Journal 01 SOilcetrall and Rockets VOLUME 6 MARCH 1969 NUMBER 3

A Description of the CMG and Its Application to Space Vehicle Control

B. J. O ’ ~ O N N O R * AND L. A. n/IORINEt The Bendix Corporation, Teterboro, N . J .

Requirements for high pointing accuracies during long missions of large spacecraft make the use of momentum exchange devices in control systems attractive for the following reasons: 1) continuous control, 2) recoverable energy source, 3) efficient control of cyclic disturbances, and 4) ease of propellant management. Of the momentum exchange devices, control moment gyros (CMG’s) have the following advantages over reaction wheels for large spacecraft: better efficiency, larger output moments, better bandwidth and dynamic range, and greater range of linearity. In this paper, particular emphasis is placed on development and implementation of CMG control laws for the Apollo Telescope Mount control system. The kinematic equa- tions of motion for the CMG’s are combined with their servo characteristics to describe their rigid-body dynamics.

Nomenclature c c c = electronic cross compensation G1(P), G3(P) = electronic stabilization network H = angiilar momentum J = polar moment of inertia subscripted for

Ki, K3 M~c-A)

ME = electromagnetic moment MR = MR(c--B) = reaction moment of a CMG on the vehicle 6, 6 = gimbal angle and angular rate E = error momentiim = H T ~ ~ - HT

wz., wz = gimbal rates relative to vehicle and inertial space, respectively

specific valiies = d.c. gains of G1(P) and G3(P), respectively = moment exerted on the inner gimbal by the

outer gimbal

Subscripts

-4, B, C = inner and outer gimbal spaces aiid base space, respectively

I , = inertial (nonrotating) aiid vehicle spaces, respectively

j = j th CMG K = commanded s = system T 1, 3

= total, for CMG cluster = 1 and 3 pivots (inner and outer gimbals), re-

spectively (see Fig. 2)

Presented as Paper 67-589 at the AIAA Guidance and Flight Dynamics Conference, Huntsville, Ala., August 14-16, 1967; submitted October 3, 1967; revision received November 6, 1968. The writers wish to acknowledge the technical assistance of the following people for the preparation of this paper: P. R. Kurzhals of Langley Research Center, and S. Seltzer, D. Schultz, C. Man- del, and H. Thomason of Marshall Space Flight Center.

* Assistant General Manager, Navigation and Control Divi- sion. Associate Fellow AIAA.

f Senior Engineer, Navigation and Control Division. Member ATAA.

Introduction

EWTON’S second law of angular motion states that the N total external moment M acting on a system is propor- t’ional to the time rat.e of change of angular momentum of t.he system Hs with respect to inertial space:

M = ( H s ) i (1) If the system momentum consists of vehicle and control momenta HV and Hr, then integrating Eq. (1) yields

J M d t = HE = HT + H v - Hsoi (2) where Hso+ is the initial value for the syst.em. If the con- troller does not expel mass, then changes in HT can be used to balance M as well as to change the spacecraft attitude by varying Hv.

The use of momentum exchange is desirable for the follow- ing reasons: 1) continuous control, 2) a recoverable energy source, 3) efficient control of cyclic disturbances, and 4) ease in management of propellant expulsion for continuous distur- bances. Items 1 and 2 result from the use of electrical energy as the prime source of power. From Eq. (2) it can be seen that the momentum-exchange device can handle cyclic torques on a continuous basis over long time periods. Constant external moments will saturate the momentum-exchange de- vice, requiring the removal of momentum by use of ambient fields (e.g., the gravity gradient) or the expulsion of propel- lants; however, the propellant expulsion can be accomplished at convenient times.

Momentum control is implemented presently by reaction wheels or control moment gyros (CMG’s). A reaction wheel has a high-inertia rotor that can be accelerat.ed by an electro- magnetic motor. A torque that changes the device’s angular momentum reacts on the vehicle through the motor’s electro- magnetic field. By contrast, a gimballed CMG wheel ro- tates at constant speed (providing a const.ant angular-momen-

225

226 B. J. O'CONNOR

SURFACE OF MAXIMUM MOMENTUM

I H I I = 1 % I "1

Fig. 1 Variable momentum effect of t w o CMG's, and momentum volume.

tum magnitude), but is capable of variable orientation rela- tive to the spacecraft. A single const'ant-momentum-mag- nitude device is not sufficient, however; Eq. (2) requires a change in magnitude as well as direction of HT. Thus, two or more CMG's are needed (see Fig. la) . The maximum mo- mentum possible in the system is obtained when all CMG momentum vect,ors are parallel, and thus momentum ex- change is possible for all CMG cluster momenta within a momentum volume (Fig. lb) . The shape of this volume de- pends on the particular CMG cluster configuration, and the condition of saturat'ion is the surface of this volume.

A cluster of CMG's offers the following advantages over a reaction wheel when large maximum moments are required for control of a large vehicle: I) the CMG wheel operates a t one speed for which the efficiency can be optimized; 2) larger maximum-moment control is obtained easily wit'h modest turning rates of the large constant momentum (without great complexity, t,orque ranges of 2000: 1 can be obtained); 3) better bandwidt'h characteristics are obtained from an angu- lar velocity-t'ype gimbal servo (the reaction wheel rotor time constant is large bccause of its physical characteristics and thus 1imit.s it's bandwidth); and 4) a linear approach toward saturation is possible, since the gimbal control rates never approach a saturat'ed st,atc. (The reaction wheel as it ap- proaches its maximum rate tends to become iionliiiear.)

A double-gimbal CMG is a two-degree-of-freedom gyro- scopic device (Fig. 2 ) comprising a constant-speed wheel held in an inner gimbal, which is coupled to an outer gimbal through the (I) pivot perpendicular to the wheel spin vector. The outer gimbal is held to the base through the (3) pivot per- pendicular to the (1) pivot. The (1) and (3) pivots are driven by geared d.c. motor torquers.

The inner gimbal ( A j space), outer gimbal (Cy space), and base (Bj space) coordinate systems are defined in Fig. 2; all spaces are parallel when the pivot angles are zero, as shown. The (1) pivot angle, Slj, is positive when t,he j t h CMG inner gimbal is rotated in a positive direction about the l l a j coordinate vector. The (3) pivot angle, S3j, is positive when the j th CMG out'er gimbal is rotated in a posit'ive direc-

J. SPACECRAFT AND L. A. MORINE

tion about the 1 S c j coordinate vector. The matrix that trans- forms a vector fromjth CMG inner gimbal space t'o its equiva- lent in base space is

C S 3 j [-s83jcSlj] [SSajsSlj] [ ::i::] = [ S83j [CS8jCSl j l [C83jS81jl ] x [ Z] (3) HaBj 0 SSlj C S l j

where c = cosine and s 3 sine. The CMG gimbal motion can be controlled by one of t,he

following basic modes (control variables) : torque, rate, or position. I n the torque mode, current is applied to the pivot torquers, and the resulting torque is balanced by a reaction torque caused by the gyroscopic action; hence, a reaction moment is applied to the vehicle. This mode is open-loop and degenerates if there is appreciable friction on the gimbal pivots, and thus is basically not. usable. The rat,e mode can be implemented in a closed-loop manner via rate servo: the CMG momentum vector is made to rotate at a commanded rate, and a reaction torque, MR = -a X H, is thus produced by gyroscopic action. This mode is implemeqted easily in analog form and is used for the Apollo Telescope Mount (ATM) CMG's. In the posit.ion mode, the gimbal pivot, angles are controlled to match the calculated desired gimbal angles; since calculat.ion of the desired gimbal angles is com- plex, this mode is not desirable.

A typical attitude control system (Fig. 3) with CMG ac- tuators (which basically defines the ATM control system)2 is a position servo with attitude rate stabilization. The (opti- cal) position sensor measures the error between the actual vehicle attitude and the desired attitude. This error is com- bined with the vehicle angular rate in the vehicle cont.ro1 law, which is basically a proportional-plus-differential controller and determines the desired command moment MK to be ap- plied to the vehicle. The command moment is processed by the CMG cont'rol law, the output's of which provide the input commands to the CMG gimbal servos. The latter commands drive the individual morneiitum vectors of each CMG. The resulting reaction moments are forcing functions that change the vehicle attitude and attitude rate, thereby closing the control loop. Thus, two system funct.ions involve the CMG: t'he CMG control-loop function, which consists of the CMG control law and the CMG configuration that will effect de- sired command moments on the vehicle; and the momentum management function, which consists of aheration of the ex- ternal moment (and hence the external moment'um) acting on the vehicle so that the CMG cluster momentum remains in- side its momentum envelope.

Properties of the CMG control law and CMG configura- tions can vary radically depending on the app1ication.l Generally, the desired total transfer functions for both CMG blocks in Fig. 3 satisfy the identity M K = Mv. Physically,

CHG ATTITUOE R A T E S I

A T T I T U O E MOMENT CONTROL

C O N f l C U R A T I O N RCFERENCE

@ I MANAGEMENT

I SATURATION 1 MEASUREMENT I I

VEHICLE A T T I T U D E

Fig. 2 Double-gimbal CMG. Fig. 3 Vehicle attitude control with CMG's.

MARCH 1969 THE CMG AND SPACE VEHICLE CONTROL 227

Izv

atg Iz + v

I n

c) One double-gimbal and one single-gimbal CMG

a) Three single-gimbal CMG’s

I Y v YV

I H

b) T h e e single-gimbal CMG pairs d) Two double-gimbal CMG’s

Fig. 4 Some basic configurations for 3-axis control.

this is acconiplished by the cross moments from one CMG being cancelled by another CMG in the cluster configuration with a net resultant moment equal to the command moment. Complete caiicellation usually is not practical because of t.radeoffs involving size, weight, power, simplicity of control law (major cause of cross coupling), momentum utilization (a function of expected moment disturbances), degree of re- dundancy required by the system, the number of axes to be controlled, and the expected external disturbances (sizing). The configurations that exhibit good momentum utilization (Le., allow the total momentum to be directed along any axes) generally require a more complicated control law to re- duce cross coupling. Figures 4 and 5 illustrate five of the many basic configurations that can be considered for 3-axis control, and Table 1 compares these configurations in terms of system considerations.

The perfect cont.rol law produces a reaction moment from t,he CMG, MB, equal to the command moment: ME = MK. This equation can be implemented either by open- or closed- loop techniques on either a moment basis:

or momentum basis:

HK - H T C ~ G - H T C M G O + ( 5 ) where HTCMG is the change in momentum state of the CMG

configuration from zero time, and HTCMG0+ is the initial mo- mentum state of the CIVIC configuration. Let us define H T = HTCMG + HTCMGo+, and H K ~ ~ ~ = -HK. Then Eq. (5) becomes

(6) H T = - H K = H K ~ ~ ~

Fig. 5 The Sixpac configuration of 3 double-gimbal CMG’s designed for the Apollo Telescope Mount (ATM) system.

228 B. J. O’CONNOR AND L. A. MORINE J. SPACECRAFT

Fig. 6 Apollo Space Station, courtesy of Marshall Space Flight Center.

The forms of the control laws or combinations of control laws will depend on the particular CMG cluster configurations.

A simple method of momentum management is to use a re- action control system, but mission life is limited because of the expulsion of fuel. A more attractive method is to use ambient fields such as the gravity gradient, aerodynamic, magnetic field, etc. to alter the long-term momentum im- pressed on the vehicle.

The first step in momentum management is saturation measurement, i.e., the measurement of the controller mo- mentum components in a given reference coordinate system (usually the vehicle principle control axes) and the combining of these components so as to determine and display the state of the momentum controller relative to the controller momen- tum volume. The complexity of this function varies widely from simple limit switches to full momentum volume indica- tion. The next step is to compute the desaturation necessary to effect the desired momentum state to which the controller is driven and to determine the initial and final desaturation commands. The computations of the desired momentum state can be as simple as commanding the total momentum to zero (likely for random disturbances) or as sophisticated as optimizing the momentum coiitrol capability by accounting for past momentum profiles. Finally, the required desatura- tion must be implemented. Simple schemes employ electri- cal caging of each CMG gimbal angle in the controller cluster to a prescribed value. More sophisticated methods use the control-loop fuiictioiis to drive the CMG’s, in concert, to at- tain the desired momentum in a momentum feedback loop; such a scheme is used in the ATM/CMG control system.

ATM/CMG Attitude Control System

The ATM pointing control system uses the Sixpac3 CMG cluster configuration (Fig. 5 ; also referred to as the Langley Configuration). Figure 6 depicts one of the ATM vehicle

MOMENTUM MANAGEMENT

INITIAL CONDITION I- MOMENT ON

VEHICLE MR

t

I” Fig. 7 HT-position control law.

Table 1 Comparison of CMG configurations of Figs. 5 and 6 with constraints for 3-axis control

Size Redmi- Config- and Alomentum dancy, uratioiia weight6 Powerc utilizationd %d

Fig. 4a 2 1.5 67% each axis 0 Fig. 4b 5 3 33% each axis 100 Fig. 4c 1 1 100% for 2 axes 0

Fig. 5 4 1 .5 100% for all axes 100

50% for 1 axis Fig. 4d 3 1 100% for all axes 2.5

a On the figures, SCMG means single-gimbal CMG; CMG means double-

The soale is for comparison in proportion to actual weight of 1 CMG and

C This scale is for comparison in proportion to actual power of 1 CMG and

d Percent of total momentum or (redundancy) of total control.

gimbal CMG.

1 SCMG.

1 SCMG.

configurations. The three CMG’s of the Sixpac are oriented with respect to the vehicle coordinates as shown in Figs. 5 and 6 when all gimbal angles are zero. This configuration pro- vides full momentum utilization in any direction and allows redundant operation consisting of any two CMG’s in the ac- tive state and the third CMG not operating. The momentum control that offers the least cross coupling and the least com- plexity is implemented in the form of the H-vector control law (Fig. 7 ) and can be used directly for desaturation with minimal mode switching. The MKCMG is integrated to yield an H K , , ~ ~ . The initial conditions are determined by the momentum managemelit control. This command is com- pared with the actual HT, and the error E is used as a modified command CMG steering law to generate 6 rate commands to alter the 3 CMG momentum vectors. The resulting gimbal positions are used to determine the Hr that closes the loop.

For slowly varying commands, this control law approxi- mates the ideal control law. It is an angular-momentum position controller. Each component of MxcMG is integrated separately to form the H K ~ ~ ~ vector, which is compared to its respective component of the total cluster HT to obtain E,

which is then processed by the CMG cross-product steering law given for the j t h CMG by

(7 )

The output of the steering law drives the CMG gimbal velocity servos and alters HT so that E vanishes. The reac- tion moment of an ideal cluster of CMG’s on the vehicle is

O Y A ~ = K,sL.H, X E

ME = - [dHr/dtv + ozv X Hrl (8)

where an ideal CMG is defined as one that has instantaneous gimbal servo loops and no inertia reaction moments, and where

3

3 = 1

If the vehicle rates are small, then ON X HT in Eq. (8) can

H T E Hj (9)

be neglected, and the reaction moment is expressed as

Since the angular-momentum vector of a CMG is constaiit in Aj space, the time derivatives in Eq. (10) are zero, so that

3

j= 1 ME = OVA X Hi (11)

Substituting (6) into (lo), expanding the triple cross- product terms, and assuming all CMG momentum magiii- tudes to be H yields

RIARCH 1969 THE CMG AND SPACE VEHICLE CONTROL 229

Fig. 8 Implementation of the HT-position closed-loop controller.

An ideal first-order equation for the control of HT is

dHtll./’dtv KE (13) This controller is stable and e is negligible if K is sufficiently large over the desired frequency range.

Equations (12) and (13) are quite similar aiid the steady- state E for Eq. (12) will be negligible for all cases except when all three vectors H,, Hz, and H3 are colinear. When this occurs, HT is zero about one axis.

Colinearity occurs when either all three CMG angular- momentum vectors are aligned in the same direction (“323” configuration) or when two are aligned and the third is in opposition to them. (“1H” configuration). The former ( 3 H ) configuration represents the CMG system a t its maximum capacity, and it must be desaturated by utilization of con- trolled external moments. The 1H configuration is avoided by the use of a distribution law4 which consists of bias rate commands on one or more of the CMG’s without changing H T.

The vectors HI, Hz, and H3 are first expressed in their respective CMG inner gimbal spaces as

The complete controller is diagrammed in Fig. 8.

H, = H1u3 j = 1 , 2 , 3 (14)

Each vector is transformed from its CMG inlier gimbal space to its base space by a resolver chain mounted on its CMG pivots, as shown by the lower coordinate transfornia-

tion. The components of HI, Hz, and H3 in CMG base space are then suitably connected to the X , Y , and 2 summing amplifiers to account for their base orientations relative to the vehicle reference frame. The output H T of each summing amplifier is theii compared with input HTK to obtain that component of E. (HT can be used also as the controller mo- mentum measurement for the momentum management func- tion.) The vector e is transformed from vehicle space to each CMG inner gimbal space, or “A” space by means of a resolver chain shown by the upper coordinate transformation to yield

3

i = l (15) e = l i A j E i A j j = 1, 2, 3

Substitution of (15) into (7) yields the required inner gimbal rate of the j t h CMG:

O Y A K ~ = K s L ( ~ I A ~ € ~ A ~ - ~ s A ~ E I A ~ ) (16)

where E ~ A ~ and E ~ A ~ are the input command signals to the (1) and (3) pivots of the j t h CMG.

The angular velocity of the inner gimbal of thej th CMG can be expressed as a function of relative gimbal angle rates as

(17) Assume that the jth CMG has two gimbal velocity servos

81j = K S L E ~ A ~ 83j = - K S L E I A ~ seeslj (18)

O Y A ~ = 1 1 ~ ~ 8 1 ~ + 1 ~ ~ ~ 6 s ~ sin8lj + 1 3 A j & j cos&,

that are infinitely fast and described by

230 B. J. O'CONNOR AND L. A. MORINE J. SPACECRAFT

The implementation of this HT control law exhibits superior performance in a mariner similar to the HT control law. The equations for the open-loop-momentum-rate control law caii be obtained by removing the feedback HT from Eq. (20), but it suffers from wide gain variations and significant cross coupling. The closed-loop form of the H T control law offers no advantage over the closed-loop HT control law and re- quires additional computation of HT for momentum manage- meiit.

FRAME

INNER FRAME

MOMENTS OF THE (11 AN0 (31 O C MOTOR

CMG Dynamic Characteristics Fig. 9 Schematic of the (1) or (3) pivot torque motor.

Then, substitution of Eqs. (18) into Eq. (17) yields the iiiiier gimbal rat.e of the j t h CMG

OVA? = K S L ( ~ I A ~ E ~ A ~ - ~ Z A ~ tanSljclAj - 13AjE1Aj) (19)

As seen by Eqs. (18) and (19), the vectorsovAKi and OVA^ dif- fer by the component. along the vector. This extra com- ponent has no effect on the controller, as can be seen by the equivalent expression resulting from the substitution of either Eq. (16) or (19) into Eq. (11). Equations (12-18) are implemented in the top half of Fig. 8. The secant term can be approximated by a constant gain since the inner pivot angle has a limited range of *7O0.

The HT controlIer can operate in a backup mode with any two CMG's operating without major switching. When only two units are operating, the open-loop gain of the controller is only + of that with three CMG's operating; however, the total closed-loop gain is unaltered.

An alternative means of controlling the reaction moment (momentum rate) on a spacecraft in either open-loop or closed-loop fa,shion is to control directly the rate of change of the controller momentum. The closed-loop control can be obtained by differentiating the expression for e and following similar steps to those given previously, yielding

The kinematic equations of motion of a C-MG are deriv- able directly from Newton's second law of angular motion, given by

M = ( & / & ) I (23)

where L is the total angular momentum of the CMG including that of the gear train. Performing the differentiation, assum- ing a gear train, as shown in Fig. 9, and assuming that the gyroscopic moments of the gimbals themselves are negligible compared with the gyroscopic moments of the wheel, the equations of motion of the CMG expressed in outer gimbal space are given by

NgMm + J M R ( ~ + Ng)Ngbmc = J ~ ~ ~ I A I A + H sin&wzC2c - H C O S ? ~ , W ~ C , ~ (24)

NgiVJm + J M R ( ~ + Ng)NgbzBsB = H COS&WIA~A + J3bz~2c + J B ~ z c ~ ~ (25)

where H is the magnitude of the wheel angular momentum, N , is the pivot gear ratio, the subscript j has been dropped since only one CMG is considered, X E ~ , JIE> are the electro- magnetic moments supplied by the (I) and (3) pivot torquers, J I M R i s the polar momeiit of inertia of the torquers about their respective spin axes, and

Fig. 10 Electronic and kinematic signal flow graph of a double-gimbal CMG.

MARCH 1969 THE CMG AND SPACE VEHICLE CONTROL 231

w IN RADIANSISEC

0 m

Z -6 w

2 -12 A

I 4-18

-24

OF VEHICLE ATTITUDE

w IN RA?DIANS/SEC

. . Fig. 11 Closed-loop frequency responses 81/81K without a)

and with b) cross compensation.

Ja = siii261[(J~33 f Jo) - (Jazz f J R ) ] (28)

where J R , Jo are the polar moments of inertia of the gyro wheel about its spiii axis and diameter axis, respectively; J~11, Jazz, J A 3 3 are the polar moments of inertia of the inner gimbal exclusive of the gyro wheel about the 11~, 1 2 A , and 1 : ~ axes, respectively; Jca3 is the polar moment of inertia of the outer gimbal only about the 13c axis; and UIAka , U I C ~ ~ ,

uIB~B (k = 1, 2, 3) are the component angular rates of the iiiiier gimbal, outer gimbal, and base, respectively.

The reaction nioments of the CMG onto the base expressed in outer gimbal coordinates are given by

. ~ ~ R ( c - R ) ~ c = - J l W 1 A l A - H Sill&WiCzC H COS8iWrcaC (29)

M R ( c - - B ) ~ C = H sinBiwiAla - J3Ljic3c (30)

- M R ( c - - B ) ~ C = - H C O S ~ I W I A ~ A - J2Ljic8c (31) where

JI = Jii - Ng(1 + LV~)JMR, J z = J33 - N g ( 1 + Ng)Jlrr~ (32) The equivalent moments expressed in base space can be ob- tained by transforming Eqs. (29-31) via a rotational trans- formation about the I3c pivot.

The ilTM CMG’s are controlled by forcing their gimbal angular rates to match command gimbal angle rates in a closed-loop manner.

81 w i A l A - Wiclc 8 3 E ~zc , c - u i B 8 B (33)

(34)

(35)

By defiiiitioii the gimbal ang rates are

The electronic signals to the gimbal torquers are given by

ME, = G~(P)(&, - 8,) - (GCCl/Ng)83

M R ~ = G3(P)(&, - 83) + (Gcc,/Ng)81

aiid

The terms involving G,,, and G,,, are electronic cross coupling incorporated to isolate the two servo axes.

Figure 10 is a signal-flow diagram of the total system char- acteristics described in Eqs. (24-35) and includes the expres- sion of the reaction moments on the vehicle in base coordi-

nates. The electronic transfer functions are shown by dashed lines; the kinematic transfer functions by solid lines. The numerical values of the parameters are as follows: N = 2000 ft-lb-see, J l l , J 3 3 = 17.66 and 21.05 ft-lb-sec2, re- spectively, N g = 56, G1(P) = 1575 (0.05P + 1)/(5P + 1) ft-lb-sec/rad, G3(P) = 1925 (0.05P + 1)/(5P + I) ft-lb-see/ rad, and G,,, = Gee3 = 0.707H = 1414 ft-lb-see. The re- sponses of the. servo.loops can be obtained by solving these equations for 61 and &; i.e.,

81/81, = NgG1(PJ33 + N,GS)/A (36) and

83/83, = NQGa(PJn + N,Gl)/A (37) where the characteristic equation is given by

The frequency responses of Eqs. (36) and (37) for the ATM CMG without electronic cross coupling (G,,, = G,,, = 0), and with electronic cross coupling set a t G,,, = G,,, = H cos (a/4) = 0.707H1 are shown in Fig. 11. Ideally, the electronic cross coupling should be equal to H cos& for complete de- coupling of the loop; however, because of the limited freedom of the inner gimbal (*70°), a constant electronic cross coupling is desirable. If this cross coupling is set a t 0.707H, the loops are sufficiently decoupled, as indicated in Fig. l l b .

Conclusions

The efficiencies aiid dynamic characteristics of clusters of two or more double-gimbal control-moment gyros (CMG’s) make the use of momentum exchange a practical method for accurate attitude control of large spacecraft over long periods of time. The particular cluster of CMG’s to be used will depend on tradeoff considerations that involve size, weight, power, allowable control-law complexity, momentum utiliza- tion, degree of redundancy, etc. For the ATM application, the Sixpac configuratioii of 3 double-gimbal CMG’s exhibits the most desirable characteristics because of its redundancy (100%), momentum utilization, and yet moderate size, weight, and power characteristics.

Extensive analyses of the control law for the ATMICMG control system indicate that either the closed-loop momentum vector position (HT) or closed-loop rate (HT) control laws is a good choice for the CMG control laws. These control laws exhibit reasonably constant gain and minimum cross cou- pling. From an implementation standpoint, the HT control law requires less complexity and provides for a maximum utilization of the hardware.

References

1 “MORL Mechanical Systems, Volume XIV, Stabilization and Control,” Rept. SM46806, NASA Contract NAS 1-3612, Sept. 1964, Douglas Aircraft Corp.

2 Seltzer, S., Schultz, D., and Chubb, W., “Attitude Control and Precision Pointing of the Apollo Telescope Mount,” Journal of Spacecraft and Rockets, Vol. 5, No. 8, Aug. 1968, pp. 896-909.

3 Kurzhals, P. R. and Grantham, C., “A System for In- ertial Experiment Pointing and Attitude Control,” T R R-247, 1965, NASA.

4 Kennel, H., “Individual Angular Momentum Vector Dis- tribution arid Rotation Laws for Three Double-Gimballed Con- trol Moment Gyros,” TMX-53696, Jan. 1968, NASA.