Dynamics and control of a space station based Tethered Elevator System

Acta Astronautica Vol. 29, No. 6, pp. 429-449, 1993 Printed in Great Britain.All rights reserved

0094-5765/93 $6.00+ 0.00 Copyright © 1993PergamonPress Ltd

DYNAMICS A N D CONTROL OF A SPACE STATION BASED TETHERED ELEVATOR SYSTEMI" V. J. MODI and S. BACHMANN Department of Mechanical Engineering, The University of British Columbia, Vancouver, B.C., Canada V6T 1W5 and A. K. MISRA Department of Mechanical Engineering, McGill University, Montreal, Quebec, Canada H3A 2K6 (Received 12 November 1991; received for publication 9 December 1992)

AImtract--A mathematical model is proposed for studying planar dynamics of a space station based Tethered Elevator System. The model accounts for finite dimensions of the station, offset of the tether attachment point from the station's mass center, and crawling motion of the elevator to or from a platform supported by the fixed length tether. The tether, assumed massless but elastic, is modeled as a double pendulum, while the elevator, end platform and moving offset attachment are treated as point masses. The system center of mass is assumed to follow an arbitrary elliptic orbit. The governing equations of motion, obtained using the Lagrangian procedure, are coupled, nonlinear, nonautonomous, and rather lengthy. Numerical results are given for the rigid tether model with the system mass center following a circular orbit. Simulation of the uncontrolled dynamics suggests that elevator maneuvers can excite unacceptably large amplitude station and tether pitch oscillations, which persist due to the absence of damping. An optimal Linear Quadratic Regulator control strategy is applied in conjunction with two distinctly different types of actuators: (i) thruster control (T=, Tv); and (ii) offset-thruster control (Ex, T~); in presence of the station based momentum gyro output 21./,. Results show that both the schemes are effective during stationkeeping, but the elevator thruster (T=) is required during its rapid retrieval. A hybrid controller which utilizes both the thruster and offset strategies offers advantages from safety considerations.

1. INTRODUCTION The three-body tether problem with mass transfer between end bodies via a crawling space elevator has several potential applications for the space station. They include isolation of the station dynamics from a tethered materials processing laboratory as suggested by von Tiesenhausen [1], and cargo transfer from a large scientific tethered platform or the Shuttle. Several authors including Liu [2], Lorenzini [3] and Misra et al. [4] have studied the in-plane three-body tether dynamics using different models. In these investigations the three bodies were treated as point masses and offset of the tether attachment point from the station center of mass was neglected. Furthermore, the tether was assumed to be massless. Except for Lorenzini [3], who included the effects of tether longitudinal elasticity in the system, the tether was considered rigid. Lorenzini examined the deployment and stationkeeping maneuvers in detail, and proposed control of the tether longitudinal oscillations

?Paper IAF-91-355 presented at the 42nd Congress of the International Astronautical Federation, Montreal, Canada, 7-11 October 1991.

with passive spring-dashpots tuned to the natural frequency of each tether segment. The system lateral oscillations were damped by application of independent reel control laws at the end masses. This paper derives governing equations of motion for the three-body tether system in-plane dynamics accounting for inertia of the space station and offset of the tether attachment point from the station center of mass. This offset, which couples the station and tether dynamics together, can be exploited to achieve control of the system. A thruster control scheme is also proposed and used as a comparison. The focus here is to study the tether and station dynamics while the elevator is either stationary or crawling along the tether towards the station, and to stabilize the system with active control. Main features of the formulation are presented first, followed by parametric analysis of the system response using typical numerical values. Next, the equations are linearized about their nominal equilibrium state, and effectiveness of the Linear Quadratic Regulator (LQR) based control strategies assessed. Details of the energy formulation, nonlinear governing equations and their linearized version for the control study are presented in Appendices I, II and III, respectively.

429

430

V.J. MODI et al, Platform,ms

/

12

il

Orbit

Elevator, m 2

.

~ , Offset

xo

C.M.

l

I Inertia] Frame //X,.

A

Space Station,too

~ YB

mmm

\ \ \ Perigee

Fig. 1. Geometry of the space station based Tethered Elevator System.

2. FORMULATION OF THE PROBLEM Figure 1 shows schematically the system under study. It consists of a space station based tethered platform and an elevator moving along the tether. The tether may be deployed away from or towards the Earth. The system is in an arbitrary elliptic orbit. The station and the tether are free to undergo rigid body rotations in the plane of the orbit. Furthermore, the tether may undergo axial oscillations. Four reference frames are chosen to describe the system dynamics. The inertial frame F l is located at the Earth's center. Orbit radius vector f¢ and true anomaly 0 specify position of the system mass center with respect to the inertial frame. The orbital frame F o and the system frame Fs have their origins at the system center of mass. The orbital frame's axes Xo, Yo remain aligned with the local horizontal in the flight direction and inward local vertical, respectively. The system frame, constrained to remain parallel to the station based body frame F B, measures the pitch

libration of the station (~,). The body fixed frame, with its origin at the station's center of mass, has its axes coincide with the principal axes of the station. denotes a shift in the center of mass of the system from the space station's mass center. Positions of the tethered platform and the elevator are expressed relative to the body frame by the tether pitch angles • , 3'; tether segment lengths It and 12; and offset x. 6 denotes angle between the two tether segments. Tether lengths l~ and/2 are specified functions of time due to the elevator moton and/or retrieval of the tether. 2.1. K i n e m a t i c s

The objective here is to obtain mathematical expressions for position and velocity of an arbitrarily located mass element with respect to an inertial frame. The position of a mass element dmi on body i is given by J~, = ~ + ~:,, i = O, 1,2,3,

431

Tethered elevator where ?~ locates the element on the ith body with respect to F s centered at the system's center of mass, a n d / ~ is the position vector from the Earth's center. Here, the subscript i varies to correspond to the station (i = 0), trolley (i = 1), elevator (i = 2), and the platform (i = 3). Differentiating with respect to time, the inertial velocity of the mass element takes the form

~,= ~c+fi × e , + ~ ,

Substituting the expression for Ri and noting the system center of mass constraints, the kinetic energy may be written as

TT=½M(~¢.rc)+ ~

+~ JMkOt d t ] =

where:

( ~ x f,). ( ~ x F,)dM

To+ TL+ T s ~ +

+ft.

Fix a t ]

dM,

r~sF,

where: = system librational velocity with respect to F I , (~ -I-O)k_,

O6/Ot = time rate change of f~ with respect to Fs, and ~ is the system's orbital velocity. The position vector :~ may be expressed as

To = orbital energy, T L = librational energy of the system, TSF = energy due to motion relative to the system frame, TLSF= energy due to coupling of the system libration and angular momentum. Here, the coupling energy TLsF and librational energy T L can be rewitten as:

~i= :~+ ~, where: :~ = position vector of the mass element on body i with respect to F B, 5 = position vector from F s to F B, i.e. shift in the system center of mass from that of the station.

TLSF= ~ ' HSF;

~

f, d M = 0 ;

where M is the total mass of the system. Here the operators d/dt and d/dt may be used interchangeably since the center of mass always remains at the system frame's origin. For a massless tether and point masses m~ (i = 1, 2, 3), the position and velocity, respectively, of the system center of mass relative to Fs are given by:

~l = -- E miii/M; i=1

-- = Ot

t=l

m~

M.

Details for the position and velocity vectors :~, d, O:dcgt and O~/Ot are given in Appendix I. It may be noted that the quantities 1~and [2 which appear in the velocity expressions are treated as independent in the energy formulation (this allows for the case of tether length variation through reel mechanisms). For a rigid tether of fixed length, the constraint Ii = - / 2 due to the elevator motion applies. This condition was substituted only after the equations of motion were derived.

2.2. Kinetic energy The kinetic energy due to translational motion in the system is given by

fiTIJl~/2;

where/lsF represents the system angular m o m e n t u m with respect to Fs, and [Y] is the system inertia dyadic with respect to the system frame given by

[Y] = fM [(e,. f,) [/] - ?,Qx] dM,

The vector ~ and velocity O8/dt are found from the center of mass constraints: f~dM=0;

TL =

where [I] is an identity matrix. Note that the orbital energy of the system is decoupled from that due to the attitude motion. The position vector f~ = f~ + ~i is now substituted into the kinetic energy expression obtained before. Recognizing that ~ and ~a/Ot are constants with respect to the mass integrals, the system center of mass constraints are applied to the expanded energy equation which lead to further simplification of the terms involved. Finally, note that for a rigid space station whose body fixed frame F B is located at the station's center of mass:

0:o

fm 0:0din0 = 0; ~ - = 0 . After the substitutions indicated, the terms in the kinetic energy expression may be evaluated as follows:

\Ot'-~)dM=i~,~'-~)m'- \Ot'Ot/l /

fM

06\

3

[Y] = __.Imo [(Q"/O)[J] --:°:°~]dm° 3

+E

[(6" 6)[Y]

-

::T]m,

L=I

-- M[(~i • ~i) [/] -- &iT];

432

V.J. MOP! et al.

where the only integration required yields the station inertias Ix.0, ly.0, I~,0and l~y,o(Ixy,Oassumed to be zero). Furthermore, due to the planar motion about the Zs axis, the librational energy TL reduces to ~'~T[,.~]~ =

I~(6 + 61)2,

where I~ is the total system inertia about the Z s axis of the frame F s. It is apparent that the kinetic energy has contributions from two sources: (i) the energy of the system relative to the space station based body frame (f~ terms); and (ii) the energy contribution due to the shifting mass center (a terms). The energy expressions involving f~, aP~/dt, fi, and Od/St may be evaluated quite readily. 2.3. Potential energy The potential energy of the system is given by the sum of the gravitational and elastic energy contributions,

u=uo+uE. The gravitational potential energy, neglecting oblateness of the Earth but accounting for gravity gradient, is given by U~ = - Gmo J M R~ where Gmo is the gravitational constant for Earth, 3.986 x 1014m3/s2. Expanding /~1 by the binomial theorem ( 6 / i ~ . 1) and substituting into the mass integral allows the potential energy, to a third order in 0"i/f~), to be written as Uc

=

G m, M _ 1 Gm~ 3 r-~ ~~ tr[~'] + ~ _ _ ~ [ J 1~o,

where ~, = sin $ / + cos ~,j_') represents direction cosines of the unit vector along f, with respect to the system frame. The first term in the above expression can be recognized as the Keplerian potential, while the remaining terms arise due to finite dimensions of the system. The gravitational potential energy expression may be further simplified as follows. For tether point masses mi (i = 1, 2, 3) and the space station modeled as a thin rectangular plate, t r [ J ] = Ix + Iy + I~ = 2I~. In addition, for planar motion, ~[oa]~o = sin 2 ~kI~+ cos 2 ~kly-- sin 2~blxy, where the inertias Ix, Iy, I~, l~y are the components of the system inertia dyadic relative to F s. Similar to the kinetic energy, there are two independent contributions to the gravitational energy: from motion relative to the station frame and that due to the shifting center of mass. To account for the elastic strain energy UE, the tether segment lengths Ii and 12 are expressed as: l,=l, +l,,,

i=1,2,

where l~o and 1~, are the commanded and stretched lengths, respectively, of the tether segment i. If reel mechanisms are not used, the commanded tether length above corresponds to the specified elevator position. The elastic strain energy is given by 1 2

vE = - X kA, 2 ill

where EA is the product of the tether elastic modulus and cross-sectional area, and k~ = EA ~lie is the elastic stiffness of the tether segments. Details of the kinetic and potential energy expressions are presented in Appendix I. It may be pointed out that the symmetry evident in both the energy expressions due to the two types of contributions mentioned above holds only for an open chain type topology. In general, the expression representing contribution from the shifting mass center contains a larger number of terms than that from motion relative to the station frame. This is because of coupling with parameters involved in t~ and d~/~t. However, these terms (indicated by the /a~s in the Appendices) account for only a small percentage of the total energy. For a typical configuration considered in the numerical example, it amounted to < 7 % . When reduced, the energy expressions compared precisely with those for the particular case considered in Ref. [4]. 2.4. Equations o f motion Using the Lagrangian procedure, the governing equations of motion can be obtained from

at

where qi and Q~ represent the ith generalized coordinate and force, respectively, and {q} = (~b,~t, ?, x, Ii,/2) r for offset control, {q} = (~, ~t, ?, Ii,/2) T for thruster control (offset locked). For the inextensible tether model (1~,= 0, l~--/~), the tether length equations are not integrated since the tether lengths are known through specified elevator motion, and are used only to evaluate the tether tensions T~, /'2. The generalized forces Q~ are evaluated using the principle of virtual work,

6 Wqi = O.i" gqi, where 6Wqj is the work done associated with a virtual displacement 6q~, and ~i are the external forces, assumed constant during the displacement (Appendix I). I n satellite applications it is convenient to use the true anomaly 0 as an independent variable instead of

Tethered elevator time. Using primes to denote differentiation with respect to 0, and substituting the transformation

d = 0 d . d2 0 2 ( d 2 dt ' d-~ = \ d t 2

2e sin 0 d ) 1 + e cos 0 '

leads to a set of six ordinary, coupled, nonlinear and nonautonomous second-order differential equations which may be expressed in the form

where: [M] = n × n system mass matrix, {F(q, q')} = vector of terms involving displacements and velocities such as gravitational, Coriolis, and centripetal contributions, nxl. The nonlinear equations of motion used in simulating the system dynamics are given in Appendix II. In general, the effect of librational and vibrational motions on the orbital dynamics is small unless the system dimensions are comparable to t:c. Hence, the orbit can be represented by the classical Keplerian relations: h2 r¢=Gm¢(l+ecos0)' f20=h, where h is the angular momentum per unit mass of the system and e is eccentricity of the orbit. For the special case of a circular orbit considered in detail here, e = 0. Furthermore, ~c = 0" = 0; 02 = Gmelf3; and ?, = Re + H, where Re is the mean Earth radius (6378 km) and H is the orbit altitude. Linearization. It was necessary to obtain a linearized version of the governing equations of motion

(

for a control study using the LQR procedure. Since the station products of inertia have been neglected, the equations were lincarized about the gravity gradient equilbrium orientation, (~, 5, ~, ~) = 0. The linearized equations may be expressed in the matrix form [M]{q"} + [C]{q'} + [Kl{q} + {P} = [][]l{u}, where:

{M]{q"} + {F(q, q')} = {Q(q)},

Retrieval Disturbance

433

[M] = [K] = [C] = [][]]= {~} =

mass matrix (n x n), stiffness matrix (n x n), gyroscopic matrix (n x n), control influence matrix (n x r), vector of terms not associated with the generalized coordinates or control inputs (n x 1), {u} = vector of control inputs (r x l).

In the inextensible tether model where the lengths Ii, 12are known through specified motion of the elevator, n -- 4 including the offset variable x. Also, r = 3 for the offset or thruster controllers and r = 4 for the hybrid controller. In the absence of time-varying effects due to elevator retrieval (or orbit eccentricity), [M] and [K] are real, symmetric, positive definite and semi-definite matrices, respectively, while [C] is skew symmetric. The vector {~} and the gyroscopic matrix [C] have Coriolis terms which vanish during the elevator stationkeeping mode. The nonzero elements of the linearized equations, after the independent variable transformation and the substitution e = 0 for circular orbit, are presented in a matrix form in Appendix III. With the tether attachment point fixed at x -- 0, the fourth row and column of all matrices vanish.

P +

[ Nonlinear = Station-TES Dynamics

State Vector

x={q}

+

Control Influence Matrix ~

~

~

Regulator Reference I n p u t

r={0} Control Vector

u =

Tr Me

Fx

Optimal Gains

( Gravity Gradient Attitude )

Fig. 2. Schematic of the state feedback control loop for the nonlinear, station based, Tethered Elevator

System.

434

V . J . MODI et al. (}

Unstable Pole Due To Offset Pole A

I

x~

~

~ 4 5 seAllct::ti°n Neutrally Stable Poles

\

Closed Loop Poles

Open Loop Poles ~=[Alx

Fig. 3. Schematic of the open and closed loop pole locations for the linearized tethered elevator system.

10 o

0

-20 tV

v

40 O~°

o ,~j

i~J

-40 -5 10

40

-y°

T~0N 0

0 0

015

1

0

0.5

Orbits Fig. 4. Thruster control during elevator stationkeeping: It = /2 = 5 km; ~0 = ~0 = 30 = 5°.

Tethered elevator

5

435

0 M~ ,Nm

-20 5

~,N

O~°

0 0 -5

10

6

7°

0

i

0.5

o

0.5

Orbits Fig. 5. Thruster control during elevator stationkeeping:

3. OPTIMAL CONTROL The previously mentioned linear, coupled, nonautonomous dynamic equations can be written in the state-space form as = A x + Bu + P,

where x is the N x 1 (N = 2n) state vector of generalized coordinates and their corresponding velocities, and u is the r × 1 control vector of thruster, offset forces and the momentum wheel torque. A is the N × N system dynamic matrix, B is the N x r control influence matrix, and P is the N x 1 vector of timevarying forcing functions due to the elevator motion:

[01

[0]

[BI= M_,~ ; {P}= _M_lp . The vector {P}, which does not affect the determination of the optimal state feedback gain matrix, can

[ l = I2 =

500 m; ~'0 = % = 6o = 5°.

be compensated for in a feed-forward manner. Here, it is considered simply as a disturbance in the feedback loop (Fig. 2).

3.1. Control strategies Three control strategies are considered to stabilize the system. They are referred to as: (i) thruster; (ii) offset-thruster; and (iii) hybrid control procedures. The corresponding control input vectors are: {u} -- (T~, TT, M , ) T for thruster control, {u} = (Fx, Tv, M , )T for offset-thruster control, {u} -- (T~, Ty, M , , Fx)T for hybrid control, where T~, Tv are the forces normal to the tether segments 11, 12 from thrusters mounted on board the elevator and end platform, respectively; M , is the station momentum wheel torque; and F x is the actuator force required to offset the tether attachment point (Fig. 1). Note, the actuator is considered to be anchored to both the station and trolley supporting

V.J. MODI et aL

436

the tether attachment point. The hybrid control scheme represents a combination of the first two. It starts out with the thruster control when the elevator is away from the station but switches over to the offset control when the elevator is close to the station. This scheme would avoid the firing of thrusters near the station and thereby eliminate the dynamic effects due to plume impingement. The system is controllable if and only if the rank of the controllability matrix [~] = N. Here ~ is the N x (N x r) matrix given by

trollability matrix at discrete time intervals and in all the cases the system was found to be controllable.

[~] = [ B i A B i A~B ! " " ~ A ~ - I B ] .

The optimal control a that minimizes this criterion is given by u = - R-1BTSx = - G x ,

Controllability of the time-varying system was established numerically by checking the rank of the con-

3.2. L Q R theory For each control strategy, linear regulator theory was used to obtain optimal state variable feedback. The infinite-time regulator was used, with the associated performance index being given by J = f ~ (xXQx + uXRu) dt.

10

10

¢°

M¢~ Nm

0

5

L/ v

-10 5 Ego

%=o

L/-

0

100 10

T ,N

7°

0

0

-

k/

V-

75

~

0

10

Fz,N

X~ m

A

v

- 50

0

L/

0

0.5

i

0

0.5

1

Orbits Fig. 6. Offset-thruster control during elevator stationkegping: Ii =/2 = 5 km; ~b0= ~ = 6o = 5°.

Tethered elevator

437

5 o

0

0

I

20

10 7°

0

-20 Fx,N

6

0

X)in

-3 0

0.5

1

0

0.5

1

Orbits Fig. 7. Offset-thruster control during elevator stationkeeping: 1~=/2 = 500 m; M, ~. 0; ~0 = ~ = 60 = 5°.

where G is the optimal state feedback gain matrix, Q >I 0 is the state penalty matrix, R > 0 the control penalty matrix, and S > 0 the solution to the algebraic or steady-state Ricatti matrix equation, SBR- IBTS -- SA -- ATS - Q = o. One classical approach to solution of the algebraic Ricatti equation requires determination of the eigenvectors of the associated 2N x 2N Hamiltonian matrix [H], where -A T

.]"

The standard LQR procedure involves selecting the state and control weighting matrices Q and R by trial and error such that the system's closed-loop poles, given by the eigenvalues of the matrix [ A - BG], or, alternately, by the eigenvalues of H with negative real parts, are moved to desirable locations in the complex plane. Here, Q and R can be diagonal matrices. No cross-coupling elements were used in this analysis. It

may be noted that the closed-loop poles can be moved to the left of a vertical line passing through - h by replacing the system dynamic matrix [A] with [A + h l] in the Ricatti equation. As an alternative, Sunkel and Shieh [5] have utilized a modified LQR procedure combined with regional pole-placement in the design of an optimal space station momentum management controller. This technique computes itcratively the state penalty matrix Q to place the closed-loop poles in the desired region of the complex plane (Fig. 3). The method reportedly reduces the computational time for the Ricatti equation solution by utilizing the matrix sign function, where sign(H) = lim ~[Hk+ H f 1]. k~ov

It may be noted that the Ricatti regulator approach does not have an explicit constraint equation to set limits for the thruster magnitudes, tether tensions and the offset. Instead these values were kept within reasonable limits by adjusting the weights in the state

V . J . MODI et al.

438

and control penalty matrices. An underlying requirement for successful controller operation is that the tether segments never become slack, i.e. :/'1, T2 > 0, and these values were checked accordingly. 4. EIGEN ANALYSIS

Given a linear state-space dynamic model in the form flAx + B u the system stability is determined entirely by the location of the poles. Position of the system zeros, which are unchanged by the feedback, only affect the speed of the transient response. A dynamic system is stable if all of its poles have negative real parts. The

open loop poles are given by the eigenvalues of the dynamic matrix A, while the closed loop poles are given by the eigenvalues of the matrix A - BG as previously noted. Figure 3 shows schematically location of the linearized system's open and closed loop poles. It is immediately apparent that the open loop system with the tether offset locked is neutrally stable with all the poles located on the imaginary axis. For the reference case of stationkeeping at l~ =/2 = 5 kin, the poles are located at +_1.732j, _+5.303j and _+27.257j. In absence of any damping in the system, the tether oscillations following an elevator maneuver will persist indefinitely. The open loop offset system is unstable due to a positive eigenvalue with the addition of the offset variable x. The eigenvalues are

1 Me

0

,Nm

0 0

5

T

=0

2

10

T~,N ,yo

0 -2

0

2O X~m

0

~

F~,N 0

v

A

v

-- 20

0

j

0.5

0.6

1

0

0.5

1

Orbits Fig. 8. Offset-thruster control during elevator stationkvcping: lm= 12= 500m; m2; m3 = 500, 1000kg; ~o = % = 60 = 5 °.

Tethered elevator

Elevator

439 Elevator Speed

V~

Platform

exponential phase

constant velocity phase

lint Tether

If /t

'~

L

exponential

phase

Space Station

to

Fig. 9. Geometry of the elevator retrieval maneuver. + 1.732j, + 5.312j, +28.816j and -t-27.223 on the real axis. The unstable eigenvalue moves further to the right as the tether length or elevator mass is increased. Both the systems are stable, however, with the addition of the state feedback, the exact pole locations are determined by the state and control penalties. Typical state penalties were 101° for the angular dispalcements, 105 for the angular velocities and 10° for the offset displacement and velocity. Control penalties were varied between 10 -5 and 10 -1° . The weighting combinations were adjusted so that the dominant closed-loop poles were moved to the left of - 1 and, approximately, to the sector +45 ° from the real axis (Fig. 3). It is well known that if the system poles lie within that region, the dynamic response exhibits well-damped, rapid convergence properties. 5. RESULTS AND DISCUSSION The governing equations of motion were cast into a first order form suitable for numerical integration. The equations were expressed as

[M]{q"} =

{ - F(q, q') + Q(q)},

and the accelerations q" determined in terms of the state variables. The integration routine used was IMSL D G E A R which utilizes an Adam's multistep method with local error of size h 5, where h is the integration step-size. The typical simulation parameters used in the analysis were as follows: Space station mass (m0) 200,000 kg Station dimensions (A x B) 120 x 100 m Station inertias (ix,0, ]y.o,iz.0) (24, 16.67, 40.67) x 107 k g . m s Trolley mass (ml) 500 kg Elevator mass (m2) 5000 kg 10,000 kg Platform mass (ms) Tether length (It) 10 km Max. elevator speed (Vmax) 5 m/s Orbit altitude/period (H/T) 400 km/92.6 min The values for the station inertias correspond to those of a uniform plate of mass 200 tons and dimensions 120 x 100 m (A x B, Fig. 1). The amount of information obtained through a planned variation of the tether length, constituent masses, elevator speeds, closed-loop pole locations, initial conditions, etc., is rather extensive. For conciseness, only a few

V.J. MODI et al.

440

of the typical results useful in establishing trends are presented.

5.1. Stationkeeping Figures 4-8 show controlled system response in the stationkeeping phase with the elevator located at the midway point between the station and the platform. Various parametric effects, including those of the tether length, and elevator and end platform masses, are investigated. The system is given an initial disturbance of 5° in each of the three degrees of freedom, i.e. ~b0, ~t0, 50 = 5°. For a 10 km tether this represents a very large (1300 m) lateral displacement of the end platform relative to the station.

I0

Ymax

5.1.1. Thruster control. Figure 4 shows the thruster control with a 10 km tether and the tether attachment point fixed at x = 0. The large station pitch of 11 ° is caused by the moment due to the tether's tension and initial deviation from the local vertical. Here the controller requires a full 1 orbit to restore equilibrium. The controller impulses, evaluated by numerically integrating the control efforts with Simpson's rule, were found to be 33 and 48 kNs for the ~t and 2~thrusters, respectively, and 13 kNms for the station momentum wheel. Effect of tether length. Figure 5 shows effect of reducing the tether length during the elevator stationkeeping phase with the thruster control strategy.

= 5 m/s

= 1 m/s

II, km 0

¢°

0

,'

•

15 0 -15 100

~o 0 -I00 10

~

7° 0 -1.5

-I0

vvvvvvv

0.5

2 TI,kN

0

2

0.3

72 k N 0

0

1

2

(a)

0 Orbits

2

4

(b)

Fig. 10. Uncontrolled elevator retrieval from 10kin: (a) V ~ = 5m/s; (b) V ~ = 1 m/s, all initial conditions are zero.

Tethered elevator

15

441

10

¢o 0

0

10 0.5

OL°

0 T1 ~ k N

- 10

0.4

WVWV .

0

T2 ~kN

- 10 0 0

2

4

0

2

4

Orbits Fig. 1I. Uncontrolled elevator retrieval from 10 km at 5 m/s with the tether nutation dampers, all initial conditions are zero. Here, the tether length is taken as I kin. The system controlled response is essentially unchanged, but the control efforts required are smaller with the shorter tether, as expected. 5.1.2. Offset-thruster control. The offset-thruster controller (Fig. 6) returns the system to gravity gradient equilibrium orientation in about 0.6 orbit. Even with a l0 km tether, the small offset motion of around 12m is seen to substantially improve the controlled performance. Effectiveness of the control was found to improve further as the tether length is reduced. By comparison with the thruster controlled case, the total demand on the controller was correspondingly higher, at 27 and 80 kNs for the offset and thruster forces, respectively. Now, however, only 8 kNms were required from the station momentum wheel. The results seem to suggest that the system's speed of response is directly governed by the controller's expenditure of energy. Addition of the offset capability is seen to produce two effects: (i) the elevator tether angle ~ is instanAA 29/b--D

taneously reduced; and (ii) amplitude of the station pitch response and the corresponding momentum wheel effort are also decreased. Effectiveness of the control for the ~ degree of freedom was found to improve further as the tether length is reduced. At shorter tether lengths, the offset strategy is particularly effective in controlling the tether dynamics. On the other hand, a higher tether tension produces a larger rotational torque on the station when the tether attachment point has an offset. Thus, the offset strategy is more effective in restoring the station's pitch attitude at longer tether lengths and with a more massive elevator and platform. In fact, the station CMGs can be eliminated entirely without any loss in controlled performance. Effect of tether length. Figure 7 shows effect of reducing the tether length during the elevator stationkeeping phase with the offset-thruster control scheme. Here, the tether length is taken as 1 kin. The magnitude of the station momentum wheel torque M, was so small (<0.001 Nm) that its time history is

V.J. MODI et al.

442

unimportant and hence not shown. Clearly, the system controllability with the offset scheme is unaffected by the absence of the station CMGs. Note that the control efforts for Tv and Fx are reduced at the shorter tether length, as expected. Effect of elevator and platform masses. Figure 8 shows effect of reducing the elevator and platform masses with the offset-thruster control. Here, the tether length is again 1 kin, but the elevator mass m2 is reduced to 500 kg and the end platform mass m3 to 1000 kg. Note that all the control efforts, particularly T~, are correspondingly reduced in magnitude. As expected, a similar effect is observed with the thruster control. Note that the amplitude of the offset motion here exceeds 20 m. Note also the smooth variation of the parameters involved.

5.2. Elevator retrieval For retrieval of the elevator, a three phase exponential-constant-exponential velocity time history, symmetric about the tether midpoint, was used. Thus for example, the retrieval maneuver can be represented as:

12= +el2, t0
/2=Vm x,a

ll= --cl,, lint<12•lf, where, 10, lexp, lint, and If are the initial, exponential, intermediate, and final lengths of the tether segment, respectively. Also, Vm~xis a specified maximum elevator crawling velocity and the parameter c = Vm~/l~p. In all simulations, the elevator was retrieved to within

j

0

10 ll, km

/

il ~ I n / s

-5 40 8

M e , Nm

¢°

0

0

~

M

- 40 50

5

T~,N

OL°

0

0

-5

- 60

5

120

T.y,N

7°

0

0

-80 0

0

0.5

0.5

1

Orbits Fig. 12. Thruster control during elevator retrieval from 10 km: Vm~= 5 m/s; ao = ~0-- 5%

Tethered elevator

443

1

11 ~ k m

-5

¢°

15

0

M@INm -1

0 0

3

To, N

Ot°

0

-10 20

1

T~%N

7°

0

0

-1

-20

Control Switchover

4

/gN

0

x~m

-2 --4

0

0.5

1

A

V

F=,N

0

v

0

0.5

Orbits Fig. 13. Hybrid control during elevator retrieval from 1 km: Vm~= I m/s; all initial conditions are zero.

10 m of the station (10 = 10 m), while the exponential length used was lap = lt/10 (Fig. 9). 5.2.1. Uncontrolled retrieval. Figure 10 attempts to emphasize importance of retrieval rate on the system response. With the tether length of 10 km, the elevator crawls towards the station from its initial position near the platform. Two retrieval rates for the elevator are considered: maximum speed of 5 m/s [Fig. 10(a)]; and 1 m/s [Fig. 10(b)]. All the initial conditions are zero. For the former case, the maneuver is completed in 0.62 orbit. Note, the uncontrolled station executes a large pitching motion which reaches a maximum amplitude of around 18°. Although the end platform swing angle with respect to the vertical 0') remains relatively small ( ~ 8°), the

elevator swings wildly ( + 100°) following the maneuver and would actually hit the station. This motion of the tether segment 11 induced steep, dangerous variations in the tensions TI, T2 with peaks reaching 2.8 kN or rendering the tether slack at some other instants. Further increase in the retrieval rate caused the elevator to wrap around the station. An increase in the elevator mass had a similar effect. The situation improves somewhat at a lower retrieval rate of I m/s [Fig. 10(b)]. Now the maneuver requires 3.1 orbits to complete. Both the station pitch (~,) and the tether swing motions (~t, 7) are dramatically reduced. The effect of reduction in the elevator mass and deployment rate were found to be similar (not shown). As can be expected, due to reversal in

V.J. MODI et al.

444

the direction of the Coriolis force during deployment, the initial station pitch was negative. The large postmaneuver oscillations were again confined to the short tether length of 10 m, i.e. to the tether segment 12. Similar to the retrieval case, an increase in the deployment speed beyond about 5 m/s caused the end platform to wrap completely around the elevator. In all the cases, a rather dramatic postmaneuver transfer of the system's energy into the short tether segment is apparent. 5.2.2. Nutation damper. A nutation damper was added in order to investigate the effect of an energy dissipation mechanism in the system. Here, damping moments in the tether pitch degrees of freedom ~ and ), were taken to depend on both the angular displacements and velocities, as given by:

5.3.2. Hybrid control. Figure 13 shows the system response with the hybrid controller. Here, the elevator is retrieved from 1 km over 0.23 orbits at a maximum speed of 1 m/s. Initially the offset is locked and the ~-thruster operates until the elevator exponential deceleration phase is reached. For zero initial conditions, T~ is found to be essentially constant in magnitude during the elevator's constant velocity phase. At the switchover point, the offset control commenced and the ~-thruster is no longer fired. The scheme proves to be equally effective as the previous one but has the advantage of eliminating problems associated with plume impingement at the terminal stage of the elevator retrieval.

M~= K~ + C.~',

Based on the analysis of the space station supported tethered elevator system with offset of the attachment point, the following general conclusions can be made:

M~= ~

+ C~',

where K~, I~, C~, C~ are damping coefficients suitably nondimensionalized. Figure l I shows effect of the nutation damper during uncontrolled elevator retrieval from 10 km at a maximum speed of 5 m/s. Here the damping coefficients were K~ = / ~ = 0, C, = Cy = 10. The damper is effective in limiting the maximum tether pitch oscillations to 10°. In addition, the postmaneuver tether oscillations are reduced to about l ° after 2 orbits. The tether tensions are also stabilized. Furthermore, the steady state space station pitch libration is reduced (~10°), although not eliminated entirely.

5.3. Controlled retrieval The tethered system dynamics during elevator deployment and retrieval are time-varying and hence the optimal Ricatti gains must be updated continuously. In typical runs the feedback gains were updated every two to five time-steps at the start of the maneuver, and after every time-step at the start of the elevator exponential deceleration phase before the onset of the resonant excitation condition. 5.3.1. Thruster control. Figure 12 shows the thruster control during retrieval from 10 km at 5 m/s. The system is also given a sizeable initial disturbance of ct0 -- ~0 = 5°. During the maneuver, note the slight deviation from a typical well-damped response due to the presence of the P vector. Regardless, the controller is still very effective in limiting the tether and station oscillations to small values. As can be expected, the actual limits are established only by the available thruster forces and the momentum wheel torque. The system is fully at rest at the end of the maneuver. The offset control action is ineffective in controlling the tether dynamics during the elevator rapid retrieval phase because, unlike the thruster scheme which incorporates the elevator based thruster, there is no direct compensation for the large sidewise Coriolis force.

6. CONCLUDING REMARKS

• Rigid body librations of the station and tether are strongly coupled. Inertia of the station must be accounted for when assessing the dynamic interactions of the elevator and tether motions on the attitude of the space station. • Deployment and retrieval of an elevator can affect the system dynamics significantly. With an elevator mass of 5 tons, a retrieval speed of 5 m/s rendered the uncontrolled system unstable. At the relatively slow speed of 1 m/s, the system still developed sustained post-maneuver oscillations. Due to the inherent absence of damping in the uncontrolled system, any disturbances encountered would persist indefinitely. • Both the thruster and offset control strategies implemented here are effective during stationkeeping. The offset scheme, which becomes more effective in restoring the station pitch attitude as the tether length is increased, can be used successfully without a station based control momentum gyro's. Conversely, effectiveness of the offset strategy in damping the tether pitch motion improves as the tether length is reduced, i.e. as the elevator approaches the station, thus making the hybrid scheme particularly useful during the terminal stages of the retrieval. An elevator based thruster, however, is still required during the rapid retrieval phase.

Acknowledgements--The investigation reported here

was

supported by the Natural Sciences and Engineering pc_ search Council of Canada, Grant No. A-2181; and the Centers of Excellence Program, Grant No. IRIS/C-8, 555380. REFERENCES

1. G. yon Tiesenhausen, Tethers in space-birth and growth: a new avenue to space utilization. NASA, TM-82571 (1984).

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445

4. A. K. Misra, Z. Antler and V. J. Modi, Attitude d y n a m i c s o f three-body tethered systems. AIAA 25th Aerospace Sciences Meeting, Reno, Nev., Paper A I A A 87-0021 (1987). 5. J. W. Sunkel a n d L. S. Shieh, A n optimal m o m e n t u m m a n a g e m e n t controller for the space station. AIAA Guidance, Navigation and Control Conference, Boston, Mass., Paper 89-3473-CP (1989).

2. F. C. Liu, O n d y n a m i c a l f o r m u l a t i o n of a tethered satellite system with m a s s transfer. AIAA 23rd Aerospace Sciences Meeting, Reno, Nev., Paper 85-0117 (1985). 3. E. C. Lorenzini, A three-mass tethered system for m i c r o - g / v a r i a b l e - g applications. AIAA /AAS Guidance, Navigation and Control Conference, Williamsburg, Va, Paper 86-1990 (1986).

APPENDIX

I

Details o f Kinematics and Potential Energy Expressions I. 1. Position and velocity vectors T h e position vectors t:i relative to the station based b o d y frame are:

~0= px_i + p~!; e, = x_r- ~i, t:2 = (x + It sin a)i - (.fi + l, cos a)j, ~3 = (x + It sin ~t + / 2 sin ~)i - (fi + l, cos a + / 2 cos ~')1, where ff is a position vector to the m a s s element dm~ on the station. T h e center of m a s s shift vector 3

a =

-

Z m,51M

is:

1

t~ = -- ~

[ml23X d-

1

m23lt sin a + m3l2 sin ),]i + ~ [ m m f i + m231t cos a + mflzcos ~]],

where mqk = m~ + ms + rag, etc. T h e c o r r e s p o n d i n g velocity vectors ~f~/dt relative to the station frame are: e3~0

d~t

0--7=0; ~ ? = ~i,

e3t~: ~ t = (It sin a + It ~ cos a + £)i - (It cos a - It ~ sin a)j,

d¢___2 ~t = (/1 sin a + I t 4 cos a + / 2 sin ? + 12f cos ~ + £ )i - (1~ cos a - I t ~ sin a +/~ cos ? - / 2 f sin ~)i. T h e velocity of the shifting center of m a s s

a~

--=

Ot

-

~

a~i./

2~m,~'M/

if

is given by: aa ot

I ~/[mt2~x + m~3([t sin a + 1~a cos ~) + m~(l~ sin r + I ~ cos r)]i + 1 [m2~([t cos ~ - It a sin a) + m~ (/2 cos ? - / 2 ~ sin ~,)]i. / }, /

1.2. Kinetic energy

T~ = ' M ( ~ + ~O ~) + ~(~ + 0)2{I~.0 + m,23(1 - g, )(x 2 + )72) + m23(I -- P2)l~ + m3(1 --/h)l~ + 2[m23(1 - #t )ll (x sin ct + )7 cos ~¢) + m3(1 -- It I )12 x (x sin ~, + y cos ~) + m 3(1 -- #2 )l112 cos ~ ]}

+ ~{rn,~(1 - #, ) ~ + z ~ ( 1 - ~,~)(~ + t,~a ~) + m3(l -- t q ) ( ~ + t ~ ~) + 2[m23 (1 - / ~ l )x (/i sin at + I l a cos a) + m 3( 1 - ~l )-r (It sin 3' + / 2 ~ cos ~,) + m3 ( 1 - g2 ) [(1~/2 + 1,12a~)cos 6 + (I,/2 a - / 2 1 , ¢)sin tJ ]]} + (~ + 6t){m,23(1 -/h).¢33 + m23(1 -/a2)12a + m,(I

--/,/3)1~)

+ m23 (1 - gl ) [.~ll cos a + 1, (.~ sin a - x cos a) + Ii a ( x sin a + .~ cos -)1 + m3 (1 - g l ) [-~/2cos ? + / 2 (P sin y - x cos ~,) + / 2 : (x sin ~, + p cos ?)l + m3(l - g2)[l, lz(a + : ) c o s 6 + (I, - 121210sin 6]}.

V . J . MODI et al.

446

L3. Potential energy

u~= GmeM ro

Gme {~(Ixo + lyo + Iz.o) + rn~23(1 - #1 )( x2 + )72)m23( 1 -/~2)1~2 + m3(l - #3) 12 3 re

+ 2[m23 (1 - #m )1~ (x sin ~t + )7 cos ~t) + m 3 (1 - / ~ t )12 (x sin ), + )7 cos y) + m 3 (1 - #2 )1~ l 2 cos 6 ]}

3Gm c

+ ~

- {sin 2 ~b{lx.o + m~23(1 - #, ))72 + m2~( 1 _ #2)12 cos: ~t + rn~(1 - / q ) l ~ cos 2 ~,

+ 2[m23 (1 --/h)lm)7 cos ct + m3 (1 -/~1 )1~)7 cos ), + m 3 (1 - #:)1~ l2 cos ~t cos 7 ]} + cos 2 ~k {ly.o + m~2~(1 - / z I )x 2 + m23 (I - #2)l~ sin: ct + m3(1 - #3)1~ sin 2 ), + 2[m23 (1 - #1)11 x sin ct + m 3(1 - #~)12 x sin y + m~ (1 - #: )l~ 12 sin ~ sin ),]} + sin 2q/{l~r.o + ml~3 (1 - ~tI )x)7 + m23 (1 -/.12 )12 sin ~t cos ~t + m3(1 - #3 )12~ sin 7 cos ), + tn23 (1 - / ~ l )l, (x cos ct + )7 sin ct)m 3 (1 - / ~ )l 2 (x cos "~ + )7 sin ~,) + m 3 (1 -/~2)lm/2 sin(,',' + ~,)} }. H e r e the shifting center o f m a s s energy c o n t r i b u t i o n s are given by:

#1 = mt23/M, #2 = m23M, ~3 = m3 M. This n o t a t i o n yields s y m m e t r y in the ~, y a n d l,, /2 equations.

L3. Generalized forces Q , = M~, + Fxr)7 + T~()7 cos ct + x sin ~t + 1~) + Tr()7 cos ~ + x sin y + 11 cos t$ + / 2 ) ,

Q~= T~I1+ Trl~ cos 6,

Qx= F~+ T~cos ~ + T~ cos ~,, Qt,= - T~, i = 1 , 2 .

APPENDIX

II

Nonlinear Equations o f M o t i o n --Platform pitch [~k" --fe(~b' + 1)]{Iz.o + m123(I -- ,tit ) ( x 2 + )72) + m23(1 --/.t2)l 2 + m3(1 --/.t3)l 2 + 2[m23 (1 -- #1 )ll (x sin ct + )7 cos ct) + m 3 (1 -- #t )12 (x sin ~ + ~ cos y ) + m 3 (1 -- #2)1t/2 cos 6 ]}

+ ~t" --fe~'){m23 (1 -- #l )ll (x sin a + )7 cos ~) + m23 (I -- #2)/2 + m3(1 -- #2)/t/2 cos 6 } + (~'" --f~)"){m3(1 - - / h )/2( x sin y + fi cos 3') + m3(1 -- #2)11/2 cos di + m3(1 -- #3)122} + ( x " --f~x'){m123(l -- ~1))7 + m23(1 - - / h ) l l cos ~ + m3(1 -- #1)/2 cos y}

+ (l~' - f J i ) { m 2 3 (1 - #~)(y sin ~t - x cos tx) - m3(1 - #2)/2 sin 6} + (l~ - f J ' 2 ) { m 3(1 - ~1 )07 sin 7 - x cos "~) + m3(1 -- #2)11 sin 6 } + 2(~b' + I){ml23(1 --/.t I )xx" + m23(1 -- #2)ll l~ + m3(l -- #3)121~ + m23(1 -/a~ )[l I x ' sin + 1~ (x sin a + )7 cos ~) + l I ~t'(x cos • - fi sin ct)] + m 3 (1 - #t)[/2 x ' sin 3' + l~ (x sin y + ~ cos y )

+ 12y'(x cos y - )7 sin y)] + m3(l - / ~ 2 ) [ ( l x1~ + 12l~)cos 6 + I 112(0t' - ),')sin 6]} + {2m23(1 - #2)111~ a ' + 2m3(1 -/~3)121~y" + m23(1 - #t )[21~ a'(x sin a + )7 cos or) + it a ,2 (x cos a - )7 (x sin a)] + m 3 (I - ~l)[21~ ~,' sin y + fi cos ),) + 12~ '2(x cos y -.17 sin y)] + m 3 (1 - #2)[2(121[a' + l,/~),')cos 6 + I 112(a '2 - y ' 2 ) s i n 6]} + f ~ {~ sin 2~k [Ix,0 - Iy,o + mr23072 - x : ) + rn23 (1 -/~2)l~ cos 2~ + ra 3(! - #3)122 cos 2y + 2[m23 ( 1 - / h )l~ 07 co s ~t - x sin ,', ) + m 3( 1 - #~ )l 2 07 cos ~ - x sin 3' ) + m3 ( 1 - #2 )l~ 12 cos(or + y )] } + 3 cos 2~, [l,y.0 + m~23 (1 -/.t~ )xp + m23 ( I - u2)12 sin ~t cos ~ + m3(1 - g3)12 sin y cos y + m23 (l - #t )l~ (x cos a + ~ sin ,~) + m 3 (! - ~t)12 (x cos 7 + )7 sin 3' ) + m~ (1 - #2)1~ I2 sin(,', + 3,)}

Q,

Tethered elevator

447

a--Tether pitch [¢" - f . ( ¢ ' + 1)] {m~s (1 - g~)l~ (x sin a + 37 cos • ) + m2s (1 - Pa )1~ + ms (1 - #2)1~/2 cos 6 }

+ (~" -f~a')[m~(1 -/~)1~] + ()," = f , y ' ) [ m 3 ( 1 - #2)It 12cos 6] + (x" -fcx')[m~3(l -/.q)l t cos ~] + (l~ -fel~)[m3(l -/./2)11 sin 6 ] + 2 ( ¢ ' + 1) {m23(I - ~ )It x ' sin ~x + m2s (1 - / a 2 ) l ~1~ + m s ( l - / t 2 ) l m[1~ cos 6

l~y' sin 6]} - ( ¢ ' + 1)2 {ms (I - / t 2 ) l ~/2 sin 6 + m23 (I - tt~)l~ (x cos • -.17 sin ~,)}

-

+ 2m23 (1 - ~2)1~ l~ a ' + m s (1 - ~2)1~ [21~y' cos 6 - 12y'2 sin 6] + f ~ { - 2[m23 (1 -/a~ )1~(x cos a - 37 sin a) + m3(1 - pa)lm/2 sin 6] 3 sin 2 ¢ [2m23 (1 -/.t~ )l t 37 sin ~ + m23 (1 - / . t 2)ll~ sin ~ + 2m s (1 - ~2)lz/2 sin ~ cos y]

-

+ 3 cos 2 ¢[2m2s(1 -

lt~)l~x cos a + m23(1 - pa)l~ sin 2~

+ 2m 3(1 - / . t 2)1t I2 cos a sin y] + 3/2 sin 2 ¢ [m23(1 - ~ )l~ 07 cos • - x sin ~x) + m2s(1 -/.t2)l ~ cos 2a + ms(1 - #2)lt l2 cos(0~ + y)]}

Q, y-Tether pitch [¢" - f ~ ( ¢ ' + 1)1 {ms (1 - t~t )/2 (x sin y + p cos y) + ms (1 -/~2)1~/2 cos 6 + ms(1 - ~u3)l~} + (~" - f . ~ ' ) [ m 3 ( 1 -/~2)l,/2 cos 61 + (~," - f d ) l m 3 ( 1

-/~s)l~l

+ (x" - f ~ x ' )[ms(Â - ~t)l~ cos r] + (1~' - f ~ l ~ )[m3(l - ~2)12 sin 6] + 2 ( ¢ ' + 1){m3(1 - #t)12x' sin r + m3(1 - ~2)12[1~ cos 6 + 1 ~ ' sin 6] + m s ( l -

lt~)l~l~}

+ ( ¢ ' + 1)2{m3(1 - ~)1~ 12 sin 6 - ms(1 - #~)12 (x cos r -37 sin r)} + m3(1 - pa)l~[2l~x' cos 6 + 11~'2 sin 6] + 2ms(1 --

I~s)121;T'

+ J ~ { - 2 [ m s (1 - / ~ t )12(x cos y - 37 sin 7) - m3(1 - ~u2)l~/2sin 6] - 3 sin 2 ¢ [2m 3(1 - ~ )1237 sin 7 + 2m3 (1 -/~2)1~ 12cos ~x sin 7 + m3 (1 - / . q ) l ~ sin 2~] + 3 cos 2 ¢ [2m 3(1 -/.t~ )12x cos 7 + 2m3 (1 - #2)Ira l 2 sin ~, cos y + m s (1 -/~3)1~ sin 27] + 3/2 sin 2 ¢ [m s (1 - #t )/2 07 cos 7 - x sin ),) + m3 (1 - #2)It/2 cos(~ + y) + m3 (1 - #3)1~ cos 2y]} =Q~ 02'

x---Offset [¢" -f~(¢" + I){m123(I -/~l )37 + m23(I - #l )Itcos • + m3(l -/~i)12 cos y} + (~" -f~')[m23(l - ~i)l~ cos ~] + (~" -fd)[m3(l

- ~t)12 cos ~]

÷ (x" -f~x') [m123(I - #l )] + (I~'-fJ~)[m23(I - #i )sin ~] + (I~ -f.l~)[m3(l - # O s i n y] + 2(¢' + I){m23 (I -/~t) (I~ cos cc - Il~' sin ~) + m 3(I -/~l) (I~cos y - 12y' sin y)} - (¢' + l)2{m12s(1 -/~1)x + m23(I - #l )Itsin • + ms(l -/x i)/2sin y}

+ {m23 (I - #i)[21~,~'cos ,I - Iia'2 sin ~] + m3 (I -/~t )[21~y' cos y -/2 y'2 sin y]} +f*~ { -2[m123 (I - #l )x + m23 (I - #i )llsin • + m3(l -/~l )/2sin y] + 3 cos 2 ¢ [mi2s(I - #i )x + m23 (I - ~l )llsin ~ + ra3(I - #i)12sin y] + 3/2 sin 2¢ [mt23(l - #l )37 + ra23(I - #t )Itcos • + m s(I - #l )/2cos y]} Qx

If--Tether length [¢" -f~(¢' + l)][m2s(l -/~i )07 sin • - x cos ~) + m3(l -/~)I 2 sin 6] - (Y" -f~Y')[m3(l -/~2)12 sin 6] + (x" -f~x') [m23(I - #i )sin ~]

+ (l~'-folD[m,3(1 - ~2)] + (l~ -fJD[m3(l - ~2)cos 6] - 2(¢' + I){m2s (I -

~)x' cos • + m23 (I -/~2)I~~' + m23 (I -/~2) [/2y' cos 6 + I~ sin 6 ]}

- (¢' + I)2{m23(I -/~i )(x sin ~ + 37 cos ~) + m2s(l -/~2)I~ + ms(1 - #2)12cos 6} - {m23 (I -/~2)It,~'2+ ms(1 -/~2)[21~y" sin 6

+ I~" sin 6 + 12y'2cos 6]}

- f~ { - 2[m2~ (I -/~t )(x sin ,~ + 37 cos ,~)+ m23 (I -/~2 )I~+ ms (I - #2 )12cos 6 ]

+ 3 sin2 ¢ [m23 (I -/~ )y cos :, + m23 (I -/~2 )I~cos2 • + ms (1 -/~2 )/2cos • cos y ] + 3 cos 2 ¢[m2s(1 -/~t)x sin ~ + m23(I -/~2)I~ sin2 • + m3(I - Pa)12sin ~ sin ~] + 3/2 sin 2 ¢ [m2~ (I -/~t )(x cos • + 37 sin • ) + m2s (l - ~)Iz sin 2~ + m 3 (I -/~)/2 sin(~ + y)]}

448

V. J. MOD! et aL

la--Tether length [~k" - f c ( ~ b ' + 1)Jim3(1 -- #1)07 sin y - x cos ~,) + m 3 ( l - #2)11 sin di] -- #2)11 sin 6] + (x" - f ~ x ' ) [ m 3 ( 1

+ (u" - f ~ ' ) [ m 3 ( 1

-- # 0 s i n ~,]

+ [1'( - f ~ t l )[m3(1 - #2)cos 6] + (l~ - f J ' 2 ) t m 3 (1 -- #5)1 -

2(~,' + 1){m3(l - # l ) x ' cos ~ + m3(1 - #3)12r" + m 3 ( l -//.-z) [11n,' cos 6 - I~ sin 6]}

-

(~k' + 1)2{m3 (1 - #1 ) (x sin ~, + )7 c o s ~,) + m 3 (1 - #2)1, cos 6 + m 3(1 - #3 )/2 }

+ {m3(1 -- #2) [21~ ~ ' sin 6 -- 11~,2 c o s 6] -- m3(1 -- #3)12y '2} + f ~ { - 2Ira 3 (1 -- #1 ) (x sin y + )7 c o s ~,) + m 3 (1 -- #2)ll cos 6 + m 3 (1 - #5)/2] + 3 sin 2 $ [m 3 ( 1 --

# I ))7 cos ~ + m3 (1 - #2)11 cos ~ cos ~, + m 3 ( 1 -- #3)12 c o s 2 ~']

+ 3 c o s 2 ~, [m 3 (1 - #1 ) x sin ? + m 3 (1 - #2 )ll sin :, sin ~, + m 3 (1 - #3 )11 sin2 ~'] + 3/2 sin 2~b [m 3 (1 - #1 ) ( x cos ~ + )7 sin ~,) + m3 (1 -- #2)12 sin(~, + y) + m3 (1 -- #3)12 sin 2~,]}

Q~ where: f~=

2e sin 0

1

l+ecosO'

J~" - l + e c o s O

APPENDIX

HI

Linearized Equations of Motion 1I. 1. M a s s matrix [M], Mij = Mj., Ml.t = 1~,0 + m~23(1 -- #l ))72 + m23( 1 _ #2)1 ~ + m 3 ( l _ #3)1 ~ + 2[m23 ( 1 _ #t)I~Y + m3(1 -- #l)l::fi + m 3 ( l -- #2)11/2] , Mr,2 = m,3 (1 - #1 )11 Y + m23 (1 - #2)l~ + m 3 (1 - #2)11/2, MI,3 = m3 (1 - g l )12)7 + m3 (1 - #2 )11/2 + m3 (1 -- #3)1], Mi,4 = m123(1 -- #t))7 + m23(1 -- #1)ll + m 3 ( l -- #1 )12, M2,2 = m23(1 -- #2)1~,

M2.4 = m23(1-- #l )ll,

M3,3 = m3(1-- #3)l~, M3,4 = m 3 ( l -- #t )la,

M2,3 = m3(I - / h ) l 1 1 2 ,

M4,4=mlz3(1--#l).

11.2. Gyroscopic matrix [C] CLI = 2[m23 (1 -- #1)l~P + m3(l -- #~ )l'2)7 + m23(I -- #2)ll 1~ + m 3 ( l -- #2)1.>1'2 + m3(1 -- #2)(I K1"2+ 121~)],

CI,2 = 2[m23(1 - #t)l~fi + m23(1 - #2)111~ + m 3 ( l - #2)121~] CI,3 = 2[m3(1 -- #1)1'2~ + m3(1 - #2)lll'2

+ m3(1 -- #3)121'2],

C2,t = 2[m23(1 - #2)11 l~ + m3(1 - #2)ll 1'2], C3,1 = 2[m3(1 -- #2)12/~ + m3(1 -- #3)12l'2], C4,1 = 2[m23(1 -- #,)l~ + m 3 ( l -- #1)1'2], Cz. 2 = 2m23(1 - #2)111~,

C2. 3 = 2m3(1 - #2)111'2,

6"3.2 = 2m3(1 - #2)/21;,

C3, 3 = 2 m 3 ( l -- #3)121'2,

G.2 = 2m23( 1 - # , ) l ~ ,

C4. 3 = 2m3(1 - #1)1'2.

11.3. Stiffness matrix [If] Ki. l = 3{I~. o -- ly.o + m123(1 -- # i ) . ; 2 + m23(1 -- #2)112 + m 3 ( l -- #3)122 + 2[m23(1 -- # j ) l l Y + m 3 ( l -- #l)laY + m3(1 --#2)1t/2]}, K,.2 = 3[m23(1 - #1 )/,Y + m23(1 - #2)1~ + m 3 ( l - #2)1112] + m23(l - #,)I~'Y + m3(1 - #2)(121'~ - Ii 1'~), KI.3 = 3[m3(1 -- #1 )lay + m3(1 -- #2)ll/2 + m3(1 -- #5)122] + m3(1 -- #1 )I~Y + m3(1 -- #2)(11 l~ -- 121[), KL4 = 3[m,23(1 -- #,)Y + m23(1 -- #1)11 + m3(1 -- #1)12] -- m23(I -- # t ) l l ' -- m3(1 -- # t ) l ~ , /('2,1 = 3 [m 25( 1 -- #1 )ll Y + m23 ( l -- #2 )l 2I + m 3 ( 1 -- #2 )11 I2], /(2, 2 = 3[m23 (1 -- #1)11)7 "b m23 ( l -- #2 ) 12 "-b m 3 (1 - #2 )1112] -- m3 ( l - - ~ ) 1 1 1 ~ ,

/(3, I = 3[m3(1 -- #1)12)7 + m3(1 -- #2)1112 + m3(1 -- #3)12], K3,3 = 3[m3(1 -- #t)12fi + m3(1 -- #2)ll/2 + m 3 ( l -- #3)/2] -- m3(1 -- #2)121~', /(4, I = 3[m123 (l -- #1 ))7 + m23 (1 --#1 )11 + m3 (l -- #1 )12],

Ka.3=m3(1--#l)lll'~, K4.2 = m23 ( i - # Ol'~ ,

K3,2=m3(1-#Olal[, K4,3 = m3 ( 1 - - #t )l'~ .

Tethered elevator

11.4. Nonautonomous vector {P} Pl = 2[m23( 1 --/zl)l~Y + m3(1 -- #t)l~)~ + m23(1 -- #2)111~+ m3(l -- #2)(111~ + 121~)+ m3(l -- #3)121~] + 31xy.O, P2 = 2[m23(1

--

~2)111~+ m3(l - #2)lll~],

/>3 ~ 2[m3(1 - la2)12l~+ m3(1 - g3)121;], P4 =

2[m23(1

-

#l)l~ + m3(1 -- #1)1~].

11.5. Control influence matrix [B] Defining the acceleration vector {q"} = (~ ", ~", ~", x") T, and the control vector {~} = {u/O 2} = (T~, TT, M , , Fx)T: ~ . , = y + l,, B,.2 = Y + It, ~1.3 = 1, ~ , -----l~, ~2.2 = I,, /~3.2 = 12, J~4.1 = 1, ]~4,2 = |, J~4,4 = ["

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