characteristics of candidate reaction wheel configurations
TRANSCRIPT
Characteristics of Candidate Reaction Wheel Configurationsfor Satellite Attitude Control
1) Introduction
Attitude control of most existing satellites is performed through groups ofreaction wheels, which exchange momentum and produce torque about fixed rotoraxes. When the wheels are mounted in simple arrangements (ie. three identicalwheels oriented along mutually orthogonal axes), the resulting momentum and tor-que envelopes, which determine the maximum actuator authority that can be pro-jected in a given direction, are straightforward to visualize (this case will result in acube-shaped envelope, since each wheel independently spans a single coordinate).When more complicated reaction wheel configurations are considered, however, theconsequent envelope shape becomes a correspondingly more complex polyhedron,which can indeed be quite difficult to directly visualize. Geometric arguments mightbe applied to divine the critical characteristics of particular wheel ensembles; suchtechniques can be readily exploited for symmetric mounting protocols. When theconfigurational symmetry is broken after degradation or failure of at least one mem-ber wheel, however, the envelope also looses symmetry, and its determination canbecome more difficult.
This report describes a straightforward means of structuring the envelope calcu-lation as a linear programming problem. This formulation is completely tolerant ofvariations in configuration design. Wheels can be added, failed, degraded, etc. inany manner, orientation, or quantity; the linear program does not require any mount-ing symmetry (although it does generally assume the configuration to span thedimensions of the control space).
Because the authorities and limits of each wheel are considered independently,envelopes can be calculated for configurations that blend wheels of different specifi-cation. The ability to arbitrarily specify different lower and upper bounds on actua-tor authority can even enable incorporation of a reaction wheel which is able todeliver more torque (or momentum) in one direction (ie. "+") than the other (ie. "-");
this option may be useful in describing unique failure modes.
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In addition to defining reaction wheel configurations which can project sufficent
authority along controlled axes, other considerations (such as power requirements,weight, volume, and cost) can impact determination of an optimal wheel arrange-ment. The power requirement is the only additional item in this list that is
addressed here; alternative mounting protocols will be proposed based on theirpower and control advantages, the ultimate selection of which depends upon sizing,mass, and financial considerations. Granted, one might be able to compose a mul-ti-dimensional optimization to find the best wheel setup considering the simultane-ous effect of all relevant factors; such an effort, however, is well beyond the scope
of this report.
2) Envelope Calculation
Since a reaction wheel projects momentum and produces torque about thesame fixed axis, both torque and momentum envelopes may be produced throughapplication of the same technique. The equations presented in this section will becast in momentum (h) space, and later generalized to torque (T) coordinates.
Calculation of the momentum envelope may be interpreted as solving for themaximum momentum projection possible along a given direction (), subject to the
Aconstraint that the momentum components orthogonal to r are zero. Quantitatively:
N
a) Maximize: hi ri=1
(1) Nb) Subject to: ^ x r
i=1
c) Limited by: I hi I hmax
Where...
N = # Reaction wheels consideredhi = Rotor momentum of wheel #i
hmax = Peak rotor rate
A hat "A" denotes a unit vector.
An underscore "_" denotes a vector of full magnitude.
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Equation (1) is a linear programming problem set up to solve for the momentumA A A A
envelope. Cost coefficents are hi .r, activity vectors are hi x r, and the decisionvariables are the signed magnitudes of rotor momenta, hi.
Although the equality constraint (lb) is expressed in terms of three-componentvectors, it is only a two-dimensional relation, since it merely states that the total
Asystem momentum in the plane orthogonal to r is zero. The redundant degree offreedom may be removed by expressing constraint (b) in the plane perpendicularto r . Equation (1) becomes:
N
a) Maximize: (a ij r )h i
b) Subject to: Z ha), = i=1
c) Limited by: -hmin. < hi hmaxi
Where...
Aa, = Unit vector in direction of rotor axis #i
(A) Cto A to A(ai)A = Vector a projected into plane orthogonal to rhi = Signed magnitude of rotor angular momenta
-hmini, hmaxi = + bounds on rotor momenta
The above relationship is a 2-dimensional linear programming problem which issolved directly to yield the rotor momenta hi that project the ensemble momentum
A Amaximally along direction r (termed hiIA). By solving this relation for a set of rspherically distributed in all directions, a momentum envelope is generated:
N
(3) henv 1J = (hble) aii=1
Note: The magnitude of henv is also the objective evaluationof the optimal solution to Eq. (2).
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By replacing all momentum-related (h) quantities in Eqs. (2) and (3) with torqueparameters (T), the torque envelope may be calculated by an identical process. Theonly physical constants that actually would require change are the bounds in Eq.(2c). If all wheels are identical, with ,,in, and r,,, identically proportional to hm,, and
h,,x, the momentum envelope may be re-scaled to form the torque envelope; arepeat of the linear programming procedure is unnecessary.
As indicated earlier, the bounds in Eq. (2c) may be specified differently for eachreaction wheel, and hmi,li, is not required to equal h,,,. This option allows consider-
able flexibility in mixing wheels with different hardware characteristics, and may beused to examine attitude control capability under unusual failure modes.
The routine "DLPRS" from the IMSL math library [1] was used to solve Eq. (2).The current release of this software, however, seems to encounter occasional diffi-culty in obtaining an accurate solution. In the cases examined, DLPRS seemed toarrive at the correct optimal solution approximately 98% of the time. Occasionally,however, it produced results which violated the equality constraint (2b) and/or deci-sion variable bounds (2c). These points were easily filtered out by checking theDLPRS solution against the constraints; if a signi'icant violation was found, the signsof an activity vector, its cost coefficent, and its decision variable were reversed, andthe DLPRS call was repeated. This second try generally allieviated the constraintviolation; if not, another activity vector was chosen for negation, and the processwas repeated.
The IMSL software also experienced an occasional problem in starting the line-ar programming simplex procedure; it occasionally would abort, claiming that theconstraints (2b,2c) were "infeasable", when this obviously was not the case. Thisseemed to be due to an occasional inability to account for the bipolar nature of thedecision variables (ie. thay are allowed to go both positive and negative in this cal-culation). Whenever this condition was detected, an activity vector and its associ-ated quantities were negated (as discussed above), and the DLPRS call wasrepeated. The new attempt generally avoided the difficulty and started simplexwithout further problem.
3) Graphic Portrayl
Applying the results of Section 2 will yield a map of maximum projectablemomentum (he,,) as a function of spherical coordinates and p (ie. r), defining a3-dimensional polyhedron as indicated in Section 1. The simplest means of qualita-
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tively evaluating characteristics of this envelope is to render it in a series of 3Dviews. Unfortunately, pre-packaged software to draw 3D solids as a function of and p do not seem to currently exist in usable form at CSDL, so a simple solid-drawing routine was written expressly for this purpose. This section includes aquick description of the techniques used; more detail may be found in [2].
The methods introduced in Section 2 yield a table of values h,(e, 0), or alterna-tively, h,,,(r). These points must first be rotated and translated to account for theviewing position. The observer is assumed to be located at position v and is look-ing into the origin. Vector v is broken into polar and azimuthal rotations ( y, ¢pv )and distance (d= Iv ); all points h,,, must be transformed accordingly:
(4) h' ev = Zd - R( - v) Rz( - )v) henv
Where...
R = Rotation about x-axisI Z = Rotation about z-axiszd = Vector distance to eye coordinates; ie. [ O,O,d ]
The above rotations map the z-axis into the viewing direction v ; the viewingtranslation is then achieved by adding z ,, which now possesses only a z-compo-nent. The envelope may then be mapped into a 2-dimensional perspective projec-tion (thereby producing "screen" coordinates) by inversely scaling the x and ycomponents by their associated z value:
(henv)x() - (h'env)x
(5) (hy (h'env)y~(hen~ (h'env)z
In all examples presented here, the points h are calculated over a uniform(8, p) spherical "web" (8 runs 0 -, 7 in 80 steps, P runs 0 - 2n' in 80 steps). Vec-tors are drawn between each point h,,, and its nearest neighbor (displacing 8 or pby one increment). Hidden line elimination is accomplished by testing the dot prod-uct between the outward normal to the envelope surface at the point being viewed
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and the vector to this point from the viewing location. If the projection is positive,the line is assumed to be hidden, and is not drawn. Quantitatively:
To test line between h'env(eo, 0) and h'env(&o + A, o0):
(6) Define: N = [ h'env(eo + A, 0) - h'env(eo, Po)] x [ h'env(o, Po + A) - h'env(eo, po)]y = J th'env(0o, 0)
If and only if y < O, draw line.
Note that this simplified hidden-line procedure may be successfully applied herebecause the envelope polyhedra for reaction wheel systems are always convex.This approach won't be able to detect all hidden surfaces in a more complex object;ie. the momentum envelope for a single-gimballed CMG system (which is puncturedby funnel-shaped cavities).
In addition to the 3D perspective views, 2D projections onto the coordinateplanes (xz, yz, and xy) are also drawn. These plots are presented in absolute coor-dinates (ie. ft-lb & ft-lb-sec), thus can be used for quantitative comparison withrequirements. No "web" lines are drawn in these figures; only projections (notincluding any perspective effect) of the envelope points (h',,) are plotted, producing
2D "shadows" of the 3D solid.
4) Comparison With Mission Objectives
In order to maintain vehicle attitude control, the torque and momentum envel-opes of candidate wheel configurations must contain the maximum respective tor-ques and momenta expected to be required during satellite operation. For thesatellite considered in the examples presented in this report, the maximum objec-tives were assumed to form cylinders in both torque and momentum space. Themaximum torque requirements were assumed to run + 1.5 ft-lb in azimuth (z) and+ 1.5 ft-lb in elevation (xy plane). This describes a cylinder of height 3 ft-lb (cen-tered on the z-axis and bisected by the xy plane), with radius 1.5 ft-lb (in xy). Thesetorque requirements are somewhat large for a spacecraft controlled via reactionwheels; they arise primarily from the need to stabilize the vehicle during the rapidslew of an onboard antenna (this estimate arises primarily from the need to slew theantenna by 150 in azimuth and 5 in elevation within 17.5 sec.; the 1.5 ft-lb result wascalculated by scaling previous simulation data [3] to this expanded slew time).
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The momentum requirements range 0 to + 100 ft-lb-sec in azimuth (z) and 15ft-lb-sec in elevation (xy plane), defining another cylinder centered on the z-axis, butextending only to positive z (height = 100 ft-lb-sec), with radius 15 ft-lb-sec. Thelarge azimuth momentum storage is needed to offset the momentum of the rotatingantenna (plus a small margin for environmental torque buildup); the total momentumof the spacecraft-plus-wheels system will then always be close to zero, and Eulerprecession torques (assuming an Earth-fixed attitude) will be negligible. The orthog-onal + 15 ft-lb-sec elevation requirement is due to integration of assumed environ-ment torques (which are expected to be quite small for this vehicle), together withsome allowance for the peak momentum required in this plane while slewing theantenna and/or vehicle.
The spacecraft could by momentum-biased in azimuth, allowing the wheels toArange + 50 ft-lb-sec along z (rather than 0 to 100 ft-lb-sec). This will always result in
an uncancelled 50 ft-lb-sec system momentum, and since the azimuth axis willrotate inertially in an Earth-fixed orientation, this excess momentum must corre-spondingly counter-rotate in spacecraft coordinates to prevent vehicle precession.This situation defines a considerably different momentum requirement, which canno longer be represented by the cylinder discus.ed above; the surface of maximumneeded momentum will now essertially define a sphere of radius 50 ft-lb-sec (plus asmall margin for integration of environmental and slewing torques). This case is notconsidered in the examples presented in this report; all investigations refer to the100 ft-lb-sec cylinder, which, · because of its associated zero net system momentum,is less demanding of the onboard attitude control system and easier to implement.The momentum-biased approach, however, could be considered for operation undera failure mode that prevents the full 100 ft-lb-sec from being achieved in azimuth.
The appropriate constraint cylinders are plotted over the momentum and torqueenvelopes in the projection plots presented in this report (the cylinders project intorectangles or squares on the xz and yz planes; they project into circles on the xyplane). The regions internal to the cylinder which are not inside the envelope aredarkened, allowing one to rapidly ascertain areas in operational momentum or tor-que space where available vehicle control authority will be inadequate and/or satu-rated. Since these are projection plots, merely noting the constraint areas lyingoutside of the envelope "shadow" is inadequate; the algorithm which darkens theuncontrolled regions compares the envelope and constraint in three dimensions,thus projects these zones to their full extent.
The ratio of uncontrolled constraint volume (cylinder volume lying outside theenvelope) to total constraint volume (complete cylinder volume) is also listed with
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the projection plots to provide a quantitative estimate of performance for the wheelconfiguration under examination. These volumes are integrated in uniform spheri-cal coordinates as the envelope is being calculated, as described below for momen-tum (and analogously for torque):
Define: out (, P) = 0 If henv,(, p) > h(, )
eout (8, (p) = hc(e, () - henv(8, P) If henv(e, /) < hc(8, p)
[Where hc(e, p)M Distance from origin to constraint cylinder at 8, c]
(7) Z t2out (8, p) sin
Ratio total = ,hc(O, cp) sin e
e,p
5) Estimating Power Requirements
The power consumed by a single reaction wheel of the type considered heremay be described by [4]:
IP = K112 + (K2 + K3w, )i + K4
Where...(8) Ii = Current consumed by wheel #i (ie. aTi)
Tji =Torque applied by wheel #iw i = Rate of wheel #i
The constants in Eq. (8) are defined [5] below:
a = 14.8 Amp/ft - lbsK1 = 0.15 n
(9) K2 = 2.26 VoltsK3 = 0.00958 Watts/Amp - RPM
K4 = 15 Watts
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Because power is an increasing function of T1, maximum power will be encount-ered at points on the torque constraint cylinder (which represent peak require-ments). In the examples presented here, it is also assumed that the peak azimuthmomentum requirement of 100 ft-lb-sec has been absorbed by the reaction wheelsystem (ie. the antenna is rotating at its maximum fixed rate); this increases thepower consumption due to the linear wi term in Eq. (8).
In order to calculate total system power needed for peak projections in alldirections, the spacecraft torque request must be distributed amoung all reactionwheels in the configuration. Each power contribution (Pi) must then be calculatedvia Eqs. (8) and (9), and summed. Ideally, one would want to use a minimum-powerdistribution policy in this calculation. Since, in the torque region of interest here(under 1.5 ft-lb), the linear K2 and K3 terms dominate in Eq. (8), linear programmingcould potentially provide the ability to generate a nearly power-optimal solution. Inthis case, one would minimize the net current consumed by the wheel ensemble,suggesting an objective function formed from the sum of the absolute torque valuesfor each wheel. Incorporating an absolute value into the objective function used bythe IMSL routine DLPRS [1] would be somewhat difficult. Other extended linearprogramming packages (eg. [6]) provide this ab;lity, but are difficult to adapt to thistask due to other application-specific features.
As a comprimise, a pseudo-inverse procedure [7] was used to solve the torquedistribution problem. Since it minimizes the solution's 2-norm (sum of squared tor-ques), it does not directly calculate the "power-optimal" solution (which would comecloser to minimizing the sum of linear torque absolute values), although it shouldnonetheless generally achieve an acceptable low-power, near-optimal result.Because the pseudo-inverse is not able to directly account for bounds on decisionvariables (as is possible with linear programming), some solutions that push thewheel configuration near to its limit may produce individual wheel torques aboveallowed bounds. The power calculated in these cases, however, should not be toofar removed from the power derived using a properly bounded solution, thus thesevalues are retained.
Consumed power is calculated in this fashion at all sampled points on the tor-que constraint cylinder. These power values are then histogrammed (after havingtheir probability weighted by sin8 to account for the directional inhomogeneity of theuniform , P distribution), allowing determination of the maximum power, averagepower @ peak torque, and most probable power @ peak torque (the latter twoquantities assume that all points on the torque constraint cylinder can be com-manded with equal probability).
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6) Applications and Results
This section presents a performance analysis of several proposed reactionwheel configurations. Results are obtained by applying the techniques and criteriadescribed in this report to the mission constraints proposed in Section 4. The 3Denvelope renderings are presented only for qualitative consideration, thus are notscaled in absolute units (the accompanying axes are unlabeled). The torque andmomentum projection plots, however, are respectively scaled in ft-lb and ft-lb-sec(as labeled), thus may be used to quantitatively ascertain the extent of availablecontrol authority. Because all wheels are assumed to be identical (with maximumtorque and momentum limits symmetric about zero) and no wheels are consideredto be running at rate limits, the shapes of all torque envelopes will be identical tothose of their corresponding momentum envelopes. Accordingly, only one set of 3Denvelope drawings need be presented per configuration; the pictures apply in bothtorque and momentum space under these conditions.
Two 3D views are generated for each configuration; one as seen from an"oblique" position suspended in the first quadrant at equal angles, and the otheragain in the first quadrant, but located slightly above-left of the x-axis (the x-axislength is extended in this view to enable it to be seen emerging from the envelope).All views look directly into the origin.
Wheel momenta are assumed to range from 40 to 65 ft-lb-sec, and wheel tor-ques are assumed to run 1.2 to 1.6 ft-lb; specific values are listed on the relevantfigures.
The first set of examples investigates a configuration proposed in [8], whichconsists of four wheels mounted with axes along the diagonals of a symmetric pyra-mid inclined at 380 to the elevation plane, plus one wheel mounted directly alongthe azimuth axis (see Fig. 1). 3D views of the envelope shape are drawn in Fig. 2;one sees that an elongated "crystalline" dodecahedron is produced. As indicatedearlier, this polyhedral shape is characteristic of reaction wheel envelopes. It isgenerally produced by solutions which contain all wheels except two at maximumlimits; these remaining two are operated at intermediate values in order to achievethe desired equality constraint, and thereby "span" a planar face of the polyhedron.
The momentum projections (assuming 50 ft-lb-sec wheels) are given in Fig. 3.The envelope is seen to amply enclose the constraint cylinder; no saturated regionsare evident. The torque projections (assuming 1.3 ft-lb per wheel) are given in Fig.
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4; again we see that the torque objectives are completely enclosed in all views,leaving no uncontrolled regions.
The next example assumes the same mounting configuration, but "fails" the azi-muthal wheel (ie. it is fixed at zero momentum and zero torque). Fig. 5 shows the3D envelope views; again we see a dodecahedron, yet it is no longer extendedalong the azimuth axis due to failure of the azimuthal wheel. Momentum and torqueprojections (still assuming 50 ft-lb-sec and 1.3 ft-lb per wheel) are shown in Figs. 6and 7, where it is seen that the envelope retains the ability to contain the controlrequirements (although it is a close fit at the azimuthal extremes of the constraintcylinder), indicating that the azimuthal wheel is fully redundant.
The following example also assumes an identical wheel definition. In this case,A A
however, a pyramid wheel is failed (the wheel pointing between the -- x and -y
axes is chosen). The 3D envelope views (Fig. 8) again depict a polyhedral structure,however much of the symmetry has been destroyed, and the surface appears to beskewed more-or-less in a direction orthogonal to the axis of the failed wheel. Theeffect is obvious in the momentum and torque projection plots (Figs. 9 & 10), wherethe skewness and loss of symmetry are readily apparent. The envelopes are defi-nitely seen to have shrunk significantly when compared to the no-failure case (Figs.2 -, 4), and the lack of symmetry has twisted and reduced the remaining volumesufficiently to prevent much of the desired control objective from being enclosed.The loss of configurational symmetry can deal a significant blow to the utility of theresulting envelopes; compare this result to that of the previous example, which losta wheel while staying symmetric, thereby retaining full control. The ability of anasymetric wheel configuration to project its momentum into arbitrary directions isseverely curtailed, as will be elaborated upon in other examples.
From Figs. 9 & 10, it is seen that very little momentum objective volume is unre-alizable (due to the surviving azimuthal wheel), but approximately 12% of the opera-tional torque volume will be unreachable, which may indeed impact missionscheduling (locations of control loss are evident from the shaded areas inside theconstraint cylinder projections).
In order to reduce the volume of the uncontrolled regions in the single-failurecase and reduce the peak torques required from the reaction wheels (1.3 ft-lb isapproximately a factor of two over the maximum output of the largest operatingwheel [used in Space Telescope]), other mounting configurations were investi-gated. Because the azimuth wheel was seen to be completely redundant in the pre-vious configuration, it was incorporated into a symmetric 5-wheel pyramid, which is
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examined in the following series of tests. Since an inclination angle of 420 was
found to yield best constraint coverage in the 1-failure case (420 was also found to
require least power in the fully-operational configuration), it was chosen as standardfor all 5-wheel pyramid examples, as depicted in Fig. 11.
Fig. 12 shows the 3D envelope for this configuration with all wheels operational.Again we see a polyhedron, however more faces are present, due to the increasedcomplexity of the mounting arrangement. From the projection plots in Figs. 13 & 14,we see that the constraint cylinder is entirely contained within the envelope in bothmomentum (40 ft-lb-sec/wheel) and torque (1.2 ft-lb/wheel).
Fig. 15 shows the 3D envelope for the 5-pyramid with one wheel failed (again,A
the wheel inclined closest to the -x direction is removed). As expected, we see asmaller, asymmetric polyhedron, which has developed a skew away from the azi-muth (z) axis. The consequences of these effects are evident in the momentumprojection plot given in Fig. 16. The peak momentum allocated per wheel isassumed identical to the 40 ft-lb-sec used in the previous test, however we now seethat 29% of the momentum constraint volume lies outside the envelope (as indi-cated by the dark region drawn inside the cylinder). Increasing the wheel momentato 60 ft-lb-sec causes the envelope to grow somewhat (Fig. 17), leaving only approx-imately 2% of the momentum constraint unreachable. The momentum allocationwas raised to 65 ft-lb-sec in Fig. 18, where we now see the constraint fully coveredby the momentum envelope.
Fig. 19 shows the torque projection plots for this configuration; a maximum of1.2 ft-lb per wheel is assumed, as was shown to be adequate in the no-failure case(Fig. 14). The situation is considerably different here; 4.6% of the torque constraintvolume is now unreachable. This is significantly better than the 12% of outside vol-ume encountered when a pyramid wheel was failed in the analogous 4-pyramid plus1-azimuth case @ 1.3 ft-lb/wheel (Fig. 10); the symmetry of the 5-pyramid configura-tion has resulted in superior tolerance to a single pyramid wheel failure.
As seen in the Fig. 19, the saturated locations are on opposite upper and lowerextremes of the constraint cylinder. Increasing the wheel torque capability to 1.5ft-lb (Fig. 20) lowers the unreachable volume to 0.4%; the opposite cylinder cornersstill protrude slightly from the envelope boundry. If the wheels are each able to tor-que up to 1.6 ft-lb, the unreachable corner region shrinks to 0.07% of the total con-straint volume, as seen in Fig. 21.
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Since the maximum torque per wheel of 1.6 ft-lb required in the previous exam-ple to span the entire operational constraint (with one wheel failed) remains signif-icantly above the latest off-the-shelf capability, a symmetric 6-wheel pyramidconfiguration (Fig. 22) was examined. An inclination of 380 to the elevation planewas found to provide optimum envelope coverage with one wheel failure, along withminimum full-up power dissipation. Fig. 23 shows the 3D envelope with all wheelsoperational; the polyhedron is quite similar to the 5-pyramid case (Fig. 12), howeveradditional faces appear due to the increased combinatoric possibilities created bythe added wheel. Envelope projections are shown in Figs. 24 & 25, where we seethat the momentum constraints (with 40 ft-lb-sec per wheel) and torque constraints(with 1.2 ft-lb per wheel) are amply satisfied.
The 3D views of the 6-pyramid envelope with one wheel failure (again, thewheel directed closest to - ) are presented in Fig. 26, where some skewness andlack of symmetry can be noted. Looking at the momentum projections (Fig. 27), aless distorted envelope is seen (when compared to the 5-pyramid with one failure),but insufficient capability exists with 40 ft-lb-sec per wheel to cover the top of theconstraint cylinder, leaving 5.4% saturated volume. Raising the peak wheel allo-cation up to 45 ft-lb-sec is sufficient to cover the entire cylinder volume in this case,as shown in Fig. 28. Torque projections are shown in Fig. 29, where it is noted thata peak wheel torque of 1.2 ft-lb is now sufficient to cover nearly the entire con-straint.
The above results show a definite advantage in controlling the 1-failure situationwith a 6-wheel pyramid; maximum required torques drop from 1.6 to 1.2 ft-lb perwheel, which is much closer to the specifications of existing devices. These exam-ples also indicate the role that configurational symmetry plays in specifying envel-ope coverage; the greater skewness and "distortion" of the 5-wheel/I-failedenvelope adversely affected constraint coverage at least as much as the generalshrinkage of the envelope resulting from the wheel loss. The symmetry is not quiteso strongly broken in the 6-wheel/i-failed case, thus the envelope is not so dis-torted, and its coverage is superior.
This point is illustrated further in the final two examples, which again use the6-pyramid configuration (Fig. 22), but now examine the effect of two simultaneouswheel failures. All plots assume each wheel to be capable of attaining 45 ft-lb-secmomentum and 1.2 ft-lb torque.
Fig. 30 shows the 3D envelope arising from the failure of two adjacent wheels.Considerable envelope shrinkage and distortion is noted. These effects cause 47%
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of the momentum constraint to be missed (Fig. 31), and cut off-18.5% of the torque
constraint (Fig. 32).
Fig. 33 shows the 3D envelope resulting from the failure of two opposing
wheels. Although the envelope is truncated about two axes (ie. it begins to
approach a "lozenge" shape), it still retains considerable symmetry and remainsA
peaked in the azimuthal (z) direction. The superior constraint coverage of this
unskewed envelope is evident in the projection plots (Figs. 34 & 35). Although the
same number of wheels were failed as in the previous example, we now miss only
1% of the momentum constraint and 3.3% of the torque constraint. Indeed, the
higher envelope symmetry retained in this case provides a better fit to the con-
straint cylinder, resulting in projection of more control authority into the region of
desired objectives.
The move to a 6-wheel configuration immediately generates concern over
increased power requirements. Eqs. (8) and (9) indicate that the 6-wheel configura-
tion will dissipate an extra 15 Watts of quiescent power at zero torque. This
increase is not very significant. The evolution of required power with increasing tor-
que, however, is more difficult to ascertain. Sirce the added wheel will generally
decrease the torque and momentum load carried per device, the non-linear K, and
w-dependent K3 terms in Eq. (8) will act to lower the total configuration power
requirement. This gain may indeed offset the 15 Watt K4 contribution, and possibly
lead to even greater savings.
In order to derive quantitative power estimates, the techniques outlined in Sec-
tion 4 were applied to the 5- and 6-wheel pyramid configurations. The wheels were
assumed to have absorbed the peak 100 ft-lb-sec antenna momentum as discussed
in Section 4. Fig. 36 shows resultant data for the 6-pyramid @ 38 °. The upper plot
depicts the required total power as a function of the torque-request elevation angle
for a uniform sample of points on the constraint cylinder surface. Points on the tor-
que constraint cylinder represent maximum desired torque in any given direction,
thus these calculations model the maximum configuration power expected to be
expended during normal satellite operation. Since the envelope and constraint cyl-
inder shapes are essentially invarient under azimuth ( ) rotation, the only
expected angular dependance of peak power will be in elevation ( ), as plotted in
Fig. 38a. This figure shows two distinct peaks at 450 and 1350, corresponding to tor-
que commands oriented toward the constraint cylinder radius at its upper and lower
ends, thereby representing simultaneous application of maximum azimuth and ele-
vation torques ( 1.5 ft-lb per axis were used as commanded torque maxima
throughout this report). The power burnt in these peaks will approach 400 Watts.
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The smallest required power occurs when torquing directly along the azimuth axis(at zero elevation); this value of approximately 325 Watts is also approached whentorquing in the elevation plane (ie. at 90° elevation).
In order to enable ready derivation of useful quantities, all data taken across theconstraint cylinder are histogrammed, as in Fig. 38b. The maximum histogram entryat 405 Watts arises from data taken in the twin peaks of Fig. 38a. The high-powertail of the distribution arises from lower portions of these peaks, the low7power tailcomes from samples taken near zero elevation, and the bulk of the distribution isdue to samples requesting torques at elevation angles around 90° or close to zero.By comparing calculated power demands from various wheel configurations, onecan ascertain which will prove problematic from the power standpoint.
The torque requests generated for this data were weighted to be uniform overspherical coordinates (as discussed in Section 4), entailing that torque commandsare equally probable in all directions. Mission requirements which impose a direc-tional bias can potentially move the most probable and mean values. Since all dataassume identical conditions, however, a comparison of these values is still valid.
Table 1 summarizes results for 5- and 6-wheel pyramid configurations. Lookingat the upper portion (all wheels operational), we see that the 6-wheel configurationrequires 20 Watts less power on average and at maximum. Indeed, the effects onEq. (8) that were mentioned earlier were more than able to cancel the added 15Watts from the extra wheel. A smaller gain (only 10 Watts) is seen in the most prob-able power; these points are generally at lower torque, however, hence don't see asmuch improvement through Eq. (8).
The lower portion of Table 1 summarizes the power parameters arising fromthese configurations with one wheel failed. Because of the higher momentum andtorque load per wheel, these values are considerably larger. The symmetryretained in the 6-pyramid/1-failure configuration (as well as the extra momentum &torque available from the additional wheel) has enabled it to beat the5-pyramid/1-failure configuration by 57 Watts on average, 185 Watts at maximum,and 30 Watts at most probable (the envelopes and power distribution loose ele-vation symmetry in this case, thus larger tails are created).
Because the pseudo-inverse is used to distribute torques in these calculations,no bounds are enforced on the output of each wheel, thus individual torques mayrise above saturation levels, potentially introducing a positive bias into these latterresults. The total momentum, however, is still assumed to be equally distributed
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amoung surviving wheels in the failure case; since the configurational symmetry isnow broken, this will not apply in reality, causing the actual power dissipated underfailures to increase even further, and introducing a negative bias into these results.
7) Conclusions and Implications
This study has shown that graphical display of the envelope polyhedra can beuseful in determing reaction wheel configurations that fit pre-determined missionconstraints.
For the particular satellite under investigation, it was shown that all wheel con-figurations considered (4-pyramid plus 1-azimuth, 5-pyramid, and 6-pyramid) wereable to answer all mission constraints with minimal sizing (1.2 ft-lb of maximum tor-que per wheel and up to 40 ft-lb-sec momentum storage). Operating with one wheelfailed, however, presents a different picture. Although the 4-plus-1 configurationpossessed a completely redundant azimuth wheel (which could be failed with littleeffect), the failure of a pyramid wheel can substantially reduce needed torque mar-gin. The failure tolerance was seen to be somewhat better in the symmetric 5-py-ramid orientation, but a capability of achieving up to 1.6 ft-lb of torque and 65ft-lb-sec of momentum per wheel was needed to meet all mission requirements.The symmetric 6-wheel pyramid exhibited the best overall performance, with nearlyfull mission requirements retained after one failure, assuming peak capacities of 1.2ft-lb and 45 ft-lb-sec per wheel. The fully-operational 6-wheel configuration was alsoseen to require 20 Watts less peak power (averaged about all angles) than neededby the 5-wheel system. Data was presented indicating that this power margin wouldwiden additionally in favor of the 6-wheel setup when operating under a single-wheel failure. Optimum pyramid skew angles (which minimize power requirementand maximize envelope coverage of the torque constraint cylinder in the single fail-ure case) were found to be 420 for the 5-pyramid and 380 for the 6-pyramid.
The break in the configurational symmetry caused by wheel failures can skewand distort the shape of the momentum and torque envelopes, causing a substantialreduction in capability to fit a symmetric operational constraint (such as the torque& momentum cylinders considered in this report). This effect contributes to thesuperior performance displayed by the 6-wheel configuration under one failure; theperturbation to the envelope symmetry is not as severe as in the 5-wheel/1-failuresystem. This effect was directly demonstrated in examples assuming simultaneous2-wheel failures in the 6-wheel configuration. Under failure of 2 adjacent wheels,much of the objective space was uncontrollable (due to the large configurational
17
III
symmetry break and subsequent envelope skew), yet after failure of 2 opposingwheels, considerably more objective space could be achieved (since configurationalsymmetry is retained).
A few final comments should be made on the interpretation of the envelope stu-dies (ie. Figs. 1 -+ 35). In the momentum projections, the dark areas (which aredrawn over the constraint cylinder where it protrudes from the envelope) signifyregions of momentum saturation. If the vehicle state is in a saturated region, thesatellite will experience a steady-state rotation; since the vehicle can spend consid-erable time in these areas (ie. when storing the full antenna momentum), momen-tum saturation must be avoided, therefore the momentum envelope is required tocover the entire expected constraint. Since this doesn't push the state-of-the-art inreaction wheels (the worst case encountered needed 65 ft-lb-sec per wheel to coverthe momentum constraint with a single failure), it isn't considered to be a problem.
Tradeoffs, however, may prove more relevant when considering the torque situ-ation. The dark areas drawn over the constraint cylinder in the torque plots indicateregions of torque saturation. When using the wheel configuration to null antenna-in-duced torques (which were used to size the cyl;nder), an operation that places thevehicle state in a dark area will-cause the vehicle to move. The excess torqueencountered in this case will be essentially proportional to the distance of the dis-turbance torque from the torque envelope; ie. the distance which the desired torqueprotrudes into the dark area (which can be estimated from these plots). Since onlya modest fraction of the constraint cylinder corners are darkened in most of the one-failure cases examined, the residual torques after saturation will be small, yieldinglittle satellite motion. If this motion can be tolerated (again, it only occurs in a fail-ure case), operation in a saturated region may be allowable. Simulations of flexiblemode excitation and calculations of maximum vehicle displacement under typicalmaneuver scenerios should be first performed before accepting such strategies.
As indicated at the close of the previous section, however, peak power require-ments can rise substantially when operating with wheel failures. This could makesaturated operation of the wheel configuration undesirable; in this case, therequired torques may be reduced (causing the constraint cylinder to shrink), result-ing in longer maneuver durations (keeping the satellite steady) or increased satellitedisturbance.
To conclude the above discussion, operation of an under-driven reaction wheelarray (ie. a 5-pyramid with one failure and less than 1.5 ft-lb available per wheel)can either result in basebody motion and large power requirements, or force longer
18
· _____I�_____________�____
maneuver durations (ie. slower antenna slews). These effects may be digestable,but must be carefully considered in simulation and mission planning. The config-uration which best covers the entire torque requirement after one failure is the 6-py-ramid running 1.2 ft-lb wheels. This setup exhibits the lowest power requirement,best envelope coverage and power response after one failure, and pushes least onexisting reaction wheel technology. A summary of envelope coverage for all wheelconfigurations examined in this report is given in Table 2.
LIST OF REFERENCES
1. "DLPRS Routine", IMSL MathChapter 8, pg. 888, April 1987.
Library User's Manual Version 1.0, Volume 3,
2. Newman, R.M. Sproull, R.F.,McGraw-Hill Computer Science
Principles of
Series, 1973.Interactive Computer Graphics,
3. Fallek, Scott, "Platform Study Status", Presentation given at McDonnell Douglas,Huntington Beach, CA., Aug. 21, 1987.
4. A Trade of Momentum Control Concepts for the Naval Research Laboratory,Honeywell Inc., Sperry Space Systems Division, Glendale, AZ, June 25, 1987.
5. Dixon, Wayne, Honeywell Inc., Sperry Space Systems Division, Glendale, AZ,personal communication, Sept., 1987.
6. Paradiso, Joseph A., "A Highly Adaptable Steering/Selection Procedure forCombined CMG/RCS Spacecraft Control", Detailed Report, CSDL-R-1835, March,1986.
7. "LSGRR Routine", IMSL Math Library User's Manual Version
Chapter 1, pg. 286, April 1987.
8. Levenson, Marv, Spacecraft ACS Actuators, Presentation givenSummer of 1987.
1.0, Volume 1,
at NRL during
19
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· Wheel rates = heq w/(Nwsinfl)
h req = Max. requested momentum (100 ft-lb-sec)a w = 12.5 RPM/ft-lb-secNw = Number of wheels in configuration/ = Pyramid skew angle (42 ° or 38 °)
* Wheel torques computed for 6400 points uniformly distributedacross maximum constraint cylinder.· Pseudoinverse calculation used· Torque limits and wheel rate saturation not enforced
* Results.(in Watts):5-Wheel 6-Wheel
All Wheels OperationalPower at Zero torque: 75 90Avg. Power Max. requested torque: 362 344Max. Power @ Max. requested torque: 425 405MP. Power @ Max. requested torque: 346 336
One Wheel FailedPower at zero torque: 60 75Avg. Power @ Max. requested torque: 435 378Max. Power @ Max. requested torque: 725 540MP. Power @ Max. requested torque: 361 331
57
TABLE II) Summary of Envelope Investigations
Cases Examined: %
* 4-Pyramid (38° ) + 1 Azimuth Wheel· Momentum Envelope (50 ft-lb-sec/wheel) .. 0...... 0.* Torque Envelope (1.3 ft-lb/wheel) .... 0.
* 4-Pyramid (38 ° ) + 1 Azimuth Wheel; Fail azimuth wheel
* Momentum Envelope (50 ft-lb-sec/wheel) .......... 0.* Torque Envelope (1.3 ft-lb/wheel) ............... 0.
* 4-Pyramid (38 ° ) + 1 Azimuth Wheel; Fail pyramid wheel* Momentum Envelope (50 ft-lb-sec/wheel) .......... 0.11* Torque Envelope (1.3 ft-lb/wheel) ............... 12.
* Symmetric 5-Wheel Pyramid (42° )
· Momentum Envelope (40 ft-lb-sec/wheel) ......... 0.* Torque Envelope (1.2 ft-lb/wheel) ............... 0.
· Symmetric 5-Wheel Pyramid (42°); Fail one wheel* Momentum Envelope (40 ft-lb-sec/wheel) .......... 29.* Momentum Envelope (60 ft-lb-sec/wheel) . .. 1.9. Momentum Envelope (65 ft-lb-sec/wheel) . ....... O.· Torque Envelope (1.2 ft-lb/wheel) ............... 4.6* Torque Envelope (1.5 ft-lb/wheel) ............... 0.40· Torque Envelope (1.6 ft-lb/wheel) ............... 0.07
* Symmetric 6-Wheel Pyramid (38 °)
. Momentum Envelope (40 ft-lb-sec/wheel) .......... 0.• Torque Envelope (1.2 ft-lb/wheel) . .............. 0.
* Symmetric 6-Wheel Pyramid (38°); Fail 1 wheel· Momentum Envelope (40 ft-lb-sec/wheel) .......... 5.4· Momentum Envelope (45 ft-lb-sec/wheel) .......... 0.· Torque Envelope (1.2 ft-lb/wheel) ................ 0.02
* Symmetric 6-Wheel Pyramid (38°); Fail 2 adj. wheels
· Momentum Envelope (45 ft-lb-sec/wheel) .......... 47.· Torque Envelope (1.2 ft-lb/wheel) ................ 19.
* Symmetric 6-Wheel Pyramid (38°); Fail 2 opp. wheels· Momentum Envelope (45 ft-lb-sec/wheel) ........... 1.0· Torque Envelope (1.2 ft-lb/wheel) ................ 3.3
58
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