Analytical Investigation of the Dynamics

of Tethered Constellations in Earth Orbit (Phase II)

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Contract NAS8-36606

Quarterly Report #4

For the period 22 December 1985 through 21 March 1986

Principal Investigator

Dr. Enrico C. Lorenzini

March 1986

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
Analytical Investigation of the Dynamics

of Tethered Constellations in Earth Orbit (Phase II)

Contract NAS8-36606

Quarterly Report #4

For the period 22 December 1985 through 21 March 1986

Principal Investigator

Dr. Enrico C. Lorenzini

Co-Investigators

Mr. David A. Arnold

Dr. Mario D. Grossi

Dr. Gordon E. Gullahorn

March 1986

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
CONTENTS

Page

Abstract 3

Figure Captions 4

SECTION 1.0 INTRODUCTION 5

2.0 TECHNICAL ACTIVITY DURING REPORTING PERIOD AND PROGRAM

STATUS 5

2.1 A Particular Case Of "G-Tuning." Reaching The Zero-G Acceleration Condition On The Middle Mass 5

2.2 Middle Mass Travelling Erom One Tether End To The Other 16

2.3 Concluding Remarks 27

3.0 PROBLEMS ENCOUNTERED DURING REPORTING PERIOD 27

4.0 ACTIVITY PLANNED FOR THE NEXT REPORTING PERIOD 28
Page 3

Abstract

The "g-tuning" maneuvers of a 3-mass, vertical tethered system are dealt

with in this quarterly report. In particular, the case of reaching a zero-g

acceleration level on board the middle mass from a non-zero initial condition is

analyzed. A control law that provides a satisfactory transient response is

derived.

The constellation dynamics in the case of the middle mass travelling from

one tether tip to the other is also investigated. Instabilities that take place

at the end of the maneuver are analyzed and accommodated by devising suitable

damping algorithms.

RKECEDiNG P&GE .BLAJtC i$2T
Figure Captions

Page 4

Figure 2.1.1 Geometry of the 3-mass tethered system at the

beginning and at the end of the "zero-g-tuning" maneuver.

Figure 2.1.2 a-i Dynamic response of the 3-mass tethered system during

a transfer maneuver, along the tether, of the middle

mass from the Space Station to the system C.M.

("zero-g-tuning" maneuver).

Figure 2.2.1 a-m Dynamic response of the 3-mass tethered system

during a transfer maneuver, along the tether, of the

middle mass from the Space Station to the opposite

tether end.

Figure 2.2.2 Tether tension vs. time in tether #2 in the case of

the transfer maneuver of the middle mass from the Space

Station to the opposite tether end. The tension

profile obtained by adopting an adaptive control logic

is compared to the non-adaptive control strategy.
Page 5

1.0 INTRODUCTION

This is the fourth 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. This report covers the

period from 22 December 1985 through 21 March 1986.

2.0 TECHNICAL ACTIVITY DURING REPORTING PERIOD AND PROGRAM STATUS

2.1 A Particular Case Of "G-Tuning." Reaching The Zero-G Acceleration Condition On The Middle Mass.

The analysis of "g-tuning" in the 3-mass, vertical tethered system was

started in Quarterly Report #3. In that report two different cases of "gtuning" were analyzed in order to devise a control law that gives a satisfactory

transient response. More simulations have been performed during this reporting

period aimed at improving the transient response during the maneuver. The particular case of "g-tuning" that has been recently investigated is related to the

achievement of zero-g acceleration condition on board the middle mass starting

from a non-zero acceleration initial condition.

Since the gravity gradient field is linearized in our present computer

code, the zero-g point coincides with the system C.M. The maneuver described in

this section starts with the middle platform placed 10 m away from the reeling

mechanisms on the Space Station. Subsequently the platform is moved toward the

C.M. position of the system pertaining to the final configuration of the constellation. The middle mass, therefore, has to reach the final C.M. position

with zero velocity and zero acceleration. For this simulation the modified
Page 6

hyperbolic tangent control law adopted in Quarterly Report #3 for the "g-tuning"

(equations 2.6.1) has been improved. That control law has been further modified

in order to have a slower acceleration phase at the beginning of the maneuver.

The new control law is therefore as follows:

£1 = £10 + A£c [tanh(at)]7

£2 = f-20 - A£c [tanh(at)]7 (2.1.1)

The exponent 7 > 1 slows down the acceleration phase and speeds up the deceleration phase. The net result is very positive since the previous control law had

a too rapid acceleration phase and a too slow deceleration phase. In the simulation, shown later on, a = 1/1000 sec"1

 and 7 = 4 have been adopted. The

position of the system C.M. at the end of the maneuver is estimated beforehand

in order to compute A£c. With reference to Figure 2.1.1, A£c is given by:

~ £cio (1 + 2

1 + 2 Tlt/(EA)

 v '

where £cMif is the distance of the system C.M. from mi at the end of the maneuver, £clo is the initial controlled tether #1 length and Tit is the final tension in tether #1.

We also have:

cnif = m3) (2.1.3)

a _ CCIQ + lc2Q /2 i

tot'f

 1 -f (£cio + £c2 * \ • •

where £tot,t is the overall constellation length at the end of the maneuver and
Page 7

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Initial Stage Final Stage

Figure 2.1.1
Page 8

(ml+ma) . £cj.o and £C20 are respectively given by:

-10

1 + 2 T10/(EA)

(2.1.5)

, _ £tot,0 ~

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C2° ~ 2 T20/(EA)

where TIO and Tjo are the initial tension in tether #1 and tether #2, which are

respectively given by:

£CMIO

(2.1.6)

(£tot,o — £OMIO)

In equations (2.1.5) it is assumed that at the beginning of the maneuver we have

£1,1 = £it and £2d = £2t, where £ld and £2d are the damper lengths for damper #1

and §2 respectively, while £lt and £2t are the tether stretches for the associated tether segments. The conditions tld = £it and £2d = £2t imply that the

longitudinal dampers are tuned to the longitudinal tether frequencies at the

initial tether lengths. It will be shown later on that this is true during the

entire maneuver because of the adaptive control logic adopted, in this particular simulation, for the longitudinal dampers.

The initial distance between the system C.M. and mi is given by:

. ni3£tot,o •+

mtot

.

(2.1. /)

The last unknown quantity in equation (2.1.2) is the final tension in tether #1

that is given by:
Page 9

Tlf = 3n2meqi,3 /*.t,f (2.1.8)

We can therefore compute the value of A£c on the basis of the initial geometry

and masses of the constellation. In the present simulation, with the platform

masses as shown in Quarterly Report #3, an initial length for tether #1 £10 =

10m and an initial overall length of the constellation £tot,o = 10 km we obtain

A£e = 302.041 m. This simulation has been run in a way slightly different from

the simulations reported in the previous quarterly report. A station-keeping

phase (no displacement of the middle mass) of 1000 sec precedes the maneuver in

order to reduce the effect of imperfect initial conditions. Moreover the longitudinal dampers are adaptive: the gains are varied with time in such a way that

the damper is constantly tuned to the associated tether longitudinal frequency

(it changes with length) and the damping coefficient is .9. Nevertheless, it is

evident from the results shown in Figure 2.1.2 a-i that the 1000 sec stationkeeping phase is not long enough to damp out completely the initial longitudinal

oscillations. No further investigation, however, has been performed on this

peculiar station-keeping phase characterized by a very short tether length for

tether #1 and high frequency longitudinal oscillations. Figure 2.1.2a shows the

length variation of tether #1 vs. time. The initial station-keeping phase and

the smooth start of the maneuver are clearly shown in the figure. The longitudinal damper tether length for tether #1, shown in Figure 2.1.2.b, has the same

shape as the tether length variation because the damper is adaptive. Figures

2.1. 2c and d show the in-plane angle vs. time and the 9-6 phase plane respectively. The 6 variation is very small and a small residual angular velocity is

still present at the end of the simulation. This residual velocity would be

damped out completely in a longer simulation run. The lateral deflection of the

middle mass € vs. time is shown in Figure 2.1.2e while the phase plane e-e is

shown in Figure 2.1.2f. It is evident from these figures that the e oscillation
Page 10

is perfectly damped out at the end of the simulation. Figures 2.1.2g and h show

the vertical and the horizontal component respectively of the acceleration measured at the middle mass. As mentioned before, the transient oscillations due to

imperfect initial conditions are not completely abated at the end of the station-keeping phase. It is possible that the delay introduced by the numerical

integration in adjusting the frequency of the longitudinal damper to the longitudinal oscillation frequency has a role in the transient response during station-keeping. Since the delay is comparatively more significant at high-frequency (short tether length) than at low frequency it is plausible that these

transient oscillations disappear when the tether is lengthened. All these hypotheses have not been verified because the initial transient response does not

affect the final steady state. The two above mentioned figures show clearly

that the acceleration components at the end of the run go to zero as expected.

Note that a perfect zero condition at the end of the simulation run is due to

the absence of steady state perturbations (e.g. Jj gravity term and air drag) in

this simulation. Figure 2.1.21 shows the tension of tether #1 vs. time while

the tension in tether #2 is not shown because it is of no significance.

As a conclusion to this "zero-g-tuning" simulation we must point out that

this simulation run can be also viewed as an alternative deployment strategy of

the 3-mass constellation. In the alternative deployment sequence the end mass

is deployed from the Space Station following a conventional deployment maneuver,

for example, like the one adopted for the TSS satellite. Subsequently the

middle mass is moved along the tether from the Space Station to the zero-g point

of the system according to the strategy explained above.

A possible improvement to the control law developed in this section is the

development of a closed loop control law in which the distance between the

middle mass and the system C.M. is fed back into the control law. This modifi-
Page 11

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Page 15

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cation makes the system response more insensitive to perturbations.

2.2 Middle Mass Travelling From One Tether End To The Other

The middle mass of a 3-mass vertical tethered constellation can be used to

service the end mass. In this application the middle mass is moved from the

Space Station to the end mass and vice versa in order to transfer materials

between the two platforms. In this section the motion of the middle mass from

the Space Station to the end mass is analyzed. This case is the most critical

of the two from the point of view of the instabilities which take place when the

middle mass approaches the end body since the end mass is less massive than the

Space Station. The control law adopted in this maneuver is like the one represented in equations (2.1.1) of the previous section. The procedure for computing the value of the length variation A£c is also similar to the one described

in the previous section if the distance from mass #1 to the final position of

the system's center of mass £CMIF is replaced by the distance between mass #1

and the final position of the middle mass £jf. Two cases have been analyzed:

in the first case the longitudinal dampers are adaptive while in the second case

the dampers are non-adaptive. Both simulations start with an initial tether

length for tether #1 equal to 10m. An initial tether length different from zero

accounts for a deployment boom between the middle mass and the Space Station;

at the same time it makes possible the efficient integration of the equations of

motion at the start of the simulation. The longitudinal oscillation frequency

is in fact dependent upon the tether length and increases to infinity for zero

tether length. In both simulations the value of the time constant I/a for the

hyperbolic tangent control law is 2000 sec while the exponent 7 has been assumed

equal 4. In the non-adaptive case the longitudinal dampers are tuned to the
Page 17

average tether length of 5000 m. Therefore the dampers have the best efficiency

when the middle mass is crossing the mid-way point between the Space Station and

the end mass. In both simulations the maneuver is preceded by a station-keeping

phase of 1000 sec in order to abate the transient motion due to imperfect initial conditions. However, since the non-adaptive longitudinal dampers are inefficient for short tether lengths the abatement of the oscillations due to the

initial conditions is better in the case of the adaptive control. The dynamic

response of the system in the case of the adaptive control is shown in Figures

2.2.1a-m. Figure 2.2.la shows the tether length variation of tether ttl vs.

time. The tether length variation of tether #2 is the complement to the total

tether length. Figure 2.2.1b shows the tether velocity of tether #1 vs. time

while Figure 2.2.Ic depicts the length variation of the longitudinal damper

associated to tether #1. Since the damper is adaptive, its length variation is

proportional to the associated tether length variation. Figure 2.2.Id and e

show respectively the in-plane angle of the constellation vs. time and the phase

plane 6-8. Both figures indicate that the overall in-plane oscillation is very

stable and well damped during the maneuver. The maximum in-plane angle does not

exceed 3.5°. Figures 2.2.If and g depict the lateral deflection (e) of the

middle mass with respect to the line through mi and mj vs. time and the phase

plane e-e respectively. We can conclude that the lateral oscillation e is also

very stable and well damped. The damping is provided by the feed-back control

loop associated with the reeling mechanism. Figures 2.2.Ih and i represent the

vertical and horizontal component respectively of the acceleration measured on

board the middle mass. It is evident from these two plots that the vertical

acceleration component of the middle mass reaches the value consistent with the

final offset from the system C.M. while the horizontal acceleration component,

perturbed from rest conditions by the middle mass motion, tends to zero at the

end of the maneuver. Figures 2.2.1J and k show the tension in
Page 18

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Page 20

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Page 21

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Page 22

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Page 23

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Page 24

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tether #1 and tether #2 respectively. The tension in tether #2 shows clearly

the frequency increase of the longitudinal oscillations when the tether length

is shortened. Figure 2.2.11 is the expanded side view of the trajectory of the

middle mass (a negative z meaning outwards) while Figure 2.2.1m is an isometric

plot of the same trajectory.

The simulation without the adaptive control logic is similar to the previt

DUS simulation except for the longitudinal oscillations in tether #2. Such

oscillations show a tendency to diverge. A comparative plot of the tension in

tether #2 is drawn in Figure 2.2.2. In this figure the solid line represents

the adaptive case while the dashed line is the non-adaptive case. The two lines

do not overlap because of a slight difference in the final tether length at the

end of the simulations. However the figure clearly shows the better performance

obtained by using the adaptive control logic. In the present simulation the

diverging oscillations for the non-adaptive case are not critical because of the

relatively slow time constant adopted in the control law that allows enough time

for the damper to damp out the oscillations. A faster time constant, however,

Would create a much more critical situation if a non-adaptive control logic is

adopted.
Page 26

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2.3 Concluding Remarks

The newly modified hyperbolic tangent control law provides the capability

for fast and smooth transfer maneuvers of the middle platform along the tether.

Two important maneuvers have been dealt with in detail in this report. The

first maneuver consists of the achievement of "zero-g" conditions on board the

middle mass starting with the middle platform initially in the vicinity of the

Space Station. The second maneuver consists of a transfer maneuver of the

middle platform from one tether end (Space Station side) to the opposite one.

An adaptive control logic has been adopted for the longitudinal dampers. Such

control logic is necessary in the second type of maneuvers especially when fast

time constants are used in the control law.

3.0 PROBLEMS ENCOUNTERED DURING REPORTING PERIOD

None
Page 28

4.0 ACTIVITY PLANNED FOR THE NEXT REPORTING PERIOD

In the next reporting period we will initiate the computer implementation

of the two-dimensional multi-mass (more than 3) simulation model. Preliminary

test cases to check the computer code will also be run if the computer code

implementation proceed according to schedule.