AN ADAPTIVE NUMERIC PREDICTOR-CORRECTOR
GUIDANCE ALGORITHM FOR ATMOSPHERIC
ENTRY VEHICLES
by
Kenneth Milton Spratlin
B.A.E., Georgia Institute of Technology
(1985)
SUBMITTED IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
MASTER
OF SCIENCE
IN
AERONAUTICS AND ASTRONAUTICS
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
May 1987
©)Kenneth Milton Spratlin, 1987
Signature
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May 8, 1987
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Professor Richard H. Battin
Department of Aeronautics and Astronautics
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Professor Harold Y. Wachman
Chairman, Department Graduate Committee
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AN ADAPTIVE NUMERIC PREDICTOR-CORRECTOR
GUIDANCE ALGORITHM FOR ATMOSPHERIC
ENTRY VEHICLES
by
Kenneth Milton Spratlin
Submitted to the Department of Aeronautics and Astronautics
on May 8, 1987, in partial fulfillment of the
requirements for the Degree of Master of Science in
Aeronautics and Astronautics
ABSTRACT
An adaptive numeric predictor-corrector guidance algorithm is developed for atmospheric entry vehicles which utilize lift to achieve maximum footprint capability. Applicability
of the guidance design to vehicles with a wide range of performance capabilities is desired
so as to reduce
the need for algorithm
redesign
with each new vehicle.
Adaptability
is
desired to minimize mission-specific analysis and planning. The guidance algorithm motivation and design are presented.
Performance is assessed for application of the algorithm to the NASA Entry Research
Vehicle (ERV). The dispersions the guidance must be designed to handle are presented.
The achievable operational footprint for expected worst-case dispersions is presented. The
algorithm performs excellently for the expected dispersions and captures most of the achiev-
able footprint.
Thesis Supervisor: Dr. Richard H. Battin
Title: Adjunct Professor of Aeronautics and Astronautics
Technical Supervisor: Timothy J. Brand
Title: Division Leader, The Charles Stark Draper Laboratory, Inc.
3
N'
4
ACKNO WLEDGEMENT
This report was prepared at The Charles Stark Draper Laboratory, Inc. in support of
NASA Langley Research Center (LaRC) Task Order No. 87-43 under Contract NAS9-17560
with the NASA Lyndon B. Johnson Space Center (JSC).
Publication of this report does not constitute approval by the Draper Laboratory or the
sponsoring agency of the findings or conclusions contained herein. It is published for the
exchange and stimulation of ideas.
I hereby assign my copyright of this thesis to The Charles Stark Draper Laboratory, Inc.,
Cambridge, Massachusetts.
Kenneth M. Sratlin
Permission is hereby granted by The Charles Stark Draper Laboratory, Inc. to the Massachusetts Institute of Technology to reproduce any or all of this thesis.
I wish to take this opportunity to thank some of the many people who have helped me
during the preparation of this thesis and my stay at M.I.T.. I would like to express my sincere gratitude to The Charles Stark Draper Laboratory for making my graduate study possible through the Draper Fellowship. I wish to thank all of the members of the Guidance and
Navigation Analysis Division for their technical assistance and willingness to share their
expertise. In particular I would like to thank T. Brand, J. Higgins, and A. Engel for their
patience, assistance, and support during the preparation of this thesis and work on the
Aeroassist Flight Experiment. I wish to thank my thesis advisor, Dr. R.H. Battin, for his
assistance and in particular for his abilities as an educator. A special thanks for his encouragement and support goes to B. Kriegsman who recently passed away. He is greatly
missed. I found the opportunity to interact with the people at CSDL to be the most rewarding aspect of my studies at M.I.T..
I also wish to thank H. Stone and R. Powell of NASA/LaRC for the opportunity to work on
the ERV entry guidance problem and their assistance in answering my questions.
Most importantly, I wish to thank my parents and sisters for their unending love and
support over the years.
5
6
TABLE OF CONTENTS
Page
Section
...................... .............................. 21
1.0
INTRODUCTION
2.0
MOTIVATION ......................
...............I................. 23
2.1
Introduction ...................
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2
Dispersions ...................
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.3 Reference Trajectories
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
..........
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.4 Guidance Approach .............
3.0
GUIDANCE DESIGN
3.1 Introduction
3.2
....
33
......
33
Unit Target Vector
34
.
3.3 Commanded Attitude
Computation
.................................
35
3.4 Corrector Algorithm
36
3.5 Predictor Algorithm
39
3.5.1 Introduction
...
39
3.5.2 Equations of Moti(on
..........................................
3.5.3 Integration of the Equations of Motion
............................
Cond litions for the Predictor
40
44
..........................
46
3.5.5 Final State Error (Comnutation
..................................
46
3.5.6 Algorithm Coding
48
3.5.4
Termination
,
.....
7
4.0
3.6 Estimators .....
...........................
3.7
..............................................
Heat Rate Control
PERFORMANCE
...................
48
52
57
....................................................
4.1 Simulator .-...................................................
57
4.2 Open-Loop Footprint
57
............................................
4.3 Effect of Dispersions on Footprint
..................................
59
4.4
Estimator Performance
4.5
Closed-Loop Performance ............................
61
4.6
Heat Rate Control Performance
63
4.7 Overcontrol
.........................................
...................................
...................................................
4.8 Algorithm Execution Time
5.0
I
.....
5.1
Future Research Topics
...
63
..................................
FUTURE RESEARCH TOPICS AND CONCLUSIONS
66
.. . . . . . . . . . . . . . . . . . . . . . . . 69
. . . . . . . . . . . . . .
5.2 Conclusions ...................................................
Appendix
......... 69
71
Page
Appendix A. ERV AERODYNAMICS MODEL
..................................
Appendix B. ALGORITHM PROGRAM LISTINGS
List of References
60
..............................
133
135
211
........................................
8
.
LIST OF ILLUSTRATIONS
Page
Figure
1.
Three-View Drawing of the ERV .......................................
78
2.
Atmospheric Wind Profile ............................................
79
3.
Envelope of Density Profiles Derived from Shuttle Flights ...- : ................
80
4.
STS-1 Density Profile Comparison
........................
81
5.
STS-9 Density Profile Comparison
.....................................
6.
Multiphase Bank Angle Program for L/D = 1.5
7.
Crossrange Versus Number of Bank Steps ...............................
84
8.
Comparison of Optimum Bank Angle Programs
85
9.
Optimum Shuttle Angle of Attack Profile for Maximum Downrange .............
86
10.
Optimum Shuttle Bank Angle Profile for Maximum Crossrange
87
11.
Optimum Shuttle Angle of Attack Profile for Maximum Crossrange
12.
Landing Footprint for the ERV .........................................
89
13.
Altitude Histories for the Entry Missions of the ERV ........................
90
14.
Bank Angle Histories for the Entry Missions of the ERV .....................
91
15.
Angle of Attack Histories for the Entry Missions of the ERV ..................
92
16.
Heat Rate Histories for the Entry Missions of the ERV
......................
93
17.
Heat Load Histories for the Entry Missions of the ERV
......................
94
18.
Bank Angle Versus Velocity Profile
19.
Definitions of Downrange and Crossrange Errors ..........................
96
20.
Predicted Lift Coefficient Profile for the ERV ..............................
97
21.
Predicted L/D Profile for the ERV ......................................
98
............
............
...............
...........................
....................................
9
82
...............
..........
83
. 88
95
22.
Predicted UD versus Angle of Attack Profile for the ERV ........
23.
ERV Open-Loop Footprint with the Control Profile
24.
Time Response of the Density Filter ...................................
101
25.
Time Response of the L/D Filter ......................................
102
26.
Closed-Loop Altitude History for the Maximum Downrange Case .............
103
27.
Closed-Loop Velocity History for the Maximum Downrange Case .............
104
28.
Closed-Loop Heat Rate History for the Maximum Downrange Case
...........
105
29.
Closed-Loop Heat Load History for the Maximum Downrange Case ...........
106
30.
Closed-Loop Downrange History for the Maximum Downrange Case ..........
107
31.
Closed-Loop Crossrange History for the Maximum Downrange Case ..........
108
32.
Closed-Loop Altitude History for the Maximum Crossrange Case .............
109
33.
Closed-Loop Velocity History for the Maximum Crossrange Case .............
110
34.
Closed-Loop Heat Rate History for the Maximum Crossrange Case ...........
111
35.
Closed-Loop Heat Load History for the Maximum Crossrange Case ...........
112
36.
Closed-Loop Downrange History for the Maximum Crossrange Case ..........
113
37.
Closed-Loop Crossrange History for the Maximum Crossrange Case ..........
114
38.
Closed-Loop Altitude History for the Minimum Downrange Case
115
39.
Closed-Loop Velocity History for the Minimum Downrange Case
40.
. . .. . . . .
........................
99
100
.............
..........
116
Closed-Loop Heat Rate History for the Minimum Downrange Case
...........
117
41.
Closed-Loop Heat Load History for the Minimum Downrange Case
...........
118
42.
Closed-Loop Downrange History for the Minimum Downrange Case
..........
119
43.
Closed-Loop Crossrange History for the Minimum Downrange Case
..........
120
44.
Angle of Attack Comparison for the Maximum Downrange Case
45.
Bank Angle Comparison for the Maximum Downrange Case
46.
Angle of Attack Comparison for the Maximum Crossrange Case
47.
Bank Angle Comparison for the Maximum Crossrange Case ................
124
48.
Angle of Attack Comparison for the Minimum Downrange Case .............
125
10
.............
................
.............
121
122
123
49.
Bank Angle Comparison for the Minimum Downrange Case ..............
50.
Bank Angle Versus Time Comparison for Heat Rate Control
51.
Angle of Attack Versus Time Comparison for Heat Rate Control ..............
128
52.
Heat Rate Versus Time Comparison for Heat Rate Control ..................
129
53.
Bank Angle Versus Velocity Comparison for Heat Rate Control
130
54.
Angle of Attack Versus Time Comparison with Overcontrol
55.
Required Execution Time for the Predictor-Corrector
11
................
..............
.................
......................
...
126
127
131
132
12
LIST OF TABLES
Page
Table
1.
Characteristics of the ERV and Trajectory Entry Conditions
2.
Dispersions
3.
Dispersed Cases for Maximum Downrange Region
...................
4.
Dispersed Cases for Maximum Crossrange Region
........................
5.
Dispersed Cases for Minimum Downrange Region ...................
6.
Maximum Downrange Region Closed-Loop Results
7.
Maximum Crossrange Region Closed-Loop Results ........................
77
8.
Minimum Downrange Region Closed-Loop Results
77
Used in Performance
Study
13
73
..................
73
.................................
.............
........................
....
74
75
......
..........
76
77
14
SYMBOLS
a
= acceleration magnitude
a
= acceleration vector
a,
= inertial acceleration measured by the inertial measurement unit
A,
= total inertial acceleration vector
AOTV
= Aerobraking Orbital Transfer Vehicle
BTU
= British Thermal IJnit
c
= mean aerodynamic chord
cos(A¢)
= cosine of incremental lift for heat rate control
C'
= proportionality factor for the linear viscosity-temperature relationship
Co
= aerodynamic drag coefficient
CL
= aerodynamic lift coefficient
C,
= speed of sound
CPU
= central processing unit
CR
= crossrange
CSDL
= The Charles Stark Draper Laboratory, Inc.
det
= determinant of sensitivity matrix
DR
= downrange
DOF
= degree of freedom
ERV
= Entry Research Vehicle
fearth
=
F,
= inertial force vector
g
= gravitational acceleration magnitude
9g
= gravitational acceleration vector
flattening of oblate Earth
15
GPS
=
h
= altitude
/h
= derivative of altitude with time
h
= second-derivative of altitude with time
h,
- scale height for exponential atmosphere model
i
= unit vector
=
Global Positioning System
second zonal harmonic coefficient
JSC
= Lyndon B. Johnson Space Center
k
= term in geodetic to geocentric latitude conversion
K
= term in heat rate control equation
K
= velocity vector term in the integration algorithm
K'
= acceleration vector term in the integration algorithm
KAt
=- gain on acceleration magnitude in variable time step equation
KLID
=multiplicative
K6
= gain on heat rate error in heat rate control equation
Kb
= gain on rate of change of heat rate in heat rate control equation
KP
= multiplicative scale factor on the standard density
K,
= gain in first-order filter for density smoothing
K2
= gain in first-order filter for L/D smoothing
LID
= lift-to-drag ratio
LaRC
= Langley Research Center
m
= mass
M
=Mach Number
MrF
=
MO
= mean molecular weight of air at sea level
n.m.
= nautical mile
NASA
= National Aeronautics and Space Administration
scale factor on the nominal L/D
inertial-to-Earth-fixed transformation matrix
16
POST
= Program to Optimize Simulated Trajectories
= dynamic pressure
Q
= heat load
Q
= heat rate
Q
= time rate of change of heat rate
R
= position vector magnitude
Requator
= radius of oblate Earth at equator
R,
= inertial position vector
Rpole
= radius of oblate Earth at pole
Re
= Reynolds Number
S
= Sutherland's constant in viscosity equation
S
= aerodynamic reference area
SEADS
= Shuttle Entry Air Data System
tG,rT
= Greenwich mean
T'
= reference temperature
TM
= molecular scale temperature
Tstatc
= freestream static temperature
Twa.
1
= wall temperature
TAEM
= Terminal Area Energy Management
V
= Viscous Interaction Parameter
V,
= magnitude of inertial velocity
V,
= inertial velocity vector
VR
= magnitude of Earth-relative velocity
VR
= Earth-relative velocity vector
z
= geocentric colatitude of the postion vector
a
= angle of attack
time
17
P/
= angle of sideslip
6
= small incremental change
A
= incremental change
y
= ratio of specific heats for air
= longitude
/.
= coefficientof viscosity for air
/.z
= Earth gravitational constant
4earth
=
c.
Earth's rotation vector
natural frequency of heat rate control response
-=
= bank angle
= universal gas constant
p
= atmospheric density
= standard deviation
-r
=
time constant of filter
sx
-= damping ratio of heat rate control response
SUBSCRIPTS
aero
= aerodynamic
c
= geocentric
cmd
= command
d
= desired
des
= desired
drag
= drag term
18
e
= error
El
= entry interface
f
= final
g
= geodetic
imu
= inertial measurement unit
inplane
= projection of target unit vector into plane formed by the
position vector and the relative velocity vector
lat
= direction perpendicular to the plane formed by the position vector
and the relative velocity vector
lift
= lift term
lim
= limiting value or boundary
LID
= lift-to-drag ratio
max
= maximum
min
= minimum
nom
= nominal
perpen
= direction perpendicular to the plane formed by the position vector
and the relative velocity vector
pole
= direction of the north pole
R
= direction of the position vector
si
= sea level
std
= standard value
t
= target aim point
p
= atmospheric density
19
SUPERSCRIPTS
EF
= coordinatized in Earth-fixed coordinates
imu
= value measured by inertial measurement unit on current cycle
imu past
= value measured by inertial measurement unit on past cycle
A
= estimated or measured
= value from previous guidance cycle
20
1.0
INTRODUCTION
Routine access to space and the maintenance of a Space Station will increasingly
require greater flexibility in mission planning and the requirement for lower system maintenance costs. The launch and recovery phases of space flight have historically been the
most demanding phases of space flight and therefore require the most development effort
and investment. Mission flexibility requires more frequent launch and deorbit opportunities.
For the case of re-entry vehicles, deorbit opportunities are defined by the ranging capability
of the vehicle.
A high L/D vehicle increases the available deorbit opportunities increasing
mission flexibility.
High L/D vehicles also are of interest for over-flight missions for the pur-
pose of reconnaissance.
Entry guidance algorithms developed to date have been highly vehicle-specific
and
required great development and maintenance efforts over the life of the vehicle. These algorithms were not applicable to other vehicles without extensive modification.
This study seeks to design an adaptive entry guidance algorithm that maximizes the
usable footprint by making full use of the available vehicle capability. This algorithm should
also be easy to maintain throughout the vehicle definition phase and operational life. Minimizing the number of mission-dependent input parameters (I-loads) is desirable.
The algo-
rithm should also be easily transported to other vehicles to minimize development cost.
Transportability is accomplished by minimizing vehicle-specific features of the algorithm.
Explicit heat rate control should be provided to allow full use of the entry corridor up to the
heat rate limits.
21
This study seeks to design such an algorithm.
A candidate entry guidance algorithm is
defined for the NASA Entry Research Vehicle (ERV), but is easily adapted to other vehicles
with minimal modification. The proposed algorithm attains almost complete coverage of the
achievable footprint, while employing a simple one-phase entry algorithm with explicit heat
rate control.
Vehicle-specific features and I-loads are minimized, reducing algorithm devel-
opment and maintenance costs.
The ERV [1] is a proposed high-performance entry vehicle designed as a test bed for
future technology development in the areas of:
1.
Maneuvering entry/synergetic plane change
2.
Atmospheric uncertainties
3.
Advanced thermal protection systems
4.
Aerodynamic/aeroheating prediction
5.
Adaptive guidance and navigation
6.
Load-bearing thermostructures
The ERV is designed for deployment from the Space Shuttle, after which the ERV enters the
atmosphere for demonstration of the synergetic plane change, over-flight, and entry missions. Figure 1 on page 78 shows a three-view drawing of the ERV and the surface areas of
the aerodynamic control surfaces. Also seen is the size of the ERV in relation to the diameter of the Shuttle payload bay in which the ERV must'fit.
22
2.0
MOTIVATION
2.1
INTRODUCTION
The goal of any entry guidance algorithm is to successfully guide the vehicle to the
desired final state for the largest range of dispersions possible without violating any vehicle
constraints while also maximizing the achievable footprint.
It is also desirable to minimize
the mission and vehicle-specific aspects of the guidance algorithm so as to minimize premission analysis and planning. Transportability of the algorithm from one vehicle to another
significantly reduces guidance algorithm development effort and cost.
To maximize the footprint attainable, the guidance algorithm must follow the optimal
path to any particular point in the footprint. The algorithms developed to date for such vehicles as the Apollo capsule [2] and the Space Shuttle [3] have attempted to do this by fitting
the optimal trajectory with phases that follow important parameters (reference profiles) over
some range of conditions. These guidance algorithms were required to be computationally
efficient
because
of the
limited
on-board
computer
resources
available.
Analytic
expressions for the reference profiles allowed for low execution time and tailoring of the trajectory for vehicle-specific constraints.
For example, trajectories for these vehicles had to
be shaped to reduce and control the maximum heat rate experienced below that allowed for
the available thermal protection system materials.
23
The Space Shuttle entry guidance system employs three major modes with seven phases:
1.
Entry
a.
Pre-entry
b.
Temperature control
c.
Equilibrium glide
d.
Constant drag
e.
Transition
2.
Terminal Area Energy Management
3.
Approach and Landing
Except for the pre-entry phase which is open-loop, each phase is described by an analytic
expression relating the desired drag and altitude rate (the measured feedback terms used)
to the desired profile.
Because the algorithms are tailored for a particular vehicle and the
reference profiles do not follow the optimal profile to all points in the footprint, guidance
algorithms developed to date can not be easily adapted to other vehicles or provide full coverage of the theoretically achievable footprint.
The next generation of entry vehicles will not be so constrained due to advances in thermal protection system materials and computer technology.
For example, flight computers
are now capable of supercomputer speeds on the order of 40 million instructions per second
utilizing parallel processing architecture [4].
A different approach to guidance that attempts
to follow an optimal profile to maximize footprint capability is therefore possible.
The proposed approach is a predictor-corrector algorithm that numerically predicts the
final state for a particular control variable history and then corrects the control variable history to satisfy the specified final state constraints.
24
This approach, proposed previously for
various guidance problems, has most often been impractical because of the long trajectories
that must be predicted and the slow computer speeds.
Such an approach has been employed for the Space Shuttle Powered Explicit Guidance
(PEG) [5] used for second stage ascent and orbit insertion burns where the trajectory is
short enough to be predicted with the available computer resources. The Shuttle algorithm
numerically predicts the gravitational effects during the powered flight phase with a 10 step
integration of the 500 second trajectory.
A predictor-corrector has also been proposed for Aerobraking Orbital Transfer Vehicles
(AOTV) [6] which would utilize more advanced computers. This algorithm numerically integrates the equations of motion along a skimming trajectory through the upper atmosphere
that is approximately 500 seconds long and requires about 100 integration steps.
The trajectories flown by the ERV or any high L/D entry vehicle are typically from 30 to
100 minutes long from entry interface (400K feet) to landing, so the computational demand
for such an algorithm is very gruat early in the entry when the time to landing is long. However, because the entry is long and the vehicle has excess ranging capability for all but a
small region along the edge of the footprint, the accuracy of the early predictions need not
be as high as for the later predictions. Hence, large time steps can be used early in the predictor algorithm.
Later, when the vehicle nears the landing site, the time remaining is short,
and hence, the prediction is short. This allows the predictor-corrector to be executed more
often near landing just like the current analytic algorithms.
Throughout the entry, vehicles
using an analytic guidance algorithm with reference profiles must closely follow the reference profile if the assumed reference profile is to guide the vehicle to the correct final state.
A predictor-corrector effectively recomputes a new reference profile each time it is executed,
so the guidance execution rate can be much lower than that for analytic algorithms.
25
2.2
DISPERSIONS
Before the guidance algorithm can be designed, the possible dispersions that may affect
the trajectory must be considered.
The Shuttle entry guidance system is required to reach
the Terminal Area Energy Management (TAEM) interface with less than a 2.5 nautical mile
position error from the target aim point. The dispersions of significance to an entry vehicle
trajectory include:
1.
2.
3.
4.
Vehicle characteristics
a.
Mass
b.
Aerodynamics
c.
Maneuver rates
Environment characteristics
a.
Atmospheric density
b.
Atmospheric winds
c.
Atmospheric properties influencing aerodynamic flow regimes (temperature,
mean free path, etc.)
Initial entry state vector
a.
Velocity
b.
Flight path angle
c.
Heading
Propagation errors in navigation state vector
Of these potential dispersion sources, only the vehicle mass, aerodynamics, and the
atmospheric density and winds will be significant. By the early 1990's, almost perfect navigation can be expected through use of the Global Positioning System (GPS). If the deorbit
burn guidance and control systems are assumed to correctly guide to the navigated state
26
and there are no navigation errors, then the dispersions in the initial entry state vector are
negligible.
The vehicle mass should be known accurately, so for this study, a 3
error of +5% is
assumed. Experience from the Space Shuttle program shows that the vehicle aerodynamics
should be known to within +5% for the force coefficients on the first flight. Only the stability
derivatives and control effectiveness were missed significantly [7].
Even though the force
coefficients may be known to excellent accuracy, reduced control effectiveness can reduce
the possible trim angle of attack range reducing the maximum L/D achievable.
Therefore,
for this study, a +10% dispersion in the lift and drag coefficients is considered. It should be
noted that the first few flights of a new vehicle are usually targeted to the middle of the footprint to maximize margin and allow for accurate determination of the vehicle characteristics
before the full ranging capability of the vehicle is used. After the first few flights, the aerodynamic characteristics should be known to within a few percent, so only about a +3% dispersion must be considered.
The atmospheric dispersions were obtained from two sources. Reference [8] specifies
the atmospheric dispersions to which aerospace vehicles must be designed. The average of
the steady state winds at four geographic locations is shown in Figure 2 on page 79. This
model was incorporated into the simulator environment with a magnitude scale factor to
simulate less than worst-case winds. The wind direction was selected for each run made
with winds and held constant throughout the trajectory.
Reference [8] specifies Reference
[9] as the source for atmospheric density dispersions.
However, the recent Shuttle flights
have provided estimated density data of a quality never before available. Atmospheric density profiles derived from Shuttle accelerometer measurements of the normal force acceleration and the estimated normal force coefficient and relative velocity vector are presented in
Reference [10].
Figure 3 on page 80, taken from that report, shows the envelope of the
27
derived density profiles for the first 12 Shuttle flights. Of particular interest is the range of
dispersions
seen: -47% to +12%.
Figures 4 on page 81 and 5 on page 82 show the density
profiles for the STS-1 and STS-9 Shuttle flights.
High frequency density shear components
and constant density biases from the standard atmosphere are seen. For this study, constant density biases of +30% and the Shuttle derived density profiles from Reference [10]
were used.
2.3
REFERENCE TRAJECTORIES
The size of the footprint for a particular vehicle is determined by the range in vehicle
L/D and the constraints placed on the trajectory such as heat rate limits.
footprint correspond to the use of maximum or minimum UD.
The edges of the
Maximum downrange or
crossrange, for example, requires maximum L/D, while minimum downrange requires minimum L/D.
The determination of the optimal angle of attack and bank angle control histories for
maximum crossrange and downrange has been the topic of many papers [11] [12] [13] .
Wagner [12] used several optimization techniques to evaluate the maximum crossrange
achievable for a multiphase bank angle history flown at maximum ULD. The multiphase bank
profiles considered are shown in Figure 6 on page 83. It is seen that as the number of phases increases, the multiphase profile approaches the optimal continuous profile also shown in
this figure. It was determined that a three-phase bank angle profile as illustrated in Figure 6
achieved almost the same crossrange as a continuous bank profile.
This is shown in
Figure 7 on page 84 reproduced here from that paper. Further, as the number of phases
28
increases, the optimum bank angle profile approaches a continuous profile that is almost linear with velocity as shown in Figure 8 on page 85. It was also hown that flying at the maximum L/D maximizes the crossrange attained.
This result is confirmed in Reference [13] which utilized a nonlinear programming technique to optimize the Space Shuttle trajectory for the maximum downrange and maximum
crossrange cases.
The maximum downrange trajectory requires flying at zero bank angle
and at the angle of attack corresponding to maximum L/D as shown in Figure 9 on page 86.
The control histories for the maximum crossrange case are shown in Figures 10 on page 87
and 11 on page 88. Again, the optimal control history is the angle of attack corresponding to
maximum L/D and an almost linear bank angle profile with velocity.
Optimized trajectories for the ERV were reported in Reference [14].
These trajectories
were determined using the Program to Optimize Simulated Trajectories (POST) [15] and
imposed the following constraints on the trajectories:
1.
Maximum heat rate of 125 BTU/sq ft/sec
2.
Maximum
heat load of 150K BTU/sq ft
The achievable footprint with these constraints, reported in Reference [14], is shown here in
Figure 12 on page 89 . Subsequently, the heat load limit was increased to 175K BTU/sq ft
resulting in the larger footprint shown in Figure 12. As will be seen, these footprints omit a
large area in the minimum downrange region that is achievable within the heating constraints. Also shown is the footprint of the Space Shuttle which has a maximum hypersonic
UL/Dof 1.2 as compared with 1.8 for the ERV.
Figure 13 on page 90 shows the altitude history for the maximum downrange, maximum
crossrange, and minimum downrange cases.
29
Figures 14 on page 91 and 15 on page 92
show the bank angle and angle of attack histories for these trajectories.
Figures 16 on page
93 and 17 on page 94 show the heat rate and heat load histories for these cases.
Figure 15 shows that the constant angle of attack corresponding to maximum L/D is
flown for the edge of the footprint except for the minimum downrange case.
For the mini-
mum downrange case, the angle of attack corresponding to the minimum L/D on the back
side of the L/D curve (high drag coefficient) is flown early, followed by a ramp in angle of
attack starting at 1500 seconds after entry interface.
This ramp corresponds to the vehicle
actually turning around and flying slightly back uprange, so maximum L/D is desired later to
maximize the distance flown uprange..
The angle of attack for the maximum downrange
case is slightly greater than that for maximum L/D because this trajectory exceeds the heat
load limit
f flown at maximum L/D.
The maximum downrange region of the footprint Is
therefore limited by the heat load limit set for the ERV. If the limit were relaxed, flight at
maximum L/D would allow a longer downrange trajectory.
Figure 14 shows that the bank angle profile for maximum crossrange is approximately
linear with time which is almost linear with velocity, which suggests that a linear bank angle
profile with velocity is sufficient. The maximum downrange case has a constant bank angle
of zero which is again linear with velocity. The minimum downrange case does not have a
linear bank profile. As was mentioned previously, for this case, the vehicle turns around and
flys back uprange.
The results of these studies suggest that use of a constant angle of attack profile and a
linear bank with velocity profile will capture a large portion of the achievable footprint.
As
will be seen in the results, these profiles suffice to capture most of the footprint reported in
Reference [14] and additionally reach a large area in the minimum downrange region out-
30
side the reported footprint.
Only a small area of the reported footprint in the minimum
downrange region is unachievable.
Also of interest are the peaks in heat rate seen in Figure 16. Because the peaks in heat
rate are very short, explicit control of the heat rate should be possible in the maximum heat
rate regions without significantly impacting the giidance.
2.4
GUIDANCE APPROACH
The guidance design will attempt to maximize the size of the footprint while flying a constant angle of attack profile and a linear bank angle with velocity profile. The predictor algorithm integrates the equations of motion forward in time using the assumed control profile
and the necessary environment and vehicle models. The corrector then determines (using
multiple predicted trajectories with various control histories) the sensitivities of the final
state constraints to the control variables.
The sensitivities are then used to compute the
required control variable values to reach the desired final state conditions. Heat rate control
is provided locally during the regions of maximum heating without significantly affecting the
assumed control histories. Also, in-flight measurements are utilized to increase the accuracy of the predicted trajectories by compensating for off-nominal conditions.
Such a simple profile for the maximum downrange and crossrange cases simplifies the
modeling of the control histories in the predictor. The only remaining question is how much
of the footprint this profile will capture.
As will be seen in Subsection "4.2 Open-Loop
31
Footprint" on page 57, such a profile achieves almost complete coverage of the achievable
footprint.
Also of concern is the linearity and convergence properties of the final state constraints
with the control variables.
As will be seen, over almost all of the footprint except near the
edges, the constraints are highly linear and convergent with the control variables.
tionally, only about 75% of the achievable footprint is used to
Thus, the question of nonconvergence near the edges is avoided.
32
Opera-
nsure guidance margin.
3.0
GUIDANCE DESIGN
3.1
INTRODUCTION
This section describes the mplementational details of the guidance scheme described
In the previous section. The equations of motion and environment and vehicle characterIstics modeled in the predictor algorithm are described. The corrector algorithm to control
the final state constraints with the two available control variables is derived.
are the heat rate control and in-flight measurement algorithms.
Also derived
The heat rate control algo-
rithm provides control of the peaks in stagnation heat rate during the early portion of entry.
The in-flight measurement algorithm utilizes accelerations measured by the navigation system to more accurately model the expected environment and vehicle characteristics in the
predictor algorithm. Because the predictor-corrector algorithm is computationally intensive,
areas where significant execution time savings have been or can be realized are ndicated.
Program listings of the algorithm coded in the HAL/S computer language are presented in
"Appendix
B. ALGORITHM
PROGRAM LISTINGS" on page 135.
As will be seen, the only inputs to the guidance system are the environment and vehicle
models, the assumed control profiles, and the navigated state vector. The state vector is an
input to any guidance system. The other inputs are developed for the analysis of any new
vehicle.
Therefore, the guidance system is highly transportable between vehicles because
only the vehicle characteristics and aerodynamics model must be changed for a new vehicle.
33
3.2
UNIT TARGET VECTOR
The target aim point to which the vehicle is to be guided is specified by the longitude
and geodetic latitude of the Terminal Area Energy Management (TAEM) interface point which
occurs at 80K feet for the Shuttle.
This point is selected based on the guidance algorithm
employed during the TAEM guidance phase. TAEM guidance provides precise control of
vehicle energy during the final stages of entry to guide to a specified runway with acceptable
energy. For computational ease, the longitude and geodetic latitude are converted to a target unit vector in Earth-fixed coordinates by first computing the geocentric latitude from,
=
tan
tang)
(
)
(1)
where,
k
R=
=Rator 2
(2)
The unit target vector is then computed from,
cos(q5)
jEl =
/cos(o)
L
cos(i.)
sin(A)
(3)
sin(&,c)
Alternatively,
34
(4)
where,
where,
i, =
3.3
sin(tb,)
(5)
i,
=
cos(A) /1
i,
=
sign(A) /1-ix-i2
- i
(6)
(7)
COMMANDED ATTITUDE COMPUTATION
Because the predictor can not be executed as frequently as analytic guidance algorithms early in the entry, and because it in fact does not have to be executed as frequently,
it is necessary to update the commands sent to the vehicle autopilot more frequently than
the predictor-corrector execution rate. Typically, this would be done at the rate of current
analytic guidance algorithms, e.g., the Space Shuttle rate of .52 hz. The commanded bank
angle, ,,,
is computed for the linear bank with velocity profile as shown in Figure 18 on
page 95 from the desired bank angle,
Ad,
and the current navigated inertial velocity magni-
tude, V,,
V, - V,
vEI -
(8)
Vf
35
to yield the near-optimal linear bank with velocity profile. The desired angle of attack control history is a constant angle of attack, and therefore,
acmd =
ad
(9)
As implemented in the current design, the guidance algorithm executive is executed at 1.0
hz. The attitude commands are updated at this frequency using Eq. (8) and (9) . The predictor-corrector algorithm is executed at .02 hz. during the entire entry phase, although it is
practical to run it much more frequently late in the trajectory when the length of the trajectory to be predicted is short. The possible execution rate of the predictor-corrector for a typical flight computer is addressed in Subsection "4.8 Algorithm Execution Time" on page 66.
3.4
CORRECTOR ALGORITHM
The corrector algorithm is executed to update the commanded attitude control history to
be flown. The guidance algorithm controls to two final state constraints, downrange error
and crossrange error, using two control variables, a constant angle of attack and the intercept of the bank profile at the entry interface velocity as shown in Figure 18 on page 95 and
expressed in Eq. (8) .
Expanding the downrange and crossrange errors in a Taylor series expansion of the
control variables and neglecting the second-order and higher terms yields,
ADR,
=
OR ,
a,
d
Ad
+
aDR,
DR
+...
and
36
(10)
aCR + a d
A CRe
and
+
(11)
+
To intercept the target, the change in the constraint errors must null the predicted errors, or,
=
ADRe
A CR,
-DR,
(12)
-CR,
(13)
Equations (10) through (13) provide a set of two simultaneous equations in two unknowns,
aDR,
ODR,e
aCR,
aCR
a1d
aad
4radl
°
LA
[-DRe
L-CReJ
dj
(14)
a d
which are solved for the control variable changes required,
(
Aad =
Ad
DR,
0
od
CR, -
/ det
deDR,)
CR)I
(15)
-
aDR, CR) / det
(16)
=(,RDR
O°%J
Od
where det is the determinant of the matrix in Eq. (14). The partial derivatives are approximated by finite difference equations of the form,
aDR,
00d
DRe(0d =
=
3)
-
03 -
DR,(bd= ,)
(17)
01
There are four partial derivatives that must be evaluated. They can be evaluated from three
predicted trajectories with control histories selected as:
37
1.
C, =
a,d1
2.
a42 =
a'd +
3.
a3
=
=
(d
ad, 02
=
'd
3
=
C'd,
d +
6
d
where the primes denote the control variables from the previous guidance solution.
The
new guidance commands are then,
ad
=
=d =
a'd +
(18)
Acd
(19)
O'd + Ad
Protection must be provided for the.case where the determinant in Eq. (15) and (16) is
small or identically zero which corresponds to a loss of control authority of the control variables over the control constraints.
In this case, no change is made to the control variables,
and the guidance command from the previous cycle is used. As the vehicle approaches the
TAEM interface altitude, the control authority decreases.
Large control variable changes
become necessary to null the constraint errors in the short flight time remaining. This problem can be avoided in one of two ways. First, the guidance commands can be frozen at a
selected point before the termination altitude. For entry guidance, this approach is not preferred because the vehicle still has not landed. Alternatively, the target aim point can be
lowered below the TAEM interface altitude point at which TAEM guidance is activated. The
decreasing control authority problem is therefore reduced.
For the simulated trajectories in this report, the first approach is employed because it is
desired to evaluate guidance performance by considering the dispersions in the final state at
the TAEM interface altitude.
Because the guidance algorithm controls only the final state
and not the intermediate states, it is necessary to target for the point at which the guidance
is terminated.
38
3.5
PREDICTOR ALGORITHM
3.5.1 Introduction
The predictor algorithm is a simplified three-degree-of-freedom (3-DOF) trajectory simulator complete with models for those environment and vehicle characteristics necessary to
model the translational equations of motion of the vehicle.
Because the predictor is compu-
tationally intensive, the algorithm must be carefully designed to minimize computation, and
the coding of the algorithm in a particular computer language should make use of any language-specific features to reduce computational requirements. Also, because the corrector
only utilizes the final state vector errors to correct the control variables, only the accuracy of
the predicted final state vector need be considered in selecting those effects to be modeled.
The environmental effects of concern for the long trajectories flown by entry vehicles
over large altitude and velocity ranges are:
1.
Variation of atmospheric properties with altitude
2.
Earth oblateness effect on gravity vector
3.
Effect of atmospheric rotation with Earth on relative velocity vector
4.
Movement of runway due to Earth rotation
The vehicle characteristics of importance are:
1.
Vehicle mass
2.
Aerodynamic coefficient variation with flight regime
3.
Aerodynamic coefficient variation with angle of attack
4.
Control history during trajectory
Dispersions to be considered are:
39
1.
Vehicle mass variation from nominal
2.
Winds
3.
Atmospheric density variation from nominal atmosphere
4.
Aerodynamic coefficient variation from nominal
These dispersions can be measured in-flight because they affect the sensed acceleration
measured by the vehicle's inertial navigation system. The estimation of these dispersions is
discussed in Subsection "3.6 Estimators" on page 48.
The predictor performs the following computations upon being called by the corrector
with a desired control variable history:
1.
Initialize the predictor state to the navigated state vector
2.
Compute any ancillary parameters from the state vector
3.
Compute the total acceleration vector from the predictor state vector and the environment and vehicle models using the control variable profiles specified by the corrector
4.
Integrate the equations of motion forward in time one time step
5.
Check the predictor termination conditions
6.
a.
Repeat steps 3 and 4 if the conditions are not met
b.
Continue on to step 6 if the conditions are met
Compute and return to the corrector the final predicted state errors from the target
state vector and the predicted final state vector
3.5.2 Equations of Motion
The corrector provides a time-homogeneous navigated state vector comprised of,
1.
The GMT time tag of the state vector, tGwr
2.
The inertial position vector, R,
3.
The inertial velocity vector, V,
40
Also provided is the control variable history to be followed for the prediction. The equations
of motion to be integrated are,
d R,
dt
(20)
'IVl
dV,
dt
=
(21)
A,
The acceleration is computed from the atmosphere and vehicle models as follows,
F,
-
A,
m
-~
g,
=
+
aaeo
(22)
The gravitational acceleration, g,, is computed including the J2 term as,
9g
(23)
IR, 12
2
JR,
where,
=
Ig
'R
3
+ -
J2
2
Rquat,
,IRI
2
((15
2) iR
+ 2 z ipoe)
(24)
and,
Z
(25)
Ipole
R
=
The aerodynamic acceleration, a,,o, is computed from,
aaero
=-
ift +
alf
(26)
adr,ag drag
where,
a,,,t
CL,q S
(27)
m
41
adag =
--
m
(28)
P Vz
(29)
q
==
vVRV
VR
* VR(30)
(30)
VR
=
Vl -
(31)
,,eanrXR.
(32)
idrag =
Iv
cos(k) + ia sin(X)
ilift = (idra X i,)
h,,
jR
IR X
drag
IR X
idrg
(33)
=
(34)
I
=
(35)
IR, I
The acceleration due to lift, a,,ft,is more easily computed from,
a,,f
=
L a,
(36)
since the nominal lift-to-drag ratio, L/D, is corrected using in-flight accelerometer measurements of the actual vehicle sensed aerodynamic accelerations.
The atmospheric density, p, is computed by the atmosphere model using the position
vector, R.. The 1962 U.S. Standard Atmosphere model is employed and is described in Reference [16].
If another atmosphere model is selected as being a more accurate estimate of
the day-of-flight atmosphere, this model would replace the 1962 U.S. Standard Atmosphere
model. An operational vehicle might employ monthly or seasonal atmospheres from such
42
sources as the GRAM Atmosphere [9] or even day-of-flight measurements to more accurately model the expected atmosphere in the predictions. The level of accuracy required in
the atmosphere model will depend on the vehicle ranging capability and the amount of that
capability to be.used for a particular entry. Entries to the edges of the footprint will demand
a very accurate atmosphere model.
The aerodynamic coefficients are highly vehicle dependent. To minimize computational
requirements, they should be updated during the prediction as infrequently as possible. Of
course, the update frequency required depends on the trajectory flown and the rate of
change of the aerodynamic coefficients with flight regime change. The aerodynamic coefficient model
for the ERV is presented
in "Appendix
A. ERV AERODYNAMICS
MODEL" on
page 133.
The density, p, from the atmosphere model and the lift-to-drag ratio, L/D, from the aerodynamic model are both corrected by in-flight measurements as covered in Subsection "3.6
Estimators" on page 48. The estimated dispersions are compensated for using the following
equations,
p
D
=
(37)
Kp Ptd
KL
6
(38)
CL)
Co
where the density and lift-to-drag ratio scale factors, Kp and KL, are provided by the estimator and are held constant throughout the prediction being made.
The control history to be followed is the constant angle of attack,
angle with velocity,
. The latter is computed from,
43
ad,
and the linear bank
~b -~d
VI VI
VE,-
(39)
V,
d is the intercept of the linear bank angle profile at the entry interface velocity, VE,.
where
Because the entry interface and final velocities are not known a priori, and because small
variations in them have little effect on the predicted trajectory compared with the selected
control variables' values, the velocities are selected as constant values that cover all
expected dispersions in the entry and final velocities. These values are,
VE, Vf
=
26,000 ft/sec
1,000 ft/sec
3.5.3 Integration of the Equationsof Motion
The equations of motion are integrated using the 4th order Runge-Kutta algorithm with a
variable time step to minimize the number of time steps required to integrate the trajectory
to the final state.
The 4th order Runge-Kutta algorithm requires four evaluations of the
acceleration per time step, but permits a time step more than four times as large as an algorithm requiring only one acceleration evaluation per time step.
The Runge-Kutta solution
[17] for the differential equations of motion of the form,
dRI
d t
dt
V,
dV
f(t,
(40)
R
V)
(41)
is,
44
4t
R,(t+ At) = R,(t)+
- 6 (Ko
-'(+
-')+
2K + 2K + K,)2
)
(42)
V,(t + At) =
V,(t) +
2K'
(43)
6t (K'o +
6
--
+ 2K' 2 + K'3)
where,
Ko
K,
=
_~
=
V,
-
(V,
(44)
K'
+
K2 = (V, +
2-)
(45)
2 )
(46)
(Vt ± K )
K3
=
K'
=
f(t,
K'
=
f(t
+
K'2 =
f(t
+
K' 3 =
f(t
+ At,
R
(47)
V,)
(48)
~~
2'
RI
+ At 2'
V,
2
R
+ At K1
V, + At
2
2'
R. + At K2,
+
At
K'
(49)
2
K' 1
V, + At K' 2)
2
)
(50)
(51)
The time step is varied inversely with the total acceleration on the vehicle. This method
of time step control was selected because of its simplicity. The time step control equation is
of the form,
At =
KA,
_
(52)
IA, I
45
and the time step is limited between a minimum and maximum value,
At
=
midval(Atm,,, At,
(53)
tma),,
The optimization of the integration algorithm is important in developing a flight quality
algorithm, but is beyond the scope of this study. Higher-order integration algorithms with
time step control methods [17] may yield significant reductions in the required computation
time.
3.5.4 Termination Conditions for the Predictor
After each integration time step, the predicted state is compared with the termination
condition.
The termination condition is defined by the altitude of TAEM interface (80K feet).
Because the predicted state at the TAEM interface altitude may have a relatively large altitude rate and range rate, the predictor must be terminated accurately to provide an altitudehomogeneous set of predicted state errors.
Also, the variable time step control may allow
large integration time steps if the acceleration is low near the final state, further complicating the task of terminating accurately. Reasonable altitude homogeneity is ensured by forcing use of the minimum integration time step starting some safe altitude above the
termination altitude.
3.5.5 Final State Error Computation
The final state errors are computed from the unit target vector and the predicted final
state vector. Because the target is fixed to the Earth and moves a significant distance dur-
46
ing the long entry trajectory, the rotation of the Earth must be considered.
This is done by
transforming the final state vector from inertial to Earth-fixed coordinates with the rotation
matrix MEf which is computed from the predicted termination time, the known orientation of
the Earth at some epoch time, and the known rotation rate of the Earth. This computation is
performed in the Earth-Fixed-From-Reference subroutine of the predictor-corrector
which
may actually be a GN&C utility function also employed by the navigation principal function.
The downrange and crossrange errors are defined as shown in Figure 19 on page 96.
The errors are computed by first computing the downrange (in-plane) and crossrange (perpendicular) directions as follows,
RIF
=
MF R
(54)
REF
I=
E(55)
5REF J
VR
=
M
(56)
V
EF
/een
VEF
I. x V
-
EF X
EF
mplar'e
'EF
(t
·
'F
(57)
l
'EF
Iperpen)perpen
(58)
_
-
|j:F
_
(t:F
_ iEeFrpen)ierpen
The downrange and crossrange errors are then,
DRe
=
Requator COs 1(i
CRe
=
Re
Cs
X iE,ane)
(Ip/ene
ijF)
sign((iR F x ipa,,ne)
F)
Sign((,Ep/ane X itE
47
i/pen)
·
(peFrpenX
(59)
,nF e))
(60)
These errors have the dimensions of R,,,,tor and are converted to nautical miles for ease of
interpretation.
3.5.6 Algorithm Coding
A few comments regarding implementation of the predictor are appropriate.
The com-
putations required to update the aerodynamic coefficients are the major computational load
for the predictor.
It was found that it is not necessary to update the aerodynamics on each
of the four acceleration evaluations of the 4th order Runge-Kutta algorithm. They are therefore only evaluated once each integration time step.
The computational load could be
reduced further if they are only updated when the independent variables (altitude, viscous
interaction parameter, and Mach Number) change by a significant amount from the previous
update. Also, although not done in this implementation, the aerodynamic coefficients should
be curve-fit if possible to avoid a table lookup and interpolation implementation.
in Figures 20 on page 97 and 21 on page- 98 that the aerodynamic
coefficients
It is noted
do not change
very much below 300K feet until the Mach Number decreases below 2, so perhaps, two
tables or curve-fits would suffice instead of the thirty tables currently used.
3.6
ESTIMATORS
The final state predicted by the predictor algorithm for a particular control history is a
function of the assumed environment and vehicle characteristics.
The accuracy of the pre-
dicted final state can be increased, and hence, the guidance margin increased, if in-flight
48
measurements are utilized to make the assumed models more accurately reflect the conditions actually experienced by the vehicle.
The accelerations modeled in the predictor are due to gravity and the aerodynamic forces. The gravity acceleration can be modeled to sufficient accuracy using standard gravity
models. However, the aerodynamic accelerations are subject to significant variations due to
uncertainties in the atmospheric density, atmospheric winds, vehicle aerodynamics, and
vehicle mass. These uncertainties can be compensated for in the predictor by applying a
multiplicative scale factor to the lift and drag accelerations modeled in the predictor that is
equal to the ratio of the actual accelerations experienced to the predicted accelerations at
any point in the trajectory.
The measured lift and drag accelerations are derived from the inertial measurement
system sensed acceleration assuming a zero sideslip angle as follows,
A
adrag
=
^
-a,
- VR
(61)
IVRI
a,,
a,
=
a,
-
ad,o
9
(62)
where the inertial acceleration, a,, is computed by back-differencing the accumulated sensed
velocity counts from the inertial measurement unit,
V;mu
a,
-
V;mu paW
(63)
=
In the predictor, the aerodynamic accelerations are,
adrg
-
C mSS
12
P V2
(64)
49
D
a,,t
aa
(65)
Data from the Shuttle program [10] shows that the primary dispersion affecting the
aerodynamic acceleration is in the atmospheric density. Further, over large altitude ranges,
this dispersion can be modeled to an accuracy sufficient for the prediction process as a constant multiplicative bias. Therefore, for implementational purposes, the dispersion in the
aerodynamic accelerations due to the atmospheric uncertainties will be lumped into a density scale factor as follows,
P
K
P
=
(66)
Pstd
where,
A
A
p=
2 adrg
k~2a/
,,o,
m
,
(67)
and the values for the nominal vehicle characteristics and the nominal atmospheric density
are determined using the predictor models for the vehicle state at the time of the measurement. Because the nominal ballistic coefficient is assumed in deriving the measured density, and the measured acceleration is due to the actual ballistic coefficient, uncertainties in
the ballistic coefficient will be reflected in the measured density. The equation for the drag
acceleration in the predictor is then,
adg
=
(co
)
2 VRKp
(68)
Pptd
or substituting for Kp from Eq. (66) yields,
ad..
(cmS)
1
(69)
V2 A
Substituting for p from Eq. (67) then yields,
50
A
ado
=
(70)
ad..g
so the modeled drag is corrected for the dispersed drag coefficient, density, relative velocity,
and vehicle mass.
In general, the measured drag acceleration is a noisy signal and will exhibit short term
variations due to short lived local atmospheric dispersions [10].
Filtering of the density
scale factor is therefore necessary and is implemented using a first-order filter,
A
Kp
P
=
(1-K)
Kp + K, P
Pstd
(71)
which has a time constant , r,, of,
=At
In(1 -K)
P
(
72)
where At is the sample rate of the measured drag acceleration, and K, is the filter gain. A
similar lift-to-drag ratio scale factor is derived and applied to the lift acceleration,
a,,ft = KL
D
()
D
nom
(7'3)
adrag
dra
where,
KL
(7 4)
=
51
and,
A
A
a,
(75)
adrag
Again, filtering is necessary,
A
KL
=
(1-K
2)
KL +
D
yielding a time constant,
ZL
'
=
K2
(76)
((
)D
nom
L, of,
At
A
--__
In(1 -
(77)
K 2)
A time constant of 25 seconds was selected for both the density and UD filters. This value
filtered out the high frequency density shear components seen in the Shuttle profiles while
still providing adequate response to long term disturbances.
3.7
HEAT RATE CONTROL
The primary trajectory constraint on entry vehicles is the maximum heat rate the vehicle
can withstand. In general, the thermal protection system material is selected to withstand
52
the maximum local heat rate on any particular portion of the vehicle, and the material thickness is selected to withstand the total integrated heat load ocer the trajectory.
Accurate
pre-flight predictions of the expected heat rate during entry can significantly reduce the thermal protection system weight yielding significant performance increases for an entire mission.
Inspecting the reference trajectories in Figure 16 on page 93 shows that sharp peaks in
the heat rate occur. If these peaks are accurately controlled, and this control can be accomplished using only short term departures from the predictor assumed control history, no significant departure will occur from the desired trajectory.
Heat rate control can be accomplished using either angle of attack, bank angle, or a
combination of both. Of these, bank angle alone is preferred because a constant angle of
attack trajectory is assumed and because angle of attack changes the vehicle drag coefficient resulting in a rapid change in energy rate and a rapid departure from the desired trajectory.
Also, most entry vehicles restrict the angle of attack range during maximum heat
rate regions to reduce the area on the vehicle that must be protected from the high heat
rate. Although the ERV does not need to restrict the angle of attack range, and hence, the
guidance does not provide for such a capabilty, the restriction can be handled by replacing
the constant angle of attack control history by a reference angle of attack control history
about which a constant angle of attack bias is applied for control.
Heat rate control is accomplished by computing the incremental bank angle required to
fly along the specified heat rate boundary (assumed to be a constant heat rate for any flight
regime) and then modulating bank angle according to the guidance value or the guidance
value plus the incremental lift for heat rate control, whichever requires more lift up. Hence,
no effort is made to pull the vehicle down into the atmosphere to follow the heat rate bound--
53
ary; instead, lift up is applied if the vehicle is flying "too low". The incremental lift for heat
rate control is computed to provide a second-order control response as follows,
Kb
cos(AO) =
q
K.
.
(-
d
q
(-
m)
(78)
To fly along a constant heat rate boundary,
Qi,
=
constant
(79)
and the desired rate of change of heat rate,
is,
0
=
(de
Qd..,
(80)
so,
K,5
+
K6
_ Q +
cos(AO) =
q
q
Q - C,,(81)
The stagnation heat rate is determined using the Engineering Correlation Formula [18] for a
one foot radius reference sphere as,
17700 -'p
:
(Q
(0
(82)
0)
The time rate of change of heat rate, Q, is determined by back-differencing the heat rate
between guidance cycles,
I
=Q
(83)
Qpast.
At
The equations of moion assuming small flight path angle yield,
h
=
C
'
SCOS()
-
g
(84)
54
Considering only the perturbations due to the incremental lift, cos(AO), from Eq. (81) yields,
m
+
K-.
K
(
Qi,.) =
-
0
(85)
Proper selection of the gains K and K is accomplished by linearizing Eq. (85) in altitude
and assuming that the time rate of change of VR is small compared to the change in -.
With these assumptions,
= 17700 (V)35
10000
d ph
(86)
dh
and,
=
-j
17700
(i
R)305
d'I
dh
10000
dh
dt
(87)
Therefore, the homogeneous second-order differential equation in altitude is,
h
+
K K
h
+
K K h
=
0
(88)
where,
m
17700
10000
dh
(89)
The natural frequency and damping ratio of the second-order differential equation are,
wn =
A/K K6
(90)
K K-
(91)
2 co,
or alternatively, for a desired natural frequency and damping ratio, K6 and Kb are selected
as,
55
=
K
(92)
n
=
(K
=K
(93)
The derivative in Eq. (89) can be evaluated assuming an exponential atmosphere of the form,
= p, e- (hlhs)
(94)
yielding,
d 4p7
e- (h2 hS)
dh
(95)
2 h,
This logic is contained in the guidance algorithm in the Heat Rate Control subroutine.
The
incremental lift required for heat rate control is provided to the Attitude Command subroutine which adds it into the guidance command if it requires more lift up than the guidance
command. This occurs when the incremental lift given by Eq. (81) is greater than zero,
cos(Aq)
>
0
(96)
Appropriate values of the natural frequency and damping ratio were determined parametrically as,
o,
z;=
=
0.10 sad
(97)
sec
(98)
1.00
56
4.0
PERFORMANCE
4.1
SIMULATOR
Open-loop and closed-loop entry trajectories were simulated for the Entry Research
Vehicle (ERV) using a derivative of the 6-DOF Aeroassist Flight Experiment Simulator (AFESIM) [19] developed at The Charles Stark Draper Laboratory which is coded in the HAL computer language.
For this study, the aerodynamic model described in "Appendix A. ERV
AERODYNAMICS
MODEL" on page 133 and the wind model shown in Figure 2 on page 79
were incorporated into the AFESIM. The characteristics of the ERV [14] are listed in Table 1
on page 73. The entry conditions with which all trajectories were initialized are also listed in
Table 1 on page 73. Because only the performance characteristics of the guidance were
being evaluated, the simulator was operated in the 3-DOF mode.
4.2
OPEN-LOOP FOOTPRINT
Open-loop trajectories were run using the constant angle of attack and linear bank with
velocity profiles to determine the portion of the footprint achievable.
All trajectories were
terminated at the TAEM interface altitude of 80K feet, so the footprint can be increased about
100 nautical miles in all directions due to the range flown below 80K feet.
57
Figure
22 on page 99 shows the lift-to-drag
angle of attack,
.
ratio, L/D, for the ERV at Mach 10 versus
It is seen that maximum L/D is obtained at an angle of attack of 15
degrees. It is desirable to fly on the back side of the L/D curve (angle of attack greater than
15 degrees) so as to maximize the drag coefficient for a given UD. This reduces heating by
causing a quicker loss bf velocity early in entry than flying at the same UD on the front side
of the L/D curve. The L/D versus angle of attack curve shows the same shape with the maximum L/D at 15 degrees for all flight regimes with only a variation in the magnitude of L/D
across the angle of attack range. Therefore, angle of attack is modulated between 15 and 50
degrees for the footprint with 15 degrees corresponding to maximum UD and 50 degrees
corresponding to minimum L/D.
The open-loop footprint is shown in Figure 23 on page 100. Also shown for comparison
is the reported footprint for a heat load limit of 175K BTU/sq ft (shown earlier in Figure 12 on
page 89). That footprint included the range flown below 80K feet, hence the slight differences. It is seen that almost the entire reportedly achievable footprint is captured with the
assumed control profile. Most importantly, all of the maximum crossrange region is reached
when the range flown below 80K feet is included.
Also, most of the minimum downrange
region of the footprint was captured even though the control profiles used do not correspond
to the optimal profiles determined using POST and shown in Figures 14 on page 91 and 15
on page 92. Additionally, the footprint reported in Reference [14] does not include the large
area in the minimum downrange region that the open-loop trajectories reached. The small
area not reached in the minimum downrange region by the control profiles is relatively
unimportant because the downrange ranging capability of the vehicle can be adjusted by
changing the deorbit time. A vehicle in low earth orbit travels at about four nautical miles
per second, so downrange is easily adjusted while on-orbit.
58
Because the predictor-corrector guidance algorithm will follow the same control histories as used to generate the open-loop footprint for a nominal trajectory, the guidance algorithm can reach all of the open-loop footprint for nominal conditions.
It is seen that the
achievable footprint is bounded by the heat rate and heat load limits imposed on the ERV.
At least an additional 2000 nautical miles of ranging capability in the downrange direction
exists if the heat limits are relaxed.
4.3
EFFECT OF DISPERSIONS ON FOOTPRINT
The effect of dispersions on the achievable footprint was determined by repeating the
open-loop trajectories with the dispersions discussed in Subsection "2.2 Dispersions" on
page 26. The worst-case
(3a) dispersions
are summarized
in Table 2 on page 73. Table 3
on page 74 shows the dispersions in downrange and crossrange for three of the control histories in the maximum downrange region of the footprint.
It is seen that only variations in
the lift and drag coefficients cause significant dispersions in the final state. Also, it is seen
that the effect of a + 10% CLdispersion is the same as that of a -10% CO. dispersion. This is
expected because both dispersions cause the same increase in the vehicle L/D. The same
occurs for a -10% CL dispersion and a +10% C dispersion, both of which decrease the
vehicle UD.
The effects of the dispersions on trajectories to the maximum crossrange region of the
footprint are seen in Table 4 on page 75. Again, it is seen that aerodynamic dispersions
have the greatest effect. A dispersion that increases UD increases the range, while a dispersion that decreases UD decreases the range.
59
Table 5 on page 76 shows the effects of the dispersions on the minimum downrange
region of the footprint. The worst-case range dispersions again occur for the aerodynamic
dispersions.
4.4
ESTIMATOR PERFORMANCE
Figure 24 on page 101 shows the time response of the density filter with a 25 second
time constant for the STS-9 atmosphere. This trajectory also has dispersions of +1.9% in
C, -3.2% in mass, and a 63.8% crosswind. Therefore, the filter output does not follow the
actual density dispersion also shown in the figure. When the acceleration level is below 0.07
g's, the measurements are not incorporated, so the filter is inactive before 300 seconds and
from 600 to 850 seconds. As the velocity drops, the wind becomes a greater contributor to
the measured density error, hence the divergence in the measured density ratio starting at
1000 seconds. Figure 25 on page 102 shows the response of the UD filter with a 25 second
time constant for a -1.9% CL and a +1.9% CO dispersion. Again, the winds affect the measurement by creating errors in the navigated angle of attack, so the estimated UD ratio is
slightly in error.
The use of an air data system like the Shuttle Entry Air Data System (SEADS) could significantly improve the estimation process by providing accurate estimates of the angle of
attack, atmospheric density, and wind magnitude and direction.
More accurate estimates
will increase the guidance margin, thereby increasing the achievable footprint for dispersed
trajectories.
60
4.5
CLOSED-LOOP PERFORMANCE
Based on the results of the open-loop trajectories with dispersions, worst-case dispersions were selected for each of three regions of the footprint:
mum
crossrange,
and
minimum
downrange.
maximum downrange, maxi-
Closed-loop
trajectories
with
the
predictor-corrector guidance algorithm were then run to the three regions of the footprint.
The three target points selected for the closed-loop performance evaluation are shown in
Figure 23 on page 100. The 3a errors defined in Table 2 on page 73 were scaled such that
the total error due to multiple error sources would still represent a 3 dispersion so as to
test the guidance system for reasonably probable dispersion cases [20].
To run all disper-
sions at their 3a levels would be unrealistic.
The nominal and dispersed results for trajectories to each of the three regions are listed
in Tables 6 on page 77 through 8 on page 77 . Plots of selected parameters from these
cases are included.
Figures 26 on page 103 through 31 on page 108 present the altitude,
velocity, heat rate, heat load, downrange, and crossrange time histories for the nominal
maximum downrange trajectory.
Figures 32 on page 109 through 37 on page 114 present the
altitude, velocity, heat rate, heat load, downrange, and crossrange time histories for the
nominal maximum crossrange trajectory.
Figures 38 on page 115 through 43 on page 120
present the altitude, velocity, heat rate, heat load, downrange, and crossrange time histories
for the nominal minimum downrange trajectory.
In each of these cases, it is seen that the
heat rate does not approach the heat rate limit, so no incremental bank angle is needed for
heat rate control.
The control histories for the nominal maximum downrange trajectory and the dispersed
case listed second in Table 6 on page 77 are presented in Figure 44 on page 121 and
61
Figure 45 on page 122 . Figure 44 shows the angle of attack histories for the nominal and
dispersed maximum downrange cases.
It is seen that a two degree change in angle of
attack is required early in the trajectory increasing to four degrees by the end of the trajectory.
Figure 45 shows the bank angle histories for the nominal and dispersed maximum
downrange cases.
The bank angle required shows no change from zero degrees for this
case.
The control histories for the nominal maximum crossrange trajectory and the dispersed
case listed second in Table 7 on page 77 are presented in Figure 46 on page 123 and
Figure 47 on page 124 . Again, it is seen that a four degree change in angle of attack is
required for the dispersed case. The bank angle history shows no change for the dispersed
case from that of the nominal case.
The control histories for the nominal minimum downrange trajectory and the dispersed
case listed second in Table 8 on page 77 are presented in Figure 48 on page 125 and
Figure 49 on page 126 .
For this case, approximately a one degree change in angle of
attack is required. No change is required in the bank angle profile.
The required change in angle of attack for each of the dispersed cases shown was primarily due to the change in the vehicle UD as this was shown to be the primary dispersion
source in Subsection "4.3 Effect of Dispersions on Footprint" on page 59. The breaking point
of the guidance occurs when the vehicle does not have enough UD range to overcome the
loss in UD due to aerodynamic dispersions. As mentioned previously, the Shuttle entry guidance algorithm was required to guide to the TAEM interface aim point to within 2.5 nautical
miles of position. The results presented for the nominal and dispersed cases show that this
requirement is met with the predictor-corrector
guidance algorithm.
Also, the trajectory
plots show that the algorithm achieves this performance with very infrequent guidance
62
updates (.02 hz.) and with very small control variable changes from the nominal constant
angle of attack and linear bank with velocity profiles.
Most i nportantly, almost all of the
achievable footprint is captured using the predictor-corrector algorithm.
4.6
HEAT RATE CONTROL PERFORMANCE
The closed-loop trajectories shown previously did not require heat rate control because
the maximum heat rate experienced was significantly lower than the limit imposed on the
ERV. The time responses for bank angle, angle of attack, and heat rate for the beginning of
a typical trajectory with and without heat rate control are shown in Figures 50 on page 127
through 52 on page 129. The resulting bank angle versus velocity profile is shown in
Figure 53 on page 130. These trajectories are for the middle of the footprint where the peak
heat rate does not exceed the limit for the ERV. Therefore, for illustrative purposes, the heat
rate limit was reduced to 100 BTU/sq ft/sec.
Comparing the trajectories with and without
heat rate control, it is seen that the heat rate control takes place over a fairly long time
range, but requires a significant departure from the linear bank profile over only a very short
velocity range. The impact on the trajectory is therefore small, and the predictor-corrector
stays converged on almost the same control history even though the vehicle does not follow
the assumed control profile during the heat rate control area.
63
4.7
OVERCONTROL
For those trajectories not at the edge of the footprint, excess vehicle capability exists
that can be utilized to increase guidance margin for dispersions that may occur later in the
trajectory.
For example, a 13,800 nautical mile downrange trajectory for the ERV only
requires flying at 20 degrees angle of attack instead of 15 degrees for the nominal trajectory.
The ERV can modulate angle of attack between 15 degrees (maximum UD) and 50 degrees
(minimum UD) on the back side of the UD curve, so the modulation capability is not equally
centered about the commanded angle of attack if flying at 20 degrees.
By flying at 15
degrees (maximum UD) early in the trajectory, guidance can center the remaining guidance
capability equally about the aim point to cover dispersions in all directions, not just those
that require less UD to reach the target point. This approach is referred to here as overcontrol or command biasing.
Overcontrol can be implemented in several ways. First, the command can be biased
from the desired command when that command is not in the center of the modulation range.
As the vehicle flies a biased angle of attack, for example, the predicted final state will differ
from that for the unbiased command in such a direction that the next guidance command will
be moved in the direction opposite to the bias. By biasing in the proper direction, the command can be driven toward the center of the modulation range. If the guidance requires an
L/D higher than that in the middle of the UD range, flying at an even higher UD will drive
the required UD toward the middle. Secondly, the target aim point can be moved from the
nominal aim point early in the entry. For example, for a trajectory to the maximum downrange region of the footprint, the target aim point can be moved even farther downrange. Of
course, at some point in the trajectory, the aim point must be moved back to the desired
point.
64
The first approach was implemented in the predictor-corrector algorithm by biasing the
angle of attack by five degrees when it was more than two degrees away from 30 degrees.
The biasing was terminated at an inertial velocity of 13,500 feet per second so as to allow
the guidance to. fly the proper control history near the end of the trajectory to reach the target aim point.
Figure 54 on page 131 compares the angle of attack control history for a 13,760 nautical
mile downrange trajectory with the dispersions used for the closed-loop trajectories shown
earlier. Without overcontrol, the vehicle misses the target aim point by 19.20 nautical miles.
This occurs because the wind contribution to the dispersion increases as the vehicle velocity
drops, so the multiplicative scale factor on density does not properly model this dispersion.
As the wind contribution increases, a higher UD is required, and the angle of attack is driven to 15 degrees or maximum L/D. Because maximum L/D was not utilized earlier in the
trajectory, the vehicle did not reach the target. Late in the trajectory, the predictor-corrector
goes unconverged as control authority is exhausted, causing the angle of attack to jump
between 15 and 30 degrees.
By this point, the target aim point was unreachable anyway
due to the dispersions.
With command biasing, the commanded angle of attack early in the trajectory is that
corresponding to maximum UD or 15 degrees. It is seen that biasing drives the commanded
angle of attack to 25 degrees once the biasing is terminated at a velocity of 13,500 feet per
second or a time of 3,700 seconds.
Later, when the wind dispersion drives the angle of
attack toward 15 degrees, there is significant margin remaining, and the angle of attack is
only driven to 24.5 degrees by the dispersion. With command biasing, the miss distance at
TAEM interface is only 0.27 nautical miles.
using overcontrol.
Therefore, guidance margin is increased by
More of the theoretically achievable footprint is attainable for dispersed
65
cases, An even larger magnitude dispersion could have been handled late in the trajectory
since the angle of attack was not driven to that for maximum L/D.
Further work is needed in this area to determine the proper way to utilize overcontrol to
maximize guidance maFgin for the expected dispersions. The probability of the various dispersions occurring and the histories of those dispersions along a trajectory must be considered.
For example, if a thick" atmosphere is encountered early in the trajectory equal to
the worst-case expected dispersion, it is highly unlikely that the atmosphere will get "thicker" later in the trajectory.
Therefore, it is unnecessary to preserve guidance margin in the
direction needed to cover a "thicker" atmosphere beyond that already required for the
expected worst-case atmosphere.
Such considerations should be taken into account in the
design of the overcontrol algorithm.
4.8
ALGORITHM EXECUTION TIME
An estimate of the execution time required for the predictor-corrector algorithm was
made using the execution time estimate feature of the HAL compiler. The estimate is for the
AP101 Shuttle flight computer. Figure 55 on page 132 shows the execution time required in
seconds as a function of the time to the TAEM interface point for a maximum downrange trajectory.
It is seen that early in the entry when the trajectory to be predicted is long, the pre-
dictor requires 43.7 seconds of CPU time.
When only 500 seconds to the TAEM interface
point remains, the required time drops to 4.5 seconds. This figure can also be interpreted as
the minimum update interval for the predictor-corrector.
Also, the guidance command will
be computed and sent to the vehicle autopilot a period of time after the start of the guidance
66
cycle equal to the required execution time.
It is seen that early in the entry, a significant
delay occurs between the start of the guidance cycle and the computation of the guidance
command. This delay was not simulated in the closed-loop trajectories, but will have a minimal effect on the guidance margin because the guidance is not trying to fly a reference trajectory
like the analytic guidance algorithms.
The predictor-corrector
is numerically
computing a trajectory that will fly directly to the target aim point. Any error that builds up
between the start of the guidance cycle and the issuing of the guidance command can be
nulled easily since the entry is long, and the error will shrink as the delay decreases with
decreasing time to the TAEM interface point.
The Shuttle AP101 CPU is the product of early 1970's technology and is significantly
slower than flight computers that might be employed in future entry vehicles.
The 80C86
CPU for example is two to five times faster than the AP101 CPU, so the execution time
required shown in Figure 55 on page 132 can be scaled down by a factor of two to five.
Computers utilizing parallel processing architecture could predict the three required trajectories simultaneously in three CPUs, cutting the required execution time by a factor of three. If
scaled by a factor of four due to the faster CPU and a factor of three due to parallel processing architecture, the maximum time required drops to 3.6 seconds, and the time with 500
seconds remaining to the TAEM interface point drops to 0.4 seconds. The predictor-corrector is therefore a viable guidance scheme for future entry vehicles.
67
68
5.0
FUTURE RESEARCH TOPICS AND CONCLUSIONS
5.1
FUTURE RESEARCH TOPICS
Several topics for further algorithm development and optimization are discussed. These
are:
1.
Further reductions in CPU execution time
2.
Use of an air data system for in-flight measurements
3.
Use of overcontrol to increase guidance margin
4.
Control of more than two state constraints
Optimization of the predictor algorithm and the integration scheme can yield significant
reductions in execution time beyond that already attained.
Simplifying the aerodynamic
model can yield a great reduction in execution time and an equally important reduction in
the computer core required. The current model has 30 tables, each with 51 breakpoints over
the angle of attack range.
A curve fit of the aerodynamic coefficients over the angle of
attack range and the flow regimes would reduce the core required to store the model data
and the computations required for each lookup.
The estimator algorithm was shown to be effective in determining the dispersions from
in-flight measurements.
However, the estimator is unable to differentiate between density
dispersions and atmospheric winds. Figure 24 on page 101 showed that the multiplicative
density scale factor did not accurately model the wind contribution to the drag acceleration
because the relative contribution of the wind to the dispersion increases as the vehicle
69
velocity drops late in the trajectory. An air data system could provide an independent measurement of the atmospheric winds, improving the estimation process and increasing guidance margin by increasing the accuracy of the predicted trajectories.
The concept of overcontrol was introduced and shown to be effective for at least one
dispersed case. Further investigations should be made to determine how much overcontrol
is optimal for the expected dispersions.
It may be possible to use the sensitivities of the
constraints to the control variables to determine a proper amount of command biasing for
any particular dispersion at any point in the trajectory.
Only the downrange error and crossrange error at TAEM interface are controlled in the
current design. The vehicle energy is not controlled which can allow significant dispersions
in the ranging capability during the TAEM phase of entry. Approaches include redefining the
TAEM aim point in terms of a desired energy level or utilizing a third control variable to provide control over an energy level constraint. The Space Shuttle makes use of a split rudder
as a speedbrake to provide a large energy control capability.
Such an approach could be
utilized with the predictor-corrector by computing the sensitivity of the three constraints to
the three control variables. This would require four predictions instead of the three currently
needed, but the fourth prediction could be made only during the latter part of entry to clean
up any dispersions in energy level that occur during the entry due to dispersions. The CPU
execution time would then increase by one-third over that currently projected when the third
constraint is controlled.
70
5.2
CONCLUSIONS
A predictor-corrector
entry guidance algorithm has been demonstrated that exhibits
excellent performance and almost complete coverage of the achievable footprint. This algorithm employs a simple control variable history to achieve near-optimal guidance for the
maximum downrange and maximum crossrange trajectories.
Explicit heat rate control is
employed without significantly impacting the achievable footprint. This is achieved because
unlike previous guidance algorithms that included a long heat rate control phase with no
active targeting, the proposed algorithm always actively targets to the aim point and only
controls heat rate in the short high heat rate regions as required.
The algorithm has been demonstrated to handle atmospheric and aerodynamic dispersions within the capability of the vehicle. The required computer execution time is shown to
be within the capability of new flight computers.
Algorithm adaptability is provided through the utilization of in-flight measurements to
improve the accuracy of the predicted trajectory.
because
there
are no reference trajectories
vehicle/mission-specific
input
parameters
Algorithm maintenance is simplified
used, and there
(I-loads).
are
Transportability
a minimum
of the
of
algorithm
between different entry vehicles is provided by eliminating vehicle-specific entry phases
other than the heat rate control phase which only requires the input of a heat rate limit. The
guidance algorithm does require a vehicle aerodynamic model, but this is developed in the
normal vehicle definition phase anyway.
In summary, an entry guidance algorithm has been developed that achieves near-optimal performance while maximizing flexibility, adaptability, and transportability.
71
Although
more computationally intensive than analytic algorithms, execution of the predictor-corrector
is within the capability of current flight computers.
It is hoped that this guidance approach
will significantly reduce the development and maintenance costs for new entry guidance systems.
72
Table 1. Characteristics of the ERV and Trajectory Entry Conditions
Mass
Reference area
Mean Aerodynamic Chord
Altitude
Inertial Velocity
Flight Path Angle
Inclination
Latitude
Longitude
Vacuum Apogee
Vacuum Perigee
186.0
177.6
25.0
slugs
sq ft
ft
400,000.0
ft
25,778.843 ft/sec
-0.996 deg
28.50 deg
-28.071 deg
-69.313 deg
150.0
20.0
n.m.
n.m.
Table 2. Dispersions Used in Performance Study
Dispersion
Aerodynamics
Lift Coefficient
Drag Coefficient
Magnitude
Direction
Symbol
(%)
(deg)
CL+
CL-
+ 10%
CD+
CD-
+10%
-10%
-10%
Vehicle Properties
Mass
Atmospheric Properties
Density
Tailwind
Positive Crosswind
Headwind
Negative Crosswind
+5%
-5%
p+
P
+ 30%
TW
99%
99%
99%
99%
-30%
CW+
HW
CW-
73
61.5 deg
151.5 deg
241.5 deg
331.5 deg
Table 3. Dispersed Cases for Maximum Downrange Region
Dispersion
ad
(deg
15s
15
15
15
15
15
15
15
15
15
15
15
15
20
20
20
20
(deg
CL-
13229
0
0
0
0
CD+
13242
16810
14890
14739
14615
15089
14882
14829
0
0
0
0
CD-
M+
MP+
P
TW
CW+
0
HW
0
CW-
0
0
NOMINAL
CL+
CL-
0
0
CW-
25
25
0
0
0
0
NOMINAL
CL+
25
25
25
0
0
0
0
0
0
0
0
13849
15388
12355
12384
15711
13909
13785
13659
'14105
CDM+
0
0
14751
14802
CD+
P+
25
25
497
456
420
422
427
498
496
494
498
498
497
496
497
14817
16445
0
0
0
0
0
0
0
0
25
(n.m.)
CL+
20
20
20
20
20
20
20
20
20
25
25
25
25
25
Crossrange
(n.m.)
0
0
0
NOMINAL
Downrange
MP
TW
13901
CW+
13857
13796
13838
HW
12116
13415
10822
10858
13715
CLCD+
CDM+
463
499
344
347
491
466
460
453
475
466
464
461
463
321
439
178
183
TW
12068
11950
12339
12155
456
325
316
305
342
324
CW+
12121
321
HW
12076
12108
317
320
13161
MP+
P
CW-
74
Table 4. Dispersed Cases for Maximum Crossrange Region
7d
Dispersion
15
15
15
15
15
15
15
15
15
15
15
15
15
60
60
60
60
60
60
60
60
60
60
60
60
60
NOMINAL
CL+
20
20
20
20
20
60
60
NOMINAL
7791
1661
60
CL+
CL-
8476
7125
1966
1370
60
CD+
60
60
60
CD-
7132
8622
M+
7820
P+
7689
7934
7766
7846
7775
7580
2007
1664
1659
1662
1660
1607
1686
1673
1572
6940
7539
6355
1344
1602
1100
6363
7661
1119
1635
CW+
6963
6915
6848
7070
6933
6958
HW
6921
1345
1342
1344
1342
1317
1355
1347
1290
(deg
(deg
20
20
20
20
20
20
20
20
(deg
60
60
60
60
60
60
CLCD+
CD-
8308
9035
7602
7605
9200
1822
2149
P+
P
8197
8465
8260
8473
8313
7984
MTW
CW+
HW
CW-
M-
7761
P-
TW
CW+
HW
CWNOMINAL
CL+
25
60
p+
25
60
25
25
25
25
60
60
60
60
P
TW
25
(n.m.)
8342
8271
60
60
60
60
60
60
25
Crossrange
(n.m.)
M+
60
25
25
25
25
25
Downrange
CLCD+
CD-
M+
M-
CW-
6819
75
1506
1531
2193
1825
1819
1823
1820
1737
1890
1866
1692
1392
Table 5. Dispersed Cases for Minimum
Id
(deg
30
30
30
30
(deg
3O
90
90
90
30
30
30
30
30
30
30
90
90
90
90
90
90
90
30
30
30
30
30
30
30
(n.m.)
CL+
CLCD+
CD-
M+
MP+
P
TW
CW+
HW
CW-
80
NOMINAL
3851
80
80
80
80
80
80
CL+
CL-
4011
3680
3668
4060
3865
3865
3778
3954
3856
3841
3834
3828
CD+
CDM+
M-
80
80
80
80
80
80
P+
P
30
30
70
70
NOMINAL
30
70
70
70
70
70
70
70
70
CLCD+
30
30
30
30
30
30
30
30
30
2982
3014
2934
2914
3045
2996
2969
2914
3078
2993
2992
2996
2982
NOMINAL
30
30
30
30
30
30
30
30
Downrange
Dispersion
(deg
90
90
90
70
70
70
Downrange Region
TW
CW+
HW
CW-
4932
5262
4602
4600
5337
4949
4915
4854
5043
4934
4925
4914
4884
CL+
CD-
M+
Mp+
pP
TW
CW+
HW
CW-
76
Crossrange
(n.m.)
938
1102
778
793
1119
937
938
943
930
914
941
939
907
1041
1233
858
875
1254
1041
1041
1046
1034
1020
1044
1042
1007
1075
1279
881
898
1304
1075
1075
1078
1069
1057
1077
1074
1040
Table 6. Maximum Downrange Region Closed-Loop Results
Dispersions
CL
(%)
-
1.91.9
1.9
1.9
CD
(%)
+
+
+
+
Mass
p
(%)
(%)
TARGET POINT
1.9
1.9
1.9
1.9
NOMINAL
- 3.2 +19.1
- 3.2 STS 1
- 3.2 STS 9
- 3.2 STS11
Final State
Wind
(%)
63.8
63.8
63.8
63.8
HW
HW
HW
HW
9
(deg)
+ 9.800
A
(deg)
+ 144.700
Error
(n.m.)
-
+
+
+
+
+
+144.702
+144.701
+144.694
+144.696
+144.698
0.13
0.08
0.30
0.27
0.13
9.799
9.799
9.803
9.802
9.801
Table 7. Maximum Crossrange Region Closed-Loop Results
Dispersions
Final State
CL
CD
Mass
p
Wind
(%)
(%)
(%)
(%)
(%)
TARGET POINT
-
1.9
1.9
1.9
1.9
+
+
+
+
1.9
1.9
1.9
1.9
NOMINAL
-3.2
+19.1
- 3.2 STS 1
- 3.2 STS 9
- 3.2 STS11
63.8
63.8
63.8
63.8
CWCW
CWCW-
kg
A
Error
(deg)
(deg)
(n.m.,
- 2.300
+ 55.000
-
+
+
+
+
+
2.301
2.295
2.304
2.301
2.299
55.000
54.998
55.001
54.990
55.000
0.06
0.32
0.25
0.60
0.06
Table 8. Minimum Downrange Region Closed-Loop Results
Dispersions
CL
CD
(%)
(%)
Mass
(%)
Final State
p
Wind
0g
A
Error
(%)
(%)
(deg)
(deg)
(n.m.)
-14.900
+ 9.800
-
-14.897
+ 9.798
0.22
-14.902
+
9.801
0.13
-14.901
-14.900
-14.899
+
+
+
9.800
9.800
9.799
0.06
0.00
0.08
TARGET POINT
NOMINAL
+ 1.9
- 1.9
+ 1.9 - 1.9
+ 1.9 - 1.9
+ 1.9 - 1.9
+ 3.2
-19.1
+ 3.2 STS 1
+ 3.2 STS 9
+ 3.2 STS11
63.8
TW
63.8 TW
63.8 TW
63.8 TW
77
Figure 1. Three-View Drawing of the ERV
78
_ ___.__
SCALARWINDSPEEDV (m/sec) STEADY-STATEENVELOPESAS FUNCTIONS
OF ALTITUDEH (km) FOR TWOPROBABILITIESP (%) ENCOMPASSING
ALL FOUR LOCATIONS
9
P · 99
H
V
H
V
H
V
H
V
i
3
22
31
6
54
10
11
12
13
75
76
78
74
17
20
23
50
60
75
80
44
29
29
150
150
120
120
I
3
5
6
7
9
11
12
13
14
28
38
56
60
68
88
88
92
88
88
15
20
23
50
60
75
80
70
41
41
170
170
135
135
PERCENTILE
70
I
,U
a
.4
1
WIND SPEED(m/l
Reproducedfrom Reference(8)
Figure 2. Atmospheric Wind Profile
79
I
Figure 3. Envelope of Density Profiles Derived from Shuttle Flights
80
Figure 4. STS-1 Density Profile Comparison
81
h , kft
320
1
_
300
.
260 _i
-_
i
::
.:
,
...
:::.·:'
. :~::
_ .-~ . ~40
230__
220
__ _!':__
200
2,tO
!:
l
::~:~$~:·
ii
_
1
·.
:
E
0
'?¢!
~::~u;~:
ir
i i~
.60
.65
.70 .75 .80
.85
_
~:ig::
:~
g
_
_
T
3
i!:IIi'
:r.:
W
190____
.55
i i_
't!.:i
- 280< _
150 -
~~-
.
_I~
e·'
_
____
.90
.95 1.00 1.05 1.10 1.15 1.20 1.25 1.JO01.J5
PCPPw
Reproduced from Reference (10)
-
-- -
Figure 5. STS-9 Density Profile Comparison
82
Int
'
a
.--
METHOD1
---
METHOD2
----
POINT WHERE,
· 900
r,
80 .1
~I
2-PHASE
1.
I~~~~~~~
I
20
n
U...i
I
-
I
nO 0.8os06
4
I
U ..
0.2
,
0
rrm-
lUN
2 80li
4-PHASE
b.
:
I
C.,
1.0 6.8
0.6
0.4
0.2
0
NONDIMENSIONALVELOCITY. V
Reproducedfrom Reference (12)
Figure 6. Multiphase Bank Angle Program for UD = 1.5
83
PLaurdiri··ipitinrryliu*irr
--..-
· ··I·
x MULTIPHASE ROLLPROGRAM(JACKSON'S EQUATIONS)
o TOV
.
)
o TOh
80,000 FT
A TO *
900
*
MULTIPHASE ROLLPROGRAMS SOLVEDON
INTEGRATEDTRAJECTORYPROGRAM (METHOD1)
TOV · 0.2
a TOh
* TOh
80,00 FT
*
EULERIAN ROLLANGLEFROM SLYE'S
EQUATIONSSOLVEDON INTEGRATED
TRAJECTORYPROGRAM(METHOD2)
I
80.00 FT
STEEPESTDESCENTSOLUTION(METHOD3)
LI 7
we
_
UD
1.5
1400
E
Ca
0
1300
cm
o1200
U
1100
if ma
0
0
A
oA
A
1
2
:
-
a
A
I.I
.
.
3
4
m-l
ROLL
.I CONTINUOUS
51PROGRAMS
NUMBER OF ROLL STEPS, n
Reproduced from Reference(12)
__
Figure 7. Crossrange Versus Number of Bank Steps
84
90
80
M (METHOD1)
AM (METHOD2)
70
!lATIONAL
!
60
b 50
Z 40
.-
30
20
10
01
NONDIMENSIONAL VELOCITY, V
Reproduced from Reference (12)
_
__
Figure 8. Comparison of Optimum Bank Angle Programs
85
__
19.0
18.0
U
17.0
0
C
003
16.0
15.0
2000.
10000.
18000.
26000.
Velocity, ft/sec
Reproducedfrom Reference(13)
Figure 9. Optimum Shuttle Angle of Attack Profile for Maximum Downrange
86
0.
cm
0)
al
-25.0
a)
cu
co
)00.
rence (13)
Figure 10. Optimum Shuttle Bank Angle Profile for Maximum Crossrange
87
17.5
o,
a'
17.2
4'
16.9
0
1a
Bc
¢
.
ince (13)
Figure 11. Optimum Shuttle Angle of Attack Profile for Maximum Crossrange
88
- --
ft
2
i ft
Crossrange,
nmi
Shuttle
1
0
2
4
6
8
Downrange, nml
10
12
14x10
Reproduced from Reference (14)
Figure 12. Landing Footprint for the ERV
89
40
35
30
-
A
25
h, ft
20
15
10
5
1
2
3
4
5x103
Time, sec
Reproduced from Reference (14)
Figure 13. Altitude Histories for the Entry Missions of the ERV
90
4
__P
on'
WUU
-------
90
Bank
angle,
deg
80
m
70
_
I
_ I
60
-
I
30
20
10.
I
_
50
40
A
I \
/
w
,,
1\\
/
-
-
1x-
0
Maximum downrange
Maximum crossrange
Minimum downrange
-
,
1
.
.-
1
I I
'
I
\
I
3
Time, sec
2
_
I
-
I
v
4
-,
I
5;x103
Reproducedfrom Reference (14)
Figure 14. Bank Angle Histories for the Entry Missions of the ERV
91
Figure 15. Angle of Attack Histories for the Entry Missions of the ERV
92
f% IN
ZUV
180
--
Maximum downrange
160
---
Maximumcrossrange
---
Minimum downrange
140
120
BTU/ft2 -sec
100
80
60
40
20
0
1
2
3
4
5x10 3
Time, sec
-
Reproducedfrom Reference (14)
_ _
Figure 16. Heat Rate Histories for the Entry Missions of the ERV
93
- 3-
2uu
7
e1
180
160
140
Heat
load,
//
120
100
/I
BTU/ft 2 80
60
40
20
0
1
2
3
4
5x10 3
Time, sec
Reproducedfrom Reference(14)
Figure 17. Heat Load Histories for the Entry Missions of the ERV
94
I
Ad
Bank Angle, b
I
I
0.
Vf
VEI
Inertial Velocity, V,
Figure 18. Bank Angle Versus Velocity Profile
95
c---
it Plane
iF
perpen
tains R, & VR)
:rossrange Error
Implane
It
_
_
Figure 19. Definitions of Downrange and Crossrange Errors
96
.8
r-
.6
a = 30 °
.4
a = 15 °
.2
01~
C
-. 2. 0
2
4
M
82Cl~~
-
I
I
6
8
10
I
1--,
=0
I
t
I
I
I
I
I
.5 1.5 2.5 3.5 4.5 5.5 6.5 7.5x10 - 2
Vl
3
I
4
5
105
6x10 5
h, ft
Reproduced from Reference (14)
Figure 20. Predicted Lift Coefficient Profile for the ERV
97
___
8
6
4
a = 30 °
a = 15 °
L/D
2
0
-2
0
2
6
4
8
10
I
a""
IV
I
I
I
I
I
I
I,
x10- 2
.5.... 1.5 2.5 3.5 4.5 5.5 6.5 7.5
I
I
v
VO<>
3
4
I
5
I
6x10
5
h, ft
Reproducedfrom Reference (14)
Figure 21. Predicted L/D Profile for the ERV
98
MRCH
X
I
1
10
i
I I- I
I
l~j/~,*II
i
i i
!
i i___
.
1~
fIJ
o
LI
II
LL
c
L
i
1o
20
30
1I
In
40
50
Angle of Attack, deg
Figure 22. Predicted L/D versus Angle of Attack Profile for the ERV
99
2500
Closed-LoopCases
O -MaximumDownrange
o
Maximum Crossrange
O -Minimum Downrange
2000
Olmt 175K
E
c
/
1500
B
175K BTU/sq I1
01
a
U
1000
16000
Figure 23. ERV Open-Loop Footprint with the Control Profile
100
0
i000
2000
3000
T
Figure 24. Time Response of the Density Filter
101
SECS)
__
1
c
..ZU
l
I
I
Ii
1.25
ii
1
I_
l
KL/D
I
1.00-
=2:
1i
I
;
....
If
0.75
I
i
i| Ti4, I
---7
EL-,
;
/I
LD estimated'
I* '
I
/
tL
1
1
I
I
I 1
^
e
__
^
I
rl
I
II
1
Z000
lOc0
.-
I
! I 1 ! 11 1
.__
/
I
JI I
I1
I-Sn U
I
/ KLIDactual
I
=
1_
TUUS
T [SEC]
__
Figure 25. Time Response of the L/D Filter
102
Figure 26. Closed-Loop Altitude History for the Maximum Downrange Case
103
VI
4 VR
...
3Nr000
c
-
.,,,_..
--
-
- --
w-
--
I
l
.,-...
- -u
-
w-
L
i
7
1· --
.
ZL
l
L
T
--
-
1-1h
L
I
\1
1
1 --d
200C30
"E
-,
O
VI
"
U
\
-L
I I
100C,oi
L
V
r
iii
1
9
.
1000
-
Lv
>
0
l
L
I1
0
w
I
VR
I-
IT
L!
I
--
..L !
l
2000
3000
~000.. ,.
I
i
5000
T (SECS]
Figure 27. Closed-Loop Velocity History for the Maximum Downrange Case
104
T_111!
I~~~~~~~~~~~~~~~~~~~~~~~~
L
T :ZZ::Z
Co ,_:ZZ
I i
I
_
a)
12
C,
03
I- I
_ _I
ill I
ca
I I
i
11
i
I
i
! 1
I
i1
i
IIi
I 1 I i I ''i I
I
1i i I
i
i I
i
I i
I
Ii
i
I
I
l_ll1. I
I i I I
11
ii
.,I
I
Ii I
1000
,
I
I
i
.
I ..
111
2000
Il
l
_11\I]
1
I
1jI
I
o~l
i
i
1\1 1 1
11llll i
1{1
I11
1 111
1 1t
I I
I
01' 11
I
E
11!1 1 11J1 _
l
li!I
0
It
t
f
IIIIII in. IIII
m
SW
II
I Ij
iIi"t t t
I
l iI i II 1 I i
i i I I I
II
I iI I I I I
I I iI
I i I
.
'-
_I I l
I I
lil
J _1111
- i
i
l
TJll
-
T
i
!
k
LL
r
1
I
I
_
Ii
I III
I i
Iiii
1
:
'
.
.
1
3000
qoOo
11i
UU
E
T SECS)
Figure 28. Closed-Loop Heat Rate History for the Maximum Downrange Case
105
·------
17UOE ._
_
L
F
7
IT _
_
I_
.
.
._
,,nnn
------
._
cO
EtB
r
7-r r'
r
f
--
I17
4
G
I1L
t Jf-jl- I_
E.
25000
r. _l rI.I
7'
!L
VI
0
'!
3
_
-
_LI
l
/
I
I
1; !
1000
_
EG
3 il-I-
l
ii
'-
--L]
L
IL
L
_J
!-r E
r
j
_
E
!
I
I I t
_J !I _J I
I
_i_II
4
--
r-
4
z
I
E
C
7'
-
t rE
L
~4
l
'V-
4 L
3 L-
!
I
iii
f-
_F
IF
0
-J
- 75000
C
1r
7
7
F-
r
7
L
L
il I
Er
Vn
D
0000oo
t
rL
l:_
. _
1
50000
.1
7
.
'50000
I
r _ r7 F r
I
F
L
3 3
t iM
_J _1
-
3l
;,
_
_-4._j
! - L
-
,EE
l
-
-
l _1_J_J -i
f
-l
--4
-
L L
4 _-4---I-- .-.
4.---W
------ 4 ---
3000
f
;
*
I
l-l
FF
.__
-1I --4
__I I
L
4
..-·- ----
'4000
Il
5000
T (SECS3
-
Figure 29. Closed-Loop Heat Load History for the Maximum Downrange Case
106
-
-
150U0
7
7-
l
C 7
7!
r
i
7
7
I
r
nrnr
.----
w-
I
L
L
I
/
T
L
!
V
.
E
.M
7
.
c
0)
L
r
0,
C
t
0o
0
I
z
I
I
/t
._
_l
I
L
L
]l
i
lI 7,I!
L
LL
!
/
.
t ] t L]
;
l
l.--
~F
l
Li
11
1,
r
F
L r
ML
/
/
5000-
7]7
-i
II
I
X
/
.
0
-I ._
r
r
t
I
7,
L
i
H
! _i
I I
I
L
E
--k-/-1 -J
J -2 -.
-2-1000
2000
1 _J_J
3000
I
-
_/._J
_J.4000
I
I
-
1
4 ._
i----
,
5000
T [SECS1
Figure 30. Closed-Loop Downrange History for the Maximum Downrange Case
107
!I
__
__
7
ZUUL
7
t
L
_
u_
. _
._
_
l
r
-i
E
I
._
._
'-.
. _
u_
l
.
f
-
_
l
l
L
U,
-200(
iri
U
-SOc
>To
I
L
4-.fi
._
_
l
-
L_
_
_
i
_
S
t
-
f
I
i
i
i
i
,
-i-
.
&
|
=k
-1
-I-
+L
IUO0
Ed
TI I
l
t
I
;--,
I -;
I;
.
!
1I
- '
1
I
;I!! I!I _
I
L
-]
_J
-F
7±
2000
i;·
.,
F
l
!
d
i
I
I
I
i ! i i i i
II
I
3000
I
;
ti
-
I I1 f
11
I I IL -
-
!
.=.
I9 I-
I4
i~~~~~~_!
. .
I
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1 1f
I
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Up I : : ! : i ! : !
I I I
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L
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-1500I- -II
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Ti i
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Figure 31. Closed-Loop Crossrange History for the Maximum Downrange Case
108
Figure 32. Closed-Loop Altitude History for the Maximum Crossrange Case
109
VI
VR
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Figure 33. Closed-Loop Velocity History for the Maximum Crossrange Case
110
.c,
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r
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ao
CO
.
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=
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0
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~
-
2000
3000
tSECS)
Figure 34. Closed-Loop Heat Rate History for the Maximum Crossrange Case
111
-
w
-
|
r(co
_J
-r
T SECS)
Figure 35. Closed-Loop Heat Load History for the Maximum Crossrange Case
112
1_^^^
1500D
_
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-7-
I,
1
10000-
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Figure 36. Closed-Loop Downrange History for the Maximum Crossrange Case
113
7-
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100
z
1000
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500
E
ii
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2000
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(SECS)
_
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Figure 37. Closed-LoopCrossrange History for the Maximum Crossrange Case
114
Figure 38. Closed-Loop Altitude History for the Minimum Downrange Case
115
VI
o VR
U
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U
0
T (SECS]
Figure 39. Closed-Loop Velocity History for the Minimum Downrange Case
116
Figure 40. Closed-Loop Heat Rate History for the Minimum Downrange Case
117
7000(
I
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Figure 41. Closed-Loop Heat Load History for the Minimum Downrange Case
118
1)UUL
l
F
l
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500
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2000
T (SECSI
--
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i
Figure 42. Closed-Loop Downrange History for the Minimum Downrange Case
i
119
._1_____
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2000
_
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.
l
_ = _
1500
I
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0 TI l I
1000
l
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500
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1 1 1 1
1500
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T (SECS)
Figure 43. Closed-Loop Crossrange History for the Minimum Downrange Case
120
-
r
r
-
F
Z
,,=
Z
l
P
--
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S
I-
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(SECS)
Figure 44. Angle of Attack Comparison for the Maximum Downrange Case
121
Figure 45. Bank Angle Comparison for the Maximum Downrange Case
122
I
I
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Case
.
0
I.
ti..1
1000
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2000
T SECS]
Figure 46. Angle of Attack Comparison for the Maximum Crossrange Case
123
--
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NominalCase :]i- IitI!I l
t t lI I f
1
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Figure 47. Bank Angle Comparison for the Maximum Crossrange Case
124
I
I1
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L_
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1 I 111 I i
FI I1
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I
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spersed.
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1500
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T (SECS)
--
Figure 48. Angle of Attack Comparison for the Minimum Downrange Case
125
I
II'
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8
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f l
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!
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Figure 49. Bank Angle Comparison for the Minimum Downrange Case
126
_
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Figure 50. Bank Angle Versus Time Comparison
127
200u
SE&CS
for Heat Rate Control
eo
ZLP
-
r
--
l
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--
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I
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heal rate
47 j
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Figure 51. Angle of Attack Versus Time Comparison for Heat Rate Control
128
Q,mg-100 8TU/sqt/sec-
!
1
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1
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Figure 52. Heat Rate Versus Time Comparison for Heat Rate Control
129
_
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Figure 53. Bank Angle Versus Velocity Comparison for Heat Rate Control
130
l
,,
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Withoutovercontrol
i
lJ
LlL
4000
5000
T SECS)
.
Figure 54. Angle of Attack Versus Time Comparison with Overcontrol
131
50
40
E
30
0
x
w
20
Figure 55. Required Execution Time for the Predictor-Corrector
132
APPENDIX A. ERV AERODYNAMICS MODEL
The aerodynamics of the ERV were reported in Reference [14], and the longitudinal performance
coefficients,
C, and L/D, are shown in Figures 20 on page 97 and 21 on page 98.
Figure 22 on page 99 shows a typical UD versus angle of attack profile. This profile is for a
Mach Number of 10, but across the flow regimes, the maximum L/D always occurs at an
angle of attack of approximately 15 degrees. This data was incorporated into the aerodynamic model of the simulator and into the aerodynamic model of the predictor.
It is seen that
the aerodynamic flow regimes are a function of:
1.
Mach Number, M
2.
Viscous Interaction Parameter, V
3.
Altitude, h
The Mach Number, M, is computed from,
M
=
vV,
(99)
Cs
where the speed of sound, C,, is computed from,
c,
=
(100)
TM
The viscous interaction parameter, V, is computed from,
V
= M
(101)
R
where,
133
ct (T'
)o5
T,,,
tc + 122.1x 10-(5lrstatc)
TsaticT'
(102)
+ 122.1 x 10-(5/r')
and,
r_
=
Tatc
0.468 + 0.532
Ttc
+ 0.195
2
M2
(103)
The Reynolds Number, Re, is calculated from,
Re
=
(104)
VE
where the coefficient of viscosity for air, p, is given by,
=
(105)
C
S + T,,,,C
134
APPENDIX B. ALGORITHM PROGRAM LISTINGS
Compiled listings of the flight software principal functions for the predictor-corrector guidance algorithm as coded for use in the 6-DOF Aeroassist Flight Experiment Simulator
(AFESIM) follow. The algorithms are coded in the HAL/S computer language. The principal
functions are:
1.
IL LOAD - Values for all constants and I-loads
2.
FSW SEQ - Flight Software Sequencer
3.
ORB_NAV - Orbit Navigation Algorithm
4.
AERO_GUID - Predictor-Corrector Guidance Algorithm
At the beginning of each principal function is a description of the function and the
input/output parameters. At the end of each principal function is a cross reference table listing the program line at which each variable is referenced or computed.
135
136
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10
14
41
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14
i
0
ui
UI
e
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in
0_
4
14
,e
oa w
a
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213