Active control of synamics transfer fucntions for a flexible spacecraft

•

October, 1979

ACTIVE CONTROL OF DYNAMICS TRANSFER FUNCTIONS FOR A FLEXIBLE SPACECRAFr

by

S. C. Garg

."

ACTIVE CONTROL OF DYNAMICS TRANSFER FUNCTIONS FOR A FLEXIBLE SPACECRAFT

by

ft

Preface

This report contains the results of investigations carried out in support of the Hermes spacecraft dynamics experiments program, funded by the National Research Council of Canada under grant

S7309,

and indirectly by the Communications Research Centre of the Department of Communications, Government of Canada.

The author is grateful to Mr. James B. Hawley for conducting the numeri-cal numeri-calculations wi th enthusiasm, and to Dr. Peter C. Hughes for encouragement in completing this investigation. Any errors in theory or computation remaining in the report are certainly the author' s om..

Abstract

This report deals with flexible spacecraft consisting of a central rigid body and flexible appendages. The dyna.mics of each appendage are tre.,ated in

general, introducing transfer functions for point excitation in translation and rotation, and the transfer matrix for the entire spacecraft is thus derived. Details are then given for pitch/twist attitude control of Her.mes (CTS), deriving transfer matrices in closed form. The expressions are found useful in several respects, being devoid of modal truncation. A scheme is examined which modifies the spacecraft transfer function· actively using

• acceleration feedback. Numerical results are obtained for Hermes, and then

confirmed fram an attitude control simulation incorporating control

nonlinear-ities. The effects of' saturation are examined. Dynamics modification under

ground control is also examined, including time delays, and an experiment of this type is proposed for Hermes •

1. 2.

3.

4.

5.

6.

CONTENTS Pref ace Abstract Notation INTRODUCTION TRANSFER FUNCTIONS HERMES PrI'CH DYNAMICS 3.1 Inertance Expressions 3.2 Special Cases

3.3 Comparison with Modal Results MODIFICATION OF DYNAMICS

4.1 Acce1eration Feedback 4.2 Effects of Time De1ay

4.3 Attitude Control with Modified Dynamics SIMULATION OF AN EXPERIMENT

5.1 The Simulation Program 5.2 Simulation Results 5.3 Comparison with Theory CONCLUDING REMARKS

REFERENCES TABLES FIGURES

APPENDIX A: PASSIVE DYNAMICS MODIFICATION

APPENDIX B: DIGrI'AL FILTER DESIGN

APPENDIX C: MOT ION EQUATIONS AND INTEGRATION

APPENDIX D: SIMlJL.ATION PROGRAM LISTING

iii v 1 2 7 7 11 15

19

20 25 27 30 31 32 34 35 37

.

'

Notation

Most syIDbols used in the text are defined below, except for syIDbols used pur e ly. loc.ally.

1\:

. A(~) A(x,y, s)

B, Be

C, Cy D(s) E(s) F(s) F -e

!(E,

EJ.)

Ha(s) It(~) I I . ae "-K K(s)

Functions defined in Eq. (3 .la)

Partial-fraction c'oefficient in Appendix B Acceleration transfer function, Eq. (4 25g)

Laplace transform of acceleration at a point (x,y) on the membrane

Functions defined in Eq. (3.10) cosh as~, cosh asy

Dynamics transfer function, Eq. (4.25f) Electronics transfer function, Eq. (4.25a)

ButterwoFth Filter transfer function, Eq. (4.25h) External force on centre body

Kernel of elasticity distribution integral, Eq. (2.6) Analog transfer function in Appendix B

Discrete-time equivalent of Ha(s) Moment of inertia of rigid spacecraft

Inertia matrix (or scalar) of rigid appendage Array inertance in Section 3ff

Inertia matrix (possibly scalar) of 'rigid central body; for ~ see Eq. (3.39)

Equivalent inertia or "inertance" Inertia quantity defined in Eq. (4.18) lmaginary part of coefficient in Appendix B Constant acceleration feedback gain, lbf-sec2

. ,

Gain in ground acceleration feedback loop, bits/milli-g Acceleration feedback transfer function

~

K s

KT

M(s) N N(s) l' p(s) T

Pulse-width modulator gain, vo1tsjbit Sensor gain, deg/rad

Motor torque gain, ft-1bs/vo1t

Motor transfer function, Eq (4.25c)

Number of modes kept in truncated expansion; number of thruster excitation pulses.

Earth sensor transfer function, Eq. (4.25d) Array tension, 1bf

Pulse-width modulator transfer function Rea1 part of coefficient in Appendix B

sinh as~, sinh asy

Torque acting at the base of solar array; in Appendix B,

sampling period

Control torque (on centre body) Disturbance torque (on centre body)

Transfer matrix for generiè spacecraft, Eq. (2.19) Attitude transfer matrix, Eq. (2.19)

Torque app1ied at tip of solar arr~, in Appendix A

Matrix. transfer function coefficient; so:].uti.on of Eq. (~.10)

Similar to

IJ.'

solution of Eq. (2.11)

Lower-Case Roman a a

-a m Cl r' c r' dO Cl t Ct Defined as

Jaw/p

Vector from centre oody mass centre to appenqage attachment point

Acce1eration measured on array Root and tip damping coefficients

Defined as (12/w~) times (c;, ci)

Translation of centre body mass centre

d n

!(!.l)

fes) f' (s) f~, f_{E ,} fs k k n k' r' k' t

~r'

kt ~

IIb

m_{r, mt}

* *

m r, mt n

E'

E~ s sk sn(Y) t v(x, y, t) w wek) x

*

'i Xt' xt x. (k) ~ Y

Gain in modal ,acceleration transfer. function

Force acting on appendage at location E~

Closed-loop characteristic function df/dS, where fes) is analytic in s

Transfer functions defined in Eqs.

(3.21), (3.23), (3.25)

respectively

Order of Butterworth filter; in Appendix B, discrete-time stage number

Gains in unconstrained modal expansion of inertance

Root and tip spr~ng constants

Defined as (12/wE ) times (~~, kt)

Length of solar array Mass of central rigid body

Root and"tip piece masses

Dimensionless values of (m_r, m_t), Eq.

(3.39)

Lead-to-lag time constant ratio in E(s)

Displacement vectors to mass elements in flexible structure Laplace transform variable

Poles of H~(s) in Appendix B

Normalized mode shape deflection at y Time

Reflection of array point (x, y) Width of solar array

Input sequence to digital filter Width-wise coordinate on array

Value of x at accelerometer location; x~

=

Xt/~

Output sequence from i th block of digital filter

z Sampled-data transform variable

Impedance functions at root, tip respectively, defined in

. Eq. (3.U)

Upper Case Greek

e

Determinant defined in Eq. (3.11); time delay of sensor

in Section

4.3;

width of thruster pulses in Section

5;

in

Appendix A, def'ined by Eq. (A.7)

Rigid sI'acecraf't pitch angle in Appendix C Summation symbol

Transfer-matrix coefficientdefined in Eq. (2 .15) Similar coeff'icient defined in Eq. (2.15)

th

Natural frequency of n appendage mode

LQwer-Case Greek

a,a

_n a(y) 1)

eer)

....

-e

8 T

Angular accelerations in Appendix C

Rotational flexible deflection of solar array

Real part of pole in Eq. (B .3)

Normalized shape of nth constrained mode

Normalized shape of nth unconstrained mode

Imaginary part of pole in Eq. (B .3)

Damping factor of flexible modes

Nondimensional length coordinate,

=

y/t

Deflection vector of generic structure at r Pitch deflection of rigid centre body

,Attitude deflection "vector" of rigid centre body

Coordinate of' nth flexible mode in Appendix C

Area density of' solar array, slugs/ftE

w*

Miscellaneous

()

( ) I (')

II( )11

Butterworth filter time constant

T orque exer e a ou t d b t r~. . g~ d een ra t 1 bod y mass cen re y t b · ~ th'

lppendage

Phase angle in Appendix C

t t · · t t b · th d

Force exer ed a r~g~d cen ral body mass cen re y ~ appen age th

Unconstrained natural frequency of n mode

Nondimensional frequency, Eq. (3.39)

Derivative of ( ) with respect to t

Derivative of ( ) with respect to y, except in c;,

Ct' k;, kt

Laplace transform of ( )

Norm of ( ) in appropriate sense

Defined to be equal to

~---

---•

1 . INTRODUCTION

It has by now been amply demonstrated that structural flexibility of spacecraft designs is a necessary evil that the control system analyst must

recognize and tame. Several symposia have recently been devoted to the

dynamics and control of flexible spacecraft (Refs. 1, 2, 3). This field

now stands on its own and faces monument al challenges with the advent of

large structures in space (Refs.

4,5).

The present report is concerned with a small segment of this field: the attitude control of a rigid spacecraft body with attached flexible

appendages (Fig. 1). This problem'has received a general treatment recently

(Refs.

6, 7),

considering both natural or free motions and forced response.

The latter can be obtained as the sum of modal coordinates, each of which satisfies a second-order linear differential equation uncoupled from the

others. Similar summations may be used with assumed shape functions (Ref.

7).

In any event, the final result obta!ned is a set of simulation equations, giving a system model of large (theoretically infinite) order. This result may be obtained explicitly using continuum mechanics, or for complex struc-tures, using approximate spatial discretization (finite elements) on a digital

computer. The high-order system model poses a problem for the control--systems

analyst akin to other large-scale' systems (see Ref. 8 for the state-of-the-art

in this broader context). The knotty prob~ems of adequate minimal-order

models, parameter changes and uncertainties, proper placement of sensors and actuators, etc, are currently active research areas. However, one dominant question is that of modal truncation errors.

The viewpoint taken in this report has two important features. First, it does not aim essentially at simulation equations, although they can be

derived herefrom in a straightforward manner. Likewise, modal expansions

are not introduced. Instead, flexible appendages are represented by transfer functions (more precisely, transfer matrices) relating the force/torque to the translation/rotation at the point of attachment to the rigid central body. For simple geometries, closed-form expressions can be obtained which ideally are devoid of truncation error. The second feature of this report is that active modificaticn of the spacecraft transfer matrix is considered. The best scheme advanced herein for this task uses acceleration feedback and

actuators located on the' central body. Conceptually, this feedback control

can be bepter thought of as being distinct from the attitude control function, .

at least for linear systems where the' principle of superposition holds.

The abave two methods are examined below in detail for a simple, yet

practically important situat1oll. This is the pitch/twist dynamics and .

attitude control of Hermes (alias CTS - Communications Technology Satellite), shown in Fig. 2. This problem has been analyzed by modal expansion methods

in Refs. 9 and 10. These methods were also used in Ref. 11 for the design

of passive dampers. The ' .de'S,igb. of Ref. 11 considered two modes. By

com-parison, Section 3 of this report shows a solution method that is simpler,

while eliminating modal truncation error. The non-modal approach is next applied to another modification of the structural dynamics - stiffening of

pitch/twist dynamics by acceleration feedback~ The advantages of this

approach stem from the fact that the analysis (modal ·or non-modal) done for

a particular sub-structure need not be repeated when additions or changes are made to it. In the particular case exemplified here, further advantages

accrue from the simplicity of the closed-form solution, which was derived with much less effort than needed for a modal analysis.

The balance of this report is organiz,ed as follows. The next section

develops transfer matrix expressions for the general class of spacecraft

depicted in Fig. 1. Section

3

gives specif'ic dynamical development of this

theory for pitch/twist dynamics of Hermes (Fig. 2). Section

4

considers

mödification of this interaction, in the sense of added passive damping and artificial stiffening using acceleration feedback. Computer simulation

results for this situation, including a detailed discrete nanlinear simulation

of the attitude control system of Hermes, are given in Section

5,

and show

good agreement with theory. A particularly interesting aspect is the effects and limitations due to nonlinearities, chiefly saturation, in the control system.

The original int ent for this work was to plan and conduct adynamics modification experiment in space on Hermes, using the spacecraft telemetry and command links, the on-board sensors and actuators, and additional signal processing in a ground-based computer. This concept introduces a time delay into the problem which is not negligible compared to the period of a flexible mode. The dynamics of a flexible spacecraft with a closed-loop cantrol system

and time delays is, to the best of the author' s knowledge, analyzed bere for the first time. The non-modal approach permits a direct at.tack on this problem

in the frequency domain. Closed-loop poles are found as the solution to a

complex transcendental equation. Detailed computer simulation is again

resorted to. The analysis, simulation results and comparison with theory are

all given in Section

6.

The final Section

7

is a dist.illation of the report

and its conclusions.

2. TRANSFER FUNCTIONS

Consider the abstract spacecraft model depicted in Fig. 1. of a rigid body, to which N flexible bodies (termed appendages) The following assumptions, removable if necessary, are made for analysis:

It consists are attached. the ensuing

1. The spacecraft is not spinning and is actively controlled in attitude.

2. The structural vibration frequencies are much higher than the mean orb i t

rate Wo'

3. The torques and forces applied to the centre'body (rigid) are dominant,

i.e., actuators are located here, although sensors':::may be on the appe'

n-dages.

The last assumption is usually satisfied in practice, although the principles remain applicable when i t is violated. The second assumption is similarly conventional and it is made to enable concentration on flexibility effectsj for example, attitude-dependent gravitational torques are neglected. The problem thus defined is similar to the one tneated in a general manner in Ref. 12. We shall follow the same spirit of treatment for generality. In particular , each elastic body will be characterized by an "el asticity dis

tri-bution" !\~, !:J.). This is an integral operator relating a force

t

at position

..

](:) =

J

K(E,

E~) i(E~)dEJ.

(2.1)

A

The 3x3 matrix f'unction F contains '.. inf'ormation on boundary conditions as

well as the governing partial dif'f'erential equations (the appendage being

-treated as a continuum). A volume integral is implied in (2.1). For appen-dages with a f'inite number of' degrees of' f'reedom, an equivalent representation using sums can be derived. The well-known reciprocity theorem f'or linearly

elastic structures implies that ."

(2.2)

The quantity

KCE,

EJ.) can aJ.so be interpreted as the def'lections

]CE)

resulting

throughout the elastic body in response to a point f'orce

l

at location

El'

It

should be noted that in (2.1), 5 ref'ers to the elastic def'lection and not to the totall d.isplacement. This mëans that rigid-body modes of' the appendage are excluded. The f'lexible body thus has three translational and three rotational

degrees of' f'reedom. Thil3'~also implies that

(2.3)

. f'or all

(E,

E~), where a suitable matrix norm is meant.

Considering now the central rigid body, let its mass centre undergo a .

displacement dOet) and let the body undergo an inf'initesimal (hence, vectorial) rotation 9. Let there be Nappendages, each connected to the centre body by

a.rigid point connection, i.e., one that transmits a f'orce ~ and a torque

Tl. to the centre body, localized at the point of' attacbment. The motion

equa-tions f'or the centre body are simply

"0 = F

_r.

_!l2

i Ilb~ -e

(2.4)

.. _i ~~ = T _e

_{- r.}

T

where Fe' Te are external f'orces, torques respectively. It is convenient to take Laplace transf'orms of' these equations, since algebraic, rather than dif'f'erential, relationships result. In the physical application it is known

that all responses of' interest are pbounded, and hence the Laplace transf'orms

exist. Denoting transf'orms by tildes and assuming zero initial conditions (customary in obtaining transf'er f'unctions) yields

,....,,...., ~

T_ S2 9

=

T -

r.

T

-Turning at tention now to a single flexible appendage, Fig. 3 shows the force and torque acting at the attachment point. The six degrees of freedom

referred to above are now constrained by the motion (dO, e) of the centre body. Let us denote the force.t per unit mass as

!

(I.e.-;

i

=

!

dm) ,and write the integral in (2.l) as a Stieltjes in~egral with respect to the mass distribution m(~). Then

]c.~)

=

,r

KC!.,

!.~) !(!.~)

dm ' A

(2.6)

where again a volume integral is meant. In the absence of any external torques or forces on the appendage other than

$,

T (the index i is dropped

for brevi ty), the following "inertial force fieldiT defines f over the body, where !.~ is measured relative to the attachment point:

-(2.7)

where a is a vector specifying the attachment point of the appendage relative to a reference fixed in the centre body at its mass centre. Substituting (2.7) into (2.6), af ter Laplace transforming both, yields

]'(i)

+ S2

,r

K(!.,

!.l.)

~(!.~)

dm A

=

,

{

-s2,-[K(!.,

!.~)dm

}(ao _

~x

j)

A + { S2

J

KC!.,

!.l.)~

x dm }

i

,

A (2.8)

This represents' an inhomogeneous integral equa tion for

'5

with 'a symmetrie, positive definite kernel. Under these conditions it is-known that a solution exists if the terms in ( } are bounded. Using the principle of superposition for this linear equation, we may write

or

where ~ and ~2 are solutions respectively to the following equations:

!2(E) + S2

J

!(E,

E~)

!2<.!JJdm

= {

S2

J

!(E,

E~)E1xdm

}

A A

(2.11)

The s.econd equation above , is simi1ar to en integra1 equation obtain.ed in

Ref. 12 for the unconstrained mode shapes of the entire spacecraft. In

principle, solutions of these integral equations, Y~(r, _{s) and Y2(r, s)}

can be obtained. Note that these wi11 be independënt-of the spacecraft

motion, end hence can be obtained once for each type of appendage separate1y. It now remains to re1ate these equations to the spacecraft motion. For this purpose, a balance of forces at the attachment point is taken:

i.

= -

J

!(E~)dm

(2.12)

A

However, a combination of (2.7), (2.9) yie1ds:

(2.13:):

Upon stibstituting this into (2.12) and recognizing ma as the tota1 appendage mass, one can write

where

"

",

"

,_

.

~!::'

ma! +

J

!~(E)dm

A

1'e

!::,

J

{!2(E) - EX -

!~(E)~x

} dm

A

(2.14 )

(2.15)

The matrices <ttl end <Pe are again specific to each appendage' (denoted by a

superscript i -wh en rlecessary) and can be found direct1y once y~ end Y2 are

known. These are a1so functions of s. The torque T cen be sImi1ar1y

obtained by balencing moments about the attachment point:

i

= -

J

!..~x I(E~)dm

A

(2.16)

where once again,

f

is given by (2.13). At this point it is appropriate to

la

D. -

J

EX EX

dm

A

(2.17)

Upon carrying out the operations in (2.16) one obtains a re1ationship simi1ar to (2.14): (2.18) where !8

~

la

+

J

EX !2(::)

dm +

J

_Ex

a

X

dm A A

(2.19)

~

D.

r

EX

(b.(r) + ,!}dm .U A

The matrices !8 and!d can be obtained direct1y from !~, Y2 and are specific

to each appendage. Co11ecting together equations (2.5),

T2.14)

and (2.18)

final1y yie1ds the results:

1 ,...,

- T

2 ·e s

(2.20)

The solution of these equations gives orientation af the centre rigid body, (as made in Ref. 12), the two sets of but in general they are coupled.

the motion of the mass centre and the Under appropriate symmetry assumptions equations in (2.20) can be uncoup1ed,

It should be ernphasized that equations (2.20) are true regardless of

the methods used to arrive at the transfer matrices

$i,

28' !d, !8' Several

possibi1ities exist:

(a) Exact analYtical solution of integra1 equation form in s-domain,

(b) Exact analytical-so1ution of governing differentia1 equation plus

boundary conditions. in s-domain.

(c) Approximate numerical solution app1ied to cases (a) and/or (b), as

feasib1e for pa~icular appendages.

(d) Exact or appr9ximate solutions based on expansions of def1ections in

terms of natural modes of each appendage (" constrained 11 modes),

Approximate solutions may be based on Ritz type methods,

The main point is that from the frequency-domain characteristics of eaèh appendage, determined sepa.rate1y in advance, one can obtain the response

. - - - -- - - -- --- - - -- - -- --

-of.the assembledstructure. There are clear advantages in this approach

in Cases (a) to (c) above. The Tesponse to ,a given force or torque input

can be easily determined using transforms • . When Te' Fe are generated by

an on-boardcontrol system, the stability and performance of the closed-loop system can be of ten analysed directly in the frequency domain. For design and synthesis purposes, frequency-domain optimal methods also may be appropriate (Ref. 13).

A number of applications and extensions of this approach are cöncep-tually straightforward. Some of these will be illustrated in the following sections. The important extensions are noted below:

(a) Closed-loop control system analysis, e.g. stability tests in frequency

domain.

(b) Closed-loop control system design and optimization in frequency domain.

(c) Spacecraft configurations like Fig. 1 with non-rigid connections, and

discrete mass, springs, damper elements.

(d) Dynamics of structures where flexible bodies are interconnected with point connections, including closed loops.

(e), pynamics of a chain of flexible bodies.

(f) Control system analysis with sensors mounted on flexible bodies.

Some of these are the subject of current research and will be reported on

separately.

3 • HERMES FrrCH DYNAMICS

This is an old problem to which a new approach is given in this

section. The Hermes spacecraft and its idealization from a continuum .

structural mechanics viewpoint have been detailed elsewhere (Refs.

9,

10, 14

J.

The pitch attitude motion interacts predominantly with symmetric twisting of the two solar panels. In this mode there is negligible motion of thé space-craft mass centre. The idealized schematic of one solar array is shown in

Fig.

4.

The root and tip pieces are rigid bodies, and the boom and the

"blanket" supporting the solár cells are treated as a uniform beam and a tense membrane respectively. The "elevation arms" linking the centre body to the root piece are modelled as a discrete spring and damper at the root. The boom in torsion acts like a discrete spring between the tip piece and the centre body. A rotational damper is also added at the tip as' shown in

Fig.

4.

This is a more gener al model of pitch/twist in'beraction than studied

previously. The membrane tension is, however, still assumed to be uniform

over a chord (along the x~axis). An x-y-z reference ~rame with origin at

the appendage attachment point is also shown in Fig.

4.

(

3.1 Inertance Expressions

It is more direct to start with the governing differential equations

however, equivalent, The tension P is constant in the y-direction, and is zero in the x-direction, Corisidering a small mass element on the blanket

(rrdxdy) and its force balance yields th~ eq~ation

~

.

v"

=

-cr(xë - ij)

where primes denote derivatives wi~h respect to y,

We

can substi tute

v(x, y) = -nx(y) . _(3.2)

to obtain

It:t" -

r:rvii

=

ow

è'

To this must be added l)oundary conditions arising from th,e motion equations of the root and tip pieces. For the root piece, we have

w/2

...1:

₁₂ m w_r 2(ë

+

ä(o)) =-k' _r a(o) - c' _r à(o)

+!

_w

'r

-x a'(o)dx (3.4) -"w/2

where the last term arises from the torque due to the flexible blanket.

Carrying out the integration and cancelling the factor (w3_{/l2), one gets}

where

k == l2k'/w2

r r '

An exactly analogous derivation for the tip piece yields

where

c

=

l2c '/w2

. t t

(3.5)

(3.6)

Equations

(3.3), (3.5), (3.6)

are the motion equations, which when

Laplace-transformed (iero ini tial conditions) yield:

d~

..

-,v

ci1

(0) + (m s2 + + k )&'(0) m s2e + p - c s = 0 r - dy r r r

'"

ci1

mt s2e + p dy (J,) + (m ts 2 + CtS + kt)cX( J,) = 0 (3.7)

Note that s is now a parameter and hence

a

is only a function of y.

Hence-forth, no special symbol wi11.be used to distinguish Laplace transforms

when this is c1ear from the context (e.g., whenever s is present). The solution of {3. 7) is straightforward:

a

= A sinh asy + B cosh asy - e

::::;>

a'

=

as(A cosh asy + B sinh asy)

Applying the boundary condi~ions leads to two simultaneous -equations for

A and B, which can be wri tten as

asP m S2 + C s +k r r r

C:)

=

c s +k (m_t s2

:r

CtS + kt)S _{(mt s}2 + CtS + kt)C r r e CtS + kt + asPC + asPS (3.9)

where S

=

sinh aJ,s; C

=

cosh aJ,s. The solution for A and B can be obtained

exp1i?itly: (3.10) wher.e z

=

m S2-+ C S + k . r r r r' (3.11)

The next step is to find the relationship· between torque and e, i.e. the

quantities corresponding to ~, ~e' Td, Te in equation (2.20). To do this

one must realize that there are two symmetrical appendages, one on each side of the centre body. Assuming perfect symmetry and that the boom centrelines

.§. of equation (2.20) are uncoup1ed. Our primary interest here is in !e'

,. the transfer funetion between torque and rotation e, which in this case is a scalar. To find i t we need the torque about the-attachment point:

t

T

=

1~

ow3

J

(ä +

ë)

dy +

1~mrw2(ä(

0) + 'e) +

1~

mtw2(à(t) + 'è) o

(3.12)

upon substituting the solution·(3.10) , (3.11) in a Lap1ace-transformed version of this equation and simp1ifying, one obtains af ter some reduction:

where

and Ae, Be, S, C are as defined in (3.9) to (3.11). The expression in [ ] above can be recognized as the inertia transfer function Ie. An a1ternati. ve name for this is the "equivalent inertia" IaeC s), also termed the "inertance":

T(s)

=

I _ae (s) • s 2e(s) _. (3.14 )

where Iae(s) is c1ear"from' equation (3.13).

An important property of Iae(s) expected on .an intuitive basis is that as s -+ 0 it should approach the appendage (rigid) moment of inertia. This is because for torque inputs of very low frequency the solar array would behave essential1y as a rigid body. This is useful check on the results

obtained above. Af ter taking 1imits careful1y, especial1y for % indeter-minate forms, one ean show that

1im

Ae

=

0,

s-+O 1im B s-+O

e

=

1 (3.15)

This is p1ausible if one notes from (3.8) that then 0: == 0, i . e. no .. f1exib1e deflections. Considering now Iae, the [ ] term in equation (3.13), the limit as s -+ 0 is dominated by the ( )Be term, and one obtains

ot

where Ia is the rigid (pitch) inertia moment of one array. Thus we can think of the inertance Iae as being constant for a rigid body, but depen-dent on the "frequency" s for a f1exib1e body. This is conceptual1y useful.

The inertance expres sion for the entire spacecraft the second of equation (2.20). Taking into account two dages in symmetrical twist, the dO term is zero (due to so that I (s) e I (s )

=

L + 21 ( s) e ~ ae that is,

Once again, 'IeeO)

=

I, the tota1 spacecraft inertia.

can be obtained fram,

symrnetrica1 appen-symmetry), and Tl.

=

T2,

8 8

(3.17)

It should be emphasized that the above result has been obtained with much greater ease than is the case using moda1 ana1ysis. The latter invo1ves finding the mode shapes and natural frequencies, the appropriate orthogona1ity condition, and another modal expansion to find the transfer function. More-over, in the present case the' transfer function has been obtained in c10sed form rather than as an infinite sumo In the deve10pment 1eading to (3.18) we have also generalized in a straightforward manner the results of Section 2, to inc1ude non-rigid connections to the centre body and discrete dampers or

springs. These extensions are not so easy in the moda1 approach. Further comparison of modal and non-moda1 approaches appears later in this section. 3.2 Special Cases

Simp1ified results for Iae(s) are now presented, for severa1 special cases of the above prob1em. The first case is the one treated in Refs. 9

and 10. i,

Case A - 'Simplest Model

In this and fo110wing two cases, the root damper and spring are absent, in the,sense Cr

= 0, kr

~oo. Since the latter implies a rigid cannection to the centre body, there is no 10ss of generalityin a1so setting mr

=

0, for then the rigid root piece inertia can be inc1uded as part of Ib. In Case A we further de1ete the tip spring and damper, so that

=k =c =0

t t '

It is c1ear that the appropriate condition on kt is kt

=

0, not 00. With the

1·

(Al~

'

)

k

~oo k

=

~C

+ asPS

=

_{mt s2c} + asPS

r ' r

Fram these expressions it is easi1y seen that

( aPS + mtsC ). Ae

= -

f ~ D,. - S + a.PC '

. mts This equation defines f~.

Case B - Tip Spring Added

B = 1

e

(3.20)

(3.21)

This case was analyzed in Ref. 10. Here the va1ues (3.19) still app1y except that kt is nonzero. The 1imits taken in (3.20) now yie1d

(3.22)

Hence

(3.23)

Case C - Tip D~er Added

This case was analyzed previous1y in Ref. 11 which obtained design criteria for rotational dampers. However, this was possib1e on1y with truncation error

(two modes were used). Although design criteria are not obtained here, the present non-modal approach permits exact and direct eva1uation of inertance. One obtains

(3.24)

and hence where A =-f

e

3' B

e

=

1 asPS +.m ts2C + (CtS + kt)(C - 1) f3 lJ. -(mt s2 + CtS + kt)S + asPC (3.25)

In all the above three cases, the inertance of a single appendage can be written as follows, from (3.12):

I _ae (s) =

~

₁₂ w2

r

~

_as (S + fel - c)} + mt(C - fS)] (3.26) where the appropriate factor f must be substituted. I t is noted that cases A, B, C have very similar results. There are two other cases of interest which are given below.

Case D - No Dampers

Here we have cr

=

Ct

=

0, and one obtains

(3.27)

lJ.

=

asPCz

t + a2s2p2. S - (zr ,ZtS + asPCz ) r

The inertance expression can be evaluated similarly to (3.26). The main significance of this case is that on the imaginary axis (s

=

jw),

Ae

and

Be

are purely imaginary. It then follows from (3.18) that le(j ) is purely real and hence, that the p·oles and zeroes of le(s) are all on the imagiriary axis. This property applies also to cases A, B which are special cases of the present one. lt is only the case without dampers which can be conven-iently treated by means of modal expansions, since the conventional modal approach does not apply if dampers are present. The non-modal approach is not thus restricted.

Case

E -

Root Spring Only

This case .is of interest in the dynamics of Hermes when the flexibility of the elevation arms is taken into account. Setting kt

=

Ct

=

Cr

=

°

yields directly,

A 6

_e

= (mtsEc + asPS)k r B

_e

6

=

-(m_ts2s + asPC)k r whence A = - f B

e

l.

e

Upon same simplification it is found that

k r Be

=

,

m S2 + k - asPfl. r r A

=

-fB

e

l.

e

(3.28) (3.29)

In these two cases, inertance expressions can be obtained from (3.18) . Deflections and Accelerations

The discussion so far has emphasized the response of the spacecraft to torques exerted on the centre body. In this connection it must be stated that forces and torques due to the environment which do not change appreciably with flexible deflections, but do act on the flexible appendages, can also be included in the above development by transferring them to a force/torque at the mass centre of the rigid centre body. Within the approximations made here, forces/torques due to solar radiation pressure and Earth's gravitational field can be treated in this manner.

The main deflection quantity is the twist angle a(y)., which from (3.8),

(3.10) iS',given bY,the Laplace transform:

ä(y, s)

=

(Ae(s) sinh asy + Be(S) cosh asy - l)e(s) (3.30)

For cases A, B, C this can be written as

ä(y, s)

=

(cosh asy - (1 + fes) sinh asy) )e(s)

wherethe factor f( s) should be substituted for each case. of the root and tip pieces ~e

a(o)

and

aCt)

respectively, the pitch rotation of the centre body, including so-called effects.

The acceleration at a point on the membrane is

am(s) (x, y, t) = -xë(t) - :xä(y, t) (3.31) The rotations and edescribes flexibility (3.32a)

which in transform form becomes

;3:' (s)

= -

xs 2(

e

+

ê1)

m

For cases A, B, C a simpler form is:

~m(s)

=

xs 2[cosh asy - f(s)sinh asy]e(s)

(3.32b)

(3.33)

(3.34)

These expressions yield the transform of the acceleration and/or deflection functions. In principle, the time history can be obtained by inverse trans-formation, and in most cases this can be done using the "fast" discrete Fourier transform. An alternative is to obtain simulation equations using either the modal approach or partial fraction expansions of the inertance (3.18) •

3.3 Comparison with Modal Re~ults

The ease with which the above inertanceexpressions have been obtained, and in closed form, contrasts with the relatively lengthy development leading to modal expansions. For the simplest cases A, B, the fOllowing results were derived in Refs. 9 and 10:

(3.35)

= s2e : I

r

1 + "\ s2Icn

1-:1.

L

S2 + W 2

n

where I is the rigid-body inertia for the spacecraft. Similar expansions can be derived for other response quantities such as the defle~tion or acceleration at a point, as discussed in Ref. 14.

In the ab ave expansions,

(nn,

Kn) are the 'constrained' natural fre-que~cies and 'gains' respectively. In effect, constrained modes are appen-dage-only modes in which the centre body is constrained to be immobile. The quantities (wn, k_{n ) are the natural} frequencie~. and gains respectively for

'unconstrained' modes in which the entire spacecraft is free to move. These two types of modes will be abbreviated as C-modes and U-modes respectively. The notion of 'gains' is not defined here except by the manner in which

they appear in the transfer functions (3.35).

As one might expect from a comparison of (3.35) and (3.18), the frequencies (nn' Wn) should respectively be the poles and zeroes of the inertance Ie(s). Of course, thisis valid only in the absence of dampers, since (3.35) cannot be written otherwise. Consider now the poles and

zeroes of Ie(s) in the simplest case, Case A. This can be compared with Ref. 10, Section 2, the root piece inertia mr w2_{/12 is inc1uded as part of}

the rigid body. Then

Upon substituting f

=

f1 from (3.21) and simp1ifying, this gives

This result, incidenta11y, also ho1ds fOr Cases B and C if the apprapriate f is inserted. Expecting po1es and zeroes of Iae(s) on the imaginary axis, we set s

=

jw and note that

sinh(jx) = jsinx, cosh(jx)

=

cosx

. to obtain the result

(mtw cos awt +

aF

sin awt)aw s

I (j w)

=

~---"-=---:r---':---:""

ae 12aw(aP cos aw] - ~w sin aw])

Fo11owing Ref. 10, we introduce the nondimensiona1 variables:

w*

=

w

Jawi,2

_{p ,}

In nondimensional terms, (3.38) can be written as

*

( 1 ) sinw* + mt w* cosw* Iae(jw*)

=

12 aw3

t

.

* . " w*( cosw* - mt w* sinw*) (3.38) (3.40)

The first point to note here is that Iae(-jw*)

=

Iae(jw*). This means that all zeroes and po1es occur in conjugate imaginary pairs. Secbnd1y, when mt

=

0, the zeroes of Iae(s) are w*

=

117r, i.e. the natura1 frequencies of the membrane a10ne in tension. This expresses the fact that at the nië m-brane natura1 frequencies, no torque is required to a1low motion of the appendage. The same interpretation app1ies to zeroes of Iae(s) when mt

f

0.

Finally, the po1es of Iae(s) should ~e, by definition, the C-mode frequencies. The poles are the roots of

*

cosw*

=

m w* sinw*

t (3.41)

Consider now the inertanee of the entire spaeeeraft. As in Eq. (3.17), this beeomes

Making use of (3.37), this result simp1ifies to:

(3.42)

where f~(s) is defined in Eq. (3.21). Onee again, for the cases where dampers are absent (A, B for examp1e), we expect to find p01es and zeroes of' Ie( s) on the imaginary axis. Henee, putting s

=

jw and emp10ying the nondimensiona1 terms in (3.39), one obtains af ter some reduction, or a1ter-native1y direet1y from (3.40), the f0110wing:

*

(

1 ) [ sinw* +

ID.t

w* eos#

=

b

crw3~

*

. w*( cos# - mt # sin#) +

6~

]

(3.43)

Again it is e1ear that Ie{j#)

=

Ie(-j#) and we expeet imaginary conjugate p01es and zeroes.

The p01es of Ie{ s) are seen to be the sane as those of Iae{ s), i. e. , the natura! f'requencies of the appendages. Upon writing the attitude response as

(3.44)

it is c1ear that at the po1es of I~(S), in this case the frequencies

(mn),

there is no attitude motion (e

=

0) for any torque app1ied to the eentre body. In a Rense, the entire torque is "absorbed" by the appendages. Although (3.31) seems tO.,imp1y that ex

=

Oa1so, Le., no appendage def1ec-tion, this is fa1se since at the poles, l' ~ po. Thus the appendages twist

but the centre body does not rotate.

From (3.44) it is a1soseen that the zeroes of Ie(s) are the resonant f'requencies of' the entire spacecraft, i.e., the U-mode frequencies

(Wh},

at which an app1ied torque e1icits unbounded response. From (3.43) the zeroes of Ie( s) are the roots of "

* . * *

(sinw'* + mt w* eosW*) +

6I

_b w*( cos# - mt # sinw*)

=

0

This can be arranged to yie1d

This equation is the same as Eq. (3.46) of Ref. 10, derived there for the unconstrained frequencies (wn}.

The natural mode shapes cana1so be derived from the inertance approach. Forexample, at t-he U-mode frequencies, Ie(w*)

=

0, so that from (3.43),

(3.2l), f~(w*)

=

.:..j6~w*. Substituting this into (3.31) and making the transfer to nondimensional variables, one finds that if S

=

8ncoswnt, the response a(y) is an(y)coswnt, where

(3.46)

In other words, this is the expression for an'u.nconstrained" mode shape. It agrees with Eq. (3.~3) of Ref. 10. Note that since the amplitude

Sn

was not specified, one is free to "normalize" the mode shapes in an arbitrary manner, for example by setting Sn

=

1. It is not even necessary to use the

same norma1ization for all modes.

The constrained mode ' shapes can be obtained simi1ar1y. As shown above, the C-mode frequencies are the po1es of Ie(s), and also of Iae(s). From (3.40) it is seen that at these frequencies, f -+ 0 0 . In the def1ection

equation (3.31) then, the f term is dominant. Assuming a suitab1e norma1iza-tion, the appendage mode shape is proportional to f sinh asy:

(3.47)

This agrees with Eq. (3.22) of Ref. 10.

The non-moda1 inertance exprèssions can a1so be used to obtain direct1y the modal expansions of Eq. (3.35). For examp1e, equating (3.42) to the first expansion (based on C-modes) of (3.35) gives

(3.48)

It has a1ready been estal:>lished that the po1es of this Ie occur at S2

=

-nn.

2; hence, (Kn} :,are simp1y the coe'fficients in a partia1 fraction expansion of'

(3.49). Tt is noted that Eq. (3.48) a1so provides proof of convergence of the infinite sum on the 1eft-hand side, except of course at po1es of Ie(s) • Also, the (nn} are simp1y the roots satisfying Eq. (3.41). Now, multip1y Eq. (3.48) by (S2 + nn2) to find that

aw3_{(s2 + ~2)(aPS + mtsC)}

6as· ( aPC + mtsS)

The right-hand side is a % indeterminate form which can be eva1uated by

(3.50)

s=j~

In order to use the nond.iinensi.onal results obtained above, we change the variables from, nnto w* (n6nQ.:iméris~on(Ü) ;getting finally

,

*

sinW* +_. ,m *cosUJ* _{t - ,}

.

.

_(rot

".

*

'

"

.

*

.

+,1) sin<A1*' +, m_t, UJ*cosUJ*

(3.51)

At this point we note 'f'rom ('3.4;1.)' that mtw*siriUJ*

= cosUJ*, and the resultant

identity

Using this identity 1n:(3~5l) 'finally yields the result

(3.52)

.

'

This is exactly the same result asEq. (3.76) of Ref. 10, which was obtained via IDodal analysis.

A similar approach can be used to Ob tain the U-mode gains kn . From Eqs. (3.'42)' and (3.35) we 'obtain

, '

[

üW3 " aPS +

mt-sc

:

J

-

,

~

_

1

~ + 6as . a.PC + mtsS , -, Ï - (3.53)

where the

(Wn}

are the U-mode frequencies, i.e., the roots (in dimensional form) of Eq. (3.45). Thesame approach of partial fraction expansions can be taken . However, ihe 'details are lengthy and hence will not be reproduced here.

In summary, the non~modal~thod leads directly to a torque/angle transfer function, the inertance. It also gives mode shapes and partial fraction, infinite-series expansions tor simulation purposes if desired. The poles and zeroes of Ie(s) have been shown to be the (nn} and (Wh}

re spe ctively, as expected on physical grounds, and the (Kn} have been obtained from a modal approach. However, the chief advantage of non-modal methods is in the ease with which they permit the transfer functions

of sub-structures to be camained, as seen more fully in the next section.

4.

IDDIFJ;CATION OF DYNAMICS

This section extends the non-modal approach to active dynamics modi-fication. The closed-form expressions derived for Hermes dynamics above will prove useful in this respect.

The gener al principle of active dynamics mcx:lification is i11ustrated in F~g., 5. A f1exib1e structure is stibjected to loca1ized force and/or torque actuators. The measured 'outputs' of the structure are divided into two group~. The control outputs go to a control system, for examp1e attitude control or shape control. Another set of outputs, termed loca1 outputs, are 'fed back, te the torque/force actuators via a suitab1e control 10gic. The, result is'that the distant control system may be designed to control a much better behaved system than the origina1 f1exib1e structure. This

two-part'solution to the control prob1em may in fact be more practicab1e than ,designing a control system for the original structure • The dynandcs modification logic and the force/torque actuators are taken to be 1inear, but the control logic may be non1inear if desired. Examp1es of modification are velocity feedback (i.e. active damping) and acce1eration feedback (i.e. active 3tiffening).

In the remainder of this section wi11 be considered modification of the Hermes pitch dynamics by active means through the on-board attitude control system. Acnve modification via te1emetry links and a ground-based processor (in addition to the normal attitude control system) wi11 be considered in Section

6.

Passive Modification

This type of dynamics change is best achieved by adding discrete springs and/or dampers at a suitable position. The purpose of adding damping, of course, wi11 be to move the dynamics po1es into the 1eft ha1f-p1ane from the imaginary axis. An exa.mp1e of passive dynandcs modi-fication is avai1ab1e in going from Case A in Section 3 to Case B (discrete spring added at the tip) and again in going to Case C (discrete damper added at the tip). The exact analysis of this type of modification using infinite-sum modal expansions is quite intractab1e (even for the re1ative1y simp1e Hermes pitch dynamics) • Yet the present approach makes this simp1e. This topic is treated in Appendix A.

4.1 Acce1eration Feedback

Consider now adynamics modification scheme of the type shown in Fig. 5. When this concept is app1ied to the pitch/twist dynamics of Hermes, the "loca1 outputs" readi1y availab1e are acce1erometers which measure the twist acce1eration. A b10ck diagram of acce1eration feedback intended to modify Hermes ' dynamics is given in Fig.

6.

The dynamics transfer function without the acce1eration loop K(s) is simp1y Ie~(s), the inverse inertance. We shal1 concentrate on the dynamics loop and 1eave the attitude control system unti1 later. It is assumed that the desired feedback torque can be direct1y applied to the spacecraft.

In order to proceed further we need the transfer functions between acce1eration and input torque. In the modal approach, a general type of transfer function for the acce1eration A(x,y) at a point on the blanket is

A(x,

y;

s)

(1 + s (y))k s2 n n é~ + W 2

]

.

T(s) I (4.1)

where sn{Y) are normalized mode shapes in twist. The chief advantage of the modal method is that the form of the transfer function is the same for virtually all single-input single-output condi tions • This is to be expected since one has, in effect, a partial-fraction expansion of the true transfer function.

For the non-modal approach, a transfer function can be written from Eq.

(3.33).

Combining this with

(3.18),

(4.2)

where Ie(s) is the inertance transfer function. For the case of excitation of a single appendage, the inertance to use is Iae(s), given by

(3.13),

(3.14).

For the case of excitation of the entire spacecraft, Ie(s) fram

(3.18)

should be used. Once again, we halTe a closed-form trapsfer function which can be expanded in partial fractions if desired . . Clearly, the veloc~ty

transfer function is simply s-~ times that given by

(4.2).

For Cases A, B and C, where the root connectióri is rigid,

Ä(x, y, s)

=

-x(cosh asy - f(s)sinh asy]s28

(4.3)

where s28 is related to torque via Ie or Iae, as .appropriate. Let ~s now assume that the acceleration is fed back·through.a input, single-output control system with transfer function K(s). Th~ ~otion equations are then

Ie(s)s2e = T{s)

=

Tc(s)

(4.4)

where Tc(s) is an external control torque, and Cy , Sy are obvious shorthand notations. The transfer function from 8 to Tc becomes therefore

I'(s)

=

I (s) + x K(s)(C - fS )

e e y y

(4.5)

Depending on K(s), the dynamics of the spacecraft can be modified to any desired inertance, in principle. However, in general, one has still an infinite set of poles and zeroes. There will also be practical limitations

arisin~ from the torque magnitudes that can oe applied. If K(s) is such that Ie(s) has imaginary poles and zeroes, ane can speak of modified natural frequencies, etc., otherwise not.

For illustrative purposes, let us consider Case A which was detailed in Section

3.

Guided by the Hermes configuration, we assume that accelero-meters are mounted at the corner of each array, where y

=

t,

x

=

Xt.

Furthermore, let K(s)

=

K, a constant gaine From

(3.42),

we have

I ( ) T_ + ~ . aPS + msC

e s

=

-0 oas aPC + msS

Simplification of the acceleration transfer function

(4.3)

yields, at

y=~,x=Xt:

Using this expression, or directly from (4.5), we obtain

, aw3_{( aPS + mtsC) + 6a2pKxtS}

Ie(s)

=

1b + 6as(aPC + mtsS)

(4.6)

(4.7)

(4.8)

With a constant gain feedback, evidently this inertance also has imaginary

poles and zeroes. It is clear that the poles, i.e. the constrained fre-.

quencies nn are unchanged by acceleration feedback. The zeroes of Iè(s) ar.e

the modified unconstrained frequencies

wh,

i.e. the resonant frequencies

of the entire closed-loop system. At this point it is assumed that a torque proportional to acceleration can be applied; this assumption will

be relaxed later. The zeroes of

(4.8)

are the roots of

Tbis characteristic equation can be converted to nondimensional form, as

done in Section 3.3. Using the notation defined by (3.39), and noting that

a2p

=

aw~ and dividing throughout by (jaw3a.P), this equation takes the

nondimensional form:

* * * *

61b w*(cosw* - rot w* sinw*) + (sinw* + rot w* cosw*) + _{6xt K*} w*

=

0 (4.10)

*

where Xt = Xt/ ~, and K* = K/ow3 _{is a nondiroensional gain.}

This equation contains oscillatory terms of constant, linear in w*,

and quadratic in w*, amplitudes, as well as the linear term w*. Except

for the last term, it is identical to the spacecraft characteristic equation

preceding (3.45). The sum of the first two terms can be visualized as an

oscillation in w* with amplitude increasing as w*2, as shown in Fig. 7.

The intersections of this oscillation with the axis w*

=

0 are the

unmodi-fied frequencies; the roodiunmodi-fied frequencies are its intersections with the

line .C-6xt K*)w*. Due to the amplitude w*2, there always exist a countably

infinite set of intersections. However, by increasing K* to large values,

the first intersection, i.e. the lowest frequency w~, can be made as high

of frequencies' seen here is a new phenomenon in dynamics . Increasing wl corresponds intuitive1y to ,stiffening, of the structure by active feedback. In practice, one expects that high' values of K wi11 require large torques to be app1ied, and so a practical limit on K exist,s.

Modal Approach

It is interesting to compare the qUa1itative results d:>tained above with the results predicted by using moda1 transfer functions. The acce1era" tion transfer function '(4.1) can be written (wit,h x = Xt, Y

=

t) as

00

.

Ä(

_{xt '}

t,

s)

=

_{-xt [}

1

+L

n=l d S2 n S2 + W 2 . n ] (Til) (4.11)

where dn

.

=

(i

+ _{sn(y))kn . The unmodified}·transfer function from torque to 8 is 00 k s2 - [ 1 + \ ' n

J

(TjI) - , ~ S2 + hl 2 n=l n (4.12)

Both these expansions are based on unconstrained modes. Making the torque proportional to acce1eration, and using the shorthands (dn ) and (kn ] for the brackets in (4.11) and (4.12) respective1y, we obtain

T =Tc - K,xt(dn1(T!I)

(4.13) T = { 1 +

~t (dn

:

JT~Tc

Using this in (4.12),

,.

(4.14)

wh~ch 'is t~e lp..ödified t'xa.nsf.et f$.ëtiçn" The simp1ified notation in this

equation IDasks the great complexity of obtaining results for modified frequencies, etc., since (dn ] and (kn1 are both infinite' sums of fractions as in

(4.11).

To proceed any further, we need to truncate the moda1 expans ions to a finite number, N. The error of our conc1usiQns wi11 be reduced as N

increases. Unfortunate1y, even for N = 2 analytical results become tedious. Keeping just one mode (N = 1), Eq. (4.14) may be written as

or (kl. + 1) s 2 + Wl. 2 S28

=

-(I + Kxt + KxtdJ.) s2 + (I + Kxt) Wl. 2 ,.., • T c (4.16)

The zeroes of this transfer function are the same as those of (4.12), indicating that the constrained (appendage-only) frequency is unchanged by feedback, in complete agreement with the non-modal resu1t. We now

write (4.16) in the form

(4.17) where

(4.18)

In this form it is c1ear that Ol. is independent of K. The unconstrained frequency

wi

(a zero of the inertance) and the inertia parameter If are changed. For posi tive K,

wi

approaches Wl./Jï + dl. as K -Ho. This va1ue

may be higher or lower than wl. depending on dl.' so that "stiffening" does

not always occur. For negative K, at K

=

-I/xt w~ is zero, a fact which mak~s attitude control difficult. For

K

=

-I/xt(dl. +

1),

one finds that

wi

=

00 but this is insi~nificant since then If

=

O. A significant special case occurs wh en Ol.

=

wl.' for it reduces the inertance to a constant and makes the spacecraft effective1y rigide From (4.17), (4.18) it can b e

shown that in this case

(4.19)

The sign of K for "rigidization" of the spacecraft depends on dJ., since always kl.

>

O. This rigid behaviour of the spacecraft is in fact mis-1eading and is not predicted by the non-moda1 approach.

It is helpful to consider brief1y the case when an arbitrary number of modes is ~ept. For N modes, let us define the following abbreviations:

Expressing the transfer functions (4.11), (4.12) in this notation,

substi-·tuting the resulting expressions in (4.14) and simplifying leads to the modified transfer function:

r

m

s) + S 2 Z k. 7T. (S)

s2'8 -

J J

- (I + Kx_t)7T + Kxts Z d.7T.

- J J

(4.21)

Comparing the munerator of this expression with that of (4.12), it is clear that the zeroes of these two expressions are identical. This shows that the constrained (appendage-only) frequencies are unaffected by feedback. This agrees With the non-modal result, and with the one-mode case as well. The modified unconstrained frequencies are simply the poles of (4.21). In principle, these can be found by sOlving the equations obtained from setting the denominator of (4.21) to zero. Hawever, it is clear that the non-modal approach makes the solution feasible, in simple form.

Numerical Results

A constant-gain feedback of acceleration has been shown to result in stiffening of the structure, affecting only the frequency. To illustrate tlie .:magnitude of this effect, the unmodified zeroes of the inertance,

(WnJ,

were first found for a set of parameters typical of Hermes, shown in

Tahle 1. . Wi tl1 zero feedback (K = 0), the first five ~ I S are Jas shown in

Fig.

8.

Consider now both positive and negative values of K, in units. of ft-lbf/ft/sec2 , i.e. lbf-sec12 • The modified characteristic equation

(4.9)

can be solved by a Newton-Raphson technique in the complex plane. Since Eq.

(4.9)

is analytic, straightforward differentiation gives the (complex) derivative. The initial guesses simply .correspond to the frequencies with K

=

O. The results of this calculation are also shown in Fig.

8.

It is clear that the frequencies may be either lowered (greater flexibility), or raised (rigidization)·. It is not surprising to find that the first mode undergoes the greatest change in frequency, since i t contributes most

heavily to the total acceleration. In Fig.

8,

of course, no damping is considered.

/

4.2 Effects of Time Delay

It is instructive to consider how the dynamics changes when a time delay is present along with a constant-gain acceleration feedback. This delay may represent processing time, or delays due to sensor lags and signal conditioning (sample and hold, etc.) . . The author is not aware of any

conventional (i.e., modal) analyses of dynamics in the presence of ti~e

delays, and believes that the advantages of a frequency-domain approach are apparent here.

The analysis proceeds by noting that a pure time delay has a simple transfer function, exp(-Ts), where T is the delay. Thus, in Eq.

(4.5),

we, substitute

K(s)

=

Ke -Ts (4.22)

The resulting equations for the modified inertance is similar to (4.8):

, ' aw3_{(aPS + mtsC) + 6a.2FKe-}TS_x tSi 6as(aPC + n1 tSS) (4.23)

The important thing te note is that the zeroes of this expression no langer lie on the imaginary axis, although 'the poles still, do. However, both poles and zeroes ~till' 90cur in complex conjugate pairs. The zeroes of

I~(s), Le. the poles 01 the', attitude response', are solutions to:

(4.24)

This modified characteristic eqUati~n,has c~lex root$. ,The real part maybe }Ositive or negative dependingon the phase of the exp(-Ts) term. The simplicity of including delaY'isclear from the tremendous similarity between Eqs. (4.8)" (4.23 )and' (4, .9), (4.24), respecti vely. The effect of

delay on stabili ty is 'illustrated. next. , " '

Numerical Results

The value of the deiay time T mus~ be ch~sen to illustrate its quali-tative effects .One pössÜ>le ctpplica~ion of these results is to modification of dynamics by meçms' of>ground control, where the telemetered acceleration is multiplied by'K

and

fed 'back 'to an on-board torque generator. This type of exper~ent wasplanned for Hermes, as discussed more fully later. The

effective,round~trip,processing delay is at least 1 sec., and may be up to

, 2 sec.. due 'tb addi ti'Ünal delays,caused by sampling and holding data to fill a "frame" on

the

spacecrart, befare transmittal to ground.

, For the, spacecraft parameters listed in Table

4.1

and corresponding to the modal frequency results in Fig. 8, the first five roots of (4.23)

for T =' 1 sec. are piotted in Fig.

9.

The dyn,amics poles are now complex. Once again it is fo~d that higher modes are affected less for the same gain. Even more important, the effects of delay are stabilizing for same modes, but unstabilizing for others. There exists no K in Fig. 9 for which all five roots are in the left half-plane.

The case where T =-2 sec. is similarly plotted in Fig. 10. Comparing this with Fig.

9,

one finds that even the qualitative effect of a delay is changed for some modes, especially the hig~er ones. In particular, for Mode

4

the qualitative trend is reversed in going from T

=

1 sec. to T

=

2 sec.

r---

---

---

---

---

.---modified due to finite bandwidth of the accelerameters. Indeed, on Hermes,

an anti-aliasing filter is used af ter the accelerameter .outputs" When

including finite bandwidth of the 'sensors, it is alsö desirable to model

actuators more realistically. This is done qui te directly, as shown below. 4.3 Attitude Control with Modified Dynamics

Consider an experiment al set-up as depicted in Fig. 11, where

ACEA PWM DYN

ACCEL

NESA

== Attitude control electronics assembly == Pulse-width modulator

== Spacecraft dynamics (~odified)

== Accelerometers

=

Non-spinning Earth sensor assembly

and telemetry/command links are shown by wavy lines.·

. .

The design approach used in pitch was to design a controller based on a rigid spacecraft, with the inner loop absent. The effect of flexibility on what might be termed "rigid-controller" poles is of obvious interest. Even more so is the effect when the inner loop is closed.

Simplified transfer functions for the various blocks in Fig. 11 are given below:

ACEA: _{E(s) == ( : +} TS +1) (4.25a)

TS +n

PWM: p(s) ==K _p (4.25b)

Wheel: M(s) =

K.r

(4.25c)

NESA: N(s) = K e-/:::'s _s (4.25d)

Ground Processor: K(s) = Ke -Ts (4.25e)

The model for the PWM is based on taking the average of a digi tal output and the motor transfer function ignores a large time constant due to the very small friction torque. The sensor is assumed to have a pure time delay 6 to account for processing. In addition, closed-form expressions are available for the dynamics and acceleration blocks:

Dynamics: D( s)

Accelerometer:

= I (s)-l.

e (4.251')