design and calibration of the x-33 flush airdata sensing ... · for 0.2 ≤ m ≤ 4.0. 5. dynamic...

NASA/TM-1998-206540

Design and Calibration of the X-33 Flush Airdata Sensing (FADS) System

Stephen A. Whitmore, Brent R. Cobleigh, and Edward A. HaeringDryden Flight Research CenterEdwards, California

January 1998

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NASA/TM-1998-206540

Design and Calibration of the X-33 Flush Airdata Sensing (FADS) System

Stephen A. Whitmore, Brent R. Cobleigh, and Edward A. HaeringDryden Flight Research CenterEdwards, California

January 1998

National Aeronautics andSpace Administration

Dryden Flight Research CenterEdwards, California 93523-0273

NOTICE

Use of trade names or names of manufacturers in this document does not constitute an officialendorsement of such products or manufacturers, either expressed or implied, by the NationalAeronautics and Space Administration.

Available from:

NASA Center for AeroSpace Information National Technical Information Service800 Elkridge Landing Road 5285 Port Royal RoadLinthicum Heights, MD 21090-2934 Springfield, VA 22161Price Code: A16 Price Code: A16

DESIGN AND CALIBRATION OF THE X-33 FLUSH AIRDATASENSING (FADS) SYSTEM

Stephen A. Whitmore,* Brent R. Cobleigh,† andEdward A. Haering, Jr.‡

NASA Dryden Flight Research CenterEdwards, California

*

Abstract

This paper presents the design of the X-33 FlushAirdata Sensing (FADS) system. The X-33 FADS uses amatrix of pressure orifices on the vehicle nose toestimate airdata parameters. The system is designed withdual-redundant measurement hardware, which producestwo independent measurement paths. Airdata parametersthat correspond to the measurement path with theminimum fit error are selected as the output values. Thismethod enables a single sensor failure to occur withminimal degrading of the system performance. Thepaper shows the X-33 FADS architecture, derives theestimating algorithms, and demonstrates a mathematicalanalysis of the FADS system stability. Preliminaryaerodynamic calibrations are also presented here. Thecalibration parameters, the position error coefficient ( ),and flow correction terms for the angle of attack ( ),and angle of sideslip ( ) are derived from wind tunneldata. Statistical accuracy of the calibration is evaluatedby comparing the wind tunnel reference conditions to theairdata parameters estimated. This comparison isaccomplished by applying the calibrated FADSalgorithm to the sensed wind tunnel pressures. When theresulting accuracy estimates are compared to accuracyrequirements for the X-33 airdata, the FADS systemmeets these requirements.

Nomenclature

a flush airdata sensing geometry coefficient

A angle of attack triples algorithm coefficient

εδα

δβ

Vehicle Aerodynamics Group Leader, Aerodynamics Branch.†Aerospace Engineer, Aerodynamics Branch.‡Aerospace Engineer, Aerodynamics Branch.Copyright 1998 by the American Institute of Aeronautics and

Astronautics, Inc. No copyright is asserted in the United States underTitle 17, U.S. Code. The U.S. Government has a royalty-free license toexercise all rights under the copyright claimed herein for Governmentalpurposes. All other rights are reserved by the copyright owner.

1American Institute of Aero

A/D analog-to-digital conversion

first angle of sideslip triples algorithm coefficient

A0(M ) zero’th order angle of attack calibration coefficient

A1(M ) first order angle of attack calibration coefficient

A2(M ) second order angle of attack calibration coefficient

A3(M ) third order angle of attack calibration coefficient

b flush airdata sensing geometry dummy variable

B second angle of sideslip triples algorithm coefficient

first angle of sideslip triples algorithm dummy variable

B0(M ) zero’th order angle of sideslip calibration coefficient

B1(M ) first order angle of sideslip calibration coefficient

B2(M ) second order angle of sideslip calibration coefficient

B3(M ) third order angle of sideslip calibration coefficient

CTE coefficient of thermal expansion

second angle of sideslip triples algorithm dummy variable

DFRC Dryden Flight Research Center

ESP electronically scanned pressure module

f [M ]subsonic subsonic Mach number function

A'

∞

∞

∞

∞

B'

∞

∞

∞

∞

C'

∞

nautics and Astronautics

f [M ]supersonic supersonic Mach number function

FADS flush airdata sensing

FailOp fail-operational (airplane operates as if there were no failures)

M estimation algorithm geometry matrix

MAAF Michael Army Air Field, Tooele, Utah

MAFB Malmstrom Air Force Base, Great Falls, Montana

MSE mean square error

M free stream Mach number

NASA National Aeronautics and Space Administration

p port pressure, lb/ft2

P free stream static pressure, lb/ft2

q weights on measured pressures

qc impact pressure, lb/ft2

Q pressure weighting matrix

RCC reinforced carbon-carbon

RLV Reusable Launch Vehicle

RMS root mean square

TPS thermal protection system

UPWT Unitary Plan Wind Tunnel, Langley Research Center, Hampton, Virginia

x state vector

X2 mean square fit error, (lb/ft2)2

angle of attack, generic, deg

e local angle of attack sensed by FADS, deg

true free stream angle of attack, deg

angle of sideslip, generic, deg

e local angle of sideslip sensed by FADS, deg

true free stream angle of sideslip, deg

1 quadratic solution 1 for angle of sideslip, deg

2 quadratic solution 2 for angle of sideslip, deg

angle of attack correction term, deg

angle of sideslip correction term, deg

pressure difference for triples algorithm, lb/ft2

ratio of specific heats

position error calibration parameter

M variation of with Mach number at

e, e = 0°

first order coefficient, fit of with angle of attack

second order coefficient, fit of with angle of attack

first order coefficient, fit of with angle of sideslip

second order coefficient, fit of with angle of sideslip

cone angle of FADS port, deg

clock angle of FADS port, deg

local flow incidence angle, deg

standard deviation

dummy variable used in angle of sideslip root analysis

chi-squared distribution

FADS geometry term,

Superscripts, Subscripts, and Mathematical Operators

i port index

j port index

(j) iteration index

k port index

^ FADS estimate

Det[.] matrix determinant

[.]-1 matrix inverse

[.]T matrix transpose

[.]x gradient with respect to vector x

partial derivative with respect to x

Introduction

The primary goal of the Single-Stage-to-OrbitTechnology program is to radically reduce the cost ofaccess to space, and the X-33 advanced technology

∞

∞

∞

α

α

α

β

β

β

β

β

δα

δβ

Γ

γ

ε

ε εα β

εα1 ε

εα2 ε

εβ1 ε

εβ2 ε

λ

φ

θ

σ

Ψ

χ2

Ω θ ε θ2sin+

2cos

∇∂ . ][∂x

----------

2American Institute of Aeronautics and Astronautics

demonstrator is the centerpiece of this effort. The X-33design is a 53-percent scale model of the LockheedMartin VentureStar * Reusable Launch Vehicle (RLV)and is based on a lifting-body design which features twolinear aerospike rocket engines. The autonomous-flightX-33 will launch from Edwards Air Force Base atEdwards, California and land at one of two sites:Michael Army Air Field (MAAF) in Tooele, Utah, orMalmstrom Air Force Base (MAFB) in Great Falls,Montana. The vehicle is designed to achieve a peakaltitude near 300,000 ft and speeds of greater than Mach12. After atmospheric re-entry, the X-33 returns to Earthwith an unpowered horizontal landing. A comparison ofthe X-33 and VentureStar vehicles is presented infigure 1.

Because the X-33 is to perform an unpowered landing,knowledge of the dynamic pressure, angle of attack, andsurface winds is critical, so that the terminal area energymanagement (TAEM) can ensure that the target runwaybe reached under a wide variety of atmosphericconditions. Direct feedback of angle of attack and angleof sideslip are also required for gust load alleviation onthe vehicle airframe during the ascent of the flight. Thus,it was determined early in the X-33 program that the fullairdata state, including Mach number, angle of attack,angle of sideslip, dynamic pressure, airspeed andaltitude, would be a flight-critical requirement for boththe RLV and the X-33.

To achieve the TAEM and ascent airdata requirementsthe airdata system must be operational betweenMach 0.20 and Mach 4.0 and meet the following 1-accuracy requirements.

1. Mach Number: ± 5.0 percent accuracy for2.5 ≤ M ≤ 4.0, ± 2.50 percent accuracy for0.6 ≤ M < 2.5, ± 0.015 absolute error betweenM = 0.20 and M = 0.60,

2. Angle of attack: ± 1.5° absolute accuracy for firstthree flights, ± 0.50° thereafter,

3. Angle of sideslip: ± 0.5° absolute accuracy for allflights.

4. Geopotential altitude: ± 200 ft absolute accuracyfor 0.2 ≤ M ≤ 4.0.

5. Dynamic Pressure: ± 15 lb/ft2 for 0.2 ≤ M ≤ 4.0.

To determine the best means of meeting the airdatarequirements, NASA Dryden Flight Research Center(DFRC) performed a feasibility study to compare the

*Lockheed Martin, Inc., Mountain View, California

performance and cost of a Flush Airdata Sensing (FADS)system to a set of deployable probes similar to thatsystem installed on the space shuttle. This studyconcluded that a FADS system was more economicalby a factor of approximately two. Two issuesmade the probe-based system prohibitively expensive:1) integration onto the X-33 airframe, and 2) systemcalibration. The FADS system requires no deploymentmechanisms and can be integrated directly onto thevehicle nosecap with no movable parts. Because theFADS system does not probe the flowfield, but insteaduses the natural contours of the forebody, the flow field ismuch cleaner and is easier to calibrate. An additionaladvantage of the FADS system is that it offers thepotential to sense airdata on ascent, an option notavailable to the probe-based system. Based on the resultsof this study, Lockheed Martin Skunkworks, Palmdale,California, selected the FADS system in favor of thedeployable probes.

The DFRC FADS design builds on work whichoriginated in the early 1960's with the X-15 program,1

continued at NASA Langley,2, 3 and Dryden FlightResearch Centers4, 5 in the 1970's and 1980's, andrecently concluded flight testing of an onboard real-timesystem in the early 1990's.6, 7 The FADS concept, inwhich airdata are inferred from nonintrusive surfacepressure measurements, does not require probing of thelocal flow field to compute airdata parameters. Thisinnovation allows the extreme hypersonic heatingcaused by the small radius of a flow-sensing probe to beavoided, which extends the useful range of the airdatameasurement system to the hypersonic flow regime. Thispaper describes the X-33 FADS system design, developsthe aerodynamic model which relates the airdataparameters to the measured pressures, and derives thecomputational algorithms used to compute the airdatafrom the pressure measurements. Wind tunnelcalibration and validation of the FADS system is alsopresented.

Flush Airdata Sensing Aerodynamic Model

The fundamental concept of the FADS system is thatairdata parameters can be estimated from flush surfacepressure measurements. To perform this estimation, theairdata states must be related to the surface pressures byan aerodynamic model that captures the salient featuresof the flow, and is valid over a large Mach number range.To be useful, the model must be simple enough to beinverted in real-time so that the airdata parameters can beextracted. To solve the problem of describing a complexflow scenario with a simple model, the FADS

σ

∞∞

∞ ∞

∞

∞


aerodynamic model was derived as a splice of the closed-form potential flow solution for a blunt body,8 applicableat low subsonic speeds; and the modified Newtonianflow model,9 applicable at hypersonic speeds. Bothpotential flow and modified Newtonian flow describe themeasured pressure coefficient in terms of the localsurface incidence angle. To blend the two solutions overa large range of Mach numbers, a calibration parameter( ) was allowed for. This parameter must be empiricallycalibrated to allow for the effects of flow compression,body shape, and other systematic effects such as shockwave compression or Prandtl-Meyer expansion on theforebody. The resulting model is6

(1)

In equation 1, is the flow incidence angle betweenthe surface normal at the i’th port and the velocity vector.The incidence angle is related to the local (or effective)angle of attack, ( ) and angle of sideslip, ( ) by6

(2)

In equation 2, the cone angle ( ) is the total angle thenormal to the surface makes with respect to thelongitudinal axis of the nosecap. The clock angle ( ) isthe clockwise angle looking aft around the axis ofsymmetry starting at the bottom of the fuselage. Thesecoordinate angles are depicted in figure 2. The remainingparameters in equation 1 are impact pressure (qc) and thefree stream static pressure ( ). Using these four basicparameters ( , , qc, and ) most other airdataquantities of interest may be directly calculated.

In addition to the calibration for , the local flowincidence angles ( and ) must be related bycalibrations to the true free stream flow angles. Theseflow-angle calibrations account for such additionalsystematic effects as bow shock flow deflection andbody-induced upwash and sidewash. For the X-33 apreliminary set of calibrations is derived using windtunnel data. The wind tunnel tests and the results of thecalibrations are discussed in detail in the Calibration ofthe Aerodynamic Model section.

The X-33 Flush Airdata Sensing Pressure Matrix

Since the simple model of equation 1 is derived frompotential and Newtonian flow around a blunt body, it ismost valid near the vehicle stagnation point. Thus, themost desirable location for the FADS pressure matrix isthe reinforced carbon-carbon (RCC) nosecap which is

used for the vehicle thermal protection system (TPS) andis in the stagnation region. The nosecap is hemisphericalwith a radius of 47.58 in. and extends longitudinally aftfrom the nose tip 20.8 in. Aft of the RCC nosecap anRCC skirt is used to protect the windward side of thevehicle, and a honeycomb metallic TPS is used for theleeward sides.

Measurement Locations

The number of measurements in the X-33 pressurematrix was selected as a compromise between the needto accurately measure the flow conditions at the nose andthe cost of locating ports on the vehicle. Since there arefour airdata states and a calibration parameter to beestimated, at least five independent pressuremeasurements must be available to derive the entireairdata state. Using five sensors to estimate the airdata isequivalent to a higher order spline fit and results in anestimating algorithm which is sensitive to noise in themeasured pressures. Providing an additional sixthsensing location mitigates the noise sensitivity, increasesredundancy options, and results in a system which givesoverall superior performance.

Figure 3 shows the FADS measurement locations. Fivemeasurement locations are on the nosecap, and a sixthmeasurement location is on the carbon-carbon skirt. Thelayout along meridian lines allows the calculations forangle of attack to be decoupled from the calculations forangle of sideslip using the triples algorithm, the real-timeairdata algorithm for the X-33. The mathematicalproperties of the triples algorithm will be discussed indetail in the Flush Airdata Sensing Estimation Algorithmsection, and the algorithm equations are derived in theappendix. The system layout is designed to give goodsensitivity for local angles of attack varying from –20° to45°, and angles of sideslip of up to ± 20°. The nominalclock and cone angles of the X-33 FADS ports aretabulated in Table 1.

ε

pi qc θi ) ε θi(2sin+(2 ) ] P∞+cos[=

θi

αe βe

θi ) αe ) βe ) λ i ) (cos(cos(cos=(cos

βe ) φi ) λ i )(sin(sin(sin+

αe ) βe ) φi ) λ i )(sin(cos(cos(sin+

λ

φ

P∞αe βe P∞

εαe βe

Table 1. X-33 FADS ports, clock andcone angles.

Port no. , deg , deg

1 180 20

2 270 20

3 0 0

4 90 20

5 0 20

6 0 45

φi λ i


Pneumatic Layout

Each X-33 flight-critical measurement subsystemmust have a fail-operational (FailOp) capability. That is,the subsystem can tolerate one failure anywhere in thesystem software or hardware and still produce a usableresult. The FADS design exploits the built-in redundancyof the pressure port matrix to achieve FailOp capabilitywith dual redundant system hardware. This dualredundancy is achieved at each surface measurementlocation by installing a plug with two surface ports.

Figure 4 shows an exploded view of the plug design.To survive the peak stagnation temperature ofapproximately 2000° F, the FADS plugs are fabricatedfrom C-103 niobium alloy.10 The C-103 alloy has amelting point in excess of 4000° F and a maximumworking temperature of approximately 2500° F. Anadditional advantage of using niobium alloy for thesurface plug is that the alloy has a coefficient of thermalexpansion (CTE) which is very close to the CTE ofcarbon-carbon.11 This match of the thermal expansionproperties ensures that hot gas leakage around the plug isunlikely to occur. The relatively close match of the twomaterials expansion coefficients also ensures that unduethermal stresses are not imparted to the RCC nosecapwhen high heating loads are applied. The plug designwas derived from an adaptation of the Space ShuttleEntry Airdata System3 which used niobium inserts withexcellent success.

On each plug the two surface ports are connected tothe pneumatic tubing using a high-temperature titaniumalloy braze. The individual surface ports are plumbed tothe measurement sensors using approximately 8–12 ft ofpneumatic tubing with an approximate inside diameterof 0.21 in. The pneumatic response of this arrangementwas analyzed extensively along the X-33 flighttrajectory. Based on these analyses, it was determinedthat the pneumatic latencies in the system would notintroduce significant errors into the airdata estimates.

Computational Architecture and Redundancy Management

A schematic of the computational architecture andredundancy management scheme for the X-33 FADS ispresented in figure 5. The design of the X-33 FADSsystem provides a total of twelve surface pressuremeasurements; however, the pressures from the dualredundant pressure ports are always analyzedindependently. Defining data flow path I as the set ofcomputations which use the grouping of the six upperand outboard pressure measurements on each plug, and

data flow path II as the set of computations which usesthe six lower and inboard pressures, then the FailOpcapability of the system will always be ensured byselecting the computational path with the minimummean square fit error. The mean square fit error iscomputed as

(3)

where, the ^ designates a quantity estimated by theFADS algorithm. This redundancy management schemeselects the system with the best overall fit consistency,and allows for a soft sensor failure (one which is notdetected by the hardware diagnostics) to occur withoutdegrading the performance of the system.

If the mean square error (MSE) of the output flow pathis normalized by an expected population variance (thatis, by the expected range of fit error that is allowable fora system with no failures) then the MSE becomesdistributed as and is a good indicator of the absolutesystem health. This chi-squared approach was developedand successfully employed by Whitmore, et al12, 13 onprevious flight tests. Expected values for the fit errorranges are still to be determined from additional windtunnel tests.

The Flush Airdata Sensing Estimating Algorithm

The aerodynamic model of equations 1 and 2 isinherently nonlinear in terms of the state parameters andcan not be directly inverted. A nonlinear regressionmethod in which the equations are recursively linearizedand inverted using iterative least squares has beendeveloped and successfully flight tested.14 Problemswith the stability of the algorithm were encountered,however, and the algorithm required special softwarepatches to maintain stability. Because the FADS is to beused for closed-loop flight control on the X-33, thenonlinear regression algorithm was determined to be toorisky and was abandoned.

A new solution algorithm was developed for the X-33FADS. A better solution algorithm is offered by takingstrategic combinations of three sensor readings that willdecouple the flow angularity states from the static and

X2

qi Pimeasured( )

q( c θ2i αe, βe( )cos[ –[

i 1=

6

∑

ε+ θ2i αe, βe( ) ]sin P∞ ]+

2

qii 1=

6

∑----------------------------------------------------------------------------------------------------=

χ2


impact pressure states and the aerodynamic calibration.Detailed derivations of the FADS estimator equations arepresented in the appendix. For completeness, portionsnecessary to explain the flow of the algorithm arerestated here.

Angle of Attack Triples Estimator

By strategically taking the differences of three surfacepressures (referred to in this paper as “triples”) thepressure related states (qc, , and ) are eliminatedfrom equation 1. The resulting pressure equation is

(4a)

where

(4b)

and pi, pj, and pk are the pressures used in the data triple.In equation 4 the local angle of attack can be decoupledfrom the local angle of sideslip using only pressuresaligned along a vertical meridian where = 0° or 180°.In this geometry arrangement, terms related to angle ofsideslip fall out of equation 4. For | | ≤ 45° the solutionfor is

(5a)

where

(5b)

When | | > 45° the correct solution is given by thecomplement of equation 5(a)

(5c)

For the X-33 there are four possible independentcombinations of -triples on the vertical meridian. The

clock and cone angles corresponding to these triplescombinations are presented in Table 2.

The output angle of attack estimate is determined asthe mean of the values computed using the fourindividual triples. This averaging procedure provides ameasure of noise rejection for the estimator. Clearly, ifone of the ports along the vertical meridian is deemedunusable and is weighted out of the algorithm, then onlyone valid triple remains for computing the angle ofattack.

Angle of Sideslip Estimator

Once the local angle of attack has been estimated, thenthe angle of sideslip may be evaluated using anycombination of the available ports, other than theobvious set in which all three ports lie on the verticalmeridian. The result, derived in the appendix, is aquadratic equation in .

(6a)

where

(6b)

and

P∞ ε

Γ ik θ2j Γ ji θ2

k Γkj θ2i 0=cos+cos+cos

Γ ik pi pk–( ) Γ ji p j pi–( ) Γkj pk p j–( )=,=,=

φ

αα

α 12--- A

B---1–tan=

A Γ ik λ2j Γ ji λ2

k Γkj λ2isin+sin+sin=

B Γ ik φj λ j λ jcossincos=

+ Γ ji φk λk λkcossincos

+ Γkj φi λ i λ icossincos

α

α 12--- π A

B---1–

tan–

=

α

Table 2. Angle of attack triples, clock and cone angles.

Triple no.i,

degi,

deg j, deg

j,deg

k, deg

k, deg

1 0 0 0 20 180 20

2 0 0 0 20 0 45

3 0 0 180 20 0 45

4 0 20 180 20 0 45

φ λ φ λ φ λ

βtan

A' β 2B' β C'+tan+2

tan 0=

A' Γ ikb j2 Γ jibk

2 Γkjbi2 + +

=

B' Γ ika jb j Γ jiakbk Γkjaibi+ +

=

C' Γ ika j2 Γ jiak

2 Γkjai2 + +

=


(6c)

Equation 6 has two solutions. However, unlike thesolution for angle of attack these solutions cannot bereduced to a single obvious choice for the angle ofsideslip roots. Determining which root is correctdepends on the port arrangement used to determine theangle of sideslip. Since only four angle-of-sideslip portsare required for a redundant measurement system, thenumber of possible angle-of-sideslip triples to beconsidered is reduced to a more manageable number byremoving the upper 20° port (port no. 1) on the verticalmeridian. When this port is eliminated the number ofavailable triples reduces to seven. The list of triples to beconsidered is tabulated in Table 3.

As developed in the appendix, port arrangements 2and 5 have mathematical singularities at 0° for certainvalues of . Near these singularities all three pressuresin the triple have nearly the same value, and the triplesequations become ill-conditioned. Away from thesesingularities, the solution for is given by the rootwhose absolute value is closest to zero. However, asderived in the appendix, near the singularity the solutionis indeterminate, and the proper solution cannot bedistinguished.

For port arrangement 2 the singularity occurs at0° . For port arrangement 5 the singularity is at18.2° . For the X-33 FADS design, which requiresfour ports for the estimator, the singularity problem ismanaged by a simple algorithm in which port number 6

= 0°, 45° is used only when is outside ofthe range from 17° to 20°. Similarly, only when isbetween the range from 17° to 20°, then is port number 5 = 0°, 20° used as the fourth angle-of-sideslipport.

As with the angle-of-attack algorithm, the outputangle of sideslip estimate is determined as the mean ofthe values computed using the four individual triples.Clearly, if one of the ports along the vertical meridian isdeemed unusable and is weighted out of the algorithm,then only one valid triple remains for computing theangle of sideslip.

Mach Number, Static Pressure, and Impact Pressure Estimator

Once the values of and have been determined,then the incidence angles at all of the ports can beevaluated, and only , , and qc remain as unknownsin the pressure equations. As derived in the appendix, is implicitly a function of and qc and the resultingsystem of equations is nonlinear. The solutions for and qc must be extracted iteratively. Defining thematrices

(7)

the original flow model (eq. 1) can be used to develop aniterative estimator of the form

(8)

The subscript (j) refers the result of the j'th iteration.The notation M(j) refers to the matrix of equation 8, with

being evaluated, using the values for Mach numberresulting from the previous iteration. The qi terms areweights which have a nominal value of 1.0. Setting the

Table 3. Angle of sideslip triples, clock and coneangles.

Triple no.

i, deg

i, deg

j, deg

j, deg

k, deg

k, deg

1 0 0 90 20 270 20

2 0 20 90 20 270 20

3 0 0 90 20 0 20

4 0 0 270 20 0 20

5 0 45 90 20 270 20

6 0 0 90 20 0 45

7 0 0 270 20 0 45

a ijk α λ ijk coscos=

+ α λ ijk φ ijk cossinsin

b ijk λ ijk φ ijk sinsin=

φ λ φ λ φ λ

βeαe

βe

αeαe

λ φ, αeαe

λ φ,

αe βe

ε P∞ε

P∞P∞

M j( )

θ1 )(2cos

+ ε j( ) θ1 )(2sin

1

. .

. .

. .

θ6 )(2cos

+ ε j( ) θ6 )(2sin

1

Q =

q1 . . . 0

. . .

. . .

. . .

0 . . . q6

,=

qc

P∞ j 1+( )M j( )

TQM j( )

1–M j( )

TQ

p1

.

.

.

p6

=

ε


value of qi to zero, weights the i'th pressure reading outof the algorithm. Equation 8 is derived in detail in theappendix.

Given qc and , Mach number can be computedusing normal one-dimensional fluid mechanicsrelationships. Subsonically, Mach number can becalculated directly using isentropic flow laws, where for

= 1.4

(9a)

Supersonically, the solution is computed using theRayleigh pitot equation which is derived from adiabaticnormal shock wave relationships.15 For = 1.4

(9b)

Equation 9(b) is solved using a Taylor seriesexpansion and a reversion of series to solve for Machnumber.16

For high Mach numbers, a very significanttemperature rise occurs across the bow shock wave andthe fluid no longer behaves as a calorically perfect gas.The large temperature rise across the shock wave causesvibrational modes of the air molecules to excite, whichdraws energy away from the flow. Thus the stagnationtemperatures actually encountered are considerablylower than those which would be computed if perfect gascalculations with = 1.4 were used.17

As a result of this nonadiabatic flow behavior, someerror is expected to occur in the Mach numbercalculation using equation 9(b). Fortunately, becausepressure is a mechanical quantity and depends primarilyupon the mechanical aspects of the flow, influencescaused by high-temperature gas properties aresecondary. Numerical analyses of equation 9(b) haveshown that the error introduced is less than 0.2 percentfor Mach numbers below 4.0.15, 18, 19 This error isacceptable for the X-33 design.

Numerical Stability of the Flush Airdata Sensing Algorithm Iteration

Since equations 8 through 9 are to be implemented aspart of a real-time airdata estimation algorithm, it isessential that potential instabilities in the iterationscheme be identified. In general it is not possible to

analyze the stability of nonlinear equations with twounknowns; however, a linearized stability analysisdetermines the behavior of the system with respect tosmall disturbances. For a discrete iteration, eigenvalueswith magnitudes less than 1.0 indicate stability20 for thecorresponding eigenmode. As the magnitudes of theeigenvalues approach unity, the system is relatively lessstable, and more iterations are required for convergence.Eigenvalues with magnitudes greater than unity indicatethat the iteration will diverge.

For the FADS algorithm there are two eigenvalues. Asderived in the appendix, these eigenvalues are primarilya function of Mach number and, to a much lesser extent,angle of attack. The individual values of qc and didnot affect the stability of the iteration, only their ratio issignificant. This conclusion is very important, as itallows the stability characteristics of the system iterationto be analyzed independently of specific trajectory.Figure 6 shows the eigenvalue magnitudes as a functionof Mach number and angle of attack. One eigenvalue isalways zero and has no effect on the stability of thesystem. The other eigenvalue has a nonzero magnitudeand determines the convergence properties of thealgorithm.

Subsonically, the nonzero eigenvalue is stable, butapproaches neutral stability at Mach 1.0. This nearlyneutral stability causes the algorithm to take a largernumber of cycles to converge transonically. As presentedin the Evaluation of the System Accuracy section, thisslow convergence also causes the overall systemaccuracy to be somewhat degraded near Mach 1.0. Formoderate supersonic Mach numbers, the eigenvaluemagnitude drops rapidly and the algorithm againbecomes stable. As the Mach number increases beyondMach 4, the stability of the algorithm is reduced untileventually the iteration becomes unstable beyondMach 8. The likely cause of this reduced stability is thatequation 9(b) has become so ill-conditioned that smallchanges in the ratio of qc to produce large changes inthe free stream Mach number. Fortunately, belowMach 4, where airdata are required by the vehicle flightcontrol system, iteration stability does not appear to be aproblem.

Algorithm Startup

At very high Mach numbers, even though the and solutions will still be valid, the stability problems

presented in the last section require the iterativealgorithm to be shutoff to avoid numerical divergence.When this algorithm shutoff is applied, a reliable method

P∞

γ

qc

P∞------- 1 .2M∞

2+[ ] 3.5

1–=

γ

qc

P∞-------

166.92M∞7

7M∞2

1–[ ]2.5

-------------------------------- 1–=

γ

P∞

P∞

αeβe


must be available for bringing the algorithm back on-linewhen the Mach number decreases back to safe limits.One approach relies on the use of inertial Mach numberto initialize the algorithm. Another simple, but effectiveapproach requires no outside information forinitialization and is presented here. The methodpresented here was developed and flight tested byWhitmore, et al6 (1995).

A block diagram of the startup algorithm is depicted infigure 7. A prescribed startup Mach number ispreselected and hard-coded. For the X-33 trajectory thedefault startup Mach number is 4.0. When the startupalgorithm is initiated, the transducers are polled and aninitial pressure data set (the target) is obtained. Using thetarget pressure set, and are evaluatedindependently of Mach number using the triplesalgorithm. With the resulting values for and andthe target pressure data, the FADS algorithm iteration isused to compute a new estimate for static pressure, Machnumber, impact pressure, and .

The aerodynamic model is then used to generate apredicted pressure distribution corresponding to thecomputed airdata set. The resulting biased pressuredistribution is linearly perturbed toward the target set bya small increment (currently 1/100 of the distancebetween the target set and the predicted pressure set).Using the perturbed value, a new estimate of the Machnumber is evaluated using the FADS iterative algorithm.A new pressure prediction corresponding to theperturbed airdata state is evaluated and the progressionalong the line from the predicted pressures to the targetpressures is repeated again until the target set is reachedor the maximum number of increments have beenperformed.

Calibration of the Aerodynamic Model

There are three calibration parameters which must beevaluated for the X-33 FADS system: the position error( ), the angle of attack flow correction angle ( ), andthe angle of sideslip flow correction angle ( ). For thepreliminary design these calibration data were obtainedfrom wind tunnel data. A two-percent model of the full-scale X-33 was instrumented with pressure tapscorresponding to the desired X-33 locations and thentested over a wide Mach number, angle of attack, andangle of sideslip range. To obtain the preliminarycalibrations, the model was tested in two NASA Langleywind tunnels (1) the 16-Foot wind tunnel21 and (2) theUnitary Plan wind tunnel (UPWT).22 A description ofthe test model, the test instrumentation, and the testconditions follows.

X-33 Wind Tunnel Model and Test Instrumentation

The two-percent X-33 model was manufactured formeasuring total vehicle forces and moments andmeasuring nosecap surface pressures. The 21 nosecapsurface pressures included the locations of the X-33FADS ports and were used to determine the FADSaerodynamic calibrations. The wind tunnel model wasmachined from a solid piece of aluminum and the nosecone was made of stainless steel to minimize damagefrom tunnel contaminants. All FADS pressure ports weredrilled normal to the surface with a diameter of 0.04 in.

The surface pressures were sensed through anelectronically scanned pressure (ESP) module whichproduces a time-multiplexed analog output with up to32 channels available for each module. The ESP waslocated approximately 5 in. aft of the nosecap and wasplumbed to the various surface ports using flexiblepneumatic tubing. Because of the short line lengthsinvolved, pneumatic lag in the pressure tubing wasconsidered negligible. The ESP sensor was a ±10 lb/in2

differential module with a manufacturers accuracy ofbetter than ± 0.1 percent of the full-scale reading. Thetime-multiplexed analog outputs from the ESP weretagged and sampled using a 16-bit analog-to-digital(A/D) conversion system.

During the wind tunnel tests, zero-shifts in the ESPmodule were frequently monitored and later applied tothe data. Additionally, spare ESP ports were plumbedtogether to monitor the drift in the individual sensors inthe ESP unit during the tests. If, for a particular testpoint, the drift rates were observed to be excessive, thenthe test point was repeated. Test points in which any ESPdrift exceeded 2.0 lb/ft2 were not used. The referencepressure source for ESP was located outside of the windtunnel and was sensed using a highly accurate absolutepressure transducer. The reference pressure transducerfeatured a capacitive transduction technology, with anaccuracy of ±0.25 lb/ft2. Outputs from the sensor weresampled using the 16-bit A/D system.

Wind Tunnel Test Facilities and Test Conditions

The Langley 16-foot wind tunnel is a closed-circuit,single-return, continuous-flow, atmospheric tunnel witha Mach number range from 0.2 to 1.3. The tunnel has anoctagonal, 16 ft slotted test section. Run limitations areoften imposed near transonic Mach numbers because ofthe lack of porous walls that absorb reflected shockwaves. The small size of the X-33 model, however,allowed Mach numbers of 0.95 and 1.05 withoutreflected shock wave interference. Angles of attack

αe βe

αe βe

ε

ε δαδβ


ranged from –10° to 24° and angles of sideslip from –10°to 16°. Angle of sideslip sweeps were done at 0° and 8°angle of attack, and angle of attack sweeps were done at–8°, –5°, 0°, 5°, and 8° angles of sideslip.

The UPWT is a closed-circuit, continuous-flow,variable density supersonic wind tunnel with two 4 × 4 fttest sections. One test section has a Mach number rangeof 1.5 to 2.9 and the other has a range of 2.3to approximately 4.5. The angle of attack ranged from–10° to 26° and angle of sideslip from ±10°. Angle-of-sideslip sweeps were done at 0° and 8° angle of attack. Angle of attack sweeps were done at –8°, –5°, 0°, 5°,and 8° angles of sideslip. The Reynolds number wasnearly constant at 2 × 106 per ft. As with the previous16-ft tunnel tests, runs were conducted with and withouta grit ring installed aft of the FADS ports, with littlevariation in the pressure distribution observed.

Table 4 summarizes the ranges of conditions andconfigurations which were tested.

Evaluation of the FADS Calibration

The calibration parameters were estimated from thewind tunnel data using measured pressures to estimate

e and e by means of the triples algorithm. Results for

e and e were used with wind tunnel referenceconditions for and qc to predict surface pressures atthe measurement locations. The residuals between thepredicted and measured pressures were then used tocalculate for each test point using linear regression.Finally, trends in the residuals were curve fit to producethe calibrations. A description of the three componentparts of the calibration follows, in detail.

Angle-of-Attack Calibration

The angle-of-attack calibration relates the local angleof attack to the free stream value. Residuals between thewind tunnel reference conditions and the estimates for

e were curve fit with third order polynomials in e togive the correction factor ( ). The coefficients of the fitwere scheduled as a function of . The resultingcalibration function is

(10)

The true angle of attack is evaluated by subtracting thecorrection factor ( ) from the indicated value. Thecalibration data and the corresponding curve fits for are presented as a function of Mach number in figure 8.The polynomial fits to the wind tunnel data are excellent.Also notice that the curves flatten out with increasingMach number, a result of diminishing flow deflections atincreasing Mach numbers.

Angle-of-Sideslip Calibration

In a manner identical to the angle-of-attackcalibration, the angle-of-sideslip calibration relates thecorrection to the local angle of sideslip sensed by theFADS to the free stream value. The wind tunnel residualdata were fit with third order polynomials in with thecoefficients scheduled as a function of Mach number.The resulting calibration function is

(11)

The calibration data and the corresponding curve fitsfor are presented in figure 9.

Position Error (ε) Calibration

The position error calibration parameter adjusts theFADS pressure model for changes in the incidence angleand Mach number. The wind tunnel data were curve fitwith second order polynomials in angle of attack andangle of sideslip, and the coefficients scheduled as afunction of Mach number. The resulting calibrationmodel is

(12)

The calibration curves for at zero are presentedin figure 10(a). The variations of with at zero are shown in figure 10(b). The parameter (at zero and ) is plotted as a function of the free stream Machnumber in figure 10(c).

It is interesting to note that the trends shown infigures 10(a) and 10(b) are remarkably similar. In bothfigures, 10(a) and 10(b), the data form a family ofparabolic curves. The peak of each parabola ( )increases in magnitude and shifts to increased angle of

Table 4. Range of wind tunnel test conditions.

Tunnel Mach range,

deg,

deg

16-ft0.25 to

1.2–10 to

24–10 to

16

UPWT1.6 to 4.50

–10 to26

–10 to10

α β

α βα β

M∞

ε

α αδα

M∞

δα A0 M∞[ ] A1 M∞[ ]α e+=

+ A2 M∞[ ]α e2

A3 M∞[ ]α e3

+

δαδα

βe

δβ B0 M∞[ ] B1 M∞[ ]β e+=

+ B2 M∞[ ]β e2

B3 M∞[ ]β e3

+

δβ

ε εM M∞[ ] ε α1M∞[ ]α e εα1

M∞[ ]α e2

+ +=

+ εβ1M∞[ ]β e εβ1

M∞[ ]β e2

+

ε βeε βe αe

εM αeβe

εM


attack as Mach number increases. The curves in figures10(a) and 10(b) have negative concavity subsonically,almost no concavity transonically, and positiveconcavity supersonically. Also, at close to supersonicMach numbers, approaches zero, an observationwhich confirms that the FADS flow model is consistentwith modified Newtonian flow theory.

Evaluation of the System Accuracy

The accuracy of the calibrated FADS system wasanalyzed using an analysis-of-variance23 method. In thismethod, the calibrations for , , and and also themeasured pressure distributions were used to generateairdata estimates with the FADS algorithm for each windtunnel test point. Assuming that the measured windtunnel reference conditions represent a truth set,quantitative measures of the overall system accuracy area product of the residuals between FADS algorithmestimates and the wind tunnel reference conditions. Theresiduals include errors in the wind tunnel referenceconditions, and are considered to be representativemeasures of the absolute system accuracy.

Note that the calibration curves were derived usingonly data from angle of attack sweeps at zero angle ofsideslip and angle of sideslip sweeps at zero angle ofattack. The error analysis, however, includes residualdata that were obtained during test points whichincluded both nonzero angle of attack and nonzero angleof sideslip.

Figure 11 shows the available wind tunnel databasefor angle of attack and Mach number and typicalMichael Army Air Field (MAAF) and Malmstrom AirForce Base (MAFB) flight trajectories. Clearly, many ofthe wind tunnel points are far outside of the actual flightconditions to be encountered during the X-33 flight tests.Introducing these unrealistic data into the residualanalysis would likely give error estimates which are toolarge. Thus, the following procedure was used to reducethe wind tunnel data sets to a more realistic andmanageable size.

1. The outer boundaries of the minimum andmaximum angles of attack and angles of sideslip tobe encountered as a function of Mach number weredetermined for typical MAAF/MAFB launch andTAEM flight trajectories.

2. Next, the union of the outer boundaries wasdetermined and ±5° angle of attack and ±3° angleof sideslip were added to give the domain space

over which the residuals would be analyzed. Thisdomain space allows for considerable variationaway from the nominal trajectory.

3. All test points which exhibited more that 2.0 lb/ft2

of drift in the ESP readings were eliminated fromthe domain space.

4. Within this domain space, root-mean-square(RMS) averages of the residuals for the angle-of-attack and angle-of-sideslip test points wereevaluated as a function of Mach number.

5. Finally, the resulting RMS averages were fared andtabulated as a function of Mach number to producea set of three simple 1- error models with Machnumber as the independent variable, and , ,and outputs. Figure 12 displays these RMSerror estimates as a function of Mach number.

Figure 12(a) shows the percentage of Mach numbererror and the accuracy requirement for the X-33 FADS.The Mach error shows a sizable error jump at transonicspeeds, but does not exceed the error limit. The angle-of-attack and angle-of-sideslip error plots are displayedwith the accuracy requirements in figures 12(b) and12(c). Below Mach 4.0 the errors for both parameters arewithin the accuracy requirements of the X-33 airdatasystem.

Note that in examining figure 12(a) in more detail,growth in the Mach number error percentage (bothtransonically and above Mach 2.5) corresponds closelyto the growth in the nonzero eigenvalue (fig. 6). Thisresult is not entirely surprising. The eigenvaluesrepresent a measure of the numerical conditioning of thesystem. Transonically this numerical ill-conditioning iscaused by the inflection point at which the flow switchesfrom subsonic to supersonic. This inflection pointcorresponds to a point of maximum slope in the Mach number curve (fig. 10(c)). At supersonic speedsthis ill-conditioning is caused by the rapid growth in thestrength of the bow shock as Mach number increases.The result is that large changes in the free stream Machnumber produce only small changes in the local pressuredistribution. Conversely, small changes in the measuredpressure distribution make for large changes in theestimated free stream Mach number. This ill-conditioning with respect to Mach number is anunavoidable consequence of the flow physics and iscommon to all airdata measurement systems at transonicand supersonic Mach numbers.

εM

ε δα δβ

σδM δα

δβ

εM


Summary and Concluding Remarks

The X-33 FADS design is presented in this paper. TheFADS concept, in which airdata are inferred fromnonintrusive surface pressure measurements, does notrequire stagnation of the surface flow. This innovationextends the useful range of the airdata measurementsystem to the hypersonic flow regime. The FADS systemuses a matrix of pressure orifices on the vehicle nose toestimate airdata parameters.

The FADS system design exploits the built-inredundancy in the pressure port matrix to achieve FailOpcapability with only dual redundancy in the systemhardware. The design produces two independentmeasurement paths for the airdata and the redundancymanagement scheme selects the measurement path withthe best overall fit consistency. This scheme allows forfailure, either detected or undetected, to occur anywherein the system without degrading the overall performance.

Presented here is the aerodynamic model that relatesthe airdata parameters to the measured pressures. Alsopresented is the derivation of the estimating algorithm.The mathematical properties of the algorithm wereanalyzed in detail. At near hypersonic Mach numbers,the stability of the algorithm is reduced because the bowshock wave has become so strong that small disturbancesare highly amplified. Eventually the iteration becomesunstable, at close to Mach 8 and beyond. This ill-conditioning with respect to Mach number is anunavoidable consequence of the flow physics.Fortunately, below Mach 4, where airdata are requiredfor the vehicle flight control system, potentialmathematical singularities and iterative instabilities inthe nonlinear estimating algorithm can be prevented byusing simple logic.

Because of the problems with stability at highsupersonic Mach numbers, the iterative algorithm mustbe shutoff in order to avoid numerical divergenceproblems. A reliable method for bringing the algorithmback on-line was presented. This simple creepingsolution method insures that the algorithm does notexceed the small perturbation limits of the stabilityanalysis.

There are three calibration parameters which must beevaluated for the X-33 FADS system: the position error( ), the angle of attack flow correction ( ), and theangle of sideslip flow correction ( ). For thepreliminary X-33 design these data were evaluated fromwind tunnel calibrations. A two-percent full-scale X-33

model was instrumented and tested over a variety ofangles of attack and Mach number ranges in two separatewind tunnels. Data were obtained for Mach numbersranging from 0.25 to 4.5, angles of attack from –10° to25°, and angles of sideslip from ±10° to 16°.

The accuracy of the system was evaluated by applyingthe calibrations and the FADS algorithm to generateairdata estimates for each wind tunnel test point usingthe measured pressure distribution. Assuming that thewind tunnel reference conditions represented a truth set,the statistical accuracy was evaluated by computingresiduals between the reference set and FADS estimatesand then compiling the results into a set of 1- errormodels. The error analysis concluded that the FADSmeets the accuracy requirements for the X-33 vehicle.

Appendix: Mathematical Analysis of the Flush Airdata Sensing Algorithm

As discussed in the main text, the airdata states arerelated to the surface pressures by an aerodynamicmodel that captures salient features of the flow, and isvalid over a very large Mach number range. Theresulting model takes the form

(A-1)

where

(A-2)

Defining the terms

(A-3)

equation A-2 reduces to

(A-4)

The Triples Formulation

Using equation A-1 to take strategic combinations ofthree pressures (triples) the impact pressure, staticpressure, and calibration parameter are decoupled fromthe local angle of attack and angle of sideslip.

ε δαδβ

σ

pi qc θi( ) ε θi( )2sin+

2cos[ ] P∞+=

θi( )cos αe( ) βe( ) λ i( )coscoscos=

+ βe( ) φi( ) λ i( )sinsinsin

+ αe( ) βe( ) φi( ) λ i( )sincoscossin

ai αe λ i αe λ i φicossinsin+coscos=

bi λ i φisinsin=

θi( )cos ai βe( ) bi βe( )sin+cos=


(A-5)

Expanding equation A-5 and defining

(A-6)

equation A-5 reduces to

(A-7)

Substituting in from equation A-4, dividing by and collecting terms, equation A-7

becomes

(A-8)

In equation A-8, the local angle of attack and angle ofsideslip have been written as a function of the measuredpressure only, the impact pressure, , and static pressurehave been decoupled.

Angle of Attack Solution

Angle of attack is decoupled from angle of sideslipusing a meridian solution, in which, along the verticalmeridian = 0, ±180°, and equation A-8 reduces to

(A-9)

Factoring out of equation A-9,

(A-10)

Expanding the squares, the result is a quadraticexpression in ,

(A-11)

Equation A-11 can be reduced further by noting that

and

(A-12)

Substituting equation A-12 into equation A-11,

(A-13)

pi p j–

p j pk–-----------------

qc θ2i ε θ2

isin+cos

P∞+

qc θ2j ε θ2

jsin+cos

P∞+–

qc θ2j ε θ2

jsin+cos

P∞ +

qc θ2k ε θ2

ksin+cos

P∞+–

----------------------------------------------------------------------------- =

=

1 ε–( ) θ2i ε+cos

1 ε–( ) θ2j ε+cos

–

1 ε–( ) θ2j ε+cos

1 ε–( ) θ2k ε+cos

–

--------------------------------------------------------------------------------------------------------

= θ2

i θ2jcos–cos

θ2j θ2

kcos–cos---------------------------------------

Γ ik pi pk Γ ji p j pi Γkj pk p j–≡,–≡,–≡

Γ ik θ2j Γ ji θ2

k Γkj θ2icos+cos+cos 0=

β2 β ≠ ± 90°( )cos

Γ ik a j b j βtan+[ ] 2 Γ ji ak bk βtan+[ ] 2+

+ Γkj ai bi βtan+[ ] 20=

ε

φ

Γ ik α λ j α λ j φjcossinsin+coscos[ ] 2

+ Γ ji α λ k α λ k φkcossinsin+coscos[ ] 2

+ Γkj α λ i α λ i φicossinsin+coscos[ ] 20=

α α≠ ± 90°( )cos

Γ ik λ j α λ j φjcossintan+cos[ ] 2

+ Γ ji λk α λ k φkcossintan+cos[ ] 2

+ Γkj λ i α λ i φicossintan+cos[ ] 20=

αtan

Γ ik λ2j Γ ji λ2

k Γkj λ2isin+sin+sin[ ] α2 tan

+ 2[Γ ik λ j λ j φj Γ ji λk λk φk cossincos+cossincos

+ Γkj λ i λ i φi ] α tancossincos

+ Γ ik λ2j Γ ji λ2

k Γkj λ2icos+cos+cos[ ] 0=

λ2cos 1 λ2

sin–=


k Γkj λ2icos+cos+cos

= Γ ik 1 λ2jsin–( ) Γ ji 1 λ2

ksin–( )+

+ Γkj 1 λ2isin–( ) Γ ik Γ ji Γkj+ +=

– Γ ik λ2j Γ ji λ2

k Γkj λ2isin+sin+sin( )

= pi pk–( ) p j pi–( ) pk p j–( )+ +

– Γ ik λ2j Γ ji λ2

k Γkj λ2isin+sin+sin( )

= Γ ik λ2j Γ ji λ2

k Γkj λ2isin+sin+sin( )–


k Γkj λ2isin+sin+sin[ ] α 1–

2tan[ ]

+ 2[Γ ik λ j λ j φjcossincos

+ Γ ji λk λk φkcossincos

+ Γkj λ i λ i φi ] αtancossincos 0=


Defining

(A-14)

Equation A-14 is written in the simple form

(A-15)

The correct solution is extracted from equation A-15by noting that

(A-16)

Substituting equation A-16 into equation A-15, andmultiplying through by ,

(A-17)

and the solution for is

(A-18)

For | | ≤ 45°, equation A-18 picks the correct root;when | | > 45° then the correct root is given by thecomplement of equation A-18

(A-19)

Angle of Sideslip Solution

Given the solution for angle of attack, a proceduresimilar to that of the previous section is followed forangle of sideslip. Expanding the squares in equation A-7and solving for like terms in ,

(A-20)

Defining

(A-21)

Equation A-20 reduces to a quadratic equation in

(A-22)

The two solutions to equation A-22 are

(A-23)

Root Selection Criteria for Angle of Sideslip

The relationship of to is developed

by taking the product of the roots

(A-24)

Rearranging equation A-24 gives the relationship of

the two roots

(A-25)

For several specific port arrangements relevant to the

X-33 design, the relationship between the two roots can

be written in closed form. Three of these special

arrangements will now be discussed.

Root Selection Criteria for the Triples on Lateral Meridian

Along the lateral angle-of-sideslip meridian,

the outer port clock angles are ± 90°, and

, thus equation

A-24 reduces to

A Γ ik λ2j Γ ji λ2

k Γkj λ2isin+sin+sin( )=

B Γ( ik φj λ j λ jcossincos

+ Γ ji φk λk λk Γkj φi λ i λ i )cossincos+cossincos

=

A α 1–2tan[ ] 2B αtan+ 0=

α 1–2

tan α α2cos–

2sin

α2cos

-----------------------------------=

= 1 2 α2

cos–

α2cos

--------------------------- 2αcos

α2cos----------------–=

α2cos

A 2α 2B α αsincos+cos–

= A 2α B 2αsin+cos– 0=

α

α 12--- A

B---1–tan=

αα

α 12--- π A

B---

1–tan–=

βtan

Γ ikb j2 Γ jibk

2 Γkjbi2

+ +[ ] β2tan

+ 2 Γ ika jb j Γ jiakbkc Γkjaibi+ +[ ] βtan

+ Γ ika j2 Γ jiak

2 Γkjai2

+ + 0=

A' Γ ikb j2 Γ jibk

2 Γkjbi2 + +( )=

B' Γ ika jb j Γ jiakbk Γkjaibi+ +( )=

C' Γ ika j2 Γ jiak

2 Γkjai2 + +( )=

βtan

A' β 2B' β C'+tan+2

tan 0=

β1 β2tan,tanB'A'-----–

B'A'-----

2 C'A'-----–±=

β1[ ]tan β2[ ]tan

β1 β2tantanB'A'-----–

B'A'-----

2 C'A'-----–

+=

× B'A'-----–

B'A'-----

2 C'A'-----––

C'

A'-----=

β1tanC'A'-----

1β2tan

-------------- C'

A'----- π

2--- β2–

tan= =

aijk α λ ijk bijk,coscos = λ ijksin±=


(A-26)

Thus, along the lateral meridian the relationshipbetween the two solutions of equation A-22 is

(A-27)

The solution for is plotted as a function of andthe local angle of attack in figure A-1. Even for angles ofattack as high as 50° the two angles of sideslip solutionsare nearly 70° out of phase; so for triples along thelateral meridian, selecting the root with | | closest tozero will always produce the correct solution for .

Off-Meridian Triples, Symmetric Arrangement

Assume ports i and j lie on the lateral meridian andport k lies on the vertical meridian, then

(A-28)

and for a symmetric arrangement, , and

reduces to

(A-29)

Noting that equation A-29 canbe reduced further

(A-30)

Substituting equation A-30 into equation A-25, therelationship between the two roots becomes

(A-31)

For the X-33 design, there are two possible symmetricoff-meridian angle of sideslip triples combinations,

(A-32)

For the first set of coordinates the solution for is plotted as a

function of and the local angle of attack infigure A-2(a). Notice that as approaches 18.207°, theangle of attack at which all of the incidence angles areequal, a singularity exists at = 0. Thus, for angles ofattack near 18.207°, and angles of sideslip values nearzero, the system is highly ill-conditioned. This ill-conditioning makes selecting the correct rootimpossible. Figure A-2(b) presents the same analysis asfigure A-2(a), except now the second set of coordinates

is used. A singularitynow exists when = 0° and = 0°.

Clearly, using the = 45° port is the preferablechoice for the off-lateral meridian. If one assumes thattrue angle-of-sideslip limits to be experienced are lessthan ± 20°, then for ≤ 17°, and ≥ 20°, a root-closest-to-zero criterion is valid for selecting the correctroot. For values of between 17° and 20°, the

system is indeterminate, and

C'A'-----

Γ ika j2 Γ jiak

2 Γkjai2 + +

Γ ikb j2 Γ jibk

2 Γkjbi2 + +

-----------------------------------------------------------=

= α2 Γ ik λ2j Γ ji λ2

k Γkj λ2icos+cos+cos


k Γkj λ2isin+sin+sin

---------------------------------------------------------------------------------------

cos

= α2cos–

β1tan α β2π2---–tan

2cos=

β2 β1

ββ

φiπ2--- ai⇒ α λ icoscos= =

bi λ isin=

φjπ2--- a j⇒ α λ jcoscos= =

b j λ jsin–=

φk 0 ak⇒ α λ k α λ ksinsin+coscos= =

bk = 0

λ i λ j=C'A'-----

C'A'----- α

2cos=

λ j2 Γ ik Γkj+( ) Γ ji λk α λ ksintan+cos( )2

+cos

λ2j Γ ik Γkj+( )sin

----------------------------------------------------------------------------------------------------------------×

Γ ik Γkj+ Γ ji ⇒–=

C'A'----- α2

cos=

Γ ji λ j

2 Γ ji λk α λ ksintan+cos( )2+cos–

Γ ji λ2jsin–

--------------------------------------------------------------------------------------------------×

= α 2

cos

λ2jsin

----------------- λ j2 1

α 2

cos----------------- α λ k α λ ksinsin+coscos( )2

–cos

= α λ j2 α λ k–( )2

cos–cos2

cos

λ2jsin

-----------------------------------------------------------------------

β1tanα λ j

2 α λ k–( )2cos–cos

2cos

λ2jsin

--------------------------------------------------------------------- =

π2--- β2–tan×

λ i λ j λk, , 20° 20° 45°,, and,=

λ i λ j λk, , 20° 20° 20°,, =

λ i λ j λk, , =20° 20° 45°,, β 2

β1α

β

λ i λ j λk, , 20° 20° 20°,, =α β

λk

α α

αλ ijk 20° 20° 45°,, =


instead the triple is used.With this substitution then, a root-closest-to-zerocriterion always selects the correct root.

Off-Meridian Triples, Asymmetric Arrangement

The root locations for one special case asymmetricport arrangement can be analyzed, that is thearrangement where = 0. Consider the case where,

(A-33)

Following the same procedures as before,

(A-34)

Factoring out

(A-35)

Collecting terms, equation A-35 reduces to

(A-36)

The pressure dependent term, , can be removed

from equation A-34 by substituting in from equation

A-5. Defining

(A-37)

then equation A-37 reduces to

(A-38)

and finally

(A-39)

For the X-33 design, there are four possibleasymmetric off-meridian angle of sideslip triplescombinations,

λ ijk 20° 20° 20°,, =

λ j

φiπ2---± ai⇒ α λ icoscos= =

bi λ isin±=

φj 0 λ j, 0 a j⇒ αcos= = =

b j 0=

φk 0 ak⇒ α λ k α λ ksinsin+coscos= =

= α λ k–( )cos

bk 0=

C'A'----- =

Γ ik α Γ ji α λ k–( ) Γkj α λ2icos2cos+2cos+

2cos

Γkj λ2isin

-------------------------------------------------------------------------------------------------------------------

α2cos

C'A'----- α

Γ ik Γ ji

α λ k–( )2cos

α2cos

------------------------------- Γkj λ2icos+ +

Γkj λ2isin

------------------------------------------------------------------------------------

2

cos=

= α

Γ ik Γ ji

α λ k–( )2cos

α2cos

------------------------------- Γkj 1 λ2isin–( )+ +

Γkj λ2isin

-------------------------------------------------------------------------------------------------

2

cos

C'A'----- α2

cos=

Γ ik Γkj+( ) Γ ji

α λ k–( )2cos

α2cos

---------------------------- Γkj λ2isin( )–+

Γkj λ2isin( )

-----------------------------------------------------------------------------------------------

×

= α2Γ ij Γ ji+

α λ k–( )2cos

α2cos

------------------------------- Γkj λ2isin( )–

Γkj λ2isin

---------------------------------------------------------------------------------------

cos

= α2 Γ ji

Γkj-------

α λ k–( )2cos

α2cos

------------------------------- 1–

λ2isin

--------------------------------------------- 1–

cos

Γ ji

Γkj-------

Ψ

α λ k–( )2cos

α2cos

------------------------------- 1–

λ i2

sin---------------------------------------------≡

C'A'----- α

Γ ji

Γkj------- Ψ 1–

2cos=

β1tan αΓ ji

Γkj------- Ψ 1–

2 π2--- β2–tancos=


(A-40)

and

(A-41)

For the first sets of coordinates , = [20°, ± 90°], [0°, 0°], [45°, 0°]

the solutions for are plotted as a function of andthe local angle of attack in figure A-3(a). In the range of| | < 25°, it is interesting to note that the secondsolution is nearly independent of the value of the firstsolution. For = 90°, –80°, and = –90°, 80°.

Figure A-3(b) presents similar results for the solutions for the second set of coordinates ,

= [20°, ± 90°], [0°, 0°], [20°, 0°].Again, in the range of | | < 25°, it is very interesting tonote that is identical to the value produced by theprevious set of coordinates ( = 45°). Clearly, the termsinvolving are canceled out of equation A-38. As aresult, for the asymmetric configuration a root-closest-to-zero criterion will always select the correct solution.

Angle-of-Sideslip Solution Summary

The quadratic equation, A-22, has two roots, with noguarantees that they be in general orthogonal. However,several conclusions can be reached for special geometryarrangements which X-33 exploits in its design. Theseconclusions are:

1. For triples which lie on the lateral meridian, thesolutions are nearly orthogonal, and for | | < 25°,using a root-closest-to-zero selection criterion willalways select the correct solution.

2. For off-meridian triples arrangements, only thesymmetric arrangement presents a problem. Forthis arrangement, there exists a singularity point atwhich all three incidence angles are equal. In thevicinity of this singularity, the system is highly ill-conditioned. This ill-conditioning makes thesystem indeterminate and selecting the correct rootis impossible without prior information.

3. For the X-33 configuration, using the lower45° port to complete the symmetric triple puts thesingularity at 18.207° local angle of attack. Usingthe lower 20° port puts the singularity at 0° localangle of attack.

4. If one assumes that true angle-of-sideslip limits, tobe experienced, are less than ± 25°, then for the

= 20°, 20°, 45° triple, for ≤ 17°,a ≥ 20°, a root-closest-to-zero criterion is valid forselecting the correct root. For values of between17° and 20°, the = 20°, 20°, 45° systemshould be considered indeterminate, and the

= 20°, 20°, 20° triple should be usedinstead. If this substitution is made, then a root-closest-to-zero criterion will always be valid forselecting the correct root.

5. For the asymmetric off-meridian configuration, thealternate solution will always hover around± 80°. Thus the root-closest-to-zero criterion isalways valid.

Static Pressure, Impact Pressure, and Mach Number Solution

Once the local values of and have been solvedfor, then the incidence angles at all of the ports can beevaluated, and only , , and qc remain as unknownsin the pressure equations. Assuming that n pressuremeasurements are to be used, the original model(eq. A-1) can be written in matrix form with static andimpact pressure broken out as variables:

(A-42)

Ignoring for now, the fact that is implicitly afunction of static and impact pressure, equation A-42 canbe solved for the static and impact pressure usingweighted linear regression.19 The result is

λ i φi,[ ] λ j φj,[ ] λ k φk,[ ], ,

= 20° 90± °,[ ] 0° 0°,[ ] 45° 0°,[ ], ,

λ i φi,[ ] λ j φj,[ ] λ k φk,[ ], ,

= 20° 90± °,[ ] 0° 0°,[ ] 20° 0°,[ ], ,

λ i φi,[ ]λ j φj,[ ] λk φk,[ ] ,

β2 β1

β

φi β2 ≅ φ i β2 ≅

β2λ i φi,[ ]

λ j φj,[ ] λk φk,[ ] ,β

β2λk

λk

β

λ ijk α

αλ ijk

λ ijk

β

α β

ε P∞

p1

.

.

.

pn

θ1( )+

2cos

ε θ1( )2sin

1

. .

. .

. .

θn( )+

2cos

ε θn( )2sin

1

qc

P∞ =

ε


(A-43)

where . In equation A-43,the qi's are weights which have a nominal value of 1.0.Setting the value of qi to zero, weights the i'th pressurereading out of the algorithm. Equation A-43 looks like aclosed form solution, but the terms on the right-handside of equation A-43 are implicitly a function of and qc. Thus equation A-43 is actually nonlinear and thesolutions for and qc must be extracted iteratively.Defining

(A-44)

where , equation A-43is written as an iterative estimator

(A-45)

In equation A-45 the subscript (j) refers the result ofthe j'th iteration. The matrix M(j) is defined inequation A-44 with being evaluated using the valuesfor Mach number resulting from the previous iterationfor and qc .

Algorithm Stability Analysis

Since equation A-45 is to be implemented as part of areal-time airdata estimation algorithm, it is essential that

potential instability regions be identified. In general it isimpossible to analyze the stability of nonlinear equationswith two unknowns; however, a linearized stabilityanalysis will determine the behavior of the system withrespect to small disturbances. Defining the terms

(A-46)

and recalling that is a function of Mach number, andimplicitly a function of static and impact pressure, then

(A-47)

and equation A-42 can be re-written as

(A-48)

Using the definitions of equation A-46 and A-47, theiterative estimation equation can be written in the form

(A-49)

In equation A-49 is the estimate after the

j + 1'th iteration, and is the matrix of

equation A-47 evaluated using the result from the j’th

iteration. Subtracting equation A-49 from A-48, and

expanding M[x] in a Taylor series about and

neglecting terms higher than first order in the

perturbations, equation A-49 reduces to

(A-50)

Defining the error vectors,

(A-51)

Equation A-51 becomes the linearized error equationfor the iteration

qc

P∞-------

=

qii 1=

n

∑ qi Ωi( )i 1=

n

∑–

qi Ωi( )i 1=

n

∑– qi Ωi( )2

i 1=

n

∑

Ωi( )qiPii 1=

n

∑

qiPii 1=

n

∑

qii 1=

n

∑ qi Ωi( )2

i 1=

n

∑ qi Ωi( )i 1=

n

∑2

–

------------------------------------------------------------------------------------------------------------

Ωi θi( )2 ε θi( )2sin+cos=

εP∞

P∞

Mj( )

Ω1j( )( ) 1

. .

. .

. .

Ωnj( )( ) 1

Q,

q1 . . . 0

. . .

. . .

. . .

0 . . . qn

= =

Ωij( ) θi( )2 ε j( ) θi( )2

sin+cos=

qc

P∞-------

j 1+( )M

j( )TQM

j( )1–M

j( )TQ

p1

.

.

.

pn

=

ε

P∞

Z

p1

.

.

.

pn

x qc

P∞≡,≡

ε

M x[ ]

Ω1 1

. .

. .

. .

Ωn 1

≡

Z M x[ ] x=

Z M xj( )[ ] x

j 1+( )=

xj 1+( )

M xj( )[ ]

xj( )

∇ Mxj( )

x xj( )

–( )[ ] x M xj( )[ ] x x

j 1+( )–( ) 0=+

xj( )

x xj( )

–( )≡

xj 1+( )

x xj 1+( )

–( )≡


(A-52)

and the eigenvalues (roots) of the characteristicequation23

(A-53)

will determine the linear stability of system. The

elements of the Jacobian, , are evaluated as

(A-54)

In equation A-54, the parameter is the

sensitivity of the calibration parameter to Mach number

and is determined by numerically differentiating the

calibration parameter ( ) with respect to Mach

number. The parameters , and are evaluated

by differentiating the isentropic flow equation (eq. 9(a))

for subsonic flow, or the Rayleigh pitot equation

(eq. 9(b)) for supersonic flow. For = 1.4, the resulting

expressions are

(A-55)

where the Mach number derivative is

(A-56)

Substituting equation A-55 into A-54, the Jacobianmatrix reduces to

(A-57)

Equation A-57 is dependent only on the free streamMach number and the local incidence angles; thus for agiven port geometry, the stability of the system(eigenvalues equation A-53) is dependent only on thefree stream Mach number and the local flow incidenceangles. This conclusion is very important, as it allows thestability characteristics of the system to be analyzedindependently of trajectory.

References

1Cary, John P. and Earl R. Keener, Flight Evaluationof the X-15 Ball-Nose Flow-Direction Sensor as an Air-Data System, NASA TN D-2923.

2Siemers, Paul M., III, Martin W. Henry, and James B.Eades, Jr., “Shuttle Entry Air Data System (SEADS)–Advanced Air Data System Results: Air Data Across theEntry Speed Range,” Orbiter Experiments (OEX)Aerothermodynamics Symposium, CP 3248, Part 1,Apr. 1995, pp. 49–78.

3While, D.M., Shuttle Entry Air Data System(SEADS) Hardware Development, Vol. II, History,NASA CR-166044, Jan. 1983.

∇ Mxj( )

x[ ] xj( )

M xj( )[ ] x

j 1+( )+ 0=

Det M xj( )[ ]

TM x

j( )[ ] 1–

M xj( )[ ]

T∇ Mx

j( )x

λ I–

0=

∇ Mxj( ) x

∇ Mxj( )

x =

qc θ21

∂ε j( )

∂M∞-----------

∂M∞

∂qc

----------- sin qc θ2

1∂ε j( )

∂M∞-----------

∂M∞

∂P∞-----------

sin

. .

. .

. .

qc θ2n

∂ε j( )

∂M∞-----------

∂M∞

∂qc

----------- sin qc θ2

n∂ε j( )

∂M∞-----------

∂M∞

∂P∞-----------

sin

∂ε∂M∞------------

εM∂M∞∂qc

------------∂M∞∂P∞------------

γ

∂M∞∂qc

------------ 1

P∞∂f∂M--------

---------------------=

∂M∂P∞----------

qc–

P∞2 ∂f

∂M--------

---------------------=

∂f∂M--------

subsonic

1.4M∞ 1 0.2M∞2

+2.5

=

for M∞ 1≤( )

∂f∂M--------

supersonic

1168.45M∞6

2M∞2

1–

7M∞2

1–3.5

--------------------------------=

for M∞ 1>( )

∇ Mxj( )

x =

qc

P∞------ θ2

1∂ε j( )

∂M∞----------- 1

∂f∂M----------------

qc

P∞------–

2

θ21

∂ε j( )

∂M∞----------- 1

∂f∂M----------------sinsin

. .

. .

. .

qc

P∞------ θ2

n∂ε j( )

∂M∞----------- 1

∂f∂M----------------

qc

P∞------–

2

θ2n

∂ε j( )

∂M∞----------- 1

∂f∂M----------------sinsin


4Larson, Terry J., Stephen A. Whitmore, L. J.Ehernberger, J. Blair Johnson, and Paul M. Siemers, III,Qualitative Evaluation of a Flush Air Data System atTransonic Speeds and High Angles of Attack, NASATP-2716, Apr. 1987.

5Larson, Terry J., Timothy R. Moes, and Paul M.Siemers, III, Wind-Tunnel Investigation of a FlushAirdata System at Mach Numbers From 0.7 to 1.4,NASA TM-101697, Apr. 1990.

6Whitmore, Stephen A., Timothy R. Moes, and TerryJ. Larson, Preliminary Results From a Subsonic HighAngle-of-Attack Flush Airdata Sensing (HI-FADS)System: Design, Calibration, and Flight Test Evaluation,NASA TM-101713, Jan. 1990.

7Whitmore, Stephen A., Roy J. Davis, and JohnMichael Fife, In-Flight Demonstration of a Real-TimeFlush Airdata Sensing System, AIAA Journal of Aircraft,vol. 33, no. 5, Sept.–Oct., 1996, pp. 970-977.

8Currie, I. G., Fundamental Mechanics of Fluid,McGraw-Hill Book Company, New York, 1974.

9Anderson, John D., Jr., Hypersonic and HighTemperature Gas Dynamics, McGraw-Hill BookCompany, New York, 1989.

10Boyer, Howard E. and Timothy L. Gall, MetalsHandbook , Desk Edition, American Society for Metals,Metals Park, Ohio, pp. 16–23, 16–25.

11Diefendorf, Russell J., “Continuous Carbon FiberReinforced Carbon Matrix Composites,” EngineeredMaterials Handbook , Volume 1: Composites, AMSInternational, Metals Park, Ohio, 1989, pp. 911–924.

12Whitmore, Stephen A., Development of a PneumaticHigh-Angle-of-Attack Flush Airdata Sensing System,SAE-912142, Sept. 1991.

13Whitmore, Stephen A., and Timothy R. Moes,Failure Detection and Fault Management Techniquesfor a Pneumatic High-Angle-of-Attack Flush AirdataSensing (HI-FADS) System, NASA TM-4335, Jan. 1992.

14Whitmore, Stephen A., Roy J. Davis, and JohnMichael Fife, In-Flight Demonstration of a Real-TimeFlush Airdata Sensing (RT-FADS) System,AIAA-95-3433, Aug. 1995.

15Shapiro, Ascher H., The Dynamics andThermodynamics of Compressible Fluid Flow, Volume I,John Wiley Sons, New York, 1953, pp. 83–88, 154.

16Haering, E. A., Jr., Airdata Calibration Techniquesfor Measuring Atmospheric Wind Profiles, Journal ofAircraft, Vol. 29, No. 4, Jul.–Aug., 1992, pp. 632–639.

17Whitmore, Stephen A. and Timothy R. Moes,Measurement Uncertainty and Feasibility Study of aFlush Airdata System for a Hypersonic FlightExperiment, AIAA-94-1930, June 1994.

18Huber, Paul W., Hypersonic Shock-Heated FlowParameters for Velocities to 46,000 Feet Per Second andAltitudes to 323,000 Feet, NASA TR R-163, LangleyResearch Center, Hampton Virginia, Dec. 1963.

19Franklin, Gene F., and J. David Powell, DigitalControl of Dynamic Systems, Addison-WesleyPublishing Company, Reading, Massachusetts,1980.

20Capone, Francis J., Linda S. Bangert, Scott C.Asbury, Charles T. L. Mills, and E. Ann Bare, The NASALangley 16-Foot Transonic Tunnel: HistoricalOverview, Facility Description, Calibration, FlowCharacteristics, and Test Capabilities, NASA TP-3521,Sept. 1995.

21Jackson, Charlie M., Jr., William A. Corlett, andWilliam J. Monta, Description and Calibration of theLangley Unitary Plan Wind Tunnel, NASA TP-1905,Nov. 1981.

22Bendat, Julius S. and Allan G. Piersol, RandomData: Analysis and Measurement Procedures, Wiley-Interscience, a division of John Wiley & Sons, Inc., NewYork, 1971.

23Freiberger, W.F., ed., The International Dictionaryof Applied Mathematics, D. Van Nostrand Company,Inc., Princeton, New Jersey, 1960.


Figure 1. A comparison of the VentureStar and X-33 vehicles.

Figure 2. Clock and cone angle definitions.

67 ft

128 ft 68 ft

Gross lift takeoff weight: 2,186,000 lb

Payload area

Payload area

VentureStarTM X-33

Gross lift takeoff weight: 289,000 lb

10 ft

127 ft

40 ft

971004

λφ XY

971005

Z

Surface normal

at i th port


Figure 3. X-33 surface measurement locations.

Figure 4. Exploded view of X-33 FADS pressure plug design.

0°16.25 in.

Left side view(looking inboard)

Nosecap centerof curvature

FADS port installation plugs

RCC skirt/ chin panel

5

3

2

6

4

1

33.86 in.

971006

~84 in.

RCC nosecap

Metallic TPSFront view

20°

20°

45°

φ

λ

16.25 in.

971007

Pressure fittings

Lock wire

Locking nut

Pneumatic tubing (0.21 in. ID, niobium)

High-temperature braze

Built-up padRCC skin

0.88in.

Threaded backside

C-103 niobium alloy plug

Pressure port

Plug lip

Plug-face

0.20 in.

1.75 in.

0.30 in.


Figure 5. X-33 FADS computational architecture.

Figure 6. Stability of the FADS Mach number iteration.

FADS algorithm

X2(l) =

6Σ

i = 16Σ

i = 1qi

qi[Pi – qc2[Ωi (α, β)] + P∞]2^ ^ ^

Computational path l

X2(l) < X

2(ll)

Output airdata

True False

971008

X-33 nosecap and skirtP1u

P1LP3uP40

P2 i

FADS algorithm

X2(ll) =

6Σ

i = 16Σ

i = 1qi

qi[Pi – qc2[Ωi (α, β)] + P∞]2^ ^ ^

Computational path ll

^ ^

P4 iP3L

P2 0

P6L

P5u

P6uP5L

1 2 3

Root 1, α = 0°Root 1, α = 50°Root 2

4 5 6 7 8 9

α = 50°

α = 0°

10

1.75

2.00

1.50

1.25

1.00

.75

.50

.25

0

Eigenvaluemagnitude

Freestream Mach number

Zero eigenvalue

Unstable algorithm

Stable algorithm

971009


Figure 7. The FADS startup algorithm.

ε

ε

M∞

Predefined startupMach number

Generate startup data

Triplesalgorithm

FADS qc, P∞iteration

FADS qc, P∞iteration

FADS aero-dynamic model

1/N

Measuredpressures(target set)

p1, p2, ...

p1, p2, ...

p1, p2, ...^

p1, p2, ...^ ^

^

p1, p2, ...^ ^

X

+ –

X

+ +

δp1, δp2, ...

i = 1, 2, ... N

971010

αe, βe,^^

αe, βe,^^

αe, βe,^^

αe, βe,^^ qc, P∞, ε^ ^

qc, P∞, ε^ ^


Figure 8. FADS angle-of-attack wind tunnel calibration.

Figure 9. FADS angle-of-sideslip wind tunnel calibration.

δα,deg

αe, deg971011

–20 –10

M∞ = 0.25

M∞ = 0.60

M∞ = 0.80M∞ = 0.90M∞ = 0.95

M∞ = 1.05M∞ = 1.20

M∞ = 1.60M∞ = 4.50M∞ = 2.00

0 10 20 30 40–6

–4

–2

0

2

4

6

8βe = 0°

16-ft, M = 0.2516-ft, M = 0.6016-ft, M = 0.8016-ft, M = 0.9016-ft, M = 0.9516-ft, M = 1.0516-ft, M = 1.20UPWT, M = 1.60UPWT, M = 2.00UPWT, M = 4.50

δβ,deg

βe, deg971012

–20 –15 –10 –5

M∞ = 0.25

M∞ = 0.60

M∞ = 0.80M∞ = 0.90M∞ = 0.95

M∞ = 1.10

M∞ = 1.60M∞ = 4.50M∞ = 2.00

0 5 10 15 20 25–4

–3

–2

0

–1

1

2

3

4

5 αe = 0°16-ft, M = 0.2516-ft, M = 0.6016-ft, M = 0.8016-ft, M = 0.9016-ft, M = 0.9516-ft, M = 1.0516-ft, M = 1.10UPWT, M = 1.60UPWT, M = 2.00UPWT, M = 4.50


(a) Variation with angle of attack.

(b) Variation with angle of sideslip.

Figure 10. FADS position error coefficient wind tunnel calibration.

ε

αe, deg971013

–20 –10

M∞ = 0.25

M∞ = 0.60

M∞ = 0.80

M∞ = 0.90

M∞ = 0.95

M∞ = 1.05M∞ = 1.20

M∞ = 1.60M∞ = 2.00

M∞ = 4.50

0 10 20 30 40–1.5

–1.0

–.5

0

βe = 0°

16-ft, M = 0.2516-ft, M = 0.6016-ft, M = 0.8016-ft, M = 0.9016-ft, M = 0.9516-ft, M = 1.0516-ft, M = 1.20UPWT, M = 1.60UPWT, M = 2.00UPWT, M = 4.50

ε

βe, deg971014

–20 –15 –10 –5

M∞ = 0.25M∞ = 0.60

M∞ = 0.80

M∞ = 0.90

M∞ = 0.95

M∞ = 1.10M∞ = 1.05

M∞ = 1.60M∞ = 2.00M∞ = 4.50

0 5 10 15 20–1.0

–.8

–.9

–.7

–.5

–.6

–.4

–.3

–.2

–.1

0 αe = 0°16-ft, M = 0.2516-ft, M = 0.6016-ft, M = 0.8016-ft, M = 0.9016-ft, M = 0.9516-ft, M = 1.0516-ft, M = 1.10UPWT, M = 1.60UPWT, M = 2.00UPWT, M = 4.50


(c) Variation with Mach number.

Figure 10. Concluded.

Figure 11. A comparison of the FADS wind tunnel points and X-33 trajectories.

0 1–.7

–.6

–.5

–.4

–.3

–.2

–.1

0

2 3 4

εM

αe, βe = 0°

Mach number971015

2.50 .5 1.0

16-ft

UPWT

Descent

Ascent

1.5 2.0 3.0 3.5 4.0

16-ft wind tunnelUPWTMAFB trajectoryMAAF trajectory

4.5 5.0

30

20

10

0

–10

–20

–30

α true,deg

Mach number971016


(a) Percentage Mach number error.

(b) Angle of attack error.

Figure 12. X-33 FADS error estimates.

0

10

1 – σ error boundaryAllowable 1 – σ error

1 2 3 4 5

Machnumbererror,

percentage

Mach number971017

0.2

.3

.4

.5

.6

1 2 3 4 5

Angle ofattack error,

deg

Mach number971018

1 – σ error boundaryAllowable 1 – σ error (includes 0.2 deg systematic bias)


(c) Angle of sideslip error.

Figure 12. Concluded.

Figure A-1. Lateral meridian triples: A comparison of β solutions.

0

.1

.2

.3

.4

.5

.6

1 2 3 4 5

Angle ofsideslip error,

deg

Mach number971019

1 – σ error boundaryAllowable 1 – σ error (includes 0.2 deg systematic bias)

0–20

–75

–50

–25

0

25

50

75

–10 10 20

β1, deg

αe = 0°

αe = 0°

αe = 50°

αe = 50°

β2,

deg

971020


(a) Ports 2, 4, and 6.

(b) Ports 2, 4, and 5.

Figure A-2. Comparison of β solutions, off-meridian triples, symmetric arrangement.

0–20

–75

–50

–25

0

25

50

75

–10 10 20

β1, deg

α = 18.207°α = 18.207°β2,

deg

971021

α = 50°

α = 21° α = 19°

α = 18°

α = 18°

α = 17°

α = 15°

α = 0°

α = 19°

α = 21°

α = 50°

α = 17°

α = 15°

α = 0°

0–20

–75

–50

–25

0

25

50

75

–10 10 20

β1, deg

β2,

deg

971022

α = 50°

α = 5°

α = 1°

α = 0.1°

α = 0.1°

α = 0.1°

α = 0.1°

α = –1°

α = –5°

α = 1°

α = 5°

α = 50°

α = –5°

α = –1°


(a) Ports 2 or 4, 3, and 6.

(b) Ports 2 or 4, 3, and 5.

Figure A-3. Comparison of β solutions, off-meridian triples, asymmetric arrangement.

0–20

–75

–50

–25

0

25

50

75

–10 10 20

β1, deg

β2,

deg

971023

φj = –90°

φj = 90°

0–20

–75

–50

–25

0

25

50

75

–10 10 20

β1, deg

β2,

deg

971024

φj = –90°

φj = 90°


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NSN 7540-01-280-5500 Standard Form 298 (Rev. 2-89)Prescribed by ANSI Std. Z39-18298-102

Design and Calibration of the X-33 Flush Airdata Sensing (FADS)System

242-33-02-00-23-00-TA3

Stephen A. Whitmore, Brent R. Cobleigh, and Edward A. Haering

NASA Dryden Flight Research CenterP.O. Box 273Edwards, California 93523-0273

H-2219

National Aeronautics and Space AdministrationWashington, DC 20546-0001 NASA/TM-1998-206540

This paper presents the design of the X-33 Flush Airdata Sensing (FADS) system. The X-33 FADS uses a matrix ofpressure orifices on the vehicle nose to estimate airdata parameters. The system is designed with dual-redundantmeasurement hardware, which produces two independent measurement paths. Airdata parameters that correspond to themeasurement path with the minimum fit error are selected as the output values. This method enables a single sensor failureto occur with minimal degrading of the system performance. The paper shows the X-33 FADS architecture, derives theestimating algorithms, and demonstrates a mathematical analysis of the FADS system stability. Preliminary aerodynamiccalibrations are also presented here. The calibration parameters, the position error coefficient ( ), and flow correction termsfor the angle of attack ( ), and angle of sideslip ( ) are derived from wind tunnel data. Statistical accuracy of thecalibration is evaluated by comparing the wind tunnel reference conditions to the airdata parameters estimated. Thiscomparison is accomplished by applying the calibrated FADS algorithm to the sensed wind tunnel pressures. When theresulting accuracy estimates are compared to accuracy requirements for the X-33 airdata, the FADS system meets theserequirements.

εδα δβ

FADS Calibration, Flush airdata sensing (FADS), X-33, RedundancyManagement, Reusable Launch Vehicle (RLV)

A03

36

Unclassified Unclassified Unclassified Unlimited

January 1998 Technical Memorandum

Available from the NASA Center for AeroSpace Information, 800 Elkridge Landing Road, Linthicum Heights, MD 21090; (301)621-0390

Presented at the AIAA Aerospace Sciences Meeting and Exhibit, January 12–15, 1998, Reno, Nevada.AIAA-98-0201

Unclassified—UnlimitedSubject Category 06

design and calibration of the x-33 flush airdata sensing ... · for 0.2 ≤ m ≤ 4.0. 5. dynamic...

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