NASA Technical Memorandum 4775

Development of a Closed-Loop

Strap Down Attitude System

for an Ultrahigh Altitude

Flight Experiment

Stephen A. Whitmore, Mike Fife, and Logan Brashear

January1997

NASA Technical Memorandum 4775

Development of a Closed-Loop

Strap Down Attitude System

for an Ultrahigh Altitude

Flight Experiment

Stephen A. Whitmore

Dryden Flight Research Center

Edwards, California

Mike Fife

Massachusetts Institute of Technology

Cambridge, Massachusetts

Logan Brashear

University of California

Los Angeles, California

National Aeronautics and

Space Administration

Office of Management

Scientific and Technical

Information Program

1997

I

DEVELOPMENT OF A CLOSED-LOOP STRAP DOWN ATTITUDE

SYSTEM FOR AN ULTRAHIGH ALTITUDE FLIGHT EXPERIMENT

Stephen A. Whitmore

NASA Dryden Flight Research Center

Edwards, California

Mike Fife

Massachusetts Institute of Technology

Cambridge, Massachusetts

Logan Brashear

University of California

Los Angeles, California

Abstract bank angles. Effects of wind shears were evaluated and,

for most cases, can be safely ignored.

A low-cost attitude system has been developed for an

ultrahigh altitude flight experiment. The experiment uses

a remotely piloted sailplane, with the wings modified for

flight at altitudes greater than 100,000 ft. Mission a

requirements deem it necessary to measure the aircraft

pitch and bank angles with accuracy better than 1.0 ° and b

heading with accuracy better than 5.0 °. Vehicle cost

c

restrictions and gross weight limits make installing a

commercial inertial navigation system unfeasible, d

Instead, a low-cost attitude system was developed using E

strap down components. Monte Carlo analyses verified

that two vector measurements, magnetic field and F

velocity, are required to completely stabilize the error

equations. In the estimating algorithm, body-axis GUk+l

observations of the airspeed vector and the magnetic GVk+I

field are compared against the inertial velocity vector and

a magnetic-field reference model. Residuals are fed back

to stabilize integration of rate gyros. The effectiveness of Gzk + 1

the estimating algorithm was demonstrated using data

from the NASA Dryden Flight Research Center Systems H

Research Aircraft (SRA) flight tests. The algorithm was

I

applied with good results to a maximum 10° pitch and

"Aerospace Engineer, Senior Member AIAA

_Graduate Student, Student Member AIAA

*Graduate Student

Copyright © 1997 by the American Institute of Aeronautics and

Astronautics, Inc. No copyright is asserted in the United States under

Title 17, U.S. Code. The U.S. Government has a royalty-free license to

exercise all rights under the copyright claimed herein for Governmental

purposes. All other rights are reserved by the copyright owner.

K

k

lat

long

Nomenclature

quaternion component 1

quaternion component 2

quaternion component 3

quaternion component 4

expectation operator

state equation matrix function, deg

airdata error covariance matrix, (ft/sec) 2

inertial velocity error covariance matrix,

(ft/sec) 2

magnetometer error covariance matrix,

(micro-Tesla) 2

altitude, ft

identity matrix

ume integral of roll rate, deg

time integral of pitch rate, deg

ume integral of yaw rate, deg

airdata/velocity correction gain matrix

time index

latitude, deg north

longitude, deg west

1

American Institute of Aeronautics and Astronautics
M[.]

mJ4

Pza

Pk+l,"k

e_+l/k+l

P

q

Ok+l

O.r

IIQ,II

r

T

t

to

U

V

W

Z

8

At

n

®

0

K

(0

Iloall

V

direction cosine matrix

i, jth component of direction cosine matrix

state error covariance matrix, previous

data frame, deg-"

predicted state error covariance

matrix, deg 2

corrected state error covariance

matrix, deg 2

roll rate, deg/sec

pitch rate, deg/sec

state equation error covariance

matrix, (deg/sec) 2

quaternion vector

norm of the quaternion vector

yaw rate, deg/sec

magnetic field reference datum,

micro-tesla

time, sec

initial time, sec

airspeed vector, ft/sec

inertial velocity vector, ft/sec

wind velocity vector, ft/sec

magnetometer measurement, micro-tesla

angle of attack, deg

angle of sideslip, deg

vector error, deg

sample interval, sec

angular velocity matrix, deg/sec

attitude vector, deg

pitch angle, deg.

magnetometer correction gain matrix

state transition matrix

bank angle, deg.

yaw angle, deg.

integrating factor matrix, deg

angular velocity vector, deg/sec

norm of the angular velocity integral, deg

gradient vector

[]_

[]e

[ ]_'k

[ ]k.1/_

[ ]k+l,k+l

[ 1_,,,

[ 1/o.8

[].

[] ....

[]r

[']

[']

Acronyms

ADC

DFRC

FCS

GPS

INS

MOS

RT-FADS

SRA

Superz_ripts and subscripts

vertical (+ down) component of

Earth-axis vector

east component of Earth-axis vector

state estimate from previous data frame

state estimate-based open-loop integration

state estimate after correction by

magnetometer or velocity data

lateral component of body-axis vector

longitudinal component of

body-axis vector

north component of Earth-axis vector

normal component of body-axis vector

matrix transpose

state estimate after magnetometer

correction

time derivative, l/sec

airdata computer

Dryden Flight Research Center

flight control system

Global Positioning System

inertial navigation system

model output statistics

real-time flush airdata sensing

Systems Research Aircraft

Introduction

Interest in ultrahigh altitude aircraft for atmospheric

sampling and remote Earth-sensing is growing, and

several aircraft are currently in the process of proving

the feasibility of extended duration flight to a maximum

altitude of 80,000 ft. Requirements for flights as high as

an altitude of 120,000 ft have been identified. These

flight regimes are difficult to design for, because

research into low-Reynolds number, high-subsonic

aerodynamics has been very limited. Although some

basic airfoils for this flight regime have been analyzed

and tested in wind tunnels, 1.2fundamental data on entire

vehicle aerodynamics, flight mechanics, and flight

performance are lacking.

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A preliminary design study was undertaken at the

NASA Dryden Flight Research Center (DFRC) recently 3

with the objective of finding a satisfactory method for

achieving trimmed flight at an altitude of 100,000 ft. The

study for a high-altitude flight experiment examined

several possible techniques for achieving this objective.

The study examined the feasibility of using a highaltitude balloon to tow a remotely piloted sailplane to an

altitude of 100,000 ft, where it would be released and

flown back to a lakeside landing on Rogers dry lakebed

at Edwards Air Force Base, California.

The preliminary study concluded that the most feasible

approach is to use a commercially available sailplane,

with the wings modified for the low-Reynolds number,

high-subsonic Mach number flight. For the ultrahigh

altitude flight experiment, no propulsive power plant

exists and this provides a low-noise environment for

studying low-Reynolds number transition phenomena.

A portion of the unswept, untapered wing will serve as

a test section for examining flow transition physics

under these conditions. Figure 1 shows the ultrahigh

altitude vehicle.

The current mission plan requires a nosedown

balloon release, with booster rockets used to generate

pseudolift to turn the vehicle from its nosedown

configuration to level flight. After sufficient Mach

number, (approximately Mach 0.65) for free flight at

high altitudes has been achieved, the rocket pack is

jettisoned. At launch and during the transition

maneuver, accelerations along the vertical axis

are substantial. Figure 2 shows a schematic of the

mission concept.

The unpowered aircraft will collect airfoil

aerodynamics and vehicle performance data from launch

to a lakebed landing at Edwards Air Force Base. The

principal research objectives of the ultrahigh altitude

flight experiment program are:

1. To validate high-altitude airfoil design

methodologies by measuring airfoil and vehicle

characteristics at low Reynolds numbers and

high subsonic Mach numbers in a low-turbulence

flight environment.

2. To establish a test bed aircraft for ultrahigh altitude

flight research.

Onboard measurements include boundary-layer velocity

profiles from total pressure rakes at several streamwise

locations, chordwise pressure distributions, the

boundary-layer laminar-to-turbulent transition state,

airfoil section drag (fixed wake rake), and flight

mechanics data such as local angle of attack, free-stream

airdata, linear accelerations, angular rates, and the

aircraft attitudes. The experiment will use an onboard

data acquisition system, and data will be telemetered to

section

3 78

Top view

i

Front view

I

in. 78.03

t_t_ in.

960644

Figure 1. The ultrahigh altitude vehicle.

3

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.__Alrcraft release

i \

\,..Ascent \ Transition to

_''_',_ _orizontal fligh_ 1" _ _1 .... __... "_ Test maneuvers

/ RocketI)_==::_-urn "_ _ / r-.__ --------_

( to achieve

_. horizontal flight _ \"- _-_-_-'----_.--.--_-" J"\

Bsllo \_

Rocket pack \

I

release for recovery /

f

J

¢.

Edward's AFB =_- -A_rc_-a_;ndTng

Roger's Dry Lake _ ......

960645

Figure 2. Schematic of mission concept.

a ground-based recording station. Flight test maneuvers

will consist of stabilized turns to achieve higher than 1-g

trim angles of attack and constant lift coefficient or

Mach-number descents.

To accomplish the research objectives of this

experiment, measuring the absolute pitch and roll

orientation of the aircraft to better than 1.0° and a

resolution of 0.5 ° at the high altitudes is necessary. For

pilot navigation, measuring the vehicle heading with an

absolute accuracy of better than 5.0 ° with a resolution

of 1.0 ° is desirable. These attitude requirements result

from the narrow speed range allowable along the flight

envelope at these extreme altitudes. Flying too fast

causes the airfoil to develop shocks, causing potential

separation and loss of lift. At excessive speeds, a flutter

boundary may also be approached. Flying too slowly

causes the aircraft to approach stall speed. Typically, an

improper attitude will translate into an unacceptable

speed change within approximately 5-10 sec. This speed

change either drives the pilot into a "speed-induced"

longitudinal oscillation or reduces the pilot's attention to

nothing but speed control. For this program the tolerance

requirements for the test points are restrictive, and

precise attitude information is required to ensure that

multiple flight conditions can be met simultaneously.

The stated requirements for attitude and heading could

be achieved using state-of-the-art inertial navigation

systems, both gimballed and strap down, 4 but the costs of

these systems are considered excessive for the ultrahigh

altitude flight experiment program. Gross vehicle weight

at high altitudes is of concern, and the extra weight

penalty caused by adding a full inertial navigation

system (INS) was considered undesirable. Furthermore,

the desired accuracy requirements will not be met by

open-loop integration of strap-down rate gyros.

To circumvent this problem, a simple lightweight,

low-power consumption, strap-down attitude system

concept was developed for this program and is detailed

in this paper. In this system concept, body-axis

observations of the airspeed and magnetic-field vectors

are compared with the known measured inertial velocity

vector and known magnetic-field vectors (in Earthrelative coordinates) to provide a "virtually inertial"

reference that is used to infer an attitude error. This error

is then fed back to correct and stabilize the rate-gyro

integration. The system has the stability of gimballed

attitude systems, but relies on low-cost strap-down

components to gather the required information. The

system performance is analyzed for the launch trajectory

using Monte Carlo s error simulations. Effects of

instrumentation bias and random errors are analyzed.

Particular attention is paid to what type of feedback is

required to ensure full-loop closure and, hence,

algorithm stability.

4

AmericanInstituteof Aeronauticsand Astronautics
Dataderivedfrom the NASA DrydenSystem

ResearchAircraft(SRA)flight tests6 areusedto

demonstratethe effectivenessof the estimating

algorithm.Thealgorithmwasappliedto datafrom

several flightswithgoodresultsachievedforupto 10°

pitchandrollattitude.Reasonsforestimatedegradation

athighattitudesarediscussed. Effectsofwindshearson

attitudeestimatesareevaluatedusingrawindsonde

weatherballoonmeasurements.

Coordinate Definitions

Figure 3 shows the coordinate definitions used in this

report. The Euler angles describe the aircraft body-axis

orientation (in longitudinal, lateral and normal

coordinates, with respect to the local tangent plane of the

Earth and true north) in north, east, and down

coordinates. The direction cosine matrix that allows

transformation from Earth axis to body axis, is the

product of three successive rotations: 6

Here, 0 is the pitch angle, ¢ is the roll rate or bank

angle, and _ is the yaw or heading angle. The angular

velocity of the aircraft is typically measured by

gyroscopic instruments fixed to the aircraft body axis.

The angular velocity vector (p is the roll rate, q is

the pitch rate, and r is the yaw rate) expressed in body

axis, is related to Earth axis by a transformation similar

to equation 1:6

1 [cosO (sin(l)sinO) cosOsinO

= c--osO/ cos¢cosO -sinOcosO

sin@ cos O

i] (2)

Proposed Attitude System

Equation 3 can be directly integrated to give the Euler

angles, given a known initial condition. But because

equation 3 is neutrally stable, bias or systematic errors in

M[O, ¢, _1 = I'0 cos¢0sine01FCoS00So01 1

0 -sine cos¢_] [_sin0 0 cos0_]

cos_ sinq/ 0

-sinw cosw 0

0 0 1

I ( cos 0 cos q/) ( cos 0 sin V) -( sin 0 )

= ](sinCsin0cos_F-cosOsin_) (sin0sin0sinw+ cos¢cos_) (sin0cosO = {mi, j}

t.(cos@sin0cos_ + sinCsin_¢) (cosOsin0sinw- sin0cosw) (cos¢cos0

4 -- --/-'J-- 7" ...... "]'_ ___ -- Horizontal

Vertical

960646

(I)

Figure 3. Axes and coordinate definitions.

5

American Institute of Aeronautics and Astronautics
the rate-gyro measurements cause the integration to drift

from the true attitudes as a function of time. Thus, loop

closure is required to get a stable attitude measurement.

For conventional inertial systems, a stable reference is

provided by a gimballed platform. Payload limitations of

the vehicle and large vertical accelerations at launch

make this approach unfeasible for this ultrahigh altitude

flight experiment, as described earlier.

The system proposed for the ultrahigh altitude vehicle

uses strap-down components that are low in cost, do not

require a stable member to perform the integration, and

are insensitive to large accelerations in the vertical axes.

Body-axis angular rates are integrated and stabilized

using measurements of airspeed, inertial velocity, and

magnetic-field vectors. As will be shown in the results

and discussion section, two vector measurements,

magnetic field and velocity, are required to achieve

complete three-attitude stability. The approach compares

body-axis observations of the magnetic field with a

magnetic-field datum derived from a mathematical

model 7 and the airspeed vector with the inertially sensed

velocity to infer an attitude error. This error signal is fed

back to stabilize the gyro integration. Because

closed-loop stability is achieved with strap-down

components, this system may be referred to as a

"virtually inertial" attitude system. Table 1 shows the list

of required measurements, the data sources, and

approximate sample rates at which the data will be used

for typical applications.

Required accuracy for the various sensing components

will be developed by the simulation studies to be

presented in the results and discussion section. Figure 4

shows the basic system layout along with the structure

of the estimation algorithm (to be developed in the

next section).

Table 1. List of required measurements.

Measurement Axis system Source Update rate

Angular rates, p, q, r

Magnetic field vector, Z

Airdata, U., _, I]

Inertial velocity, V,, V,, Vj

Aircraft lat, long, H

Magnetic field datum, T

body axis

body ax_s

body axLs

Earth axLs

Earth ax_s

Earth axis

Rate gyro, 3-axis 20-50 Hz

3-axis magnetometer 1 Hz

Airdata probe/boom 1-10 Hz

Global Positioning System (GPS) 1-10 Hz

Global Positioning System (GPS) 1 Hz

Spherical harmonic model 1 Hz

r

Rate gyro

(3-exls)[p]

k+llk+l

Magnetometer

(:)-axis)

k+l

Major frame

JAt k

k+llk

+__

+ False

i

LSv]

Zx, Zy, Z z

Magnetometer Tn, Te Td

correction I-----]

Spherical

harmonic

model

GPS

reclever

probe

,e--Latitude, longitude, altitude

Figure 4. Schematic of the ultrahigh altitude flight experiment attitude system.

_0_7

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American Institute of Aeronautics and Astronautics
Development of the Estimation Algorithm

This section develops a closed-loop estimation

algorithm for the strap-down (nongimballed) attitude

system. Measurement errors from the independent

systems, (such as airspeed, magnetometer, inertial

velocity, and angular rates) are assumed to be

uncorrelated. The algorithm emulates the form of the

Kalman filter, modified for asynchronously arriving data

(the fundamental Kalman assumes that all data from the

various measurement sources arrive simultaneously).

For the types of disparate data sources to be used for

ultrahigh altitude flight systems (for example,

magnetometer, Global Positioning System (GPS),

analog), a frame-by-frame algorithmic approach is too

restrictive. To circumvent this difficulty, the estimation

algorithm is implemented as a two-step predictor/

corrector filter where the prediction-step (quaternion

integration of the rate-gyro data) is performed at a fixed

rate, typically 25-50 Hz, and the correction step is

performed whenever fresh velocity and magnetometer

data are available, typically 1-10 Hz.

Quaternion Formulation of the Rate Integrator

As is typical of all modern INS systems, to eliminate

problems with infinite angular rates caused by the

nosedown initial attitude at launch, the estimation

algorithm formulates the problem in terms of quaternion

parameters. 4.8 In the quaternion transformation the

orientation is written as a 4-space vector with the

magnitude being constrained to always be unity. Using

mij for the elements of the direction cosine matrix (eq. 2),

then the quaternions can be evaluated by

ax E

m23 - m32

4d

m31 - m13

4d

m12 - m21

4d

i_/l + roll + m22 +m33

tlaxll2= _ (3)

This quaternion substitution transforms the kinematics

equations (eq. 2) to a linear form,

Ii] =5 1[:q: -p]000q 0]i -q -r

q(t) = o_(p, q, r) q(t) (4)

and the direction cosine matrix becomes

M Iilla2+d2-b2-c22(ab+cd) 2(ac-bd) 1 = [ 2(ab-cd) b2+d2-a2-c 2 2(bc+ad) I

L 2(ac+bd) 2(bc-ad) c2+d2-a2-b2J

(5)

The transformation from quaternions to Euler angles is

sin0 = 2(bd-ac)

2(ab + cd)

tan_ - 2(a2 + d2 ) _ 1

tan_ - 2(bc+ad)

2(c 2 + d 2) - 1

(6)

Solution of the Integrating Equation

Equation 4 is solved by the integrating factor 9 method:

Qx(t) = e_t 11( p, q, r) dt Qx(to) (7)

0

The integrating factor is

fl-r 0pr dt

e q

0 " to --p 0

-- --q -r

[ o / = e -Ir/2 0 Ip/2 Iq/21 _etl(P,q,r, t)

k-l p/2 -I q/2 -I r/2

(8)

where

"21r_t= r(t)dt lq_ t= q(t)dt lp_ _ p(t)dt (9)

0 2 ' 2 0 2 ' 2 o 2

and can be written in closed form by expanding the

exponential in a Taylor series 9

f_ _2 _3

e = I+_+-_-+-_-. +... (10)

and noting that

14 (11)

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The matrix, I, is the identity matrix. Collecting terms

and simplifying gives

e° = cos l+l_-_sm f/_q_(t, t0) (12)

resulting in the homogeneous linear equation

Qx(t) = _(t, to)Qx(to) (13)

The integrator is made recursive over a small time

step, At, using angular rates averaged over the interval

Qxk + I = _k + l, kQx k (14)

Equation 14 is the "one-step prediction equation," and

is always norm preserving. That is, the quaternion

magnitude will always be equal to unity, and one does

not have to re-normalize the quaternion values after each

integration cycle. This ensures greater numerical

accuracy and is a unique result not published in

navigation literature.

Correction of Integ-rated Data

At the end of each integration frame, the algorithm

corrects for drift instabilities by using the differences

between the magnetic-field and airspeed vectors,

measured in body axes and the magnetic-field datum and

inertial velocity vector, measured in Earth-relative axes.

A particular state error correction is performed only

when a fresh measurement is available. To allow for this

asynchronous operation, the correction step is

partitioned into two parts: a magnetometer correction,

and a velocity correction.

The magnetometer correction is performed first. In

this step, differences between the observed and expected

magnetic-field vector are fed back to stabilize the openloop integration,

I_] =[i3 +Kk+'IZk+_-MI_I Tk+'] (15a)

k + I/k+ I k+ l/k [._l.Jk + Ilk

Here, the matrix _k÷_is a "Kalman-styte" gain matrix,

Tk+_ is the magnetic-field reference datum in Earth

coordinates, and Zk+_ is a three-axis magnetometer

measurement in body axis.

The vector

Ii]k + l/k

is the predicted state estimate based on open-loop

integration over one integration cycle, and the vector

I_lk + l/k + l

is the state estimate resulting from the magnetic-field

vector correction. The matrix M[.] is the transformation

matrix for rotation from Earth-to-body axis based on the

predicted state parameters

Zlong

Zlat

LZnor_

= M T e

t_Vdk + Ilk

(15b)

If no fresh magnetic-field data are available, then the

magnetometer correction step is ignored.

After the magnetometer update, the result is corrected

using newly acquired velocity data (if available):

+ l/k+ I

--" Ii]k+l/k+l

+ Kk+ 1[ gk +

I-M

(16)

k+l/k+l

Here, the matrix Kk+) is the gain matrix, Ug+_is the

body-axis airspeed vector, Vk+x is the inertial velocity

measurement (from the GPS), and Wk+x is the vector

representing the local atmospheric winds. As with the

previous correction, the vector

Ii]k + l/k + l

is the new state estimate. As with the magnetometer

correction step, if fresh velocity data are unavailable,

then the velocity correction step is ignored.

With no direct measurement of the vehicle attitudes

available, the wind terms in equation 16 are difficult to

sense in real time. Conceptually, the wind vector can be

estimated by comparing inertial velocities to airdata

8

American Institute of Aeronautics and Astronautics
measurements. Butbecausetheinertialvelocityvector is

sensed in the Earth-relative axis system and the airdata

velocity vector is sensed in the body-axis, the vehicle

attitudes are implicit in the velocity comparison.

This implicitness requires that the attitudes and winds

are iteratively estimated. In the iterative method,

the attitudes would first be estimated assuming no

winds. Using these attitude estimates, the winds are

estimated from the differences between the inertial

and airdata velocity vectors. The attitudes are then

reevaluated iteratively using the estimated winds.

Two primary difficulties exist with the iterative method.

First, such a complex iterative scheme is very difficult

to run in real time; and second, because the attitude

equations are highly nonlinear, no guarantee exists

that the iterative equations will converge. This

nonlinearity makes the iterative algorithm unsuitable for

flight-critical navigation.

For simplicity, the winds will either be measured by

weather balloon or ignored altogether. As described later

in the Results and Discussion section, the winds can

generally be measured with good steady-state accuracy

by rawindsonde weather balloonsJ ° but these data are

subject to diurnal (time) and spatial variation errors.

Ignoring the winds, or using wind measurements with

large errors, will result in biases in the attitude estimates;

however, these errors will not affect the stability of the

algorithm. The effects of wind shear will be illustrated in

the Results and Discussion section using data from the

SRA flight tests.

Computation of the Gain Matrixes

No optimal method exists for selecting the gain for a

nonlinear system of equations; however, if the gain is

selected to emulate the form of the Kalman gain

matrix, 11 a robust closed-loop algorithm results. The

resulting formula for the gain matrixes are as follows:

Magnetometer G_ha:

_k+l = [Pk+l/k] V M Tk+ 1]T

L:] T

Tk+l +Gzk+

-I

(17)

Velocity Gain:

xI[]l/_][E:] V _ M 1+ vV6]Mv,+, [P,+ v,+, Gv,+,

[:]

.-I

I:l V:]1

+M GVk+l M r

L_/Jk+I/_.I LYJk+I/t-+I

(18)

In equations 17 and 18, the operator [ ]v is the matrix

transpose, P_+ l/k is the error covariance of the

integrated state estimate, Pk+ l/k+l is the error

covariance of the state estimate after the magnetometer

correction, Gzk+l is the errorcovariance of the

magnetometer measurement vector, Guk + I is the error

covariance of the airspeed measurement vector,

and Gvk + 1 is the error covariance of the inertial

velocity measurement vector. The predicted attitude

gradient matrixes are

Viii M Tk + i]

Vo3M r:] 3M r:t

/ LVJ T LWJr

= / @_ k+l _ k÷!

t

+,+If

LWJT ]

(19)

and

v[i]M w+. iI

V+[:l

= --_ VI: +

++F1

L_Jv

i i_¢ *+

L_3V

I _ _:+ '](20)

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American Institute of Aeronautics and Astronautics
The forms of the gain matrixes determine that

very little correction occurs to the integrated state if

the measurement (magnetometer and velocity vectors)

covariances are large. Conversely, if the integration

error covariance is large, then the filter applies a

large correction based on the measured data. In this

case, the gradient is a sensitivity matrix describing

how much correction should be applied to the integrated

state vector based on the magnetic-field and velocityvector feedbacks.

Computation of the Error Covarianc¢ MatriCes

A mechanism for evaluating and propagating the error

covariances from one data frame to the next must be

developed for the filter to be complete. Again, because of

the highly nonlinear nature of the problem, an exact

expression cannot be developed tbr the covariance

propagations. Instead, approximate expressions are

derived from the equations of motion, linearized about

the state estimate from the previous data frame.

The error covariance of the integrated state estimate is

derived indirectly from the basic kinematics equations.

Equation 2 is written here in vector form as

0 = F[O]¢0 (21)

where

O= , co=

1 Ic°sO

=c-osoL

i]' and

sinCsin0 cosCsin0

cos ¢ cos 0 -sin ¢ cos 0

sine cos¢

(22)

By integrating equation 22 over one data frame using

explicit _2 differencing to approximate the time

derivative, the result is

f x

Ok. l/k = Okk+ At_F[Oklk]OJk) (23)

Taking perturbations and collecting terms gives the

following error equation:

8®k + 1/k = [I + At Vo(F[Ok/t]t.Ok)]8Ok/k

+ At F[Ot/k]8o _

(24)

Defining the covariance matrices

Pk + l/k = E E8Ok+ l/kSOkT+ 1/k 1'

I TI Pk/k = E 80k/kSOk/k ,

Qk = E[80_kS0_kT_, etc.

(25)

and assuming that the current input error &.ok is

uncorrelated with the current state error 80_k, then the

prediction error covariance update equation can be

written approximately as

Pk + l/k

+ At2F[O_/klQtFr[Ot/t,I

(26)

After the correction steps are performed, the error

covariances of the state estimates are reevaluated in a

similar manner by propagating the error through

correction equations. The resulting propagation

equations are as follows:

magnetometer covariance update:

[ 1] Pk+l/_+l ---- l-kk+ 1 V O M Tk._l Px-+ l/t: (27)

velocity covariance updaI¢:

Pk÷l/k+l = (28)

Equations 2, 4, 15, 7, 27, 18, 20, 28, 19, 21, and 29 are

the collected algorithm equations. The equation set is

repeated recursively until the data stream is exhausted. If

at a particular time a new measurement is unavailable,

then that correction step is not performed, and the state

and covariance matrixes remain unchanged. Figure 4

shows the basic algorithm structure as discussed earlier.

10

American Institute of Aeronautics and Astronautics
Results and Discussion

This section presents results that evaluate the

performance of the estimating algorithm. The system

performance is analyzed for the ultrahigh altitude flight

experiment launch trajectory using Monte Carlo error

simulations. The estimation algorithm is also

demonstrated using flight data derived from the SRA

flight tests. Effects of wind shears on the attitude

estimate are evaluated using these data.

Six-Degree- 0f-Freedom Monte Carlo Errors Analyses

The performance of the attitude estimation algorithm

was evaluated by Monte Carlo error simulations

generated using the NASA Dryden six-degree-offreedom piloted simulation for the ultrahigh altitude

aircraft. Typical balloon-release maneuvers were flown

to give representative trajectories for the vehicle launch.

These data were then corrupted by bias and random

errors, magnetic compass lags, and measurement

latencies to simulate the types of real-world

measurement errors that would be encountered. Bias

errors were introduced into the various measurements at

the beginning of each simulation run using a random

number generator and a specified bias standard

deviation. When the values for the biases were set for a

particular data run, they were held constant throughout

the remainder of the run. Random errors were generated

in a similar manner, except these errors were added to the

various data sources at each data frame.

The effects of systematic errors such as compass lags

or data latencies were evaluated using various filters that

modeled these effects. Also, the piloted launch was

repeated several times with different initial launch

conditions for heading and initial angular rates.

Magnetometer data were generated using a spherical

harmonic model of the Earths magnetic field 7 with the

aircraft longitude, latitude, and altitude as inputs to the

magnetic-field model. These data were then rotated to

body axes and corrupted with bias, random, and compass

lag errors. The compass lag errors were added to

simulate the induced errors caused by crossing lines of

magnetic flux as the aircraft changes pitch attitude or

heading attitude. Relative to the expected magnetometer

errors, the magnetic-field reference model is extremely

accurate, and the resulting reference data are assumed to

be known without error.

For each simulation run, the corrupted data were used

by the attitude-estimating algorithm to generate Euler

angle estimates. Comparisons of the estimated attitude

angles with the actual attitude angles (from the

simulation) give quantitative measures of the system

performance. To obtain ensemble averages of the errors

induced by random and bias error components, the

simulations were run repeatedly and model output

statistics (MOS) were generated. The measurement

requirements to achieve the 1.0 ° attitude accuracy of the

program were determined by this method. Particular

attention was paid to determine when a given set

of measurements provided closed-loop stability to

the algorithm.

Figures 5 through 9 show results of a typical

simulation run. Figure 5 shows the basic launch

trajectory, where time histories of the airdata parameters

(airspeed, angle of attack, altitude, and the roll, pitch,

and yaw rates) are presented. Figure 5 shows the

uncorrupted and corrupted data that was actually used to

perform the analyses. Table 2 presents the bias and

random error standard deviations that were used in this

simulation run.

The integration is performed at a rate of 25 samples/

sec, the magnetometer data are assumed to be available

5 times/sec, and the velocity data are obtained every 1

sec. The corrupted magnetometer data were lagged using

a second-order filter with a time constant of 1.25 sec, and

a damping ratio of 0.8.

Figure 6 shows a comparison of the true Euler angles

with the attitudes derived from open-loop integration of

the corrupted angular-rate data. Clearly, the integration is

Table 2. Bias and random error standard deviations of measurement data.

Measurement Bias error Random error

Angular rates, p, q, r

Magnetic field vector, Z

Airdata, V_, e¢, [5

Inertial velocity, Vn, Ve, Vj

Magnetic field datum, T

-0.81, 0.72, -0.77 deg/sec

-1.7, 1.1, 0.74 percent

-2.2 ft/sec, -0.14 °, -0.15 °

-1.3,-2.6,-3.7 ft/sec

no error

_+0.25 deg/sec

.+.5.0%

+_5.0 ft/sec, _+0.25 °, _+0.25 °

+3.0 ft/sec

no error

11

American Institute of Aeronautics and Astronautics
Airspeed,

ft/sec

700

600

5O0

400

30O

200

100

0

-100

0 12 24 36 48 60 72 84 96 108 120

Time, see

960_S

(a) Airspeed.

True airspeed

----- Noise corrupted

airspeed

7.00

5.75

4.50

3.25

Angle of

attack, 2.00

deg

.75

-.50

-1.75

-3°00

12 24 36 48 60 72 84 96 108 120

Time, sec

96o649

(b) Angle of attack.

True angle of

attack

Noise corrupted

angle of attack

Altitude,

ft

112,000

109,500

107,000

104,500

102,000

99,500

97,000

94,500

92,000

12 24 36 48 60 72 84 96 108 120

Tlme, sac

960650

(c) Altitude.

Figure 5. Ultrahigh altitude flight experiment launch trajectory time histories.

True altitude

-- -- - Noise corrupted

altitude

12

American Institute of Aeronautics and Astronautics
Roll rate,

deg/sec

1.500

1.125

.750

.375

0

-.375

-.750

-1.125

- 1.500

--If !

0 12 24 36 48 60 72 84 96 108 120

Tlme, sac

960651

(d) Roll rate.

True roll rate

---- - Noise corrupted

roll rate

Pitch rate,

deg/sec

6

4

2

0

-2

-4

-6

-8

: i : : ; ; ;

..... i_ i .... _ i Ii _ i

A ........ t = .....

0 12 24 36 48 60 72 84 96 108 120

Time, sec

960652

(e) Pitch rate.

True pitch rate

--- - Noise corrupted

pitch rate

Yaw rate,

deoJsec

1.500

i.....Iit .....if',: ̧,

--ih r _-ff_-i 1[

i i![l' il ,,l'_i

l[ili ,,,,I!!t':.,_,tti 1 thl i q l

,.,. .... ,,...... ii........ .......... ,,....... ___

-.375 .--- I_, I!,i!;!!i!i,i;i!lli

-1.125 ..........

- 1.500

0 12 24 36 48 60 72 84 96 108 120

Time, sac

96O653

(f) Yaw rate.

Figure 5. Concluded.

True yew rate

Noise corrupted

yaw rate

13

American Institute of Aeronautics and Astronautics
Pitch

angle,

deg

3O

15

0

-15

-30

-45

-60

-75

-90

0 12 24 36 48 60 72 84 96 108 120

Time, sec

960654

(a) Pitch angle

True angle

-- Open-loop estimate

Bank

angle,

deg

2O

1°Io........ .... • _1o......__ ................... ! i iii

-20

-40 .................. _ ..... .......................

-50 .....:....................... :............. i........... _ ;

- 60 [ :

0 12 24 36 48 60 72 84 96 108 120

Time, sec

960656

(b) Bank angle

True angle

-- Open-loop estimate

Heading

angle,

deg

70 I _ i :

60 I ........i........

.................... _.......... r..................... ;.......... .........

True angle

-- Open-loop estimate

0 12 24 36 48 60 72 84 96 108 120

Time, sac

960656

(c) Heading angle.

Figure 6. Launch Euler angle comparisons: true attitudes versus open-loop estimates.

14

American Institute of Aeronautics and Astronautics
unstable, and the three attitude angles wander from the

true attitudes. Figure 7 shows similar comparisons,

except the attitudes have been estimated using the

magnetometer data for partial system loop closure.

Clearly, the performance is now much better, having

only small drifts on pitch and bank angles and a more

substantial drift in the heading angle. The system is not

yet stable, and the stringent requirements of the ultrahigh

altitude flight experiment are not met.

Figure 8 shows the same comparisons, except now

only the airspeed and inertial velocity differences have

been used to partially close the loop. Again, the system is

fairly stable in the pitch and yaw axes, but drifts badly in

the roll axis. Because airdata provided only small

amounts of information regarding the roll of the vehicle,

this result is expected. Figure 9 shows the Euler angle

comparisons where magnetometer, airspeed and velocity

differences have been used for loop closure. Here, the

system loop has been completely closed, and the

computations are stable in all three axes. Almost

immediately after launch, the estimates converge to

the true values. The error equations have been

fully stabilized.

Clearly then both the magnetometer and airspeed and

inertial velocity stabilizations are required for a

completely reliable system. Based on the accuracy and

resolution specifications stated earlier, the Monte Carlo

analyses MOS were used to assess the maximum bias

and random errors allowable to meet the ultrahigh

altitude flight program requirements. Table 3 shows

estimates of these data for the fully stable system.

Measurement latency limitations have yet to be

established, but based on the simulation runs, these

requirements are very relaxed and can be easily achieved

with conventional instrumentation methods.

The key to achieving this instrumentation robustness is

in using the two independent attitude references, the

magnetic and velocity-vector fields, to completely close

the loop on the error equation so that a stable set of

(linearized) error equations results. Even in the presence

of sizable measurement biases, the system errors will not

grow as a function of time.

Evaluation of Aglitu_te-Estimating Algorithm

Using Systems Research Aircraft Flight Data

As mentioned in the introduction, the effectiveness of

the estimating algorithm under real flight conditions will

be demonstrated using data derived from the SRA flight

tests. 13The SRA is an FFFF-18 aircraft and is an early

model equipped with a high-quality mechanical-gyro

INS. _4The attitudes from this INS system are used as the

reference standard, or "truth model," for this analysis.

Estimated accuracies of the INS attitudes are better

than _-+0.25° in pitch and bank angle and _+0.5° in yaw

angle. Other outputs from the INS include linear

accelerations, angular rates, inertial velocity, flight

path and ground track, and aircraft latitude, longitude,

and altitude.

The F/TF- 18 is also equipped with an additional set of

angular rate sensors that are used as feedback sensors for

the flight control system (FCS). 14The FCS rate gyros,

although efficient dynamically, are subject to small bias

errors and systematic drifts. Because FCS rate gyros are

independent from the INS (although clearly not as high

in quality as the INS angular rate outputs), the FCS data

were used for the roll, pitch, and yaw rate inputs to the

attitude algorithm. This use of FCS rate gyros avoids any

perceived incestuousness in the analysis. Estimated

accuracies of the rate data are better than +-0.1 deg/sec for

bias error, and _+0.2 deg/sec for random error.

Two airdata systems are available on the SRA: the

ship system airdata from the airdata computer (ADC), 14

or the research real-time flush airdata sensing

(RT-FADS) system. At low angles of attack, the

RT-FADS system and ADC have equivalent levels of

accuracy. At greater than 32° angle of attack, the ADC

systems stop operating, whereas the RT-FADS system

has been demonstrated to be accurate to a maximum

48 ° angle of attack. L3 For this reason, the RT-FADS

airdata were used in this analysis. Estimated accuracies

Table 3. Maximum bias and random errors allowable for the measurement data.

Measurement Bias error Random error

Angular rates, p, q, r

Magnetic field vector, Z

Airdata, U=, et, ]3

Inertial velocity, V,, Ve, Vd

+-0.1, +_1.50, +-2.5 deg/sec

+_10.0 percent

_10.0 ft/sec, ---0.4 °, -__-)-0.4 °

+ 10.0 ft/sec

+_.2.0deg/sec

+_.20percent

+15.0 ft/sec, +-0.5°, _+0.5°

+-15.0 ft/sec

15

American Instituteof Aeronautics and Astronautics
30

15

-lS

Pitch

angle, -30

deg

- 45

-60

-75

-90

0 12 24 36 48 60 72 84 96 108 120

Time, sac 960657

(a) Pitch angle.

True angle

-- Open-loop

estimate

Bank

angle,

deg

200

150

100

5O

0

- 50

-100

-150

- 200

0 12 24 36 48 60 72 84 96 106 120

Time, sac ;60658

(b) Bank angle.

True angle

-- Open-loop

estimate

Heading

angle,

deg

2OO

150

100

50

0

- 50

-100

-150

- 200

0

.... !!

12 24 36 48 60 72 84 96 108 120

Time, sac

96(3659

True angle

-- Open4oop

estimate

(c) Heading angle.

Figure 7. Launch Euler angle comparisons: true attitudes versus - magnetometer only - estimates.

16

American Institute of Aeronautics and Astronautics
30

15

0

-15

Pitch

angle, -30

deg

-45

-60

-75

-90

12 24 36 48 60 72 84 96 108 120

Time, eec

96o660

(a) Pitch angle.

True angle

-- Open-loop estimate

Bank

angle, -20

deg

-40

-60

-80

-100

0 12 24 36 48 60 72 84 96 108 120

Time, lec 960661

(b) Bank angle.

True angle

Open-loop estimate

70

60

50

40

Heading

angle, 30

d_

20

10

0

-10

0

True angle

-- Open-loop estimate

12 24 36 48 60 72 84 96 108 120

Time, lmc

_6o662

(c) Heading angle.

Figure 8. Launch Euler angle comparisons: true attitudes versus airdata only estimates.

17

American Institute of Aeronautics and Astronautics
30

15

0

-15

Pitch

angle, -30

deg

-45

-60

-75

-90

12 24 36 48 60 72 84 96 108 120

Time, sec

960663

(a) Pitch angle.

True angle

-- Open-loop estimate

150 |. ....... ....... i ....... _.... i .......

100 .............. i .... i ........................

Bank 51 _ ....

angle,

deg

-100 til ill ...... : .... -150 ....... _ _ _ _ ..... ,- _ _ _ ..... :.... .... _ ....

-200 L

0 12 24 36 48 60 72 84 96 108 120

Time, aec

96o664

(b) Bank angle.

True angle

-- Open-loop estimate

2OO

1,50

100

50

Heading

angle, 0

deg

-50

-100

-150

-2OO

12 24 36 48 60 72 84 96 108 120

Time, lee

960665

(c) Heading angle.

True angle

-- Open-loop estimate

Figure 9. Launch Euler angle comparisons: true attitudes versus magnetometer estimates.

18

American Institute of Aeronautics and Astronautics
of the RT-FADS system 13 are 3-5 ft/sec in airspeed,

and +0.25 ° in angle of attack, and angle of sideslip. The

system has been tested and found to be reliable to a

maximum of Mach 1.60.

Again, to avoid any perceived incestuousness in the

analysis, the inertial velocity measurements from the

INS were not used in this analysis. Because, the SRA

was not equipped with a GPS, inertial velocity estimates

were obtained using C-band radar tracking data. In this

procedure, space-positioning data are numerically

differentiated to give velocities relative to the Earth-axis

system. For radar elevation angles greater than 10°, the

random accuracies in the radar velocity data are believed

to be approximately +4-8 ft/sec and are of necessity

unbiased (because of the numerical differentiation). 13

Time skews resulting from the differentiating filters

were removed before processing the data in the

attitude algorithm. Although not truly a strap-down

measurement, these C-band data are a good

approximation of the type of data that would be acquired

by an onboard GPS system.

Three-axis magnetic-field measurements were not

available on the SRA vehicle. Instead, the F/TF-18 SRA

has a single-axis magnetic compass available as a part of

the cockpit instrumentation, and this instrument was

used to generate a pseudo 3-axis magnetometer data set.

The estimated compass accuracy is +1 ° in bias, and

+5 ° random. For a given latitude and longitude, the

reference magnetic-field datum vector (from the

spherical harmonic model) was rotated about the vertical

axis until the vector angle from magnetic north matched

the compass heading. Similarly, a pseudo pitch angle

was generated from the radar flightpath angle and the

angle-of-attack measurement. This angle was used to

rotate the magnetic-field vector to the proper pitch

attitude. As no data for bank angle were available

independent of the INS measurements, the magneticfield vector was not rotated about the roll axis. To prevent

erroneous data corrections caused by this lack of

roll information, the magnetometer correction was

performed only when the absolute value of the predicted

bank angle was less than 10°. This lack of bank angle

information at high roll angles presents a less-than-ideal

scenario for the estimating algorithm.

Flight 540 Res_alts

Results from SRA flight 540 will now be presented.

Calculations are performed from takeoff to landing and

are typical of results from other SRA flights. All of the

data used are from the sources described in the previous

section. Figure 10 shows time histories of the airdata

parameters airspeed, angle of attack, and altitude, and

the angular-rate parameters roll, pitch and yaw rate.

As with the attitude analyses presented earlier in the

paper, figures 11 and 12 show comparisons of the

estimated attitudes and INS attitudes. Figure 1! shows a

comparison of the INS-derived Euler angles with the

attitudes derived from open-loop integration. The rategyro integration was performed at a rate of 25 samples/

sec. Clearly, the open-loop estimates are unstable, and

the Euler components wander around the INS values

throughout the flight. Figure 12 shows attitude

comparisons where corrections have been applied to

compass, velocity, and airdata once every second to

stabilize the rate-gyro integration. As predicted by

the earlier Monte Carlo simulations, the error equation is

stabilized and no significant long-term attitude

drift occurs.

Figure 13 shows a subset of the data shown in

figure 12, but with an expanded time scale to more

clearly show the level of agreement between the INS and

estimated attitudes. For bank angles less than 10°, the

agreements are within the specified requirements for this

experiment (errors less than :£-0.5° in pitch axis and +1.0 °

in roll). For roll angles larger than 10°, the estimates

degrade in both pitch and roll axes, with pitch angle

errors as large as -3 ° and roll angle errors as large

as-10 °. As mentioned previously the deviation from

INS attitudes is not surprising given that for this analysis

no magnetic field correction is applied for bank angles at

greater than 10°. The ultrahigh altitude flight experiment

heading accuracy requirement of +_5° is met throughout

the entire flight. Analyses of data from other SRA flights

have produced similar results.

The Effects of Wind Shear on the Attitude Estimates

As mentioned earlier, the wind velocity will be

typically ignored by the ultrahigh altitude flight

experiment algorithm because measuring in real time is

difficult. In the presence of significant wind shears, the

result of ignoring the winds is a bias or systematic error

in the attitude estimates. Figures 14 and 15 show the

effect of wind shear using data from the SRA flight 540.

For this flight, the atmospheric winds were measured

before and after flight using rawinsonde _1 weather

balloon soundings, interpolated as a function of time to

minimize the effects of diurnal variations in the winds.

During the flight, the vehicle rapidly climbed from an

altitude of 10,000 ft to an altitude of 40,000 ft, resulting

in a large change in the easterly component of the wind.

Figure 14 shows time histories of the horizontal wind

components and altitude. The full feed back (magneticfield, airdata, and velocity) attitude-estimating algorithm

was run for this segment of data with and without

the measured winds being included in the computations.

19

AmericanInstitute of Aeronauticsand Astronautics
1100

Airspeed, 600

fUsec

100

0 400 800 1200 1600 2000 2400 2800 3200 3600 4000

Time, eec _oO666

(a) Airspeed.

Angle of

attack,

deg

5O

25

400 800 1200 1600 2000 2400 2800 3200 3600 4000

Time, sec

960667

(b) Angle of attack.

50,000

Altitude,

ft 25,000

400 800 1200 1600 2000 2400 2800

Time, sec

32OO 36OO

(c) Altitude.

Figure 10. SRA flight 540 time history, takeoff to landing.

400o

960668

20

American Institute of Aeronautics and Astronautics
Roll rate,

deg/sec

75

I i i :

I

'["' ! .... r'

-75

0 400 800 1200 1600 2000 2400

Time, sec

2800 3200 3600

(d) Roll rate.

4000

_0_9

Pitch rate,

deoJsec

o i

-15 _

0 400 800 1200 1600 2000 2400

Time, sec

(e) Pitch rate.

2800 3200

i

3600 4000

960670

Yaw rate,

deg/eec

10

-10

0

i r , J i

1200 1600 2000 2400 2800

Time, sec

(f) Yaw rate.

Figure 10. Concluded.

i

3200 3600 4000

960671

21

American Institute of Aeronautics and Astronautics
100

INS attitude

Open loop estimate

Pitch

angle,

(leg

-100

/

I

400

r I i i

1200 1600 2000 2400

Time, sec

, B i

800 2800 3200 3600

(a) Pitch angle.

40O0

960672

200

INS attitude

-- Open loop estimate

Bank

angle, 0

deg

- 200

k

0 400

i i ! !

800 1200 1600 2000 2400 2800 3200 3600 4000

Time, sec 960673

(b) Bank angle.

200

Heading

angle, 0

deg

-200

INS attitude

Open loop estimate

0 400 800

i

1200 1600 2000 2400 2800 3200 3600 4000

Time, sec 960674

(c) Heading angle.

Figure 11. SRA flight 540 Euler angle comparisons INS to open-loop-estimates.

22

American Institute of Aeronautics and Astronautics
Pitch

angle,

deg

3O

0 i

U -20

0

T400 800 1200 1600 2000

Time, eec

(a) Pitch angle.

INS attitude

-- Open loop estimate

I

2400 2860

i i

3200 3600 4000

960675

Bank

angle,

deg

I

i

-75

0

75

0_ __

400 800

INS attitude

Open loop estimate

I i i i

1200 1600 2000 2400 2800 3200 3600 4000

Time, sec

960676

(b) Bank angle.

200

Heading

angle, 0

deg

-20O

INS attitude

Open loop estimate

f_

!

J

0 400 800 1200 1600 2000 2400 2800 3200

Time, eec

36OO

(c) Heading angle.

Figure 12. SRA flight 540 Euler angle comparisons, INS to closed-loop estimates.

4OOO

960677

23

American Institute of Aeronautics and Astronautics
16

12

8

4

Pitch

angle,

deg 0

-4

-8

-12 .............. : ................... :..... _...... - :

-16 :

1300 1350 1400 1450 1500 1550 1600 1650 1700 1750 1800

Time, sec

960678

(a) Pitch angle.

INS attitude

-- Closed loop estimate

40

30 ......:

20

10 .....

Bank

v

angle, 0

deg

i

-10 ....... ; ............... ,

-20 ................ _ : ....

-30 ............

i

i

- 40

1300 1350 1400 1450 1500 1550 1600 1650 1700 1750 1800

Time, sec

960679

(b) Bank angle.

INS attitude

-- Closed loop estimate

150

125

100

75

Heading

angle, 50

deg

25

0

-=5 ........:........ :............. i ............. I.............. i..........i..... i ..........

- 50

1300 1350 1400 1450 1500 1550 1600 1650 1700 1750 1800

Time, se¢

96C680

(c) Heading angle.

Figure 13. SRA flight 540, expanded scale Euler angle comparisons, INS to closed-loop estimates.

INS attitude

-- Closed loop estimate

24

American Institute of Aeronautics and Astronautics
16

Velocity,

ft/sec

900 930 960 990 1020 1050 1080 1110 1140 1170 1200

Time, sec 9e_el

(a) Northerly wind component

-5

Velocity,

ft/sec -25

-45

i

i i i i

i i i i

i i i i

900 930 960 990 1020 1050 1080 1110 1140

Time, sec

(b) Easterly wind component

1170 1200

960682

50,000

Altitude,

ft 25,000

0

900

i , _ i i

930 960 990 1020 1050 1080 1110 1140 1170 1200

Time, see 960683

(c) Altitude

Figure 14. SRA flight 540 wind shear components and altitude time histories.

25

American Institute of Aeronautics and Astronautics
Figure 15 shows the resulting estimates. Clearly, the

stability of the computations has not been affected, but

small changes in the attitude estimates are noted. The

magnitudes of the induced errors in the pitch and

heading angles are within the specified accuracy

requirements of the program (less than 0.5 ° for pitch

and 2.0 ° for heading). The accuracy limits are exceeded

slightly for the bank angle, with induced errors as large

as 2.5 ° occurring. But as mentioned previously, because

no roll information was provided by the single-axis

compass, this result is not significant. Use of a three-axis

magnetic-field measurement would likely reduce the

bank angle errors to within the accuracy requirement

limits. To be safe, if significant wind shears along the

flightpath are anticipated, then wind tables from

prelaunch balloon soundings could be loaded into the

flight computer and used in the attitude computations.

Summary_ and Concluding Remarks

A low-cost strap-down architecture has been

developed to estimate closed-loop attitudes for the

ultrahigh altitude flight test experiment. In this system,

body-axis observations of the airspeed and magneticfield vectors are compared with the measured inertial

velocity and magnetic-field datum vectors to provide a

"virtually inertial" reference that is used to infer attitude

error. This error is then fed back to correct and stabilize

the rate-gyro integration. The system has the stability of

gimballed systems, but relies on strap-down components

to gather the required information.

The system performance was analyzed for the

ultrahigh altitude vehicle launch trajectory using Monte

Carlo error simulations. These simulations verified that

2O

lr i

............................................................................ i ................................. i

14 ............... ................_.......... ,........ _......... ........ ......... _.............

11

angle, 8

deg

5

2 ......... -_....... _...... -_....... _...... _........ ;..............

-1

-4

Attitude

i

900 930 960 990 1020 1050 1080 1110 1140 1170 1200

Time, aec

960684

(a) Pitch angle.

Attitude estimate

without winds

-- Attitude estimate

including winds

Attitude

angle,

deg

-4

-8

-12

-16

-20

-24

9OO

m

............................. _........................... t ................ ............... '_............

930 960 990 1020 1050 1080 1110 1140 1170 1200

Time, sec

960685

Attitude estimate

without winds

Attitude estimate

including winds

(b) Bank angle.

Figure 15. SRA flight 540, the effects of wind shear on the attitude estimates.

26

American Institute of Aeronautics and Astronautics
Attitude

angle,

deg

-103

-170

9O0

i i

930 960 990 1020 1050 1080 1110 1140

Time, sec

(c) Heading angle.

Figure 15. Concluded.

1170 1200

960686

Attitude estimate

without winds

Attitude estimate

including winds

two vector measurements, magnetic field and velocity,

are required to achieve complete three-attitude stability.

Based on the accuracy requirements of the ultrahigh

altitude vehicle, the simulations were used to establish

accuracy requirements for the basic measurements.

These requirements are not stringent and can easily be

achieved using standard instrumentation. The two

independent attitude references completely close the

loop on the error equation so that a stable set of

(linearized) error equations results and the errors of this

system will not grow with time.

The effectiveness of the estimating algorithm under

real flight conditions was demonstrated using data

derived from the NASA Dryden Flight Research Center,

Systems Research Aircraft flight tests. The algorithm

was applied to data from several flights with flight

conditions to a maximum 30 ° pitch attitude with good

results. Effects of wind shears on the attitude estimate

were evaluated using these data, and it was concluded

that the winds can be safely ignored by the estimating

algorithm in most cases. To be safe, if significant wind

shears along the flightpath are anticipated, then wind

tables from prelaunch balloon soundings could be loaded

into the flight computer and used in the attitude

computations.

References

IDrela, Mark, "Transonic Low-Reynolds Number

Airfoils," Journal of Aircraft, vol. 29, no. 6, Nov.-Dec.

1992.

2Toot, Peggy L., "Summary of Experimental Testing of a

Transonic Low Reynolds Number Airfoil," Low

Reynolds Number Aerodynamics Conference, Notre

Dame, June 1989, pp. 369-380.

3Murray, James F., Moes, Timothy R., Norlin, Ken,

Bauer, Jeff, Geenen, Robert J., and Mouiton, Bryan J.,

"A Piloted Simulation Study of a Balloon-Assisted

Deployment of an Aircraft at High Altitude," NASA

TM- 104245, Jan. 1992.

4Regan, Frank J. and Anandakrishnan, Staya M., "ReEntry Vehicle Dynamics," AIAA Education Series,

American Institute of Aeronautics and Astronautics,

New York, 1993.

5James, F., Monte Carlo Theory and Practice

Reports on Progress in Physics, vol. 43, Sept. 1980,

pp. 1145-1189.

6Etkin, Bernard, Dynamics of Atmospheric Flight, John

Wiley and Sons, New York, 1972.

7Langel, L. A., Mundt, W., Barraclough, D. R., Barton,

C. E., Golovkov, V. P., Hood, P. J., Lowes, F. J., Peddie,

N. W., Qi, GUI-Zhong, and Quinn, J. M., "International

Geomagnetic Reference Field," 1991 revision, Journal

of Geomagnetism and Geoelectricity, vol. 43, no. 12,

1991, pp. 1007-1012.

8Battin, Richard H., "An Introduction to the Mathematics

and Methods of Astrodynamics," AIAA Education

Series, American Institute of Aeronautics and

Astronautics, New York, 1987.

9Rade, Lennart, and Westergren, Bertil, Beta

Mathematics Handbook, 2nd ed., CRC Press, Boca

Raton, FL, 1990.

27

American Institute of Aeronautics and Astronautics
_°Ehernberger, L. J., Haering, Edward A., Jr., Lockhart,

Mary G., and Teets, Edward H., "Atmospheric Analysis

for Airdata Calibration on Research Aircraft," AIAA

Paper 92-0293, Jan. 1992.

_Meditch, J. S., Stochastic Optimal Linear Estimation

and Control, McGraw-Hill Book Co., Inc., New York,

1969.

12Franklin, Gene E and Powell, J. David, Digital

Control of Dynamic Systems, Addison Wesley

Publishing Co., Reading, Massachusetts, 1980.

13Whitmore, Stephen A., Davis, Roy J., and Fife, John

Michael, hi-Flight Demonstration of a Real-Time Flush

Air Data Sensing (RT-FADS) System, NASA

TM-104314, Oct. 1995.

14F/TF-18A Naval Air Training and Operating

Procedures Standards (NATOPS) Manual, Reference #

A l-F18AA-NFM-000, Feb. 1980.

_SHaering, Edward A., Jr. and Whitmore, Stephen A.,

FORTRAN Program for the Analysis of Ground-Based

Radar Data; Derivations and Usage, Version 6.2, NASA

TM-3430, 1995.

28

American Institute of Aeronautics and Astronautics

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January 1997 Technical Memorandum

4.TITLE AND SUBTITLE 5. FUNDING NUMBERS

Development of a Closed-Loop Strap Down Attitude System for an

Ultrahigh Altitude Flight Experiment

S. AUTHOR(S)

Stephen A. Whitmore, Mike Fife, and Logan Brashear

7.PERFORMINGORGANIZATION NAME(S)ANDADDRESS(ES)

NASA Dryden Flight Research Center

P.O. Box 273

Edwards, California 93523-0273

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration

Washington, DC 20546-0001

WU 537-10-40

8. PERFORMING ORGANIZATION

REPORT NUMBER

H-2140

10. SPONSORING/MONITORING

AGENCY REPORT NUMBER

NASA TM-4775

11. SUPPLEMENTARY NOTES

Presented at the 35th Aerospace Sciences and Exhibit, January 6-9, 1997, Reno, Nevada.

12a. DISTRIBUTION/AVAILABILITY STATEMENT

Unclassified--Unlimited

Subject Category 06

13. AB_iHACT (Maximum 200 words)

12b. DISTRIBUTION CODE

A low-cost attitude system has been developed for an ultrahigh altitude flight experiment. The experiment uses

a remotely piloted sailplane, with the wings modified for flight at altitudes greater than 100,000 ft. Mission

requirements deem it necessary to measure the aircraft pitch and bank angles with accuracy better than 1.0 °

and heading with accuracy better than 5.0 °. Vehicle cost restrictions and gross weight limits make installing a

commercial inertial navigation system unfeasible. Instead, a low-cost attitude system was developed using

strap down components. Monte Carlo analyses verified that two vector measurements, magnetic field and

velocity, are required to completely stabilize the error equations. In the estimating algorithm, body-axis

observations of the airspeed vector and the magnetic field are compared against the inertial velocity vector and

a magnetic-field reference model. Residuals are fed back to stabilize integration of rate gyros. The

effectiveness of the estimating algorithm was demonstrated using data from the NASA Dryden Flight Research

Center Systems Research Aircraft (SRA) flight tests. The algorithm was applied with good results to a

maximum 10° pitch and bank angles. Effects of wind shears were evaluated and, for most cases, can be safely

ignored.

14. SUBJECTTERMS

Attitude estimation, Euler angles, Kalman filter, Quaternion, Ultrahigh altitude

17. SECURITY CLASSIFICATION 18. SECURITY CLASSIFICATION 19. SECURITY CLASSIFICATION

OF REPORT OF THIS PAGE OF ABSTRACT

Unclassified Unclassified Unclassified

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32

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AO3

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