space shuttle entry landing navigation analysis · 2020. 3. 22. · contract no. nas 9-12968 for...

THE ANALYTIC SCIENCES CORPORATION

NASA CR-

TR-302-2

SPACE SHUTTLE ENTRY ANDLANDING NAVIGATION ANALYSIS

31 July 1974

(NASA-CR-1417 6 5) SPACE SHUTTLE ENTRY AND

N75-23654]

LANDING NAVIGATION ANALYSIS Final Report

(Analytic Sciences Corp.), 251 p HCSCL 23C Unclas

G3/ 1 7 21640

Prepared under:Contract No. NAS 9-12968

for

NATIONAL AERONAUTICS AND SPACE ADMINISTRATIONJohnson Space Center

Houston, Texas

Prepared by:

Harold L. JonesBard S. Crawford

Approved by:

Raymond A. Nash, Jr.Arthur Gelb

THE ANALYTIC SCIENCES CORPORATION6 Jacob Way

Reading, Massachusetts 01867

https://ntrs.nasa.gov/search.jsp?R=19750015582 2020-03-22T21:11:51+00:00Z

THE ANALYTiC SC!ENCES CORPORATION

ABSTRACT

A navigation system for the entry phase of a Space Shuttle

mission is evaluated in detail. The navigation system is

an aided-inertial system which uses a Kalman filter to mix

IMU data with data derived from external navigation aids.

Adrag pseudo-measurement used during radio blackout is

treated as an additional external aid. The Kalman filter

has a variable dimension of between 6 and 15 staL'e. A

comprehensive truth model with 101 states is formulatedand used to generate detailed error budgets at several sig-nificant time points -- end-of-blackout, start of final ap-

proach, over runway threshold,and touchdown. Sensitivitycurves illustrating the effect of variations in the size of

individual error sources on navigation accuracy are pre-sented. In addition, the sensitivity of the navigation sys-tem performance to filter modifications is analyzed. The

projected overall performance is shown in the form of

time histories of position and velocity error components.The detailed results are summarized and interpreted, and

suggestions are made concerning possible software im-

provements.

Ii


ACI NOWLEDGEMENT

The authors acknowledge the guidance and assistance of the

Contract Technical Monitors, Messrs. R. T. Savely and

E. R. Schiesser of the Mathematical Physics Group of the

Mission Planning and Analysis Division of NASA/JSC. As-

sistance and cooperation were also received from Messrs.

B. F. Cockrell and E. W. Henry of the same group. The

authors would also like to thank Mr. E. M. Duiven of TASC

for his many useful suggestions during the course of this

work.

iii

THE ANALYTIC SCIENCES COPOR ATION

TABLE OF CONTENTS

PageNo.

ABSTRACT

ACKNOWLEDGEMENT

1. INTRODUCTION 1-1

1.1 Objectives and Scope 1-1

1.2 Relationship to Other Efforts 1-5

1.3 Approach 1-5

1.4 Organization of the Report 1-7

2. TRAJECTORY, MEASUREMENT SCHEDULES, AND METHODOLOGY 2-1

2.1 Nominal Trajectory 2-1

2.2 Measurement Schedule 2-3

2.3 Covariance Equations 2-5

3. TRUTH MODELS 3-1

3.1 Drag-Update Model 3-2

3.1.1 States and Error Sources 3-2

3.1.2 Truth Model Data Base 3-5

3.2 System E Model 3-15

3.2.1 States and Error Sources 3-153.2.2 Truth Model Data Base 3-16

4. FILTER COVARIANCE AND GAIN CALCULATIONS 4-1

4.1 Filter Structure 4-1

4.2 Drag-Update Filter 4-4

4.3 System E 4-9

5. RESULTS 5-1

5.1 Drag-Update Phase Evaluation 5-1

5.1.1 Overall System Performance 5-2

5.1.2 Baseline Error Budget 5-5

5.1.3 Sensitivity Curves: Drag-Update Phase 5-9

5.2 System E 5-14

5.2.1 Overall System Performance 5-15

iv


TABLE OF CONTENTS (Continued)

PageNo.

5.2.2 Detailed Error Budgets 5-19

5.2.3 Sensitivity Curves: System E 5-25

5.:2.4 Performance Sensitivity to Filter Modifications: System E 5-30

5.3 Discussion5.3.1 Drag-Update Phase5.3.2 System E

6. CONCLUSION

APPENDIX A - COORDINATE FRAMES AND TRANSFORMATIONS A-1

APPENDIX B - COMPUTATION OF SPECIAL OUTPUT QUANTITIES B-1

APPENDIX C - TRUTH MODEL MEASUREMENT MATRIX FOR DRAG C-1

ACCELERATION PSEUDO-MEASUREMENT

APPENDIX D - NON-STANDARD ATMOSPHERE MODEL FOR SHUTTLE D-1

ENTRY NAVIGATION

D. 1 Non-Standard Density Model D-1

D.2 Wind Model D-9

APPENDIX E - TRUTH MODEL STRUCTURE FOR SHUTTLE ENTRY AND E-1

LANDING NAVIGATION STUDIES

E.1 Drag-Update Phase E-1

E.2 System E E-11

APPENDIX F - DRAG-UPDATE ERROR CONTRIBUTION TIME F-1

HISTORIES

APPENDIX G - SYSTEM E ERROR CONTRIBUTION TIME HISTORIES G-1

REFERENCES R-1

V

THE ANALYTIC SCIENCES CORPURATION

LIST OF FIGURES

Figure Page

No. No.

1.1-1 Baseline Orbiter Landing System 1-2

1.3-1 Realistic System Performance Projections 1-6

2.1-1 Shuttle Entry Ground Track for Reference Mission 3B 2-2

2.1-2 Approach and Landing Ground Track for Reference Mission 3B 2-2

2.3-1 Truth Model Structure 2-7

3.1-1 Deviation of Navigation Filter Density Models from 1962 Standard 3-9

Atmosphere

3.1-2a Time-Varying Density Bias for June-July Given as Percent Depar- 3-11

ture from 1962 Standard Atmosphere

3.1-2b Time-Varying Density Bias for December-January Given as Percent 3-11

Departure from 1962 Standard Atmosphere

3.1-3a Mean Zonal (Westerly) Wind in July for Reference Mission 3B Entry 3-12-

3.1-3b Mean Zonal (Westerly) Wind in January for Reference Mission 3B .3-12

Entry

4.1-1 Filter Covariance Program Flow Chart (Drag-Update Phase and 4-2

System E)

4.3-1 Schedule for System E Filter States 4-10

5.1-1 Drag-Update Phase Overall Performance: Position 5-3

5.1-2 Drag-Update Phase Overall Performance: Velocity 5-3

5.1-3 Sensitivity of Crossrange Position Errors at 130,000 ft to Gyro Bias 5-10

Drift

5.1-4 Sensitivity of Vertical Position and Velocity Errors at 130,000 ft 5-12

to Accelerometer Scale Factor Errors

5.1-5 Sensitivity of Vertical Position Errors at 130,000 ft to Density Time- 5-13

Varying Bias

5.1-6 Sensitivity of Downrange Position Errors at 130,000 ft to Density 5-13Time-Varying Bias

5.2-1 System E Overall Performance: Position 5-16

5.2-2 System E Overall Performance: Velocity 5-16

5.2-3 System E Overall Performance During Final Approach: Position 5-17

vi


LIST OF FIGURES (Continued)

Figure PageNo. No.

5.2-4 Sensitivity of Vertical Position and Velocity Errors at Touchdown 5-27to Radar Altimeter First-Order Markov Error

5.2-5 Sensitivity of Crossrange Position Error at Runway Threshold to 5-28MLS Azimuth Bias

5.2-6 Sensitivity of Downrange Position Error at Runway Threshold to 5-28MLS DME Bias

5.2-7 Sensitivity of Downrange Position Error at Runway Threshold to 5-29MLS DME Timing Bias Error

5.2-8 Sensitivity of Downrange Velocity Error at Runway Threshold to 5-29Gyro Bias Drift Error

5.2-9 Sensitivity of Crossrange Velocity Error at Runway Threshold to 5-30Gyro Bias Drift Error

A-1 Use of Transformation Matrices A-4

C-I Definition of Perturbations in Unit Relative Velocity Vector and C-5Specific Force Vector

D-la Median Densities Given as Percent Departure from U.S. Standard D-4Atmosphere, 1962 During June-July at 80 0N, 60 0N, 450 N, and 300N

D-lb Median Densities Given as Percent Departure from U.S. Standard D-4Atmosphere, 1962 During December-January at 80 0N, 60 0N, 45 0N,and 30 0 N

D-2 Histogram of Monthly Mean Zonal (Westerly) Wind Components (fps) D-10for 40*N and an Altitude of 200,000 ft

D-3a Mean Zonal (Westerly) Wind in July for Reference Mission 3B Entry D-11

D-3b Mean Zonal (Westerly) Wind in January for Reference Mission 3B D-11Entry

- D-4 Season-averaged Standard Deviation of Zonal Wind Component as a D-12Function of Altitude

D-5 Correlation Functions for Headwind VH(r) and Crosswind VC(r) as D-15Defined by Eqs. (D-12) and (D-13)

E. 1-1 Drag-Update Phase Truth Model System Matrix Structure E-2

vii


LIST OF FIGURES (Continued)

Figure Page

No. No.

E.1-2 Orientation of Gyro Axes E-7

E.2-1 System E Truth Model Structure E-13

E. 2-2 TACAN-Shuttle Geometry E-16

E.2-3 MLS Azimuth-Shuttle and MLS DME Shuttle Geometries E-18

E. 2-4 MLS Elevation-Shuttle Geometry E-18

viii


LIST OF TABLES

Table PageNo. No.

1.1-1 Candidate Systems 1-3

2.2-1 Key Events for Drag-Update and System E Evaluations 2-3

2.2-2 Sensor Locations With Respect to Touchdown Point 2-4

3.1-1 Drag-Update Truth Model States and Error Sources 3-4

3.1-2 Drag-Update Truth Model Data Base for IMU-Related Error Sources 3-7

3.1-3 Drag-Update Truth Model Data Base for Drag-Related Error Sources 3-8

3.2-1 System E Truth Model States and Error Sources 3-13

3.2-2 System E Truth Model Data Base (Where Different from the Drag- 3-17Update Phase)

4.2-1 Filter State and Measurement Error Statistics forDrag-Update 4-5Phase

4.2-2 Coefficient Values for Atmospheric Density Models 4-8

4.2-3 Drag-Update Phase Filter-Indicated Performance 4-9

4.3-1 Filter State and Measurement Error Statistics for System E 4-11

4.3-2 Description of Gain Computation Algorithms for Final Approach 4-1.2and Landing

4.3-3 System E Filter-Indicated Performance 4-13

5.1-1 Summary of Drag-Update Filter Performance 5-4

5.1-2 Baseline Error Budget for Drag-Update Filter 5-7

5.1-3 Drag-Update Phase Alternative Contributions 5-8

5.2-1 System E Baseline Error Budget -- 12,000 ft 5-21

5.2-2 System E Baseline Error Budget -- Runway Threshold 5-22

5.2-3 System E Baseline Error Budget -- Touchdown 5-24

5.2-4 Navigation Errors at Runway Threshold Due to Gyro Bias Drift 5-32Errors

ix /

/r


LIST OF TABLES (Continued)

Table PageNo. No.

D-1 Correlation Function Coefficients for Sum of Diurnal and Short- D-6Term Density Deviations From Long-Term (Seasonal) Average

D-2 Recommended Correlation Function Coefficients for the First- D-7Order Markov Process PM(R) in the Non-Standard AtmosphereModel

D-3 Spatial Correlation Distances and Standard Deviations for Iso- D-14tropic Winds (VH(R) and VC(R) in Non-Standard Wind Model))

x


1. INTRODUCTION

This document describes the results of the second part of a

study conducted by TASC, involving detailed error models and perform-

ance evaluations of candidate Space Shuttle navigation systems. The re-

sults of the preceding part were reported in Ref. 1. This study is one

element of a general effort led by NASA to establish operational charac-

teristics of the Space Shuttle Orbiter systems. Considerable progress

has been made during the past two years in the portion of this effort re-

lated to navigation. The "baseline systems" treated herein are the result

of a continuing process of design, evaluation, and refinement, in which

individuals representing a number of organizations have participated.

1.1 OBJECTIVES AND SCOPE

The general purpose of this study is to aid in the evaluation

and design of the multi-sensor schemes proposed for the Orbiter navi-

gation system. The scope of the effort described herein covers the

entry, energy management, and approach and landing phases of the

mission -- from an altitude of 400,000 ft to touchdown. Figure 1.1-1

illustrates the current baseline approach and landing system operation.

A single TACAN station provides external range and azimuth measure-

ments beginning with the termination of radio blackout, and ending when

the MLS (Microwave Landing System) signals are acquired in the vicin-

ity of the runway. The MLS signals include range, azimuth, and ele-

vation angle measurements. Some key elements not shown in the figure

1-1


R-14200

ORBITER

SMILS ELEVATION

MLS DME AND .._TACANAZIMUTH

Figure 1.1-1 Baseline Orbiter Landing System

are the onboard IMUs (Inertial Measurement Units), altimeters, and com-

puters. The latter are programmed to mix together (filter) the IMU data

and the externally derived data. The algorithms used to accomplish this

function are based on optimal filtering schemes developed by Kalman

and others (Ref. 24).

Specific objectives of this effort are:

SDefine and develop mathematical models describ-ing all potentially significant sources of error foreach navigation system considered.

SDevelop a detailed, quantitative understanding ofthe contributions of individual error sources tooverall system performance.

* Present results in a form which will help NASAchoose or specify hardware elements or theircharacteristics.

SEvaluate software approaches to the navigationfilter design problem.

1-2


Table 1.1-1 summarizes the main features of six candidate systems

which have been evaluated in detail -- four in the previous part of this

TABLE 1.1-1

CANDIDATE SYSTEMS

EXTERNAL FILTER USE OF DRAGIMU AIDSAIDS UPDATING

PART I

2 DMEs

Baro Altimeter 24-State

System A KT-70 ILS (3 deg) Multi-Phase Filter No

Radar Altimeter

2 DMEs

Baro Altimeter 6-State.System B KT-0 (3 deg) Square-Root Filter No

Radar Altimeter

One-Way Doppler22-State

System C KT-70 Baro Altimeter Multi-Phase Fiter No

Radar Altinieter

1 DME TACAN

1 VOR :.--

System D KT-70 ILS (Localizer 23-State YesOnly) Multi-Phase Filter

Baro Altimeter

Radar Altimeter

PART H

Drag-Update KT-70 None' "13State

Filter Unified FlRer

1 DME T :TACAN

1 VOR

System E KT-70 MLS 15 4 6-State Yes

Baro Altimeter Unified Fiter

Radar Altimeter

Acronyms:

IMU - Inertial Measurement UnitDME - Distance Measuring Equipment frange)ILS - Instrument Landing System

p0 0 P (elevation and azimuth)o4,1 MIS . - .Microwave Landing System

(range, elevation, and azimuth)VOR - VIIF Omnidirectional Range (azimuth)

1-3


study and two in the current part. The latter correspond to two phases

of the entry and landing trajectory using the current baseline system,

as follows:

* The drag-update phase, covering the timefrom entry interface (400,000 ft) to the endof radio blackout (assumed here to be130,000 ft).

o The approach and landing phase, covering thetime from end-of-blackout to touchdown.

In the first phase, a 13-state drag-update filter is used to blend in drag

"pseudo-measurements," as discussed in Chapters 4 and 5 and Appendix

C. In the second phase, a variable-dimensioned System E filter is used

to blend the inertial data with the externally derived data. Both of these

filters are manifestations of a single "unified" filter structure whose

maximum state dimension is 15. With the measurement sequence and

groundrules established for this study, the maximum number of active

states used at one time is 15; the minimum is 6 -- during final approach.

The principal computational results of the study are the error budget

tables given in Chapter 5. These tables list separate contributions to

system error due to individual error sources, or small groups of error

sources, at four critical trajectory points:

* The end of radio blackout (130, 000 ft)

* The beginning of the terminal approach (12,000 ft)

* The runway threshold, and

* The touchdown.

1.2 RELATIONSHIP TO OTHER EFFORTS

Other efforts which relate to the present study fall into three

categories as follows:

1-4


* Past multi-sensor navigation studies by TASCin which the methodology employed herein wasdeveloped.

* Past studies by others involving Shuttle filter

design, IMU error propagation, and sensor error

modeling.

* Ongoing and future efforts to establish operationalcharacteristics of the Shuttle navigation system.

A discussion of the first two items above may be found in Section 1.2 of

Ref. 1 and in Ref. 2. Highlights involve TASC's work for the Air Force

on the CIRIS (Ref. 3) and CLASS (Ref. 4) programs; filter design studies

by Lear of TRW (Ref. 5) and Kriegsman, Muller, and others of the Charles

Stark Draper Laboratory (Refs. 6 through 9); a "one rev" Space Shuttle

mission IMU error study by Clark and Mitchell (Ref. 10); and the

periodically-revised NASA -document (Ref. 11) describing various navi-

gation subsystem characteristics and error models.

The relationship of this study to ongoing and future efforts in-

volves its impact on software and hardware decisions to be made by

NASA and its major contractors. The software decisions involve methods

for mixing inertial data with externally-derived data -- such as the choice

of filter states and parameters. The hardware decisions involve the selec-

tion of the onboard and ground-emplaced devices needed to obtain the ex-

ternal data, as well as the specification of particular hardware

characteristics, required survey accuracy, etc. The detailed results

generated in this study provide a sound quantitaiive basis for rational de-

cisions.

1.3 APPROACH

The general approach and mathematical techniques described

in Sections 1.3 and 2.4 of Ref. 1, were employed in essentially the same

1-5


way in this part of the study. (Some important differences in the de-tailed truth model structure, required for the drag-update evaluation,are described in Appendix C.) The overall methodology used is brieflyreviewed here with the aid of Fig. 1.3-1. The upper half of the diagramrepresents the recursive solution of the filter covariance propagate andupdate equations. These are solved once for a given trajectory and meas-urement schedule. Certain elements of the filter dynamics and measure-ment matrices are functions of the Shuttle position and velocity vectorsand the relative geometry between the Shuttle and the ground-navaidantenna locations. The outputs are the time histories of the filter-indicated performance and the Kalman filter gain matrices. The latterare stored on tape and called a "gain file. " The lower half of the diagramrepresents the recursive solution of the linear system covariance equa-tions. These are solved repeatedly to produce an error budget, the samegain file being used each time. The trajectory-dependent matrices de-scribing the real-world error model are of much higher dimension thanthe corresponding filter matrices. In individual error budget runs,

m-STATE FILTER .R-4288FILTER DESIGN COVARIANCE (OPTIMISTIC)MODEL EQUATIONS FILTER-INDICATED

(NONLINEAR) PERFORMANCE

TIME-VARYING ERROR BUDGETFILTER GAINS Error

REALISTICALLY Source System Errorsn-STATE SENSITIVITY PROJECTED SYSTEM Group ele21 3 . * *

"TRUTH MODEL" COVARIANCE PERFORMANCE 1 IDnEm) EQUATIONS2 r(LINEAR)

Figure 1.3-1 Realistic System Performance Projections

1-6


specific elements of input matrices, corresponding to specific error

sources, are set to non-zero values, with all other elements set to zero.

The output time history of the system error covariance matrix is then a

statistical measure of the effect of that particular error source or small

group of error sources, and generates one row of the error budget table.

When the entire table is filled in, an overall system performance projec-

tion is calculated from the detailed error-source-by-error-source break-

down.

1.4 ORGANIZATION OF THE REPORT

Chapter 2 describes the approach.and landing trajectory, and

the measurement schedules used in evaluating the drag-update and the

System E navigation performance. The trajectory is based on a one-orbit

polar mission beginning and ending at Vandenberg Air Force Base.

Chapter 3 outlines the truth model states and error sources

used in studying the two mission phases. The associated mathematical

details are relegated to Appendices C, D and E. (Appendix D provides

a useful discussion of non-standard atmospheric density and wind mod-

els.)

Chapter 4 defines the filter states, parameters, and algorithms

used to compute filter-indicated performance and to generate the filter

gain sequences, which were filed on magnetic tape.

Chapter 5 presents the major results of the study. For the

two mission phases treated, the parameters defining error source statis-

tics are listed and error budgets, detailing contributions to system er-

ror at important trajectory times, are given. Overall system performance

1-7


Sprojections, in the form of rms position and velocity error time histor-

ies, are plotted. (Detailed computer printouts, showing individual con-

tribution time histories are given in Appendices F and G.) Sensitivity

curves showing the effects of variations in the size of major error

sources are presented. For the approach and landing phase, the effect

of alternate filter configurations on overall system performance is also

given. A general discussion of system performance, comparison with

previous results, and filter design is included.

A brief summary of results, conclusions, and recommended

future studies is provided in Chapter 6.

1-8


2. TRAJECTORY, MEASUREMENT SCHEDULES,AND METHODOLOGY

This Chapter describes the approach and landing trajectory and

the measurement schedules used in evaluating the entry navigation sys-

tem. Section 2.3 defines the basic terminology and states the linear co-

variance equations used in evaluating candidate systems.

2.1 NOMINAL TRAJECTORY

The reference trajectory was provided on magnetic tape by JSC

(Ref. 12). The tape consists of the entry portion of the Shuttle reference

mission 3B. This is a one-orbit abort mission initiated with a launch

from Vandenberg Air Force Base into a 100 nautical mile circular orbit

with an inclination of 1040. The trajectory from entry interface

(400,000 ft altitude) to touchdown is illustrated in Fig. 2. 1-1. The

drag-update phase begins over Greenland and terminates over northern

California. System E operation begins at this point and terminates at

touchdown.

The ground track for the reference trajectory approach and

landing is shown in greater detail in Fig. 2. 1-2. The final approach be-

gins at an altitude of approximately 12,000 ft. The final approach glide

slope of 240 is maintained until a flare maneiuver is initiated at an alti-

tude of 1, 360 ft -- approximately 27 sec before touchdown. Touchdown

is 1,000 ft past the runway threshold.

2-1


1-12206

I- -BEGINNING OF DRAG

UPDATE PHASE

ENTRY INTERFACE

i E ND OF DRAG

0.UPDATE PHASE O

Figure 2.1-1 Shuttle Entry Ground Track for Reference Mission 3B

ALTITUDEG0.000FT

lIWE-TO-GO3 MIN

SUMLS2 AZIMUTH

EOME

OP UMLS ELEVATION

MIUIN

12000 FT

o.0W FT

AST POSITION ImIm

Figure 2.1-2 Approach and Landing Ground Track for Reference Mission 3B

2-2/


2.2 MEASUREMENT SCHEDULE-

The drag-update phase of the entry begins at the time of the

first drag acceleration pseudo-measurement (see Section 3.1). This

pseudo-measurement occurs at the time at which the deceleration mag-

nitude reaches 0. 1g. The drag-update phase ends at the time at which

blackout terminates -- which is assumed to occur nominally at an altitude

of 130,000 ft. Both times are indicated in Table 2.2-1 for the reference

trajectory.

TABLE 2.2-1

KEY EVENTS FOR DRAG-UPDATEAND SYSTEM E EVALUATIONS

Elapsed Time to -RelativeEvent Time Touchdown Altitude Velocity

(sec) (sec) (ft) (ft/sec)

Entry Interface, TO 0. 1946.5 399,989 26059.

'irst Drag Measurement, TDRAG 312. 1634.5 269,691 26072.

Beginning of Drag-UpdatePhase

First TACAN Measurement, TTACAN 1432. 514.5 129,904 6121.End of Drag-Update PhaseBeginning of System E

First Baro Altimeter Measurement, TBA 1550. 396.5 99,676 3695.(No Drag Measurement)

First MLS Azimuth and Elevation 1836. 110.5 20,002 583.Measurements, TMLS(No TACAN Measurement)(No Baro Altimeter)

First MLS DME Measurement, TDM E 1846. 100.5 18,051 581.

Reduce Filter Dimension, TSW 1874. 72.5 11,862 569.Switch from 2 sec Update Cycleto 0.5 sec Update Cycle

First Radar Altimeter Measurement, T 1943.5 3. 12 317.(No MLS Measurement)Switch from 0.5 sec UpdateCycle to 0.1 sec Update Cycle

Touchdown 1946.5 0. 0 290.

2-3


System E operation begins at the termination of the drag-updatephase. The key events for System E are the times at which the first andlast.measurement from each sensor are processed, the times at whichthe dimension of the System E Kalman filter changes, and the times atwhich the measurement update rate changes. These events are indicatedin Table 2.2-1. The MLS and TACAN antenna locations relative to thetouchdown point are indicated in Table 2.2-2.

TABLE 2.2-2

SENSOR LOCATIONS WITH RESPECTTO TOUCHDOWN POINT

Downrange Crossrange(ft) , (ft)

MLS Elevation 0 250

MLS Azimuth andMLS DME 15,000 250

TACAN 6,500 250

Above 100, 000 ft the System E sensors are TACAN and the dragpseudo-measurement. Between 100,000 ft and MLS acquisition at 20,000 ft,the drag pseudo-measurement is replaced by the baro altimeter. BothTACAN and the baro altimeter are terminated at 20,000 ft.

The azimuth angle and the elevation angle segments of MLS areacquired at an altitude of 20,000 ft; the DME segment is acquired at18,000 ft. MLS is the only external aid used between the altitudes of20,000 ft and 12 ft -- at 12 ft the Shuttle is nominally over the runwaythreshold. MLS is terminated at the runway threshold and the radar alti-meter is switched on for the final landing sequence.

2-4


The basic measurement update cycle for the drag-update phase

and for System E is 2 sec -- one measurement from each operating sen-

sor is processed every 2 sec. The interval between updates is reduced to

0.5 sec at the beginning of the final approach. It is further reduced to

0.1 sec at the runway threshold.

2.3 COVARIANCE EQUATIONS

Detailed system error budgets are generated in Chapter 5 by

solving a set of linearized system covariance equations. Solution of

these equations requires that a gain file of the measurement gains used

in the navigation filter be generated. The relationship between the

filter model from which the filter gains are computed, and the truth

model from which the error budget is computed, is summarized in

this section. Although covariance equations alone are sufficient to es-

tablish this relationship, the state equations for the two models are also

presented.

The filter model (FF, HF) upon which the navigation filter is

defined is a low-order approximation of the navigation system dynamics

and of the real-world environment in which the filter is designed to oper-

ate. The filter, which may be nonlinear, generates an m-dimensional

state estimate, F' which propagates between measurements according

to

F(t +1) F G MV(t+ 1 ) (2.3-1)k

where G is the control matrix, 4DFk is the filter state transition matrix,

and AV is the computed velocity change over the interval. If the filter

dynamics matrix, FF(t), is constant between tk and tk+, 4Fk is given by

2-5


Fk = Lxp F.F k k k (2. 3-2)

The system measurements, z, are used to update the filter state estimates

by first generating the measurement residual vector

S (tk) = z(tk) - HF(tk) x(tk) (2.3-3)

and then using the filter update equation

x (tk) = (t ) + KF(tk) 8_zstk) (2.3-4)

where KF(tk) is the filter gain matrix and HF(tk) is the filter measure-

ment matrix.

Generation of a detailed error budget does not require thatEqs. (2.3-1) through (2.3-4) be solved; all that is necessary is to generatethe filter gain matrices. If the navigation filter is suboptimal, an algo-rithm for computing the gains must be provided. If the navigation filteris a Kalman filter, the gains satisfy

-1

KF (tk) PF(t)H(tk)T HF PF(tk) HF(k) +RFkJ

(2. 3-5)

where PF(tk) is the filter covariance matrix* before the measurementupdate and RFk is the measurement noise covariance matrix used in thefilter model. The filter covariance propagates from one measurement tothe next by

PF(tkl) OFPF(tk) FT F (2.3-6)+ Fk F F + OFk (2.3-6)

* The filter covariance is the filter-generated estimate of the variance ofxF ; it is not necessarily the true variance of x F

2-6


where QF is the process noise covariance matrix used in the filter model.kThe filter covariance after the measurement update is

PF(tk ) = PF() -KF (tk ) HF(t) Pk(t) (2. 3-7)

In order to apply Eqs. (2. 3-5) through (2. 3-7); PF(0), RFk, and QF mustbe provided for the filter model along with (FF, HF). k

The truth model (FS, HS) upon which the error budget is basedis a detailed model of both the navigation system dynamics and the real-world environment. The truth model includes all important error sourceswhich affect the navigation system performance, including those whichwere omitted from the filter model. Because the error budget is con-cerned with navigation errors, the truth model can be linearized about thenominal system state., This yields a set of linearized -covariance equationsfor the system errors.

Figure 2.3-1 is a conceptual representation of the mathematicalstructure of all truth models used in this study. An n-dimensional system

1 -o8047

,8 s

SAMPLER

F5 Ks: AK, (see text)

REAL WORLD ERROR MODEL (n-dimensionol)

Figure 2.3-1 Truth Model Structure

2-7


error state vector, 8xs, is defined which includes both the filter state

estimation errors and the navigation error sources. The error state vec-

tor propagates between measurement times according to the linear, time-

varying differential equation

8 s S(t) = F(t) 6x(t) + W(t) (2.3-8)

where F (t) is the system error dynamics matrix and w S is a zero

mean, gaussian process noise with

E [wS (t)S (t)T] QS 8 (t -) (2.3-9)

The measurement residual vector is the same as in Eq. (2. 3-3), but it

can be expressed in terms of the system error state vector and a random

measurement noise

8zs(t k ) = HS(tk) 8xS (tk) + v(tk) (2.3-10)

where HS(tk) is the system measurement matrix and VS(tk) is a zero mean,gaussian white measurement noise with

E s(tk)v S(tk)T] =RSk (2.3-11)

After the filter update using Eq. (2.3-3), the system error state vector is

6xs(t ) = _XS(t ) + Ks(t) zS (tk) (2.3-12)

where K(t k ) is a gain matrix with n rows.

Ks(tk) = AKF(tk) = - KF(tk) -m rows\L O}(of zeros

(2.3-13)

//2-8

2-8 //


A is an n x m matrix linearly relating the m filter states to the n truth

model states.

The system error budgets presented in Chapter 5 are concerned

with the variance of the error state vector 6x S . Given the initial condi-

tion

PS(0) = E [6xs(0) 8 x S(0)T (2. 3-14)

the truth model covariance propagates between measurements by

PS+1 k S+ T + k (2.3-15)PsA+1) 1)k Ps(t) k+ QSk

where 4k is the error state transition matrix. If FS(t) is constant be-

tween tk and tk+l, then

k = exp [FS(tk) (tk+l - tk)] (2.3-16)

The discrete process noise matrix in Eq. (2.3-15) satisfies

tk+1

k= exp [F(k+ l-r)] exp [FS(tk+i- )] T dr

(2. 3-17)

Truncated matrix exponential series are used in evaluating the transition

and discrete noise matrices defined above. For update of the system co-

variance matrix at tk

2-9


Ps(t) = I - KS (tk) HS(] s(t ) [I - Ksk) HS T

+ Ks (tk) Rs Ks T

(2.3-18)

Equations (2. 3-15) and (2.3-18) are time-varying, linear equa-

tions in the system error covariance, PS(t). This fact allows easy com-

putation of overall system performance by root-sum-squaring separate

contributions and easy development of the sensitivity curves. The results

in Chapter 5 are based upon repeated solution of Eqs. (2.3-13) through

(2.3-18).

2-10


3. TRUTH MODELS

This section describes the "truth models" used in evaluatingthe entry navigation system. * Each truth model description includes alist of error sources, a detailed mathematical structure, and a data base.The overall mathematical structure is a set of linear differential andalgebraic equations describing system error propagation, as in Eqs.(2. 3-13), (2. 3-15) and (2. 3-18). Most of the detailed structure is givenin terms of sub-matrices of FS and HS, the system error dynamicsmatrix and measurement matrix, defined in Section 2. 3. The data baseis the set of numerical values used to represent real-world error sourcestatistics.

The analysis of the current baseline Space Shuttle entry navi-gation system is undertaken in two phases:

* Drag-update phase

* System E approach and landing

The drag-update phase commences at 400,000 ft and terminates at theend of the blackout - nominally 130,000 ft. The approach and landing isinitiated at 130, 000 ft and terminates at touchdown. The only navigationhardware elements common to both phases are the KT-70 IMU and theAP-101 computer. Detailed error budgets for each phase are generatedin Chapter 5. The truth models used to compute the error budgets areoutlined in this chapter.

* By "truth model," we mean a mathematical model of all potentially sig-nificant sources of error and -the way they affect system performancein the real world.

3-1/


3.1 DRAG-UPDATE MODEL

The computer navigation program for the drag-update phase is

based upon a 13-state Kalman filter (see Chapter 4). The only "external"

aid operating during the drag-update phase of entry is a drag "pseudo-

measurement." The pseudo-measurement is a drag acceleration meas-

urement constructed from the IMU accelerometer outputs. The "meas-

ured" drag acceleration is compared with the expected drag acceleration

computed from a nominal atmospheric density model, a nominal estimate

of the Shuttle drag coefficient, CD, and the currentestimate of position

and velocity. Although drag acceleration is not a state variable in the

Kalman filter,* a measurement matrix, HF,is constructed which relates

the difference between the measured and expected drag accelerations to

the error in the position and velocity estimates. The Kalman filter util-

izes this H F to update the position and velocity estimates (Section 4.2).

The corresponding measurement matrix, HS, for the truth model is de-

rived in Appendix C.

3.1.1 States and Error Sources

Incorporation of a drag pseudo-measurement into the drag-

update filter requires both the pseudo-measurement and a drag acceler-

ation prediction based upon the current filter state estimate. The pseudo-

measurement of drag acceleration is defined in Appendix C as

meas = VR (t) AV(t)j/At (3.1-1)

where LVR (t) is a unit vector in the direction of the computed relative

velocity, and AV(t) is the velocity change Over the last At sec as

The filter does include one "drag correlated error" state designed toaccount for non-standard atmosphere and aerodynamics, etc.

3-2


* measured by the IMU. The computed drag acceleration based upon the

filter estimates of position and velocity is

I A 2q pre(t) C (t) P (t V 2 (t) (3.1-2)pred C (t) Rc c

where CDc(t) is the computed drag coefficient for the Space Shuttle, A is

its cross-sectional area, and m is its mass. pc(t) and VRc(t) are the

computed atmospheric density and relative velocity, respectively. The

difference between qmeas(t) and qpred(t) is the residual which the drag-

update filter uses to improve the filter state estimates [see Eq. (2.3-4)]:

6zs(t)= meas (t) - pred(t) (3.1-3)

The accuracy of the drag-update filter is determined by the

accuracy of the state estimate propagation between updates, and the accu-

racy of qmeas (t) and qpred(t) used in the updates. The error in the

propagation is due to the standard IMU error sources. The error in

qmeas(t) is due to the standard IMU acceleration error sources and to

factors which contribute to errors in the computed unit relative velocity

vector, iVR (t), such as:

* position errors

* velocity errors

* atmospheric winds

The error in qpred(t) is attributable to these last three error sources,and in addition to:

* atmospheric density modeling errors

* drag coefficient modeling errors

The measurement sensitivities to the various error sources are developed

in Appendix C. The truth model states and other error sources used in

evaluating the drag-update filter are listed in Table 3. 1-1, which divides

them into three categories:

3-3


TABLE 3. 1-1

DRAG-UPDATE TRUTH MODELSTATES AND ERROR SOURCES

NUMBER NUBER OFERROR SOURCE NAMES OF

STATES ERROR SOURCES

I. ESTIMATED STATES ANDUNCORRELATED NOISES

Group 1.

Position Errors - 3 3Velocity Errors 3 3Platform Misalignments 3 3Accelerometer 'Biases' (First-Order 3 3Markovs)

Correlated Measurement Error (First-Order Markov)

Drag 1 1

Uncorrelated Measurement Noise

Drag. - 1

INS Quantization Noise - S

U. NON-ESTIMATED, IMU-RELATED STATES

Group la, True Platform Misalignments 3 3

Group 2. Accelerometer True Biases 3 3

Group 3. Accelerometer Scale 3 3Factor Errors

Group 4. Accelerometer Misalignments 6 6Group 5. Accelerometer Nonlinearities 3 3 -

Group 6. Gravitational Deflections and 3 3Anomalies

Group 7. Gyro True Bias Drifts 3 3Group 8. Gyro Mass Unbalances 6 6Group 9. Gyro Anisoelasticities .3 3

Ill. NON-ESTIMATED, DRAG RELATED STATES

Group 21.. Non-Standard Density

1962 Standard Atmosphere Error 1 1

Time-Varying Bias 1 1

-P Group 22. Non-Standard Wind

Q A , 49 Westerly (Time-Varying Bias) 1 1Headwind (First-Order Markov) 1 1

Crosswind (Second-Order Markov) 2 1

Turbulence (First- and Second- 5 3Order Markovs)

Group 23. Non-Standard Aerodynamics

First-Order Markov I 1

Totals 59 60

3-4


* -The 13 estimated states and uncorrelated measure-ment and process noises.

o Non-estimated states related to the inertial system.

* Non-estimated states related to the drag accelera-tion pseudo-measurement (q (t) and q pr(t)).meas pred

Error budget results corresponding to the first category aregenerated using a truth model structure (FS, HS) which is similar to thefilter error model structure -- the principal difference. is in the definitionof HF and HS. Results corresponding to the other two categories aregenerated using a higher-dimensional truth model, which contains thebasic 13-state structure plus other states representing time-correlatederror sources not modeled explicitly in the filter design. For the base-line navigation system, these additional error sources are divided into atotal of 23 groups. (The group numbers are consistent with those used inRef. 1), Only those groups which affect the drag-update phase are sum-marized in Table 3.1-1.

The drag-update truth model requires a 13 x 13 filter dynamicsmatrix, FF, and a 59 x 59 system matrix, FS . These matrices, aswell as the transformation matrix A, are defined in Appendix E. Themeasurement matrices HS and HF, as defined in Appendix C and Sec-tion 4.2, respectively, complete the definition of the truth model.

3.1.2 Truth Model Data Base

In this section numerical values are assigned to the drag-updatetruth model matrix elements describing error source statistics. Most ofthese values are elements of the following matrices:

PS(O) the initial system (real-world) errorcovariance matrix

3-5


QS the system process noise matrix

R - the pseudo-measurement noisevariance

In addition, however, there are elements of FS which must be defined.

Typically, these are diagonal elements of FS which define the correlation

times of markov processes. These elements of FS and corresponding

elements of PS(0), HS, and QS are normally chosen together to define

markov processes with desired properties. If the markov processes are

nonstationary, the appropriate elements of FS, HS, and QS may be

time-varying. The details of these manipulations are found in Appendices

C, D, and E.

Values are not given in this section for the Group 1 (estimated)

first-order markov states. These are the estimated platform misalign-

ments, acceleration errors, and drag correlated error. The truth model

structure distinguishes between these estimated states and the "true" states

modeled in Groups la through 23. The time constants associated with the

Group 1 states are identical to those assigned by the drag-update filter

(Section 4.2), but the truth model assigns them a zero initial covariance

and no process noise.

The elements of FS associated with the standard IMU error

sources (Groups la through 9) are assigned in Appendix E. The remain-

ing task associated with these error sources is to determine PS(O) at

entry interface.

The 9x9 portion of PS(0) associated with position, velocity,and true misalignment errors was based upon Table C-III-c in Ref. 10.

The 9x 9 matrix represents error statistics at entry interface following

a "one-rev" mission in which pure inertial navigation is used from launch

3-6


through to entry interface. The IMU model used in Ref. 10 is the KT-70;the assumed prelaunch alignment technique consists of gyrocompassing forazimuth alignment and accelerometer tilt leveling. The 9x9 error covari-ance matrix given in Ref. 10 was corrected to reflect the calibration andalignment errors at launch as given in Ref. 13 -- these errors are sum-marized in Groups I through 4 and 7 through 9 in Table 3. 1-2. Reference10 did not consider accelerometer nonlinearities (Group 5) or gravity de-flections and anomalies (Group 6).

TABLE 3.1-2

DRAG-UPDATE TRUTH MODEL DATA BASEFOR IMU-RELATED ERROR SOURCES

Error Source Standard Deviation Data Source

Group 1. INS Quantization Error 1.0 cm/sec Ref. 13

Group 2. Accelerometer Biases 50 pg Ref. 13Group 3. Accelerometer Scale Factor 40 ppm Ref. 13Group 4. Accelerometer Misalignments 15 sec Ref. 13

Group 5. Accelerometer Nonlinearities 3 .5 p g/g2 Ref. 13

Group 6. Gravity Deflections and 8 sec Ref. 14Anomalies' 67 mgal

Group 7. Gyro Bias Drifts 0.015 deg/hr Ref. 13Group 8. Gyro Mass Unbalances 0.025 deg/hr/g Ref. 13Group 9.. Gyro Anisoelasticities 0.025 deg/hr/g2 Ref. 13

Sea level values are specified here. The truth model database uses the sea level values and the model in Ref. 14 toprovide standard deviations and correlation times as afunction of altitude.

The truth model states for gravity deflections and anomaliesare considerably more detailed than they appear to be in Table 3.1-2.The standard deviations specified correspond to the errors at sea levelfor a crude on-board gravity model. The model in Ref. 14 provides forattenuation of the magnitude of the errors as a function of altitude, e. g.,

3-7


at 250, 000 ft the standard deviations have been attenuated by a factor of

three. The model also relates the correlation distance to altitude.

The error sources in Groups 21 through 23 can be divided into

two categories:

* time-varying biases

* stationary or nonstationary markov processes

Models for non-standard atmospheric density errors (Group 21) and non-

standard wind errors (Group 22) are developed in Appendix D. For both

time-varying biases and the nonstationary markov processes, Appendix D

assigns a variance for the individual error sources as a function of alti-

tude, latitude, and season -- in the latter case correlation times are also

expressed as a function of altitude, latitude, season, and flight-path angle.

The parameter ranges for these error models are summarized in Table

3.1-3.

TABLE 3.1-3

DRAG-UPDATE TRUTH MODEL DATA BASEFOR DRAG-RELATED ERROR SOURCES

Error Source Standard Deviation Correlation Time Data Source(Distance)

Group 21. Non-Standard Density

1962 Standard Atmosphere Error 0 - 21% -- FIg. 3.1-1

'Time-Varying Bias 4 - 49% '-- g. 3.1-2

First-Order Markov 2 - 6% (14 - 270 nm) Table D-2

Group 22. Non-Standard Wind

Westerly (Time-Varying Bias) 0 - 295 ft/sec -- Fig. 3.1-3

Headwind (First-Order Markov) 40 - 110 ft/sec (270 nm) Fig. D-4Crosswind (Second-Order Markov) 40 - 110 ft/sec (270 rm) Fig. D-4Turbulence (First- and Second- 13 ft/sec (1750 ft) Appendix D

Order Markovs)

Group 23. CD Variations 5% 100 sec

3-8


Only turbulence and CD (drag coefficient) variations are modeled

in Table 3.1-1 as stationary markov processes; the range of values for the

nonstationary processes is indicated in the table. The correlation time for

the CD variation has not been documented in the literature. In composing

the drag-update error budget, CD variations were assumed to be either a

stationary markov process with a correlation time of 100 sec (Group 23) or

a white random sequence (drag measurement noise in Group 1). Three of

the error sources are modeled in Table 3.1-1 as time-varying biases

rather than true random processes. These are:

* errors in the drag-update filter modelof the 1962 Standard Atmosphere

* season- and latitude-dependent (time-varying) density bias

* westerly wind

The models developed in Appendix D for these error sources are sum-marized in the remainder of this section.

Altitude profiles for the drag-update filter density modeling er-ror are shown in Fig. 3.1-1 for two different onboard density models --

30 R-12913

/"0 I.\ERM MODELI- ERM MODEL

-20

10000 130.000 200,000 250.000 300.000ALTITUDE (It)

Figure 3. 1-1 Deviation of Navigation Filter Density Modelsfrom 1962 Standard Atmosphere

3-9


the models are defined in Section 4.2. The density modeling errors are as

great as 21% at 145, 000 ft for the one-term model and 7% at 250, 000 ft

for the four-term model. Navigation errors due to both models are deter-

mined in Chapter 5.

The time-varying density bias exhibits both latitude- and

season-dependence. Altitude profiles for both summer and winter are shown

in Figs. 3. 1-2a and 3. 1-2b respectively as percent deviations from the 1962

Standard Atmosphere. Dashed lines on the profiles denote the mean density

deviations corresponding to the reference mission 3B altitude-latitude pro-

file. In both instances the deviations are greatest when the Shuttle is at high

altitudes and high latitudes. Both models are analyzed in Chapter 5.

The mean westerly wind is also both season- and latitude-

dependent. The season-dependence is shown in Fig. 3.1-3. Figure 3.1-3

was constructed explicitly for the reference mission 3B trajectory, it is notdirectly applicable to nonpolar entry trajectories. For reference, the stan-

dard deviation for the westerly wind is also indicated in Fig. 3.1-3. Thisis the same standard deviation indicated for the headwinds and crosswinds

in Table 3.1-3. Again, navigation errors resulting from both profiles areanalyzed in Chapter 5.

The non-standard atmospheric density and non-standard wind

models summarized in this section provide a realistic statistical descrip-tion of the individual drag-related error sources which affect the drag-

update filter performance. Certain physical correlations, such as the cor-relations between wind and short term (first order markov) density devia-

tions, were omitted from the error models; however, it is not expected thatinclusion of these correlations would alter the performance analysis in

Chapter 5 significantly. The correlations can be added for applications inwhich a more detailed model of atmospheric properties is necessary.

3-10


'300

30N 450 N 800

N 600 NI

250

D1

S200 -

I-I

150 -

- - -RECOMMENDED SUMMERMEAN DENSITY DEVIATIONFOR REFERENCE MISSION 3B

0 25 50

DEPARTURE FROM 1962 STANDARD DENSITY (%)

Figure 3.1-2a Time-Varying Density Bias for June-JulyGiven as Percent Departure from 1962Standard Atmosphere

300 R-12036

80N 600N 450 N 30N/

" .250

200

Op Poop' QU. ...

. ..\ ,-J

150

----- RECUOMFMENDED:WINTERMEAN DENSITY DEVIATION

SI FOR REFERENCE MISSION 38

-50 -25 0 ' 25

DEPARTURE FROM 1962 STANDARD DENSITY (%)

Figure 3.1-2b Time-Varying Density Bias for December-January Given as Percent Departure from1962 Standard Atmosphere

3-11

-THE ANALYTIC SCIENCES CORPORATION

R-12144

- - - 10 BOUNDSFOR HEADWINDS& CROSSWINDS

S50 -

-VELOCITY (ft/sec)

0 200

1% \

150

-300 -20 100 200 100

VELOCITY (ft/sec)

Figure 3.1-3a Mean Zonal (Westerly) Wind inJuly for Reference Mission 3BEntry

3-12

---- -- a BOUNDSFOR HEADWINDS& CROSSWINDS

250

-100 0 100 200 300 400

VELOCITY (ft/$ec)

Figure 3.1-3b Mean Zonal (Westerly) Wind inJanuary for Reference Mission3B Entry

3-12


3.2 SYSTEM E MODEL

The external aids used in System E are the drag pseudo-

measurement, TACAN, a baro altimeter, a radar altimeter, and MLS.With minor exceptions, the measurement sequence defined in Chapter 2is structured such that one range measuring device, one bearing measur-ing device, and one altimeter are always operating -- during MLS, altitudeis derived from elevation angle and range measurements. The computernavigation program includes a variable-dimension Kalman filter with be-tween 6 and 15 states. A total of 19 different state variables are used inthe filter.

3.2.1 States and Error Sources

The truth model for System E includes the truth model statesand other error sources of the drag-update phase, and also includes statesand error sources required to model the accuracy of the additional exter-nal navigation aids. The truth model states and other error sources usedin evaluating the System E filter are listed in Table 3.2-1, which dividesthem into three categories:

4 The 19 estimated states and uncorrelated measure-ment and process noises.

* Non-estimated states related to the inertial system.* Non-estimated states related to the external navi-

gation aids.

Error budget results for System E are generated similarly tothose for the drag-update phase. The principal difference is that the truthmodel structure (FS, HS) for Group 1 is variable-dimensioned in a man-ner similar to that of the navigation system Kalman filter. Results

3-13 /


TABLE 3.2-1

SYSTEM E TRUTH MODEL STATESAND ERROR SOURCES

. NUMBER NUMBER OFERROR SOURCE NAMES OF ERROR

STATES SOURCES

I. ESTIMATED STATES ANDUNCORRELATED NOISES

Group 1.

Position Errors 3. 3Velocity Errors 3 3Platform Misalignments 3 3Accelerometer 'Biases' (First- 3 3Order Markovs)

Correlated Measurement Errors(First-Order Markovs)

Drag I1

TACAN 2

Baro Altimeter I 1

Radar Altimeter 1 1

MLS 3 3

Uncorrelated Measurement Noise

Drag - I

TACAN - 2

Baro Altimeter - 1

Radar Altimeter - -

MLS - 3

INS Quantization Noise - 3

II. NON-ESTIMATED, IMU-0PR~ U.A l RELATED STATES

O0P 0_ QUJ Group la. True Platform 3 3Mlsalignments

Group 2. Accelerometer True Biases 3 3

Group 3. Accelerometer Scale Fac- 3 3tor Errors

Group 4. Accelerometer Misallgn- 6 6ments

Group 5. Accelerometer Nonlin- 3 3earities

Group 6. Gravitational Deflections " 3 3and Anomalies

Group 7. Gyro True Bias Drifts 3 3

Group 8. Gyro Mass Unbalances - 6 6

Group 9. Gyro Anisoelasticities 3 3

S3-14


TABLE-3.2-1 (Continued)

SYSTEM E TRUTH MODEL STATESAND ERROR SOURCES

NUMBER

ERROR SOURCE NAMES NUMBER OFOF ERRORSTATES SOURCES

Ill. NON-ESTIMATED, EXTERNAL AIDRELATED STATES

Group 10. TACAN Range Bias Error 1 1First-Order Markov

Group 11. TACAN Range Scale Factor 1 1

Group 12. Baro Altimeter Errors

Bias 1 1

Scale Factor 1 1

First-Order Markov 1 1

Static Defect 1 1

Group 13. MLS Time-Varying BiasesRange Bias 1 1

Range Scale Factor 1 1

Azimuth 1 1

Elevation 1 1

Group 14. MLS Second-Order Markov

Range 2 1

Azimuth 2 1

Elevation 2 1

Group 15. Radar Altimeter Errors

Bias Error 1 1


Group 16. TACAN Survey Errors 6 6

Group 17. MLS Survey Errors 6 6

Group 19. TACAN Bearing Bias Error 1 1

First-Order Markov

Group 20. MLS Timing Errors

Bias 3 3

Group 21. Non-Standard Density

1962 Standard Atmosphere Error 1 1

Time-Varying Bias 1 1


Group 22. Non-Standard Wind

Westerly (Time-Varying Bias) 1 1

Headwind (First-Order Markov) 1 1

Crosswind (Second-Order Markov) 2 1

Turbulence (First- and Second- 5 3Order Markovs)

Group 23. Non-Standard Aerodynamics 1 1First-Order Markov

Totals 100 105

3-15


corresponding to Groups la through 23 are generated using a higher-dimensional truth model, which contains the basic variable-dimensioned

Kalman filter plus other states representing time-correlated error sourcesnot modeled explicitly in the filter design. The group numbering system isconsistent with that used in the drag-update phase evaluation.

The System E truth model requires a variable-dimension filterdynamics matrix, F F , and a 100 x 100 system matrix, FS. Thesematrices, as well as HF, HS, and A, are defined in Appendix E.

.3.2.2 Truth Model Data Base

The general discussion in Section 3.1.2 pertaining to the drag-update phase data base applies to System E and is not repeated here. Theprincipal difference is that System E includes truth model states and errorsources related to TACAN, a baro altimeter, a radar altimeter, and MLSwhich did not appear in the drag-update phase data base. The necessaryadditions to the drag-update data base are made in this section.

The initial covariance matrix, PS(0), for System E is the finalcovariance matrix at 130, 000 ft for the drag-update phase analysis. Theerror sources'for System E which are not present in the drag-update phaseinclude measurement noise (Group 1), and biases and correlated randomerrors (Groups 10 through 20) for the new sensors. The numerical valuesselected for these error sources are summarized in Tabie 3.2.2.

The uncorrelated measurement noises in Group 1 are used tomodel receiver noise from a variety of sources. For the radar altimeter,the principal source is a quantization error; for TACAN range, it is theability to estimate the return pulse arrival time, etc. The measurementnoise variances for each sensor are constant.

3-16

TABLE 3.2-2 -I

(. SYSTEM E TRUTH MODEL DATA BASE(WHERE DIFFERENT FROM THE DRAG-UPDATE PHASE) Z

E ERROR SOURCE STANDARD DEVIA7TON TIME CONSTANT DATA SOURCE

-- Group 1. TACAN Range Measurement Noise 100 ft -- Ref. 11TACAN Bearing Measurement Noise 6 mrad -- Ref. 11Baro Altimeter Measurement Noise 5 ft -- Ref. 4Radar Altimeter Measurement Noise 1 ft - Ref. 1MLS Azimuth Measurement Noise . 0. 1 mrad -- Ref. 15 ZMLS Elevation Measurement'Noise 0.1 mrad -- Ref. 15 0MLS Range Measurement Noise 17 ft -- Ref. 15 M

Group 10. TACAN Range Markov 385 ft 300 see Ref. 11Group 11. TACAN Range Scale Factor .100 ppm " Ref. 11 0Group 12. Baro Altimeter Errors " • " J)

Bias 100 ft - Ref. 4Scale Factor 3% of alt ".- Ref. 1Static Defect . 1.52 x 10- 4 ft/ft2 /sec 2

Ref. 1First-Order Markov 20 ft 100 see Ref. 4

Group 13. MLS Time-Varying Biases ZAzimuth 0.4 mrad - Ref. 15Elevation 0.4 mrad * Ref. 15Range Bias 8 8ft .- Ref. 15Range Scale Factor 400 ppm - Ref. 15

Group 14. MLS Second-Order Markov

Azimuth 0. mrad 4 see Ref. 15Elevation 0.2 mrad 4 see Ref. 15

. Range . 17 ft 4 see Ref. 15

Group 15. Radar Altimeter Errors

Bias 1 ft * Ref. 1First-Order Markov 1 ft 1.5 see Ref. 1

Group 16. TACAN Survey Errors 1 ft * Ref. 16

Group 17. MLS Survey.Errors 1 ft : Ref. 16

Group 19. TACAN Bearing Markov 6 mrad 2x104 ft/VR Ref. 11

Group 20. MLS Timing Errors 50 msec - Ref. 15


The TACAN range correlated errors are separated into a

first-order markov, scale factor, and survey errors (Groups 10, 11, and

16, respectively). The markov process represents the combined errors

in calibrating both the airborne and ground equipment. The scale factor

components represent the inaccuracies in calibrating the index of refrac-

tion. Values ranging from 10 parts per million to 100 parts per million

have been mentioned in the literature, varying with the extent to which

knowledge of local atmospheric conditions and measurement geometry is

used in the calibration procedure. Survey errors represent inaccuracies

in knowledge of the ground transponder locations relative to the runway.

Since, in this case, the devices are nearby, small survey errors are ex-

pected.

The TACAN bearing correlated errors are modeled as first-

order markov and survey errors (Groups 19 and 16, respectively). The

markov process is due primarily to a bending of the radiated constant

bearing lines due to multipath effects. The time constant for the error

is proportional to the Space Shuttle relative velocity and is somewhat longer

than that suggested in Ref. 11. The survey errors are treated as distinct

from the TACAN range errors because different antennae are involved.

Baro altimeter correlated errors (Group 12) are separated into

bias, markov, scale factor, and static defect components. Values for the

bias and markov components are taken from Ref. 4, which treats a baro-

inertial-transponder system involving overflight of a transponder. In that

case (the CLASS filter evaluation study) it was found to be important to in-

clude the markov state to account for moderately rapid changes in local

weather conditions. Values for the scale factor and static defect compo-

nents are taken from Ref. 1, which discusses them in the context of the

Space Shuttle landing problem.

3-18


Models for the radar altimeter correlated errors (Group 15)

have not been well established. The selection of bias and first-order

markov errors was made in order to account for instrument errors and

terrain variations. A scale factor error was not included in the model

because of the low altitude at which the altimeter is to be used.

The basis for the MLS correlated error models (Groups 13, 14,and 17) is Ref. 15. The errors for each of the three sensors are separa-

ted into bias and second-order markov errors. In addition, a scale factor

error is modeled for the MLS DME. The biases in the MLS azimuth and

elevation measurements are attributable to errors in the antennae sweep

angles and to angle pickoff errors; the bias in the MLS DME measurement

is primarily due to hardware timing delays. The MLS DME scale factor

error is a function of the measurement signal-to-noise ratio. For all

three sensors, the second-order markov error models multipath effects.

The four sec "correlation time" chosen for the second-order markov satis-fies the model given in Ref. i5.

Measurements from each of the external sensors are subject totiming errors if the measurements are not accurately time tagged. Inmost instances the effect of the timing error can be included in the instru-ment bias model. A timing delay for MLS is included in the truth model(Group 20) to provide an indication of the magnitude of timing-induced

errors relative to other error sources.

/3-19

3-19 /


4. FILTER COVARIANCE ANDGAIN CALCULATIONS

Two of the preliminary steps required prior to performing er-

ror budget calculations are the detailed definition of the filter which forms

part of the system and the preparation of a filter covariance program.

This program contains the covariance update and propagation algorithms

specified by the filter designer and generates a sequence of gain vectors

corresponding to a particular trajectory and measurement schedule.

This sequence is saved on tape and repeatedly used by the truth model

covariance program in generating the system error budget. Presented

below are the specific algorithms used to represent the filter gain cal-

culations corresponding to the navigation system evaluated in this study.

The detailed structure of the two phases of the navigation sys-

tem -- the drag-update phase and System E -- are presented separately

in Sections 4.2 and 4. 3, respectively. The general structures of the

navigation filter and of the filter covariance program are presented in

Section 4.1.

4.1 FILTER STRUCTURE

Figure 4.1-1 is a macro flowchart indicating the overall organ-

ization of the filter covariance program. The principal inputs are the

starting time, t 0, the initial filter covariance matrix, PF(0) , and the

nominal Space Shuttle trajectory. The principal output is the file of gain

4-1


START ft-407d

INITIALIZE:AID LOCATIONS,TRANSFORMATIONS,READ t'0 ,P(0)

LOOP

CALL TRA) JPOSVEL., V, - TRAJECTORYFILE

STME YES PUNCH P

NO

NEWNAV AID YES REASSIGN NAV AIDACOUIRED BIAS STATES

NO

YESt- t hsw REDUCE P, TO 66

NO4

YES

FILTER COVARIANCEUPDATE

FILTER TRANSITIONMATRIX ANDPROCESS NOISE

FILTER COVARIANCEPROPAGATION

NO t TD

END

Figure.4.1-1 Filter Covariance Program Flow Chart(Drag-Update Phase and System E)

4-2


vectors to be used by the truth model covariance program. At the alti-

tudes of 130,000 ft (t = TDME) and 12,000 ft (t = TSW), the current

PF matrices are punched on cards. This gives the capability of re-

starting the program at the end of blackout, and the beginning of the final

approach and landing, respectively.

The navigation filter is a variable-dimension Kalman filter with

between 6 and 15 states. A maximum of three of the states are bias states

associated with the operating external navigation aids (TACAN, etc.). As

each new navigation aid is acquired, the filter is restructured and bias

states are associated with that navigation aid. The exception is that no

bias states are associated with the radar altimeter.

The navigation filter is implemented using the discrete time

covariance formulation with exact process noise. The covariance propa-

gation between measurements is accomplished using the "average G"'

numerical integration filter (Ref. 17). The covariance matrix is propa-

gated only to the beginning of the final approach -- which occurs at a

nominal altitude of 12,000 ft (t = TSW). Below 12,000 ft, three differ-

ent filter configurations are analyzed. The configuration indicated in

Fig. 4.1-1 computes MLS gains below 12,000 ft based upon a 6 x 6 sub-

matrix of PF saved from 12,000 ft.-- a pre-stored gain schedule is used

for the radar altimeter. There are no covariance reinitializations duringentry.

At each measurement time, a sequence of filter gains are

computed corresponding to a sequence of scalar measurements. For

every measurement except radar altimeter measurements, the gain vec-

tor is calculated using

F(tk) = F(t) HF (tk) F PF(t) H (tk) u70 + o (4.1-1)

4-3


where HF is the appropriate scalar measurement matrix, PF(tk) is thecovariance just prior to the measurement, u 7 0 is a scalar measurement

underweighting factor,* and a 2 is the assumed measurement noise var-mlance.

At altitudes above 12, 000 ft, the filter gains are used to se-quentially update the filter covariance in accordance with

PK( = PF(t) -KF(t k) HF( PF(t) (4.1-2)

The covariance is then propagated forward to the next measurement tim-ing using the standard formula presented in Section 2.3:

PF(tl) = 'FkPF(tl) . k + Fk (4.1-3)k *k k

Equations (4.1-2) and (4. 1-3) are not executed below 12, 000 ft.

4.2 DRAG-UPDATE FILTER

The navigation filter for the drag-update phase is a 13-stateKalman filter. The general structure of the filter is:

State Numbers Variables

1 - 3 Position4 - 6 Velocity7- 9 Misalignment angles

10 - 12 Acceleration errors13 Drag correlated error

*The scalar u7 0 is normally unity, but has the value 1.2 during theearly portion ob the trajectory (see Ref. 5).

4-4


The time constants and measurement noise for states 7 - 13 are sum-

marized in Table 4.2-1.

TABLE 4.2-1

FILTER STATE AND MEASUREMENT ERROR STATISTICSFOR DRAG-UPDATE PHASE*

Correlated Error (Filter State) MeasurementNoise

Standard Correlation StandardDeviation Time Deviation

(sec)

Misalignment. 1 mrad 600

Acceleration 0.0017 ft/sec 60

Drag 307% of drag 60 '5% of drag

States 7 - 13 are generalized error states and do not corre-

spond precisely to physical error sources. The misalignment states

differ from true IMU misalignment angles in that they do not have an in-

finite time constant; the acceleration states differ from true acceler-

ometer biases because of the time constant and because they are defined

in an inertial coordinate frame (Appendix E). The drag correlated error

state is intended to account in a general manner for all error sources

which directly affect either the drag pseudo-measurement or the pre-dicted drag computation. The rationale behind selecting the values inTable 4.2-1 was to yield a filter covariance matrix which is reasonably

consistent with the true navigation errors.

An initial covariance matrix for the entry interface at 400,000 fthas been provided by CSDL (Ref. 18). The matrix is the filter covariance

*Data from Ref. 7 and personal communication with B. Kriegsman ofCSDL.

4-5


.matrix at entry interface for the Shuttle reference mission 3A abort. This

is a single orbit mission launched from Vandenberg Air Force Base into a

50 by 100 nautical mile orbit with an inclination of 1040. The filter co-

variance matrix was propagated from launch with no reinitializations.

The filter state noise matrix used to propagate the filter co-

variance during the drag-update phase is defined by

At2 AtSV 4 SV2

0

AtSv 2 SV

OF k (4.2-1)k 0"

00 S13

where At is the update interval (2 sec), sV is a 3 x 3 matrix

2a 0 0q

s V = 0 aq 0 (4.2-2)

q0 0 2

and sj, j =7 ... 13, are scalars

. 2(1 e2t - /j) (4.2-3)

The value of aq suggested by CSDL personnel is

2= 0.001 (ft/sec)2

Oq

4-6


The values of a. and r. are the entries in Table 4.2-1 for the appro-J J

priate filter states.

The only external aid used during the drag-update phase is

the drag acceleration pseudo-measurement defined in Appendix C -- one

pseudo-measurement is processed every 2 sec. The measurement matrix

used in the gain computation and to update the filter covariance was taken

from Ref. 19:

T T drag correlated-b c 2i9 random variable

IF(t) = qpred(t) h VR(t) 0 1 (4.2-4)

where iRc(t) and IVRc(t) are computed unit vectors along the vertical and

the relative velocity, respectively, VR(t) is the computed relative velocity,

and hse is a scale height (see below). The predicted drag

pred (t) CD Pc(t) VR 2 (t) (4.2-5)c c.

is based upon onboard models of the aerodyhamic drag coefficient and the

atmospheric density.

The onboard drag coefficient model from which CD (t) is com-

puted was provided in tabular form by JSC (Ref. 20). The model is a func-

tion of Mach number and angle of attack. The values of the cross-sectional

area, A, and the mass, m, used by JSC to generate the reference trajec-

tory were also.provided.

The onboard atmospheric density model for the navigation fil-

ter was assumed to be a simple exponential

'h (t)h

pc(t) = POe S (4.2-6)

4-7


where h c(t) is computed altitude. Two different variations of the model

were considered:

* Single values of hsc and p0 were chosen(1-term model).

* Values of hsc and p0 were chosen for differentaltitude layers (4-term model).

The models were obtained from Refs. 5 and 21 , respectively. The co-

efficient values used are summarized in Table 4.2-2. The resulting er-

rors in modeling the 1962 Standard Atmosphere were illustrated in

Fig. 3.1-1.

TABLE 4.2-2

COEFFICIENT VALUES FORATMOSPHERIC DENSITY MODELS

Model 1-Term Model 4-Term Model

Altitude 100,000- 100,000- i50,000- 200,000- 230,000-270,000 ft 150,000 ft 200,000 ft 230,000 ft 270,000 ft

Scale HeightScale Height 24018. 22104. 26590. 26170. 21206.h Sc(ft)

Sea LevelDensity Level 0.00215 0.00306 0.00097 0.00110 0.00860Density

PO (slugs/ft3)

The filter covariance program based upon the 13-state filter

outlined above has been exercised over the trajectory and measurement

schedule outlined in Chapter 2. The time history of the filter-indicated

performance is summarized in Table 4.2-3. RMS values of position and

velocity components in the V frame (vertical, 'downrange, crossrange; see

Appendix A) are given at key times -- at entry interface, just before and

4-8


TABLE 4.2-3

DRAG-UPDATE PHASE FILTER-INDICATED PERFORMANCE

Event RMS Position Errors (ft) RMS Velocity Errors (ft/sec)(see Time

Table 2.2-1) (sec) V DR CR V DR CR

TO 0 5571. 26073. 7774. 27.52 5.40 4.31

312. 4975. 27447. 8523. 29.48 5.05 1.06TDRAG 312. 4123. 26497. 8524. 28.53 4.29 1.05

T 1432.- 4885. 16780. 9034. 18.34 15.70 16.51TACAN

after the first drag pseudo-measurement, and at the end of the drag-update

phase. The results given in Table 4.2-3 represent what the navigation sys-

tem performance would be if the 13-state filter model were a completely

accurate representation of real-world dynamics. The results given in

Section 5.1 are a projection of the performance of the 13-state filter in a

59-state model of the real world.

4.3 SYSTEM E

The navigation filter for System E is a variable-dimension

Kalman filter with between 6 and 15 states. The general structure of the

filter is:

State Numbers Variables

1 - 3 Position4 - 6 Velocity7 - 9 Misalignment angles

10 - 12 Acceleration errors13 Baro altimeter, drag, or elevation

correlated error14 Range correlated error15 Azimuth correlated error

4-9/


States 1 through 12 are defined precisely as they were for the drag-

update filter. States 13 through 15 are correlated measurement errors.As a new sensor is acquired, one of the measurement error states isreinitialized and associated with that sensor -- the exception is that nobias states are associated with the radar altimeter. The precise filter

state schedule is indicated in Fig. 4. 3-1. The correlation times forstates 7 through 15 and the measurement variances are summarized inTable 4.3-1.

R-11206

ALTITUDE 130.000 FT 100.000 FT 20,000 FT 12.000 FT 12 FT TOUCHDOWN

I I I I I I

.1- POSITION AND VELOCITY

7- 12 MISALIGNMENTS AND ACCELERATION

13 DRAG UPDATE J6IAROALTIMETER, 1 MLS ELEVATION

14 TACAN DME d MLS DME

s "TACAN VOR MI S AZIMUTH ,

Figure 4.3-1 Schedule for System E Filter States

The initial covariance matrix for the System E filter is thefilter covariance matrix at the end of the drag-update phase. The pro-cess noise matrix, QF , satisfies Eq. (4.2-1) through (4.2-3) with amaximum of 15 states. The measurement update interval is 2 sec untilthe beginning of the final approach and landing (t = TSW) at 12,000 ft.The update interval is then reduced to 0.5 sec. At the radar altimeterturn-on (t = TRA), the update interval is further reduced to 0.1 sec.

Computer duty cycle constraints require that during thefinal approach and landing:

4-10


TABLE 4.3-1

FILTER STATE AND MEASUREMENTERROR STATISTICS FOR SYSTEM E

Correlated Error (Filter State)Measurement Noise

CorrelationStandard Time Standard

Measurement Deviation (sec) Deviation

Misalignments 1 mrad 600

Acceleration 0.0017 ft/sec2 60

Drag 30% of drag 60 5% of drag

Baro Altimeter 3% altitude 200 0.03% altitude

Radar Altimeter 1 ft a 0.1 ft

TACAN DME 275 ft 400 90 ft

TACAN VOR 6 mrad 400 6 mrad

MLS DME 35 ft o 24 ft

MLS Azimuth 0.4 mrad .0 0.2 mrad

MLS Elevation 0.4 mrad o 0.2 mrad

* The System E filter estimate only positionand velocity (6 states)

* The filter covariance propagation be eliminated

Elimination of the covariance propagation implies that, below 12, 000 ft,

the filter gains must be computed from a suboptimal algorithm. A num-

ber of different algorithms have been proposed. The salient features of

the algorithm selected for System E are summarized in Table 4.3-2.

Descriptions are also provided of two alternative algorithms analyzed

in Chapter 5.

Above 12,000 ft, the System E measurement matrices for all

external navigation aids are of the form:

HF [h h 0 0 h ] (4.3-1)F p vI I h

4-11


TABLE 4.3-2

DESCRIPTION OF GAIN COMPUTATION ALGORITHMSFOR FINAL APPROACH AND LANDING*

SYSTEM E FILTER ALTERNATIVE I ALTERNATIVE 2

* 6 states * same as SYSTEM E * same as SYSTEM E* no PF propagation filter except: filter except:* HF computed for MLS * MLS gains computed • 0 MLS gains are thoseand radar altimeter from Kalman algorithm computed at 12,000 ftand radar altimeter [Eq. (4. 1-1)] with 15 x 15 (azimuth and elevation* MLS gains computed P saved from 12,000 ft gains scaled by range-from Kalman algo- (only 6 elements of KF to-go)

rithm [Eq. (4. 1-1)] used)with 6 x 6 P savedfrom 1 2 , 00Fft

* gain schedule for radaraltimeter provided byCSDL

* MLS and IMU bias esti-mates at 12,000 ft usedto compensate data

where hp is the 1 x 3 partial derivative of the measurement with respectto the nominal position, h is the partial derivative with respect to thevnominal velocity, and hb is the partial derivative with respect to the biasstates. Below 12,000 ft, HF is of the form:

HF = [hp hv] (4.3-2)

For all external navigation aids except the drag pseudo-measurement,the first six elements of HF are identical to those of HS in the truthmodel. The equations for HF are developed in Appendix E.

All three gain computation algorithms were provided by CSDL (Ref. 9).

4-12


The pre-stored gain schedule for the radar altimeter (Ref. 21)

is used to update vertical position and velocity only. The gains for verti-

cal position are

k = 1. -3.75 (t - TRA) t -TRA < 0. 2 secp (4. 3- 3)

0.25 t - TRA > 0. 2 sec

and the gains for vertical velocity are

k = -0.1 + 1.5 (t- T RA) t-T RA 0.2 secv (4. 3-4)0.20 t -TRA > 0.2 sec

The filter gain matrix KF(t) is then

k iR (t)

KF(t) c (4.3-5)kv_ R (t)

where iRc(t) is the unit vector along vertical.

The filter covariance program based upon the System E filter

outlined above has been exercised over the trajectory and measurement

schedule outlined in Chapter 2. The time history of the filter-indicated

performance is summarized in Table 4.3-3. RMS values of the position

and velocity components in the V frame (vertical, downrange, cross-

range; see Appendix A) are given at key times -- before and after the

first measurement from each external navigation aid -- down to 12,000 ft.

Below 12,000 ft, PF is not computed, and hence, no filter-indicated per-

formance is available. The results given in Table 4.3-3 represent what

the navigation system performance would be-if the System E filter model

4-13


TABLE 4.3-3

SYSTEM E FILTER-INDICATED PERFORMANCE

Event RMS Position Errors (ft) RMS Velocity Errors (ft/sec)(see Time

Table 2.2-1) (sec) V DR CR V DR CR

1432. 4885. 16780. 9034. 18.34 15.70 16.51

TACAN 1432. 4684. 6742. 6063. 15.81 14.32 14.23

1550. 4080. 1013. 1742. 13.49 8.39 10.66

BA 1550. + 2744. 790. 1481. 9.75 6.98 10.27

1836. 589. 311. 264. 2.60 4.18 2.80

1836.+ 90. 136. 185. 2.12 2.37 2.78

1846. 91. 167. 168. 1.73 1.98 2.76TDME+ 1846.+ 25. 35. 32. 1.29 1.78 1.55

TSV 1874. 18. 36. 19. 0.40. 0.81 0.46

1943.5 -- - - - - -

RA 1943.5 - -

TD 1 9 4 6 . 5 - - - -- -

(Table 4.3-1) were a completely accurate description of real-world dynam-

ics. The results given in Section 5.2 are a projection of the performance

of the System E filter in a 100-state model of the real world.

4-14


5. RESULTS

This chapter presents detailed results for the entry naviga-

tion system. Overall performance curves are given for the drag-update

filter and for System E, showing time histories of position and velocity

errors due to all sources combined. In each case "Baseline Error Bud-

get" tables are given, showing the contributions of individual error

sources, or small groups of error sources, at certain times. Smaller

tables of "Alternative Contributions" are provided, allowing the reader

to see the effects of different truth model assumptions. Sensitivity curves

illustrating the effect of variations in the rms value of a given major er-

ror source (or group of error sources) on overall system performance

are also given. For System E, the effect of alternative filter configura-

tions on overall system performance is also evaluated. Section 5.3 pro-

vides a general summary, including discussions of how particular groups

of error sources contribute to the errors in each of the mission phases

and of filter states which might be safely deleted or altered.

5.1 DRAG-UPDATE FILTER EVALUATION

The navigation error results presented below are for the. drag-

update phase of the reference mission 3B entry as defined in Section 2.1.

The drag-update phase ends 1432 sec after entry interface at a nominal

altitude of 130,000 ft. The only external navigation aid during this phase

is the drag acceleration pseudo-measurement; the measurement schedule

was defined in Section 2.2. A file of gain values, generated using the

5-1


filter covariance program outlined in Sections 4.1 and 4.2, has been

used repeatedly to compute detailed error contributions corresponding

to the drag-update phase of this particular trajectory and measurement

schedule.

5.1.1 Overall System Performance

Figures 5.1-1 and 5.1-2 present overall performance curves

for the drag-update filter, showing position error components and velo-

city error components, respectively. The curves plotted represent rms

errors due to the combined effects of all error source groups in the base-

line error budget. They were generated by root-sum-squaring individual

contributions at two-minute intervals (tabulated in Appendix F). At

T D when the first drag pseudo-measurement is made, the root-sum-DRAG'

square calculation was performed both before and after the update. Thus,

the jumps which occur in certain component errors at this time are ac-

curately shown. Otherwise, the curves are faired-in over the two-

minute intervals between the calculated points.

At the time of the first drag pseudo-measurements, the verti-

cal and downrange components of both position and velocity errors in-

crease. The downrange errors remain essentially constant at the new

levels; the vertical errors continue to increase for several hundred

seconds before beginning to decline. Examination of the individual er-

ror contributions (Appendix F) verifies that the jump is primarily due

to the effect of non-standard density errors (Group 21) on the pseudo-

measurement. This suggests that the drag-update filter overweights

the drag acceleration pseudo-measurement during the initial portion of

the trajectory.

5-2


R-14079

e VERTICAL ERROR (UlDDOWNRANGE ERROR IV)ACROSSRANGE ERROR (W)

0

0

0 200 400 600 800 1000 1200 140G 1600DAG ELAPSED TIME (sec) AC

Figure 5.1-1 Drag-Update Phase Overall Performance: Position

t-14078

0 VERTICAL ERROR (U)0 DOWNRANGE ERROR IV)a CROSSRANGE ERROR (W)

to

0 200 400 600 800 1000. 1200 1400 1600

ELAPSED TIME (sec)

Figure 5.1-2 Drag-Update Phase Overall Performance: Velocity

5-3

-THE ANALYTIC SCIENCES CORPORATION

Figures 5.1-1 and 5.1-2 indicate the performance of the drag-

update filter, but they do not indicate the overall effectiveness of the

filter in limiting the growth of navigation errors during entry. This in-

formation is provided in Table 5. 1-1 by comparing the total projected

drag-update filter performance at 130,000 ft with the performance of a

pure inertial navigator* at the same altitude. The truth model for the

inertial navigator consists of the important IMU-related states in the

drag-update phase truth model (Groups la through 5 and 7 through 9),

plus the INS quantization noise. The performance was obtained by cor-

recting the data in Table D-III-c of Ref. 10 for the truth model data base

given in Table 3.1-2.

TABLE 5.1-1

SUMMARY OF DRAG-UPDATE FILTER PERFORMANCE

/ ONE a ERRORS AT 130,000 ft

Position (ft) Velocity (ft/sec)

V DR CR V DR CR

Total Projected Drag-Update 8389. 14700. , 9723. 20.97 6.70 14.51Filter Performance

Pure Inertial Navigator 14756. 21196. 12184. 42.01 9.75 14.44Performance

Filter Indicated Drag-Update 4885. 16781. 9035. 18.34 15.70 16.51Performance

The only significant difference between the mechanizations of

the drag-update filter and the pure inertial navigator is that the drag-

update filter uses drag acceleration pseudo-measurements to improve

its state estimates. Table 5.1-1 indicates that the navigation errors for

*A pure inertial navigator uses only an initial state estimate and accum-ulated IMU outputs to obtain a current state estimate.

5-4


the drag-update filter are approximately two-thirds as great as the cor-

responding pure inertial navigator errors. The drag acceleration pseudo-

measurement provides an estimate of instantaneous vertical position only;

the filter uses correlations and trajectory geometry to achieve the indi-

cated improvements in the downrange and crossrange channels. Table

5. 1-1 verifies that the drag acceleration pseudo-measurement is a poten-

tially valuable navigation aid and that the candidate drag-update filter ef-

fectively limits the growth of navigation errors during entry.

Table 5.1-1 also contains the filter-indicated drag-update fil-

ter performance at 130,000 ft (see Section 4.2). Comparison with the

total projected drag-update filter performance indicates that the filter

is quite optimistic in its estimate of the vertical position errors -- the

true errors are nearly twice as large as the filter expects. It is reason-

able to suspect that this optimistic outlook results in an underweighting

of. the pseudo-measurements during the final portion of the drag-update

phase, but more information than Table 5.1-1 provides is necessary to

confirm this suspicion. Regardless, it would seem desirable to modify

the filter to bring the filter-indicated performance more in line with the

total projected performance.

The observations made in this section suggest that the drag-

update filter can be "tuned" to improve its performance. All the filter

parameters are subjects for optimization and sensitivity studies. Sev-

eral prospective modifications are discussed in Section 5.3.

5.1.2 Baseline Error Budget

System error covariances have been computed for each group

of error sources over the entire drag-update phase of the reference

5-5


. trajectory. The total amount of data generated is quite large. In orderto summarize important results in one table of manageable size, a"snapshot" view of conditions at the end of blackout (130, 000 ft) is pre-sented in Table 5.1-2. Table 5.1-2 is the Drag-Update Baseline ErrorBudget showing rms estimation errors in the position and velocity com-ponents at 130, 000 ft. Each value is the rms contribution of the errorsource or sources indicated in the left-hand column; it comes from a com-puter run in which the parameters for that error source are set at truthmodel values -- and all others are zero. The entries in each column areroot-sum-squared at the bottom of the table to generate the total projected

performance of the 1 3-state drag-update filter, The total projected per-formance is compared at the bottom of the table with the filter-indicatedperformance and with the inertial navigator performance. Table5.1-3 lists the Dra-Update Phase Alternative Contributions howing howdifferent error source models would produce other contributions to posi-tion and velocity errors at 130,000 ft.

Particular numbers have been circled in Table 5 .1-1 to focusattention on major contributors. The rule followed in deciding whichnumbers to circle was to include, in any column, every number whosemagnitude is greater than 20% of the RMS total for that column. Thus,every circled number contributes at least 2% to the RMS error for theappropriate column.

Examination of Table 5.1-2 reveals that the major contributorsto navigation errors at the end of the drag-update phase are:

* accelerometer scale factor error (Group 3)* gyro bias drift (Group 7)* non-standard atmospheric density modeling

errors (Group 21)

5-6


TABLE 5.1-2

BASELINE ERROR BUDGET FOR DRAG-UPDATE FILTER

RMS NAVIGATION ERROR AT 130,000 ftERROR SOURCE VALUE POSITION (ft) VELOCITY (ft/sec)

V DR CR V DR CRI. Estimated States

Uncorrelated Noise 5% of drag 1038. 995. 442. 2.97 0.46 0.82

Quantization Error 0.0328 ft/sec 165. 176. 167. 0.20 0.23 0.20

IT. IMU-Related States

2. Accelerometer Biases 50 pg 648. 3.72

3. Accelerometer Scale Factor 40 ppm(6 5

4. Accelerometer Misalign- 15 sec 1092. 1536. 1259. 3.34 1.61 2.57ments

5. Accelerometer Nonlinearities 3.5 pg/g 2 0. 0. 0. 0.00 0.00 0.00

6. Gravity Anomalies and 67 mgal - 110. 176. 112. 0.28 0.23 0.25Deflections 8 sec

7. Gyro Bias Drifts 0.015 deg/hr E3)i

8. Gyro Mass Unbalances 0.025 deg/hr/g 949. 480. 559. 1.95 1.14 1.34

9. Gyro Antsoelasticities 0.025 deg/hr/g 2 49. 130. 80. 0.23 0.20 0.41

m. Drag-Related Sources

21. Non-Standard Density

1962 Standard Atmosphere 1-term 0.87. 2.02Modeling Error 9 0.7 2.0Time-Varying Bias , Winter 2.61

First-Order Markov Table D-1 405. 421. 300. 1.10 0.14 0.32

22. Non-Standard Wind

Westerly Winter 558. 407. 229. 1.73 0.14 0.52

Crosswind Table D-3 229. 210. 86. 0.68 0.13 0.18Headwind

23. Non-Standard Aerodynamics 5% of CD 705. 889. 415. 1.74 0.40 0.55

First-Order ?vlarkov

Total Projected Performance 8389. 14700. 9723. 20.97 6.70 14.51

Pure Inertial Navigator 14756. 2119. 12184. 42.01 9.75 14.44Performance

Filter-Indicated Performance 4885. 16781. 9035. 18.34 15.70 16.51

5-7


TABLE 5.1-3

DRAG-UPDATE PHASE ALTERNATIVE CONTRIBUTIONS

RMS NAVIGATION ERROR AT 130,000 ft

ERROR SOURCE VALUE POSITION (ft) VELOCITY (ft/sec)

V DR CR V DR CR

Group 21. Non-StandardDensity

1962 StandardAtmosphere 4 term 2530. 4868. 2540. 5.35 0.54 1.54Error

Time-Varying Summer 3026. 5320. 2733. 6.71 0.71 1.81Bias

Group 22. Non-StandardWind

Westerly Summer 443. 321. 197. 1.48 0.10 0.26

The IMU-related contributions are primarily due to large initial errors at

400,000 ft -- velocity errors for Group 3 and misalignments for Group 7.

The atmospheric density contributions are primarily due to the error in

modeling the 1962 Standard Atmosphere and the season- and latitude-

dependent (time-varying) bias. The importance of both these density

sources could conceivably be diminished by improving the onboard at-

mospheric density model; but the emphasis should be on including the

time-varying bias.

It is important to note that the IMU-related contributions to all

three position component errors and to vertical velocity errors are com-

parable in magnitude to the drag-related contributions. This suggests

that, considering performance at the end of blackout only, the drag-

update filter is doing a reasonable job of mixing inertially-derived in-

formation with the drag acceleration pseudo-measurement. However,

the observations made in Section 5.1.1 indicate that improved perform-

ance should be attainable by modifying the filter gains. More detailed

5-8 /


discussion of the important error mechanisms and possible ways to re-

duce some of the contributions is given in Section 5. 3.

Detailed tabulations of the error contribution time histories

are given in Appendix F. For every row in Tables 5.1-2 and 5.1-3,

there is a page in Appendix F, which is a reproduction of a computer

printout page summarizing important results from one error budget run.

5.1.3 Sensitivity Curves: Drag-Update Phase

This section contains several curves illustrating the sensitivity

of drag-update phase performance to variations in error source statis-

tics. These "fixed-filter" sensitivity calculations answer the question:

'"What is the effect of an unknown variation in the rms value or values of

an error source or group of error sources?" These calculations can be

made easily, given the type of error budget information summarized in

Table 5.1-2, because the appropriate error covariance equations are

linear.

All of the example curves given in this section correspond to

major contributors to the system performance. Similar sensitivity

curves may be constructed for any error source group for which error

budget data exists in Table 5.1-2. Sensitivity curves corresponding to

error sources which produce minor contributions when their nominal

values are assumed are quite flat and of little interest.

To illustrate the means by which the data points for the sen-

sitivity curves were calculated, an example will be given. The sensi-

tivity of crossrange position at 130, 000 ft to gyro bias drift is shown

in Fig. 5.1-3. The baseline data point for the example is the total

5-9


25000 R-14197

. 20000-

0 VALUE COMPUTED /S. IN THE EXAMPLE /

615000-o " PURE INERTIALI- NAVIGATOR0

uJ BASELINE010000 - VALUEzO )

u 5000 -

0 0.01- 0.02 0.03 Q04 0.05

GYRO BIAS DRIFT (deg/hr)

Figure 5.1-3 Sensitivity of Crossrange Position Errorsat 130, 000 ft to Gyro Bias Drift

crossrange position error for the drag-update filter (9,723 ft). The totalincludes the effect of a 0.015 deg/hr bias drift about each platform axis.The contribution of these bias drifts is shown in Table 5.1-2 to be6,373 ft. To compute the effect of a 0.03 deg/hr gyro drift, the 0.015deg/hr bias drift contribution is removed from the total system errorand replaced with an error which is twice as large. Thus, the cross-range position error is

c = [(9,723) 2 - (6, 373)2 + (12,746) ]= 14,710 ft

This result is indicated in Fig. 5.1-3 as a boxed-in point. The dashedline through the origin is the asymptote approached by the total sys-tem error curve as the gyro bias drift contribution becomes large and

5-10


dominates all others. The remaining points on the curve are obtainedas illustrated above.

For reference purposes, Fig. 5. 1-3 also includes the sensi-tivity of the pure inertial navigator (see Section 5. 1-1) to gyro bias drift.The two curves intersect at a gyro bias drift error of 0.027 deg/hr, i.e.,if the gyro bias drift error is greater than 0.027 deg/hr and all otherIMU-related error sources assume their baseline values, a pure inertialnavigator provides better crossrange position estimates than the candidatedrag-update filter. The drag-update filter has increased the crossrangeposition performance sensitivity to gyro bias drift errors. The followingexample provides an instance for which the drag-update filter decreasesperformance sensitivity to a dominant error source.

Figure 5.1-4 shows the sensitivity of vertical position andvelocity errors to variations in the accelerometer scale factor error.The major contributors to vertical position error are of roughly com-parable magnitude so that the total projected error is not particularlysensitive to any given component. Accelerometer scale factor error isthe dominant contributor to vertical velocity error, however, and anincrease of the scale factor error to 80 ppm increases the vertical vel-ocity error by 54 %. The pure inertial navigator performance would bedegraded by 70%, however, and Fig. 5.1-4 indicates that the drag-updatefilter has decreased vertical channel performance sensitivity to accel-erometer scale factor errors.

The time-varying density bias is latitude - and season-dependent. The contribution of this error source to the total projectederror could be decreased by making the filter onboard density modelboth latitude- and season-dependent. Figures 5.1-5 and 5.1-6 illustratethe sensitivity of vertical position error and downrange position error,

5-11/


R-14190

ca40000O

*U

03000 PURE INERTIALNAVIGATO.

0.2000

"j . .11- BASELINE1000_ / VALE'

0 25 50 75 100 125ACCELEROMETER SCALE FACTOR (ppm)

a) Position

S100- R-14191

IC /PURE INERTIALU .. i AVIGATOR

>7>60 -/

, 40

U

U b) Velocity

> 20

Figure 5.1-4 Sensitivity of Vertical Psition and Velocity E

0 25 50 75 100 125ACCELEROMETER SCALE FACTOR (ppm)

b) Velocity

Figure 5. 1-4 Sensitivity of Vertical Position and Velocity Errorsat 130,000 ft to Accelerometer Scale Factor Errors

5-12


R-1419215000

0O

UJ 10000- BASELINEZ VALUE0

O .,-

5000-

LI

0 50 100 " 150 200

DENSITY BIAS (percent of Fig.3.1-2o)

Figure 5.1-5 Sensitivity of Vertical Position Errorsat 130,000 ft to Density Time-Varying Bias

R-14193- 30000

O

LU

z0 20000 -- -BASELINE -

•O VALUE

LU

zoc 10000 -

0 50 100 150 200

DENSITY BIAS (percent of Fig.3.1-2a)

Figure 5.1-6 Sensitivity of Downrange Position Errorsat 130,000 ft to Density Time-Varying Bias

5-13


respectively, to this error source. The curves were arrived at by scal-

ing the time-varying bias defined in Fig. 3.1-2a by the indicated amount.

The figures indicate that a large improvement in the density model ac-

curacy would not result in a significant improvement in navigation ac-

curacy if the filter gains for the pseudo-measurement were not modified.

It should be emphasized that the sensitivity analyses made in

this section are for a "fixed-filter" configuration. The sensitivity curves

would be altered if the filter were modified. As an example, if the fil-

ter were modified to increase the gains for the pseudo-measurement,

the total projected performance would become much more sensitive to

the time-varying density bias. For such a filter, an improvement in

the onboard density model could result in a significant improvement in

performance.

5.2 SYSTEM E

The navigation error results presented in this section are for

the approach and landing phase of the reference mission 3B entry as de-

fined in Section 2. 1. System E starts at the termination of blackout,

nominally 130, 000 ft, and ends at touchdown. The measurement sched-

ule for the external navigation aids is specified in Table 2.2-1. The

filter covariance program defined in Sections 4.1 and 4.2 has been used

to generate gain files for each of the three filter configurations sum-

marized in Table 4.2-3. The gain file for the System E filter was used

repeatedly to compute detailed error budgets at three significant time

points:

* initiation of the final approach and landingsequence (12,000 altitude; t = TW)

5-14


* the runway threshold (12 ft altitude; t = TRA)* touchdown (t = TD)

The gain files for the alternative System E filters were used to estimate

the sensitivity of the projected navigation errors at the runway thresh-

old to filter modifications.

5.2.1 Overall System Performance

Figures 5.2-1 and 5.2-2 present overall performance curves

for System E, showing position error components and velocity error

components, respectively. The curves plotted represent rms errors

due to the combined effects of all error source groups in the baseline

error budget. They were generated by root-sum-squaring individual con-

tributions at one-minute intervals (tabulated in Appendix G). At TTACAN,

when the first TACAN measurements are made, and at other key times,such as TBA TMLS, TRA, and TDME, the root-sum-square calcula-

tion was performed both before and after update. Thus, the large jumps

which occur in certain component errors at these times are accurately

shown. Otherwise, the curves are faired-in over the one-minute inter-

vals between the calculated points.

At the time of the first TACAN measurements, TTACAN, all

three position errors decrease significantly -- the greatest improvement

is in downrange position. The only significant velocity change is a de-

crease in vertical velocity error. Both position and velocity errors

then decrease rapidly over the approximately two minutes between

TACAN acquisition and baro altimeter acquisition. This improvement

in all error components is in contrast to the performance of three of the

5-15


1-14075

O VERTICAL ERROR (U)o DOWNRANGE ERROR (V)A CROSSRANGE ERROR (W

12000

O

z 8000

t-

400• -.

1400 1500 1600 1700 1800 ) 1 1900 200

TTACAN A . ELAPSED TIME (sec) T, gToE TD.

Figure 5.2-1 System E Overall Performance: Position

2- 1-14076

0 VERTICAL ERROR (U)8 DOWNRANGE ERROR (V)A CROSSRANGE ERROR 1W)

20

15

5-

14001 L00 I 1600 1700 1800 t t 1900 It 2000

TTACAN TaA rtSTrVo w TRTELAPSED TIME (sec

Figure 5.2-2 System E Overall Performance: Velocity

5-16


systems studied in Ref. 1. It suggests that a filter covariance matrix

propagated through the orbital and drag-update phases of the mission gen-

erates correlations between error sources which are valuable in proces-

sing measurements during the early post-blackout portion of entry.

The System E navigation errors decrease steadily down to the

acquisition of the azimuth and elevation segments of MLS at TMLSposition errors for the final portion of entry are plotted on an enlarged

scale in Fig. 5.2-3. At TMLS, there is a large decrease in all three

R-14077600

0 VERTICAL ERROR (U)o DOWNRANGE ERROR (V)A CROSSRANGE ERROR (W)

500

400 -

0

uj 300

0

200

100-

t1800 TMS TDME 1850 Tsw 1900 T T 1950

ELAPSED TIME (sec)

Figure 5.2-3 System E Overall PerformanceDuring Final Approach: Position

5-17


.position error components, most notably in vertical position. The de-crease in both downrange and crossrange errors is misleading. Theazimuth and elevation segments of MLS do not decrease the range-to-goerror, but at TMLS, the Space Shuttle heading angle (Fig. 2.1-2) is suchthat both the crossrange and the downrange position errors contain atransverse component (with respect to the runway) which the azimuthmeasurement reduces. As the Space Shuttle completes its turning man--euver, the range-to-go error is transferred from the crossrange to thedownrange channel -- thus accounting for the growth of downrange posi-tion error prior to TDME. Following the MLS DME acquisition, the range-to-go error is reduced and the position navigation errors become essen-tially the navigation accuracy of MLS.

The growth of the crossrange and downrange velocity errorsfollowing the initiation of the final approach and landing sequence at Tindicates a major difficulty with the System E filter. At TW, the navi-gation filter must switch to a suboptimal operation mode. The System Efilter computes gains after TSW based upon a 6 x 6 submatrix of Pwhich preserves the correlations between position and velocity errors.Figure 5.2-2 indicates that this is not sufficient; it is also necessary topreserve the correlations between the MLS errors and the navigation er-rors. The alternative System E filters summarized in Table 4.2-3 at-tempt to account for these correlations; their performance is summarizedin Section 5.2.3.

Figure 5.2-2 indicates that the crossrange and downrangeposition errors increase after the radar altimeter is switched on at TRA'The increase is not due to the radar altimeter gains, but rather to thefact that MLS is simultaneously switched off. Thus after TRA, the navi-gation system has no heading reference and no range reference. The

5-18

THIE ANALYTIC SCIENCES CORPORATION

crossrange and downrange position errors would not increase if MLSremained on until touchdown.

5.2.2 Detailed Error Budgets

System error covariances have been calculated for eachgroup of error sources over the reference mission 3B trajectory from130,000 ft (t = 1432 sec) to touchdown (t = 1946.5 sec). Detailed errorbudgets showing rms position and velocity errors were generated at threesignificant time points. Each entry in the error budgets is the rms con-tribution of the error source or sources indicated in the left-hand column;it comes from a computer run in which those errors alone are set at truth-model values -- and all others are -zero. The overall System E perform-ance at the indicated time is given at the bottom of each table and comparedwith the Space Shuttle landing navigation specification given in Ref. 22.*The filter-indicated performance is also presented for the error budgetat 12,000 ft; the System E filter does not compute a performance measurebelow 12,000 ft.

Particular numbers have been circled in the error budgetsto focus attention on major contributors. The rule followed in decidingwhich numbers to circle is the same rule employed in the drag-updatephase analysis. In each column, every number whose magnitude is greaterthan 20% of the RMS total for that column is circled. Thus, every circlednumber contributes at least 2% to the RMS error for the appropriate col-umn.

These landing specifications are guidelines, not inviolable require-ments. The vertical velocity specification, in particular, may ulti-mately be relaxed.

5-19


The initialization of the final approach and landing sequence

occurs nominally at 12,000 ft, t = TSW. At this time, the navigation

filter is required to switch to a "suboptimal" operating model. Table,5.2-1 presents a detailed error budget in order to evaluate the System Eperformance prior to the configuration change.

The measurement schedule outlined in Table 2.2-1 indicatesthat at TSW the navigation filter will have processed 38 sec of MLS azi-muth and elevation measurements and 28 sec of MLS DME measure-

ments. A general observation from Table 5.2-1 (and from Fig. 5.2-3)is that this is a sufficient length of time to permit the filter to improveits navigation accuracy to a desirable level. The position errors at TSWare almost entirely a function of MLS accuracy -- the major contributorsare the biases and second-order markovs. If the filter continued tooperate in an "optimal"mode, the vertical and crossrange position er-rors would decrease nearly linearly as the range to the elevation andazimuth antennae, respectively, decreased; 'and it is reasonable to sus-

pect that, if necessary, MLS could satisfy the touchdown specificationsin position without requiring a radar altimeter. The velocity errors atTW indicate a continuing effect of error sources other than MLS --IMU-related sources and TACAN -- but the downrange and cross-

range velocity errors already satisfy the touchdown specifications.

The detailed error budget at the runway threshold -- 3 sec-onds from touchdown -- is presented in Table 5.2-2. The error budgetwas taken prior to processing the first radar altimeter measurement.The filter configuration used between 12,000 ft and the runway thresh-old is the suboptimal System E filter (see Table 4.2-3). In addition tousing a 6 x 6 covariance matrix to compute MLS gains, the suboptimalfilter uses the MLS bias estimates and IMU misalignment estimates

5-20


TABLE 5.2-1SYSTEM E BASELINE ERROR BUDGET -- 12,000 ft

RMS NAVIGATION ERROR AT 12,000 ftERROR SOURCE VALUE POSITION (f) VELOCITY (ft /sec)V D CR V DR CRI. Uncorrelated Noise

TACAN Range 100 ft 0.3 0.8 0.4 0.03 0.06 0.05TACAN Bearing 6 mrad 0.4 1.0 0.5 0.05 0.05 0.08MLS Azimuth 0.1 mrad 0.4 1.4 3.9 0.01 0.04 0.13MIS Elevation 0. 1 mrad 2 9 1.1 0.2 0.03 0.14MS DME 17 ft (Z 1.5 a 0.05Baro Altimeter 5 ft 0.0 0.0 0.0 0.00 0.00 0.00Quantization Error 0.0328 ft/sec 0.1 0.1 0.1 0.03 0.03 0.03

II. IMU-Related States

2. Accelerometer Biases 50og 0.4 0.4 0.4 0.09 0.143. Accelerometer Scale Factor 40 ppm 0.2 0.2 0.1 0.02 0.04 0.094. Accelerometer Misalign- 15 sec 0.5 0.5 0.4 0.08 0.20ments

5. Accelerometer Nonlin- 3 .5pg/g 2 0.0 0.0 0.0 0.00 0.00 0.00earities

6. Gravity Anomalies and 67 mgal-Vertical Deflections 8 sec 0.2 0.2 0.i 0.04 0.03 0.05

7. Gyro Bias Drifts 0.015/hr 0.4 1'.2 1.58. Gyro Mass Unbalances 0.025A/hr/g 0.4 0.6 0.59. Gyro Anisoelasticitiles 0.025'/hr/g 2 0.0 0.1 0.1 0.02 0.09 0.04

mI. External Aid-Related Sources10. TACAN Range Bias Error 385 ft 2.4 6.9 2.311. TACAN Range Scale Factor 100 ppm 0.0 0.1 0.0 0.00 0.02 0.0012. Baro Altimeter Errors

Bias 100 ftScale Factor 3% 4Static Defect 1.52 x 10- 0.3 0.1 0.1 0.05 0.05 0.08

ft/ft2 /sec 2

First Order Markov 20 ft13. MIS Time-Varying Biases

AzImuth 0.4 mrad 0.0 2.8. < 0.02 0.07 0.15Elevation 0.4 mrad 3.3 0.3 (Q 0.07 0.03Range Bias '8 ft 2.6 7.3 1.0 0.00 0.01 0.03Range Scale Factor 4 00 ppm [ (Q 2.5 0.07 0.14 0.0814. MLS Second-Order Markov

Azimuth 0.2 mrad 0.8 2.5 : 0.02 0.08Eleation 0.2 nrad 1.9 0.4 (00.05 0.03Range 17 ft 3.5 1.2 . 0.04

16. TACAN Survey Errors l ft 0.0 0.0 0.0 0.00 0.00 0.0017. MLS Survey Errors 1 f 0.8 0.4 1.0 0.00 0.01 0.0019. TACAN Bearing Bias 6 mrad 1.0 3.2 2.1 0.11 0.1720. MLS Timing Bias Errors

Azimuth 50 msec 0.1 0.7 2.5 0.00 0.01Elevation 50 msec 3.9 0.2 0.2 0.01 0.02 0.00Range 50 msec <Z3> Q 3.5 0.06 0.18 0.141., 21.-23. Drag-Related Errors 1.8 1.6. 0Z.6 0.04 0.20Total Projected Performance 20.4 37.3 21.4 0.53 1.09 0.96Filter-indicated Performance 18.5 35.6 19.2 0.40 0.80 0.46Touchdown Specifications 3.0 80.0 4.7 0.20 3.00 2.00

5-21


TABLE 5.2-2

SYSTEM E BASELINE ERROR BUDGET -- RUNWAY THRESHOLD.

RMS NAVIGATION EIRItOR AT RUNWAY THIESIlOIDERROR SOURCE VALUE POSITION (ft) VELOCITY (ft/sec)

V 1)11 CR V DRi CRI. Uncorrelated Noise

TACAN Range .100 ft 0.0 0.9 0.2 0.09 0.27 0.33TACAN Bearing 6 mrad 0.0 1.2 0.3 0.14 0.19 0.61MLS Azimuth 0.1 mrad 0.0 1.4 0.03 0.05 0.16MLS Elevation 0.1 mrad 0.2 0.9 .0.1 0.18 0.06 0.13MLS DME 17 t ( t. 0.2 0.20 . 0.09 0.18

SBaroe Altimeter 5 ft 0.0 0.0 0.0 0.00 0.00 0.00Quantization Error 0.0328 ft/sec 0.0 0.2 0.1 0.09 0.09 0.16

II. IMU-Related States

2. Accelerometer Biases 50pg 0.1 1.1 0.7 0.503. Accelerometer Scale Factor 40 ppm 0.1 0.2 0.2 0.16 0.24 0.584. Accelerometer Misalignments 15 sec 0.1 1.2 0.6

5. Accelerometer Nonlinearities 3.5pg/g 2 0.0 0.0 0.0 0.00 0.00 0.006. Gravity Anomalies and Verti- 67 mgal-

cal Deflections 8 sec 0.1 0.4 0.1 0.21 0.14 0.33cal Deflections 8 sec

7. Gyro Bias Drlits 0.0150/hr () 3.4 6 4 )8. Gyro Mass Unbalances 0.0250/hr/g 0.1 1.4 0.79. Gyro Antsoelasticities 0.025°/hr/g 2 0.0 0.4 0.1 0.12 0.42 0.30

III. External Aid-Related Sources

10. TACAN Range Bias 385ftError 385 0.1 0.8 (0 O 6

11. TACAN Range ScaleFactor 100 ppm 0.0 0.3 0.0 0.02 0.11 0.04

12. Baro Altimeter Errors

Bias 100 ftScale Factor 3%Static Defect 1.52 x 104 0.0 0.2 0.2 0.06 0.35 0.51

ft/ft2 /sec 2

First Order Markov 20 ft

13. MLS Time-Varying Biases

Azimuth . 0.4 mrad 0.0 0.2 ( 0.02 0.02 0.10.Elevation ,0.4 mrad 0 2 0.0 0.18 0.03 0.12Range Bias 8 ft 0.1 0.1 0.02 0.01 0.04Range Scale Factor 400 ppm 0.1 5.1 0.0 0.07 0.02 0.15

14. MLS Second-Order Markov

Azimuth 0.2 mrad 0.1 2.8 0.05 . 0.09 0.33Elevation 0.2 mrad <O 0.3 c 0.10 0.24Range. . 17ft EJ (1.2 0.2 0.20 0.13 0.18

16. TACAN Survey Errors 1 ft 0.0 0.0 0.0 0.00 0.00 0.0017. MLS Survey Errors 1 t 1.2 1.0 0.01 0.01 0.0019. TACAN Bearing Bias 6 mrad 0.1 4.0 0.8 4 0.4320. MLS Timing Bias Errors

Azimuth 50 msec 0.0 0.4 0.6 0.02 0.16 0.18Elevation 50 msec 0.2 0 6 0.1 0.01 0.01 0.13Range 50 msec 0.3 O 0.2 0.11 0.05 0.05

1., 21.-23. Drag-Related Errors 0.1 2.5 0.5 0.30Total Projected Performance 1.7 28.7 8.1 1.59 3.58 5.76Touchdown Specifications 3.0 80.0 4.7 0.20 3.00 2.00

5-22


saved from TSW to compensate the MLS measurements and to resolve

the accelerometer outputs into the proper coordinates.

The position errors at the runway threshold reflect the expec-

ted continuing improvement in vertical and crossrange errors as the Space

Shuttle approaches the MLS elevation and azimuth antennae -- the major

contributors to position errors continue to be the MLS biases and second-

order markovs. The vertical and downrange position errors satisfy the

touchdown specifications, but the crossrange position error is somewhat

larger than desired. The magnitude of the crossrange error is primarily

due to the placement of the MLS azimuth antenna 15, 000 ft downrange

from the nominal touchdown point. If the antenna were 10, 000 ft from

the touchdown point, however, the crossrange position error would be

close to the touchdown specification.

The disconcerting aspect of Table 5.2-2 is the significant

growth of the velocity errors relative to their magnitude at 12, 000 ft

(t = TSW). The major contributors remain the same as in Table 5.2-1,but the contributions due to the IMU-related error sources and TACAN

have increased considerably. In particular, the contribution of gyro

bias drift alone exceeds the touchdown specification for crossrange er-

rors. The relatively larger downrange and crossrange velocity error

contributions are a consequence of the filter's inability to estimate azi-

muth misalignment prior to TSW, but the growth of the velocity errors

is due to an inappropriate selection of filter velocity gains after TW.

The error budget at touchdown is presented in Table 5.2-3.

In order to emphasize vertical channel errors, the table considers only

three general error categories:

5-23


TABLE 5.2-3

SYSTEM E BASELINE ERROR BUDGET -- TOUCHDOWN

RMS NAVIGATION ERROR AT TOUCHDOWNERROR SOURCE VALUE POSITION (ft) VELOCITY (ft/sec)

V DR CR V DR CRI. Uncorrelated Noise

Radar Altimeter' '1 ft 0.4 0.0 0.0 - 0.00 0.00RSS for Other ExternalNavigation Aids 0.0 Q 2.5 0.06 0.37 0.77

II. IMU-Related States 0.0 0.07

III. External Aid-RelatedStates

15. Radar Altimeter StatesBias 1' ft ( 0.0 0.0 0.00 0.00 0.00First-Order Markov 1 ft 0 0.0 0.0 0.00 0.00

10.-14, 16.-23. RMS forOther External Navigation 0.0 270.04Aids

Total Projected Performance 1.5 31.5 21.4 0.61 3.73 5.92'Touchdown Specifications ' 3.0 80.0 47 0.20 3003.0 80.0 4.7 0.20 3.00 2.00"

* radar-related error sources* IMU-related error sources* other external aid-related error sources

The radar altimeter is switched on at the runway threshold and MLS isswitched off. Thus for the three seconds remaining until touchdown,the principal activity in the downrange and crossrange channels is thepropagation of the velocity errors into the position errors. The princi-pal contributors to the downrange and crossrange channel errors are thesame as at the runway threshold.

5-24


The only significant contributors to the vertical channel er-

rors are the radar altimeter-related error sources. The measurement

noise contributes to both the position and the velocity error; the bias con-

tributes to the position error only. The contribution of the first order

markov alone exceeds the vertical velocity touchdown specification. Any

of the data values in the truth model data base (Table 3.2-2) may change

as hardware selections are made, but the radar altimeter error models

are particularly likely to be revised. It should be noted, however, that the

error model for the radar altimeter reflects instrument errors and ter-

rain-dependent errors so that an accurate instrument would not automati-

cally yield small vertical channel errors. Sensitivity curves relating the

vertical channel errors to the radar altimeter error models are provided

in Section 5.2.3.

The System E total projected performance is out-of-spec in

all velocity components and in crossrange position. The crossrange

position error can be reduced significantly by using MLS down to touch-

down; the vertical velocity specification conceivably could be relaxed

sufficiently enough for the System E performance to be in-spec. The

principal question mark is the acceptability of the large downrange and

crossrange velocity errors. The difficulty appears to be software-

rather than hardware-related. Section 5.2.4 considers filter modifica-

tions for improving the velocity performance.

5.2.3 Sensitivity Curves: System E

This section contains several curves illustrating the sensitivity

of System E performance to variations in error source statistics. The

curves for vertical channel errors were produced using the error budget

data at touchdown taken from Table 5.2-3; the curves for downrange and

5-25


crossrange channel errors were produced using the error budget data atthe runway threshold taken from Table 5.2-2. The method of generatingthe data for these curves is discussed in Section 5. 1.3.

Similar sensitivity curves may be constructed for any errorsource group for which error budget data is available in Tables 5.1-2 and5.1-3. All of the error curves in this section correspond to major con-tributors to system performance. Sensitivity curves corresponding to er-ror sources which produce minor contributions are quite flat.

The vertical position and velocity error sensitivity to radaraltimeter markov error is shown in Fig. 5.2-4. The relevant touchdownspecifications are included in each of the graphs. Figure 5. 2-4a is identi-cal to the vertical position sensitivity to radar altimeter bias. It is appar-ent from Fig. 5.2-4b that the 0.2 ft/sec vertical velocity specificationcannot be met even if the markov error is zero.

The sensitivity of the crossrange position error at the runwaythreshold to MLS azimuth measurement bias is shown in Fig. 5.2-5. Thefigure indicates that the touchdown specification could be met at the run-way threshold only if the bias were reduced to less than 0.2 mrad. Analternative not considered in Fig. 5.2-5 would be to move the azimuthantenna closer to the nominal touchdown point.

The principal contributors to downrange position errors at therunway threshold are the MLS DME second-order markov and the MLSDME timing bias error. Sensitivity curves for these two error sourcesare presented in Figs. 5.2-6 and 5.2-7, respectively. It does not appearlikely that either error source could become large enough to violate thetouchdown specification.

The downrange and crossrange velocity errors at the runwaythreshold are due primarily to the IMU-related states. Figures 5.2-8

5-26


R .R-14188

O TOUCHDOWN SPECIFICATION

BASELINE

BA 18

U oo- /

0 VALUE

-TOUCHDOWN SPECIFICATION

o /

0 0.5 1 1.5 2 2.5RADAR ALTIMETER MARKOV ERROR (It)

b) Velocit

Figure 5.2-4 Sensitivity of Vertical Position and Velocity Errors

5-275-27


R-1419420

0

uJ 15

O "Z

0

U. 0BASELINE

n • TOUCHDOWN SPECIFICATIONo 5

0 0.2 0.4 Q6 08 0.10MLS AZIMUTH BIAS (mrod)

Figure 5.2-5 Sensitivity of Crossrange Position Error atRunway Threshold to MLS Azimuth Bias

R-14186100

STOUCHDOWN SPECIFICATIONr 80

O0P 601 60 7

5-28

z< 40-

MLS DME BIAS Ut)

Figure 5.2-6 Sensitivity of Downrange Position Error atRunway Threshold to MLS DME Bias

5-28


100 R-14187

80 TOUCHDOWN SPECIFICATION

0

S60

060

0

40 BASELINEW VALUEz0

20 - /

0 s50 100 150 200 250MLS DME TIMING BIAS ERROR (msec)

Figure 5.2-7 Sensitivity of Downrange Position Error atRunway Threshold to MLS DME TimingBias Error

R-1419610

-oil px "

,U 4

TOUCHDOWN SPECIFICATION

Z 2-

0 01 0.02 0.03 0.04 0.05GYRO BIAS DRIFT (deg/hr)

Figure 5.2-8 Sensitivity of Downrange Velocity Error atRunway Threshold to Gyro Bias Drift Error

5-29"/


and 5.2-9 present the velocity error sensitivity curves for gyro bias

drift errors. The curves indicate that the velocity errors may become

quite large if the gyro bias drift error exceeds the baseline value of

0.015 deg/hr by a significant amount. Similar curves can be drawn for

accelerometer bias errors, accelerometer misalignments, and gyro mass

unbalances; except that the curves will be somewhat flatter.

R-1419512

S//-- /o -

o BASELINE

di 6- VALUE

LYO BI PAG IS/POoR //

-

Figure 5.2-9 Sensitivity of Crossrange Velocity Error atRunway Threshold to Gyro Bias Drift Error

5.2.4 Performance Sensitivity to FilterModifications: System E

The overall performance curves in Section 5.2.1 indicate that,

after switching to a suboptimal operation mode at T , the System E fil-

ter cannot further improve the velocity estimates. instead, the down-

range and crossrange errors iricrease significantly during the final

5-30


Sapproach and landing sequence. The study summarized in this section

was undertaken to determine whether this velocity divergence is a general

property of all suboptimal Space Shuttle navigation filters, or whether fil-

ter configurations which yield a significant improvement in performance

can be devised.

The particular filter configurations analyzed are the System E

filter and the alternative System E filters as defined in Table 4.2-3. The

three filters are identical 15-state Kalman filters doirn to initiation ofthe final approach and landing sequence at TS After T they are

6-state (position and velocity) suboptimal filters. The principal differ-

ences after TSW are the MLS gain computation algorithms:

* System E filter gains are computed fromKalman algorithm with6 x 6 covariance savedfrom TS.

* Alternative 1 gains are computed fromKalman algorithm with 15 x 15covariance saved from TS(only six elements of gainvector are used).

* Alternative 2 - gains are the optimal gainscomputed at TW (only sixelements of gain vector areused; azimuth and elevationgains scaled by range-to-go)

The sensitivity analysis performed in this section is refer-

enced to the System E filter and to the error budget at the runway thresh-

old for that filter (Table 3.2-2). Gain files were generated for each of thealternative filters using a modified filter covariance program (see Sec-

tion 4.1). The gain file for each alternative System E filter was used to

determine the contribution of a :single error source to the total projected

performance for that filter. The error source selected was gyro bias

5-31


* drift (Group 7), which is the dominant contributor to the System E filter

downrange and crossrange velocity errors. By comparing the effects

of gyro bias drift on the performance of the three filters, it is then pos-sible to make a general estimate of the performance of the alternative

filters.

The navigation errors at the runway threshold due to gyro biasdrift are presented in Table 5.2-4 for the System E filter and for the twoalternative filters. The position errors for all three filters are com-

parable; the important differences are in the velocity errors. These aresummarized below.

TABLE 5.2-4

NAVIGATION ERRORS AT RUNWAY THRESHOLDDUE TO GYRO BIAS DRIFT ERRORS

POSITION ERRORS VELOCITY ERRORSFILTER (ft) (ft/sec)

V DR CR V DR CR

System E Filter 0.3 3.4 1.8 1.06 2.44 4.15

Alternative 1 0.6 3.5 1.3 1.62 0.59 2.90

Alternative 2 0.3 1.2 1.8 0.27 0.73 0.81

The downrange and crossrange velocity errors for Alternative 1represent a significant improvement over the performance of the System Efilter, but the vertical velocity performance is worse. It is reasonable tosuspect that Alternative 1 would yield similar changes in the velocity er-rors due to the other major IMU-related error sources. If this occurred,the total projected performance at the runway threshold for Alternative 1would satisfy the touchdown specifications in downrange velocity, but would

5-32


still exceed the specifications in crossrange velocity and vertical velo-

city. Since vertical velocity information is provided by the radar alti-

meter once the runway threshold has been passed, the decrease in

vertical velocity. accuracy for Alternative 1 may not be important. Thus,

it appears that Alternative 1 is an improvement over the System E fil-

ter.

The more interesting aspect of Table 5.2-4 is the performance

of Alternative 2. Alternative 2 yields a factor of four improvement in the

accuracy of all three velocity components. If similar improvements were

realized in the velocity errors due to the other major IMU-related error

sources, the total projected performance at the runway threshold for Al-

ternative 2 would satisfy all touchdown. specifications except in vertical

velocity -- and the vertical velocity estimates would be better than those

provided by either of the other filters. Thus, from a navigation accuracy

vantage point, Alternative 2 appears to be superior to both Alternative 1

and to the System E filter.

It is not obvious why Alternative 2 appears to perform better

than Alternative 1, but it is clear that both should perform better than

the System E filter. At TSW, the velocity and position errors are highly

correlated with the MLS biases. The 15 x 15 covariance at T reflects

this correlation and the filter gains for the state estimate update at T

are selected accordingly. Thus, after TSW both Alternative 1 (saved

15 x 15 covariance) and Alternative 2 (saved gains) account for MLS

biases. The System E filter, however, saves only a 6 x 6 covariance

and therefore it cannot account for the correlation between the navigation

errors and the MLS biases. The net result is that the System E filter

makes a poor selection of velocity gains.

5-33


It should be emphasized that the sensitivity analysis presented

in this section is based upon navigation errors from a single error source.

The analysis indicates that the alternative filters do not experience the

velocity estimation difficulties which plagued the System E filter, but

this indication must be verified through a detailed error budget before a

final performance determination can be made.

5.3 DISCUSSION

Sections 5.1 and 5.2 presented a description of the navi-

gation filter performance during the drag-update phase and the ap-

proach and landing phase, respectively. The results are summarized

in this section. A review is made of the dominant error sources for the

two mission phases and of their effect upon the filter performance. The

manner in which the various filter states contribute to the filter per-

formance is then discussed and possible software modifications to im-

prove the performance are presented.

5.3.1 Drag-Update Phase

The overall performance of the drag-update filter was evalu-

ated in Section 5. 1. 1 via comparison with the performance of a pure

inertial navigator. The analysis indicated that the drag-update filter,as presently designed, yields navigation errors approximately two-

thirds as great as those for the pure inertial navigator. This result

verifies that the drag acceleration pseudo-measurement is a potentially

5-34


valuable navigation aid and that the drag-update filter effectively limits

the growth of navigation errors during entry. However, a detailed study

of the overall performance indicates that a significant improvement in

performance may be obtainable through filter design changes. Formu-

lation of the most promising design changes is a consequence of the analy-

sis to follow.

An error budget provides a detailed breakdown of the contri-

butions of individual error sources to the total navigation error at a

particular time -- the error budget presented in Section 5.1.2 corre-

sponds to the end of the drag-update phase. The major contributors to

the navigation error at this time-point are:

* accelerometer scale factor error (Group 3)

* gyro bias drift (Group 7)

* non-standard atmospheric density modelingerrors (Group 21)

Analysis of the individual error contribution time histories given in

Appendix F reveals that the IMU-related contributions are primarily

due to large initial errors at 400,000 ft -- velocity errors for Group 3

and misalignments for Group 7. The atmospheric density contributions

are primarily due to errors in modeling the 1962 Standard Atmosphere

and to a season- and latitude-dependent bias.

The time histories in Appendix F also provide a second

important piece of information: The relative importance of the major

error contributors is time-dependent. The IMU-related error sources

(Groups la through 9) are the only contributors to navigation errors

prior to the first drag acceleration pseudo-measurement (t = TDRAG).

In the several minutes following TDRAG, the drag-related error sources--

5-35 /


primarily Group 21 -- are the principal contributors to vertical posi-tion errors. During the latter portion of the drag-update phase, theIMU-related error Sources again become dominant in all error com-ponents. This exchange of dominance between IMU-related and drag-related error sources is not necessarily undesirable -- it could be afunction of changes in the trajectory geometry, acceleration profiles,etc. -- however, to the extent that it is due to a poor selection of filter.gains, it should be eliminated.

The analysis presented in Section 5.1.1 indicates that thedrag-update filter overweights the drag pseudo-measurements for thefirst several minutes after TDRAG' The evidence to support this ob-servation is that:

* the vertical position error increases significantlyafter the first drag-update at TDRAGDRAG* drag-related error sources become the principalcontributors to vertical position error for thefirst several minutes after TDRAG"

On the other hand, there are three related indications that the drag-update filter underweights the drag pseudo-measurements during thelast portion of the drag-update phase:

Sthe accuracy of a single drag pseudo-measurementincreases as altitude decreases, i.e., the rmsmagnitude of the drag-related error sources de-creases is altitude decreases (see Figs.: 3. 1-1through 3.1-3)

* the filter-indicated performance at TDM E isoverly optimisticSthe relative error contribution of IMU-related

error sources increases over the final minutesprior to TDME.

DM5-36

5-36


On the basis of these observations, it seems desirable to modify thefilter gain sequence such that the gains in the first portion of the drag-

update phase are decreased, and the gains in the latter portion are in-

creased.

The drag-update filter is a 13-state Kalman filter. The firstsix states are navigation states (position and velocity); the last sevenstates are error states (platform misaligments (3), acceleration errors(3), drag-correlated error (1)). The error states are generalized stateswhich are used to describe the type of error sources to be encountered

and to prevent the filter covariance from becoming unrealistically small,i.e., to prevent the filter from generating overly optimistic perform-ance predictions. As an example, drag-related error sources are dom-inant contributors to navigation errors during the drag -update phase.

The drag-update filter uses the drag-correlated error state to estimatethe net effect of these error sources on the drag acceleration pseudo-measurement. If the estimate is accurate enough, the filter uses it todiminish the effect of the drag-related error sources on the positionand velocity estimates. Even if the estimate is not accurate, however,the presence of the error state prevents the filter from assuming thepseudo-measurement is more accurate than it actually is.

The most straightforward mechanism for modifying the dragpseudo-measurement gains is to change the parameters associated withthe drag-correlated error state. Increasing the variance of the drag-correlated state has the effect of decreasing the drag pseudo-measurementgains. The gain modifications suggested in the preceding paragraphs cantherefore be effected by assigning an altitude dependence to the varianceof the white noise associated with the drag-correlated state, e.g., thevariance could be 40% of drag at high altitudes and 20% of drag at low

5-371


altitudes. Equivalently, the variance could be maintained at 30%0 for

all altitudes, but the filter measurement matrix [Eq. (4.2-4)] could

be modified to

- T . T

HF(t)= pred(t) h VR(t) 0 0 ' a(h) (5.3-1)

where a(h) is a decreasing function of altitude. It is possible that

neither of these modifications will bring the filter-indicated performance

at TDM E into accord with the total projected performance. In this case,

it may also be necessary to increase the process noise for the filter.

A second possibility for improving the performance of the

drag-update filter is to improve the accuracy of the onboard atmospheric

density model. The improvement could involve either a more accurate

model of the 1962 Standard Atmosphere, i.e., the four-term model, or

the selection of a season- and latitude-dependent model to minimize the

importance of the time-varying bias. The Alternative Contributions

Table (Table 5.1-3) indicates that the four-term model for the 1962

Standard Atmosphere is not significantly better than the one-term

model; therefore, inclusion of a season- and latitude-dependence in the

onboard density model is recommended. The sensitivity analysis in

Section 5.1.3 indicates that a significant improvement in navigation ac-

curacy cannot be achieved by improving the filter onboard density model

unless accompanying modifications in the filter gains are made. If the

error in the drag pseudo-measurement is reduced, the gains must be

increased to take advantage of the improvement, etc. The previous par-

agraph outlines several mechanisms for increasing the gains.

The performance of the drag-update filter is a function not

only of the filter gains, but also of the selection of a measurement

5-38

THE ANALYTIC SCIENCES CORP RATION

matrix HF(t) for the drag acceleration pseudo-measurement. Compari-son of HF(t) as defined in Eq. (4.2-4) or Eq. (5. 3-1) with the first 13components of HS(t) as defined in Eq. (C-42)

velocityposition errors errors

T I I I+ L - ), 2_i L ,0 i ,Hs(t)= q(t) h + t -) 0 1

(5. 3-2)reveals that there are several terms which appear in HS(t) but do not ap-pear in HF(t). These terms represent dynamic relationships which af-fect the pseudo-measurement, but which were not included in the filtermodel. The additional terms in the position components are due tominor dynamic effects and can be ignored; however, the two terms inthe velocity components are of the same order of magnitude. The2i/ R(t) term relates the velocity estimation error to the errorin computing the predicted drag; the FL/q(t) term relates the velocityestimation error to the error in computing the drag component of theaccelerometer outputs. If one of these terms is included in HF(t), theother should be also.

The most important terms in HF are the position componentterms; the -i T/hsc term reflects the basic sensitivity of the pseudo-measurement to vertical position errors. A tradeoff study was under-taken to establish the importance of the velocity component terms. Thisstudy considered three different definitions of HF(t):

* HF(t) as defined in Eq. (4.2-4).* HF(t) as defined in Eq. (4.2-4), but with the-FL/q(t) velocity term added.* H (t) as defined in Eq. (4.2-4), but with novelocity terms.

5-39


The filter covariance program was used to generate a gain file corre-sponding to each definition of HF(t), and navigation errors due to at-mospheric density modeling errors (Group 21) were computed for eachgain file. There was no discernible difference between the performanceof the three different filters. This suggests that the velocity componentterms of HF(t) are unimportant and could be omitted.

A last comment should be made concerning the definitions ofthe error states used in the drag-update filter. If the drag-update filteruses an error state to estimate a correlated error source, or if it re-quires an error state to develop the proper correlations between naviga-tion error components, the presence of that state -is important forimproving the filter performance. If the filter uses the state only toprevent itself from becoming overly optimistic in its performance pre-dictions, however, the function of that state might be performed adequatelyby an appropriately defined process noise.* The drag acceleration pseudo-measurement is of sufficiently poor quality that the only error state usedby the drag-update filter to estimate correlated error sources is the dragcorrelated error state. Thus, if only performance during the drag-updatephase is considered, it is possible that a 7-state drag-update filter couldbe used. The possibility exists, however, that the misalignment and ac-celeration error states establish correlations in the filter covariancematrix during the drag-update phase which are important in the earlyportion of the approach and landing (System E) phase. This possibilityis discussed further in the following section.

The advantage of using process noise in lieu of an error state is a re-duction in the computational requirements for the filter.

5-40


5.3.2 System E

The overall performance of System E was evaluated in Section

5.2.1. The analysis indicated that, during the initial portion of the ap-

proach and landing, the System E filter effectively uses the drag accel-

eration pseudo-measurement, TACAN, the baro altimeter, and MLS to

improve navigation accuracy. During the final approach and landing se-

quence, however, the velocity estimates become degraded to the extent

that the System E filter violates the touchdown accuracy specification in

all three velocity components. The degradation appears to be a function

of the particular filter configuration studied, rather than an intrinsic lim-

itation of the TACAN-MLS-radar altimeter landing system. Suggested

modifications for improving the filter performance are a consequence of

the analysis to follow.

Section 5.2.2 provides detailed error budgets for the System E

filter at three significant time points:

* initiation of the final approach and landingsequence (t = TSW; Table 5.2-1)

* the runway threshold (t = TRA; Table 5.2-2)

* touchdown ( t = TD; Table 5.2-3)

From the end of blackout (t = TTACAN ) until TW, the System E filter

is a 15-state Kalman filter. Thus, the error budget at TSW is represen-

tative of the maximum potential navigation accuracy of a TACAN-MLS

landing system at that point. After TS, the System E filter is a 6-state

suboptimal filter and this suboptimality contributes significantly to the

navigation errors. The important difference between the last two error

budgets is in the vertical channel errors. The error budget at the run-

way threshold was made immediately prior to the first radar altimeter

measurement. Comparison of the two error budgets permits an evalua.-

tion of the radar altimeter performance to be made.

5-41 /


The total projected navigation error at TSW reveals that

the System E filter has already reduced the downrange and cross-

range velocity errors to within the touchdown specifications. The fil-

ter also provides good vertical channel estimates, but this is not

particularly important since the vertical channel errors at touchdown

are expected to be dominated by the radar altimeter error sources.

There are indications that, if the filter continued to operate in an "opti-

mal" mode, the TACAN-MLS landing system could satisfy the touchdown

specifications in all components except vertical velocity.

The "suboptimal" System E filter imp:roves the position esti-

mates during the final approach and landing sequence. At the runway

threshold, both the vertical and downrange position errors satisfy the

touchdown specifications, but the crossrange position error is some-

what larger than desired. The magnitude of the crossrange error is

primarily due to the placement of the MLS azimuth antenna 15, 000 ft

downrange from the nominal touchdown point. If tihe antenna were

10,000 ft from the touchdown point, the crossrange position at the run-

way threshold would be close to the touchdown specification.

The effect of the switch to a suboptimal filter is evident in

the significant growth of the velocity errors during the final approach

and landing. At the runway threshold, all three velocity components

violate the touchdown specifications. If the IMU-related errors are

greater than the values assumed for the baseline error budget, the sen-

sitivity analysis in Section 5.2.3 implies that the velocity errors at the

runway threshold may be considerably greater than indicated in the

error budget.

The touchdown error budget reveals that the downrange and

crossrange position errors begin to increase once the runway threshold

5-42


is crossed. This increase occurs because the measurement schedule

given in Table 2.2-1 requires MLS to be switched off at the runway

threshold. If MLS were used through touchdown and roll-out, this in-

crease in downrange and crossrange position errors would not occur.

Comparison of the touchdown error budget with that at the run-

way threshold indicates that the radar altimeter reduces the vertical vel-

ocity error significantly, but that the error still violates the touchdown

specification. It should be noted that the radar altimeter error model

used to generate the error budget includes both instrument- and terrain

variation-related errors. If the hardware specifications require a more

accurate altimeter, and if the effect of terrain variations on the radiated

signal are shown to be small, the sensitivity analysis in Section 5.2.3 can

be used to estimate the vertical channel errors corresponding to the im-

proved error model; however, it is not apparent that the radar altimeter will

be accurate enough to meet the vertical velocity touchdown specification.

The major contributors to the total projected navigation errors

are the same for all three error budgets -- the only exception is that the

radar altimeter error sources dominate the vertical channel errors at

touchdown. Otherwise, position errors are dominated by the MLS cor-

related error sources. Some smoothing of the MLS second-order markov

errors is accomplished up to TSW , but essentially the System E

filter is "riding the MLS correlated errors," e.g., the downrange posi-

tion error in all three error budgets is approximately the rms value of

the MLS DME bias, scale factor, and second-order markov error magni-

tudes. The significant improvement in crossrange and vertical position

errors as the Shuttle nears the runway is a geometric effect -- the cross-

range position error due to an MLS azimuth bias is a linear function of

the range to the antenna, etc.

5-43


The velocity errors in the three error budgets indicate a con-

tinuing effect of error sources other than MLS. In addition to the MLS

correlated error sources and the MLS DME measurement noise, the

major error contributors include gyro bias drift, gyro mass unbalance,and TACAN bias errors. An important aspect of the error budgets

is that the relative importance of these error sources changes as the

Shuttle nears the runway. At TSW, the MLS-related and non-MLS-

related error sources are of approximately- equal importance. At the

runway threshold, however, the non-MLS-related error sources are

clearly dominant. As mentioned earlier, the dominant contributors to

vertical velocity errors at touchdown are the radar altimeter error

sources.

The sensitivity analysis undertaken in Section 5.2.4 confirmed

that the velocity estimate degradation during the final approach and land-

ing sequence is a software problem rather than a limitation of the hard-

ware accuracy. The System E filter does not properly account for strong

correlations between the navigation errors and the MLS biases. As a

consequence, it makes a poor selection of the velocity components of the

MLS gains. The sensitivity analysis indicates that either of the alterna-

tive filters analyzed in Section 5.2.4 would yield significantly better navi-

gation accuracy than the System E filter. Preliminary indications are

that the scaled-gain filter (Alternative 2) is the best of the three filters

studied, but this observation should be verified through a detailed per-

formance analysis.

The System E filter utilizes a total of 13 different error states --

platform misalignments (3), acceleration errors (3), and one bias state

associated with each of the external navigation aids except the radar alti-

meter (7). The filter is not able to generate accurate estimates of any of

5-44/


* the navaid biases; however, the filter does attempt to generate bias esti-mates. In addition, the filter develops correlations between the naviga-tion errors and the bias states which appear to be extremely importantin computing gains for the measurement updates. It is recommended thatall of these navigation aid bias states be retained in the System E filter.

The system E filter uses the three platform misalignmentstates to compensate for the effects of IMU-related -error sources. Thevertical error state, which corresponds to azimuth misalignment, isused less than the other two components, but all three misalignmenterror states appear to perform an important function for the System Efilter. In contrast, the three acceleration error states are not used bythe filter. Correlated acceleration error sources have less effect thanplatform misalignments on the overall navigation system performanceduring entry, and consequently the System E filter is less able to observeand estimate such error sources. It appears that the three accelerationerror states could be eliminated without degrading the System E perform-ance, but this possibility should be verified through simulation.

5-45


6. CONCLUSIONS

6.1 SUMMARY OF FINDINGS

A navigation filter for the entry navigation system of the Space

Shuttle Orbiter has been evaluated in detail. The baseline navigation

system assumed for the study is an aided-inertial system with external

data provided by a drag acceleration "pseudo-measurement," TACAN, a

baro altimeter, MLS, and a radar altimeter. Prior to the final approach,the navigation filter is a variable-dimension Kalman filter with between

12 and 15 states; during the final approach it is a 6-state suboptimal

filter. Comprehensive truth models representing all potentially signi-

ficant error sources have been formulated and used to generate detailed

error budgets and sensitivity curves. In addition, the effect of several

major filter modifications upon the overall navigation system perform-

ance has been analyzed. A detailed summary of the results is contained

in Section 5.3.

The major findings of the study are as follows:

* The drag acceleration pseudo-measurement is apotentially valuable navigation aid during radioblackout. It is capable of limiting the growth ofnavigation errors which would appear in a pureinertial system. In addition, it establishes cor-relations in the state estimates which are valuablefor post-blackout navigation.

* The navigation filter utilizes the drag accelerationpseudo-measurement effectively, but a furtherimprovement in performance seems possible.Filter modifications should be undertaken to de-crease the pseudo-measurement gains during

6-1


the first portion of radio blackout and increase themduring the last portion.

* Improvement of the onboard atmospheric densitymodel used for the drag acceleration pseudo-measurement could result in a significant improve-ment in navigation accuracy at the end of radioblackout. Specifically, the model should reflectseasonal and latitudinal density variations. Thefilter gain structure should also be modified toreflect the model improvement.

* The TACAN-MLS-radar altimeter landing systemappears capable of providing the desired navigationaccuracy at touchdown in all components except verti-cal velocity. The ability to meet the vertical velocityspecification is dependent upon the accuracy of theradar altimeter, the amount of radar altimeter dataavailable for processing, and the effect of terrainvariations on the radar altimeter signal.

* The navigation filter is not capable of meeting thetouchdown specifications for velocity. Prior to theinitiation of the final approach, the filter uses theexternal navigation aids effectively to improve navi-gation accuracy. During the final approach, however,the velocity estimates diverge.

* A study of alternate filter configurations during thefinal approach suggests that simple modifications ofthe navigation system can be made which would per-mit the touchdown accuracy specifications to be metin all position components and in both downrange andcrossrange velocity.

* MLS azimuth should be used through touchdown androllout to minimize the importance of crossrangevelocity estimation errors.

A number of comments and suggestions regarding the choice

of filter states and filter design parameters are given in Section 5.3.

The error states associated with the various external navigation aids

appear to be instrumental in permitting the filter to accommodate cor-

related errors in the measurements; however, some of the parameter

6-2/


values associated with these states should be modified. The misalign-

ment states are used by the filter in compensating for IMU-related error

sources, but the acceleration error states appear to be unnecessary.

These and other suggested filter modifications must be verified through

simulation.

6.2 RECOMMENDED FUTURE STUDIES

It isrecommended that, as the design of the Space Shuttle

navigation system is refined, similar detailed error budgets be gener-

ated to verify that the navigation system satisfies the mission require-

ments. The present study encompasses the entry phase of a Space

Shuttle mission from entry interface at 400,000 ft to touchdown. Sim-

ilar studies related to other Space Shuttle mission phases such as boost

and rendezvous are also desirable.

The study of alternative filter configurations presented in

Chapter 5 illustrates an important application of the error budget con-

cept in filter design efforts. Given a detailed error budget for a base-

line filter, the performance of alternative filters can be analyzed for

a limited set of the most important error sources. Once a "good"

design is approached, its performance can be verified using the full

set of error states included in the truth model. This design technique

can be used to increase the scope of future error budget analyses.

The System E analysis in Chapter 5 emphasized the sensi-

tivity of vertical channel navigation errors at touchdown to radar alti-

metry errors -- either instrument errors or the effect of terrain

variations upon the radiated signal. Accurate models for radar alti-

metry errors are not presently available. An effort should be made

6-3


to improve these models in order to obtain more accurate navigation

system performance estimates at touchdown.

6-4


APPENDIX A

COORDINATE FRAMES AND TRANSFORMATIONS

There are several coordinate frames whose relationship to each

other had to be defined as part of the development of the Space Shuttle

evaluation tools. These coordinate systems may be divided into three

distinct groups:

* Systems which remain fixed in inertial space

* Systems which are fixed to and rotate with theearth

* Systems which are defined by the vehicle'sposition and velocity vectors.

It is the purpose of this appendix to define all of the necessary coordinate

systems and their orientation with respect to each other.

The inertially fixed coordinate system group is composed of

two reference frames: the I and the P frames. The I coordinate frame,*

with unit vectors 1xi' 1 , and 1 , is the system in which the naviga-

tion error analysis is performed and is also the system in which the

Space Shuttle trajectory data was furnished. The x axis is along the

intersection of the ecliptic and the mean earth equatorial plane (the

equinox) at the beginning of Besselian Year 1950. The zI axis is along

the mean earth rotation axis of the same date.

* The I coordinate frame is referred to in Ref. 12 as the Basic Referencecoordinate frame.

A-1


The second inertially fixed coordinate system is that of the

nominal platform axes, the P coordinate frame. This frame has unit

vectors 1 , andp 3 (see Fig. E-2). The transformation fromP- 1 2 =P 3

I to P is taken from Ref. 12.

-0.70098874 0.39396469 0.59448010

T / = 0.71294726 0.40805269 0.57026239 (A-1)

-0.01791596 0.82358049 -0.56691639

The two earth-fixed coordinate frames are given the initials G

and R. The G frame (1x , 1 ) is an earth-centered reference sys-G -YG -z

tem with its z axis along the true rotational axis of the earth. The xG

and yG axes lie in the equatorial plane with xG passing through the

Greenwich Meridian. The transformation from the I frame to the G frame

is

Cos 0Et sin l t 0 II0.9209857522 0.3895881588 -0.0025121443

TG = -sin Et cos 0Et 0 -0.3895865546 0.9209891768 0.0011192091

0 0 1 0.0027496883 -0,0000520780 0.9999962182

(A-2)

where OE is the earth's spin rate and t is elapsed time (t =0 at

400,000 ft).

The R system (Up, Downrange, Crossrange) is located with

respect to the landing sight and runway and is the system in which navi-

gation aid locations and survey errors are defined. The system has

axes R, D, and C. with R assumed parallel to the local vertical at the

landing sight. The D axis is pointed down the runway (approximately

southeast; see Fig. 2.1-2) and C is normal to R and D. The transforma-

tion from the I to the R system is

A-2


0 0 1 -sin I 0

TR = sin 8 cos O 0 -sin PT cos T -sin A sin XT cos q TG/

-cos e sin 8 0 cos PT cos XT cos o sin XT sin AT

(A-3)

where

0 = 136.00 rotation of runway east of north

XT = 239. 43470 longitude of the touchdown point

PT = 34.72190 latitude of the touchdown point

There are three vehicle position and velocity related coordinate

systems used in the Space Shuttle evaluation programs. The first to be

discussed is a locally level, L frame, with one axis pointed towards north.

This system is used to relate westerly winds to the inertial system. The

unit vectors which define this system may be written in terms of the

vehicle's position vector (R) and the North Pole vector (l) as follows:

1 = R/IRi (A-4)-u

1 =1 x 1u/I x 1 (A- 5)-e -z u/ -z -u

n= 1 u x 1 (A-6)-n -u -e

with these unit vectors defined it is easily shown that the transformation

from the L system to the I system may be written as

* The quantity a(i) means the vector a coordinatized in the I frame.

A-3


The remaining two coordinate systems (U and V) are related

to the vehicle's inertial and earth relative velocity vectors (V and V R )-Rrespectively, as well as position. The U frame is the one in which the

data of Ref. 10 is defined and that data must be rotated to the I system.

The V frame is used to output position and velocity errors and platform

misalignments in a physically meaningful coordinate system, and to re-

late gravity errors and headwinds and crosswinds to the inertial system.

The U system has unit vectors 1 - 1 and which may be evaluated

using the expressions below.

u = _R/IRI (A-8)U (

w =R X V/IjR x. VI (A-9)-w

1V 1W IU (A-10)

The transformation from U to I is given by the following equation.

TI = 1 VI 1 (A-11)

The V system's unit vectors (Iu, 1v' 1 ) are defined in thesame manner as the U systems except that earth relative rather than

inertial velocity is used. That is:

1 = R/IRI (A-12)-u

1 = RxV /IRxVI (A-13)

1 = 1 x 1 (A-14)-V -W --u

in like manner TI/V is given by Eq. (A-15) below.

A-4


I/v = (u - 0) "( IV (A- 15)

This completes the definition of thThi s completes the definitin f the coordinate frames used in the SpaceShuttle programs. Figure A-i shows pictorially the way in which all thesesystems interrelate in order to produce the desired entry and landingerror budgets for the Space Shuttle mission.

REFERENCE 10 DATA -i6046PIATFORM RESULTS ATERROR SOURCES hs4OOiflIP FRAME) (u F .I

TRAJECTORyTAE

HEADINDSCROSswNDS

INITIALFILTER FLTER

COVRIANCE COVARIANCE GAINFE MAINE(1 |FRAME) ROC 'C RAM T N

AID

ITV/ lYin T ft. .

% .W ;OUTPUT DATA D. C

WVRAME OUTPUT DATA(R FRAMfI

Figure A-1 Use of TUse of Transformation Matrices

A-5


APPENDIX B

COMPUTATION OF SPECIAL OUTPUT QUANTITIES

In addition to the output that is obtained from a typical naviga-

tion error analysis program, such as position, velocity, and platform

alignment errors in the navigation coordinate system, several special

quantities were requested for the Space Shuttle study. These included

position, velocity, and platform alignment errors in the runway coordin-

ate system (R) and the coordinate system (V) which is referenced to the

position and earth relative velocity vector. The .rms values of the error

in computing altitude rate (8h), velocity magnitude (8vR), flight path

angle (8 v), and track angle (84) were also desired. These quantities

are tabulated at the end of each error analysis run for specific output

times, and the tables are included in Appendix C. The equations re-

quired to calculate these errors and their statistics are derived in this

appendix.

The rotation of the position, velocity, and platform alignment

errors from the inertial (I) to either the R or V coordinate systems

is complicated only by the fact that the major interest here is in earth

relative rather than inertial velocity errors. The error analysis pro-

gram determines the error in computing inertial velocity. This difference

is easily adjusted for by the application of Eq. (B-1) which relates inertial

to earth relative velocity errors.

.8VR = _v -_ x 8r (B-I)

B-1


Thus, the errors in position, earth relative velocity, and platform align-

ment may be related to the inertial errors using Eq. (B-2).

6r i I o i 0 6r 6r---- ------------- ----------- ------ ----

0 0 0E a

6v = - 0 0 0 6v = M 6v-R E (I) E -RI

IW

S0 0 0

L- - --- --- --- ----- I--------------6'_(p) 0 , 0 i 60_(p) [p)

(B-2)

If the covariance matrix associated with 8r(I), 8VR ) , and 88 (p) is

defined to be PI (the output of the navigation error analysis program),

then a similar matrix PE may be evaluated using Eq. (B-3).

PE = ME PIM (B-3)

The coordinate rotations TI/R, TI /V and TI/p, defined in Appendix A,may now be used to obtain the desired errors and/or error statistics in

the R and V coordinate frames. This is done using Eqs. (B-4) through

B-7).

(IR) R/I .6 SR -)I I

I () (B-4)

:T IIuaT " )B-2


= MRPE M

" I6r TV/I 0 a 0 r 6rr(v) "Tv/I 5-(I) r(I )

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

6v o TV/I 0 6v M 61RI I

6 0 0 TV/I TI/P 6 (p) (p)

(B- 6)

PV E T (B-7)

The derivation of the equations used to evaluate altitude rate,

velocity magnitude, flight path angle, and track angle errors are some-

what more difficult and are derived, one at a time, below. The error in

each of the four quantities is the difference between the actual value and

the value which would be computed using the navigator's estimate of posi-

tion and earth relative velocity.

Altitude rate (h) is defined as

R*Vh = (B-8)

R I

and the calculated value (h ) is evaluated using Eq. (B-9) below.

= c (B-9)c IR-I

However:

8r = R-R - (B-10)- -c

B-3


8V = - V (B-li)R = R -R

8h = h -h (B-12)

If Eqs. (B-10) through (B-12) are inserted into Eq. (B-9), the following

equation results:

(R - 8 r) (V - 8 v )(R - Sh = . (B-13)

[(R - 8r) (R - 8r)]

Equation (B-12) may now be expanded and only first-order terms main-

tained. The result is an equation relating 8h to 8 r and 8vR .

) x) (V x)R -(I)

v6 = 6v I= I v3 0 MrI vR

-O(~p) _ 6"(P)

(B-14)

The variance of 8h (c 82) may be calculated using Eq. (B-14) below,The8

all the terms of which have been defined in other places in this appendix.

2 Ta = M PE T (B-15)

a1 0 -a 3 a2

If a() = a2 then(a(I) X a3 0 -a 1

23 -a 2 aI 0

B-4


The velocity magnitude error equations are derived using the

same logic as that which was employed to obtain the altitude rate error

equation. Equstions (B-15) and (B-16) show the definition of velocity

magnitude and the means of computing it.

V = (V-R V)R (B-16)

VR = R VR = VR 8 vR (B - 17)

With the aid of Eqs. (B-11) and (B-17) as well as the definition of Eq. (B-16)

the following equation for 8 vR may be written:

6r -(I)

)I OO r

OR o o v Mv ovR (B-18), v (I) (I)

16 (p) 6! (p)

In addition, the variance of 8vR may be calculated using Eq. (B-19)

below.

2R E= TM (B-19)

The equations which define flight path angle (y) and track angle

(4) are given below.

YIR -sin y = (B-20)

R e

oRQV4 VR . e

B-5


where the unit vector along the earth's axis, 1 is needed to defineR

u = J (B-22)

1 xu

z x [ (B-23)

n = ux e (B-24)

The equations used, in an onboard computer, to compute , and 1 are thesame except that the computed values of u, e, and n as well as VR arerequired. This leads to the interesting problem of trying to computequantities in a coordinate frame which itself must be computed. Thusthere are two causes of error in the real time computation of y and :errors in computing VR and errors in computing R, which result in er-rors in the computation of u, e, n. Before expressions for 8 y and 681are evaluated, it is required that u, , and n be evaluated. Thederivation follows exactly the logic used thus far and the algebra will notbe presented here. The results, however, ar Eqs. (B-25) through (B-27)below.

1 ji- ii---u -l MU r (B -25)1 R - 8 q a bu ar (B()-

6eI = u -() e (z ) 6u(1) = Me r(i ) (B-26)

Z_) (u() ) 6e( ) - (e() x) = M 6r() (B-27)

With Eqs. (B-25) through (B-27) defined, it is quite straight-forward to derive expressions for 8 and 61 . The equation used tocompute y' is:

B-6


-R -csin C c (B-28).

VR C

However, the definitions

a y= 7- e (B-29)

8u = u - u (B-30)-c

along with (B-11) and (B-25) may be substituted into (B-28) to obtain the

following expression for 6 .

see. y = ... .0 "8Vay= R (H-I) ) Or ) : X R

( x(B) -31)1R2 v MP 1 (B-32)

Or

y y

Finally, the expression for 64 and a 2 may be shown, with

some effort, to be given by Eqs. (B-33) and (B-34).

B-7


6r)--)

2 (R- R !! )-R( R R RIR x i)L M) 0 -R0 - [ )2 + )2 R ) ) ) 0 R

S.P)

' P) (B-33)

.2 =M PE MT (B-34)

This completes the development of the equations used to evaluate

the special output quantities for the Space Shuttle landing system evalua-

tion.

B-8


APPENDIX C

TRUTH MODEL MEASUREMENT MATRIXFOR DRAG ACCELERATION PSEUDO-MEASUREMENT

The drag update filter used in the Space Shuttle entry navigation

system during blackout employs an atmospheric drag pseudo-measure-

ment in order to improve the on-board estimate of position and vel-

ocity. The Kalman filter used in the computer navigation program

requires the definition of a measurement matrix HF for the pseudo-

measurement. If the performance of the drag update filter is to be

evaluated using a "truth model", the measurement matrix HS for the

truth model must also be defined. The measurement matrix associated

with a truth model similar to that defined in Appendix E is derived in

this appendix.

The atmospheric drag pseudo-measurement is defined to

be a drag acceleration measurement constructed from the IMU accel-

erometer outputs. The measured drag acceleration qmeas(t). is

related to the state vector of the truth model by the matrix of first

partial derivatives Hmeas. The expected drag acceleration qre(t)is computed from a nominal atmospheric density model, estimates of

the shuttle's aerodynamic coefficients, and the current estimate of

position and velocity. qpred(t) is related to the state estimate vector

generated by the truth model by the matrix of first partial deriva-

tives Hpred It is shown in this appendix that HS can be defined in

terms of H and Hmeas pred -

C-1


The drag acceleration pseudo-measurement extracted from

the accelerometer outputs is

meas R(t) - AV(t)/At = 1VR(t) .c(t)l (C-l)c c

where VR (t) is a unit vector in the direction of the computed relative

velocity and f (t) is the measured specific force. The expected drag-cacceleration is

1 A 2(t) = CD (t) A- p(t)VR (t) (C-2)

C C

where CD (t) is the computed drag coefficient for the Space Shuttle, Ac

is its cross-sectional area, and m is its mass. p (t) and VRc(t) are the

computed atmospheric density and relative velocity, respectively. The

difference between qmeas(t) and qpred(t) is the residual

S(t) = qmeas()- pred(t) (C-3)

which appears in the update equation for the truth'model error state

vector

as(t = l s(t-) .Ks(t) 6zS(t) (C-4)

where _Sx(t) is the estimation error prior to the update and 6s(t+)

is the estimation error after the update.

A linear covariance analysis requires that qS(t) be expressed

as a linear function of 8x (t ). The drag accelerations q eas(t) and

pred(t) in Eq. (C-3) can first be referenced to the true drag accelera-

tion q(t)

C-2


(t) q(t) - (t)

qmeas meas

qpred(t) = (t) 6 qpred(t) (C-5)

where q(t) is the solution to either Eq. (C-1) or Eq. (C-2) in the special

case of perfect measurements, perfect state estimates, and precise at-

mospheric and drag coefficient models. The perturbations 8qmeas(t)

and 8qpred(t) are then related to 8 x () by the first partial derivative

matrices H (t) and H (t), respectivelymeas pred

meas(t) = Hmeas LES meas

6pred(t) = Hpred(t) 6(t ) + pre(t) (C-6)

where meas (t) and pred(t) are zero mean white Gaussian sequences

with variances

E[ (t) M (t) T] Rmeas measmeas

E [ pred(t) pred(t)] = RSd (C-7)

These Gaussian sequences are used to model accelerometer quantization

errors, uncorrelated errors in the estimates of atmospheric density and

the drag coefficient, etc.

The measurement matrix HS(t) for the truth model can be

defined by

Hs(t) = H (t) - H (t) (C-8)S pred meas

C-3


SWith the composite zero mean white Gaussian sequence r7S (t) defined

.s,(t) A= meas p(t) pred(t)

E [ S (t) ?IS ( R (C-9)

the residual 8 zS in (C-3) then becomes

8S (t) = qpred(t) - meas

Hs(t) 8 x (t) + 1 (t) (C-10)

The update in Eq. (C-4) can then be expressed in the desired form

6s (t)= S) - Ks(t) HS(t) 6 (t) KS(t) S(t) (C-11)

This formulation of the update equation requires an explicit computation

of HS(t) from Eq. (C-8).

The perturbation 8qmeas(t) in the measured drag accelera-

tion can be determined by considering small perturbations about the.

nominal values in Eq. (C-1). These perturbations are depicted graphic-

ally in Fig. C-1. f(t) is the resultant of the lift and drag accelerations

on the Space Shuttle

L(t) = R(t) x ((t) x -R(t))

a(t) = - (VR(t) f(t)) iVR(t) (C-12)

Clearly iVR(t) f(t) must always be negative. Hence, the absolute value

in Eq. (C-1) can be ignored in defining the perturbation

C-4


,. R-12038

IVR,

IVR .

Figure C-1 Definition of Perturbations in Unit RelativeVelocity Vector and Specific Force Vector

8qmeas(t)= - R(t) 1(t) + .R(t)" 6f(t) (C-13)

where

6VR(t) = VR(t - VR (t)

6f (t) = f (t) - (t) (C-14)

The error 6f (t) in measured acceleration can be attributed to

the standard IMU acceleration error sources defined in the drag-update

phase truth model (Table 3. 1-1)

* platform misalignments (Group la)

* accelerometer errors (Groups 2through 5)

* quantization errors (Group 1)

Since the drag acceleration pseudo-measurement is formed directly from

the accelerometer output, 6f(t) in Fig. C-1 is the same acceleration er-

ror which drives the velocity components of. 8x(t) in the truth model

C-5


.(see Appendix E). The perturbation 8kR(t) is defined in Fig. C-I tobe perpendicular to iVR(t). Thus the principal effect of using the com-puted normalized relative velocity vector -VR (t) in the drag accelera-

tion pseudo-measurement is that a fraction of cthe Space Shuttle lift accel-eration is resolved into the pseudo-measurement.

The relative velocity of the Space Shuttle with respect to theatmosphere is

VR(t) = (t) - E x R(t) - V (t) (C-15)

where 0E is earth rate and V W(t) is the atmospheric wind. The unit-Evector iVR(t) is then

V(t) - E x R(t) - YV(t)VR(t) = I_v(t) - _E x R(t)- V(t)I (C-16)

-The computed relative velocity is a function of computed inertial positionand velocity only

R (t) = VC(t) - E x R (t) (C-17)C

and the unit vector is

V (t) -0 x R (t)-IVR W K c E -- c (C- 18)

c ct) = -c -E -c

Rc (t) and _Vc(t) are related to the position and velocity components of8x (t) by

6v(t) = V(t) -.V (t)

C-66-r(t) 11(t- Rc(t) (C-19)

C-6


The relationship of VR (t) to -R(t) can be determined by

-substituting Eq. (C-19) into Eq. (C-16):

S V(t) + 8v(t) - _E X (R(t) + 8r(t)) - Vw(t))R(t) c E

1VR c(t) + bv(t)- E x (R(t) + 6r(t)) - YV(t)I

(C-20)

With the definition

8vR(t) = (t) - x (t) -E (t (t) (C-21)

it can be shown that a first-order expansion of Eq. (C-20) yields

6 R(t) - VR (t) 6!R(t) •-,R(t) t)+ (t) jR(j -x t)x R (t/ (C-22)

With Eq. (C-14), the above can be solved for biVR

(t)/ VR (t) * 6 R(t)

-VR IV (t) - x Rc(t) - VR (t) l(t) - x R(t) (-23)

•L -E . - .c_ . . . E . . - c .. . -

The second term in Eq. (C-23) is minus the component of the first term

along iVRc(t). It follows that 6iVR(t) has no component along !VRc(t) and

therefore is perpendicular to .VR (t). By the small angle approximation

implied by the first-order expansion, 6IVR(t) is also approximately

perpendicular to iVR(t) as was noted in Fig. C-1.

Equation (C-23) and the standard IMU acceleration error sources

defined in Table 3.1-1 permit H meas(t) in Eq. (C-6) to be written. The

partitions indicated below correspond to similar partitions of the error

C-7


state vector 6 S(t) in Appendix E. To simplify the notation, it is as-sumed in Eq. (C-24) that the non-standard wind components have beentransformed to inertial coordinates.

position velocity true platform accelerometererrors errors misalignments biases

Rmeas EL(gE x) FVR I/P FP R3 6 13 16 19

accelerometer accelerometer accelerometerscale factor errors misalignments nonlinearities winds[ T T1 -1 F3 I TT/, T- :0 O- j0.--R RT/p F F5 O 0-

22 28 31 49 58 59

(c-24)

where

F (t) = T(t)L IV (t) - x R c(t)

fT_ R (t) VR (t)T (C-25)C C

Iyc(t) -PE x RC(t)

0

S = o (C-26)

TI/p = transformation matrix from platform toinertial reference coordinates

C-8


0 -f3 f2

F = f3 0 -fl (C-27)

-f2 fl 0

fl 0 0

F3 = 0 f2 0 (C-28)

S 0 f3

f f 0 0 0 0

F4 = 0 0 fl f3 0 0 (C-29)

o o o o i

f2 0 0

F = 0 f2 0 (C-30)

o 0

The elements of F F , and F 5 are the components of the specificforce f(t) coordinatized in platform coordinates.

The computation of HS(t) requires that both Hmeas(t) andHpred(t) be determined; therefore it is necessary to determine

C-9


6 qpred(t) by considering small perturbations about the nominal val-

ues in Eq. (C-2)1( A 2 1 A (t)V2(t)

pred(t) = CD(t) (t) R (t) + CD(t) t p(t)V (t)(C-31)

CD(t) AP (t) V (t). 8 R(t)

where

6p(t) = p(t)- pct)

6 CD(t) = CD(t) CD (t)C

6R(t) = VR(t) - VR (t)C

S6v(t) - DE x 8r(t) - YV(t) (C-32)

The expression for 6qpred(t) can be simplified by writing it in terms of

the true drag q(t)

6 CD(t) 6 p (t) 2 IR(t) - 8 VR

pred(t) = q(t) CD(t) +p(t) 2.VR(t)8v(t) (C-33)red DCD R

Relating 6 p(t) and 8CD(t) to xS (tC) requires that both the computational

models for the atmospheric density and drag coefficient be known and

also that their true values be known.

Assume that the atmospheric density model from which Pc(t)

is computed is a simple exponential function of computed altitude h (t)C

The perturbation 8 CD(t) is assumed to include perturbations in A

and M. A more precise notation would be (CD(t) A

C-10


h (t)c

hPc(h) = Poe sc(C-35)

and that the true atmospheric density satisfies

h(t)h

p(R) poe sc + pf(R) (C-36)

where h(t) is true altitude. In the above po and hsc are constants andPf(R) is the difference between the true atmospheric density and the atmos-pheric density predicted by the model. A first-order expansion for 6p(t)similar to that used for 6bVR(t) yields

R (t ) 8r(t)Sp (R) -(t) + Pf(R) C-37)

where iR(t) is the unit vector along R(t).

One possible choice of the computational model of the drag co-efficient is a quadratic function of the computed angle of attack a (t)

CDac C + a 2 c (C-38)

where CO, C1, and C2 are constants. The true drag coefficient can thenbe referenced to the model

CD(, M) = CD (ac)+CD (aM,t) (C-39)

where a(t) is the true angle of attack and M(t) is the Mach number. If theerror in ac(t) is a negligible contributor to the error in CD( a ) then afirst-order expansion of 8 CD(t) is c

6 CD(t) = CD (, M, t) (C-40)

C-11


If this assumption is not true, 8 CD(t) must be derived based upon the

computational algorithm for c(t) and Eqs. (C-38) and (C-39).

Equations (C-32) through (C-40) permit H red(t) to be writ-

ten. The partitions indicated below correspond to similiar partitions of

the error state vector 68 (t-) coordinatized in an inertial reference co-ZSordinate system. Two of the non-zero partitions below (for density and

drag coefficient errors) correspond to partitions of H in Eq. (C-34)measwhich were zero. Hpred(t) as given in (C-41) assumes that the drag cor-

related error (Group 1), non-standard density errors (Group 21), and

non-standard aerodynamics errors (Group 23) are defined as percent

deviations from their nominal values. It is also assumed that the non-

standard wind components have been transformed into inertial coordinates.

position velocity dragerrors errors correlated error

T .i .. Ta.R 2 E X) 2 - R& VR -0 1 0H red(t) q(t) -h- (t) 1pred h. .(

3 6 12 13 46

non-standard non-standaarddensity error wind aerodynamics

model errors

j 2 RI1 1 1 - 0

49 58 59

(C-41)

This completes the derivation of the truth model measurement

matrix Hs(t), which is obtained by substituting Eqs. (C-24) and (C-41)

into

H (t) =H (t) -H (t) (C-42)S pred meas

C-12/


This definition of HS(t) properly relates the error in the drag accelera-

tion pseudo-measurement to the error sources included in the truth

model. If additional error sources not considered in this appendix are

included in the truth model, and if these error sources affect the pseudo-

measurement, the corresponding additional non-zero partitions of Hs(t)can be computed using the approach presented above.

C-13


APPENDIX D

NON-STANDARD ATMOSPHERE MODELFOR SHUTTLE ENTRY NAVIGATION

The drag update filter used in the Space Shuttle entry navigationsystem during blackout requires an atmospheric density model in orderto compare accelerometer outputs with predicted drag acceleration.This comparison is used to improve the onboard estimate of positionand velocity. The accuracy of the drag-update filter is degraded byboth atmospheric winds and the difference between the true atmosphericdensity at a given altitude and the density predicted by the atmosphericdensity model. These error sources are referred to as non-standardatmosphere errors - by definition, the standard atmosphere is thenominal atmosphere model used in the drag-update filter. In order tostatistically evaluate the performance of the drag-update filter, a reas-onable statistical description of the atmosphere in the 100,000 ft to270,000 ft altitude range is required. The non-standard atmosphere

model developed in this appendix is consistent with the limited amountof data available in the open literature.

D. 1 NON-STANDARD DENSITY MODEL

The atmospheric density model used in the drag update filtermechanization is some function of altitude pc(h ). As an example, itmay be a simple exponential model ..

D-1


hc(t)

hPc(hc) = poe sc (D-l)

where hc(t) is computed altitude. If the true atmospheric density p(R)

is a function of latitude and longitude as well as altitude, then the difference

6p(R) = p(R) - pc(hc) (D-2)

is the non-standard atmospheric density error. The non-standard atmos-

pheric density model developed in this appendix is comprised of three

components

p (R) = 6 2 (R) + PB(Ry + pM(R) (D-3)

where p6 2 (R) is a standard reference model, PB(R) is a position-

dependent bias, and pM(R) is a first-order markov process. The threecomponents of p(R) in Eq. (D-3) are defined such that the statistical

properties of 6 p(R) are similar to those that can be expected from anactual flight test.

The most commonly used reference for atmospheric densitymodels is the 1962 Standard Atmosphere (Ref. 36). The model consists

of a single density versus altitude table with data points at 500 ft inter-

vals. The component P6 2 (R) in Eq. (D-3) is defined to be a linear inter-polation of the 1962 Standard Atmosphere.

The true atmosphere density may differ from P6 2 (R) for avariety of reasons. The most significant density variations in the alti-tude range of interest are:

* Long-term deviations due to season changes

D-2


* Diurnal deviations due to thermal heatingand tidal interactions, and

* Short term deviations due to weather patterns.

Each of these effects has a correlation time considerably longer than the

entry flight time of the Space Shuttle; therefore, the principal time-

dependent deviations from P6 2 (R) during reentry are a function of the

spatial distributions of the density deviations. The long-term deviations

at different altitudes and latitudes have a strong spatial correlation andcan be modeled by the position-dependent bias PB(R). Diurnal and short

term density deviations do not exhibit strong spatial correlations and arebest modeled in Eq. (D-3) by the markov process PM(R).

Models of the long-term density deviations are contained in

U.S. Standard Atmosphere Supplements, 1966 (Ref. 37). This referencedefines separate density versus altitude profiles for each season at sev-eral different latitudes. As a rule, the most significant differencesfrom the 1962 Standard Atmosphere are in the winter and summer pro-files. From experimental data, Cole (Ref. 29) has determined probabilitydistributions for percent departure from the 1962 Standard Atmosphere asa function of latitude for both the December-January and the June-Julytime periods. The median departures determined by Cole are presentedin Fig. D-1. The data in Fig. D-1 corresponds well with the supple-mentary atmospheres of Ref. 37.

During entry, the Space Shuttle will not pass through black-

out at a constant latitude except for near-equatorial orbits. In modelingthe effect of long-term density deviations on the drag-update filter per-formance, therefore, it is necessary to consider the particular altitude-latitude profile being flown. This can be done by scribing that profileonto Fig. D-1. This procedure is illustrated by the dashed lines in

D-3


3oo

30N 45N 80N 60*N

IOI

250

150 /

- - - RECOMMENDED SUMUERPROFILE FOR p0 (RI FORREFERENCE MISSION 38

26 50

DEPARTURE FROM 1962 STANDARD DENSIfY (%)

Figure D-la Median Densities Given as Percent Departurefrom U.S. Standard Atmosphere, 1962 DuringJune-July at 80 0 N, 60 0 N, 45 0 N, and 30 0 N

.250

ORIGINAL PAqE ISOF POOR QUALITY

. - 6I SO .

RECOMMENDED WINTER|I P ROFILE FOR P0 € FO(REFERENCE MISSON 38

*60 -25 0 25

DEPARTURE FROM 1962 STANDARD DENSITY (2%

Figure D-lb Median Densities. Given as Percent Departurefrom U.S. Standard Atmosphere, 1962 DuringDecember-January at 80 0 N, 600 N, 45 0 N, and 30 0 N

D-4


Fig. D-1 for the reference mission 3B entry from a 1040 inclina-tion orbit with a landing at Vandenberg AFB. This trajectory crosses269,000 ft at 741N and ends blackout at 130,000 ft and 37.3 0 N. For theindicated seasons, the two dashed lines in Fig. D-1 provide a reasonable

model of the long-term density deviations to be encountered on the ref-erence trajectory.

The position-dependent bias PB(R) is defined to be one oftwo altitude-density profiles - a summer profile corresponding to the

dashed line in Fig. D-la and a winter profile corresponding to the dashedline in Fig. D-lb.* This choice of PB(R) is conservative in that thelong-term density deviation expected along this trajectory for a springor fall mission would be smaller than PB(R). The model is preferable

to treatment of the long-term density deviation as a Gaussian randombias, however, because for four to six months of every year, the long-term. density deviations can be expected to have the amplitudes shown.Guassian statistics are not satisfactory for describing phenomena inwhich large positive or negative values are more likely (or nearly aslikely) as small values.

Data on the cumulative magnitude of the diurnal and short-term density deviations from P6 2 (R) is available from two sources -the distribution functions computed by Cole (Ref. 29) and spatial corre-lation functions computed by Justus and Woodrum (Ref. 33 and 34). Inaddition, Cole (Ref. 30) and Daniels (Ref. 32) provide estimates of spa-tial correlation distances.- Although there is a latitude dependence citedin the literature, it is not of major importance and can be ignored.

Table D-1 summarizes the relevant information.

For a different reference trajectory, PB(R) should be defined for theappropriate altitude-latitude profile.

D-5


TABLE D-1

CORRELATION FUNCTION COEFFICIENTS FOR SUM OFDIURNAL AND SHORT-TERM DENSITY DEVIATIONS

FROM LONG-TERM (SEASONAL) AVERAGE

Source Justus and Woodrum Cole Daniels

Altitude 115,000- 210,000- 113,000 ft 210,000 ft 230,000 ft all altitudes210,000 ft 280,000 ft

Standard deviation 2% 8% 2.75% 6% 6%(% of averagedensity)

Vertical displace- 11 nm 11 nm 14 nm 14 nm 14 nm -ment correlationdistance

Horizontal displace- 54 nin 54 nm 270nm* 270nm 270 nm 270 nmment correlationdistance

*Data was inferred from statements that density deviations forhorizontal displacements greater than 600 nm are uncorrelated.

The entries in Table D-1 are reasonably consistent except

for the horizontal displacement correlation distances. Little informa-tion on this parameter is available and the different authors admit touncertainty in their estimates. The estimates attributed to Daniels andto Cole are made from macro-scale observations and are consistentwith wind horizontal correlation distances computed by Buell (Ref. 28)..The recommended design parameters for pM(R) are summarized in

Table D-2.

With the standard deviation aM, and the correlation distancesTZ and rX determined from Table D-2, definition of PM(R) requiresthat the form of its correlation function be established. If the changein PM(R) resulting from an altitude displacement AZ'is independentof the change from a horizontal displacement AX, -the correct choicefor the correlation function is

D-6


TABLE D-2

RECOMMENDED CORRELATION FUNCTION COEFFICIENTSFOR THE FIRST-ORDER MARKOV PROCESS M (R)IN THE NON-STANDARD ATMOSPHERE MODEL

Altitude 130,000 ft 2 10,000 ft 230,000 ft 280,000 ft

Standard deviation, aM 2% 6% 6% 8%(% of standard ref-erence plus bias)

Vertical displacement 14 nm 14 nm 14 nm 14 nmcorrelation distance,Th

Horizontal displacement 270 nm 270 nm 2 70 nm 270 nmcorrelation distance,

'X

(h X) = E M (R ) PM(R + Ah + AX)62R ,+ PB(R) p62 (R + Ah + AX) +PB(R + Ah + AX)

Ah2 AX2( 2 2. X (R) e ' X

(D-4)

Since only density variations along the shuttle flight path are of concern,AX and Ah are related by

Ah = AR sin y

AX = AR cosy (D-5)

where V is the flight path angle and AR is the net position displace-ment. The exponent in (D-4) can therefore be simplified

D-7


2 2 .22+ 2) =AR (+ cos2 f AA

' h TX h X R

(D-6)

With TR as defined in Eq. (D-6), the two-dimensional correlation func-tion CM(Ah, AX) can then be replaced by the one-dimensional correla-tion function

AR

2 TRCM(AR) = (R) e (D-7)

Eq. (D-7) implies that PM(R) satisfies the first-order dif-ferential equation

b(R) = - pM(R) + 7M(R) (D-8)R

where -QM(R) is a zero mean white Gaussian noise with a variance of

2 2 aM (R)Er [-(R) R) (D-9)TR

This equation can be expressed in discrete form, but it is important toremember that the independent variable by which gk(R) is defined inEq. (D-8) is position rather than time. The state transition matrix for

PM(R) is therefore

AR

Q(AR) = e R (D-10)

This completes the development of the non-standard atmos-pheric density model. The model is defined as a sum of three terms

D-8


in Eq. (D-3). The first term P6 2 (R) is the standard atmospheric density

reference model and defines the density as a function of altitude only. The

second term PB(R) adds the necessary latitude and season dependence to

the model; recommended profiles are given in Fig. D-1. The last term

PM(R) adds the diurnal and short-term dependence to the model; a recom-

mended model for this term is defined by Eq. (D-8) and the values given

in Table D-2.

D.2 WIND MODEL

The second item required for the non-standard atmosphere

model is a non-standard wind model. The drag-update filter mechaniza-

tion may not include a wind model, although it may account for turbu-

lence by increasing the estimate of the pseudo-measurement variance.

With this possible exception, the non-standard wind error is equal to

the non-standard wind model. Because wind is a vector quantity, the

model must describe both the wind direction and its velocity. This re-

sults in a model which is more complicated than the non-standard atmos-

pheric density model. A second complication is that the wind at a given

location is analyzed in East-West and North-South coordinates in the

literature, but the spatial correlation of the wind encountered by a re-

entry vehicle is analyzed in terms of headwinds and crosswinds. This

implies that much of the data must be transformed into trajectory-

oriented coordinates before it can be used to define the non-standard

wind model.

The non-standard wind model defined in this appendix is com-

posed of four components

Vw(R, t) = V (R) + V H(R) + V (R) +V T(t) (D-11)

where

D-9


Yzw(R,t) = a westerly (zonal) wind, modeled as analtitude-dependent biasVH(R) = a headwind along the relative velocityvector of the Space Shuttle;_modeled as a

first-order markov processVc(R) = a crosswind in the horizontal plane and per-pendicular to the relative velocity vector ofthe Space Shuttle, modeled as a second-ordermarkov process, andVT (t) = turbulence along the longitudinal, lateral andvertical axes of the Space Shuttle; modeledas first- and second-order markov processes.

The characteristics of the non-standard wind which most greatly affect theperformance of the drag update filter can be accounted for by developingstochastic models for the magnitudes of these four components.

Average winds are generally modeled as East-West (zonal)and North-South (meridional) components. The zonal wind is the onlycomponent with sufficient spatial correlation to warrant treatment asa bias in the non-standard wind model. Groves (Ref. 31) has tabulatedmonthly means for the zonal wind in the northern hemisphere as a func-tion of altitude and latitude. A sample histogram corresponding to200,000 ft and 400 N is given in Fig. D-2. From the skewed distribution

44 R-12039

SI JAN33

SDECM.

O 2.I JUL SEP OCT FEB2D ----- -----

AUG JUN MAY APR MAR NOV

-250 -150 -50 50 150 2-0 35 0MEAN ZONAL WIND (feet per second)Figure D-2 Histogram of Monthly Mean Zonal (Westerly)

Wind Components (fps) for 400 N and an Altitudeof 200,000 ft

D-10


in Fig. D-2 it is clear that the average zonal wind cannot adequately be

modeled as a Gaussian random bias. A more reasonable approach is

that previously adopted for the long-term density deviations, i.e., to

define VZW(R) deterministically using the January and July zonal wind

profiles. This approach corresponds roughly with a worst-case average

zonal wind model. Figure D-3 presents the July and January profiles

for VZW(R) based upon the 1040 inclination orbit and a landing at

Vandenberg AFB. For different trajectories, the necessary profiles

for V (R) can be taken from Tables 7a and 10 of Groves (Ref. 31).

The zonal wind component encountered during reentry can

differ from the appropriate monthly mean by a significant amount. The

available data on the standard deviation is summarized in Tables 15 and

17 of Ref. .31 as a function of altitude, latitude, and month. The latitude

3 , ,,, 300-- - to BOUNDS FROM

FIGURE D-4 lo BOUNDS FROMFIGURE 04

\ 150 tSD

1 \" '

E /00 0 100 2W 300 400

6-

VELOCITY VELOCITY (ft/wcj

Figure D-3a Mean Zonal (Westerly) Figure D-3b Mean Zonal (Westerly)Wind in July for Ref- Wind in January forerence mission 3B Reference Mission 3BEntry Entry

D-11


* dependence is not significant and can be ignored. Figure D-4 presents

the season-averaged standard deviations as a function of altitude only -

this is in contrast to the season and trajectory dependence of Vzw(R).

As an indication of the likely range in magnitude of the zonal wind com-

ponent, la bounds derived from Fig. D-4 are shown in Fig. D-3.

R-12037

300WITH THE ISOTROPIC WIND APPROXIMATION,

THIS GRAPH YIELDS oH FOR THEHEADWIND VH(R)

AND OC FOR THE CROSSWIND VC (R)

250

O

S200

V150

0 25 50 75 100 125 150

STANDARD DEVIATION (feet per second)

Figure D-4 Season-averaged Standard Deviation ofZonal Wind Component as a Functionof Altitude

The data in Fig. D-4 must be used to define the standard dev-

iation of the trajectory-oriented wind components VH(R) and VC(R).

Before this definition can be made, however, it is necessary to deter-

mine the standard deviation of the meridional wind component. The

meridional wind is characterized at a given latitude, longitude, and

D-12


alitude by a small monthly mean and a larger standard deviation (Ref. 31).The monthly mean does not exhibit strong spatial correlation so that themeridional wind encountered along a reentry trajectorrelcan be assumedto be unbiased. Data on the standard deviation of the meridional wind isscarce, but that supplied byGroves indicates that hedata in Fig. D-4 canalso be used for the meridional wind. The advantage of this result is thatit permits wind variations to be modeled as isotropic i.e., at a givenaltitude, the standard deviation of the ind component in an arbitrary

(horizontal) direction is a constant which is independent of the direction.In particular, the standard deviations of VH(R) and VC(R) are given byFig. D-4.

In order to complete the definition of the spatial correlationfunctions of VH(R) and Vc(R), it is necessary to determine the spatialcorrelation coefficients Justus and Woodrum (Ref. 34) have experi-mentally determined spatial correlation coefficients in the desired alti-tude range, but their analysis includes only the high frequency portion ofthe wind spectrum Consequently, the standard deviations of the randomprocesses which they analyzed are less than the corresponding values inFig. D-4. Buell (Ref. 28) determined the horizontal correlation coeffi-cient for the full wind spectrum, but at an altitude of 20, 000 ft. Bothsets of results are summarized in Table D-3. If Buell's results areextended to the appropriate altitudes, it is possible to show that thetwo sets of data are consistent - a random process defined by Buell's

displacement correlation coefficient and the standard deviations in Fig.D-4 has nearly the same expected change for a small horizontal dis-placement as is predicted by Justus and Woodrums model. This con-sistency implies that Buell's results can reasonably be used in thedesired altitude range. can reasonably be used in the

D-13


TABLE D-3

SPATIAL CORRELATION DISTANCES ANDSTANDARD DEVIATIONS FOR ISOTROPIC WINDS

VH(R) AND Vc(R) IN NON-STANDARD WIND MODEL)

Source Justus and Woodrum BuellAltitude 150,000- 210,000-

210,000 ft 280,000 ft all altitudes

Horizontal displacement 27 nm 55 nm 270 nmcorrelation distance TX

Standard deviation for 20 ft/sec 59 ft/sec Fig. 1-4horizontal correlationfunction oH, OC

Vertical displacement 3.4 nm 5.5 nmcorrelation distance

Standard deviation for 6.5 ft/sec 16 ft/sec -vertical correlationfunction

The horizontal displacement coefficient 7X chosen for VH(R)and VC(R) is the 270 nm value from Buell. The small standard devia-tion which Justus and Woodrum associated with the vertical displacementcoefficient indicates that the horizontal winds at different altitudes arestrongly correlated. Compounded with the fact that Space Shuttle re-entry trajectories have small flight path angles, this implies that thevertical displacement variation in VH(R) and VC(R) can be ignored.

Given that the spatial correlation functions for VH(R) andVC(R) are functions of horizontal displacement only, they can beexpressed in terms of Fig. D-4 and the chosen value for TX. Buellmakes an important distinction between the two correlation functions.The headwind VH(R) is modeled as a first-order random process withthe correlation function

D-14


CH(X) = 2 e X (D-12)

but the crosswind is modeled as a second-order random process with the

correlation function

AX

2 TXCc(AX) = C (1 - rX AX) e (D-13)

The standard deviations crH and oC are the appropriate entries fromFig. D-4. These two functions are illustrated in Fig. D-5. The negativevalue for CC(AX) for a displacement greater than 600 nm is a consequence

of the cyclonic nature of weather patterns - a reentry vehicle crossing alow pressure region in the northern hemisphere will experience cross-winds first from port and then from starboard.

R-120,401.0

0.8 , CORRELATION DISTANCE IS 270 nm

l STANDARD DEVIATIONS ARE GIVEN BY FIG. 0-4

0.6

O

0.4

N \HEADWIND

. 0.2a CROSSWINDO02

0

-0.20 200 400 600 800 1000 1200 1400 1600

HORIZONTAL DISPLACEMENT. Ax (nm)

Figure D-5 Correlation Functions for Headwind VH(R)and Crosswind V (R) as Defined by Eqs.(D-12) and (D-13R

D-15


Equation (D-12) implies that VH(R) satisfies the first-orderdifferential equation

VH(R) = -T VH(R) + Hi(R) (D-14)

On the other hand, VC(R) is the output of a second-order differential

equation

WC(R) = FCWC(R)+ GCi C(R)

VC(R) = HW (R) (D-15)

where

2-- 1

F =C 12 0

X

G =

HC = [ 1 0 ] (D-16)

The inputs in Eqs. (D-14) and (D-15) are zero mean white Gaussian noiseswith variances of

D-16


22 2 2

E [?H(R) ] x

2 4 oE [tC(R) ] = (D-17)

In all the above equations the independent variable is horizontal displace-

ment. In the discrete case, the state transition matrices are

AX

OH(AX) = e TX

QC(AX) = eFC AX (D-18)

The last component of the non-standard atmospheric wind modelis the turbulence component VT(t ) . Turbulence in the free atmospherecan be modeled by the Dryden spectrum (Ref. 32). The Dryden spectrumfor turbulence along the relative velocity vector is

L 2

TVs)2 (D-19)VR(t) 1 + L2 s 2 /VR(t) 2

and the spectrum for turbulence perpendicular to the relative velocityvector is

L.2 1+ 3 s 2L 2/VR(t) 2

Ti 2V (t) D2 (D-20)1+ s2 R 2

D-17


. where VR(t) is relative velocity and L and aT are design parameters.

The suggested values of L and aT during blackout are

L = 1750 feet

aT > 13 ft/sec (D-21)

These are the values used in the definition of VT(t).

Random processes with the spectra given by Eqs. (D-19) and(D-20) can be defined by passing white noise through the appropriatelinear system. The turbulence along the relative velocity vector isdefined by the first-order system

VR(t)WTV(t) = WTV(t) + ? (TV

V (t) = L (t) (D-22)

and the turbulence perpendicular to the relative velocity vector is definedby the second-order system

WT (t) = FT WT . (t) + GT 9T±(t)

VTl(t) = HTi W T(t) (D -23)

where

0 1

FT = 2 (t) VR(t)-2-R-

L2 L

D-18

THE ANALYTIC SCIENCES.CORPoRATION

G

2

-

'IT - L C3- (D-24)

The inputs to Eqs. (D-22) and (D-23) are zero mean white Gaussian noiswith variances es

E t El (t2 -- )2 2- T(D-25)

It should be noted that the independent variable in the aboveequations is time. This is in contrast to the models developed forveVH(R) and VC(R). The discrete version s of Eq. (D-22) and Eq. (D-24)respectively, are versions f E. (D-22) and Eq. D-24)

VR(t)

OTV(At) e Land (D-26)

T(At) e FTA (D-27)

This completes the development of thenon-standard windmodel. The model is defined as a sum of four terms in Eq. (D- 11).Th e first term VZW(R) is the average zonal (westerly) wind as a func-tion of altitude, latitude, and season; recommended profiles are givenin Fig. D-3. The terms V(R) and VC(R) (headwind and crosswind,respectively) add the long period wind variations due to diurnal effects

D-19


and weather patterns; recommended models and values are given in

Eqs. (D-14) and (D-15) and Table D-3, respectively. The last term

VT(t), is a function of time only and adds the high frequency wind com-

ponents. VT(t) is expressed in vehicle coordinates and is modeled by

a longitudinal component VTV(t) and a transverse component VT (t);

recommended models and values are given by Eqs. (D-22), (D-23)

and (D-24), respectively.

D-20


APPENDIX E

TRUTH MODEL STRUCTURE FOR SHUTTLE ENTRYAND LANDING NAVIGATION STUDIES

The basic truth model structure is showni in Fig. 2. 3-1. Thetruth model (FS, HS) is related to the filter model (FF, HF) by a trans-formation matrix A. Equations for HS and HF during the drag-update

phase were developed in Appendix D and Section 4.2, respectively; FS,FF, and A for the drag-update phase are developed in this appendix.

In addition, all five matrices are developed for System E.

E.1 DRAG-UPDATE PHASE

The drag-update phase truth model requires a 13 x 13 filterdynamics matrix, FF, and a 59 x 59 system dynamics matrix FS . Theonly external navigation aid is the drag acceleration pseudo-measurement.

Figure E. 1-1 presents the overall structure of the F S matrixfor the drag-update phase. The upper-left partition of FS is the 13 x 13matrix F 1, whose elements define the dynamic interaction betweenthe Group 1 error states. This sub-matrix of FS corresponds* to thefilter matrix, F F . The horizontal row of sub-matrices, F, la through

* F 1 i and FF are identical except that states 7 through 12 are expressedin fhe I frame in FF and in the P frame for F 1 1 . The definition of Ain Eq. (E. 1-21) provides the necessary transfdrmation between the tworepresentations.

E-1

THE ANALYTIC SCIENCES CORPO RATION

'-14198

ESTIMATED NON-ESTIMATED NON-ESTIMATEDSTATED NON- ESTIMATED EXTERNAL-AIDS EIM -RELATED STATES RELATED STATES3

0

F F ,o F1,2 F ' ... " F

13 0 0

F,10a IFo,2 a.3 Flo9F2,2 0

F3,3

ORIGINAL P

0 F22 ,22

59 0 . F23,23146 59

Figure E.I- I Drag-Update Phase Truth ModelSystem Matrix Structure

F9 defines the effects of the non-estimated, IMU-related error sources(Groups la through 9) on the velocity errors (states 4 through 6). The sub-matrices along the main diagonal, Fla la through FF23 define the dynamics la la through Fg9 9 and F21,21 through

23,23' define the dynamics of all non-estimated, correlated error sources.For a group of random-constant error sources, this submatrix is zero.

The submatrices F are defined indetailbelowgroup-number designations given in Table 3. 1-1.

E-2


Group 1 - Estimated States (See also, Section 4.2)

0 Io o0 0

Gr 0 TI/ Fp I 0

0

(. 1- 10 0 0 8 0 00 s9

F1,1 0 0 s1O 0

G .00 s12

1s = - =7 ... 13

and T7 "" 713 are the drag-update filter time -constants assigned inTable 4.2-1. Also

r 3 (E. 1- 3)

p = gravitational constant

I = 3 x 3 identity matrix

S=. position vector (in I frame;see Appendix A)

E-3


0 -f f

Fp = f3 - f 1 (E.1-4)

-f2 fl 0

f2 = specific force vector in theP frame

L3j

T /p 3 x 3 matrix which transforms0 MAL PA(e; I/P vector in the P (platform) frame

Q OR QUALUT to the I (navigation error analysis)frame; see Appendix A. See also,the discussion of the A matrix, atthe end of this section.

Group la - True Platform Misalignments (3 states)

F la TI/pFp 3 F a,a = [0]3 x3 (E.15)

Group 2 - Accelerometer True Biases, (3 sensors, 1 state each)

F 1 , 2 = [TI/P] 3 ; F la2 = F 2 = [0]3x3 (E.1-6)

Group 3 - Accelerometer Scale Factor Errors(3 sensors, 1 state each)

F 1 , 3 = [TI/P F 313X 3 ; Fla, 3 F3, 3 =[013 3

(E. 1-7)

E-4


where

fl 0 0

F3 = 0 f2 0 (E.1-8)

0 0 f

Group 4 - Accelerometer Misalignments(3 sensors, 2 states each)

F14 = [T/p F 4 3 x 6 Fla,4 = [0] ; F4 = [011,4 3x6 3x6 ' 6x6

(E. 1-9)

where

f2 f3 0 0 0 0

F4 = 0 0 f f3 0 0 (E.1-10)

.0 0 0 0 fl f2

Group 5 - Accelerometer Nonlinearities(3 sensors, 1 state each)

F1,5 = [TI/pF 5] ;F = FF1,5 I/P 5 3x3 Fla, 5 F = [01 (E.1-11)

3x3 .3x 3

where

f 2 0

O f 0F5 2 (E. 1-12)

0 0 f2

E-5


Group 6 - Gravity Deflections and Anomaly(in the vehicle relative velocity coordinate system (V frame))

-i/ru 0 0

F1,6 [TV 3; Fla,6= [O13 x3; 6,6 = 0 -1/"v 0

0 0 -l/w W

(E. 1-13)

where

01IGINAL PAGE s = du/VRPOO R QUALTy = d /VR

V vR

w = dw/VR

and du, d , dw are correlation distances which are expressed in tabular

form as a function of altitude (Ref. 14).

Group 7 - Gyro Bias Drifts (3 sensors, 1 state each)

F lai = [I] 3 3 ; F 1 7 = F 7 7 = [ 0 1 3x3 (E.1-14)

Group 8 - Gyro Mass Unbalances (3 sensors, 2 states each)

fl 2 0 0 0 0

Fla,8 0 0 2 3 0 0 F1,8 = [0]3 x6 ; F8,8 = [0]6 x 6

0 0 0 0 f3 f2

(E.1-15)

E-6


The elements of F 1 , 8 and Fl, 9 below, are determined by the gyro in-

put and spin axis directions shown in Fig. E. 1-2.

VERTICAL -so044AT LAUNCH

i . INPUT AXISa * SPIN AXISo * OUTPUT AXIS

GYRO I

GYRO 3 DOWNRANGEAT LAUNCH

02 P3 ' (SOUTH)

GYRO2 0

P2

Figure E. 1-2 Orientation of Gyro Axes

Group 9- Gyro Anisoelasticity, (3 sensors, 1 state each)

flf2 0 0

Fla, 9 = 0 f2 f 3 0 ; F1,9 =F9,9 = [] 3x3

0 0 f 2 f 3

3x3

(E.1-16)

E-7


Group 21 - Non-Standard Density (3 error sources, 3 states)

0 0 0 1962 Standard Atmospheremodeling error

F21, 21 = 0 0 0 time-varying bias

VR0 0 - -- markov

TR- (E.1-17)

where TR is determined as a function of altitude and flight path angle inAppendix D.

Group 22 - Non-Standard Wind (4 error sources, 9 states)

0 0 0 0 westerly

VR0 - 0 0 headwind

0 o - V v R

F22 crosswind

0. O 0

0 FT turbulence

(E. 1- 18)

where Tx is given in Table 3.1-3. Equations for turbulence dynamicsare defined in Appendix D, but the time constants are so short and themagnitude of the disturbance is so small that they are not used in thetruth model. Instead, the assignment

E-8


FT = [0]5 5 (E.1-19)

is made and turbulence is included as a portion of the 5% drag accelera-

tion measurement noise ii, Group 1.

Group 23 - Non-Standard Aerodynamics (1 error source, 1 state)

F = - markov (E.1-20)23,23 = CD

where TCD is given in Table 3.1-3.

A Matrix

The 59 x 13 matrix relating the drag-update filter states to the

truth model states has the form

A = (E.1-21)

where A' is a 13 x 13 submatrix

6 9 12 13

ORIGINAL PAGE IS 66 6 0 0 0OF POOR QUALITy

0 Tp/I 0 0

A' = 0 T 0 (E. 1-22)

0 0 0 Ilx1

The filter states 7 through 12 (misalignments and accelerations) are de-

fined in the inertial (I) frame. The corresponding truth model states in

E-9


Group 1 are defined in the platform (P) frame in order to relate them

more readily to Groups la and 2. Thus, the transformation I to P ap-

pears in the A matrix. The drag-update filter system matrix is related

to Fl, 1 in Eq. (E.1-1) by

FF = A' F1,1 (A')-1 (E. 1-23)

The markov processes in Groups la through 23 in the drag-

update phase truth model require that the elements of the system pro-

cess noise matrix QSk associated with those states be non-zero. If

all the non-zero markov processes are stationary, then the continuousprocess noise matrix Q defined in Eq. (2. 3-9) is related to FS and

P (0) by

QS = - FS PS (0) - Ps(0) FST (E.1-24)

The discrete process noise is then related to QS by

tik+1

Qk = exp [Fs(tk T)] Q, exp FS(tkl- T)]T dT (E.1-25)

To illustrate the application of Eqs. (E. 1-24) and (E. 1-25), ifthe ith truth model state is a stationary first-order markov process, acorrelation time, Ti, for the state is assigned in one of Eqs. (E.1-1)through (E.1-20) and a variance, a. , is defined in either Table 3.1-2or 3.1-3. The corresponding elements of Pg(0), FS, and QSk are

PSH(0) = a2 (E.1-26)

S = -- (E.1-27)

E-10


2acQSk Ti (E.1-28)

Similar results are obtained for stationary second-order markov proces-ses, although the expression for QSk depends upon the precise structureof the 2 x 2 submatrices of PS(0) and F S associated with the markovprocesses.

Constant error sources can be considered a special case ofEqs. (E.1-26) through (E.1-28) with . = and FSii = QSki i = 0.Nonstationary markov processes with slowly-varying parameters canbe modeled by recomputing QSki i as and a. change. For time-ki I Ivarying biases which have no associated dynamics, such as the time-varying density bias (Group 21) or the mean westerly wind (Group 22), thecorresponding components of FS and QSk are set equal to zero and thecorresponding elements of Ps(t) are obtained directly from the appro-priate profiles in Appendix D.

E.2 SYSTEM E

The System E truth model requires a variable-dimensionednavigation system Kalman filter dynamics matrix, FF , and a truth modelsystem dynamics matrix, FS . The maximum possible dimensions areassumed in this appendix, i.e., FF and F S are 15 x 15 and 100 x 100,respectively. The discussion in Section 4.3 indicates which states of FFshould be removed when the Kalman filter has fewer than 15 states. Therelationship developed below between FF and FS then indicates whichstates should be removed from FS.

E-11


Figure E. 2-1 presents the overall structure of the F S and HSmatrices for System E. The structure of FS is similar to that of FSfor the drag-update phase. The eight system measurement matrices

(row vectors) are outlined in the lower half of Fig. E.2-1. For all but

the drag pseudo-measurement (HS8 ), the first 15 elements are the same

as those of the corresponding filter measurement matrix. Other non-

zero submatrices, H., define the effects of the correlated error sources

(Group j) in question on the measurement residual, 6zi , in question.

The only submatrices of F S which have changed from the

drag-update phase are F 1 1 and the new block diagonal matrices

F ... F20,20 These submatrices and H. . are defined below us-10,10 20,20 -,j

ing the group number designations given in Table 3.2-1. In addition, the

filter measurement matrices HF .. HF2 are also defined; HF is de-1 8

fined in Section 4.2.

Group 1 - Estimated States (see also Section 4.3)

Two possibilities exist:

* Fl, 1 is the upper left 6 x 6 submatrix of Fl, 1as defined in Eq. (E. 1-1).

* F1 1 is as defined in Eq. (E.1-1), except thatstates 14, and perhaps 15, may be added. Theonly non-zero elements of F1 , 1 associated withthese states are the diagonal elements s14 and

s15 , respectively.

Group 10 - TACAN Range Bias Error(1 station, 1 state)

F1010 = [0]1 x 1 (E.2-1)

H1 1 0 = 1 (E.2-2)

E-12


R-14199

NON- ESTIMATEDESTIMATED NON-ESTIMATED EXTERNAL-AIDSTATES IMU-RELATED STATES RELATED STATES

3 0

F6a F2 1,3 F1, 9

F1 1

F Il

0

15Flol a Flo,2 F 3 Fla 0

F2,2 0

F3 e3

0

0 _

0

QO . . F2 2,2 2

F23,2 31000

15 48. ioo

8s5 [ i I o Is,to 1s,t I o I j,so I o

8 I' I I 0,1 o us,3o o,'[" r olI, o 1 ,rzlOl ,,ol o]

s ." ( I o I o I o I %s,71 I o 01 0486 -[" 4 0 I o , 4 Is l 0 o

H 8, . [ I o o 0I1,13H.,14 O 0 0 j 0 3nI, . l IIo o saI01 u, l ,s7o I1 ,oI o ]

S"~Zq. (C-42)

Figure E.2-1 System E Truth Model StructureORIGINAL PAGE ISOF POOR QUALITy

E-13


Group 11 - TACAN Range Scale Factor (1 station, 1 state)

111,11= [-1/sf]l x (E.2-3)

H11, 11 -P1 (E.2-4)

where rsf is defined in Table 3.2-1 and j P1 is the range to the TACANstation.

Group 12 - Baro Altimeter Errors (4 error sources, 4 states)

0 0 0 0 bias

0 0 0 0 scale factor2,2 (E.2-5)F12,2 0 0 -1/ra 0 markov

0 0 0 0 static defect

3,12 = [ h 1 VR2 (E.2-6)

where r a is defined in Table 3.2-2, h is altitude, and VR is the Shuttlerelative velocity with respect to earth.

Group 13 - MLS Time-Varying Biases(azimuth bias, elevation bias, range bias,

range scale factor; 4 states)

F13,13 = [0]4 x4 (E.2-7)

H4,13 = [1 0 0 0] (E.2-8)

H6,13 = [ 0 1 0 0] (E.2-9)

H7,13 = [ 0 0 1 IAl] (E.2-10)

E-14


where A is the range to the MLS DME antenna.

Group 14 - MLS Second-Order Markovs (3 error sources; 6 states)

2 1

a 'a "a a 0 azimuth1

-- 0Ta

2 1

'e 'e

F 140 1 0 elevation14,14 -- 0

2 1

'DME T'DME

0 0 range1 0

"DM E

ORIGINAL-PAGE ISOF POOR QUALITY

H4, 1 4 =[ 1 0 0 0 ] (E.2-12)

H6,14 =[ 0 1 0 I- 0 ] (E.2-13)

7,14 = [ 0 0 1 0 ] (E.2-14)

where ra, re, and TDM E are assigned in Table 3.2-2.

Group 15 - Radar Altimeter (2 error sources, 2 states)0 oF15, 15 = [ (E.2-15)

ra

I,15 = [1 1 ] (E.2-16)

where Tra is defined in Table 3.2-2.

E-15


Group 16 - TACAN Survey Errors(2 antennae, 3 components each)

F 1 6 , 16 = [O]6x 6 (E.2-17)

HPR D C 0 0 0 (E.2-18)1, 16 1Pll 1 I i l I

H = 0 0 0 I2R - 2 C (E. 2-19)2,6 I I !.s I, 1

where PR P1D' PiC are the components in the R frame of the vector

P (see Fig. E. 2-2); p 1 is the relative position vector from TACAN to theShuttle. u2R, U2C , 2D are the components of the unit vector u 2 ' per-pendicular to pl and lying in the horizontal plane (see Fig. E.2-2).

The vector sl is the projection of Pl onto horizontal plane.

SHUTTLE -14202POSITION

hP

S 2

TACAN*- X1---- - ---

RUNWAY

Figure E.2-2 TACAN-Shuttle Geometry

E-16


Group 17 - MLS Survey Errors(3 antennae; 3 components each)

F 17 ,17 = [01]9 x (E.2-20)

H4, 17 [-R -'B -( I o_ (E.2-21)

H6, 17 [ o I -sP - -'8 C 0 ] (E.2-22)

I-P[ AR PAD PAc (E.2-23)R7,17 00(E.2I-) IA

where 8~ , 8 o, and 8aC are the components in the R frame of the

vector 8a (see Fig. E.2-3) defined by

(A XC)8a = (E.2-24)I-X 2

IA x S "

8 ' 8 pD, and 8 pC are the components in the R frame of the vector

j. (see Fig. E.2-4) defined by

- x a

8P = 2 (E.2-25)

LL E Xa2

and PA , PA ' PAC are the components in the R frame of PAR AD A-A*

Group 18 - Not Used for System E

Group 19 - TACAN Bearing Bias Error (1 error source; 1 state)

F 9 = - 20 1 (E.2-26)19, 19 20,000 1 x 1

= [ 1 ] (E.2-27)

E-17


SHUTTLE R-14 201POSITION

p A

\-MLS AZIMUTH\ AND DME

\ - -.-- - -- -

RUNWAY

Figure E.2-3 MLS Azimuth-Shuttle andMLS DME-Shuttle Geometries

O'avjAL j

R-14203

M.LSSHUTTLE E ELEVATION oPOSITION / e

RUNWAY

Figure E.2-4 MLS Elevation-Shuttle Geometry

E-18


Group 20 - MLS Timing Errors (3 error sources; 3 states)

F20,20 = [0]3 x 3 (E.2-28)

H4,20 = [ 0 ] (E.2-29)

H7,20 = [0 0 IAI] (E.2-30)

where o&, p, A are the azimuth rate, elevation rate, and range rate,respectively, of the Shuttle relative to the appropriate MLS antenna.

The only elements of FS and HS which have -not been definedare the filter measurement matrices HF * HF

1* V

3 6 A 12HF 1 = [ -H,16 0 0 J 0 1 0 ] (E.2-32)

HF 2 = [ - H 2 16 0 j 0 0 0 1 ] (E.2-33)

HF3 = [2 iR3 0 0 l O 0 ] (E.2-34)

HF [ -H4,17 I 0 j 0 0 0 1 ] (E.2-35)

HF5 = [1 R2 3 0 (E.2-36)

HF6 = [ -H6,17 0- 0 1 0 0 ] (E.2-37)

HF = [ - 17 0 0 0 0 0 ] (E.2-38)

where iR1 , iR2 , and iR3 are the components in the I frame of the unitvector in the vertical direction.

E-19


A Matrix

The A matrix for System E is a 100 x 15 matrix relating theKalman filter states to the truth model states. A has the form

A = (E.2-39)

where A' is a 15 x 15 matrix which differs from A' as defined inEq. (E. 1-22) only in that the Ilx 1 submatrix is replaced by a 13x3 sub-matrix.

E-20


APPENDIX F

DRAG-UPDATE ERROR CONTRIBUTIONTIME HISTORIES

This appendix presents the time histories associated with the

error budget results summarized in Section 5.1. The data is presented in

tabular form with each table indicating the rms errors in position and

velocity, and the rms values of the platform alignment estimates, in boththe R (runway)* and V (relative velocity) coordinate frames (see Appendix A),which result from a specific error source or group of errors. The rms

errors in altitude, velocity magnitude, flight path angle (y), and track

angle (0) are also presented. In addition, the rms value of each of states

10 through 13 is given at 130,000 ft.

The time points in each table correspond to entry interface at

400,000 ft (t = 0 sec), just before and after the first drag acceleration

pseudo-measurement (t = 312 sec), and every 60 seconds thereafter until

just before the first TACAN measurement (t = 1432 sec). The magnitudes

and mathematical description of the error sources are given in Chapter 3.

Units are in feet, feet/sec, radians, and radians/sec.

Note that the order of the runway coordinate printout is (Vertical, Cross-range, Downrange) in Appendix F and (Vertical, Downrange, Crossrange)in Appendix G.

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APPENDIX G

SYSTEM E ERROR CONTRIBUTIONTIME HISTORIES

This appendix presents the time histories associated with theerror budget results summarized in Section 5.2. The data is presented intabular form with each table indicating the rms errors in position andvelocity, and the rms value of the platform alignment estimates, in boththe R (runway)* and V (relative velocity) coordinate frames (see AppendixA), which result from a specific error source or group of errors. Therms errors in altitude, velocity magnitude, flight path angle (y), andtrack angle (¢) are also presented.

The time points in each table correspond to every minute be-tween the end of radio blackout and MLS acquisition, and to a shortertime interval thereafter. Time points are also included just before andafter each external navigation aid is activated. Thus there are two rowsof data for each of the following times (see Table 2.2-1):

TTACAN = 1432 sec

TBA = 1550 sec

TMLS = 1836 sec

TDM E = 1846 sec

TRA = 19 4 3 .5 sec

Note that the order of the runway coordinate printout is (Vertical, Cross-range, Downrange) in Appendix F and (Vertical, Downrange, Crossrange)in Appendix G.

G-1


A single row of each table corresponds to the initiation of the final ap-

proach and landing sequence at an altitude of 12,000 ft:

T = 1872 sec

The magnitudes and mathematical description of the error sources are

given in Chapter 3. Units are in feet, feet/sec, radians, and radians/sec.

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1550.00 1.0E-sOOOE-10 1.000E-10 1. 0Io0E-10 1.0001-10iooi 1.000E-1 1.000-10 1OE-10 1.0E-010001-10 1.000E-101550.00 1.075E#01 1.465E+00 2.462E#00 3.3271-02 1.593E-02 8.444E-03 2.235E-07 5.902E-07 5.504E-07 3,33s6E-02 1,572E-0200 1610.00 9.262E#00 1.119E*00 3.561t400 2,976E-02 9.3211-03 1.450L-02 2.!)13E-07 1.099E-06 1.029E-06 2.999E-02 1.291E-021670.00 S.2371.00 7.460E-01 3.026E#00 1.9951-02 5.510F-03 1,7671-02 3,629E-07 1.413L-06 1.287E-06 2,005L-02 8,847E.031730.00 5.194F+00 2.5791-01 3,525E+00 1.9291-02 5.164E-03 1.812L-02 2.675E-07 1,125L-06 7,3081-07 1.9351-02' 4,585E-031790.00 4,522E#00 5#362E-01 9.798E.01 2.109E-02 '.6511-03 6.814E-03 2,655E-07 6.045L-07 7.959E-07 2.1111-02 5.674E-03 flp1836.00 8.915E.00 8.424E-01 0.1551-Oj 3.538E.02 6.370E-03 9,0'JSL-03 4.405E-07 2.330E-06 1.178E-06 3,542E-02 Z.030E.o2 z~ , 836.00 1,216E-01 5,998E-01 1.214E.01 1.333E-02 6.600E-03 1.218E-02 3.472E-07 7.7711-07 1.4401-06 1.337E-02 5*O2iE.03 nj~:'0184J6.00 -3.24J6E-01 6,988E-01 5.722E-02 1,358E-02 6.349E-03 8.6511-03 6.323E-07 9.575E-07 1.497E-06 1.36it-02 2,731E:03 ni1846.00 4.325E-02 8,2101-03 9.859E-03 9.320E-03 2.096E-03 3#161E-03 6.809E-07 1.405-06 .1..261E-06 9.327L-03 2.220E-03 c1880.00Q1894.001804.00

0u

1920.00

1934,00 31943,50>1943.50 71940.501945.500

1946,50

TIME1 u V w u V NU V w GAMMA ERROR PSI ERROR

1432.001432.001090.001550.00 1.000E-10 t.8000-10 1.000toto 1.000f-10, 1.000E10 1.000E-10 1.000E-10 I.o00E-10 1.000E10 2.884Z-14 2,892E-141550.00 1.071t.01 2,259E+00 2.001E+00 3,293E-02 1.816E-02 3#79?E103 2,267E-07 4#093E-07 6,945E-07 9.976E.06 1,0811-061610.00 9.236E+00 3.1601.00 2.104E*00 2.963E-02 1.647E-02 5,809L-01 2,092EwO7 3.bb8-07 1.4611-06 1.3051-05 2,359Em061670,00 5.229E#00 2.015E#00 2.397[+00 1,990Ew02 1,185E-02 1.0281.02 3.6021-07 7.159--7 1.772E-06 1.3611-05 9.079E-061730.00 5.1911.00 1.793E.00 3.050E+00 1.927E-02 1.OOOF-02 1.598E-02 2.700E-07 5,913E-07 1.206E-06 2.0321-05 10653E-051790.00 4.521E+00 2.603E-01 1.092[+00 2,111E-02 7.325E-03 3.718L-03 2.8781-07 6.895L-07 7.22bE-07 2.9101-05 5,190E-061836.00 8,912E*00 6,77.7E-01 6.939E-01 3.540E02 1.010E-02 4,424E-03 41,566E-07 1.176E-06 2.330E-06 4.987E-05 7.5701-061836.00 1.240E-01 3.356E-01 5.112E-01 1.338E-02 9.218E-03 1,0281-02 3.0177E-07 1,591E-06 3.797E-07 2.506E-05 1.758E-0518.*46.00 3.2771-01 0.917E.01 4,978E.01 1.362E.02 3.663E-03 I,Dc3E-02 6,364E-07 1.419E-06 1,067E-06 2,318L-05 1,777E-051806.00 4.330E-02 4.3.48E-03 1.1901-02 9@327E-03 2.745E-03 2.590E-03 6.864E-07 1,500E-06 1.143E-06 1.587E-05 'J,590E-061860.001870.001880 .001890, 001900.001914.00THSSGETOTALNOPR TD1920.00THSSGETOTALNOPRTD1930.001943.501943,501940501905.501948.50

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GROUP LS RADAR ALTIMETER BIAS

PO3ITION ESTIMATE ERROR VELOCITY ESTIMATE ERROR PLATFORM TILT ESTIMATE ALTITUDE VELOCITYmTIME R 0C R 0C R D C RATE ERROR MAG ERROR

c 1032.001431.00r1090.00 -~ 1550.00

-1610.00(71670.001730.00 0

1806.00Z

184600

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1934.00

1943,50 060OOE*00 0.000E~00 040001+00 00000f+00 0. .OOOE#00 00OO0E+00 0.OOOE+00* 0*000E+00 0.OOOE'00 0,000e.00 0,000E+001903.50 1.00OE+00 2 31SE-03 2.159E-03 1.00OE-01 2.73SE-oo 1,726.E-04 0,0O0E+00 0.000E+00 0.OOOE+Oo 10OO0E-01 2.166E-031940.50 9T97TE-0I Z.*22SE-03 2.156E-03 5,41SE-02 1.634F-04 7.770E-05 0.OOOE+00 0,OOOE+00 0.OOOE+00 5.418E-02 8.817E-041905.50 9*912E-0l 2.216E-03 2,2201-03 1*707E-02 To7S0E-05 1,744E-06 0@00OOE+00 0,OOOE+00 O.OOOE+00 1.707E-OZ2 ,176E-041946.50 9.966E-01 2ot89E-03 2.270E-03 4975?E-03 'i.937E-o5 2.818E-05 0.OOOE+00 0.OOOE+00 0.000E+00 44787E-03 8.40E.OS

TIME 'U V w U V w U V N GAMMA ERROR PS1 ERROR

1432.001432.001090.0015500001550.001610.001670.001730.001790.001836.001836.001846.001506.0001860.00

N. 1870.001880,001894.001900.0019101.001920.001930.001943.50 0:0001+00 o0oo00E400 0,000E+00O0.OOOE#00 0.OOO0"00 000OOE000 0.OOOE*00 0s000E+00 0.OOOE+00 0,000E+00 0,000e+001903.50 1..000E+00 1.0SE-05 2.154E-03 1.000E-0i 0.307E*05 4.537E-05 0.OOOEtO0O .000E;00 0.000E+00 30177E-04 1.442E-071940.50 9;797r-ol 1.249E-0,3 lt987E-05 5#41$E-02 3.96OC-o5 44034E-05 0.OOOEeO0O .O00E+0o 0.OOOE+00 1.768E-04 1,316E.071905.50 9.91ZE.01 3.704E-05 S.849E-05 1.707E-oz 3.881E-05 3.886E-05 0.000E+00 0.OOOE+00 O*000E100 59722E-05 1.303E-071906.50 9.966E-01 4.786E-05 7.532E-OS 4.76TE.03 3.872E-05 3.862E-05 0.000E+00 09000E+00 0.000E+00 1*627E-0S 1.313E-07

GROUP 15 RADAR ALTIMETER FIRST-ORDER MARKOV mM POSITION ESTIMATF ERROR VELnCITY ESTIMATE RROR PLATFORM TILT ESTIMATE ALTITUDE VELOCITYTIME R 0 c P D C p D C RATE ERROR MAG ERROR

1432.00

1432.00

1590.00Isso.oo

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1790,001836.00

21836.00 .1846.0018 06.00

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1945.50 9,790E-01 Pl.9E-03 2,177E-01 5.77E-01 1,214E-03 1.232E-03 1.O0OE-10 1.00OL-10 1.000E-10 5,477E-01 5.708E-03 0196.50 1.023E.o00 2.?75E-03 2.290E-03 5.234E-01 1.149E-03 1.182E-03 1.000E-10 1OOOE-10 1.000E-10 5.234E-01 4933E-03 ZTIME U U U

TIME u v w

GAMMA ERROR PSI ERROR1432.001432.001090.001550.001550,001610.001670,001730.001790.00183b,001836.00

1806.001806.001860.001874.001884.001890.001900.003914.001920.00193a.0013. 50 l OnOE-o toOE-io 1%000E-16 1.O00E-1O 1. 00010 1.O00E-10 1.000E-10 1,O00E-10 1.000E-10 3.17TE-13 3.178E-13"103.50 1,000Eo0 1.34SE-OS 2%134E-03 1.000E-01 4.307E-05 4.537E-05 1.OOOE-10 1,000E-10 1.000E-10 3,177E-04 1.OQZE-O71900.50 9276E-01 1.278E-05 2031f05 5.0 E.01 3,9004Fo05 4,081E05 .000E-10 1.000E10 1,000E-10 1.66E-03 1.332E-071905.50 9790E-01 3.898E-05 6152E-05 5.47E-01 4.936F-05 5T724E-05 1.000E-0 1.000E-10 1.000E-10 1.836E-03 1.919E-071946.50 1.023E+00 6,21E-05 1.009E[-0 5.234E-01 5.432E-05 6.545E-05 1.000E-10 1,000E.10 1000E-10 1,779E-03 2.22QE-07

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GRlOUP 21 FIRST-OnDER MAflKOV DENSITY DEVIATIONS

CPOSITInN ESTIMATF ERROR VILOCITY ESTIMATE ERROR PLATFORM TILT ESTIMATE ALTITUDE VELOCITY

TIME. D. C R0 C R D C, RATE ERROR HAG ERROR -

193P.00 3;8711+A2 2.197E*OP. 4'830F+0? 1*09SE400 1.80IF-01 2.90it-0i 1.930F.06 7.462E-06 3.806E06b 9.78Eo .bE001432.00 .3.IlF0 .A51E#01 7.3t9E.0j A.62?E-01 9.037E-02 2.016E-01 3.040OE-06 1.2'4bE-05 3.b57E-06 8.S62Em01 1.554JE-01 C

541490.00 3.675F*o2 9.161E+01* 3.683E*01 q.222E-01 2.773E-01 2.040E-01 5,22SE-06 1,494E-05 6.120E-.06 .9.2to1E-01 2,788E-01

*Tzme U V wU V NU V w GAMMA ERROR P31 ERROR n)S01432.00 4;051F+02 i.72?4E+02 2;103E.0e i.096E+00 1.Boo!'.o1 3.006E-01 1.853E--0b 6.739E-06 5.0OLJE.06 I.605E.04 5.'J32E.05 z

1432.00 3,669F+02 3..156E,6. 6,3'4E+01 8.59SE-01 1:731E-01 1.562E-01 3.390F.-Ob 1.028E-05 7.797E-06 1.404E"0u 2.580t-Cs

149000 3.6S8E*02 5.729E.01 6.648F~o1 9.139E-01 3..303F-01 I..603E-01 .5.424~E-06 7.1'JSE-06 1,44E-05 .1.99SE-04 3.149E-OS m0

GROUP 22 CROS3wIN0 AND HEADW14D -

POSITION ESTIMATF ERROR VELOCITY ESTIMATE ERROR PLATFORM TILT ESTIMATE ALTITUDE VELOCITY0TIME R D C. R D C R D RATE ERROR MAC ERROR

1832.00 209~8F.002 1.035E*02 2;102F+0j 602Z0E-01 7.243F-A2 2.01 E-01 7.08bE-0? 5.02.0E-06 .1*660E-06 6o.88E-01 1.123E.01

1032.00 2.1OIt"02 5.137E+00 21382F+0j 5.58SE-oj 3.552.E-02 t110J6E-OI 1.836F.-06 8.567E-06 1.b201-0b 5,535E-01 1,06bE-010

1090.00 2.499E402 5.281E401 1.401E+01 6.799E-01 1.792F-01 1.752E-01 3.'J59E-06 1.069E-05 6.008E-06 6.82SE-01 2.25ZE-01 0

WTIME U V U V u GAMMA ERROR PSI ERROR

143P.60 2,28'JE4802 2.o.9qrEo02 a,374f+0j 6.782E-01 l13u.2F-01 1,819E-01 7.503E-07 .3.S6qE-06. 3.893E-06 i.OITE-04 [email protected] 2,10oE*02 6.341E+00 2,0741E+0 1 5.54a6E-0I 1.2ua5E-61 9.907E.-02 2.097E-06 6jobSE-06 6.178E-06 9.lo'JE-05 1.676E-051490.00 2.487E+02 4.192E.01 '.307E.ol 6.73if.01 2.632E-01 5.?57E-02 3.617E-06 4.216E-06 1.146E-05 1,469E-04 1.04J4E-05

GROUP 23 DRAG COEFFICITENT VARIATIONS

POSITION ESTIMATF ERROR VELOCITY ESTIMATE ERROR PLATFORM TILT ESTIMATE ALTITUDE VELOCITY

TIME R 0 CR D C . R D C RATE ERROR MAC ERROR

1032.00 2%6A3E+03 1*600OE+03 3;680403O 7.992E+06 1;opbr.00 2.lstE*00 1.Io3E-65 s.549E-OS 2.556E-05 7,047E*00 1,373E+00

1032.00 2,5?.'E~n3 1.019E.02 5,7O1E+07 6,154E+00 5,656E-01 'S.535E.00 2.309E-05 9.127E-05 2.45&JEP05 6.082E+00 1.2~42E+001090.00 2.630F+03 6.136E+02. 2.199F+02 6.748E#00 J.8AOE+00 1.680E+00 3,694E-05 1.091E-04~ 5.320E.05 6,775E+00 2,179E+00

TIME U* V U V w U .V w GAMMA ERROR PBI ERROR

1432.00 2,818E+03 3.573E*03 l;619F.+01 T.qsl1E4.0 1.605F#00 2o188E+00 10108E-05 4.315E-05. '.325E-65 1.158E-03 4.031E-04

1432.00 2,S311+03 2.S90E00 '&,796E*Op 6.115E+00 I.L416E+00 1.07uL*00 2.575F-05 6,T15E-05 6.553E05S 9,999E-04 1.814~E-04.1490.00 2 .617F+03 4,242E+02 5.60?E*02 6,681t.00 2.555E~o0 8.503E-01 3.854E-05 4.823E-05 1.109E-04 i.458E.03 10670E-04


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Aerospace and Electronic Systems, September 1973.

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33. Justus, C.G. and Woodrum,A., "Short and Long Period Atmo-spheric Variations between 25 and 200 km," NASA CR-2203,February 1973.

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35. Cole, A.E. and Kantor, A.J., "COESA TaskGroup I Report onProposed 50 to 100 km Revision to U.S. Standard Atmosphere1962," AFCRL-72-0692, November 1972.

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