A Steering Law for a Roof Type Configuration of Single Gimbal Control Moment Gyro System

A STEERING LAW FOR A ROOF TYPE CONFIGURATION OF SINGLE GIMBAL CONTROL MOMENT GYRO SYSTEM

Tsuneo Yoshikawa* NASA Postdoctoral Research Associate Systems Dynamics Laboratory Marshall Space Flight Center Huntsville, Alabama, U.S.A.

ABSTRACT A steering law is proposed for a roof type configuration Single Gimbal Control Moment Gyro (SGCMG) system for attitude control of the Large Space Telescope (a payload of the Space Shuttle). Although various steering laws have been developed for SGCMG systems, they have some drawbacks because of singular states of the system. The proposed steering law minimizes the effect of singular states by using a new momentum distribution scheme that makes all the singular states unstable. This scheme is formulated by treating the system as a sampled-data system. 1. INTRODUCTION Control Moment Gyro (CMG) systems are effective actuators for attitude control of various spacecrafts(l, 2). For Skylab, a three double gimbal CMG system was used successfully(3, 4, 5). As a candidate for attitude control actuator of the Large Space Telescope (1ST; a payload of the Space Shuttle)(6, 7) and the High Energy Astronomy Observatory (HEAO)(8), four Single Gimbal (SG) CMG systems have recently been investigated by several researchers (9, 10). Two configurations of 4-SGCMG systems have been proposed so far. One is a pyramid type configuration (i.e., the gimbal axis normal to the faces of a square-based pyramid; see Figure 1). The other is a roof type configuration (see Figure 2). At first, the pyramid type configuration was investigated extensively and various steering laws were proposed(9). In this configuration, however, the singular states (the state at which the torque output axes associated with each of the SGCMG's in the system are coplaner and the system cannot respond to out-of-plane commands(lO)) cause serious difficulties and each of these proposed steering laws has shown some unsatisfactory performance for certain situations. As a configuration in which the difficulty of singularities would be less serious, the roof type configuration was investigated and at the same * On leave from Automation Research Laboratory, Kyoto University, Kyoto, Japan.

time the OMEGA (Optimum Momentum Exchange by Gimbal Alignment) steering law was proposed(lO). The main feature of this configuration is that it consists of two pairs of gyros with each pair sharing a common gimbal axis direction. Although the OMEGA steering law gives rather good control performance, it still has the defect that stable singular states exist. A stable singular state is defined as a singular state which can be maintained by setting the input torque command to zero. The existence of such states is not desirable because of the following reason: If a certain torque command is given when the system is in, or near, a singular state, there will be large control deterioration, and the existence of stable singular states makes the possibility of occurrence of this control deterioration larger. In this paper, for four SGCMG systems with the roof-type configuration, a steering law is proposed which does not have the above-mentioned defect. This steering law is obtained by regarding the CMG system as a sampled-data system and providing a new momentum distribution scheme. The basic procedure of the steering law is as follows. At each sampling instant, first, the desirable total momentum at the next instant is calculated from the present total momentum and torque command. Second, according to the predetermined momentum distribution scheme, the desirable total momentum is distributed into two desirable momenta for the two CMG pairs. Third, a desirable combination of gimbal angles which realizes these desirable momenta is calculated for each CMG pair. Fourth, the desirable gimbal angles are compared with the present gimbal angles to select a best way to attain the desirable gimbal angles. Finally, a gimbal rate command is calculated and if the calculated rate command exceeds the hardware limit, the rate command is modified not to exceed the hardware limit. An important feature of this steering law is that, because of a new momentum distribution scheme, it does not result in any stable internal singular state. Therefore, the possibility of the occurrence of unfavorable torque command when the CMG system is at, or near, a singular state is negligible. Moreover, it will be shown by computer simulations

361

that even if such an nufortunat~ situation happened, control deterioration would be very small. Another feature is that, since this momentum distribution scheme treats the momentum directly, any reasonable momentum distribution can easily be realized.

h can also be expressed as IZ h hII2

h

In the f o llowing sections, the physical explanations and selection of constants for the simulations are done with r ega rd to the LST. Almost the same argument can be presented for the HEAO .

j

' (in YIYIIY-coordinate system)

hU-hIU Notice that h is determined uniquely if a ~ [al;az:Ct3:a41T l.s. given, but that

~

is

g~nerally

2. ROOF TYPE CMG SYSTEM

not determined uniquely even if h is given.

Figures 2 and 3 show the roof type configuration of a four SGCMG system and its mounting arrangement relative to the LST . Each CMG angular momentum (and torque) vector is restricted to the plane I or 11, which is skewed relative to the vehicle Y-Z plane by the angle f3 .

The input (control) variable of the CMG system is the rate of the gimbal angle E (= £), th e output is the torque, i. e. , the rate of th e tot a l momentum t = ~, and the purpose of this system is t o make i equa l to the torque command ic . 3. OMEGA STEERING LAW

The angular momentum vector of the i-th CMG is denoted by hi' i=1,2,3,4. In addition to the coordinate system (XYZ), we also use coordinate systems (XtYIZI) and (XIIYIIZII)' which are shown

In this section th e OMEGA steering law is described briefly(lO~ Let

in Figure 2, for the pairs (hI' h2) and (h3' h4), respectively. The angle between the rotation axis of the i-th CMG and XI for i=1,2 (or XII for i=3,4)

Y2

(al+aZ)/Z'

01

( a 3 +a4) /Z,

Oz

(3 )

Then from (l)

is called the i-th gimbal angle and is denoted by a i (in degrees). Figure 2 shows the state where ai =O', i=1,Z,3,4. Let us assume that hi's have the same magnitude h* and define

hI

hI = hl+h2 hII = h3+~ Then

By using an approximation to the inverse of the coefficient matrix in the above equation , one can obtain the following equation.

hr=[hIl:hIZ:OlT, (in XIYIZI-coordinate system) hII=[hII1;hII2:0]T, (in XIIYIIZlI-coordinate system)

[::1

where the superscript T denotes transpose and

1

Zh*

[-,,(")"OY, ',('I -~c(ol)cosYl

)00",

I] .b

~c(ol)sinYl

where ~ s (0 1 )

(1)

={

s eco 1 ' i f 101- 90 1>E sec£ ,

i f 101-901'i E

(4 ) Sc(ol) The total momentum h of the CMG system is given by

={

csco l' i f 1011> E csc£ ,

h

L

4

i=l

h. = h +h

-~

-I-II

i f 1011 fE

and E is an arbitrary small number. Deriving th e same equation for pair 11, adding them, and replacing ~ by the torque command ic, we ob tain the following equation.

[hxihy;hzlT, (in XYZ-coordinate system) (2) where

Yl = ~s(01)[-sinYltc3l+cosYltcll/(2h*)

sin i3(h I2 +h IIZ )

01 =-sc (01) [cosY 1 tc31 +sinY 1 tell ! (2h,,,)

362

(Sa)

y2 =

I; s

(6 2)[ -s iny 2tc32+cosY 2 tc211(2h*) (5b)

82 =-l;c(62) [cosY2tc32+sinY2tc21 1 (2h*)

stable singular state (because this state gives f = 0). Control deteriorations due to singular states will be illustrated in s ec tion 6. 4. NEW STEERING LAW

where

and (6)

In order t o determine the distribution of tc3 between tc31 and tc32' a distribution function

For the case where a digital computer is used to obtain the desirabl e gimbal rate from the torque command !C, it is impossible to make! equal to !c at all times. Hence the treatment of the system as a sampl e d-d1iI:'a syst'em is appropriate in this case. This treatment also plays an important role for the development of a steering law in later sections. Let T be a time variable, 6 be a sampling period, geT) and ~(T) be g and ~ at time T.

f(ql' q2) is introduced where It will be simpl e and prac tical to impose the restriction

=.j 4h,~2_hIl

ql

hIl lX I'

xl

q2

h IIl /x 2 ,

x2 = /4h/-h II /

(7) ~ (T ) = !i(n6 ),

It is r equ ired that f(ql' q2 ) = 0 should give the desired distribution. In order to make f(ql' q2) converge to zero, a linea r correction feedback is introduced;

f

= -IJ f

(n+l)6,

n = 0,1,2, ...

This corresponds to using a zero-order hold . The output torque then satisfies the following equation: (14 )

(8)

where IJ is an arbitrary constant. The solution of (6) and (8) gives th e distribution be tw ee n tc31 and t c32 ' In Crenshaw(lO) the following function is proposed as f:

Then t c 31

xl[(5-4ql)tc3-lJx2fl/s

(9 )

tc32

x2[(5+4q2)tc3+IJxlfl/s

(10)

where (11 )

From (3), we have

~ is replaced by !:c)

y1 - 61 '

(12 )

)-2 - 62 ,

Hence the OMEGA ste e ring law is given by the following thr ee steps. 1). Calculat e tc31 and tc32 by (1), (7), (9), (10) and (11) . 2). Calculate Yl , Y2' and 82 by (4) and (5). 3). Calculate!:c by (12).

Since the torque command is usually obtained from the obs e rved position and rate of the spacecraft and since the spacecraft is exposed to unknown disturbances, it is impossible to know the future !c(T), T>n6 exactly at the present time n 6 . The refore, it is natural to assume that the purpose of the system is to make the average of the output torque during the coming 6 period, t(T), n6fTf(n+l)6, equal to !c(n6) where n = 0,1,2, ... The left hand side of (14) is th e momentum which is transf e rred from the CMG system to the spacecraft and which in turn causes the change of angular rat e of the spac ec raft. Hence the above purpose of the system can also be inte rpreted as to change the rate of the spacecraft for a desirable amount during each sampling interval. Hereafter, a(n6) and h(il6) will be denoted by ~Cn) and g(n) fo~ the sake-of simplicity. In the following a steering law is proposed which will fulfill the above describ e d purpose. This steering law consists of the following five steps. Assume that the present time is T=n6. ~ : Given the present gimbal angle ~(n) and torque command !c(n), calcul a te the desired total momentum gd at the np.xt sampling instant:

!!d

'h,

As is shown by Crenshaw(lO), there are two types of singularities: o (a) al = -~ = ±90°, or a 3 = -a4 = ±90 (b)

n6 fT<

!at -~ I =

= .!!(n)+6!c (n)

i

[hdl hd2 ;hd31; (in YIYIIY-coordinate system) (15) where hI2 Cn)

(13 )

1~-a41 = 180

0

In cas e of OMEGA steering law, singularities of type (a) are stable . For example the state {al = -~ = 90', ~ = -a4 = cos-ICO.8) ~ 3i} is a

(in YIYIIY-coordinate h II2 (n)

g(n) [

363

hIl Cn)-h IIl (n)

system)

h n (n)

h*(cosal(n)+cosa2(n»

hI2(n)

h*(sinal (n)+s inCl 2(n»

Then we decide as follows: Choose (aI) if 2 2 ell +e I 2

(16)

~

2 2 eI3 +e I4

hln (n)

h*(cosCl3(n)+cosCl4(n»

and choose (b I ) otherWise.

h II2 (n)

h*(s inCl3 (n)+s inCl4 (n»

For pair 11, define (all)' (b II ), eIll ~ e n4, and follow the same procedure.

~: Obtain the desired momentum distribution between two momentum vectors ~I and ~II at the next sampling interval:

hIl = fI(~' ~(n»

(17)

hIll = -fII(~' ~(n»

(18)

hI2 = hdl

(19 )

hIl2 = hd2

(20)

~:

Calculate the command rate EC : if r'max;;rg (26 )

where (rl', r 2 ') = {(e ll /{;., e I 2/{;.)' i f (a I ) is chosen (e

where fI and fII are given by the momentum distribution scheme which will be described in the next section. Calculate the pairs of gimbal angles (ClIl' ClI2) and (ClIIl' ClII2) which give the momentum vectors ~I and ~II respectively (see Figure 4). As can be seen easily from Figure 4, Cl Il and Cl I2 are given by

(r3"

YI+o I

(21)

Cl 12

YroI

(22 )

where Y = {-tan-l(hIl/hI2)+90Sgn(hI2)' i f h fO I2 I 90sgn(h n )[sgn(h U )-1],

{,o-,.n-

°1

l (h !(4h.'-h I I

'»'

if h

I2

(23 )

=O

i f h I <2h*

(24 )

i f hI~2h*

0,

and sgn(' ) is a sign function defined by sgn(a) =

r

i f a=O

-1,

i f a
I4

/{;. ), if (b I ) is chosen

r4') = {(e Ill / {;., eIl2/ (;. ), i f (an) is chosen

r'max = max (Irl'l '

h'l '

Ir 3'1 '

ri' is the gimbal rate of the i-th CMG required to realize the desirable gimbal angles obta1ned in Step 3. The meaning of (26) is that if any of ri"s e xceeds the maximum limit of the gimbal rate rg (which is determined by CMG hardware), th e n gimbal rates rl'~r4' are limited proportionally in order to minimize the effect of undesirable output torque du e to the hardware restriction. This completes one cycle of calculation of the command rate Ec from ~(n) and !c(n). At time (n+l){;., upon obtaining the new state ~(n+l), a new cycle begins from Step 1.

In this section a momemtum distribution scheme b e tween hIl and hIll' that is, a specification of a pair of functions fI and fII' is given. Le t xl and x2 be

~: For pair I, select the better way between the following: (a I ) to bring Cl l to all and

aI2 or (b I ) to bring Cl l to Cl I2 and Cl2 to Cl Il .

Let

(28) These values express capability of the two pairs in producing mom~ntum in the direction of Y-axis when h I2 =hdl , hII2=hd2' Using these variabl e s we specify fI(~' ~(n»

ell

mod (Clll-Cll (n», eI2

eI3

mod (Cln-Cl2 (n», e I4

mod(Cl I2 - a 2(n»

(25)

mod (ClI2-Cll (n»

fII(~' ~(n»

= x l h d3 / (xl+x2)+g =

(29)

x2hd3 / (xl+x2)+g

where the first terms in the right hand sides denote a proportional distribution of hd3 , and g of the second term represents an additional ( or excessive) distribution. Notice that for any g, fI and fn satisfy

where mod (Cl) is defined by mod (Cl) = { Cl, CI-360sgn(CI) ,

(27)

and rg is the maximum limit of the gimbal angle rate.

aIll and aII2 are also given by similar equations.

Cl2 to

Ir 4'1)

5. l-KlMENTUM DISTRIBUTION SCHEME

i f a>O

0,

/{;., e

(e II 3/{;.' e II 4/{;.)' if (b II ) is chosen

~:

an

I3

i f IClI~180 i f IClI >180

364

g = Therefore we can select any value for g.

0.25xlx2{0.9cos[90hd3/(xl+x2») +(·.(2-0.9 )cos 2 [90hd 3 / (xl +x2») }

gb

(30)

0.2xlx2cos[90hd3/(xl+x2»)

Then we select one of ga and gb as g* in the following way.

f gc +x I hd3 / (xl +x 2 ) otherwise

(31 )

where gc ={(0.5+k l )ga+(0.5-k l )gb' if g*=ga at the last cycle (32 )

(0.5-k l )ga+(0.5+k l )gb' if g*=gb at the last cycle 0~klfO . 5

(for example kl = 0.2).

A desirable momentum distribution (f r *, frr*) which corresponds to g* is given by fr* = x l hd3 /(xl+x2)+g*

(33 )

frr* = x2 hd3/(xI+x2)-g* Figure 5 shows a schematic diagram of fr* and frr* as a function of hd3 for a given pair of xl and x 2 . Now let us specify a relation between g and g*. The additional distribution g(n) at the present state ~(n) is given by g(n)

=

h

n

(n)

-xl (n)[ h n (n )-h III (n») /[ xl (n )+x2 (n»)

(34)

where xl(n) = /4h 2-h 2(n) * II2

Using g(n), we specify g as follows:

i f Ig*-g(n)l~gmax

where

(35 ) (36 )

and k2 is a constant (for example k 2 =0.5). The above eq'llAtfOJ!l'meat!\s that, although the additional distribution g at the next sampling instant should be as close to g* as possibl e , its change in one sampling interval should be larger than a maximum allowable change ~ax. The constant k2 determines a degree of instability of the singularities. Therefore, if the torque command is ke pt zero, any initial state converges to a desirable momentum distributlon state in a finlte time . As was mentioned in Section 3, singular states satisfy (a) or (b) of (13). This means that a singular state ShOUld give hrl~O or hrrl=O. On the

if gc-x2hd3/(xl+x2)fhd3

and

g *,

g(n)+gmaxsgn(g*-g(n», if Ig*-g(n)l>gmax

First, a desirable g, g* will be given as a function of hd3' xl and x2. Then the value of g will be given as a function of g* and the present state a(n). This two stage approach is taKen to make the present additional distribution g(n) converge to the desirable additional distribution g* without too much change for each sampling interval. g* is given as follows. Let us define two candidates for g*; ga

i

other hand, the d e sirable momentum distribution (fr*, frr*) does not have any point which satisfies hrl=O or hrrl=O due to two jumps (see Figure 5). Therefore, all the singular states cannot be stable. This instability causes quick movement out from singular states and will make the possibility very small of facing unfavorable torque command just at th e singular state. Moreover, even if an unfavorable torque command is applied just when the syst em is in a singular state, the control deterioration is very small. This will be shown by digital simulation results in Section 6. A realistic situation in which serious control deterioration may occur will be the passage of the system state through a neighborhood of a singularity caused by a fluctuating torque command. This fluctuation is unavoidable because of various disturbance torques and sensor noises. A model of such a situation is shown in Figure 6 where a small sinusoidal torque command in YZ plane with a little bias in Y axis is applied to the system at zero momentum state. This sinusoidal torque command forces the system state to pass near, or hit, a singularity. Now let us show by an intuitive argument that the hystereses introduced at two jumps of fr* and frr* keep the control deterioration very small in this situation. rn Figure 5 the passage of the system close to a singular state corresponds to the passage of the value of hd3 close to one of two jump points. According to the fluctuation of torque command, the values of xl and x2 will also fluctuate. rf there is no hysteresis (that is, kl=O in (32», this fluctuation may cause a fluctuation of g* between ga and gb· This causes control deterioration due to the hardware limit on gimbal rate. By the introduction of hysteresis, however, most of the fluctuation of g* between ga and gb could be avoided, thus keeping the control deterioration to a minimum. Because of these two features of this steering law, instability of the singular states and the hystere-

365

sis of the desirable momentum distribution, we can say the following: If any control deterioration occurs due to a singularity, keep the torque command zero and in a short time the trouble will be gone. Candidates for gi', ga and gb of (30), have been selected as a simple pair which satisfies the following requirements. 1. The value of ga should be ~ for xl=x2=2 and hd3=0 in order to give the gimbal angles {01=03=45~ ° 2 =°4=-451 for the zero momentum stationary state. These gimbal angles are desirable for the LST because the possible output torques in Y and Z directions are balanced.(ll)

Figure 10 shows the response to the same fluctuating torque command in the case of OMEGA steering law. It is clear by comparing Figures 9 and 10 that the proposed steering law keeps the period of control deterioration to a minimum (it should be noticed that slewing of 0 3 and 04 for 180~ cannot be avoided). Any steering law for the roof type configuration with no hysteresis will suffer to a certain extent from such a control deterioration as that in Figure 10. 7. DISCUSSIONS Because of Step 4, this steering law redestributes away from undesirable initial states in a very fast way.

2. ~gb;ga~min(xl' x2)(x l +x2-l h d3 1 )/(xl+x2) (37) for 0~xl~2, 0~x2~2, Ihd3Ifxl+x2' xl+x2;<0.

3. ga and gb should be smooth. 6. COMPUTER SIMULATIONS A program for computer simulation of the CMG system with proposed steering law was made using BASIC language. A minicomputer HEWLETT PACKARD MODEL 9830A was used. Values of constants are selected as follows. 6=30 (deg),

to =2 (sec), k l =0.2,

h*=l (normalized) r g =2 (deg/sec)

k 2 =0.5

Some resluts of computer simulations are given in Figure 7-10. Instability of a singular state is illustrated in Figure 7 where the torque command was changed to try to stop the system at the singular state but the attempt failed. The effect of unfavoralbe torque commands at a singularity is shown in Figure 8-(a). The torque command of magnitude 0.01 in the direction of Yaxis was suddenly changed to the direction of minus XII-axis (most unfavorable direction) just when the system reached the singularity. The figure shows a delay of 12 seconds in the response of output torque to the command torque. Generally, the smaller the magnitude of unfavorable torque command, the smaller the delay in output torque. In the case of OMEGA steering law, the delay is independent of the magnitude and it is 45 seconds as is shown in Figure 8- (b) . Figure 9 shows the response to a fluctuating torque command which is composed by adding X-directional sinusoidal component to that in Figure 6:

1

O.0025Sin(10T) !c(T) =

0.005sin(5T)+o.0005

(38)

r

0.005cos(5 T ) 0'

I

Even if the length of the sampling interval is very large, control performance of our steering law has no theoretical error as far as the control purpose stated in section 4 is concerned, except when it is impossible to meet the torque command because of the gimbal rate limitation or momentum saturation. This might be an advantage if changes of sampling interval during the operation of CMG system is desired for some reason. It should be noticed that a longer sampling interval may cause a control deterioration due to fewer up-dating of the torque command. So far, we have discussed our steering law with one specified desirable momentum distribution (f I *, f *) given by (33). The answer to the II question what the desirable momentum distribution is, might vary depending on each engineer and on the capability of the CMG hardware used. One of the features of our steering law is that, since our momentum distribution scheme treats the momentum directly, it can easily realize any reasonable momentum distribution. To illustrate this, it was attempted to give a control performance similar to that of OMEGA steering law. The desirable additional distribution g~ was selected as

Simulation results for this g* have shown a control performance which is quite similar to that of the OMEGA steering law. The computing requirements will now be discussed. According to the author's experience the length of the program for our steering law is about 1.2 times that for the OMEGA steering law when a proportional limit on rate command (like (26) and (27» is included in the OMEGA steering law. Since our steering law is given in five steps each of which has very simple physical meaning, it will be easy to modify or to simplify it in accordance with various requirements in practical applications without changing its main concept. When this point is taken into consideration, computing requirement of our steering law is very reasonable. 8. CONCLUS ION

.0 I

The initial state is ::=[21.36 : -21.36" ;81.88 ; -8l.88~lT which corresponds to ~(0)=[0:1.58:01T.

A steering law is proposed for roof type configura-

366

tion of Single Gimbal CMG system. This 'teering law is obtained by L-ega rdLng tne CM\.; sy~ Le", as a. sample d-data system and providing a new momentum distribution scheme .

(9) B. G. Davis: A comparison of CMr. Qteering laws for high energy astronomy observatories (HEAOs). NASA TM X-64727, Marshall Space Flight Center, July, 1':172.

This scheme LS designed to bring any state of the system to a Q~ate with q predetermined rlpsirable momentum distribution . This desirable momentum distribution has two jumps with hystereses around singular states. It is analytically shown that these jumps make all the singular sta-t es unstable and that these hystereses make the system relatively insensitive to singularities.

(10) J. W. Crenshaw : 2-SPEED, a single gimbal CMG ,,~t;t .. cte {,<'Introl systems. Northrop Services, Inc., TR-243-11J9, 1972. (11) T. Yoshikawa: A _Qteering law for a roof type configuratLou for a single-gimbal control moment gyro system . NASA TM X-64907, Marshall Space Flight Center, Dec., 1974.

With these two features it is expected that the steering law will give a control performance which is good enough for practical applications. Results of preliminary comput:· r simulations entirely support this pxpectatiG~. ACKNOWLEDGMENTS The author would like to express his sincere thanks to Dr. M. T. Borelli and Mr. H. Kennel of the Systems Dynamics Laboratory at the Marshall Space Flight Center for their discussions and help in writing this paper. This research was performed while the author held an NRC Resident Research Associateship. Figure 1.

REFERENCES

Pyramid-type s1ngle-gimbal CMG configuration.

(1) A. D. Jacot and D. J. Liska: Control moment gyros in attitude control. J. Spacecraft and Rockets, 1, l3l3-13LO (1966). (2) A. L. Greensite: Analysis and design of spac e vehicle flight control systems. Spartan Books, New York (1970). (3) B. J. O'Conner and L. A. Morine: Description of the CMG and its application to space vehicle control. AlAA Guidance, Control, and Flight Dynamics Conference, Paper No . 67-550, Aug., 1967. (4)

w.

(5)

w.

B. Chubb, D. N. Schultz, S. M. Seltzer: Attitude control and precision pointing of Apollo Telescope Mount. J. Spacecraft and Rockets, 2 (1968).

B. Chubb, H. F. Kennel, C. C. Rupp, and S. M. Seltzer: Flight performance of Skylab attitude and pointing control system. AlAA Mechanics and Control of Flight Conference, Paper No. 74-900, Aug., 1974.

Figure 2.

(6) Large Space Telescope Phase A Final Report, Volume V. NASA TM X-64726, Marshall Space Flight Center, Dec., 1972. (7) A. D. Jacot and W. W. Emsley : Assessment of fine stabilization problems for 1ST. AlAA Guidance and Control Conference, Paper No. 73881, Aug., 1973. (8) Conceptual design of a high energy astronomy observatory. NASA TM X-53976, Marshall Space Flight Center, Feb., 1971.

367

Roof-type slngle-gimbal CMG corlfiguration.

TELESCOPE AXIS X

TIME

TIME

a.

Torque command.

X

z

SOLAR WING

Figure 3.

Mounting arrangement

of roof-type CMG system relative to LST.

b. Transition of total momentum.

V,

Figure 6 . A model of fluctuating torque command and

VII

corresponding transition of total momentum.

'oo

Figure •• Glmbalangleaou'

0[2'

am' and° U2

for giveD ~ and .!!.no

.

1

-

~

.~ j

a

""

,-----

r- - - - -

:

.Jr .,.,

I

•

a'l

-..

1---+---0----+---0----+---0----+-----.. TI... (....

...

"" ·'20

I I

I

.,,,,

-d>

a1

I

I

_ _ _ _ -lI

", - - - - j

OA;

I

." I ,oo

I >DO

"'0'

'-'111

~

r-----I

U g- = Ka at the Last cycle.

h. If g.

::t.

0

"'O~ I

", a.

lun

'b at the last cycle.

Figure 5. Schematic diagram of the deairable momeotum distrlbutloo.

368

I

..,

~~J § ..., .. :=

I

...

I ,oo

I

'oo

I

.:. c::::1::J

I >DO

.,. .....

... ...

r===\

I

I

I

I

'10

./

'20

SlNGUURITY

-

~

-

~

~ <

z < < ~

/

,'20

·,so

"~

-0.01

:

...

..,

00;1

I '00

<'>

......

TORQUE COMMAND

q>

V

....

c ...... ""'" p

""'"

')

/

" ..... ~ .... to .... > ...... "'"

I

~ TOROU' COM ..... ND

C:--

I '00

-4.01

o

!"

...

JOO

,

.,

I

OO~ r~C'>.","","""""'<:'j {-'

...

I

I

JOO

200

JOO

"

I

: I 1 200

001

"

I I TORQUE COMMAND

0 .01

~j

200\

'00

·120

JOO

200

'00 I

.,

" ...

.~

Q

..

~

TIME IlK!

~

.

~

...

JOO

200

~

.

\\:r-v

uo

:'

.,

00; 1

..

lOO

001

100

/

TORQUE COMMAND

..",..,4'C7( JOO

200 TlME(~d

TIME (MC)

Figur@ 8 - (8).

figure~ .

IttHIVUIIS c Lt) a OUl:tUolt ll ll; t o r4uU l'o m m;uld,

Effect of Wlfavorable torque commaDd at singularity

(command magnitude'" o. ol).

180

;: 120 SI NGULARITY

/

....ro.•,

//' """

//i ....,""\

"./

/

··.•i

60

'I TIM( (slOe)

200

300

100

'00

200

,

TI ME (s@c)

300

'00

-60

\.

./\

- 120

./

\/" -180

l,

ZOO

100

300

400

'00

I---+--~,:"""-",'C~Jt::====!'==:::=7'::: I """( ~

'-----OO'QU, COtt1ANO lV 1

100 t

z

1

!

1

1

200

<

JOO

400

1

'00

TORQUE COfof1ANO

f---+--~':--"" J.1_-4-~:--
-0.01

lOO

200

400

300

TI ME (s ec)

rlM[ (se c) Ylg u r .. 10. R... pon.~ to 11 {Iu. t ... jI,t ing [ t'rlll" Id"' )

'igure 8-(b), !fftCI o f unfavorable l"'rque COWNIOO . t .I ngu l.lrlty (O MEGA .I"f'rlng I ... ).

369

lOlmlAnd