Nonlinear control of tethered satellite system oscillations

Nonlinear

Analysis,

Theory,

Methods

&Applications, Vol. 30, No. 6. pp. 3867-3878, 1997 Proc. 2nd World Congress of Nonlinear Analysrs 0 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved

0362-546X/97

PII:SO362-546X(97)00114-4

NONLINEAR

CONTROL

OF ‘TETHERED OSCXLLATIONS MONICA

Dipartimento

SATELLITE

$17.00 + 0.00

SYSTEM

PASCA

di Ingegneria Strutturale e Geotecnica, Universit& di Roma “La Sapienza” via Eudossiaua, 18 - 00184 Roma - Italy

Kq XV&S and phmses: Tethered satellite system, uonlinear plmov stability, averaging method

oscillations,

nonlinear

control, Lia-

ABSTRACT The control of in-plane transversal vibrations of a tethered satellite system by means of a longitudinal force exerted by subsatellite thrusts is analyzed. A one-mode model is derived for describing the inplane dynamics of such a system. On the basis of this model, two different nonlinear (and simple to be implemented) control laws are proposed for the stabilization of the tethered satellite system. The Liapunov stability theory is used to show the g,lobal asymptotic stability of the station-keeping phase of the tethered satellite system, subjected to either control laws. The effectiveness of the proposed control laws is shown by means of both analytical arguments and simulation runs. 1. INTRODUCTION The problem of control of a tethered satellite system (briefly TSS) has been addressed in many papers and works, since the idea of such a system was introduced in 1974 [l], especially in relation to the deployment and retrieval phases of such a system. As far as the deployment phase is concerned, the Small Expendable Deployment System (SEDS) missions have shown that tether librations can easily occur [2]: in the SEDS-I mission, the tether librations were not controlled and they reached a maximum value of 57” with respect to the local vertical; on the contrary, in the SEDS-II mission, the tether librations at the end of deployment were stabilized to within 4” by means of an active feedback control based on shortening and lengthening of the tether proportionally to the in-plane libration angle in order to dissipate energy at each libration cycle [3]. Less attention has been devoted to the control of the station-keeping phase; nevertheless, tether oscillations can still occur, being excited by external perturbation. Since the preliminary feasibility analyses [4], the problem has been addressed by considering a simple lienar modle of the tethered system, which included only rigid motion of an unextensible tether subjected to linear gravity graon the dient. Lately, Liangdong and Bainum [5] h ave pointed out the relevance of tether flexibility stability of the pendular motion, even controlled through length variation. In the present work, without having the aim of designing a control law for a specific mission, the control of nonlinear free oscillations during station-keeping phase is tackled! taking into account both tether mass and elasticity in the system model. The nonlinear dynamic boundary-value problem for a massive elastic tether subjected only to formulation, showing the rather gravity forces was studied by Wang et al. [61] with a Hamiltonian 3867

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complicated nature of this problem even in the absence of atmosphere; the station-keeping configurations were derived as relative equilibria corresponding to symmetry reduction: and their stability was attacked by applying the reduced energy-momentum method. An elastic continuous model has been adopted for the tether by the present author in [7] and the feee linear dynamics have been deeply studied. Within the linear framework, the longitudinal and transversal motions in the orbital plane are coupled by gyroscopic forces, while they are decoupled from the out-of-plane displacements. When a nonlinear kinematic model is considered, the 3D motion components are coupled by gyroscopic and elastic forces [8]. Due to the coupling from gyroscopic forces and from kinematic nonlinearities, a longitudinal control force applied at the subsatellite tether end can be effective in reducing transverse oscillations. Linear and nonlinear velocity feedback control laws have been tested in [9, 101 on coupled in-plane oscillations under in-plane and out-of-plane disturbances; that a linear velocity feedback, while not producing damping effects! modifies the nonlinear terms with satisfactory effects on the stability regions; on the contrary, a quadratic enhanced velocity feedback produces cubic damping terms which have been shown to be able practically to suppress system oscillations. In the present paper, a one-mode model is derived for describing the in-plane dynamics of such a system. According to this model and on the basis of previous researches, two different nonlinear (and simple to be implemented) control laws are proposed for the stabilization of the station-keeping phase of the tethered satellite system. The global asymptotic stability of the system, subjected to each of the proposed control laws, is demonstrated by means of the Liapunov stability theory. The rate of convergence of the proposed control laws is analyzed with reference to both analytical asymptotic evaluations (averaging method) and numerical simulations, with a good agreement. 2. EQU.4TIONS

OF MOTION

The mechanical system under consideration is a model of a tethered satellite system flying in high atmosphere, thus implying negligible atmospheric aerodynamic effects. The tether is modeled as an elastic continuum with mass under variablle tension, due to the variation of the gravity gradient, which is the difference between the gravitational and the centrifugal forces, while the orbiter and the subsatellite are modeled as two point masses [7]. Denote the orbiter mass by m.,, the subsatellite mass by mp, the tether length by E, and the linear mass density of the tether by p. Furthermore, in the present analysis, the case of “short” tethers (i.e. E 5 2 lo4 m) connected to a massive orbiter, such as the Shuttle in the TSS-1 mission configuration, is addressed; under these assumptions, the orbiter mass is much greater than the subsatellite and the tether ones and, consequently, such a value, of E implies that p < 1 and y N 1, where

(2.1) The center of mass of the system can therefore be approximated with the orbiter center of mass. Two reference frames, (OXYZ) and (SZYZ), are defined: (OXYZ) is an inertial fixed reference system with origin in the earth centroid, while (SJ~Z) is rotating with the orbiter; its origin is located at the orbiter center of mass. The orbiter is assumed to rotate in the XY plane at an invariant distance R, from the earth centroid, with constant angular velocity 0 (see Fig. 1). The systems dynamics are described in the rotating reference system (Szyz). For each time t E R, let Z(.?,t), c(s,t) and Z_(s,t) be the coordinates in the frame (Szz/z) of a tether infinitesimal element of curvilinear abscissa S E [0, E]; whence, (5(O) t), y(O, t), %(O: t)) and (Z(E, t), ?J(E,t), r(l, i)) are the coordinates in (SZYZ) of the orbiter and of the subsatellite, respectively. The steady-state

Second World

Fig.

configuration

of the system is characterized

Congress

1.

of Nonlinear

Analysts

3869

Fteference system

by

(,(S,t)ll/(S,t),t(S,t:l) = (cqS),O,O) . vtc 92,

(2.2)

where 20 (9) is the solution of the static problem depending on tether elasticity and gravity gradient. Under the assumption of small displaceme-nts (C(Z, t), V(S, i); G(S, t)) from the steady-state configuration (20(S). O,O), the coordinates (i(S, I?)%y(.F, i). Z(5, t)) o f a tether infinitesimal element of curvilinear abscissa S E [0, I] are .?(S, 2)

=

cio(S) + ii(i, i) ,

Q(,S,i)

=

6(&i)

qs,q

=

zo(9, i).

Time t, curvilinear abscissa S and displacement components dimensionless by means of the following definitions:

s := s. t:=nt, I ’

(2.3a)

1

(2.3b) (2.3~)

(11(9, t), 2;(9, t), I&( S, t)) are rendered

q(s) := ; Zo(ls) ,

u(s, t) := ; ii(h. ;,

1 , u(s: 1.):= f G(Es:;, , W(S,t):=CW(ES.;).

Let superscript ’ denote derivative with respect to curvilinear abscissa .s and denote derivative with respect to time t. If confusion camiot arise, explicit dependence on the argument variable will be omitted. As already determined in (11, 81 by means of the Lagrangian approach, the nonlinear equations of motion of the mechanical system under consideration are: ii-2ti-u-2

(1 +;xo)”

1 (?I’)2 + (uQ2 (v’12 + Cd2 ’ = o (2.4a) .- -ru2 IL-I2 - u’ 2 11’ ? ( ( 1)

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6+2Ti-v+

where <:=lIRo area. Equations

+

(1 +:x0)”

a2

*2

,.;

1

-iii

(( u,

--

(1+7;ro)3

Congress

_

of Nonlinear

(v’J2

+

Analysts

(WV2

J

_

u/2vl

’ =

1

()

(2.4b)

)>

1’1’:‘..;‘2)~,-~,2-,~/=~,

(2.4~)

((

and ct2:=pLR2E2/EA,

with E being the Young module

(2.4) have to be solved on the basis of the following

and A the tether

boundary

cross section

conditions:

u (0,t) = 0 ,

(2.5a)

0 (0, t) = 0 )

(2.5b)

w (0, t) = 0 )

(2.5~)

u (1, t)

;!1,t)--2u(1:t)-u(l,t)-2~+Ero(1))3

+y

u/(l 9t)+ (~‘(1,t))2+(,‘(lIt))2 2

( _

uI (1 t) _ (u’ (11 ill2 (

’

+ (w’ (1, t)j2 2

uI (1 t) 1

+ f

c

(t)

= o

(2.5d)

,

4

G (11t) + 25 (1, t) - ?J(11t) + cl-+ Ezo (1, # +“r

u’(l

t)>

((

(~‘(1,t))“+(w’)2(llt)

w (1, t) ti(lst)+

v’(1,t)-uu/2(1

2

+y

=o

l)

(~‘(1,t))2+(.W’(lrt))2

’

((

t)

1

v,(l t)-

(1+
t)v’(l

,I(;

2

)

(2.5e) ’

t) ’

-u’2 (1, t) 70’ (1, t)) = 0 where fC(t) is the dimensionless longitudinal control force applied at the subsatellite end, as defined in the following. Boundary conditions (2.!ja-2.5c) are the consequence of the smallness of the mass ratio p. The analysis of the linearized motion equations, derived from eqns. (2.4) together with boundary conditions (2.5), performed in [7], h as shown that the fundamental frequency of the longitudinal motion is much greater than the one of the transverse motion, while its amplitude is of greater order than that of V(S, t). As a consequence, the inertial and the gravity gradient terms can be neglected in eqn. (2.4a) and in eqn. (2.5d) and a condensation procedure can be adopted [8]; function ~(3, t) is computed by integration of the resulting equation, on the basis of corresponding boundary conditions, and substituted in eqns. (2.4) and (2.5), thus obtaining two integral-differential equations: i; -

[fv’]‘+ fcv”

w-

3Ezov

- ~s$[$,‘)2+(w’)2)dn

-

= 0 ,

[fw’]‘f(l-3
tti

(1,t)w’-70’

[:6&t)

.‘-u’~‘2tid,]’

s s 1 2ddu

(2.6a)

‘+f+“=O,

1

and the relevant

boundary

conditions:

21(0, t) = 0 ,

(2.7a)

w (0, t) = 0 ,

(2.7b)

Second World Congress of Nonlinear Analysts ii+yfu’-33E()v-

J,’ & ((u’)’

+ (w’)‘)

3871

da + 2 ti 71’ - y fc u’] s=l = 0 )

[ii,+yfw’+(1-3Ez0)w+21,zu’-yf,w’]s_l=0, where

(2.7~) (2.7d)

f(s):-02 l d(s) - l ’ v’sEp,q, zgq-

(2.8)

is the dimensionless tether tension. Furthermore, the same linear analysis [7] h as shown that the in-plane longitudinal and transverse oscillations are coupled by means of the gyroscopic forces, while the out-of-plane oscillation is decoupled from the in-plane ones. When non1inea.r terms are considered, the in-plane and out-of-plane oscillations are coupled; nevertheless, it can be observed that an in-plane oscillation can still occur by itself if no perturbation is given in the out-of-plane direction; the opposite does not hold. In the following, a one-degree-of-freedom model of the in-plane oscillation is considered. Further analysis will be necessary to take into account the effects of the modes neglected to derive the one-degree-offreedom model and of the coupling between the in- and out-of-plane vibrations. 3.

Consider

CONTROL

the following

AND

STABILl:TY

modal expansion

OF

IN-PLANE

of t,he in-plane

OSCILLATIONS

transversal

displacement:

where pi(s):=Aisin

(Xis)

, i E4 ) i 2 1)

(3.2)

with A, being suitable real normalization constants and Xi > 0 being the eigenvalues of the linearized form of eqns. (2.4) under boundary conditions (2.5) about the steady-state configuration, are the orthonormal eigenfunctions associated to the natural frequencies in, determined in [7]; in particular, the first mode (corresponding to Xi) is a pendular mode, while the successive ones are characterized by AZ N (i - 1) 7r, (1: E 2,1. > 1) and this relation’becomes even more true as the mode number i increases. For some 2. E 4, i 1 1, and for all t E R, a one-mode description of the mechanical system corresponds to the assumption that the difference o(s,t) - cpi(s)qi(t) is small in a certain norm, i.e. the spatial distribution of the transversal displacement V(S. t) is well approximated by gi(s)q%(t). Without sake of generality, q%(t)and i%(t) can be assumed to be measurable. As a matter of fact, if the transverse displacement and velocity of the end-point of the tether (i.e., ~(1, t) and V(1, t)) are measured, since ppi(l) # 0 and ~(21, t) N ppi(s)qi(t), then ni(t) E # and ii(t) P # are therefore measured. More efficient procedures can be adopted to determine the behavior of the generalized coordinates q?(t) and ii(t) from the measurements of the displacement and velocity of a selected point. The one-mode motion equation is determined by means of the Galerkin procedure, which leads to: i-z(t) + u,‘%(t)

+ a, ii(t)%(t)

+ c&%(t) fc(t) = 0 ,

where ai = ht/m,zr and mi, bi and c, are suitable constants depending on the considered whose expressions can be found in [9], apart from ci = s,’ cp:(s)cp:(s)ds . For the sake of simplicity, index %will be omitted in the following on variable q.

(3.3) eigenfunction,

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The control laws proposed in the present tethered system, are the following: f=(t)

of Nonlinear

paper

Yc q(t)i(t)

=

for the stabilization

- G(t)

Yc sign [&i(t)1

fee(t) =

Analysts

of the first modes of the

(3.4a)

3

- Q(t)

cz

(3.4b)

,

where yc is a real constant which will be chosen properly in order to guarantee the asymptotic stability of the closed-loop system, and fast transient response. The two closed-loop systems obtained by the feedback connection of eqn. (3.3) with eqns. (3.4) are (3.5a) (3.5b) respectively. For both the closed-loop systems (3.5), the stability properties are stated subsequent Theorem 1 by means of the same candidate Liapunov function: v = $,2+

notice that V is a globally

positive

definite

g2;

and radially

Compute

the total time derivative

in the

(3.6)

unbounded

THEOREM 1 For each positive yc, (q, 4) = 1(0,0) is a globally of both closed-loop systems (3.5). Proof.

and proven

function

asymptotically

of V along the dynamics

of q and q’. stable equilibrium

of (3.5a) and (3.5b),

point

respectively:

ri = --Yc IQ-l2 li12,

(3.7a)

G

(3.7b)

=

--Yclql lil

From equations (3.7), it follows in both cases that v is globally negative semi-definite. Thus, taking advantage from the classical stability theorems, the following statements can be given. 1. Since V is globally positive definite and e is globally negative semi-definite, Stability Theorem of Liapunov (Theorem 25.1 of Hahn [12]), the equilibrium point of both (3.5a) and (3.5b) is stable.

by the First (q, 4) = (0: 0)

2. Since the largest invariant subset M of the set S:= {(q, i) : e = O> is constituted by the only point (q, 4) = (0,O) for both closed-loop systems (3.5), then-by Theorem 26.1 of [12], the point (q, 4) = (0,O) is attractive, whence, by Theorem 26.2 of [12] (q, 4) = (0,O) is an asymptotically stable equilibrium point of (3.5a) and1 (3.5b). 3. Since V is radially unbounded, then by Theorem is globally asymptotically stable.

26.3 of [12], the equilibrium

point

(q, 4) = (0,O)

0

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Table 1. S!/stern parameters 0 ( set-’

6.652

IO6

1.1624

EA )

1O-3

(21

1

(kg

(II,,

10,j

2

4. PERFORMANCE

lo4

5.76

k-1)

1O-3

WI

2.8

lo5

500

ANALYSIS

In order to evaluate the effectiveness of the proposed control laws in terms of convergence velocity, numerical analyses have been performed. The mechanical parameters of the system under analysis) which resemble those of the NASA-AS1 TSS-1 and TSS-1R projects, are given in Table 1; the corresponding values of the dimensionless parameters are N 2 = 1.111. lo-’ and y = 0.2304, having introduced the approximation ,3 = 0, previously discussed. It must be pointed out that, when the analysis is limited to the one-mode model previously described, and the orthonormal eigenfuctions am,, as derived in [7], and whose time frequencies and eigenvalues are listed in Table 2 for the first three modes, are considered, coefficients o, in eqn. (3.3) are identically zero. Therefore, if only quadratic nonlinearities are retained, a single mode under no control behaves according to the linear equation of motion. Consequently, if an in-plane perturbation is given to the undisturbed linear configuration along the local vertical, the system oscillates with stationary periodic vibrations with an amplitude related to the imposed initial conditions. As the proposed control laws, which have been shown to be able to make the equilibrium configuration (4, 4) = (0.0) a globally asymptotically stable point, are applied, the system oscillations due to an initial perturbation are damped down to the stationary position. The corresponding convergence velocity depends not only on the value of the control gain, but also on the control law. Table 3 lists the test cases here presented. For both control laws. the first two in-plane modes and two different control coefficient values are reported. As already pointed out, the first mode corresponds to an almost pendular motion while the second mode is close to the first mode of an elastic taut string. Figure 2 shows the different time histories for the two values of the control gain yc (case A and case B in Table 3) when the control law (3.4a) is applied, starting from the same initial disturbance. It can be observed that this type of control implies a strong initial reduction of the oscillation amplitude followed by a slow decreasing of the same amplitude towards the stationary position. The same behavior is recovered if the second mode is considered (Fig 3); as the first mode is characterized by

Table 2. Liiiear freqliencies and eigenvalues

1.73097 12.03152 23.66345

0.46233 3.21317 6.31963

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control

law

mode

Congress

of Nonlinear

3. Test

Analysts

parameters

control

Yc

law

mode

eqn. (3.4a) eqn. (3.4a)

2 2

Yc

/I

Case Case Case Case

A B C D

eqn. eqn. eqn. eqn.

Fig.

(3.4a) (3.4a) (3.4b) (3.4b)

2. First

1 1 1 1

1.0 21.0 1.0 . 2.0

106 106 10’ 10’

mode

stabilization

Case Case Case Case

under

E F G H

eqn.

(3.4b)

nonlinear

control

CaseE CaseF

Fig.

3. Second

mode

stabilization

under

2

eqn. (3.4b)

nonlinear

control

2

law (a)

-___.

law (a)

1.0 10’ 2.0 . 107 2.5 lOI 5.0 101

Second World Congress of Nonlinear Analysts quite a long period (around 3.6 times the orbital period), flexible mode rather than to reduce the libration one.

shorter

3875

time is required

to stabilize

the

0.006

CaaeCcase n -__O-CO4

0.002

m

0

4.002

o.m4

o.ca 5

10

IS

20

25 1

Fig. 4. First mode stabilization

30

35

40

4s

so

under bang-bang control law (b)

0.006 CasBG CaseH

---_

0.001

0.002

v

0

4.m2

Fig. 5. Second mode stabiliza,tion

under bang-bang control law (b)

Control law (3.4b), which, as ai = 0, has the characteristics of a bang-bang control law, seems to be more effective in stabilizing either the pendular or the flexural mode (see Figg. 4 and 5), in terms both of required time interval and of residual oscillation amplitude. The maximum value of the control force does not depend on the oscillation amplitude, but only on the control gain and on the mode under analysis, through coefficients ci, which assume the following values, cl = 0.1087 and cg = 5.2521, for the pendular and flexura.l-type mode, respectively. Figure 6 shows the control action time history for cases A and C, representative of a pendular-type oscillation control; as the

Second World Congress of Nonlinear Analysts

3876 I5

10

5

I 0

3 P i5 k 5

.5

-10

.15

-20

-25

-30

,L 0

5

10

15

20

25 1

30

35

40

45

50

Fig. 6. Control force f, for the two different coutrol laws oscillation amplitude decreases, control force a) is reduced, thus reducing its effectiveness too, while control force b) maintains its alternating values throughout the all process duration. The different effectiveness of the two control laws can be clarified by making use of an asymptotic solution of eqn.(3.3) which governs the oscillation of the dynamical system with a single degree of freedom. In particular, the averaging method, as adopted by Nayfeh and Mook [13] to analyze the effects of different types of damping of the response of one-degree-of-freedom linear systems, is considered. Same results are recovered by applying the method of multiple time scales [13]. For the system uncontrolled oscillation, for which “ic = 0, the solution of eqn. (3.3) can be written as q7(t):=a

cos (uJJit+ ti)

(4.1)

where a and 4 are constants representing the initial amplitude and phase, respectively. When active control is activated, i.e yc # 0, the solution can still be expressed in the same form (4.1), provided that a and 1c,are considered to be functions of time t rather than constants; it can be assumed that their variation is much slower in time than the oscillating component ti,t. As a consequence, the variation of amplitude and phase can be averaged over the period of an oscillation, thus obtaining the following equations describing the slow variation of amplitude and phase, for both the control laws eqn. (3.4a) and eqn. (3.4b):

& = --‘Yc1 27rw,

oZnsinB 9 [q(G), i(d)] &I ,

J

2?r 4

=

-&c

1

J 0

cosfig [q(d),i(

de

(4.2) (4.3)

where 6:=i~?,t+$), q(G):=acos(ti) and i(G)::= --i~,a sin(d) [13], and 9 [q. i] :=q2 4 in the case of control law (3.4a) and 9 [q3i] :=q sign,[q cj] f or control law (3.4b). For both control laws it results 4 = 0, thus, up to this order of approximation, the frequency is not affected by the damping effect of the control, in accordance with numerical results from previous plots; on the contrary, the amplitude is

Second World

10

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15

20

of Nonlinear

Analysts

’ 25 t

35

30

Fig. 7. Oscilla.tion amplitude

shown to decay with

the following

3877

40

45

decay

laws: a

1

a0

((1 + &a$)

a -=

exp (-$Tct)

a0

(4.4a) 3 :

(4.4b)

While in the case of (3.4b), the where au is the oscillation amplitude due to initial conditions. oscillation amplitude decays exponentially with time, as in the case of linear damping, according to law (4.4b) that depends on the mode frequency but not on the initial condition, when control law (3.4a) is apphed, the amplitude decays only algebraically, with an explicit dependence on the initial oscillation amplitude. Furthermore, the initial decay is faster in case a) than in case b), especially for large initial amplitudes, but it becomes slower as the oscillation amplitude is reduced, as depicted in Fig. 7. These analytical approximate results show a good agreement with the results of numerical integration of the equation of motion previously described. 5. CONCLUSIONS A longitudinal force applied at one end of a tethered system can be effective in stabilizing the straight equilibrium configuration along the local vertical, under in-plane disturbances. The Liapunov stability of this system under two different cant rol laws is analyzed on a one-degree-of-freedom model, derived by means of a Galerkin procedure in terms of the system linear eigenfunctions. A quadratic (displacement-velocity) and a bang-bang feedback control, are shown to be capable of making the equilibrium point (q, 4) = (0,O) a globally asymptotically stable point. Numerical simulations and an approximate analysis by means of the averaging method, have shown that, among the proposed control laws, the bang-bang control law can be more effective than the quadratic law, in terms of convergence velocity, due to better dissipative qualities.

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ACKNOWLEDGMENTS This work Technological

has been partially Research (MURST

supported 40%).

by the Italian

Ministry

of University

and Scientific

and

REFERENCES 1. COLOMBO a new

G., GAPOSCHKIN

tool

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Study

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M., PIGNATARO M. & LUONGO A., Three-dimensional o,f Guidance, Control and D?JTUZ~~CS 14 , 312-320 (1991). F., Nonlinear (1994).

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free vibrations M.,

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D.T.,

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Berlin Oncillations.

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New

York

& Sons,

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.fol

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Stability and Control of Transversal and Computation 70, 343-360 (1995).

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P.M.. Effect of Tether Flexibility o,f Gurdance, Control and Dynnmics

11. PASCA M., Stability and Dynamics of Tethered Systems, di Roma “La Sapienza”, Rome (in Italian) (1991). 13. NAYFEH

Pmc.4th,

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S.J. & SHIH C.W., On the Dynamics Annl?/sis 127 , 297-318 (1994).

8. LUONGO A. & VESTRONI, nn.d Vibrations 175, 299-315

10. PASCA Orbiting

for Micro.-g/Variable-g (1985).

E.C., A Three-Mass Tethered System and Dynamics 10: 242 ~249 (1987).

5. LIANGDONG L. & BAINUM Stability and Control. .lournal

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