Document
NASA/TM-97-206318
Linearized Poststall Aerodynamic and
Control Law Models of the X-31A Aircraft
and ComparisonWith Flight Data
P. C. Stoliker, John T. Bosworth, and Jennifer Georgie Dryden Flight Research Center Edwards, California National Aeronautics and Space Administration Dryden Flight Research Center Edwards, California 93523-0273 December 1997
NOTICE
Use of trade names or names of manufacturers in this document does not constitute an official endorsement of such products or manufacturers, either expressed or implied, by the National Aeronautics and Space Administration.
Available from: National Technical Information Service NASA Center for AeroSpace Information 800 Elkridge Landing Road 5285 Port Royal Road Linthicum Heights, MD 21090-2934 Springfield, VA 2216 l Price Code: A16 Price Code: A 16
ABSTRACT
The X-31A aircrafthasa uniqueconfigurationthatusesthrust-vectorvanesandaerodynamic control
effectorsto provide an operatingenvelopeto a maximum70° angleof attack,an inherentlynonlinear portion of the flight envelope.This reportpresentslinearizedversionsof the X-31A longitudinal and lateral-directionalcontrol systems, with aerodynamic modelssufficient to evaluatecharacteristics in the poststallenvelope at 30°, 45°, and60° angleof attack.The modelsarepresented with detail sufficientto allow the readerto reproducethe linear resultsor perform independent control studies.Comparisons betweenthe responses of the linear modelsand flight data are presentedin the time and frequency domainsto demonstrate the strengths andweaknesses of the ability to predicthigh-angle-of-attack flight dynamicsusing linear models.The X-31A six-degree-of-freedom simulationcontains a programthat calculateslinear perturbationmodelsthroughoutthe X-31A flight envelope.The modelsincludeaerody- namicsandflight controlsystemdynamicsthatareusedfor stability, controllability, andhandlingquali- ties analysis.The modelspresentedin this reportdemonstrate the ability to provide reasonable linear representations in thepoststallflight regime.
NOMENCLATURE Acronyms HARV High Alpha Research Vehicle MATV Multi-Axis Thrust Vectoring TEF trailing-edge flaps Symbols A state derivative matrix ALFC filtered angle-of-attack command, deg ALFCO delayed angle-of-attack command, deg ALFX processed angle-of-attack feedback, deg B control derivative matrix BETC commanded angle of sideslip, deg BETX processed angle of sideslip, deg C state observation matrix CALFX cosine of ALFX D control observation matrix DAFB summation of feedback compensation to differential trailing-edge flaps, deg DALF feedback error between commanded and sensed angle of attack, deg DBET error between commanded and sensed angle of sideslip, deg DBETDXR sideslip command from rudder pedals, deg DDEFC commanded differential trailing-edge flap deflection, deg DECAN canard deflection, deg DECANC commanded canard deflection, deg DECAN_IL inner-loop feedback to canard, deg DERUDC commanded rudder deflection, deg DEVQ pitch thrust-vector deflection, deg DEVQCL pitch thrust-vector deflection command, deg DEVR yaw thrust-vector deflection, deg DEVRCL yaw thrust-vector deflection command, deg DPE error between stability-axis roll rate and command, deg/sec DQE error between flightpath pitch rate and command, deg/sec DRE error between stability-axis yaw rate and command, deg/sec DRFB summation of feedback compensation to rudder, deg DRPF normalized rudder command from flight data DRUD rudder deflection, deg DTED differential trailing-edge deflection, deg DTES symmetric trailing-edge deflection, deg DTES_IL inner-loop feedback to trailing-edge flaps, deg DTESC commanded symmetric trailing-edge flap deflection, deg DTR rt/180, ract/deg FDWGT0 ratio of estimated thrust to estimated weight FDWGTINV inverse of FDWGT0 FFCOMP angle-of-attack feedforward compensation, deg FKAPPA rudder to thrust-vectoring effectiveness multiplier FZETA rudder fade multiplier gravitational acceleration constant, 32.2 ft/sec 2 g GODVK gravitational acceleration constant divided by velocity, deg/sec HIALO angle-of-attack command to canard gain, deg/deg HRKBEO side force for each angle-of-sideslip ratio, g/deg HURBEO angle-of-sideslip command-to-rudder gain, deg/deg HURPKO roll-rate command normalized by velocity-to-rudder gain, ft HURPPO roll acceleration-to-rudder gain, deg/(deg/sec 2) HURRPO yaw acceleration-to-rudder gain, deg/(deg/sec 2) HXIBEO angle-of-sideslip command-to-aileron gain, deg/deg HXIPKO roll-rate command normalized by velocity-to-aileron gain, ft roll acceleration-to-aileron gain, deg/(deg/sec 2) HXIPPO HXIRPO yaw acceleration-to-aileron gain, deg/(deg/sec 2) moment of inertia about the x axis, slug-ft 2 lxx xz product of inertia, slug-ft 2 lx z moment of inertia about the y axis, slug-ft 2 moment of inertia about the z axis, slug-ft 2 Izz KADEO angle of attack-to--trailing-edge flap gain, deg/deg KBKAO angle of sideslip-to-thrust-vectoring gain, deg/deg KBXIO angle of sideslip-to-aileron gain, deg/deg KBZEO angle of sideslip-to-rudder gain, deg/deg KDECO multiplier for ratio of canard from trailing-edge flaps, deg/deg KDEVQO pitch thrust-vectoring gain, deg/deg KPKKA roll rate-to-thrust-vectoring gain, deg/(deg/sec) KPKXI roll rate-to-aileron gain, deg/(deg/sec) KPKZE roll rate-to-rudder gain, deg/(deg/sec) KQDEO pitch rate-to-trailing-edge flap gain, deg/(deg/sec) KRKKA yaw rate-to-thrust-vector gain, deg/(deg/sec) KRKXI yaw rate-to-aileron gain, deg/(deg/sec) KRKZE yaw rate-to-rudder gain, deg/(deg/sec) KXIO0 multiplier for ratio of thrust vectoring to aileron, deg/deg KZETA thrust vectoring-to-aileron multiplier, deg/deg m mass, slug MSALFX negative sine of ALFX stability-axis acceleration, g II L 1l longitudinal acceleration at the center of gravity, g xcg longitudinal acceleration at the sensor location, g nxinu NXS sensed body-axis longitudinal acceleration, g lateral acceleration at the center of gravity, g n ycg lateral acceleration at the sensor location, g n v in u NYKC commanded lateral acceleration, g normal acceleration at the center of gravity, g nzcg NZC body-axis normal acceleration command, g normal acceleration at the sensor location, g H • Zl?lll
NZKC
commanded stability-axis normal acceleration, g
NZ30D
computed normal acceleration at 30 ° angle of attack, g body-axis roll rate, deg/sec
P
PDT
derived roll acceleration, deg/sec 2
PDTFB
feedforward compensation for the lateral axis, deg
PHIF
filtered bank angle, rad
PKC
stability-axis roll-rate command, deg/sec
PKCDVK
roll-rate command normalized by velocity, deg/ft
PKCF
pilot roll-rate command, deg/sec
PKCMAX
maximum stability-axis roll-rate command, deg/sec stability-axis roll rate, deg/sec
Pstab
PS
sensed body-axis roll rate, deg/sec
PSTAB
stability-axis roll rate, deg/sec
q body-axis pitch rate, deg/sec
Q pitch rate, deg/sec
QBWGTO normalized dynamic pressure
QEC flightpath pitch-rate command, deg/sec
QS
sensed body-axis pitch rate, deg/sec r body-axis yaw rate, deg/sec RDT derived yaw acceleration, deg/sec 2 RDTFB feedforward compensation for the directional axis, deg REC commanded stability-axis yaw rate, deg/sec stability-axis yaw rate, deg/sec rstab RS sensed body-axis yaw rate, deg/sec RSTAB stability-axis yaw rate, deg/sec s Laplace transform variable SALFX sine of ALFX T flight control computer frame rate, 0.02 sec TCNREF reference aerodynamic normal force curve TDECCRU canard pitch trim, deg TDETA pitch trim, deg TIME time reference for pilot inputs to simulation, sec TSDQBDY dynamic pressure ratio TVFAC 1 thrust-vectoring fade multiplier for lateral-directional axes TVFAC2 thrust-vectoring fade multiplier for the longitudinal axis TVFAD thrust-vectoring engagement multiplier TVFB summation of feedback compensation to yaw thrust vectoring, deg U control input vector V velocity, ft/sec VINV inverse of velocity, l/(ft/sec) VKO true airspeed, ft/sec x state vector derivative of the state vector Y output vector discrete transform variable z (X angle of attack, deg angle of sideslip, deg 7 flightpath elevation angle, deg _canard canard deflection, deg differential trailing-edge flap deflection, deg 5dtef leading-edge flap deflection, deg 8&f rudder deflection, deg 8rud symmetric trailing-edge flap deflection, deg 8tef pitch thrust-vector plume deflection, deg _tvvp _tt,t'l' yaw thrust-vector plume deflection, deg pitch angle, deg P flightpath bank angle, deg 7t constant, 3.141592654 bank angle, deg Sign Conventions Angle of attack Positive noseup Positive nose left Angle of sideslip Canard deflection Positive trailing-edge down Differential flap Positive right trailing-edge down (right - left)/2.0 Lateral acceleration Positive out right wing
Lateral stick Positive
right roll
Pitch rate Positive
noseup
Pitchstick Positive
aft (noseupcommand)
Pitch thrust-vectorcommand Positive nosedown
Roll rate Positive
right wing down
Ruddersurface Positive
trailing-edgeleft
Positive
Symmetricflap trailing-edgedown
Yaw rate Positive
noseright
Positive noseleft
Yaw thrust-vectorcommand
INTRODUCTION
Regardless of the flight regimeto be explored,linear andnonlinearsimulationshavebeenusedas
tools in the designandtest processes. Nonlinearsimulations,including piloted simulations,havelong beenusedfor flight control systemcheckout,verification andvalidation of operationalflight software, test missionplanning, andpilot training. Linear models(which includethe flight control system,rigid- body aerodynamics, actuatordynamics,feedbacksensors, andfilters) haveproven to be an invaluable tool for the analysisof newor modifiedflight controlsystems, whetherthe control systemdesignis per- formedusingclassicalroot-locusmethodsor moderncontroltheories.Linear simulationsalsoprovidea cost-effectiveandtimely tool for obtainingsurveysof stability,control,andhandlingqualitiescharacter- istics throughoutthe flight envelope. Thesemodelshavean importantrole in the early stagesof control
systemdevelopment or controllaw revisionsandhavebeenshownto bevaluablewhenvalidatedagainst
flight testdata.l
As control systemand computercapabilitieshaveadvanced, aircraft havecontinually enterednew
flight regimesandthe necessityfor evaluationof linear modelshascontinued.The latestgenerationof developmental or experimental aircraft hasinitiatedtheinvestigationof controlledflight beyondthe stall angleof attackfor the wing, or thepoststallregime.The capabilityfor sustained andcontrolledflight in this regime has beenprovidedby integratingmultiaxis thrust vectoringinto the control laws2 for the X-31A aircraft,the F-18High Alpha Research Vehicle(HARV), andthe F-16Multi-Axis ThrustVector- ing (MATV) aircraft. Thrust vectoringhas also beendemonstrated with two-dimensionalconverging nozzlesfor the YF-22 aircraft3 andthe F-15 ShortTakeoff andLandingDemonstrator. 4 Initial applica- tionson the F-18HARV andX-31A aircraftusedhigh-temperature nickel-based steelandcarbon-carbon paddles,respectively,to deflect the thrust-vector plume.Recently,rapid advances in enginetechnology haveallowedthe incorporationof axisymmetricthrustvectoringinto productionengineswith little or no penaltiesin aircraft weight or systems.
TheX-31A aircraft is a recentexampleof a poststall-capable aircraftusinga "first generation"thrust- vectoring capability. The vehicle is stabilizedand controlledby a full authority, fly-by-wire control systemthathasintegratedpitch andyaw thrustvectoringwith the aerodynamic control surfaces. Linear modelswereusedextensivelyin the initial controlsystemdesign, 5wherea linearquadraticregulatornon- zero set-pointtracker methodologywas used.During flight tests,parameter identification resultswere usedto modify the aerodynamic database. 6 Linearmodelsgenerated from themodifiedaerodynamic data were used to develop control system modification that allowed envelope expansion to proceed to 70 ° angle of attack and 265 kn poststall entry speed. Subsequent efforts used the linear models for an in-flight simulation of an aircraft with reduced vertical tail size 7 and a high-angle-of-attack handling qualities investigation. 8 This paper discusses the linear models and validation of the models with flight data for the X-31A aircraft. Three flight conditions have been selected for presentation. These flight conditions provide the ability to examine 1-g flight at 30 °, 45 °, and 60 ° angle of attack for both the longitudinal and lateral- directional axes. These cases provide a representative sampling of the poststall flight envelope. The rigid- body aerodynamics are calculated using linear perturbation methods of the wind-tunnel and parameter- estimation-modified data six-degree-of-freedom base. The linear models are compared with flight test data in the time and frequency domains.
AIRCRAFT DESCRIPTION Two X-31A aircraft were built by Rockwell International (Downey, California) and Daimler-Benz Aerospace (Germany) using joint funding from the Advanced Research Projects Agency and Germany's Federal Ministry of Defense. The aircraft (fig. 1) is a single-seat fighter configuration with an empty weight of approximately 12,000 lbm that uses a single GE-F404-400 engine (General Electric, Lynn, Massachusetts). The wing planform is a double-delta with an inboard leading-edge sweep of 56.6 ° and an outboard sweep of 45 °. The wing area, span, and mean chord are 226.3 ft 2, 22.833 ft, and 12.35 ft, respectively. Figure 2 shows an aircraft three-view drawing. Tables 1 and 2 show the physical character- istics and accelerometer locations for the aircraft. A more detailed aircraft description has previously been published. 8 Four trailing-edge flaps on the wing can be deflected symmetrically for pitch control and differen- tially (left and right side) for roll control. The inboard and outboard trailing-edge flaps are geared together on each side of the aircraft. The leading-edge flaps are scheduled to deflect symmetrically as a function of angle of attack. An all-moving canard was added to meet the desired instability level for maneuverability and to meet the requirement for aerodynamic recovery from extreme angles of attack.
The vertical tail contains a rudder for directional control at less than 40 ° angle of attack, Pitch and yaw moments can be generated by the three thrust-vector vanes (fig. 3). Table 3 shows the control surface characteristics. The engine inlet lip is moveable and is deflected as a function of angle of attack.
Table 1. Physical characteristics of the X-31A aircraft.
22.833 ft Wing span 226.3 ft 2 Wing area Wing leading-edge sweep: inboard 56.6 deg outboard 45 deg 12.35 fl Mean aerodynamic chord 12,168 Ibm Vehicle empty weight 4,000 lbm Maximum fuel capacity 23.6 ft 2 Canard area Table2.Accelerometer locationsof theX-31A aircraft.
Accelerometer Fuselage station, Buttockline, Waterline, in. in. in.
Normal 191.625 5.225 111.672
Lateral 191.625 5.550 111.672
Axial 198.680 5.550 111.672
Table3. Controlsurfacecharacteristics.
Positionlimit, Ratelimit,
Control surface deg deg/sec
Canard
-70,20 ±60
-40,0 ±25
Inboardleading-edge flaps
-32,0 ±25
Outboardleading-edge flaps
±30 ±60 or ±80*
Trailing-edgeflaps
Rudder
±30 ±80
-48,35 ±60**
Thrust-vectoring vanes
Higher rate allowed forhigher engine power settings.
Paddle-rate limitresults inapproximately 40deg/sec plume deflection-rate limit.Paddle limit allowed +_ 15 °plume deflection.
RIGID-BODY AERODYNAMIC MODEL DESCRIPTION Linear rigid-body aerodynamic models were obtained by solving for steady-state trim points and using finite differences to generate the linear equations of motion. The trim condition was determined by using an iterative search technique to determine deflections of the aerodynamic and thrust control effec- tors, angle of attack, pitch angle, and thrust to obtain steady-state flight at the desired condition. For each combination of effector position, angle of attack, and thrust, the forces and moments were computed using the full six-degree-of-freedom nonlinear equations of motion with a full envelope aerodynamic database. The aerodynamic database used in the simulation incorporated modifications to the wind-tunnel data using increments calculated using parameter estimation techniques and flight data. 6 The linear perturbation equations of motion were formulated in the following state space form: = Ax+Bu (1) y = Cx + Du (2) The coefficients in the matrices were obtained using a linearization technique that calculates numeri- cal perturbations about the trim condition. The perturbations were ±1 ft/sec for velocity, ±1 ° for angles of attack and sideslip, ±1 deg/sec for body rates, + 1° for attitudes, and ±1 ° for control-effector deflections.
Lateral-Directional Control System Figure 12 shows a block diagram of the lateral-directional linear model of the X-31 control laws. The lateral stick input is scaled by the maximum stability-axis roll-rate command, PKCMAX, to the stability- axis roll-rate command, PKC. The PKCMAX is a function of dynamic pressure, angle of attack, and estimated thrust to ensure that the thrust-vector vanes can generate enough control moment to coordinate a turn. The rudder pedals command the angle of sideslip, which is scaled for the maximum angle-of-side- slip command. The maximum angle-of-sideslip command is a function of true airspeed, angle of attack, and dynamic pressure. The rudder-pedal command authority is faded from 1.0 to 0.0 between 30 ° and 45 ° angle of attack. This fade is caused by the loss of rudder effectiveness as angle of attack increases.
The primary feedbacks for the lateral-directional flight control system are the sensed body-axis roll rate, PS, sensed body-axis yaw rate, RS, and processed angle of sideslip, BETX. Bank angle, 0, is used for gravity compensation. The BETX is obtained from a blended combination of inertial measurements and sideslip from the flight test noseboom flow vane. Figure 13 shows the linear model for this function. Fig- ures 14 to 16 show the filters required for roll rate, yaw rate, and bank angle.
Figure 17 shows the calculations for the feedback parameters and includes the stability-axis transformation for the rates, the gravity compensation, and the generation of the yaw-rate command.
Sensed body-axis roll and yaw rate are converted to the stability-axis roll and yaw rate by the follow- ing equations: (15) Pstah = P * cos(a)+ r • sin(or) (16) rstab = r, cos(O_)-p, sin(R) Figure 18 shows the implementation of the conversion between body- and stability-axis rates in terms of X-31A control system variables: {17) PSTAB = PS • cos{ALFX)+ RS * sin(ALFX) (18) RSTAB = RS * cos(ALFX)-PS * sin(ALFX) The error between stability-axis roll rate and command, DPE, is obtained from the difference between the stability-axis roll-rate, PSTAB, and PKC. Similarly, the error between commanded and sensed angle of sideslip, DBET, is obtained from BETX and the commanded angle of sideslip, BETC. The commanded stability-axis yaw rate, REC, is obtained from the following stability-axis lateral acceleration equation: 5 (19} nycg = r • (V/g) • (/t/180)- sin(0 ) • cos(],) Rearranging terms in the equation results in the following: {20) r = (nycg + sin(0 ) • cos{y)) • {g/V) • (180/rt) Linearizing the gravity term (as reflected by the flightpath angle terms) reduces to be equal to the bank angle, q_: (21) r = (n3,cg+(_) * (g/V) * (180/rt) The calculationof the yaw-ratecommand requiresthe definitionof the lateralacceleration command, which canbe calculatedusingthe following relationship: (22) nycg = [(drag-thrust • cos(or)) • sin([3)]/(m • g) This equation represents the contributions of the normalized (drag/(m. g)) and thrust (thrust/(m • g)) components. The normalized drag component is estimated by a table lookup value based on flight condition. Estimated thrust is calculated using flight condition and sensed engine parameters.
The equation is simplified by using the small angle approximation for the sine function and replacing the angle of sideslip, [3, with the BETC. In terms of X-31A control system variables, the commanded lateral acceleration, NYKC, can be expressed as follows: (23) NYKC = HRKBEO • QBWGT0-FDWGT0 • cos(ALFX)) • BETC(lz/180) Thus, the REC can be expressed in terms of X-31A control system variables shown in figure 19: (24) REC = (NYKC+O) * (g/VKO) • (180/rt) Angular accelerations caused by the gravity terms are compensated by a feedforward command (fig. 20).
The gravity contribution is differentiated and transformed into the stability axis.
The three feedback error signals (DPE, DRE, and DBET) are passed through a gain compensation (fig. 21). Figure 22 shows forward-path compensation gains. The feedback and feedforward compensa- tion paths are combined to provide commands to the differential trailing-edge flaps, the rudder control surfaces, and the yaw thrust-vector system (fig. 23). Figures 24 to 26 show the filtering and actuator models for the differential trailing-edge flaps, rudder, and yaw thrust vectoring.
SELECTED FLIGHT CONDITIONS FOR LINEAR MODELS Flight conditions were selected to provide the opportunity to examine the poststall characteristics of the X-3 IA aircraft and the unique control configuration provided by the addition of thrust vectoring as a control variable. The flight conditions provide a survey of 1-g characteristics at 30 °, 45 ° and 60 ° angle of attack. Table 4 shows the three longitudinal and three lateral-directional cases presented in this report.
Tables 5, 6, and 7 show the trim surface positions, weights, and inertial characteristics for each case.
Table 4. Trim conditions for the six linear models.
Target angle Angle of Load True Case of attack, attack, Altitude, factor, Mach velocity, no. deg deg ft g no. ft/sec Input 1 30 29.9 34,900 0.93 0.373 363 Pitch doublet 2 30 24.8 24,000 1.90 0.435 444 Yaw/roll doublet 3 45 46.1 30,800 0.69 0.270 268 Pitch doublet 4 45 38.4 22,700 1.33 0.326 334 Roll doublet 5 60 59.9 31,600 0.73 0.263 260 Pitch doublet 6 60 59.2 21,300 0.50 0.174 179 Roll doublet Table 5. Trim surface positions.
Canard Symmetric flap Case position, position, no. deg deg 1 -30.9 1.8 2 -23.3 -2.5 3 -39.9 -2.1 4 -35.4 -3.9 5 -42.7 -4.2 6 -40.3 -6.2 Table 6. Mass properties descriptions.
Case Weight, lxx, lvv, I..,.._ lxz, no. /bin slug- ft 2 slug-ft 2 slug-fl 2 slug-fl 2 1 14,500 3,110 35,400 36,200 -224 2 14,100 3,060 35,300 36,100 -209 3 15,000 3,180 35,500 36,300 -242 4 14,200 3,080 35,300 36,100 -214 5 13,600 3,010 35,100 36,000 -192 6 13,600 3,010 35,100 36,000 -192 Table 7. Center-of-gravity locations.
Case Fuselage station, Buttock line, Waterline, no. in. in. in.
1 268.8 0.0 97.4 2 269.6 0.0 97.0 3 269.3 0.0 98.1 4 270.1 0.0 97.1 5 271.0 0.0 96.5 6 272.0 0.0 96.5 State space models are presented for the longitudinal and lateral-directional axes for each of the cases.
Tables 8 and 9 show the flight control system gains scheduled as a function of flight condition for all six cases. Tables I0 to 15 show the state space matrices for the linearized airframes.
Table8.Control systemgainsfor the longitudinalcases.
Gain Case1 Case3 Case5
CALFX 0.867 0.693 0.501
FDWGTINV 5.560 2.493 2.580
GODVK 5.077 6.882 7.102 HIALO 1.020 1.141 1.166 KADEO 1. 177 0.781 0.792 KDECO -0.758 -1.051 -1.299 KDEVQO 0.205 0.278 0.299 KQDEO 0.754 0.682 0.672 NZ30D 0.933 0.658 0.619 SALFX 0.498 0.721 0.866 TCNREF 0.016 0.014 0.003 TDECCRU -1.400 0.000 -0.595 TDETA -0.037 -0.066 -0.109 TVFAC2 1.017 1.000 1.000 TVFAD 0.976 1.000 1.000 Table9. Controlssystemgainsandconstants for thelateral-directionalcases.
Gain Case2 Case 4 Case 6
CALFX 0.908 0.783 0.512 DBETDXR 4.385 2.038 0.000 FDWGT0 0.618 0.610 0.592 FDWGTINV 1.619 1.640 1.690 FKAPPA 0.000 0.030 0.016 FZETA 1.000 0.434 0.000 GODVK 4.147 5.515 10.300 HRKBEO - 12.639 3.026 25.555 HURBEO 0.462 1.955 -6.024 HURPKO -32.501 -63.611 -36.634 HURPPO 0.125 0.420 -0.411 HURRPO -2.601 -5.879 -8.630 HXIBEO -0.852 -2.184 -1.609 HXIPKO -8.940 28.608 16.141 HXIPPO -0.218 -0.635 -0.619 HXIRPO -0.269 -0.104 -0.157 KBKAO -0.147 -0.759 -0.906 KBXIO -0.787 -1.171 0.669 KBZEO -0.997 -0.966 -0.001 KPKKA 0.042 0.190 0.441 KPKXI 0.187 0.787 0.715 KPKZE 0.286 0.256 0.000 KRKKA 0.105 0.530 0.401 KRKX1 0.153 -0.778 -0.936 KRKZE 0.735 0.610 0.000 KXIO0 0.080 -0.011 0.001 KZETA 3.405 6.080 0.000 MS ALFX -0.418 -0.622 -0.859 QBWGTO 0.090 0.053 0.016 SALFX 0.418 0.622 0.859 TSDQBDY 0.190 0.322 1.076 TVFAC 1 1.000 1.000 1.000 TVFAD 1.000 1.000 1.000 VINV 0.002 0.003 0.006 Table10.Longitudinal statespacematricesfor 30° angleof attack.
A Matrix (4 by 4)
-0.2592E+00 0.1293E+01 0.3081E-01 0.4940E-02
0.1000E+O1 -0.6681E-O1 -0.2386E--01 0.1689E-01
0.0000E+O0 -0.2097E+00 -0.6462E-01 -0.5493E+00
0.0000E+O0 0.0000E+00
0.1000E+O 1 0.O000E+O0
B Matrix (4 by 4)
0.1856E-01
0.7888E+00 -0.1324E+01 -0.1740E+01
-0.1039E-01 -0.4400E-01 0.4200E-02 -0.1806E-01
-0.5900E-02 -0.1619E+00 --0.1480E-01 -0.6571E-01
0.0000E+00 0.0000E+O0 0.0000E+O0 0.0000E+00
C Matrix (8 by 4)
O. 1000E+O 1 0.O000E+O0 0.0000E+00 0.0000E+00
0.O000E+O0 0.1000E+O 1 0.0000E+00 0.0000E+00
0.O000E+00 0.O000E+O0 0.1000E+01 O.0000E+00
0.O000E+O0 0.O000E+O0 0.0000E+00 0.1000E+01
0.0000E+O0 0.1532E-01 0.5000E-02 -0.2000E-04
-0.9100E-03 0.1985E-01 0.5110E-02 0.0000E+00
0.0000E+O0 0.5! 00E-03 0.5600E-03 0.6000E-04
0.2300E-03
-0.3200E-03 0.5400E-03 0.5000E-04
D Matrix (8 by 4)
O.O000E+00 0.0000E+00 0.0000E+00 0.0000E+00
O.0000E+O0 0.0000E+00 O.O000E+00 0.0000E+00
0.0000E+00 0.O000E+00 0.0000E+00
0.0000E+O0
O.0000E+O0 0.0000E+00 O.O000E+00 0.0000E+O0
O.1870E-02 O.1002E-01 -0.4900E-03 0.4100E-02
0.4630E-02 0.5390E--02 -0.4200E-03 -0.1990E-02
0.8600E-03 -0.5000E-04
-0.8100E-03 O.O000E+O0
0.3500E-03 0.8100E-03 -0.8200E-03
0.1120E-02
Table11.Lateral-directional statespacematricesfor 30° angleof attack.
A Matrix (4 by 4)
0.O000E+O0
0.7904E+00 -0.3420E+02
-0.6926E+00
-0.6763E+00 0.0000E+O0
-0.3457E+00
-0.8387E-01
0.7172E-01
-0.1177E+00
-0.9033E+00
0.4208E+00
-0.1038E-01
0.O000E+00
-0.1209E+00
0.1000E+01
B Matrix (4 by 3)
0.2209E+01 0.1022E+01
-0.2471E+02
-0.4375E+01
-0.1025E+01 -0.1827E+01
0.4497E-01
0.3700E-01 0.3229E-01
0.0000E+O0 0.0000E+00
O.0000E+00
C Matrix (6 by 4)
0.0000E+00 0.0000E+00
0.0000E+00
O.IO00E+O1
0.0000E+00 0.0000E+O0
0.1000E+01
0.0000E+O0
0.1000E+01 O.O000E+00
0.0000E+00 0.0000E+00
0.1000E+01
0.0000E+O0
0.O000E+00 0.0000E+00
-0.4000E-04
0.1080E-02 -0.2133E-01
0.3000E-03
-0.4000E-04
0.3300E-03 -0.4653E-01
-0.1600E-03
D Matrix (6 by 3)
0.0000E+00 0.0000E+00 0.0000E+00
0.0000E+00
O.O000E+00 0.0000E+00
0.O000E+O0
O.0000E+O0 0.O000E+00
0.O000E+00 0.O000E+00
O.O000E+O0
0.7780E-02 0.1084E-01
0.8920E-02
0.2800E-02 -0.3930E-02
-0.1118E-01
Table12.Longitudinalstatespace matricesfor 45° angleof attack.
A Matrix (4 by 4)
0.3061E-01 0.4640E-02
-0.2313E+00 -0.6293E-01
-0.3459E-01 -0.3272E-01 0.1642E-01
O.1000E+O 1
-0.1472E+00 -0.1059E+00 -0.5546E+00
0.0000E+O0
O.0000E+00 0.O000E+O0 O.0000E+O0
O.1000E+01
B Matrix (4 by 4)
0.5252E+00 -0.5492E+00 -0.9660E-02 -0.2894E+01
-0.5730E-02 -0.1339E-01 0.1840E-02 -0.3184E-01
-0.4440E-02 -0.1526E+00
-0.4120E-02 -0.8698E-01
0.0000E+00 0.0000E+O0 0.O000E+00
0.0000E+00
C Matrix (8 by 4)
0.1000E+01 0.0000E+00 0.0000E+O0 0.O000E+00
O.0000E+00 0.1000E+01 0.0000E+00 0.O000E+O0
0.O000E+O0
0.0000E+00 0.O000E+00 0.1000E+01
0.O000E+00 0.0000E+00 0.1000E+OI
0.0000E+00
0.1024E-01 0.5120E-02 -0.3000E-04
0.0000E+00
-0.7900E-03 0.1001E-01 0.5230E-02 -0.2000E-04
0.0000E+00 -0.1290E-02 0.5700E-03 0.4000E-04
0.3500E-03 -0.1250E-02 0.5600E-03 0.4000E-04
D Matrix (8 by 4)
0.0000E+00 0.0000E+O0 0.O000E+O0 O.O000E+00
0.0000E+O0 0.0000E+O0 0.O000E+O0 O.O000E+O0
0.0000E+O0 0.0000E+O0 0.O000E+O0 O.O000E+00
0.0000E+O0 0.0000E+O0 0.O000E+O0 0.O000E+00
0.6700E-03 0.3300E-02 -0.9000E-04 0.6630E-02
0.2520E-02 0.1360E-02 -0.1200E-03 -0.3570E-02
0.5100E-03 -0.4700E-03 -0.2900E-03 0.4000E-04
-0.1400E-03
0.1900E-03 -0.2800E-03 0.1860E-02
Table13.Lateral-directional statespace matricesfor 45° angleof attack.
A Matrix (4 by 4)
O.0000E+00
-0.1070E+02
-0.1356E+01
0.1630E+01
0.0000E+00
0.3274E+00 0.1938E+01
-0.4152E+00
0.9599E-01
-0.7589E-01
-0.7738E+00
0.6175E+00
-0.2700E-03
0.O000E+00
-0.4160E-02
0.1000E+01
B Matrix (4 by 3)
0.9682E+00
0.1034E+00
-0.4571E+01
-0.4269E+01
-0.7325E+00
-0.3659E+00
0.5786E-01
0.1455E-01
0.1016E-O1
0.O000E+00
0.0000E+00
0.0000E+00
C Matrix (6 by 4)
0.O000E+00
0.0000E+00
0.O000E+O0
0.1000E+01
O.0000E+00
0.0000E+00
0.1000E+01
0.0000E+00
0.0000E+O0
O.IO00E+01
O.0000E+00
0.0000E+00
O. 1000E+01
0.O000E+O0 0.O000E+00
O.O000E+O0
-0.7940E-02 -0.4000E-04
O.1820E-02
-0.6400E-03
-0.4000E-04
-0.8110E-02
0.2050E-02
-0.8100E-03
D Matrix (6 by 3)
0.0000E+O0
0.0000E+00
0.O000E+00
O.0000E+00
O.0000E+O0
0.0000E+00
0.O000E+O0
O.0000E+O0 0.O000E+00
0.0000E+00
0.0000E+O0 0.0000E+00
0.2640E-02 O.1050E-O 1
0.1840E-02
-0.4030E-02
0.1000E-03
-0.2470E-02
Table 14.Longitudinal statespacematricesfor 60° angleof attack.
A Matrix (4 by 4)
-0.1859E+00 -0.7335E--01 0.3132E-01 0.6300E-02
O. 1000E+O 1 -0.2695E-01 -0.2634E-01 0.6718E-O 1
0.0000E+00 0.3972E-01 --0.1478E+00 -0.4699E+00
O. 1000E+01 0.0000E+O0 0.0000E+00 O.0000E+O0
B Matrix (4 by 4)
0.6008E+00 -0.3392E+00 --0.4820E-02 -0.2884E+01
-0.5420E-02 -0.2617E-01
-0.5320E-02 0.7700E-03
-0.4770E-01 -0.7037E-01 -0.2840E-02 -0.2022E+00
0.0000E+00 0.0000E+00 0.0000E+O0 0.0000E+O0
C Matrix (8 by 4)
0.1000E+O 1 O.0000E+00 0.0000E+00 0.0000E+00
0.0000E+00 0.1000E+01 0.0000E+00 0.0000E+00
O.O000E+O0 0.0000E+O0 0.1000E+01 0.0000E+00
0.O000E+00 0.0000E+O0 0.0000E+00 0.1000E+01
O.O000E+00 0.1040E-02 0.5660E-02
-0.2000E-04
-0.6600E-03 0.7700E-03 0.5770E-02
0.0000E+00
O.O000E+O0 0.1110E-02 0.6100E-03 0.5000E-04
0.2500E-03 0.1160E-02 0.5800E-03 0.4000E-04
D Matrix (8 by 4)
0.0000E+O0 0.O000E+O0 0.0000E+00 O.O000E+O0
0.0000E+O0 0.O000E+O0
0.0000E+O0 0.0000E+00
0.O000E+00
0.0000E+00 0.0000E+O0 0.0000E+00
O.O000E+O0 0.O000E+O0
0.0000E+O0 0.0000E+00
0.1670E-02 0.2270E-02 0.2000E-04 0.7290E-02
0.3830E-02 0.1050E-02 0.O000E+00
-0.3100E-02
-0.8000E-04 -0.4500E-03 -0.1400E-03
0.4000E-04
-0.5000E-03 -0.2200E-03 -0.1300E-03 0.2050E-02
Table 15.Lateral-directional statespace matricesfor 60° angleof attack.
A Matrix (4 by 4)
-0.3655E+01 O.0000E+00
-O.1701E+00 0.2849E+00
0.5032E+00 O.0000E+00
-0.7870E-02
--0.2376E-01
--0.4192E-01 O. 1475E+00
-0.5142E+00
0.8575E+00
O.0000E+O0 -0.1633E-01
0.6898E+00
O.1000E+01
B Matrix (4 by 3)
O.1091E+O1
0.2000E-04
-0.1609E+01
-0.3600E-03 -0.4220E+01
0.6701E-01
0.1121E+00
0.0000E+00
-0.4490E-02
0.O000E+O0
0.O000E+00
0.O000E+00
C Matrix (6 by 4)
0.O000E+O0
0.0000E+00
0.1000E+01 0.0000E+00
0.O000E+O0
0.1000E+01 0.0000E+00
0.0000E+00
0.O000E+00
O.0000E+00 0.1000E+01
0.O000E+00
0.1000E+01
0.0000E+O0 0.0000E+O0
0.O000E+00
-0.3000E-04
-0.1300E-03 0.1190E-02
-0.9000E-04
-0.3000E-O4
0.5000E-04 0.5000E-03
-0.3800E-03
D Matrix (6 by 3)
0.0000E+00
0.0000E+00 0.0000E+00
O.0000E+O0
0.0000E+O0 0.0000E+00
0.0000E+00 0.O000E+00
0.0000E+00
0.O000E+00
O.0000E+00 0.0000E+00
0.O000E+00 0.1089E-01
-0.4400E-03
-0.1300E-02 O.O000E+O0 -0.3640E-02
LINEAR MODEL AND FLIGHT DATA COMPARISONS Time-domain comparisons are shown in this section for each of the selected flight conditions. Pilot inputs recorded in flight were used as inputs to the simulations to provide the time-domain comparisons.
Unfortunately, no frequency sweeps were performed during the X-3 IA poststall flight testing. The longi- tudinal pitch doublets, however, provided sufficient excitation to produce reasonable frequency responses when passed through a fast Fourier transformation algorithm. Standard linear methods were used to calculate frequency responses from the linear models for the same flight conditions. Fast Fourier transformation of the roll doublets generally did not provide reasonable results: however, adequate frequency content existed for one case to generate a comparison for the lateral-directional axes at 45 ° angle of attack.
Longitudinal Comparisons Pitch doublets were performed at the three selected flight conditions. The ALFC was used as input to the linear models. The ALFC was measured downstream of the nonlinear elements in the pilot command path. The most noticeable nonlinearity is a 25-deg/sec rate limit imposed by the flight control system on the pilot command. Figures 27 to 29 show the response of the vehicle compared with the response of the linear model to the pitch doublets. For all three cases, the response of the linear models correlates well with the flight-measured responses. For the 45 ° and 60 ° angle-of-attack cases, the linear model required less control surface and thrust-vector deflection to achieve the same vehicle motion (fig. 28(b)). Two potential sources exist for the difference: nonlinearities in the aerodynamics or control system, or a differ- ence between the modeled and actual control effectiveness. For example, the linear model uses a control surface effectiveness based on +_1° deflection from the trim point, and surface deflections of larger mag- nitudes can have a varying effectiveness over the range of deflection.
A comparison between a nonlinear simulation and the flight data for the 45 ° angle-of-attack case shows good correlation, although a bias exists between the flight and simulation trim deflections (fig. 30). This comparison shows that the linearization process caused the differences shown in figure 28.
Further study of the 45" angle-of-attack case shows several reasons for the differences seen in the surface deflections. Figure 31 shows a comparison between the eigenvalues at 40 ° and 45 ° angle of attack. At the high angles of attack, the basic airframe longitudinal characteristics change from an unstable divergence to a nearly neutrally damped oscillation over a small change in angle of attack. To account for these changes in dynamics, the flight control system gains are also a strong function of angle of attack. The nonlinear simulation shows how the angle of attack-to-trailing-edge flap gain, KADEO, and the pitch rate-to-trailing-edge flap gain, KQDEO, vary throughout the maneuver at 45 ° angle of attack (fig. 30).
The shape of the canard trace (fig. 28) is strongly influenced by the forward path command to the canard. Figure 32 shows the canard position commanded by the forward path. The output of the forward path is a function of the delayed angle-of-attack command, ALFCO, which is a nonlinear element. The linear models represent this element by a gain (canard pitch trim, TDECCRU), which is the slope of the curve shown in figure 32. As can be seen in figure 32, the slope between 40 ° and 50 ° angle of attack is approximately 0.0, and between 35 ° and 40 ° angle of attack, the slope is -1.2. The original linear model has a calculated gain of 0.0. Despite these nonlinear characteristics, the linear models provide a reason- able representation of the aircraft response at high angles of attack over the frequency range of interest, 0.3 to 20 rad/sec.
Figures 33 to 38 show the frequency response of the linear models compared to results obtained from fast Fourier transformation of flight-measured data. The responses of pitch rate and angle of attack caused by angle-of-attack command are shown. An unexpected benefit of the rate limiting on the longitu- dinal pilot command path was that better frequency responses were obtained. The sharp comers intro- duced by the rate limiting caused a broader range of frequencies to be excited. The comparisons of the frequency responses show that the linear models produce a reasonable representation of the vehicle closed-loop behavior at all angles of attack.
Lateral-Directional Comparisons Time-domain comparisons were made for roll doublets at the three selected flight conditions. As with the longitudinal axis, the nonlinearities of the stick shaping were avoided by using a measurement of the shapedpilot PKC as input to the linear models. Figures 39 to 41 show the response of the vehicle compared with the response of the linear model to the roll doublets. In general, the time history matches show good correlation with flight-measured responses. The angle-of-sideslip responses do not correlate as well as the other response parameters. The control laws were designed to produce no angle of sideslip during the roll stick input, and the angle-of-sideslip command caused by rudder pedal was reduced to zero at 45 ° angle of attack and greater. As a result, the angle-of-sideslip excitation caused by the pilot inputs is on the same order of magnitude as the angle of sideslip caused by disturbances. As with the longitudinal doublets, the amount of control surface required to achieve the same vehicle response was not well-predicted by the linear models.
Figure 42 shows the frequency response of the linear model at 45 ° angle of attack compared to results obtained from fast Fourier transformation of flight-measured data. This case was the only lateral- directional case that had sufficient time at the target angle of attack to extract a frequency response.
Although extracting a smooth transfer function from the flight data was not possible, the comparison with the linear model shows reasonable agreement.
CONCLUDING REMARKS Linear models of the X-31A aircraft have been presented for six poststall flight conditions. Sufficient descriptions of the flight control system and state space representations of the aerodynamics have been included so that the linear models can be reproduced by the reader. The purpose has been to provide vali- dated aerodynamic and control system models for the unique poststall portion of the flight envelope, using thrust vectoring as an additional control effector.
The poststall flight regime is a very nonlinear environment; however, the results and models presented in this report demonstrate that local linearization techniques can be used and do provide a reasonable representation of the airframe and control system. The successful flight results of the X-31A aircraft demonstrate that the use of linear models for control system design is an appropriate strategy for the high-angle-of-attack regime.
Flight data comparisons with the linear models have been presented for the l-g flight conditions to demonstrate that these models are representative of the flight test vehicle. Comparisons have been made in both the time and frequency domains. In general, the response measurements from flight correlated well with the linear model responses. The surface inputs required to achieve these responses did not cor- relate as well. The differences observed were mostly attributable to the sensitivity of the aircraft dynam- ics and control system gains to changes in angle of attack.
The frequency response correlations for the longitudinal axis show surprisingly good agreement, considering that a tailored input such as a frequency sweep was not used. The lateral-axis frequency response comparison demonstrated that the linear model is a reasonable representation of the actual aircraft in flight.
Dr3,den Flight Research Center National Aeronautics and Space Administration Edwards, California, Januar3, 23, 1997 REFERENCES 1Bosworth, John T., Linearized Aerodynamic and Control Law Models of the X-29A Airplane and Comparison With Flight Data, NASA TM-4356, 1992.
2Flynn, Billy, Rogers E. Smith, and Ed Schneider, "Thrust Vectoring: A New Dimension," Canadian Aeronautics and Space Journal, vol. 41, no. 4, Dec. 1995, pp. 171-178.
3Clark, C. and M. Bernens, "High Angle-of-Attack Flight Characteristics of the YF-22," AIAA 91-3194, Sept. 1991.
4Bursey, R. and R. Dickinson, "Flight Test Results of the F-15 SMTD Thrust Vectoring/Thrust Reversing Exhaust Nozzle," AIAA 90-1906, July 1990.
5Beh, H. and G. Hofinger, "X-3 IA Control Law Design," Technologies for Highly Maneuverable Aircraft, AGARD CP-548, 1994, pp. 13-1-13-9. (Available from DTIC as AD 280 271.)
6Weiss, S., D. Rohlf, and E. Plaetschke, "Parameter Identification for X-31A at High Angles of Attack," Fourth High Alpha Conference, NASA CP-10143, 1994.
7Bosworth, John T. and P. C. Stoliker, The X-31A Quasi-Tailless Flight Test Results, NASA TP-3624, 1996.
8Stoliker, P. C., High-Angle-of-Attack Handling Qualities Predictions and Criteria Evaluation for the X-31A, NASA TM-4758, 1997. (Distribution authorized to U.S. Government agencies and their contrac- tors; other requests shall be referred to WL/FIMS Wright-Paterson AFB, Ohio 45433-6503.)
EC 94 42478-1 Figure 1. X-31A aircraft in poststall flight.
Surface dimensions Wing Canard Vertical Area, ft2 (m 2) 226.3 23.6 37.6 (21.0) (2.2) (3.5) 2.3 3.2 1.2 Aspect ratio Weight, Ibm (kg) Empty 12,000 (5,450) Maximum 16,200 (7,350) 14.6 ft (4.5 m) _1 11.6ft (3.5 m)---_ 96022_
(2:3m) I 43.3 ff
(13.2 m) Figure 2. Three-view drawing of X-31A aircraft.
Figure 3. Arrangement of thrust-vector vanes.
Summation of feedbacks [ [TIME, ALFC] z---:E_.gs I Pilot input Lag filter
_ o.o_z ÷ oI A_
m DECAN DECANC ALFC ALFX + = m_ ÷ ÷ DTESC --_ I+a "-4 I+1 NXS "-'t I÷1 II II .X Summation of feedbacks to TEF ' DEVQi DEVQCL QS Actuators X-31A Feedback dynamics compensation (figs. 9-11 ) (fig. 5) _iixin un Sign change
A--
Feedback filters 970780 (figs. 6-8) Figure 4. Longitudinal control system linear model.
.a b _.2
°'! E
°°
Di °°
<
t
+ I Figure 5. Feedback compensation for the longitudinal axis.
Sum Prefilter 67.23 msec time delay _._ 26046.7 _.] 0.6385z + 0.3615 U
ALFX
IVl s2÷ 1721s ÷---_60,6 71-1 - z--_ /
I _ " " Processed [ angle-of-attack .......
I=1
0.325z 2 + 0.024z + 0.276 ___ feedback z 2 - 0.743z + 0.368 Notch filter Sensor Antialiasing 9.75 msec 970782 lag filter time delay Figure 6. Processing for angle-of-attack feedback.
Pitch-rate lead-lag filter z - 0.58 _ 2.4z- 1,98 _ [_ 27074 s 2 + 225s + 27074 H 0.5125z + 0°4875 L_ 0.255z2 +0.099z + 0.211 z Ivl z 2-0.81z + 0.375 QS 9.75 msec Pitch-rate q Antialiasing Sensed body-axis Body-axis filter time delay notch filter pitch rate pitch rate 970783 Figure 7. Filtering for pitch-rate feedback.
[_ 0.45z + 0.55 _._ Z -nxinu 11 msec NXS Longitudinal time delay Sensed acceleration body-axis at the sensor longitudinal location acceleration 970784 Figure 8. Filtering for axial-acceleration feedback.
Actuator Lead-lag filter dynamics s 2 + 136.2s + 19881 H 19881 ECA_NC z - 0.6798 1.._ 4900 I I s2 + 273S + 4900 D Smoothing filter Actuator dynamics Canard Commanded deflection canard deflection 970785 Figure 9. Filters and actuator models for the canard.
Lead-lag Actuator filter dynamics
H t
z-0.6798 _ 4900 26896 "I 166.7 .I I I 0.3202z s 2 + 273s + 4900 s 2 + 175.8s + 26896 s + 166.7 DTESC ' Smoothing filter Actuator dynamics DTES Commanded Symmetric symmetric trailing-edge trailing-edge deflection deflection 970786 Figure 10. Filter and actuator models for the trailing-edge flaps.
Actuator Lead-lag dynamics filter 34328.7 82.634 _ 1 I s 2 + 262.245s + 34328.7 s + 82.634 ' 0.3202z s 2 + 273s + 4900 EVQ_CL z - 0.6798 _ t 4900 i DEVQ D Smoothing filter Actuator dynamics Pitch Pitch thrust-vector thrust-vector deflection command deflection 970787 Figure 11. Filter and actuator models for pitch thrust vectoring.
Pilot roll-stick input Summation of feedback and feedforward I [TIME' PKCF] r_ [_um_T compensations (fig. 23) Feedback Pi_u"dder-pedal _ I P) ;_J gains DTED P (fig. 21) PDTFB DDEFC input _ m r PS DPE ma DEVR + + X x <o [ II II •x >, I_)
.E,x (
PHIF m Feedback BETC filters -----'--m Actuators X-31A (figs. 13-161 dynamics (figs. 24-26) Feedback calculations Feedforwsrd (fig. 17) gains (fig. 22) 970788 Figure 12. Lateral-directional control system linear model.
3O Sum Prefilter 67.23 msec time delay I
_.J _ L.J 0 _85z ÷0.361, LJ
0.08 L__I-'_
_ BETX J_ I--I s2+ 172.1S +-'_---26046.7 I--I -- z'---_ / / - Processed .ng,reof [ " ........
angle of sideslip [ ._ 0.325z 2 + 0.024z + 0.276 ] I sideslip
,0 ]_ 27074 _ 0.51._z±0.4675
z 2 - 0.743z + 0.368
'' -_-I s2÷22.÷270741-1 z÷o I-I V
Notch filter 970789 Sensor lag Antialiasing filter 9.75 msec time delay Figure 13. Processing for angle-of-sideslip feedback.
Roll gyro 25 Hz notch filter 8.9 msec time delay PS s 2 + 225s + 27074 H 0"4513s2+15"265+12590H 0"555z+O'445_S 2 + 99.57S + 12500 z Sensed Body-axis Antialiasing filter body-axis roll rate roll rate q 0.208Z + 0.064z + 0.158 I ,=.-I 2.3z- 1.52 1.0Z 2 - 1.01Z + 0,44 _] Z - 0.22 Roll-rate notch filter Phase advance filter 970790 Figure 14. Filters for roll-rate feedback.
8.9 msec Phase advance filter time delay 0.555z + 0.445 Z s 2 + 225s + 27074 r_l 27074 H I 1.0z 2 - 0.738z + 0.353 z - 0.3 Yaw-rate notch filter RS Yaw gyro Body-axis Sensed yaw rate body-axis yaw rate 970791 Figure 15. Filters for yaw-rate feedback.
z 4 _ 0.6385z + 0.3615 PHIF 67.23 msec Degrees-to-radians Bank time delay conversion Filtered angle bank angle 970792 Figure 16. Filters for bank-angle feedback.
D
PKC Stability-axis Stability-axis transformation I Sum Roll-rate roll-rate command (fig. 18) ] error P, PS -_ -_-_kS Sensed body-axis I I- roll rate V RST
D -
RS Sum Yaw-rate Sensed error body-axis yaw rate _ - I F DBET Sum BETX Angle-of-sideslip error Processed angle of sideslip
_D
PDT v Derived roll u acceleration PHI Filtered bank tgle
_D
m RDT Gravity Derived yaw compensation acceleration (fig. 20) BETC [ Commanded I angle of s des p Stability-axis yaw command estimation (fig. 19) 970793 Figure 17. Feedback calculations.
RS Sensed body-axis ID_ yaw rate RSTAB Stability-axis yaw rate 970794 Figure 18. Stability-axis transformation for lateral-directional feedbacks.
m + a Commanded stability-axis v + yaw rate PHIF Sum Filtered bank 970795 angle Figure 19. Calculation of stability-axis yaw-rate command.
Differentiation Derived yaw acceleration 970796 Figure 20. Linear model of gravity compensation.
÷
DAFB DPE + Summation of feedback Stability-axis w!
roll-rate Sum compensation to differential error TEF DRE DRFB Stability-axis + Summation of feedback yaw-rate error Sum compensation to rudder l ---I_ + I i
D
v + DBET Sideslip TVFB error Summation =,,.-- + of feedback Sum compensation to yew thrust vectoring 970797 Figure 21. Lateral-directional axes feedback gain compensation.
D
PKCDVK Roll-rate + command normalized !
by velocity b..-= : .t.
v _1_+ PDTFB PDT Feed forward compensation Derived roll --I_+ for the acceleration lateral axis Sum
D _1_+
RDT Derived yaw _ + acceleration m
---D
--_i.
RDTFB Feedforward compensation
D
for the BETC directional axis Commanded Sum angle of sideslip 970798 Figure 22. Lateral-directional axes forward-path compensation.
@
PDTFB Feedforwsrd compensation for the lateral axis C ded
@
tial DAFB on Summation of feedback compensation to differential TEF
½
DRFB Summation
:D
of feedback compensation Sum Sum DERUDC to rudder Commanded rudder deflection
D
TVFB Summation of feedback DEVRCL compensation Yaw Sum to yaw thrust thrust-vector deflection command
vec l
RDTFB _FKAP_ Feedforwsrd compensation for the directional axis 970799 Figure 23. Summation of feedback and feedforward compensations.
Lead-lag filter Actuator dynamics 0.3202z s 2 + 273s + 4900 s + 166.7 s 2 + 175.8s + 26896 DEF_C z-0.6798 H 4900 H 166.7 H 26896 D_TE D D Smoothing filter Actuator Commanded dynamics Differential differential trailing-edge TEF deflection deflection 970800 Figure 24. Filters and actuator models for the differential trailing-edge flaps.
Lead-lag filter Actuator dynamics 0.3202z s 2 + 273s + 4900 I I s + 58.8 I ] s2 + 126.588s + 23716 D Smoothing filter Actuator Commanded dynamics Rudder rudder deflection deflection 970801 Figure 25. Filters and actuator models for the rudder.
Actuator dynamics Lead-lag filter z - 0.6798 _ 4900 _ 34328"7 _ 82"634 _DEV R s 2 + 273s + 4900 s 2 + 262.245s + 34328,7 s + 82.634 0.3202z Actuator dynamics Smoothing filter Yaw Yaw thrust-vector thrust-vector deflection deflection 970802 command Figure 26. Filters and actuator models for yaw thrust vectoring.
-- Flight data ----- Linear simulation r a" .......... J .... s.............. ' .................. _ .................
3O _,_ i s ", ...... _ .................. _ .... __ _:. .:_ _. --'.
Angle of ...... : .......... _.,_ ........ :.................. ' .................
attack, • J deg a, t I * q • _ st i 2O .......... _---ff--1 .....................................................
"'"" _-iAngle-of-attack commar d r [ .................. i ..... s r : ' -- Pitch rate, . $ . .
degJsec : t : : -5 -10 Pitch attitude, deg 1.00 .95 Normal .90 acceleration, g .85 .80 .75 0 1 2 3 4 Time, sec 970803 (a) Comparison of linear simulation response with flight data for a pitch doublet at 30 ° angle of attack.
Figure 27. Comparison between flight and simulation data at 30 ° angle of attack.
-15 _'_'_ _ Flight data ///ii /- _ Linear simulation - 2O Canard deflection, ..............//a I/ ............ _ -- - -'_ i,:................................
deg - 25 ............ I- I----- ......... _'_-_.'_-_L ............... : .................
- 30
Z' i i
TEF command, deg -5 -10
/" ! i i
...... l, z - _ -: .................. : .............. _ .........
Pitch thrust-vector command,
_,_ -7-- i
deg 1 2 3 4 Time, sec 970804 (b) Comparison of linear simulation response with flight data for a pitch doublet at 30 ° angle of attack.
Figure 27. Concluded.
6O i : o°_ • , _ Flight data ', ' ," ----- Linear simulation 5O Angle of 4O attack, deg
".,i ,," i !"., ,,'" ""
3O ......... -"-": -,'-'!!.-_Angle.of.ettack comman_di ............ : - - - :'-':'-" ................. :_ ............
, i d_'% p , , ............ : ............ :---//-_-- : ........... : ............ :...........
............ i ............ ;-/ ................... : ............ ; ..... _ ....
' ;S
Pitch rate, _ __ .__tl ............ ,_\\ _ . , _ /7 __ deg/sec -_.,_ ,,, , /I , : \% , .,. /i_ -'.,,q.._,_ , -5 ........ _._ ..... _ ........... : ---_c...... _ ........ :- ........
, \ I.
-10 -:- , , - _ ............ : ............
i i i i -15 4O ..................................... >*_- --'- "_"_. ................. i............
........... \_ ......... ,..... /---_ ...... _.-::: ...... :___ ::;;_ ....
Pitch attitude, deg 3O .80 ; ::11, "" "--. _/-'---.
.75 Normal .70 acceleration, g .65 .60 0 2 3 4 5 6 Time, sec 970805 (a) Comparison of linear simulation response with flight data for a pitch doublet at 45 ° angle of attack.
Figure 28. Comparison between flight and simulation data at 45 ° angle of attack.
- 30 i JJ'_ : __, Flightdata - 35 Canard deflection, - 4O deg - 45 "... : I : : I I ' , _.
- 50
TEF X _ ,f _,,,# X', ,q
deflection,
............ -_ ....:_/! ....... , ........ ,%;,/£;/ ..... \-,p
deg -5 -10 -f\ __ ,/t, Pitch thrust-vector ............ ,___\ ........ ,_..... _ ........... ,_\- ..... //_ _ _,, :.,, ..... fl - command, deg
\.;:,,4 %,7
-5 -10 0 1 2 3 4 5 6 Time, sec 970806 (b) Comparison of linear simulation response with flight data for a pitch doublet at 45 ° angle of attack.
Figure 28. Concluded.
i , ! . _ t i ...... , ..... :--- _'_- _ --: ...... _-_'---_ ', .' -- - - '-,:- 2:-- ".-" Angle of attack, 50 ........ • - - -i ...... • ............ _ .... •_s r 1 deg • • , sw i q ............ + ............ ,........... , ........... 'r........... i...........
Angle-of-attack command s S i 4O .......... ' ............. ' ........... ' ........... _ .......... #1 [g-h-td-ata ....
...... Linear simulatio 5O , i , q ............ : ............ :__ __ ........... _........... : ............
: i/ i :
_._ ............. _
Pitch rate, , t , : : deg/sec -5 .......... i........ _--!........... : .......... _........... : ........
-10 I + L I -15 r ' 3O I -- i ! i .......... _ .......... : ........... _-_._:-_- : _...-J- ............
Pitch attitude, deg ............. _;, ....i___/:___ _ ....................
1.10 1.05 ............ i............ i ........... ! ........... '- ........... i.......
1.00 ; /I "_-\ _,_ -_'_ Normal / , \ ', acceleration, .95 ...... '- - -/- ........ ', ......
g .90 .85 + - - i .80 0 2 4 6 Time, sec 970807 (a) Comparison of linear simulation response with flight data for a pitch doublet at 60 ° angle of attack.
Figure 29. Comparison between flight and simulation data at 60 ° angle of attack.
- 25 Flight data ------ Linear simulation J - 3O - 35 Canard deflection, ............ :'--i -_ ......... i...................... : .........
deg - 4O - 45 - 5O TEF deflection, deg --5 r- -10 10 ' ' Pitch : .t , A /_
deg , _', : //"/ _' W- "_
-10 0 1 2 3 4 5 6 Time, sec 97o8o8 (b) Comparison of linear simulation response with flight data for a pitch doublet at 60 ° angle of attack.
Figure 29. Concluded.
60 ' ' "'" _ Flight data . ,_" ,, ', -- -- -- Nonlinear simulation 50 ..................... , .................................... ; .......
of Angle attack, 40 ,.- ........
deg 2O - 25 , , , - 30 ................ /- ............. ' ........... ' ....................
deflection, Canard deg _--454035 _iii _ ........... "- ! ...... " ........ "_ ...... _" ; ......
- 50 10 , , , TEF 0 - - - _ - - - deflection, "_-- ; \ ; / deg 5 i .... \ .... - -/................ -'3,, -j-%,L ....... ",.-r .....
\ I -10 ................... T .........................................
-15 ' ' ' 1.0 .,% .,,,.- _ / : :, ..... !.......... !-/-" ...... i__ _':__--_.
.9 .......... :--,. ...... :--- , , KADEO, , / "l I ;/ deg/deg z , I. , t" .8 .......... .r ................ ii -_ ._-- _._ _- J ......................
,.F b-, q .,.i , q .7 I .9 ....
.8 .......... : .......... ; .......... :..................... '_ .........
KQDEO, ---- ' _ deg/(deg/sec) ,'"'_ _- _'_ _'_ .7 ....... _-,,.':"_-_ ....... : ..... _-', - - -: ........ 7_ ...................
.... : ' _-.,..L__ ._. _ , , .6 0 1 2 3 4 5 6 Time, sec 970809 Figure 30. Comparison of nonlinear simulation response with flight data for a pitch doublet at 45 ° angle of attack.
Angle .5 ] 1 I ¸ L I I 1 of attack C) 40 ° ,4 -- [] 45 ° .3 [] .2 .1 Imaginary [] 0 0 axis, 0 rad/sec --,I -- -- ,2 -- [] --,3 --.4
I
L L I
--,5 .8 .0 - .8 -'.6 -'.4 - .2 0 .2 .4 .6 - 1.0 Real axis, rad/sec 9708_0 Figure 31. Comparison of bare airframe longitudinal axis roots at 40 ° and 45 ° angle of attack.
-15 --- Canard ............. ' ...... - ............ , ....... i ....
position - 20 command, deg i .... : .....
-25 - 30 - 35
t
- 40 - 45 0 10 20 30 40 50 60 70 -10 Delayed angle-of-attack command (ALFC0), deg 9708_ Figure 32. Feedforward gain from angle of attack to canard.
........... Flight data d ___'.__ ', : : ; ', : : ...... Linear simulation, - 2O Gain, dB - 4O _-_- _ , , , , , , , , , , , - 100 "- .- - -:- -, - -:- -i- J-'. ......... _---7---;'_--_- _-_-_-: ...............
, i , i .... _, .......
Phase >., :, angle, - 200 deg
............... ,, r\
- 3OO : i : : [ : : : ; ; i : i - 400 1 10 Frequency, rad/sec 970812 Figure 33. Frequency response comparison between linear simulation and flight data for ALFX/ALFC at 30 ° angle of attack.
Flight data -5 Gain, dB -10 i i , , b , , , ' ' ' ' __ _,_ -15 , , , , , b , ....
LI
- 2O
II Ill
5O Phase - 5O angle, deg - 100 - 150 , , , , , , , , i , i , \ L , , i , \ , , , , , , i , , \ , , , _ , , - 2OO 1 10 Frequency, rad/sec 970813 Figure 34. Frequency response comparison between linear simulation and flight data for Q/ALFC at 30 ° angle of attack.
Gain, dB -IO0 Phase angle, _ 200 deg - 30O - 400 1 10 Frequency, rad/sec 970814 Figure 35. Frequency response comparison between linear simulation and flight data for ALFX/ALFC at 45 ° angle of attack.
-- Flight data -10 Gain, dB - 2O i , i 4 , .... , _ , , , , , , --30 ...... _---' -'--'--_'_ ........................ _--_--' _-_-_ ...........
, , , , , , , , , , , , , , , , I J I J * L J I h L _ I I + h I
'°°1 ' i ,ii,, I
I
Phase angle, deg - 100 ,\ , \ \ \ - 2OO \, 1 10 Frequency, rad/sec 970815 Figure 36. Frequency response comparison between linear simulation and flight data for Q/ALFC at 45 ° angle of attack.
Flight data i i t , , b , b i i , 0 - - - _ ..... '_- -- '_- - _'- - '_- _ .... Linear simulation :- - 2O Gain, dB - 4O _..: :____ : i _............................................ ,,,i _ ....... _i i _ i i i i , i , i , , _ , i , i , i i l _ , _ , , t
t!
- 100 - 400 1 10 Frequency, rad/sec 970816 Figure 37. Frequency response comparison between linear simulation and flight data for ALFX/ALFC at 60 ° angle of attack.
5O .... Flight data ' --- Linear simulation l
i!iiii
-I0 Gain, dB - 2O - 3O Phase ......
angle, ' ......
deg _100 ...... i__.' _i__',_'_i ..............................
! ,: !!!i ,: ,: : ,' i l %/_
i : : : :: : : : :'' ' ' ' ' ' ' _t -200 ...... ;-- "- -i- -i- _ -' ........................ i- - _- -' - "- _ -: ............ \ .... , , , , , , , , , 1 10 Frequency, rad/sec 970817 Figure 38. Frequency response comparison between linear simulation and flight data for QIALFC at 60 ° angle of attack.
Flight data ----- Linear simulation ; /.-,. i--,\ i ............. _ .............. :-_:\--'_.... ,_ ....... /--k_-: .............
Angle of ' II! \ x ' /,' \l, r"" sideslip, : // \ _ : ,# \_. , deg : J -2 ............. _ ................................. ._-_-:'_ __ _ -: .............
2O Roll rate,
__ ....=___--:_.. ,_, // \,---.._
deg/sec -- ' ' _ I ............. _i ................. .... .......... .......
- 20 - 4O 2O Yaw rate, deg/sec -10 - 2O - 30 - 40 - 50 Bank - 60 angle, deg - 70 \ / \ / ............................................. _\_/ ........ ,.............
- 80 - 90 0 2 4 6 8 10 Time, sec 970818 (a) Comparison of linear response with flight data for a roll doublet at 30 ° angle of attack.
Figure 39. Comparison between flight and simulation data at 30 ° angle of attack.
.... ' Linear simulation i ............ i .............. * ..............
, - ........ , ....... _. 7-_ _. ..............
Lateral acceleration, g
_.i !
.Jff'v"_ \x, _.Jl "_\_ Differential 0 ____:>,S ......S;--:--'_, ........
TEF deflection, deg -1( 2O Rudder deflection, 0 deg -10 - 20 ...... ..... _- ............... Yaw thrust-vector command, ____ __-_-__.__ .... ._'___./_/5...... ....' _, __ .Z_ / .................
deg - 2 , , , _ L- -4 ............................. i ...................................
-6 0 2 4 6 8 10 Time, sec 97oe _ (b) Comparison of linear simulation response with flight data for a roll doublet at 30 ° angle of attack.
Figure 39. Continued.
Flight data 4O Stability-axis 20 roll-rate command, _ _--_--_ L!nea.r_simu!a_t!on_ ......... i .............. i-- .....................
deg - 20 - 40 1.0 .5 Rudder pedal command -.5 - 1.0 i 0 2 4 10 Time, sec 970820 (c) Comparison of linear simulation response with flight data for a roll doublet at 30 ° angle of attack.
Figure 39. Concluded.
; ; , -- Flight data , . _. .
------ Linear simulation : : /" \ ', .............. r .............. r .............. r ......... -_ ----_- t ..............
,, ,,," \ : .__ :_ ...... I.,: .... _'_ ..............
:_./ \: ..--,, Ii \, X Angle of sideslip, deg
" 7....... \---:/
.............. L .............. L_ - - ..........
-1 -2 2O r, : ; /., i ............. ' .............. ' ............... _ ....... ,, .............
.... - _ xx : : Roll rate, deg/sec ............. '- ............. r ........................... r ..........
-10 - 20 - 30 20 r . ' I I 10 ..... _ ..................... ; ............ ; .................
Yaw rate, deg/sec ................
_,o} i i 'J : Bank angle, .............. '_.............. ;- ....... ................. "%'\ i' ............. / deg -10
.............. _.............. _.............. !..... ;"'":- --!.......... I
-20 0 2 4 6 8 10 Time, sec 970821 (a) Comparison of linear simulation response with flight data for a roll doublet at 45 ° angle of attack.
Figure 40. Comparison between flight and simulation data at 45 ° angle of attack.
"41 _ Flight data' , i II
.2 ------ Linear simulation , Lateral 0 acceleration, g --,2 -,4 q Differential TEF ........... _ ....... / ..........
deflection, ____ _ ... I _i I I __ _ _ I / deg -5 r -- - .......................... .............. ...............
-10 -15 Rudder deflection, 0 deg -5 -10 -15 10 ; 5 .............. _ .............. _ ..................................
Yaw 0 .... ' ............
thrust-vector commend, deg : ', i ". i -10 -15 0 2 4 6 8 10 Time, sec 970822 (b) Comparison of linear simulation response with flight data for a roll doublet at 45 ° angle of attack.
Figure 40. Continued.
4O Flight data ----- Linear simulation 2O Stability-axis roll-rate command, deg - 2O - 40 1.0 Rudder pedal command L I I I 2 4 6 8 10 Time, sec 970823 (c) Comparison of linear simulation response with flight data for a roll doublet at 45 ° angle of attack.
Figure 40. Concluded.
-- Flight data ' , ----- Linear simulation / '_"--""x Angle of sideslip, -1 deg \ ' / .............................. -_\_ ............ _..._ ____ -2 -3 10 ........... , ............ , ............ , ....... ._-___J ....... _._ _ _ _ ...........
Roll rate, deg/sec ..................... , ....... _1 _,_..
;iiii iii
-5 -10 2O Yaw rate, I0 _-."-'_" .................................................... i i deg/sec 0 _ ', "',_,. , ._j- " , : -10 ............ "- ......... :.... .._i_ _ _ _:............ , ........... - ...........
- 20 0 ....
-10 ............ _ ..................................... _ ........................
Bank ----_-.-.._ : ', '. ._" "" .......
........... _...... :............ :............ , ...... p<__ ___: ............
angle, -20 -- deg ' " ' ' _ / ' - 30 , - .......... S,._ ....... , ........... ,_ .......................
_ 40 0 1 2 3 4 5 6 Time, sec 970824 (a) Comparison of linear simulation response with flight data for a roll doublet at 60 ° angle of attack.
Figure 41. Comparison between flight and simulation data at 60 ° angle of attack.
.15 Flight data ' ----- Linear simulation , . , .10 .05 Lateral acceleration, 0 g - .05 - .10 k i , h , - .15 Differential TEF deflection, deg -10 -15 -1,8 - 1.9 Rudder deflection, deg -2.0 -2.1 Yaw thrust-vector command, deg -5 -10 0 1 2 3 4 5 Time, sec 970825 (b) Comparison of linear simulation response with flight data for a roll doublet at 60 ° angle of attack.
Figure 41. Continued.
2O -7__--y_n_.r_'_u_'__t_i_on ..... !............ ! / \ ....... i............
Stability-axis roll-rate command, deg
!!!! !!! !i!iiiiiii
-10 - 20 .016 .014 ........... , ........................ , ............ r ............ ...........
.012 Rudder pedal .010 command .008 .006 .004
o 1 2 3 4 S 6
Time, sec 970826 (c) Comparison of linear simulation response with flight data for a roll doublet at 60 ° angle of attack.
Figure 41. Concluded.
6O i ....... Flight data -5 Gain, dB -10 -15 , , , , , , , = , i , , , _ , , , , , , , , , , , , , , , 0 .......
;i, _
-so : !!i!!: ....... _..... _ i i !! _!i ........... i
\ .......... i\\ , ,
phase i : !:_ !ii i i ! i i i i_ i
angle,deg -100 ! --i- i -!- i -!i i ! i ! i _ .......... :i ..... i! --!--i ii i: ! ! _-_x_ i ! ! ! \\\\ ....... : _so --'---: -_--'--:-"-: .......... ' ...... --- _---:--"--:-;-"-: ........ \i'i .... : ! :\\\\\\ '/ - 200 ......
1 10 Frequency, rad/sec 970827 Figure 42. Frequency response comparison between linear simulation and flight data for PSTAB/PKC at 45 ° angle of attack.
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1, AGENCY USE ONLY (Leave blank) 2. REPORT DATE 3. REPORTTYPE AND DATES COVERED December 1997 Technical Memorandum 4. TITLE AND SUBTITLE 5. FUNDING NUMBERS Linearized Poststall Aerodynamic and Control Law Models of the X-31A Aircraft and Comparison with Flight Data WU 529-30-04 6. AUTHOR(S) Patrick C. Stoliker, John T. Bosworth, and Jennifer Georgie 8. PERFORMING ORGANIZATION 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) REPORT NUMBER NASA Dryden Flight Research Center P.O. Box 273 H-2194 Edwards, California 93523-0273 10. SPONSORING/MONITORING 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) AGENCY REPORT NUMBER National Aeronautics and Space Administration NAS A/TM-97-206318 Washington, DC 20546-0001 11. SUPPLEMENTARY NOTES 12a. DISTRIBUTION/AVAILABILITY STATEMENT 12b. DISTRIBUTION CODE Unclassified--Unlimited Subject Category 08 13. ABSTRACT (Maximum 200 words) The X-31A aircraft has a unique configuration that uses thrust-vector vanes and aerodynamic control effectors to provide an operating envelope to a maximum 70 ° angle of attack, an inherently nonlinear portion of the flight envelope. This report presents linearized versions of the X-31A longitudinal and lateral-directional control systems, with aerodynamic models sufficient to evaluate characteristics in the poststall envelope at 30 °, 45 °, and 60 ° angle of attack. The models are presented with detail sufficient to allow the reader to reproduce the linear results or perform independent control studies. Comparisons between the responses of the linear models and flight data are presented in the time and frequency domains to demonstrate the strengths and weaknesses of the ability to predict high-angle-of-attack flight dynamics using linear models. The X-31A six-degree-of-freedom simulation contains a program that calculates linear perturbation models throughout the X-31A flight envelope.
The models include aerodynamics and flight control system dynamics that are used for stability, controllability, and handling qualities analysis. The models presented in this report demonstrate the ability to provide reasonable linear representations in the poststall flight regime.
14. SUBJECTTERMS 15. NUMBER OF PAGES Aircraft flight control system, Linear aerodynamics model, Rigid-body dynamic 16. PRICE CODE model, State space, X-31A airplane A04 17. SECURITY CLASSIFICATION 18. SECURITY CLASSIFICATION 19. SECURITY CLASSIFICATION 20. LIMITATION OF ABSTRACT OF REPORT OFTHIS PAGE OF ABSTRACT Unclassified Unclassified Unclassified Unlimited NSN 7540-01-280-5500 Available from the NASA Center for AeroSpace Information, 800 Elkridge Landing Road, Standard Form 298 (Rev. 2-89) Prescribed by ANSI Std Z39-18 Linthicum Heights, MD 21090; (301)621-0390 298-102