A Predictor-Corrector Guidance Algorithm Design for a
Low L/D Autonomous Re-entry Vehicle
by
Carla Haroz
B.S. Aerospace Engineering
B.A. Russian Language and Literature
The University of Texas at Austin, 1996
Submitted to the Department of Aeronautics and Astronautics in
partial fulfillment of the requirements for the degree of
Master of Science in Aeronautics and Astronautics
at the
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Peter Neirinckx
Charles Stark Draper Laboratory, Inc.
Technical Supervisor
Certified by ..................................... ..........................
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Richard H. Battin
Professor of Aeronautics and Astronautics
Thesis Supervisor
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2
A Predictor-Corrector Guidance Algorithm Design for a Low L/D
Autonomous Re-entry Vehicle
by
Carla Haroz
Submitted to the Department of Aeronautics and Astronautics
on December 18, 1998, in Partial Fulfillment of the
Requirements for the Degree of Master of Science in
Aeronautics and Astronautics
Abstract
The Precision Landing Reusable Launch Vehicle (PL-RLV) is a low L/D, 2-stage craft
with a mission plan that calls for low cost, speedy retrieval, and quick turn-around-times
for successive flights. A guidance scheme that best adheres to these goals and captures the
vehicle's capability is desired. During re-entry, the PL-RLV's second stage, the Precision
Landing Vehicle 2 (PLV-2), will perform a reversal maneuver. This thesis concentrates on
a possible re-entry guidance scheme for the PLV-2during the terminal phase, the time
from the completion of the reversal until the landing system parachutes are deployed.
A simple bank-to-steer algorithm is suggested. The angle of attack is trimmed, and the
bank angle (or bank rate) remains as the only means for control. The algorithm controls
the time history of the vehicle's bank angle and tunes the bank angle history to meet landing and fuel requirements. This versatile guidance approach employs predictor-corrector
methods. The guidance scheme presented generates a possibility of bankrate profiles
within limitations that could be used for target acquisition. Selection of a robust target
location and the nominal bankrate profile which will yield a minimum target miss are
investigated. Testing shows the trade-offs between fuel cost and landing capability. Dis-
persion testing with winds and density are also performed.
The predictor-corrector combination can yield target miss distances on the order of hundreds of feet or less. Open-loop and closed-loop results display the guidance system's
ability to capture the PLV-2's capability in the presence of dispersions while still meeting
system requirements.
Thesis Supervisor: Dr. Richard H. Battin
Title: Professor of Aeronautics and Astronautics
Technical Supervisor: Peter J. Neirinckx
Title: Member Technical Staff, The Charles Stark Draper Laboratory, Inc.
4
Acknowledgements
There are many people that I would like to thank that have made my experience at MIT and
Draper Labs educationally broadening, challenging, and enriching.
Thank you to Tim Brand, who gave me the opportunity to work at Draper, and to Peter Neir-
inckx for supervising during the thesis process. Thank you also to Lee Norris, Doug Fuhry, and
George Schmidt for their guidance and advice.
I have been honored to be in the classrooms of two of the greatest minds in Orbital Mechan-
ics, Dr. Richard Battin and Dr. Victor Szebehely. A special thanks to both of them for presenting
the beauty of planetary motion to me and for bringing the history of US Space Exploration alive.
A big thanks to my professors at the University of Texas who encouraged me to attend MIT
and who are always there for advice and support: Dr. Hans Mark, Dr. Wallace Fowler, and Dr.
Robert Bishop.
Thank you to all my friends at Draper Labs - Gregg "TMG" Barton for all the encouragement
and cookies!, Chris D'Souza for checking up on me to make sure I was still alive in my cubicle,
Chris "Sparkster" Stoll for constantly battling the gremlins in my computer, Jenn "KB" Hamelin
for the chats, Ed Bachelder for the tete-e-tete's and love of Trader Joe's Chocolate, and to all my
Draper fellow friends with whom I made it through classes, work, and fun with - Christina, Atif,
Nate "Shenckenstein", Geoff, Pat, Chisolm, and my favorite softball team, the Draper Monkeys!!
Also, a thank you smile to my encouraging friends at the DLR German Space Center - Manfred,
M. Klimke, M. Reichart, and Dr. Seiboldt; to Shaun, Carolyn, Benno, Tobias, Jonathan, Santiago,
and my NASA buds, Terry, Greg, and Andy.
Thank you to Nick Nuzzo, for all the love and support, thesis empathy, late night Draper dinners, Swing dancing in the halls, and our thesis getaway island adventure in Greece. S'agapo.
The biggest thank you goes to my family: Mom, Dad, Lezlie, Tammy, and Kim. To my older
sisters, you can officially stop calling me your "LITTLE" sister now. Mom and Dad, you have
always encouraged me to follow my dreams and shoot for the stars. I continue my journey on the
road less traveled knowing that you are always there for me. I love you.
5
This thesis was prepared at the Charles Stark Draper Laboratory, Inc.
Publication of this thesis 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 stim-
ulation of ideas.
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=
Carla
"
S.
Haro
Carla S. Haroz
6
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Table of Contents
1 Introduction ..........................................
15
1.1 Problem Definition...........................................
16
1.2 Autonomous Bank-to-Steer Guidance Challenges ..........................................18
1.3 Thesis Overview ............................................................................................. 19
1.4 Chapter Breakdown .........................................................................................19
21
2 Mission Overview and Requirements .......................................
2.1
2.2
21
22
Flight Profile .......................................
PLV-2 Aerodynamic and Mass Properties .......................................
2.3 Coordinate Frames ........................................
3 Simulation Code Structure....................................
3.1 Simulation Environments ...........................................................................
3.2 General Design Features of the Re-entry Guidance .......................................
3.3
4
5
6
7
25
31
31
31
PLV-2 Entry Guidance Code Definition..........................................................33
34
3.4 Functions of the Initial Subphases ........................................
38
3.5 Terminal Subphase..................................
38
3.6 Landing System Phase ........................................
40
3.7 Atmospheric Models...................................
3.8 Vehicle Uncertainty and Environment Dispersion Sources.............................42
45
Guidance Design ..................................................
4.1 Guidance Scheme Definitions..................................................
45
49
4.2 Trajectory Control..................................................
4.3 Predictor..........................................................................................................50
4.4 Corrector ........................................
57
Nominal Bank Rate Profile Selection ........................................
65
5.1 Profile Generation ............................................................................................66
5.2 Bank Rate Bin Definition.................................................................................67
5.3 Example Profiles ........................................
69
5.4 Nominal Profile Selection
..........................................
72
Robustness Testing .......................................
77
77
6.1 Range Capability
.
......................................
82
.................................
6.2 Nominal Target Robustness Results ...........
6.3 Footprint Range ........................................
85
Guidance Performance.....................................
91
7.1 Predictor and Corrector Performance ........................................
91
99
7.2 Fuel Cost vs. Landing Performance.......................................
7.3
Corrector Performance On Nominal Profile ..................................................
112
7.4
7.5
Bin Number Selection ................................................................................
Effects of Atmospheric Dispersions ........................................
113
118
8 C onclusions ......................................................................................................
127
Appendix A Analytical Study of the PL-RLV Re-entry Guidance .............................129
. .............................. 157
Appendix B Acceleration Model for Bank Maneuvers
Appendix C Nominal Profile Simulation Plots................................................
165
7
References
..........................................................................................................................
8
171
List of Figures
Figure 1.1: The PL-RLV Concept Drawing ....................................
17
Figure 2.1: Mach vs. Trim Angle of Attack During PL-RLV Descent ..........................24
Figure 2.2: Aerodynamic Parameter History During PL-RLV Descent........................ 24
Figure 2.3: Relationship Between Inertial, LVLH, Body, and Velocity Axis Systems [8]
27
Figure 2.4: Relationship Between Bank Angle and Lift................................................. 28
Figure 2.5: Relationship Between Body Frame and LVLH Frame [1] ........................... 29
Figure 2.6: Relationship Between North/East and Crossrange/Downrange Frames ......30
Figure 3.1: Re-entry Guidance Outputs..............................................................
32
Figure 3.2: Sample Bank Angle Profile..............................................................
39
Figure 3.3: Example Footprint .
.............................................................
41
Figure 3.4: Altitude vs. Density, Typical PLV-2 Descent..............................................41
Figure 3.5: Altitude vs. Speed of Sound, Typical PLV-2 Descent .................................42
Figure 4.1: Re-Entry Guidance Predictor Calls ............................................................. 51
Figure 4.2: Closed-Loop Predictor/Corrector Guidance .................................................52
Figure 4.3: Predictions for Corrector Guidance in Entry Phase .....................................58
Figure 4.4: Minimum Miss Guidance for Reversal Stage .............................................
62
Figure 5.1: Flow Chart for Selection of the Nominal Profile .........................................65
Figure 5.2: Terminal Phase Bins..............................................................
67
Figure 5.3: Bankrate Profile Generation, Cycling Through First Bin ............................68
Figure 5.4: Open-Loop Footprint of Landing Locations ................................................71
Figure 5.5: Location of Target Selected Within Footprint .............................................74
Figure 5.6: Acceptible Bank Rate Profiles (#1-#9) Generated for New Target .............75
Figure 5.7: Acceptible Bank Rate Profiles (#10-#14) Generated for New Target ......... 76
Figuie 6.1: Target A, B, C, and D Locations Within the Footprint ...................
..........
78
Figure 6.2: Bank Rate Profile for Target A ..............................................................
79
Figure 6.3: Bank Rate Profile for Target B..............................................................
Figure 6.4: Bank Rate Profile for Target C.............................................................
80
81
Figure 6.5: Bank Rate Profile for Target D ..............................................................
82
Figure 6.6: Range Capability of Test Cases, Final Miss Within 10,000 ft.....................83
Figure 6.7: Range Capability of Test Cases, Final Miss Within 1,000 ft.......................84
Figure 6.8: Footprints at Time= 0 seconds in Terminal .................................................86
Figure 6.9: Footprints at Time= 5 seconds in Terminal .
.......................................
86
Figure 6.10: Footprints at Time= 10 seconds in Terminal ............................................ 87
Figure 6.11: Footprints at Time= 15 seconds in Terminal ............................................ 87
Figure 6.12: Footprints at Time= 20 seconds in Terminal ............................................ 88
Figure
Figure
Figure
Figure
6.13: Footprints at Time= 30 seconds in Terminal .............................................
6.14: Footprints at Time= 40 seconds in Terminal ............................................
7.1: Bank Angle Profile for Case 7.1 .............................................
7.2: Terminal Phase Bank Rate Profile for Case 7.1 ...........................................
88
89
93
93
Figure 7.3: Open-loop Horizontal Prediction Error: Case 7.1 .......................................94
Figure 7.4: Open-loop Horizontal Prediction Error: No Pre-terminal Bin Knowledge: Case
7.1............................................
95
9
Figure 7.5: Horizontal Prediction Error, Closed-Loop Terminal Phase .........................96
Figure
Figure
Figure
Figure
Figure
7.6: Open-loop Horizontal Prediction Error: Case 7.2 ........................................
7.7: Closed-loop Horizontal Prediction Error: Case 7.2......................................
7.8: Terminal Phase Bank Rate Profile for Case 7.2...........................................
7.9: Closed-Loop Target Miss Distances for the 14 Profiles ............................
7. 10: Corrections to the Bank Rate Profile - Case 1..........................................
97
98
98
100
101
Figure 7.11: Trajectory Path - Case 1 ...........................................................................102
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
7.12:
7.13:
7.14:
7.15:
7.16:
7.17:
7.18:
7.19:
7.20:
7.21:
7.22:
7.23:
7.24:
7.25:
Trajectory Path - Case 1 ...........................................................................
Prediction Error- Case 1 ...........................................................................
Corrections to the Bank Rate Profile - Case 2.........................................
Trajectory Path - Case 2 ............................................................................
Trajectory Path - Case 2 ...........................................................................
Trajectory Path - Case 2 ......................................................................
Corrections to the Bank Rate Profile - Case 3 .......................................
Trajectory Path - Case 3 ...........................................................................
Trajectory Path - Case 3 ...........................................................................
Trajectory Path - Case 3 ........................................................................
Corrections to the Bank Rate Profile - Case 4.........................................
Trajectory Path - Case 4 .......................................................................
Trajectory Path - Case 4 ...........................................................................
Trajectory Path - Case 4 ...........................................................................
102
103
104
104
105
105
106
107
107
108
109
109
110
110
Figure 7.26: Closed-Loop Prediction Error: Offset From Nominal Target ................. 112
Figure 7.27: Terminal Phase Bank Rate Profile: Offset From Nominal Target ..........113
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
7.28:
7.29:
7.30:
7.31:
7.32:
7.33:
7.34:
7.35:
7.36:
7.37:
7.38:
Open-loop Landing Locations for Large Bin #'s of 3, 4, 5, and 6 ........... 115
Open-loop Footprint Given Maximum Small Bins of 10, 20, 30, 35, 36, & 40116
119
Altitude vs. Density: Summer Case ........................................................
119
Altitude vs. Density: Winter Case ........................................................
Altitude vs. North Wind: Summer Case ................................................... 120
120
Altitude vs. North Wind: Winter Case ..........................................
121
Altitude vs. East Wind: Summer Case ..........................................
121
Altitude vs. East Wind: Winter Case ...........................................
122
Altitude vs. Vertical Wind: Summer Case ..........................................
Altitude vs. Vertical Wind: Winter Case.....................................122
125
Dispersion Miss Distances (1000 ft): Summer . .................................
Figure 7.39: Dispersion Miss Distances (6000 ft): Summer......................................... 125
Figure 7.40: Dispersion Miss Distances (1000 ft): Winter .......................................... 126
Figure 7.41: Dispersion Miss Distances (6000 ft): Winter ........................................... 126
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
........................
A. 1: Single-Switch Program Schematic ...............
A.2: X, Y, Z Axes and Bank Angle q Defined ...........................................
A.3: Final Positions When T_max = 40 sec ............................................
A.4: Final Positions When T_max = 30 sec ...........................................
A.5: Final Positions When T_max = 20 sec ...........................................
.......................................
A.6: Final Positions When T_max = 10 sec ...
A.7: Final Positions When T_max = 5 sec ............................................
A.8: All Final Positions, Tswitch = 10 sec ...........................................
A.9: All Final Positions, Tswitch = 5 sec ..........................................................
10
130
131
135
136
136
137
137
139
139
Figure A.10: Final Positions, Initial Bank Angle = 90 deg ..........................................
140
Figure A. 11: Resulting Final Positions, Each Parameter Tweeked One at a Time...... 142
146
Figure A. 12: Accessible Target Case . ............................................................
Figure A.13: Another Solution to Case 1, Target Acquisition ..................................... 147
Figure A. 14: Case 1 With Gains Set to .50............................................................
............................................................
Figure A.15: Case 1 With Gains Set to 1.0
148
148
Figure A. 16: Figure 12. Crossing the Inaccessible Zone.............................................. 150
Figure A.17: Non-instantaneous Bank Rate Change ....................................................152
.. 153
1....................
Figure A. 18: Non-zero Initial Positions ...........................................
Figure A. 19: Non-zero Initial Velocities .............................................................
153
Figure A.20: Wait Times between Bank Rate Changes Schematic.............................. 155
Figure B. 1: Acceleration Model Bankrate vs. Time .....................................................
157
.......................... 160
Figure B.2: Profile Redesign Due to Time Constraints ...........
165
Figure C.1: Altitude Profile ............................................................
Figure C.2: Earth Relative Velocity Profile .............................................................
166
Figure C.3: Relative Flight Path Angle Profile............................................................ 166
167
Figure C.4: Dynamic Pressure Profile .............................................................
167
.......................................
Figure C.5: Dynamic Pressure x Alpha Profile .
168
Figure C.6: Heating Rate Profile .............................................................
Figure C.7: Stagnation Point Temperature Profile .......................................................168
169
1.........
......... ....................................
Figure C.8: Acceleration Profile .........
Figure C.9: Mach Number Profile ............................................................
169
Figure C. 10: Angle of Attack Profile...........................................................
170
11
12
List of Tables
Table 2.1: Modeled Aerodynamic and Mass Properties of the PL-RLV
Table 2.2: PLV-2 Mass Properties 23
Table 3.1: Guidance Subphase Transition Criteria 34
Table 5.1: Bin Generation Conditions for Sample Run
Table 5.2: Initial Target and PLV-2 Setting 70
Table
Table
Table
Table
69
6.1: Target Coordinates in ECEF and North/East Frame 78
7.1: Case 7.1 Simulation Conditions 92
7.2: Closed-Loop Results for the 14 Profiles 99
7.3: Fuel Cost and Miss Distance Comparisons 111
Table 7.4: Large Bin Number Range Testing 114
Table 7.5: Small Bin Size Testing Results 117
Table 7.6: Dispersion Testing Results: Summer Case 123
Table 7.7: Dispersion Testing Results: Winter Case 124
13
22
14
Chapter
1
Introduction
With the new era of satellite constellations, the advent of space station construction, and renewed planning for Mars missions, comes an increased interest in Reusable
Launch Vehicles (RLV's). Commercial RLV's must be able to handle particular challenges
that expendable launch vehicles do not need to meet. RLV's are necessarily more complex
than expendables and thus investigations of design tradeoffs are all the more important.
Some RLV mission requirements can be critical, such as the desire to land the vehicle
within a target area that is easily accessible to support and launch facilities, ground crews,
and transportation. To ensure frequent deployment opportunities, RLV's must be able to
operate under relatively quick turn-around-times between launches. Also, a desire for
complexity reduction in one design area can often sacrifice system performance.
A specific guidance design for a RLV, hereafter refered to as the PL-RLV (Preci-
sion Landing Reusable Launch Vehicle), is presented in this thesis. Structural design simplification has yielded a low L/D (Lift-over-Drag) vehicle. However, this simplification is
traded against the already sensitive guidance and landing requirements of the RLV's reentry and landing phases. This trade-off requires an in-depth analysis into the precision
and robustness of any re-entry guidance design. This thesis project designs and analyzes a
particular re-entry autonomous guidance scheme for the PL-RLV during the terminal
phase of descent, the time between a reversal maneuver and the landing system parachute
deployment. A simple bank-to-steer algorithm meets both the vehicle constraints and
landing requirements.
15
1.1 Problem Definition
One of the most important aspects of a RLV is the ability to land at a predeter-
mined target which is located close to the processing facility. As opposed to the mid-ocean
splashdown techniques of the Apollo days, a more sophisticated landing scheme must be
developed. This scheme could also be more simplistic than the Space Shuttle's guidance,
which relies on three control variables for its precision, but at a significant design and
structure cost [2]. Furthermore, due to its simplicity, the new landing scheme could be utilized in the future for an entry vehicle flying missions on distant planets. In that case, it
would be imperative that the guidance scheme provide a precise landing since the survival
of the manned or unmanned mission could hinge upon the precision of the ship's landing
location.
It is desired that the PL-RLV return as close to the launch site as possible - within a
few miles. This will enable quick turn-around-times, more launches, less cost for retrieval,
and more opportunities to deploy payloads into space. The PL-RLV is shown below during the ascent phase in Figure 1.1.
16
~·~i~__"MIRMn
;6001
Figure 1.1: The PL-RLV Concept Drawing
The central theme and challenge of this project lies in designing a terminal guidance algorithm that captures most of the vehicle's physical capability, is controllable, and
is robust to environmental variations. An early examination of precision landing schemes
and an investigation into their cost/risk trade-offs are justified since their results impact
many aspects of a program, including: mission architecture and objectives, landing site
selection, navigation infrastructure, trajectory design, vehicle loads and structure, and
environmental knowledge requirements.
Preliminary studies indicated that the trade-off between design complexity and
guidance performance of the PL-RLV proved favorable with a bank-to-steer guidance
scheme. See Appendix A for one of these early studies.
17
1.2 Autonomous Bank-to-Steer Guidance Challenges
Bank-to-steer guidance, similar to the re-entry guidance used for the Apollo mission, utilizes only a single control parameter: vehicle bank angle. The bank angle is essentially the angle the lift vector points with respect to the vertical in the plane perpendicular
to the velocity vector at a trim angle of attack and zero sideslip. Commanding and executing a change in bank angle steers the vehicle to the target. Since this guidance scheme is
constrained to only one control parameter, a considerable reduction in the structure and
software complexity of the design results if the scheme is successful. The price for this
simplicity, however, could cause a reduction in guidance capability and landing performance. The challenge lies in creating a bank-to-steer design that meets both the mission
performance requirements and is robust in its response to in-flight dispersions.
Atmospheric uncertainties in air density and wind information pose a significant
challenge to the trajectory control of an autonomous guidance scheme with a single control parameter and for a low L/D body. The navigation uncertainties on guidance performance are typically small in comparison, and therefore will not be addressed in this study.
The precision landing problem is further complicated upon inclusion of an unguided landing phase which exposes the vehicle to wind effects after landing system deployment,
which is the case for the PL-RLV, or possible hazard detection maneuvers during the ter-
minal phase.
To successfully direct the vehicle to the desired landing location, the re-entry guidance scheme must be able to construct a realistic bank versus time profile, predict the
resulting landing locations, and correct the bank plan if necessary. The profile must also be
18
able to conform to the RCS jet specifications and utilize only the fuel available. Additional
trajectory control parameters, such as angle-of-attack modulation or landing system
deployment time, could also be considered in the development of a guidance scheme, but
are not researched in this study.
1.3 Thesis Overview
The concentration of this thesis is the development of a precision autonomous
guidance scheme for the low L/D PL-RLV to be used during the terminal phase. Several
bank-to-steer schemes were designed using a predictor/corrector coding structure. Draper
Laboratory's CSIM simulation framework of the PL-RLV was used to test this scheme.
It is hoped that with minor modifications, the design will serve as a model for those
missions that will take place on other celestial frontiers in the future, specifically the safe
transportation of human explorers to the surface of Mars.
1.4 Chapter Breakdown
Chapter 1 provides introductory information on this thesis. Chapter 2 defines the
PL-RLV mission and the vehicle's properties and requirements. Chapter 3 introduces the
entry guidance algorithm software structure. Chapter 4 defines the re-entry guidance
schemes and the predictor/corrector algorithms utilized. Chapter 5 introduces the selection
of the nominal bank profile, and chapter 6 contains the nominal design robustness testing.
The guidance design performance, fuel reduction versus landing performance results,
19
optimization of the terminal phase design, and the effects of atmospheric dispersions are
all presented in Chapter 7. Chapter 8 is devoted to overali results and conclusions to the
thesis, as well as suggestions for future work.
20
Chapter 2
Mission Overview and Requirements
Knowledge of the "big-picture" which surrounds the task at hand is always a key
factor in the production of a simulation design. This chapter provides a brief overview of
the PL-RLV's flight profile, as well as an introduction of certain vehicle parameters which
have influenced the design of the PL-RLV. Coordinate frame definitions are provided in
this chapter.
2.1 Flight Profile
The PL-RLV is a fully reusable launch vehicle consisting of two stages: the PLV-1
(Precision Landing Vehicle, the first stage) and the PLV-2 (the second stage). At lift-off,
the PLV- 1 helps to boost the PLV-2 on its trajectory path.
The specific phases of the mission's descent phase will be described in Chapter 3.
At an appropriate time after lift-off, the PLV-2separates from the PLV-1, fires its engine,
and continues on its own journey. The PLV-1 guidance is not investigated in this thesis.
The PLV-2's ascent continues to the desired low-earth-orbit (LEO) or middle-earth-orbit
(MEO). Orbital maneuvers are accomplished by orbital maneuvering system (OMS)
engines.
While in orbit, the satellite housed inside the PLV-2 is deployed. Following
deployment, the PLV-2 performs orbital adjust maneuvers. At the appropriate time, the
21
PLV-2 performs a pitchover maneuver and fires the OMS engines to deorbit. Upon atmospheric entry, the PLV-2 guidance will fly the craft on a path toward the target until the
landing system deploys.
2.2 PLV-2 Aerodynamic and Mass Properties
Crucial in mission design is the knowledge of the craft's aerodynamic and mass
properties. The modeled aerodynamic and mass properties of the PL-RLV used in the
guidance design are shown below in Table 2.1.
In the guidance simulation, the vehicle is modeled as a point mass of 923.19 slugs
with an effective aerodynamic area of 201 ft2 . A maximum bank angle and bank rate were
introduced into guidance control in order to prevent unrealistic modeling of the vehicle's
motion. Table 2.2 gives a more in-depth mass property listings for the PLV-2.
Point Mass
923.19 slugs
Vehicle Aerodynamic
Reference Area
201 ft 2
Average Lift / Drag Ratio
.1
Maximum Bank Angle
175 deg= Entry Phase
180 deg = Rev/Terminal
Maximum Bank Rate
10 deg/sec
Velocity at
974.25 ft/sec
Landing System Deploy
Table 2.1: Modeled Aerodynamic and Mass Properties of the PL-RLV
22
Table 2.2: PLV-2 Mass Properties
The coefficient of lift, C1 , and the coefficient of drag, Cd, are both functions of the
vehicle's angle of attack and Mach number. These aerodynamic parameters are independent of the trajectory. The following plots are typical examples of the PL-RLV's aerodynamic history as it flies to the ground. The large jump in the Cd and C1 plots near 75,000 ft
occurs when the landing system parachute is deployed.
23
Figure 2.1: Mach vs. Trim Angle of Attack During PL-RLV Descent
a,
3-
0
0.2
0.4
0.6
1.2
0.8
1
Lift and Drac Coefficients
1.4
1.6
1.8
Figure 2.2: Aerodynamic Parameter History During PL-RLV Descent
24
2
2.3 Coordinate Frames
Several different coordinate system reference frames are used throughout this simulation. These frames are used in the design and analysis of the entire system. The following coordinate frames were used in the PL-RLV simulation:
Inertial Frame
The Inertial reference frame is non-rotating with an earth-centered origin. The X I
axis points through zero longitude at time zero (epoch), the Z I axis through the North
Pole, and the YI axis points in the direction perpendicular to the XI and Z I axes in order to
complete a right handed set. Integration of the vehicle's state (equations of motion) is performed in this frame.
Earth Centered Earth Fixed (ECEF)
The ECEF frame rotates about the inertial frame z-axis at a fixed rotation rate
defined by the earth's constant rotation rate (without precession). The inertial and ECEF
frame are co-aligned at tirne=O.The origin is the earth's center and the axes are defined as
follows: the Xecefaxis points through zero geodetic longitude and latitude, the Zecefaxis
through the North Pole, and the Yecefaxis completes the right-handed set. The landing site
is given and some guidance calculations are performed in this frame.
25
Local Vertical Local Horizontal (LVLH)
This coordinate frame is used a great deal in trajectory design. The origin is the
vehicle's center of gravity, the Zivlh axis (local vertical) points towards the earth's center,
the Yvlh axis is the cross product of Zlvh and the vehicle's earth-relative velocity vector,
and the Xlvlh axis (local horizontal) is defined by the cross product of Y1vlhand the Zvlh
axes.
Body Frame
The body frame is fixed to the vehicle center of gravity with the Xb axis along the
longitudinal axis (nose positive) of the PLV-2, the Zb axis pointing "down" through the
lateral axis of the PLV-2, and the Yb axis completes the right handed set. The rotations
between this frame and the LVLH frame express the typical aerodynamic control parameters: yaw, pitch, and roll.
Velocity Frame
The velocity frame has the X v axis defined along the vehicle's earth-relative velocity vector, the Yv axis a result of the cross product between the gravity vector and X,. The
Z. axis completes the right handed set.
The relationships between the Inertial, LVLH, Body, and Velocity frames are
shown in Figure 2.3 below. The angles X, y, a, and 0 are defined as the co-latitude, flight
path angle, angle of attack, and the pitch angle of the craft.
26
I
,I
Xb
a
C
I
Zlvlh
I
*Zv
XI
Figure 2.3: Relationship Between Inertial, LVLH, Body, and Velocity Axis Systems [8]
The figure below shows a different view of the Body and Velocity Frames. Most
importantly for this thesis is the definition of the vehicle bank angle,
4.The angle between
the Z v axis and the lift vector is the bank angle. The bank angle rotates around the earthrelative velocity vector, Xv.
27
__
Vector
Xb
!
v
Zv
__
Figure 2.4: Relationship Between Bank Angle and Lift
Figure 2.5 below illustrates the relationship between the Body and Local Vertical
Local Horizontal Frame. The vehicle attitude is defined by
4,
0,
A;
the roll, pitch, and yaw
angles of the craft. When a change in bank is commanded, the maneuver is executed by
changing the roll and pitch angles.
28
Xb
-
IsI
\
I
Zb
Yb
Figure 2.5: Relationship Between Body Frame and LVLH Frame [1]
Crossrange-Downrange Vehicle Trajectory Frame
This coordinate frame is used to express the landing-point to target miss components. The frame itself is tied to the vehicle trajectory instead of the target. A full lift-up
trajectory, a trajectory with no reversal, is used as the reference and defines the frame. The
crossrange direction is perpendicular to the trajectory at the landing location, while the
downrange direction lies along the trajectory groundtrack. This frame ties more closely
the trajectory controls on the miss components, i.e. more in-plane lift increases the coordi-
nate along one axis, while more out-of-plane lift increases the coordinate along the
orthogonal axis.
North/East Frame
The North/East Frame is used for plotting miss errors once the PLV-2 lands. A
cross product between the Zecefand the landing location vector in the ECEF frame yields
29
the definition of the East vector, E. A cross product between the landing location (ECEF)
vector and the unit vector in the E direction defines the North direction, N. The relationship between the crossrange/downrange
and north/east frames are shown in Figure 2.6.
N
Udr
Lift-Up
_
I
E
Landing Point
Trajectory
Ground Track
Ucr
Trajectory Ground Track
Figure 2.6: Relationship Between North/East and Crossrange/Downrange Frames
30
Chapter 3
Simulation Code Structure
This chapter provides a breakdown of the code and design features used for the
PL-RLV's descent simulation. Characteristics of each phase encountered during the
descent trajectory are defined briefly for basic understanding of the simulation. Particular
attention will be given to the simulation of the teminal phase. Sources of dispersions that
need to be addressed in simulation design are also presented.
3.1 Simulation Environments
A C-based 3-DOF simulation environment (C-SIM) was used as the framework to
develop the guidance simulation code and run the PLV-2's trajectory simulations. The C-
SIM recognizes point mass assumptions and guidance commanding. The guidance commanding in this thesis is concerned with bank angle commands and bank rate histories for
control purposes. The bank angle commands themselves are limited to avoid unreasonable
vehicle motion and bank rate commands for the vehicle. In this thesis, the vehicle is
assumed to be aerodynamically trimmed at every point on the trajectory.
3.2 General Design Features of the Re-entry Guidance
The re-entry guidance is a closed-loop predictor/corrector algorithm with the
downrange and crossrange target miss distance as the constraints. The original guidance
31
algorithms for the entry and reversal phases were designed for Draper Laboratory by Doug
Fuhry [4]. The vehicle bank angle, the time of a single symmetric bank reversal, and a
bankrate profile are the controls. There is no explicit control of any other parameters, such
as convective heating rate, aerodynamic load, or landing system parachute deployment
time. Figure 3.1 illustrates the guidance output flow. The re-entry guidance produces bank
angle and bank rate commands, as well as the commanded time of the reversal and estimated landing system parachute deployment time, to re-entry control. A bank reversal is
always commanded, and it is always executed as a lift-up bank reversal (one which passes
through 0 degrees bank).
II
F
4,Bank
--.
Angle
I
4dot, Bank Rate
m = Output Mass
t___-par)Parachlte
Denlov Time
---------
I
Figure 3.1: Re-entry Guidance Outputs
Since the guidance has no knowledge regarding mass decrements as a result of
Reaction Control System (RCS) firings, the output mass is a constant value equal to the
initial value provided by onorbit guidance. Guidance targets a geographic location in
32
Earth-centered/Earth-fixed (ECEF) coordinates at the time of landing system parachute
deployment. The re-entry guidance also allows uplinks of atmospheric density, speed of
sound, and ECEF wind vector profiles.
3.3 PLV-2 Entry Guidance Code Definition
The PLV-2's re-entry guidance is divided into 5 main phases: pre-entry, entry,
reversal, terminal, and landing system [4]. Until the vehicle reaches 400,000 ft, it is considered to be in the pre-entry stage. Once the craft has passed this altitude and has a specific force greater than the I-load, the code turns to the entry stage. The reversal stage
begins at a commanded time determined by the predictor/corrector algorithms found in the
entry phase. Immediately after the PLV-2 reversal is complete, the terminal guidance
phase begins.
The terminal phase and predictor/corrector algorithm is the concentration of the
thesis and will be discussed in greater detail in Chapter 4. When the vehicle reaches a
velocity of 975.00 ft/sec, a landing system parachute is deployed. The landing system
phase continues until the desired altitude for the simulation termination is achieved. A
5,000 ft altitude was chosen to be the termination value of the simulation. From this point,
it is assumed that the craft will travel on a relatively straight ECEF path down to the
ground. Table 3.1 summarizes the criteria for phase transitioning.
33
Transition Criteria
Guidance Subphase
pre-entry
entry
reversal
Default at cyclic guidance initiation
Geodetic altitude < 400kft (I-load) AND
Measured specific force > (I-load)
Current time within .1 sec (I-load) of
commanded bank reversal start time
terminal
landing system
Current bank within 0.05 deg (I-load) of
commanded bank angle at reversal end
Velocity <= landing system parachute
deploy value (975.00 ft/sec, mission
load), approx 75kft
Table 3.1: Guidance Subphase Transition Criteria
3.4 Functions of the Initial Subphases
Each subphase of the descent profile is responsible for certain guidance commands. These commands are responsible for guiding the craft on a trajectory that will ultimately yield a minimum target miss. The requirements of each phase are described below
[4].
Pre-Entry Phase
During this phase, the commanded bank angle is set to a mission load value. The
proper bank angle rate command sign is computed as well. A positive bank rate sign is
indicative of a positive initial bank angle, and a negative sign indicates a negative initial
34
bank angle. The pre-entry guidance also sets the maximum commanded bank angle rate to
a mission load magnitude times the computed sign.
Entry Phase
During the first pass in the Entry Phase, a function (CRDRlanding) is called to
compute the Crossrange/Downrange Coordinate Frame Reference axes based upon the
predicted stabilization parachute deploy positions of full lift-up and full lift-down trajectories. It is important to note that these defined axes are invariant during re-entry. The sign of
the initial bank command is computed based upon the crossrange offset of the targeted
landing site.
Guidance during this phase consists of a trajectory prediction to estimate the parachute deploy time using the predictor inputs: current guidance bank angle, the state vector
(position, velocity, and current time), the reversal start time commands, and the terminal
phase bankrate profile. If this estimated position is not within 500 ft (tolerance) of the target, two additional trajectory predictions with parameter perturbations will be run; one
with the bank angle perturbed by 2 deg (mission load) and one with the reversal start time
perturbed by 5 sec (mission load). The numerical partial derivatives of the constraints with
respect to the guidance control parameters using the target miss distances from the three
trajectory predictions are then computed. Corrections to the control parameters are generated based upon the predicted target miss and the numerical partial derivatives.
35
The bank reversal start time is forced to begin before the estimated landing system
parachute deployment time less the time it takes for reversal execution. The bank angle
magnitude is limited between 0 and 180 deg (mission loads). The two guidance control
parameters, commanded bank angle or reversal time, are checked to make sure they are
not outside the proscribed limits. If saturation of only one control parameter is present, a
single iteration on the remaining free parameter is performed to reduce target miss. If saturation of either or both control parameters is detected, guidance computes a new, achievable offset target point.
A guidance solution is iterated upon until one of the following three criteria is met:
(1) Three iterations completed
(2) Predicted target miss < 500 ft
(3) The commanded bank angle correction < 1 deg AND
commanded reversal time correction < 1 sec
During entry, the proper bank angle rate command sign is computed and this rate is
set to a mission load magnitude times the computed sign. Another parameter that guidance
is responsible for calculating is the time at which the velocity reaches 975.00 ft/sec. This
time is designated as the commanded landing system parachute deployment time.
36
Reversal Phase
The reversal phase consists of a closed-loop predictor/corrector algorithm with the
horizontal target miss distance as the only constraint and the bank angle immediately following bank reversal completion as the only control. The first trajectory prediction estimates the landing system parachute deployment position using the current guidance
command. The second trajectory prediction is made to estimate the stabilization parachute
deployment position for a trajectory with a bank angle perturbed by 2 deg. The minimum
achievable target miss distance is estimated using a linear approximation of the horizontal
miss distance as a function of bank angle. If the predicted miss distance is more than 100
ft (Mission-load) greater than the minimum miss, guidance computes a correction to the
bank angle in order to achieve the minimum miss.
Just as in the entry phase, the bank angle magnitude is limited between 0 and 180
deg. Saturation of the commanded bank angle is checked as well. The guidance solution is
iterated upon until one of the following three criteria is met:
(1) Three iterations completed
(2) Predicted target miss < 100 ft
(3) The commanded bank angle correction < 1 deg
The proper bank angle rate command sign is calculated, and the commanded bank angle
rate is set to a mission load magnitude times this computed sign.
37
3.5 Terminal Subphase
When the bank angle is within .05 degrees (Mission-load tolerance) of reversal
completion, guidance enters the terminal phase. The time between the end of the reversal
and the estimated beginning of the landing system parachute deployment is time spent in
the terminal phase. A bank rate profile separated into bins of time is selected off-line. The
selection of this nominal bank rate profile will be discussed in Chapter 5. A closed-loop
predictor/corrector
algorithm with the horizontal target miss distance as the only con-
straint and the bank rates in each bin as the controls is used in the terminal phase. If the
predicted miss distance is more than 100 ft (Mission-load) from the target, guidance computes a correction to the bank angle in order to achieve a smaller miss distance.
The bank angle magnitude is limited between 0 and 180 deg. The maximum
change in the bank rate from one bin of time to the next is limited by the bank acceleration, which is set at 2.5 deg/sec 2 . The guidance solution converges if the predicted miss is
less than 100 ft.
3.6 Landing System Phase
The terminal phase ends its predictions and corrections at a time, 1 second (Mis-
sion-load tolerance) before the commanded landing system parachute deploy time. From
this time on, the commanded bank rate is held at zero, and no predictions/corrections
are
made. When the PLV-2reaches a velocity of 975.00 ft/sec, the landing system parachutes
are deployed. Guidance is no longer active. A constant bank angle, the final bank angle
38
from the end of the terminal phase, is assumed at this point on. The simulation remains in
this phase until the PLV-2 has reached the altitude of the target.
A typical bank angle profile for a simulation run is shown in Figure 3.3. The different phases are clearly distinguishable. Here a +45 deg bank is commanded through preentry and entry. The reversal phase begins at approximately 533 seconds when a -45 deg
bank is commanded. Once the reversal is complete, the terminal phase begins. In the terminal phase, several different bank changes can be commanded, as illustrated in this
example. The last bank angle of terminal becomes the bank angle through the landing system phase. In this example, the simulation run reaches the termination conditions near 860
seconds.
BankAngleProfile
a,
2'
-C
a,
co
-
0
100
200
300
400
500
Time(sec)
600
700
Figure 3.2: Sample Bank Angle Profile
39
800
900
3.7 Atmospheric Models
The US62 atmosphere model [10] was the main model employed in this thesis
study. The atmospheric density, speed of sound, and pressure are computed by the atmosphere model using the position vector of the PLV-2 as the model's input. The parameter
values at 400 ft altitude increments through the descent phase are saved in table form for
access by guidance. A quick table look-up of the appropriate atmospheric parameters for a
certain altitude is used by the predictor model.
To model the atmosphere even more precisely, monthly or seasonal atmospheres
for the desired target location were generated from the GRAM95 atmosphere model [7].
Day-of-flight measurements could also be taken for greater accuracy in the predictions
during actual flight. The required accuracy level of the atmosphere model depends on the
PLV-2's ranging capability (ability to adapt to a change in target) and on the amount of
capability (the precision of the range ability) to be used for a particular entry. A footprint
of possible landing locations given different bankrate profiles can be generated at one
point in time to determine the ranging capability. Figure 3.4 below shows an example
open-loop footprint of these possible landing locations generated at one instant in time at
the start of the terminal phase. Chapter 5 will describe the footprint characteristics in
detail. The ranging capability is not very robust on the edge of this footprint. For those
entries landing on the edge of the footprint, a very accurate atmosphere model is necessary
[9]. For over time, the locations particularly on the edge become inaccessible in the presence of unexpected dispersions. The importance of an accurate atmosphere model will be
justified through the robustness and dispersion testing in Chapters 6 and 7.
40
Figure 3.3: Example Footprint
Figures 3.4 and 3.5 below show a typical sampling of the atmospheric parameters
during re-entry of the PLV-2.
4
X10S
Altitude vs. Density
3.5
3
~~~~~~~~....
.........
2. 5
..........
.......
.....
...
....
.............
F=_2
§
..
.
.
.
..
.
5
1
0.!t
!I
I
0.01
0.02
0.03
0.04
3
Densitvsluajft 1
0.05
0.06
0.07
Figure 3.4: Altitude vs. Density, Typical PLV-2Descent
41
Figure 3.5: Altitude vs. Speed of Sound, Typical PLV-2 Descent
3.8 Vehicle Uncertainty and Environment Dispersion Sources
In the design of a flight simulation, sources of uncertainty and dispersions must be
addressed in order to correctly model the actual conditions that the PLV-2 will be sub-
jected to. The two most prominent categories of possible uncertainties and dispersions are
listed below [9]:
1. Vehicle Uncertainties
- Aerodynamics, (major source)
- Mass, (minor source)
- Navigation, (minor source)
- Maneuver Rates, Control and Modeling of Actual Vehicle Attitude
42
2. Environmental Dispersions
- Atmospheric Density Variations, (major source)
- Atmospheric Wind Variations, (major source)
- Other Atmospheric Properties, such as the temperature, (minor source)
- Gravity, (minor source)
The vehicle aerodynamics and atmospheric properties are the most important characteristics to be modeled precisely in the simulation. In reality, navigation errors can arise
when the state vectors (position, velocity, and time) are sampled from the true environment by navigation, and are then delivered to guidance at each cycle. This simulation is
modeled as if all minor sources are zero or perfectly modeled.
43
44
Chapter 4
Guidance Design
This chapter presents different guidance scheme options, and illustrates the reasons behind the PLV-2's particular guidance scheme selection. The bank-to-steer control
of the craft is discussed. The predictor and corrector for each phase is defined as well.
4.1 Guidance Scheme Definitions
The PLV-2 will have to undergo some form of control if it is to land within a
desired minimal distance from the target. Historically, there have been two main forms of
guidance control of an entry vehicle; reference trajectories and trajectory control profiles
[2]. The latter was chosen for the PLV-2 guidance scheme. A look into the reference trajectory scheme, utilized by Apollo and the Shuttle, will illustrate the reasons this scheme
was not selected for the PLV-2's mission.
A reference trajectory is a complete course of travel that a craft should follow as it
makes its way to the earth given nominal conditions. The control tries to keep the vehicle
flying, within certain bounds, on this assumed optimal path that has been predetermined.
By staying within a certain corridor of the controlled parameters, the vehicle is indirectly
led to the target. In order to ensure the successful flight of an entry vehicle, great attention
must be given to the energy state of the vehicle. In the past, the reference trajectory
method has proven to provide a good means for control while maintaining energy state
limitations. These limitations are definitely a primary concern for manned missions. In
45
fact, the Space Shuttle and the Apollo missions both utilized trajectory referencing to control their crafts. The downfall of this method comes into play if the craft deviates a large
amount from the reference trajectory itself. This occurs when the conditions during entry
vary greatly from the expected.
The entry guidance schemes for the Apollo and Space Shuttle are described in
Todd Dierlam's thesis written at Draper Laboratory [2]. He mentions several defining
qualities of each scheme. The Apollo guidance, designed for a craft with a low L/D=.3,
was required to maintain a 3 sigma accuracy of 15 nautical miles in track and range from
the desired landing site. Due to the concern for the human crew, the energy state associ-
ated with the heat shield was of prime importance. A reference trajectory was determined
prior to the flight, in order to meet the downrange, heating, and g-load requirements. In
Apollo's case, one control variable, the bank angle, was used to define and maintain the
reference trajectory. Vehicle lift was directed by the reaction control system with a limited
amount of fuel usage. In order to meet crossrange requirements, guidance commanded
reversals by varying the bank angle sign based upon the current crossrange error. One
advantage of the Apollo design, a low L/D craft, was the generation of a small amount of
lift compared to the amount of drag. This feature caused a reduction in the trajectory
length which, in turn, causes the vehicle to be subjected to less atmospheric dispersions.
Less exposure to possible dispersions results in greater landing accuracy. The Apollo
guidance design was not chosen for the PL-RLV. The PL-RLV calls for a more precise
landing scheme than Apollo.
The Space Shuttle requires a more precise landing than Apollo. This craft, with an
L/D = 4.0, has to deorbit from LEO and land as precisely as an airplane does on a runway
46
[5]. Three control variables, angle of attack, bank angle, and a speed brake, are used by the
Shuttle's guidance to maintain the pre-determined reference trajectory. Crossrange control
comes from bank reversals, although these reversals are not based upon the crossrange
error, but on the difference between the current heading and the heading to the target. Like
Apollo, the reference trajectory should result in a flight within the heating and g-load limitations. The profile which results in the desired downrange and which also avoids exceeding any structural and life-support limitations is the one chosen to be flown by the Space
Shuttle's guidance. A guidance scheme with less control variables than the Shuttle's
scheme is desired for the PL-RLV.
The PL-RLV, a craft with a low L/D = .1, is also designed for precision landing. It
is desired to land within 1 nautical mile of the target location even in the presence of wind
variations. The single control is the bank angle. Landing within this proximity of the target
is of utter importance, for the mission chosen for the study stresses the issue of quick, easy
retrieval of its crafts, and a fast turn-around-time between launches. Also, a land-locked
launch location would be feasible if the PL-RLV could land reliably within the 1 nautical
mile boundary. Since the PL-RLV mission has a payload deploying craft with no humans
on board, the vehicle energy state limitations are not as strenuous as they would be for a
manned mission. The heat load and heat rate can be higher for unmanned missions. For
these reasons, a guidance scheme using trajectory control profiles, not reference trajectories, was designed for the PL-RLV re-entry vehicle mission.
When using the trajectory control profile approach, a predictor-corrector algorithm
is utilized. The predictor-corrector takes a reference bank and bankrate profile to recompute the control parameters at each guidance cycle. Since the control parameters are
47
updated constantly, guidance is able to adapt to the atmospheric variations as well as those
variations in the vehicle conditions with great success. The predictor-corrector algorithm
is responsible for the computation of the control corrections necessary to update the trajectory. These corrections are based on the predictions of the final conditions obtained when
the assumed profile is flown to the ground. This is the main difference between the control
profile approach and the reference trajectory approach. Trajectory control depends on the
predicted final landing location, while the reference trajectory method controls the craft by
keeping other parameters within a corridor of values.
Every guidance scheme design cannot be perfect, however. One disadvantage to
the trajectory control profile approach is the fact that the vehicle is guided based solely
upon the final predicted state. This lack of concern for the intermediate states could violate
guidance constraints, such as maximum g-load, even though the desired final conditions
are achieved. Also, a poorly designed predictor-corrector algorithm can lead to inefficiency and result in computationally intensive loads.
The PL-RLV mission, however, is still conducive to a trajectory control profile
guidance scheme, as will be shown in this thesis. The trajectory control profile approach
was chosen for the PLV-2 due to the adaptability of the guidance. The PLV-2 may encoun-
ter wind and other atmospheric variations that would be easily dealt with using this design.
The PLV-2's predictor-corrector algorithm works accurately and models the environment
well. With the PLV-2design, the trajectory can be corrected to yield the best possible solution for the PLV-2 path while in flight.
48
4.2 Trajectory Control
Upon entry, the PLV-2 will experience both gravitational and aerodynamic forces.
T, update the trajectory while still meeting constraints, it is necessary for the entry vehicle
to control the two components of the aerodynamic force; lift and drag. These forces are
indirectly controlled by the bank angle control of the PLV-2.
On most entry vehicles the amount of lift is more directly controlled by the angle
of attack in combination with the bank [2]. The angle of attack can be modulated with the
use of a body flap, reaction control jets, or by center of gravity movement. In the PLV-2's
case, a body flap would add weight and complexity to the vehicle design. Modifying the
angle of attack with reaction control jets would also increase the fuel cost substantially.
Preliminary studies state that control of the craft could be difficult when the center of gravity is shifted [2]. Controlling the magnitude of the lift with the angle of attack becomes
less desirable in the face of these drawbacks. Another option was chosen for the PL-RLV
design in which the PLV-2is flown at a constant trim angle of attack which is determined
by the center of gravity placement. The trimmed angle of attack reduces the complexity of
the design.
Trajectory control is also possible by varying the direction of lift generated instead
of controlling the magnitude of the lift. This is accomplished through the rotation of the
vehicle, and thus its lift vector, about the atmosphere-relative velocity vector. A bank
maneuver can be commanded to rotate the vehicle. RCS jets are only means for this task
on the PLV-2. Since the PLV-2design maintains a trim angle of attack, a trade-off presents
itself - guidance simplicity versus fuel cost. The simplicity of the guidance is a good
49
choice in the PLV-2's case as its relatively small size and minor maneuvers will not cause
a large amount of fuel expenditure.
Drag, the second aerodynamic force to be concerned with, can be directly altered
with changes in the vehicle surface area or by modulating the coefficient of drag [2].
Unfortunately, these methods, such as making structural additions, add complexity to the
vehicle. Simpler vehicle designs call for a more indirect method of altering the drag component of the aerodynamic force. Indirect drag control is accomplished for the PI-RLVby
using the bank angle to vary the vertical lift on the vehicle. Bank angle control allows con-
trol authority over both lift and drag during descent, without adding undue weight or complexity to the vehicle by requiring angle of attack control.
4.3 Predictor
As mentioned previously, trajectory profile control methods make use of a predic-
tor-corrector. The predictor algorithm used in this thesis is a 3-DOF trajectory simulator.
The predictor is responsible for numerically integrating the vehicle's translational equations of motion forward in time by using the vehicle model, the atmospheric model, gravity, and the commanded bank profile. It is necessary to match these models extremely well
with the environment in order to predict the final states precisely and yield robust corrector
performance.
50
4.3.1 Predictor Flow
The predictor is called upon through three phases of the descent: entry, reversal,
and the terminal phase, as shown in Figure 4.1.
FPredictor
- _ W
Figure 4.1: Re-Entry Guidance Predictor Calls
Figure 4.2 illustrates the logic flow of the predictor-corrector algorithm. The vehicle state and parameter variations are fed into the predictor. The predictor then integrates
the equations of motion until the termination condition is met. The final state at the landing location is compared to the desired landing location, and a 2-D error vector is computed. A corrector is then inplemented that modifies the bank control profile in order to
lessen these final errors, and it sends out the converged control commands for the vehicle
to follow. This process continues during each guidance cycle.
51
Figure 4.2: Closed-Loop Predictor/Corrector Guidance
The entire bank rate profile from entry to the stabilization phase is given to the predictor at each guidance cycle. That is to say, the entry phase already knows what the bank
rate profile will be when the vehicle is flying the terminal phase. Initially, this information
is found "off-line" in order to find the most suitable bank rate profile for the initial deorbit
state (at 400 kft) and the desired landing site. The selection of a suitable nominal profle is
discussed in Chapter 5.
4.3.2 Equations of Motion
The predictor itself is a 3-DOF trajectory simulator that integrates the equations of
motion from the current state until the final conditions are met. The basic equations of
motion to be integrated are as follows:
d= v
dt
d9V=
dj= artarl
52
(4.1)
(4.2)
The initial conditions of each prediction are given by the current estimates of position and velocity from navigation:
o =
(tn)
(4.3)
PO =
U"",)
(4.4)
The total acceleration of the vehicle is then computed from the sum of the gravitational and aerodynamic accelerations:
(4.5)
atotal = agrav +alift+ adrag
The acceleration due to lift is calculated using the bank angle,
4, in
the following
manner,
Ujift = (sint)Ufperp
+ ( COS)norm
(4.6)
where the unit vector normal to the plane containing the position vector and the
airmass-relative velocity is denoted uperp,and the unit vector normal to the airmass-relative velocity and that is in the plane containing the position vector and the airmass-relative
velocity is denoted Unormn
=
where
(CLQSref)
CL = the coefficient of lift (a function of Mach and angle of attack)
Q = the dynamic pressure
Sref = PLV-2 aerodynamic reference area
53
(4.7)
m = PLV-2 mass (from Navigation)
The dynamic pressure, Q, is defined by,
(4.8)
Q = 2pv2,
where the relative velocity, vrel, is the magnitude of the airmass-relative velocity vector,
and p is the airmass density. The airmass-relative velocity vector itself can be found by
subtracting the wind velocity vector from the earth-relative velocity vector of the vehicle.
Finally, the airmass-relative velocity vector is converted to the inertial frame.
The acceleration due to drag, is determined from:
(QSref'CD
adrag =- -m
where
(CDSrefPara)'
JVrel
(4.9)
eACD+ Sref
Cd = the coefficient of drag (a function of Mach and angle of attack)
CdSrefPara= landing system parachute Cd times
the reference area when deployed
Lastly, the gravitational acceleration is simply calculated using:
agrav =
(n
where the unit position vector in the inertial frame is defined by
54
(4.10)
Dr
r,.
4.3.3 Integration of the Equations of Motion
The translational equations of motion, Equations 4.1 and 4.2, are integrated using a
3rd order Runge-Kutta algorithm for integrating the position, and a 2nd order RungeKutta algorithm for integrating the velocity. A time step of .04 seconds was chosen as the
time step of the trajectory's integration to the final state. For an adequate prediction of the
environment this time step need not be .04 for all phases. The differential equations of
motion
aRO
aVo - _~tow
-t
= f(t,
(4.11)
Vo
Ro, Vo)= agravO
(4.12)
are integrated by stepping through the following process [4,6]:
1. Assume contact acceleration is zero. Calculate the gravitational acceleration.
agravo = f(to, Ro, Vo)
(4.13)
2. Update the position by using the old gravitational acceleration.
R = Ro+ VoAt+ (agravo)At
(4.14)
3. Compute gravitational acceleration based on intermediate state.
agravl = f(t 1, R1, Vo)
55
(4.15)
4. Update velocity using the verage gravitational acceleration.
V = Vo + (agrav + agravO)At
(4.16)
5. Correct the position using the old and new gravitational accelerations.
Ri
= -
+
t 2
1(agravI -agravO)
~t2
-
(4.17)
(4.17)
4.3.4 Termination Conditions for the Predictor
After each integration time step, the predicted state is compared with the termination condition. The termination condition corresponds with the target's altitude for that
simulation. A more exact assessment of the final landing errors can be made if the simulation ends at the target's altitude. The error is then 2-D, and can easily be calculated. Time
step discrepancies between the environment and the predictor can cause nonhomogeneity
in the altitude termination states. It is necessary for the predictor to terminate accurately,
for the predictor's final state values provide the state errors that are then dealt with in the
corrector algorithm. An error in the predictor can propagate through the corrector, and
cause the corrector to try to correct for errors that do not exist.
4.3.5 Final State Error Computation
The final state errors are computed in the crossrange/downrange
frame. The cross-
range/downrange coordinate frame system is described in Section 2.5. The final horizontal
position error
56
Rerror -=Rpred-Rtarget
(4.18)
is computed by calculating the crossrange and downrange directions as follows:
CRmiss = Rerror
iCR
DRmiss = Rerror iDR
(4.19)
(4.20)
where the predictor's position vector is the final predicted inertial position and the
target vector is the also in the inertial reference frame.
The horizontal error in position is then given by:
ErrOrhorizontal=
(CRmiss)
2
+ (DRmiss) 2
(4.21)
4.4 Corrector
The original entry and reversal phase corrector algorithm was designed by Doug
Fuhry [4]. The corrector's task is to use the predicted horizontal landing errors to update
the bankrate profile to be flown. The resulting profile should yield a minimal final position
error. Three different correctors, each with their own constraints and controls, were used
in the three main stages of the re-entry. While in the entry phase, the downrange and cross-
range target miss distance comprised the two error signals while the two controls consisted
of the time for the reversal maneuver to begin, t
and the bank angle of the reversal, 0o.
Once the craft goes into the reversal phase, the horizontal target miss distance becomes the
only error signal, and the single control parameter is the final bank angle of the reversal.
Upon entering the terminal phase, the two target miss distances in the crossrange and
57
downrange directions are the error signals, and the bank rates in each of the bins comprise
the controls. The different corrector algorithms found in each phase will be described next.
4.4.1 Entry Phase Corrector
In this phase of the re-entry flight, three predictions are made: a nominal prediction, a prediction with t tweaked only, and a prediction with 0 tweaked only.
Y
) Reversal Time tweaked
First Run Case (Nominal)
weaked
Figure 4.3: Predictions for Corrector Guidance in Entry Phase
Prediction#1: t == ttrrnom,
=
_nom=> ACR,
ADR
Prediction#2: t = trrnom + tm, O = o0_nom==> ACRI,ADRI
Prediction #3: tff = trr nom, po = 40 _nom+ Zo ==> ACR 2, ADR2
ACRo and ADRo can be found from data obtained after running the nominal case,
where
58
The nominal prediction is used to estimate the landing system parachute deploy
time by implementing the current bank reversal angle and reversal start time commands. If
this estimated position for the landing system parachute deployment is not within 500 ft of
the target, the two cases with tweaked parameters are predicted. In Prediction #3, the guidance bank angle is perturbed by 2 degrees; and in Prediction #2, the time for reversal is
altered by 5 seconds.
These three trajectory predictions yield target miss distances that are used to compute the numerical partial derivatives of the constraints with respect to the control parameters. The bank angle and reversal time corrections are computed using a Taylor Series
expansion which neglects all terms of second order and higher.
The linear approximations of the changes in crossrange and downrange miss distance are shown below:
ACRo = (a
At,,+(
aR)A O
+
Ao=
,a--rtrr
59
( )A q)
(4.22)
ADRo = (aDR)At + (DR)O
(4.23)
The numerical partials defined by the three predictions are determined by:
aCR
atr,,
aCR
o
CRmissl - CRmisso
t rr,,,
- trr
(4.24)
CRmiss 2 - CRmisso
=
0(2) - o(o)
aDR
atr
DRmissl - DRmisso
trrl - trrO
(4.26)
aDR
o-
DRmiss2 - DRmisso
0(2) - O(O)
(4.27)
where
( CRmissl-CRmisso) = CRmiss from Prediction 2 (the tweaked reversal time) CRmiss from Prediction 1 (the nominal case)
(trrl - tO) = Reversal Start Time from Prediction 2 - Reversal Start Time from Prediction 1
(0o(l) - 0o(o)) = Reversal Bank from Prediction 2 - Reversal Bank from Prediction 1 etc.
With the knowledge of these partials and the nominal cross-range and down-range
miss distances, the controls corrections can be calculated as shown below:
aCR aCR
[Ati
LAoi
arr4
ACRO1
aDR DR
o4o
ADROj
Latrr
(4.28)
The new increments and gains are then subtracted from the old parameters to yield
the new reversal time and bank angle at reversal:
60
trrnew)
4O(new) =
trrzold)-
(K)(Atrr)
O(old)-(K)(AOo)
(4.29)
(4.30)
The gains (K) are evenly weighted factors on the correction terms.
4.4.2 Reversal Phase Corrector
During the reversal portion of the re-entry trajectory, a closed-loop predictor/corrector algorithm was designed with one constraint, horizontal miss distance, and one control, the fixed bank angle of the reversal phase. A nominal prediction is made, as well as a
prediction in which the reversal phase bank angle is perturbed by 2.0 degrees. The correc-
tor then estimates the minimum target miss distance that is achievable by using a linear
approximation of the horizontal miss distance as a function of bank angle. The search for
the minimum miss distance is defined below. This technique was developed by Doug
Fuhry [4]. Figure 4.5 illustrates the minimum miss guidance scheme.
It is assumed that the predicted target miss distance as a function of the reversal
bank angle, gp(), is a continuous, smooth function, and that a single minimum exists over
the possible range of bank angles.
0o
=> nominal prediction
01 = o0+4&
=> perturbed prediction
PO(-o)
=> predicted target miss using nominal bank
2l(l1)
=> predicted target miss using perturbed bank
§S = _l --
=> difference between the miss vectors
61
Figure 4.4: Minimum Miss Guidance for Reversal Stage
The unit vector of the minimum miss distance is calculated and is then used to
define the difference between the nominal miss and the minimum miss distance.
apmin= UNIT(Ae x (o x p,))
Pmin = (Po Upmin)pmin
APmin =
(4.31)
(4.32)
min - Po
(4.33)
The bank correction to be added to the bank angle is defined by:
A
= SIGN(Apo
Ami)(I
62
n)
(4.34)
4.4.3 Terminal Phase Corrector
Once in the terminal stage of re-entry, the closed-loop predictor/corrector algorithm changes again to better handle the characteristics of this phase. Within this stage,
there are two requirements, crossrange and downrange miss distances, and the bank rates
in each bin represent the controls. The original terminal phase corrector was designed by
Dr. Chris D'Souza [3].
The crossrange and downrange errors can be calculated using the Taylor Series
expansion of the control variables, which can neglect the second and higher order terms as
shown below:
ACR =
aCR
aCR
aDR
aDR
-R(ABR,) +
R (ABR2)...
ADR = a--(ABR 1) + a-.R(ABR 2) +
...
aCR
+a
aDR
--+
(ABR,)
(4.35)
(ABR,)
(4.36)
36 bins are chosen for testing purposes; thus, there would be 36 different controls.
It can be seen right away that in any case where there are more than 2 bins, there are more
unknowns than equations. The underdetermined system's desired corrections for each bin
are determined from the minimum norm pseudoinverse that minimizes the root-meansquare (rms) bankrate deviation from the current bank profile:
The A matrix is defined by the partials of the constraints (miss distances) with
respect to the controls (bank rates).
63
aCRI aCR 2
A
aCR]
''
aBRof
aDR aDR 2
aDR
[
aBR
,RIaBR 2
(4.37)
The partials themselves are found from:
aCRI
aBRI
(CRmiss)ominal - (CRmiss)S 1 nl perturbed
(BankRate)inl,
(4.38)
nominal- (BankRate)Blnl. perturbed
ABRI
ABR 2 = A (AAT)-I
CR
(4.39)
~~BR.
where ACR = (Predicted Landing Position)cR - (Target Position)cR
ADR = (Predicted Landing Position)DR - (Target Position)DR
ABR1 = (Bank Rate in Bin #l)nominal - (Bank Rate in Bin #l)ne w , etc
To obtain the new bankrate for each individual bin, the correction term is subtracted from the original bankrate in that bin. For the first bin, this would look like:
(BankRate)Bin I,new
(BankRate)Bin I nominal- BR
(4.40)
After all the bins have their new, corrected bank rate, a prediction is made using
these updates.
64
Chapter 5
Nominal Bank Rate Profile Selection
Chapter 3 briefly discussed the characteristics of the terminal phase guidance
scheme. This chapter will expand on that knowledge and present an in-depth look at the
bankrate bin generation technique used in the terminal phase of the PL-RLV simulation.
The selection of the optimal nominal bank rate profile will be discussed as well. Figure 5.1
below illustrates the steps in the selection process. These steps will be defined in the fol-
lowing sections.
Figure 5.1: Flow Chart for Selection of the Nominal Profile
65
5.1 Profile Generation
In the terminal phase, we are interested in determining all the possible bank rate
profiles that will land the PLV-2 near the target. The nominal bankrate profile will be
selected from these profiles. The first step in choosing a profile is to choose an appropriate
target. An off-line code is used to generate all possible bank profiles. These profiles are
run open-loop at a one point in time to produce a footprint of landing locations. This foot-
print illustrates the area that the PLV-2can land within. A target can be selected within this
footprint. The optimal target selection location will be explored in Chapter 6.
Once a target is selected, the profile generation process repeats. New profiles are
generated and run for the new target. A corrector is present in all phases up until terminal.
Upon entering the terminal phase, an open-loop predictor produces a footprint of landing
locations and when desired also saves acceptable profiles, typically chosen as those with a
horizontal target miss distance of less than 3000 ft.
After running the initial code and investigating all possible bank rate profiles
yielded, these profiles can be chosen from to be used as the nominal profile in a closedloop simulation of the terminal phase. Given the nominal profile, the terminal phase uses a
predictor and corrector scheme to reduce the target miss distance even further.
66
5.2 Bank Rate Bin Definition
As part of the guidance design, the terminal time profile is broken up into several
"bins"- large and small. The off-line code is used to define and generate these bins. Large
bins contain several smaller bins. These small bins are divided into equal slots of time.
Within these large time bins, the bank rate will stay constant. Each large bin, overall, will
consist of a bank rate that falls between the minimum and maximum limitations. The maximum number of bins is chosen as 36, and the number of large bins is chosen as 5. These
numbers allow for a broad range of bankrate profile possibilities, and are small enough not
to utilize too much computation time. The small bins are used for the controls in the
closed-loop terminal phase, as shown in Section 4.4.3.
Figure 3.1 below illustrates how the time line between the end of reversal and the
beginning of landing system parachute deploy can be separated into individual bins. The
time steps are represented by "dt" and the bankrates in each small bins are "br".
Terminal Bank Rate Profile - Bin Breakdown
rcy,
T. nro
T
Bin 1
Bin 2
Ad'
T ar
''
"
T nrae
"
''
Bin 3
t0-tt
Q
M.
W
4"
I
I 4I I
Y
LV
:4
'i
mV
Il
N
I 1
"'
A
T
d
rap
'''
.W
Bin 4
Bin 5
-HHHI
illHll
-Max
Bank
Rate
I
f
dt(8)
An
\ .ULl)
br(8)
--
br(1)
m
.
~
Terminal
Start
/ Small Bins
Time
. _
_-
Figure 5.2: Terminal Phase Bins
67
Min
Bank
Rate
Terminal
Finish
Figure 3.2 below illustrates how all the possible bank rate profiles are incremented
in order to ensure that every possible bin combination will be defined. All bins start out at
the minimum bank rate, and a prediction is made with that profile. Next, the last bin
increases its bankrate by the allotted increment, while the other bins remain at the minimum bankrate. A new prediction is made, and the cycling continues until all combinations
have been created. By looping through all possible bank rate values in all the possible
bins, the off-line code yields all the possible bank rate profiles for the PLV-2to fly during
the terminal phase and also produces the capability footprint.
All of these profiles are flown by the predictor, but ultimately, only those profiles
that yield a final landing location within 3000 feet of the target will be considered suitable
candidates for the nominal bank profile. Those profiles that are actually acceptable have a
range of generated fuel cost and the horizontal target miss distances to choose from.
Beginning Profile Generation
- Max
Bank
Rate
j
0
W
-0
c
Cu
1
U
co
j~
Terminnl
Start
-
Time
_ sl
-Min
Bank
Rate
Terminal
Finish
--
--
Figure 5.3: Bankrate Profile Generation, Cycling Through First Bin
68
5.3 Example Profiles
A sample simulation run is used to illustrate the nominal profile selection process.
Table 5.1 lists the conditions for the bin generation of this example simulation run. These
conditions are fed to guidance. The bank rate search limit is kept below the maximum
bank rate to reduce the risk of bank rate saturation. The bank rate search increment is sufficient to generate a large number of profiles.
Bank Acceleration
2.5 deg/sec 2
Number of Large Bins
5
Max Number of Bins
36
Maximum Bin Bank Rate
+10 deg/sec (0.17453 rad/sec)
Bank Rate Search Limit
+9.1673 deg/sec (.16 rad/sec)
Bank Rate Search Increment
2.2918 deg/sec (.04 rad/sec)
Table 5.1: Bin Generation Conditions for Sample Run
The entry and reversal phase for these profile generations are all closed-loop. It is
not until the terminal phase that the predictor generates the open-loop profiles. Thus, an
inital target value should be set. Table 5.2 below defines the initial conditions for the PLV2 and the target that are used during entry and reversal for this sample run
69
Target
PLV-2
XECEF
-1.27838717442284e7 ft
1.40469665694540e+06 ft
YECEF
1.24061145352144e7 ft
1.30789611824149e+07
ZECEF
-1.09520604018682e7 ft
-1.67303063866579e+07 ft
Altitude
ft
400,000.0 ft
4995.0 ft
XVelocity
-2.5428430228e+04 ft/sec
Yvelocity
4266.949332075 ft/sec
ZVelocity
1753.2389192229 ft/sec
Table 5.2: Initial Targetand PLV-2Setting
At the terminal phase, the predictor flies all the profiles generated. No corrector is
implemented here. Figure 5.3 below shows an example of the footprint of all possible
landing locations that results when each profile is run through the predictor. A +45 to -45
degree bank reversal was commanded here. The landing location at (0,0), called the "noaction point", represents the profile with no terminal phase bankrate action. If the initial
conditions, such as position and velocity, to the simulation are different the footprint will
have a different origin location and shape. The size of the footprint directly correlates to
the altitude at which the open-loop simulations are run. The higher the altitude, the larger
the accessible area. Chapter 6 demonstrates this characteristic of the footprint.
70
Fia
4
c
x 104
Site
Ladn
oll
It
fies
Final LandingSites of AllProfiles:1--O
I
4
2
0
z
-2
-4
-6
-n
--
- 10
.
.
-8
_
I
__ -
-
-6
-4
East(ft)
I
I
-2
0
2
x
4
-
Figure 5.4: Open-Loop Footprint of Landing Locations
The footprint provides many possibilities for target selection. Chapter 6 will discuss the best regions within the footprint to choose the target position. Once the targeted
position is chosen, the off-line code can generate possible bank rate profiles that get close
to the new target.
The target miss distance for each profile is computed from the rss value of the
crossrange and downrange miss distance. Those profiles that result in a miss distance less
than 3000 ft are acceptible for selection as possible nominal profiles. Given these acceptible profiles, the corrector will be able to reduce the miss distance considerably. There are
other limitations in the possible bank rate combinations, however. The maximum change
in the bank rate is defined by:
71
MaxABinBR = SmallBinSize x BankAccel
(5.1)
A profile will not be generated for those combinations that yield a change in bank rate
larger than the maximum defined above. For example, if the first change in bank rate from
large bin 1 to large bin 2 is greater than the maximum possible change in bank rate, that
bin combination is skipped, and no profile generation will occur.
ABinBR 1 > MaxABinBR
(5.2)
Further selection is based upon fuel. The amount of time it takes to make a maneuver change directly correlates to the amount of fuel that is necessary for the maneuver.
More bank rate changes mean a greater amount of fuel that must be expended. The fuel
cost was modeled by the time it took the bank rate to change from one bin to the next.
The fuelcost is modeled as a function of the sum of the bank rate in each bin
divided by the maximum bankrate limit, and is also a function of the time in each bin:
9L~k.
+
MaxBRin(i)!
FuelCost = 1
+ MaxBR)
n(i)(5.3)
i= I
5.4 Nominal Profile Selection
A nominal bank rate profile can then be selected from the acceptible profiles generated in the off-line code. Selection is based upon minimizing either the miss distance,
72
minimizing the fuel cost, or a compromise between the two. The degree to which a profile
consumes the physical capability of the vehicle is tightly correlated to the degree of banking activity in the profile. Profiles that tend to utilize the entire range of bank angles have a
greater access to the achievable landing footprint. For good landing performance, the
nominal profile should be chosen such that the vehicle bank rate is rarely zero. However,
this maneuvering requires more fuel and thus cuts into fuel perfomance.
Once the corrector is utilized, the target miss distance will be minimal, but maybe
at the cost of changing the bank rates in each bin. These maneuvers use fuel. When flying
a particular mission, the optimal criteria from this trade-off should be predetermined. As
mentioned previously, the initial problem for the guidance design to tackle is the selection
of the target point within the initial footprint. This target point should achieve the best
trade-off between fuel usage and robustness to dispersions. Chapter 6 will discuss the
robustness testing of target selection within the footprint.
As an example, a target within the footprint was chosen as seen in Figure 5.4
below. Figure 5.5 shows the 14 possible bank rate profiles that come within 3000 ft of the
chosen target in this example run. It is assumed that any one of these profiles could reach
the target once the corrector is implemented. The optimal one for the mission at hand
should be selected based upon fuel usage and robustness criteria.
It is important to note that the fuel cost for these open-loop profiles was calculated
for the full profile. The entire open-loop terminal profile might not be completed, as the
stabilization time can vary once the corrector is implemented. Since some of the profiles
will not be completed, many of the profiles become identical. For example, if the terminal
73
phase ends within 60 seconds, Profile #2 and Profile #3 are identical. Closed-loop testing
of the profiles will provide a more accurate final target miss distance and fuel cost. The
closed-loop testing of the 14 profiles can be seen in Chapter 7.
FinalLandingSites of AllProfiles
x 10'
I
I
t
I0
z
-2
-4
__l.I
-10
-10
-8
-6
-4
East(ft)
-2
0
Figure 5.5: Location of Target Selected Within Footprint
74
2
x10o'
Profi 1 MIDeWl-2296.142
FuelCost- 18 304
0
O*
-00
,
-0. 1
to
0
20
_nc
..
,
. .I.
30
40
50
6o
70
Profll 2 M lt
2391315FuelCot * 16.5256
Profile 4 MDbt-.
290 642 Fuel Cost
80o
17.0858
01
005.
'
.
.
..
...
.-
0
-00
'
10
20
I
30
40
50
60
70
80
90
Time(sec)
Profile7 MisDsl. 2475.062
FuelCost· 17.9784
O
Figure 5.6: Acceptable Bank Rate Profiles (#1-#9) Generated for New Target
75
Pmle tOM10lgd 2666362 Ful C st 16.3218
015-
I
I
I
I
i
0.1
005
0
20 P10 30
20
30
o10
40
IO u
e
4 70.14Ful Co
70
7
.
O.
-0.02
.
-.. AI
1I
0
10
20
30
40
90
so
..
50
60
70
80
90
80
I0
Potle 12MuDist. 2725679Fuel Cost 15.5579
1
_
_
-
I005
0
Q
-005
-.
"0.I
10
20
30
40
50
60
70
Tme(.ec)
Pmfile
13bIsssD-1166.749
FuelCost=13.4429
l .
A7
.
.
.
.
.
.~-
-""2
0
2-0.02
0
0
10
I
20
I I
30
40
50
60
70
Promfe
14 OssOit=
2936.181
Fuel
Cos= 11.3279
0.02
I
I
I
60
70
80
90
80
90
0
a
0
.
C,1
-0.02
0
D
ca
0
10
20
30
40
50
Tone(sec)
Figure 5.7: Acceptable Bank Rate Profiles (#10-#14) Generated for New Target
76
Chapter 6
Robustness Testing
This chapter discusses the corrector's capability within the footprint of possible
landing locations. The size of the footprint over time is also studied. Four different target
areas are investigated in order to characterize performance in those regions of the footprint.
6.1 Range Capability
Once the initial footprint is determined, the target for testing purposes can be chosen from anywhere within that region. Testing of the footprint is performed to successfully
select a target which will achieve fuel cost limitations and maximize the capability of the
PLV-2. Off-line testing of the footprint will enable preflight selection of the target.
Four different targets, A, B, C, and D, were selected in different regions of the
footprint in order to test the vehicle capability. Figure 6.1 below shows the location of Target A, B, C, and D. The target value at the origin (0,0) of the footprint plot is the resulting
landing location if there were no maneuvers performed during the terminal phase.
77
Figure 6.1: Target A, B, C, and D Locations Within the Footprint
The target locations in ECEF and North/East coordinates are given in Table 6.1
below.
Target A
ii
- .-- - - - -
ZECEF (ft)
gog~3a
.
.
'4
_
A
j
A
Target D
Target C
i Target B
s
_
_
_
1.277055661
-1.2793245034
e+07
-1.2763761472
e+07
-1.271515701
+07
.2451867278
+07
1.2415871446
e+07
1.2435381223
e+07
1.2441237349
e+07
1.091585896
+07
-1.0930181969
e+07
-1.0942369776
e+07
-1.0991867543
e+07
.266886e+04
2.5794145e+04
1.1420108e+04
-4.697762e+04
-4.68234e+02
-3.492335e+04
-7.300804e+04
4.196583e+04
e+07
..
.
Table 6.1: Target Coordinates in ECEF and North/East Frame
78
Given the initial bank rate profiles to reach these targets, it is of interest to see the
range of capability that each target location can achieve. This is accomplished by using the
initial bank rate profile, but the target location is moved further and further away from the
original target. The guidance scheme can redirect its maneuvers to try to reach these new
locations. At some point, the scheme will not be able to make enough corrections to be
able to hit the desired targeted location. This range of capability is directly related to the
original target's location within the footprint and to the ability of the guidance algorithm..
Four cases which come from the example simulation run in Chapter 5 are explored below.
6.1.1 Case 1 - Target A
Target A lies near the outer edge of the footprint. The bank rate profile to reach this
target is shown below in Figure 6.2.
Figure 6.2: Bank Rate Profile for Target A
79
The desired target location was increased to distances further and further from the
original target. Figure 6.6 shows the range capability from Target A's location. With the
original profile as an input, the guidance scheme was only able to capture other targets that
were located within a small area near Target A.
6.1.2 Case 2 - Target B
Target B lies north of the footprint's origin. The bank rate profile to reach this target is shown below in Figure 6.3.
I
Time (sec)
Figure 6.3: Bank Rate Profile for Target B
Figure 6.6 shows the range capability from Target B's location. With the original
profile as an input, the guidance scheme was able to capture other targets that were located
over a wide area of the footprint.
80
6.1.3 Case 2 - Target C
Target C lies near the center of the footprint. The bank rate profile to reach this target is shown below in Figure 6.4.
0.04
Bankrate Profile for Target C
r
0.03
0.02
0.01
I
.. . .. .. . .. . .. . .. ... . . ... .. .. ... .. . .. .. .. ... .. .. .
0
.. .. . . .. . .. .. .. .. .. . ... ... . . . .. . .. .. ... .. . ... .. . .. ... . .. ... .. . .. ... . .. .. . .
-0.01
. . .. ... . .. .. .. ..
-0.02
. .. . .. .. .. .. .. .. ... ..
...........
. . . .. .. ... . .. .. .. .. .. .
-0.03
-0.04
. .... . .
I
10
.
.
20
.
30
. .. .. .. . .. . . ... ..
... . ... .. . . .. . ... .. . . ... .. . ... . . .. .. . ..
.
.
40
50
Time '"'~'
I- c-
I
60
.
70
I
80
90
Figure 6.4: Bank Rate Profile for Target C
Figure 6.6 shows the range capability from Target C's location. With the original
profile as an input, the guidance scheme captured other targets that were located within the
center of the footprint.
6.1.4 Case 3 - Target D
Target D is centrally located in the lower region of the footprint. The bank rate profile to reach this target is shown below in Figure 6.5.
81
I
Tlmo sol
__
Figure 6.5: Bank Rate Profile for Target D
Figure 6.6 also shows the range capability from Target D's location. With the original profile as an input, the guidance scheme was only able to capture other targets within
a small circular region around Target D.
6.2 Nominal Target Robustness Results
The resulting range capabilities for the three cases are shown below.
82
4
F
in
v 1n4
I
St
of
Al
P
i
Final Landing Sites of All Profiles
9
0
z
-10
-8
-6
-4
East (ft)
_~~~~~~~~~~~~~~~~~~~~~~~~~
2
0
-2
0
x 10
Figure 6.6: Range Capability of Test Cases, Final Miss Within 10,000 ft
83
4
t4
V
IA
Fina Lanin Site of AlIroi
~~~~~~~~~~~~~~~~~~~~~~~~~....
Final Landing Sites of All Profiles
z
-10
-8
-6
-4
-2
0
East (ft)
--
,~~~-
-w
-
~
2
x 104
-
Figure 6.7: Range Capability of Test Cases, Final Miss Within 1,000 ft
Target A was on the edge of the footprint, and thus had a small range of capability.
Less maneuverability
within the proscribed bank limits is available when a target is
located near the edge of possible landing locations. Although located in the lower central
region of the footprint, Target D's range capability was limited by the edge as well. Target
B's location enabled the capture of a wide range of landing locations for both miss conditions. Target C was located in the center of the densest section of target locations. With
this initial location, the guidance scheme was able to maneuver to a central range of final
locations. In this example, it is obvious that Target B has the largest range of capability,
84
which is a desirable characteristic of target selection. Another characteristic, the footprint
range through time, must also be considered before target selection is final.
6.3 Footprint Range
In selecting the nominal target, its location within the achievable open-loop footprint must be considered, as well as how this footprint changes through time. During the
mission, the target location might need to be altered, or in-flight dispersions could happen
far enough along the trajectory to make it impossible to hit the target. A target location
with robust capability potential throughout time is the most desireable option.
The area attainable by the PLV-2 changes as the descent time increases. If the
open-loop footprint is generated near the beginning of the terminal phase, it will be much
larger in area than if the footprint is generated 30 seconds into the terminal phase. Figures
6.7 - 6.13 below show the resulting footprints generated at different times during the terminal phase: 0, 5, 10, 15, 20, 30, and 40 seconds into terminal. The original target locations (A,B,C,D) are left on the figures for comparison in the footprint size changes.
85
Final LandingSite$of AllPofles: 10
FinalLandingSitesof All Profiles:.0
X 10
a 10'
I
4
2
0
-2
-4
-6
-8
-10
i
-8
.
*
-6
-4
i *I
-2
East (ft)
I
0
2
x 10'
Figure 6.8: Footprints at Time= 0 seconds in Terminal
4
v 41
FinalLandina
Sitesof AllProfiles:
t=5
i
Figure 6.9: Footprints at Time= 5 seconds in Terminal
86
4
FinalLandingSitesofAll Profiles:t10
P x 10
_ _
4
2
0
. . ... .
-2
-4
-6
-_IR 0
-10
*
-8
.
*
-6
-4
-2
East(ft)
- - --- -
0
~
~ ~ ~
2
x 10
I`
Figure 6.10: Footprints at Time= 10 seconds in Terminal
Figure 6.11: Footprints at Time= 15 seconds in Terminal
87
Figure 6.12: Footprints at Time= 20 seconds in Terminal
_·
i
Final LandingSitesof All Profiles:1t=30
x 10'
4
2
0
Z
z
-2
-4
-6
-R
-10
I
-8
I
-6
-4
East()
I
-2
0
2
x10'
Figure 6.13: Footprints at Time= 30 seconds in Terminal
88
_
_ __
10'
,
fl
........
_
..
.. . ..... ... ........ ........ .............. ....... ........ ..............
4
2
__
FinalLanding
SitesofAUProfile:1t40
......... ............ ................ ................ ................................
0
~, ......
.. .
.... ....... ...
.....
.......
-2
-4
.
_~~~~~~~~~~~~~~~~·
.
:I
,
.
.
.
·
·
·
_·····
-6
-A
-10
-8
-6
-4
East(ft)
-2
0
2
x 10'
Figure 6.14: Footprints at Time= 40 seconds in Terminal
As seen in the figures, the sooner the open-loop footprint is generated, the larger
the attainable area. This feature becomes extremely important in target selection. Dispersions could arise along the PLV-2's descent trajectory. Early on, the corrector can easily
handle these dispersions and direct the PLV-2 to the target. A dispersion arising near the
end of the terminal phase, however, could result in a target miss if the target is not selected
well. The craft could simply be incapable of performing the bank manuevers necessary to
reach the target. Figure 6.13 shows the footprint after 40 seconds in terminal. Those
"lobes" are the only regions that could be accessible at that point and time.
The best choice for target selection from within the footprint can be seen from the
range testing and the dispersion robustness testing. All of the characteristics of the footprint should be taken into consideration to find a suitable target to steer towards. These
89
results display the basic,target selection criteria. Target B is the best option given its range
capability and presence within the footprint over time. This target will be used for the
remainder of the test cases presented unless otherwise noted.
90
Chapter 7
Guidance Performance
The performance of the predictor and corrector for several cases are presented in
this chapter. This chapter also explores the guidance performance in three specific areas:
fuel cost versus landing performance, optimizing the bin numbers, and the effects of atmospheric dispersions. The corrector's performance in each of these cases will be shown.
7.1 Predictor and Corrector Performance
As mentioned previously, the predictor's ability to accurately model the actual trajectory path is of utter importance. The more exact the predictions, the better the corrections will be for steering the PLV-2 towards the desired target. Cases will be presented to
show the predictor and corrector fidelity throughout the entire simulation and also particularly in the terminal phase.
7.1.1 Case 7.1 - Target Set to Actual Open-Loop Landing Location
Case 7.1 is an open-loop case with the target set to the actual landing location. The
corrector is not utilized. Table 7.1 below shows the initial and final state vectors as well as
the phase times during the simulation for Case 7.1.
91
Initial Inertial Position (ft)
1.404696656945e6
1.30789611824149e7
-1.67303063866579e7
Initial Inertial Velocity (ft)
-2.542843022765e4
4266.94933207518261042
1753.23891922291841183
Time at Start (sec)
0.0
Final Inertial Position (ft)
-1.35308617120651e7
1.15858368540514e7
-1.09531053051175e7
Final Inertial Velocity (ft)
-623.7359138965
-1177.1030328138
179.515068968216
Time at Finish (sec)
854.24
Target ECEF (ft)
-1.27833779269691e7
1.24056875285493e7
-1.09531053051175e7
Entry Start Time (sec)
133.04
Reversal Start Time (sec)
533.04
Terminal Start Time (sec)
543.04
Stabilization Start Time
709.28
(sec)
Table 7.1: Case 7.1 Simulation Conditions
Figure 7.1 presents the bank angle history for this test case. A +45 degree bank is
initially commanded until the reversal time (533.04 sec) when a -45 degree bank is com-
manded. The terminal phase starts at 543.04 seconds into the simulation. The bank rate
profile for the terminal phase can be seen in Figure 7.2.
92
Figure 7.1: Bank Angle Profile for Case 7.1
4
,
U.l
n
0.05-.
ci
i
0-0.05-0.1
750
7nn
. 1-N
6,w
650 -
20
600
550
*-_
0.1
: 0.05
ED O
.
-0.05
n -0.1
I
~--
.
.
Hr \--
i
10
Bin Number
.
. .
500 0
.
time (ec)
3
.
. ' i .
, .' '. '. . . .
- ...-
InitialProfile
Final Profile
02468 1012141618202224262830323436
.
.
.
.
.
.
.
-
Bin Number
Figure 7.2: Terminal Phase Bank Rate Profile for Case 7.1
93
4
Figure 7.3 shows the error between the actual landing location and the predicted
landing location. For Case 7.1, the actual landing location is the target location. In this
case, the pre-terminal phases know about the terminal bank rate profile and the predictor
uses this knowledge in its calculations. During entry, the error is on the order of 1250 feet,
but by the end of terminal, the error is near 70 feet. The jumps near 220 and 310 seconds
are due to jumps in the atmosphere tables. The reversal can be seen near 533.04 seconds.
The error increases slightly at one point in the terminal phase when the bank rates change
from negative to positive values. A small timing error when the bank rates change could
cause this bump, but since it is on the order of 20 feet, it can be neglected at this level.
HorizontalMissvs. Time
I
I
300
400
I
I
I
1200 I-
1000 I-
F
800
'a
r
0
600
400
200
I
v-
100
200
500
600
700
800
900
Time secl
Figure 7.3: Open-loop Horizontal Prediction Error: Case 7.1
94
Figure 7.4 below shows the same test run as Case 7.1, but the pre-terminal phases
do not know about the terminal bank rate profile. A zero bankrate profile is assumed after
the reversal. The error at the beginning of entry is on the order of 22,000 feet, and at the
landing system parachute deploy, the error is near 70 feet. The difference between Figure
7.3 and 7.4 shows how important the knowledge of the bin activity is for the predictor.
Predicting with a better accuracy at the beginning of the trajectory will enable the guidance scheme to handle dispersions encountered along the way with better success.
Horizontal
Missvs. Tine
I
I
I
x 10
I
2
-
I
I
I
L
.
1.5
.
......
'a
8
1
0..5
/'
100
200
I
I
300
400
-
I
500
-
600
700
800
900
Timerf .el
Figure 7.4: Open-loop Horizontal Prediction Error: No Pre-terminal Bin Knowledge:
Case 7.1
Predicting the actual landing location with such a small error gives the corrector a
sound base to work from. Figure 7.5 shows the same simulation run, but with the terminal
phase corrector implemented.
95
Horizontal
Missvs.Time
Time secl
Figure 7.5: Horizontal Prediction Error, Closed-Loop Terminal Phase
With the terminal phase corrector on, the target miss distance at the end of terminal
is 61 feet. Only minute corrections were necessary for this run, as the target position was
initially set at the real open-loop landing location. These plots show how well the predic-
tor and corrector work together to minimize the landing location error.
7.1.2 Case 7.2 - Target Offset From Open-Loop Landing Location
Case 7.2 is an open-loop case with the target offset from the actual landing location.
The
new
target
in
ECEF
coordinates
is
[-1.27829366885425e+07
1.24074256101389e+07 -1.09515194167939e+07] ft. The corrector is not utilized. Figure
7.6 below shows the horizontal predictor error with respect to the actual landing location
and the target location.
96
PLV-2Descent
Rf'W~
is
·
I
2500lF
I
·
· ·· ·I·-
·
s
W
WRT
Actu
-- -
WRT
Target
7:.T...
.. . .. . . .
.........
.
.
......
...
...
Z2000
g 1500
•
S
0
x
1000 .
500
il
...
.
100
.
.
.
200
300
400
.
.
500
Time(sec)
.
600
.
.
700
800
900
Figure 7.6: Open-loop Horizontal Prediction Error: Case 7.2
Once the corrector is utilized, the prediction error decreases. Figure 7.7 below
shows the closed-loop response. The predictor is now able to predict the actual landing
location within 100 feet. With corrections, the final miss distance is 1135.5687 feet. Figure
7.8 displays the terminal phase bank rate changes that were necessary for target acquisition.
97
PLV-2 Descent
6
0
'aCo
e
ao
0
r
0O
Time (sec)
Figure 7.7: Closed-loop Horizontal Prediction Error: Case 7.2
BankRate Corrections
n
005
a0
,·
- 005
40
time (sec)
V
BinNumber
0.1
20.05,
mu X
Xu
. ..
0.0
0-
~r :
. Initial
Profile
_;__
l
Final Profile
c -0.05
m -0.1
.
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36
Bin Number
Figure 7.8: Terminal Phase Bank Rate Profile for Case 7.2
98
7.2 Fuel Cost vs. Land'ng Performance
As mentioned in Chapter 5, there is a trade-off between fuel cost and landing performance when selecting a profile. Four different open-loop profiles will be investigated to
show how the closed-loop guidance scheme performs under different conditions. The
cases chosen to investigate come from the acceptable profiles presented in Chapter 5.
The closed-loop performance for all the profiles is shown below in Table 7.2.
Profile Number
Fuel Cost (sec)
Target Miss
Distance (ft)
1
11.033
95.23
2
11.304
16.24
3
11.41
89.76
4
11.51
68.05
5
10.82
97.81
6
11.58
91.73
7
10.98
1432.206
8
10.64
103.98
9
10.62
47.019
10
10.97
75.97
11
10.23
54.39
12
10.756
926.43
13
9.92
71.81
14
10.31
641.06
Table 7.2: Closed-Loop Results for the 14 Profiles
99
All but four of the cases come within 100 feet of the target. All of the cases come
within 1433 feet of the target. All but one profile (Profile #9), had the target miss distance
decrease. The small increase in Profile #9's miss distance is due to predictor error and the
different start time for the landing system phase once the corrector was implemented. Figure 7.9 below shows the closed-loop miss distances in relation to the target.
_
Closed-LoopMissDistances
"^A
15C
100
50
x#2
:
0
x#9
+
x#10
z
-50
x
#8
x#
x#13
11
#6
x
_.5
-100
-150
-0
-200
-150
-100
-50
0
50
EastMiss(ft)
100
150
200
Figure 7.9: Closed-Loop Target Miss Distances for the 14 Profiles
7.2.1 Case 1 - Initial Conditions: High Fuel Cost, Low Miss Distance
The first case, Profile #1, has an initial high fuel cost of 18.306 seconds, and a low
miss distance of 2296.142 ft. When the profile is used as the nominal bank rate profile in
closed-loop simulation, the resulting bank rate changes occur. See Figure 7.10. With the
100
presence of the corrector, the final miss distance is 95.23 ft at a fuel cost of 11.033 seconds. Figures 7.11 and 7.12 show the PLV-2's path towards the target as well as the pre-
dicted landing locations along the way. Most of the predictions are located within 100 feet
of the target.
__ _
Bankrate
Corrections
0.1
3 0.05
O
<
r1z-0.05
m
-0.1
750
___
700
time(sec)
600
0
Bn Number
13,
:av,
- - -
InitialProfile
Final ProfileI
ra:
cc
M
By
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36
Bin Number
Figure 7.10: Corrections to the Bank Rate Profile - Case 1
101
40
Figure 7.11: Trajectory Path - Case 1
PLV-2 Descent
2.!
-
a, 1.5
1
0.5
3000
2
3000
Nor
EastMiss (ft)
Figure 7.12: Trajectory Path - Case 1
102
Figure 7.13 shows the prediction error for Case 1.
PLV-2 Descent
SPAR
2500
~I ~~~I
I
---
WRTActua
WRT
Targqt
2000
C
s
.2
21
C 1000
r
0
500
n
630
640
I
I
I
I
650
660
670
Time(sec)
680
690
I
I
700
710
Figure 7.13: Prediction Error- Case 1
7.2.2 Case 2 - Initial Conditions: Average Fuel Cost, Average Miss Distance
The second case, Profile #2, has an initial average fuel cost of 16.5256 seconds,
and an average miss distance of 2391.315 ft. Figure 7.14 shows the resulting bank rate
changes in closed-loop simulation. With the presence of the corrector, the final miss dis-
tance is 11.304 ft at a fuel cost of 16.24 seconds. Figures 7.15 and 7.16 show the trajectory
path and the predicted landing locations.
103
BankrateCorrections
(U
4
T
21
Ir
5I
40
0.15
0.1
0.1-_.:
' 005
a
Initial Profile
Final Profile
0m-0.05
.
0 2 4 6 8 1012141618202224262830323436
Bin Number
Figure 7.14: Corrections to the Bank Rate Profile - Case 2
Figure 7.15: Trajectory Path - Case 2
104
C
PLV-2Descent
<:
r-
30
F
EastMiss(ft)
Figure 7.16: Trajectory Path - Case 2
The prediction error for Case 2 is shown in Figure 7.18.
PLV-2 Descent
I
.
I
I
'
i
90
.
WRTAci
WRTati
Act
I---
I:
..
WRTTarg .
70
ri 60
C
0
. ....
......
N 40
I0
30 . ..
.....
... . . ... . . . I . ....
I
10
......
I
.
. .....-..
...
....
I
600
. . .-
.........
. ...... II
20
.....
650
700
-
-
.
.. ... _ .
-
-
-
..
-
-
-
-
-
-
......
....
:
:
:
750
Time(sec)
800
850
Figure 7.17: Trajectory Path - Case 2
105
.
...
900
7.2.3 Case 3- Initial Conditions: Low Fuel Cost, High Miss Distance
The third case, Profile #14, has an initial low fuel cost of 11.3279 seconds, and a
high target miss distance of 2936.181 ft. Figure 7.18 shows the resulting bank rate changes
in closed-loop simulation. With the presence of the corrector, the final miss distance is
641.06 ft at a fuel cost of 10.31 seconds. Figures 7.19 and 7.20 show the trajectory path
and the predicted landing locations.
BankrateCorrections
020.1
0-
40
time(sec)
0.2 . . . .
a 0.1
BinNumber
.-
.
Cu2
e -02.1
.
4 6
.
.
.
.
InitialProfile
.t
Final Profile
.
0 2 4 6 8 1012141618202224262830323436
-
Bin Number
---
Figure 7.18: Corrections to the Bank Rate Profile - Case 3
106
Figure 7.19: Trajectory Path - Case 3
.
PLV-2 Descent
2.5
2.5\
.,.
~
*-
..
1ol.5-
300021
20 ,
-
f
2000
·
ooo
...\ .
.-
-p...
30
.
.
-..
000. '
2000
000
2000
NorthMiss(ft)
3000
-
1000
-3000
__
__
Figure 7.20: Trajectory Path - Case 3
107
Figure 7.21 displays the prediction error for Case 3.
PLV-2Descent
./,
i
I
I
I
WRTActua
WRTTargEt
- -- - - - 600
500
..·
w
c 400
.
I
I,
I
I
I
.'
I
200
*I
*
r
I
:
100
I'
I
/
il
O1
600
650
&
Il
700
I
,
750
Time(sec)
800
850
900
Figure 7.21: Trajectory Path - Case 3
7.2.4 Case 4- Initial Conditions: Low Fuel Cost, Low Miss Distance
The fourth case, Profile #9, has an initial low fuel cost of 14.2069 seconds, and a
low target miss distance of 2.5433e-4 ft. Figure 7.22 shows the resulting bank rate changes
in closed-loop simulation. With the presence of the corrector, the final miss distance is
47.109 ft at a fuel cost of 10.62. Figures 7.22 and 7.23 show the trajectory path and the
predicted landing locations.
108
Figure 7.22: Corrections to the Bank Rate Profile - Case 4
Figure 7.23: Trajectory Path - Case 4
109
L
PLV-2Descent
lo8;
,,
........
x lo'
.
.......
-
· ·· · ·
V ............ 1.. ...:'·
.
-1min
.
· · ·· · ·
.
'
:....
·· ·
· · ··
·· · ·
·· · ·· ·
··· .. · · .. :··
·· ··-
:·min
·
-0
'
'
0
'.,,,,
-
:
"
: ......
,,.
-' '
-1.000.
-2000
NorthMiss(t)
0
3000
''
X
1000
...
2000
-2000
-3000
East Miss ()
Figure 7.24: Trajectory Path - Case 4
PLV-2 Descent
0
.2
6
°
no
I
0.
0
640
650
660
670
680
Time (sec)
690
700
Figure 7.25: Trajectory Path - Case 4
110
710
720
7.2.5 Fuel Cost vs. Landing Performance Discussion
Four different initial profiles resulting in a range of fuel cost and target miss values
were shown. The corrector was able to lessen the target miss distance in all but one case.
Those cases with the most bin activity and bank rate changes suffered in fuel cost. Profile
#9 is the optimal selection for the nominal profile, as it resulted in a small fuel cost, and
achieved a minimal target miss value. When the open-loop profiles are generated, the one
with the smallest miss distance will inevitably be one with a small closed-loop miss distance as well. The closed-loop fuel cost does not vary greatly between the different profiles, but a profile like #9 which as an initially low miss distance and fuel cost will have a
small closed-loop fuel cost as well.
Figures for other characteristics of the nominal profile simulation, such as the heat
loading and dynamic pressure profiles, are given in Appendix C.
Open-loop
Miss Distance
Closed-loop
Miss Distance
Open-loop
Fuel Cost
Closed-loop
Fuel Cost
(ft)
(ft)
(sec)
(sec)
Profile #1
2296.142
95.23
18.3604
11.033
Profile #2
2391.315
16.24
16.5256
11.304
Profile #14
2936.181
641.06
11.3279
10.31
Profile #9
2.5433e-4
47.019
14.2069
10.62
Table 7.3: Fuel Cost and Miss Distance Comparisons
111
7.3 Corrector Performance On Nominal Profile
The nominal profile target was offset by .01 deg East in order to evaluate the corrector's performance on the nominal profile (Profile #9). The new target coordinates in the
ECEF frame are:
[ - 1.27953660797014e+07
1.2413594041297e+07 -1.0930143171118e+07]
ft.
Figure 7.26 shows the predictor error with respect to the actual landing point and
also the target point. The target was acquired in this case with a 50.2198 ft miss distance.
The changes to the bank rate profile can be seen in Figure 7.27.
PLV-2 Descent
earn
...
.
.
ant
I( -
WRTActu
WRT Target
- - -
2000
o
i 1500
2SD
0
M
oo
*8
o
500
-
n
600
Id,
..
650
II.~~~~~~~~~~~~~~~
.
.-...
-
700
-
.
.I
750
Time (secl
I
800
850
900
Figure 7.26: Closed-Loop Prediction Error: Offset From Nominal Target
112
BankRateCorrections
x,<
U.Ut
^ 0.04
:e 0.02
cc
0
c -0.02
-0.04
750
____
40
05...............
I..-
Initial Profile
Final Profile
-0.05
0 2 4 6 8 1012141618202224262830323436
BinNumber
--
Figure 7.27: Terminal Phase Bank Rate Profile: Offset From Nominal Target
7.4 Bin Number Selection
The number of large bins dictates the amount of bank rate profile possibilities that
can be generated by the open-loop code. A range of large bin numbers from 3 to 6 were
tested in order to find a good balance between the number of profiles generated with an
emphasis on reducing the computational load. The maximum amount of small bins was
left at 36. Table 7.4 lists the large bin number, the amount of open-loop profiles generated
within limitations for each large bin number, and the number of profiles that fall within the
acceptable range. The input target for this study is the same target for the original footprint. The coordinates are given in Table 5.2.
113
Large Bin Number
# Total Profiles
# Acceptable Profiles
3
85
1
4
381
1
5
1731
6
6
7887
32
Table 7.4: Large Bin Number Range Testing
From the footprint plots below, it can be seen that a large bin number of less than 5
results in a very small range of landing locations. A large bin number greater than 5 is
dense with landing possibilities, however a larger bin size would take up too much computation time with little to gain in the profile selection compared to the bin size of 5. A large
bin amount of 5 covers a wide range of profile possibilities, yet does not strain the computational load.
114
Sitmol #4 Pto6.a I.SSis.3. 8Sf.
FlmI L8K84V.Q
*
36
-
I. o'
avow
Fhl
drt
~
Ses
ot
mP.l
LlnlI.
3, St
K· 36KK-
s
FIhi Lndng S of Al Pfi.
lS l
* 4, 8Bi . 36
Lk.
KK
*4
2
.
x~r
l
m
·
'
t
g
",.m
Ks
88
~~~
V
_4
.n-
4
..
-v-
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Figure 7.28: Open-loop Landing Locations for Large Bin #'s of 3, 4, 5, and 6
In the closed-loop code, the amount of small bins affects the fidelity of the guidance scheme. A correction can be made on each bin individually. A range of the maximum
number of small bins was investigated as well. In one set of test, the number of large bins
was left at 5, while the maximum number of small bins used in the open-loop testing varied from 10, 20, 30, 35, 36, and 40. As seen in the figures blow, when 36 bins are used, the
footprint essentially covers the same area as when 30-40 bins are used. Thus, the capability when using 30-40 bins is the same. The more small bins, the more controls available
115
once the corrector is utilized. A maximum bin number of 36 offers a good balance
between the open-loop and closed-loop capability.
__
, lo'
Fll LdinG 6Set d A Pl.U
1
L
__
. SBh s. 10
r
l AI P.Io e: U.L,
Fxu L.ndig
st
o
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lo'
0
Figure 7.29: Open-loop Footprint Given Maximum Small Bins of 10, 20, 30, 35, 36, & 40
116
Closed-loop testing was also performed to justify the choice for maximum number
of bins. The nominal profile was flown, but the time increments were changed such that
the number of bins was ranged from 18, 36, 72, and finally to 108. Table 7.5 below shows
the final miss distances for the same simulation run (Target B, Profile 9) given different
maximum small bin numbers.
Number of
Small Bins
Horizontal Miss Distance
from Target (ft)
18
44.29
36
47.019
72
2.574757019 e4
108
1.993549189 e4
Table 7.5: Small Bin Size Testing Results
The greater number of small bins, the smaller the bin times. Each bin time can be
given a correction to better steer the PLV-2to the target. But, if the bin times are too small,
there will not be enough time for the bank maneuvers to complete. For those bins greater
than 36, the horizontal miss distance was large. For a bin number of 18, there was a small
reduction in the miss distance of three feet. Since a greater number of bins will handle dispersions better, this reduction in miss distance due to a smaller number of bins is not the
best option for the bin number. The balance can be made by selecting the small bin number of 36.
117
7.5 Effects of Atmospheric Dispersions
As the PLV-2 re-enters the atmosphere, it will be subjected to dispersions caused
by wind, density changes, and pressure changes. An atmospheric modeling program,
GRAM-95, was used to generate dispersion data. GRAM-95 data also provides an insight
to monthly and seasonal variations. Case 4 (Target B, Profile 9) was used as the nominal
case to test the guidance scheme robustness to different atmospheric dispersions in two
different seasons, winter and summer. Figures 7.30- 7.37 below show the sampled mean,
1-sigma,and 2-sigma wind variation and density variation profiles for test cases.
118
x 10~~
x 10'
Altitude vs. Density/Mean
Summerl Case
Denslty~~~Ill
Aftftudevs. Density/MeanDensfty:Summer Case
i
0.5
06
0.7
0.8
0.9
1
I.1
3
Densityslugs/ft1
1.2
1.3
1.4
Figure 7.30: Altitude vs. Density: Summer Case
Figure 7.31: Altitude vs. Density: Winter Case
119
1.5
Figure 7.32: Altitude vs. North Wind: Summer Case
Figure 7.33: Altitude vs. North Wind: Winter Case
120
Figure 7.34: Altitude vs. East Wind: Summer Case
I
n
s
Altitudevs. East Wind:WinterCase
S
M
R
0
EastWind ft/secl
Figure 7.35: Altitude vs. East Wind: Winter Case
121
I
A1le..i
.
%!A.41~1 tUAin,
C)..
e
Vertical
Windftt/secl
I
Figure 7.36: Altitude vs. VerticalWind: Summer Case
I 1l
s
L
__
Altitudevs.Vertical
Wind:WinterCase
0
a0
,
't
D
Vartil
Wnd
Aft/s1
Figure 7.37: Altitude vs. Vertical Wind: Winter Case
122
The terminal phase dispersion testing results are given in Table 7.6 and 7.7. Only a
few of the cases did not come within 1 nautical mile (-6000 feet). Most of the cases came
within 100 feet of the target even in the face of dispersions. In general, the mean and 1-
sigma cases had lower target miss distances than the 2-sigma cases. The East wind dispersions caused the greatest miss distances.
Dispersion
CRmiss (ft)
DRmiss (ft)
Nominal
-33.5030
-32.9906
East Wind - mean
-728.1376
-3708.5116
East Wind - lo
-1010.6408
-5658.0987
East Wind - 2
3113.3935
-9759.5360
North Wind - mean
1.0227
-55.8933
North Wind - l
-184.6070
47.0619
North Wind - 2a
-315.0263
-76.6906
Vertical Wind - mean
-33.5008
-32.9992
Vertical Wind - l
-34.5593
-26.8857
Vertical Wind - 2c
-27.2190
-24.3609
Density - mean
-23.9225
94.0316
Density - lo
4617.7372
-3846.9591
Density - 2(c
-276.1927
-2377.3139
Summer Case
Table 7.6: Dispersion Testing Results: Summer Case
123
Dispersion
CRmiss (ft)
DRmiss (ft)
Nominal
-33.5030
-32.9906
East Wind - mean
-3.25027859e4
-8.39587232e4
East Wind - la
-2922.0533
4123.0254
East Wind - 2
-102.3566
-912.2753
North Wind - mean
-26.8952
-30.8330
North Wind - l
-250.1215
55.0452
North Wind - 20
123.5137
-1432.2408
Vertical Wind - mean
-33.4941
-32.9920
Vertical Wind- lo
-34.5368
-34.5368
Vertical Wind - 2c
-24.5079
-26.4444
Density - mean
-6687.2872
2018.8684
Density - 1a
-75.7021
-243.6869
Density - 2
-214.2973
-3046.1677
Winter Case
Table 7.7: Dispersion Testing Results: Winter Case
Figures 7.38 - 7.41 below show the miss distances in relation to the target and the
nominal case. The Summer test cases generally had lower miss distances than those in the
Winter. This would be expected as the winter time would yield greater atmospheric variations.
124
SummerDispersionTest:ng:Miss Distance
0
=2
o
La
0
DownrangeMiss (ft)
Figure 7.38: Dispersion Miss Distances (1000 ft): Summer
SummerDispersionTesting:Miss Distance
a0,
oi
be
-\
N
0
0
I'j
O
DownrangeMiss (ft)
Figure 7.39: Dispersion Miss Distances (6000 ft): Summer
125
(
Winter Dispersion Testing:MissDistance
1uuu
nnA
.
........
'
'
" "
-
jr.
800
.
·
../
600
!. -" ~,,"..
.East
.. ..
9
200
ii
0
/
Vertical Wind
Density
0
/O
400
:
Wind
North
Wind
'.o
Nominal
.
.
;
\C
I
U -200
C
-400
-600
-800
.
-
-A
,
,
,
-1000
-,100 -800 -600 -400
...
i
-200
0
200
DownrangeMiss(t)
.
400
r
600
800
1000
Figure 7.40: Dispersion Miss Distances (1000 ft): Winter
WinterDispersionTesting Miss Distance
CE
t
co
2P
ru,
DownrangeMiss(ft)
Figure 7.41: Dispersion Miss Distances (6000 ft): Winter
126
Chapter 8
Conclusions
A robust bank-to-steer guidance scheme was designed that could be used for a
reusable launch vehicle designed for precision landing, such as the PL-RLV.The presence
of the terminal phase bins allows for more diversity in target acquisition. Bank rate profiles can be chosen preflight based upon fuel cost requirements and the minimum target
miss distance desired. Several different bank rate profiles can lead to the target, so profile
selection is at the discretion of the mission design.
It was proven that the pre-terminal knowledge of the nominal profile can reduce
the pre-terminal predictor error on orders of approximately 20,000 feet. This reduction
will allow for better handling of dispersions that the craft will encounter throughout its
descent.
The landing footprint, all bank rate profile possibilities, and the partials of the con-
straints with respect to the controls can all be calculated off-line. This off-line code allows
for minimal computation load. A more accessible target can be chosen preflight from the
footprint generated as well. The bin number testing provides good insight as to bin number
selection for future simulations.
The predictor/corrector pairing proved to work well. When given a nominal openloop profile, the closed-loop guidance can minimize the target miss error to less than 100
feet. Even in the presence of atmospheric dispersions, the guidance scheme was robust.
127
The nominal profile reached the target within requirements when tested with wind and
density variations of 1 and 2-sigma.
All of these advantages to the designed guidance makes this scheme a pliable
option for any bank-to-steer entry vehicle.
Suggestions for Future Work
In regards to the designed guidance scheme, additional testing could be performed
on:
1) The use of bank angle profiles instead of bank rate profiles
2) Modulating the bin length
3) Additional dispersions
In order to provide the most accurate modeling of the PLV-2's flight, a 6 DOF simulation could be created, as well. An acceleration model could also be implemented to
help model the bank changes more precisely. Appendix B provides a sample acceleration
model which could be implemented [4].
This thesis provides a robust guidance scheme design that can be utilized. The suggestions for future work can be used for improvement on the design. Future work should
combine advantages from all possible approaches in order to provide the most optimal
guidance scheme.
128
Appendix A
Analytical Study of the PL-RLV Re-entry Guidance
I. Purpose
The re-entry guidance requirements for the PL-RLV's second stage (PLV-2) present a
complex problem due to the constraints placed on the vehicle and its performance.
Upon
re-entry from orbit, the PLV-2 will experience control until landing system parachute
deployment.
The vehicle is also required to land in a specified area which adds complex-
ity to the problem. This analysis makes certain simplifications to the problem in order to
gain insight used to guide the development of the re-entry guidance algorithms. The reentry guidance task is to predict the bank rate profile effect on landing locations, correct
the bank rate plan, and direct the vehicle to the desired landing location.
This study is
only the first of several planned analyses each exploring certain guidance algorithms.
II. Set Up
Given the current position and velocity from navigation as well as a target position, this
simulation calculates the best possible combination of two bank rates (ThetadotA &
ThetadotB) and a time at which to switch instantaneously from the first to the second bank
rate (t_switch) to minimize landing error. The simulation begins at a specified altitude and
controls flight down to an altitude of 70,000 ft. After this point, the landing system para-
129
chute deploys, all automated control ceases, and the vehicle will enter the coast phase.
See Figure A. 1 below.
Start Altitude
-~~~
4L
A, - , -Re
LAt= ~l
IlIl
70,000 ft
.r
0
_
A
(a
Coast Interval
At = 108.5 sec
_·
I
tswitch
0
-
---
,
tv
·,
tmax
--
..
iF
·
l'
5,000 ft
Z Z
Z
L-
Figure A.1: Single-Switch Program Schematic
130
w
j~
- - - - - - - --
III. Equations of Motion Implemented
The coordinate system for this simulation is defined as follows:
-
I
Y
Y
0 = bank
X
Figure A.2: X, Y, Z Axes and Bank Angle O Defined
Assumptions and Initial Conditions:
* Falling straight down in the Z direction (Simulation does not actually include gravity,
but simply time spent in a particular bank phase)
*
O0 = Initial Bank Angle = 0 [rad]
* Instantaneous bank rate change
·
a = 11.28 ft./sec 2 , horizontal acceleration in bank direction, due to lift
Vxo = VyO= 0 ft/sec , VzOis irrelevant since this simplified simulation is not concerned
with the vertical channel
·
X = Yo =
ft
* t_max = 20 sec, time until landing system parachute deploy
*
t_init = 0 sec
131
The equations of horizontal motion used during bank flight are defined as follows:
-At +O
=
(A.1)
o
o
(A.2)
d(At) = dt
(A.3)
a. = a cosO
(A.4)
ay = a
(A.5)
At = t-t
sinO
v, = a fcos(- At+ o)dt
a cosOO
V=
.
sinAt +
= a
=
v
-a
=
coso
0
a sinG o
. cosAt + (CIx)
sin(
-(3
r =
-a - cos oO
(0)
ry=
2
2
cos0At +
sin 0At + (Cly)
(A.8)
(A.9)
H
a- sin0 o
2
sin At + (Clx) · At + (C2x)
(A. 10)
(6)2
-a- cos0 o
a-sin
sin At+
2
(G)2
O)dt
a sin 0·
cost+
(A.7)
Ato)dt
a-sin(6-A+
c
(A.6)
o
-cosAt
(6)2
where Clx, C2x, Cly, and C2y are all constants.
132
+ (Cly) ·At +(C2y)
(A. 11)
Propagation to the ground after landing system parachute deploy (at 70,000 ft) is calculated with these equations below:
Final Position(X) = PositionX(70,000 ft) + VelocityX(70,000 ft) x tcoast
Final Position(Y) = PositionY(70,000
ft) + VelocityY(70,000 ft) x t_coast
The approximate ranges of final positions can be determined by looping through the following steps for values of ThetadotA and ThetadotB:
*
Use Equations of Motion (EOM) and ThetadotA on the interval, t = t_init - tswitch,
to determine conditions at t_switch.
*
Update bank angle, assuming instantaneous bank rate change.
·
Use updated bank angle and ThetadotB EOM on the interval, t = t_switch - t_max
*
Finally, propagate to ground during coast interval (t-108.5 sec) using the propagation
equations. T_coast, the approximate time for the vehicle to coast from 70,000 ft. to
5,000 ft., was determined from the nominal profile obtained during previous PL-RLV
testing.
133
IV. Aim-Point Envelopes
A program was generated to calculate all possible final positions for a given range of
ThetadotA, ThetadotB, and t_switch. These contours approach "bow-tie" shaped contours
about the origin as time-to-go becomes small. (See Figures A. 3, 4, 5, 6, and 7 on the following pages)
Bow-tie regions define where the vehicle can land and also the landing areas that are
impossible to reach given the initial conditions. It is important to note that all bow-tie
contours in this study had an initial bank angle of zero, which is aligned with the initial
non-zero acceleration vector. Given a different initial bank angle, the contours would
rotate about the origin by the initial bank angle, thus sweeping out all areas around the origin when all possible initial banks are looked at. An example of initial bank angle effect
can be see in Figure A.10.
Each final location is not necessarily reached by just one combination of the parameters.
This simulation proved the ability to reach the same aimpoint with different sets of parameters (ThetadotA, ThetadotB, and t_switch). All areas covered within the defined bow-tie
contour are accessible. The complete density accessibility of aimpoint locations is insured
by minute tweeking of parameter incrementation (e.g. t_switch = 1.456 instead of 1.5)
That is to say, all final locations within the contour can be selected as aimpoints.
134
The following parameter ranges were chosen since they result in all final positions within
the parameter limits:
ThetadotA = -15 to +15 in steps of 1 deg/sec
ThetadotB = -15 to +15 in steps of 1 deg/sec
T_switch = 0 to t_max in steps of 1 sec
An initial bank angle of 0 deg was used for the contours found in Figures A.3, 4, 5, 6, and
7. The initial bank angle direction lies along the horizontal X-axis for these cases.
Final Positions,Lmax - 40 sec
It .
no
.-
.
.
. .
.
.
40
20
c
C
._
C
.;
-20
-40
miPS! I
-40
-
-30
-20
-10
.
.
. ·
0
10
20
30
Distancein the X Direction[kft]
40
Figure A.3: Final Positions When T_max = 40 sec
135
50
60
Final Positions, tmax
- 30 sec
:1
I
C
I
a
I
i
.c
CD
C
0
-30
-20
-10
0
10
20
Distance in the X Direction [ktt]
30
40
50
Figure A.4: Final Positions When T_max = 30 sec
Final Positions, t_max = 20 sec
An
I
'4U
I
I
I
.... .
30
I
I
. . ..
I
. .
20
C
0
=
a)
S
o
10
-c
0)
-
a)
c -10
.
. .
. .
. .
. .
Cu
.5
-20
. . .
. . . .
-30
__I·I
-40
v-40
-40
. . .
.
..
.
I
I
-30
. . . .
-20
-10
.
.
.
.
[
0
-
.
..
.
.
.
.
.
. .
I
10
.
.
.
.
I
20
30
Distance in the X Direction [kft]
__
Figure A.5: Final Positions When T_max = 20 sec
136
.
40
Final Positions, tmax
= 10 sec
40!
30................................... .........................
FE 20
... . . .... . .. I... .. . .. .. . .
10 ................................................
10
O
.,
8
.
.
.
I
.............................
. . . :
.
. .
4
. I I
:
[
I.
-20
-30
-40
... .......
-30
-20
-10
......
,
0
10
... .......... ...... .: . ..
20
30
40
Distance in the X Direction [kft]
Figure A.6: Final Positions When T_max = 10 sec
Final Positions, tmax
-5 sec
I
.. .; . ..
30
.. . . . . . . .
.
... ..
20
.. .. ... · .··..
I
0
10
...:......
0
c
.. . . . .
... .. .
-1,
.. . ..
...
...
..
-20
... . .
-30
-0
,
-40
-30
[
-20
I.
-10
I
0
10
Distance in the X Direction [kft]
20
Figure A.7: Final Positions When T_max = 5 sec
137
30
40
Contour Observations
By increasing the time until drogue deploy (increase in t_max), the contour area increases
and begins to close in around the target. The inaccessible zone becomes smaller, thus
increasing the range of final positions. As t_max decreases, the contour area decreases
and widens the opening around the target. The inaccessible zone becomes larger, thus
limiting the possible final positions. Small holes resulted in the high t_max contours.
These holes are thought to vanish by tweeking certain parameters (i.e. time or bank rate
increments), and thus they are not true inaccessible zones.
Most of the "upper" half of the contour, located in the positive X and Y quadrant, is the
result of +ThetadotA's and + ThetadotB's. The "lower" half results mainly from (-ThetadotA's) and + ThetadotB's . A +ThetadotA and B usually makes up the upper half, while
a -ThetadotA and B compose the lower half of the contour. There is some overlap around
the zero y-axis when the magnitude of ThetadotA is much smaller than ThetadotB (e.g.
ThetadotA = 5 deg/sec and ThetadotB = -15 deg/sec) or when the switch time only allows
ThetadotA to occur for a short time (e.g. t_switch = 1 sec). When both ThetadotA and
ThetadotB are 0 deg/sec, the final positions lie along the zero y-axis.
The original test case, tmax
set to 20 sec, was investigated further to define characteris-
tics of the contour. Setting t_switch = 5 sec for this test case results in final positions that
cover almost all areas possible, except the small outer edge slivers of the tswitch
case. Shown in Figures A.8 and A.9 below.
138
= 10
All Final Positions, Tswitch-1 0 sec
I
.CD
I
I
I
I~~~~
20
15
*
+r3"
10
' 5
-5
BiG
-10
-15
-20
_-2
I
--10
-5
·
0
I
~~I
I
5
10
15
Distance In the X Direction [kft]
20
i
I
25
30
__
Figure A.8: All Final Positions, Tswitch = 10 sec
All Final Positions, Tswitch-=5sec
.
20
15
.
-
,4-
'
.
-.
'
-.
.'
-..
: ~~~~~~~~~~..'*'
*. ,..-·~~~~~~~~~~~~···
'~~~~
'
.
·
·
'':i' r'
· :'
"
.-o
C
. .- .- .. · ',' . .
·
-5o
e~~~~
' ,',',','...
·
I -10
-15
I
~I
~
~
~
~
°
.
~
o
~
.- o
'-
~~~~~~
~
'.
· .~.1
~~~~~~i
-20
_or
-10
-5
0
5
10
15
Distance in the X Direction [kft]
20
Figure A.9: All Final Positions, Tswitch = 5 sec
139
25
30
As mentioned previously, a change in the initial bank angle does not alter the form of the
contour, but rotates the contour. The test case in Figure A. 10 was given the same conditions as Figure A.5, but an initial bank angle of 90 deg was used instead of 0 deg. Other
factors, such as the addition of an atmospheric model, will likely transform the contours.
In the effort to reduce the original simplifications and therefore generate a more precise
simulation, future work will analyze the effects of the atmosphere and other factors on the
contours.
Final Positions, t_max = 20 sec
JA
,a
I
I
I
I
I
I
I
I~~''
I
I
I
I
I
3C
20 k
10 p
E
..............
0
...
C
c
.......
.10
.....
.......
.. -
...
._
0
:
.
...
-20
.....
.
..
...
......
...
.
.
:
.
.
.
..
.
.
.
..
-
.
...
:
.
.
-30
.
:
.
.
.
....
-
....
:
.
.
.
.
A
Ni
40
I[[
I.
-30
-20
I.
I.
.
I.
-10
0
10
Distance in the X Direction [kft]
20
30
Figure A.10: Final Positions, Initial Bank Angle = 90 deg
140
40
V. Predictor/Corrector Basics
The main goal of the single-switch simulation is to calculate the most desirable combination of the three parameters (ThetadotA, ThetadotB, and t_switch) that will result in a final
position close to the target position. These combinations must also lie within the limitations of the parameters. All bank rate limits were set between ± 15 deg/sec; and the time
interval from t_init to t_max (drogue deploy) was set to 20 sec. except when testing the
effects of an increased or decreased t_max (i.e. tmax = 40,30,10,5 sec.)
A predictor/corrector was added to calculate the final positions resulting from the given
input. The corrector program in turn checks to see if the predictor's final positions are
close enough to the target position. In these simulations, a + 200 ft difference in the two
positions was considered acceptable. If the final position is not close enough, the corrector tweeks each parameter one at a time, noting the resulting final positions for each case.
(See Figure A. 11 below) The sensitivities for tweeking each parameter were set at:
inc_thetadotA = .01 deg/sec
inc_thetadotB = .01 deg/sec
inc_tswitch = .02 sec
With this data produced by the different test cases, the corrector implements a least
squares method in order to find the changes needed for each parameter. These changes are
141
then added to the old parameter values in an effort to steer the new final positions closer
and closer to the target.
_
__
_
_
Y
X
(4) ThetadotA tweeked
adotB tweeked
Figure A.11: Resulting Final Positions, Each Parameter Tweeked One at a Time
Once the multiple test case data is supplied, the corrector's steps are as follows:
AX =
(A. 12)
A 8 +st,y- At
(A.13)
B+
8yA
SOB
where:
tS
X. A A +
x
AX = Target_X - Position_X(l)
AY = Target_Y - Position_Y(1)
(1) = the first run case as shown in Fig.A. 11
142
(2), (3), (4) = tweeked cases as shown in Fig.A. 11
ax
S
8x(3) - Sx( 1 )
F6B
(A.14)
esb(3)- 8B(1 )
Sx = Sx(4)- Sx(1)
seA
Sx _
Et s
(A.15)
seA(4)- 8A(1)
x(2) 8ts(2)-
x(1 )
t,( 1)
,etc.
*
Solve for ThetadotA, ThetadotB, and At s , Least Squares Method
*
Add new increments and gains to the three parameters:
(A.16)
New ThetadotA = Old ThetadotA + (Gain)(A ThetadotA) , etc.
The gains used in these simulations were as follows:
K_ThetadotA = .1
K_ ThetadotB = .1
K_Tswitch = .1
__
Gains were added due to the non-linearity of the problem, as indicated by preliminary
simulations. Further work will investigate these values.
143
Run through loop until within a certain acceptable radius of the target
2
2 + (TargetY - NewAimPointY)
error2 = (TargetX - NewAimPointX)
For these simulations, error < 200 ft = stop, close proximity to target.
144
Case 1: Predictor / Corrector Reaches Target Within Parameter Limits
Test case 1 shows an example of target acquisition within the parameter bounds. Each "+"
shows the predictor/corrector's path to the optimal landing location, beginning with the
location defined by the initial parameters and ending with the target location.
The target location in this case (5000 ft , 15000 ft ) can be found within the "upper" part of
the contour. The initial conditions defined a location (- 7000 ft, 13500 ft) in the "upper"
part of the contour as well. (Note: the plot's axes show the location within the contour.)
This case demonstrates the corrector's path to the target while trying to optimize the
parameter combinations. The initial and final conditions for case 1 are listed below. Note
that all final parameters are within the parameter limits. (See page 5 for parameter limits)
Initial Guess
Final
ThetadotA
=
5.0 deg/sec
6.389 deg/sec
ThetadotB
=
15.0 deg/sec
12.976 deg/sec
t_switch
=
10.0 sec
10.0 sec
Note that t_switch did not change, indicating that the landing location is most sensitive to
changes in ThetadotA and B than to changes in t_switch within this region of the contour.
Different regions of the contour are sensitive to different parameters, as the following
cases will show.
145
Case 1: Achieving The Target Within The Parameter Limits
15.2j
1
,
I
4............
I
...............
14.8
..
14.4.
.
14.
14
....
*...................................'.....
.......
_.
13.4
5 3
5!.6
--
6.5
Distance in the X Direction [(kttJ
7
7.5
--
Figure A.12: Accessible Target Case
To demonstrate that more than one solution for target acquisition is possible, an identical
case to case 1 was run, but tswitch
was changed from 10 to 11 seconds. In this case, the
target was also achieved within the parameter limits.
Final
Initial Guess
ThetadotA
=
5.0 deg/sec
6.6273 deg/sec
ThetadotB
=
15.0 deg/sec
12.8949 deg/sec
t_switch
=
11.0 sec
11.0 sec
146
Target Acquisition with T_switch - 11
K
F5
[a
'I
.G
co
5
Distance in the X Direction [kft]
Figure A.13: Another Solution to Case 1, Target Acquisition
Changing the parameter gains can result in different corrector paths to identical target
locations. The parameter gains were each set to .10 in case 1 (shown on page 13). Gains
of .5 and 1.0 were also tested, and their results are shown below. In general, as the gain
was increased, the final bank rates were increased slightly. The higher the gain, the faster
the speed of convergence as well.
Gain = .5
Gain = 1.0
Final ThetadotA
=
6.3902 deg/sec
6.3915 deg/sec
Final ThetadotB
=
12.9773 deg/sec
12.9794 deg/sec
Final t_switch
=
10 sec
10 sec
147
Figure A.14: Case 1 With Gains Set to .50
Nominal C
1.
-a.with GOIln.
1.0
~
-~~
~
~
~~~~~~~~~~~~~~~~
-
.
1'6
. I
.G 14.6
.
.
14
13.5
'
14
.
65
_
5.5
Distanc
_
a
in the X Direction
_
_
e.6
[kIt]
_
7
7.5
--
Figure A.15: Case 1 With Gains Set to 1.0
Since the gains of 1.0 reach the target, the possibility that the gains are unnecessary arise.
Although the target is acquired in this case, the overshoot is noticeable and could prevent
the corrector from converging given a different target/initial guess setup.
148
Case 2: Target Placed Within the Inaccessible Zone
In Case 2, the target (8000 ft, 0 ft) was placed inside the contour's inaccessible area of
Figure A.5 in order to verify that the target could not be achieved. Initial parameters and
the resulting parameters for this case were:
Initial Guess
Final
ThetadotA
=
5.0 deg/sec
313.77 deg/sec
ThetadotB
=
15.0 deg/sec
14.310 deg/sec
t - switch
=
10.0 sec
1.889
sec
-- ---
Although the corrector did converge on a final set of parameters that reached the target,
final parameter values were grossly out of range, thus proving that a target cannot be
reached if located in the "inaccessible zone" of the contour within the limits of the parameters.
Case 3: Target Placed Across the Inaccessible Zone
In Case 3, initial conditions were given for a position at the "top" of the contour (~ 7000 ft,
13500 ft) in Figure A.5, and the target position (5000 ft, -15000 ft) was given a location at
the "bottom" of the contour. Initial parameters and the resulting final parameters were as
follows:
149
Initial Guess
ThetadotA
Final
5.0 deg/sec
ThetadotB
15.0 deg/sec
t_switch
10.0 sec
154.00 deg/sec
-5.56 deg/sec
6.7841 sec
Figure A. 16 shows the predicted aimpoint path as the corrector tries to close in onto the
target. For cases such as this, the corrector forces the predicted aimpoint to enter the
inaccessible zone, where the parameters, as in Case 2, are driven out of range. Upon re-
entering the accessible zone near the target, the corrector is forced to use parameters out of
the desired range even if they achieve the target.
_
_
_
Case 3: Crossing the Inaccessible Zone
C
.o
a
0
-c/
S0
C.
-
-4
-2
0
2
4
Distance in the X Direction [kft]
b
Figure A.16: Figure 12. Crossing the Inaccessible Zone
150
u
'1
The ability of the corrector to reach the target with a different solution than the desired one
(all parameters in range) proves that there are multiple solutions to target acquisition and
that the corrector is inadequate in cases where the initial guess is poor. A new scheme for
the corrector, in its elementary stage, is currently being analyzed in order to avoid problems like Case 3 points out. This corrector adds a second stage after the corrector has converged that modifies the parameters towards their desired ranges while holding the
aimpoint position constant.
VII. Current Work (March 1997)
The single-switch simulation provided preliminary insight on parameter combinations,
their resulting effect on final positions, and final position contour bounds. Current simulations in progress update the single-switch simulation to a more realistic representation of
the re-entry process. The results yielded from the single-switch simulation and the current
simulations listed below will help mold re-entry guidance algorithm development for the
PL-RLV.
The current simulations in progress include:
·
Non- instantaneous change of bank rate:
Intending to simulate the use of RCS thrusters, the bank rate profile now incorpo-
rates bank acceleration over certain times, as seen in Figure 13. Over these times, the
evaluation of a Fresnal Integral will be necessary for an analytic predictor.
151
Figure A.17: Non-instantaneous Bank Rate Change
*
Addition of an Atmosphere / Drag Model
·
Investigation of Initial Condition Effects
Tests have already shown that non-zero initial positions result in translation of
the final position by the values of X0 and YO. Similarly, non-zero initial velocities do not
change the final position contour shape, but push the positions farther in the velocity vector's direction. Figure A. 18 shows the results of adding an initial X and Y of 10,000 ft to
the nominal test case found in Figure A.5. Figure A. 19 displays the results of adding an
initial X and Y velocity of 100 ft/sec to the nominal case in Figure A.5. It is easy to see
that the initial positions translate the contour of Figure A.5 by 10,000 ft in both the X and
Y directions. The initial velocities push the original contour in the velocity vector's direction as shown in Figure A.19.
152
Figure A.18: Non-zero Initial Positions
_ _
InitialX and Y Velocities=100, T_switch = 5
r
ac
o
C
0
C
Distancein the X Direction[kft]
Figure A.19: Non-zero Initial Velocities
153
With the addition of an atmosphere model, it is expected that the final position
contours will change shape and warp slightly when the initial bank angle and velocities
are not aligned.
Other areas to be investigated include:
*
Perfected corrector scheme to handle all cases
*
Work towards the reduction of the number of iterations needed by the corrector
*
Investigation of "Wait" Time Model
The "wait time" model (see Figure A.20) allows for coast times before ThetadotA
implemnent
n, in-between ThetadotA and ThetadotB implementation, and even after
ThetadotB. This model significantly increases the dimension of the problem. The wait
times and times for the bank rates do not necessarily have to equal the maximum time
until drogue deploy. Test cases investigated thus far set the parameters within these limits:
ThetadotA: 2.5, 5.0, 7.5, 10 deg/sec
ThetadotB: +ThetadotA
t= 0 sec
tA must be at least 5 sec
tB must be at least 5 sec
154
I
i
-
e
7ZZZZZ~
-
- ~---
V/JIlL! on
XtA
tw-- tB
Figure A.20: Wait Times between Bank Rate Changes Schematic
155
s
Ia
ta
156
Appendix B
Acceleration Model for Bank Maneuvers
I. Acceleration Model Defined
Due to time constraints, this thesis assumed instantaneous response to changes in bank.
For future investigation, an acceleration model could be employed for the reversal and
even the terminal bank rate changes. A model is suggested below for the acceleration during the reversal phase [4]. This design can also be followed for the acceleration during the
terminal phase bank rate changes.
Figure B.1: Acceleration Model Bankrate vs. Time
where
he = the commanded bank angle
~0 = the bank acceleration from to to t 1
157
~i = the bank acceleration from t to t 2 , which is zero
;2 = the bank acceleration from t2 to t3 , which is -~o
;3 = the bank acceleration after t3 , which is zero
The time differences are defined by:
At I =
- to
(B. 1)
At2 = t 2 - t o
(B.2)
At 3 = t3
(B. 3)
-
to
The bank angle and bank rates are defined as a function of time in each phase as:
Acceleration Phase
¢(t) = 0 + Io(t-t0
)
(B
4)
to)
(B.- i)
o(At )2 + )c(t- t )
(B.6)
~(t) = o(t-
Coast Phase
¢(t) = 0
+
c
(B.7)
¢(t) = ¢°+ Ib0(Atl)+ kc(t2 - tl) + I 2(t - t2)2
(t)=2t. 21.t2
(B.8)
¢(t) = 4oAtl + 4 2 (t- t2)
(B.9)
¢(t) = doAt =
Deceleration Phase
158
The acceleration model should be placed in the predictor calculations. The bank
angle and bank rate calculations will be different depending on the actual time when the
predictor is called. A prediction call made during the entry phase will model all of the
acceleration, coasting, and deceleration phase of the maneuver. A call made within the
reversal will depend on the time from reversal start to yield the bank angle and bankrate
calculations. The following subsections define the bank angle and bank rate calculations
when called for at different times: prior to reversal, during the acceleration phase, during
the coast phase, and during the decent phase [4].
II. Prior to Reversal
If the current time is before the reversal time, the switch times can be calculated by
first finding the time it takes to accelerate to the commanded bank rate and the amount the
bank angle changes during the acceleration phase.
Atacc
c
Aacc = 5¢o(Atacc),
2
159
(B.1 0)
(B.11)
If the change in the bank angle during the acceleration phase is greater than half of
the total maneu ver angle desired (IAaaccl> 0o), the acceleration model should be redefined
as follows [4]:
Figure B.2: Profile Redesign Due to Time Constraints
Acceleration occurs over half of the maneuver and deceleration occurs over the
second half.
(B. 12)
At = At2
At3 = 2At,
(B.13)
If there is enough time to perform the maneuver, the original profile design should
be used. The switch times can be found from:
At
= Atacc
160
(B.14)
At=At,2
At 2 = At +-°,.
- (i(o)(A,)
2
(B.15)
(~o)(at,)
At3 = At, + At2
(B.16)
III. Current Time in Acceleration Phase of Reversal
When the current time is in the acceleration phase of the reversal, the change in the
bank angle that remains to be completed and the time since the start of the reversal need to
be calculated. The current time and bank angle are the references.
A =
cp-
(B.17)
ef
At = te f -trevstart
(B.18)
}ref = ' 0At
(B.19)
The current bank rate is given by:
It is necessary to calculate the bank change during the remaining acceleration and also the
bank change during deceleration from the commanded bankrate.
A4tacc=
Adc
=
o(At, - At) + ~,ef(Atl - At)
o(Atl)(At
3-
At 2) + I)2(t3
161
- t2)
(B.20)
(B.21)
If (IAacc + Adecl > IA~I ) there is not enough time to complete the acceleration to the
commanded rate and to decelerate to a zero rate at the commanded bank angle. In this
case, the switch times are calculated as:
At = At -
+
.
2jO
2()+A
*
(B.22)
At 2 = At I
(B.23)
At3 = 2At -At- + lref
(B.24)
If (IAacc + OAdec!< IAOI) then there is enough time to complete the maneuver, and the
switch times are calculated as:
At = Atl +
IA1- IA4acc+ AodeI
At3 = At2 + _5
0
162
(B.25)
(B.26)
IV. Current Time in Coast Phase of Reversal
During the coast phase, the total bank change that remains and the time since the
start of the reversal are defined as:
A = c-r,
At = t
(B.27)
(B.28)
f -re trevstart
The switch times can then be found from:
(B.29)
At 3 = At 2 -ref
at 2 = at + a + 2
(B.30)
f
If (At2 - At) < 0 , then the desired end conditions cannot be reached with the maneuver
modeled. It is necessary to start decelerating right away. The switch times become:
(B.31)
At 2 = At
2~ /(e2
rf
At3 = At 2
163
(B.32)
V. Current Time in Deceleration Phase of Reversal
When the current time is in the deceleration phase, the final switch time can be calculated from:
(B.33)
At = Atref - Atre vsIart
At 3
-
At
re2f
42
2
(ref) )+
il42
2
(B.34)
'tZ
If radicand of quadratic equation is negative, the commanded bank angle will not be
reached prior to reversal of the bankrate sign. The maneuver end should be set to the time
at which bankrate passes through zero.
3~ re2
At 3 = At-
164
(B.35)
Appendix C
Nominal Profile Simulation Plots
The following plots were generated for the nominal profile trajectory presented in this thesis. (Target B, Profile #9)
Figure C.1: Altitude Profile
165
I
0
100
200
__
300
_
400
500
Time(seci_
_
600
700
800
_
Figure C.2: Earth Relative Velocity Profile
_
_ _
PLV-2:Descent
i
Ii
mi
Figure C.3: Relative Flight Path Angle Profile
166
900
__
Figure C.4: Dynamic Pressure Profile
PLV-2: Descent
90o0
I
I
I
I
8000
x q-alpha = 8575.4957
7000
*
0
*t121785.31
ft.
0 z 652 sec
6000-.
5000
. .
.
.
,-
II
5000..:..
Di
'O!r
5.
gI4000
3
.
ii3000
.......
......
.......
2000 ................
.
1000_ /.
1
,1
0
100
200
I
300
,
400
500
Time {lsec
I
,
600
700
800
..1- ---
Figure C.5: Dynamic Pressure x Alpha Profile
167
900
Figure C.6: Heating Rate Profile
L3VU
i
,
i
PLV-2:Descent
,
.
i
,
.
I
X
3000
ft
2500
a
gh
S. 2000
.
.
.
.
J.i.
.
.
I
.
. . .
.
I.
.
.
d
&1500
oI
9x
S:i
o
v,
1000 ......
I.....
I...
... ;...
500
I
0
100
I
200
300
I
400
500
Time (seC)
I
600
700
800
Figure C.7: Stagnation Point Temperature Profile
168
900
_
_
PLV-2:Descent
e
i;
.5
Il
u
Ti
(.tn eia
I -LN
Tima (secl
Figure C.8: Acceleration Profile
Figure C.9: Mach Number Profile
169
OW
1Io
n
PLV-2: Decent
14
LB .
2
0
.... ......,..... ......
... .. .. ...
200
100
,
,
,
,
_10
...
I
i !Alphat
I
400
300
.
!
500
·
!
600
700
....
.
800
90C
Tms (sec)
.3 .
2........
O.25
:; ...
; ......: .....
;......
.........; ..... .. ;, .......
....
0
I
u
0.115
I
.
0 1.
.
.
0.c05
.....
nI "'
.w
0
.
i
boo
100
200
300
l
.
400
500
Time(sec)
600
700
Figure C.10: Angle of Attack Profile
170
L.
800
900
--
References
[1]
Barchers, J. D., Entry Guidance for Abort Scenarios, S.M. Thesis, Department of
Aeronautics and Astronautics, MIT, June 1997.
[2]
Dierlam, T. A., Entry Vehicle Performance Analysis and Atmosperic Guidance Algorithm
for Precision Landing on Mars, S.M. Thesis, Department of Aeronautics and Astronautics,
MIT, June 1990.
[3]
D'Souza, C. N., PhD., Notes and Personal Correspondence, Draper Laboratory, Jan. - Dec.
1998.
[4]
Fuhry, D. P., Simulation Code, Notes, and Personal Correspondence, Draper Laboratory,
Jan.- Dec. 1998.
[5]
Harpold, J. C., C. A. Graves, Jr., Shuttle Entry Guidance, Journal of the Astronautical
Sciences, Vol. 27, No. 3, July-Sept 1979, pp. 239-268.
[6]
Hildebrand, F. B., Advanced Calculus for Applications, Prentice-Hall Inc., Englewood
Cliffs, New Jersey, 1976.
[7]
Justus, C. G., et al., The NASA/MSFC Global Reference Atmospheric Model -1995 Version
(GRAM-95).
[8]
Regan, F. J., S. M. Anandakrishnan, Dynamics of Atmospheric Re-Entry, AIAA Publishing,
Washington, D. C., 1993.
[9]
Spratlin, K. M., An Adaptive Numeric Predictor-Corrector Algorithmfor Atmospheric Entry
Vehicles, S.M. Thesis, Department of Aeronautics and Astronautics, MIT, May 1987.
[10] US Standard Atmosphere 1962, Prepared by the National Aeronautics and Space
Administration, United States Air Force, and United States Weather Bureau.
171
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