Mass conservative attitude control systems for interplanetary spacecraft. Gilbreath, David Slagle

Mass conservative attitude control systems for interplanetary spacecraft. Gilbreath, David Slagle
Calhoun: The NPS Institutional Archive
Theses and Dissertations
Thesis Collection
1967-09
Mass conservative attitude control systems for
interplanetary spacecraft.
Gilbreath, David Slagle
Monterey, California. U.S. Naval Postgraduate School
http://hdl.handle.net/10945/12614
I
£?^
<£>lu&&fcATH,0.
LIBRARY
SCHOOL
NAVAL POSTGRADUATE
93940
CALIF.
MONTEREY,
TABLE OF CONTENTS
Section
I.
II.
III.
IV.
V.
Page
INTRODUCTION
11
ATTITUDE CONTROL REQUIREMENTS
12
Internally Caused Torque
12
Externally Caused Disturbance Torque
13
Dynamic Requirements of ACS
14
ATTITUDE CONTROL SYSTEMS
16
Solar Radiation Torque
16
Active ACS
19
Reaction Wheel
21
Reaction Sphere
24
Control Moment Gyro
26
Momentum Dumping
39
COMPARISON OF ACS
41
Power and Energy Comparison
43
Weight Comparison
47
Angular Impulse Capacity Comparison
47
Pointing Accuracy Comparison
48
Reliability Comparison
48
CONCLUSIONS
BIBLIOGRAPHY
50
51
APPENDIX A.
Gravity Gradient Torque
52
APPENDIX
B.
Equations of Angular Motion
54
APPENDIX
C.
Brushless D. C. Motor Driven Inertia Wheel Built
by Sperry Farragut Company for NASA
57
Theoretical, Magnetically Suspended and Torqued
Reaction Sphere
59
APPENDIX D.
LIST OF FIGURES
Page
Figure
1.
Sun's Radiation vs Distance from the Sun
17
2.
Radiation Absorption and Reflection
18
3.
Solar Torquers and Energy Collectors
20
4.
Electrostatically Supported Magnetically Torqued
Reaction Sphere
25
5.
Single Degree of Freedom Control Moment Gyro
27
6.
Twin Single Degree of Freedom Control Moment Gyro
29
7.
Two Degree of Freedom Control Moment Gyro
32
8.
Twin Two Degree of Freedom Control Moment Gyro
36
9.
Summary of Equations
42
Three Twin Control Moment Gyros
46
10.
SYMBOLS
A
-
Angular impulse, ft- lb- sec
D
-
Damping torque per unit relative angular speed, ft-lb-sec/rad
E
-
Energy, ft- lb
H
-
Angular momentum about center of mass, ft- lb- sec
I
-
Mass moment of inertia, ft-lb-sec
P
-
Power, watts
S
-
Absolute angular speed of ACS spin axis, rad/sec
T
-
Torque, ft- lb
t
-
Time, sec
X
-
Vector cross product operator
^
-
Vehicle absolute angular speed, rad/sec
-
Angular speed of ACS rotating element relative to vehicle,
rad/sec
-
Unit vectors along vehicle x, y and
spectively
St
i,
j,
k
2
z
principal axes, re-
NOTATION
Letters with overbars are vector quantities.
are scalar.
quantity.
Unbarred quantities
A dot over a quantity denotes the time derivative of that
TERMINOLOGY AND ABBREVIATIONS
ACS
-
Attitude control system/ systems
DOF
-
Degree/degrees of freedom
CMG
-
Control moment gyro /gyros
ME
-
Momentum exchange
SYSTEM
-
Denotes the entire spacecraft
VEHICLE
-
That part of the system separate from a smaller rotating
component of the system
SUBSCRIPTS
control torque
c
-
Control; e.g., T
D
-
Disturbance; e.g., T n
M
-
Torque motor; e.g., T^, torque motor torque
R
-
About the spin axis of the rotating element; e.g., H
angular momentum of ACS about the spin axis
v
-
Vehicle; e.g., H
x,y,z
-
Vehicle principal axis; e.g., T
x axis
,
,
,
disturbance torque
,
angular momentum of vehicle
,
control torque about the
SECTION
I
INTRODUCTION
Future space travel will involve substantial increases in vehicle
size and trip length.
Flight paths presently under consideration are
low thrust, minimum energy trajectories with long free fall stages.
A
typical proposed round trip is the Mars flyby which requires about 700
It is for such long duration flights that the attitude control
days.
systems (ACS) discussed in this paper might be employed.
Torque produc-
ing jet thrusters have been utilized in all manned space flights to date
and are optimum for short duration flights.
For the longer voyages con-
templated, the amount of fuel required for attitude control thrusters
becomes an appreciable percentage of total system weight and storage
space.
It is this factor that has prompted the search for a substitute
for mass expulsion ACS.
The mass conservative ACS considered in this thesis are described
in Sec.
Ill and compared in Sec.
IV.
Immediately following is a dis-
cussion of attitude control requirements.
11
SECTION II
ATTITUDE CONTROL REQUIREMENTS
Need for attitude control may arise from the following:
1.
Vehicle orientation for use of main thrust engines, solar cells, navi-
gation and communication equipment, etc.
2.
Docking
3.
Crew and component limitations on vehicle angular velocity.
Orientation for scientific measurements may well be the most demanding in terms of frequency and accuracy.
Interplanetary trajectories
usually involve midcourse guidance; some programs require several velocity
changes with corresponding demands on navigational information.
The abil-
ity of data gathering instruments to compensate for vehicle attitude
error is often quite limited; the Orbiting Astronomical Laboratory concept allows a vehicle pointing error of 0.1 arc second.
Torque magni-
tudes are mentioned later in this report.
Control torque serves to give the vehicle the desired angular velocity and orientation and to counter disturbance or unwanted torque.
Dis-
turbance torques are conveniently categorized by origin as follows.
Internally caused torque
.
Interaction between components of a
spacecraft system does not affect the total angular momentum of the system, but it can change the angular motion of the vehicle; e.g., the
stopping or starting of a rotary pump does not change the angular momentum of the system but it does torque the vehicle.
Gyroscopic torque
results from precession of the momentum of rotating machinery.
Sources
of these torques include tape recorders and motion of antennas, crew and
stored liquid.
For the Apollo vehicle, 100 seconds of arc per second
represents a typical disturbance from crew movement,
12
[l]
Externally caused disturbance torque
.
Sources of external disturb-
ance torque are:
1.
Gravity gradient
2.
Solar radiation pressure
3.
Magnetic field
4.
Micrometeorite impingement
5.
Thermal radiation from vehicle
6.
Gas leaks, out gassing of vehicle material
7.
Unwanted torque from ACS
8.
Unwanted torque from main thrust engines
The facing of the same side of the moon toward earth is due to gravity gradient torque.
in Appendix A.
The salient features of this torque are described
As well as being dependent on system orientation and
mass distribution, gravity gradient torque decreases rapidly with in-
creasing distance between bodies.
The importance of gravity gradient as
a disturbance or control torque gives way to solar radiation pressure at
about 600 miles from earth.
Therefore, although gravity gradient has
been successfully employed for earth satellite stabilization, it is of
little concern for interplanetary travel.
Discussion of solar radiation torque is given in Sec. III.
Like gravity gradient, magnetically induced torque is important
only in the near vicinity of earth or other bodies with surrounding magnetic fields.
Quantitative assessment of micrometeorite induced torque will re-
main difficult until more data are accumulated on the sizes, velocities
and distribution of these particles.
Although the sizes and collision
frequency are expected to be low, the engaging speeds are such as to
13
impart considerable momentum to a spacecraft.
The extent to which mete-
orites produce unbalanced torque is, of course, a function of vehicle
geometry.
Protruding solar energy collectors and antennas render space-
craft more susceptible to this torque.
Inefficiencies in spacecraft energy conversion devices necessitate
The torque produced by thermal
thermal radiation from the vehicle.
radiation is minor and readily design controlled.
Gas leaks and outgassing of vehicle surface material would normally
cause very small torques.
ACS non-zero pointing error results from imperfect attitude sensors
and/or control systems which cause oscillations or limit cycles about the
desired orientation.
Momentum exchange (ME) ACS produce unwanted torque
due to gyroscopic coupling; this phenomenon is discussed later.
During powered flight, directional control of the main engine thrust
vector by nozzle gimballing, or other means, affords two axis attitude
control; however, supplemental attitude control may be necessary to meet
accuracy requirements.
Dynamic requirements of ACS
.
For a given system, pointing accuracy
and response time constitute the dynamic capability of the ACS.
Since
response time is a function of control torque, the latter parameter, being more convenient and general, is used for comparison purposes.
The governing general equation of angular motion for a system of
masses reduces to simplified equations in limiting cases.
and listing of these equations, Nos.
The derivation
1-9, are in Appendix B.
These equa-
tions appear in this paper without lengthy explanation and bear the same
identifying number as given in Appendix B.
equations are developed in context.
Equation (10) and subsequent
An assumption made throughout is
14
in response to control or disturbance torque, the vehicle behaves
that,
as a rigid inelastic body;
i.e., the effect on vehicle motion of compo-
nent interactions is neglected.
These interactions are accompanied by
transfer of kinetic to thermal energy and must be considered when dealSpin stabilization is an unlikely
ing with spin stabilized vehicles.
choice for interplanetary flight due to the need for fixed orientation
for the use of solar energy collectors, antennas and other apparatus.
The dynamics of rigid body angular motion is described by the following equation.
T-flM u/x 4-(I»-Iyy)u;ybUzjr
+ [Iyyli/y +fl xy
where:
"Izz)^
[XJZ]1
T = sum of control and disturbance torques
I =
\JJ
mass moment of inertia
= vehicle angular speed
The coordinate system is fixed in the vehicle and aligned with the vehi-
cle's principal axes.
The kinematical equations relating angular dis-
placement, velocity and acceleration and response time require a coordi-
nate transformation to Eulerian Angles or their equivalent.
For the
purposes of this thesis it suffices to note that response time is a function of acceleration, and this acceleration is torque generated as indicated in Eq.
(3).
15
SECTION III
ATTITUDE CONTROL SYSTEMS
ACS may be termed passive if they do not require energy or mass ex-
penditure.
In the active category are jet thrusters and momentum ex-
Passive control torque is obtained from some of the
change devices.
same environmental sources that cause external disturbance torque;
namely, gravity gradient, magnetic field and solar radiation.
For rea-
sons previously set forth, only the last of these is significant for
interplanetary travel.
Solar radiation is also spatially variant but
not so unfavorably, as indicated in Fig.
1.
Although not the only electromagnetic
Solar radiation torque .
radiation in our solar system, the sun's emissions dwarf those of all
others combined and need be the only radiation considered.
Incident
radiation is either absorbed or reflected, the reflected part being diffuse or specular.
Figure
2
illustrates the three possibilities.
The
radiation force on a surface is the time derivative of the incident mo-
Using Einstein's mass energy equivalency:
mentum.
Tr 4lmH
where
E is the
- <L(I)
,
incident energy and
is the speed of light.
c
With P f
defined as the pressure of fully absorbed radiation perpendicular to the
surface, the expression for solar pressure is:
2
p -fep(\-s)coss -h(/+sf)cos eJpf
where 9 and the unit vectors N and
/°
and
S
+(i-s<*)pf cos©
h
"t
s/nei
are shown in Fig. 2.
The symbols
are defined as follows:
0<P<\
=
fraction of incident radiation reflected
0<S<
=
fraction of reflected radiation reflected specularly
I
16
—
/
s- /ooo
J-
\A
I-*
V
3
/oo
i*
*>*
H
/o
'
1
1
1
*
3
1
1
f
H
Distance f>oyn Sun
^
r~
—
?
kvux/o"
Figure
1
Sun's Radiation versus Distance from the Sun
17
•/
Absohpt
I
Specula/-
oyi
wmiim
reflection
Diffuse
mnrnrm
V
N
Figure
2
Radiation Absorption and Reflection
18
reflection
If
and p = S=l, the maximum pressure of 2 Pf is realized.
=
earth's distance from the sun
ft-
- 9»? X
to
At an
lb/'ft\
Torque generated by solar pressure will, of course, be dependent
on moment arm and control surface area as well as the parameters in the
pressure equation.
The following example illustrates the order of mag-
nitude of solar torque.
A non-absorbing, specularly reflecting panel,
with a moment arm of 10 ft and oriented normal to the
10 ft by 10 ft,
sun's radiation at an earth's distance from the sun, will exert a torque
of 19.6 x 10"
ft-lb.
Figure
3
shows some conceptual designs that have
appeared in the literature; Figs. 3(b) and 3(c) are self aligning energy
It seems likely that if solar torque is utilized for atti-
collectors.
tude control, it will be in the form of such dual purpose devices.
Active ACS .
Active ACS are mass expulsion and momentum exchange
In the latter group are reaction wheels, reaction spheres
(ME) devices.
and control moment gyros (CMG)
.
The three ME ACS provide torque by
changing the angular momentum of a mass within the vehicle.
Their
torque is "internal"; therefore, any change in system angular momentum
is due to external disturbance torque.
The first two devices effect
momentum exchange between vehicle and ACS by changing the angular speed
of a wheel or sphere.
The CMG maintains constant angular speed and
effects momentum exchange by altering the direction of the rotating ele-
ment's spin axis.
In equation form the ME ACS are described by:
Reaction wheel or sphere:
otfu+uV)
l-
CMG
:
-J
c —o
s-o
(8)
olt
S-=0
TC = -SX^ =-IR (sx2)
(9)
19
Movable
Solar vane
panels
Hinge
Vehicle
(b)
Aligned
Reflector^/
Reflector
^Collector
Fn5r
FORCE
L--4
wll
tl>
Misqliflned
Reflector
Aligned
Boom
(c)
vehicle
Swivel points
Collector
Parabolic
reflectors
Figure
3
Solar Torquers and Energy Collectors
20
where:
Tc
= control torque
Hr
= ACS angular
S
= absolute angular
Si
momentum
velocity of the ACS spin axis
= angular speed of the wheel or sphere relative to the
vehicle
W%
= component of vehicle angular velocity about the ACS
spin axis
A more detailed description of the three basic ME ACS follows.
Reaction wheel
.
A reaction wheel is a motor driven wheel.
The
motor exerts a control torque on the vehicle and an equal and opposite
torque on the wheel.
cle.
The torque axis is fixed with respect to the vehi-
Three such wheels with mutually perpendicular spin axes provide
complete attitude control; normally the spin axes would be parallel to
vehicle principal axes to reduce the coupling evident in Eq.
(3).
For a reaction wheel parallel to the vehicle x axis, combining Eqs.
(3)
and (8) yields:
Tc=I
j?
(-d-0; x )
=TP
+IxyLU x +(Izi-Iyy)LU y LVz
(10)
)
where Tp is disturbance torque, and the assumption is made that the moments of inertia of the vehicle approximate those of the system.
For
the case of steady pointing Eq. (10) reduces to:
Tc -lK Sl-TD
(11)
For attitude changes in interplanetary space, the control torque re-
quired for reasonable response time is normally much greater than dis-
turbance torque, a possible exception being the disturbance torque resulting from misalignment of the main engine thrust vector.
tion, Eq.
(10)
With this assump-
becomes, for the case of commanded reorientation:
k=I«(A-lM=£wltf*
+(lzz-Iyy)UUylUz
21
(12)
:
Power required, less motor, bearing and windage losses, is, using
Eq.
(10):
Energy required, less the losses previously noted, is, using Eqs.
(10)
and (13):
I,
(-Q-cux )iux Jt
jj
= 5t
[(Ji2
-uy, 2 f-(ii,-u;x/
^ +[
fc
u;x Jt
(14)
The maximum angular impulse absorbed by the wheel, assuming the wheel
initially at rest and using Eq.
/W
x
= fc<# =
(10),
is:
r,f(A-^)<#*i»JW
,
»
UJ*V
where the assumption is made that
1
£l^
/T71AX
< 15)
,
Optimization of any ME ACS consists of arriving at the proper blend
of the following objectives, commensurate with cost and reliability
specifications
1.
Maximize control torque
2.
Maximize angular impulse capacity
3.
Minimize power and energy
4.
Minimize weight
5.
Minimize space
With respect to a reaction wheel the following points evolve:
1.
Objective 4 and the need for large wheel moment of inertia dictate a wheel with its mass concentrated in the rim.
2.
Objectives
3
and 4 are in direct conflict with one another;
22
for a given control torque, decreasing 1^ necessitates a higher
wheel angular acceleration with resulting increase in power and
energy.
Reference
T2~J
accounts for this compromise by minimiz-
ing an "equivalent weight" which is a function of power as well
as weight.
This function is a measure of the relative impor-
tance of power and weight and is representative of the need for
a systems approach to ACS selection and design.
The equations that have been derived for a reaction wheel are for a
single wheel.
The three wheels required for three axis control would,
in general, all be spinning.
Precession of this momentum by vehicle
angular velocity results in gyroscopic torque.
set of reaction wheels is, according to Eq.
c
Lit
The torque, then, for a
(6):
~SXH R
(6)
S=rO
where, in this case, H R is the net momentum of the three wheels and
SrCD
in Eq.
.
Normally the first term would be predominant.
(6)
is,
The second term
in general, unwanted since it increases the power re-
quired for attitude changes; however, it does afford rate damping for
steady pointing.
Damping serves to reduce oscillations or limit cycling
caused by electromechanical lag in the motor-wheel-vehicle combination.
Eddy current, viscous and hysteresis damping have been suggested.
Ref-
erence [3] claims weight and size advantages for the hysteresis damper.
Their design weighs one pound, provides 1.5 ft-lb per rad/sec and consists of permanent magnets attached to the controller housing on either
side of a thin vane protruding from the wheel rim.
Damping torque re-
sults from local magnetization polarity changes in the vane material
23
Appendix C is a brief description
as it moves relative to the magnets.
of a reaction wheel built for NASA.
Reaction sphere
.
The reaction sphere has the singular capability,
among ME devices, of providing torque about any axis.
Due to weight and
moment of inertia considerations it takes the form of a spherical shell.
Lacking mechanical supports it may be suspended by gas bearings, a magnetic field, or an electric field.
Gas bearing suspension causes high
viscous losses and limited positioning force.
If the sphere has equal
moments of inertia and the torquing imparts no radial forces, positioning force is necessary only to counter vehicle linear accelerations and
centrifugal forces due to vehicle angular speed.
Reference
[4]
suggests
magnetic suspension in the form of three orthogonal pairs of servo controlled electromagnets surrounding the sphere.
Positioning force results
from interaction between the applied magnetic field and eddy currents
induced in the spherical shell.
of suspension.
Some drag torque results from this type
Electric field suspension is the only means that does
not cause drag torque on the sphere; however, positioning force is re-
stricted by the limited voltage gradient maintainable between housing
and sphere.
Torquing the sphere magnetically has been proposed.
Reference
[5]
suggests an electrostatically suspended, electromagnetically torqued
sphere as pictured in Fig. 4.
As with magnetic suspension, torque re-
sults from interaction between the applied field and eddy currents
induced in windings in the sphere's shell.
In this design there are
three orthogonal stator windings in the shell.
rotates, "dragging" along the sphere.
The magnetic field
Eight electrode areas on the
shell and four pairs of electrodes in the housing afford four independent
24
Figure 4
Electrostatically Supported Magnetically
Torqued Reaction Sphere
25
positioning forces.
The electrodes ideally cover the sphere surface
except for motor stators.
If a reaction sphere is magnetically torqued and suspended,
is unwanted interaction between the magnetic fields.
Reference
there
[4]
recom-
mends such an arrangement with the claim that this interaction is negligible.
A description of this device is in Appendix D.
Although the reaction sphere has theoretical advantages, it has not
reached the hardware stage because of large scale problems associated
with sphere suspension and torquing.
Control moment gyro
The final and most promising ME ACS is the
A CMG may have one or two degrees of freedom; both
control moment gyro.
will be discussed.
.
The device produces torque by precessing the angular
momentum of a wheel.
7^
S,
- -sxh = -l (sxR)
r
R
(9)
the absolute angular velocity of the CMG spin axis,
is the vector sum
of vehicle angular velocity and the angular velocity of the spin axis
relative to the vehicle.
motor as shown in Fig.
The latter motion is effected by a torque
5.
The angular speed of the wheel is held constant by a drive motor
which, aside from initial speed buildup, needs to overcome only bearing
and windage losses.
In Fig.
5
the torque motor torque, Tw, acts on the
gimbal; an opposite and equal torque acts on the vehicle.
tion will be followed in subsequent figures.
natural and artificial damping.
speed about the output axis;
S,
This conven-
The damper represents
S2 is the component of vehicle angular
is the gimbal angular speed.
angular directions assumed in Fig.
5,
26
For the
the gimbal equation of motion is:
Cfixec/J
C>nba.l
Figure
5
Single Degree of Freedom Control Moment Gyro
27
JM -D6 tS£ H R -IS,
where
I
(16)
;
is the moment of inertia of the gimbal and rotor about the
s
torque motor axis.
is a gyroscopic torque.
H
S
2
D6
is the damping
R
torque, where 6 is the gimbal angular speed relative to the vehicle.
Un-
wanted torques on the vehicle are those of the damper and the torque and
drive motors; normally these would be much less than the output torque
S-.H
R
.
A more serious problem arises from the output axis not remaining
fixed with respect to the vehicle.
As mentioned previously it is desir-
able to align the output axis parallel to a vehicle principal axis; the
CMG output axis remains perpendicular to the fixed torque motor axis and
the wheel spin axis.
Therefore, to minimize coupling, such a device
must be restricted to small gimbal displacements.
Three axis control is
provided by a CMG for each vehicle principal axis.
As with reaction wheels, such a configuration must contend with
unwanted gyroscopic torque due to precession of the wheels
vehicle angular velocity.
1
momenta by
Since, in general, the CMG possesses a higher
angular momentum than the reaction wheel, the problem is more severe
with the former.
The twin gyro configuration shown in Fig. 6 provides a fixed output
axis relative to the vehicle and substantially alleviates the problem of
the preceding paragraph.
The three axes shown in Fig.
6
are orthogonal.
Not shown are the two drive motors that maintain the two wheels at a
constant and equal speed.
The torquers are coordinated so that both
wheels are precessed an equal amount at the same rate.
The neutral
positions of the angular momentum vectors are along the spin reference
axis pointing inward.
28
o
u
c
X
o
I
4)
O
•
o
"2
U
c
o
o
vO
B
o
X)
u
</)
L
01
o>
J-l
00
•H
\
00
o
00
c
•H
CO
c
•H
29
The absolute angular rate about the torque motor axis of each spin
axis is the sum of its angular speed relative to the vehicle, 6
,
and
the component of vehicle angular speed about the torque motor axis.
For
this configuration it is convenient to consider separately the gyro-
scopic torque effected by each of these angular rates.
6 H_
The gyroscopic torque,
,
is of equal magnitude for both wheels.
The components of these torques along the spin reference axis are equal
and opposite; the components along the output axis are additive and to-
gether equal:
=2H^ecos9
Tc
(17)
The reactance torques on the vehicle from the torquers and dampers are
equal and opposite, thus eliminating a minor source of unwanted torque.
The output axis is permanently parallel to a vehicle principal axis.
With the twin gyro configuration the wheel momenta cancel to some extent;
the net momentum is 2H^ sin
and is directed along the output axis.
Ve-
hicle angular speed about the spin reference and torque motor axes acts
on the net momentum to produce unwanted gyroscopic torque about the
other axis, respectively.
An increase in 9 has two detrimental effects:
The net momentum increases, causing larger coupling torques, and control
torque, Eq.
(17), decreases.
Therefore a gimbal limit prior to the
o
9 = 90
zero torque position is suggested; reference
r
,
[lj
limited 9 to
±60°.
The equation of motion for an individual gimbal is comparable to
Eq.
(16).
The output axis is again assumed parallel to the vehicle x
axis.
Ta-De-U^H*cos6-u^H«£wd*Id
(18)
;
30
where the signs of the third and fourth terms are dependent on the directions of UJX and
9
,
UJy
,
respectively, and the relative angular acceleration,
approximates the absolute gimbal angular acceleration.
quantities are the same as in Eq.
are observed, Eq.
Tm
CL
16
-\~UUxHr,COsQ
The other
If the suggested gimbal limits
(16).
(18) can be closely approximated by:
Power required, less motor losses, for
.
both torquers is:
P~zTh
e
=z(ie+iux H R cos6)e
d9)
As with the single CMG, drive motor torque for constant speed operation
is necessary only to counter bearing and windage torques.
To preclude
unnecessarily large drive motors, a long build up period is desirable.
The CMG described in reference
\l\
was allowed two hours to reach oper-
ating speed; yet the power required for this acceleration was twenty
times that needed for constant speed operation.
Energy required for both torque motors is, using Eqs.
(17)
and (19):
E=2^(l'6mH*cosb)eJt -I(b]-e]) i-^TiLux Jt
(20)
The maximum angular impulse absorbed, assuming the wheel spin axes
are initially aligned with the spin reference axis, is, using Eq.
kmn-zH^ecoseJi
=2/yR s/A/£^ ax
(17):
(21)
A twin controller for each vehicle principal axis would be necessary
for complete attitude control.
A two DOF CMG is shown in Fig.
ible.
7;
the wheel drive motor is not vis-
An additional torquer and gimbal freedom enable this CMG to pro-
duce gyroscopic control torque about two perpendicular axes.
31
The
'
Parallel
Z
to vehicle
axis
*-J
Damper
Parallel to
A
T
M2
vehicle
%.
Torque motors—
8/
£/;f
/0
^
o/
v Parol lei
to vehicle
X axis
Figure
7
Two Degree of Freedom Control Moment Gyro
32
Y
axis
Sfi^/
following discussion of the two DOF CMG utilizes the directions of angular motion arbitrarily assumed in Fig.
7.
As with the single DOF CMG,
control torque is effected by precessing wheel angular momentum.
How-
ever, with this device the gyroscopic torque is transmitted primarily
through the torque motors; the single DOF CMG control torque was conveyed
to the vehicle by the gimbals and their supports.
For example, control
torque about the x axis is obtained by the inner gimbal axis torque motor
rotating the inner gimbal.
The resulting gyroscopic torque is
H
S
2.
component about the x axis is
S
sin F
H
R
the
;
and is transmitted to the
The component S«H R cos F
vehicle via the outer gimbal axis torque motor.
is perpendicular to the two gimbal axes and is the only control torque
not conveyed to the vehicle by a torque motor.
bal by its torque motor causes a torque,
S..H-
Rotation of the outer gimsin
that is totally
,
|
about the inner gimbal axis and must be transmitted to the vehicle by the
In referring to torque motor induced
inner gimbal axis torque motor.
gimbal rotation, the assumption is made that
$\
—§
*»>Cr
S^ — P.
The interdependency of the two torquers and gimbal rotations is evident in the gimbal equations of motion.
^-Df-StHrSIMPsI,?!
(22)
,
where
I,
and I2 are gimbal plus wheel moments of inertia about the outer
and inner axes, respectively; I~ is constant and
Equation (22) shows that as
diminish.
When
1
varies with
I
/
1
departs from 90°, the gyroscopic torques
differs from zero, the torque about the inner gimbal
axis is no longer parallel to the vehicle y axis.
ments are undesirable and indicate
a.
33
Both of these develop-
need for gimbal limits; reference
[6]
suggests P
both be limited to one radian deflection from their
and
initial displacements of P
0=0°.
= 90° and
Remembering that the
motor torque, as shown, is acting on the gimbals, the control torque on
the vehicle is:
f&foCOSPCftstf +TMl SlH(t>)T
HRCosrcos^f(DPfS
rs2
+
(23) yields:
(22) and
Combining Eqs.
(23)
7
/
/y /e
s/A//
An order of magnitude analysis of Eq.
damping terms 00
the term
S
2
H R cos
7c*
and
/
DP
sin
fI2 S2 ) s 'N^]k
(
24 >
(24) permits the dropping of the
If the suggested gimbal limits are observed,
.
These considerations result in:
is negligible.
7
[SiH g SIN/ -I,sll
- [(S W R SlN/ , fI
4
l
S2 )COS0)
J
+ [StHKCDSrcosb+i^HzSlNP +Iz
where the gyroscopic terms
SoHpSin/
S l )sm<i\l
and
S-,
H R sin
(25)
/
cos
predominate.
Determination of the angular impulse capacity of the two DOF CMG is
facilitated by considering the components of the absolute gimbal rates,
and
S
x
S
2
.
S,
=
+LU*
Sl -f' t VUyC0S(f> i-UUz SIN^
(26)
)
34
Wy
where
and UJz
have the proper algebraic sign.
Since control torque
exists primarily about the vehicle x and y axes, the impulse about only
Saturation of the two DOF CMG occurs when
these axes is computed.
and
0=0°,
If the gimbals are initially at
are reached.
max
/
r
.
= 90° and
the maximum angular impulses accumulated about the x and y axes
are, using Eqs.
A*^
(25)
and (26):
=ftSt//«SWr-I
l
4
l
)Jt
=-^C05^
tH R $(lX)yCOst+LUiSmt>)SlNNt
-SlCi+^Jt
(27)
- Hr ^uux ^inP cost dt
-Ti^Cr+d/yCosj-WytsiNt
+uuzSiN<l>+lx)z 0cos$)costc/'t
A look at the relative magnitudes of the factors comprising Eqs.
(28)
and
in Eq.
(28)
x
average
/
lOy
are small numbers.
The term
is unintegrable in present form;
is assumed.
(27) and
H R is by far the largest number; 1^,
leads to reduction in terms.
l>0
I2 ,
(28)
-//*
^ SUvTcosQc/t
for illustration purposes an
Equations (27) and (28) can now be approximated
by:
^Wx" "^ CM/
ih£>,
+H« $(Vy«>st +LUzSl»<t)siNrJt
(29)
where the first term on the right side of each equation usually predominates.
A twin two DOF CMG configuration similar to Fig.
cancelling of unwanted torques.
6
provides similar
Figure 8 is a schematic of such a device
showing only the torque motors, momentum vectors and appropriate axes.
35
spin
to
referee
vehicle
Figure
z
ax/*, />*"* //•/
axis
8
Control Moment Gyro
Twin Two Degree of Freedom
36
As with the twin single DOF CMG,
the deflections of the wheel spin axes
from the spin reference axis are mirror images.
The directions of torque
and angular motion shown in Fig. 8 are for control torque about the vehiAs with the previous twin CMG, it is convenient to examine
cle x axis.
separately the gyroscopic torque caused by vehicle angular velocity and
gimbal angular motion relative to the vehicle.
The following analysis
applies to Fig. 8; the primed quantities denote the properties of the
upper wheel.
Mr - HR J"cos
RR =
rl tSiA/Ps/A/0T +SiNPcos<l>k]
H R fcosri
Net H R
f sinT
sin$1 -SinT Qos(b£\
-ZH R [cosrl +SinTsin4
fj
(31)
_J
r
- -r\cos<f>7 +s//y0/f]
PXHK +D(Hr -2Hk P\sinPI
-cosPziN<f>lj
(32)
Equation (32) describes the net gyroscopic control torque produced by
rotation of the inner gimbals by the inner gimbal axes torque motors.
If
/
are each limited to one radian deflection, sin/
and
and the y axis component in Eq.
(32)
is negligible.
»
cos/ sin
The torque compon-
ents of the inner gimbal torque motors about the y axis cancel; the com-
ponents about the
z
axis are additive and together equal
2 Tv«2
sin
.
Presumably control torque from a single controller would be generated
about one axis at a time.
0=0,
the x axis,
and
T^
Thus, for the case of control torque about
is effecting r
ciable gyroscopic torque to the vehicle.
small, the
z
axis component is negligible.
37
and not transferring appre-
Since
T(^
2
and sin
are both
,
A similar analysis for control torque about the y axis is accomThe
plished by reversing the direction of the upper x axis motor torque.
resulting gyroscopic control torque is:
$xS*-^X/yR
= -2H R ^SINFzos<t>
<
J
33 >
In general, unwanted gyroscopic torque will be caused by vehicle angular
velocity precession of the net controller momentum.
The twin configura-
tion and gimbal limits serve to reduce the net momentum and therefore the
unwanted torque.
Neglecting damping torques and gimbal accelerations, the angular impulse capacities about the x and y axes are, using Eqs.
Awx-sfawJt
(33):
*2H*$r<;iNrJt = -zH*cosrmLy[
siNtw
=-2Af/rsw/; wll5e
As expected, Eqs.
(34) and
<34>
< 35 >
;
where, again for purposes of illustration, in Eq.
been assumed.
(32) and
(35)
(35) an average
has
/
show essentially double
the angular impulse capacities of the single wheels described by Eqs.
(29) and
(30).
Torque motor power, less motor losses, for the case of control torque
about the x axis, where
0=0
,
is:
Px=r(TMti-TM Z )
(36)
For the directions assumed in Fig. 8, the following holds:
Thi
=I 2 S 2
+ S,HR SINP
(22)
38
Si
=
S2
= rtLUyC0S^ -bUz SW0
+
w* =
UIK -
S2 - r "M^y C05
S2 -
F-UJy
COS<f)
-
s,
W
z
-UUz
SIN
<()
SlA/fl
Using these relationships Eq.
(36) becomes:
—2Flz lUz£INf
where the term
ser order of magnitude of
lU2 SIN
has been dropped because of the les(p
A similar analysis for the case
.
of control torque about the y axis yields:
Py-$(Th
t
fT«,)
-Imfi+WyHfiCosQ
sinPJ
(38)
Torque motor energy, less motor losses, for control about the x
axis is, using Eqs.
(32) and
^-llzCPPdt
+
(37):
Czt
,
t
\A)*H*SiNf
dt
=I
z
(/\-r*)+C Tx UUx dt
(39)
Similarly, for control about the y axis the torque motor energy is:
Ey
=
I,
(&'#,)
+[TyWyh
(40)
Three twin two DOF CMG could be arranged to provide control torque
from two controllers about each vehicle principal axis.
Momentum dumping .
The term, momentum dumping, refers to the removal
of accumulated angular impulse from ME ACS.
39
For the reaction wheel or
sphere it is a matter of decelerating the rotating element; for any of
the CMG discussed it involves aligning the wheel spin axes with the spin
reference axis.
Saturation is caused by long acting unidirectional dis-
turbance torque such as may result from solar pressure, meteorites or gas
Presumably any ME ACS would be designed with sufficient impulse
leakage.
capacity to handle all attitude commands and oscillatory disturbances.
Dumping may be effected by solar pressure or jet reaction torque.
Since
solar torque is very small, it would necessarily be a continuous acting
system that would prevent the ME ACS from reaching saturation.
Jet reaction appears the more likely candidate; it is more reliable
and easier to mechanize.
With jet reaction dumping there is a trade off
between Amax of the ME device and the fuel required for the reaction
jets; as
A.
ax is increased, the frequency of dumping decreases.
\iax is P ro P ortiona l t0 H R =
I
R
^
Since
for all ME ACS, increase in Amax will
require additional weight in the form of a larger wheel or bigger drive
motor to maintain higher
H
.
40
SECTION IV
COMPARISON OF ACS
The following is a discussion of the relative merits of three of
the ME ACS.
Reaction sphere and solar torque ACS are excluded.
reaction sphere is omitted because it has yet to prove feasible.
The
Its
theoretical advantage is that it accomplishes with a single rotating
element that which requires three of either of the other type of ME ACS.
In addition to suspension and torquing difficulties, disadvantages of the
reaction sphere are:
The required torquing power is comparable to that
of the reaction wheel and therefore suffers the same disadvantage that
is noted in the next section; momentary power interruption for a magneti-
cally or electrostatically suspended sphere results in total failure.
Solar torque is not compared since, with the control surface sizes pres-
ently deemed practical, it is insufficient for commanded attitude changes.
Bases for comparison are:
power and energy, weight, angular impulse
capacity, pointing accuracy, reliability, size and cost.
Sizes of the
ME ACS are not compared since, with the exception of the reaction sphere,
they do not vary appreciably.
Cost is also not considered.
The comparisons, in general, are not numerical; such comparison
would require numerical evaluation of the weights, efficiencies and reliabilities of the torque and drive motors.
The ACS compared are the reac-
tion wheel, twin single DOF CMG and twin two DOF CMG.
It is assumed
that the superior performance of the twin configurations compared to the
single CMG controllers would discourage the selection of the latter.
Each ACS is assumed to consist of three controllers oriented parallel to
vehicle principal axes.
The applicable equations are listed in Fig.
for ease of reference.
41
9
1
4.
-K
•f-
5
nji^»
~T&
*+-»
/
-J
v
3 V
*
O
—x m-j
o
5'
i
1
*
* —
w
+
'
0/
4.
«
•<D
O
o
»
J?
<u
1
m|<m
+
-i-
+
•^5?
5
3
O
H
1*?
*
?
I
*
S
s
—
'
^
1
+
'?•
:<D
2
-v.
3
JE
&\
*.
U
IK
M
H
CM
"~"5
CM
5
£
'a-
5
J*
(D
X
d
~
3-
O
U
2
s
»/>
a:
<M
«M
?
« OS*
P R °
CD
2
u
O
2
i/»
z-4
Xof
H
_^_
<^
«b
,-*»
_,_
2
o
1^
o
t.
o
•<D
U
c
-H
CM
R
o
^
3
+->
-2
.5
oo
o r
q
.5
<-»
3
•f
42
Q,
ex.
o
a
o
u
o
3
-h
u.
us
o
z
Q
O
Power and energy comparison
In comparing ME ACS, power and energy
.
need not be differentiated if, for a given control torque and vehicle
angular velocity, the power is always the same; i.e., the power required
Reaction wheels
for the ACS is independent of the state of the ACS.
violate this condition; power is proportional toil
DOF CMG behave identically in this respect; Eqs.
fact that the term,
2
^W^U^COS
&
.
The single and two
(17) and
(19)
and the
is predominant show that CMG power
,
is essentially independent of gimbal position.
Maximum power is a criti-
cal factor in spacecraft systems; the varying power of a reaction wheel
system for identical dynamic response is undesirable.
It is illustrative to observe the fraction of total power that goes
into torquing the vehicle; the remainder serves to accelerate ACS components.
For the reaction wheel this fraction is, using Eqs.
IR (ji-w„).a
(12) and
-ft
For the twin single DOF CMG, using Eqs.
(17)
and (19) and neglecting
drive motor power, this fraction is:
—
TxlUx
Hr LUx
COS 6
^
.
» »
since
TO « HrLUx COS 6
Similarly for the twin two DOF CMG the
.
fractions for the x and y axes are, respectively:
LP
+H e ujx siA/P
~
l
and
H F LUySlHPCOs4>
^
.
43
(13):
where again drive motor power has been neglected.
This omission is jus-
tified if drive motor power is negligible compared to the power expended
torquing the vehicle, the latter quantity being dependent on the required
torque and the vehicle angular speed about the axis in question.
Drive motor power is minimized by using high quality bearings, carefully balancing wheels and surrounding the rotating element with a light
gas at low pressure.
scribed in reference
The 50 ft-lb of control torque, two DOF CMG, de[l\
required 18 watts to maintain 12,000 rpnu
A
twin two DOF CMG producing 50 ft-lb would require 43 watts for torquer
and drive motors if the vehicle were rotating at 0.1 rad/sec about the
torque axis.
A reaction wheel with a 100 per cent efficient drive motor,
no windage and bearing losses, and equal wheel size and torque could not
exceed
6
rpm if limited to 43 watts; and it would reach that speed in 0.01
seconds from rest.
The reaction wheel can compensate to a limited extent for its low
efficiency by employing regenerative braking; i.e., when the direction
of control torque and wheel speed are such as to require deceleration of
the wheel, the drive motor can generate its own armature current down to
a certain speed.
Although some of the kinetic energy stored in the
wheel is regained, the overall efficiency does not approach that of the
CMG.
To summarize the power and energy comparison of the three ME ACS:
1.
The single and two DOF CMG have identical energy conversion effi-
ciencies.
2.
As evidenced by Eqs.
(14),
(20),
(39) and
(40)
and the power
fractions, the major part of the energy input to a reaction wheel goes
into changing the kinetic energy of the wheel; the only appreciable
44
energy loss in a CMG, outside of initial speed buildup, is motor, bearing and windage loss of the drive motor.
3.
The reaction wheel is competitive, from a power and energy
standpoint, when either infrequent use or small torque and low angular
impulse capacity are needed.
and Amax dependency on
tively.
-0.
The latter condition results from power
as described by Eqs.
(13)
and (15), respec-
The first consideration reflects the constant drive motor power
required for the CMG; a reaction wheel, when not in use, idles at a
lower speed than an equivalent CMG, thereby requiring less power during
This advantage can be negated somewhat by the following:
such periods.
With regard to the twin single DOF CMG shown in Fig.
6,
the spin refer-
ence axis of an individual controller can be arbitrarily oriented in a
plane perpendicular to the output axis without affecting the net angular
momentum or the control torque.
Figure 10 represents three identical
twin controllers with spin axes aligned along their respective spin ref-
erence axes.
Vectors A, B, C, A
tum vectors of the six wheels.
1
,
B
1
and C
1
represent the angular momen-
The output axes of the three controllers
are parallel to vehicle principal axes.
Since the angular momenta of
the six wheels are equal in magnitude, the particular orientation of
Fig.
A', B
10 yields:
1
and
C'
A+B+C
=
A
and
1
+B +C'
1
=
.
If A, B and C or
are aligned with their respective spin axes as shown and
are brought to zero magnitude at equal rates, there is no net torque on
the vehicle and the net momentum of the controllers remains zero.
procedure halves drive motor power.
This
The reduced capability would nor-
mally be adequate for countering environmental disturbance torque.
preceding arrangement is applicable to the twin two DOF CMG as well.
45
The
axes
are torque
X} y
ar\d z
axes
and are parallel To
pKucipa.!
output
ve/ucle
axes
y
afalle/ to
axis
Figure 10
Three Twin Control Moment Gyros
46
coa's
Weight comparison .
The rotating elements, torque motors and drive
motors are the primary components contributing to the weight of ME ACS.
As noted earlier, reaction wheels require larger 1^ than do CMC,
for the
same torque and power; the larger I_, of course, means increased weight.
This difference is offset to a varying degree by the gimbals and extra
motors required for
CMC
Thus, from a weight comparison standpoint, the
reaction wheel improves with decreasing control torque and becomes com-
petitive when the larger weight of the wheel is offset by the weight of
the gimbals and additional motors of the CMG.
The two types of CMG compare in much the same fashion as CMG in general compare with reaction wheels.
With an ACS composed of three twin
two DOF CMG, each axis can be torqued by two controllers.
Therefore, an
ACS consisting of three twin two DOF CMG requires half as much angular
momentum per wheel as a similar system of single DOF CMG; this allows a
wheel weight saving for the two DOF system.
Although single and two DOF
CMG require the same power, the latter lacks the torque multiplication
of the single DOF CMG.
The gearing necessary to achieve the higher
torque results in extra weight for the two DOF system.
Thus the two
DOF CMG system is lighter than the single DOF system when the wheel
weight saving offsets the additional weight of the gearing and the extra
gimbal and torque motor.
This situation arises with larger vehicles
which require greater H^ for larger torques and angular impulse capacities.
Angular impulse capacity comparison .
The angular impulse capacity
of a reaction wheel is equal to the maximum angular momentum of the
wheel.
The impulse capacity of a CMG would also equal the H^ of the
device were it not for gimbal limits imposed.
47
This better "efficiency"
v
of the reaction wheel is more than offset by
J its typically
J
J low H
R max
,
'
which is a result of power considerations; from a power standpoint the
reaction wheel cannot be torqued at the speeds of a comparable CMG.
A significant advantage of the two DOF CMG is the fact that its
angular impulse capacity is nearly double that of the single DOF CMG.
Pointing accuracy comparison .
As noted earlier, net momentum af-
fords gyroscopic rigidity, thereby improving pointing accuracy.
The
variable net momentum of a reaction wheel system is comparable to that
of either twin CMG system; the individual wheel momenta of the CMG are
much greater than those of the reaction wheels, but a large percentage
of the CMG momenta is cancelled by the twin configuration.
of CMG have identical pointing accuracy.
The two types
CMG and reaction wheels are
subject to similar electromechanical lag in the controllers.
In summary, there is no significant inherent difference in pointing
accuracy capability between the three ACS.
Reliability comparison .
The three ME ACS have in common the poten-
tially dangerous situation of a rapidly spinning mass supported with a
minimum of physical contact for extended time periods.
Since actual
hardware is not being evaluated, the relative reliability of the ACS may
The reaction wheel, single DOF CMG
be discussed only in general terms.
and two DOF CMG increase in complexity in the order listed.
The CMG
differ only in the extra gimbal and torquer of the two DOF CMG.
Since
the motors and bearings of ME ACS are likely failure areas, the addi-
tional torquer of the two DOF CMG is a significant disadvantage.
The
constant and higher speeds of CMG make bearing failure more likely with
them than with reaction wheels.
The twin CMG configuration affords a
limited amount of redundancy, providing failure of one wheel or
48
associated equipment does not incapacitate the other half of the controller.
The twin two DOF CMG system provides further redundancy, in
that control torque is available about all axes with one controller com-
pletely failed.
The varying complexity of the three ME devices also
appears in the gathering of control logic information from the control-
With the reaction wheel, wheel speed is the only parameter moni-
lers.
tored.
For a single two DOF CMG, the control system must maintain con-
stant wheel speed and monitor the angular motion of two gimbals.
Therefore, despite the ability of the CMG systems to function while
partially failed, a rigorous reliability analysis of actual hardware
would probably find the reliability varying inversely with the complexity;
i.e., the reaction wheel is the most reliable and the two DOF CMG
is the least reliable.
49
SECTION V
CONCLUSIONS
The observations made in Sec. IV lead to the following conclusions:
1.
Solar pressure torque is insufficient for attitude control but
may be utilized to prevent the saturation of ME ACS.
2.
Reaction spheres have theoretical advantages, but there are
mechanization problems still to be solved.
3.
If power is critical and response time limited,
reaction wheels
must be relegated to relatively small vehicles; for larger vehicles the
torque needed to achieve reasonable response time requires prohibitive
power from a reaction wheel ACS.
4.
For still larger vehicles the two DOF CMG has significant weight
and angular impulse capacity advantages, but, with its increased complexity,
it is the least reliable of the three ME ACS compared.
5.
Selection of a mass conservative ACS, assuming cost is not per-
tinent, therefore hinges on vehicle size, required reliability and the
relative priorities placed on ACS weight, power and angular impulse capacity.
50
BIBLIOGRAPHY
1.
Simulator Study of Precise Atti Lopez, A. E., and J. W. Ratcliff.
tude Stabilization of £i Manned Spacecraft by Twin Gyros and Pulse
NASA TND-1645,
Modulated Reaction Jets . Ames Research Center.
1964.
2.
Yarber, G. W.
K. T. Chang, J. Kukel, B. F. McKee, C. S. Smith,
Control Moment Gyro
A. F. Anderson, C. J. Bertrem and S„ Tarhov.
Optimization Study
Garrett Airesearch Manufacturing Division.
NASA CR-400, 1966.
,
.
3.
Wheeler, P. C.
R. G. Nishinaga, J. G. Zaremba and H.
Evaluation of a Semi Active Gravity Gradient System
NASA CR-594, 1966.
,
.
4.
Williams.
TRW Systems.
L.
Hering, K. W., and R. E. Hufnagel.
"Inertial Sphere System for Complete Attitude Control of Earth Satellites," American Rocket
Society Journal XXXI (August, 1961), pp. 1074-1078.
,
5.
and M. H. Smith.
"Capabilities and Limitations of
Ormsby, R. D.
Reaction Spheres for Attitude Control," American Rocket Society
Journal XXXI (June, 1961), pp. 808-812.
,
,
6.
"The Control Moment Gyroscope,"
Dohogne, J. R., and R. F. Morrison.
Sperry Engineering Review (Spring, 1965), pp. 33-40.
7.
Reaction Wheel with Brushless D. C. Motor Drive .
Casaday, W. M.
NASA CR-388, 1966.
Sperry Farragut Company.
8.
Bell, M. W. J.
"An Evolutionary Program for Manned Interplanetary
Explorations," Journal of Spacecraft and Rockets IV (May, 1967),
pp. 625-630.
,
9.
Purser, P. E., M. A. Faget and N. F. Smith. Manned Spacecraft
Engineering and Design . New York: Fairchild Publications Inc.,
;
1964.
10.
Merrick, V. K., and F. J. Moran. The Highly Coupled System - A Gen eral Approach to the Passive Attitude Stabilization of Space
Vehicles . Ames Research Center. NASA TND-3480, 1966.
11.
Roberson, R. E.
"Torques on a Satellite Vehicle from Internal Moving Parts," Journal of Applied Mechanics XXV (June, 1958), pp.
196-200.
,
12.
Cannon, R. H.
"Gyroscopic Coupling in Space Vehicle Attitude Control Systems," Journal of Basic Engineering LXXXIV (March, 1962),
,
p.
13.
81.
"Control Moment Gyros in Attitude
D., and D. J. Liska
Control," Journal of Spacecraft and Rockets III (September,
1966), pp. 1313-1320.
Jacot, A.
u
,
51
APPENDIX A
GRAVITY GRADIENT TORQUE
The important aspects of gravity gradient torque are illustrated by
a model consisting of two spheres of equal mass connected by a
massless
rod and subject to an inverse square gravitational field.
Notation:
m,=
nu = m = mass of rod connected spheres
M
= mass of attracting body
G
= universal gravitational constant
T
= gravity induced torque
Newton •s law:
5 hQn
~
P.
^
F - ££x
2
T=4[F s,N fo-'O-F.siN(0**,j
*
-
h.
SINfo-Q
1
Jl
2
F2
h
SIN0
rsiN<t>
>
SIN(0+fi,)
F,\rSlNJ
<IN$
- jGM»*J?rSlN<t 1
y;
The gravity gradient torque is zero when
orientation, except for the unstable
52
is zero.
For any other
= 90° position, a torque exists.
This example may be generalized for an arbitrary body as follows.
A body
subject to an inverse square gravitational attraction experiences a
torque tending to align its axis of minimum moment of inertia with the
gravitational field.
53
I
x
APPENDIX B
EQUATIONS OF ANGULAR MOTION
Symbols used only in this appendix:
m-
/J-
=
mass of i'th particle
=
position vector from system center of mass to i'th
particle
The reference coordinate system is Cartesian with origin at the system
center of mass and axes colinear with system principal axes.
The system
is assumed rigid except for rotation of ACS devices.
f>c
- Xi
+ Vl T +
I
k
z-L
UJ
=
LU* I f
W
y
J + LUz k
Definitions:
Ix y - 1
y x
- ~ L
Ixz - Izx = -
/7yi
L
X-L yc
4 /m
lyz -Izy = - \rrni
ff
= I
/?
X /W:
L
Xi 2 L
yi Zj.
?i
Equation derivations:
<\
=
+ lux^
it
Of
H-I/w L
j
RyJ + zj)u; x -x £
X'l^x f(xj
+-f-/£
- UJXPl
foe
a
rigid
body
=0
+z*)wy -y£
y£ ou y -
z u>k]
.
tf=LxU/xI ilyylUyT + Izz
J
UUz k
54
£
2
•
u>*j
-»-pz. *,
ua -2 L
yL ujy ^x-
fy^Jkj
(1)
Newton's Second Law as applied to angular motion:
Combining Eqs.
T=
and (2):
(1)
[ixx LUx + (Izz -Jyy) UUy
life]
I
+ [iyyWy *(Iwf-I«)lMiU&] 7
+
Jlzz U/ 2
+ Clyy "Ixx) bUx UJQ
(3)
it
Equations for system composed of vehicle and ME ACS
T^
=
= H
Hv f H R
.
(4)
;
where T E is torque external to the system as opposed to internal torque
generated by interaction of system components, E^ and H^ are the angular
momenta of the vehicle and wheel, respectively.
In the absence of exter-
nal torque:
H
—
—
= HV
¥ H r = COHSTMT
*
*
Tc = H v = -HK
(5)
where T c is control torque on the vehicle generated by the ME ACS.
Combining Eqs.
Tc=where
S
from
\JJ
(2)
and (5):
"
Jt\.
SxH *
(6)
>
is the absolute angular velocity of the ACS spin axis;
when the spin axis moves relative to the vehicle.
S
Since
moment of inertia of the rotating element about its spin axis, is
55
differs
I
R
,
the
typically much larger than the element's moment of inertia about its
other two principal axes:
HP
where
^
\JJ
K
IR (SL+0JK)
(7)
}
is the component of vehicle angular velocity about the spin
axis of the ACS.
For a reaction wheel or reaction sphere, using Eqs.
(6)
and (7):
= -I,
->s-o
(8)
Jit
5
For a control moment gyro (CMG)
,
=0
_Q.
»
[/Je
l
using Eqs.
(6)
and
(7)
Tc = -SXH* = -I R (s xsl)
(9)
56
APPENDIX C
BRUSHLESS D.C. MOTOR DRIVEN INERTIA WHEEL BUILT BY SPERRY FARRAGUT COMPANY FOR NASA.
General description .
is hermetically sealed.
The motor is bidirectional.
The entire unit
Conventional commutation is replaced by photo
optical detectors and transistorized switches thus avoiding physical
contact between commutator and armature.
Energy saving regenerative
braking is employed; voltage generated in armature windings affords complete control of wheel when decelerating.
When the counter EMF decreases
to where it can no longer produce required armature current,
the system
is automatically switched to a "driving mode."
Motor wheel characteristics .
Total weight
-
13.7 lb
Size
-
1
Power required
-
40 watts max
Control torque
-
0.65 ft-lb @ 0-250 rpm
Friction torque
-
0.023 ft-lb
HR
-
1
I
-
1.25 lb-ft
-
0-550
R
RPM
ft x
1
ft x % ft
ft-lb-sec @ 250 rpm
2
The contractor offered the following reliability prediction for the
motor-wheel combination, considering all components in series and assuming any component failure to be a complete failure:
RELIABILITY
1000 HRS
1
YR
3
YRS
Motor and wheel actually constructed:
98%
84%
59%
Identical device with high reliability
parts:
99%
89%
71%
57
The preceding reliability estimate is based on the following power
level operation:
5
peak power
1
per cent of the time, half of peak power
per cent of the time and 6.9 per cent peak power 94 per cent of the
time.
58
APPENDIX D
THEORETICAL, MAGNETICALLY SUSPENDED AND TORQUED REACTION SPHERE [4]
Spherical shell radius
9.8 in
Spherical shell thickness
.2 in
Spherical shell material
aluminum
Spherical shell weight
23.4 lb
SI
@ maximum T
54 rad/sec
Maximum torquirtg power
18 watts
Maximum T c
.103 ft- lb
Torquing coils weight
66.2 lb
Effective suspension "spring constant"
3
5.72 x 10"
lb/in
Suspension power
8.4 watts
Suspension coils weight
11.1 lb
Total weight excluding housing and electronics
101 lb
Total power
26.4 watts
59
INITIAL DISTRIBUTION LIST
No.
Copies
1.
Defense Documentation Center
Cameron Station
Alexandria, Virginia 22314
20
2.
Library
Naval Postgraduate School
Monterey, California 93940
2
3.
Commander, Naval Air Systems Command
Department of the Navy
Washington, D. C.
20360
1
4.
Chairman, Department of Aeronautics
Naval Postgraduate School
Monterey, California 93940
2
5.
Dr. Cameron M. Smith
Department of Aeronautics
1
Naval Postgraduate School
Monterey, California 93940
6.
LT David S. Gilbreath, USN
Route 1, Box 440
Oak Harbor, Washington
1
7.
Dr. Allen E. Fuhs
Department of Aeronautics
1
Naval Postgraduate School
Monterey, California 93940
60
UNCLASSIFIED
Security Classification
DOCUMENT CONTROL DATA
(Security claeeitlcstlon ol
ORIGINATING ACTIVITY
I.
title,
•
R&D
body ol mmetrmct and indexing annotation mull be entered whan the overall report
(Corporate author)
2«.
i»
cleat 11 Led)
REPORT SECURITY CLASSIFICATION
UNCLASSIFIED
Naval Postgraduate School
Monterey, California
26 CROUP
REPORT TITLE
3.
MASS CONSERVATIVE ATTITUDE CONTROL SYSTEMS FOR INTERPLANETARY SPACECRAFT
DESCRIPTIVE NOTES (Type
4-
o/ report anrf Inchielve date*)
Thesis, M.S. in Aeronautical Engineering, Sept 1967
3-
AUTHORfSJ
CLmet nam*. Hret name, Initial)
GILBREATH, David
S.
REPORT DATE
«
7a.
TOTAL NO. OF RACES
58
• a.
ORiaiNATOR'S REPORT NUMBERfSJ
September 1967
$m.
b.
CONTRACT OR GRANT
PROJECT
NO.
OTHER REPORT
tfiia
AVAILABILITY/LIMITATION NOTICES
*ni
11.
Hi
'
-*
OP REFS
13
NO.
9b
10.
7*. NO.
£
j
ii
i
M
in
yj^nt^^oj^fm^i
m
SUPPLEMENTARY NOTES
12.
NO(S) (A ny other number* that may be aatigned
report)
*
mM
SPONSORING MILITARY ACTIVITY
Naval Air Systems Command
IS.
ABSTRACT
Attitude control requirements for interplanetary space are discussed,
Spacecraft attitude control systems, excluding mass expulsion devices,
are described.
Solar pressure, reaction spheres, reaction wheels and
control moment gyros are analyzed as sources of control torque.
A comparison is made between the reaction wheel and two types of control
moment gyros on the basis of weight, power consumption, momentum absorption capability and reliability.
The control moment gyros are shown to
be the most promising for larger vehicles.
DD
FORM
1
JAN
64
1473
UNCLASSIFIED
61
Security Classification
UNCLASSIFIED
Security Classification
KEY WO RDS
RO
L E
WT
Attitude control systems
Control moment gyro
Reaction wheel
*
a*
'.:
DD
AT.. 1473
S/N 0101-807-682
(back)
.
*i*S4»-,*i(^..
tanujt»-< «*-i-< *•»*
w
.
,4m-- **»
mi
UNCLASSIFIED
Security Classification
1
62
A-31 409
thesG424
^Pi! Y KN0X
LIBRARY
miiiiinii
"
3 2768
004148064
Dudley knox
library"
m
mmn
Mm
m
BIBB
IIMB
mwBR
MHiWK
Hiii
mi
-
1HH
mm
m
::
iiH
MKwii
> '.
,
NftHffi
mn
IB m
Rljoffi
'«
IK
ill
XK»
fflM
MSHK
NHN
m
1
Was this manual useful for you? yes no
Thank you for your participation!

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Download PDF

advertisement