of the estimation
of the parameters using the model with
the colored process noises 213
CHAPTER 5 SUMMARY OF CONCLUSIONS AND SUGGESTIONS
FOR FURTHER RESEARCH 226 5.1 Summary of conclusions 226 5.2 Suggestions for Further Reseach 228
Reference 228
APPENDIX A THE LAGRANGE'S EQUATIONS FOR QUASI-COORDINATES 229
APPENDIX B THE MASS AND STIFFNESS MATRICES OF
A RECTANGULAR PLATE ELEMENT 240
APPENDIX C THE FISHER INFORMATION MATRIX OF
A DETERMINISTIC SYSTEM 249
ACKNOWLEDGEMENT
The r e s e a r c h r e p o r t e d in t h i s t h e s i s i s a project of the Subject Group of S t a b i l i t y and C o n t r o l of the F a c u l t y of Aerospace E n g i n e e r i n g , D e l f t University of Technology, under the supervision of p r o f . d r . i r . O.H. Gerlach, my promotor. Many friends and collaborators of t h i s faculty have contributed t o the r e a l i s a t i o n of t h i s t h e s i s . In p a r t i c u l a r , I would l i k e to acknow ledge:
d r . i r . J . A . Mulder and i r . P. Ph. van den Broek, my mentors, for t h e i r i n s p i r i n g and invaluable suggestions and the constant d i s c u s s i o n s which have been of much h e l p i n o b t a i n i n g a deeper u n d e r s t a n d i n g of the s t u d i e d problems,
mr. H. Lindenburg and mr. P . J . Verkerk for t h e i r kind help during the design of the software package,
mrs . J . van D e v e n t e r - G i l l e , miss V. van der Veen, mr. P . J . Cornelissen and i r . H. de Jonge for t h e i r friendly support during the m o d i f i c a t i o n s of the typed t h e s i s with the word processor,
mr. J . C . Eggens-Dijkshoorn who typed the manuscript and mr. J.A. Jongenelen and mr. W. Spee who drew the diagrams,
the Chinese Academy of Sciences and the Delft University of Technology for t h e i r support which makes the study in t h i s beautiful country p o s s i b l e .
Qi Ping Chu
f l e x i b l e spacecraft.
The f i n i t e element method i s applied to generate a l i n e a r mathematical model of a g e n e r a l f l e x i b l e s p a c e c r a f t in a r b i t r a r y o r b i t s . R e l a t i v e l y few p a r a m e t e r s govern the behaviour of the individual elements in the model. A subsequent reduction of the order of the model was o b t a i n e d by means of a q u a s i - s t a t i c a p p r o x i m a t i o n of t h e higher frequency c h a r a c t e r i s t i c modes. Different types of modelling e r r o r s r e s u l t i n g from the o r d e r r e d u c t i o n and t h e d i s c r e t i z a t i o n a r e d i s c u s s e d . S t o c h a s t i c models a r e proposed t o ap proximate these modelling e r r o r s . Consequently, the models t o be used for parameter e s i m a t i o n take the form of linear dynamical systems with unknown process noise. Some new algorithms for maximum likelihood parameter e s t i m a t i o n a r e d e v e l o p e d . These algorithms take account of possible c o r r e l a t i o n s between process and measurement n o i s e s . Simulated measurements a r e used t o v e r i f y the a p p l i c a b i l i t y of the maximum likelihood parameter estimation a l gorithms to the i d e n t i f i c a t i o n of f l e x i b l e s p a c e c r a f t . I t i s demonstrated t h a t for the p a r t i c u l a r f l e x i b l e spacecraft used in the simulation, a l l of the unknown parameters can be e s t i m a t e d from the s i m u l a t e d measurements simultaneously with the time h i s t o r i e s of the system model s t a t e s .
1
CHAPTER 0
LIST OF SYMBOLS AND COORDINATE REFERENCE FRAME DEFINITIONS
0.1 List of Symbols
A assembled coupling matrix for the rotational degrees of freedom of the rigid main body and the flexible substructure degrees of freedom defined by Eq. (2.62) a width of a plate elenment
A augmented system matrix defined by Eq. (3.49) A condensed coupling matrix defined by Eqs. (3«8) and
(3.183) .
A. coupling matrix associated with element i defined by Eq. (2.48)
a, matrix defined by Eq. (3-41)
A coupling matrix associated with the master degrees of freedom defined in Eq. (3.184)
A original high dimensional system matrix defined in Eq. (2.91)
A reduced system matrix defined in Eqs. (3.16) and (3.30)
A coupling matrix associated with the slave degrees of freedom defined in Eq. (3.184)
b length of a plate element
B augmented system input matrix defined by Eq. (3.51)
sag,
b. matrix defined by Eq. (3.42)
B original high dimensional system input matrix defined in Eq. (2.91)
B reduced order system input matrix defined in Eqs. (3.16) and (3.30)
shape function matrix r e l a t e d to the r o t a t i o n a l degrees of freedom
shape function matrix r e l a t e d t o the t r a n s l a t i o n a l degrees of freedom
observation matrix defined in Eq. (3.32) observation matrix defined in Eq. (3.32) observation matrix defined in Eq. (3.1) observation matrix defined in Eq. (3.1)
o r i g i n a l high dimensional damping matrix, augmented feed-formard matrix
o r i g i n a l high dimensional vector of the assembled degrees of freedom of f l e x i b l e s t r u c t u r e s
condensed damping matrix defined in Eqs. (3.10) and
( 3 . 1 8 1 )
displacement vector of the k node in the t o t a l displacement vector d of the f l e x i b l e substructures displacement vector at node k of element i
assembled displacement vector of element i
displacement vector r e l a t e d to the master degrees of freedom
rotational displacement vector of an element
displacement vector related to the slave degrees of
freedom
t r a n s l a t i o n a l displacement vector of an element shaping f i l t e r matrix defined in Eq. (3.^3) Young's modulus of a material
k i n e t i c energy of a spacecraft
k i n e t i c energy of f l e x i b l e substructures
3
k i n e t i c energy of the r i g i d main body of a spacecraft t o t a l p o t e n t i a l energy of a spacecraft
p o t e n t i a l energy due to s t r u c t u r a l deformations p o t e n t i a l energy of element i due t o deformations p o t e n t i a l energy of the undeformed spacecraft unknown disturbance vector defined in Eq. (3.30) t o t a l force vector acting on the f l e x i b l e
substructures defined in Eq. (2.84)
e x t e r n a l force vector on the r i g i d main body of a. spacecraft
damping force vector of f l e x i b l e s u b s t r u c t u r e s d i s t r i b u t e d force vector acting on a f i n i t e element defined in Eq. (2.69)
discrete force vector acting on a finite element defined in Eq. (2.74b)
displacement function matrix in the finite element method
original high dimensional system matrix reduced order system matrix
augmented noise input matrix
first order gradient vector of the likelihood function matrix defined in Eq. (3.44)
matrix defined in Eq. (3.44)
augmented observation matrix defined by Eq. (3.55) assumed input level
reduced order observation matrix defined by Eq. (3«40) spacecraft inertia matrix
inertia matrix of flexible substructures
inertia matrix defined by Eq. (2.52)
inertia matrix of element i defined by Eq. (2.51) original high dimensional stiffness matrix, steady state kalman matrix
nonsteady state Kalman gain matrix at time step k+1 condensed stifness matrix defined in Eqs. C3-11) and (3.180)
stiffness matrix of element i
stiffness matrix related to the master degrees of freedom
coupling matrix between K and K
=mm =ss
one stage prediction gain matrix in the Kalman filter for the system with correlated process and measurement noises
stiffness matrix related to the slave degrees of freedom
negative logarithm of the likelihood function o r i g i n a l high dimensional mass matrix defined by Eq. (2.60)
total number of degrees of freedom of flexible sub structures, dimension of the Fisher information
matrix, total numbr of parameters to be identified in the system
condensed mass matrix defined by Eqs. (3.9) and (3.179)
mass matrix of element i dimension of u
mass matrix related to the master degrees of freedom coupling matrix between M and M
=mm =ss number of the master degrees of freedom number of the slave degrees of freedom
5
mass matrix related to the slave degrees of freedom dimension of w
dimension of z
number of selected elements in the flexible substructures, number of observation samples number of nodes in an element
total number of nodes on flexible substructures n x 1 zero vector
n x n zeor matrix n x m zero matrix
steady state covariance matrix of the state estimates assembling matrix between the degrees of freedom of element i and the total degrees of freedom of flexible substructures
nonsteady state a priori covariance matrix of the state predicts
nonsteady state a posteriori covariance matrix of the state estimates
parameter vector to be identified in Chapter 2
coupling matrix between the translational diplacements and the rotational displacements of the undeformed spacecraft
number of total measured variables
coupling matrix between the translational and rotational displacements of the undeformed substructures
number of measured acceleration variables coupling matrix between the translational and rotational displacements of the rigid main body coupling submatrix defined in Eq. (2.5^)
number of measured translational displacements of the rigid main body
qv
qe
%
number of measured v e l o c i t i e s of the r i g i d main body number of measured a t t i t u d e angles of the r i g i d main body
number of measured angular v e l o c i t i e s of the r i g i d
r_ _ w , r -b.p.b ' =b,p,b
main body
R. , vector from the origin of the rigid body fixed -b,o,b
reference frame to the origin of the flexible
substructure local reference frame, represented in the rigid body fixed reference frame
3h ' ?K' vector from the origin of the rigid body fixed reference frame to the origin of the flexible
substructure local reference frame, represented in the flexible substructure local reference frame and the associated skew symmetric matrix of elements
vector from the origin of the rigid body fixed
reference frame to a particle in the rigid main body, represented in the rigid body fixed reference frame and the associated skew symmtriic matrix of elements Fisher information matrix in the i iteration
element of the Fisher information matrix at the i row and the j column
vector from the origin of the inertial reference frame to the origin of the rigid body fixed reference frame vector from the origin of the inertial reference frame to a particle in the rigid main body of spacecraft vector from the origin of the flexible substructure local reference frame to the undeformed particle, represented in the rigid body fixed reference frame vector from the origin of the flexible substructure local reference frame to the undeformed particle, represented in the flexible substructure local reference frame and the associated skew symmetric matrix of elements Si
Su
El.b r-i,p -o,po,b -o,po,l ' -o.po.l7 •po.p.b R -po,p,l
h
=b,l T =e =1,0 =o,b u U =n v V. V (k + 1 Ik) =e 'vector from the undeformed partiele to the deformed
partiele, represented in the rigid body fixed reference frame
vector from the undeformed partiele to the deformed
partiele, represented in the flexible substructure local reference frame
transformation matrix from the original high
dimensional vector of degrees of freedom to the condensed vector of degrees of freedom
thickness of a plate element
transformation matrix between the master degrees of
freedom and the slave degrees of freedom
external torque vector on the rigid main body of a
spacecraft
transformation matrix from the rigid body fixed
reference frame to the flexible substructure local reference frame
elementary transformation matrix defined in Eq. (3.3)
transformation matrix from the inertial reference
frame to the orbital reference frame defined by Eq. (0.2)
transformation matrix from the o r b i t a l reference frame to the r i g i d body fixed reference frame defined by Eq. (0.3)
d e t e r m i n i s t i c input vector n x n unit matrix
measurement noise vector defined by Eq. (3.57)
matrix defined in Eq. (2.^7)
steady state covariance matrix of the prediction
errors (innovations)
nonsteady state covariance matrix of the prediction
V c o v a r i a n c e m a t r i x of measurement n o i s e s d e f i n e d by Eq. ( 3 . 6 3 )
V c o v a r i a n c e m a t r i x of p r o c e s s n o i s e s defined by Eq. ( 3 - 5 6 )
V c o v a r i a n c e m a t r i x of process and measurement n o i s e s =wv
d e f i n e d by Eq. ( 3 . 6 5 )
V , covariance matrix of measurement noises defined in
=vi
Eq. (3.59)
v measurement n o i s e vector defined i n Eq. ( 3 . 3 2 )
W c o u p l i n g m a t r i x between the t r a n s l a t i o n a l degrees of freedom of the r i g i d main body and the f l e x i b l e
s u b s t r u c t u r e degrees of freedom defined by Eq. ( 2 . 6 3 ) w p r o c e s s n o i s e v e c t o r d e f i n e d by Eq. ( 3 . 5 3 ) w e s t i m a t e d p r o c e s s noise v e c t o r W. i s u b m a t r i x of m a t r i x W defined by Eq. ( 2 . 4 6 ) x ( k ) s t a t e v a r i a b l e v e c t o r a t t h e k time s t e p x ( k + l | k ) one s t a g e p r e d i c t i o n of s t a t e s x ( k + l | k + 1) s t a t e e s t i m a t e v e c t o r y ( k + 1 | k ) p r e d i c t i o n e r r o r ( i n n o v a t i o n ) v e c t o r jj(k) measurement v a r i a b l e v e c t o r a t t h e k time s t e p £ ( k + l | k ) one s t a g e p r e d i c t i o n o u t p u t v e c t o r z s t a t e v a r i a b l e v e c t o r of a s h a p i n g f i l t e r d e f i n e d i n Eq. ( 3 . 4 3 ) a damping c o e f f i c i e n t r e l a t e d t o t h e mass m a t r i x d e f i n e d i n Eq. ( 2 . 8 6 ) a ( i ) f a c t o r g e n e r a t e d by a l i n e m i n i m i z a t i o n s e a r c h i n t h e Gauss-Newton o p t i m i z a t i o n procedure a t t h e i i t e r a t i o n
9
damping coefficient related to the stiffness matrix defined in Eq. (2.86)
d e t e r m i n i s t i c input d i s t r i b u t i o n matrix defined by Eq. (3.68)
noise input distribution matrix defined by Eq. (3.69) offset angle of flexible appendages in Chapter 2 damping ratio of the i mode defined in Eq. (2.88) pitch angle of spacecraft rigid main body
parameter to be identified in Chapter 3, attitude
angle vector defined in Eqs. (2.82) to (2.84) maximum likelihood estimates of parameters
A. y.
j eigenvalue of Fisher infomation matrix Poisson r a t i o of a material
mass density of the material of f l e x i b l e substructures element of the variance matrix V « at the i row and
=v1
.th
j column
element of the variance matrix V at the i row and =w
. t h , j column
transition matix of the system matrix defined by Eq. (3.67)
r o l l angle of the spacecraft r i g i d main body matrix defined by Eq. (3.140)
yaw angle of the spacecraft r i g i d main body
angular velocity vector of the spacecraft r i g i d body fixed reference frame r e l a t i v e t o the i n e r t i a l
reference frame and the associated skew symmetric matrix of elements represented in the r i g i d body fixed reference frame.
angular velocity of the spacecraft o r b i t
skew symmetric matrix of w defined in Eqs. (2.82) t o (2.84)
ƒ dm integration with respect to mass within the i
Ei
element of a f l e x i b l e substructure
ƒ dm i n t e g r a t i o n with respect to mass within the r i g i d main
b
body of the spacecraft,
ƒ dv i n t e g r a t i o n with respect to volume within the i Ei N i-1 N
n
i-1element of a flexible substructure
summation of a. sine i = 1 till i = N
l
n a. multiplication of a. sine i = 1 till i = N
11
0.2 Coordinate Reference Frame Definitions
The reference frames to be used in t h i s study a r e as follows.
1. The r i g i d main body fixed reference frame F . .
We d e f i n e f i r s t a r i g i d main body fixed reference frame system, which i s represented by the unit vector t r i a d ( i . , j . , k . ) with origin 0. . This body a x i s system w i l l be assumed to be fixed t o the r i g i d main body of the f l e x i b l e s p a c e c r a f t . The origin 0. i s fixed in the r i g i d main body.
2. The o r b i t a l reference frame F . o
The o r b i t a l r e f e r e n c e frame i s d e f i n e d by t h e u n i t v e c t o r t r i a d ( i , j , k ). The origin of the o r b i t a l reference frame 0 i s located on the
o o o o nominal spacecraft o r b i t , i points to the o r b i t a l d i r e c t i o n , k p o i n t s to
t h e c e n t e r of t h e e a r t h and j completes the orthogonal system (normal to the o r b i t p l a n e ) .
3. The i n e r t i a l reference frame F . .
The fundamental coordinate system, t o which a l l motions must be r e f e r r e d , i s t h e i n e r t i a l reference frame. In i t s most general sense, i t i s a coordinate system fixed with respect to the s t a r s . However, p r a c t i c a l s i t u a t i o n s d i c t a t e o n l y , t h a t the i n e r t i a l frame be a r e f e r e n c e c o o r d i n a t e set which guarantees the required accuracy over the time i n t e r v a l of i n t e r e s t . For the problem c o n s i d e r e d in t h i s s t u d y , it i s s u f f i c i e n t to s e l e c t a coordinate frame with o r i g i n 0. at the origin of the o r b i t a l reference frame 0 and one a x i s directed along a fixed c e l e s t i c a l d i r e c t i o n , such as the f i r s t point of Aries, one other axis would be normal t o the o r b i t a l plane and t h e t h i r d a x i s would complete t h e o r t h o g o n a l set (in the orbit p l a n e ) . The i n e r t i a l reference frame can be represented by t h e u n i t v e c t o r t r i a d ( i . , j , k ) with o r i g i n 0. (see Fig. 0 . 1 ) .
In particular, when the discussed flexible spacecraft has symmetrical struc ture, the origin of the inertial reference frame can be considered as the
o r i g i n of the body fixed reference frame, i . e . , the t r a n s l a t i o n a l d i s p l a c e ment of the spacecraft from the o r i g i n of the o r b i t a l reference frame 0 to the o r i g i n of the body fixed reference frame 0. remains zero (see Ref. 7 and Chapter 2 i n t h i s s t u d y ) . This i n e r t i a l reference frame will be considered for a special symmetric f l e x i b l e s a t e l l i t e in t h i s study as well.
The t r a n s f o r m a t i o n s from the i n e r t i a l r e f e r e n c e frame F. to the o r b i t a l reference frame F and from the o r b i t a l reference frame F to the r i g i d body
o o fixed reference frame F. are now required. When defining the o r i e n t a t i o n of
a body with respect t o a r e f e r e n c e frame, a s e r i e s of pure r o t a t i o n s i s used, and t h i s r e s u l t s i n an orthogonal transformation. The associated r o t a tions are called Euler angles and they uniquely determine the o r i e n t a t i o n of the body. For the transformation from the o r b i t a l reference frame F to the r i g i d body f i x e d r e f e r e n c e frame F , the Euler a n g l e s a r e d e f i n e d a s follows:
1) r o t a t i o n about the k a x i s by the yaw a n g l e ip, p o s i t i v e i n t h e clockwise d i r e c t i o n ,
2) r o t a t i o n about the intermediate j axis by t h e pitch angle 6, p o s i t i v e in the clockwise d i r e c t i o n , and
3) r o t a t i o n about the f i n a l i axis by the r o l l angle <)>, p o s i t i v e i n t h e clockwise d i r e c t i o n .
The i n e r t i a l reference frame F. , the o r b i t reference frame F and t h e body fixed reference frame F. are presented in Fig. 0 . 1 .
The transformations can be obtained from Fig. 0.1 a s :
=o,b =i,o i .
l
13
where T. i s the transformation matrix between the i n e r t i a l reference frame = 1 , 0
F. and the o r b i t a l reference frame F , T . i s the transformation matrix be-1 o =o,b
tween the o r b i t a l r e f e r e n c e frame F and the r i g i d body f i x e d r e f e r e n c e frame F. , and i . j . k. and i . j . k. are the components of a unit vector in
= b b b b 1 1 i
the body f i x e d r e f e r e n c e frame and t h e i n e r t i a l r e f e r e n c e f r a m e , r e s p e c t e v e l y .
The matrix T. i s given by: = 1 , 0 =i,o COS (ü) t + 8 ) 0 0 0 sin(u t + 9 ) 0 0 0 1 0 sin(ü3 t + 6 ) 0 0 0 cos (u) t + 9 ) 0 0 ( 0 . 2 )
in which w i s the o r b i t a l angular velocity and 8 i s a constant angle. The matrix T . i s given by:
=o,b
T K = =o,b
cos8cos<(> cos8simp - s i n e sin<|)sin8cost|>-cos<|)sinijj sin<j>sin8sin<|;+cos<|>cosi(i sin<j>cos9 cos<{>sin8cos<ji+sin<j>sin8 cos<f>sin9sini|>-sin<|>cosijj cos<(icos8
( 0 . 3 )
If the Euler a n g l e s a r e very s m a l l , the transformation matrix T . can be simplified a s : =o,b 1 ¥ 6 * 1 - * -8
♦
1 ( 0 . H )i*. The f l e x i b l e s u b s t r u c t u r e l o c a l reference frame F .
To develop the e q u a t i o n s of motion of the f l e x i b l e substructures of the s p a c e c r a f t , we define c e r t a i n f l e x i b l e s u b s t r u c t u r e l o c a l r e f e r e n c e frames as follows.
For any f l e x i b l e s u b s t r u c t u r e , the origin 0.. i s f i x e d a t some p o s i t i o n i n t h e s u b s t r u c t u r e , and t h e u n i t v e c t o r s i , j and k p o i n t t o s u i t a b l e d i r e c t i o n s (see F i g . 0 . 3 ) .
The transformation from the body fixed reference frame Fb to the flexible
substructure local reference frame F.. is:
* T, b,l
-ï
b J'b ~kb _ — (0.5)where Ï . , j , and k are the components of a u n i t v e c t o r r e p r e s e n t e d in F1 and T denotes a transformation matrix between F and F. .
= b , 1 i o
I t should be noted here that in t h i s study t h e f i n i t e element method ( s e e Chapter 2) w i l l be applied to analyse the dynamical motion of the f l e x i b l e s u b s t r u c t u r e s . This means that f l e x i b l e s u b s t r u c t u r e local, reference frame,
in f a c t , w i l l be located in each element t o be discussed.
v J i > J o
Fig. 0.1 The inertial reference frame F., the orbital reference
frame F and the body fixed reference frame ? definitions,
15
Fig. 0.2 Transformation from the inertial reference frame F to the body fixed reference frame F .
Fig. 0.3 The flexible substructure local reference frame F, definition.
CHAPTER 1
A GENERAL INTRODUCTION
1.1 The F i e l d of I n v e s t i g a t i o n
In t h e e a r l y days of space e x p l o r a t i o n , when s p a c e c r a f t were s m a l l , mechani c a l l y s i m p l e and r e l a t i v e l y compact, they were c o n s i d e r e d a s r i g i d b o d i e s f o r t h e purpose of p r e d i c t i n g t h e i r dynamical behaviour i n s p a c e . This p r a c t i c e , however, was soon found t o be i n a d e q u a t e i n many c a s e s .
E x p l o r e r I , a s the c l a s s i c a l example, did not p e r s i s t i n t h e i n t e n d e d s t a t e of s p i n about i t s a x i s of symmetry, b u t soon tumbled o v e r . The e x p l a n a t i o n f o r t h i s a n o m a l o u s b e h a v i o u r l a y i n t h e f l e x i b i l i t y of t h e s m a l l w i r e t u r n s t i l e a n t e n n a s p r o t r u d i n g from t h e c y l i n d r i c a l h o u s i n g of t h e v e h i c l e , s e e [ 3 ] . Had t h e s a t e l l i t e been a t r u l y r i g i d body, i t would have m a i n t a i n e d t h e s p i n n i n g motion i m p a r t e d t o i t d u r i n g l a u n c h . Because of energy d i s s i p a t i o n induced by t h e motion of. t h e f l e x i b l e a n t e n n a s , the s p i n n i n g motion was u n s t a b l e and t h e s a t e l l i t e ended up r o t a t i n g about i t s symmetric a x i s , i . e . t h e a x i s which i s normal t o the l o n g i t u d i n a l a x i s .
Great s t r i d e s i n t h e u n d e r s t a n d i n g of the d y n a m i c s of f l e x i b l e s p a c e c r a f t have been made s i n c e 1 9 5 8 , when E x p l o r e r I gave an e a r l y warning of the dangers of t r e a t i n g space v e h i c l e s a s r i g i d b o d i e s . H o w e v e r , d e s p i t e t h e ' b e s t e f f o r t s of e n g i n e e r s , numerous s u r p r i s e performances have been g i v e n .
In a d d i t i o n t o E x p l o r e r I , t h e f o l l o w i n g s a t e l l i t e s , a r e e x a m p l e s .
- A l o u t t e I ( 1 9 6 2 ) , w h e r e r a p i d s p i n d e c a y due t o s o l a r t o r q u e on t h e t h e r m a l l y deformed v e h i c l e was e x p e r i e n c e d ,
- 1963-22A a g r a v i t y g r a d i e n t s t a b i l i s e d s p a c e c r a f t was s u b j e c t e d t o e x c e s s i v e l i b r a t i o n s , due t o boom bending a r i s i n g from s o l a r h e a t i n g , - Explorer XX ( 1 9 6 4 ) , which behaved i n t h e same way a s A l o u t t e I ,
- 0G0 I I I ( 1 9 6 6 ) , c o n t r o l l e d by r e a c t i o n w h e e l s , which developed e x c e s s i v e a t t i t u d e o s c i l l a t i o n s due t o c o n t r o l system i n t e r a c t i o n w i t h f l e x i b l e booms,
- 0V1-10 ( 1 9 6 6 ) , which behaved s i m i l a r l y t o 1963-22A, and
- TACSAT I ( 1 9 6 9 ) , a d u a l s p i n s t a b i l i z e d v e h i c l e , w h i c h d e m o n s t r a t e d a n u n e x p e c t e d l i m i t c y c l e d u e t o e n e r g y d i s s i p a t i o n i n t h e b e a r i n g a s s e m b l y .
17
The examples of ' f l e x i b l e ' s p a c e c r a f t quoted so far have been extreme in t h e i r behaviour. Nevertheless, they have indicated the meticulous a t t e n t i o n which must be brought t o bear on any new s p a c e c r a f t design i n terms of dynamic a n a l y s i s at the design s t a g e , if d i s a s t e r s are t o be avoided.
The c u r r e n t t r e n d i s toward i n c r e a s i n g l y f l e x i b l e s p a c e c r a f t , as i s i n d i cated by the following examples.
The Canadian Communication Technology S a t e l l i t e (CTS) c a r r i e s two solar panels 1.2m x 7.3m, to generate 1.2kw e l e c t r i c power. The Radio Astronomy Explorer (RAE) S a t e l l i t e used four 230m antennas for detecting low frequency s i g n a l s . Among the European spacecraft exhibiting f l e x i b i l i t y , GEOS, ISEE-B and OTS, can be m e n t i o n e d . From t h e s e examples i t may be concluded that modern spacecraft consist of s t r u c t u r a l subsystems, some e s s e n t i a l l y r i g i d , such a s the r i g i d main body of the s a t e l l i t e , and some f l e x i b l e , such as s o l a r a r r a y s , antennas, e t c . I t a l s o can be e x p e c t e d t h a t i n t h e f u t u r e l a r g e r and l a r g e r s p a c e c r a f t w i l l be placed in o r b i t . Due t o mass l i m i t a t i o n , these space vehicles will be extremely f l e x i b l e . Examples are the NASA space s t a t i o n and the European space s t a t i o n (COLUMBUS).
When s p a c e c r a f t a r e c o n s i d e r e d as f l e x i b l e space s t r u c t u r e s , the d i f f i c u l t i e s to analyse the system will be increased, because of the complexity of the vehicle dynamics. The main problem areas a r e :
1. Dynamical modelling of the given f l e x i b l e space s t r u c t u r e . 2. Determining the system u n c e r t a i n t i e s or parameters.
3- Designing the control system.
The problems of f l e x i b l e spacecraft will be of major concern i n space a t t i t u d e c o n t r o l in many forthcoming missions. Therefore, a l a r g e amount of work r e l a t e d t o the areas of modelling and control of f l e x i b l e spacecraft i s being done by many space r e s e a r c h e r s . The comprehensive survey publication by Modi, s e e [ 3 ] . a l o n e mentions over 200 p u b l i c a t i o n s a s s o c i a t e d w i t h m o d e l l i n g and c o n t r o l of f l e x i b l e s t r u c t u r e s in space. However, the iden
t i f i c a t i o n of parameters of models of f l e x i b l e s p a c e c r a f t i s a d d r e s s e d r e l a t i v e l y r a r e l y . T h e r e a p p e a r s t o be only a very s m a l l number of references in which t h i s p a r t i c u l a r i d e n t i f i c a t i o n problem i s a d d r e s s e d i n some d e t a i l .
The parameter i d e n t i f i c a t i o n of a f l e x i b l e spacecraft i s a subject as impor t a n t as the parameter i d e n t i f i c a t i o n of a a i r c r a f t , which has been studied since 1960's, see the works of Gerlach [1] and [2] (1964, 1970), Mulder [17]
and [18] (1973, 1979), Mehra [20] (1973), e t c . The reason for the importance of parametér i d e t i f i c a t i o n for a f l e x i b l e spacecraft i s , that the parameters of a f l e x i b l e spacecraft model might be changeable in space, s i n c e :
1 . the mass of the spacecraft may be dissipated due to the operation of the control system,
2. the s t r u c t u r e of the spacecraft may be deployed or r e c o n f i g u r e d a f t e r launch,
3. the shape of the s p a c e c r a f t , e . g . long beam antennas or o t h e r f l e x i b l e a p p e n d a g e s , may c h a n g e due t o the s u r r o u n d i n g c o n d i t i o n s of the spacecraft, such as imposed by s o l a r heating, e t c .
All the e f f e c t s , as metioned above, lead to changes of the parameters of the f l e x i b l e spacecraft system model. To design a control system of the s p a c e c r a f t , the system model to be used should be as accurate as p o s s i b l e , since the performance of the control system depends on the accuracy of the model. This means that the parameters in the system model should a l s o be a c c u r a t e . Many papers on general system i d e n t i f i c a t i o n have been presented. The works of Gupta ( 1 9 8 1 ) , H e n d r i c k s , Rajaram, Kamat and J u n k i n s (1981!), Taylor (1985), Chen, Kuo and Garba ( 1 9 8 3 ) , Heylen and van' Honacker (1983) and Berman (1979, 1983) are relevant t o the study presented in t h i s r e p o r t . Gupta [4] g i v e s an i n t r o d u c t i o n i n which the system i d e n t i f i c a t i o n of f l e x i b l e space s t r u c t u r e s i s mentioned. Hendricks [5] finds mass, s t i f f n e s s and damping m a t r i c e s , from the measurements of a c c e l e r a t i o n , v e l o c i t y and displacement a t a l l d e g r e e s of freedom, u s i n g t h e l e a s t squares method. Taylor, see [ 6 ] , uses the d e t e r m i n i s t i c maximum likelihood estimation method t o find the unknown parameters and suggests using p a r t i a l d i f f e r e n t i a l equa t i o n s to construct the dynamical model, in o r d e r t o reduce the number of unknown p a r a m e t e r s . Chen, s e e [ 7 ] , [ 8 ] , discusses an i t e r a t i v e approach, based on c a l c u l a t i n g the Jacobian matrix, which measures the s e n s i t i v i t i e s of the system p a r a m e t e r s . Berman, see [ 9 ] , [10], finds mass and s t i f f n e s s matrices ' n e a r e s t ' to the a p r i o r i model matrices, which s a t i s f y c e r t a i n o r t h o g o n a l i t y c o n s t r a i n t s with r e s p e c t to measured f r e q u e n c i e s and mode shapes. Heylen, see [ 1 2 ] , follows the approach of Berman t o o b t a i n a f i r s t c o r r e c t i o n t o the system m a t r i c e s . The correction i s augmented by a second modification, using eigenvalue s e n s i t i v i t i e s .
All the papers mentioned above, seek best estimates for the elements of the system mass, damping and s t i f f n e s s matrices or t h e c a n o n i c a l forms (modal
19
c o o r d i n a t e s ) for t h e s e m a t r i c e s . Unfortunately, when the o r i g i n a l f l e x i b l e s p a c e c r a f t of continua i s descretized by u s i n g t h e f i n i t e element method, t h e o r d e r of the dynamical models w i l l usually be very high (higher than 200 i s common), except for very simple s t r u c t u r e s with only few e l e m e n t s . T h e r e f o r e , l a r g e dimensional matrices of mass, damping and s t i f f n e s s will r e s u l t in the dynamical models of f l e x i b l e s p a c e c r a f t . If a l l elements of these matrices have to be estimated, the number of these parameters will be come unmanageable, see [ 1 ] , [ 5 ] , and [ 6 ] , even when the c a n o n i c a l forms
(modal coordinates) a r e used to reduce the number, see [ 6 ] .
Taylor, see [ 6 ] , suggests that the f l e x i b l e s p a c e c r a f t dynamics s h o u l d be d e s c r i b e d by p a r t i a l d i f f e r e n t i a l e q u a t i o n s , t o greately reduce the number of unknown parameters. But, the p a r t i a l d i f f e r e n t i a l e q u a t i o n s may be con v e r t e d t o ordinary d i f f e r e n t i a l equations and the d i s t r i b u t e d parameters in the p a r t i a l d i f f e r e n t i a l equation will be d i s c r e t i z e d a s a l a r g e number of c o n s t a n t p a r a m e t e r s during the conversion, see [ 2 6 ] . Moreover, most of the algorithms in parameter i d e n t i f i c a t i o n , s t a t e estimation and system optimal c o n t r o l design are based on the system model in ordinary d i f f e r e n t i a l equa t i o n form, see e . g . [ 4 ] , [ 5 ] , [ 6 ] , [ 7 ] , [ 8 ] , [ 9 ] , [ 1 0 ] , [ 1 1 ] , [ 1 2 ] , [ 2 4 ] , [ 2 5 ] , [ 2 6 ] , [28] and [ 2 9 ] ) .
On the other hand, from the control system design point of view, v e r y high order models w i l l always be very d i f f i c u l t to use to design the control sys tems, due to l i m i t a t i o n s in computer word l e n g t h , memory s i z e , speed and n u m e r i c a l a c c u r a c y . In other words, the order of the o r i g i n a l l y high order system must be reduced for control system design.
I f the o r d e r of the system i s reduced, c e r t a i n modelling e r r o r s will creep up in the reduced order model. The parameter e s t i m a t i o n a l g o r i t h m s must h a n d l e t h e system w i t h l a r g e modelling e r r o r s . The r e f e r e n c e s , mentioned above, a r e a l l working with the system model of high o r d e r and assume t h a t t h e r e are no modelling e r r o r s in the system model. The models to be used for parameter i d e n t i f i c a t i o n are actually d e t e r m i n i s t i c models of high o r d e r . However, these papers give a p o s s i b i l i t y to i d e n t i f y the parameters for high order system models. Once the parameters are obtained by u s i n g t h e s e a l g o r i t h m s f o r a high order system model, that model can be used to reduce the order of the system for system control design.
In a system model of reduced order, modelling e r r o r s are unavoidable. The i d e n t i f i c a t i o n algorithms, as used for d e t e r m i n i s t i c systems a d d r e s s e d i n
t h e l i t e r a t u r e , may not t r e a t t h i s p a r t i c u l a r problem since they only work well in systems without modelling e r r o r s or w i t h very small m o d e l l i n g e r r o r s , see [ 4 ] , [13] and [ 1 8 ] .
1.2 Problem Statement
From Section 1.1, i t may be concluded t h a t the area of parameter i d e n t i f i c a t i o n of f l e x i b l e spacecraft i s an important subject of r e s e a r c h . According t o the d i s c u s s i o n s i n S e c t i o n 1 . 1 , t h e main problems a d d r e s s e d i n t h e l i t e r a t u r e , possibly t o be solved in t h i s area are as follows:
1. The dynamical model of a f l e x i b l e s p a c e c r a f t should be a lower order system, not only for the parameter e s t i m a t i o n but a l s o for the s t a t e estimation and system control design.
2. The number of parameters to be estimated should be reasonably low, such that i t can be managed by the parameter estimation algorithm.
3. The parameter estimation algorithms must bé able t o handle the system with modelling e r r o r s r e s u l t i n g from order reduction. In other words the algorithms must be i n s e n s i t i v e t o modelling e r r o r s .
4. The c h a r a c t e r i s t i c s of the o b t a i n e d model should be a s u f f i c i e n t l y a c c u r a t e r e f l e c t i o n of the c h a r a c t e r i s t i c s of t h e r e a l f l e x i b l e s p a c e c r a f t . The c r i t e r i o n t o measure the correspondence should not only be based on the estimated model parameters but also on the system s t a t e estimates.
The above mentioned problems may be c h a r a c t e r i z e d as problems o c c u r i n g m a i n l y due t o c o m p u t a t i o n a l l i m i t a t i o n s i n the a p p l i c a t i o n of system
i d e n t i f i c a t i o n .
One p o s s i b l e c l a s s i f i c a t i o n of i d e n t i f i c a t i o n methods i s based on the type of implementation, see [ 1 3 ] , v i z . :
1. e x p l i c i t methods, and
2. parameter adjustment methods.
This c l a s s i f i c a t i o n i s useful from the computation point of view, e s p e c i a l l y with respect t o the c o s t s . In g e n e r a l , e x p l i c i t i d e n t i f i c a t i o n methods a r e l e s s accurate than parameter adjustment method, but are computationally more
21
e f f i c i e n t and some times are indispensable t o obtain i n i t i a l parameter e s t i m a t e s , see [ 1 4 ] ,
This study w i l l focus on the maximum l i k e l i h o o d e s t i m a t i o n method, one of the advanced parameter adjustment methods, to estimate the parameters of complex systems.
The method of maximum l i k e l i h o o d was developed by F i s h e r (1912, 1921). Although the basic ideas date back to Gauss (1809), the method has been fur t h e r expanded by many r e s e a r c h e r s over the l a s t 20 y e a r s , see e . g . [13]» [ 1 4 ] , [ 1 5 ] , [ 1 6 ] , [17], [ 1 8 ] , [19], [ 2 0 ] , [ 2 1 ] , and [ 2 2 ] .
The r e a s o n of using the maximum likelihood method to estimate the f l e x i b l e spacecraft parameter may be summarized as follows.
1. I t i s a very general method for parameter estimation (Aström [ 1 5 ] ) , since parameters esmated from t h i s method i n c l u d e s t r u c t u r a l p a r a m e t e r s of f l e x i b l e spacecraft, i n i t i a l conditions of system s t a t e variables as well as the noise s t a t i s t i c s (means, variances and covariances of measurements and p r o c e s s n o i s e s ) . The s t a t e estimation of a f l e x i b l e spacecraft can a l s o be obtained from a Kalman f i l t e r in the maximum likelihood approach. 2. The maximum likelihood method provides a lower bound on the variances of the estimated parameters and provides the models of the measurement and process noise disturbances for the means, variances and covariances. 3. Provided that there are no modelling e r r o r s or t h a t the modelling e r r o r s
a r e c o r r e c t l y modeled, the parameter e s t i m a t e s a r e c o n s i s t e n t and asymptotically unbiased, e f f i c i e n t and normal (Eykhoff [ 1 3 ] ) .
The e x t e n d e d Kalman f i l t e r may a l s o be used to estimate the parameters in the system model, see [ 4 ] . However, the properties of the process n o i s e and measurement n o i s e (means, v a r i a n c e s and covariances) can not be obtained from the extended Kalman f i l t e r .
However, the standard maximum likelihood estimation algorithm, as discussed i n the mentioned l i t e r a t u r e , can not be d i r e c t l y employed in the a p p l i c a t i o n t o f l e x i b l e spacecraft, since the measurement instruments or transducers in the f l e x i b l e s p a c e c r a f t may c o n s i s t of a c c e l e r o m e t e r s which d e t e c t t h e s p e c i f i c f o r c e s on t h e f l e x i b l e appendages rather than the pure kinematic a c c e l e r a t i o n s . The difference between the measured s p e c i f i c f o r c e and t h e r e q u i r e d a c c e l e r a t i o n i s the component of gravity along the s e n s i t i v e a x i s of the t r a n s d u c e r . Assuming the a t t i t u d e of the t r a n s d u c e r r e l a t i v e t o t h e l o c a l v e r t i c a l to vary only very slowly, f l u c t u a t i o n s in the s p e c i f i c force
w i l l be very n e a r l y i d e n t i c a l t o t h e f l u c t u a t i o n s in t h e k i n e m a t i c a c c e l e r a t i o n s . Therefore, in the following i t will be assumed that the a c -celerometers measure kinematic a c c e l e r a t i o n s rather than s p e c i f i c forces. When the s p e c i f i c f o r c e s on a f l e x i b l e spacecraft are used as part of the measurements, the process and measurement noises will be c o r r e l a t e d , as w i l l be shown in Chapter
3-Including t h i s p a r t i c u l a r problem in the f l e x i b l e spacecraft model, in t h i s report the following work will be done.
1. The f i n i t e element method i s applied to generate a mathematical model of a f l e x i b l e spacecraft. Relatively few parameters govern the behaviour of the individual elements in t h i s model. A reduction in the complexity of the c a l u l a t i o n s is obtained by carefully reducing the order of the model. Finally the parameters to be estimated are obtained by using the maximum likelihood estimation method.
2. Different types of modelling e r r o r s are d i s c u s s e d in some d e t a i l . They a r e due t o t h e o r d e r r e d u c t i o n and the d i s c r e t i z a t i o n . Some applicable modelling error models are given.
3. As mentioned above, a s u i t a b l e o r d e r r e d u c t i o n algorithm i s used, to reduce the order of the o r i g i n a l high dimensional dynamical model of t h e f l e x i b l e s p a c e c r a f t . The r e s u l t i s a reduced order model applicable to parameter i d e n t i f i c a t i o n .
4. Some new a l g o r i t h m s of maximum likelihood estimation are developed, to solve the problems associated with c o r r e l a t e d p r o c e s s and measurement noises, based on different c o n s i d e r a t i o n s .
A l i m i t a t i o n of t h i s s t u d y i s , t h a t o n l y l i n e a r s y s t e m models a r e c o n s i d e r e d . This means that only small p o s i t i o n and a t t i t u d e e r r o r s of the spacecraft, and small d e f o r m a t i o n s of t h e f l e x i b l e s u b s t r u c t u r e s of t h e spacecraft a r e discussed.
1 .3 Organization of the Thesis
The kernel of the t h e s i s c o n s i s t s of the Chapters 2 through 4. In Chapter 2, a mathematical model of general f l e x i b l e s p a c e c r a f t dynamics w i l l be o b t a i n e d , u s i n g f i n i t e element a n a l y s i s . To discuss the model in d e t a i l , a
23
p a r t i c u l a r s a t e l l i t e with two large f l e x i b l e solar panels w i l l be c o n s i d e r e d . Large numbers of parameters will occur. But they a l l depend r e l a t i v e l y few key paramters in the dynamical model. As a consequence the number of the p a r a m e t e r s to be e s t i m a t e d i n the model given i n t h i s Chapter remains manageable.
In Chapter 3. the order of the model discussed in Chapter 2 will be reduced by a s u i t a b l e algorithm. Modelling e r r o r s from the order r e d u c t i o n w i l l be i n v e s t i g a t e d . In t h i s c h a p t e r , t h r e e d i f f e r e n t a l g o r i t h m s , a s s o c i a t e d parameter estimation in a system with c o r r e l a t e d p r o c e s s and measurement n o i s e s , w i l l be obtained based on different c o n s i d e r a t i o n s .
In Chapter H, the description of computer programs of the t h r e e a l g o r i t h m s w i l l be g i v e n . They a r e followed by some t e s t c a c u l a t i o n s . Simulated measurement d a t a of the p a r t i c u l a r f l e x i b l e s p a c e c r a f t d i s c u s s e d i n Chapter 2 w i l l be generated for parameter e s t i m a t i o n . Two different models of modelling e r r o r s , caused by the d e s c r e t i z a t i o n and order r e d u c t i o n , w i l l be a p p l i e d t o t h e reduced system model and r e s u l t s of parameter estimation w i l l be given.
In a d d i t i o n , some a u x i l i a r y r e s u l t s o b t a i n e d from the main t e x t of t h i s report are given in appendices.
References
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' I d e n t i f i c a t i o n of L a r g e F l e x i b l e S t r u c t u r e M a s s / S t i f f n e s s and Damping from On-Orbit E x p e r i m e n t s ' , J.. Guidance and C o n t r o l , March-A p r i l , 1984, pp 244-245.
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1.7 C h e n , J . C . and Wada , B..K. ' M a t r i x P e r t u r b a t i o n f o r S t r u c t u r a l Dynamics A n a l y s i s ' , AIAA J o u r n a l , V o l . 15, No. 8, pp 1 0 9 5 - 1 1 0 0 , Aug., 1979.
1.8 Chen, J . C , Kuo, C.P. and Garba, J . A . , ' D i r e c t S t r u c t u r a l Parameter I d e n t i f i c a t i o n by model T e s t R e s u l t s ' , 24th S t r u c t u r e s , S t r u c t r a l Dynamics and M a t e r i a l s C o n f e r e n c e , AIAA/ASME/ASCE/AAS-, P a r t 2 of p r o c e e d i n g s , May, 2 - 4 , 1983.
1.9 Berman, A. and F l a n n e l y , W.G., ' T h e o r y of I n c o m p l e t e Models of Dynamic S t r u c t u r e s ' , AIAA J o u r n a l , Vol. 9, No. 8, Aug., 1971, pp
1481-1487.
1.10 B e r m a n , A. 'Mass M a t r i x C o r r e c t i o n Using a I n c o m p l e t e S e t of Measured M o d e s ' , AIAA J o u r n a l , V o l . 1 7 , No. 10, O c t . , 1 9 7 9 , pp
1147-1148.
1.11 Berman, A. and Nagy, E . J . 'Improvement of L a r g e A n a l y t i c a l Model U s i n g T e s t D a t a ' , AIAA J o u r n a l , V o l . 2 1 , No. 8, A u g . , 1983, pp
1168-1173.
1.12 H e y l e n , W. a n d Van H o n a c k e r , P. 'An Automated T e c h n i q u e f o r I m p r o v i n g Modal M a t r i c e s by Means of E x p e r i m e n t a l l y O b t a i n e d Dynamic D a t a ' , 2 4 t h S t r u c t u r e s , S t r u c t u r a l Dynamics and M a t e r i a l s C o n f e r e n c e , AIAA/ASME/ASCE/AAS, P a r t 2 of P r o c e e d i n g s , May, 2 - 4 ,
1983-1.13 E y k h o f f , P . ' S y s t e m I d e n t i f c a t i o n - P a r a m e t e r a n d S t a t e E s t i m a t i o n ' , John Wiley and Sons, 1974.
25
1.14 Akaike, H. ' C a n o n i c a l C o r r e l a t i o n A n a l y s i s of Time S e r i e s and t h e Use of an I n f o r m a t i o n C r i t e r i o n ' , In ' S y s t e m I d e n t i f i c a t i o n -Advances and Case S t u d i e s ' , Academic P r e s s , 2 7 - 9 6 .
1.15 A s t r ö m , K . J . 'Maximum L i k e l i h o o d and P r e d i c t i o n E r r o r M e t h o d s ' , In ' T r e n d s and P r o g r e s s i n S y s t e m I d e n t i f i c a t i o n ' , E d i t e d by P . Eykhoff, pp 145-168.
1.16 Nahi, N . E . , ' E s t i m a t i o n Theory and A p p l i c a t i o n s ' , John W i l e y and Sons I n c . 1969.
1.17 M u l d e r , J . A . ' A i r c r a f t P e r f o r m a n c e M e a s u r e m e n t i n N o n s t e a d y F l i g h t ' , I n t . P r o c . 3rd IFAC Symposium on ' I d e n t i f i c a t i o n and System Parameter E s t i m i n a t i o n ' , N o r t h - H o l l a n d , Amsterdam 1973 PP 1131-1145.
1.18 M u l d e r , J . A . ' A n a l y s i s of A i r c r a f t P e r f o r m a n c e S t a b i l i t y a n d C o n t r o l M e a s u r e m e n t s ' , AGARD L e c t u r e S e r i e s , No. 104 Parameter I d e n t i f i c a t i o n , 1979.
1.19 S o n n e v e l d , P. a n d Mulder, J . A . 'Development and I d e n t i f i c a t i o n of Multicompartment f o r t h e D i s t r i b u t i o n of A d r i a m y c i n i n t h e R a t ' , J o u r n a l of P h a m a c o k i n e t i c s and B i o h a r m a c e u t i c s , V o l . 9, No. 5 , 1981.
1 . 2 0 M e h r a , R.K. ' C a s e S t u d i e s i n A i r c r a f t Parameter I d e n t i f i c a t i o n ' , I n t . P r o c . 3rd IFAC Symp., The H a g u e / D e l f t , The N e t h e r l a n d s , 1 2 - 1 5 J u n e ,
1973-1.21 G u p t a , K.N. and M e h r a , R.K. ' C o m p u t a t i o n a l A s p e c t s of Maximum L i k e l i h o o d E s t i m a t i o n and Reduction i n S e n s i t i v i t y C a l c u l a t i o n s ' , IEEE T r a n s a c t i o n s on A u t o m a t i c C o n t r o l , V o l . A C - 1 9 , N o . 6 , December, 1974.
1.22 Sage, A.P. and Melsa, J . L . 'System I d e n t i f i c a t i o n ' , M c G r a w - H i l l , 1971.
1.23 Thieme, G ' A t t i t u d e Control of Large F l e x i b l e S p a c e c r a f t ' , AGARD-CP-350, 1984, pp 18-1 - 1 8 - 1 5 .
1.24 Wu, Y.W., R i c e , R . B . , J u a n g , J.N. ' C o n t r o l of Large F l e x i b l e Space S t r u c t u r e Using Pole Placement Design T e c h n i q u e s ' , J . Guidance and C o n t r o l , Vol. 4 No. 3, May-June, 1981 pp
298-303-1.25 K o s u t , R . L . , S a l z w e d e l , H. and Emami-Neaini, A. 'Robust C o n t r o l of F l e x i b l e S p a c e c r a f t ' , J . G u i d a n c e a n d C o n t r o l , V o l . 6 , No. 2 , M a r c h - A p r i l , 1983 PP 104-111.
1.26 Benhabib, R . J . , Iwens, R.P. and J a c k s o n , R.L. ' S t a b i l i t y of L a r g e S p a c e S t r u c t u r e C o n t r o l S y s t e m s U s i n g P o s i t i v i t y C o n c e p t s ' , J . Guidance and C o n t r o l Vol. 4, No. 5 , S e p t . - O c t . , 1981, pp 487-493. 1.27 H e r t o g , G. d e n , 'De V l a k k e E i g e n b e w e g i n g e n van een E e n v o u d i g e
S a t e l l i t e c o n f i g u r a t i e met F l e x i b e l e Z o n n e p a n e l e n ' , ( i n D u t c h ) M a s t e r t h e s i s , D e p a r t m e n t of A e r o s p a c e E n g i n e e r i n g , D e l f t U n i v e r s i t y of Technology, D e l f t , The N e t h e r l a n d s , 1984.
1.28 S c h a e c h t e r , D.B. 'Optimal Local C o n t r o l of F l e x i b l e S t r u c t u r e s ' , J . Guidance and C o n t r o l , Vol. 4, No. 1, J a n . - F e b . , 1981 pp 2 2 - 2 6 . 1.29 M e i r o v i t c h , L. a n d B a r u h , H. ' O p t i m a l Control of Damped F l e x i b l e
Gyroscopic S y s t e m ' , J . Guidance and C o n t r o l , Vol. 4, No. 2, M a r c h -A p r i l , 1981, pp 157-163.
27
CHAPTER 2
MATHEMATICAL MODELLING OF FLEXIBLE SPACECRAFT DYNAMICS
2.1 Introduction
As mentioned in Chapter 1 , t h i s study focuses on the parameter i d e n t i f i c a t i o n of f l e x i b l e s p a c e c r a f t . The p a r a m e t e r s to be i d e n t i f i e d occur in a m a t h e m a t i c a l model of t h e s p a c e c r a f t . To allow i d e n t i f i c a t i o n , a s u i t a b l e form of the mathematical model must be developed. The development of such a model i s the subject of the present c h a p t e r .
Accoding to [ 1 ] , mathematical models of f l e x i b l e spacecraft can t a k e e i t h e r of two d i f f e r e n t forms:
1) p a r t i a l d i f f e r e n t i a l equations with d i s t r i b u t e d model p a r a m e t e r s ( s e e , e . g . , [ 2 ] , [ 3 ] , [1] and [ 5 ] ) , and
2) ordinary d i f f e r e n t i a l equations ( s e e , e . g . , [6] and [ 7 ] ) .
In t h i s s t u d y , the second form w i l l be applied, since most algorithms in parameter i d e n t i f i c a t i o n , s t a t e e s t i m a t i o n and system c o n t r o l d e s i g n a r e based on system models in terms of o r d i n a r y d i f f e r e n t i a l equations (see, e . g . , [ 1 5 ] , [ 1 6 ] , [ 1 7 ] , [ 2 0 ] , [ 2 1 ] , [ 2 2 ] , [ 2 3 ] and [ 2 4 ] ) and p a r t i a l d i f f e r e n t i a l e q u a t i o n s may be converted to ordinary d i f f e r e n t i a l equations a s well ( s e e , e . g . , [ 1 9 ] ) .
One p o s s i b i l i t y to generate the mathematical model of a f l e x i b l e spacecraft in terms of ordinary d i f f e r e n t i a l e q u a t i o n s i s based on the a n a l y s i s of f l e x i b l e space s t r u c t u r e s using the ^ f i n i t e element method'. F i n i t e element techniques for s t r u c t u r a l analysis a r e widely used and a c c e p t e d . The b a s i c t h e o r y , developed and augmented by many researchers over the l a s t twenty y e a r s , is available in a number of comprehensive t e x t books, such a s [ 8 ] , [9] and [ 1 0 ] .
In the f i n i t e element method, a modified s t r u c t u r e system consisting of d i s c r e t e ( f i n i t e ) elements i s s u b s t i t u t e d for the a c t u a l continuum. There are s e v e r a l methods to obtain the e q u a t i o n s of motion of f l e x i b l e s p a c e c r a f t dynamics in f i n i t e element form. For example, L i k i n s [ 6 ] used Newton's method to obtain both r i g i d body and f l e x i b l e appendage equations of motion. Nguyen and Hughes [7] applied the xangular momentum p r i n c i p l e ' t o obtain the r i g i d body motion and used Lagrange's equation to e s t a b i l i s h expressions for
t h e f l e x i b l e a p p e n d a g e m o t i o n . The e q u a t i o n s of motion of f l e x i b l e spacecraft in t h i s chapter w i l l be more general as compared t o ,the work of Nguyen and Hughes [ 7 ] , in t h a t the t r a n s l a t i o n a l displacements of the r i g i d main body of the spacecraft will be included. Furthermore, the model t r e a t e d i n t h i s chapter can be applied for the spacecraft with any kind of f l e x i b l e substructures which a r e not necessarily t o be symmetrical s t r u c t u r e s s i n c e t h e r i g i d body f i x e d reference frame i s selected not to be the mass center of the r i g i d main body of the s p a c e c r a f t , see Section 0.2.
This chapter i s organized as follows.
To employ Lagrange's formulation of the equations of motion, expressions for the k i n e t i c energy and p o t e n t i a l energy must be obtained. They w i l l be given in Sections 2.2 and 2 . 3 . A g e n e r a l l i n e a r dynamical model of a f l e x i l e s p a c e c r a f t will be obtained in s e c t i o n 2 . 4 . The model as derived in s e c t i o n 2.4, is generic in the sense t h a t i t i s valid not only for t w o - d i m e n s i o n a l f l e x i b l e s t r u c t u r e s but a l s o f o r t h r e e - d i m e n s i o n a l f l e x i b l e s t r u c t u r e s . However, for a spacecraft with two-dimensional-symmetrical panels, the model can be s i m p l i f i e d . In t h i s s t u d y a s a t e l l i t e with two symmetrical s o l a r panels will also be examined. The model for t h i s particular s p a c e c r a f t w i l l be discussed in Section 2.5 in some d e t a i l .
The selection of the p a r a m e t e r s t o be i d e n t i f i e d i n f l e x i b l e s p a c e c r a f t dynamical models i s the f i n a l purpose of t h i s c h a p t e r . I t will be shown t h a t , u s i n g t h e p r o c e d u r e d e s c r i b e d i n t h i s c h a p t e r , t h e number of parameters to be i d e n t i f i e d can be reduced t o a r e a s o n a b l e value. This r e s u l t i s d i f f e r e n t from o t h e r r e c e n t l y a v a i l a b l e s t u d i e s i n which t h e a u t h o r s t r y t o e s t i m a t e a l l t h e elements in the so-called modal m a t r i c e s , for example, [15], [16] and [ 1 7 ] . The l a t t e r studies are expected to lead t o many d i f f i c u l t i e s because of the unmanageably large number of unknown model parameters, even if canonical forms are used [ 1 7 ] . In f a c t , i f the f i n i t e element method i s employed as the method t o obtain the dynamical model of a f l e x i b l e spacecraft, a large number of the vunknown parameters' mentioned i n [ 1 5 ] , [ 1 6 ] and [ 1 7 ] can be o b t a i n e d d i r e c t l y from the f i n i t e element a n a l y s i s . I t will be shown in Chapter 4, t h a t due to t h i s r e d u c t i o n of t h e number of unknown p a r a m e t e r s , the. d i f f i c u l t i e s with respect t o parameter i d e n t i f l a b i l i t y as addressed in [ 1 5 ] , [16] and [17] may be circumvented.
29
2.2 The Kinetic Energy of the Spacecraft
The kinetic energy of a flexible spacefraft consists of the kinetic energy of the rigid main body and the kinetic energy of the flexible substructures. In this section these two parts of the kinetic energy separately will be discussed separately.
2.2.1 The kinetic energy of the rigid main body
Considering a particle p with mass dm in the rigid main body, the kinetic energy of the particle relative to the assumed inertial reference frame, see Section 0.2, can be written as:
1 "T
dE, . = ^ r • r. dm , (2.1)
kb 2 -l,p -l,p
where dE. . is the k i n e t i c energy of the p a r t i c l e , r , i s a vector from the
K D — X f p
o r i g i n 0. in the i n e r t i a l reference frame F. to t h e d i s c u s s e d p a r t i c l e p (see F i g . 2.1) and the " * ' means d i f f e r e n t i a t i o n with respect t o time. From F i g . 2 . 1 , the vector r . can a l s o be decomposed a s :
~i ,p
r . = r . . + rw , (2.2)
- i , p - i , b - b , p
where r . . i s a vector from the o r i g i n 0, in the i n e r t i a l reference frame F. t o t h e o r i g i n 0. in the body fixed reference frame F. and r. i s a vector
b b -b,p from 0. to the discussed p a r t i c l e p (see F i g . 2 . 1 ) .
Since F. i s moving r e l a t i v e to F . , the d i f f e r e n t i a t i o n of rw with respect
b i ' -b,p K
t o time i n F. can be represented as:
