Performance Optimization Study of a Common Aero Vehicle Using a Legendre Pseudospectral ...

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Performance Optimization Study of a Common Aero
Vehicle Using a Legendre Pseudospectral Method
by
Kimberley A. Clarke
B.S. Aerospace Engineering, Pennsylvania State University, 2001
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
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2003
© Kimberley A. Clarke, MMIII. All rights reserved.
The author hereby grants to MIT permission to reproduce and
distribute publicly paper and electronic copies of this thesis
document in whole or in part.
Author ...........................................
,epVtnnt ofAeronautics and Astronautics
May 23, 2003
Certified by.... ..................
Anil V. Rao, Ph.D.
Senior Member of the Technical Staff
The Charles Stark Draper Laboratory, Inc.
Technical Supervisor
Certified by ............ ................
Jonathan P. How, Ph.D.
Professor, Department of Aeronautics and Astronautics
Thesis Advisor
Accepted by ......... ......................
Edward M. Greitzer, Ph.D.
H.N. Slater Professor of Aeronautics and Astronautics
Chair, Committee on Graduate Students
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
AEROBRES
LIBRARIES
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Performance Optimization Study of a Common Aero Vehicle
Using a Legendre Pseudospectral Method
by
Kimberley A. Clarke
Submitted to the Department of Aeronautics and Astronautics
on May 23, 2003, in partial fulfillment of the
requirements for the degree of
Master of Science in Aeronautics and Astronautics
Abstract
The problem of performance optimization of a Common Aero Vehicle (CAV) is
considered. In particular, the CAV is modeled as an unpowered high lift-to-drag
ratio Earth penetrating re-entry vehicle. The CAV mission design problem is to
determine a steering command that takes the CAV from a known initial state to
a target on the surface of the Earth while optimizing a given performance index
and satisfying all of the constraints imposed during flight. The CAV mission de-
sign problem is formulated as an optimal control problem. The optimal control
problem is transformed to a nonlinear programming problem using a Legen-
dre Pseudospectral Method. The nonlinear programming problem is then solved
using a sparse nonlinear optimization algorithm. Once a solution to the CAV
mission design problem is obtained, three main studies are conducted. First,
the accuracy of the Legendre Pseudospectral Method is evaluated and the de-
sirable characteristics of the solution to the CAV mission design problem are
defined. Second, a study is conducted to demonstrate the effect of the param-
eters on the performance of the CAV. This parametric study demonstrates the
use of the Legendre Pseudospectral method as a design tool and provides in-
sight to the behavior of the CAV. Third, a preliminary investigation is performed
on the real-time application of the Legendre Pseudospectral Method to the CAV
mission design problem. This real-time analysis includes an assessment of the
robustness of the solution to realistic environmental disturbances.
Technical Supervisor: Anil V. Rao, Ph.D.
Title: Senior Member of the Technical Staff
The Charles Stark Draper Laboratory, Inc.
Thesis Advisor: Jonathan P. How, Ph.D.
Title: Professor, Department of Aeronautics and Astronautics
3
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Acknowledgments
I am very grateful for everyone who has made the completion of my masters
degree possible. Without the unique network of the Draper staff, MIT faculty,
family, and friends, I would not be where I am today.
I would like to thank the Charles Stark Draper Laboratory for providing me
with the funding and support necessary for the completion of my degree from
MIT. In particular I would like to thank the GCB2 staff as well as the Education
Office. I would also like to individually thank Doug Fuhry and Anil Rao. Doug,
even though we only worked together briefly, I learned a lot from you. Special
thanks to Anil Rao for the guidance and support not only on my project, but also
with my job search. It has been two years of laughter, frustration, and growth.
Oh and I will especially miss your corny, but funny engineering jokes.
Thanks to the MIT professors and the entire Aero/Astro staff. The most
amazing part about studying at MIT is the intelligence of the professors and
their first hand experiences that are integrated into the classroom. Professor
How, I am grateful for your patience and thank you for being my thesis advisor.
Furthermore, I would like to thank professors Battin and Ramnath for being a
reference for me on job applications.
Now to my MIT friends, these past two years have been years of personal
growth. Each and everyone of you has expanded my horizon and I thank you for
that. In particular, I would like to thank the "forget your lunch Friday" Draper
crew who I have directly shared the past two years with. Christine, thanks for the
bathroom breaks, Thursday night dinners, and most importantly, the girl time.
Jen, thanks for the kick-board chats, Friday morning breakfast, and trips to do-
nate blood. "Coach" Geoff, thanks for bringing out the child in me by playing
games while waiting in line for rides at "Great Adventure" and by stopping on a
six hour car trip to ride go-carts. We have come a long way since Texas. Stephen,
thank you for being my e-mail buddy, introducing me to Strongbad, and nor-
malcy. Dave Benson, thank you for attending our review sessions, teaching me
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how to make bread, and being my party buddy. Daveed, thanks for the ping-
pong breaks, late night e-mails, and 6 a.m. breakfast. Heidi, thanks for listening
to my complaints, sailing, and your sanity. Stuart, thanks for all of your help
and good luck with your music career. To the first year Draper fellows, Dave,
Steve, and Drew, thanks for the fresh faces and I wish you the best of luck.
I would also like to thank those who rescued me from my graduate studies.
To my roommates, Sarah and Libby, thanks for providing me with food, clean
clothes, and clean dishes these past couple of months. To "the girls" from Penn
State, I want to thank you for your open ears and for understanding why I have
not kept in touch recently. Nick, thanks for picking me up when I lost motiva-
tion and always knowing the right thing to say. Preston, thanks for moving to
CT, leading me through the trees on ski trips, spooning, and most importantly,
for making me laugh. Parker, thanks for giving me something to smile about
despite my frustrations with writing my thesis. Last but not least, I would like
to thank my family for their love and support. Thanks for putting up with my
moods and helping me through these past two years. I could not have asked for
more.
This thesis was prepared at The Charles Stark Draper Laboratory, Inc., under
Internal Research and Development, Project Advanced Guidance and Trajectory
Design, 03-2-5037.
Publication of this thesis does not constitute approval by Draper or the spon-
soring agency of the findings or conclusions contained herein. It is published
for the exchange and stimulation of ideas.
Kim berley A. Clarke.............................. .............
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Contents
1 Introduction 17
1.1 Motivation ......................................... 17
1.2 Common Aero Vehicle ............................ 19
1.3 Mission Design Problem .......................... . 21
1.4 Mission Design Approach ........................... 23
1.5 Research Objectives .............................. 24
1.6 Thesis Overview ................................ 25
2 Common Aero Vehicle Problem Formulation 27
2.1 Overview .......................................... 27
2.2 Dynamic Model ................................ . 28
2.2.1 Coordinate System .......................... 28
2.2.2 Equations of Motion ............................. 29
2.3 Boundary Conditions ............................. 34
2.4 Path Constraints ................................ 35
2.5 Perform ance Index .............................. . 36
3 Optimal Control: Problem Formulation and Solution Methods 39
3.1 Overview .......................................... 39
3.2 Optimal Control Problem ............................. 40
3.2.1 Dynam ics ................................ 40
3.2.2 Path Constraints ............................... 41
3.2.3 Boundary Conditions ............................ 41
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3.2.4 Performance Index . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.2.5 General Form of an Optimal Control Problem .......... .42
3.3 Methods for Solving Optimal Control Problems . . . . . . . . . . . . . 43
3.3.1 Analytic Methods for Solving Optimal Control Problems . . . 43
3.3.2 Numerical Methods for Solving Optimal Control Problems . 48
3.4 Direct Transcription of Optimal Control Problem Via Pseudospec-
tral M ethods .................................. 50
3.4.1 Pseudospectral Methods .......................... 50
3.4.2 Legendre Pseudospectral Method . . . . . . . . . . . . . . . . . 52
3.5 Summary of Optimal Control ........................ 59
4 Numerical Optimization Study of the Common Aero Vehicle Problem
Using the Legendre Pseudospectral Method 61
4.1 Overview .......................................... 61
4.2 Discretization via the Legendre Pseudospectral Method ........ 62
4.2.1 Optimization Vector ........................ . 62
4.2.2 Discretization of the Dynamic Constraints ............ .65
4.2.3 Discretization of the Path Constraints and the Terminal Con-
straints ..................................... 66
4.2.4 Discretization of the Performance Index ............. 68
4.3 Common Aero Vehicle Nonlinear Programming Problem ....... .69
4.3.1 Summary of the Common Aero Vehicle Nonlinear Program-
ming Problem ................................. 69
4.3.2 Structure of the Common Aero Vehicle Nonlinear Program-
ming Problem ................................. 71
4.3.3 Scaling of the Common Aero Vehicle Nonlinear Programming
Problem ...................................... 72
4.4 Numerical Optimization via SNOPT ....................... 74
4.4.1 Description of SNOPT ........................... 75
4.4.2 User Requirements and Options for SNOPT ........... .76
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4.5 Numerical Optimization Study ....................... 77
4.5.1 Specification of the Required Inputs . . . . . . . . . . . . . . . . 77
4.5.2 Determination of an Adequate Number of Nodes . . . . . . . 81
4.5.3 Choice of Weighting Factors Used in the Performance Index . 88
4.6 Summary of the Numerical Optimization Study . . . . . . . . . . . . . 97
5 Parametric Optimization Study of the Common Aero Vehicle Problem 99
5.1 Overview .......................................... 99
5.2 Key Features of the Trajectory and Control ............... 100
5.3 Effects of Dynamic Pressure on the Trajectory and Control ..... .105
5.4 Effects of the Stagnation Point Heat Load on the Trajectory and
Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.5 Effects of the Lift-to-Drag Ratio on the Trajectory and Control . . 117
5.6 Summary of the Parametric Study ..................... 123
6 Preliminary Study of the Real-Time Application of the Legendre Pseu-
dospectral Method 125
6.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
6.2 Common Aero Vehicle Flight Simulation . . . . . . . . . . . . . . . . . 126
6.3 Assessment of the Accuracy of the Legendre Pseudospectral Method 129
6.4 Sum m ary.................................... . 134
7 Conclusions 137
7.1 Summary...................................... .137
7.2 Conclusions .................................. . 139
A Notation 143
B Matrix Derivatives 145
C Constraint Jacobian and Objective Gradient Derivation 149
C.1 Constraint Jacobian ................................. 150
C.2 Objective Gradient .................................. 176
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D Initial Guess 179
E Earth Relative Downtrack and Crosstrack 183
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