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Description  |
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BACKGROUND OF THE INVENTION
Thermal elastic shock (TES) that spacecraft can experience is attributed to
a disturbance torque created by the presence of a sunlit and a shadowed
section of the orbit plane. The darkened region is formed as the Earth
moves between the satellite and sun, thus eclipsing the satellite view of
the sun. The eclipsed portion of the orbit is referred to as the umbra and
the transition to and from this region is called the penumbra. Thermal
elastic shock is a result of a rapidly changing temperature difference
between the hot (sun pointing) and cold (anti-sun pointing) surfaces of a
large flexible appendage extending away from the spacecraft. The
temperature offset across the two surfaces quickly decreases at umbra
entrance when the solar heat flux is turned off in a stepwise fashion,
thus subjecting the flexible structure to a rapid cooling effect. In a
similar respect, the thermal gradient quickly increases as the appendage
experiences rapid heating during umbra exit when the solar flux turns on
in a stepwise manner. The rapid change of the thermal gradient causes the
hot surface of the structure to bend due to either thermal compression
(sunset) or thermal expansion (sunrise). The thermally induced bending
motion of the flexible member generates a disturbance torque which is then
transferred back onto the space vehicle core-body.
To minimize or eliminate the TES disturbance torque, feasible solutions,
other than the solution offered by this invention, do exist. One obvious
solution is to alter the material properties of the disturbing member so
that the thermal gradient at umbra entrance and exit is not so dramatic.
However, this solution must be pursued during the design phase of a
spacecraft before the hardware development begins. If the hardware phase
is already underway, redesigning the structure will be quite costly.
Furthermore, prior to the research leading to this invention a detailed
TES disturbance model was unavailable to determine the effect of the TES
disturbance on the spacecraft attitude pointing performance. Thus an
accurate upper bound of the allowable thermal gradient to maintain
pointing accuracy was also unavailable.
Another solution is to impose structural constraints on the appendage to
increase rigidity and structural damping. As in the case of the first
alternative solution, this solution also requires additional analysis to
determine the required stiffness to minimize pointing errors introduced by
the TES disturbance. Although this solution may not be as costly as the
first alternative approach, some additional costs would be involved.
Again, the use of the TES disturbance model is needed to produce estimates
of the vehicle attitude pointing performance.
A final approach would be to use feed forward compensation techniques in
the vehicle attitude control system to minimize the effect of the TES
disturbance. Although this solution seems to be the easiest and cheapest
to employ, it can be quite deceiving. This approach would require a
software change in the vehicle on-board computer and some resource to
obtain thermal gradient measurements. The TES disturbance equations would
be an integral part of the software needed to predict the magnitude of the
TES disturbance. However, current reaction wheel assemblies (RWA's) reach
their maximum torque authority responding to the TES disturbance as
evidenced and predicted for some spacecraft (i.e. TOPEX, LANDSAT). Thus
the feed forward torque prediciton method may provide little or no
compensation. An actuator capable of generating larger torques would be
necessary for the feed forward compensation technique to be valid.
Unfortunately, current RWA designs do not produce enough torque and new
RWA's would be costly to develop. Large torque actuators, such as a
control moment gyroscope (CMG), could provide the necessary torque to
eliminate the TES disturbance, but CMG's are bulky, heavy and require a
large power input. Also, for small satellites, CMG's are not a practical
solution.
All of the preceeding described solutions would only be valid for the
individual spacecraft under consideration. Furthermore, these solutions
would require the use of the previously mentioned TES torque equations.
The spacecraft thermal disturbance control system invention overcomes
these previous problems. With this invention it is possible to effectively
counter the effects of thermal elastic shock without changing the
materials used in the manufacture of portions of the spacecraft. Moreover,
this invention does not require the redesign of the spacecraft. In fact,
it is possible to retrofit many spacecraft to use this invention without
any extensive redesign or increases in cost. The localized active control
system solution of this invention on the other hand, would still require
the use of the TES disturbance model, but would not be limited to a
specific vehicle.
SUMMARY OF THE INVENTION
This invention relates to thermal disturbance control systems and more
particularly to thermal disturbance control systems for spacecraft.
It is accordingly an object of the invention to provide a spacecraft
thermal disturbance control system that effectively compensates for
thermal disturbances to which the spacecraft is subjected.
It is an object of the invention to provide a spacecraft thermal
disturbance control system that effectively reduces unwanted thermal
torque on the spacecraft.
It is an object of the invention to provide a spacecraft thermal
disturbance control system that permits the spacecraft to be accurately
positioned.
It is an object of the invention to provide a spacecraft thermal
disturbance control system that permits the spacecraft to be accurately
maintained in position.
It is an object of the invention to provide a spacecraft thermal
disturbance control system that reduces undesired torques on the
spacecraft resulting from spacecraft structure that is exposed to a
temperature differential.
It is an object of the invention to provide a spacecraft thermal
disturbance control system that reduces undesired torques on the
spacecraft resulting from the spacecraft structure being partially exposed
to sunlight.
It is an object of the invention to provide a spacecraft thermal
disturbance control system that does not involve the main spacecraft
attitude control system.
It is also an object of the invention to provide a spacecraft thermal
disturbance control system that is particularly useful for spacecraft that
have structures with large areas that are subjected to sunlight-shadow
transitions.
It is also an object of the invention to provide a spacecraft thermal
disturbance control system that is particularly useful for spacecraft that
have non-symmetric projecting structures.
It is an object of the invention to provide a spacecraft thermal
disturbance control system that does not require any significant changes
to the structure of the spacecraft.
It is an object of the invention to provide a spacecraft thermal
disturbance control system that does not require changes to the materials
used in constructing the spacecraft.
It is an object of the invention to provide a spacecraft thermal
disturbance control system that does not require any significant additions
to the spacecraft structure.
It is an object of the invention to provide a spacecraft thermal
disturbance control system which provides a solution to a thermal
disturbance problem where currently there is no effective solution.
It is an object of the invention to provide a spacecraft thermal
disturbance control system that does not add significant weight to the
spacecraft.
It is an object of the invention to provide a spacecraft thermal
disturbance control system that is simple in its construction.
It is also an object of the invention to provide a spacecraft thermal
disturbance control system that is simple in its operation.
It is also an object of the invention to provide a spacecraft thermal
disturbance control system that is very reliable.
These and other objects are obtained from the present spacecraft thermal
disturbance control system that includes a plurality of temperature
sensors located on a portion of a spacecraft that projects from the main
portion of the spacecraft and control means including a microprocessor
operatively connected to the plurality of temperature sensors for
receiving temperature information from the temperature sensors and for
providing commands based upon the received temperature information. The
spacecraft thermal disturbance control system also includes a reaction
wheel assembly operatively connected to the control means for receiving
commands from the control means and providing a torque based upon the
received commands to counter the torque induced as a result of a thermal
gradient being applied to the projecting portion of the spacecraft.
BRIEF DESCRIPTION OF THE DRAWINGS
The invention will be hereinafter more fully described with reference to
the appended drawings in which:
FIG. 1 is a side elevational view of a spacecraft panel and associated
structure illustrating the effect of thermal disturbance on the
spacecraft;
FIG. 2 is a perspective view of the spacecraft thermal disturbance control
system located on a spacecraft solar array panel;
FIG. 3 is a block diagram of the spacecraft thermal disturbance control
system illustrated in FIG. 2; and
FIG. 4 is a functional block diagram illustrating how the spacecraft
thermal distrubance control system of FIGS. 2 and 3 functions and certain
equations used during the functioning.
DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT
Referring first to FIG. 1, a portion of a prior art spacecraft appendage is
illustrated and is designated generally by the number 10. The spacecraft
structure 10 comprises a spacecraft solar panel 12 whose inner end portion
14 is rigidly attached to the shaft 16 that is in turn connected to the
main portion of the spacecraft designated by the number 18 with only a
portion thereof illustrated for clarity. The normal position of the solar
panel 12 is that illustrated by the dashed lines. Then when the solar
panel 12 is subjected to uneven heating due to the action of sunlight
represented by the arrows 20 incident upon the panel 12 in an uneven
manner this causes portions of the panel 12 to expand and become distorted
as indicated by the panel 12 that is shown in solid lines. This distortion
represented by the solid line panel 12 results in a torque represented by
the arrow and the letter T in FIG. 1 being applied to the main portion of
the spacecraft 18 through the shaft 16. This torque T can have very
undesireable effects upon the spacecraft 10 including causing the
spacecraft 10 to move from its desired location in space.
Obviously, the undesired effects of unwanted torque T can have serious
adverse consequences including preventing a spacecraft from completing its
planned mission. At the very least, it could cause the spacecraft to
expend important fuel to regain or maintain its desired position in space.
This can result in shortening the useful lifetime of the spacecraft. These
undesired effects can be eliminated or greatly reduced through the use of
this invention.
As illustrated in FIG. 2, the spacecraft thermal disturbance control system
is designated generally by the number 22 and comprises a thermal sensor
network 24, a micro-computer 26 and a small reaction wheel assembly (RWA)
28. The thermal sensor network 24 comprises a plurality of distributed
thermal sensors strategically mounted on each face 30 and 32 of the
spacecraft solar panel 34. As illustrated, the thermal sensors are
arranged so that there is a first series of thermal sensors designated by
the number 36 on the one side or face 30 of the solar panel 34 and a
second series of thermal sensors designated by the number 38 on the other
face or side 32 of the solar panel 34.
Each thermal sensor of the first series of thermal sensors 36 is connected
individually to the micro-computer 26 as illustrated in FIG. 2 by the
electrical leads 40 and 42 that electrically connect thermal sensors 36 to
the microcomputer 26 and the electrical leads 44 and 46 that electrically
connect the thermal sensors 38 to the microcomputer 26. It should be noted
that although only a few of the thermal sensors 36 and 38 are illustrated
as being connected by the respective leads 40 and 42 and 44 and 46 to the
microcomputer 26, all of the thermal sensors 36 and 38 would be
individually electrically connected to the microcomputer 26, but the other
leads have been omitted for clarity and since they are not necessary for
an understanding of the invention or to teach one skilled in the art how
to practice the invention.
As illustrated in FIG. 2 the microcomputer 26 is electrically connected to
the RWA 28 by the electrical conductor cable 48 and both the microcomputer
26 and the associated RWA 28 are located within the generally rectangular
hollow housing 50 and connected through the housing 50 to the inner end
portion 52 of the spacecraft solar panel 34 so that the reaction wheel
assembly 28 is capable of exerting a torque, represented by the arrow and
letter T.sup.1, upon the inner end portion 52 of the solar panel 34. The
inner end portion 52 of the solar panel 34 is connected to the main
portion of the spacecraft designated by the number 54 by the rigid shaft
56 and the connected rigid projecting members 58 and 60 that are connected
to the shaft 56 and the inner end portion 52 of the solar panel 34. It
should be noted that only a portion of the spacecraft main portion 54 is
illustrated for clarity.
FIG. 3 sets forth the spacecraft thermal disturbance control system 22,
previously illustrated in FIG. 2, in block diagram form and provides
further details on the composition of the microcomputer 26. As illustrated
in FIG. 3, the thermal sensors in the series of thermal sensors 36 and 38
are identical and each thermal sensor comprises a thermistor 62 and a
resistor 64 that are connected in parallel. Each of the thermal sensors 36
and 38 are connected electrically to an analog multiplexer 66 that is part
of the microcomputer 26 as represented by the leads 40 and 44. The thermal
sensors 36 and 38 are also connected to a common lead 68 via the
respective leads 70 and 72 and this common lead 68 is connected to a
constant electric current source 74 that is in turn connected to the
analog multiplexer 66 by the lead 76.
The analog multiplexer 66 is also connected to a programmable peripheral
interface (PPI) circuit 78 that also forms part of the microcomputer via
the lead 80. The output from the analog multiplexer 66 that contains
information from the series of thermal sensors 36 and 38 is sent as
represented by the arrow to a scaling circuit 82 and then the output of
the scaling circuit 82 that is an analog voltage is sent to an analog to
digital (A/D) converter 84 as represented by the arrow. Both the scaling
circuit 82 and the A/D converter 84 also form part of the microcomputer
26. As indicated by the arrow, the output of the A/D converter is sent to
the PPI circuit 78 that also receives and sends information, as
represented by the arrows, from and to a common bus 86 that also forms
part of the microcomputer 26.
As also illustrated, the memory 88, the programmable interval timer (PIT)
90 and the interrupt 92 each provide inputs represented by the respective
arrows to the common bus 86. It will be noted that the memory 88 comprises
both a random access memory (RAM) and a read only memory (ROM) and that
the PIT timer 90 receives information represented by the arrow from a
clock 94 that also forms part of the microcomputer 26 and that the PIT 90
also provides an input represented by the arrow to the interrupt 92. A
microprocessor 96, associated co-processor 98 and a remote interface unit
(RIU) 100 also form part of the microcomputer 26 and as illustrated by the
various arrows the microprocessor 96 receives information from the clock
94, the bus 86 and the co-processor 98 as well as the RIU 100. The
microprocessor 96 as indicated by the arrows also provides information to
the bus 86 and to the RIU 100. The RIU 100 would in turn as indicated by
the arrows be receiving information from and transmitting information to
the spacecraft communication system 102 or the like located exterior to
the spacecraft.
As illustrated in FIG. 3, the microcomputer 26 also has a second PPI
circuit 104 and a digital to analog (D/A) converter 106 and as illustrated
by the arrows, the PPI circuit 104 provides information to both the bus 86
and to the D/A converter 106. As indicated by the arrow, the D/A converter
106 provides an anlog voltage input to the RWA represented by the block
numbered 28 and as indicated the RWA 28 provides an applied torque and a
return power signal. The microcomputer 26 also includes the power
converter 108 that, as indicated by the arrow, receives power from an
external source such as the battery on the spacecraft (not shown). The
power converter 108, as indicated by the arrows, also provides an input to
the bus 86 and to the return power signal. It will also be noted, as
indicated by the arrow, that the external source of power such as the
spacecraft battery (not shown) provides power directly to the RWA 28 to
provide its required operational power.
The flexibility built into the software design will allow on-orbit
adjustments to accommodate the true nature of the TES disturbance torque.
The complete system functional block diagram is depicted in FIG. 4. An
inspection of FIG. 4 shows that the system uses the thermal sensor network
24 to provide temperature measurements at various locations on the solar
array. The measurement data are then fed into an optimal estimation
computer routine designated by the block numbered 110. This routine is
based on the Gauss-Markov theorem and has been used extensively in the
fields of meteorology and oceanography. An optimal estimate of the thermal
gradient that exists across the panel at a desired location is then used
to predict the magnitude of TES disturbance torque using the detailed
analytical model that will be hereinafter developed that provides the
torque prediction equations represented by the block 112. The generated
torque prediction Tp is used to drive the RWA 28. The applied torque Ta
from the RWA 28 that combines with and counteracts the torque caused by
TES that gives a greatly diminished net torque Tnet. As indicated in FIG.
4, temperature gains, Kp, Kr, and Ka are included in the software to
tailor the system response as indicated by the blocks 114, 116, and 118.
The optimal estimation routine equations set forth in FIG. 4 are developed
as follows. A linear form of the observations is assumed and can be
expressed as:
.theta..sub.i =.THETA.(r,s)+.epsilon..sub.i (1)
For i=1, . . . , N where
.phi..sub.i .ident.i.sup.th measurement
.epsilon..sub.i .ident.i.sup.th measurement error
N.ident.total number of observations
.THETA.(r,s).ident.scalar variable at position(r,s)
Furthermore, the assumption is made that the measurement errors are
un-correlated and independent of .theta.. Under these assumptions, the
Gauss-Markov theorem, provides the resulting estimation equation given as
follows:
##EQU1##
where
.THETA..ident.estimated mean of the observations
A.sub.ij .ident.covariance between all pairs of observations
C.sub.zi .ident.covariance between the estimate and the i.sup.th
observation
and the associated error matrix, C.sub.e given as:
##EQU2##
The estimated mean of the observations is computed such that the sum of
the weighted measurements equal zero and is determined by the following
equation:
##EQU3##
The previous three equations are thus used to provide an optimal estimate
of a solar array temperature at a prescribed location and the error
associated with the estimate.
The key to implementing the optimal estimation technique is the
determination of both the C.sub.x matrix and an analytic weighting
function to scale the variance of the data. The weighting function is
necessary to compute numerical values for the C.sub.xi and A.sub.ij
matrices. The C.sub.x matrix is generally unknown but can be approximated
by the variance of the given data set. The numerical computation of
C.sub.x is generated using the following equation:
##EQU4##
where .sigma..sub..phi. is the standard deviation of the measurement data
given as:
##EQU5##
The last term on the right hand side of equation (5) accounts for
uncertainties associated with the estimated mean.
A weighting function is selected to weight the measurements according to
their spatial location with respect to one another and to the desired
position of the estimate. This type of weighting function can be used as a
first cut statistical model given no a priori knowledge of the data
statistics. The estimation technique can, however, easily accommodate more
complex statistical models if desired. An analytical expression for the
weighting function is given as:
W.sub.ij =0.2(.gamma.-r.sub.ij.sup.2
-s.sub.ij.sup.2)exp.sup.-.sqroot.r.sbsp.ij.spsp.2.sup.+s.sbsp.ij.spsp.2(8)
where
.gamma..ident.measurement degradation factor
r.sub.ij .ident.scaling parameter between the i.sup.th and j.sup.th
observations in the r direction
s.sub.ij .ident.scalling parameter between the i.sup.th and j.sup.th
observations in the s direction
The parameter .gamma. is introduced to change the quality of the
observations. If .gamma. is set equal to 5.0, then a maximum correlation
of 1.0 will exist when the condition i=j is satisfied in the equation (8)
set forth above. As .gamma. linearly decreases, the maximum attainable
correlation also decreases in a linear fashion. The scaling parameters,
r.sub.ij and s.sub.ij, are calculated using the following equations:
##EQU6##
The variables r.sub.scale and s.sub.scale can be specified to determine an
effective range of data influence (de-correlation scale) or set to the
dimensions of the spatial area over which the measurements are confined.
For the present invention, the latter condition is assumed. The variables
r.sub.i and s.sub.i denote the spatial location of the i.sup.th
observation while the variables r.sub.j and s.sub.j indicate the spatial
position of the j.sup.th point. Thus, given the weighting function, a
spatially weighted covariance can be computed between the point of
estimation and the measurements, C.sub.xi, and between the observations
themselves, A.sub.ij. The calculation of C.sub.xi can be expressed by:
C.sub.xi =W.sub.xi .sigma..sub..phi..sup.2 (11)
where the subscript x is used to denote the desired
location,(r.sub.x,s.sub.x), of the estimate, while the weighted
observation matrix, A.sub.ij, is determined from the following equation:
A.sub.ij =W.sub.ij .sigma..sub..phi..sup.2 +.sigma..sub..epsilon..sup.2
.delta..sub.ij (12)
where .delta..sub.ij is the Kronecker delta function expressed as:
##EQU7##
and .sigma..sub..epsilon. the standard deviation of the error.
The torque prediction equations set forth in FIG. 4 are developed in the
following manner. The following equation gives a rough approximation of
the radius of curvature for a rod like structure such as that illustrated
in FIG. 1 when it is exposed to a temperature difference .DELTA.T:
##EQU8##
The above relation holds true for both the top and bottom rods (i.e.
pb.apprxeq.pb.sub.top .apprxeq.pb.sub.bottom) under the assumption that
the separation distance is small, d<<p.sub.b.
The mass distribution about the fixed end of the slender rod is given by:
##EQU9##
where
L=rod length
M=rod mass
If the rod length and mass are divided into n pieces then a discrete
representation for the rod inertia becomes:
##EQU10##
where the i.sup.th mass element is expressed as:
##EQU11##
By factoring out the mass term, the equation (16) above can be rewritten
as an equivalent inertia distribution (lumped mass) given as:
##EQU12##
where l.sub.i is of the form:
##EQU13##
Let I.sub.b be an equivalent inertial displacement defined as the rod
inertia, in the equation previous to the one above, multiplied by the
angle through which the structure rotates relative to the constrained end.
The structure is assumed to deform in a circular arc of radius pb under
the influence of a uniform temperature gradient applied over the entire
length. Then for the i.sup.th inertia element and the bend angle of the
i.sup.th mass element:
##EQU14##
The arc length each element of mass travels is given by:
s.sub.i =l.sub.i .theta..sub.i (21)
Rearranging and substituting the expression for arc length, equation (21),
into equation (20) above yields:
##EQU15##
The change in radius of curvature associated with each l.sub.i increment
is expressed as follows:
.DELTA.pb=pb-pbcos(.phi..sub.i) (23)
where the curvature angle, .phi..sub.i, associated with the i.sup.th
inertia length element is given by:
##EQU16##
If the assumption is made that each inertia length increment is small
compared to the radius of curvature, such that the ratio li/pb<0.176, then
the curvature angle .phi..sub.i, is less than 10 degrees. Under this
assumption, the curvature angle for the i.sup.th element can be simplified
as:
##EQU17##
If it is further assumed that each mass element only travels through a
small angle, .theta..sub.i <10 degrees, then the arc length, is
essentially linear and equal to the change in radius of curvature given by
the third equation above. This approximation can be stated as:
s.sub.i .apprxeq..DELTA.p.sub.b (26)
Thus from the previous two equations (23) and (25) and equation (26) above,
##EQU18##
Substituting the arc length equation (27) above into the derived
expression for I.sub.b, equation (22) yields:
##EQU19##
The previous equation defines a general analytical expression to describe
the dynamic motion of a slender rod of finite thickness given an applied
thermal gradient. This equation can also be used to represent the movement
of thin beams and thin flat plates since each geometric figure shares a
common expression for the inertia about one end.
Intuitively, the first and second time derivatives of the previous equation
(28) would yield the momentum and torque time histories describing the TES
disturbance. The resulting torque function calculated using the previous
equation only captures the latter part (i.e. the exponentially decaying
step) of the overall disturbance. This is because I.sub.b is a
multi-valued function at t=.tau. which yields a step discontinuity at
t=.tau. for the first time derivative of I.sub.b. Neglecting the step
discontinuity in the second time derivative eliminates the impulsive term
present in the expected structure of the disturbance torque. Thus, in
order to capture the disturbance torque function which completely
characterizes the observed on-orbit attitude response, fundamental
continuous theoretical functions are derived to explain the underlying
mathematics behind the TES phenomena. In the following development a
continuous function for I.sub.b, analogous to the discrete I.sub.b
expression given in the previous equation (28), will be developed.
Let I.sub.b be defined as a continuous function of the form:
I.sub.b =.alpha.[.mu.(t+.infin.)-.mu.(t-r)]+f(t-r).mu.(t-r)(29)
where
##EQU20##
and u(t-r) is a unit step function defined as:
##EQU21##
In order for I.sub.b to be continuous at t=r, the following matching
condition must hold true:
f(t-r).vertline..sub.t=r =.alpha. (31)
Momentum is defined as the first derivative of I.sub.b with respect to
time, i.e. dI.sub.h /dt. Computing the first time derivative of equation
(29) above gives the following expression for momentum:
##EQU22##
where u(t-.tau.) is a Dirac function described as:
##EQU23##
The second derivative of equation (29) with respect to time, yields the
equation for the torque:
##EQU24##
It is assumed that all terms containing the derivatives of the unit step
function u(t+.infin.), i.e. u(t+.infin.) and u(t+.infin.), have no
contribution to the dynamics at t.gtoreq.0. Thus, the previous momentum
and torque equations (32) (34) reduce to the following form:
##EQU25##
The doublet, u(t-r) in the above equation (36) which is multiplied by the
function f(t-r) can be expressed as two terms given as:
f(t-r)u(t-r)=f(0)u(t-r)-f(0)u(t-r) (37)
Finally, if the matching condition, equation (31), requiring I.sub.b to be
continuous at t=.tau. and the doublet equation (37) above are applied to
both the momentum and torque equations, these expressions further reduce
to the following:
H.sub.b =f(t-r)u(t-r) (38)
T.sub.b =f(t-r)u(t-r)+f(t-r)u(t-r) (39)
The fundamental relationships given by these equations (38) and (39) can be
used to compute the TES disturbance momentum and torque using the time
derivatives of the known function f(t-r). An inspection of equation (38)
above indicates that the momentum is comprised of the first time
derivative of the function f(t-r) multiplied by a unit step function.
In order to determine the functional form of f(t-r), a time dependent
analytical relationship for the thermal gradient is assumed. Using an
analytical function to represent the thermal gradient is advantageous in
that the derivatives may be computed continuously rather than by an
approximate first-order difference technique. A relatively simple
exponential function has been chosen to represent the thermal gradient.
The analytical thermal gradient function for umbra entrance is expressed
as:
##EQU26##
where
k.ident.decay constant
b.ident.temperature bias
c.ident.temperature scaling
q.ident.temperature constant
r.sub.sunset .ident.umbra entrance time constant
A similar expression is used for sunrise and is given by:
##EQU27##
with r.sub.sunrise being the umbra exit time constant. The parameters used
to calculate .DELTA.T.sub.sunset and .DELTA.T.sub.sunrise can be
prescribed to match the predicted thermal response of the structure.
Substituting previously developed equations (14) and (40) into equation
(28) yields the following expressions for I.sub.b at sunset:
##EQU28##
Substituting previously developed equations (14) and (41) into the
discrete equation (28) for the inertial displacement yields the following
expressions for I.sub.b at sunrise:
##EQU29##
It can be seen that equation (42) above is equivalent to the previously
developed equation for the continuous inertial displacement (29) with:
##EQU30##
Likewise, it is evident that equation (43) above is equivalent to the
equation (29) for the continuous inertial displacement with:
##EQU31##
If the thermal gradient is represented as .DELTA.T without any subscript,
then a general expression for the function f(t-r) can be formulated as:
##EQU32##
It is evident that this equation (48) is equivalent to the previous
equation (28) for the discrete inertial displacement with pb replaced by
the previous equation (14) for the radius of curvature. This result makes
sense as the physical relationships used to derive the discrete equation
must be contained in the mathematics.
Thus, the momentum and torque equations can be computed using the equations
(38) and (39) for H.sub.b and for T.sub.b, along with the equation (48)
for f(t-r). They are as follows:
##EQU33##
where .DELTA.T and .DELTA.T are the first and second derivatives of the
thermal gradient, .DELTA.T, with respect time.
The TES disturbance momentum and torques can be evaluated using these
equations for any one of three geometric shapes. These include slender
rods, thin beams and thin flat plates. The momentum and torque expressions
are entirely dependent on the applied thermal gradient and its successive
derivatives along with the mass and material properties of the selected
geometric object.
The spacecraft thermal disturbance control system 22 is made and used in
the following manner. The dependence of the TES disturbance torque model
set forth in FIG. 4 on the successive derivatives of the thermal gradient
motivates the need for accurate temperature determinations. To accurately
measure the true thermal response of the solar panel 12, two types of
thermal sensors, thermistor and platinum resistance thermometers (PRT)
known in the art should be utilized for the respective thermistor 62 and
resistors 64. Fully integrated (i.e. signal conditioned) thermistor are
accurate to about .+-.4 degrees centigrade (C) while the accuracy of a
fully integrated PRT is approximately .+-.1 degree C.
The microcomputer 26 illustrated in FIGS. 2 and 3 is made from standard
space qualified components known to those skilled in the art. The
microcomputer 26 is also made and tested using standard techniques known
to those skilled in the art. When the microcomputer 26 is in use it
functions in the following manner. Alternating thermal measurements are
sampled and input into the computer 26 using an analog multiplexer and a
PPI 78. Each observation is scaled between 0 and 5 V through the scaling
circuit 82 and then digitized using the A/D converter 84. The digitized
voltage measurement is then input to the bus 86. The bus 86 also contains
connections to the memory 88 with both ROM and RAM, a clock 94, a PIT 90
with interrupt 92, a microprocessor 96 and a second PPI 104 which outputs
the commanded wheel voltages. Housed within the memory module is the
computer code containing the voltage to temperature conversion factor, the
previously developed optimal estimation routine equations, the previously
developed TES disturbance torque equations and the RWA torque to voltage
conversion. The PIT 90 is used to set the sample rate for the measurement
data and select the control system execution rate at which voltage
commands are issued to the RWA 28. The microprocessor 96 performs the
mathematical calculations and also communicates with the host spacecraft
through a RIU 100. The clock 94 keeps the microprocessor 96 active and the
co-processor 98 helps to speed the numerical computations. A power supply
(not shown), provided by the host vehicle, is connected to a power
converter 108 which provides the bus 86 and the RWA 28 with the necessary
power for operation. The digital voltage output is sent through a D/A
converter 106 to generate an analog voltage which is then used to drive
the RWA 28 electronics.
It is estimated that 64K of memory 88 will satisfy the needs of this
system. One of the following microprocessor 96 chips, the 8086, 80286 or
80386 supplemented with the proper co-processor 98 chip, 8087, 80287 or
80387, is required for this system. The selection of the microprocessor
chip will determine the bit size of the bus 86. For example, the 8086 chip
uses a 16 bit bus while the 80386 processor utilizes a 32 bit bus. The
microprocessor 96 chip size is a function of the desired control system
execution rate and the amount of mathematical operations needed to
perform. For this control system, the amount of mathematical computations
is a function of the quantity of thermal sensors, n, being employed since
the optimal estimation routine performs the inversion of two n.times.n
matrices. The most economical design would be to use the 8086 for the
microprocessor 98 with the 8087 for the coprocessor 98.
The RWA 28 is a self contained package which contains the reaction wheel,
the torque motor and the wheel drive electronics. The wheel dri | | |
