CSDL-T-902

AN ADAPTIVE GUIDANCE ALGORITHM

FOR AN AERODYNAMICALLY ASSISTED

ORBITAL PLANE CHANGE MANEUVER

by

Capt. James A. Blissit

May 1986

{NASA-CR-177296) AN ADAPTIVE GUIDANCE N86-26S33

ALGORITHM FOR AN AERODYNAMICA1LY ASSISTED

OBBITAL PLANE CBAKG E MANEUVER H.S..Thesis

(Draper (Charles Stark) Lab., Inc.) 229 p Onclas

HC A13/HF A01 , CSCL 09B G3/61 43592

The Charles Stark Draper Laboratory, Inc.

Cambridge, Massachusetts 02139
AN ADAPTIVE GUIDANCE ALGORITHM FOR AN AERODYNAMICALLY ASSISTED

ORBITAL PLANE CHANGE MANEUVER

by

James A Bli ss i t Jr.

B. S., United States Air Force Academy

(1975)

SUBMITTED IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE

DEGREE OF

MASTER OF SCIENCE IN

AERONAUTICS AND ASTRONAUTICS

at the

MASSACHUSETTS INSTITUTE OF TECHNOLOGY

JUNE 1986

James A. Blis s it Jr., 1986

Signature of Author

Approved by

Certified by

Department of Aeronautics and. Astronaut ics

fTa^ 16, 1986

_ ^

\J Ti

Technical S

Timothy J. Brand

Technical Supervisor ,CSDL

/(_

' Professor Antonio L. Elia s

Thesis Supervisor,

Department of Aeronautics and Astronautics, MIT

Accepted by

Professor Harold Y. Wachman

Chairman, Department Graduate Committee
AN ADAPTIVE GUIDANCE ALGORITHM FOR AN AERODYNAMICALLY ASSISTED

ORBITAL PLANE CHANGE MANEUVER

BY

JAMES A. BLISSIT JR.

Submitted to the Department of Aeronautics and

Astronautics on 16 May 1986 in partial fulfillmen t of the

requirements for the Degree of Master of Science

in Aeronautics and Astronautics

ABSTRACT

Using analysis results from the POST trajectory optimization program, an adaptive guidance algorithm is developed to compensate for density, aerodynamic and thrust perturbations during an atmospheric orbital

plane change maneuver. The maneuver offers increased mission flexibili -

ty along with potential fuel savings for future reentry vehicles.

Although designed to guide a proposed NASA Entry Research Vehicle, the

algorithm is sufficiently generic for a range of future entry vehicles.

The plane change analysis provides insight suggesting a straight-forward

algorithm based on an optimized nominal command profile. Bank angle,

angle-of-attack, and engine thrust level, ignition and cutoff times are

modulated to adjust the vehicle's trajectory to achieve the desired endconditions. A performance evaluation of the scheme demonstrates a capabilit y to guide to within 0.05 degrees of the desired plane change and

five nautical miles of the desired apogee altitude while maintaining

heating constraints. The algorithm is tested under off-nominal conditions of ±30% density biases, mwo density profile models, ±15% aerodynamic uncertainty, and a 33% thrust loss and for various combinations of

these conditions. Based on fuel comparisons with results of the optimization program, the guidance scheme offers a nearmoptimum solution

without the complexity of real-time optimization.

Technical Supervisor: Mr. Timothy J. Brand

Title: Division Leader, Charles Stark Draper Laboratory,

Inc.

Thesis Advisor: Dr. Antonio-L. Elia s

Title: Assistant Professor of Aeronautics and Astronautics
ACKNOWLEDGEMENT

The production of this thesis was influenced by quite a few people

and it is only fitting that I should start with an expression of my

thanks and appreciation to them.

First and foremost, without the continuous support of my family

(Maria, Carmen, and Jessica), I could have never made it to this satisfying point of my career. To them, I send my sincere thanks for everything and my love, always.

Thanks to my advisor, Professor Antonio Elias, and my Draper Supervisor, Tim Brand, for their support, advice and guidance. To the rest

of the technical staff at Draper, especially Bob Richards with his continuous willin g assistance and advice, Bernie Kriegsman with his constant words of encouragement, Steve Bauer who got me out of the office

often enough to keep my in-office time efficent and Florie Fabiano who

is responsible for making this document look as great as it does, I say

thanks for everything. Draper has certainly been an enlightening place

to work and a highligh t of my time at MIT.

Thanks to Howard Stone, Dick Powell, and the rest of the Vehicle

Analysis Branch at NASA Langley Research Center who provided a great

place to start my research experience and continued to provide assistance, support, and advice throughout my effort.
A special thanks to the men and women of our space program who

through their efforts have provided a great deal of motivation to myself

and I am sure, countless others. The price of their efforts can be

quite high as the tragedy of "Challenger" has reminded us all , but the

rewards of success can be truly sensational and beyond our imagination.

Thus, I dedicate this paper to the crew of "Challenger" and all the men

and women who follow them in our quest for the stars. Good luck!

Finally , I would lik e to extend a special thanks to one person in

particular. Like most things I have encountered as I have moved along,

this thesis is very much a product of the work and effort that was put

into it. Whether it sinks or floats wil l be left to history, but the

contribution that it has played in my overall education is already fixed

and can not be modified or deleted. Therefore, I say thanks for the

dedication and effort that I exerted to make this a worthwhile contribution, because this time has surely been a rewarding experience.

Thanks!

This report was prepared at the Charles Stark Draper Laboratory,

Inc. under Contract NAS1-18061.

Publication of this report does not constitute approval by the Draper Laboratory or the National Aeronautics and Space Administration

(NASA) of the findings or conclusions contained herein.
I hereby assign my copyright of this thesis to The Charles Stark

Draper Laboratory, Inc. Cambridge, Massachusetts.

James i. Bli ss11, JR

Permission is hereby granted by the Charles Stark Draper Laboratory,

Inc. to the Massachusetts Institute of Technology to reproduce any or

all of thi s thes i s.
TABLE OF CONTENTS

CHAPTER PAGE

1 . INTRODUCTION 16

1.1 Thesis Contribution and Scope 17

1.2 Thesis Overview 18

2. PROBLEM BACKGROUND 20

2.1 Introduction 20

2.2 Concept Justification 24

2.3 Vehicle Description 27

2.4 Baseline Trajectory 32

2.4.1 Background and Strategy 32

2.4.2 Trajectory Development and Understanding 34

2.5 Guidelines and Requirements 44

3. PROBLEM DEVELOPMENT 48

3.1 Introduction 48

3.2 Guidance Characteristics and Concepts 49

3.3 Control Variables and Pairings 53

4. TRAJECTORY ANALYSIS 56
CHAPTER PAGf

4.1 Introduction 56

4.2 Method of Analysis 58

4.2.1 Objective 58

4.2.2 Specific Approach 59

4.2.3 Limitations 60

4.3 Overall Results 62

4.4 Detailed Results 64

4.4.1 Density Perturbations 66

4.4.2 Aerodynamic Uncertainties 74

4.4.3 Heat-Rate Variation 82

4.4.4 Single-Engine Failure 84

4.4.5 Multipl e Off-Nominal Conditions 89

5. GUIDANCE SCHEME 108

5.1 Introduction 108

5.2 Guidance Algorithm 111

5.2.1 Entry Phase 111

5.2.2 Cruise Phase 113

5.2.3 Exit Phase 116

5.3 Guidance Rationale 119

5.3.1 Entry Phase 119

5.3.2 Cruise Phase 120

5.3.3 Exit Phase 122
CHAPTER PAGE

6. GUIDANCE ALGORITHM TESTING 124

6.1 Introduction 124

6.2 Evaluation Criteria and the Nominal Simulation 125

6.3 Simulation Test Results 129

6.3.1 Single Off-Nominal Condition Tests 129

6.3.2 Multipl e Off-Nominal Condition Tests 163

7. SUMMARY, CONCLUSIONS, AND TOPICS FOR FURTHER RESEARCH 177

7.1 Summary 177

7.2 Conclusions 178

7.3 Topics for Further Research 179

7.3.1 Node Control 180

7.3.2 Plane Change Trajectory Improvements 180

-7.3.3 Heat-Rate Control 181

' 7.3.4 Minimum Fuel 181

7.3.5 Apogee Control 182

LIST OF REFERENCES 183

APPENDIX

A. POST 187

A.1 Introduction 187

A.2 POST Description 187
P.HAPTER PAGE

A.3 Trajectory Generation Example 190

A.3.1 Input Fil e Description 206

A.3.2 Output Description 209

B. DRAPER ERV COMPUTER SIMULATOR AND GUDIANCE ALGORITHM 224

B.1 Simulator Description 224

B.2 Simulator and Guidance Algorithm Computer Program . 224
List Of Illustrations

PAQf

2-1 Candidate missions for the entry research vehicle .......... 23

2-2 Typical entry profile for an ERV synergetic plane change

followed by reentry ........................................ 23

2-3 Comparison of fuel requirements for all-propulsive maneuvers

and synergetic maneuvers to produce an orbital inclination

change of the ERV .... ...................................... 26

2-4 Comparison of AV requirements for all-propulsive maneuvers

and synergetic maneuvers to produce an orbital inclination

change of the ERV .......................................... 26

2-5 Optimum convective heat-rate for synergetic orbital

inclination change using the ERV ........................... 28

2-6 Artist's concept of the ERV ................................ 28

2-7 Three-view drawing of the ERV .............................. 31

2-8 Inboard profile of the ERV ................................. 31

2-9 Altitude profiles for synergetic plane changes

using the ERV .............................................. 36

2-10 Ang I e-of -attack profiles for synergetic plane change

using the ERV ............................................... 36

2-11 Bank profiles for synergetic plane change using the ERV .... 37

2-12 Engine throttling profiles for synergetic plane change

using the ERV .............................................. 37

2-13 Heat-rate profiles for synergetic plane change

using the ERA .............................................. 38

2-14 Heat-load profiles for synergetic plane change

using the ERV .............................................. 38

2-15 POST synergetic plane change altitude profile and generic

10
FIGURE PAGE

phase names 39

4-1 Heat-rate profiles for +30% density, -15% CL and +15% CD ... 94

4-2 Dynamic pressure profiles for +30% density, -15% CL

and +15% CD 96

4-3 Bank angle profiles for +30% density, -15% CL and +15% CD .. 98

4-4 Engine throttling profiles for +30% density, -15% CL

and +15% CD 99

4-5 Heat-rate profiles for -30% density, -15% CL and +15% CD ... 101

5-1 Guidance algorithm phases 110

5-2 Synergetic plane change guidance system summary 112

6-1 Altitude profiles for QDOT = 80 and 125,

nom i na1 cond i t i ons 130

6-2 Vehicle velocity profiles for QDOT = 80 and 125,

nom i na1 cond i t i ons 131

6-3 Angle-of-attack profiles for QDOT = 80 and 125,

nom i na1 cond i t i ons 132

6-4 Bank angle profiles for QDOT = 80 and 125,

nom i na1 cond i t i ons 133

6-5 Engine throttling profiles for QDOT = 80 and 125,

nominal conditions 134

6-6 Heat-rate profiles for QDOT = 80 and 125,

nominal conditions 135

6-7 Inclination change profiles for QDOT = 80 and 125,

nom i na1 cond i t i ons 136

6-8 Apogee and perigee altitude profiles for QDOT » 80 and 125,

nom i na1 cond i t i ons 137

6-9 Altitude and vehicle velocity profiles for QDOT = 80,

+50% dens i ty bias 141

6-10 Angle-of-attack and bank angle profiles for QDOT = 80,

+50% dens i ty bias 142

6-11 Engine throttling and heat-rate profiles for QDOT = 80,

+50% dens i ty bias 143

6-12 Inclination change, apogee and perigee altitude profiles

for QDOT = 80, +50% density bias 144

11
FIGURE PAGF

6-13 STS-4 dens i ty prof i1e 147

6-14 CSDL density shear model profile 148

6-15 Altitude and vehicle velocity profiles for QDOT =125,

sts-4 density shear profile 151

6-16 Angle-of-attack and bank angle profiles for QDOT =125,

STS-4 dens i ty prof i 1 e 152

6-17 Engine throttling and heat-rate profiles for QDOT =125,

STS-4 dens i ty prof i 1 e 153

6-18 Inclination change, and density perturbation profiles for

QDOT = 125, STS-4 density profile 154

6-19 Altitude and vehicle velocity profiles for QDOT = 125,

-25% CL and +25% CD 158

6-20 Angle-of-attack and bank angle profiles for QDOT = 125,

-25% CL and +25% CD 159

6-21 Engine throttling and heat-rate profiles for QDOT = 125,

-25% CL and +25% CD 160

6-22 Inclination change, apogee and perigee altitude profiles

for QDOT = 125, -25% CL and +25% CD 161

* 6-23 Altitude and vehicle velocity profiles for QDOT = 125,

single-engine failure at level-off 165

6-24 Angle-of-attack and bank angle profiles for QDOT = 125,

single-engine failure at level-off 166

6-25 Engine throttling and heat-rate profiles for QDOT = 125,

single-engine failure at level-off 167

6-26 Inclination change, apogee and perigee altitude profiles

for QDOT =125, single-engine failure at level-off 168

6-27 Altitude and vehicle velocity profiles for QDOT =125,

-30% density, -15% CL and +15% CD, single-engine failure

at level-off 171

6-28 Angle-of-attack and bank angle profiles for QDOT = 125,

-30% density, -15% CL and +15% CD, single-engine failure

at level-off 172

6-29 Engine throttling and heat-rate profiles for QDOT = 125,

-30% density, -15% CL and +15% CD, single-engine failure

at level-off 173

6-30 Inclination change, apogee and perigee altitude profiles

for QDOT =125, -30% density, -15% CL and +15% CD,

12
OfiilRL £Afi£

single-engine failure at level-off 174

A-1 Sample of POST output data for cruise phase 211

A-2 Sample of POST output data for circularization phase 212

A-3 Sample of POST trajectory summary 213

A-4 Baseline altitude profiles 215

A-5 Baseline vehicle velocity relative to the atmosphere

prof i 1 e :. 216

A-6 Baseline angle-of-attack profiles 217

A-7 Baseline bank angle profiles 218

A-8 Baseline engine throttling profiles 219

A-9 Baseline heat-rate profiles 220

A-10 Baseline heat-load profiles 221

A-11 Baseline dynamic Pressure profiles 222

A-12 Baseline inclination angle profiles 223

B-1 ERV top-level computer simulation and guidance

algorithm flowchart 228

13
List of Tables

TABLE PAGE

4-1 Density bias "Open-Loop" results with POST 68

4-2 Density bias optimization results for QDOT = 80 70

4-3 Density bias optimization results for QDOT = 125 71

4-4 Density bias optimization results for QDOT = 125

with entry parameters fixed 72

4-5 Aerodynamic uncertainty "Open-Loop" results with POST 76

4-6 Aerodynamic uncertainty optimization results

for QDOT = 80 77

4-7 Aerodynamic uncertainty optimization results

for QDOT = 125 78

4-8 Aerodynamic uncertainty optimization results for QDOT = 125

with entry parameters fixed 79

4-9 Single-engine failure "Open-Loop" results with POST for

QDOT = 125 86

4-10 Single-engine failure optimization results for QDOT = 125 .. 88

4-11 Multiple off-nominal "Open-Loop" results with POST for

QDOT = 125 91

4-12 Multipl e off-nominal optimization results for QDOT = 125

with entry parameters fixed 93

4-13 Multipl e off-nominal optimization results for QDOT = 125

with entry parameters fixed 102

4-14 Multipl e off-nominal optimization results for QDOT = 125 ... 105

4-15 Multipl e off-nominal optimization results for QDOT = 125 ... 106

6-1 Density bias test results for QDOT * 80 139

6-2 Density bias test results for QDOT « 125 140

6-3 Density shear test results for QDOT = 125 150

6-4 Aerodynamic uncertainty test results for QDOT = 80 156

6-5 Aerodynamic uncertainty test results for QDOT = 125 157

6-6 Single-engine failure test results 164

14
TABLE PAGE

6-7 Multipl e off-nominal test results for QDOT = 80 169

6-8 Multipl e off-nominal test results for QDOT = 125 170

15
CHAPTER 1

INTRODUCTION

A potential joint NASA/Air Force atmospheric reentry research flight

experiment is being considered for the 1990 time frame. This fligh t

experiment wil l help provide the technology base for development of

future reusable earth-to-orbit transportation systems and Air Force

aero/space vehicles. The experimental vehicle is envisioned to be carried aloft and deployed in orbit by the Space Shuttle. After executing

a deorbit procedure, the vehicle wil l reenter the atmosphere to perform

a variety of maneuvers and experiments and eventually glid e to a landing

site.

One of the more interesting maneuvers is the synergetic orbital

plane change. A synergetic plane change is a maneuver in which a change

in the orbital inclination is accomplished through a combination of

aerodynamic and propulsive forces rather than through propulsion alone

(17). The guidance system presented here is designed to guide a winged

vehicle such as the proposed NASA Entry Research Vehicle (ERV) during

the atmospheric phase of a synergetic plane change mission. This mission involves a deorbit maneuver and aerodynamic turn in the atmosphere

combined with a powered cruise and exit back into orbit. The guidance

16
system takes the vehicle from entry interface (400,000 feet), through

the atmospheric maneuvering turn and back to orbit.

1 .1 THESIS CONTRIBUTION AND SCOPE

This thesis present the problem analysis, development, and testing

of a guidance algorithm for a synergetic plane change maneuver for

NASA's ERV. The motivation for this effort was the design analysis

requirements for the development of the guidance scheme. The major contribution is the new level of insight on how to efficiently perform this

maneuver in the presence of density and vehicular perturbations. The

guidance algorithm is a summation of this insight and satisfies NASA's

requirement for a generic guidance scheme. The testing of the algorithm

demonstrates the effectiveness of the scheme and indirectly adds support

or proof to the analysis results. Hence the parts fit nicely together

as a complete monograph with each part both resting on and supporting

the other parts.

The guidance algorithm is only one possibility of many guidance

schemes that could and will , as demand requires, be developed to satisfy

the mission. However, it completely satisfies the stated NASA requirements at this conceptal phase of ERV development. In addition, the

algorithm offers a basic scheme to modify and refine as the ERV program

17
progresses and a standard to measure future schemes against. In short,

the guidance algorithm is a simple straight-forward application of the

results from the synergetic plane change trajectory analysis.

The algorithm testing was conducted on a tmree degree of freedom

computer simulator developed at The Charles Stark Draper Laboratory

(CSDL) for the NASA ERV contract. The algorithm testing is by no means

exhaustive but for the time and money constraints, it does offer a good

validation of the guidance algorithm.

1.2 Thesis Overview

In the remaining six chapters, this thesis details the problem

development and analysis along with guidance algorithm development and

testing. Chapter 2 presents background information for the ERV and synergetic plane change problem to include concept justification, vehicle

description, an explanation of the baseline trajectories, and basic NASA

guidelines and requirements. Chapter 3 contains a brief discussion of

the initia l problem development and formulation. Chapter 4 details the

problem analysis with a computer optimization program called "POST".

This analysis provides insight into the difficulty of the guidance problem along with specific means of handling density, aerodynamic and propulsive uncertainties. Chapter 5 presents our final iteration of the

guidance scheme along with rationale leading to the scheme.

18
The evaluation criteria and test results for the algorithm are contained in Chapter 6. Chapter 7 summarizes the work covered in this thesis and discusses what can be done to expand on the analysis insight and

possible algorithm improvements. Appendix A provides a brief explanation of "POST" along with an example trajectory generation. A brief

description and flowchart for the Draper Laboratory ERV simulator and

guidance algorithm computer programs are presented in Appendix B.

19
CHAPTER 2

PROBLEM BACKGROUND

2.1 Introduction

During the past few years, there has been resurgency of national

interest in vehicles that can maneuver in the atmosphere whil e returning

from orbit and vehicles that can fly in the atmosphere at hypersonic

speeds for sustained periods of time. This rebirth of interest in

hypersonics has resulted both from the success of the Space Shuttle and

from a developing awareness of the potential benefits that can be

derived from vehicles with the capability to operate between the limit s

of existing aircraft and spacecraft. Examples of the types of maneuvers

that are projected for this new class of vehicles are presented in Figure 2-1. Most of the projected requirements are dictated by either a

need for flexibility offered by a large landing footprint or by a

requirement for specific entry maneuvers such as a synergetic plane

change (4).

For vehicles that operate in space or in the upper reaches of the

atmosphere, an orbital plane change offers significant flexibility for

20
Earth overflight and space operations. Specifically, the studies of the

late 1960's indicated that the same type of high lift-to-drag ratio

(L/D) vehicle required for the synergetic plane change maneuver could

also perform Earth-return missions with large cross range capability,

could perform evasive maneuvers, and could have considerable militar y as

well as civilia n uses (17). This new class of vehicles wil l dictate new

requirements for the vehicle's thermal protection systems (TPS) and

thermo-structures design. The plane change maneuver wil l require a TPS

designed for a high heat-rate whil e the large landing foot print wil l

require a high heat-rate and heat-load capability (6). The Space Shuttle provides a good starting point but to achieve the potential fuel

efficiency and large landing foot print wil l require an improvement in

TPS capabi1ity.

Addressing the technology deficiencies for these future systems wil l

provide a significant challenge for the aerospace community. The synergetic or atmospheric plane change wil l be one of the more difficul t missions for future entry vehicles. Analysis to date have shown that

performance of the maneuver wil l require sustained flight at or near

entry mach numbers at altitudes over 200,000 feet. This is the fligh t

regime where the predicted aerodynamic trim of the Space Shuttle orbiter

was in error by as much as nine degrees body-flap deflection (4). Whereas the shuttle currently flies a tightly tailored trajectory, the

fligh t envelope for the ERV wil l be broadened to allow the vehicle to

21
maneuver during entry. The results from Space Shuttle flights presented

in Reference 4 have shown density variations as large as 60 percent in

the altitude regime where the synergetic plane change maneuver occurs.

This large density variation is random, can not be predicted and must,

therefore, be treated as an uncertainty in the estimated aerodynamics

and aeroheating used in the vehicle design.

With the ERV, NASA is attempting to address these issues and develop

a building block approach to handling the difficulties. The ERV is

designed to demonstrate atmospheric maneuvers that are beyond the capabilit y of existing spacecraft and aircraft. Presented in Figure 2-2 is

a typical altitude time profile for an entry research vehicle fligh t

where first a synergetic turn is performed, and then the vehicle performs a normal entry (4). As shown in the figure, the entry vehicle is

released from the shuttle, performs a deorbit burn, and then descends

into the atmosphere to an altitude of approximately 220,000 feet. There

the engines are started, and the vehicle cruises at a mach number of 25

until its exit point. It then uses a combination of angle-of-attack and

throttle to reboost into orbit while completing the plane change. After

remaining on orbit for several revolutions, the vehicle deorbits and

performs an entry and landing.

In order to accommodate the large density variations and vehicle

aerodynamic uncertainties referred to earlier, the new vehicle wil l be

22
ERV

Quick descent ~^\ \ \ /

V^-i^V

^ VtnT '

/

/Synergetic

/ plane change

'

down ran9e

\

S.

mid west

Figure 2-1. Candidate missions for the entry research vehicle [4].

Launch

Altitude100 000 ft.

Orbital return

2 •

I I

Outer limit Earth's atmosphere

60nmi (364 800 ft)

J2p_qOO_tt

Synergetic turn and controlled

thrusting, At = 23min 20° plane change^ Landing^

i i i i i i i i _i

123456 7 8 9 1 0 1 1

Ground elapsed time (GET), hrs

Figure 2-2. Typical entry profile for an ERV synergetic plane change

followed by reentry [4].

23
required to have a sophisticated adaptive guidance and navigation system. This system must be able to sense the atmospheric conditions and

provide necessary control inputs in real-time to enable the vehicle to

accurately achieve its end-conditions and remain within the vehicle's

flight boundaries. The best previous example is the Space Shuttle.

However, the shuttle worked around these uncertainties by designing into

the guidance algorithm large margins for error. This resulted in a

reentry trajectory with less concern for maneuvering efficiency than is

possible with the ERV and often the solution to handling the uncertainties (e.g., roll reversals) resulted in other problems lik e attitude

instability of the vehicle (8). The ERV is designed to demonstrate the

technologies required to perform a synergetic plane change mission and

other maneuvering entries. Thus, an acceptable guidance system must be

able to efficiently compensate for the discussed uncertainties while

safely executing the maneuver.

2.2 Concept Justification -

The strongest argument for a synergetic plane change is the opportunity for a significant reduction in the amount of fuel required for an

orbital plane change maneuver. Many studies have shown that for a large

plane change in low-earth orbit, a synergetic plane change wil l use less

fuel than an al1-propulsive maneuver. The initia l motivation for this

work was the recognition that orbital plane changes required very large

24
character! si tc velocities UV) . As pointed out by Caudra and Arthur

(18), the AV required for a single-impulse 60 degree plane change is as

large as that required to place a satelite in low earth orbit in the

first place. The fuel required for the 12,000 pound ERV to perform a

20-degree orbital inclination change is shown in Figure 2-3, and the

corresponding velocity change requirement is shown in Figure 2-4.

Results are shown for an in-orbit, al1-propulsive maneuver and for two

synergetic plane change maneuvers, one constrained to a maximum convective heat-rate referenced to a one-foot radius sphere of 80 BTU/ft2-sec

and one constrained to a maximum heat-rate of 125 BTU/ft2-sec. For the

ERV to perform an all-propulsive plane change requires 7100 pounds (Ibs)

of propel lant. The same plane change using an aerodynamic turn limited

by heating to 80 BTU/ft2-sec requires 6400 pounds of propellant. If the

ERV can withstand a heating rate of 125 BTU/ft2-sec, the fuel savings

are even greater, and only 5500 Ibs of fuel would be required (6). It

should be kept in mind however, that significant fuel savings are only

realized for a vehicle that already requires the added weight for TPS

and wings. Or as Maslau concluded "... the results indicate that the

synergetic made can be more desirable than in-orbit propulsion only if

the aerodynamic performance must be provided for other reasons..." (20).

Further analysis done at NASA Langley Research Center (LaRC), determined that there is an optimum heat-rate depending on the amount of

desired plane change. Figure 2-5 provies a curve of this heat-rate

25
8rx103

7

6

Fuel

required, 4

Ib

3

2

1

o All propulsive

D Max heat rate 80 BTU/ft2-sec

OMax heat rate 125 BTU/ft2-sec

10

Ai, deg

15 20

Figure 2-3. Comparison of fuel requirements for all-propulsive maneuvers

and synergetic maneuvers to produce an orbital inclination

change of the ERV [4].

10rx103

g

8

7

6

AV, fps 5

4

3

2

1

o All propulsive

D Max heat rate 80 BTU/ft2-sec

O Max heat rate 125 BTU/ft2-

_i i i

sec

10

Ai, deg

15 20

Figure 2-4. Comparison of Delta V requirements for all-propulsive

maneuvers and synergetic maneuvers to produce an orbital

inclination change of the ERV [4].

26
verses inclination relationship for the ERV for angles through 20

degrees. Specifically, the results indicated that 125 BTU/ft2-sec was

nearly the optimum heat-rate for a 20 degree plane change and a higher

heat-rate of say 135 BTU/ft2-sec wil l actually require more fuel (6).

Thus, the maximum allowable heat-rate wil l significantly influence the

vehicle's fuel efficiency and, as it wil l be shown later, the vehicle's

trajectory and response to perturbations.

2.3 Vehicle Description

The candidate Entry Research Vehicle shown in Figure 2-6 is 25 feet

long w.i th a wing span of 13.9 feet. It is designed to occupy half of

the Shuttle cargo bay which is 60 feet long and 15 feet in diameter.

The vehicle has a distinct wing-body design with a blended wing-body

interface to increase the overall hypersonic performance of the vehicle.

The wing has a reference area of 177.4 square feet. Conventional elevens are used for pitch and roll control, and tip-fin controllers provide the yaw stability augmentation and control. A body flap assists in

trimming the vehicle hypersonical 1 y. Reaction control system jets are

used to control the vehicle in the upper atmosphere. The jets are

placed in the nose of the vehicle and in the tip fins. Figure 2-7 is a

three-view drawing of the ERV with the basic vehicle dimensions and

relationship with the Shuttle payload bay envelope. The fineness ratio

of the fuselage is approximately six, and its volume is 300 cubic feet.

27
130r

120

110

q.

BTU/ft2-sec

100

90

80

10

Ai, deg

15 20

Figure 2-5. Optimum convective heat-rate for synergetic orbital

inclination change using the ERV [6].

28
ORIGINAL PACE-IS

OF POOR QUALITY

29
Figure 2-8 shows an inboard profile of the ERV. The payload bay housing

many of the experiments is located in the nose of the vehicle where

atmospheric sensing can be performed with minima l engine combustion byproduct contamination. The avionics are located behind the payload compartment. To minimiz e the center-of-gravity shift as propellant is

depleted, the fuel and oxidizer tanks are positioned around the center

of gravity which is located at 67 percent of the body length. The electrical system and batteries are positioned between the fuel tanks,

approximately at the center of gravity. Propulsion is provided by three

Marquardt R-40-B rocket motors, each providing 1100 pounds of thrust

with an ISP of 295 seconds. The engines are aligned with the vehicle's

longitudinal axis and thrust through the vehicle's center of gravity.

The vehicle's launch weight from the Shuttle is estimated to be

11,430 pounds. 6000 pounds of propellant is carried to provide the

nearly 7000 feet-per-second AV required for a 20 degree synergetic plane

change maneuver with the Marquardt engines. The leading-edge sweep

angle is 72.5 degrees. An NACA four-digit series symmetrical airfoi l

mection is used for the wing and its hypersonic maximum L/D is approximately two. Complete coefficient of lif t (CL) and drag (CD) data is

provided in Appendix A. The thickness ratio was varied to maintain a

ha If-inch radius over the entire leading edge of the wing. The geometric shape of the wing was designed with the intention of reducing the

aerothermodynamic heating on entry. A heat pipe, in addition to new

30
Theoretical wing area = 177.40 ft2

Body flap area = 7.50 ft2

Eleven area per side = 6.07 ft2

Aileron area per side = 0.88 ft2

Tip fin area per side = 4.16 ft2

Shuttle payload

bay envelope

Figure 2-7. Three-view drawing of the ERV [5],

Plan View

Sym

about

Delta wing

Tip fin

Elevon

Rocket

3- engine

^-Body flap

»W*>>*« / -«»^fc Aileron

vol 42 ft3 L Oxidizer

tank

vol 42 ft3

Side View _, x . . Electrical system _ . . . /ON

Avionics, and batteries^ ^<=k«' *»*TM «)

Ref line -**-» ^~^ i^^l^^Tip fin

- Landing

skid

^-Body wing

"" interface

Figure 2-8. Inboard profile of the ERV [5].

31
thermal protection systems and materials, wil l be used to accommodate

excessive temperatures (6).

2.4 Baseline Trajectory

2.4.1 Background and Strategy

The vehicle analysis branch at LaRC defined a baseline or nominal

synergetic plane change trajectory. This baseline trajectory was developed by combining the best ideas and concepts that have emerged since

1961 when London's paper first convincingly demonstrated a significant

performance gain with a synergetic plane change. A brief listin g of

these basic concepts are shown below as an aid to understanding the

vehicle's design and baseline trajectory. A more detailed discussion of

these concepts are in reference 17.

1. For plane changes of less than 15 degrees, the all-propulsive

maneuver is generally more efficient.

2. An L/D of at least two is required to offer a significant advantage over the al1-propulsive plane change, and it is desirable to

maximize vehicle L/D.

32
3. Turns made at maximum CL (I.e., high angle-of-attack) are quicker

and hence more fuel efficient than the more gradual maximum L/D

turns. This improved efficiency is primarily a result of orbital

mechanics. A plane change made at the intersection of two orbits

(ie., the node) produces all inclination change whereas a turn at

the orbit apex (90 degree from node) produces no inclination

change, only a node shift. Hence, for maximum inclination change

and minimum node shift, the turn should be centered over the node

in the shortest duration possible.

4. A large reduction in total heat load can be achieved by carrying

out a short duration, high angle-of-attack maneuver rather than

the slower maximum L/D turn.

5. A thrusting turn (aerocruise) offers significant advantages over

the glidin g turn (aeroglide) when the desired plane change is

more than 10 degrees. This relationship is especilly true when

lower wall temperature (heat-rate) limit s are imposed.

6. Aerocruise vehicles use continuous thrusting sufficient to balance aerodynamic drag and maintain an altitude high enough to

allow optimum bank angles throughout the turn without exceeding

heat-rate 1im i ts.

33
7. For the high angle-of-attack aerocruise vehicle, assuming thrust

along the longitudinal axis, a significant portion of the plane

change results from out-of-plane thrust due to the high angle-ofattack and bank attitude of the vehicle during the turn.

8. For the quick, high angle-of-attack turn, the vehicle is banking

throughout the maneuver, and the exit should be timed such that

the plane change is accomplished just as the vehicle exits the

atmosphere to minimize fuel.

9. For each amount of plane change, there is a heat-rate that offers

optimum fuel efficiency. If this heat-rate can be achieved, the

resulting trajectory is more a combination aeroglide-aerocruise.

A part of the turn is achieved on the glidin g entry and the

remaining plane change is achieved during the powered exit with

almost no level-flight phase.

2.4.2 Trajectory Development and Understanding

Using these basic concepts, R.W. Powell and J.C. Naftel of NASA

Langley Research Center developed a baseline synergetic plane change

trajectory for a reference heating rate of both 80 and 125 BTU/ft2-sec.

These trajectories were generated on their three-degree-of-freedom Program to Optimize Simulated Trajectories (POST). References 1,2, and 3

34
offer a detailed description of this program while Appendix A provides a

brief description of POST along with an example of trajectory generation

for the synergetic plane change maneuver.

The strategy for both baseline plane change trajectories are similar. First a deorbit burn is performed. Once the vehicles enters the

atmosphere and the heating rate reaches the desired cruise value, the

engines are restarted and throttled to maintain a nearly constant vehicle velocity. At the end of this cruising phase the engines are commanded to maximum thrust, and the vehicle exits the atmosphere. Bank

angle and angle-of-attack are used during the entire atmospheric phase

so that the required inclination change occurs during the entry, cruise

and exit legs. Figures 2-9 thru 14 show profile plots for altitude,

angle-of-attack, bank angle, engine throttling, heat-rate (QDOT), and

heat load histories for both the 80 and 125 BTU/ft2-sec 20 degree synergetic plane change trajectories (6). Figure 2-15 shows a typical POSTgenerated altitude plot and the generic phase names that wil l be used

during the plane change trajectory discussion. These profiles are presented here as basic reference material that should be helpful during

the discussion in the remainder of this chapter. Appendix A provides

more details into the generation of these trajectories whil e Chapter 4

wil l cover these trajectories and their plots in detail.

35
h,ft 30-

Max heat rate,

BTU/ft2-sec

125

3x103

Figure 2-9. Altitude profiles for synergetic plane change using

the ERV [6].

Ai = 20°

40

30

a,deg 20

10

Max heat rate,

BTU/ft2-sec

125

80

1 2

Time, sec

3x103

Figure 2-10. Angle-of-attack profiles for synergetic plane change

using the ERV [6].

36
Ai = 20° Max heat rate,

0

-25

a, deg -50

-75

-100

(

BTU/ft2-sec

125

80

ii i

» \

» \ J

i i i

) 1 2 3

Time, sec

3x10^

Figure 2-11. Bank angle profiles for synergetic plane change using

the ERV [6].

1.00r

.75

Throttling .50

.25

Ai=20°

Max heat rate,

BTU/ft2-sec

125

80

1 2

Time, sec

3x103

Figure 2-12. Engine throttling profiles for synergetic plane change

using the ERV (0-off, 1 maximum) [6].

37
q,

BTU/ft2-sec

150

125

100

75

50

25

Ai=20° Max heat rate,

BTU/ft2-sec

125

80

1 2

Time, sec

3x103

Figure 2-13. Heat-rate profiles for synergetic plane change using

the ERV [6].

Heat

load,

BTU/ft2

Max heat rate,

8TU/ft2-sec

125

80

1 2

Time, sec

3x103

Figure 2-14. Heat load profile for synergetic plane change using

the ERV [6].

38
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Lang ley's objective was to obtain the most fuel efficient plane

change trajectory (within the adopted strategy) by allowing POST to find

optimum values for carefully selected guidance parameters (independent

variables). The 12 independent variables used for the baseline trajectories are listed below with a brief description of how POST uses the

parameter.

1. Geocentric Latitude: The latitude at which a deorbit burn is

started.

2. Perigee Altitude; The conically predicted orbital parameter

which determines when the deorbit burn is stopped. This determines

the AV of the deorbit burn and thus the position and velocity of the

vehicle at entry interface.

3. Entry Angle-of-Attaek; Angle-of-attack from atmospheric entry

to level-off.

4. Entry Bank Angle: The bank angle from atmospheric entry to level-off.

5. Cruise Angle-of-Attack; Angle-of-attack for the cruise phase of

flight.

40
6. Cruise Acceleration: The acceleration maintained during the

cruise phase of flight.

7. Exi t Inclination: The inclination angle where the exit is

started.

8. Exit Bank Angle; The bank angle during the powered part of the

exi t.

9. Exit Angle-of-Attack; The angle-of-attack during the powered

part of the exit.

10. Apogee Altitude: The conically predicted orbital parameter

which determines when the rocket engines are cutoff. This determines when the vehicle can fly unpowered back to the desired orbital

altitude.

11. Unpowered Exit Angle-of-Attack; The angle-of-attack after the

engines are cutoff during the exit phase.

12. Unpowered Exit Bank Angle; The bank angle after the engines are

cutoff during the exit phase.

The constraints and targeting parameters include:

41
1 . Maximum heat-rate

2. Desired inclination angle

3. Desired final orbital apogee altitude

4. Maximum throttle setting

It should be noted that total heat-load is not considered a constraint

for the relatively short plane change maneuver.

The optimizing parameter is final vehicle weight in the desired

final circular orbit. Thus, POST attempts to minimize the amount of

fuel used from the start of the deorbit burn until the ERV is back in

the desired circular orbit. During this time, the 12 guidance parameters (free parameters) are adjusted to achieve the targeting parameters

of inclination angle, apogee altitude, and maximum throttle setting. In

addition, as a means of controlling heat-rate and achieving the cruise

acceleration, POST can respectively vary bank angle and throttle setting

during the level-off and cruise phase.

To simplify discussion here and throughout the remainder of this

thesis, an abbreviated notation wil l be used to represent a heat-rate of

80 or 125 BTU/ft2-sec. Typically, the numbers 80 or 125 wil l be used

42
without units as stand-alone adjectives to describe trajectories or

cases, or as objects (e.g., heat-rate of 80 or heating rate of 125).

The more complete notation wil l be used where any uncertainty of meaning

might exist or occassionally as a reminder.

To provide additional understanding of the baseline trajectories, a

more detailed explanation of the previously presented trajectory plots

wil l be made. Figure 2-9 provides an altitude verses time profile

starting and ending at 400,000 feet (maximum extent of atmosphere). For

the synergetic plane change with a design maximum convective heat-rate

of 80, the ERV initiates level-off approaching the design heat-rate, and

the engines are ignited. However, for the case with the design maximum

heat-rate of 125 BTU/ft2-sec, the ERV phugoids at a heat-rate of approximately 100 BTU/ft2-sec and then continues to glide (engines off) until

the design maximum heat-rate is reached. Once the design maximum heatrate is reached, the level-off is started and the engines are ignited.

For the 125 case, the exit maneuver begins almost immediately. For the

80 case the cruise phase continues significantly longer before throttling to 100% and boosting back into orbit. These differences are the

result of a higher heat-rate allowing the vehicle to fly deeper into the

atmosphere, thus generating higher lif t to turn the vehicle quicker. As

stated earlier, the quicker the maneuver can be performed, allowing the

vehicle to remain close to the orbital node, the more efficient the

maneuver.

43
Figure 2-10 presents angle-of-attack profiles that are reasonably

simila r for the 80 and 125 cases with length of the cruise phase making

the biggest difference. The bank profiles in Figure 2-11 are again simila r for the two cases with the spike just prior to 1000 seconds due to

the ERV's heat-rate controller adjusting the bank angle to maintain the

designed maximum heat-rate during level-off. Figure 2-12, the engine

throttling profiles, displays the high percentage of maximum throttle

required for level-off and cruise for the 125 case as compared to the

relatively low throttle setting required for the 80 case. In Figure

2-13, the "double-peaked" heat-rate (QDOT) curve for the 125 case stands

out quite clearly from the "Table-top" curve of the 80 case. Finally ,

the heat-load profiles in Figure 2-14 demonstrate the significantly

higher total heat-load for the 80 case.

2.5 Guidelines and Requirements

NASA contracted with the Charles Stark Draper Laboratory to devolop

a guidance algorithm to satisfy the broad requirements for a future

entry vehicle. As stated earlier, the new class of vehicles wil l

require a sophisticated adaptive guidance and navigation system. This

system must be able to sense the actual conditions and provide necessary

control inputs in real-time to enable the vehicle to accurately achieve

it's end-conditions and remain within the vehicle's flight boundaries.

44
A primary objective in the development of the guidance system for

the ERV was that it be flexible. Many past guidance algorithms have

lacked flexibilit y because of severe computational constraints. With

the availabilit y of more advanced fligh t computers, the intent is to

devise a guidance law offering more flexibilit y even if it is more computationally intensive.

In addition, the guidance system should not require exhaustive premiss ion planning. It should explicitl y calculate on-board the desired

trajectory or guidance commands instead of relying on elaborate tables

which are highly mission dependent. Overall, this desire for greater

flexibilit y and independence is in response to the need for increased

operational performance to go along with the improved capabilities of

future space vehicles.

Other general guidelines include:

1. Explici t control of vehicle heat-rate

2. Maximum abilit y to handle dispersions

3. High accuracy

45
4. Investigate the use of both angle-of-attack and bank angle as

control variables.

5. Control capability for powered and unpowered maneuvering during

the atmospheric flight phase.

6. Minimiz e fuel usage

7. Usable in vehicles with low roll rates

8. Transportable between vehicles.

Specifically, the plane change guidance system should be able

to:

1. Achieve an inclination change of 20 ± 0.1 degrees and a final

apogee altitude of 160 ± 5 nautical miles (nm).

2. Demonstrate acceptable performance for a heat-rate constraint of

either 80 or 125 BTU/ft2-sec.

3. Demonstrate acceptable performance with atmospheric dispersions

(±30% density biases along with density shear models), aerodyna46
mic uncertainties (±15% for all CL and CD combinations), and propulsion performance loss of one of its three engines.

One caveat should be pointed out concerning the requirements. The

longitude of the ascending node (LAN) is not specifically required to be

controlled. It is important to remember that an orbital plane is usually defined in terms of inclination and longitude of the ascending node.

This defines a single orbital plane with respect to the equator and the

vernal equinox. Thus, a change from mne orbital plane to another

orbital plane will , in general, cause a change in both of these parameters. However, for the purpose of this thesis, only an inclination

change wil l be controlled. Longitude of the ascending node wil l be monitored to determine the magnitude of any shift.

47
CHAPTER 3

PROBLEM DEVELOPMENT

3.1 Introduction

The objective is to develop a guidance system to take the ERV

through the atmospheric phase of a synergetic plane change mission. The

powered fligh t guidance for accomplishing the deorbit and recircularization burns is well developed and documented. Starting with a well-defined reference trajectory, a- specific set of end-condition

requirements, one limitin g constraint (maximum heat rate), and a large

number of control parameters, the problem of developing a guidance

scheme seemed rather straight forward. However, it was not immediately

clear as to which combination of control variables provided the most

efficient solution. Some type of analysis was done for each combination

considered. However, some of the decisions made about the nominal guidance scheme were based on heuristic arguments rather than precise numei—

ical comparisons. Naturally, where possible, both means were used to

arrive at conclusions, however, numerical results alone can be misleading if they are not fully understood. Where appropriate, information

48
from other sources was used and referenced, rather than data generated

specifically for this study.

Still , with the nominal baseline trajectories that are presented in

Chapter 2, the problem is quite simply: how to develop an algorithm to

use the vehicle's control variables to achieve a 20 degree plane change

and a 160 nm apogee altitude in the presence of dispersions? Thus, it

was a matter of choosing a basic guidance concept and a pairing of

available control variables with the conditions to be controlled.

3.2 Guidance Characteristics and Concepts

Guidance systems span a spectrum of adaptability or flexibility from

very rigi d like an open-loop guidance system to a highly flexible system

which continually generates an optimum trajectory from the current position to the desired end-conditions based on the latest knowledge. As

stated in the previous chapter, the task was to develop a guidance system that is as flexible as possible within the constraints of the

expected fligh t computer capability.

In addition, NASA desired an explicit type of guidance system not

based on a reference trajectory requiring a great deal of pre-mission

planning and loading of data. In terms of explicit guidance concepts,

three general types of systems are available:

49
1. Perturbation

2. Predictor/Corrector

3. Real-time Optimization

Perturbation guidance generally relies on the explicit solution of

simplified equations of motion. Typical perturbation guidance trajectories are equilibrium glide, constant drag, and constant heating rate.

Often a relationship between the end-condition and the type of trajectory can be determined. For perturbation guidance, guidance commands are

composed of two parts—nominal and perturbed. In a broader sense, the

nominal could be a nominal reference command or a nominal reference trajectory profile. In either case, a nominal is calculated each guidance

cycle to achieve mission objectives. The perturbed part is used as

feedback to bring the vehicle back to the nominal. This has been used

by the Johnson Space Center for Shuttle entry and various aerobraking

schemes to force a vehicle to fly a drag acceleration profile to meet

mission objectives. The advantages for this type of scheme are:

1. Lower sensitivity to initia l conditions than an open-loop scheme

2. Guidance parameters are calculated explicitl y

50
3. Low computational load

4. Fligh t experience on Shuttle

While the disadvantages include:

1. No direct control on the end conditions

2. A lack of mission flexibilit y

3. Scheme is usually very vehicle specific

A predictor/corrector scheme on the other hand, generally works with

more accurate equations of motion. The guidance continuously computes a

new trajectory to go from the present position to the end-conditions.

The sensitivity of the end-conditions to changes in the control variables is determined by multipl e numerical intergrations of the trajectory. Once the sensitivity is determined, the current control profile can

be corrected to satisfy the end-conditions. Advantages for this type of

scheme are:

1. Sensed dispersions can be easily incorporated into the predictor

to improve the performance during the remaining part the mission.

51
2. The scheme is less vehicle specific and thus can be transferred

between vehicles more easily because changes in the vehicle model or

aerodynamic data base do not change the guidance scheme.

3. The predictor can use a conservative vehicle model to allow

additional margin for dispersions.

4. Very flexible in terms of using different control variables

Whil e the disadvantages include:

1. A higher computational load

2. Possible convergence problems

3. Not especially useful for control of boundary conditions (like

heating rate) that are unrelated to end-conditions

4. More difficult to verify and validate

Real-time optimization provides the most desirable guidance solution. At each guidance cycle a trajectory is calculated to hit the

end-conditions and minimize a cost function such as fuel usage. The

advantages include extreme flexibilit y and dispersion compensation capa52
bilit y while minimizin g the costs. The disadvantages are just a magnification of the predictor/corrector disadvantages of high computational

load along with reliabilit y and convergence concerns.

In the final analysis, a hybrid offered the best hope of satisfying

all of NASA's guidelines. A perturbation guidance scheme is used to

maintain the heat-rate boundary whil e a predictor/corrector is used to

achieve the end-conditions. As a means of achieving at least a nearly

optimum solution, insight gained from our trajectory analysis that wil l

be covered in Chapter 4 is used to develop techniques which allow efficient handling of perturbed situations.

3.3 Control Variables and Pairings

As a final step of breaking down the guidance problem, the degrees

of freedom and the constraints or end conditions that must be satisfied

with these control variables should be discussed. The objectives for

the plane change are:

1. Final orbital inclination angle control

2. Final apogee altitude control

3. Heat-rate constraint

53
4. Minimization of fuel usage

Whil e the parameters available to handle these conditions are:

1. Angle-of-attack

2. Bank angle

3. Engine thrust level (adjustable throttle setting)

4. Engine start time

5. Engine cutoff time

Simple accounting points out the fact that there are more control variables than trajectory objectives. Our approach to the problem was to

explore the use of all parameters but use only what was necessary. This

provides a straight forward scheme that can be more easily modified for

later developments or other vehicles.

Based on several sensitivity studies conducted by R. Richards of

CSOL (19), the effectiveness of angle-of-attack and bank angle was

determined to be very simila r during the last two-thirds of the plane

change trajectory in controlling inclination. In addition, either con54
trol variable was shown to be capable of correcting inclination errors

of up to 30% during the exit phase. The same group of studies indicated

that final apogee altitude could be easily control with engine cutoff

time. It is important to remember however, that because of the relatively high angles-of-attack during the powered fligh t phases, a significant amount of the turn and hence inclination change is directly

attributable to engine thrust. Therefore, engine cutoff time displays a

large coupling effect on final inclination. In-spite of this effect,

the pairing of bank or angle-of-attack with inclinatio n change and

engine cutoff with apogee altitude offered a simple, reasonable approach

to the problem.

For control of heating rate and fuel usage no simple study offered a

clear indication of what control pairings offered the best means of control. However, based on the author's experience with POST, it seemed

possible to gain the additional insight into the problem through this

avenue.

55
CHAPTER 4

TRAJECTORY ANALYSIS

4.1 Introduction

Although the concept of performing an orbital plane change manuever

by using aerodynamic forces has been seriously considered and worked on

since the early 1960's(17), this thesis offers a new level of knowledge

for the optimum control of the maneuver in the presence of density and

vehicular perturbations.

As discussed in Chapter 3, it became quickly apparent that some

method was needed to provide insight into which of the various options

was the preferred way to guide the ERV. Having some experience with

POST, it was felt that at least a certain amount of that insight could

be gained by simply perturbing the nominal conditions (POST initia l conditions) and studying the initia l and the optimized trajectory generated

by POST. If nothing else, we could obtain the scope of the problem

(i.e., the net effect of the perturbations on the end-conditions) for

the required density, aerodynamic, and thrust perturbation/uncertainty

56
criteria. Furthermore, analysis of the optimized solutions generated by

POST could help determine a generalized method for correcting the perturbation errors in an optimum manner without the complexity of continuously determining the optimum solution. That is, a means to optimize

the guidance process without using real-time optimization. As it turned

out, not only did the work with POST offer a simple means to size the

problem, but it offered enough insight to logically and directly develop

a guidance law which easily met all the required conditions stated in

Chapter 2.

Overall, the basic plane change strategy and the analysis results

obtained with POST appear to be generally applicable and sound. The

performance evaluation results of Chapter 6 strongly validate most of

the conclusions reached during our analysis. Still , it is important to

remember that our trajectory analysis is based on POST and the nominal

plane change trajectories that Langley developed with POST. Thus, any

unique properties of how POST optimized or any changes or errors in the

basic plane change strategy could effect the results and conclusions

contained herein.

The remaining part of this chapter wil l discuss the methods and

results of our trajectory analysis with POST. After discussing the

methods, overall analysis results wil l be covered. This wil l be followed by a detailed discussion of the scope of the problem and the

57
optimization/compensation results for density perturbations aerodynamic

uncertainties, single-engine failure and various combinations of these

conditions on the nominal trajectory.

4.2 Method of Analysis

4.2.1 Objective

As previously discussed, our objective with POST was to obtain the

type and size of trajectory errors that were caused by changes in density, aerodynamic characteristics and available thrust. In addition, we

were seeking ways to optimize or compensate for these changes. To

achieve this objective, we started with the nominal guidance parameters

(independent variables) but off-nominal conditions and observed the ini -

tia l and final trajectory generated by POST during its optimization process.

As an aid to understanding this method, a brief explanation of how

POST operates is in order. Using POST as an optimum trajectory generator, it goes through an iteration process to determine the best solution. Starting with the initia l conditions and guidance parameters it

generates an initia l trajectory based solely on those inputs. POST then

tries to improve the trajectory by changing the independent variables

based on numerically determined partial derivatives. On each successive

58
iteration, it repeats the process of changing the guidance parameters

based on the partial derivatives and compares the resulting end-conditions with the desired conditions. This process continues until the

desired conditions are obtained. At the same time, POST is trying to

minimize or maximize a cost function. Thus, the final trajectory wil l

not only achieve the desired end-conditions but do it in an optimum manner.

4.2.2 Specific Approach

All the trajectory analysis discussed in this chapter was performed

with the POST computer program. The program was started with the ERV in

a 160 nm circular orbit. The specific trajectory objective was to

achieve a 20 degree synergetic inclination change using a maximum heatrate of either 80 or 125 BTU/ft2-sec and return to a 160 nm circular

orbit with the maximum vehicle weight (i.e., minimum amount of fuel consumed) .

Starting with the 12 optimum independent variables for the nominal/

baseline trajectories (see Chapter 2 and Appendix A for additional

information), various amounts and combinations of density and aerodynamic perturbations and/or single engine failure were added to the nominal

conditions. During the first cycle through (first iteration of the

optimization process), POST would use the nominal independent variables

59
to generate a trajectory. The end-condition results of this initia l

trajectory were the "qualified" open-loop results (not a true open-loop

as wil l be discussed in the next subsection) caused by the perturbations. These results provide an indication of the net effect of the

perturbation on the trajectory.

After the first iteration, the independent variables are adjusted to

find an optimum trajectory for the off-nominal conditions. Analyzing

the manner in which the independent variables were adjusted for various

perturbations, certain trends stood out that offered insight into how

the ERV could best be guided.

The final step in our analysis process was to systematically "fix"

or "freeze" some of the guidance parameters (e.g., the four entry variables) at their nominal value. By fixing the entry variables we were

able to gain insight into how to optimize under what we felt were "realworld" constraints. In other words, you should not be able to adjust

your entry parameters before you have actually had time to sense that

entry conditions are not nominal. Thus, insight was gained on how to

optimize various sections of the trajectory in addition to the trajectory as a whole.

4.2.3 L imi tati ons

60
Before discussing the results of our trajectory analysis, three lim -

itations or qualifiers on POST should be covered. The first qualifie r

concerns the 12 free parameters (independent variables). The analysis

discussed in this chapter used the same free parameters that were used

in developing the baseline trajectories. Therefore, the plane change

trajectories discussed are really only optimum to the degree that POST

was allowed to adjust trajectory parameters. Still , based on the knowledge of how Langley developed the baseline trajectories and the work

done for this thesis, the baseline trajectories do represent a very good

approximation of the "true" optimum solution.

The second limitatio n deals with the manner in which POST optimized

for off-nominal conditions. As POST iterates to an optimum trajectory,

it has apriori knowledge of perturbations. This is in contrast to a

real-time guidance system that can only estimate what future perturbations wil l be. Therefore, not all methods used by POST to optimize are

directly applicable to a real-world guidance algorithm. The process of

"freezing" the entry parameters as discussed in the last subsection, was

one approach used to overcome this limitation.

The third qualifier deals with the way the baseline plane change

trajectories were defined and how POST handles certain constraints.

Specifically, the strategy for the baseline trajectories was to maintain

a maximum design heat-rate and an optimum vehicle acceleration during

61
the cruise phase. It should be pointed out that the results of numerous

POST runs indicated that the optimum vehicle acceleration in all practicality does not vary and is almost zero (0.0036 ft/sec2). Thus, a near

constant velocity is maintained which (as it wil l be shown later)

implies a nearly constant altitude to maintain a constant heat-rate. To

handle these constraints, POST is allowed to vary bank angle to control

heat-rate and throttle setting to maintain vehicle velocity during the

cruise phase. The heat-rate and constant velocity (actually a near zero

acceleration) constraints are necessary to maintain the same type of

trajectory from case to case, but they must be considered when interpreting the trajectory analysis results. Specifically, it is important

to realize that POST wil l always be able to adjust bank angle and throttle setting during the cruise phase. Therefore, as suggested earlier,

the initia l trajectory generate by POST during its optimization process

is not really a true open-loop trajectory. Still , even with these limi -

tations, this initia l trajectory offers some valuable insight on the

effects of the perturbations, and the results wil l be labeled as "openloop" results.

4.3 Qveral1 Results

The results of our trajectory analysis with POST are listed below as

a means of overview. Each specific point wil l be discussed in the following sections. It is important to remember that whil e the following

62
results are generally applicable to the plane change problem, they must

be considered with the qualifiers and limitations discussed above. That

is, while the listed results are not uniquely attributable to POST, they

are restricted to the plane change problem as previously defined.

1. "Open-Loop" Results: The single-engine failure case caused the

greatest error in end-conditions, with roughly a 25% error in final

inclination, while the ±15% aerodynamic uncertainty produced a range of

5-20% error in final inclination. The ±30% density perturbations, surprisingly, caused almost no end-condition errors.

2. Heat-Rate Control: .Proved to be very important for minimizin g fuel

usage and density compensation. In addition, the trajectory's sensitivity to perturbations was a function of its maximum heat-rate.

3. Angle-of-Attack: Remained almost nominal except during the cruise

phase where it is adjusted to equalize the thrust-drag balance.

4. Cruise Acceleration: Does not significantly change under any tested

conditions and is practically zero. Fixin g it at zero (i.e., constant

velocity) has no observable effect.

5. Earlv Corrections: From a fuel optimization standpoint, corrections

should be made as early as possible. However, "fixing" the four entry

63
parameters only resulted in a maximum loss of 100 pounds of fuel efficiency.

6. Time-to-Exi t: Significant variations in exit inclination were made

to compensate for the various perturbations. With the entry parameters

fixed, the overall trend was to adjust the exit point and then exit

using nominal exit parameters.

7. Bank Angle Versus Angle-of-Attack: Typically bank was a more efficient (in terms of fuel usage) means of correcting for perturbations

than angle-of-attack.

8. Fuel Usage! Several items significantly effect the fuel usage.

Aerodynamic characteristics, not surprisingly, had the largest effect

followed by the time-to-exit, maximum heat-rate, and the transition from

entry to the heat-rate boundary during cruise.

9. Exit Parameters: Tended to remain at near nominal values. In fact,

angle-of-attack and bank angle after engine cutoff never significantly

changed from the nominal values.

4.4 DETAILED RESULTS

64
In the following subsections, the "open-loop" and

optimization/compensation results for each perturbation type and perturbation combinations are discussed in detail. The "open-loops" results

consist of the "open-loop" errors and the final vehicle weight for the

specified conditions. The "open-loop" errors indicate how much deviation (from the desired value) in final inclination, apogee altitude or

maximum throttle setting for a perturbed trajectory using nominal guidance parameter values (see Section 4.2). As a reminder, the desired

end-conditions values are:

1. Fina l inclination change - 20 degrees

2. Fina l Apogee Altitude - 160 nm

3. Maximum throttle setting - 100%

The final vehicle weight provides an idea of how the perturbations

affected fuel usage. However, significant "open-loop" errors wil l

affect the usefulness of this result. As reference weights, the final

vehicle weight (in a 160 nm circular orbit) for the baseline 80 and 125

trajectories is 5262 Ibs and 5873 Ibs respectively.

The optimization/compensation results consist of the optimized guidance parameter values and the best final vehicle weight for the speci65
fied conditions. Recall that these optimized guidance parameter values

are the best independent variable values that POST can generate thru its

optimization process for the specified conditons. Using these optimized

independent variables as POST guidance parameters or commands wil l produce a trajectory to achieve to desired end-conditions with the best

final vehicle weight. By comparing these off-nominal optimized guidance

parameters with the nominal guidance-parameters (also optimized for nominal conditions), insight can be gained on how to compensate for the

dispersions in an optimum manner. Comparing the final vehicle weights

(nominal verses off-nominal conditions), the net effect of the perturbation in terms of fuel usage can be obtained.

4.4.1 Density Perturbations

Trajectory analysis results for density perturbations indicated that

heat-rate control offered "automatic" density compensation. This result

was the most surprising result of our analysis. From an initia l study

it was determined that density dispersions would cause significant

inclination and apogee altitude errors (19). In fact, this earlier

study suggested that the extreme cases of density perturbations (±30%)

could result in 4 to 6 degrees (20-30%) of inclination error. However,

as the results in the next subsection show, the ±30% density perturbations produced also no POST "open-loop" errors. This implies that

either the trajectory is unaffected by density perturbations and our

66
initia l study was incorrect or there is actually some automatic density

compensator in the "open-loop" trajectory.

4.4.1.1 "Open-Loop" Results

To obtain the scope of the problem for density perturbations, POST

runs were made using a 30% high density bias, a 30% low density bias and

a "worst case" density situation where the density was 30% high until

the start of the exit phase and then it switched to 30% low. The three

cases were run for plane change missions using a maximum heat-rate of 80

and 125. Overall, as the results in Table 4-1 demonstrate, density variations had almost no effect on the end-condition results for the "openloop" trajectory. In fact, for a heating rate of 80 BTU/ft2-sec with a

30% low density atmosphere, the resulting trajectory meets the desired

end-conditions more accurately and with less fuel than under nominal

(standard atmosphere) conditions. Even for the "worst case", the endconditions were just outside of NASA's accuracy criteria.

One final point that warrants mention concerning the "open-loop"

results shown Table 4-1. For two of the cases, the maximum throttle

setting exceeded 100%. To understand the significance of this point,

further discussion on how the throttle setting is determined is in

order. Except during the cruise phase, the throttle is fixed at either

100% or zero. During the cruise phase, to maintain the cruise accerler67
Table 4-1. Density bias "open-loop" results with post.

Condition

QDOT = 80

Nominal

QDOT = 80

+30%

QDOT = 80

-30%

QDOT = 125

Nominal

QDOT = 125

+30%

QDOT = 125

-30%

QDOT = 125

+30% Until

Exit Then

-30%

Inclination

Error

(deg)

0.004

0.004

-0.001

0.001

0.129

-0.214

-0.670

Apogee

Altitude

Error

(nm)

-0.27

-0.28

0.07

0.20

0.39

0.09

-5.38

Maximum

Throttle

Error

(%)

0

0

0

0

0.80

0

0.83

Final

Vehicle

Weight

(Ibs)

5262

5253

5269

5873

5847

5904

5978

NOTE: A negative sign indicates the error was below the desired value.

68
ation, POST has the freedom to adjust the throttle. Initially , POST has

the freedom to exceed 100% but during the optimization process it must

eliminate the excess (real-world necessity) as one of its end-condition

constraints. Therefore, the maximum throttle error is simila r to incli -

nation and apogee errors — some sort of adjustment wil l be needed to

compensate for the error. Whil e in both cases shown in Table 4-1, the

throttle error was less than 1%, it is the first hint that during

cruise, nominal angle-of-attack and 100% thrust wil l not maintain a constant velocity under some conditions.

4.4.1.2 Optimization/Compensation Results

As might be expected the optimized trajectory results for density

perturbations do not provide much insight since the "open-loop" errors

were so small. As the summary of the results shown in Tables 4-2 thru 4

indicate, many of the parameters do not change at all . The few changes

that are made do not really stand out until analyzing the data for a 125

heat-rate with the four entry parameters frozen (Table 4-4). In these

cases, the data suggests that the best way to correct is with a variation in exit inclination or a "time-to-exit" routine. However, before

dealing with smaller details of the results, the basic question of why

such large density biases result in such small end-condition errors

should be covered.

69
Table 4-2. Density bias optimization results for

QDOT = 80 BTU/ft2-sec.

Guidance Parameter

Name (Units)

Geocentric Latitude (deg)

Perigee Altitude (nm)

Entry Angle-of -Attack (deg)

Entry Bank Angle (deg)

Cruise Angle-of -Attack (deg)

p

Cruise Acceleration (ft/s )

Exit Inclination (deg)

Exit Bank angle (deg)

Exit Angle-of -Attack (deg)

Apogee Altitude (nm)

Unpowered Exit

Angle-of -Attack (deg)

Unpowered Exit

Bank Angle (deg)

Final Vehicle Weight (Ibs)

Condition

Nominal

0.957

30.0

32.2

-67.8

30.8

0.0036

47.4

-59.1

21.8

193

8.03

-68.5

5262

+30%

Same

Same

Same

Same

Same

Same

Same

Same

Same

Same

Same

Same

5253

-30%

Same

30.1

33.5

-63.5

31.3

Same

47.2

-60.8

Same

192

8.05

-68.7

5293

NOTE: (1) "Same" indicates the parameter value did not change from

the nominal value.

(2) "Fixed" indicates the parameter was not allowed to change

during the optimization process.

(3) A negative sign indicates a left bank angle.

70
Table 4-3. Density bias optimization results for QDOT = 125 BTU/ft2-sec,

Guidance Parameter

Name (Units)

Geocentric Latitude (deg)

Perigee Altitude (nm)

Entry Angle-of -Attack

(deg)

Entry Bank Angle (deg)

Cruise Angle-of -Attack

(deg)

Cruise Acceleration

(ft/s2)

Exit Inclination (deg)

Exit Bank angle (deg)

Exit Angle-of -Attack

(deg)

Apogee Altitude (nm)

Unpowered Exit

Angle-of -Attack (deg)

Unpowered Exit

Bank Angle (deg)

Final Vehicle Weight (Ibs)

Condition

Nominal

-0.958

27.6

27.2

-69.2

25.4

0.0036

40.6

-65.7

20.4

165

8.13

-69.8

5873

+30%

Same

27.4

26.8

-71.1

25.3

Same

40.1

-66.1

20.3

Same

Same

Same

5941

-30%

Same

27.5

26.8

-71.4

25.3

Same

40;5

-66.4

Same

Same

Same

Same

5943

+30% Until

Exit Then

-30%

Same

27.5

27.4

-69.4

25.1

Same

40.9

-65.8

20.6

169

8.12

-69.7

5842

NOTE: (1) "Same" indicates the parameter value did not change from the

nominal value.

(2) "Fixed" indicates the parameter was not allowed to change

during the optimization process.

(3) A negative sign indicates a left bank angle.

71
Table 4-4. Density bias optimization results for QDOT = 125 BTU/ft -sec.

with entry parameters fixed.

Guidance Parameter

Name (Units)

Geocentric Latitude (deg)

Perigee Altitude (nm)

Entry Angle-of -Attack (deg)

Entry Bank Angle (deg)

Cruise Angle-of -Attack (deg)

o

Cruise Acceleration (ft/s )

Exit Inclination (deg)

Exit Bank angle (deg)

Exit Angle-of -Attack (deg)

Apogee Altitude (nm)

Unpowered Exit

Angle-of -Attack (deg)

Unpowered Exit

Bank Angle (deg)

Final Vehicle Weight (Ibs)

Condition

Nominal

-0.958

27.6

27.2

-69.2

25.4

0.0036

40.6

-65.7

20.4

165

8.13

-69.8

5873

+30%

Fixed

Fixed

Fixed

Fixed

25.2

Same

39.2

-67.1

20.3

166

Same

Same5893

-30%

Fixed

Fixed

Fixed

Fixed

25.2

Same

40.2

-67 -.3

20.3

166

Same

Same

5884

NOTE: (1) "Same" indicates the parameter value did not change from

the nominal value.

(2) "Fixed" indicates the parameter was not allowed to change

during the optimization process.

(3) A negative sign indicates a left bank angle.

72
The most reasonable explanation is that POST is doing something

automatically to control one thing and ends up also compensating automatically for density variations. Closer examination and tests clearly

pointed out that the POST "heat-rate" controller (necessary to maintain

the same plane change strategy) was also providing a built-i n density

compensator. Examination of Chapman's heat-rate (QDOT) equation

QDOT = 17600/Rn (P/pSL)°-5 (VR/VC)

 3 15

where Rn is the vehicle's nose radius, p and pSL are the current and sea

level atmospheric density, Vc is the reference circular orbital velocity, and VR is the relative vehicle velocity, offers an explanation. By

controlling to a given heat-rate you are actually controlling to a density-velocity combination. There are infinit e combinations of velocity

and density to give the desired heat-rate, but with density providing

the predominate deacceleration force during reentry, a straight change

in density wil l only change the geometric altitude where the deacceleration profile occurs and eventually you wil l hit the heat-rate at roughly the same velocity-density combination. In other words, the vehicle

wil l basically fly through the same density profile, but at a different

geometric altitude. Thus, the heat-rate controller insures the same

density profile which insures the same plane change results. So, while

providing for safety of the vehicle, you are also getting compensation

for density perturbations.

73
4.4.2 AERODYNAMIC UNCERTAINTIES

As discussed in Chapter 2, the ERV is expected to perform the plane

change maneuver in the region where previous aerodynamic predictions

have been significantly in error. Therefore, the capability to handle

uncertainties in the predicted aerodynamics is important for an accurate

guidance system. To obtain the "open-loop" and

optimization/compensation results, the following four variations of

aerodynamic characteristics were considered.

1. +15% CL and CD (Both High)

2. -15% CL and CD (Both Low)

3. +15% CL and -15% CD (High Aero)

4. -15% CL and +15% CD (Low Aero)

where CL is the coefficient of lif t and CD is the coefficient of drag.

These four cases represent the four possible extremes for a maximum

aerodynamic uncertainty of ±15%.

74
4.4.2.1 "OPEN-LOOP" RESULTS

The effects of the ERV's ±15% aerodynamic uncertainties produced a

range of inclination errors of 5-20% depending of the combination of CL

and CD as shown in Table 4-5. Overall, two important conclusions can be

made. First, aerodynamic perturbations, unlike density perturbations,

produce significant open-loop errors (e.g., fuel and final inclination)

and wil l require some compensation to minimize their effects. Second,

the problem of having sufficient power during level off was again highlighted as potentially serious because the +15% CL and CO case (QDOT -

125) required nearly 120% of maximum throttle to level-off using nominal

guidance parameters. Some adjustment in a vehicle control parameter

wil l be needed to correct this problem.

4.4.2.2 QPTIMIZAIQKI/CQMPENSATIQN RESULTS

Analyzing the results shown in Table 4-6 thru 8, four specific

observations or insights are apparent concerning (1) fuel usage, (2)

angle-of-attack (3) optimum location for corrections, and (4) sensitivity of the 80 and 125 heat-rate cases to the aerodynamic uncertainties.

On fuel usage, the results of Tables 4-6 thru 8 fal l into two categories. For the case where CL and CD are either high or low (no net

change of L/D), it is possible to compensate for the perturbations with75
Table 4-5. Aerodynamic uncertainty "open-loop" results with post.

Condition

QDOT = 80

Nominal

QDOT = 80

+15% CL & CD

QDOT = 80

-15% CL & CD

QDOT = 80

+15% CL

-15% CD

QDOT = 80

-15% CL

+15% CD

QDOT = 125

Nominal

QDOT = 125

+15% CL & CD

QDOT = 125

-15% CL & CD

QDOT = 125

+15% CL

-15% CD

QDOT = 125

-15% CL and

+15% CD

Inclination

Error

(deg)

0.004

0.322

-0.397

-0.043

0.0387

0.001

1.070

-1.232

-2.411

-3.681

Apogee

Altitude

Error

(nm)

-0.27

7.52

-3.33

25.5

-12.1

0.20

-0.50

-0.86

4.96

-7.00

Maximum

Throttle

Error

(%)

0

0

0

0

0

0

19.7

0

0

0

Final

Vehicle

Weight

(Ibs)

5262

5306

5073

5996

4543

5873

5721

6041

6926

5691

NOTE: A negative sign indicates the error was below the desired value.

76
Table 4-6. Aerodynamic uncertainty optimization results for

QDOT = 80 BTU/ft2-sec.

Guidance Parameter

Name (Units)

Geocentric Latitude (deg)

Perigee Altitude (nm)

Entry Angle-of -Attack (deg)

Entry Bank Angle (deg)

Cruise Angle-of -Attack (deg)

2

Cruise Acceleration (ft/s )

Exit Inclination (deg)

Exit Bank angle (deg)

Exit Angle-of -Attack (deg)

Apogee Altitude (nm)

Unpowered Exit

Angle-of -Attack (deg)

Unpowered Exit

Bank Angle (deg)

Final Vehicle Weight (Ibs)

Condition

Nominal

0.957

30.0

32.2

-67.8

30.8

0.0036

47.4

-59.1

21.8

193

8.03

-68.5

5262

+15% CL

-15% CD

0.956

29.2

32.1

Same

30.2

Same

47.5

-59.7

21.6

186

8.04

-68.7

6029

-15% CL

+15% CD

0.959

30.5

32.5

-67.4

30.9

Same

47.3

-58.3

22.1

201

8.00

-68.4

4521

NOTE: (1) "Same" indicates the parameter value did not change from

the nominal value.

(2) "Fixed" indicates the parameter was not allowed to change

during the optimization process.

(3) A negative sign indicates a left bank angle.

77
Table 4-7. Aerodynamic uncertainty optimization results for

QDOT = 125 BTU/ft2-sec.

Guidance Parameter

Name (Units)

Geocentric

Latitude (deg)

Perigee

Altitude (nm)

Entry Angle-ofAttack (deg)

Entry Bank Angle

(deg)

Cruise Angle-ofAttack (deg)

Cruise Acceleration (ft/s )

Exit Inclination

(deg)

Exit Bank angle

(deg)

Exit Angle-ofAttack (deg)

Apogee Altitude

(nm)

Unpowered Exit

Angle-of -Attack

(deg)

Unpowered Exit

Bank Angle (deg)

Final Vehicle

Weight (Ibs)

Condition

Nominal

-0.958

27.6

27.2

-69.2

25.4

0.0036

40.6

-65.7

20.4

165

8.13

-69.8

5873

+15%

CL & CD

Same

27.8

26.2

-68.5

23.3

Same

Same

-65.3

Same

164

8.12

Same

5892

-15%

CL & CD

Same

27.5

27.4

-69.6

Same

Same"

41.5

-66.3

20.5

166

Same

Same

5819

+15% CL

-15% CD

-0.948

29.4

26.7

-73.3

25.3

Same

41.4

-69.8

20.3

166

8.12

Same

6682

-15% CL

+15% CD

Same

28.3

27.7

-70.4

Same

Same

44.2

-65.8

20.9

172

8.12

-69.7

4886

NOTE: (1) "Same" indicates the parameter value did not change from the

nominal value.

(2) "Fixed" indicates the parameter was not allowed to change

during the optimization process.

(3) A negative sign indicates a left bank angle.

78
Table 4-8. Aerodynamic uncertainty optimization results for

QDOT = 125 BTU/ft2-sec. with entry parameters fixed.

Guidance Parameter

Name (Units)

Geocentric Latitude (deg)

Perigee Altitude (nm)

Entry Angle-of -Attack (deg)

Entry Bank Angle (deg)

Cruise Angle-of -Attack (deg)

2

Cruise Acceleration (ft/s )

Exit Inclination (deg)

Exit Bank, angle (deg)

Exit Angle-of -Attack (deg)

Apogee Altitude (nm)

Unpowered Exit

Angle-of-Attack (deg)

Unpowered Exit

Bank Angle (deg)

Final Vehicle Weight (Ibs)

Condition

Nominal

-0.958

27.6

27.2

-69.2

25.4

0.0036

40.6

-65.7

20.4

165

8.13

-69.8

5873

+15% CL

-15% CD

Fixed

Fixed

Fixed

Fixed

24.8

Same

41.4

-72.1

20.1

168

Same

Same

6584

-15% CL

+15% CD

Fixed

Fixed

Fixed

Fixed

24.6

Same

42.5

-74.3

21.3

185

8.11

-69.7

4981

NOTE: (1) "Same" indicates the parameter value did not change from

the nominal value.

(2) "Fixed" indicates the parameter was not allowed to change

during the optimization process.

(3) A negative sign indicates a left bank angle.

79
out a net effect on fuel usage. For the cases where you have either

high aero or low aero, the optimized trajectory had a roughly 10-15% net

increase or decrease in fuel usage or about 700 pounds of fuel. Neither

of these observations are unexpected but should be kept in mind when

evaluating the final guidance algorithm.

Concerning angIe-of-attack (alpha) control, the results of Tables

4-6 thru 8 fall into two categories. First, for the case of both high

CL and CO in Table 4-7, a change in alpha was able to eliminate the maximum throttle error (not shown in table). Thus, angle-of-attack is a

good candidate for handling thrust deficiencies. Second, throughout the

optimization process for aerodynamic uncertainties, bank angle was

adjusted significantly more than alpha to eliminate inclination errors.

This is especially apparent in Table 4-8 where the entry parameters are

not allowed to change. Examination of the partial derivatives of incli -

nation with respect to alpha and bank (not shown in tables), indicates

that the effectiveness of alpha verses bank angle is about the same.

However, the fuel efficiency is apparently not the same because the

optimization process adjusts almost strictly with bank angle during

entry and exit. For example, Table 4-8 shows that during the powered

exit phase, bank and alpha for nominal conditions are respectively -65.7

and 20.4, while for the high aero case, bank and alpha are -72.1 and

20.1. In other words, a change of 6.4 degrees in bank but only 0.3

degrees in alpha is used to optimize for the off-nominal condition.

80
This implies that bank control is more fuel efficient for aerodynamic

perturbation corrections than alpha.

Concerning the optimum location for corrections, one general trend

runs through all the results thus far (density and aerodynamic). Looking at where POST modified its guidance parameters to correct for errors

caused by perturbations, it is apparent that an "ear 1ier-is-better"

logic prevails. That is, most of the changes in the independent variables are made at the front end of the trajectory. In addition, the

size of these changes tend to hide their true significance since the

effectiveness of corrections (sensitivity) is much higher on the front

end of the trajectory. Particularly surprising was how littl e the exit

parameters were changed by the optimization process. This observation

suggest that corrections for perturbations errors should be made early

and then exit nominally. Bear in mind though that POST "knows" about

the perturbations before hand and can correct early. This apriori knowledge is not normally available "real-world", thus the results are not

completely applicable to a real situation. Table 4-8 shows the optimized parameters after "fixing" the first four. This requires POST to

optimize for perturbations without the abilit y to adjust the entry

parameters. Just as before though, the results show most of the adjustments were made as soon as possible (i.e., during the cruise phase),

and then the exit is made with a near nominal command profile. Thus,

for fuel conservation, corrections should be made as early as possible.

81
As a final observation, the 125 BTU/ft2-sec trajectories are much

more sensitive to perturbations than the 80 BTU/ft2-sec trajectories.

This trend is apparent in the density perturbation results, but is especiall y highlighted in the aerodynamic uncertainty results. This insight

wil l be further expanded in the next subsection.

4.4.3 HEAT-RATE VARIATION

As discussed in Chapter 2, the guidance algorithm should be able to

handle plane change maneuvers at a heating-rate of both 80 and 125.

During the early trajectory analysis work, both 80 and 125 BTU/ft2-sec

heat-rates were examined. However, a review of Table 4-1 and 4-5 indicates the relative insensitivity of a 20 degree plane change done at a

heating rate of 80 versus 125 to both density and aerodynamic perturbations. Specifically, for the 80 case, the perturbations have almost no

effect (i.e., less than 2%) on final inclination, littl e effect on fuel

usage (10% in the most extreme cases) and only a moderate effect on the

final apogee altitude. In contrast, the "open-loop" errors for the 125

case are significantly larger.

The only exception to this trend is with apogee altitude. However,

the apparent exception to the sensitivity trend is not really as it

appears. It can be explained by the large difference in engine cutoff

altitudes for the two trajectories. Examination of Figures A-8 and A-1

82
shows that the engines were cutoff at 255,000 feet for the 80

BTU/ft2-sec trajectory but at 281,000 feet for the 125 trajectory. This

difference in cutoff altitude magnifies the perturbation effects resulting in a much larger apogee error for the 80 trajectory. In other

words, the perturbation effects are not really larger for the 80 case,

they just have longer to propagate. Whil e this particular difference in

the trajectories must be considered in developing a guidance algorithm,

the 125 heat-rate trajectory wil l drive most of the design requirements.

Actually, the sensitivity trend is not surprising in ligh t of the

work done earlier at NASA LaRC which demonstrated that each plane change

had an optimum heat-rate (using fuel usage as an efficiency criterion)

(6)• From that study, it was shown that for a 20 degree plane change, a

heating rate of 125 was near-optimum for the ERV, while a heating rate

of 80 was significantly less efficient. Thus it is reasonable that density and aerodynamic perturbations would have a larger effect on the

optimum heat-rate trajectory (125) than on other trajectories..

A more quantitative explanation can be made by comparing the dynamic

pressure profiles shown in Figure A-11 for the 80 and 125 trajectories

(nominal conditions). Not only does the 125 trajectory reach a much

higher maximum value but the time-integrated total dynamic pressure is

significantly higher (area under the curve). Considering the direct

relationship between aerodynamic forces and dynamic pressure,

83
aerodynamic - dynamic x aerodynamic x surface

forces pressure coefficients area

the sensitivity differences between the 80 and 125 trajectories is

straight foward. The 125 plane change uses less fuel because it uses

more aerodynamic forces to assist in the plane change. Therefore, perturbations that effect the aerodynamic forces (e.g., density, CL or CD)

wil l cause a more significant impact on the 125 trajectory.

The final guidance scheme must demonstrate satisfactory performance

for both heat-rates, and Chapter 6 provides that demonstration. However, in the interest of clarity of discussion and because the 125 heatrate is generally the more difficult case, the remaining discussion of

this chapter wil l deal only with the 125 case.

4.4.4 SINGLE-ENGINE FAILURE

The purpose for considering engine failure is more a question of

safety than anything else. That is, we do not expect to experience an

engine failure as we do expect density and aerodynamic dispersions. So

the first question is can the ERV be safely returned to orbit even if

the desired plane change is not completed. To obtain that answer, two

cases were considered that were thought to be the two worst possibilities for single-engine failure. The first case was where failure

occurs at the start of the level off phase and the second case at the

start of the exit phase. In both cases, the thrust from one of the

ERV's three engines was eliminated for the duration of the mission.

84
4.4.4.1 "OPEN-LOOP" RESULTS

As might be expected, the loss of an engine during the plane change

has by far the largest effect on end-condition errors, but the maneuver

can stil l be accomplished. Table 4-9 shows the "open-loop" errors for

the two, single-mngine failure cases. The inclination and apogee errors

are initiall y the same for both situations because the exit command is

based on an inclination angle. However, the loss of an engine at the

start of the level-off phase is the more difficul t case because of the

power deficiency as highlighted by the maximum throttle error of 50%.

The other significant error is the roughly 25% error in final inclination. The fact that too much plane change was obtained might seem surprising unless you realized that, "open-loop", the exit wil l be at the

same nominal inclination. Therefore, with a slower climb back to orbit

(less power), the extra time in the atmosphere wil l produce a greater

turn and hence a greater plane change.

4.4.4.2 OPTIMIZATION/COMPENSATION RESULTS

Table 4-10 presents the results for the optimized trajectory with a

single-engine failure. Analysis of these results brings out two recurring observations and one new piece of insight.

85
Table 4-9. Single engine failure "open-loop" results with post for

QDOT = 125 BTU/ft2-sec.

Condition

Nominal

Single Engine

Failure at

Level Off

Single Engine

Failure at

Exit

Inclination

Error

(deg)

0.001

4.651

4.651

Apogee

Altitude

Error

(rim)

0.20

-2.82

-2.82

Maximum

Throttle

Error

(%)

0

49.9

0

Final

Vehicle

Wei ght

(Ibs)

5873

5092

5092

NOTE: A negative sign indicates the error was below the desired value.

86
First of all , concerning angle-of-attack control, the results demonstrate that a thrust deficiency can be corrected with alpha. As stated

before, some sort of scheme wil l be needed to correct a power deficiency, and angle-of-attack is capable of handling even this worst case of

an engine out. Furthermore, this means of compensation for a loss of

power is evidently fuel efficient since the optimized results for an

engine failure (Table 4-10) show a final vehicle weight that is almost

the same as for nominal conditions.

The second recurring trend is the preference for bank instead of

alpha for inclination corrections. This trend is especially evident in

Table 4-10 when comparing bank and alpha during the exit phase for an

engine failure at level-off. When the entry parameters are fixed, the

optimum means of compensation is to change the bank by 3.5 degrees but

alpha by only 0.1 degrees. Again, the sensitivity of bank and alpha are

almost the same (the partial derivatives are not shown), but evidently,

the fuel efficiency is not, because bank is strongly favored.

Finally , the new bit of insight deals with determination of the time

or point to exit. The results of Table 4-10 indicate that exit inclination was adjusted during the optimization process as much as 2.3

degrees. This represents a significant modification to the trajectory

and can be explained by the basic plane change strategy. Recall from

Chapter 2 that as much of the plane change should be done during entry

87
Table 4-10. Single engine failure optimization results for

QDOT = 125 BTU/ft2-sec.

Guidance Parameter

Name (Units)

Geocentric Latitude (deg)

Perigee Altitude (nm)

Entry Angle-of -Attack

(deg)

Entry Bank Angle (deg)

Cruise Angle-of -Attack

(deg)

Cruise Acceleration

(ft/s2)

Exit Inclination (deg)

Exit Bank angle (deg)

Exit Angle-of -Attack

(deg)

Apogee Altitude (nm)

Unpowered Exit

Angle-of -Attack (deg)

Unpowered Exit

Bank Angle (deg)

Final Vehicle Weight (Ib)

Condition

Nominal

-0.958

27.6

27.2

-69.2

25.4

0.0036

40.6

-65.7

20.4

165

8.13

-69.8

5873

Single Engine Failure

At Level -

Off

-0.961

28.7

25.6

-69.6

20.4

Same

38.3

-65.9

Same

166

Same

Same

5827

At Exit

Same

27.7

26.6

-68.0

Same

Same

39.7

-62.8

20.3

167

Same

Same

5827

At Level -

Off

Fixed

Fixed

Fixed

Fi xed

20.6

Same

38.8

-62.2

20.3

Same

Same

Same

5732

NOTE: (1) "Same" indicates the parameter value did not change from the

nominal value.

(2) "Fixed" indicates the parameter was not allowed to change

during the optimization process.

(3) A negative sign indicates a left bank angle.

88
and exit to minimize fuel usage. Thus, this shift in exit just represents the optimization process adjusting to achieve this objective. A

simila r capability wil l probably be needed for an efficient guidance

scheme.

4.4.5 MULTIPLE OFF-NOMINAL CONDITIONS

Having discussed analysis results on each individual type of perturbation, it is also worthwhile to discuss various combinations of perturbations. Although, littl e additional insight was gained by combining

various perturbations, the fact that these combinations did not cause

any new problems is in itself, a very useful insight.

4.4.5.1 "OPEN-LOOP" RESULTS

Although a large number of perturbation combinations are possible,

the following five combinations were chosen to scope the problem.

1. 30% high density and high aero

2. 30% high density and low aero

3. 30% low density and high aero

89
4. 30% low.density and low aero

5. 30% low density, low aero, and single-engine failure at

level-off

The "open-loop" errors for these runs are summarized in Table 4-11. For

these cases, the trajectory problems caused by the combination of perturbations did not produce any errors larger than had been seen previously and in some cases the perturbations seem to neutralize each other.

Overall, inclination errors were within 25% of nominal, apogee errors

withi n 10%, and maximum throttle errors withi n 23%.

4.4.5.2 OPTIMIZATION/COMPENSATION RESULTS

With but one exception, the insight gained from the results of optimizing the five cases are mainly a repeat of what was discussed in the

previous sections. The new point wil l be discussed in detail whil e the

old points wil l only be briefly covered.

4.4.5.2.1 HEAT-RATE CONTROL

4.4.5.2.1.1 TRANSITION TO CRUISE

90
Table 4-11. Multiple off-nominal "open-loop" results with post for

QDOT = 125 BTU/ft2-sec.

Condition

Nominal

+30% Density

+15% CL

-15% CD

+30% Density

-15% CL

+15% CD

-30% Density

+15% CL

-15% CD

-30% Density

-15% CL

+15% CD

-30% Density

-15% CL

+15% CD

Single Engine

Failure

Inclination

Error

(deg)

0.001

-2.362

-4.471

-2.523

-0.166

-4.711

Apogee

Altitude

Error

(nm)

0.20

5.29

-13.1

4.34

-2.58

8.04

Maximum

Throttle

Error

(%)

0

0

0

0

11.9

22.8

Final

Vehicle

Weight

(Ibs)

5873

6922

5819

6938

5158

5915

NOTE: A negative sign indicates the error was below the desired value.

91
Our analysis showed that the transition from entry to the cruise

phase could effect the plane change fuel efficiency by as much as seven

or eight percent of the total fuel required for a plane change.

4.4.5.2.1.1.1 THE PROBLEM

While analyzing the results, for multipl e off-nominal conditions, we

realized that a couple of the trajectories had lost the characteristic

(for the 125 case) "double-peak" in the heat-rate verse time curve.

Further analysis indicated that if the "double-peak" (see Figure 2-13 or

A-9) is eliminated or altered, extra fuel wil l be required to complete

the mission. For example, the results in Table 4-12 show that the optimized trajectory for 30% high density and 15% low aero has a final vehicle weight at 4839 pounds. However, by slightly modifying the

trajectory (the modification wil l be covered in the next subsection),

the heat-rate "double-peak" is restored, and the final vehicle weight is

now 5281 pounds for a net fuel savings of roughly 450 pounds. The heatrate verses time profiles for the two cases are shown in Figure 4-1.

Further study suggests two reasons for this fuel savings, both a benefit

of the relatively high dynamic pressure during this phase of flight.

The first reason can be seen by comparing the plots in Figure 4-2

which show the dynamic pressure profiles for the unmodified (original)

trajectory and the modified trajectory (both trajectories are optimized

92
Table 4-12. Multiple off-nominal optimization results for

QDOT = 125 BTU/ft^-sec. with entry parameters fixed.

Guidance Parameter

Name (Units)

Geocentric Latitude (deg)

Perigee Altitude (nm)

Entry Angle-of -Attack (deg)

Entry Bank Angle (deg)

Cruise Angle-of -Attack (deg)

2

Cruise Acceleration (ft/s )

Exit Inclination (deg)

Exit Bank angle (deg)

Exit Angle-of -Attack (deg)

Apogee Altitude (nm)

Unpowered Exit

Angle-of -Attack (deg)

Unpowered Exit

Bank Angle (deg)

Final Vehicle Weight (Ibs)

Condition

Nominal

-0.958

27.6

27.2

-69.2

25.4

0.0036

40.6

-65.7

20.4

165

8.13

-69.8

5873

+30% Density

-15% CL and +15% CD

Unmodified

Fixed

Fixed

Fi xed

Fixed

25.3

Same

44.2

-69.2

21.4

180

8.12

-69.7

4839

Modified

Fi xed

Fixed

Fixed

Fixed

23.1

Same

35.8

-66.5

Same

168

Same

Same

5281

NOTE: (1) "Same" indicates the parameter value did not change from

the nominal value.

(2) "Fixed" indicates the parameter was not allowed to change

during the optimization process.

(3) A negative sign indicates a left bank angle.

93
ERV CRUISING SYNERGISTIC PLANE CHANGE Q-DOT=125

q .

BTU/ft2-sec

160

140

120

100

80

60

40

20

0

Unmodified Transition

(Lead-point = 115 Btu/ft2-sec)

350 700 1050 1400 1750 2100

Time, sec

ERV CRUISING SYNERGISTIC PLANE CHANGE Q-DOT=125

Modified Transition

(Lead-point = 120 Btu/ft2-sec)

q .

BTU/ft2-sec

350 700 1050 1400 1750 2100

Figure 4-1. Heat-rate profiles for +30% density, -15% CL and +15% CD.

94
as previously explained for 30% high density and 15% low aero). The

curves are simila r except the first curve does not extend above 60

lbs/ft2 in dynamic pressure, while the second curve has a section that

goes almost to 85 lbs/ft2. Overall, the maximum and the time-integrated

total dynamic pressure is less for the unmodified trajectory. This situation is simila r to the comparison between the 80 and 125 BTU/ft2-sec

trajectories that was covered in Section 4.6. The lower dynamic pressure results in lower aerodynamic forces, hence less turning capability,

which results in more of the plane change being done with propulsive

forces. Thus, a simila r fuel difference is seen between the unmodified

and modified trajectories as between the 80 and 125 trajectories.

A second explanation for the fuel difference has to do with the

plane change strategy itself. Recall from previous explanations of this

strategy, that during entry as the vehicle approaches the maximum heatrate, a level-off is started. For the unmodified trajectory, the level-off was started as the heat-rate curve was approaching what should

have been its first heat-rate peak. In fact, closer examination of Figure 4-1 (unmodified plot) shows a slight change in the slope at 115

BTU/ft2-sec. For the modified trajectory the level-off was not started

until the heat-rate curve was approaching a second peak.

By starting the level off at the first heat-rate peak, the vehicle

reaches the maximum heat-rate boundary earlier than necessary, and the

95
ERV CRUISING SYNERGISTIC PLANE CHANGE Q-DOT=125

Ib/ft2

120

105

90

75

60

45

30

15

Unmodified Transition

(Lead-point = 115 6tu/ft2-sec)

350 700 1050 HOC

Time, sec

1750 2100

ERV CRUISING SYNERGISTIC PLANE CHANGE Q-DOT=125

Ib/ft2

120

105

90

75

60

45

30

15

0

Modified Transition

(Lead-point = 120 Btu/ft2-sec)

I I LU-KI I I I I I I I I I I I I I I I I I I I I I I I I I I KU I I I I I I I I I I I I I I I I I I I I I

0 350 700 1050 1400 1750 2100

Time, sec

Figure 4-2. Dynamic pressure profiles for +30% density, -15% CL and

+15% CD.

96
engines are used earlier than necessary. Recall that the engines are

started as a part of the level-off. A comparison of the plots in Figure

4-3 shows how for the unmodified trajectory, the ERV is initiall y commanded to increase its bank angle (driven onto the heat boundary) before

decreasing the bank angle to join the heat-rate boundary. A comparison

of the plots in Figure 4-4 shows how the engines are started 210 seconds

earlier for the unmodified trajectory but only cutoff 95 seconds earlier. This results in a larger time-integrated total, which translates

into a fuel difference between the two trajectories. In essence, you

are throwing away this phase of free, relatively hig h dynamic pressure

which could be used to give a high turning rate at no cost in propellant. In addition, the higher dynamic pressures that are obtained by

delaying your level-off (modified trajectory) are never obtained. Thus,

the problem that was discussed in the first explanation.

It should be pointed out that this alteration of the heat-rate curve

also occurred for the 15% low aero case. This helps explain the large

decrease (from nominal) in final vehicle weight that is shown in Table

4-7. In other words, this transition to cruise problem is not

restricted only to multipl e off-nominal conditions.

4.4.5.2.1.1.2 THE SOLUTION

97
ERV CRUISING SYNERGISTIC PLANE CHANGE Q-DOT=125

deg

Unmodified Transition

(Lead-point = 115 Btu/ft2-sec)

20

0

-20

-40

-60

-80

-100

-120

-140rt I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I 11 I I I I I I I I I I I I I I I I I I

0 350 700 1050 1400 1750 2100

Time, sec

ERV CRUISING SYNERGISTIC PLANE CHANGE Q-DOT=125

deg

20

-20

-40

-60

-80

-100

-120

-140

Modified Transition

(Lead-point = 120 Btu/ft2-sec)

ti1 1 I 1111111 1 M i i1 1 11 1 1111111 1 I 1111 1 11111 1 1 1 1111 1 11 1 I I I 1111 1

350 700 1050 1400

Time, sec

1750 2100

Figure 4-3. Bank angle profiles for +30% density, -15% CL and +15% CD.

98
ERV CRUISING SYNERGISTIC PLANE CHANGE Q-DOT=125

1 6

1.4

1.2

1.0

Unmodified Transition

(Lead-point = 115 Btu/ft2-sec)

D h i i i i i i i i I i i i i li i i i i I i i i i i i i i i I i i li i i i i i i I i i i i i i i i i I i i i i i i i i i

0 350 700 1050 HOO 1750 2100

Time, sec

ERV CRUISING SYNERGISTIC PLANE CHANGE Q-DOT-125

1.6

1.4

1.2

1.0

< .8

Ul

.6

.4

.2

0

Modified Transition

(Lead-point = 120 Btu/ft2-sec)

0 350 700 1050 1400 1750 2100

Time, sec

Figure 4-4. Engine throttling profiles for +30% density, -15% CL and

+15% CD (0 - OFF, 1.0 - MAXIMUM).

99
The difference between the modified and unmodified trajectories was

achieved by changing the lead-point used by the heat-rate controller.

For a dynamic environment, to avoid overshooting the desired heat-rate,

a controller uses a value that is different than the desired heat-rate.

This value can be called the lead-point value or simply lead-point. For

all previous 125 trajectories, the lead-point was set at 115

BTU/ft2-sec. The only change for the modified trajectory was to raise

the lead-point to 120 BTU/ft2-sec. This allowed the restoration of the

"double-peaked" heat-rate curve and the 450 pound fuel savings.

This simple lead-point modification could not full y restore the

first heat-rate peak for all perturbation combinations, but it did provide a progressive improvement in fuel usage (i.e., the more restoration, the more fuel savings). Figure 4-5 provides a comparison of a

modified and unmodified trajectory as an example of partial restoration.

The scales for the plots are not the same so an exact comparison is difficult. However, it is evident that some restoration is made. A comparison of the final vehicle weights in Table 4-13 indicates that nearly

200 pounds of fuel was recovered for this example.

There is another possible modification that could be used since the

lead-point modification does not always work. The loss of fuel efficiency is caused by either a lead-point that is too low or off-nominal

conditions that raise the occurrence of the first heat peak up to the

100
ERV CRUISING SYMERGISTIC PLANE CHANGE Q-DOT=125

q .

BTU/ft2-sec

150

125

100

75

50

25

0

Unmodified Transition

(Lead-point = 115 Btu/ft2-sec)

1111111111111111111111111111111111111 1

300 600 900 1200

Time, sec

1500 1800

ERV CRUISING SYNERGISTIC PLANE CHANGE 0-DOT=125

BTU/ft2-sec

160

HO

120

100

80

60

4-0

20

Modified Transition

(Lead-point = 120 Btu/ft2-sec)

ObH-fl I I I I I I I I I I II I I I I I I I I I I I I I I I I I I I I I I lH+4-l I I I I I I I I II I-U4-I-I

350 700 1050 1400 1750 2100

Time, sec

Figure 4-5. Heat-rate profiles for -30% density, -15% CL and +15% CD.

101
Table 4-13. Multiple off-nominal optimization results for

QOOT = 125 BTU/ft^-sec. with entry parameters fixed.

Guidance Parameter

Name (Units)

Geocentric Latitude (deg)

Perigee Altitude (nm)

Entry Angle-of -Attack (deg)

Entry Bank Angle (deg)

Cruise Angle-of -Attack (deg)

p

Cruise Acceleration (ft/s )

• Exit Inclination (deg)

Exit Bank angle (deg)

Exit Angle-of -Attack (deg)

Apogee Altitude (nm)

Unpowered Exit

Angle-of -Attack (deg)

Unpowered Exit

Bank Angle (deg)

Final Vehicle Weight (Ibs)

Condition

Nomi nal

-0.958

27.6

27.2

-69.2

25.4

0.0036

40.6

-65.7

20.4

165

8.13

-69.8

5873

-30% Density

-15% CL and +15% CD

Unmodified

Fixed

Fixed

Fixed

Fixed

24.1

Same

41.2

-67.1

20.6

168

Same

Same

5024

Modified

Fixed

Fixed

Fixed

Fixed

24.0

Same

37.2

-70.9

19.9

173

8.12

Same

5222

NOTE: (1) "Same" indicates the parameter value did not change from

the nominal value.

(2) "Fixed" indicates the parameter was not allowed to change

during the optimization process.

(3) A negative sign indicates a left bank angle.

102
nominal lead-point. Thus, the lead-point needs to be carefully chosen

to avoid the maximum level of the first heat peak for off-nominal conditions. However, if this is not possible, active control to maintain the

first heat peak below the lead-point might be possible. This option

wil l be discussed in Chapter 7.

4.4.5.2.1.2 HEAT-RATE UNCERTAINTY

It has been shown several times that the efficiency of the synergetic plane change is very much a function of heat-rate and how accurately

it can be controlled. In fact, for the latest example, a modification

of the lead-point by only 5 BTU/ft2-sec offered a significant fuel savings. Chapman's equation, defines heat-rate as a function of velocity

and density. Therefore, to achieve accurate heat-rate control, accurate

velocity and density measurements are a necessity. Previous schemes

have used other control boundaries such as constant drag because it can

be measured quite accurately and avoids the uncertainty in density.

However, flying a constant drag level does not insure that heat-rate is

accurately controlled or constrained. To remain withi n a heat-rate constraint, a constant drag level stil l must be far enough from the heatrate limi t to account for the uncertainties. Therefore, no advantage is

gained by using a controller other than heat-rate. The bottom lin e is

you stil l must deal with the heat-rate uncertainty if you wish to

103
achieve the performance gains offered by a maneuver lik e the synergetic

plane change.

4.4.5.2.2 ANGLE-QF-ATTACK CONTROL

Tables 4-12 thru 15 indicate, as have the previous optimized tables,

that there is very littl e variation between the optimized angles-of-attack for nominal conditions and the perturbated conditions except during

the cruise phase. However, during cruise, the angle-of-attack is significantly adjusted in the cases where there was an "open-loop" maximum

throttle error. In all such cases, enough modification in the cruise

alpha was made to eliminate the throttle error at no cost in terms of

fuel usage.

4.4.5.2.3 BANK ANGLE VERSUS ANGLE-QF-ATTACK

The preference for bank angle over angle-at-attack for adjustments

during entry or exit is again highlighted in Table 4-12 and 13. In all

four off-nominal cases, the exit bank angle is increased anywhere from

0.8 to 5.2 degrees while angle-of-attack at most (except for one unmodified case) is decreased by 0.5 degrees. Again, the partial derivatives

(not shown in the tables) indicate that the effectiveness of angle-ofattack and bank angle are about the same. Thus, the preference for bank

104
Table 4-14. Multiple off-nominal optimization results for

QDOT = 125 BTU/ft2-sec.

Guidance Parameter

Name (Units)

Geocentric Latitude (deg)

Perigee Altitude (nm)

Entry Angle-of -Attack (deg)

Entry Bank Angle (deg)

Cruise Angle-of -Attack (deg)

p

Cruise Acceleration (ft/s )

Exit Inclination (deg)

Exit Bank angle (deg)

Exit Angle-of -Attack (deg)

Apogee Altitude (nm)

Unpowered Exit

Angle-of -Attack (deg)

Unpowered Exit

Bank Angle (deg)

Final Vehicle Weight (Ibs)

Condition

Nominal

-0.958

27.6

27.2

-69.2

25.4

0.0036

40.6

-65.7

20.4

165

8.13

-69.8

5873

+30% Density

+15% CL

-15% CD

Same

Same

27.4

-70.1

Same

Same

42.4

-66.7

Same

163

Same

Same

6559

-15% CL

+15% CD

Same

27.8

28.0

-69.0

24.6

Same

43.7

-66.5

20.6

171

Same

Same

4941

NOTE: (1) "Same" indicates the parameter value did not change from

the nominal value.

(2) "Fixed" indicates the parameter was not allowed to change

during the optimization process.

(3) A negative sign indicates a left bank angle.

105
Table 4-15. Multiple off-nominal optimization results for

QDOT = 125 BTU/ft2-sec.

Guidance Parameter

Name (Units)

Geocentric Latitude (deg)

Perigee Altitude (nm)

Entry Angle-of -Attack

(deg)

Entry Bank Angle (deg)

Cruise Angle-of -Attack

(deg)

Cruise Acceleration

(ft/s2)

Exit Inclination (deg)

Exit Bank angle (deg)

Exit Angle-of -Attack

(deg)

Apogee Altitude (nm)

Unpowered Exit

Angle-of -Attack (deg)

Unpowered Exit

Bank Angle (deg)

Final Vehicle Weight (Ib)

Condition

Nominal

-0.958

27.6

27.2

-69.2

25.4

0.0036

40.6

-65.7

20.4

165

8.13

-69.8

5873

-30% Density

+15% CL

-15% CD

-0.960

28.1

27.2

-69.0

25.5

Same

42.9

-67.8

20.5

164

Same

Same

6474

-15% CL and +15% CD

Same

27.5

26.7

Same

24.1

Same

40.6

-67.1

20.5

168

Same

Same

5124

1 Engine

Out

-0.959

28.0

21.8

-68.3

22.8

Same

44.9

-66.1

20.8

169

8.12

-69.7

4882

NOTE: (1) "Same" indicates the parameter value did not change from the

nominal value.

(2) "Fixed" indicates the parameter was not allowed to change

during the optimization process.

(3) A negative sign indicates a left bank angle.

106
angle implies that bank angle is more fuel efficient in compensating for

the perturbations in question.

4.4.5.2.4 EARLY CORRECTIONS

Whil e the results of Table 4-12 thru 15 are not as clear as the previous results, there is stil l a trend of making adjustments early and

flying the remaining trajectory with near nominal guidance parameters.

Again the effeciveness of alpha and bank changes is much higher during

entry so the preference for early changes is somewhat hidden by the size

of the changes.

4.4.5.2.5 TIHE-TQ-EXIT

Tables 4-12 thru 15 show a significant variation in the exit incli -

nation. Actually, this result is not especially surprising since the

closer a plane change is done to the node and the quicker it is performed, the more efficient it tends to be. Therefore, the variation in

exit inclination is just a result of the optimization program adjusting

for changes in speed and location of the plane change for the perturbed

conditions. Thus, a guidance scheme wil l certainly need to make simila r

types'of adjustments to obtain a reasonable degree of fuel efficiency.

107
CHAPTER 5

GUIDANCE SCHEME

5.1 INTRODUCTION

Based on the guidance concepts discussed in Chapter 3 and the trajectory analysis covered in Chapter 4, a guidance scheme designed to

guide the ERV through the atmospheric phase of a synergetic plane change

maneuver is suggested. This guidance scheme by design produces a trajectory very similar to the plane change trajectory generated by POST.

As shown in Chapter 4, this trajectory provides a very good baseline

that requires relatively few modifications to maintain its optimality

even in the presence of significant density and vehicular perturbations.

Therefore, the suggested algorithm mirrors the strategy used in POST

with a few simplifications (e.g., no optimization cycle) and several

technique refinements. The resulting trajectory can be divided into

three phases: (1) An entry phase where angle-of-attack and bank are

held constant until pullout, (2) A powered cruise phase where the maximum heating rate is maintained and the time to start the exit phase is

determined and (3) The exit phase where apogee and inclination angle

are controlled with thrust and bank angle respectively. Figure 5-1

108
shows an altitude profile, for a target heat-rate of 125 BTU/ft2-sec and

nominal conditions, with the three trajectory phases labeled. Withi n

these three phases are three primary guidance controllers that form the

hybrid guidance system discussed in Chapter 3. (1) A heat-rate controller using perturbation guidance techniques, (2) A time-to-exit controller using predictor logic, and (3) An end-condition controller using

predictor/corrector techniques. Fuel is not explicitly optimized. The

remaining two sections of this chapter wil l discuss in detail the guidance algorithm and the rationale for choosing the suggested scheme.

109
H

OFFSETX FOR VRR 05 = RLT (FEET)

CRUISE

-•X I.5000E250000--

150000

0 500 1000

JOB JHB2335R 04/25/85 13:15

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POWERED

1500 2000

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QDOT=125 NOMINflL CONDITIONS - RLPHR CONTROL TEST

H 1 1 1 1 1 1 1 1 1 1

Figure 5-1. Guidance algorithm phases.

110
5.2 GUIDANCE ALGORITHM

This section details the guidance algorithm by individuall y discussing each of the three trajectory phase. Figure 5-2 provides a summary

of the guidance system and should be helpful as the algorithm is covered. All the symbols used in this chart are explained in the following

subsections.

5.2.1 ENTRY PHASE

The entry phase is a constant attitude glidin g maneuver that starts

at atmosphere interface (400,00 feet) and ends at the pul l out point for

the cruise phase. A deorbit burn is made to achieve the desired velocity and position at 400,00 feet. The deorbit burn, perigee altitude,

angle-of-attack, and bank angle are determined pre-mission to provide

the most fuel efficient entry conditions for nominal conditions. The

entry angle-of-attack and bank angle are set at 400,00 feet and remain

constant throughout the entry phase. The pullout point is defined as

the point where the vehicle reaches 90% of the desired heat-rate. Heatrate (QDOT) is calculated on board using Chapman's equation.

QDOT = 17600 (pm/PSL)°-5 (VR/26000)3-

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112 ,
where pra is the measured atmospheric density, pSL is the sea-level atmospheric density, and VR is the vehicle's velocity relative to the atmosphere in feet/second2. The density measurement is determined by direct

measurement from an air data system lik e the Shuttle Entry Air Data System. Velocity is obtained from the inertia] measurment unit (IMU) .

5.2.2 CRUISE PHASE

The cruise phase begins when the heat-rate reaches 90% of the

desired heat-rate and ends when exit is commanded by the time-to-exit

predictor. During this phase, the ERV transitions from a descending

glid e to a powered, level-flight, constant-velocity cruise to maintain

the desired heat-rate. The level-off is initiated by lowering the

angle-of-attack to the nominal cruise value and adding power as necessary to offset drag. The cruise condition is maintained by modulating

primaril y engine thrust, but also angle-of-attack as necessary to maintain a constant velocity. Throttle setting is commanded such that

thrust equals sensed drag or

THROTTLE = DRAG/(THRUST x COS (a))

where a is angle-of-attack. If thrust is insufficient to cancel drag

(e.g., single-engine failure), then throttle is fixed at 100% and a is

adjusted in 0.5 degree increments until thrust is able to offset drag.

1 13
5.2.2.1 HEAT-RATE CONTROL

With velocity fixed, altitude and hence heat-rate are controlled by

modulating bank angle. Heat-rate is again defined by Chapman's

equation. Heat-rate control is achieved by flying the vehicle at a density appropriate to the current velocity. The desired density is then

PO = PSL QDOT0

2/( (17600) (VR/26000)3-15)2

where the "D" subscripts represent desired values. So, given a velocity

and desired heat-rate, a desired density can be calculated and converted

to a desired density attitude (hp) with a simple exponential model. The

heat-rate controller equation is a standard second order controller

defining a desired vertical acceleration (av) such that

av = k,Ahp + k2Ahp' + Ahp"

where k1 and k2 are constants. Ah , Ah ', and Ah " are the difference

between desired and current density altitude, density altitude rate, and

density altitude acceleration respectively. Desired altitude rate and

altitude acceleration are determined by back differencing. The desired

altitude acceleration is smoothed with a low pass filter. The current

114
altitude rate is the vertical component of the vehicle velocity and this

is back differenced to determine current altitude acceleration.

Finall y the desired vertical acceleration can be converted to a commanded bank angle (4>c) such that

Vev -

 av (180/w)/(TAN(*pPW)) asv

where *prev is the previous bank angle and asv is the vertical component

of the sensed nongravi tational acceleration (19).

5.2.2.2 TIME-TQ-EXIT

Whil e the heat-rate controller is maintaining a constant heat-rate,

a predictor is run every ten seconds to see how much of the desired

plane change could be accomplished using nominal exit controls and conditions if the exit was begun at current time. Specifically, the guidance system integrates the vehicle state ahead to 400,000 feet assuming

nominal alpha, bank and 100% thrust to determine the final inclination.

When the predicted inclination angle (i ) is greater that the desired

inclination angle (iD) then exit is started. There is an additional

constraint on determining the time to begin the exit. For heat control,

any bank angle greater than the current commanded bank angle wil l cause

overheating of the vehicle. Thus when determining the time to begin the

115
exit, the algorithm uses the smaller of current or nominal exit bank

angle in the integration to 400,000 feet.

5.2.3 EXIT PHASE

The exit phase begins when exit is commanded by the time-to-exit

predictor and ends when the vehicles leaves the atmosphere at 400,000

feet. During this phase, thrust is increased to 100%, angle-of-attack

is set at the nominal exit value, bank angle is modulated to achieve the

desired final inclination, and final apogee altitude is controlled by

engine cutoff. However, heat-rate control must be continued until the

vehicle climbs to an altitude where heating is no longer a concern.

Therefore, during the first minutes of exit, if heat control requires a

smaller bank angle than inclination control, heat control takes priority. Once heat-rate commands a bank angle greater than inclination control, then inclination control takes priority (19).

5.2.3.1 INCLINATION CONTROL

During the final part of the exit phase, the inclination at 400,000

feet is controlled by modulating bank angle. A predictor/corrector guidance routine predicts a bank angle that, if held constantly throughout

the remaining exit phase, wil l achieve the desired inclination angle.

116
The predicted bank angle is determined by finding a linear relationship between the final inclination and bank angle. The predictor algorithm numerically integrates the trajectroy ahead to 400,000 feet to

determine the final inclination Oold) • Another intergration is made

with a slightl y different bank profile (^npey + 8<l>) and the new inclination dnew) is determined. A partial derivative of inclination with

respect to bank angle (di/d<t>) is calculated such that

(inew

With the sensitivity of changes in inclination to changes in bank angle,

the change in bank angle (6<f>D) to achieve the final inclination (iD) can

be determined as

Then the commanded bank angle is the previous bank angle plus the update

or

*- = *„„«>, + 8<J>n

If the update is large, then non-linear effects may require the process

to be iterated.

117
5.2.3.2 APOGEE ALTITUDE CONTROL

The final apogee altitude is controlled by engine cutoff time. The

engine cutoff is commanded when the vehicle has sufficient velocity to

coast to the desired apogee altitude (haD) . For simplicity , the initia l

apogee prediction is done assuming the vehicle is moving conically

(point mass gravity and no atmosphere). The expression for the apogee

of any conic orbit can be written in closed form. However, since the

engine cutoff occurs in the atmosphere, the conic apogee (hac) cutoff

altitude is biased to compensate for the residual atmospheric drag

between cutoff and apogee. So, the cutoff time is when

nac

 = nao + has

where haB is apogee bias. To compensate for perturbations, this bias

must be adjusted using the predictor/corrector's estimation of apogee

altitude (hap) at 400,000 feet such that

(ha)€ = hap - haD

haB = haB + K (ha) £

where (ha) e is apogee error and k is a constant. This process is

repeated if apogee error is greater than five nautical miles.

118
5.3 GUIDANCE RATIONALE

5.3.1 ENTRY PHASE

The reasoning for going with constant attitude or open-loop is based

on the overall results of Chapter 4. Specifically, the results in

Tables 4-8, 4-12 and 4-13, where the entry parameters were fixed, indicated that the cost was at most 100 pounds of fuel. In actuality, the

possibly of achieving a 100 pound savings by adjusting entry parameters

was remote since the optimization program achieved this result with

apriori knowledge. In fact, we could visualize situations where a correction for one deviation is made on the front-end while it might be

reversed on the tail-end of the trajectory. At the same time, allowing

the errors to build up and trying to fix them at the end did not seem

reasonable either. First, it goes completely against what our trajectory analysis indicated was the most fuel efficient method, and second, it

might over tax the guidance system to try and compensate too late.

Thus, we compromised and chose to begin closed loop guidance at the

start of the cruise phase. The final consideration during entry was the

lead-point for the heat-rate controller. Once again a compromise

approach was chosen. To avoid exceeding the maximum heat-rate, an adequate lead-point for the heat-rate is needed. However, as shown in

Chapter 4, if the lead-point is too low, it is possible to cutoff the

119
first heat-rate peak which can cost up to 450 pounds of fuel. Based on

test results, 90% of the maximum heat-rate was determined to be a reasonable compromise (19).

5.3.2 CRUISE PHASE

The rationale for control of heat-rate and time-to-exit are a result

of the baseline plane change strategy and our trajectory analysis. The

optimization analysis showed that fuel could be minimized by doing the

pul l out and cruise at maximum heat-rate. Also, the time in the atmosphere could be reduced by doing as much of the plane change during the

exit as possible.

5.3.2.1 HEAT-RATE CONTROL

The control of heat-rate during the cruise phase is very important.

Overheating wil l jeopardize vehicle safety and underheat ing wil l cost

additional fuel. In order to maintain tight control over heat-rate,

perturbation guidance was chosen.

Heat-rate, based on Chapman's equation, is only a function of density and velocity. Therefore, of the available control parameters discussed in Chapter 3, only angle-of-attack, bank angle and engine thrust

are possible controller candidates. Engine thrust has no direct effect

120
on density so it could only be paired with velocity. Bank angle and

angle-of-attack can be used to control density or velocity. Analysis of

the POST results indicated that bank angle was always able to control

the heat-rate and hence density but the nominal angle-of-attack along

with maximum thrust (e.g., single-engine failure) was not always able to

maintain a constant velocity. -Therefore, it was reasonable to use

angle-of-attack and engine thrust to maintain velocity. Because of the

multipl e influence of angle-of-attack (affects CL.CD and the component

of thrust aligned with drag) and for simplicity it seemed more appropriate to use angle-of-attack only as necessary and primaril y modulate

thrust to maintain velocity. Therefore, the final scheme sets a nominal

angle-of-attack and adjusts thrust as necessary to maintain a constant

velocity. If thrust is insufficient to cancel drag (i.e., maintain

velocity), then angle-of-attack is adjusted unti l drag is offset.

Recall that this reduction of alpha is beneficial for two reasons: (1)

The CD is lowered as angle-of-attack is reduced, thereby reducing drag

and (2) A reduced alpha increases the component of thrust in the direction of drag. Separately, bank angle is modulated to control altitude

which controls density which in turn controls heat-rate.

5.3.2.2 TIME-TD-EXIT

The rationale for the time-to-exit predictor is a result of the trajectory analysis work in Chapter 4 and an earlier study by R. Richards

121
(19). The trajectory analysis results generally showed that fuel is

minimized for the entire trajectory by adjusting the exit point and

using near nominal parameters during exit. The study that Richards conducted showed that fuel was minimized for the exit phase when the ERV

started the exit at a point where it could just make the end-conditions

using approximately 90 degrees of bank and nearly nominal angle-of-attack. However, this study assumed no perturbations were encountered

during exit and left very littl e room to compensate for any future perturbations. Recall from Chapter 4, optimum exit bank angles for all

cases considered with POST ranged from 65-72 degrees. Thus, the scheme

of exiting when the predictor concluded you could make the end-conditions with nominal exit controls and conditions seemed to offer a near

optimum solution whil e providing some extra margin to compensate for

future perturbations.

5.3.3 EXIT PHASE

The rationale for the end-condition controller during the exit phaes

is a result of the discussion in Chapter 3 on control variable pairings

and analysis of the POST results covered in Chapter 4. For simplicity ,

we decided to make a one-to-one pairing of each control variable and

targeting parameter and if possible, handle each pair separately even

though there are coupling effects. Likewise, even though there is one

more control variables than items to constrain, we chose not to go with

122
the added complexity and convergence concerns unless the extra variable

was needed or offered some clear advantage. Recall from Chapter 4,

angle-of-attack was hardly used during exit and consistently used less

than bank angle for fuel efficiency reasons. Thus, we chose to use bank

angle only as an inclination controller and engine cutoff for apogee

control. Hence, angle-of-attack is left as an additional control variable, completely free for future guidance requirements.

123
CHAPTER 6

GUIDANCE ALGORITHM TESTING

6.1 INTRODUCTION

To evaluate the performance of the plane change guidance scheme discussed in Chapter 5, many simulations are made using various operating

conditions. Although the guidance scheme is the same, the characteristic trajectory for a heating rate of 80 as compared to 125 BTU/ft2-sec

is quite different. Therefore, they are treated as separate cases for

the various operating conditions.

The entire plane change process entails using the Earth's atmosphere

to change the orbital plane. Therefore, if the atmosphere is not a nominal atmosphere with standard density values, the guidance algorithm

must stil l be able to do its job. Many test simulations are made with

off-nominal atmospheric conditions to verify that the algorithm does, in

fact, perform as desired. Among these tests are density bias tests,

where the nominal density is increased or decreased by a constant factor, and density shears, where the trajectory passes through a varying

124
density profile model. Various values of density biases and different

shear models are simulated to evaluate the algorithm's performance.

Another test of the algorithm is to change the aerodynamic characteristics of the ERV. This is done by using various combinations of CL

and CD. A final test was made to determine the algorithm's response to

a single-engine failure. This was done by eliminatin g the thrust produced by one of the three ERV engines.

After testing each off-nominal condition individually , several combinations are tested as well. For example, a thin atmosphere is tested

in combination with low aerodynamic characteristics. Some of the combinations of off-nominal .conditions yield a more realistic simulation overall , as there are bound to be multipl e variations in a real environment

at all times. However, some of the combinations probably provide a more

severe test of the algorithm than would be reasonable to expect the ERV

to encounter. The scheme was not specifically designed to handle these

cases but are included to demonstrate its robustness.

6.2 EVALUATION CRITERIA AND THE NOMINAL TRAJECTORY SIMULATION

There are several quantities that are useful in evaluating the performance of the guidance algorithm. The main goal of the atmospheric

maneuvering is to change the inclination of the orbital plane. If the

125
algorithm does not guide the ERV to the desired inclination, it is not

doing the job. Therefore, one of the most important quantities is the

amount of inclination change that was accomplished during the atmospheric maneuver. The algorithm has been designed for a nominal plane change

of 20 degrees. If the actual plane change in a test run is withi n 0.1

degrees of that nominal value, the test wil l be considered successful.

The closer the inclination is to the nominal value, the better, as less

fuel wil l be required to correct the orbit.

Another important criterion is the final apogee altitude. The algorithm has been designed for a nominal target apogee of 160 nautical

miles. If the actual apogee in a test run is withi n 5 nautical miles of

that nominal value, the test wil l be considered successful. Once again,

the closer the apogee is to the nominal value, the better as less fuel

wil l be required to correct the apogee.

Since the potential fuel savings is one of the main reasons for performing the maneuver, fuel use is important and is also a consideration

in the performance evaluation. For each test run, the fuel required to

circularize the orbit at the target apogee height is computed assuming

that engine burns are made at the apogee and then at the raised perigee.

Also included in this calculation is the fuel required to correct any

inclination error. Comparing the total fuel required to obtain the 160

126
run circular orbit to the nominal trajectory simulation run provides one

measure of the algorithm performance.

In addition, a comparison with the fuel used in the optimized trajectory generated by POST for the same conditions provides an evaluation

of how well the algorithm performed in comparison to the best it could

be expected to perform. The only caveat being, the fuel results from

POST shown in Chapter 4 and the summary of results for this chapter are

for a 1976 standard atmosphere. The CSOL simulations used a 1962 standard atmosphere. These models are very simila r below 290,000 feet.

Therefore, the comparison between POST and the CSDL simulation fuel

results whil e not exactly comparable should offer some vali d insight.

Also computed is the fuel required to correct the perturbed orbit's

longitude of the ascending node (LAN) to the nominal. Although no

attempt was made to control the node shift, the node does shift during

the atmosphere phase of the trajectory. This is because the inclination

change was spread out across the node, and unless it was balanced evenly

before and after the node there is going to be some node shift. For the

CSDL nominal 80 and 125 BTU/ft2-sec trajectories the LAN shifted from

zero to 3.91 and 1.44 degrees respectively. Therefore, all simulations

wil l use this LAN as the final desired LAN. Whil e the fuel to correct

the note shift is typically small, a large value wil l indicate that the

algorithm's performance could possibly be improved. As a point of ref127
erence, the final LAN for the POST 80 and 125 nominal trajectories (1962

atmosphere) was 3.90 and 1.44 degrees respectively.

The final important criterion used in the performance evaluation is

maximum heating rate. Since there is a limi t to the heating rate that

the ERV can withstand, a run which allows the heating rate to exceed its

limi t has safety of fligh t problems. On the other hand, if the algorithm allows the vehicle to operated below it maximum limit , fuel efficiency wil l be hurt because of the correlation between heat-rate and

plane change performance (6). Although exceeding the maximum heating

rate limi t would probably cause a much worse consequence than degraded

plane change efficiency, for simplicity , an actual heating rate withi n 5

BTU/ft2-sec of the desired heating rate, high or low, wil l be considered

acceptable and the test successful. As mentioned in Chapter 2, total

heat load for this type of plane change is not considered a constraint

and is not included as a performance criterion.

The CSDL 80 and 125 BTU/ft2-sec nominal trajectory simulation runs

wil l be used as the primary basis for comparison. For these runs the

inclination change is 20.007 and 20.006 degrees respectively, and the

final apogee altitude is 158.0 and 158.5 nautical miles. The final

vehicle weight in a 160 nm circular orbit with the complete plane change

is 5450 and 5961 pounds respectively, and the maximum heating rate is 80

and 126 BTU/ft2-sec. For reference, Figures 6-1 thru 8 present plot

128
profiles for the 80 and 125 nominal synergetic plane change trajectories

generated by the CSDL computer simulation using the plane change guidance algorithm. These figures should be useful for comparison with the

figures of the various test runs presented in the following sections.

6.3 SIMULATION TEST RESULTS

A wide variety of off-nominal conditions are simulated to evaluate

the guidance algorithm's performance. An example of the resulting trajectory plots wil l be presented for each type of test condition and

where some new information is contained in the plotted data. Tables

wil l summarize the test results in each subsection that follows.

6.3.1 SINGLE OFF-NOMINAL CONDITION TESTS

The following subsections wil l present the results for tests which

involve one off-nominal condition while all other values are nominal.

The tests include density biases, density shears, aerodynamic uncertainties, and single-engine failure performance.

6.3.1.1 DENSITY BIASES

The nominal density levels are multiplied by a constant factor in

this set of tests to determine how thin or thick an atmosphere the guid129
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ance algorithm can handle. A -30% atmosphere is one which is constantly

30% below normal density, whil e a +30% atmosphere is 30% more dense, or

thicker, that a normal atmosphere. When the atmosphere is too thin, the

ERV does not turn as fast, so the plane change tends to take longer and

use more fuel. On the other hand, too thick an atmosphere turns the

vehicle quicker and uses slightly less fuel. Overall, the algorithm

performs excellently in atmospheres ranging from 50% of normal to 150%

of normal. Table 6-1 and 2 contains the results of these test runs.

Figures 6-9 thru 12 show the plot profiles for the QDOT = 80 trajectory

with a +50% density bias.

From these test results two points should be noted. First, a comparison of the final vehicle weights for the CSDL and POST simulations

suggests that the designed guidance algorithm performs as well or better

(e.g., QDOT = 80) than POST. Recall however, from earlier in this chapter, that the CSOL and POST trajectories were generated using different

atmospheric models. For the 125 trajectories, the vehicle weights compare very close. For the 80 BTU/ft2-sec trajectories there is a significant difference in vehicle weights (about 200 Ibs). It should be

remembered that the 80 trajectory stays significantly higher in the

atmosphere (approximately 35,000 feet), and the difference between the

atmospheric models is more significant for the higher altitudes. Therefore, the comparison between CSDL and POST vehicle weights may not be as

valid for the 80 trajectories.

138
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Second, the test results for both the 80 and 125 BTU/ft2-sec trajectories indicate that as the density bias increases so does the shift in

the longitude of the ascending node. The 125 trajectory shows the larger node shift reflecting its higher sensitivity to density perturbations. Still , the node shift is relatively small considering the size

of the tested density biases.

145
6.3.1.2 DENSITY SHEARS

Density shears are introduced along the plane change trajectory to

evaluate the guidance algorithm's performace flying through regions of

changing density factors or levels. Two types of density shear models

are used for testing, the density shear experienced by the shuttle on

the STS-4 mission and a derived model. The STS-4 model is shown in Figure 6-13 and the derived model in Figure 6-14 as a normalized density

verses altitude plot. The density is normalized by the 1962 standard

atmospheric density. The shear models are mechanized as a table of density multiplier s verses altitude.

The algorithm has no difficulty with density shears. The density

shears encountered during the descending portion of the trajectory, are

simply flown through, as the ERV approaches level-off, the automatic

compensation of heating rate that was discussed in Chapter 4 tends to

automatial1y adjust the cruise altitude to correct any effects caused by

the density variations. Even when the shears are encountered during

exit, the bank controller is easily able to correct for the density vari ations.

In fact, the density shears were so benign, a ±30% density bias was

added to the STS-4 shear during cruise. The objective was to show that

the guiance would respond to the changing density and guide the ERV back

146
STS-4 DENSITY PROFILE

1.15

1.10

1.05

1.00

M

<0

24.0 26.0 28.0 30.0 32.0 34.0 36.0

0.95

0.90

X10'

Figure 6-13. STS-4 density profile.

147
CSDL DENSITY SHEAR MODEL

1.3 +

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1.2

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1.05

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200 220 240 260 280 300 320 340 360 380 400

ALT (ft) X103

Figure 6-14. CSDL density shear model profile.

148
to the reference heating rate. The density bias was added as the vehicle flew below 225,000 feet. This caused a step change in density and

heating rate that does not reflect real-world conditions. However, the

guidance algorithm very rapidly drove the ERV back to the reference

heating rate. Table 6-3 contains the results of a sample of the tested

cases. The heating rate values in the parentheses are for cruise equilibriu m conditions and are a better reflection of the algorithm's performance for the shear tests with the added density bias. Figures 6-15

thru 18 show profile plots for a 125 trajectory with a STS-4 density

shear profile.

6.3.1.3 AERODYNAMIC UNCERTAINTIES

This test is useful for determining how well the guidance algorithm

can perform if the vehicle's modeled aerodynamic characteristics are in

error. This error is simulated by inputing a straight bias for CL and

CD to be used throughout the plane change trajectory. Guidance was

never given any knowledge of the CL and CD shift. Whil e there are an

unlimited possible combinations of CL and CD errors, the testing was

limite d to the same four combinations of ±15% CL and CD used in Chapter

4, plus two extreme cases of ±25% CL and CD error to demonstrate the

robustness of the guidance system. It should be noted that the 15%

cases included two cases with no net change of L/D and two cases with a

L/D change (one increased and one decreased).

149
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Overall, the algorithm handled the uncertainties well and exhibited

no unexpected performance loss/gain. Tables 6-4 and 5 summarize these

results and offer a comparison with the performance loss/gain that was

seen with POST. Figures 6-19 thru 22 show the plot profiles for a QDOT

= 125 test with -25% CL and +25% CD.

Several points are worth noting:

1. The fuel usage for both the 80 and 125 trajectories was very

simila r to that calculated by POST under the same conditions.

2. For the performance cases with no change in L/D (e.g., +CL and

+CD) , the final vehicle weight is almost the same as the nominal

V

vehicle weight for both the QDOT 80 and 125 trajectories. However,

in terms of LAN shift, the results were opposite. The QDOT = 80

trajectory had a large node shift whil e the 125 trajectory showed

almost no shift.

3. For the cases with a change in L/D (e.g., +CL and -CD), the

final vehicle weight reflected that change with a corresponding

increase or decrease for both QDOT = 80 and 125. However, in terms

of LAN shift, the results were again opposite. The 80 trajectories

showed almost no node shift while the 125 trajectories showed a

large node shift.

155
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4. The 25% low vehicle performance case (QDOT = 125) lost the characteristic "double-peak" heat-rate curve as shown in Figure 6-21.

This test and one other multipl e off-nominal test were the only

cases that lost the "double-peak" suggesting that the 90% of maximum

heat-rate was an adequate lead-point.

5. The 15% low vehicle performance case (QDOT = 80) failed to meet

the apogee altitude criterion. Recall from Chapter 4 that for a

heat-rate of 80 BTU/ft2-sec engine cutoff occurs much lower in the

atmosphere than for a heat-rate of 125. This makes adequate compensation for large perturbations more difficul t for a QDOT = 80 trajectory. Overall, the apogee error does not represent a significant

fault in the guiidance algorithm but some improvement might be needed.

6.3.1.4 SINGLE-ENGINE FAILURE

In this test, the algorithm's abilit y maintain control of the vehicle, complete the plane change and return safely to orbit is determined.

Two tests are conducted with single-engine failure at a different

location for each test. We felt the start of the level-off phase and

the start of the exit phase represented the two worst possibiI ties.

Guidance was given knowledge of the engine failure in both cases (i.e.,

162
the predictor/corrector used a reduced thrust model). Overall, the

algorithm was able to compensate for the engine failure in both cases

and stil l obtain the desired plane change and final apogee altitude.

Table 6-6 summarizes the results while Figures 6-23 thru 26 show the

profile plots for QDOT = 125 with single-engine failure at level-off.

Three points are worth noting:

1. The CSOL plane change algorithm was able to achieve almost the same

fuel results as the POST simulation.

2. A comparison of Figure 6-1 and 6-23 shows the longer mission time

for the QDOT = 125 single-engine case. The increased time to complete

the mission caused a larger LAN shift than with the nominal mission.

Thus, a significant fuel penalty would be paid to achieve the same

orbital plane as the nominal run.

3. The QDOT = 80 case was almost totally unaffected by the single-engine failure. Although not shown, the mission time was nearly nominal

and as the results show in Table 6-6 , very littl e LAN shift occurred.

6.3.2 MULTIPLE OFF-NOMINAL CONDITION TEST

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174
By combining two or more of the off-nominal conditions discussed in

Section 6.3.1, further tests of the algorithm's performance can be made.

The purpose here is to explore the combinations which yield useful

information in the determination of what the guidance algorithm can and

cannot do. The results of all of the combinations tested are summarized

in Table 6-7 and 8.

Several points are worth noting:

1. All the test cases that had some combination of low vehicle performance and single-engine failure (e.g., -30% density, -15% CL and +15% CD,

and single-engine failure) experienced bank control saturation during

level-off. Figures 6-27 thru 30 show the profile plots for one such

case. The "common thread" for all cases was the significant change in

cruise angle-of-attack. As a result, bank angle was saturated for about

30 seconds and a heat-rate overshoot occurred. Still , the heat-rate

remained withi n the performance criterion. Delaying the angle-of-attack

adjustment by 30 or 40 seconds after level-off would probably eliminate

this bank saturation.

2. The same test cases that experienced bank saturation also lost

velocity during the initia l part of the exit phase. However, this loss

of velocity had no significant overall effect as the vehicle was able to

complete the intended maneuver.

175
3. Four of the five test combination cases for a heat-rate of 80

BTU/ft2-sec failed to meet the apogee altitude criterion. However, only

two of these cases (+30% density with 15% low aero and -30% density with

15% low aero) were significantly outside the criterion. Inaddition, the

combination of off-nominal conditions may be too severe of a test (i.e.,

unrealistic). Still , as discussed earlier, these results point out a

need for improvement in the apogee altitude predictor.

4. Table 6-8 shows again how comparible the CSDL guidance system

results (in terms of fuel) are to the POST results.

5. Several of the QDOT = 125 test showed a significant shift in the

LAN. For the combinations involving high aero, the final vehicle weight

to achieve the nominal inertial plane is almost nominal. However, for

the combinations with low aero, the vehicle weight to achieve the nominalm orbit was significantly below nominal.

176
7.2 CONCLUSIONS

1. A guidance system has been developed for the ERV synergetic

plane change maneuver.

2. The recommended plane change algorithm is a hybrid of perturbution guidance and predictor/corrector guidance. Perturbation guidance

is used to control heat-rate during cruise, and the predictor/corrector

guidance is used to control the end-conditions.

3. The guidance system explicitl y controls the end-conditions and

wil l guide to any specified heat-rate, inclination and apogee combination.

4. The guidance system provides accurate control over heating rate

within its knowledge of density and velocity. Due to the importance of

heat-rate control (both from a performance and a vehicle safety point of

view), an accurate, reliable measurement/estimate of density and velocity should receive top priority.

5. The guidance system has demonstrated the ability to handle ±30%

density biases, two density shear profiles, ±15% aerodynamic uncertainties, a single-engine failure, and various combinations of the above.

178

PRECEDING PAGE BUNK NOT FILMED
The guidance system is adaptive in the sense that it wil l respond to

these dispersions and maintain control over heat-rate and the desired

targeting parameters.

6. The aerodynamic uncertainties caused the largest uncontrollable

performance effect (fuel usage) and therefore represent the largest

uncertainty in efficiency of the ERV synergetic plane change maneuver.

7. Simulated plane change missions using this guidance system have

nearly the same total fuel usage as trajectories generated by an optimization program.

8. Overall, with this guidance algorithm in its software the ERV

can be expected to accurately maneuver through the earth's atmosphere

for the programmed synergetic plane change and thereby demonstrate capabilitie s that wil l be desired on future operational vehicles.

7.3 TOPICS FOR FURTHER RESEARCH

As with most all challenging tasks, time never seems available to

explore all the possible options nor to deal with all the opportunities

that surface during the investigation. Listed below are five such items

which seem to warrant further study and/or work.

179
7.3.1 NODE CONTROL

While the objective of this task was to provide a guidance system to

handle an ERV during a plane change, the ultimate desire wil l surely

involve a system to maneuver into a specific inertia! plane. Therefore,

in addition to inclination control, control of the longitude of the

ascending node would also be required. While we have addressed the node

shift in Chapter 6 by including the fuel required to correct any undesired shift, it is not clear that both inclination and node can not be

efficiently controlled during the atmospheric phase. In other words,

would the total fuel for handling both in the atmosphere be less than

the current scheme where only inclination is actively controlled in the

atmosphere and any node shift is corrected on-orbit.

The current algorithm has available three possible control variables, (1) the length of the cruise phase, (2) angle-of-attack during

exit, and (3) engine throttling during exit. These would seem to offer

a means of controlling the node. However, it is also unclear if the

added complexity is worth any possible fuel savings.

7.3.2 PLANE CHANGE TRAJECTORY IMPROVEMENTS

During our analysis of single-engine failure performance, some of

our results seemed to suggest that some performance gain could be made

180
by allowing the thrust to vary during the exit phase. In theory, this

concept could be tested with only two or three POST runs. Likewise,

there are other possible parameters that have been fixed during our

plane change analysis that if allowed to vary or change more often could

offer some performance improvements. For example, a few POST runs were

made where alpha and bank angle were allowed to change two additional

times during entry, resulting in a small gain of roughly 100 pounds of

fuel. Thus, it seems possible that a "more optimum" trajectory exists.

7.3.3 HEAT-RATE CONTROL

Our final scheme used a lead-point for our heat-rate controller that

wil l stil l modify the first heat-rate peak under some perturbed conditions causing fuel efficiency to suffer. A method to avoid this was

mentioned in Chapter 4 but never tested. Specifically it might be possible to actively control heat-rate such that the first heat-rate peak

is not eliminated or significantly modified. The resulting fuel savings

could be as high as 8% of the total fuel used or roughly 450 pounds for

a 20 degree plane change.

7.3.4 MINIMUM FUEL

One advantage of the predictor/corrector guidance scheme is the

availabilit y of fairly accurate predictions of end-conditions lik e fuel

181
remaining. This information could be used to make adjustments to the

intended trajectory (i.e., reduced the amount of desired plane change)

as necessary to insure the ERV returns to orbit with the minimum

requi red fuel.

7.3.5 APOGEE ALTITUDE CONTROL

Several of the test cases in Chapter 6 for multipl e off-nominal conditions, failed to guide to the apogee altitude withi n the desired accuracy criterion. All of the cases were for a heat-rate of 80 BTU/ft2-sec

and most of them involved off-nominal aerodynamic conditions. It might

be possible to improve the apogee accuracy by providing an improved

estimate of the aerodynamic characteristics to the predictor/corrector

routine.

182
6. Powell, R.W., Naftel, J.C. Cunningham, N.J., "Performance Evaluation Of An Entry Research Vehicle," AIAA Paper Mo. 86-0270, AIAA

24th Aerospace Sciences Meeting, Reno, NA, January 1986.

7. Stalony-Dobrzanski, J., "Re-entry Guidance And Control Using Temperature Rate Fligh t Control System"

8. Bradt, J.E., Hardtla, J.W., "An Adaptive Guidance Algorithm For

Aerospace Vehicles", AIAA Paper No. 85-1917.

9. Cervisi,. R.T., "Analytic Solution For A Cruising Plane Change

Maneuver," AIAA Paper No. 83-2095, AIAA 10th Atmospheric Fligh t

Mechanics Conference, Gatlinburg, TN, August 1983.

10. Clauss, J.S., Jr. and Yeatman, R.D., "Effect Of Heating

Restraints On Aeroglide And Aerocruise Synergetic Maneuver Performance,

 M__jlflUO13J^_Ilf__5|iSiei£af_L_^nji__fioiJs£jL5j. Vol. 4, August

1967, pp. 1107-1109.

11. Dickmanns, E.D., "The Effect Of Finit e Thrust And Heating Constraints On The Synergetic Plane Change Maneuver For A Space

Shuttle Orbiter - Class Vehicle," NASA TN-0-7211, October 1973.

184
12. Joosten, B.K. and Pierson, B.L., "Minimum Fuel Aerodynamic

Orbital Plane Change Maneuvers," AIAA Paper 81-0167, AIAA 19th

Aerospace Sciences Meeting, St. Louis, Missouri, January 1981.

13. Hull , D.G., "Optimal Reentry and mlane-Change Trajectories,"

Paper 81-156, AAS/AIAA Astrodynamics Specialist Conference, Lake

Tahoe, Nev., August 1981.

14. Ikawa, H. and Rudiger, T.F., "Synergetic Maneuvering of Winged

Spacecraft For Orbital Plane Change," Journal Of Spacecraft And

Rockets. Volume 19, November - December 1982, pp. 513-520.

15. Parsons, W.D., "Analytic- Solution Of The Synergetic Turn." Journal Of space Craft And Rockets. Volume 3, No. 11, November 1966,

pp1675-1678.

16. Bate, R.R., Mueller, D.D., and White, J.E., Fundamentals of

Astrodynamics. Dover Publications, Inc., New York, 1971, pp. 396-

407.

17. Walderg, G.D., "A Review of Aeroassisted Orbit Transfer", AIAA

paper 82-1378, AIAA 9th Atmospheric Fligh t Mechanic Conference,

San Diego, California, August 1982.

185
18. Candra, E. and Arthur, Paul D., "Orbit Plane Change by External

Buring Aerocruise", AIAA Paper 65-21, AIAA 2nd Aerospace Sciences

Meetings, New York, NY, January 1965.

19. Kriegsman, B.A., Richards, R.T., Brand, T.J., "Guidance and Navigation System Studies for Entry Research Vehicle," CSDL-P-2864,

Cambridge, Massachusetts, Apri l 1986.

20. Maslen, S.H., "Synergetic Turns with Variable Aerodynamics",

ylQUfryal of Spacecraft and Rockets. Vol. 4, November 1967, pp

1475-1482, Cambridge, Massachusetts.

186
APPENDIX A

POST

A.1 INTRODUCTION

This appendix provides the details on the primary tool that was used

to conduct the plane change analysis. The formulation of the plane

change problem involves many nonlinearities, beginning with the

equations of motion and heat transfer and skin temperature computations.

In this situation, the value of linearized analysis is quite limite d

and, at times, even misleading. Therefore, most of the analysis and

synthesis work was accomplished with the aid of digita l simulations.

Specifically, this appendix wil l provide details on the Program to Optimize Simulated Trajectories (POST) and an example of trajectory generation for a nominal 20 degree plane change as an illustration of POST.

A.2 POST DESCRIPTION

The following was taken almost verbatim from Reference 1. POST, a

generalized point mass, discrete parameter targeting and optimization

program, provides the capability to target and optimize point mass trajectories for a powered or unpowered vehicle operating near a rotating

oblate planet. Developed for the vehicle analysis branch, space direc187
torate, NASA Lang ley Research Center by the Martin Marietta Corporation,

it was intended to provide the capability to simulate and optimize

ascent and reentry trajectories for shuttle type vehicles. However, the

generality of the program also allows various other types of vehicles to

be simulated. POST has been used successfully to solve a wide variety

of atmospheric flight mechanics and orbital transfer problems. The generality of the program is evidenced by its N-phase simulation capability, which features generalized planet and vehicle models. This flexible

simulation capability is augmented by an efficient discrete parameter

optimization capability that includes equality and inequality constraints. POST was originall y written in Fortran IV for the CDC 6000

series computers. However, it is also operational on the IBM 370, and

Univac 1108 computers.

One of the key features of POST is an easy to use NAMELIST-type

input procedure. This feature significantly reduces input deck set-up

time (and costs) for studies that require the normal large amount of

input data. In addition, the general applicability of POST is further

enhanced by a general-purpose discrete parameter targeting and optimization capability. This capability can be used to solve a broad spectrum

of problems related to the performance characteristics of aerospace

vehicles.

188
The basic simulation flexibilit y is achieved by decomposing the trajectory into a logical sequence of simulation segments. These trajectory segments, referred to as phases, enable the trajectory analyst to

model both the physical and the nonphysical aspects of the simulation

accurately and efficiently. By segmenting the mission into phases, each

phase can be modeled and simulated in a manner most appropriate to that

particular fligh t regime. For example, the planet model, the vehicle

model, and the simulation options can be changed in any phase to be compatible with the level of detail required in that phase.

Every computational routine in the program can be categorized

according to five basic functional elements. These elements are: The

planet model, the vehicle model, the trajectory simulation model, the

auxiliar y calculations module, and the targeting and optimization module. The planet model is composed of an oblate spheroid model, a gravitational model, an atmosphere model, and a winds model. These models

define the environment in which the vehicle operates. The vehicle model

comprises mass properties, propulsion, aerodynamics and aeroheating and

a navigation and guidance model. These models define the basic vehicle

simulation characteristics. The trajectory simulation models are the

event-sequencing module that controls the program cycling, table

interpolation routines, and several standard numerical integration techniques. These models are used in numerically solving the translational

and rotational equations of motion. The auxiliary calculations module

189
provides for a wide variety of output calculations. For example, conic

parameters, range calculations, and tracking data are among the many

output variables computed. The targeting and optimization module provides a general discrete parameter iteration capability. The user can

select the optimization variable, the dependent variables, and the independent variables from a list of more than 400 program variables. An

accelerated projected gradient algorithm is used as the basic optimization technique. This algorithm is a combination of Rosen's projection

method for nonlimear programming and Davidon's variable metric method

for unconstrainted optimization. In the targeting mode, the minimum

norm algorithm is used to satisfy the trajectory constraints. The cost

and constraint gradients required by these algorithms are computed as

first differences calculated from perturbed trajectories. To reduce the

costs of calculating numerical sensitivities, only that portion of the

trajectory influenced by any particular independent variable is reintegrated on the perturbed runs. this feature saves a significant amount

of computer time when targeting and optimization is performed (1). A

more complete description and addtional information can be found in the

formulation, utilization, and programmers manuals (1), (2), and (3).

A.3 TRAJECTORY GENERATION EXAMPLE

The baseline plane change trajectories were primaril y developed by

R.W. Powell and J.C. Naftel of NASA Langley Research Center. However,

190
the author with help and assistance from Powell and Naftel made modifications and some improvements to the POST baseline trajectory input deck

for his specific application. The following 14 pages are presented as

an example of a POST input deck necessary to produce the nominal 20

degree plane change trajectory with a heating rate of 125 BTU/ft2-sec.

The complete ERV CL and CO data is included as a part of the input deck.

The data is set up in three types of tables corresponding to three correlation parameters: (1) altitude (ALTITO) , (2) viscous (VINV) , and (3)

mach number (MACH). Each correlation parameter table is broken into a

number of sections for CL and CD (e.g., ALTITO = 300,000' and 537,000').

Withi n each section the CL or CD values are catalogued for angles-of-attack 0 to 60 degrees (1 degree increments).

Following the input code is a brief description of the input conditions, the various phases of the trajectory, and the independent variables that were used while optimizing the amount of fuel required for

the plane change along with the constraints or dependent variables that

were used in generating the plane change trajectory with POST. Most of

the notation used in this appendix is the same notation that \s used by

POST. However, where possible both the POST and the more generic notation used in the main body of this thesis is shown.

191
P 0 S T - MMC 3D TARGETING AND OPTIMIZATION PROGRAM. REVISED 052181 , PROBLEM NO. 1

COPYRIGHT 1981 MARTIN MARIETTA CORPORATION

LSSEARCH

PERT = O1..1,3».01,.00001,6* 01,

PCTCC=.01,

SRCHM-4,

OPT=1,

OPTVAR=6HWEIGHT,

OPTPH=1000,

NDEPV=3,

DEPVR=6HINC .6HALTA .6HXMAX4 ,

DEPVAL=48.5,160,1.0,

DEPTL=.10,5,.01.

DEPPH=1000.1000,1000,

NINDV=12,

INDVR=5HCRITR,5HCRITR,6HALPPC1,6HBNKPC1,

6HALPPC1.6HDPVLS1.5HCRITR.5HDBANK,6HDALPHA,5HCRITR.6HALPPC1,6HBNKPC1,

INDPH=50,55,100,100.105,105,180,180,180,200,200.200,

U=- 9.27.27,-64,25,.004,40,-66,20,170,8,-50,

$

LSGENDAT

PRNC=1,

NPC(9) = 1,NPC(22) = 1 ,ETAPC(1)=0,

TITLE=OH*ERV CRUISING SYNERGISTIC PLANE CHANGE 0-DOT=125',

EVENT* 1,

SREF=177.40.

LREF=25.,

FESN=1000,

NPC(2)=4,

WGTSG=11430,

NPC( 1)=3,

NPC(15)=1,

NPC(3)=5,

NPC(5)=0,

NPC(8)=0,

NPC(12)=0,

NPC(16) = 1 ,

ISPV=295.0,

DT=100,

IGUID(1)=3,0,1.

ALTP=160,

ALTA=160,

PINC=1000.

NPC(25)=3,

PRNT(91)=5HXMAX1.5HXMIN2.5HXMAX3.5HXMIN4.5HXMAX4.5HBETAI.6HTIMRF1.6HVIDEAL,

6HVINV ,6HINC .6HPSTOP ,

MONX=6HHEATRT,6HALTITO,6HDYNP ,6HETA

INC=28.5,

TRUAN=150.,PGCLAT=-19,

BETA=0,

$

LSTBLMLT TVC1M=3,AE 1M = 3,CLM= 5 ,CDM= . 5 ,CLHM=.5,CDHM=.5,CLVM=.5,CDVM= . 5 $

LSTAB TABLE=5HTVC1T,0,1100$

L$TAB TABLE=6HAE1T ,0,1.4 $

LSTAB

TABLE«6HCLHT ,2.5HALPHA.6HALTITO,

3.00000000E+05,

,51 .2,1 ,1 ,1,1,1,1,1,1 ,

0. , 1 .54130435E-02,

2.00000000E+00, 4 .26161832E-02,

4.00000000E+00, 6.96270748E-02,

6.00000000E+00,

8.00000000E+00,

1.OOOOOOOOE+01,

1.20000000E+01

1.40000000E+01

1.60000000E+01,

1 80000000E+01, 4.09199247E-01,

2.00000000E+01, 4.71840614E-01,

1 .03075910E-01,

1 .43063966E-01,

1.83045852E-01,

2.36086535E-01,

2.89287518E-01,

3 46924729E-01,

1.00000000E+00,

3.00000000E+00,

5.00000000E+00,

7.00000000E+00,

9.00000000E+00,

1.10000000E+01.

1.30000000E+01,

1.50000000E+01,

1.70000000E+01,

1.90000000E+01,

2.10000000E+01,

2.90433333E-02,

5.61415108E-02,

8.30790323E-02,

1.23070807E-01,

1.63055590E-01,

2.09544373E-01,

2.62668672E-01,

3.15940152E-01,

3.78013237E-01,

4.40476852E-01,

5.05921147E-01,

1-

2-

3-

4-

5-

6-

7-

8-

9-

10-

1 1-

12-

13-

14-

15-

16-

17-

18-

18-

19-

20-

21-

22-

23-

24-

25-

26-

27-

28-

29-

30-

31-

32-

33-

34-

35-

36-

37-

38-

39-

40-

41-

42-

43-

44-

45-

46-

47-

48-

48-

48-

48-

48-

49-

50-

51-

52-

53-

54-

55-

56-

57-

58-

59-

60-

61-

192
ORIGINAL PAGE STOP POOR QUALITY

2.20000000E+01,

2.400000OOE+01.

2.60000000E+01,

2.80000000E+01,

3.00000000E+01,

3.20000000E+01,

3.40000000E+01.

3.60000000E+01,

3.80000000E+01,

4.00000000E+01.

4.20000000E+01,

4 40000000E+01,

4.60000000E+01,

4.80000000E+01,

5.00000000E+01,

5.37000000E+05,

0

2.00000000E+00,

4.00000000E+00,

S.OOOOOOOOE+00,

8.00000000E+00,

1.OOOOOOOOE+01,

1.20000000E+01,

1 40000000E+01,

1 60000000E+01,

1.80000000E+01,

2.OOOOOOOOE+01,

2.20000000E+01,

2 40000000E+01,

2 60000000E+01.

2.80000000E+01,

3.OOOOOOOOE+01.

3.20000000E+01.

3.40000000E+01,

3.60000000E+01.

3.80000000E+01,

4.OOOOOOOOE+01,

4.20000000E+01,

4 40000000E+01,

4.60000000E+01,

4.80000000E+01,

5.OOOOOOOOE+01,

$

L$TAB

TABLE=6HCDHT ,2

3.00000000E+05,

0.

2.00000000E+00,

4 OOOOOOOOE+00.

6.00000000E+00,

8 OOOOOOOOE+00,

1.OOOOOOOOE+01.

1.20000000E+01,

1 40000000E+01,

1 60000000E+01,

1.80000000E+01,

2.OOOOOOOOE+01,

2.20000000E+01,

2 40000000E+01,

2.60000000E+01,

2.80000000E+01,

3.OOOOOOOOE+01,

3.20000000E+01,

3 40000000E+01,

3.60000000E+01,

3 80000000E+01,

4.OOOOOOOOE+01,

4.20000000E+01,

4.40000000E+01,

4.60000000E+01,

5 39782977E-01,

6 07475313E-01,

6.75698509E-01,

7.44588487E-01,

8.15700000E-01,

8.87540000E-01,

9.59380000E-01,

1.02692000E+00,

1.09016000E+00,

1 15340000E+00,

1.20220000E+00,

1.25100000E+00,

1.28949000E+00,

1.31767000E+00,

1.34585000E+00,

9.20449438E-03,

1 27396078E-02,

1 70659970E-02,

2.15988761E-02,

2.33543284E-02,

2.97500000E-02,

3.85700000E-02,

4.73900000E-02,

5.69150000E-02,

6.71450000E-02,

7.73750000E-02,

8.83450000E-02,

9.93150000E-02,

1.10300000E-01 ,

1 .21300000E-01,

1 32300000E-01 ,

1 42510000E-01,

1.52720000E-01,

1.62160000E-01,

1 70830000E-01,

1 79500000E-01,

1 85910000E-01,

1.92320000E-01,

1.97300000E-01,

2.00850000E-01",

2.04400000E-01,

2.30000000E+01,

2.50000000E+01,

2.70000000E+01,

2.90000000E+01,

3 10000000E+01,

3.30000000E+01,

3.50000000E+01,

3.70000000E+01,

3 90000000E+01,

4.10000000E+01,

4.30000000E+01,

4.50000000E+01,

4.70000000E+01,

4.90000000E+01,

1.OOOOOOOOE+00,

3 OOOOOOOOE+00,

5.OOOOOOOOE+00,

7.OOOOOOOOE+00,

9.OOOOOOOOE+00,

1 10000000E+01,

1 30000000E+01,

1.50000000E+01,

1.70000000E+01.

1.90000000E+01,

2 10000000E+01,

2.30000000E+01,

2.50000000E+01,

2.70000000E+01,

2.90000000E+01,

3 10000000E+01,

3.30000000E+01,

3.50000000E+01,

3.70000000E+01,

3.90000000E+01,

4.10000000E+01,

4.30000000E+01,

4.50000000E+01,

4.70000000E+01,

4.90000000E+01,

5.73567038E-01,

6.41499793E-01,

7.10064539E-01,

7.7926164SE-01,

8.51620000E-01,

9.23460000E-01,

9.95300000E-01,

1.05854000E+00,

1.12178000E+00,

1 17780000E+00,

1.22660000E+00,

1.27540000E+00,

1.30358000E+00,

1.33176000E+00,

1 08389264E-02,

1 48316963E-02,

1 94088435E-02,

2.24467141E-02,

2.63200000E-02,

3.41600000E-02,

4.29800000E-02,

5.18000000E-02,

6.20300000E-02,

7.22600000E-02,

8.28600000E-02,

9.38300000E-02,

1.04800000E-01,

1.15800000E-01,

1.26800000E-01,

1.37405000E-01 ,

1 47615000E-01.

1.57825000E-01,

1.66495000E-01,

1.75165000E-01,

1.82705000E-01,

1.89115000E-01,

1.95525000E-01,

1.99075000E-01,

2.02625000E-01,

,5HALPHA,6HALTITO,51,2,1,1,1.1.1,1.1,1.

9 73826087E-02.

1 02212214E-01,

1 07470748E-01,

1 16979753E-01.

1.30511331E-01.

1 44054585E-01,

1 67867737E-01,

1 91448482E-01.

2.22711371E-01,

2.61490092E-01,

2 99887372E-01,

3.54901625E-01 .

4 10871250E-01,

4.74809028E-01,

5.46673604E-01,

6.19675000E-01.

7 15715000E-01,

8 11755000E-01 ,

9. 15945000E-01,

1 02828500E+00,

1.14062500E+00,

1.26684500E+00,

1 39306500E+00,

1.52469000E+00.

1.OOOOOOOOE+00,

3.OOOOOOOOE+00,

5.OOOOOOOOE+00,

7.OOOOOOOOE+00,

9.OOOOOOOOE+00,

1.10000000E+01,

1 30000000E+01.

1.50000000E+01,

1-70000000E+01,

1.90000000E+01,

2.10000000E+01,

2.30000000E+01,

2.50000000E+01,

2.70000000E+01,

2 90000000E+01,

3.10000000E+01,

3.30000000E+01,

3.50000000E+01,

3.70000000E+01,

3.90000000E+01,

4.10000000E+01,

4.30000000E+01,

4.50000000E+01 ,

4.70000000E+01,

9.97333333E-02,

1.04797122E-01,

1.10219355E-01,

1.23743898E-01,

1.37281670E-01,

1 55992797E-01,

1 79684720E-01,

2.03163258E-01,

2 42151422E-01,

2.80733524E-01,

3.27071201E-01,

3.82984404E-01,

4.38574793E-01,

5.10838124E-01,

5.82326141E-01,

S.67695000E-01,

7.63735000E-01,

8.59775000E-01 ,

9.72115000E-01,

1.08445500E+00,

1.20373500E+00.

1.32995500E+00,

1.45617500E+00,

1.59320500E+00,

62-

F3-

64-

65-

66-

67-

68-

69-

70-

71-

72-

73-

74-

75-

76-

77-

78-

79-

80-

81-

82-

83-

84-

85-

86-

87-

88-

89-

90-

91-

92-

93-

94-

95-

96-

97-

98-

99-

100-

101-

102-

103-

104-

104-

105-

106-

107-

108-

109-

110-

111 -

112-

113-

114-

115-

116-

117-

118-

119-

120-

121-

122-

123-

124-

125-

126-

127-

128-

129-

130-

193
4.80000000E+01,

5.00000000E+01,

5.37000000E+05.

0.

2.00000000E+00,

4.0OOOOOOOE+00,

6.00000000E+00,

8.00000000E+00,

1.00000000E+01,

1.20000000E+01,

1.40000000E+01,

1.60OOOOOOE+01,

1.80000000E+01.

2.00OOOOOOE+01,

2.20000000E+Ot.

2.40000000E+01,

2.60000000E+01,

2.80000000E+01,

3.000OOOOOE+01,

3.20000000E+01,

3.40000000E+01,

3.60000000E+01,

3.80000000E+01,

4.00000000E+01,

4.20000000E+01,

4 40000000E+01,

4.60000000E+01,

4.80000000E+01,

5.00000000E+01,

$

LSTAB

TABLE=6HCLVT .2

5.00000000E-03,

0.

2.00000000E+00,

4.00000000E+00,

6.00000000E+00,

8.00000000E+OO,

1.00000000E+01,

1.20000000E+01,

1.40000000E+01,

1.60000000E+01,

1.80000000E+01,

2.00000000E+01,

2.20000000E+01,

2 40000000E+01,

2.60000OOOE+01,

2 80000000E+01,

3 OOOOOOOOE+01,

3.20000000E+01,

3.40000000E+01,

3 60000000E+01,

3 80000000E+01,

4.OOOOOOOOE+01,

4.20000000E+01,

4 40000000E+01,

4.60000000E+01,

4 80000000E+01,

5.OOOOOOOOE+01,

1.03000000E-02,

0.

2.00000000E+00,

4.00000000E+00,

6.00000000E+00,

8.00000000E+00,

1.OOOOOOOOE+01.

1.20000000E+01,

1 40000000E+01,

1 60000000E+01,

1.80000000E+01,

1.6S172000E+00,

1.79875000E+00,

6.84452809E-01,

7.24948627E-01,

7.73231049E-01,

8.44880153E-01,

9.66479159E-01,

1.08697500E+00,

1.21427500E+00,

1.34157500E+00,

1.46999400E+00,

1.59953200E+00,

1.72907000E+00,

1.85840200E+00.

1.98773400E+00,

2.11653500E+00,

2.24480500E+00,

2.37307500E+00,

2.50044500E+00,

2 62781500E+00,

2 75382000E+00,

2 87846000E+00,

3.00310000E+00,

3. 12228000E+00,

3.24146000E+00,

3.35755500E+00,

3.47056500E+00,

3.58357500E+00.

4.90000000E+01, 1 73023500E+00,

1.00000000E+00,

3.00000000E+00.

5.00000000E+00.

7.00000000E+00,

9.00000000E+00,

1.10000000E+01,

1.30000000E+01,

1.50000000E+01,

1 70000000E+01.

1.90000000E+01,

2.10000000E+01,

2.30000000E+01,

2.50000000E+01,

2 70000000E+01,

2 900000OOE+01,

3.100000OOE+01,

3.30000000E+01,

3.50000000E+01,

3 70000000E+01,

3 900000OOE+01,

4 10000000E+01,

4.30000000E+01.

4 50000000E+01,

4.70000000E+01,

4.90000000E+01,

7.03390696E-01,

7.48390113E-01,

7.99140136E-01,

9.06022390E-01,

1.02663500E+00,

1.15062500E+00,

1.27792500E+00,

1.40522500E+00,

1.53476300E+00,

1.66430100E+00,

1.79373600E+00,

1.92306800E+00,

2.05240000E+00,

2.18067000E+00,

2.30894000E+00,

2.43676000E+00,

2.56413000E+00,

2 69150000E+00,

2.81614000E+00,

2.94078000E+00,.

3.06269000E+00,

3.18187000E+00,

3.30105000E+00,

3.41406000E+00,

3.52707000E+00,

5HALPHA.6HVINV ,51 ,8 ,1,1,1,1,1,1,1,1,

1.46464286E-02,

4.25473300E-02,

7.02738964E-02,

1.04420102E-01,

1 45075276E-01,

1.85721212E-01,

2.38744784E-01,

2.91953236E-01,

3.50041208E-01,

4.13206959E-01,

4.76757143E-01,

5.44962558E-01,

6 13558139E-01,

6.82654131E-01.

7 52433468E-01,

8.27800000E-01,

9.00010000E-01,

9.72220000E-01,

1.03987000E+00,

1.10296000E+00,

1 16605000E+00,

1.21374000E+00,

1.26143000E+00,

1.29905500E+00,

1.32661500E+00,

1.35417500E+00,

1 48928571E-02,

4.25916877E-02,

7.01326577E-02.

1.04138937E-01,

1.44689774E-01,

1.85230435E-01,

2.37959444E-01,

2.90882813E-01,

3.48780309E-01,

4.11833621E-01,

1.00000000E+00,

3.00000000E+00,

5.00000000E+00,

7.00000000E+00,

9.00000000E+00,

1 10000000E+01,

1.30000000E+01,

1 50000000E+01,

1 70000000E+01,

1.90000000E+01,

2 10000000E+01,

2 30000000E+01,

2.50000000E+01,

2 70000000E+01,

2 90000000E+01,

3 10000000E+01,

3.30000000E+01,

3.50000000E+01,

3.700000OOE+01,

3 90000000E+01,

4 10000000E+01,

4.30000000E+01,

4.50000000E+01,

4.70000000E+01,

4.90000000E+01,

1.00000000E+00,

3.00000000E+00,

5.00000000E+00,

7.00000000E+00,

9.00000000E+00,

1.10000000E+01,

1.30000000E+01,

1.50000000E+01,

1.70000000E+01,

1.90000000E+01,

2.86227845E-02,

5.64287515E-02,

8.40882353E-02,

1.24748993E-01,

1.65399262E-01,

2.12207849E-01,

2.65327825E-01,

3.18617669E-01 ,

3.81573117E-01,

4.44936796E-01,

5.10854598E-01 ,

5.79199794E-01,

6.48030103E-01,

7.17458692E-01 ,

7.89700000E-01,

8.63905000E-01,

9.36115000E-01,

1.00832500E+00,

1.07141500E+00,

1. 13450500E+00,

1.18989500E+00,

1.23758500E+00,

1.28527500E+00,

1.31283500E+00,

1.34039500E+00,

2.87657296E-02,

5.63785969E-02,

8.38588235E-02,

1.24415788E-01,

1.64961229E-01,

2.11568432E-01,

2.S4398919E-01,

3.17407519E-01,

3.80257749E-01,

4.43502191E-01,

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ORIGINAL PAGE 13

OF POOR QUALITY

2.OOOOOOOOE+01.

2.20000000E+01,

2.40000000E+01,

2.60000000E+01,

2.80000000E+01,

3.00000000E+01,

3.20000000E+01.

3.40000000E+01,

3.60000000E+01,

3.80000000E+01,

4.00000000E+01,

4.20000000E+01,

4.40000000E+01,

4.60000000E+01,

4.80000000E+01,

5.00000000E+01.

1.48000000E-02,

0.

2.00000000E+00,

4.00000000E+00,

6.00000000E+00,

8.00000000E+00,

1.00000000E+01,

1.20000000E+01,

1 40000000E+01,

1 60000000E+01,

1.80000000E+01,

2.00000000E+01,

2.20000000E+01,

2.40000000E+01,

2.60000000E+01,

2.80000000E+01,

3.00000000E+01,

3.20000000E+01,

3.40000000E+01,

3.60000000E+01,

3.80000000E+01,

4.00000000E+01,

4.20000000E+01,

4.40000000E+01,

4.60000000E+01,

4.80000000E+01,

5.00000000E+01,

2.08000000E-02,

0.

2.00000000E+00,

4 OOOOOOOOE+00,

6.00000000E+00,

8.OOOOOOOOE+00,

1.00000000E+01,

1.20000000E+01,

1.40000000E+01,

1.60000000E+01,

1 80000000E+01,

2 OOOOOOOOE+01,

2.20000000E+01,

2.40000000E+01,

2.60000000E+01,

2.80000000E+01,

3.OOOOOOOOE+01,

3.200<56000E+01,

3.40000000E+01,

3.60000000E+01,

3 80000000E+01,

4.OOOOOOOOE+01,

4.20000000E+01,

4.40000000E+01,

4.60000000E+01,

4.80000000E+01,

5.OOOOOOOOE+01,

4.75253163E-01,

5.43145521E-01,

6.11440817E-01,

6.80353476E-01,

7.50032232E-01,

8.25275000E-01,

8.97415000E-01,

9.69555000E-01,

1.03714000E+00,

1.10017000E+00,

1.16320000E+00,

1.21132000E+00,

1.25944000E+00,

1.29737000E+00,

1.32511000E+00,

1.35285000E+00,

1.48928571E-02,

4.25225253E-02,

7.00035747E-02,

1.03926982E-01,

1 44368098E-01,

1.84800000E-01,

2.37542553E-01,

2.90472643E-01,

3 48269013E-01,

4 11123819E-01,

4.74357143E-01,

5.42217826E-01,

6.10479318E-01,

6 79290655E-01,

7.48812100E-01,

8.23900000E-01,

8.95990000E-01,

9.68080000E-01,

1.03566000E+00,

1.09873000E+00,

1 16180000E+00,

1.21008000E+00,

1.25836000E+00,

1.29641000E+00,

1 32423000E+00,

1 35205000E+00,

1 48857143E-02,

4.26460606E-02,

7.02564706E-02,

1.04186043E-01,

1.44515184E-01,

1.84839738E-01,

2.37510927E-01,

2.90380495E-01,

3 48096819E-01,

4 10840679E-01,

4.73957143E-01,

5.41777826E-01,

6.09999318E-01,

6.78723589E-01,

7.48103126E-01,

8.23050000E-01,

8.95110000E-01,

9.67170000E-01,

1.03471000E+00,

1.09773000E+00,

1 16075000E+00,

1.20923000E+00,

1.25771000E+00,

1.29589000E+00,

1.32377.000E+00,

1.35165000E+00,

2 10000000E+01,

2.30000000E+01,

2.50000000E+01,

2 70000000E+01,

2.90000000E+01,

3 10000000E+01,

3.30000000E+01,

3.50000000E+01,

3.70000000E+01,

3.90000000E+01,

4. 10000000E+01,

4.30000000E+01,

4.50000000E+01,

4 70000000E+01,

4.90000000E+01,

1.OOOOOOOOE+00,

3. OOOOOOOOE+00,

5.OOOOOOOOE+00,

7.OOOOOOOOE+00,

9 OOOOOOOOE+00,

1.10000000E+01,

1.30000000E+01,

1.50000000E+01,

1 70000000E+01,

1.90000000E+01,

2 10000000E+01,

2.30000000E+01,

2.50000000E+01,

2.70000000E+01,

2.90000000E+01,

3.10000000E+01,

3.30000000E+01,

3.50000000E+01,

3.70000000E+01,

3.90000000E+01,

4.10000000E+01,

4.30000000E+01,

4.50000000E+01,

4.70000000E+01,

4 90000000E+01,

1.OOOOOOOOE+00,

3.OOOOOOOOE+00,

5.OOOOOOOOE+00,

7 OOOOOOOOE+00,

9 OOOOOOOOE+00,

1.10000000E+01,

1.30000000E+01,

1 50000000E+01,

1 70000000E+01,

1 90000000E+01,

2 10000000E+01,

2 30000000E+01,

2.50000000E+01,

2.70000000E+01,

2.90000000E+01,

3 10000000E+01,

3.30000000E+01,

3.50000000E+01,

3.70000000E+01,

3.90000000E+01,

4 10000000E+01,

4 30000000E+01,

4.50000000E+01,

4.70000000E+01,

4.90000000E+01,

5.09193591E-01,

5 77230735E-01.

6.45768041E-01,

7.15111494E-01,

7.87275000E-01,

8.61345000E-01,

9.33485000E-01,

.00562500E+00,

.06865500E+00,

13168500E+00,

.18726000E+00,

.23538000E+00,

1.28350000E+00,

1 31124000E+00,

1 33898000E+00,

2.87297051E-02,

5.62785680E-02,

8.37021505E-02,

1.24148841E-01,

1 64585063E-01,

2 11145700E-01,

2 63986169E-01,

3 16998496E-01,

3 79646234E-01,

4 42695922E-01,

5.08280279E-01 ,

5.76287000E-01,

6 44787010E-01,

7 13969013E-01,

7.85940000E-01,

8 59945000E-01,

9 32035000E-01,

1 00412500E+00,

1.06719500E+00,

1.13026500E+00,

1 18594000E+00,

1 23422000E+00,

1.28250000E+00,

1 31032000E+00,

1.33814000E+00,

2.87880965E-02.

5.64669212E-02,

8.40193548E-02,

1.24351260E-01,

1.64677967E-01,

2.11148206E-01,

2 63923105E-01,

3 16879323E-01,

3.79419351E-01,

4 42355049E-01,

5.07860279E-01,

5.75827000E-01 ,

6.44287010E-01,

7 13332239E-01,

7 85160000E-01,

8.59080000E-01,

9.31140000E-01,

1.00320000E+00,

1.06622000E+00,

1 12924000E+00,

1.18499000E+00,

1.23347000E+00,

1 28195000E+00,

1.30983000E+00,

1 33771000E+00,

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195
2.45000000E-02,

0.

2.OOOOOOOOE+00,

4.OOOOOOOOE+00,

6.00000000E+00,

8.00000000E+00,

1.00000000E+01,

1.200000OOE+01,

1 40000000E+01 ,

1.60000000E+01,

1.80000000E+01,

2.00000000E+01,

2.20000000E+01,

2.40000000E+01,

2.60000000E+01,

2.80000000E+01,

3.00000000E+01,

3.20000000E+01,

3.40000000E+01,

3.60000000E+01,

3.80000000E+01,

4.00000000E+01,

4.20000000E+01,

4.40000000E+01,

4.60000000E+01 ,

4.80000000E+01 ,

5.00000000E+01,

3.73000000E-02,

0.

2.00000000E+00,

4.00000000E+00,

6.00000000E+00,

8.00000000E+00,

1.00000000E+01,

1.20000000E+01,

1.40000000E+01,

1.60000000E+01,

1 80000000E+01,

2.00000000E+01,

2.20000000E+01,

2.40000000E+01,

2.60000000E+01,

2.80000000E+01,

3.00000000E+01,

3.20000000E+01,

3.40000000E+01,

3.60000000E+01,

3.80000000E+01,

4 OOOOOOOOE+01,

4.20000000E+01,

4.40000000E+01,

4.600OOOOOE+01,

4.80000000E+01,

5.OOOOOOOOE+01,

5 70000000E-02,

0.

2.00000000E+00,

4.00000000E+00,

6.00000000E+00,

8.00000000E+00,

1.OOOOOOOOE+01,

1.20000000E+01,

1.40000000E+01,

1.60000000E+01,

1 80000000E+01,

2.OOOOOOOOE+01,

2.20000000E+01,

2.40000000E+01,

2.60000000E+01,

2. 80000000E+01,

1 49043165E-02,

4.25933840E-02,

7.01096489E-02,

1.04000741E-01,

1.44358409E-01,

1.84710526E-01 ,

2.37416705E-01,

2.90317019E-01 ,

3.47993097E-01,

4.10631327E-01,

4 73645392E-01,

5 41402120E-01,

6.09531081E-01,

6.78192470E-01,

7.47538288E-01,

8.22275000E-01,

8.94325000E-01,

9.66375000E-01,

1.03393000E+00,

1.09699000E+00,

1.16005000E+00,

1.20853000E+00,

1.25701000E+00,

1.29523000E+00,

1.32319000E+00,

1 35115000E+00,

1 48971223E-02,

4.28394170E-02,

7.05992072E-02,

1 04571655E-01,

1.44853617E-01,

1 85129694E-01 ,

2.37593114E-01,

2.90249170E-01,

3.47718108E-01,

4 10179134E-01 ,

4.73006826E-01,

5 4Q789620E-01,

6.08816258E-01,

6.77340073E-01,

7 46512333E-01,

8.20400000E-01,

8.92350000E-01,

9.64300000E-01,

1.03183500E+00,

1 09495500E+00,

1.15807500E+00,

1.20675500E+00,

1.25543500E+00,

1.29379000E+00,

1 .32182000E+00,'

1.34985000E+00,

1.52669O65E-02,

4.27977440E-02,

7.01564892E-02,

1.04128818E-01 ,

1.44795713E-01,

1.85445852E-01,

2.37810337E-01,

2.90362002E-01,

3 47704080E-01,

4.10021703E-01.

4.72708874E-01,

5.40566855E-01,

6.08385743E-01,

6.76708562E-01,

7.45670728E-01,

1.00000000E+00,

3.00000000E+00.

5.00000000E+00,

7.00000000E+00,

9.00000000E+00,

1.10000000E+01,

1.30000000E+01,

1.50000000E+01,

1.70000000E+01.

1.90000000E+01,

2.10000000E+01,

2.30000000E+01,

2.50000000E+01.

2.70000000E+01,

2.9000OOOOE+01 ,

3.10000000E+01 ,

3.30000000E+01,

3.50000000E+01 ,

3.70000000E+01,

3.90000000E+01,

4.10000000E+01,

4.30000000E+01,

4.50000000E+01,

4.70000000E+01,

4.90000000E+01,

1.OOOOOOOOE+00,

3.00000000E+00,

5.OOOOOOOOE+00,

7.OOOOOOOOE+00,

9.OOOOOOOOE+00,

1.10000000E+01,

1.30000000E+01,

1.50000000E+01 ,

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1.90000000E+01 ,

2.10000000E+01 ,

2.30000000E+01,

2.50000000E+01 ,

2.70000000E+01 ,

2.90000000E+01,

3.10000000E+01 ,

3.30000000E+01,

3.50000000E+01,

3.70000000E+01 ,

3.90000000E+01 ,

4.10000000E+01 ,

4.30000000E+01,

4.50000000E+01 ,

4.70000000E+01,

4.90000000E+01 ,

1 OOOOOOOOE+00,

3.OOOOOOOOE+00,

5.OOOOOOOOE+00,

7.OOOOOOOOE+00,

9.OOOOOOOOE+00,

1.10000000E+01,

1.30000000E+01,

1 50000000E+01,

1.70000000E+01,

1.90000000E+01,

2.10000000E+01.

2.30000000E+01,

2.50000000E+01,

2.70000000E+01,

2 90000000E+01.

2.87745553E-02,

5.63694737E-02,

8.38193548E-02,

1.24180355E-01,

1.64535082E-01,

2.11037060E-01,

2.63844748E-01,

3 16829811E-01,

3 79262375E-01,

4.42094127E-01,

5.07535986E-01,

5.75403454E-01,

6.43776860E-01,

7.12783016E-01,

7.84405000E-01,

8.58300000E-01,

9.30350000E-01,

1 00240000E+00,

1.06546000E+00,

1.12852000E+00,

1.18429000E+00,

1.23277000E+00,

1.28125000E+00,

1.30921000E+00,

1.33717000E+00,

2.88954178E-02,

5.67382775E-02,

8 44279570E-02,

1.24713465E-01,

1.64992305E-01,

2.11335123E-01,

2.63899138E-01,

3.16639623E-01,

3.78900000E-01,

4.41549826E-01,

5.06972694E-01,

5.74740415E-01,

6 43009091E-01,

7.11844464E-01,

7.82590000E-01,

8.56375000E-01,

9.28325000E-01,

1 00027500E+00,

1.06339500E+00,

1.12651500E+00,

1.18241500E+00,

1.23109500E+00,

1.27977500E+00,

1.30780500E+00,

1.33583500E+00,

2.90579245E-02,

5.64950000E-02,

8.37876344E-02,

1.24464626E-01,

1.65122632E-01,

2.11602557E-01,

2.64064789E-01,

3.16698491E-01,

3.78813888E-01,

4.41321795E-01,

5.06855488E-01,

5.74413238E-01.

6.42476240E-01,

7.11109952E-01,

7.80381934E-01,

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196
3.00000000E+01,

3.20000000E+01,

3.40000000E+01.

3.60000000E+01.

3.80000000E+01.

4.00000000E+01.

4.200CXX)OOE+01,

4.40000000E+01,

4.60000000E+01,

4.80000000E+01,

5.00000000E+01,

7.44000000E-02,

0.

2. OOOOOOOOE+00,

4. 00000000 E +00,

6.00000000E+00.

8.00000000E+00,

1.00000000E+01,

1.20000000E+01,

1.40000000E+01,

1.60000000E+01,

1.80000000E+01,

2.00000000E+01,

2.20000000E+01,

2.40000000E+01,

2.60000000E+01.

2.80000000E+01,

3.00000000E+01,

3.20000000E+01,

3 40000000E+01,

3.60000000E+01,

3.80000000E+01,

4.00000000E+01,

4.20000000E+01,

4.40000000E+01,

4 60000000E+01.

4.80000000E+01,

5.00000000E+01,

$

LSTAB

TABLE=6HCDVT ,2

5.00000000E-03,

0

2.00000000E+00,

4. 00000000 E +00,

6.00000000E+00.

8.00000000E+00,

1.00000000E+01,

1.20000000E+01,

1 40000000E+01 ,

1 60000000E+01 .

1.80000000E+01,

2.00000000E+01,

2.20000000E+01,

2 40000000 E +01 ,

2.60000000E+01,

2 80000000E+01 ,

3.00000000E+01 ,

3.20000000E+01,

3.40000000E+01,

3.60000000E+01.

3.80000000E+01,

4.00000000E+01,

4.20000000E+01.

4 40000000E+01 ,

4.60000000E+01 ,

4.80000000E+01 ,

5. 00000000 E +01,

1 03000000E-02,

0.

8 18075000E-01 ,

8.89955000E-01,

9.61835000E-01,

1 .02932500E+00,

1.09242500E+00,

1 . 15552500E+00,

1 .20438SOOE+00,

1.25324500E+00,

1.29173000E+00,

1 .31984000E+00,

1.34795000E+00,

1 54130435E-02,

4.26161832E-02,

6.96270748E-02,

1 .03075910E-01 ,

1 .43063966E-01 ,

1 .83045852E-01 ,

2.36086535E-01,

2.89287518E-01 ,

3.46924729E-01,

4.09199247E-01,

4 718406.14E-01,

5.39782977E-01,

6.07475313E-01 ,

6 75698509E-01,

7 44588487E-01 ,

8.15700000E-01,

8.87540000E-01,

9.59380000E-01,

1 .02692000E+00,

1 09016000E+00,

1 15340000E+00,

20220000E+00,

.25100000E+00,

.28949000E+00,

.31767000E+00,

34585000E+00,

,5HALPHA,6HVINV

4 61507143E-02,

5 33939295E-02,

6.09642117E-02,

7.23717058E-02,

8.74465792E-02,

1 02536364E-01,

1 .28157208E-01 ,

1 53509668E-01 ,

1 .85489352E-01 ,

2.23915035E-01 ,

2.61911224E-01 ,

3 17964993E-01 ,

3.73429491E-01 ,

4.37095319E-01 ,

5.08915933E-01 ,

5.85725000E-01,

6 82605000E-01,

7.79485000E-01,

8.84770000E-01,

9.98460000E-01,

1 . 11215000E+00,

1 .24043000E+00,

1.36871000E+00,

1 .50210000E+00,

1 .64060000E+00,

1 .77910000E+00,

4 65000000E-02,

3

3

3

3

3

4

4

4

4

4

1

3

5

7

9

1

1

1

1

1

2

2

2

2

2

3

3

3

3

3

4

4

4

4

4

,5

1

3

5

7

9

1

1

1

1

1

2

2

2

2

2

3

3

3

3

3

4

4

4

4

4

1

3.10000000E+01,

3.30000000E+01,

3.50000000E+01,

3.70000000E+01,

3.90000000E+01,

4.1000000CE+01,

4.30000000E+01,

4.50000000E+01,

4.70000000E+01,

4.90000000E+01,

1.00000000E+00,

3 OOOOOOOOE+00,

5 OOOOOOOOE+00,

7 OOOOOOOOE+00,

9.OOOOOOOOE+00,

1.10000000E+01,

1.30000000E+01,

1.50000000E+01,

1.70000000E+01,

1 90000000E+01,

2 10000000E+01,

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2.50000000E+01,

2.70000000E+01,

2.90000000E+01,

3.10000000E+01,

3.30000000E+01,

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4.30000000E+01,

4.50000000E+01,

4.70000000E+01,

4.90000000E+01,

8.54015000E-01,

9.25895000E-01,

9.97775000E-01,

1.06087500E+00,

1.12397500E+00,

1 .17995500E+00,

1.22881500E+00,

1 27767500E+00,

1 30578500E+00,

1.33389500E+00,

2.90433333E-02,

5.61415108E-02.

8.30790323E-02,

1.23070807E-01,

1.63055590E-01,

2.09544373E-01,

2.62668672E-01,

3.15940152E-01,

3.78013237E-01,

4.4047S852E-01,

5.05921147E-01,

5.73567038E-01,

6 41499793E-01,

7 10064539E-01,

7 79261646E-01,

8.51620000E-01,

9.23460000E-01,

9.95300000E-01,

1.05854000E+00,

1.12178000E+00,

1 17780000E+00,

1 .22660000E+00,

1 27540000E+00,

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1.33176000E+00,

,51,8,1,1,1,1,1 ,1,1,1,

1.OOOOOOOOE+00,

3.OOOOOOOOE+00,

5 OOOOOOOOE+00,

7.OOOOOOOOE+00,

9.OOOOOOOOE+00.

1 10000000E+01,

1.30000000E+01,

1.50000000E+01,

1 70000000E+01,

1.90000000E+01,

2. 10000000E+01 ,

2.30000000E+01,

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2.70000000E+01,

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3.10000000E+01,

3.30000000E+01,

3.50000000E+01,

3.70000000E+01,

3.90000000E+01,

4.10000000E+01,

4.30000000E+01,

4.50000000E+01,

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4.90000000E+01,

4.97237216E-02,

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6.48411765E-02,

7.99070381E-02,

9.49898290E-02,

1. 15383294E-01,

1.40864190E-01,

1 66098496E-01,

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2.42963689E-01,

2.89909726E-01,

3.45800284E-01,

4.00865361E-01,

4 73108038E-01,

5.46885000E-01,

6.34165000E-01,

7.31045000E-01,

8.27925000E-01,

9.41615000E-01,

1 .05530500E+00,

1.17S29000E+00,

1.30457000E+00,

1 43285000E+00,

1.57135000E+OOi

1 70985000E+00,

1.OOOOOOOOE+00, 5.02267470E-02 ,

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197
2.00000000E+00,

4.00000000E+00,

6.00000000E+00.

8 OOOOOOOOE+00,

1.OOOOOOOOE+01,

1.200000OOE+01,

1 400OOOOOE+01,

1.60000000E+01,

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2.OOOOOOOOE+01,

2.20000000E+01,

2.40000000E+01,

2.60000000E+01,

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3.OOOOOOOOE+01,

3.20000000E+01,

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3.80000000E+01,

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4.20000000E+01,

4.40000000E+01,

4 60000000E.+01,

4.80000000E+01,

5.OOOOOOOOE+01,

1.48000000E-02,

0.

2.OOOOOOOOE+00,

4.OOOOOOOOE+00.

6.OOOOOOOOE+00,

8.OOOOOOOOE+00,

1.OOOOOOOOE+01,

1.20000000E+01,

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4.20000000E+01.

4.40000000E+01,

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2.08000000E-02,

0.

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1.OOOOOOOOE+01,

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1.20477513E-01,

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2.50557573E-01,

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3.54099027E-01,

4 09451134E-01,

4.81857392E-01,

5 55820000E-01,

6.43020000E-01,

7.39560CXX)E-01,

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198
3.40000000E+01,

3.60000000E+01,

3.80000000E+01,

4.00000000E+01,

4.20000000E+01,

4.40000000E+01,

4.60000000E+01.

4.80000000E+01.

5.00000000E+01.

2.45000000E-02,

0.

2.OOOOOOOOE+00,

4.OOOOOOOOE+00,

6.OOOOOOOOE+00,

8.OOOOOOOOE+00,

1.00000000E+01,

1.20000000E+01,

1 40000000E+01,

1.60000000E+01,

1.80000000E+01,

2.00000000E+01,

2.20000000E+01,

2 40000000E+01.

2.60000000E+01.

2.80000000E+01,

3.00000000E+01,

3.20000000E+01,

3.40000000E+01,

3.60000000E+01,

3.80000000E+01,

4.00000000E+01,

4.20000000E+01,

4.40000000E-KM,

4.60000000E+01,

4.80000000E+01,

5.00000000E+01,

3.73000000E-02,

0.

2.00000000E+00,

4.OOOOOOOOE+00.

6.00000000E+00,

8.00000000E+00,

1.00000000E+01,

1.20000000E+01,

1.40000000E+01,

1 60000000E+01,

1.80000000E+01,

2.00000000E+01,

2 20000000E+01.

2 40000000E+01,

2.60000000E+01,

2.80000000E+01,

3.00000000E+01,

3.20000000E+01.

3.40000000E+01,

3.60000000E+01,

3.80000000E+01,

4.00000000E+01,

4.20000000E+01,

4.40000000E+01,

4 60000000E+01,

4.80000000E+01,

5.00000000E+01,

5.70000000E-02.

0.

2.00000000E+00,

4.00000000E+00.

6.00000000E+00,

8 OOOOOOOOE+00,

1.00000000E+01,

7.87830000E-01,

8.92775000E-01,

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1.OOOOOOOOE+00,

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9.OOOOOOOOE+00,

1.10000000E+01,

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1.OOOOOOOOE+00,

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5.OOOOOOOOE+00,

7.OOOOOOOOE+00,

9 OOOOOOOOE+00,

1 10000000E+01,

1.30000000E+01,

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2.10000000E+01,

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3.30000000E+01,

3.50000000E+01,

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4.90000000E+01,

1.OOOOOOOOE+00,

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5.OOOOOOOOE+00,

7 OOOOOOOOE+00,

9.OOOOOOOOE+00,

1.10000000E+01,

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8.75500000E-02,

9.36107527E-02,

1.06818543E-01,

1.20058947E-01,

1.39894716E-01,

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1.13312500E+00,

1.25936500E+00,

1.38560500E+00,

1.20000000E+01, 1.53029614E-01,

1.40000000E+01 1.79110908E-01,

1.60000000E+01, 2.11730466E-01,

1.80000000E+01, 2.50722271E-01,

2.OOOOOOOOE+01, 2.89311263E-01,

2.20000000E+01. 3.44728057E-01,

2.40000000E+01, 4 00618446E-01,

2.60000000E+01, 4.64530231E-01,

2 80000000E+01, 5 36430331E-01,

3.OOOOOOOOE+01, S.10875000E-01,

3.20000000E+01, 7.07125000E-01,

3.40000000E+01, 8.03375000E-01.

3.60000000E+01,

3.80000000E+01,

4.00000000E+01,

4.20000000E+01,

4.40000000E+01,

4.60000000E+01, 1 51744500E+00,

4 80000000E+01, 1 65488500E+00,

5.00000000E+01, 1 79232500E+00,

7.44000000E-02,

0 , 9 73826087E-02,

2.00000000E+00,

4.00000000E+00,

6.00000000E+00,

8.00000000E+00,

1-OOOOOOOOE+01 ,

1.20000000E+01,

1.40000000E+01,

1.60000000E+01,

1 80000000E+01,

2.OOOOOOOOE+01,

2.20000000E+01,

2 40000000E+01,

2.60000000E+01,

2.80000000E+01,

3.OOOOOOOOE+01,

3.20000000E+01,

3.40000000E+01,

1.02212214E-01,

1 07470748E-01,

1. 16979753E-01,

1.30511331E-01,

1 44054585E-01,

1 67867737E-01,

1 91448482E-01,

2.22711371E-01,

2.61490092E-01,

2 99887372E-01,

3.54901625E-01,

4.10871250E-01,

4.74809028E-01,

5.466-73604E-01 ,

6.19675000E-01,

7.15715000E-01,

8.11755000E-01,

3.60000000E+01, 9.15945000E-01,

3.80000000E+01, 1 02828500E+00,

4.00000000E+01, 1 14062500E+00,

4.20000000E+01 , 1.26684500E+00,

4.40000000E+01, 1.39306500E+00,

4.60000000E+01, 1.52469000E+00,

4.80000000E+01, 1 66172000E+00,

5.OOOOOOOOE+01, 1 79875000E+00,

$

L$TA8

TABLE=6HCLT ,2,5HALPHA,6HMACH

2.00000000E+00,

0. ,1 50099291E-02,

2.00000000E+00, 4.28950063E-02,

4.0000OOOOE+00, 7.06300112E-02,

S.OOOOOOOOE+00, 1.04932768E-01,

8.00000000E+00, 1 45876120E-01,

1.OOOOOOOOE+01,

1.20000000E+01,

1.40000000E+01,

1.60000000E+01,

1.80000000E+01,

2.OOOOOOOOE+01,

1 86806494E-01,

2 40249601E-01,

2.93890793E-01,

3.52517061E-01,

4.16320651E-01,

4.80513993E-01,

2.20000000E+01, 5.49448757E-01,

2.40000000E+01, 6.18675980E-01,

2.60000000E+01, 6.88337692E-01,

2.80000000E+01, 7 58606494E-01,

3.OOOOOOOOE+01, 8.33875000E-01,

3.20000000E+01, 9.06355000E-01,

3.40000000E+01, 9 78835000E-01,

3.60000000E+01, 1 04627500E+00,

1.30000000E+01,

1.50000000E+01,

1.70000000E+01,

1.90000000t+01,

2.10000000E+01,

2.30000000E+01,

2.50000000E+01,

2.70000000E+01,

2.90000000E+01,

3.10000000E+01,

3.30000000E+01,

3.50000000E+01,

3.70000000E+01,

3.90000000E+01,

4 10000000E+01,

4.30000000E+01,

4.50000000E+01,

4.70000000E+01,

4.90000000E+01,

1.00000000E+00,

3.00000000E+00,

5.00000000E+00,

7.00000000E+00,

9.00000000E+00,

1 10000000E+01,

1.30000000E+01,

1.50000000E+01,

1.70000000E+01,

1.90000000E+01,

2.10000000E+01 ,

2.30000000E+01 ,

2.50000000E+01,

2 70000000E+01,

2.90000000E+01,

3.10000000E+01,

3.30000000E+01,

3.50000000E+01,

3.70000000E+01,

3.90000000E+01,

4.10000000E+01,

4.30000000E+01,

4.50000000E+01,

4.70000000E+01,

4 90000000E+01,

1.66099872E-01,

1.92067547E-01,

2.31279783E-01,

2.70064175E-01,

3.16470018E-01,

3.72772850E-01,

4.28277686E-01,

5.00577188E-01,

5 72100550E-01,

6.59000000E-01,

7.55250000E-01,

8 51500000E-01,

9.64150000E-01,

1.07680000E+00,

1.19624500E+00,

1.32248500E+00,

1 44872500E+00,

1.58616500E+00,

1 72360500E+00,

9 97333333E-02,

1.04797122E-01,

1.10219355E-01,

1.23743898E-01,

1.37281670E-01,

1.55992797E-01,

1.79684720E-01,

2.03163258E-01,

2.42151422E-01,

2.80733524E-01,

3.27071201E-01,

3.82984404E-01,

4.38574793E-01,

5.10838124E-01,

5.82326141E-01,

6.67695000E-01,

7 63735000E-01,

8.59775000E-01,

9 72115000E-01.

1.08445500E+00,

1.20373500E+00,

1.32995500E+00,

1.45617500E+00,

1.59320500E+00,

1.73023500E+00,

,51,5,1,1,1,1,1 ,1.1,1,

1.OOOOOOOOE+00,

3.00000000E+00,

5.OOOOOOOOE+00,

7.OOOOOOOOE+00,

9.OOOOOOOOE+00,

1.10000000E+01,

1.30000000E+01,

1.50000000E+01,

1.7OOOOOOOE+01,

1 90000000E+01 ,

2.10000000E+01,

2.30000000E+01,

2.50000000E+01,

2.70000000E+01,

2.90000000E+01.

3.10000000E+01,

3.30000000E+01,

3 5OOOOOOOE+01 ,

3.70000000E+01,

2.89746738E-02,

5.67781969E-02.

8.44550802E-02,

1.25406276E-01,

1.66342736E-01,

2.13501041E-01,

2.S7047553E-01,

3.20775655E-01,

3.84367397E-01,

4 48371244E-01,

5.15038280E-01,

5.83997671E-01,

6.53475410E-01,

7.23384751E-01,

7 95500000E-01,

8.70115000E-01,

9.42595000E-01,

1.01507500E+00,

1.07747500E+00,

-- 549-

-- 550-

-- 551-

-- 552-

-- 553-

-- 554-

-- 555-

-- 556-

-- 557-

-- 558-

-- 559-

-- 560-

-- 561-

-- 562-

-- 563-

-- 564-

-- 565-

-- 566-

-- 567-

-- 568-

-- 569--

-- 570-

-- 571--

-- 572-

-- 573-

-- 574--

-- 575--

-- 576--

-- 577--

-- 578--

-- 579--

-- 580-•

-- 581--

-- 582--

-- 583--

-- 584--

-- 585--

-- 586--

-- 587--

-- 588--

-- 589--

-- 590-•

-- 591--

-- 592--

-- 593--

-- 594--

-- 595--

-- 596--

-- 596--

-- 597--

-- 598--

-- 599--

-- 600--

-- 601--

-- 602--

-- 603--

-- 604--

-- 605--

-- 606--

-- 607--

-- 608--

-- 609--

-- 610--

-- 611--

-- 612--

-- 613--

-- 614--

-- 615--

-- 616--

-- 617--

200
3.80000000E+01,

4.00000000E+01,

4.20000000E+01,

4 40000000E+01,

4.60000000E+01,

4 80000000E+01.

5.OOOOOOOOE+01,

4.00000000E+00,

0.

2.OOOOOOOOE+00,

4.OOOOOOOOE+00.

6.OOOOOOOOE+00,

8.OOOOOOOOE+00.

1.OOOOOOOOE+01,

1.20000000E+01,

1 40000000E+01,

1.60000000E+01,

1.80000000E+01,

2.OOOOOOOOE+01.

2.20000000E+01,

2.40000000E+01,

2.60000000E+01,

2.80000000E+01.

3.OOOOOOOOE+01,

3.20000000E+01,

3.40000000E+01,

3.60000000E+01.

3.80000000E+01,

4.OOOOOOOOE+01,

4.20000000E+01,

4.40000000E+01.

4.60000000E+01,

4.80000000E+01,

5.OOOOOOOOE+01,

6.00000000E+00,

0.

2.00000000E+00.

4.00000000E+00,

6.00000000E+00,

8.00000000E+00,

1.OOOOOOOOE+01.

1.20000000E+01,

1.40000000E+01,

1.60000000E+01,

1.80000000E+01,

2.OOOOOOOOE+01.

2.20000000E+01,

2 40000000E+01,

2.60COOOOOE+01,

2.80000000E+01,

3.OOOOOOOOE+01,

3.20000000E+01,

3.40000000E+01,

3.60000000E+01,

3.80000000E+01,

4.OOOOOOOOE+01.

4.20000000E+01,

4.40000000E+01,

4.60000000E+01,

4.80000000E+01,

5.OOOOOOOOE+01,

8.00000000E+00,

0.

2.00000000E+00,

4 OOOOOOOOE+00,

6.OOOOOOOOE+00,

8.OOOOOOOOE+00,

1.OOOOOOOOE+01,

1.20000000E+01,

1.40000000E+01,

1 .10867500E+00,

1.17107500E+00,

'. .21798500E+00,

1.26489500E+00,

1 30180500E+00,

1.32871500E+00,

1.35562500E+00,

1.50142857E-02,

4.28866499E-02,

7.05905405E-02,

1.04803473E-01,

1.45612371E-01,

1 86412987E-01,

2.39718792E-01,

2.93204541E-01,

3.51643699E-01,

4.15242169E-01,

4.79231741E-01,

5.47888405E-01.

6.16870327E-01,

6.86340645E-01,

7.56478773E-01,

8.31850000E-01,

9.04540000E-01,

9 77230000E-01,

1 04486500E+00,

1 10744500E+00,

1 17002500E+00,

1 21707500E+00,

1.26412500E+00,

1.30118000E+00,

1.32824000E+00,

1.35530000E+00,

1.49071429E-02,

4.27687657E-02,

7.04641892E-02,

1.04558611E-01,

1 45141818E-01,

1.85721212E-01,

2.38885159E-01,

2.92227089E-01,

3.50492985E-01,

4.13879816E-01,

4.77645392E-01,

5.45994252E-01,

6.14711666E-01,

S.83931330E-01,

7.53824783E-01,

8.29125000E-01,

9.01495000E-01,

9.73865000E-01,

1.04158500E+00,

1.10465500E+00,

1.16772500E+00,

1.21512500E+00,

1.26252500E+00,

1.29981500E+00,

1.32699500E+00,

1.35417500E+00,

1 49071429E-02,

4.27069521E-02,

7.03444595E-02,

1 04463800E-01,

1.45147273E-01.

1 85821212E-01,

2.38894971E-01,

2.92150162E-01,

3.90000000E+01,

4 10000000E+01,

4.30000000E+01,

4.50000000E+01,

4.70000000E+01,

4.90000000E+01,

1.OOOOOOOOE+00,

3.OOOOOOOOE+00,

5.OOOOOOOOE+00,

7.OOOOOOOOE+00.

9.OOOOOOOOE+00,

1 10000000E+01,

1.30000000E+01,

1 50000000E+01,

1 70000000E+01,

1.90000000E+01,

2.10000000E+01,

2.30000000E+01,

2.50000000E+01,

2.70000000E+01,

2.90000000E+01,

3.10000000E+01,

3.30000000E+01,

3.50000000E+01,

3.70000000E+01 ,

3.90000000E+01,

4.10000000E+01,

4 30000000E+01,

4.50000000E+01,

4 70000000E+01,

4.90000000E+01,

1 . 13987500E+00,

1. 19453000E+00,

1.24144000E+00,

1.28835000E+00,

1.31526000E+00,

1.34217000E+00,

2.89755020E-02,

5.67561237E-02,

8.43951872E-02,

1.25209091E-01,

1.66013591E-01,

2.13041412E-01,

2.66441048E-01,

3.20006015E-01,

3.83391185E-01,

4.47190821E-01,

5.13595007E-01,

5.82316391E-01,

6 51542300E-01,

7.21322823E-01,

7.9354OOOOE-01,

8.68195000E-01,

9 40885000E-01,

1.01357500E+00,

1.07615500E+00,

1 13873500E+00,

1.19355000E+00,

1 24060000E+00,

1.28765000E+00,

1.31471000E+00,

1.34177000E+00,

1.OOOOOOOOE+00,

3 OOOOOOOOE+00,

5.OOOOOOOOE+00,

7.OOOOOOOOE+00,

9.OOOOOOOOE+00,

1.10000000E+01,

1.30000000E+01,

1.50000000E+01,

1.70000000E+01,

1.90000000E+01,

2 10000000E+01,

2 30000000E+01,

2 50000000E+01,

2 70000000E+01,

2.90000000E+01,

3 10000000E+01,

3.30000000E+01,

3 50000000E+01,

3 70000000E+01,

3 90000000E+01,

4 10000000E+01,

4.30000000E+01,

4.50000000E+01,

4 70000000E+01,

4.90000000E+01 ,

1.OOOOOOOOE+00,

3.OOOOOOOOE+00,

5.OOOOOOOOE+00,

7.OOOOOOOOE+00,

9.OOOOOOOOE+00,

1.10000000E+01,

1.30000000E+01,

1 50000000E+01,

2.88626506E-02,

5.66337693E-02,

8.42652406E-02,

1.24850753E-01,

1.65431935E-01,

2.12278975E-01,

2.65535730E-01,

3.18956015E-01,

3.82136286E-01,

4.45717928E-01,

5.11832959E-01,

5.80290090E-01,

6 49250924E-01,

7 18792582E-01,

7.90945000E-01,

8.65310000E-01,

9.37680000E-01,

1 01005000E+00,

.07312000E+00,

.13619000E+00,

.19142500E+00,

.23882500E+00,

.28622500E+00,

.31340500E+00,

1.34058500E+00,

2.88311647E-02,

5 65425922E-02,

8.41176471E-02,

1.24806882E-01,

1 65485293E-01,

2 12333412E-01,

2.65501778E-01,

3. 18836842E-01,

-- 618-

-- 619-

-- 620-

-- 621-

-- 622-

-- 623-

-- 624-

-- 625-

-- 626-

-- 627-

-- 628-

-- 629-

-- 630-

-- 631-

-- 632-

-- 633-

-- 634-

-- 635-

-- 636-

-- 637-

-- 638-

-- 639-

-- 640-

-- 641-

-- 642-

-- 643-

-- 644-

-- 645-

-- 646-

-- 647-

-- 648-

-- 649-

-- 650-

-- 651-

-- 652-

-- 653-

-- 654-

-- 655-

-- 656-

-- 657-

-- 658-

-- 659-

-- 660-

-- 661-

-- 662-

-- 663-

-- 664-

-- 665-

-- 666-

-- 667-

-- 668-

-- 669-

-- 670-

-- 671-

-- 672-

-- 673-

-- 674-

-- 675-

-- 676-

-- 677-

-- 678-

-- 679-

-- 680-

-- 681-

-- 682-

-- 683-

-- 684-

-- 685-

-- 686-

-- 687-

201
1.60000000E+01,

1.80OOOOOOE+01,

2.00000000E+01,

2.20000000E+01,

2.40000000E+01,

2.60000000E+01,

2.80000000E+01,

3.00000000E+01,

3.20000000E+01,

3.40000000E+01,

3.60000000E+01,

3.80000000E+01,

4.00000000E+01,

4.20000000E+01,

4.40000000E+01,

4.60000000E+01,

4.80000000E+01,

5.00000000E+01,

1.00000000E+01,

0.

2.00000000E+00,

4.OOOOOOOOE+00,

6.OOOOOOOOE+00,

8.OOOOOOOOE+00,

1.00000000E+01,

1.20000000E+01,

1.40000000E+01,

1.60000000E+01,

1.80000000E+01,

2.00000000E+01,

2.20000000E+01,

2.40000000E+01,

2.60000000E+01,

2.80000000E+01,

3.00000000E+01,

3.20000000E+01,

3.40dbOOOOE+01.

3.60000000E+01,

3.80000000E+01,

4.00000000E+01,

4.20000000E+01,

4 40000000E+01,

4.60000000E+01,

4.80000000E+01,

5.00000000E+01,

$

L$TAB

TABLE=6HCDT ,2

2.00000000E+00,

0

2.0000OOOOE+00,

4.00OOOOOOE+00,

6.00000000E+00,

8.00000000E+00,

1.00000000E+01,

1.20000000E+01,

1.40000000E+01,

1.60000000E+01,

1.80000000E+01,

2.00000000E+01,

2.20000000E+01.

2.40000000E+01.

2.SOOOOOOOE+01,

2.80000000E+01,

3.00000000E+01,

3.20000000E+01,

3.40000000E+01,

3.60000000E+01,

3.80000000E+01,

4.000OOOOOE+01,

3.50096610E-01,

4.13196102E-01,

4.76997611E-01,

5.45741749E-01,

6.14280219E-01.

6.83160000E-01,

7.52983810E-01,

8.28400000E-01,

9.00690000E-01,

9.72980000E-01,

1.04068500E+00,

1.10380500E+00,

1.16692500E+00,

1.21444500E+00,

1.26196500E+00,

1.29943500E+00,

1.32685500E+00,

1.35427500E+00,

1.46464286E-02,

4.25473300E-02,

7.02738964E-02,

1.04440102E-01,

1 45135276E-01,

1.85821212E-01,

2.38854971E-01,

2.92070162E-01,

3.50153402E-01,

4.13310099E-01,

4.76857143E-01,

5.45100290E-01,

6 13728943E-01,

6.82840000E-01,

7 52623810E-01,

8.28000000E-01,

9.00250000E-01,

9 72500000E-01,

1.04015000E+00,

1.10320000E+00.

1.16625000E+00.

1.21394000E+00,

1.26163000E+00,

1.29923500E+00,

1.32675500E+00,

1.35427500E+00,

1.70000000E+01.

1.90000000E+01,

2.10000000E+01,

2.30000000E+01,

2.50000000E+01,

2.70000000E+01,

2.90000000E+01,

3.10000000E+01,

3.30000000E+01,

3.50000000E+01,

3.70000000E+01,

3.90000000E+01,

4.10000000E+01,

4.30000000E+01,

4.50000000E+01,

4.70000000E+01,

4.90000000E+01,

1.00000000E+00,

3.000000OOE+00.

5.00000000E+00,

7.00000000E+00,

9.00000000E+00,

1 10000000E+01,

1 30000000E+01,

1.50000000E+01,

1.70000000E+01,

1.90000000E+01,

2.10000000E+01,

2.30000000E+01,

2.50000000E+01,

2.70000000E+01,

2.90000000E+01,

3.10000000E+01,

3.30000000E+01,

3.50000000E+01,

3.70000000E+01,

3.90000000E+01,

4.10000000E+01,

4.30000000E+01,

4 50000000E+01,

4.70000000E+01,

4.90000000E+01,

3 81553468E-01,

4 45014047E-01,

5.11435919E-01,

5.80022658E-01,

6.48515876E-01,

7.17986112E-01,

7.90275000E-01,

8.64545000E-01,

9.36835000E-01,

1.00912500E+00,

1.07224500E+00,

1.13536500E»00,

1.19068500E+00,

1.23820500E+00,

1.28572500E+00,

1 .31314500E+00,

1.34056500E+00,

2.86227845E-02,

5.64287515E-02,

8 40882353E-02,

1.24788993E-01,

1 65479262E-01,

2 12313412E-01,

2 65441778E-01,

3 18736842E-01,

3.81680000E-01,

4 45037670E-01,

5 10974085E-01,

5.79354606E-01,

6.48215876E-01,

7.17646112E-01,

7.89895000E-01,

8.64125000E-01,

9.36375000E-01,

1.00862500E+00,

1.07167500E+00,

1.13472500E+00,

1.19009500E+00,

1.23778500E+00,

1.28547500E+00,

1 31299500E+00,

1.34051500E+00,

,5HALPHA,6HMACH ,51,5,1,1,1,1,1,1,1,1,

5.20985816E-02,

5.86878545E-02,

6.56436220E-02,

7.62035138E-02,

9.01804686E-02,

1 04179221E-01,

1.28481663E-01,

1.52499923E-01,

1.83271095E-01,

2.20588705E-01,

2.57431399E-01,

3.12418214E-01,

3.67112563E-01,

4.30299405E-01,

5.01899489E-01,

5.77775000E-01,

6.74845000E-01.

7 71915000E-01,

8.77890000E-01,

9 92770000E-01,

1.10765000E+00,

1.OOOOOOOOE+00,

3.00000000E+00,

5.OOOOOOOOE+00,

7.OOOOOOOOE+00,

9.OOOOOOOOE+00,

1.10000000E+01,

1.30000000E+01 ,

1.50000000E+01,

1.70000000E+01,

1.90000000E+01,

2.10000000E+01,

2.30000000E+01,

2.50000000E+01,

2.70000000E+01,

2.90000000E+01,

3.10000000E+01,

3 30000000E+01,

3.50000000E+01 ,

3.70000000E+01 ,

3.90000000E+01,

4.10000000E+01,

5.53389880E-02,

6.21274259E-02,

6.92251337E-02,

8.31889150E-02,

9.71774437E-02,

1.16369186E-01,

1.40523278E-01,

1.64416854E-01.

2.01992603E-01,

2.39066199E-01,

2.84744086E-01,

3.39869521E-01,

3.94160656E-01,

4 66208161E-01,

5.39060000E-01,

6.26310000E-01,

7.23380000E-01,

8.20450000E-01.

9.35330000E-01,

1 .05021000E+00,

1.17231500E+00,

-- 688-

-- 689-

-- 690-

-- 691-

-- 692-

-- 693-

-- 694-

-- 695-

-- 696-

-- 697-

-- 698-

-- 699-

-- 700-

-- 701-

-- 702-

-- 703-

-- 704-

-- 705-

-- 706-•

-- 707-•

-- 708--

-- 709-•

-- 710--

-- 711- -

-- 712--

-- 713--

-- 714--

-- 715--

-- 716--

-- 717--

-- 718--

-- 719--

-- 720-•

-- 721-•

-- 722--

-- 723--

-- 724--

-- 725--

-- 726--

-- 727--

-- 728--

-- 729--

-- 730-•

-- 731--

-- 732--

-- 733--

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-- 735--

-- 736--

-- 737--

-- 738--

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-- 740-•

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-- 750--

-- 751--

-- 752--

-- 753--

-- 754--

-- 755--

-- 756--

202
ORIGINAL PAGE IS

OF POOR QUALITY

4.20000000E+01.

4.40000000E+01,

4.60000000E+01.

4.80000000E+01,

5.OOOOOOOOE+01,

4.OOOOOOOOE+00,

0.

2.OOOOOOOOE+00.

4.00000000E-I-00,

6.00000000E+00.

8.OOOOOOOOE+00,

1.00000000E+01,

1.20000000E+01,

1 40000000E+01,

1.60000000E+01,

1.80000000E+01,

2 OOOOOOOOE+01,

2.20000000E+01,

2.40000000E+01,

2.60000000E+01,

2-80000000E+01,

3.OOOOOOOOE+01,

3.20000000E+01.

3 40000000E+01.

3 SOOOOOOOE+01,

3.80000000E+01.

4 OOOOOOOOE+01,

4.20000000E+01,

4 40000000E+01,

4.SOOOOOOOE+01,

4.80000000E+01,

5.00000000E+O1,

6-OOOOOOOOE+00,

0.

2.OOOOOOOOE+00,

4 OOOOOOOOE+00.

6 OOOOOOOOE+00.

8.00000000E+00,

1.OOOOOOOOE+01,

1.20000000E+01,

1.40000000E+01,

1 60000000E+01,

1.80000000E+01,

2.OOOOOOOOE+01,

2.20000000E+01,

2.40000000E+01,

2.60000000E+01,

2.80000000E+01,

3.OOOOOOOOE+01,

3.20000000E+01,

3-40000000E+01,

3.60000000E+01,

3 80000000E+01,

4.OOOOOOOOE+01,

4.20000000E+01,

4.40000000E+01,

4.60000000E+01,

4 80000000E+01,

5.OOOOOOOOE+01,

8.00000000E+00,

0.

2-OOOOOOOOE+OO,

4.00000000E+00,

6.00000000E+00,

8 OOOOOOOOE+00,

1.OOOOOOOOE+01,

1.20000000E+01,

1.40000000E+01,

1.60000000E+01,

1 80000000E+01,

1.23698000E+00,

1.36631000E+00.

1.50058500E+00,

1.63980500E+00,

1.77902500E+00,

5.01285714E-02,

5.68429723E-02,

S.39113514E-02,

7.46608989E-02,

8.89053233E-02,

1.03163203E-01,

1.27911396E-01 ,

1.52383737E-01,

1.83630995E-01,

2.21446024E-01,

2 58792833E-01,

3 14182761E-01,

3 69191082E-01,

4.3253437SE-01,

5 04125652E-01,

5.80250000E-01,

6.76860000E-01,

7.73470000E-01,

8.79060000E-01,

9.93630000E-01,

1.10820000E+00,

1.23736000E+00,

1.36652000E+00,

1 50059000E+00,

1.63957000E+00,

1.77855000E+00,

4.95500000E-02,

5.64894710E-02,

6.37693243E-02,

7.47755056E-02,

8 93225679E-02,

1.03875758E-01,

1.29066269E-01,

1 53987351E-01,

1.8S640707E-01,

2.23825344E-01,

2 61554266E-01,

3.17297990E-01,

3 72581234E-01,

4 36150653E-01,

5.07906087E-01,

5.84400000E-01,

6.81230000E-01,

7.78060000E-01,

8.83385000E-01,

9.97205000E-01,

1 11102500E+00,

1.23967500E+00,

1 36832500E+00,

1.50203500E+00,

1.64080500E+00,

1.77957500E+00,

4 74642857E-02,

5.44096725E-02,

6.16830180E-02,

7.28820838E-02,

8.78365886E-02,

1 02806061E-01,

1 .28231739E-01,

1.53388510E-01,

1.85226780E-01,

2.23547314E-01,

4.30000000E+01, 1.30164500E+00,

4.50000000E+01, A.43097500E+00,

4.70000000E+01, 1.57019500E+00,

4.90000000E+01, 1.70941500E+00,

1.OOOOOOOOE+00,

3.OOOOOOOOE+00,

5.OOOOOOOOE+00,

7.OOOOOOOOE+00,

9.OOOOOOOOE+00,

1.10000000E+01,

1.30000000E+01,

1.50000000E+01,

1.70000000E+01,

1.90000000E+01,

2.10000000E+01,

2.30000000E+01,

2.50000000E+01,

2.70000000E+01,

2.90000000E+01,

3.10000000E+01,

3.30000000E+01,

3.50000000E+01,

3.70000000E+01,

3.90000000E+01,

4.10000000E+01,

4.30000000E+01,

4.50000000E+01,

4.70000000E+01,

4.90000000E+01,

1.OOOOOOOOE+00,

3.OOOOOOOOE+00,

5.OOOOOOOOE+00,

7.OOOOOOOOE+00,

9 OOOOOOOOE+00,

1.10000000E+01,

1.30000000E+01,

1.50000000E+01,

1.70000000E+01,

1.90000000E+01,

2.10000000E+01,

2.30000000E+01,

2.50000000E+01,

2.70000000E+01,

2.90000000E+01,

3.10000000E+01,

3.30000000E+01,

3.50000000E+01,

3.70000000E+01,

3.90000000E+01,

4.10000000E+01,

4.30000000E+01,

4.50000000E+01,

4.70000000E+01,

4.90000000E+01,

1.OOOOOOOOE+00,

3.00000000E+OO,

5.OOOOOOOOE+00,

7.OOOOOOOOE+00,

9.OOOOOOOOE+00,

1 .10000000E+01,

1.30000000E+01,

1.50000000E+01,

1.70000000E+01,

1 90000000E+01,

5.34331727E-02,

6.03403329E-02,

6 75449198E-02,

8 17812121E-02,

9 60327813E-02,

1 . 15574824E-01,

1 40179175E-01,

1.64530075E-01,

2.02600462E-01,

2.40174659E-01,

2.86355912E-01,

3.41789781E-01,

3.96399589E-01,

4.68438861E-01,

5 41535000E-01,

6.28555000E-01,

7.25165000E-01,

8 21775000E-01,

9 36345000E-01,

1.05091500E+00,

1.17278000E+00,

1.30194000E+00,

1 43110000E+00,

1.57008000E+00,

1.70906000E+00,

5.29691566E-02,

6.00939834E-02,

6.75048128E-02,

8.20481720E-02,

9.65984878E-02,

1.16507664E-01,

1.41557683E-01,

1.66360150E-01,

2 04793324E-01,

2 42743561E-01,

2.89342649E-01,

3.45039520E-01,

3.99935934E-01,

4.72136528E-01,

5.45600000E-01,

6.32815000E-01,

7.29645000E-01,

8.26475000E-01,

9.40295000E-01,

1 05411500E+00,

1 17535000E+00,

1 30400000E+00,

1 43265000E+00,

1 57142000E+00,

1 .71019000E+00,

5 08882463E-02,

5.80122235E-02,

6.54117647E-02,

8.03572239E-02,

9.53196760E-02,

1 15555479E-01,

1.40840937E-01,

1.65879323E-01,

2.04447009E-01,

2.42534451E-01,

-- 757-

-- 758-

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-- 824-

-- 825-

-- 826-

203
4.80000000E+01,

5.00000000E+01,

2.00000000E+01, 2.61414676E-01,

2.20000000E+01, 3.17344993E-01,

2.40000000E+01, 3.72689491E-01,

2.60000000E+01, 4.36315319E-01,

2.80000000E+01, 5.08175933E-01.

3.00000000E+01, 5.85025000E-01,

3.20000000E+01, 6.81855000E-01,

3 40000000E+01, 7.78685000E-01,

3.60000000E+01, 8.83965000E-01,

3.80000000E+01, 9.97695000E-01,

4.000OOOOOE+01, 1.11142500E+00.

4.20000000E+01, 1.23993500E+00,

4.40000000E+01, 1.36844500E+00,

4.60000000E+01, 1.50199500E+00,

1.64058500E+00,

1.77917500E+00,

1.OOOOOOOOE+01,

0 , 4.59507143E-02,

2.OOOOOOOOE+00, 5.31539295E-02,

4.OOOOOOOOE+00, 6.06842117E-02,

6.OOOOOOOOE+00, 7.20717058E-02,

8.OOOOOOOOE+00, 8.71465792E-02,

1.00000000E+01, 1.02236364E-01,

1.20000000E+01,

1 40000000E+01,

1.60000000E+01,

1.80000000E+01,

2.00000000E+01,

2.20000000E+01,

2.40000000E+01,

2.60000000E+01,

2.80000000E+01,

3.00000000E+01,

3.20000000E+01,

3.40000000E+01,

3.60000000E+01,

3.80000000E+01,

4.00000000E+01,

4.20000000E+01,

4.40000000E+01,

4.60000000E+01,

4.80000000E+01,

5.00000000E+01, 1.77890000E+00,

2 1000OOOOE+01, 2.89349726E-01,

2.30000000E+01, 3.45120284E-01,

2.50000000E+01, 4.00065361E-01,

2.70000000E+01, 4.72348038E-01,

2.90000000E+01. 5.46165000E-01,

3 10000000E+01, 6.33440000E-01,

3.30000000E+01, 7.30270000E-01,

3.50000000E+01, 8.27100000E-01,

3.70000000E+01, 9.40830000E-01,

3.90000000E+01, 1.05456000E+00,

4.10000000E+01, 1 17568000E+00,

4.30000000E+01, 1.30419000E+00,

4.50000000E+01, 1 43270000E+00,

4.70000000E+01, 1.57129000E+00,

4.90000000E+01, 1.70988000E+00,

1.OOOOOOOOE+00,

3.OOOOOOOOE+00,

5.OOOOOOOOE+00,

1.27857208E-01,

1.53209668E-01,

1 8S172401E-01,

2.23559745E-01,

2 61512245E-01,

3 17520457E-01,

3 72931099E-01,

4.36587057E-01,

5.08456617E-01,

5.85325000E-01,

6.82185000E-01,

7.79045000E-01,

8.84350000E-01,

9.98100000E-01,

1 11185000E+00,

1.24013000E+00,

1 36841000E+00,

1.50182000E+00,

1.64036000E+00,

4.95037216E-02,

5.68850416E-02,

6.45411765E-02,

7.OOOOOOOOE+00, 7.96070381E-02,

9.OOOOOOOOE+00, 9.46898290E-02,

1 10000000E+01,

1.30000000E+01,

1.50000000E+01,

1.70000000E+01,

90000000E+01,

1.15083294E-01,

1 40564190E-01,

1.65798496E-01,

2.04423723E-01,

1.90000000E+01, 2 42587184E-01,

2 10000000E+01, 2.89488701E-01,

2.30000000E+01, 3.45329910E-01,

2.50000000E+01, 4.00336907E-01,

2.70000000E+01, 4 72622878E-01,

2.90000000E+01, 5.46455000E-01,

3.10000000E+01, 6.33755000E-01,

3.30000000E+01, 7.30615000E-01,

3.50000000E+01, 8.27475000E-01,

3.70000000E+01, 9 41225000E-01,

3.90000000E+01, 1.05497500E+00,

4.10000000E+01, 1 17599000E+00,

4.300000OOE+01, 1.30427000E+00,

4.50000000E+01, 1 43255000E+00,

4.70000000E+01,

4.900000OOE+01,

1 .57109000E+00,

1 70963000E+00,

,VALUE=28.8348843,ETAPC(1)=0,ENDPHS=1 $ --

ENDPHS=1 $

LSGENDAT

EVENT=50,CRITR=5HGCLAT,ENDPHS*1,ETAPC(1 ) = 1,NENG=1,NPC(21

ALPPC(1)=180.,

NPC(2) = 1,DT = 1 ,

$

LSGENDAT

EVENT=55,NPC(2)=4,DT=100,CRITR=6HALTP

LSGENDAT

EVENT=100,CRITR=6HALTITO,VALUE=400000,ENDPHS=1,PRNC=1,NPC(12)=1,

PINC=100,

IGUID=0.1,0,IGUID(6)=1,0,1.NPC(15)=1 ,DT=1,NPC(2)=1,NPC(5)=4,

ALPPC(1 )=37.0010556.BNKPCt1)=-69.2957708.

NPC(8)=-2,NPC(5)=4,DTIMR=1,BETA=0.,

$

LSGENDAT

EVENT'105,CRITR=6HHEATRT,PINC=20,VALUE=115,IGUID(6)=1,0,4,

NPC(9)=1,NPC(22)=1,PRNC=1,

NDEPVS'1,DEPVRS=5HVELAD,DEPTLS=.00001,INDVRS=6HETAPC1,

US=.1,MAXITS=10,

DGF(3)=6HHEATRT,IDGF(3)=1,KDG(3)=30,KRG(3)=2000,

$

LSTBLMLT $

LSTAB TABLE = 5HGDF3T,1.5HTDURP,5,1 ,1,1,

0,115,15,120,30,124,60,125,1000,125 $

LSTAB TABLE=6HGNMN3T,0,-180S

LSTAB TABLE=6HGNMX3T,0,0$

-- 827-

-- 828-

-- 829-

-- 830-

-- 831-

-- 832-

-- 833-

-- 834-

-- 835-

-- 836-

-- 837-

-- 838-

-- 839-

-- 840-

-- 841-'

-- 842-

-- 843-

-- 844--

-- 845--

-- 846-'

-- 847-'

-- 848-

-- 849-

-- 850--

-- 851--

-- 852--

-- 853--

-- 854--

-- 855--

-- 856--

-- 857--

-- 858--

-- 859--

-- 860--

-- 861--

-- 862--

-- 863--

-- 864--

•- 865--

-- 866--

-- 867--

-- 868--

•- 869--

-- 870--

-- 870--

•- 871--

•- 872--

•- 873--

•- 874--

•- 874--

875--

875--

876--

877--

878--

879--

880--

881--

881--

882--

883--

884--

885--

886--

887--

887--

887--

888--

888--

888--

204
ORlGfNAL PAGa-rs

OF POOR QUALITY

LSTAB TABLE=6HGNOM3T,0.-70.0000000,ENDPHS=1 $ -- 888-

LSGENDAT -- 888-

EVENT=180,CRITR=6HINC ,ENDPHS=1.NDEPVS'O, -- 889-

IGUIO(6)=3,0,3,DBANK=O.DALPHA=12 $ -- 890-

LSGENDAT -- 890-

EVENT=190,CRITR=5HTDURP,PINC=1000,VALUE=10,ETAPC(1)=1,ENDPHS=1 $ -- 891-

LSGENDAT -- 891-

EVENT=200,CRITR=6HALTA .ENDPHS=1,DT=10,IGUID(S)=1,0,1, -- 892-

ETAPC(1)=0. -- 893-

PRNC=10, -- 894-

$ -- 895-

LSGENDAT -- 895-

EVENT=210,CRITR=6HALTITO,VALUE=400000,NPC(2)=4,DT=100,NPC(5)=0,NPC(8)=0, -- 896-

ENDPHS=1 $ --N 897-

LSGENDAT -- 897-

EVENT=220,CRITR=6HGAMMAI,VALUE=0,NPC(2) = 1,DT=1,ETAPC(1) = 1 , -- 898-

IGUID=3,0,1,ALPPC(1)=O.BETPC(1)=0,BNKPC(1)=0,ENDPHS*1 $ -- 899-

L$GENDAT -- 899-

EVENT=1000.CRITR=6HVELI ,VALUE=25354.1388,ENDPHS=1,ENDPRB=1,ENDJOB=1 $ -- 900-

205
A.3.1 INPUT FILE DESCRIPTION

A.3.1.1 INPUT CONDITIONS

The following is a lis t of the input conditions for the nominal

plane change trajectory.

Initia l inclinatio n = 28.5 degrees

Initia l Orbit altitude = 160 nautical mil e circular orbit

True anomaly (Truan) =150 degrees

Vehicle Model:

Reference Surface Area (SREF) = 177.4 square feet

Reference length (LREF) = 25 feet

Initia l weight (WGTSG) = 11,430 pounds

Engine ISP (ISPV) - 295.0 seconds

206
Coefficients of lift (CL) and drag (CD) as a function of angleof-attack and a correlation parameter of either altitude (ALTITO), viscous interaction (VINV) , or mach number (MACH).

Spherical, rotating earth model.

1976 standard atmospheric model

A.3.1.2 TRAJECTORY PHASES

The trajectory has 11 different phases called events and are numbered: 1,50,55,100,105,180,190,200,210,220,and 1000. Each phase

represents a change in either the vehicle and/or the environmental conditions. The following is a lis t of the events and their purpose. Most

of these phases are shown on Figure A-4.

Event Mumber Purpose

1 Sets initia l conditions

50 Starts deorbit burn at best point

55 Stops deorbit burn at best point

100 Turns on atmosphere at 400,000 feet

105 Starts level-off based on heat-rate

180 Starts exit based on best inclinatio n

207
190 Completes the exit set-up by going 100% throttle

200 Cuts-off engines based on desired apogee altitude

210 Turns off atmosphere at 400,00 feet

220 Starts circularization burn

1000 Stops circularization burn when a 160 nautical mil e

circular orbit is obtained.

A.3.1.3 INDEPENDENT VARIABLES

POST has been given a number of parameters which it can vary to

optimize the problem. There are 12 variables that are what we call

exterior parameters and two interior parameters. These variables are

listed below with the associated phase/event numbers and a brief

descr iption.

208
Exterior

Independent

Variables

Names

GCLAT

ALTP

ALPPC1

BANKPC1

ALPPC1

DEPVLSI

INC

OBANK

DALPHA

ALTA

ALPPC1

BANKPC1

Interior

Independent

Variables

Names

ETAPC1

BANK

Event

Number

50

55

100

100

105

105

180

180

180

200

200

200

Event

Number

105

105

Descr i ption

Geocentric latitude for deorbit burn

Perigee altitude to stop burn

Entry angle-of-attack

Entry bank angle

Level -off angle-of-attack

Cruise acceleration

Exi t incl i nation

Exi t bank angle

Exit angle-of-attack

Apogee altitude for engine cut-off

Angle-of-attack after engine cut-off

Bank angle after enging cut-off

Descr i ption

Throttle % to maintain cruise acceleration

Bank angle to maintain heating rate

A.3.2 OUTPUT DESCRIPTION

209
As might be expected, POST has the capacity to generate a large

amount of output data and plot information. Included below are samples

of the output data and generated plots from our sample trajectory.

A.3.2.1 OUTPUT DATA

Figures A-1 thru 3 present a sample from the output data fil e for

the nominal 20 degree plane change. Figure A-1 presents the 270 parameters offered as nominal output. Specifically, the parameters are for

the start of phase 105 (cruise phase). Figure A-2 presents the same

parameter for the end of the trajectory (last frame of phase 220 and

first and only frame of phase 1000). Figure A-3 provides a summary of

the trajectory to include the independent variable values (u) , the end

condition errors (e), the optimization parameter value (P1) and a targeting parameter (P2) .

A.3.2.2 TRAJECTORY PLOTS

Shown in Figures A-4 thru 12 are the baseline trajectory plots generated by POST for a heat-rate of 80 and 125 BTU/ft2-sec. The plots for

QDOT = 125 were generated with the trajectory generation input deck

shown in Section A.3. The Figures present a time history of altitude

above the planet, relative atmospheric vehicle velocity, angle-of-attack, bank angle, Engine throttle percentage, heat-rate, heat load, and

210
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213
inclination for the nominal conditions. The time reference for the

plots (t=0) is atmospheric interface at 400,000 feet. The phase/event

divisions and numbers are shown on the altitude time history.

214
ORIGINAL PAGE IS

OF POOR QUALITY

h,

ft

ERV CRUISING SYNERGISTIC PLANE CHANGE Q-DOT=80

EVENT/PHASE NUMBERS

210

200000 tl I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I III

220

350 700 1050 1400

Time, sec

1750. 2100

h.

ft

ERV CRUISING SYNERGISTIC PLANE CHANGE Q-DOT=125

EVENT/PHASE NUMBERS

I 210

^220

ti inn i in 11

225000 —

200000

350 700 1050 1400

Time, sec

1750 2100

Figure A-4. Baseline altitude profiles.

215
ERV CRUISING SYNERGISTIC PLANE CHANGE 0-DOT=80

ft/sec

30000

28500

27000

25500

24000

22500

21000

19500

18000 -tl I I I I I I I I I I I I I I I I I I I I I I I I I 11 I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I

350 700 1050 1400

Time, sec

1750 2100

ERV CRUISING SYNERGISTIC PLANE CHANGE 0-DOT=125

ft/sec

30000

28500

27000

25500

24000

22500

21000

19500

18000 'I I I I I I I | I I I I I I I I I I I I I I II I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I

700 1050 HOO 1750 2100

Time, sec

Figure A-5. Baseline vehicle velocity relative to the atmosphere

profiles.

216
ORIGINAL PAGE IS

OF POOR QUALfTY

ERV CRUISING SYNERGISTIC PLANE CHANGE 0-DOT=80

40 P~

35 —

deg

30

25

20

15

10

5

OtUL

0

In ii Ill ii inn I I mini i mi i

350 700 1050 1400 1750 2100

Time, sec

ERV CRUISING SYNERGISTIC PLANE CHANGE Q-DOT=125

deg

40

35

30

25

20

15

10

5

o n n 11111 1 n i n 111111 1 n 1 1 n 111 1 i i n n n 1 1 n n i n 11 1 n i n 11 1

0 300 700 1050 1400 1750 2100

Time, sec

Figure A-6. Baseline angle-of-attack profiles.

217
ERV CRUISING SYNERGISTIC PLANE CHANGE Q-DOT=80

deg

deg

20

0

-20

-40

-60

-80

-100

-120

-140 11 I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I

0 350 700 1050 1400 1750 2100

Time, sec

ERV CRUISING SYNERGISTIC PLANE CHANGE Q-DOT=125

-20 —

-40 ^-

-60 ^-

-80 5~

-100 ^-

-120 —

-140 tLLJ

0

I III M I M I MM Illlll l MM Mil l Mil l III Mil

350 700 1050 1400 1750 2100

Time, sec

Figure A-7. Baseline bank angle profiles.

218
ERV CRUISING SYNERGISTIC PLANE CHANGE Q-DOT=80

1.6 p~

1.4

1.2

1.0

< -8

LJ

,6

.4

.2

0 11 I I I I I I I I I I I I I I I II I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I

350 700 1050 1400

Time, sec

1750 2100

ERV CRUISING SYNERGIoTIC PLANE CHANGE 0-DOT=125

UJ

1.6

1.4

1.2

1 0

.8

.6

.4

.2

0

0 350 700 1050 1400

Time, sec

1750 2100

Figure A-8. Baseline engine throttling profiles (1.0 = maximum),

219
ERV CRUISING SYNERGI3TIC PLANE CHANGE Q-DOT=80

q ,

BTU/ft2-sec

160

140

120

100

80

60

40

20

0 I I I Illll l Mil l I I I I I IN I II I M M I I II I I I T-H-U4-U-)

350 700 1050 1400 1750 2100

Time, sec

ERV CRUISING SYNERGISTIC PLANE CHANGE Q-DOT=125

q -

BTU/ft2-sec

2100

Figure A-9. Baseline heat-rate profiles.

220
ORIGINAL PAGE FS

-OF POOR QUALITY

ERV CRUISING SYNERGISTIC PLANE CHANGE Q-DOT=80

100000

Heat Load

BTU/ft

60000

20000 —

350 700 1050 1400

Time, sec

1750 2100

ERV CRUISING SYNERGISTIC PLANE CHANGE Q-DOT=125

160000

140000

120000

100000

Heat Load

BTU/ft:2 60000

60000

40000

20000

Kil l LU-H-f I I I I I I I I I I I I I I I I I I I I I I I I I I II I I I I I I I I I I I I

350 700 1050 1400

Time, sec

1750 2100

Figure A-10. Baseline heat-load profiles.

221
Ib/ft2

ERV CRUISING SYNERGISTIC' PLANE CHANGE Q-DOT=80

120

105

90

75

60

4-5

30

15

l-LH-f I HI I I I I I I I I I I I I I I N4J I I I I I I I I I I

350 700 1050 1400 1750 2100

Time, sec

ERV CRUISING SYNERGISTIC PLANE CHANGE Q-DOT=125

Qoo •

Ib/ft2

120

105

90

75

60

45

30

15

mil LU-Tl I I I I I I 11 I I I I I I NJj I I I 11 I I I I I I I I 11 I I I

350 700 1050 1400

Time, sec

1750 2100

Figure A-ll. Baseline dynamic pressure profiles.

222
INC,

deg

INC.

deg

ORIGINAL PAGE !3

OF POOR QUALITY

ERV CRUISING SYNERGISTIC PLANE CHANGE Q-DOT=80

60 p~

55 ^-

50 ^~

45 —

40

35

30

25

20ht jjil l i 11 1 n 1 1 il l i n 111 1 i l i i i n n ni l i n il l 11 1

350 700 1050 1400 1750 2100

Time, sec

ERV CRUISING SYNERGISTIC PLANE CHANGE Q-DOT=125

60

55

50

45

40

35

30

25

20 11 I I I I I II I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I

350 700 1050 1400

Time, sec

1750 2100

Figure A-12. Baseline inclination angle profiles.

223
APPENDIX B

DRAPER ERV COMPUTER SIMULATOR

AND GUIDANCE ALGORITHM

B.1 SIMULATOR DESCRIPTION

The ERV simulator is a three degree-of-freedom (point mass) simulator used to simulate the atmospheric phase of an orbital plane change

maneuver. A fourth order Runge-kutta integrator is used to integrate

the vehicle state. A spherical earth gravity model and a 1962 standard

atmospheric model are used. Bank angle, angle-of-attack, and throttle

are controllable with bank angle and angle-of-attack being rate limited.

Thrust is modeled as a force along the ERV's longitudinal axis through

the center of gravity. Complete ERV CL and CD data including viscous

interaction effects are used.

B.2 SIMULATOR AND GUIDANCE ALGORITHM COMPUTER PROGRAM

The computer program for the CSDL ERV simulation and guidance algorithm actually consists of a control program and a plot program that

collects and uses 14 other programs to generate the trajectory data and

plot information. Whil e some of the programs were developed or modified

224
by the author for his own application, most of the code was modified and

or developed by R. Richards based on previous in-house computer simulators at C.S Draper Laboratory (19).

The software is written in MAC, a language developed at and used

only at Draper Laboratory. It is an excellent language for dealing with

vector and matrix quantities and operations. However, because of the

unique language in which the software is written and because the total

collection of code is mearly 300 pages, the actual code is not included

(The computer code can be obtained though by contacting the CSDL NASA

Department, Guidance and Navigation Division, 10E).

To provide at least a basic understanding of the ERV computer simulator and guidance program, a listin g and brief description of the programs used are shown below. Figure B-1 provides a top level flowchart

showing how the programs are grouped and connected.

1. Control Program; Provides a top-level input/output control channel

for the various subprograms. Allows for easy changes in vehicle and

atmospheric conditions (e.g., density, aerodynamic, and engine performance) by the operator.

2. Plot Program: Uses data collected by the control program to plot

various trajectory parameters (e.g., altitude, velocity, and heat-rate).

225
3. Driver Program; Reads data from the control program and calculates

the initia l vehicle state to be used by the other subprograms. Also

collects information from the other programs and prints the desired output.

4. ERV Simulator Program; Uses the initia l ERV state information along

with atmospheric, vehicle performance, and guidance information to estimate the current vehicle state.

5. Guidance Program; Provides the guidance information produced by the

guidance algorithm described in Chapter 5m Outputs include the commanded

angle-of-attack, bank angle, and throttle setting.

6. Predictor/Corrector Program: Predictor/corrector algorithm for the

guidance program. Outputs include the estimated final inclination

change, apogee altitude, and final vehicle weight.

7. Conic Program; Performs the conic calculations for the

predictor/corrector and simulator program.

8. Kepler Program: Solves the Kepler problem for the conic program.

226
9. Atmosphere Program: Provides the atmospheric model data (e.g., density, temperature, and dynamic pressure) for the various subprograms.

10. CLCD Driver Program: Provides the ERV aerodynamic data to the simulator program. Interpolates the vehicle's CL/CD data based on angleof-attack and a correlation parameter of either altitude, viscous, or

mach number.

11. VINV Programs: Calculates the viscous correlation parameter values

for the CLCD Driver program.

12. DATA. DATA 1.2 and 3; Contains the tables of ERV aerodynamic data.

227
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