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

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(b) PARALLEL MOUNTED CMG REFERENCE CONFIGURATION (ZERO GIMBAL ANGLES)

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(c) INITIAL ORIENTATION OF CMG ROTORS

10

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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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22

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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

-, I : , ; ; 'I DI>

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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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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

III