N A S A C O N T R A C T O R

R E P O R T

N A S A C R - 2 9 0 7

AUTOMATED LANDING, ROLLOUT,AND TURNOFF USING MLSAND MAGNETIC CABLE SENSORS

S. Pines, S. F. Schmidt, and F. Mann

Prepared by

ANALYTICAL MECHANICS ASSOCIATES, INC.

Jericho, N. Y. 11753

for Langley Research Center

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION • WASHINGTON, D. C. • OCTOBER 1977

https://ntrs.nasa.gov/search.jsp?R=19780002099 2020-06-14T11:52:25+00:00Z

1. Report No.NASA CR-2907

4. Title and Subtitle

Automated Landing, Rollout,Magnetic Cable Sensors

7. Author(s)

S. Pines, S. F. Schmidt, F.

9. Performing Organization Name and Address

Analytical Mechanics Associa50 Jericho TurnpikeJericho, New York 11753

2. Government Accession No.

and Turnoff Using MLS and

Mann

tes, Inc.

12. Sponsoring Agency Name and Address

Langley Research CenterNASA .Hampton, VA 23665

15. Supplementary Notes

Langley Technical Monitor:Topical Report

3. Recipient's Catalog No.

5. Report DateOctober 1977

6. Performing Organization Code

8. Performing Organization Report No.

AMA No. 77-310. Work Unit No.

11. Contract or Grant No.

NAS1-1431113. Type of Report and Period Covered

Final14. Sponsoring Agency Code

Richard M. Hueschen

16. Abstract

This report contains a description of the simulation program used to study thelanding approach, rollout and turn-off of the B737-100 aircraft utilizing MLS anda buried magnetic leader cable as navigation aids. The results of the simulationare contained in the report and show the concept to be both feasible and practicalfor commercial type aircraft terminal area control.

17. Key Words (Suggested by Author (s)) 18. Distribution Statement

Automatic landing; Rolloff and Turnoff ;Navigation and Guidance; Magnetic Leader Unlimited - UnclassifiedCable; MLS; Kalman Filter;ComplementaryFilter Subject Category 17

19. Security Qassif. (of this report)

Uncl.20. Security Claisif. (of this page)

Uncl.21. No. of Pages 22. Price*

151 $6.75

For sale by the National Technical Information Service,Springfield, Virginia 22161

TABLE OF CONTENTS

Section Item Pag

SUMMARY lx

INTRODUCTION x

I AIRCRAFT DYNAMICS 1A. Choice of the Coordinate Systems 1B. The Aircraft Equations of Motion 4C. Landing Gear Dynamics 10

H NAVIGATION AND FILTERING , . . . 23A. Dead Reckoning Navigation , , . . . . . . 23B. Kalman Filter Implementation 26C. Error Model of the Estimated Variables 32D. Kalman Filter Data Updates 36E. Complementary Filter 37

IE THE MLS OBSERVABLES 43A. MLS Noise Model 46B. Radar Altimeter 48C. MLS Partials . 52

IV MAGNETIC LEADER CABLE SENSOR 56A. Three-Coil Sensor 56B. Error Model of the Magnetic Coil Sensor . 60

V NONCRITICAL FLIGHT PHASE NAVIGATIONAL AIDS 62A. VORTAC Range and Bearing . 62

VI GUIDANCE AND CONTROL 66A. Flight Critical Lateral Control Guidance 67B. Flight Critical Vertical Control Guidance 68C. Rudder Control for Landing and Decrab 71D. Noncritical Aileron Command Guidance 75E. Noncritical Elevator Control Guidance 77F. Airborne Throttle Command 77G. . Rollout and Turnoff Guidance . . 79H. Requirements for G. E. Whole-Word Computer to

Execute Navigation and Kalman Filtering During Rollout . . . . 85

APPENDK A 96

APPENDIX B 108

REFERENCES

iii

LIST OF FIGURES

FigureNo. Title Page

1 Runway (Inertial) Coordinates and Aircraft Position,Velocity Vectors ( R , R ) .. . ' . . ' . .'; ". . ."". . . . . 2

2a Aircraft Body Coordinate System, Control Surface Deflections,and Aerodynamic Angles 5

, ,2b Euler Angles and Rotation Rates Defined 6

3 Landing Gear Geometry 12

4 Main Gear Strut Compression Force . . . . . . . . . . . . . . . . . 16

r 5 Nose Gear Strut Compression Force . . . 17

6 Friction Coefficients 19

. 7 Navigation System Block Diagram . 24

,8 Complementary Filter for Navigation, with Body-MountedAccelerometers Using MLS Measurements .'. • ' ; . . . ' " . ' . . . . 38

9a The MLS Coordinate System . ' . . ' • . ' . . . \ . '• . . . . ' . ' / ' ^ . ... 44

9b MLS Conical Scan Time Sequence 49

, . 9c MLS Jitter, Azimuth 50

9d MLS Jitter, Elevation 51

10 Magnetic Leader Voltage Signals 57

11 Magnetic Field Equations 59

12 Flight Critical Lateral Roll Command 69

13 Flight Critical Lateral Inner Loop 70

iv

LIST OF FIGURES (Cont. )

FigureNo. Title Page

14 Flight Critical Elevator Command ............. ..... 72

15 .. Rudder Yaw Damper and Decrab Command . . . _ . . . ........ 74

16 Noncritical Aileron Command . . . . . . . . . . . . . . ...... 76

17 Noncritical Elevator Command ..... . . . . . . . . . . . . . . 78

18 Throttle Control ...... ........ . . . ' . . . . ' . ' . ' . . . . '80

19 Rollout Control Surface Slowout Command . . . . . . . . . . . . . !81

20 Rollout and Turnoff Steering Command . .'. . . . . . . . . . . .. . 84

21 Brake Pressure Command .............. . . . . . . . . 86

22 Rollout Thrust Command . . . . . . . . . . . . . . . . ... . . . . 87

23 Block Diagram of Overall Filter Mechanization . . .. . '. ..." ..... 89

24 Rollout Navigation Timing Estimates for the MCP-701 ,Computer ........................ ". . . . . 94

Al Phugoid and Short Period Modes ........... . ....... 105

A2 Dutch Roll Mode . ........ . . . . . . . . . . . . . . . . . . 106

Bl High-Speed Turnoff on Dry Runway — V . = 9. 144 m/sec ..... 113CcLXl

B2 High-Speed Turnoff on Dry Runway — V . = 12. 19 m/sec ..... Il5taxi

B3 High-Speed Turnoff on Dry Runway — V .= 15. 24 m/sec . . . . . 117IcLXl , -

B4 High-Speed Turnoff on Dry Runway — V .= 18. 29 m/sec ..... 119LcLXl

B5 High-Speed Turnoff on Wet Runway — V . = 15. 24 m/sec . . . . . 121

B6 High-Speed Turnoff on Wet Runway — V . = 18. 29 m/sec ..... 123taxi

LIST OF FIGURES. (Cont.)

Figure... No. Title

B7 High-Speed Turnoff on Wet Runway — 150° Headwind,5. 144 m/sec. , V . = 9. 144 m/s£c :J ; . '. V :. ".' Y :':\ '"'; ;' ..... 125

taxi

B8 High-Speed Turnoff on Wet Runway — 150 Headwind, ' '5. 144 m/sec. , V . - 15.24 m/sec ................ 127

' - • • = 'taXl '•'•:! 1! "' • • : • • • • : • : - • • • ' • • : " • ! • • • • • : • ; • , . : . . . :;:, ;

B9 High-Speed Turnoff on Wet Runway — 150° Headwind, .5.144 m/sec., V '. = 18.29 m/sec . .:'. '/'. . ' . ; ? ':. . . ;. . .. . . 129

taxi i

BIO High-Speed Turnoff on Wet Runway — 150° Headwind,8.716 m/sec. , V ,,= 12.19 m/sec ................ 131

• • • • : < . • • ' - • • - • . • ' • • - . -'.^'taxi'. • • / • - • ' . - • : • ' . • . • • ' • ••:.." - ; • • : : • : • : : - . . • / : . ; ; ' : " : • ' ::::. : <

,B11 High-Speed Turnoff on Wet Runway— 150 Headwind," 8.716 m/sec., V '. = 15. 4" m/sec :. '. '"". :. "". ''. '' . . ". Y'-. '1 ..... 133

taxi

B12 High-Speed Turnoff on Wet Runway — No Wind,V. .= 9. 144 m/sec ........................ 135taxi

B13 High-Speed Turnoff on Wet Runway — No Wind,V .= 15.24 m/sec ........................ 137taxi

B14 High-Speed Turnoff on Wet Runway — No Wind,V. .= 18.29 m/sec ........................ 139taxi

B15 High-Speed Turnoff on Dry Runway — 150° Headwind,Complementary Filter Used . . .................. 141

VI

LIST OF TABLES

TableNo.

1

2

3

4

5

6

7

Title

State Transit-ion Matrix "f

Square Roof Pfbce'ss-Noi'se-l^atrix:;^.,^^'^'.;^,; . . . . . . . . .

MLS Error^Mod.efsf;':^. ^.r/i .-';^5-.>.:.v. ;";.. . . . .' . . 'V'". . . .

VORTAC Error Model

GE MCP-701 Computer Instruction Execution Times . . . . . . .

Dynamic Response Comparison — B-737 . . . . . . . ... . . .

Pag<

13

34

35

' "47

64

93

107

Vll

PageIntentionally

AUTOMATED LANDING, ROLLOUT, AND TURNOFF USING

MLS AND MAGNETIC CABLE SENSORS

S. PinesS. F. Schmidt

F. Mann

Analytical Mechanics Associates, Inc.

SUMMARY

This report contains a description of the simulation program used to study the' •"""'-•' >N

landing approach, rollout and turn-off of the B737-100 aircraft utilizing MLS and a buried

magnetic leader cable as navigation aids.r.The results of the simulation are contained in

the report and show the concept to be both feasible and practical for commercial type

aircraft terminal area control.

INTRODUCTION

The use of inertial navigations in conjunction with navigation aids such as ILS,

MLS, TACAN, etc., in commercial aircraft is rapidly becoming a common state-of-

the-art technology. As part of the Terminal Configured Vehicle program (TCV),

sponsored jointly by NASA and FAA, this study was undertaken to: (1) investigate the

compatability of existing Autoland guidance laws utilizing a Microwave Landing System

(MLS) in place of the conventional ILS navigation aids during landing approach; and (2)

to extend the concept of automated flight to rollout arid turnoff for reduced runway

occupancy during normal and adverse (CAT III) landing conditions. In order to achieve

acceptable tracking performance in dry and adverse weather conditions, a buried magne-

tic runway cable and an aircraft mounted three coil magnetic pickup were investigated.

For the purpose of the study, a dynamical model of the B737 was generated in

a computer program (ALERT) to simulate the aircraft forces and moments during land-

ing approach, rollout and turnoff. A simulation of the existing Autoland guidance

ix

equations for the B737 was supplied by the Langley Research Center* Optimal Kalman

filtering, utilizing a square root formulation for onboard implementation, and a non-

optimal fixed gain complementary filter were studied. A rollout and turnoff guidance

law was generated. Both Kalman type filtering and fixed gain complementary filtering

are shown to be useable with little change to conventional guidance equations presently

in use in commercial aircraft.

This report includes a study of the requirements for the Kalman filter and the

rollout a n d turnoff guidance laws. . . .

:;I;:i AIRCRAFT DYNAMICS • = " " " " ' > ' . - • • • ' • . - < • • - • - . - • • :

. • • ' . • - . . :•• . - .• . . . - . - . - , . • . • • ' . • . : ; < - : • / . ' K':: : . : • " = i ' i "• . : ; • • v •;•' , ';.' • ' ' : • • • . • • ' • . • • • : ' (" '

. - . . . . This section contains the; derivation;of the equations,used in; the simulation,

of the dynamic model of the B-737 during .landing approach,: capture, decrab, .flareij

touchdown, rollout and.turnoff. ,;,...... . . - , . ,, ,;? • . , . ; . ;. . ' • . , : ,..- . •, ;•,;;; •••

A. Choice of the Coordinate Systems

-' ;: :; It is- customary to utilize linearized''equatibhs 'of m'otion describing '

the deviation of the aircraft from a constant-speed, fixed-flight path angleV wings-*'''

level equilibrium condition in the body-stability axis system. While this is suf-

ficient for the landing approach phase, it is inadequate for rollout and turnoff due

to large deviations in air speed, flight path angle, and sizable changes in yaw

during turnoff. In order to accommodate the broad spectrum of conditions, it was

decided to use nonlinear equations of motion in a runway Cartesian coordinate

system to describe the linear acceleration of the aircraft and to use the body axes

and the Euler angles of roll, pitch, and yaw to describe the angular acceleration

equations.

Since the nonlinear variation of the aerodynamic coefficients is small

over the range of Mach numbers and aircraft altitudes in landing, and since the

aerodynamic forces during rollout and turnoff become relatively unimportant when

compared to the landing gear and thrust forces and moments, it was decided to

use a simplified form of constant lift, drag, and moment coefficients fitted to the

B-737 for the landing simulation. Numerical values of these aerodynamic coef-

ficients were obtained from NASA Langley Research Center and are given in

Appendix A.

The orientation and sign convention for the runway inertial Cartesian co-

ordinate system is given in Figure 1. The aircraft position vector, R(x,y, z),

is measured relative to the origin fixed on the center line of the runway with the

positive x axis pointing forward along the runway center line. The z axis is

AIRCRAFT

MLSELEVATIONANTENNA

MLSRANGE

AZIMUTHANTENNA

Figure 1. Punway (Inertial) Coordinates and Aircraft Position, Velocity Vectors (R, R ).

positive down, and the y axis is positive to the right in a right-handed orthogonal

coordinate system.

The transformation from the body system to the inertial runway system is

given in terms of the Euler angles.

Let

0 = roll •

0 = pitch _ ! • '

fy = yaw (measured from magnetic north)

= runway yaw. (measured from magnetic north)

Then,

R

1 0 0

0 cos 0 sin 0

0' . -sin 0 cos 0

(la)

cos 6 •"•-,. 0 -sin 0:0 1 0

sin 6 0 cos 0

(Ib)

T3(a) =

cos a sin a 0

-sin a cos a 0

0 0 1

where

a = R (Id)

A vector in body system V is transformed into a vector in runway inertialB

system by the matrix L ,113

VI = LIB VB (2)

where

(2a)

. ^ .. »• ' • , - - ' . •It follows that the three unit-body-axes vectors a , a , and a may be expressed

, JL u <5

in the inertial system in terms of the Euler angles (0, 8, a)« "See Figs. 2a and 2b.

The a axis is along the aircraft center line, pqsitive forward. .; . . , ,

cos 6 cos a

cos 6 sin'a

-sin 8 .

The a axis is the lateral aircraft axis, pointing positive along the right wing.£l

-sin a cos 0 + cos a sin 0 sin 6

cos a cos 0 + sin a sin 0 sin 6

sin 0 cos 9

(3a)

(3b)

The a axis is the vertical body axis pointing positive downward; "'k3 .. •'.' '•. • '•• •••;.',"'

sin a sin 0 + cos a cos 0 sin Q

-cos a sin"0 + singer cos 0 sin 6

dos 0"cos-'6'

(3c)

B. The Aircraft Equations of Motion •

The aircraft inertial velocity vector, R( x, y, z) is giveri by the time

rate of change of the position vectbr • , : , - • ; . :

(4)

FOR + «a (AILERON)fLEFT AILERON SURFACE DM.I RIGHT AILERON SURFACE UP

Note:

a , a , a (unit body axisi ft O

vectors) are expressed in the

runway coordinate system

Figure 2(a). Aircraft Body Coordinate System, Control Surface Deflections, and-Aerodynamic Angles. : .

The vector velocity of the aircraft relative to the atmosphere V(v ,v , v ) is—-^J. L* «5

given by the aircraft inertial velocity" v"ectof~and the winds.

• .. V. = R - [W + G +S] (5)

where

W = constant wind vector

G = gust vector ' , ? , ' . , : :

S •= wind shear vector ./ • •

/ ' • 'The simulation of wind, shear, and gust is not derived in this report and was

supplied by the NASA Langley Research Center. /

The aerodynamic coefficients for lift, drag, and side force atre obtained

from wind tunnel data and relate to forces in the so-called aircraft stability axis

system. In this system, the lift acts normal to the aircraft relative velocity

vector, V, and in the a , a plane. The drag acts in the a , a plane andJ. -, *5 . JL O

is perpendicular to the lift in the .direction negative to the relative velocity vec-

tor, V . The side force acts along the a axis. To express the stability axes• • 2i ,

(s,, s , s ) in the inertial runway system, we have the transformationJ. £ o *. : .

where a is the angle of attack of the aircraft

.-1a = sin

a - V ) 2 + ( a - V ) 2

, . CD -

' -! VV

tana • V

Thus the lift acts along the negative SQ axis

s = '- sin a a + cos a S.o 1 o (8a)

1 > ' . - ,* •• • ';The drag acts along the negative s axis' ••• - . • • ' - . . . ' " ' - . • '1

s ...=. cos,a a +. sin a a1 " ; -1 . ' - . • ' . ' • ' O '-.'_.

(8b)

The aerodynamic side force acts along the positive s9 axis

S2 = (8c)

The total forces 'acting on the aircraft, expressed in the runway-referenced

frame, are given by -/ •.•..•: ;;, : :

where

E = q S L.IS

-CD

CY'-C '

L

. - • -+%'FXG^

FYGFrZG

^ •>"<

FXTFYT"P '"'""'

ZT

q = fpV (dynamic pressure)

S = wing reference area >

C , C , C = effective drag, side force, and lift coefficients(see Appendix A)

F^._ , Fv., , F_ _ = landing gear forces expressed in the runway ,. XOr ... YCjr ZOr .- ; . • • • ' " . - • • ' • • system . - • • / • . , - ; : . • • . , - - , . . - , ; • • • • . • . - • ^ • • . : - -

F , F _,F = thrust forces expressed in the runway systemXT Yl

(9a)

The time rate of change of the ihertial velocity-vector in;the irunway ;

frame is given by

XGFYG

•(io)

K+ — {cos a a, - sin a a_.}vm o 1 o 3%

where

A G =

k =

(gravity vector)

plane; Positive^ a -is a-o

magnitude of the thrust

"v/v. Q, ?.•;.';•=;. thrust offset Bangle in>the, S^;,o 1

rotation about the a. axis.2 " " -' '

It is convenient to integrate the angular,accelerations of the aircraft in

-the body system since tiie mass moments of inertia do not change in that frame.

The kinematic equations,relating the Euler angles and the body angular rates

(P.Q.r) are given by

6 = cos 0 q -,. sin 0 r

V = (r cos 0 + q sin 0)/cos, 6 ;• «K ,: ; r/1

= " p""+' sin"""0' "^" ".' " ' ' " ' ' ':' " ,,•;.' '•• •:• " i r - - : ' ; '?•> . ' . • • • ' •> ' • '•"-:;-/- -N.'

The differential equations for the angular accelerations in the body system

are giveifby! • • •^••v," 'v •••••'i- •--. •,••, •.. ':>:':;*•• •••••. -•••.<••••. -7 . . . . ; • „ . ;v ., ..

(lOa)

(11)

5 Cw'J --. (12)

where

'xx0

-<xz

0

IYY0

-*xz0

Tzz

: (12a)

w =

Pqr

(12b)

VMsN

. S >

' + <

LGMGN

G ^

> + <

t ' V

NM

T

N. T ".

(12c)

The aerodynamic moments, L_ , M N , in the stability frame are given inB O O , "

Appendix A. The landing gear moments,

and are derived in Section C. The moment due to thrust is given by

, 1VI . N_ , are in the body frameG (jr ;

LTMT

NT-

> = <

daTk

d

a is the moment arm of the engine thrust and is listed in Appendix A.

C. Landing Gear Dynamics

The landing gear is modeled as an airspring and damper. Strut

pression is initiated whenever the strut load exceeds a preset preload, PJi

10

For loads less than the preload, tire compression provides a linear spring (initial

tire compression). Elastic deflection of the strut and inertial acceleration of the

unsprung mass of the strut and tire are considered high frequency phenomena and

are neglected in the analysis.

In order to determine the forces and moments acting on the aircraft due

to ground reactions in landing and braking, the locations of the main gear and

nose strut with respect to the center of gravity are required. The locations of

the three extended, uncompressed gears of the B-737 are given in Ref. 1 and

illustrated in Fig. 3. The positions of the three gears are measured from an

origin at the leading edge of the MAC.

Let the coordinates of the i gear with respect to the leading edge of

the MAC be given by

x. , y. , z. (see Table 1)

During landing, rollout, and turnoff, it is assumed that the aircraft experiences

small values of pitch and roll. Large values of yaw are permitted. Thus, small

angle theory is used to model- roll and pitch motions but the full nonlinear theory

holds in yaw.

The height of the i, gear above the runway.is given by

h.Q = -z.G = -z - 0y. + 6 ( x . + eg* ) (13)

where

g = center of gravity expressed as the percentage of meanaerodynamic chord

11

(15").381 m

T1

(30.5").7747 m

CG

(28. 3')8.6258 m

(8. 58')2.6152 m

-i 1.8288 mI1

LEADINGEDGE MAC

LANDING GEAR LOCATION

CG

rt(8.77')2.6731 m

Q|(9.78')2.9809 m

EXTENDED GEAR HEIGHTS

Figure 3. Landing Gear Geometry

12. .f

TABLE 1

EXTENDED GEAR LOCATIONS

(Distances in meters from e.g.)

Right Main Gear

Left Main Gear

Nose Gear

X

-1. 8288

-1.8288 ;

8.6258

y2.6152

-2. 6152

0

z

2.9809

2.9809

2.6731

Gear No.

1

:: 2

• 3

13

Ground contact occurs whenever

h iG< 0 Z i G > 0 (13a)

Since the strut is preloaded, no strut deflection occurs until z exceeds thelljr

preload tire deflection, z. . Thus, for

._, . TiG ipL (13b)

the strut load is a linear function of z ._ .

F = - F -i pi z

'iG (14)ipL

Once the preload has been exceeded, the strut begins to compress and the force

is given by the isothermal compression law

F. = -F.i pi1 -

._ - z. TiG ipL(15)

where -t. is the bottom-out maximum deflection permitted by the strut design.

Equation (15) holds for all z._ such thatiG

ZipL * ZiG * (15a)

The strut deflection is given by

Z ~ Z "~ ZiD iG ipL (15b)

14

The damping force is generated by the strut deflection rate, z. ,

given by

The damping force of the i strut is given by

FiD = "'ID Z lDJC lv

where c. is the damping coefficient, which is modeled as a function of the

strut deflection

2'iv li iD 2i iD

where a .; a are strut damping design constants. Figures 4 and 5 illustrate-LI ^ I

the strut compression curves for the main and nose gears.

The net vertical strut force in the body frame is given by

FNi = Fi + FDi

The tire drag force may now be computed as a function of the normal

strut force and the friction coefficient. The drag force acts in the plane of the

tire and in a direction opposite to the axle velocity. We have

The tire friction coefficient, ^, is the sum of the rolling friction coefficient, jj ,R

and the friction coefficient developed by brake application, ^ . We have, for

dry runway conditions,

15

nse

Fo

£•ICO

LOA

D)

3

6

8

6

3

2

io5

8

2

• -Q_O_o_ ActualCyl

>,...

Coi

cAy^r-

Strut (Oil Compressibility andinder Expansion Included)

fied Strut Model (Considers Airnpressibility Only)

•. ..

"V^^:

rV&r^

rfrT^

i

: \:

)

" 1

Irf\

:.. - -,_

•

..

5 2510 15 20

Strut Stroke (cm)

Figure 4. Main'Gear Strut Compression Force

30 < 35

16

3

woa5CO

I I ~ T i " I~O~O—O— Actual Strut (Oil Compressibility and Cylinder Expansion Included)

—• Simplified Strut Model (Considers Air Compressibility Only)

10

:6

10

5 10 15

- Strut Stroke (cm) ,

Figure 5. Nose Gear Strut Compression Force.

MB = -5 (20)pmax

where b is the brake pressure and b is the maximum pressure available,p pmax

For wet runways, we have (from Ref. 1)

V + 100 b •M s: — — - - 2 — /2Qa)MB 10/Vr +;200 b ) v '

G pmax

/ {where V_ is the ground speed in knots. The tire friction curves are given'in'

Fig. 6. ./' . . ' ( ; • , .

.-•"' ?The guidance law for brake pressure application is given in the Guidance

and Control Section of this report. . , . . • • " ' r

Finally, we have , .' : • ' . . l

...... ... • ....... "" i.. w^,- -: ^

(21)

wher;e p. is the coefficient of rolling friction. Braking is not available.. on,ttie,"' •• R '" '• ''.' ' ' " '' . : _ • • . I-'

nose wheel so that only rolling friction prevails.

The landing gear lateral forces and steering is taken from the theory

developed by W. B. Home in Ref. 2. We first compute the tire compression

coefficient r\. •

If

1 (p^.OSp^Jw^w. d.)15

j .> . 09

18

.6

.4

.3

.2

.1

A* (DRY)

/I (WET)^"

20I I I

40 60 80

VELOCITY (KTS)

100 120 140

Figure 6. Friction Coefficients.

19

the tire .compression, is given by

If

Tj. < .09

tire compression -is-- ,„ , -, ; . . . . . .

03"*"-42 V (22s.)

6ti = 4 Wi ^i (22b)

The tire cornering force coefficient is given by

2lT (23)

R»,';i2', -..multiplying 7T/180 represents the number,of tires per,strut..

Thus,, for a single tire strut, the lactpr would be 1:..,.)>..• ,., ,v t.••. f h-.-. ,v . . . , ; - \ - ., -.- -.-i, ;-,

For

... 0875:d.

(24a)

and for

6t. "< .0875 d.

Ni = (24b)

20

The lateral tire cornering force acts normal to the plane of the'tire

and is given by

. ; I1" FLI •= N.(y. + g- e.) (25)

where e. is the ground track angle . ;

siny + rC(x. + c g )cos a - y. si• £.. = tan .-: • - • - =— — — - • ; • • " : • • ;-:.':.. .-,•:<'">• ' <r'-P (25a)

"

(a= '

and y. is the steering angle of the wheel. For the main gear, y. = 0 ; and- l . . ' ' - ' - . ' ' • • - • . - • . , ; •=•; , • : , : • : : - . . : ; • '.'.;::>-...o. - -.:;;, ..-.;> ;i.v.,. •„-..;•;•-•-; • • ' h l ' - -^ ; ' -

for the ripse wheel, y. .is supplied by the control logic.

The plane of the main tire is parallel to the a , a body frame. How- :.•. . • _ ••- . . . . ' ••- .. - \ .- i • o'

ever, the nose wheel develops a drag and cornering force which must be re-

sdlved into the bbdy axes/ Let y • , F^0 , arid 'FT Jibe, the nose wheel steering i?i• 3 [ ' Ho Lo

angle and horizontal forces developed in the plane of the nose wheel arid perpen-

dicular to it; Then we have, for the forces in the body system,

0FH3•

To obtain the moments in the body frame, we have

The lateral tire'cornering force acts normal to the plane of the tire

and is given by .

where G. is the ground track angle

y+ r(j(x. + c g*)cos a - y. sin a]. = .tan~ ' •. — - - l - l- ••-.••••.':• •.*:<•, . , • : • ' (25a)

' '

(a =

and y is the steering angle of the wheel. For the main gear, y. =0 : , and

for the nose wheel, y. . is supplied by the control logic.

The plane of the main tire is parallel to the a , a body frame. How-' . 1 o •

ever, the nose wheel develops a drag and cornering force which must be re-

s61ved intp the body axesv Let y0 , F_ , and1 F_ _-'•' b'e the riose wheel steering• ' 3 H3 ljv

angle and horizontal forces developed in the plane of the nose wheel arid perpen

dicular to it. Then we have, for the forces in the body system,

F ' = F - 8F - 6 y "FXi Hi Ni °i3 73 L3

- + f i v F (26)Yi Li Ni °i373 H3 l '

=' 0F - 0 F + F - 6 v (9F +Zi Hi . Li Ni °i3 73 v L3

To obtain the moments in the body frame, we have

21

IS

LG = Z < F Z i V F Y i Z i G >

TVL = (27)

NG =

To obtain the forces in the runway frame, we have

FXG = h>3FL3+ ' F-'COS

3

H3 Z FLi sin cr

YG . -/3

«j

ZFLi sin a + -o

Z FHi cos a (28)

o

- ZF = > FZG L Ni

10 Ci

22

to

0 , 9 , * , x, y, ii ;

Attitude Valid

(tfoay \ f ,-f ,fMounted ^ x y yAccelerometers

Rate j P.q.rGyros 1

flarevalid

f , hB,hBCAS ' ' '

Alt. Valid

MLS J A Z , El, DME

RECEIVER | Vallds

>'

"•

i

1 fr™

^ , :-

Coordinate •

Transformations•- •

Filtering '•,

t'Cborc

Transfoat

SourceSele(

V '

k ..-x, y,.z

Unate"rmationidVariable3t Mix;

A ~Pre-F liter

and j xMeasurement

.Selection :' "*>

'

*R/W. '

A . A A

x, y, zA ' . A A

x, .y , . z

1R/W lat., long., MLS

CoordinateTransformation

andComputations

Calculation.AutolandVariables

RNAVGuidance

andControl

andDisplays

Ay, Ah,.hT]DZ

Range

y, vg, uI h

AutolandControl

L Displays

7.;' :

Figure 7. Navigation System Block Diagram

where £-,_,,., is the specific force measured in the body frame by the acceler-oFHometers. Since the accelerometer and gyro data are sampled at 20Hz, we

may integrate Eq. (30) with a simple Taylor series

- - - At2

R(t + At) =' R( t ) + R(t) At + R( t ) ^-

• o *•

R(t + At) = R ( t ) + R ( t ) A t (31)

At = . 05 sec.

; In the presence of error sources cited earlier, the dead reckoning esti-

mates become useless after some time, and navigation aids and filter techniques

are required to correct for the errors. In the subsequent sections, we describe

two such filter systems. The first is the Kalman filter. This filter utilizes & , •

priori estimates of the uncertainty, in the error states to be corrected and the :

expected variance in the observations, to obtain a minimum variance estimate

of the correction to the states as a linear function of the difference between the .

observations and the expected value of the observation. ^

The second filter system is called a complementary filter. It uses fixed

gains to estimate the corrections to the position and velocity of the aircraft as a

linear function of the observation residual. Since it knows nothing about the a

priori uncertainty of the aircraft errors or the observation noise, it depends

solely upon the concept of exponentially correlated random noise to smooth out

noise. As shown in the results of the study, the complementary filter does as

well as the Kalman filter in landing approach, but does not do as well in turnoff

because of the effect of gyro misalignment biases.

25

B. Kalman Filter Implementation :

, The optimal fliter; algorithm involves a very large number of machine opera-

tions. The basic algorithm separates into the; following: , . . • . ' . . . . . . - . . . -

1. Calculations associated with a change in the time referenceo f t h e filter, arid ; • ' ' • • • ' : , ; • , • • • • • , ' , . • • - . . . - .

2. Calculations of the incremental state estimate from the"' measurement information. ' ' !

In order to.keep the number of operations within the capabilities of current air-r

borne computers, the accelerpmeters^and gyros are, sampled, at a rate of 20 Hz ,

the observations are sampled at a rate of 10 Hz , and the filtejr is, updated at a

rate of 1 Hz .

Past experience has shown that the square root implementation of the opti-

. . ; ; < mal filter algorithm (Ref. 3) can reduce the effects of numerical errors to insig-

nificant levels. The square root implementation is therefore incorporated in the

proposed design. Modeling errors are compensated for by the appropriate use

of random forcing functions. , This technique causes the more repent measurements

to be weighted higher than past measurements;..therefore, the estimate tends to

follow the more recent measurements. , . , .f - f '

The proposed filter will operate in a manner illustrated in the sketch below:

At the start of the sequence, we assume that the filter has its covariance

matrix referenced to time t . At the times t, , t , , t , measurements

26

are accepted and residual sums and partials are computed and saved in pre-

processing routines. After accepting the measurement at time t , the residual

sums and partials are transferred to arrays for processing by the filter; and the

locations used for the pre-processing are cleared for use in pre-processing the

t and subsequent measurements. The residual sums are processed by the '.K ' J . ' • - . • . • . . , - . . • • ' •

filter and the incremental state change computed in lower priority logic. When

these lower priority calculations have been completed, the logic sets markers

which cause the higher priority logic to pick up the state change and add it to the

system. Meanwhile, the lower priority logic updates the covariance matrix to

the time t and readies the filter for processing the residual sums in the next

interval. The onboard program operations for executing such Ibgic'will be de- ;

scribed in a later report. - : - . - { • •

The error state is assumed to obey a linear differential equation

dx = F dx + F ri ; • • ; • . ' . ; • • - : ' ' . ; , •', f;;-u (32)x n v

where

dx: = the n-component error state vector (n = 15)

F = an nxn matrix , •x ;F = an nxm matrixn :n = an m-vector of random forcing functions for

compensation of error growth caused by unmpdelederror sources.

An approximate solution of Eq. (32) is

27

where. . . . . , . . - , , . . . . . . . , - ; - • . . . . - . . • . . . . , . . . :

$ = the transition matrix

<t - the forcing function sensitivity matrix

u(t, ) = a constant (in the interval t, to t ) vectorK ' K K' -L

for approximating the effects of the noise vector,n, of Eq. (32)

• ' T h e covariance matrix for the error state at time t. is giveti by1C

1 ' ! >! ' - P(t. ) = W(t. ) W(t, ) ;= e[dx(t, ) dx(t. ) ] : -"•'••• (34)\ 1 _ / \ I— / \ I— r IT" If

IV IV I*. IV IV

where

W(t ) = the square root covariahce. (w is calculated inthe proposed square root implementation of theoptima! filter)

£ [ ] = the expected value operator.

We assume that u(t, ) is a random independent vector such that' " '

where U(t.) is the square root of the process noise variance.

The appropriate use of the expected value operator with Eq. (33) gives

the time update of the covariance matrix

(35)

I?

.T ,T

(36)

28

• TWe see from Eq. (36) that we can form W(.t . ) in two steps as follows: ' •' '"'•~~

TThe matrix, W(t, ) of Eq. (37), has dimension (n+m)xn. The

Householder algorithm described in R'e.f.,4 can be used to reduce this matrix to

an upper triangular form; that is, all the terms below the diagonal are zero in" .'• • • • • T

the reduced matrix. The matrix reduction algorithm leaves the product W W

invariant.

(37)

The measurement residual, Ay , ; is defined by . ,.

Ay(t) = y(t)-y(x, t). , . (38)

where y(t) = measured value

y ( x > t ) = the computed value of the measurement based, . ., -, -^on the current estimate of state indicated bythe output of the navigation equations.

It is assumed that the residual is related to the error state by

y(t) = Hdx( t ) +q ' (39)

where

q == the random error in the measurement

Since we may have an estimate of dx given by dx from other measure-

ments, 'we write Eq. (39) as ;

29

Ay = y ( t ) - H d x ( t ) = H d x ( t ) + q (40)

= our best estimate of the residual includingany past measurements whose effect is notyet included in the estimate of state carriedby the navigation equations

As was mentioned in the beginning of this section, the filter operates with

the reference time t for measurements in the interval t. to t . This

means that we estimate dx(t ) rather than dx(t). Hence we make the assump-

tion for t < t < t that

dx( t ) = $(t;t ) dx(t ) (41)1C K

Equation (40) can then be written as

Ay = y( t ) - H « ( t ; t k ) d x ( t k ) (42)

One may note that Eqs. (41) and (33) are in disagreement since the in-

fluence of the random forcing function, u ( t , ) , is ignored in Eq. (41). The

effect of this simplification is that measurements are not quite optimally weighted.

Experience has shown that this simplification is justified if the batch interval

.A. = t - t is short compared to natural periods of INS error growth (e. g., theK K"i~.l K

Schuler period). ,

We let

H = H f ( t ; t )xii K • .

K = W(t. ) W(t. )T HT /(H W(t. ) W(t. )T HT +Q) (43)m v k' v k' m / v m v k' v k' m v '

Q = e[q2]

30

Then the estimate of the error state following inclusion of the measurement is

dx(t, ) = d x ( t . ) +K Ay (44)k'a . k'b m v '

In Eq. (44), the subscript notation ( ) and ( ) means before and after inclusionb aof the measurement, respectively.

TThe square root covariance matrix, W( t ) , after inclusion of theK

measurement is given by

W(t ) = W(t ) - £ T ] / x (45)

where

T T* = W(Vb Hm

n = w ( tk)bc

x = (£T £ + Q)(i + /Q/(CTC + Q)

Equation (45) is referred to as Potter's algorithm (see Ref. 3).

m order to minimize the number of measurement operations, we will use

accumulated residuals over about a one-second period. This means that the

residual, Ay(t) of Eq. (38) , is the accumulated residual for all the measurements

of the same type in a half-second interval. The partials of the observations with

respect to the error state, H of Eq. (43), are calculated and accumulated simul-

taneously with the residual accumulation. The Q of Eq. (43) is the assumed

variance of the random noise in the accumulated residual sum.

31/

C. Error Model of the Estimated Variables

We choose for our aircraft error state a vector of 15 variables. These

are: ' ' • . ^ - : • . • - • . • ' • ' - . ' , . ' " ' - ' • . -

R error in position (3) :

R error in velocity (3)

io gyro drift rate (3) , , , ,

y bias error in observables (3)

•-.. z vertical accelerometer scale factor error (1)

w error in the horizontal wind (2)

These prove sufficient to maintain a stable continuous estimate of the aircraft

state, and to provide the Autoland guidance equations and pilot displays with

usable information over long time periods (several hours).

The time history of the 15-dimensioned error vector is described by dif-

ferential equations. The derivative of the position error vector R( t ) is given by

(46)

Let the specific force in the runway frame be { f "}_ which is given by

(47)

The ;erfor in the inertia! specific force is :given by

\ =

0

6

z

32 K,

(48)

The differential equation for the error velocity vector.is given by . :

' • • > • •>-,- . ; . ; . ^ - - R - . = , [..llj+.-le-lg. - - . - . . , - . - . (49)

where { £ }* is white noise.K

The three gyro drifts, the three sensor biases, the horizontal wind com-

ponents, and the vertical accelerometer scale factor are described as exponentially

correlated random variables, and their differential equations are"

d i ' • ' • • i ' • / 2 • '-rr x. = - — x . ' ' + a. / — £." ' ' 7 ' : '' (50)dt i T. i i V T. i v ;

i = 7 ,15

where >r. is the correlation time constant, a. is the variance, and e. is• • ' l-,1/ ViV. i : ; . i u^ iL- i ' • . • - , : . . . . ' . ' . • : . . - ; • . . . > . • • , . • • • . ! , , . • • • „ ' . . . , ' - v : i ; • • ' • ; ' : . • - . • , . l A ; . - : - / .^£v

white noise.

i . ...The, solution of the above equations, for the short time span of the filter

update, is given by

At2

= R(t) + R( t ) At + {f }_-%-\ ' L «

R(t + At) = R( t ) + (f }TAt + {£ ]~ At (51)'"•'-••• -v • • • ' ' • . . • " • . < • • ' • : • • . . .1 V, , . . . - , , • J \ i ; l . : ' . . , . ' , • • • , - i ; . „ • - , , :.,.;-.-. . . • , . i

i = 7, 15

Equation (51) provides the elements for the state transition matrices,

and $ (t ,t ). These are listed in Tables 2 and 3.U K"""l K

33

TABLE 2

STATE TRANSITION MATRIX

Nonzero Elements

*2.2'

*3,3= * \6= At

*4 ,8 = f z A t

-(At/rj

*6.7

*8.8=e *9,9=e

-(At/TAz) -(At/r^) -(At/rRG)

*10,lb=e *ll,ll=e *12.12=6

-<At/rx) -(At/Tj -(At/rz)

*13,13=e *I4i14ae *15,15=e

34

coen

TABLE 3

SQUARE ROOT PROCESS NOISE MATRIX

Nonzero Elements

= a AtR 95,2

= a AtV

= a At6,3

<& = oU_ • CO V T

7,4 .to

' • ' " . / 2 A t= a / •u - cc v T8,5 tu

= all,8

<J> = a.U12,9

= a13,10

2 A tTx Ty

= a._ 10 toz15,12 z

D. Kalman Filter Data Updates

The guidance and control functions require smooth and current data cor-

rected for the errors listed above. Since the Kalman filter updates the aircraft

corrections every second, AT , and the dead reckoning estimates of Eq. (31)sare updated every . 05 seconds, AT, the corrections to the dead reckoning

estimates are smoothed and updated as follows:

' , . - • ' " _ . • __j •- _•_ .A , . -

Let the Kalman correction vector of R( t ) and R( t ) be Ax( t ) . We de-

fine an exponential smoother, a six-element vector function, CS(t ) , to be given

by

CS(t + AT) = e S [CS( t ) -Ax(T )] + Ax(T ) (52)s s

-(AT/AT )e

with CS(0) = 0

The corrected, smoothed values of the R(t ) and R( t ) vectors are given by

R( t ) = R(t ) + CS (t) . . . • ; - . - . • . • . - . . rR (53)

* •

R(t ) = R( t ) + C S ' ( t )it .

To correct for the gyro drifts, we have the following:

1 . 'A A A . ' - • •;•

Let Ap , Aq, and Ar be the current best estimates of the gyro drift

rates and let fy, 9, and 0 be current readout of the uncorrected Euler angles.

Then, to obtain the best estimates of the Euler angles, we have

36

0( t ) = 0 ( t ) + [cos *(t) Ap( t ) + sin tf(t) Aq(t)]/cos 6 ( t )

6( t ) = 9 ( t ) + cos

c?( t ) =R

Aq(t) - sin flr(t) Ap( t )

- 0(t)] sin 9(t)

(54)

Finally, to obtain the best estimate of acceleration vector in the runway

frame, we have

R ( t ) = <

0

0

G

0

0

AX

(55)

where Az is the best estimate of the bias in the vertical accelerometer.

E. Complementary Filter

A complementary filter can be designed to provide a set of nonvarying

gains, without any relationship to the uncertainty in the aircraft state or the

biases in the INS and MLS measurements. By adjusting the complementary

filter gains to pass the relative low frequency of the true aircraft acceleration,

it is possible to filter out the high frequency noise in the MLS measurements.

The main criterion for such a filter is to provide a stable estimator, one that

decays disturbances exponentially and lets the low frequency motions persist.

A block diagram of the complementary filter is shown in Fig. 8.

Let one of the coordinates of the R(t) vector be x.(t) . The differential

equations of the complementary filter for the x. coordinate are given by

37

CO00

GYROS

ACCELERO-METERS

SPECIFIC FORCEIN BODY SYSTEM

COMPLEMENTARY FILTER FOR NAVIGATION WITH

BODY MOUNTED ACCELEROMETERS

USING MLS MEASUREMENTS

BODY TORUNWAY INERTIALTRANSFORMATION

SPECIFICFORCE INRUNWAY

INERTIALCORDS.

MLS

RECEIVER

R frITAZ KWEL few

DATA EDIT

XYZ SOFTWARE

COMPUTATION

X, Y, Z,PSEUDO^NMEASURE

RESIDUAL1

LIMIT ER

NOTE: THERE ARETHREE SEPARATE

COMPLEMENTARY FILTERSFOR x = x, y, z.

Figure 8. Complementary Filter for Navigation With Body-Mounted Accelerometers Using MLS Measurements

dHi(t) =

Tt *«<*> = K2 [XMLS - Vt)] + V*' + *i(t) <56>i

where ;

x.(t) = i component of R ( t ) [ Eq. (30)]

It is plain that if the residual x - x (t) is zero, the complementary

filter will provide the dead reckoning solution of the aircraft motion in the runway

system. Moreover, the system is linear with constant coefficients K , K , and1.-.. • • , - -L o "'•

K (the filter?gains),; with nonhomOgeneous forcing functions due to the noisy MLS

measurements, and the aircraft measurement of acceleration obtained from the gyros

and accelerometers. By choosing the values of K, , K~ , and K0 , the solution- •• '- i. & • . . o

can be designed to be stable and to filter out the high frequency noise.' i • « . . " '

• Since the MLS measurements are available only in a discrete form at

fixed intervals and the output of the accelerometers are available at fixed inter-

vals, we may. produce a pseudo MLS measurement which is continuous and is

equal to the-actual MLS measurement at the discrete sampling time.

We .model x ( t ) , the pseudo MLS, to be:;. MLiO.

<57>

We choose x (t ) so that at t = T , the observation time, the pseudo MLSJVlLo. Ois the same as the observed MLS.

39

AT - (57a)

With this, definition of the MLS measurement, _we can effect, an explicit solution

of the system of differential equations [ Eqs. (56) ] which has the required con-

dition of giving the dead reckoning solution if the residual is zero; and which

corrects the state to match the MLS observations without the high frequency

component. The solution is given by

where the constants b , b , and b .are functions of K , K , K0 , and AT1 Z * $ j 1 ^ 3

b =

a2 --

- to2)"- - sln(«

\" 2 2 2 2a (j8 - co ) - (w - ' fl AT

] e p " sin a

a[e0fAT

= K,

cos(u)AT)]

sin(coAT)Ao2 2 2

40

The constants, a, fl, co are the roots of the characteristic equation of the

system defined by Eqs. (56)

; 2 s + K 3 = o

•K «..

. ..K_ = 2 Q-/S + B +

Kg =

(60)

By choosing small positive values of a, 5, and w we can obtain a stable low

pass filter.. We recommend values of .' - .

a = .08

(61)

? T.o provide the best estimates of the aircraft position, velocity, and ac-'.'. : - v t * ' \ - ' . 'delegation in the runway frame using the complementary filter output, we have

R(t), - .

xn(T)

x31(T)

R(t) = ^

R(t) =

(62)

R (t)

41

It is to be noted that the complementary filter provides no;correction to

the errors in the estimates of aircraft attitude due to gyro drift or to the position

error due to MLS biases. • 3

42

III. : f THE MLS OBSERVABLES

The characteristics of the MLS observables are described in documents

supplied by NASA Langley Research Center • The system used is for

conical coordinates of azimuth, f$, elevation, a, and range. The range and

azimuth origin is at the center of azimuth antenna facing outward toward the ap-

proaching aircraft along the runway centerline. The elevation origin is offset

from the runway and is at the center of the elevation antenna. Thus, the Car-

tesian coordinate system for the MLS observable is the negative of the runway

inertial system in the x and z directions and in the same direction as the y

of the runway. We have, in the runway coordinate system

X "

yz

= -

MLS

x"

-yz

+

Runway

x "

-yz Antenna

(63)

where the antenna vector specifies the azimuth antenna position in the runway system.

The origin of the MLS system is at the azimuth antenna and the position of the

elevation antenna is measured relative to it. (See Fig. 9a)

Let the MLS coordinates of the aircraft be X , Y , Z

observables in the MLS coordinate system are

Then the

RANGE =

= sin'1 (-YM /RANGE)

a = sin-1

ZM " ZOE

(64)

43

AIRCRAFT

MLS

ELEVATIONANGLE

GROtJND >PROJECTION......

RANGEAZIMUTH

MLS

Figure 9(a). The MLS Coordinate System.

44

In the above, X,__, and Y_ are the coordinates of the elevation antenna fromU.ti U r j - . . . - . .

the azimuth origin, and Z is the coordinate of the elevation antenna in the

MLS system.

To invert the solution and obtain .X,, , Y,_ , Z_, from RANGE , - 8, andM M • M > r- •a, we have . '

= - RANGE (sin (65)

.whVres

ZM = M M

= xoE (s ino; )

h = (XQE2 + YQE

2) ( sin a )2 - (RANGE)2 ( cos a )2 -• 2 YM YO£( sin a j2

It should be noted that the above equations do not account for height differences of the

elevation antenna from the MLS origin which has been defined as the coordinate Z _".OE

Finally, to obtain the aircraft coordinates in the runway inertial system, we have

" X "

yZ

= ,

Runway

X

yZ

•-

Azimuth Ant.

x .M

-yM._ ' z . _

(66)

45

A. MLS Noise Model -. .

The noise characteristics are obtained from Ref. 5. For the exponential

correlated noise of RANGE, ft, and a, we have

(67)

where u ( t ) is a random number of zero mean, a. is the standard deviation

of the i measurement and

-(AT/T.)lA. = e : (68a)

For initialization

r j . ( O ) = u (0 ) (68b)

Table 4 contains the values of the noise characteristics used in the MLS error

model.

To simulate data dropout, we use a random number with uniform distri-

bution from -1 to 1 and set the observation equal to zero whenever the random

number exceeds . 98 . Thus, 2% of each observation string is lost. Since there

are three observation types, 6% of the pseudo state observations are lost. This

percentage can be varied.

46

TABLE 4

MLS ERROR MODELS

Function

Elevation

Azimuth

Range

Model

y(n) = r y 1 - A2 u(n) + Ay(n-l)

y(n) = tjl - A* u(n) + Ay(n-l)

y(n) = r y 1 - A2 u (n) + Ay(n-l)

A

-«Te

-aTe

-aTe

a(— )sec

19.100

0.971

1.013

r

-27.01x10 Deg.

-35.10x10 Deg.

(21.1 Ft.)6. 431 Meters

u(n)

IGRS

IGRS

IGRS

Note: IGRS = Independent Gaussian Random Sample

Initialization is achieved by setting y{0) = u(0).

See Figures 9(c) and 9(d) for illustrations of MLS jLtter in azimuth and

elevation.

To simulate bad data, a similar random number with a uniform distribu-

tion is used, and an average percentage of the observations is set equal to a

large humber:( 1000 times the.bias) even though the data valid flag is set true".

Thus bad data enters the system marked goo'd a small percentage of the:time. ;;

' ; • j > ?

Biases are included in the simulation. Values of the-biases are initially

chosen from a random number generator and retained for the simulation run.

The bias, values are also given in Table 4. . ; .. , , ,„

MLS "jitter" or "stagger" is simulated to produce the data train sequence

as shown on Figs, 9b, 9c, and 9d.' • ' • • ' , ' •/

To accommodate data dropout,/MLS invalids, and bad data generally,

the recommended course of action is to replace the bad data type by the air-' • . " • - - • ' ' ' ! l • . : I

craft estimate of the data missing (or .considered rejectable) by the;best esti- ;

mate of the data type based on the current values of the vehicle state. Thus, if

data is missing, we replace the bad data by range, azimuth, or elevation from• • . • • • * • * AEq. (64), using X^, YM , and••>Z "computed from Eq. (63). This mode is

still under study.

B . Radar Altimeter . : ' . ' • ' • ' ' |* - ! -'S ,' *•- - ' , •' ,'

; The radar altimeter readout is activated when the aircraft estimate of

altitude is below 150 ft. The MLS elevation readout is simultaneously de-

activated. For the purposes of this study, the error model of the radar al-

timeter is described by an exponentially correlated bias variable driven by

white noise. Thus, the radar signal readout Jis given by

48

200

SEQ. SEQ, SEQ.

I *1 I *? I

0 100

I I I I0 64J 129 207

65 143

SEQ.#2

SEQ.

~ •• • • .•'; iSEQ. SEQ,

300

291I

357

400

441

500

4-.5EQ. SEQ.

I*1'

227 293 371 447

FULL CYCLE - 592 m$ ———

.I J511 575

600 mt

592ms .

SUB-SEQUENCE

10.2 26 31 36.2 48 53.8 59.0 54.0 rm

EL FLARH AZIMUTH;

10 15 20

FLARE ELp j—

BACK AZ .GROWTH ELI (1) I

FLARE

25 30 35 40 : .45 ; ,50 5 5 ; ; : i 6 0 65ms

64 ms

10.2 26 31 36.2 38.6 53.8 59.0 64.0 ms

SUB-SEQUENCE #2::: EL FLARE AZIMUTH FLARE EL

5 10 15 20 25 30 35» • • : • . t\ 40 45

^ BASICDATAWORD#1

GROWTH12)

ELI

FLARE

50 55 60 65 ms

NOTES: (1) AUXILIARY DATA (1 WORD) OR BACK ELEVATION(2) 360° AZIMUTH

Figure 9(b). MLS Conical Scan Time Sequence.

49

eno

0.1

o

o0)

CO,

0)

-0.1200

Figure 9(c). MLS Jitter, Azimuth

0.1

oo

Q)

0O.•-MCJ

0)

W

0

-0.1100

Time (Sec)

200

Figure 9(d). MLS ,1 itter, Elevation

h(t) = - z ( t ) + bh(t) (69)

where

nrJ T-

C. MLS Partials

The Kalman filter requires the partial derivatives of the MLS observables

with respect to the 15 estimated variables.- ,

We have, for the azimuth observable,

Mbx

RANGE

ZM YMbz

RANGE ,M M

(70)v

M M

52

..- For the bias in azimuth, we have( .• • : 1

b/3 = 1bb.R .

(70a)

For the remaining 11 variables, we have

ox.= 0 (70b)

The elevation observable is

bqbx

- ZOE><VXOE>.

(YM" YOE)(Z Mby

ROE

(71)

babz

t ,

ROE

where

ROE (71a)

For the bias in elevations, we have

bb = 1a

(71b)

53

For the remaining 11 variables, we have

£ - 0

The range observable is

GRANGE XMbx RANGE

6 RANGE = [ (72)by RANGE

I 6RANGE ZMbz RANGE

For the bias in range, we have

bRANGEbb

r= 1 (72a)

For the remaining 11 variables, we have. . . . .

: ^P = Q (?2b)

For the radar altimeter, we have

^ = 1 , . (73)

For the altimeter bias, we have

bh(t = 1 (73a)bbh

54

For the remaining 13 variables, we have

.6h( t ]bx. = 0 (73b)

55

IV. MAGNETIC LEADER CABLE SENSOR

While the MLS sensor described in the previous section proves adequate

for landing approach through touchdown, problems in multipath distortion and

other biases create serious problems against relying solely on the use of MLS

in rollout and.turnoff. In order to provide a navigation sensor aid with smaller

errors, the concept of a single magnetic leader cable embedded along the center-

line of the runway and along the turnoff lane is recommended.

Initial studies of such devices began in the 1960's. Reference'6 contains

a description of a magnetic cable system and a three-coil sensor developed in

England. A series of studies for use in automotive steering along reinforced

concrete highway systems is presently sponsored by DOT. Reference 7 contains

a description of such an automotive steering aid for lateral control.

.1 :

For our purposes, the British system and experience seems to be more

applicable and is incorporated into this report. '•

A. Three-Coil Sensor • ' -

The magnetic field is generated by a single conductor set beneath the

nonconducting concrete and along the runway centerline. A detector, consisting

of three coils having an equal number of windings of conducting wire with iden-

tical characteristics, is used to generate three voltages. The coils are mounted

about a spherical, nonconductive shell in three perpendicular planes. Figure 10

illustrates the leader cable and the three coils.

56 =

DBS

LONGITUDINALAXIS OF AIRCRAFT

COIL 1

(Ref. 5)

Figure 10. Magnetic Leader Voltage Signals.

57.

Let the Cartesian components of the magnetic field be

H = (H x , Hy, Hz) (74)

as illustrated in Fig. 11. We have, for the three induced voltages,

v. - H-1 z

v ~ H cos (A*) (75)^ y

v ~ H sinj y

For the yaw deviation, we have

V3tan (A#) ^ — . (76a)

V2 "X,

: , • < ; '' . '• :• . •

and for the lateral displacement . u •

H : vAd = Az -^ = Az ——1== (76b)

-

The error in yaw and the error in lateral displacement are directly ob-

tainable from the voltages. To obtain the time rates of these quantities, which

are required for damping stability in the Autoland guidance equations, we have

chosen not to rely on differentiation of the magnetic coil signals, but to generate

the required information analytically. The equations for these functions are

given in the Guidance Section of this report.

58

Conductor

Detector.

Ground Plane

Ad

H frz

H

-e-

(Ref. 5)

Figure 11. Magnetic Field Equations.

H

59

B. jSrror Model of the Magnetic Coil Sensor ,•.f, •, . . ' • , , " i 'i.

}The distortion in signal strength is due tot several sources, as shown in

Refs. 6 and 7. c

1. ground conduction > ; (

2. nonlinearity1 * • ' . . . - , ' ' •• A,

3. phase modulation '' ' *''

The effect of ground conduction is described in Ref. 6. The effect is es-

pecially :serious:for,:concrete runways containing steel -reinforcing rods. Refer-

ence 6 recommends using a low carrier frequency (165 Hz) to minimize the

distortion effect due to ground conductivity.

Nonlinear effects are most troublesome in the lateral deviation equation

where the linear relationship between Ad and Az is relied upon in Eq. (76b).

By restricting the use of the detector to rollout and turnoff, we are assured of

operating within a narrow range of vertical displacement ( ± 2 ft. ) due to strut

compression and pitch oscillations. Thus, we can expect to operate in a linear

range during those phases of the flight where the magnetic cable is required.

Finally, phase modulation can be eliminated as a serious error source

by restricting the use of the detector to small range in Az . In general, none

of the references indicated any difficulty in obtaining a good A# signal.

For the purposes of simulating the error sources, the following equa-

tions were used to generate realistic magnetic measurements of A >&._.. and

60

where

M "true

: : (77)

... = ' Ad. (1 - .1 cos wt)—— + e "M true z Ad ., .nom J

CA j i c» ,. are white noiseAd

0) is the carrier frequency (165 Hz)

z is the nominal height of the detector above groundi and a fixed constant for the aircraft, for all landings

61

V. NQNCRITICAL FLIGHT PHASE NAVIGATIONAL AIDS

In addition to the MLS Radar altimeter and the magnetic leader cable,

which are required for the flight critical phases of landing approach, rollout,

and turnoff, we include VORTAC range and bearing devices and a baro-altimeter

as additional navigational aids for the noncritical phases of the aircraft flight

regime. These instruments have relatively large bias errors. However, since

they are used only for navigation between way points away from the terminal area,

they can provide adequate control for the gyro drift and accelerometer scale

factor error sources in conjunction with a Kalman filter.

A. VORTAC Range and Bearing

Let the VORTAC station coordinates be x^ , y , z in a level frame

at the VORTAC station. The axes are: x due north; y due west; and z

vertical, positive upward. Let the aircraft position in the VORTAC coordinate

system be x , y , z . In the measured VORTAC range, we havea - a a

+ ( ya-yV ) + (VV + br + ^ (78)

and for the measured VORTAC relative bearing

-1 yv"yaA* = V- *y - tan r—- + b + £ (79)a v

To complete the triad, we obtain the barometer measurement of altitude

- £ (80)

62

In the above, b , b , and b are biases modeled as exponentially cor-

related random errors and e , ' e , and C are white noise. The biases arer ^f hmodeled as follows:

K— b = -- .+ a I— ndt r r r V T 'r

V CThn n

where r , r, , and T, are the correlation times; a , a T , and a. are ther w n r w . nvariances; and 77 , 77. , and rj, are white noise samples. (See Table 5)

To complete the necessary equations for the Kalman filter, we derive the

partials of the three observables with respect to the 15 -element state. For the

range partial s, we have ' '

brv V6xa

5rv _ Vyvbya " rV

(82a)

r zvbz rT7a V

bbr

63,.

TABLE 5:

VORTAC ERROR MODEL

Variable

Range

Bearing

Baro-altimeter

TimeConstant

900 sec.

900 sec.

100 sec.

Variancesbias observ.

300 meters

0.7°

40 meters

40 meters

0; 7°40 meters

64

The other 11 partials are zero. For the bearing partials, we have

ya~yVbx

_ • .

(VV + ( y a"V

The other 12 partials are zero. For the barometer partials, we have

a(82c)

bbh

The remaining 13 partials are zero.

65

VI. GUIDANCE AND CONTROL

The Langley TCV program utilizes a modified Boeing 737 aircraft for its

research. It contains an automated guidance and control system for landing approach

and rollout. The simulation of the guidance laws were supplied by the Langley

Research Center. These laws utilize ILS navigation aids. In adapting the B-737

aircraft for the Kalman filter and the complementary filter utilizing MLS navi-

gation aids and a magnetic cable for rollout and turnoff, certain changes had to

be made to the Boeing Autoland guidance laws. The changes are outlined below:

1. The Boeing complementary filters were removed from the

flight critical glideslope and localizer control laws.

2. Since no requirement for turnoff guidance existed in the original

B-737 TCV aircraft, the Boeing rollout laws were modified

to permit a uniform guidance law to hold for both rollout and

turnoff.!,.' • ' . . ' - „ ' . ' ' "

In general, except for the above changes, the control laws, gains, etc. used in

the Boeing Autoland equations were maintained.

This section contains guidance and control laws for the following phases:

1. Flight critical lateral control for oncourse landing approach.

2. Flight critical vertical control in landing approach for capture

and flare.

3. Rudder control for landing and decrab.

4. Lateral control for noncritical level flight and coordinated

banked turns.

66

5. Vertical control for noncritical level flight and descent.

6. Steering laws for rollout and turnoff.

A. Flight Critical IJateral Control Guidance

The flight critical lateral guidance contains two control functions, localizer

engage arid oncourse. These are triggered by deviation criteria computed from

data supplied by the navigation filter and the desired glide slope path.

The computations required for these control modes are

= ' t ( x - x ) 2 Vy 2 +£ 2 ] * ' • ' • " • " '

RANGE 77

f y y

\

• - - x xV = ( x + y + z )

.180Pdeg = P~

where h is the extended landing gear height (see Fig. 3).VJ . . . . , ' , . • • ' : . ' . . . ' -

67

The triggers for localizer engage and capture are

L.E. is TRUE if | r? |<2.5°

0/C is TRUE if \ ' H \ < n L m > I r\ \ < . 027°/sec. ,

and 10, I < 3.° '.1 deg '

Two block diagrams describe the flight critical lateral control mode.

The first, Fig. 12, generates the aileron roll command signal as a function of

the lateral deviation from the glide slope descent path. The second, Fig. 13,i-

is the flight control lateral inner loop which generates the aileron command to

the aileron servo.

It should be noted that the inner loop logic contains an input from the rud-

:rab command, 60 _,. The guidance lo]RDCFig. 15 covering the rudder command signals.

der decrab command, 60 _,. The guidance logic for this input is given inRDC

B. Flight Critical Vertical Control Guidance

The flight critical vertical control guidance covers three modes:: capture,- :

capture plus 10 seconds, and flare. As in the case of the flight critical lateral

control guidance, the inputs for these modes are generated by the navigation filter

equations to control the vertical deviation from the glide slope trajectory.

The equations for the required inputs are given below: •!

68

152 305

1.0

25-.--

'LIM

1-h (M)457 152 305

O/C

+>'

RLIM

^COM

RLIM

L. E. =O/C =Flare =

25.10.5.

C22

L. E.O/C

= .70.6= . 988

C2 = 2.5

Dl = 25.D3 = . 16

Figure 12. Flight Critical Lateral Roll Command

deg

RDC

XOM if] T. ( TM 1 -rx/ \~yr*_r

•

0TTA/rLIM18

+ /

0/C = 4. °/secL. E., not O/C = 7. /sec

C5 =C6 =C55 =D4 =RD =

20.1.7751.1835..7

V(M/S)

KPOTL

KPOTL

185

Figure 13. Flight Critical Lateral Inner Loop

y = glide slope flight path angle (-3. )

hA

= - z

h = - z

2 2 2 "* (84)

RAN = [(x - x) +y +z ]

-i rSin L

o -Q, = qM,deg M 77 ,

where x is the glide slope intercept with the runway (x_ = 304. 8 meters)G G

The triggers for capture, capture plus ten seconds, and flare are given

below:

• C (capture) is TRUE if | /L 0 |< -108°;.. • '. ' - • - Go : -•

CP10; (capture plus 10 seconds) is TRUE ten seconds

after C is actuated

F (flare) is ; TRUE if . 416(h + 15.) + 26 h ^ 0 .

The block diagram for the flight critical vertical control guidance is

shown in Fig. 14. The output of this control law is the elevator command.

C. Rudder Control for Landing and Decrab

The rudder control guidance contains two functions: first, to provide

yaw damping during all phases of aircraft flight, including rollout and turnoff;

71

/—*r(&)-~-F

Qdeg

+

.000032

sin y

?1r- I -- '. .j te/A5 i b. :-' I

SV.J

..;

t = time of capture •*••cp • • > • • - . • / • • r . f . . . • •

ECOM -*-

Figure 14. Flight Critical Elevator Command.

secondly, to provide the rudder command for decrab immediately prior to

.-touchdown in the event of a cross wind.

;• The information required by the rudder command guidance laws is pro

I vided by the navigation filter.

' - ' _ * 180R, = r -deg TT

TT

(85)180 •

R runway TT

t, = time of decrab initiationdc

The trigger for decrab is given by

DC (decrab) is TRUE if O/C is true and h < 45.7 meters. ' . . . . r

The block diagram for the rudder control command for yaw damping and

decrab is shown in Fig. 15.

It should be noted that the rudder decrab command signal, flL.,^,,. rlJUO • • • > : • • ;

required in the flight critical lateral control guidance loop is generated in Fig. 15.

73

Rdeg +

-*

a ^des Q

Yaw Damper Loop

DC167

. !_, S EASYON

t = time of decrab initializationDC

<-5 5

-»T

EASYON

0

DC

Al = 7.A2 = .3Gl = 1.493G2 '•= .06403G3 = 305. 67

:.'Rl"-= -; . 45:R2;= : .34 .

R3'.==. 2.222 •-,-.

Figure 15. Rudder Yaw Damper and Decrab Command

D. Noncritical Aileron Command Guidance

Lateral guidance laws for trajectories between way points and for banked

turns are governed by noncritical aileron command guidance. For banked turns, :

the lateral guidance requires the following information:

R = turn radius :

SIGN = '{ • -* leftju:nI -1 right turn

x ,y = x and y coordinates of the center of the : ;w w ./ . }

turn circle in runway inertia! coordinates <

: CRTE (cross track error) = SIGN R - J (x-x )2 + ( y - y )2

; '.. . v ; (86)(y -y )y + (x -x )x• v J w w1 I TANGE (track angle error) = ; = —

•-'• ; . .-. (x -x )y - (y-y )x ,;'• w w

V = desired constant turn speed

VT

= [v2 i

* -1 / D \\ 180tan ( )T.

- -180j = 0 . — ^ ~deg 77

. The block diagram illustrating the noncritical flight aileron command is

shown in Fig. 16.

75

-aos

C l = 4 0 . ;-C2 = (C3)

2/7.12C3 = .68-(.00107) (.3048) [Airspeed (M/S)]C4 = 5. " :

C5 = 1.82 •

QPOT= .22 + 126.5 (. 3048)/AIRSPEED(M/S)

Figure 16. Noncritical Aileron Command.

E. Noncritical Elevator Control Guidance

Vertical control during the noncritical phases for trajectories between

way points and in banked turns is illustrated in Fig. 17 for the elevator command.

The information needed to supply the control equations for a banked turn

along a descent trajectory for a flight path angle of -3 is

h(t ) =, altitude at the start of the turn circleo — ' • . . , ' - . - . • •

h ^ h ( t ) - J R ntan-V*^7*c x o ' L T L ' W( t ) - y

Q

y / ' -y• o w . . w

= 0 - (87)

• i§2 • ; /'deg 77 , .

=/• 0,deg TT

F. Airborne Throttle Command , :

Thrust control is designed to maintain a given airspeed and to compen-

sate for the loss of lift during a banked turn.

The variables needed to operate the throttle command guidance are

given below: ' •

77

-aoo

*CHOP is TRUE if trimming and (e A > 0), otherwise it is false.

**I fL<0 . , f(a) = 1.

Figure 17. Noncritical Elevator Command.

-*

f ( 0 ) = - - (1 -cos 0)

V(m/s) = airspeed in meters per second obtained fromthe airspeed indicator

AV(m/s) = V - V : . . (88)CL6S

r-r *-+ *. £ ** £l '

VG = (x + y ) ;

DTVG = (xx + yy ) /VG

The block diagram for the throttle command is shown in Fig. 18.

G. Rollout and Turnoff Guidance

Rollout and turnoff guidance encompasses rudder and/or-.nose wheel steering

commands, thrust control commands for speed acceleration and taxiing, and finally,

brake pressure commands for speed deceleration and taxiing.

Rollout guidance is activated as soon as the main gear preload is exceeded.

At this instant, two commands are initiated. First, the throttle is manually set

into the idle position. This action is accomplished manually in order to prevent

accidental actuation of the throttle into idle detent prematurely. Secondly, a two-

second (2 sec.) slow-out and slow-in is initiated to bring the inflight, pre-touchdown

control surface guidance commands for the aileron, elevator, and rudder to zero

and to raise the post-touchdown rudder/nosewheel steering commands to full signal

strength. The mixing takes place within two seconds and is illustrated in Fig. 19.

79

00o

/•1s -/Al\-

V '• J

T TTVJTTrn\

+ 2

conversion

Al = ,0625Bl = . Q5^ v

K3 = 13711.4FX

_

KY =KI =KO =MX =

.2. l.:5-.'-::1.2 :

^10^ :'

.9697;1.9 •-

THRUST

TH

If AP<0 . , FX = AP

I f A P > 0., FX= ^ CMX-.l-(i THRUST - KO)/K3]AP

Figure 18. Throttle Control.

ECOM

ACOM

RCOM

RCOM(R )

ECOM

ACOM

B;

slowin .

*- 6RCOM

B

1.0

2.;t-t0 (sec.)

R

00

Figure 19. Rollout Control Surface Slowout Command.

The rollout and turnoff control is unified into a single guidance law by

requiring the aircraft to follow a single magnetic cable. A three-coil magnetic

pickup, mounted below the aircraft fuselage generates two signals; one measures

the lateral deviation of the aircraft from the cable, and the other measures the

yaw deviation of the aircraft from the cable. - The signals necessary to provide

damping stability for the lateral and yaw deviations are generated analytically,

in a manner similar to the banked turn logic in the lateral noncritical flight

guidance equations. By requiring that the magnetic cable consist of a sequence

of piecewise straight portions, connected with circular arc segments, we are

able to specify the desired ground velocity vector and the desired,yaw rate for

both the straight line and circular segments. By using the aircraft estimates of

the ground velocity vector and yaw rate, we generate the deviations for use in

guidance equations.

In order to communicate the start of each straight line or circular seg-

ment, a radio signal is given to the aircraft at the beginning of each segment.

The information contains the following:

Circular Segment '

1. radius of turn /i/ '''

2. sign of turn (+1, right turn; -1, left turn)

3. x and y (coordinates of turn center). . . ' , .w ; • , w V • . • • • . . , . . • • . , . , • : . . - • • . • , ' > , • • , ' . . • • i . . :>•.• . . ,; ,;

4. signal to start turn

Straight Line Segment

1. yaw angle of segment

2. x arid y (coordinate. . . . . .0-". °

3. signal to start straight line guidance

2. x arid y (coordinates of start of segment)

82 ? • - . : •

. .The variables required to operate the rollout and turnoff guidance equations

are listed below: > ' '

For Circular Turn: - .

• 2 • 2VG •' = (x + y

,des

Ay = - SIGN

TT

• a.

(89a)

,deg TT

For Straight Line:

Ay = cos ^ y - sin x,

R, . = 0.des

(89b)

The logic for the uniform rollout-turnoff guidance is illustrated in the block

diagram shown in Fig. 20. . -

. .The logic for the brake pressure control is designed to first assist the

reverse thrust in decelerating the aircraft to the coast speed (V ) , -and

secondly, to maintain a somewhat lower taxi speed, V . . During the first,. . . . T3.XI ' . .( • .

phase, ; 80% of maximum brake pressure is available to the brake control logic.

'83

CO

Rdeg

6RCOM(R )

Figure 20. Rollout and Turnoff Steering Command.

During the second phase, only 40% of maximum brake pressure is available ;

for. control. An anti-skid device maintains a modulating control to prevent

wheel lock and skidding. *;

The logic for the brake pressure is shown in Fig. 21.

. The rollout thrust command is illustrated in Fig.: 22. The throttle is

held in the idle detent for two seconds. Following that, the thrust guidance

calls for the specified reverse thrust. When the aircraft reaches a ground speed

of V , the throttle control is once again brought to the idle position and. *COaSt* ••"'. (

rv \f •!,. \

remains there throughout t h e taxi phase, ' j ' - - ' . . - : : ' ' ' • . • -

H. Requirements for GE Whole-Word Computer to Execute Navigation-and"•' '"' • ..; • ': • \

; Kalman Filtering During Rollout :\ - ' . , ; • • ' " ' . • ''-.-' • • ' •• ':

' • ' • ' • ' • • ' . - • ' • • f • ' : • • ' & ' ' ' •'"'. .This section contains estimates of the timing land storage requirements '

' • • ' • • • ' • ' • ' . • ' • ) - . . . : • ? " ' * " . " . ';'. ' :

for the General Electric MCP-701 computer to execute the navigation and Kalman1 1 ' ' " ' ' . " * ' : . * " ' - • ' • '

filtering tasks during the touchdown and rollout sequences. Since the navigation .

and filtering functions are,.programmable as separate and dis'tinct; modules;' esti-

mates of their timing and^storage requirements may.be made: separately, and''-" : ' . ' * • • ' • ' , ' ' ~ . V ' '

the resulting estimates may be summed to-obtain total estimates for timing k'nd .. ' • ' * " ; • ' . ^ • ' • ' .-'"' -:--;'storage. '< ll .1 ^ -. ''"• ;.; :

• ' - • • ' J .;•• ' • . . - . ' • ' " ' . / ; '

. .-Different methods were used in obtaining the 'estimates for the navigation,

function and for,the Kalman filtering function.

' . •'• FORTRAN code of the rollout navigation algorithm was used in making the

storage and timing estimates. FORTRAN operations were translated into;equivalent

85

MCP-701 operations in estimating both timing and the program-instructions

component of storage.

A more direct approach was used for the Kalman filter. The Kalman filter

is already programmed and running in single precision on the.Sperry 1819A 'com-

puter at the Ames Research Center. Therefore, the timing estimate is the actual

1819A running time scaled to reflect the relative performance of the MCP-701.

The storage requirement for the filter is essentially identical to that of the 1819A.

Kalman Filter Requirements:i • • . .

To give the estimates of the filter storage and timing requirements some; meaning, a brief description of the Kalman filter is in order. It is a three-axis

Kalman filter with segregated horizontal and .vertical channels, as depicted in

Fig. 23'(for the Sperry-1819A computer). For the application under consideration,i . !

the MODILS measurements would be replaced by the equivalent MLS measurements,

and the TACAN and MLS measurements would probably not be used simultaneously.'

Attitude and heading data and three, body-mounted accelerometers ;are used

to give the acceleration in runway reference. Existing 1819A software performs

these calculations at a 20 Hz frequency. Biases in the accelerations (estimated

by the x-y ,and z filters) are added to the accelerations and then integrated in

double precision to give the three-axis velocity vector, which is in turn integrated

in double precision to give the three-axis position vector. This is the only double-

precision logic and is performed in the block Navigation Equations in Fig. 23. The

estimated state, vector coming from this block includes the following:

3-component position vector with respect to runway

3-component velocity vector with respect to runway

3-component acceleration bias vector

88

Attitude andheading data

Body-mountedaccelerometers

Airdata

TACAN

MODILS

Radioaltimeter

_C omputer_

Dataprocessingb y 'Sperrysoftware

True airspeed

TACAN range. and bearing .„•

MODILS rangeand azimuth

X-Y Filter

Acceleration inrunway reference

Estimated

Baro altituderadio altitude

MODILS elevation

State

Estimated errors(level channels)

SmoothingLogic

Z Filter

Smooth StateEstimate I

isplays-*-!to DisplaysG and C

•Estimated errors(vertical channel)

oo

Figure 23. Block Diagram of Overall Filter Mechanization

2-component estimate of winds (:x and y )

1-component estimate of TACAN range bias

1 -component estimate of TACAN bearing bias

1-component estimate of baro-altimeter bias

The position and velocity components of the estimated state are updated from the

acceleration data at the 20 Hz frequency. All components of the state are up-

dated from the x-y and z filters at .'667 Hz. This low-frequency update causes

discrete changes in the estimated state, which are likely to be objectionable if this

estimated state were used for guidance display and control purposes. The block

labeled Smoothing Logic in Fig. 23 filters the discrete "changes from the estimated

state so that the smooth state estimate output will be free of objectionable jumps.

The x-y filter accepts the true airspeed, TACAN range and bearing, and

MODILS range and azimuth measurements. This filter has 10 states, as follows:

x - position along the runway

, y position normal to the runway

x - velocity along the runway.

y - velocity normal to the, runway ; . , . . . - . . . . . ,

a - along runway component of acceleration bias"

a , - normal to runway component of acceleration bias ,,

b - bias in TACAN range measurement ,

_.- bias in TACAN bearing measurement

w -, wind along the runway . . .• . • . X • - ' * - . . ' . . . i .- , . . • ,• j •

w - wind normal to the runway- - . ' ; . y . - . . - : . . - . . . : • . . J ' . - • • . - . • ' . . ' • - , . . . I . • :

The z filter accepts altitude measurements from the barometric altimeter

and radio altimeter and. altitude measurements calculated from the MODULS eleva-

tion data. This filter has four states:

90

z - vertical component of position

z - vertical component of velocity

a - vertical component of acceleration bias .zb, - bias in the barometric altitudeh

All raw measurements are accepted at a 10 Hz frequency. Preprocessing

algorithms perform residual and partial-derivative calculations as well as validity

checks, and the resultant good data is accumulated for a period of one-and-one-

half seconds. The accumulated information is then transferred to locations for

processing by the square-root Kalman filter algorithm. Once the filter has esti-

mated the error state for this information, the logic updates the estimated state

and performs the smoothing. This concludes the brief description of the Kalman

filter.

The total storage (instructions and common variables) of the filter, which

is mostly in single precision, is 2,825 words on the Sperry 1819A computer. If

the filter were entirely in double precision, it would require 3/053 words. These

storage requirements are essentially valid for any mini-computer, since the

assembly-language instructions for various mini-computers are very similar.

The filter consumes 18.3% of 1819A running time. The MCP-701 is felt

to be at least twice as fast as the 1819A. In particular, double-precis ion additions

are approximately twice as fast and single-precision multiplications are about four

times as fast. Therefore, it is estimated that the filter will consume about 10%

or less of MCP-701 running time if implemented mostly in single precision. A

consideration of the operations necessary to implement a double-precision multiply

function on the MCP-701 has yielded the conclusion that a worst-case factor of 4

would result if the filter were programmed entirely in double precision. It is esti-

mated, therefore, that the filter would consume about 40% or less of MCP-701

running time if implemented entirely in double precision.

91

Navigation Requirements

^ 'A detailed investigation of each phase of the rollout navigation algorithm

was performed to determine the timing and storage requirements. FORTRAN

operations were translated into equivalent MCP-701 operations (or sets of opera-

tions). Timing estimates for each phase of the rollout were then estimated by

.directly counting the required equivalent MCP-701 operations and making use of

the information in Table 6.

•« j The rollout sequence and timing, estimates are presented in Fig. 24. The

times are given in microseconds and include a 50% "safety factor" over the values

actually computed (i. e., a multiplicative factor of i. 5). Timing estimates are

given for the assumptions of all-single-precision arid all-double-precision coding,

and percentages of real-time consumed are given in parentheses assuming a 20 Hz

frequency. Two situations may occur during rollout:

;^ ; •" : ' . a) braking is completed before the turn beg ins ; , - ; , . , . ;

b) braking continues into the turn

The latter case- results in the-largestpercentage time load, of the;-rollout navigation

algorithm on the computer ( 2. 5% in single precision and 8. 6% in double precision).

, — The rollout navigation algorithm is estimated to require 702 MCP-701 in-

' structioris. The size of COMMON (where variables and constants are stored) is

estimated to be 146 words if single-precision coding is sufficient, and 277 words

if double precision is required. These values include a 50% "safety factor" over

•: the values actually computed. The total storage estimate for rollout navigation is

obtained by summing the instructions and COMMON values. In the.single-precision

mode, a total of 848 words are required, and in double precision 979 words are

required. > - . - • . . .

92

Memory AddressableInstruction

UEMORV ADDRESSABLE INSTRUCTION FORMAT LOAD INSTRUCTIONS ASSEMfLEH FORMAT 1 .

11 14

|

11 12

J_

11

MOOC

10 0 1 7 6 6 4 1 2 1 0

( M i l l I I I

INSTRUCTION Aocmu*

UNE

ICULOtLOXLOOLPK

CODE

OB10

- 182028

INSTRUCTION NAME

Load Upper RegularLoaJ Lowe* RtgitterLoad Index Register 'Load DoubleLo»l Packed

TIUEueacI

2.22.22.23J2.2

STORE INSTRUCTIONS

MODE VARIATION

TKc mod* wUtiora for memory n1i1i«Mihli inttnxtionsdt*trm the method for gtnoritinf th« «ff«etiw« mLoouon of d«.U within memory-

ARITHMETIC INSTRUCTIONS

'MODE

10.000011I1

>0

.0110011

801010101

EFFECTIVE ADDRESSGENERATION

Current PageZero Pag.IndirectZero Pap/IndirectIndeM

. • Zaro Pege/lndeji •Indirect/IndexZero Paoe/lndirect/lnde*

4TIMElu tec)

001 '10.20.21.1

LOGIC INSTRUCTIONS

BRANCH INSTRUCTIONS

STUSTLSTXSTOSPK

30384048SO

Start Upper RegietarStar* Lo«r«r RegntarSuxt Indmx RegiitarStart DoubleStait Packed

JJJ.22.13.22.4

ADD ;AOMADD

SBUS8DUPY

' OIV

58606870788088

Add to Upper RegitlerAdd to MemoryAdd DoubleSubtract from Upper RegisterSubtract DoubleMultiplyDivide

2.23.23.23.23.24.45.8

ANU. ORU

EXU'CPU

9098AOAS

And to Upper RegisterOr to Upper RegisterEzdusive Or to Upper RegisterCompare to Upper Register ' •

2.22.22.2

•2.2

JMPJMS

• MOS

8088CO

Jump UnconditionallyJump to SubroutineMemory Decrement end Skip

1.42.23.2

Non-Memory AddressableInstruction

NON-MEMORY ADDRESSABLE INSTRUCTION FORMAT

BRANCH INSTRUCTIONS

RAW*Hil^fJ

RStf5*1sai

samSfHSBN

(NSTHUCHOH NAI4E

HtgifUr SuOtTtCI i Skip OH PC

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h..,,,,H,j,M,

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INSTNUCTION

ft

7.6 6 4

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I I I I I I I

SET/CLEAR INSTRUCTIONS

SETCLFt Ofl CU*r irexvcimr

REGISTER INSTRUCTIONS

»gunr Trvnrftw

2vo R

UMd I 1.4

1.41.4t.i1.4

REGISTER COOES

INDICATOR CODES

INPUT/OUTPUT INSTRUCTIONS

UNE

URLMX«snUL

CODE

13)4C

BEGISTEB

Upper RegularLOMMT RegtctarIndei DeainerStatue RegrturUpper and Lower Register

ENBLINMB,RST

TAKESNSPUTttL

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ftitMt SoWCtMl DwiCaTT«k.« 0«t» ta Uoo>af I

Put DAJ from UHMT R*giMr

SHIFT INSTRUCTIONS

MNE

SA

ZA• AF

CFOR10LAIB

CODE

. 01

• J34667

INDICATOR

Sign of AdderZero AdderArithmetic FaultComputer Fault OitabaiDeyice Reedy

' Input/Output DitatattlLogic Indicator ALogk Indicator 8

SRZSLZSRSSRCNHUSRX

Shirt Ri0«. Eflt,.* 2,voMShift l«lt EnurZ.wo« .Shift RignL RCPMI SiyiShift R>«..l CircuUr

Shift R

Shift CountShift COMHIShilt CountShift CountNot UMdNot UMd -

CONTROL INSTRUCTIONS

14141.1

14

1J

0.3n0.2n

HALTNOT

00EO

H*ftNO Op*e-4tlO*.

NO.U.M.4

tor| "

Table 6. GE MCP-701 Computer Instruction Execution Times.93

Time in \i sec

% for 20 Hz

S=Single Precision

D=Double Precision

Touchdown

2 Second Phase

Braking Phase

Taxi-Bef ore-

Turn Phase

B

--C

D

S 837 (1.'

D 2477 (5. 0%)

S 677 (1.

D 1815 (3.

S 642 (1. 3%)

D 1766 (3.5%)

Taxi-Before-

Turn Phase Absent

S 1229 (2. 5%)

D 4287 (8.6%)

Turn Phase S 1148 (2.3%)

D 4172 (8.4%)

Taxi-After-

Turn Phase S 764 (1. 5%)

D 2207 (4. 4%)

Figure 24. Rollout Navigation Timing Estimates for the MCP-701 Computer

94

Total Requirements

The total Kalman filter and rollout navigation requirements (with a 50%

safety factor included in the navigation component) are obtained by summing the

separate requirements discussed above, assuming the worst case for the rollout

timing estimate. In single precision, the total requirements are 3,673 words

of storage and 12. 5% of real time; in double precision, the total requirements

are 4, 032 wo'rds of storage and 48. 6% of real time.

95

APPENDIX A

B-737 ,

Aerodynamic Force and Moment Goefficients

and

Dynamic Response

96 j -

I. AERODYNAMIC COEFFICIENTS

The aerodynamic coefficients are functions of the velocity of the aircraft

relative to the air stream, control surface deflections, altitude and altitude rates

of the aircraft.

We define the following quantities:

c = mean aerodynamic chord

b = wing span

S - wing reference area .-. ••, •

p = air density

6_ = elevator deflectionJli :-

ft = aileron deflectiona~ stabilizer deflection

f\ = rudder deflectionK

§ = aileron spoiler deflectionSir

6cr>oTTi\/r ~ symmetric spoiler deflectionoir OU JVL

a = fuselage offset angle

The velocity vector of the aircraft relative to the airstream in the runway

coordinate system is given by

V = R - [ W + G - f S } (A-l)

The relative velocity vector in the body system is given by

V1BV2BV3B

= LBI

{ V } (A-2)

97 '•*'»

The airspeed is the magnitude of the vector V .

.. v = |V | . . (A-3)

The dynamic pressure is given by

- , 2q = 2 pv (A-4)

The angle of attack is given by

a - sin —+ v ^

IB 3Bor (A-5)

-1 3B; • ' . : , a = tanV1B

The aerodynamic coefficients are functions of the angle of attack relative

to the mean chord line

a = a + a , (A-6)

The side force angle of attack is given by

The effective drag coefficient is given by

CD = CD1 + CD2« + <CD3+ CD4^^SPSUM+ CD6(A-8)

98

The effective lift coefficient is given by .

CL =

CL5 'STAB + CL6 *E + (CL7 + CL8 SPSUM

L9 GEAR

The effective side force coefficient is given by

Cy = Cyl ft + CY2 - ( R.COS « + r sin

^70 r- ( r , - p s i n a + r cos a)Y3 2v gust (A-10)

The aerodynamic roll moment coefficient is given by

_ _ b

CS3 to ( ~ P (CS4+ CS5 > (A-ll

99

The pitch moment coefficient is given by

M = qSco

'Mil

(A-12)(CM12 + °M13 °M14 I 6R

M(g*- .25)C

GEAR

In order to compute the yaw moment coefficient, we require the lateral

acceleration in the body reference frame. We have

U

V

wBI

The yaw moment coefficient is given by

(A-13)

Ng = qSb <{

N1 CN2 ^ 7 + CN3 I? ( Pgust + P C°S

(A-14)

The numerical values of the coefficients for the 737 aircraft were fur-

nished by NASA Langley Research Center and are listed below:

100

AIRCRAFT AERODYNAMIC COEFFICIENTS

Drag Coefficients

CD1CD2CD3CD4

CD5

CD6CD7CD8c

D9CD10

Side Force

CY1c;,0Y2CY3CY4

CY5CY6

.185

.9225

= .0329

-.1067

0.

.0075

.0573

.019

.043

= -.308

Coefficients

-1. 564

. .4871

.189

-.0487

-. 1067

.4383

Lift Coefficients

CL1 =CL2 =

CL3 =CL4 =CL5 =

CI,6 =

CL7 VCL8 =

C T AL9CL10

C ,Lll

\*t ~~L12

C =L13

1.36

6.9328

-8.0

8.208

.963

.464

-.4584

-.0258

-.058

93.

.008

-.625

.7735

Roll Moment Coefficients

csi -C =

S2CS3 =CS4 -CS5 -

CS6 =

c _S7C =

S8

-.3152

-.645

.395

.0967

.0817

.0816

.06

-.1899

cc.

cMl3

•003

~-02

~-859

•115

N2

AT3

#4

'*»

°k°»:

C

•0215

-on

• 0343

'1862

The geometric parameters for the 737 aircraft are listed below:

c = 3.41376 meters (11.2ft.)

b = 28.3464 meters (93f t . ) ,

S = 91.045 meters2 (980ft.2)

a =• 1.524 meters (5 ft.)

Mass = 36287.5kg (80,000. Ib.)

I,~; = 142572 newton meters2 (345,000. Ib. ft.2)2 2

I = 328148 newton meters (794, 062. 5 Ib. ft. )

!„ ' '= 490738 newton meters2 (U87,500. Ib. ft.2)2 2

I „= 21351 newton meters ( 51,667. Ib. ft. )

103

II. DYNAMIC STABILITY COMPARISON

In order to check the accuracy of the dynamic model of the aircraft, the

response of the plane to an elevator impulse and a rudder impulse was tested.

The response curves are,shown in Figs. Al and A2. These runs were made with,"

the automatic control equations inoperative in order to obtain the period, damping,

and the decay time to one-half amplitude of the plane for the short period, phu- .

goid, and dutch roll motions, at an airspeed of 66. 88 m/sec. (130. knots) and :

an altitude of 426. 7 meters (1400 ft.) ;

A comparison of these results was made with 737 aircraft characteristics -

supplied by Langley Research Center. The results, shown in Table 7, are in : :

good agreement. ,

104

Dynamic Response to Elevator Impulse

-i

TIME12 14 18 18 20 X 10'

It 10'

B 6 10 12 1H 16 18 20 X 10'I I I I I I I I Tl I I I I I I I

eo x to1

20 X 10'

eo x to1

Figure Al. Phugoid and Short Period Modes105

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Dynamic Response to Rudder Impulse

j - • • ' • ' . • - „ •

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20 X !0>

20 X 10>

eo x :oi

20 X 10'

106Figure A2. Dutch Roll Mode

TABLE 7

DYNAMIC RESPONSE COMPARISON — B-737

Simulation Model

Mode :

Phugoid

Short Period

Dutch Roll

. Period(sec. )

39.0

5.0

5.0

. DampingCoefficient

.098

.25

.055

V2(sec.)

43.7

4.84

9.99

B-737 Data

Phugoid

Short Period

Dutch Roll

34.0

5.39

• 4.B1 : ""

.084

.41,.....,, r-_ : . . .

43.7

5.39

9:31^

107

APPENDIX B

Computer Simulation Results for

Rollout and Turnoff

108

I. DESCRIPTION OF TEST DATA

This section contains the results of computer runs of the ALERT program

in the form of plots. Although the B-737 aircraft simulation program is capable

of executing automated banked turns and landing approaches including capture,

oncourse, flare, and decrab, the main purpose of this study was to develop and

test a guidance law for rollout and turnoff. The computer plots contained in this

section illustrate the terminal portion of..each landing from touchdown to the end

of turnoff. Turnoff, in the context of this study, is defined to be completed when

the aircraft is at a fixed desired taxi speed, at a fixed constant heading (30 yaw

relative to the landing runway centerline), and at a distance of 91. 44 meters

(300 ft.) from the runway centerline.

In order to reduce the computer run time to complete the study for a

range of wind conditions, runway conditions, and taxi speeds, the entire air-

borne program was run for different wind conditions. The aircraft variables

at touchdown were stored, and a set of rollout and turnoff runs were made

at different taxi speeds and runway conditions (dry and wet) for each fixed landing

condition.

A description of the plot variables and units is given. Each landing case

contains eleven plots. The horizontal variable is always time from touchdown

in seconds. The first plot is the aircraft thrust in newtons. (One pound of force

is equal to 4. 4482216 newtons.) The specified reverse thrust for this aircraft is

62275. newtons (14,000.#) The steady state idle thrust is 6672. newtons (1500. #)

The thrust is manually set at idle and, after a two-second slowout, the thrust

logic calls for maximum retro thrust which continues until the aircraft reaches the

required V speed, at which time the thrust logic calls for idle thrust through-COcLSt

out the taxi period.

109

The second plot is the automatic braking pressure plotted in percent of

maximum allowed pressure. 100% of maximum allowed braking pressure

corresponds to a friction coefficient, jj , of .4. Thus, the automatic braking

logic is not designed to command the full capabilities of the tire on a good dry

runway. Zero brake pressure corresponds to the condition where only rolling

friction is operative. Maximum brake pressure is available during deceleration

to V . Following that, only medium brake pressure (^ = .2) is availableCOclSl

to the automatic braking logic. During landings on wet runways, the anti-skid

device moderates the pressure to that allowable to forestall wheel lock.

The third plot is the forward acceleration in the body system, u . The2 2units are in meters/sec. . l .Og is 9. 817 m/sec. . For rapid turnoffs, a

2maximum deceleration of 5 m/sec. is considered within the limits of passenger

comfort.

The fourth and sixth plots are the x and y components of the winds plus

shear and gust. These are measured in meters/sec.

The fifth plot is the aircraft ground speed. This is measured in meters/sec.

The flat portion after the linear deceleration illustrates how well the aircraft keeps

the desired V. . speed,taxi

The seventh plot is the first plot on the second page of each case. The

variable CRTE is the cross track error, or the lateral deviation from the

buried magnetic cable. It is measured in meters.

The eighth plot is the lateral acceleration in body system. It is measured

in meters/sec. Passenger comfort requires that this variable not exceed an ab-2

solute value of . 15g's or 1.472 m/sec. (4.83ft .

mum taxi speed if fast turnoff times are required.

2 2solute value of . 15g's or 1.472 m/sec. (4.83 ft./sec. ). This limits the maxi-

110

The ninth plot is the aircraft yaw angle.

The tenth plot is the nose wheel steering angle.

The final plot is the difference between the magnetic cable yaw angle and

the aircraft yaw angle.

For the purposes of this simulation, the magnetic cable is laid out in three

segments: first, a straight segment along the runway centerline to the turnoff

position; secondly, a turn with a fixed input radius making a change in yaw of1

30 heading; finally, a straight line segment at a fixed yaw heading of 30 .

Following the plot of the eleventh variable is a short data set giving the

wind condition, the V and V . inputs, and the runway condition.coast taxi ,

111

II. SIMULATION RESULTS AND DISCUSSION

Figures Bl, B2, B3, and :B4 are acceptable fast turnoffs on good dry

runways. Figure B5 is a repeat of Run_B3_for a wet runway. For this case,

because of the inability to apply braking at high speeds, the desirable taxi speed

of 50 knots was not achieved and high lateral acceleration resulted. In order

to produce acceptable turns at high speeds on wet runways, it was found neces-r

sary to lower the V speed (the speed at which the retro thrust is switchedcoast ' rto idle) to exceed the desired V . speed by 5. knots. With this change, ac-

ZcLXl ,. ( . . ( r f

ceptable fast turns are achieved on., wet-runways., .Figure.B6 is an illustration! ••.-;. - • . . - . . , ; ~ ' ' i . • : ' • - . - - - - . -

of a good 6. 0 knot turn on a wet runway. ' ;-:

Figures B7 through B14 are all acceptable high-speed turns on wet run- :.

ways for different wind conditions. '. ' ' ' • ' • , ' ' • - • - . ' *

' ' ~ , ' » ' " *

Figure B15 is an illustration of high-speed turnoff on a good dry runway,

replacing the Kalman filter with the complementary filter in the navigation loop. -'- , i ' - "' : '

Examination of the cross track error in Fig. BI5 shows-that the complementary !

filter does a good job until the turn is completed, at which .time it ends up with .

an error in lateral deviation of 5 meters as compared with the Kalman filter ,

deviation of only 2 meters .

112

I 10*

W O1

-8

10

10 U7 V J T I I I I I 1 I

HUE (SEC)-a< SO ' 35 so so

I T I «

10 "V ISME (SEC)

20 "Vv--4S _ SO « _ 40 _ 45- _ 507 T 7 l 7 1 ^ | I I I I 1 1 7 7 1 1 1 1 1 1 1 1 1 1 1

TIME (SEC)26

10 ISTIME (SEC)

20 25 SO. _35 40 45 507 -

\

TIME (SEC)

3E

f 10»1

-

f - f

6 f .1 | . 1

5

- J —- >

10

' 1 '

to*/>_fH/-

is. , , ,

IS

-*V/ -»V-s.

20 25 SO 35 MO. . , , . , , . . . , I I . , . . . , .

TIME (SEC)20 25 30 35 40

1 1 II 1 1 1 1 1 1 I 1 1 1 I 1 1 1 1 1'~**~.r- S~-*S'

45 501 1 | 1 7 , |

45 601 1 I 1 1 1 1

Figure Bla. High-Speed Turnbff on Dry Runway

113

U

/j

_ ' ; ' - ' 1 ' ' '

HUE (SEC)-**^ 15 20 25 30 36 40 « SO1 V ' ' ' ' ' LLUJ- ' • • i • ' • i ' ' • i • • • i ' ' • i

= 0

F / v ,x~- / ^^^?<^vv^WE (SEC)rv .^5 -..- --17 is Xao -^s •• so

iN-fl i r*r i rn i i i i i i/"i i i i i r i i i i i35 HO 45 ' S(1 j: 1 1 1 1 1 i l.j I.I 1 1

I 10'

i i iirTIME (SEC)

20 25 30 35I I 1 1 I I I I I I I 1 1 I I" . Y . . . i.?.-

I E (SECT - . " . . - • • . , • ; • • - • • ; - . - . • • ; • '25 30 38 40 45 ' SO

-4

S

Ss

Figure Bib. High-Speed Turnoff on Dry Runway

114

30 Tail Wind, 5.144m/sec (10 knots)V = 25.72 tn/sec (50 knots)coast

V =15.43 m/sec (30 knots)taxiRT = 609. 6 meters (2000 ft.)

I 10«

10 15 .^_TIME (SEC)

gQ a6 30 35 HO HS '-'5.0

' ' I ' . ' • ' I

oec:UJ'

\n

^ o1

-3

c

*• 0o,

10T I M E (SEC)

30 35 . HO 15 ..-.'.; SO'

I ' ' ' M

10 ISTIME (SEC)

20 . _ , . gj 30 35 "10 45 • . '-.; SO;i r

1ft- 10 15T I M E (SEC)

20 25 30 . HO111 i r IT i •-.-T p:>.:M; T :r. i t-.i., M r . - T T i i i i i • i .1: i ; i _ -

I 10'

>• o!CD

-I

5 10. l o > l I I I I I I I

I I I

TIME CSEC 1 . ' •:, -.-16 20 25 30 35 HO • 46 - SOI I I I I I I I I I I I I I ^ 1 I 1 I I I II T. I 111 '• "1

10 16TIME (SEC)

20 25 30 35 HO H6 ''• SOI I I .1 I I III I II : I

Figure B2a. High-Speed Turnoff on Dry Runway

115

6* P fI I I I I i 1TIME ISEC)

15 20 25 30 35 , 10 / ' -_ - 15 . ,.< ;S(I 1 I I I T -I T I I I T 1 !. I I I "I I I I I I I I I 1 l.'.r I . L l

v J I 1 I I I I -I

50:

V.

w

^ a20 !5 30 35 HO , It _;_ ;SO| i i i | i i i [ i T i j r i « | i ;i i.. |i \' i 'i » i ixi i i i i i \

-?i • - ' . • 'I 10'

II

10 ' (S o3

=f=f=- . , . . , . .•:-• / , ' - • • . TIME1 (SEC) : . • ; *i5 107 tt : 20 ZS 30^ 35! y. \|QT ; ^ S5 •" >• •'y:-S(|—t—» » -J I I I [ I 1 t | I I I I I I I [ I I I | I I I | 1 I I J 7 l^.i>|

10 / ISTIME ISEC)

30 35 SO «f f

-61 10"

Figure B2b. High-Speed Turnoff on Dry Runway'

30° Tail Wind, 5.144 m/sec (10 knots)V =30.86 m/sec (60 knots)

C03.SIV =20.57 m/sec (40 knots)taxi

RT=(2500 feet)

116' ' • • C

r 10'

1°s _ io

I II I I-..1,TIME: ISECJ

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Figure B3a. High-Speed Turnoff on Dry Runway117

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118

30 Tail Wind, 5.144 m/sec (10 knots)V = 36.01 m/sec (70 knots)coast

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30° Tail Wind, 5.144 m/sec (10 knots)V =41.16 m/sec (80 knots)coast

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R =(3500 feet)

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Figure B5b. High-Speed Turnoff on Wet Runway ;

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Runway turnoff distance = (3500 feet)

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Figure B6a. High-Speed Turnoff on .Wet Runway123

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124

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Figure B7b. High-Speed Turnoff on Wet Runway

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RT = (2000 feet)

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Figure B8a. High-Speed Turnoff on Wet Runway

127

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Figure B8b. High-Speed Turnoff on Wet Runway :'

150° Head Wind, 5.144 m/sec (10 knots)V =28 .29 m/sec (55 knots)coast

V = 25.72 m/sec (50 knots)taxi

R = (3000 feet)

128

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Figure B9b. High-Speed Turnoff on Wet Runway

150° Head Wind, 5.144 m/sec (10 knots)V =33.44 m/sec (65 knots)coast

V =30.87 m/sec (60 knots)taxi

R = (3000 feet)

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Figure BlOa. High-Speed Turnoff on Wet Runway131

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Figure BlOb. High-Speed Turnoff on Wet Runway

132

150 Headwind, 7.716 m/secV = 23.15 m/sec (45 knots)coastV =20.58 m/sec (40 knots)taxi

R =(2500 feet)

Runway turnoff distance = (5000 feet)

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133

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150° Head Wind, 7.716 m/sec (15 knots)V = 28.29 m/sec (55 knots)coastV . = 2 5 . 7 2 ra/sec (50 knots)taxi

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Figure B12a. High-Speed Turnoff on Wet Runway

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Figure B12b. High-Speed Turnoff on Wet Runway

136

No WindsV =18.00 m/sec (35 knots)coastV =15.43 m/sec (30 knots)taxiR = (2000 feet)

Runway turnoff distance = (5000 feet)

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Figure B13a. High-Speed Turnoff on Wet Runway

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Figure B14b.; High-Speed Turnoff on Wet Runway

140

No WindsV = 33.44 m/sec (65 knots)coastV = 30.87 m/sec (60 knots)taxi

R^ (3000 feet)- '

I 10«

.-to . . _JLS

TIME TSEC)es go -M- — so

1^1 | i . .

-r 10'10

£ * L 10 15• "TIME

eo es 30 : 35 soo(t

•£.CO

g

V- ' i

s 10

i i r

15 SO ' _• 85 • 30 35 ^_ SO 45 50

-3|K 10'

1

X ttt-pV

IT

10 15 80' •' ' - . ' ' 85•.-..'.,--= 30 •JS . 50

-l|1 10'

. . , .i| ' ' ' I ' ' ' I ' ;l ' I

10 15TIME tSEC)

eo es 30 35 «0 45 50| I I I II I I | I I I r I I I I I I r III | I I I T 1 I I

to ts eoTIME (SEC)

es 30 35 40 so

Figure B15a. High-Speed Turnoff on Dry,rRunway

141

10 -to-nriE

T-fr 35 40 45 50I I I I I I I I I I I

tO

oIr TIME (SEC)

35 40 45 V 50i T " i

-2X L0l

o'l 10 15TIME (SEC)

80 £5 30 40 95 501 , , , 1

10 isTIME (SEC)

eo es so 35 40 45 , 50

da

-5

1 1r1"1 • i

Figure B15b. High-Speed Turnoff on Dry Runway

142

150 Head Wind, 5.144 m/sec (10 knots)V = 30.87 m/sec (60 knots)coast

V = 2 0 . 5 8 m/sec (40 knots)taxi

R =(2000 feet)

Complementary filter used

REFERENCES

1. Attri, N.S. and Dunn, R.; "NASA Request for Data in Landing Gear, Tires,

Antiskid, Thrust, Nose Wheel Dynamics, and Rudder Aerodynamics,"

Boeing Document B-8190-75-5, March 25, 1975.

2. Home, W. B. and Leland, T. J.W.; "Influence of Tire Tread Pattern and

Runway Surface Conditions on Braking Friction and Rolling Resistance of

a Modern Aircraft Tire," NASA TN D-1376, 1962.

3i Kaminski, P. G., Bryson, A. E., Jr. and Schmidt, S. F.; "Discrete Square

Root Filtering. A Survey of Current Techniques," IEEE Transactions on

Automatic Control, December 1971.

4. Duff, William G., Guarino, Charles R.; "Refinement and Validation of

two Digital Microwave Landing System (MLS) Theoretical Models",

NASA CR-132713, August 15, 1975. . , ',

5. Morgan, H. C. and England, P.; "A Taxi Guidance System for Aircraft

Using a Single Magnetic Leader Cable," R.A.E. Technical Report #66065,

February 1966.

6. Olson, K.W., Fenton, R.E. and Melocik, G. C.; "Studies in Vehicle

Automatic Lateral Control - Theory and Experiment," Ohio State

University Report EE 1-276A-16, September 1974.

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