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Page 1

Analytical Investigation of the Dynamics

of Tethered Constellations in Earth Orbit (Phase II)

Contract NAS8-36606

E •

e .

Quarterly Report # 1

For the period 22 February 1985 through 21 June 1985

Principal Investigator

Dr. Enrico Lorenzini

Co-Investiaator

Mr. David A. Arnold

Dr. Mario D. Grossi

Dr. Gordon E. Gullahorn

.Tuly 1985

Prepared for

National Aeronautics and Space Administration

Marshall Space Flight Center, Alabama 35812 

Smithsonian Institution

Astrophysical Observatory

Cambridge, Massachusetts 02138

.f

The Smithsonian Astrophysical Observatory

is a member of the

Harvard-Smithsonian Center for Astrophysics

.L
Page 2

CONTENTS

Page

SECTION 1.0 4

2.0

2.1

2.2

2.2.1

2.2.2

2.3

2.4

2.4.1

2.4.2

2.4.3

3.0

4.0

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . .

TECHNICAL ACTIVITY DURING REPORTING PERIOD AND PROGRAM

STATUS . . . . . . . . . . . . . . . . . . . . . . . . .

Introductory Remarks . . . . . . . . . . . . . . . . . .

Two-Dimensional Equations Of Motion For A Single-Axis

Vertical Constellation With Three Masses . . . . . . .

GeneralCase . . . . . . . . . . . . . . . . . . . . . .

Case Of The Middle Mass At The System Center Of Mass . .

Numerical Integration Of The Two-Dimensional Equations

Of Motion . . . . . . . . . . . . . . . . . . . . . . .

DeploymentManeuver . . . . . . . . . . . . . . . . .

Control haws . . . . . . . . . . . . . . . . . . . . . .

Simulation Results . . . . . . . . . . . . . . . . . . .

Concluding Remarks . . . . . . . . . . . . . . . . . . .

PROBLEMS ENCOUNTERED DURING REPORTING PERIOD. . . . . . .

ACTIVITY PLANNED FOR THE NEXT REPORTING PERIOE

5

5

5

7

13

15

16

16

18

29

29
Quarterly Report # 1

For the period 22 February 1985 through 21 3une 1985

Principal Investigator

Dr. Enrico Lorenzini

?Ittt

Prepared for

National Aeronautics and Space Administration

Marshall Space Flight Center, Alabama 35812

Smithsonian Institution

Astrophysical Observatory

Cambridge, Massachusetts 02138

The Smithsonian A.g t:rophysical Observatory

is a member of the

Harvard-Smithsonian Center for Astroph sics

'NA C:; -171.11) ANALYTICAL 1.hVtjTlt;,;110N

L Y:;z1" 1 UE iE i1_LhLii C (' h -Ctitiii( iv j

ih t, f,t.'1fs t, E1'i, I'L'A;L I ^udLtE-L1 y j, Lt.,

iLstItUti01., q Md.`-L: 31
Page 1

1

Analytical Investigation of the Dynamics

of Tethered Constellations in Earth Orbit (Phase II)

Contract NAS8-36606

i •

a .

Quarterly Report # 1

For the period 22 February 1985 through 21 June 1985

Principal Investigator

Dr. Enrico Lorenzini

Co-Investia^ator

Mr. David A. Arnold

Dr. Mario D. Grossi

Dr. Gordon E. Gullahorn

Jul y 1985

Prepared for

National Aeronautics and Space Administration

Marshall Space Flight Center, Alabama 35812

Smithsonian Institution

Astrophysical Observatory

Cambridge, Massachusetts 02138

The Smithsonian Astrophysical Observatory

is a member of the

Harvard-Smithsonian Center for Astrophysics
Page 2

CONTENTS

Page

SECTION 1.0 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . 4

2.0 TECHNICAL ACTIVITY DURING REPORTING PERIOD AND PROGRAM

STATUS . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1 Introductory Remarks . . . . . . . . . . . . . . . . . . 5

2.2 Two-Dimensional Equations Of Motion For A Single-Axis

Vertical Constellation With Three Masses . . . . . . . 5

2.2.1 General Case . . . . . . . . . . . . . . . . . . . . . . 7

2.2.2 Case Of The Middle Mass At The System Center Of Mass . . 13

2.3 Numerical Integration Of The Two-Dimensional Equations

Of Motion . . . . . . . . . . . . . . . . . . . . . . . 15

2.4 Deployment Maneuver . . . . . . . . . . . . . . . . . . 16

2.4.1 Control Laws . . . . . . . . . . . . . . . . . . . . . . 16

2.4.2 Simulation Results . . . . . . . . . . . . . . . . . . . 18

2.4.3 Concluding Remarks . . . . . . . . . . . . . . . . . . . 29

3.0 PROBLEMS ENCOUNTERED DURING REPORTING PERIOD. . . . . . . 29

4.0 ACTIVITY PLANNED FOR THE NEXT REPORTING PERIOD . . . . . . 30
Page 3

Abstract

This Quarterly Report deals with the deployment maneuver of a single-axis,

vertical constellation with three masses. A new, easy to handle, computer code

that simulates the two-dimensional dynamics of the constellation has been ir,,plemented. This computer code is used for designing control laws for the deployment maneuver that minimizes the acceleration level of the low-g platform during

the maneuver.
Page 4

1.0 INTRODUCTION

This is the first Quarterly Report submitted by SAO under contract NAS8-

36606, "Analytical Investigation of the Dynamics of Tethered Constellations in

Earth Orbit (Phase II)," Dr. Enrico Lorenzini, PI, and covers the period from 22

February 1985 through 21 June 1985.
Page 5

2.0 TECHNICAL ACTIVITY DURING REPORTING PERIOD AND PROGRAM STATUS

2.1 Introductory Remarks

The results of the study performed under NASA contract NAS8-35497 on the

dynamics of tethered constellations in earth orbit were very encouraging for a

single-axis vertical constellation. In the three-mass configuration the middle

mass, if appropriately placed at the orbital center of the system, can be conveniently utilized as a low-g experiment platform. A preliminary estimation of

the acceleration level achievable was around 10 -8 g when the system began its

station-keeping phase from a rest condition and the J 3 term of the gravity field

was taken into account. The above referenced report dealt with the stationkeeping of the constellation only. No indications were provided on how to

deploy the system to the final, steady state configuration. This Quarterly

Report addresses problems related to deployment of a single-axis constellation

with three masses from the initial, compact configuration to the final alignment

of the constellation with the local vertical at fully deployed tether length.

2.2 Two-Dimensional Equations Of Motion For A Single-Axis Vertical Constellation With Three Masses

The SKYHOOK program was used to simulate the station-keeping phase of the

constellation in Phase I studies. Station-keeping is easier to model than deployment so that a limited number of simulation runs were required. Deployment

of a three-mass constellation is a new topic that requires a longer and more

difficult effort. A complex computer code, like SKYHOOK, is therefore prohibitively cumbersome for a parametr±c study like this. For this reason a more
Page 6

flexible and easy to handle mathematical model has been derived to study the

deployment dynamics of a single-axis constellation with three masses.

The model is two-dimensional. This provides a good approximation of the

deployment maneuver that is dominated by forces acting in the plane of the

orbit. The model assumes three point masses generically located in space (on

the orbital plane) connected by two massless tethers of known elastic characteristics.

The center of mass of the system is supposed to ra on a circular or nearly

circular orbit but in any case the orbital motion is decoupled by the system

oscillation. Presently no aerodynamic forces are in the model but they could be

added easily. Aerodynamic forces, however, are not so important during deployment at 500 km altitude. Excessive complication of the model would be impractical since this model is developed in order to design a suitable deployment

strategy for verification by SKYHOOK.

The equations of motion were derived by using the Lagrangian f,)emulation

The Lagrangian coordinates were chosen, as shown in Figure (2.2.1), to give an

immediate representation of the system oscillations. They also have the advantage of representing oscillatory modes that are, in some cases, very weakly

coupled and therefore suitable for consistent simplifications in order to

achieve analytical results.
.e,

Page 7

2.2.1 General Case

The three masses are generically located on the plane of the orbit. With

reference to Figure (2.2.1), the position of the masses with respect to a reference frame centered at the center of mass and co-rotating with the system is

given by:

X1 = (R313 - R111 + 11) sin B - R 2 e cos B

X 2 = (R313 - R111) sin 9 + (R1 + R3 ) a cos B

X3 = (R313 - Rl ll - 1 3 ) sin B + R 2 e cos 8

21 = (R313 — R1 1 1 + 1 1) z;os 9 + R2e sin 9

Z 2 = (R313 — R 1 1 1 ) cos B — ( R1 + R3 ) a sin B

Z 3 = (R31 3 - R11 1 - 13 ) cos B + R2 e sin B (2.2.1)

where:

R1 = ml/mt,t i = 1, 2, 3 

Equations ( 2.2.1) are valid for e << ti, which is applicable in most cases.

The potential energy of the system is given by:
h

Page 8

ballast)

Lam

e

E

M

A.

Figure 2.2.1 Geometry and reference frame for a single-axis

tethered constellation with three masses.
Page 9

V = μ J: i m i /r i (2.2.2)

1

which to the second order of approximation is as follows:

V = _ i ml l 1 + z 1 _ z 1 + x1 - 3zi (2.2.3)

a 1 Il a 2a=

where μ is the gravitational constant of the earth and a is the semi - mayor axis

of this orbit of the system center of mass. After considerable rearranging, the

potential energy can be expressed as:

V = - 2 n2 { m toc[ 2a2 - 2(R,t ., - R 1 t 1 ) 2 (3 cos 1 0-1)] +

+ R2 (m i + mi ) If 2 ( 3 sin'8 - 1 ) 1 +

+ m1[

2

t 1 (3 cos 1 8 - 1) + 61 1R2E cos B sin B +

+ M31 t 3 (3 cos 2 8 - 1) - 6t 3R 2 e coe B sin 9 I

where n is the mean orbital rate.

The kinetic energy of the system is given by:

T = ^L i mil vi l'

1

where:

I vi 1' = [ x i + 0 (a - zi ) ]' + [zi + nxi]'

(2.2.4)

(2.2.5)

(2.2.6)
qs = 9; q2 = e; qj = t,; q• = t, (2.2.10)

r.

Page 10

After taking the derivative of equations (2.2.1) and substituting them into

equations (2.2.6) and (2.2.5) we finally get:

T = 2 1Mtot[02a2 - ( RJ3 - Ri l l )' - (R,t, - R,t,)' (^ - Q)^I +

+ R, (m l + m,)[i' + Ol e 2 + e '(9 — n) 2 ] +

+ m, l; + r, (9 - 0) 2 - 20R2 (1,e - t,t)I +

(2.2.7)

The Lagrangian function is easily computed as:

L = T - V (2.2.8)

The equations of motion are given by:

dt (aL) d 8L

a41

8qi

(2.2.9)

where the Lagrangian coordinates are:

The Qi 'u are the generalized forces given by:

er,

8qj (2.2.11)

In equation (2.2.11) F' 1 are the forces acting on the masses and rs the radius
Page 11

vectors, respectively given by:

F i M x1 I + z1 k (2.2.12)

F1 = — T, (cos B + c/li sin I) k — T1 (sin I — c/11 cos B) i

F 1 = [(Ti — TO cos B + (T1 /11 + T3 /1 3 ) t sin B]k + [(Ti — T.) sin B — (T1 /1 1 + Tj/l,) t cos B]i

F, - Ta (cos f — c/1 3 sin B) k + Tj (sin $ + t/1j cos B) I (2.2.13)

By substituting (2.2.12) and (2.2.13) into (2.2.11) we finally get the generalized forces as follows:

Qe = 0

- t (T1/ 1 1 * T1/11)

Ql i = -T1

Qty _ -Tj (2.2.14)

where T1 and T1 are t1--o tensions in the respective tethers.

By substituting the above into the Lagrangian equations (2.2.9) and after

performing the necessary derivatl.ons we finally get the equations of motion as

follows:

mi 1#[,,3 + t i(Ritj - Riti)] + 21 1( B -0)[11 + (R113 —R1 1 1)] +

+ 303 1 1 cos B sin B[l l + 2 (R,1, — R111)] — 3Q1 1 1Rac cos (2B)} +

^ i s i
Page 12

+ m,S i[l 3, - fj(R3 l 3 -R1 l 1 )] + 21,(d-Q)[12+ ( R i l3 - R i i ij +

+ 3O1 f 1 cos $ sin B[f 1 + 2 (Ri f ) - R l f l )] + M12 1,R 1 c cos (2B ) } +

+ R2( m :+mj)1( 4 + 2ei(0-0) - 3O 1 e 1 eoa0 sin 0] = 0

R 1 (mi + m1 ) [ i - c (d - Q) 1 - 301 1 sin 1 8 1 +

+ R 1mt,t [302 eosOsin8 ( R i l l - R l f l ) -2O(R 1 1 1 -R 1 1 1 ) ] -e (Ti/fl + T1 /1 1 ) (2.2.15)

M 11 1 1 + ( R i l l - R i l l ) - 01 (3 cos 1 8 - 1)[t, + 2 ( R i l ) - R i f,)] -

- (B - CI) l ^f^ + ( R i l l - R i f,)]- 311 1R 2 ( cos 0 sin 0 - 21IR j i) _ - T1

m 1 11, - (R i l l - R i ll ) - 01 (3 cos 1 8 - 1) [ f 1 - 2 (R i l l - R i f,) ] -

n) 1 [f 1 + (R ) C 1 - Ri f,)] + 3O1R 2 1 cos f sin 0 + 20R2i}

These equations of motion will be used to simulate the dynamics of the threcmass system during deployment as will be shown later in this report. After

further simplifications, relavnnt to odrtieulmr cases, these equations can also

provide information or. the dvnami-s of the system withoutceinq numerically

integrated.
Page 13

2.2.2 Case Of The Middle Mass At The System Center Of Mass -

In equations (2.2.15) the term R3l3 - Rl l l is the distance (along the line

connecting ml and m3 ) of the middle mass m 3 from the system center of mass. If

the middle mass is maintained constantly at the center of mass by means of an

appropriate control law then the above mentioned term and its derivatives can be

eliminated in equation (2.2.15). The distances 1 1 and L 3 can also be expressed

as a function of the overall tether length 1. From equation (2.2.13), by requiring that the component of F 3 along the line connecting ml and m3 is zero, we

get T1 = T3 = T so that equations (2. 2.15) reduce to:

m.q [L 3 B + 211 (B - 0) 3 + 303 0 cos 0 sin 01 +

+ R3 (ml + m3) If 4 + 2EE (J - n) - 302e cos 0 sin 01 = 0

R3 [j - E (B -n) 2 - 303 E sin'B] _ - ET/(Lm.q ) ( 2.2.16)

m ' [f - 03 c (3 cos'B - 1) - ( 9 - n) 3 c] -

- R2 (m l - m 3 ) [ 30 2 E cos 0 sin B + 20i] _ -T/m,

Equations (2.2.16) provide some useful information on the dynamics of the Sys-

. tem. The coupling among the rotational motion (equation in B) and the lateral

d •-flection (equation in E) is strongly reduced by placing the middle mass m 2 at

the system center of mass. In particular if the initial lateral displacement is

zero it stays zero ii spite of the system libration. This suggests, for example, that the best way to deploy the constellation without overly perturbing the

low-g platform is to design a control law that keeps the middle mass at the

system center of mass throughout the entire deployment. At the end of the

NOW-
Page 14

maneuver the mass m2 can be moved to the orbital center (a few meters below the

c.m.) in order to minimize the vertical acceleration.

If we assume that the constellation is fully deployed, that 0 and B are

equal to zero so that the tension is approximately constant and is given by T =

3m.gf1 1 C, then the E equation in (2.2.16) reduces to:

Rae + 02 E (3 — R 3 ) = 0 (2. 2.17)

From this extremely simple equation we can derive an analytic formula to compute

the free oscillation frequency of the lateral vibration of mass m 2 as follows:

f, = 2 (3/R, — 1) 1,2 0 (2.2.18)

As an example, for the constellation studied in Phase I, with: m i = 90600. kg,

m 2

4530 kg, m3 = 9060. kg, 0 = 1.1068 x 10 -3 rad/sec (altitude = 500 km) we

get:

f, = 1.45 hz ; T, = 688.4 sec (2.2.19)

Notice also that in the first approximation the lateral oscillation frequency

does not depend on the overall system length. This is because the tension is

directly proportional to f while the angle between the tethers and the line

through the end masses m, and m 3 is inversely proportional to it. The tether

length, therefore, cancels out when computing the horizontal component of the

force acting upon the middle mass.

r _
Page 15

2.3 Numerical Integration Of The Two-Dimensional Equations Of Motion 

A new software code called 3MASS has been implemented to simulate the dynamics of a 3-mass constellation by using the equations of motion previously

described. The computer program uses a fourth order Runge-Kutta integration

subroutine with variable step size. This integration has been successfully

tested in the past with a similar mathematical model. Additional subroutines

called DEPLOYS and STAKES model the tether control laws during the deployment

and the station-keeping maneuver respectively. All the geometrical and mass

properties and orbital characteristics of the constellation are read as input

parameters by the main program and passed to the integrator. The integration

subroutine, called DRKGS, calls the subroutine FCT which computes the derivatives of the Lagrangian coor-mates. DEPLOY3 or STAKES are called by ECT and

after a successful integration step the output subroutine OUTP is called. OUTP

formats the output variables for the printout and it prepares the output file

for the plotter. Subroutine OUTP also stops the program in the case of program

malfunctions or the achievement of the desired final conditions.

Typical program malfunctions are: the in-plane angle is too large so that

t

the constellation attitude is close to the horizontal plane or the integration

f step size becomes too small. This computer program is very efficient, at least

when the tethers are considered unstretchable. Normal maximum integration step

sizes are around 50 sec. These characteristics make it suitable for deriving

appropriate tether control laws for the deployment phase.
F

Page 16

2.4 Deployment Maneuver 

2.4.1 Control Laws -

Deriving a deployment control law depends on the control parameters available to the deployer and on the measurement accuracy of these parameters.

Knowledge of the in-plane and out-of-plane angular rate of the tether libration

is required for a very effective damping of librations. Simpler control laws

rely on the tether length and length rate or on length, length rate and tension.

They are respectively called rate control laws and tension control laws even if

the latter denomination is inaccurate. Rate control laws are easy to implement

and are very reliable. However, they do not damp librations: the decrease of

the libration amplitudes in this case is due to the conservation of angular

momentum while the tether length is being varied. The tether tension is, on the

contrary, related to libration angles and angular rates (well coupled to the inplane oscillation and very weakly to the out-of-plare) so that a tension control

law, if appropriately tuned to the tether longitudinal oscillation, provides a

good damping of the in-plane oscillation. A major drawback is that the tension

for short tether lengths is too low to be a reliable parameter to control. For

this reason rate control laws (with or without angular feedback) have to be used

for short tether lengths. In this study we used a rate control law for deployment mainly for two reasons: a) if low level acceleration can be achieved in

the middle mass, notwithstanding a residual oscillation of the system, then any

other control law will be more effective than this control law; b) a rate 

control law constitutes the base for any other more refined control law.

The system starts with the three masses very close together, though not

coincident, orbiting in relative equilibrium. The lower mass ; ml , is the space

station, the central mass, m l , is the low gravity facility, and the higher mass,

r
a t ti < t < I T

— 0(t — tf) I

T < t < lac

(2.4.1)

Page 17

M 3, is a ballast mass. Thereafter the natural length of each cf %he twc tethers

is derived from a control law:

where the various parameters may be different for the two tethers. Here t is

the deployed natural length, t I an initial length, I T a transition length at

which the reel begins decelerating, tsK the desired final length, and I f = L sK +

a is the length which would result at infinite time, o being a small length

chosen so that the deployment takes finite time but the cutoff is not too

abrupt. The constan t. a = 0.75 0 sin (20 c ) is chosen by analogy to a single mass

deployment where it would result in an attitude angle during deployment equal to

B C ; and fi = at T /(t f - t T ) is such that the deployment velocity is continuous

through the transition. This control law may be solved explicitly to give the

deployed natural length as a function of time. Assuming the initial time is t =

0, we get

I (t) = I I e°t

0 < t < tT (2.4.2)

I f - (t f - I T ) exp [ - ^ (t - tT ) ] tT < t < tSK

where tT = (1/n) In (tT/L I ) is the transition time and t8K = tT + (1/#) In [ (tft T )/o)] is the turn off time.

The same time constant (t,, = 1 /ac) is used for the two tethers in order to keep

the middle mass at the same location throughout the maneuver. The same consideration applies to the time constant for the deceleration phase (t,,g = 1/0).

s
Page 18

All the characterist!.c lengths, such as lr, i f , a are in the ratio ISK1/tSK3 in

order to provide simultaneous transitions in the two tethers.

6 - 2.4.2 Simulation Results -

• Numerous simulation runs have been performed after implementing the previously described control law in the new simulation program. Different values of

parameters, such as a, £r, a and esK have been tested to obtain a set of parameters that results in a well behaved deployment with a low level of acceleration

of the middle platform.

The dimensions and the masses used in these simulation runs are equal to

the values used in the final report of Phase 1. Namely they are: m l = 90600.

kg, m 2 = 4530 kg, m 3 = 9060. kg, e sK, = 909.09 m, L SK3 = 9090.9 m, orbital

altitude = 500 km. Notice that this mathematical model does not separate the

orbital center from the center of mass since the gravity field is apprcximated

to the second order. This does not detract from the model since during deployment the major perturbations on m 3 are due to the coupling of the lateral deflection with the libration and since this coupling is not substantially affected by the third order term of the gravity field. The third order terms of

the gravity field affect the vertical acceleration components so that a mass

located at the system center of mass will actually experience a vertical acceleration different from zero. The conclusion is that the results of these simulation runs are accurate for the horizontal acceleration component (the most

important during deployment) while the vertical component does not show the bias

dependent on the offset between the center of mass and the orbital center.
Page 19

The best set of cortrol parameters came out to be: L ?1 = 400. m, lT2 _

4000. m, of = 15 m, o, = 150 m, 1/a = 1874.14 sec (that implies = 20 0 ). The

deployment was completed in 5.1 hours while the simulation was stopped at 6

hours in order to plot part of a station-keeping (with no tether control). Two

sets of figures follow that show results of a successful (and improvable) deployment maneuver. Figure (2.4.1 a,b) and Figure (2.4.2 a,b) show the tether

• length and tether speed for tether No. 1 and No. 2 respectively (tether No. 2 is

called L3 to homogenize the notation in the equations of motion). Notice that

these quantities, as mentioned before, are in the ratio tSK11tSK3 and notice

also the small discontinuity of the tether speed at the end of the deployment

due to the cut off of the exponential deceleration phase. The deployment is

initialized with C 11 = 2 m and Y. 13 = 20 m. This condition can be actually

realized with two deployment booms w... are extended at the beginning of the

maneuver. These initial tether lengths affect the duration of the maneuver by

expediting the initial phase of the deployment. Figure (2.4.3) shows the inplane angle B of the constellation vs. time. Because of the choice of the

I Lagrangian coordinates (see Figure 2.2.1) Figure (2.4.3) must be read together

iwith Figure (2.4.4), which plots the lateral deflection of the middle mass. The

constellation's initial attitude is 20 1 consistent with the selected steady

I - state inclination angle during the first phase of deployment. For this reason

the in-plane angle is constant throughout the entire acceleration phase and

decreases to the final, residual oscillation amplitude during the deceleration

phase. The residual oscillation can be reduced to the steady-state value forced

by the J 2 , as shown in the final report of Phase I studies, by means of a tether

controlled station-keeping phase. It could also be reduced during the deployment deceleration phase by using a tension control law. This second level

improvement will be investigated in the next reporting period if desired by

NASA.

0, .0y Y
Page 20

oo^

I

eoo C ^

600

F 600 .'

x

-

w

^

s 400 r -,

F ^

1

k

- 900 — y

200 r- J

100 — ..^

4

D ^

0 S 1 15 2 25 3 9.`. t 45 ,, 55 6

TIME (HR51 Rue Control 1a• Oeploymn•,:

(a) Lower Mass ml

rnon = —'

k ,

6000 —_

+0co -.

—

6000

Y

7

^ S000 —

c -

V Y 1'

a X000 --

t

F ^

n 3000

FF

a

2000 r

t ^

1000 = 7

r ^

E ^

OF I a

0 5 1 15 k 25 3 35 1 4 45 5 55

6_...

TINE (HRS) Rate Control fe. Deployment

(h) Upper Klass m3

Figure 2.4.1 Tether length vs, time.

li

. j#
page 21

22

2

.Ie

16

u .14

a

- t2 c

.1

et

De

J

De

.04

02

0

0 .6 1 IS 2 25 3 36 4 45 6 6.6 6

TIME (MRS) Rate Control 1A. Deployment

(a) Lower Mass m1

22rI 'r-t-^-r--^-rr-: ^'^-1-r-rri-.-rT's^-- . I ,

F

2r

t

Ib—

k

f6--

u 1•

u

r

i

12

c

r

c

a: F

s

e^

r e^

G

^tF

2^

0F,

0 .6 1 16 2 25 3 36 4 46 6 66 6

TIME (MRS) Rate Control Law Deployment

(b) Upper Mass m3

Figure 2.4.2 'Tether speed vs. time,
Papal 22

^^,-,-T+rr' T"-' 'T'-'-'l ^-T' 1-TT^TTT ^' Tom, Trl-+-rr-r-r--rte-, -1-•-rT-T,--T•^

e0 —

15

I

to

y

w a

Z

6

I\

1

0 ^

1 1

k ^

0 6 1 16 2 2.6 7 95 ♦ 45 6 65 •6^ TIME (HRS ) Rite Control LAw Deployment

Figure 2.4.3 Constellation's in-plane angle vs. time.

Initial value = 20 0 .

I'1 -T, 1-,-r,-, T`­

r TT7 T - f-I I I I I 1 t T- T 1 - TT r,^ T 1-T 7 rl 1 .

'T T rTITT TTTT-71 7 1 11.1

a

E e

0 161

8F 16

-

a

w

t7^1

il

7.

0

E

0

I

—

3

N

-U-161-

♦E-Ie

I I .1.1 u l i. 111 1 _i_l-i ELI a u 1-I i i I I-LA 1, 1 t 1 1.1 L l.i 1 1 111J ll.i1- L t..l 1 1 1 1 1 1 A-1 1-4- k 0 6 1 16 2 26 3 ;1S 4 45 6 55 8

TIME (Hi(s) kete Cmllrol IAw Deployment

Figure 2.4.4 Middle mass lateral deflection vs. time.

Initial value = 0. 
ORIGINAL PAGE IS

OF MR QUALITY Page 23

Figure (2.4.4) shows that the lateral deflection a of the middle mass stays

very small if the deployment is initialized with zero lateral deflection. This 

Is because the coupling between lateral oscillation and libration is very weak 

when the middle platform is located at the system c.m. as mentioned before. Any

other location of mass m 2 creates larger lateral deflection and consequently a

higher acceleration level. Figure (2.4.5 a,b) show the respective trajectory of

the lower and upper mass. Negative numbers mean trajectory outwards from the

earth. Figure (2.4.6) shows the tension in tether 1 which is almost exactly

equal to the one in tether 2. Figure (2.4.7) is the horizontal component of the

acceleration of mass m 2 . The horizontal acceleration component is smaller than

the accuracy of our computer in double precision (the middle mass initial lateral deflec^;ion is equal to zero) and for this reason it must be considered only

as an order of magnitude. Vertical acceleration components and acceleration

modulus are not shown here because they are comparable to the horizontal component. Notice also that the vertical component has no bias due to the center of

mass-orbital center offset as mentioned before.

It is also interesting to see what the dynamic response and the acceleration level would be if the system starts the deployment with a small alignment

error. Some simulation runs were, therefore, performed starting the deployment

with different lateral deflections of the system. Results from a simulation run 

with an initial lateral deflection e = 0.05 m (2 inches) are shown in the

following figures. The oscillation amplitude of the lateral deflection stays

approximately the same throughout the entire simulation as shown in Figure

(2.4.8). The damping of this type of oscillation deserves further study. Damping terms were not included in the control law this time and it would be inter-
Page 24

6OJ FF

r

Ton L

G

E

1100

600

rk

4,

a

r

. 4

300 F

C

200 -

h

F

100

0

J_ i_ 1

- 110 -40 -20 0 20 40 60 60 f00 120 1e0 ]so

%1 (METERS) Ml Trsleolory Side View

(a) Lower Mass ml

r•

r

1000 -

-2000

r

r

-3000

a

6000

-6060 -

- 700• -

-6900 -

C

-11000

-1600 -140C

a

J

-7

7

1200 1000 -606 -600 -100 2011 0 1200 400 600

X9 (MFTER>) M/ Tre)erlorl Sidr V^r

(b) Upper Mass m3

Figure 2.4.5 Trajectory's side view of the lower and upper mass
7r. 14

IF 19I

v (11

'sr 1'I I

i.+ 1

i^'r 1

z^

.f 1)1

Page 25

^^-r-.Ter-Tr 7'TT rZ'T+"rT'T'T'7•T'TT'T T-r-t-rr ^

4.

e5p

= 200 —

t J

15n —

ORIGINAL PACE 13

1DO Of POOR QUALITY

a

50 —

L _

D +^^U 1 IU 1 25 3 JS ^ 15 6 65 d

TIME ( b r! M.I. Cottrol L.• Deployment

Figure 2.4.6 Tension in tether 01 vs. time.

..r....r........,( I I .I 1 .... 1 ., .(...rrt-r..I, ..T..-r.I....(..-.II'!

w

^.^, 1 II I uu

Yr l I^;1 ij^ I^ I !!^

^ I w `I! II

^ I I V ^ PI

1

r

r. . 1 I' '. I • ey ^ I !. S .. A

wr im. , 1 I+r^•• r i.lrnl I . L^pl.y menl

Figure 2.3.7 Horizontal acceleration component of the middle

mass vs. time for an initial lateral deflection equal

to zero.
Page 26

eating to see how effective n tension control law could be in damping this

oscillation out. Figures (2.4.9), (2.4.10) and (2.4.11) show the horizontal 

component, vertical component and modulus of the acceleration rin the middle

mass. All of them are of the order of 10 -6 m/sect (10 7 9) but these values are

strictly dependent on the initial lateral deflection and in any case they should

be fine-tuned (and possibly improved) with the use of a tension control law.
(MIGINAL PAGE IS .... ..

QF POOR QUALITY

.a ^ i IIi

^II

^ ;

^II iIA PI ^i I I ^ I

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j I I II I^ i 1 1, 11 1, i ! !j ;; • ,^ _

L

02

II

G4

-05

1 S 2 2 5 3 35 4 L 5 S 6

TIME (H, - ) Rate Control La . C

'

ePI Dymen.

Page 27

Figure 2.4.8 Lateral deflection of the middle mass vs. time.

Initial value = 0.05 m.

4i-6

P

I I ^ ^ I

H-6

r ' r ^ ' ^ ^ ^ I I

II

^I

^^ I I I. I^ -

_

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^ ^I f ^ I I ^ ^ I^ I

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it

I I

II ( I

Ii 1 I I '

!^ { '

II I I^ I

1E-6

IC

- 0-6

2 26 3 35 44 S 5 65 6

TIME lm p S' ROte Control IA• Dplovmn'.

Figure 2.4.9 Horizontal acceleration component of the middle

mass vs. time for an initial lateral deflection = 0.05 m.

_1T
ORIGINAL PACE IS

OF POOR QUALITY

Page 28

7--

IE-6

N I I ' ^I I

^ I ^

!S SE-7 .-

1 I

II I I I

, I .I

t I I I I I. 1 1

I I. I ' ! I

II I I I

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I ^ I

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1 I ! I I

^j III (' :^

I

I I

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I I I I; I

! I iI I

II j I i II I IIiI I (I l i

i' ^ ,I II it

(I

i ^ ^ I

III II I I + II I it

SSE_,_ li f II ^I II t -+

!! :

II II II II II I II I I

! ^ I ;. ^ I ^L,-

F

>

I,i

I!

II

-IE-6

0 5 1 15 2 25

335

4 45 5

b.. 6

TIME (MR-) Rote Control Lao Deployment

Figure 2.4.10 Vertical acceleration component of the middle

mass vs. time for an initial lateral deflection = O.OS m.

4 SE -6 7—

7^

L II

4F E

FSE-S — l i C

V

j,

2E-b

I, C^ ^ t 4F { i

I i iIII^

I^ ^ I rl ' I 1

I

N ' (I I i

_

I -

^

1t-6'--

I 1 7

L li^ I A

15 2 25 3 35 4 45 5 55

TIME IHP 1 1 Rate Control Lao Deployment

Figure 2.4.11 Acceleration modulus of the middle mass vs.

time for an initial lateral deflection = 0.05 m.
Page 29

2.4.3 Concluding Remarks -

The simple rate control law derived in this study proved to be effective in

deploying a single -axis constellation with three masses. By using the same time

constant for tether 1 and tether 2 and simultaneous transitions from one phase

to another, the middle mass m2 can be maintained at the center of mass of the

system. This strategy reduces the acceleration level induced on the middle

platform during deployment. When the deployment is over, the low-g platform can

be moved to the orbital center of the system in order to reduce the vertical

component of the acceleration. In the system under investigation in this Quarterly Report the orbital center is only 1.2 m lower than the center of mass. It

is also important to minimize the initial lateral deflection of mass m 2 with

respect to the line through ml and m3 in order to achieve a low acceleration

level.

Tension control laws should be investigated to reduce further the final

libration and most of all to understand how effective these control laws are in

abating the lateral oscillation of the system.

3.0 PROBLEMS ENCOUNTERED DURING REPORTING PERIOD.

None.
Page 30

4.0 ACTIVITY PLANNED FOR THE NEXT REPORTING PERIOD

During the nex t_ reporting period the analysis of the deployment phase of a

single-axis constellation with three masses will be further developed. Tension

• control laws will be investigated. The effect of elastic tethers will be also

considered. The tether control law that proves itself most effective for deployment will be verified by SKYHOOK or by a similar high fidelity model specialized to simulate the dynamics of three-mass constellations.

In addition to the above, the dynamics of the constellation when the middle

mass travels along the tether in between the two end masses will be studied with

the mathematical model derived during this reporting period. This model is

perfectly suitable for investigating the situation where the middle platform

travels along the tether.