CSDL-R-1901
Performance and Applications of a Hybrid
Jet Selection and CMG Steering Law Based on Linear Programming
by
Joseph A. Paradiso
October, 1986
ABSTRACT
Linear programming techniques have been applied to develop a
CMG steering law which exhibits very high adaptability to CMG
hardware failures and variations in CMG system definition. The
procedure is also capable of performing a fuel-optimal jet selection
and establishing control via a hybrid mixture of jets and CMGs.
This report summarizes the results of simulations that have been
performed to examine the capabilities and advantages of the resulting
hybrid selection/steering procedure.
INTRODUCTION
Space station attitude control will most likely be provided by
Control Moment Gyros (CMGs), while momentum desaturation and
translational control will be accomplished through Reaction Control
System (RCS) jets and/or environmental torques. Previous spacecraft
control systems have managed such dissimilar sets of actuators via
independent control laws linked only through operational requirements.
The anticipated mission requirements on a space station place a
high premium on reliability and efficiency. To meet these demands, it
is highly desirable to develop control laws which are able to derive
maximum benefit from both types of actuators (CMGs and jets), used
independently and in coordinated combination. Examples of the latter
include use of jets to augment the response of a degraded CMG system
(i.e., CMGs unable to deliver desired torque or momentum due to
hardware failure), attitude maneuvers requiring large rates,
simultaneous attitude and position control, and simultaneous
maneuvering and momentum desaturation.
A further consideration is the requirement to control across
significant vehicle configuration changes occurring during build-up,
docking, and articulation. The space station mass properties and
actuator configuration/response will evolve substantially during these
operations, requiring a capability to track such changes and
accommodate them in the control system. While desensitizing the
control system may be adequate for stability purposes, high reliability
and efficiency require an adaptive actuator management and control
strategy.
Existing CMG control and steering systems are generally subject
to drawbacks and restrictions which can prove significantly
disadvantageous for space station operation. Particular mounting
orientations and specific modes of CMG usage are often implicit in the
structure of most CMG steering laws1 Hardware saturation (i.e., peak
limits on CMG output torque and stop constraints on gimbal excursion)
is not considered in most CMG steering procedures, thus must be
2
enforced after the CMG selection has been performed. These limitations
can appreciably compromise the flexibility and accuracy of available
CMG response.
The effort summarized in this report has addressed these
challenges, and has resulted in an efficient and extremely flexible
actuator management procedure which can be used to steer a CMG system,
perform fuel-optimal jet selections, and control spacecraft via a
hybrid mixture of jets and CMGs. Prototype controllers and vehicle
simulations have also been developed to examine the performance and
features of this novel approach to actuator selection and steering.
KIS
Ah(,
ROTOR
A
AXIS
(a) SINGLE-GIMBALLED CMG
FIGURE 1: SINGLE AND DOUBLE GIMBALLED CMGs
APPROACH
The original approach taken to develop the hybrid actuator
selection process has been summarized in Reference 2 and detailed in
Reference 3. The current structure of the steering/selection procedure
is qualitatively described below.
CMGs are momentum exchange devices which create an output torque
by changing the orientation of angular momentum stored in a rotor
spinning at constant rate. The two standard CMG types are depicted in
Fig. 1. Figure la shows a single gimballed CMG (rotor constrained to
3
gimbal in a fixed plane), and Figure lb shows a double gimballed CMG
(rotor able to align with any direction given sufficient gimbal
freedom). The angular velocity of each CMG gimbal may be commanded to
yield an output torque, which is instantaneously described by the
vector product of gimbal rate and corresponding rotor momentum.
The hybrid selection process optimally commands an array of CMGs
to produce either a vehicle torque or change in vehicle angular rate.
A Linear programming algorithm is used to calculate a set of gimbal
displacements in response to a commanded rate change (in the latter
case, CMG gimbal rates are normalized such that the gimbal which moves
the farthest is run at its peak rate).
The instantaneous output torque of each CMG gimbal is used to
form a set of "activity vectors" for linear programming selection. CMG
systems generally possess more available degrees of freedom than
required for control purposes, thus often admit several possible
solutions. The linear programming process converges to the particular
solution which meets the input command while minimizing an objective
function that penalizes CMG configurations of low controllability,
thereby encouraging avoidance of problematic CMG orientations. The
objective calculation for double gimballed CMG's includes terms which
act to avoid parallel and antiparallel rotor alignment (which can lead
to singular CMG orientations and degraded control capability),
encourage inner gimbal angles to be kept minimal (large inner gimbal
angles reduce the control authority of the outer gimbal), and remove
CMGs from the vicinity of gimbal stops. Since singular states of a
single gimballed CMG ensemble are not always related to rotor
alignment, they are instead steered to directly maximize 3-axis control
authority; anti-singularity objective factors are derived from a matrix
4of MG output torque vectors representing net system controllability
4
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Each CMG gimbal may be moved in two directions (i.e., "forward"
and "backward" rotation), corresponding to positive and negative CMG
gimbal rates and/or displacements. Linear programs, however, do not
normally admit solutions containing negative decision variables; the
structure of the selection algorithm was thus modified to enable
specification of "bipolar" CMG gimbal motion, as detailed in Reference
3. Two objective coefficients are calculated per CMG gimbal,
representing the consequence of gimbal rotation in the respective
sense. Under the net objective minimization, gimbals are encouraged to
move in the directions that avoid configurational difficulty as
outlined above.
The "upper bound simplex" algorithm 5 has been adapted to perform
linear CMG selection. When solving for CMG gimbal displacements in
response to an 'input rate-change request, the simplex process
incorporates upper bounds on the decision variable, which, in this
case, represent restrictions on CMG travel imposed by gimbal stops.
These bounds are also made to decrease as the CMG system approaches
momentum saturation, thereby directly accounting for the reduced CMG
control capacity.
When solving for CMG gimbal rates in response to an input torque
request, the simplex upper bounds directly limit the decision variables
such that peak gimbal rates are not exceeded. When a gimbal stop is
approached, the corresponding upper bound is decreased to account for
the limited travel available in that direction.
Solutions derived via conventional linear programming (without
upper bounds) contain only as many non-zero decision variables as there
are dimensions in the control request (i.e., only 3 CMG gimbals will be
specified in the response to a 3-axis request), which does not
generally yield an appropriate means of commanding a multi-CMG system.
Adoption of upper bounds remedies this difficulty; solutions derived
via upper bound simplex contain as many CMG gimbals as required to
"optimally" answer the input request. If the request is too large (or
CMGs are approaching saturation), the upper-bound simplex solution will
contain "artificial variables", which indicate that the CMGs are unable
5
to provide the requested output, and suggests the introduction of jets
(possessing higher control authority), or application of an alternate
strategy, as detailed below.
Because the expression used to form activity vectors is an
instantaneous approximation to the CMG output torque, the linear CMG
selection must be iterated periodically with updated activity vectors
and objective coefficients (all CMG controllers run in this fashion; to
maintain a constant torque as the CMGs rotate, the CMG gimbal rates
must be periodically re-calculated). The CMG control software used to
produce the test results presented in this report allows the CMGs to
rotate no further than five degrees before forcing an update
selection.
A method of reducing this nonlinear "crosscoupling" effect has
been developed to improve the accuracy of linear CMG solutions to
rate-change requests. The exact momentum transfer resulting from the
set of gimbal displacements specified by the simplex solution is
calculated via a simple vector sum weighted by sines and cosines of the
gimbal angles involved6. If this actual momentum transfer is found
to differ appreciably from that requested, simplex may be re-invoked to
solve for the momentum residual, and gimbal displacements from both
solutions may be superimposed to form a substantially more accurate CMG
response.
Linear programming has been used to perform highly adaptable,
fuel-optimal jet selections in the Draper-developed OEX advanced
autopilot7, which has been successfully flight-tested 8 onboard the
Shuttle Orbiter. This capability has been retained in the hybrid
control package. Activity vectors representing jets are derived from
jet thrusts and placements (both translational and rotational
components are calculated; jet activity vectors are 6-dimensional
quantities). The simplex selection process determines a set of jet
firing times in response to an input rate-change request. In order to
converge to a minimum-burn, fuel-optimal solution, objective factors
for jets are made proportional to their fuel usage rate.
Provided that the input torque or rate change is within the
capabilities of the current CMG configuration and not large enough to
6
drive the CMG system into saturation, the linear program is performed
with jets inhibited, and simplex will produce an exclusively CMG-based
solution. If the request is too large or the CMG array has ~neared
saturation, the linear program will indicate that the desired output
cannot be realized solely with CMGs. The ability of an arbitrary CMG
array to produce a given rate change may also be quantitatively
determined by the value of a "saturation index", which defines the
ratio of the desired final CMG momentum state to the largest possible
projection of CMG momentum in the final state direction (formulating
the saturation index for a CMG system including single gimballed
devices required the development of special projection formulae, as
documented in Ref. 3). If this ratio is greater than unity, the CMGs
will be driven into saturation by the input request.
If the input request cannot be realized with CMGs alone (as
defined via the above conditions), a "hybrid" selection may be
performed in which both sets of actuators are made available for
selection, yielding simultaneous CMG displacements and jet on-times in
response to an input vehicle rate-change request. The balance of jet
vs CMG usage in hybrid selections is governed by both the mean
CMG-to-jet objective factor ratio and the CMG upper bound values.
Under the current convention, hybrid selections are generally performed
with reduced CMG upper bounds in order to realize the bulk of the
maneuver with jets and preserve CMGs for rotational trimming (thereby
avoiding lengthy maneuver durations and saturated CMGs). Jet costs are
made compatible with mean CMG costs during hybrid selections in order
to favor "cheaper" CMG activity vectors which correspond to CMG motion
in the most favorable directions. Such hybrid selections thus often
tend to "desaturate" the CMG array, and remove it from problematic
orientations.
The simplex selection is capable of solving a six-component input
request, thereby providing coordinated control of vehicle translation
and rotation. Hybrid selections made for such cases employ jets to
achieve the desired translational impulse and introduce CMGs to aid in
rotational stability.
7
The linear programming selection package is capable of
coordinating actuators under several different modes of operation.
CMGs may be selected exclusively in answer to either- input rate-change
or torque requests. Jets may be commanded exclusively in response to
input rate changes. Jets and CMGs may be handled independently (i.e.,
jets produce a requested rate-change, while CMGs are simultaneously
instructed to generate a torque); this mode is useful in desaturating
the CMGs or damping extraneous vehicle rates created by quantized jet
firings. Finally, jet and CMG activity may be coordinated under a
hybrid selection, as discussed above. The latter feature is a unique
application of the linear programming selection scheme.
Conventional CMG steering laws require the addition of an
independent "null motion" procedure to continuously calculate gimbal
rates which redistribute the CMGs into more controllable orientations
without transferring momentum to the host vehicle (torque-producing
gimbal rates, as calculated in conventional steering laws, tend to lead
the CMGs into singular orientations). When steering via linear
programming, this is achieved via the objective function; objective
minimization encourages the CMGs to avoid singular orientations as they
respond to input commands, thus no additional null rates are required.
The ability to redistribute the CMGs without torquing the vehicle may,
however, prove to be desirable; i.e., in cases where the CMGs are to be
restored from an initial sub-optimal orientation. The simplex
selection is able to calculate null rates with very little
modification; by allowing objective factors to become negative and
zeroing the input request, null gimbal rates are directly produced.
The amount of "null" gimbal redistribution present in the CMG response
to requests of finite magnitude may be increased by allowing the
objective factors to also become negative when performing conventional
selections.
If jets and CMGs are selected together during the null motion
process, coordinated jet firings and gimbal motion will be specified
such that the net CMG system cost is reduced, thereby moving the CMGs
away from problematic orientations and momentum saturation (as
8
reflected in the CMG objective factors), hence achieving a
"desaturation" effect while holding constant vehicle rates. Jet
desaturation may also be performed in a more conventional manner by
firing jets to transfer momentum into the spacecraft along the
saturation axis while simultaneously commanding the CMGs to produce a
compensating torque in the opposite direction.
In summary, the application of linear programming to CMG
selection has produced an extremely flexible CMG steering law.
Activity vectors representing linearized actuator response are kept in
a common pool which is scanned during each simplex selection. Actuator
failures may be accommodated by preventing corresponding activity
vectors from being selected by the simplex process (and eliminating the
failed devices from the objective calculations). Since each CMG gimbal
is modeled via an independent activity vector, single gimbals of
dual-gimballed CMGs may be failed (i.e., frozen at constant position),
still leaving the surviving gimbal available for selection. Gimbal
stops and peak gimbal rates are represented by bounds imposed by
simplex on selected gimbal rates and displacements; these are
considered directly in each simplex selection, and may be inserted,
removed, or re-defined at any time. Actuator response and vehicle mass
properties are represented by parameters used in activity vector
calculation which may be easily changed and updated. Since specific
CMG mounting configurations are not assumed anywhere in the selection
process, CMGs may be mounted in any orientation. A linearized
objective function is optimized in each selection, eliminating the need
of a separate "null motion" procedure to calculate gimbal rates which
avoid problematic CMG orientations. Null motion, if desired, may also
be calculated directly under simplex by requesting a zero net rate
change using a modified objective function. The same linear
programming procedure is able to perform fuel-optimal jet selections
and simultaneously command CMGs and jets in a coordinated fashion.
Jets may be used to desaturate the CMG array under null motion or by
commanding both actuators independently to transfer opposing momentum
impulses.
9
FIGURE 2: PARALLEL MOUNTED CMG CONFIGURATION
I YAW
'CH
POWER TOWER SPACE STATION
(a) SPACE STATION REFERENCE CONFIGURATIONS
-Ti
71
62 1
/2
(b) PARALLEL MOUNTED CMG REFERENCE CONFIGURATION (ZERO GIMBAL ANGLES)
h1
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(c) INITIAL ORIENTATION OF CMG ROTORS
10
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Dual-keel Space Slation
It
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This method of linear programming selection is not necessarily
limited to jets and CMGs; other devices (such as reaction wheels) can
potentially be included, provided that they may be effectively
represented by linearized activity vectors, objective coefficients, and
upper bounds.
SIMULATION RESULTS
a) Simulation Setup and Conventions
In order to test the performance of the hybrid selection/steering
procedure, it has been interfaced to a closed-loop control system based
upon the OEX Advanced Autopilot7 . This autopilot incorporates a
phase-space control law, which derives vehicle rate-change commands
YAW
t
PITCHROLL
INITIAL CMG ORIENTATION
FIGURE 3: ORTHOGONAL MOUNTED CMG CONFIGURATION
11
- 3
from a weighted sum of vehicle attitude and rate errors, providing
coordinated control of vehicle rotational states. Mass properties of
the Power Tower9 and Dual Keel 10 Space Stations (without orbiter
attached) have been used in these simulations (see Figure 2a). Double
gimballed CMGs are assumed mounted in either of two standard
arrangements (it is possible to define any mounting protocol with this
steering law). Figure 2b illustrates the "parallel" mounting
configuration (assumed by most control laws proposed for the space
station; e.g., Ref. 1), where all CMGs are mounted with outer gimbal
axes parallel. This configuration is initialized in a "T" orientation,
with all inner gimbals at zero displacement and outer gimbals initially
spaced uniformly apart such that all rotors are symmetrically oriented,
and the net CMG momentum sums to zero (as depicted in Figure 2c for a
system of 4 CMGs). The other standard double gimballed CMG mounting
setup, termed "orthogonal mounting", is depicted in Figure 3, which
shows the CMGs in their initial orientation. This mounting style was
derived from the convention used in Skylab 11 (three CMGs initially
perpendicular), with a fourth added skewed at equal angles to each of
the others. When orthogonal arrays contain over four CMGs, the
additional devices are skew-mounted in the other quadrants.
All CMGs are modeled in accordance with recent specifications
proposed for the space station12; i.e., an angular momentum capacity of
3500 ft-lb-s per rotor, and peak gimbal rates of 5 deg/s on both inner
and outer gimbals. Stops are imposed at +/-90 degs. on inner gimbal
travel, and the outer gimbal is assumed capable of continuous
360-degree rotation. "Ideal" CMGs are assumed in the simulation, and
higher-order dynamic effects (e.g., gimbal acceleration torques, servo
effects, etc.) are not currently included.
The standard arrangement assumed for single gimballed CMGs is
termed "pyramid" mounting, where CMG rotors gimbal in planes which form
12
III
the faces of a regular pyramid, as depicted in Figure 4 for a 5-CMG
array. Single gimballed CMGs are modeled with a limit of +/-180 on
gimbal rotation. Other parameters (peak rate, etc.) are identical to
those defined above for double gimballed devices.
YAW
ROLL
al
- I
PITCH
INITIAL CMG ORIENTATION
FIGURE 4 PYRAMID MOUNTED SINGLE GIMBALLED CMG CONFIGURATION
All space station models are assumed to possess 12 RCS jets,
which operate at a nominal thrust of either 75 lb (Power Tower) or 10
lb (Dual Keel) each, and are clustered into planar triads located at
four positions on the spacecraft (see Reference 3). While holding LVLH
attitude, all jets are oriented in the orbital plane; no direct
out-of-plane thrust is possible with this configuration. Jets are
assumed to be discrete devices, and firings are rounded to the closest
80 ms increment. No separate logic is used to introduce jets; they are
prescribed and selected by simplex as discussed in the previous
section.
13
b) Verification of Features and Performance
The set of tests presented below consists of a series of
sequenced maneuver commands performed under various conditions in order
to demonstrate the capabilities and features of the linear programming
approach to CMG/RCS management. All examples in this section are
performed in an inertial environment; Euler coupling and precessional
torques due to rotation of stored CMG momentum are included as
disturbances, while aerodynamic and gravity gradient torques, which
would be present on-orbit, are ignored. Vehicle rates are initialized
to zero at the start of each test.
Example #1: Validation of Objective Function
As mentioned earlier, the objective function is used to steer the
CMGs away from problematic orientations (i.e., rotor alignments and
excessive inner gimbal angles). The first example in this series
demonstrates how the anti-alignment and inner gimbal minimization
contributions affect CMG steering.
A Power Tower model is assumed, driven by an array of 4
orthogonally-mounted CMGs (Figure 3). A series of rate increases
(0.003 deg/s each) are commanded about the vehicle's pitch and roll
axes (the yaw rate is held at zero). Five such increases are
requested, building net rates of 0.015 deg/s over the 100 s duration of
the test run. This is a significant rate, considering the sizable
vehicle inertias (106-+107slug-ft2 ) and limited CMG control
authority. Two test cases are considered; one with a complete
objective function, and another without including the anti-lineup
component.
Gimbal angles for both cases are shown in Figure 5. Results
using a complete objective are shown in the left column; both inner and
outer gimbal systems are seen to be employed in answering requests
(excessive inner gimbal swings are avoided), and jets are required to
complete maneuvers (as indicated by asterisks) toward the end of the
14
III
run. The results for the test omitting the anti-lineup component are
shown in the right column. The remaining objective components work
only to minimize the inner gimbal angles, and this is indeed what is
noted in Figure 5c; the inner gimbals are hardly used until they are
required to complete requests at the conclusion of the test.
CMG rotor alignments are portrayed in the upper portion of Figure
6. The complements of the relative angles between all possible CMG
rotor pairs are plotted (there are 6 combinations of pairs possible in
a 4-CMG system). A parallel lineup is indicated when a curve nears
+900, an antiparallel lineup is indicated when a curve nears -90 ° , and
the respective CMG rotors are orthogonal (the "ideal" case) when the
curve is in proximity to zero. When using the complete objective
function (Figure 6a), CMGs are generally seen to avoid alignment until
all move together toward saturation at the close of the test. When
lineup avoidance is not considered in the objective (Figure 6b), CMG
rotors are seen to more frequently approach alignment; at the positions
indicated on the plot, rotor pairs moved within 100 of parallel and
antiparallel alignment before saturation was reached. The extreme
minimization of inner gimbal angles attained in this case was performed
at the expense of avoiding interim CMG rotor alignments. In both test
cases, jets were automatically selected by simplex when continued CMG
performance was inhibited due to momentum saturation.
Vehicle rates are plotted in the lower portionof Figure 6. The
commanded input about the pitch and roll axes is plotted as the dotted
"staircase"; the vehicle rates (solid lines) are seen to follow this
input. Since rotor alignments are avoided, the average CMG control
authority is generally higher, and a quicker vehicle response is noted
in the run incorporating the complete objective function. The last
requested rate increase occurs after the CMGs have reached momentum
saturation, hence it is answered primarily via an RCS response; due to
their greater control authority, the vehicle is seen to respond more
rapidly when jets are employed. The vehicle maintains a constant yaw
attitude while rates are commanded around the other axes. Although a
small yaw disturbance is encountered when jets participate in maneuvers
15
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Example #2: Automatic Jet Response and Single Gimbal Failures
A major advantage of steering CMGs via linear programming is the
ease with which the system of available actuators may be dynamically
reconfigured and re-defined. Linear programming also provides a unique
ability to blend jets and CMGs under a single selection procedure;
jets are automatically introduced in response to commands which exceed
available CMG control authority. The next example demonstrates these
abilities; a mixed jet/CMG response is required when an excessive
vehicle rate increment is commanded, and the CMG ensemble is instantly
reconfigured into a single gimballed system when all outer gimbals are
failed.
This example also assumes the Power Tower model, but uses an
orthogonally-mounted 5-CMG array; four are included as depicted in
Figure 3, and a fifth "skewed" CMG is added perpendicular to CMG #4.
The command sequence consists of two rate-change requests followed by
attitude holds. The initial segment requests 0.03 deg/s about the
pitch axis, and enters attitude hold after 30 s have elapsed. Once the
vehicle rates have settled and the desired attitude has been acquired,
a rate of 0.005 deg/s is requested about the roll axis. All CMG outer
gimbals are failed after this rate is achieved; thus subsequent CMG
control must be realized exclusively with inner gimbals. After keeping
this rate for approx. 60 s, an attitude hold is commanded, and rates
are again brought to zero.
Results are given in Figure 7. Gimbal angles are plotted in the
left column; note that jets were introduced to establish and remove the
commanded 0.03 deg/s pitch rate. Since the CMG array (as mounted and
initialized in this configuration) can only provide enough momentum to
achieve approx. 0.015 deg/s along this axis, jets are required to reach
the desired value of 0.03 deg/s. Because of the restrictive upper
bounds placed upon CMG gimbal displacement in solutions including jets
(as described earlier), CMG participation is limited in responding to
18
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this request, and the bulk of maneuvering is accomplished via the
jets. This prevents the CMGs from being first driven into saturation
before introducing jets; the hybrid response to such a large input
request favors a jet-based solution in order that the CMGs are not
saturated when the final state is achieved.
Vehicle rates are given in Figure 7c. The 0.03 deg/s request is
seen to be quickly established (A) and removed (B) primarily via jets.
The attitude hold is commanded while the vehicle is coasting at 0.03
deg/s. There is no feedforward implementation of attitude commands;
instructions are executed by the controller as they arrive. Because of
this, the vehicle is made to coast at approximately -0.002 deg/s after
the attitude hold is commanded in order to compensate for attitude
errors accumulated during deceleration. This rate is removed and the
vehicle is stabilized (C) entirely via the CMG system (0.002 deg/s is
well within the CMG control margin).
After vehicle rates have damped to zero, a 0.005 deg/s rate is
requested about the roll axis. This is handled exclusively by the
CMGs, and no jets are required (see Figs. 7a and b). After this rate
is achieved (D), the outer gimbals of all CMGs are failed (i.e., frozen
at constant position and inhibited from selection {E}), and an attitude
hold is commanded (F). As seen in Figs. 7a and b, this was achieved
entirely via the inner gimbal system; no RCS assistance was required.
At point G, vehicle attitude was restored, and rates -were returned to
zero.
Vehicle attitudes are plotted in Figure 7d. The command sequence
executed in this test established attitude changes of approximately
0.85 deg. in pitch and 0.3 deg. in roll. Yaw attitude remains at zero,
as commanded.
Even though the CMG selection process can instantly adapt to
commanding single gimballed CMGs, effective singularity avoidance in
such a CMG system cannot be achieved by merely steering away from rotor
alignments; modifications to the objective which address this problem
are outlined and demonstrated in Example 7.
20
III
Example #3: Adaptation to Changing Vehicle Mass Properties
Since the vehicle mass properties are incorporated into the
calculation of actuator activity vectors, the selection process is
easily able to adapt to a changing vehicle configuration. This ability
is demonstrated in the following test, which initially assumes the mass
properties of the Dual Keel Space Station (approximately 2.5 times
heavier than the Power Tower), controlled by six double gimballed CMGs
mounted in the orthogonal configuration (i.e., Figure 3). Two
identical maneuver sequences are commanded. Each sequence begins by
requesting a rate of 0.0008 deg/s along the vehicle pitch axis. This
rate is held after it is achieved, and an attitude hold is commanded
after 50 s have elapsed since the start of the maneuver. The vehicle
inertia about the pitch axis is decreased by 50% before the second
maneuver begins.
Resulting vehicle rates are plotted in the lower portion of
Figure 8. The desired peak rates are attained and removed; the
feedback controller creates the small undershoots at the close of each
maneuver in order to restore vehicle attitude following receipt of the
attitude hold instruction. We see a somewhat faster, yet nearly
identical vehicle response when the pitch inertia is reduced,
indicating that the controller was able to adapt to the change in
vehicle environment without difficulty.
The corresponding CMG gimbal activity is illustrated in the plot
of outer gimbal angles presented in the upper portion of Fig. 8 (inner
gimbal angles were kept minimal by the objective function). Because
the vehicle responds more promptly to CMG motion after the inertia
reduction, the CMGs are seen to follow somewhat different trajectories
when answering each set of commands.
Example #4: Introduction of Jets After CMG Failures
In the event of hardware failures severe enough to prevent the
CMGs from answering input commands, the hybrid selection has the
21
COMMAND 2 STEPS IN PITCH; VEHICLE INERIIA CHANGE
OUTER GIMBAL ANGLES
180 -
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VEHICLE RATESPitch Inertiadecreased by 50%
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LegendRATM: ROLL
RATE: PITCH
RAIE: YAW_ _
UESIREO RAT: ROLL
OEfIRED RATE: PITCH
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FIGURE 8
22
'FullInertias
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VEHICLE RATES
LegendRATC: ROLL
RATE: PITCH
RAIE: TAW
DESIRED RATE: ROLL...! ..........OESIRED RArE: PITCH
DESIRED RATE: IAW
FIGURE 9
23
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ability to automatically introduce jets in order to recover the needed
control authority. This is illustrated in the next test, which assumes
an identical setup (6 ortho-mounted CMGs driving the Dual Keel), and
uses an identical input command sequence (two 0.008 deg/s rate commands
followed by attitude holds). The first set of commands is again
implemented nominally. Two CMGs are failed (i.e., inhibited from
selection) before the second set begins.
Outer gimbal angles and vehicle rates are plotted in Fig. 9.
Because of the limited control authority available from the truncated
CMG configuration, the rate increase and decrease cannot be attained
solely by the CMGs, hence the bulk of the second command sequence is
handled primarily via jet firings (which yield much greater torque,
hence cause considerably faster vehicle response). CMGs are still used
exclusively to damp vehicle rates and restore attitude, and the
controller has encountered no difficulty in achieving the desired
vehicle response using the truncated CMG system together with reaction
control jets.
Example #5: Dynamic Redefinition of Gimbal Stops
Most CMG steering laws are incapable of accounting for
restrictions on gimbal freedom (i.e., gimbal stops), which must usually
be imposed in an "ad hoc" fashion after gimbal rates are calculated.
The linear programming approach accounts for gimbal stops both in the
objective function (i.e., encouraging CMGs not to closely approach
stops) and directly as upper bounds on the decision variables (i.e.,
forcing the selection to limit the corresponding gimbal rotation). The
location of gimbal stops is specified as a parameter in the selection
software, and thus may be dynamically redefined at will (i.e., if a CMG
gimbal is degraded, the selection can prevent its rotation past an
arbitrarily defined angle).
The test run of Figure 10 illustrates how gimbal stops may be
easily altered under the linear programming approach. A Dual Keel
model is assumed to be driven by an array of six parallel-mounted CMGs
24
III
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(Figure 2). A command sequence is input which builds, maintains, and
removes peak rates of 0.0054, 0.0036, and 0.0018 deg/s along the
vehicle roll, pitch, and yaw axes, respectively. Two tests are
performed, the first assuming no stops on the outer CMG gimbals (as has
been conventional), and the second imposing an outer gimbal stop at
1800 (preventing rotation past this point). Outer gimbal angles and
resulting vehicle rates are plotted in Figure 10 for both cases. In
the nominal situation (left column), we see that two CMGs (#4 & #5) are
indeed rotated through 1800 in order to provide the requested rate
profile (lower plot). The CMGs are used in a different fashion after
the stops are imposed (right column). Although no outer gimbals cross
1800 (where the stops are defined), a set of vehicle rates is produced
(lower plot) which is identical to that obtained without gimbal stops.
Example #6: Demonstration of Null Motion
The term "null motion" refers to CMG gimbal redistribution that
places the CMGs into a superior orientation without torquing the
vehicle. Even though the objective function intrinsically accomplishes
such redistribution while answering input requests, the ability to
command the linear programming procedure to produce null rates and
redistribute the gimbals without creating net torque may prove
desirable (e.g., to restore controllability after the._CMGs are
initialized in a poor configuration). A null motion capability has
been developed and integrated into simplex, as demonstrated in the
following example.
This test assumes the Power Tower controlled by the standard quad
orthogonally-mounted double-gimballed CMG'array (Figure 3), and
commands a series of requests which increase the vehicle rate along the
pitch axis until momentum saturation is reached. The CMGs are
initialized in a "sub-optimal" orientation of reduced controllability
(with finite inner gimbal angles and rotor alignments), and null motion
(as discussed earlier) now attempts to move the CMGs into a better
orientation before the receipt of each input request.
26
III
Gimbal angles are shown in the left column of Figure 11. The
upper plot shows inner gimbal angles; all are at a minimum by t = 30 s
(as indicated). Excessive inner gimbal swings are otherwise seen to be
avoided until a jet firing is required at t = 135 s.
Rotor alignment plots are given in the right column of Figure 11;
the alignments are brought to a minimum by t = 30 s (as indicated). Jet
assistance is required after the CMG rotors line up in saturation at
the end of the run.
The saturation index is shown in the upper left of Figure 12; it
exceeds unity at the points where jets were introduced, indicating a
momentum saturation condition.
Quantities representing the degree of 3-axis CMG controllability
have been plotted for this run in the lower left portion of Figure 12.
The low value of CMG controllability (solid curve; termed "CMG gain")
at the start of the run (due to the initial sub-optimal orientation) is
restored to a maximum value by t = 20 s (as indicated). We see that
the objective contribution which avoids rotor alignments has indeed
maintained a high value of CMG controllability until saturation was
reached and jet assistance was required.
The upper-right plot in Figure 12 shows the net CMG objective
evaluation (corresponding to the "nonoptimality" of the CMG
orientation). This is termed the "CMG cost", and is defined such that
high cost values imply problematic CMG orientations (re. stops, inner
gimbals, lineups, etc.); lower cost orientations are preferable.
Asterisks are plotted over this curve where null motion was attempted.
Its effect is plainly to decrease the CMG cost (thus achieve a superior
CMG orientation). One sees that the relatively high cost of the
initial sub-optimal orientation was reduced drastically by null motion
which was performed at the start of the test; the minimum cost thus
achieved is at t = 20 s (as indicated), which corresponds to the "best"
physical orientation noted on the previous figures. Null motion is
suspended when it can no longer achieve any further decrease in the
system cost, or a significant command arrives from the operator or
phase-space controller.
27
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Vehicle rates are plotted in the lower right portion of Figure
12. The dotted curve represents the desired vehicle rate profile
(which is commanded to increase in a series of discrete steps); the
actual vehicle response is plotted in a heavier curve, and is seen to
follow the input commands (note that the vehicle responds much more
quickly at the last step, where jets were introduced). One sees that
vehicle rates were indeed held constant while null motion was being
performed. All rate-change commands ere applied about the pitch axis;
residual roll and yaw rates are seen to remain minimal (the vehicle
attitude is commanded to hold at zero about these axes).
Example #7: Steering Systems of Single Gimballed CMGs
Because of their mechanical simplicity, torque amplification
advantage, and lower weight, cost, and power requirements, single
gimballed CMGs are preferred over double gimballed devices in many
applications. Avoidance of singular configurations (where control is
restricted to a single axis or plane) can be considerably more
difficult, however, since singular states may not be directly related
to particular geometrical properties (such as rotor alignments) in
arbitrary single gimballed CMG configurations. A method of deriving
objective factors from the gradient of the net CMG controllability
index 3 has been developed to aid in steering single gimballed CMG
systems to maximize 3-axis control authority and avoid singular
orientations.
The next simulation example assumes the Power Tower to be driven
by a pyramid-mounted array of five single gimballed CMGs (Figure 4). A
series of rate increases (0.001 deg/s each) are commanded about the
vehicle roll and yaw axes, eventually driving the CMG array into
momentum saturation and forcing the introduction of jets. Yaw attitude
is maintained at zero. Two tests are performed; one with no
singularity avoidance in the objective, and another using
maximum-controllability steering, as outlined above.
30
III
Resulting gimbal angles and saturation indices for both tests are
shown in Figure 13. Each test exhibits similar behavior; i.e., gimbals
are moved to achieve the desired momentum transfer while avoiding stops
(CMG motion seems more responsive when gain steering is used), and CMGs
step uniformly toward momentum saturation after each request is
satisfied (jets are introduced to continue to meet commands after
saturation).
The CMG controllability index (termed "CMG gain") and resulting
vehicle rates are plotted in Figure 14. The right column shows results
generated without including anti-singularity components in the CMG
objective; significant drops in CMG gain (upper curve) occur at
t = 50 s, and again when momentum saturation is reached at t = 95 s.
These signify loss of 3-axis CMG control, and the effect of this
c6ntrol loss is evident in the vehicle rates (lower plot), where we see
that the CMG system encounters difficulty achieving the commands.
The situation changes when the CMGs are steered to maximize CMG
controllability; the large drop in gain becomes a gentle dip (the
singular state is avoided), and 3-axis CMG control is maintained until
momentum saturation is reached at t = 95 s. The identical set of
vehicle rate commands are thus answered without difficulty throughout
this test case.
Although the CMG controllability gradient aids in steering
systems of single gimballed CMGs, it can not guarantee the avoidance of
all singular orientations3 ,4. In order to attain more reliable
performance, a more sensitive objective formulation must be employed
which accounts for "global" CMG optimality and is able to detect
singular states well before they are approached.
Example #8: Dynamic Definition of Peak CMG Gimbal Rates and
Comparison With Other CMG Steering Laws
The maximum output torque of a CMG gimbal is proportional to the
peak rate at which it can be rotated. In order to extend the lifetime
of degrading CMG hardware, the ability to independently limit these
31
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maximum rates may prove necessary. The overall performance of
conventional (i.e., pseudoinverse) steering laws can suffer
considerably when driving a group of CMGs constrained to operate at
differing peak gimbal rates. Because of the separate upper bounds
imposed on each gimbal when answering torque requests, however, the
linear programming method is capable of obtaining maximum performance
from a group of CMGs limited to arbitrary peak rates. This is
illustrated in the following example.
A system of six parallel-mounted, double-gimballed CMGs (Figure
2) is assumed to control the Dual Keel configuration. Three of these
CMGs can operate up to conventional peak rates of 0.09 rad/s (as has
been previously assumed), while both gimbals of the remaining three
units are constrained to rotate at less than 0.03 rad/s. The vehicle
is commanded to increase its rate about the pitch and roll axes by an
additional 0.00035 deg/s every two seconds throughout the duration of
this test. Yaw rate is commanded to remain at zero. Two test runs are
performed; one employs the steering law of Ref. 1 (which uses an
approach "hardwired" to parallel mounting), and the other employs the
linear programming selection described in this text. Both actuator
management procedures are driven by the same control law, and both
tests use the same input command sequence (as detailed above).
Gimbal angles are given in Figure 15. Results from the run using
linear programming are shown in the left column. All-inner gimbal
angles (upper plot; inner gimbals provide the only means of controlling
pitch under this mounting scheme) are seen to increase together until
jets are required after approximately 33 s into the test. Outer gimbal
angles are shown in the lower plot; these are seen to converge toward a
common value (indicating approaching momentum saturation). In order to
continue providing the torque required to satisfy input commands, jets
are introduced shortly before the CMGs become saturated (the capacity
of the CMG array to maintain torque along the saturation axis
diminishes with the sine of the average inter-rotor half-angle as
saturation is approached). The CMGs finish just short of saturation in
this case; hybrid jet/CMG control is applied between t = 35 and 45 s,
34
III
after which the CMGs possess negligible authority about the commanded
axis, causing a transition to mostly jet-based control.
The gimbal angles plotted in the right column of Figure 15 result
from the test case that uses the steering law of Reference 1. The
general behavior seems quite similar to the linear programming
results. This steering law is directly optimized for parallel mounting
(it will not work in its present form for any other configuration), and
is subject to several constraints, one of which forces equal inner
gimbal angles, as is evident from the upper-right plot in Figure 15.
All outer gimbal angles are seen to converge to an identical value
after t = 51 s, indicating momentum saturation (since this steering
procedure alone does not possess the ability to exploit a different set
of actuators for additional authority, CMGs are driven hard into
saturation).
Vehicle rates from both steering laws are plotted in Figure 16.
Desired rates are shown as the increasing dotted staircase; actual
vehicle rates are denoted by heavier curves. The linear programming
results are plotted at left, where we see that the CMGs are able to
supply the torque needed to meet the input requests until momentum
saturation is approached, at which point jets are gradually introduced
to continue satisfying commands. The higher control authority of the
jets is evident in the quicker vehicle response at the end of the run.
Vehicle rates arising from the steering law of Reference 1 are
plotted on the right side of Figure 16. It is immediately evident that
the CMGs are now unable to supply sufficient torque to meet the input
commands (even though identical CMG hardware definitions are used).
This arises from the inability to specify independent rate limits for
each CMG gimbal; when this steering law exceeds at least one maximum
gimbal rate, all gimbal rates must be scaled down proportionally such
that every gimbal is operating within limits. In a mixed configuration
such as assumed in this example (where half of the gimbals are
constrained to run below reduced maximum rates), the response generated
from a large input torque request will often specify at least one of
the degraded gimbals at a rate above its limit, causing all gimbal
35
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rates to be scaled down accordingly and resulting in the net CMG system
being driven below capacity.
Since the steering law of Reference 1 has no provision to select
any other actuators, the available output 'torque eventually decreases
as momentum saturation is approached, and the system is unable to
realize any rates greater than 0.0065 deg/s (simultaneously about roll
and pitch). The hybrid selection gradually introduced jets to
compensate for the lower CMG torque capacity near saturation (if jets
were inhibited, however, the linear program would also align the CMG
rotors in hard saturation, causing the achieved vehicle rate to
likewise plateau).
The steering law of Reference 1 calculates null CMG
redistribution rates independently from torque-producing gimbal motion,
and sums both to form its final solution. If the redistribution rates
become significantly large, they can prematurely limit the magnitude of
available torque-producing gimbal rates, thereby causing the CMG system
to be run below capacity. The linear programming selection
accomplishes CMG redistribution by optimizing the assignment of gimbal
rates through an objective function; all solutions provided by the
simplex procedure meet the input request, and no additional
"redistribution" rates are added afterward.
The steering law of Ref. 1 exploits many properties of the
parallel-mounted double-gimballed CMG configuration i order to achieve
its level of performance. Because of this, it lacks much of the
flexibility of the linear programming approach; i.e., abilities to
assume any mounting orientation, effectively impose different maxima on
individual CMG gimbal rates, fail individual CMG gimbals, impose
arbitrary gimbal stops, and blend other actuators through hybrid
selections.
Example #9: Coordinated Control of Vehicle Translation and Rotation
Translational maneuvers will be executed during normal operation
of the space station, both periodically (i.e., orbital re-boost) and
38
III
upon demand (i.e., evasive action). Translational control authority is
derived from the onboard system of reaction control jets. Because jet
placements are generally offset from the vehicle center of mass, each
jet also generates a torque while firing. If translational jet
selections are performed without maintaining rotational control, the
vehicle attitude will drift as jets are firing, causing the direction
of translational thrust to shift correspondingly. The linear
programming jet selection can be expanded to fire up to six jets at a
time (even more could be included in a solution if upper bounds are
used), thereby enabling simultaneous translational and rotational
control. The corresponding autopilot assumes, however, that jets
instantaneously impart their impulse to the vehicle. Although this may
be a reasonable assumption for maneuvers employing short jet pulses,
the necessary firing durations will be quite long when a larger
translational rate is desired (as is certainly true during orbital
re-boost). If the applied jets are not aligned in a perfectly balanced
configuration (i.e., at equal moment arms from the vehicle center of
mass), the firing times will differ for various jets in the solution in
order to balance their mismatched rotational impulses. This generates
a finite torque throughout the burn that changes after each jet is shut
down. Although the vehicle rotational rates will reach the desired
values at the close of the maneuver (as per the 6-dimensional input
request), the dynamic torque imbalance can cause the -vehicle to exhibit
considerable undesired rotation while jets are active, leading to the
accumulation of significant attitude errors. In order to maintain the
desired vehicle attitude during translational maneuvers (and thus
control the translational thrust vector), sets of jets are generally
"cycled" on and off, compensating on average for this torque
imbalance. Unless jets are cycled very rapidly, this strategy will
generally result in a "swaying" vehicle motion during translation,
where a change in the jet firing is forced such that the direction of
vehicle rotation is reversed whenever significant attitude errors have
accumulated.
39
-- --------- 1_1_^ ____^·~1~ 11 _ __
The linear programming selection has been applied to investigate
the possibility of using CMGs together with jets in order to stabilize
vehicle rotation during translational maneuvers. The simplest method
of coordinating jet and CMG activity in this case is to perform a
6-dimensional hybrid selection; CMGs will be specified to aid in
rotational control, while translational authority will be provided
exclusively via jets. This is an effective strategy for short
translational pulses; the objective function will encourage CMG motion
that tends toward desaturation, and the RCS fuel demand may be
reduced. To build up significant translational rates on a massive
spacecraft such as the space station, however, long-duration firings
will be required, hence such hybrid selections will be of limited value
due to the large differences in control authority between CMGs and
jets.
Another strategy has been pursued to combine jet and CMG activity
during translational maneuvers. Provided that the CMGs can yield
sufficient output torques and have enough momentum capacity to maintain
this torque during significantly long jet firings, the CMGs may be
commanded independently to null the jet imbalance during translational
maneuvers and maintain steady vehicle attitude.
A set of simulations has been performed to illustrate this means
of coordinating rotational and translational control. The Dual Keel
configuration (with 10 lb thrusters) is used with anarray of six
parallel-mounted double-gimballed CMGs (Figure 2). The input command
sequence consists of a request to achieve and remove a .05 ft/s
translational rate along the vehicle x-axis. The vehicle attitude and
rotational rates are actively compensated by a proportional/integral
feedback controller which operates after the translational jet firings
are completed. Vehicle rates are nulled to within 10- 4 deg/s and
attitudes to within 3x10-4 deg. before another firing is attempted.
Two test runs are performed; one inhibiting all CMG activity during the
translational firings (CMGs are used to restore vehicle attitude
afterwards), and another using CMGs to dynamically null the jet
imbalance torques.
40
11
Gimbal angles are shown in Figure 17. The example using
uncompensated jet firings is shown in the left column, while results
employing dynamic CMG torque compensation are shown at right. In the
latter case, most CMG motion is seen to occur during the jet firings,
while in the former example, all CMG activity happens after jets have
ceased operation. Although the jet firings are on the order of 50 s
each, the CMGs do not approach saturation in either example.
Vehicle rates and attitudes are shown in Figure 18. The
uncompensated case is shown in the left column. Although vehicle rates
are seen to return to zero at the conclusion of the jet firings (as
indicated), the vehicle must be rotated additionally to null
accumulated attitude errors (of order 0.25 deg.). Vehicle rates and
attitude errors occurring under dynamic torque compensation are plotted
to the same vertical scale in the right column. Because the CMGs were
able to dynamically null nearly all torques during jet firings,
residual vehicle rates and resulting attitude errors are seen to remain
negligible. As mentioned earlier, the controller commands the second
jet firing as soon as vehicle attitudes and rates are returned to
within allotted deadband limits; since the rotational disturbance is
much smaller during the actively compensated test, any residual errors
are removed much more promptly, and the simulation is completed in less
than half of the time required without compensation (note the
difference in horizontal axis scaling).
Translational rate and relative position changes are plotted in
Figure 19. Jet firings established and removed the .05 ft/s
translational velocity without difficulty. The vehicle coasted at
constant velocity while CMGs trimmed residual attitude errors. The
shorter duration of the compensated test (as explained above) resulted
in a 5-ft vehicle displacement, vs 10 ft. for the uncompensated example
(due to the longer coast interval).
One could easily extend such compensated jet firings by stacking
one immediately after another. Attitude errors will remain minute (any
residual error can be removed by commanding a small rate offset in the
succeeding firing cycle), and the CMGs always finish in a zero-momentum
41
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state (provided that they have not torque-saturated during the
maneuver). Since small 10-lb thrusters were assumed in this example,
this strategy, as demonstrated, might be directly applicable to local
maneuvering of the space station (i.e., evasion, rendezvous). Re-boost
operation, however, may be accomplished under significantly higher
thrust (i.e., several parallel jets firing at each location), which
would create significantly larger torques. If CMGs are to be employed
for active compensation in this case, either the jets must be placed
more symmetrically about the vehicle center-of-mass (although a
well-balanced jet configuration will prove impossible to maintain in a
dynamic vehicle such as space station), or more/larger CMGs must be
employed to yield sufficient torque output and momentum capacity. It
must be noted that these simulations have been conducted on a
rigid-body model; for actual implementation, flexible modes must be
taken into account and the average jet cycling times must be placed
away from structural resonances.
This example has demonstrated how the linear programming approach
has been applied to command jets and CMGs for coordination of
translational and rotational control. The method of active torque
compensation could also be used to improve the accuracy of purely
rotational multijet firings. Given sufficient CMG authority, moderate
changes in torque due to jets turning off at staggered intervals could
be dynamically compensated by the CMGs such that the ehicle is subject
to a constant torque through the duration of the maneuver.
c) On-Orbit Performance
The preceding examples demonstrated the performance and features
of the linear programming selection/steering process by commanding a
rigid-body space station model to execute specific inertial maneuver
sequences. The next series of tests model a more realistic orbital
environment in order to examine the behavior of the selection/steering
approach as it would be implemented onboard an actual spacecraft.
A block diagram of the basic control law used to hold LVLH
attitude appears in Figure 20. An estimate of net on-orbit
45
environmental torque is derived from calculationsl 3 of aerodynamic drag
and gravity gradient components, which depend upon vehicle attitude and
orbital position. These are summed with the inertial torques due to
vehicle Euler coupling and LVLH CMG precession to form a net
environmental disturbance. This disturbance torque is combined with
feedback from the vehicle LVLH attitude error and rate (scaled by the
gains K1 and K2), and submitted to the linear programming
selection/steering package as a torque request (if jets are required,
the hybrid selection will automatically solve for a vehicle rate-change
which assumes that the input torque request is active over a
pre-specified time interval).
The vehicle controller and environment are updated at one-second
intervals in these examples. All tests use a 6-CMG double-gimballed
configuration (mounted as specified; hardware parameters were defined
10earlier) to control the rigid-body Dual Keel Space Station model
All jets are sized at 10 lb. thrust. These tests assume a circular
orbit at 400 km altitude, and all examples are 6000 s in duration
(i.e., 1.08 orbits). In order to accumulate sufficient torque over a
single orbit such that CMGs are used significantly, the vehicle
fabrication coordinates are assumed (which, adding a full set of
payloads, are offset from the principal axes by approximately -7 in
roll, -6° in pitch, and -0.5 ° in yaw). The vehicle is commanded to be
additionally rotated about pitch such that averaged gravity gradient
and aerodynamic torques are balanced over an orbital period.
The effects of environmental torques are evident in Figure 21,
which shows the LVLH vehicle rates and attitude errors developing
across an orbit for a freely drifting vehicle (note that these plots
are in orbital coordinates, and do not show orbital rates or initial
attitude offsets). The largest rates develop about the vehicle roll
and yaw axes, and arise from the action of unbalanced aerodynamic and
gravity gradient torques (respectively) together with Euler coupling.
Because of the gravity-gradient/aerodynamic torque balance in pitch
(plus lack of orbital Euler coupling), disturbances remain
comparatively low about this axis. In summary, vehicle rates are seen
46
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VEHICLE RATES
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RATE: PITCH
RATE: YAW
One Orbit
o 1000 2000 3000 4000 5000 6000
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VEHICLE ATTITUDE
48
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to approach 0.009 deg/s, and attitude errors surpass 20 deg. at the
close of an uncontrolled orbit (assuming initial conditions as
described above). This simulation serves as a baseline against which
the following examples (which stabilize vehicle attitude via the
controller sketched in Figure 20) can be compared.
Example #10: Nominal On-Orbit Operation with Parallel-Mounted CMGs
This example uses a standard parallel-mounted CMG configuration
(Figure 2) in order to stabilize vehicle attitude throughout the
orbit. Gimbal angle profiles are shown in the left column of Figure
22. Since the objective function encourages inner gimbal angles to be
kept minimal (and inner gimbals provide exclusive pitch authority for
this mounting orientation, thus require only minor adjustment over a
torque-balanced orbit), they are seen to remain small. As was noted in
Figure 21, most disturbance occurs about roll and yaw axes; thus we see
considerable outer gimbal activity in order to null these torques.
Outer gimbal angles are seen to begin converging (indicating approach
of momentum saturation) at approx. t = 2500 s, yet the system is still
sufficiently far from saturation to retain control without difficulty.
Vehicle rates and attitudes are plotted in the right column of
Figure 22. Disturbances are insignificant; LVLH vehicle rates remain
well below 0.0001 deg/s (thus are unable to be resolved in these
plots), yielding attitude errors well under one arc second (in the
actual vehicle, limited sensor resolution and environmental "noise" due
to astronaut activity and mechanical operation would preclude any such
accuracy; this idealized orbital environment is assumed here for test
purposes only). The effects of active control are evident when these
results are compared with Figure 21; the CMGs are driven to effectively
null orbital torques and maintain vehicle attitude.
Example #11: Nominal On-Orbit Operation with Ortho-Mounted CMGs
The linear programming approach to CMG steering allows any CMG
mounting protocol to be defined (this capacity is not supported by most
49
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steering laws, e.g., Reference 1). Physical constraints onboard
spacecraft may favor alternate CMG mounting schemes; these may be
specified arbitrarily under this selection/steering principle.
The following example illustrates this point by performing
on-orbit attitude control of the Dual Keel with an orthogonally-mounted
6-CMG configuration (Figure 3). Gimbal angles are shown in Figure 23
(left column). Inner gimbal angles are encouraged to remain minimal by
the objective function, and this is what is indeed seen in the
upper-left plot. Extensive outer gimbal activity is performed to
compensate the environmental torques. Attitude and rate perturbations
(right column) remain insignificant throughout the orbit, as
established in the previous example.
Example #12: Hybrid On-Orbit Operation After Two CMG Failures
Another unique feature of the linear programming approach is its
ability to coordinate the operation of different types of actuators
under a hybrid selection. At the start of this example, the Dual Keel
is controlled by six parallel-mounted CMGs; at t = 1500 s, two CMGs are
failed (i.e., both gimbals are inhibited from selection), and control
must be accomplished thereafter via the surviving CMGs and reaction
control jets.
Results are shown in Figure 24; gimbal angles ae plotted at
left. The start of the test is analogous to that of the nominal case
(Figure 22) until CMGs #1 and #2 are failed at t = 1500 s. Other CMGs
are used extensively to continue nulling the environmental torques
until they eventually converge in momentum saturation after t = 3000 s.
Attitude control between t = 3000 and 4000 s is achieved primarily via
jets; small CMG deflections about saturation are executed to trim these
firings. When input requests are directed away from saturation after
t = 4000 s, the CMGs are again able to respond to commands and once
more provide attitude stabilization (thereby removing themselves from
saturation).
52
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Vehicle rates and attitude are shown in the right column of
Figure 24. A higher disturbance level is noted during the region where
jets were required (primarily due to their impulsive action and much
higher control authority). Vehicle rates were nonetheless maintained
within 10- 4 deg/s, and attitude errors remained negligible.
Example #13: On-Orbit Desaturation of the CMG System
The previous example illustrated how jets were automatically
introduced to maintain vehicle control after CMGs were driven into
momentum saturation. Another option can be pursued when momentum
saturation is detected. Jet firings and CMG motion can be specifically
coordinated to desaturate the CMG system (i.e., lower the net stored
momentum), enabling continued CMG response to input requests.
The hybrid controller may realize jet desaturation in several
ways. The simplest means of achieving a- limited desaturation is to
merely adjust the CMG vs jet objective factors in hybrid selections,
such that extensive "favorable" CMG motion (with respect to a lower
objective evaluation) is mixed with jet firings in response to routine
requests. Another method, which has been demonstrated with some
success, involves enabling both jets and CMGs to be selected
simultaneously via the simplex-based null motion process. This will
combine favorable CMG motion with jet firings to produce a zero net
rate change, thereby driving the CMGs into a minimum-cost configuration
and generally achieving significant desaturation. These techniques are
demonstrated and discussed at length in Reference 3.
Another more conventional desaturation strategy may also be
pursued by firing the jets to apply a momentum impulse to the vehicle
equaling the stored CMG momentum, while simultaneously commanding the
CMGs to produce an opposing torque. This should ideally return the
momentum state of the CMG system to zero.
The example of Figure 25 examines an application of the latter
strategy. The Dual Keel model is used with a 6-CMG parallel-mounted
configuration (Figure 2). The vehicle is offset from equilibrium by
54
III
1.5 degrees in pitch, thus creating an imbalance between averaged
gravity gradient and aerodynamic torques which results in considerably
increased momentum loading about 'this axis. The remaining coordinates
are aligned with the vehicle principal axes, yielding otherwise reduced
environmental torques and smaller Euler coupling.
Inner gimbal angles are shown at upper left. Since the outer
gimbals cannot exert a torque about the pitch axis (as mounted in this
configuration), significant inner gimbal displacement is required to
maintain attitude. As a result, inner gimbal angles increase until
momentum saturation is approached when they near 90 degs. At this
point, the saturation condition is detected (via the saturation index
plotted at upper right), and the desaturation logic sketched above is
automatically activated.
The efficiency of this desaturation method is evident in Figure
25; while jets are firing, inner gimbal angles are promptly returned to
zero and the saturation index is dramatically decreased. CMGs are then
once more able to maintain vehicle attitude until they again load up
with momentum, eventually initiating another jet desaturation. This
process is repeated as required, creating the observed ramp response in
the CMG gimbal angles and saturation index (note that the mean time
between desaturations may be reduced in this case by firing jets to
load the CMGs with momentum about the axis opposite to the initial
saturation). Outer gimbals are used to stabilize the-vehicle attitude
about roll and yaw; since the vehicle coordinates were adjusted in this
example to minimize coupling and environmental torques about these
axes, most outer gimbal motion is dedicated to reducing jet-related
disturbances and minimizing inner gimbal crosscoupling.
Vehicle attitude is plotted in the lower right corner of Figure
25. Short spikes are seen to develop about the pitch axis during the
desaturation intervals. These are due to the mismatch of authority
between jets and CMGs; during most of the desaturation process, the
CMGs were torque-saturated, allowing the jets to produce a net torque
on the vehicle. Incidental vehicle disturbance may be reduced by
pursuing a policy that desaturates over longer intervals (i.e.,
55
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combining several smaller desaturation steps), such as the null motion
process discussed in Reference 3.
CONCLUSIONS
Linear programming has been successfully adapted to manage
arbitrarily-defined arrays of both CMGs and jets. Closed-loop attitude
controllers and environmental models have been developed in order to
drive the linear programming selection/steering process and examine its
performance under both inertial and on-orbit simulations. The
flexibility implicit in linear programming has been seen to enable
modes of CMG operation which cannot be achieved with standard (e.g.,
pseudo-inverse) CMG steering laws. Investigations have also been
conducted into the application of linear programming techniques to
steering systems of single gimballed CMGs, achieving coordinated
translational/rotational vehicle control, and automatically
desaturating the CMG sys-tem with reaction control jets.
ACKNOWLEDGEMENTS
The author wishes to thank Ed Bergmann, Harvey Malchow, Phil
Hattis and Steve Bauer for helpful discussions. The work was prepared
at the Charles Stark Draper Laboratory under Independent Research and
Development Project #207 and National Aeronautics and Space
Administration contract NAS917560.
Publication of this paper does not constitute approval by NASA of
the findings or conclusions contained herein. It is published for the
exchange and stimulation of ideas.
57
REFERENCES
(1) Kennel, H.F., "Steering Law For Parallel Mounted Double-Gimballed
Control Moment Gyros - Revision A," NASA TM-82390, Jan. 1981.
(2) Paradiso, J.A., "A Highly Adaptable Steering/Selection Procedure
for Combined CMG/RCS Spacecraft Control," Proc. of the 1986
American Astronautical Society Rocky Mtn. Guidance & Control
Conf., AAS-036, Feb. 1986.
(3) Paradiso, J.A., "A Highly Adaptable Steering/Selection Procedure
for Combined CMG/RCS Control; Detailed Report," C.S. Draper Lab,
Report CSDL-R-1835, March 1986.
(4) Schiehlen, W.O., "Two Different Approaches for a Control Law of
Single Gimbal Control Moment Gyro Systems," NASA TM X-64693,
Aug. 2, 1972.
(5) Bradley, S.P., A.C. Hax, T.L. Magnanti, Applied Mathematical
Programming, Addison-Wesley Publishing Co., Reading, Mass., 1977.
(6) Paradiso, J.A., "An Efficient OFS-Compatible Kinematic Model for
Generalized CMGs," C.S. Draper Lab. Dept. 10C Space Station Memo
85-9, March 25, 1985.
(7) Bergmann, E.V., S.R. Croopnick, J.J. Turkovich, C.C. Work, "An
Advanced Spacecraft Autopilot Concept," Journ. of Guidance and
Control, Vol. 2, No. 3, May/June 1979, p. 161.
(8) This autopilot was flight-tested on Space Shuttle missions STS
51G (June 1985) and STS 61B (Nov. 1985).
(9) "Space Station Reference Configuration Description," NASA/JSC,
JSC-19989, August 1984.
(10) Bishop, Lynda, NASA/JSC-EH, Personal communication, March 5,
1986.
(11) Skylab Program: Operational Data Book, Vols. II and IV, NASA
MSC-01549, 1971.
(12) "Space Station Advanced Development Control Moment Gyro (CMG)
Data," NASA MSFC memo ED15-85-43, July 1985.
(13) Malchow, Harvey CSDL, Personal communication, July 1986.
58
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