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 design problem is formulated as an optimal control problem. The optimal control
problem is transformed to a nonlinear programming problem using a Legendre 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 desirable characteristics of the solution to the CAV mission design problem are
defined. Second, a study is conducted to demonstrate the effect of the parameters on the performance of the CAV. This parametric study demonstrates the
use of the Legendre Pseudospectral method as a design tool and provides insight 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 donate 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 normalcy. Dave Benson, thank you for attending our review sessions, teaching me
5
how to make bread, and being my party buddy. Daveed, thanks for the pingpong 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 motivation 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 sponsoring agency of the findings or conclusions contained herein. It is published
for the exchange and stimulation of ideas.
Kim berley A. Clarke..............................
6
.............
Contents
17
1 Introduction
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
2.1 Overview ..........................................
2.2 Dynamic Model ................................
2.2.1 Coordinate System
..........................
2.2.2 Equations of Motion .............................
27
27
. 28
28
29
2.3 Boundary Conditions .............................
34
2.4 Path Constraints ................................
35
Perform ance Index ..............................
. 36
2.5
3 Optimal Control: Problem Formulation and Solution Methods
3.1
39
39
Overview ..........................................
3.2 Optimal Control Problem .............................
40
3.2.1
Dynamics ................................
40
3.2.2
Path Constraints ...............................
41
3.2.3
Boundary Conditions ............................
41
7
. . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.4
Performance Index
3.2.5
General Form of an Optimal Control Problem ..........
42
.42
3.3 Methods for Solving Optimal Control Problems . . . . . . . . . . . . .
43
Analytic Methods for Solving Optimal Control Problems . . .
43
3.3.1
3.3.2 Numerical Methods for Solving Optimal Control Problems .
48
3.4 Direct Transcription of Optimal Control Problem Via Pseudospectral Methods
3.5
50
..................................
3.4.1 Pseudospectral Methods ..........................
50
3.4.2 Legendre Pseudospectral Method . . . . . . . . . . . . . . . . .
52
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
. 62
4.2.1 Optimization Vector ........................
4.2.2 Discretization of the Dynamic Constraints ............
.65
4.2.3 Discretization of the Path Constraints and the Terminal Con66
straints .....................................
68
4.2.4 Discretization of the Performance Index .............
4.3 Common Aero Vehicle Nonlinear Programming Problem .......
4.3.1
.69
Summary of the Common Aero Vehicle Nonlinear Program69
ming Problem .................................
4.3.2 Structure of the Common Aero Vehicle Nonlinear Program-
71
ming Problem .................................
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 ...........
8
.76
77
4.5 Numerical Optimization Study .......................
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
Summary of the Numerical Optimization Study . . . . . . . . . . . . .
97
5 Parametric Optimization Study of the Common Aero Vehicle Problem
99
4.6
5.1
99
Overview ..........................................
5.2 Key Features of the Trajectory and Control
5.3
100
...............
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
5.6
117
. .
123
Summary of the Parametric Study .....................
6 Preliminary Study of the Real-Time Application of the Legendre Pseu125
dospectral Method
6.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
6.2 Common Aero Vehicle Flight Simulation . . . . . . . . . . . . . . . . . 126
6.3 Assessment of the Accuracy of the Legendre Pseudospectral Method 129
. 134
6.4 Sum m ary....................................
137
7 Conclusions
7.1
.137
Summary......................................
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
9
D Initial Guess
179
E Earth Relative Downtrack and Crosstrack
183
10
List of Figures
1-1 Common Aero Vehicle Mission Profile ...................
.
22
2-1 Earth-Centered Earth-Fixed Coordinate System . . . . . . . . . . . . .
28
2-2 Free Body Diagram of the Common Aero Vehicle .............
31
2-3
..............
Bank Angle .....................
3-1 Distribution of LGL points for a given number of nodes
33
.......
54
4-1 Sparsity Pattern of the Common Aero Vehicle Nonlinear Programming Problem .....................................
73
4-2 Angle of Attack vs. Time for M=(25, 50, 75, 100) .............
83
4-3 Bank Angle vs. Time for M=(25, 50, 75, 100) ................
83
4-4 Earth Relative Downtrack Distance vs. Earth Relative Crosstrack
Distance for 50 Nodes ...............................
84
4-5 Earth Relative Downtrack Distance vs. Earth Relative Crosstrack
84
Distance for 75 Nodes ...............................
4-6 Earth Relative Downtrack Distance vs. Earth Relative Crosstrack
Distance for 100 Nodes ............................
85
4-7 Earth Relative Speed vs. Time for 50 Nodes ................
85
4-8 Earth Relative Speed vs. Time for 75 Nodes ................
86
4-9 Earth Relative Speed vs. Time for 100 Nodes
. . .. . . . . .. . . . .
4-10Angle of Attack vs. Time for ki = (0.1, 1.0, 10, 100), k 2 = k3= 1.0
4-11 Angle of Attack Rate vs. Time for kI = (0.1, 1.0, 10, 100), k 2
86
90
=k3
91
1.0.............................................
11
4-12 Bank Angle Rate vs. Time for ki
(0.1, 1.0, 10, 100), k 2 =k
4-13 Angle of Attack vs. Time for k 2
(0.1, 1.0, 10, 100), ki
4-14 Angle of Attack Rate vs. Time k 2
4-15 Bank Angle Rate vs. Time for k2
4-16Angle of Attack vs. Time for k 3
=
=
=
1.0 .
91
1.0
92
3
=k3
(0.1, 1.0, 10, 100), ki
1.0
93
1.0 .
93
1.0
94
=
(0.1, 1.0, 10, 100), k =k
(0.1, 1.0, 10, 100), ki
3 =
k2
=
.
.
4-17Angle of Attack Rate vs. Time for k 3 = (0.1, 1.0, 10, 100), ki = k2
1.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
95
4-18 Bank Angle Rate vs. Time for k3 = (0.1, 1.0, 10, 100), ki = k 2 = 1.0.
95
5-1 Altitude vs. Energy for M=100, ki = k 2 = 1,k 3 = 0.1 . .
100
- . . .. . . .
5-2 Altitude and Dynamic Pressure vs. Time for M=100, ki = k2 =
1,k 3 = 0.1 . . . . .
.. . . .
- - -.
-.
..
. . -.
.
. . -.
.. .
101
5-3 Earth Relative Crosstrack Distance vs. Earth Relative Downtrack
Distance for M=100, ki = k2 = 1,k 3 = 0.1 .................
5-4 Angle of Attack vs. Time for M=100, k 1 =
5-5 Bank Angle vs. Time for M=100, ki = k2
1,k 3 = 0.1
102
......
1,k3 = 0.1 .........
103
104
5-6 Altitude vs. Energy for qmin = (11.97, 23.94, 35.91, 47.88) kPa . . . . 106
5-7 Earth Relative Speed vs. Time for qmin = (11.97,23.94,35.91,47.88)
kPa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5-8 Earth Relative Crosstrack Distance vs. Earth Relative Downtrack
Distance for qmin = (11.97, 23.94, 35.91,47.88) kPa ..........
108
5-9 Angle of Attack vs. Time for qmin = (11.97, 23.94, 35.91, 47.88) kPa
108
5-10Value of the Performance Index vs. Minimum Allowable Dynamic
Pressure for qmin = (11.97, 23.94, 35.91, 47.88) kPa ..........
109
5-11 Total Heat Load vs. Time for Qmnax = (1100, 1300, 1400,1700,2000,2300)
MJ/m 2
........
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .... . . . . . .
110
5-12 Heating Rate vs. Time for Qmax = (1100, 1300, 1400, 1700, 2000, 2300)
MJ/m 2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . .
11
5-13Altitude vs. Time for Qrmax = (1100, 1300,1400,1700,2000,2300)
MJ/m 2
.......
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . .
12
112
5-14 Earth Relative Speed vs. Time for Qmax = (1100, 1300, 1400, 1700, 2000, 2300)
M J/m2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5-15 Earth Relative Crosstrack Distance vs. Earth Relative Downtrack
Distance for Qmnax = (1100, 1300, 1400, 1700,2000,2300) MJ/m
2
. .
114
5-16Angle of Attackvs. Time for Qmax = (1100, 1300, 1400, 1700,2000,2300)
MJ/m 2
.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
115
5-17 Bank Angle Rate vs. Time for Qmax = (1100, 1300,1400,1700,2000,2300)
M J/m 2
. . .. . . .. . .. . .. . . . . .. . . . . .. . . . . . . .. . . . .
115
5-18 Value of the Performance Index vs. Qmax for Qmax = (1100, 1300, 1400, 1700,2000, 230
M J/m 2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5-19 Altitude vs. Energy for (L/D)max = (2, 2.1, 2.2, 2.3, 2.4, 2.5) . . . . . . 119
5-20 Earth Relative Crosstrack Distance vs. Earth Relative Downtrack
Distance for (L/D)max = (2, 2.1, 2.2, 2.3, 2.4, 2.5) . . . . . . . . . . . . 120
5-21 Earth Relative Speed vs. Time for (L/D)max = (2, 2.1, 2.2, 2.3, 2.4,2.5) 120
5-22 Stagnation Point Heat Load vs. Time for (L/D)max = (2, 2.1, 2.2, 2.3, 2.4,2.5)121
5-23 Angle of Attack vs. Time for (L /D)max = (2, 2.1, 2.2, 2.3, 2.4, 2.5) . . 121
5-24 Value of the Performance Index vs. (L/D)max for (L /D)max = (2, 2.1, 2.2, 2.3, 2.4, 2.5)12
6-1 Flight Simulation Block Diagram . . . . . . . . . . . . . . . . . . . . . . 127
D-1 Spherical Representation of Position with Respect to a Cartesian
ECEF Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
D-2 Spherical Representation of Velocity with Respect to a Set of Axes
Defined in the Cartesian ECEF Coordinate System . . . . . . . . . . . 181
E-1 Earth Relative Downtrack Plane and Earth Relative Crosstrack Plane 184
13
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List of Tables
4.1
Numerical Values Used for Numerical Optimization ..........
4.2 Numerical Values for the Bounds on the Optimization Variables
4.3
.78
79
.
Numerical Values for the Bounds on the Path Constraints ......
.79
4.4 Options Set in SNOPT ................................
4.5
80
Terminal errors produced by integration for M = (50, 75, 100) . ...
87
4.6 Results from Varying the Weighting Factors (ki, k 2 , k 3 ) . . . . . . . .
6.1
Terminal Errors from the Simulation with Perturbations
.......
97
.133
6.2 Computational Performance of the Simulation with Perturbations . 134
D.1 Values Used to Generate an Initial Guess .................
15
179
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Chapter 1
Introduction
1
Motivation
Gulf War II has brought attention to the importance of space applications on
current warfare tactics. GPS navigation, high resolution imagery, and near-real-
time missile detection via communication satellites are a few of the many critical
technological capabilities that result from space applications. U.S. and coalition
forces can gain significant advantages on the battlefield from space-based capabilities. At this point, it is evident that the U.S. can not afford the loss of space
assets or even the current time delay of months to fix or replace failed systems
[24]. Recognizing this, the Air Force Space Command (AFSPC) has shifted its
attention to "quick-response-space". The AFSPC is currently conducting an Operationally Responsive Spacelift Analysis of Alternatives (ORS/AOA) to address
the issue of responsiveness in terms of space applications [241. This analysis will
evaluate the application of ORS to military space assets for force enhancement,
space support, force application, and counterspace. In particular, the United
States currently has a high level of interest in developing global power projection capabilities because of instabilities in the international environment. While
it may be desirable to place armed forces in enemy territory, such a strategy
may be difficult to implement given current conditions. Consequently, it may
be beneficial to conduct military operations remotely. The ultimate goal of the
17
AFSPC is to develop the abilities to launch satellites within hours or days of the
given command, quickly repair a damaged system in space, and strike an enemy
anywhere on the globe in less than one hour with conventional weapons [241.
Space-based global strike refers to the ability to project power with conventional weapons from the United States to any point on the globe in less than one
hour. With this new desired ability comes the challenge of demonstrating the
technological feasibility of such an approach through vehicles capable of delivering the required conventional weapons. U.S. political and military leaders are
re-examining the entire realm of space-based capabilities along with strategic
weapons for new counteractive tactics [241. While B-2 bombers have demonstrated the ability to conduct global reach operations, the Air Force is not investing in more long-range bombers [25]. The existing intercontinental ballistic
missiles (ICBMs) and sea-launched ballistic missiles (SLBMs) are also capable of
striking any point on the globe, but they carry nuclear armed weapons. In order
to project power without the use of nuclear weapons, ICBMs and SLBMs must
be modified to carry conventional weapons and are referred to as conventional
ballistic missiles (CBMs). However, rearming the current ICBMs and SLBMs would
jeopardize the incredible accuracy and reliability of the already existing systems.
In addition, CBMs can easily be mistaken as nuclear armed weapons [251. Therefore, advanced reusable space launch vehicles are being considered as a means
for global projection.
The distance and speed requirements associated with space-based global
strike capabilities necessitates the design of a vehicle with space launch and
Earth reentry capabilities [23]. Proposed space-based global strike vehicles include the Space Operations Vehicle (SOV) and the Common Aero Vehicle (CAV).
The SOV is a fully reusable launch vehicle capable of flying sub-orbital pop-up
trajectories. This type of trajectory allows the SOV to carry a significantly greater
amount of weight through space. For example, an SOV capable of putting 6,000
lbs into orbit can carry 40,000 lbs through space in a pop-up trajectory [251.
There are four basic capabilities behind the SOV motivating its design. First is
18
the desire for aircraft like characteristics including the reliability and maintainability of today's aircrafts [8]. The second desired capability is launch on short
notice which corresponds to the feasibility of near-real-time to real-time com-
pletion of missions. The last two capabilities, expeditious deployment rates and
rapid transition between missions, reduce the cost of SOVs [8]. The CAV, currently being considered in the ORS/AOA, is a reentry vehicle that departs from
a launch vehicle or other booster and returns to Earth with the purpose of delivering weapons, payloads, or cargo to a specified location. It is essentially a
shell weighing 1300-2400 lbs (fully loaded) with a cross-range maneuverability
of at least 2400 Nautical Miles [23]. Yet to be determined is whether or not to design a steerable ballistic CAV similar to the existing Maneuvering Reentry Vehicle
(MaRV), Advanced Maneuvering Reentry Vehicle (AMaRV), or High Performance
Maneuvering Reentry Vehicle (HpMaRV) concepts. Another design option is a
fully powered steerable CAV with considerable maneuverability both in space
and in the Earth's atmosphere. Nonetheless, the purpose of the CAV is to create
rapid response for global reach from within the continental United States and to
operate under abnormal conditions [23].
1.2
Common Aero Vehicle
The deployment of a CAV involves launch, atmospheric reentry, release of cargo,
payload, or weapon depending on the mission, and disposal of the CAV. Three
different launch vehicles are currently being considered for the CAV: an expendable ground launched rocket, an expendable air launched rocket, and the SOV.
Both the expendable ground launched and air launched rockets will most likely
However, in the long-term the SOV is
be utilized for near term applications.
most desirable because of its aircraft-like operability [231. Regarding launch scenarios, the following two trajectories are being considered: a pop-up trajectory
and an orbital trajectory. As mentioned above, the pop-up trajectory allows for
more weight to be carried by the launch vehicle. The orbital trajectory refers
19
to either a low Earth orbit (LEO) or one that orbits the Earth once. After the
CAV is launched, it must re-enter the Earth's atmosphere, reduce speed, and rely
on guidance to steer the vehicle to a specified release point. At this point the
cargo, payload, or weapon is released for the purpose of either force application
or force enhancement. Force application is utilized during combat before U.S.
forces arrive in attempt to stall adversarial advances. For example, Small Smart
Munitions can locate and identify stationary or mobile targets. Force application is also used to attack highly valued, heavily protected, or time critical targets. Examples of CAV payloads used for such missions include a single Unitary
Penetrator that is used to destroy deeply-buried targets such as underground
bunkers and storage facilities; an Agent Defeat Weapon that neutralizes biological or chemical weapons; Highly Effective Area Attack Submunitions that can
attack multiple dispersed targets; and precision area attack weapons such as
Low Cost Autonomous Attack Systems used to attack moving targets [23]. Force
enhancement is used to strengthen and provide services for military operations.
For example, Unmanned Aerial Vehicles are used for reconnaissance or surveillance purposes and CAVs may provide a means of delivering urgent cargoes to
remote locations in near-real-time [23]. Finally, the CAV may either be reusable
or expendable. If the CAV is reusable, it must return to a suitable recovery area
and if it is expendable, it must be destroyed without a trace.
Many technical challenges stand between concept and development of the
CAV. These challenges include designing a thermal protection, a propulsion system, a guidance and control system, and a payload release system. The CAV
must be able to structurally withstand the forces created from rapid acceleration and deceleration as well as excessive heat build-up. The propulsion system
is crucial for reentry positioning and must be safe and reliable. The guidance
and control system must be able to accurately and reliably guide the CAV using inertial navigation and/or GPS. Finally, the CAV must be able to release the
payload without disturbing its flight [23].
20
1.3
Mission Design Problem
The particular problem considered in this thesis is the application of a CAV as
a kinetic energy weapon, where instead of using explosives, its kinetic energy
upon impact is used to destroy a ground target. Consequently, the CAV is itself
the weapon and its deployment is simplified to launch and atmospheric reentry. An unpowered bank-to-turn high lift-to-drag ratio vehicle model is chosen.
Furthermore, the particular application is that of an Earth penetrator used to
strike hardened deeply-buried targets (HDBTs). The high maneuverability of the
CAV allows for contingencies such as avoiding adversarial anti-missile missiles
(AMM's) or in flight re-targeting. The mission design problem is to steer the CAV
from a fully specified initial state at or near atmospheric entry to a specified
target on the surface of the Earth.
The mission profile for the CAV considered, as shown in Fig. 1-1, consists
of atmospheric entry, a skip maneuver, a glide maneuver, speed depletion, and
Earth impact. The skip maneuver is characterized by a rise in altitude that en-
ables the vehicle to fly in a low density region in order to make the required
range. During the skip maneuver, control authority is lost as the vehicle rises
in altitude. Thus, it is desirable to prevent the vehicle from exiting the Earth's
atmosphere. In order to restrict the maximum altitude attained during the skip
maneuver, the initial condition is taken to be at the point after atmospheric
entry, but before the altitude of the vehicle starts to increase. The maximum
altitude attained during the skip maneuver is limited by imposing a minimum
allowable dynamic pressure constraint. During a glide maneuver, the vehicle
flies along a trajectory without using much control effort. As the vehicle nears
the target, it must deplete speed in order to meet the large but bounded terminal speed requirements for striking HDBTs. Finally, the mission terminates
when the vehicle strikes the target on the surface of the Earth. Typical terminal
conditions associated with HDBTs include position accuracy to within several
meters, a speed large enough for Earth penetration, but low enough so that the
21
Figure 1-1: Common Aero Vehicle Mission Profile
vehicle does not vaporize upon re-entry, and a nearly zero angle of incidence
[26]. These terminal conditions require that the vehicle approach the target with
negative lift. However, the CAV considered has one-sided angle of attack control
(i.e. the angle of attack must remain positive throughout flight). Thus, the vehicle
must rotate and fly upside-down in order to generate negative lift. Furthermore,
the natural behavior of the vehicle is to maintain a larger terminal speed and a
larger incidence angle at impact.
The terminal conditions associated with striking a HDBT pose a great challenge for the guidance and control system due to the conflict that arises between
high maneuverability and the need to achieve such tightly prescribed terminal
conditions. Early in-flight, the high maneuverability is desirable for both reachability and contingency plans; however, as the vehicle nears the target this maneuverability becomes a liability. Small errors in vehicle attitude can produce
extremely large errors in lift force that will, in turn, drive the vehicle away from
22
the desired target. Moreover, since this type of vehicle has one-sided angle of attack control, uncertainties in the environment can further increase errors. Con-
sequently, the demands on the guidance and control systems increase greatly as
the vehicle approaches the target. In this particular application, the ability to
withstand unexpected events during flight is a critical requirement in meeting
the boundary conditions with extremely high accuracy. In an attempt to obtain a
solution robust to in-flight dispersions, it is beneficial to design a trajectory and
control that has as much control margin as possible. The control margin is the
magnitude of the difference between the actual control and the control limits.
In addition to satisfying the initial and terminal conditions, the vehicle has
thermal, structural, and operational constraints during re-entry. Thermal constraints include maximum limits on heating rate and total heat load, structural
constraints include maximum limits on sensed acceleration, and operational
constraints include limits on control authority (i.e. limits on steering and steering rate capability).
For any set of initial and terminal conditions, a wide range of feasible trajectories and controls may exist. In order to obtain the most desirable performance, it is preferable to choose a particular performance index and determine
the particular trajectory and control that optimizes this performance index. This
results in the need to determine an optimal mission plan. The optimal mission
planning problem is then described as follows: determine a steering command
as a function of time that takes the vehicle from a specified initial state to a
target on the surface of the Earth while optimizing the given performance index
and satisfying all of the constraints imposed during flight.
1.4
Mission Design Approach
The optimal mission planning problem described in the preceding section is an
optimal control problem [3, 171. In general, aerospace optimal control problems
are nonlinear and do not have analytic solutions. Consequently, a numerical
23
method must be used to obtain a solution to these optimal control problems.
Numerical methods for solving optimal control problems can be categorized as
either indirect methods or direct methods. Indirect methods solve a HamiltonianBoundary-Value-Problem (HBVP) which is often difficult to solve numerically [11].
Direct methods discretize the optimal control problem at particular time points
which leads to a nonlinear programming problem (NLP). The resulting NLP is
solved using one of the many available optimization algorithms. While direct
methods have a wider range of convergence, the control time histories are not
as accurate as those obtained via an indirect method [201. A method that combines the advantages of both indirect methods and direct methods is desirable.
Pseudospectralmethods of Ref. [12, 16] utilize an approach to solve optimal control problems that has positive attributes of both indirect and direct methods.
In a pseudospectral method, the optimal control problem is discretized at specified time points using a basis of global orthogonal polynomials. This discretization procedure provides an efficient transcription of the continuous-time optimal control problem to a NLP. The solution of the resulting NLP provides an
accurate approximation to the continuous-time optimal control problem. In this
thesis, the Legendre PseudospectralMethod of Refs. [7, 9, 10, 11] is applied to
the Common Aero Vehicle optimal mission planning problem.
1.5
Research Objectives
This thesis seeks to demonstrate the application of the Legendre Pseudospectral Method to the problem of performance optimization of the Common Aero
Vehicle (CAV). In doing so, the accuracy of the Legendre Pseudospectral method
is assessed as well as the desirable traits of the trajectory and control for the
CAV. Furthermore, a parametric study is conducted to illustrate the use of the
Legendre Pseudospectral Method as a design tool as well as to gain a better
understanding of the behavior of the CAV under discussion. Finally, a preliminary investigation is performed of the real-time application of the Legendre
24
Pseudospectral Method to the CAV. The accuracy of the Legendre Pseudospec-
tral Method is considered in regards to the simulation of the flight of the CAV
and the robustness of the control to vehicle and environmental perturbations is
considered.
1.6
Thesis Overview
Chapter 2 mathematically describes the mission design problem stated in section 1.3. The equations of motion which govern the flight of the Common Aero
Vehicle are derived and the known initial and terminal conditions are defined.
Also included in this chapter are the constraints imposed throughout the flight
of the vehicle and the development of a performance index which reflects the
goal of maximizing the control margin.
Chapter 3 provides the theory and motivation behind the mission design approach discussed in section 1.4. A formal definition of an optimal control problem is stated and from which it is seen that the CAV mission design problem
is an optimal control problem. Also included is a discussion of analytic and
numerical methods for solving optimal control problems, which motivates the
use of a pseudospectral method. This discussion leads to an overview of pseudospectral methods and is followed by a detailed description of the Legendre
Pseudospectral Method used to solve the CAV optimal control problem.
Chapter 4 demonstrates the application of the Legendre Pseudospectral method
to the CAV optimal control problem. The CAV optimal control problem is discretized and the resulting nonlinear programming problem is discussed. A brief
overview is provided of the optimization algorithm SNOPT, which is used to
solve the nonlinear programming problem. A numerical optimization study is
then conducted to determine the number of nodes required to meet the accuracy requirements of the CAV mission design problem. Also included in the
numerical optimization study is the determination of the values of the weighting factors in the performance index that produce the most desirable trajectory
25
and control.
Chapter 5 presents a parametric optimization study of the CAV optimal control problem. This demonstrates the use of the Legendre Pseudospectral Method
for both vehicle design and trajectory design. The key features of the trajectory
and control generated from the Legendre Pseudospectral Method are discussed
to provide insight on the behavior of the CAV. Parameters in the problem are
then varied to determine the effect on the trajectory and control. The characteristics of the trajectory and control, using the control margin as a performance
metric, are then evaluated.
Chapter 6 describes a preliminary investigation into the real-time application
of the Legendre Pseudospectral Method to the CAV. This is done by simulating the flight of the CAV using the control obtained from the Legendre Pseudospectral Method. The assumptions used to develop the simulation along with
a description of the simulation itself is given in this chapter. The accuracy of
the Legendre pseudospectral solution is assessed by comparing the trajectory
obtained from the optimizer to the trajectory obtained via numerical integration. Perturbations which reflect realistic model uncertainties are then added to
the simulation in an attempt to assess the robustness of the solution. Finally,
Chapter 7 provides a summary of the material presented in this thesis and the
conclusions drawn from the results obtained.
26
Chapter 2
Common Aero Vehicle Problem
Formulation
2.1
Overview
This chapter gives a quantitative description of the optimal mission planning
problem stated in the introductory chapter. Recall that the optimal mission
planning problem is to determine a steering command as a function of time that
takes the CAV from a specified 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. First, a mathematical model of the CAV, which is
an unpowered high lift-to-drag ratio vehicle in atmospheric flight, is developed.
Second, boundary conditions are specified to indicate the known initial and terminal conditions for the vehicle. Third, constraints during flight are identified
and quantified. Fourth, the desired performance index that is to be optimized is
developed.
27
2.2
2.2.1
Dynamic Model
Coordinate System
In this particular application, the target is a point on the surface of the Earth.
Consequently, it is most desirable to describe the motion of the vehicle using a
coordinate system that rotates with the Earth. Furthermore, in this research we
are interested only in vehicle performance. Therefore, it is adequate to model
the vehicle as a point mass and consider only the translational motion of the
center of mass (i.e. rotational effects are ignored). In this research, a Cartesian Earth-centered Earth-fixed (ECEF) coordinate system is used. Fig. 2-1 shows
schematically, the position, r, and inertial velocity, v, of the center of mass of
the vehicle represented in an ECEF coordinate system where 0 marks the cen-
Prime Meridian
Nv
G/
r
0 1
R
Equatorial Plane
Earth
Figure 2-1: Earth-Centered Earth-Fixed Coordinate System
ter of the Earth, N is the North Pole, and G is the location of the Greenwich
Observatory in the United Kingdom. The three principle-axis directions of the
ECEF frame are OQ, OR, and ON where OQ, OR, and ON are defined as follows.
The OQ-axis is the first principle direction, lies in the plane (OG,ON), and passes
through the equator (along the Prime Meridian). The ON-axis is the third principle direction and passes through the North Pole. Finally, the OR-axis completes
28
the right-handed system (OQ,OR,ON). Furthermore, the Earth rotates about the
ON-axis with a constant magnitude 0.
2.2.2
Equations of Motion
The three degree-of-freedom equations of motion in ECEF coordinates for a vehicle modeled as a point mass in flight over a spherical rotating Earth are derived
as follows. The position of the vehicle is given as
r = r(t) = xex + yey + zez
(2.1)
where ex is the unit vector in the direction of OQ, ey is the unit vector in the
direction of OR, and ez is the unit vector in the direction of ON. Differentiating
the position with respect to time, the absolute velocity, v, is given as
v = v(t) = ar
at
(
Noting that w =
+ w xr
(2.2)
(ex,ey,ez)
2ez, we have that
v = *ex + pje, + ez +Qez x (xex + yey + zez)
(2.3)
= (* - G2y)ex + (p + Qx)ey +
zez
Differentiating v(t) the absolute acceleration, a, is given as
a=(a(t)
(at)(ex,ey,ez)WXV
=(lzr\(r
at
at2(ex,ey,ez)
+2w x (ar)
+ w x (w x r)
2
t (ex,ey,ez)
= (k - 7p)ex + (p'+ i)ey + 2ez +
=
(
-29
-Y
2
x)ex +((jy)ey
+2k
-+
ez x ((k - Qy)ex + (p + i2x)ey + ez)
+ez
(2.4)
29
Now, let vr represent the Earth relative velocity, i.e.
(
vr
=ar ar
at
(ex,ey,ez)
= xex +
pe,
(2.5)
+ ez
= vxex + vye, + vzez
Substituting Eq. (2.5) into Eq. (2.4), the motion of the vehicle can be expressed in
terms of the position and Earth relative velocity as
at
/av
= Vr
ex,ey,e)
(,
at )(ex,ey,ez)
(2.6)
a -2w x v,
)=
-
w x (w
x r)
The following notation change is made and used in the remainder of this thesis:
)
(ex,ey,ez)
Applying Newton's second law to the vehicle, (i.e. F = ma) where m is the mass
is the absolute acceleration of the vehicle, and F is the
dt
total force acting on the vehicle, we obtain
of the vehicle, a
=d
-
= Vr
r
F
= - - 2w x vr - w x (w x r)
(2.8)
Throughout flight the vehicle is under the influence of gravitational and aerodynamic forces. The free body diagram of the vehicle shown in Figure 2-2 depicts
the individual affects of each component of the total force applied. The gravitational force is denoted by g, the lift force is denoted by L, and the drag force is
30
AL
g
Figure 2-2: Free Body Diagram of the Common Aero Vehicle
denoted by D. The total force on the vehicle is then given as:
(2.9)
F =g +L+ D
where the following equations represent each of the forces with respect to their
ECEF components.
gzez
g
= gxex+gyey+
L
=
D
= Dxex + Dye, + Dzez
Lxex + Lye, + Lzez
(2.10)
The gravitational force is inversely proportional to the square of the distance
between the center of the Earth and the vehicle given as
g = -m P r
r3
(2.11)
where p is the Earth's gravitational parameter and r = lirll 2 is the radius.
The aerodynamic model used to define the lift and drag forces is taken from
Ref. [29] which assumes air is a uniform gas. A drag polar is used and is given
as [221
CD
=
CDO + KCL
CL
=
CL,Cxo
31
(2.12)
where
CD
is the drag coefficient,
CDO
is the zero-lift coefficient of drag, K is
the drag polar parameter, CL is the lift coefficient, at is the angle of attack, and
CLa is the lift slope. The angle of attack is defined as the angle between the
Earth relative velocity and the zero lift line. It can be seen that using the above
assumptions, no lift is produced when c = 0.
The lift and drag forces are defined as follows
L
(2.13)
= LwL
D. =
-Dv
V
(2.14)
where L is the magnitude of the lift force, WL is the unit vector in the lift direction, D is the magnitude of the drag force, and v = lIVr IIis the Earth relative
speed of the vehicle. L and D are defined as
where q
=
L
= qSCL
(2.15)
D
=
(2.16)
qSCD
pv 2 /2 is the dynamic pressure and S is the reference area of the
vehicle. The atmospheric density, p, is modeled as an exponential function of
the radius as shown below
p = po exp [-(r - Re)/H]
(2.17)
where po is the density at sea level, Re is the radius of the Earth, and H is the
density scale-height [281.
The lift direction WL lies in the (r,vy)-plane and rotates with the vehicle while
32
the drag direction is opposite vr. The lift direction, WL, is computed as follows:
y
Wi
=
W2
=
V
r
x vr
rxVr1
(2.18)
W3
Wi X W 2
WL
sin 0w
2+
cOS Ow
3
where o- is the bank angle taken from Ref. [291 as depicted in Figure 2-3. Therefore,
L3
W,
W,
r
Figure 2-3: Bank Angle
the vehicle is controlled aerodynamically via a and o-. While in theory it is possible to control o( and - directly, in practice it is not possible to apply these
controls instantaneously. Consequently, it is necessary to impose rate limits on
a and -. Rate limits are imposed by augmenting the following two differential
equations to the dynamics of Eq. (2.8)
de
=
u,
J
= Uo.
(2.19)
(2.20)
where u, and u- are pseudocontrols that define the angle of attack rate and
the bank angle rate, respectively. The resulting augmented dynamics for an
33
unpowered vehicle in atmospheric flight over a spherical rotating Earth are given
in Cartesian Earth-centered Earth-fixed coordinates as
*c
=vx
9;
=v,
2
=v
X
9x + Lx + Dx + 20v, +
2
9y + Ly + Dy -2Qv±
2
m
X
y
(2.21)
m
2.3
z + Lz + Dz
Iz
=
di
=
u"
0-
=
uo.
m
Boundary Conditions
The desired trajectory steers the Common Aero Vehicle from a fully specified
initial position and velocity to a fully specified terminal position with terminal constraints on speed, the Earth relative flight path angle, and angle of attack. The Earth relative flight path angle, y, is computed from r and vr as
y
=
arcsin r vr). The initial conditions are then given as
r(to)
=
ro(2.22)
Vr(to)
=
Vr,o
while the terminal conditions are given as
=
r(tj)
rf
Vf
y(tf)
=
Yf
c(t5)
=
af
34
S=
(2.23)
2.4
Path Constraints
Flight path constraints are imposed throughout the entire trajectory. Trajectory
constraints include restrictions placed on the radius, speed, and dynamic pressure while vehicle constraints include restrictions placed on the structural load-
ing, thermal loading, and the control authority. Physically, the vehicle cannot fly
below the surface of the Earth. Therefore, it is necessary to impose an inequality
constraint on radius. Because the CAV has no propulsive capability, the speed
will never increase during re-entry. In order to enhance the performance of the
optimizer, a path constraint is placed on the speed. A dynamic pressure constraint is imposed during entry in order to maintain control authority. The CAV
has a limit on the maximum sensed acceleration it can withstand. Therefore, a
path constraint is placed on the sensed acceleration, a, which is defined as
a = VD 2 + L 2
(2.24)
During entry the vehicle absorbs heat. Because the amount of heat that the vehicle absorbs is limited by the material used in construction, a maximum allowable
heat load constraint is imposed. In this research, the heat load is taken to be the
stagnation point heat load [5] given as
Q=
to
Qdt
(2.25)
where
Q
K(PIPo)os(VIV)
3
.S
(2.26)
ve is the speed of a vehicle in circular orbit at a radius equal to the radius of
the Earth, po is the atmospheric density at sea level, and K is a known constant.
Operational path constraints include limits on the angle of attack and rate limits on the angle of attack and bank angle. The resulting inequality constraints
35
imposed during flight are given quantitatively as
r
>
Vmin :
V
0
a,nin
Uo-,min
2.5
Re
vmax
q
ii
a
amax
Q
Qmax
5
O
Uc
Uo-
(2.27)
5 amax
5 Ua,max
Uaomax
Performance Index
The CAV mission design problem includes steering the vehicle from a known initial state to a specified terminal state. Thus, it is important for the guidance and
control system to be able to reach the target. This requires a guidance and control system capable of not only steering the vehicle with precision and accuracy,
but also designing a trajectory that is capable of handling environmental disturbances experienced throughout flight. In attempt to minimize the demands on
the guidance and control systems, it is desirable to keep the controls away from
their upper and lower limits. This allows for more flexibility in the controls to
account for off-nominal perturbations experienced during flight. Defining the
control margin as the magnitude of the difference between the actual control
and the control limits, the goal of the performance index is to maximize the
control margin. Therefore, a performance index is constructed that attempts to
keep x in the middle of its capability and penalizes large control rates. Mathematically, a penalty is imposed on deviations in a from &, where 6 = VCDo/K.
This value of 6t corresponds to the angle of attack at the maximum L/D ratio
and lies in the middle of the bounds placed on the angle of attack. Minimization
of the control rates (ux, u,) keeps the controls smooth and within their allowable limits. While many possible performance indices can be constructed, the
36
following performance index is used in this thesis:
=ua
to
L
+ ks
(max /
2/
u,max
u
2]
o-U,max
dt
(2.28)
where ki, k 2 , and k 3 are positive constants. Each term is weighted by its respective maximum value for easier interpretation of the constants and squared to
account for the possibility of a negative value.
37
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Chapter 3
Optimal Control: Problem Formulation
and Solution Methods
3.1
Overview
Consider a dynamical system that is subject to constraints. Furthermore, con-
sider a system whose state can be affected by the choice of various inputs or
controls. Any input to the dynamical system that satisfies the constraints is
called a feasible control. The time history of the state that results from the application of a feasible control is called a feasible trajectory.For many problems it is
desired to determine the feasible control and feasible trajectory that optimizes a
specified performance index for a dynamical system subject to constraints. Such
a problem is called an optimal control problem.
It can be seen from Chapter 2 that the CAV mission design problem is an optimal control problem. While in principle any well-posed optimal control problem
has a solution, finding such a solution is often a difficult task. In this chapter an
overview is given of the basic theory of optimal control. Furthermore, a survey
of various numerical methods for solving optimal control problems is discussed.
Finally, the method used to solve the CAV mission design problem, the Legendre
Pseudospectral Method of Refs. [7, 9, 10, 11], is described.
39
3.2
Optimal Control Problem
An optimal control problem consists of four parts: (1) a mathematical model
describing the dynamics of the vehicle (equations of motion), (2) the boundary
conditions that specify the initial and terminal states, (3) path constraints that
are enforced during the trajectory, and (4) a performance index that measures
the optimality of the solution.
3.2.1
Dynamics
In general, a mathematical model for the dynamics of a particular system is comprised of three quantities: the state, the control, and the independent variable
(generally speaking, time). The state, denoted x(t), is a vector whose components individually define the variables that are required to describe the behavior
of the system at any instant of time. Denoting the dimension of the state by n,
the state is given mathematically as
xit)
x(t)
=
x 2 (t )
E R"
(3.1)
xn (t )
Similarly, the control, denoted u(t), is a vector whose components individually
define the inputs to the system at any instant of time. Denoting the dimension
of the control by m, the control is given mathematically as
Sui(t)
u(t) =
u 2 (t)
Um(t)
40
E R'
(3.2)
The dynamics of the system are described via a system of ordinary differential
equations of the form
i t) = f (X(t), U (t), t)
(3.3)
where i(t) is the time derivative of the state vector and f : R" x R" x R - R". In
general, the dynamics of Eq. (3.3) are nonlinear.
3.2.2
Path Constraints
Virtually all problems in dynamical systems are subject to constraints during
the evolution of the system. Such constraints are called path constraints. Denoting the number of path constraints by p, the path constraints are described
mathematically as
gmin s! g(X (t), u (t), t) s! gmax
(3.4)
where g: RI x Rm x R - RP and gmin E RP and gmax C RP are constant vectors.
3.2.3
Boundary Conditions
The boundary conditions describe events that occur at either the beginning or
the end of the trajectory. The boundary conditions are split into initial conditions that occur at the initial time, to, and terminal conditions that occur at
the terminal time, tj. Denoting the number of initial conditions by qo and the
number of terminal conditions by qf, the boundary conditions can be expressed
mathematically as
ho(x(to), to)
=
0
(3.5)
hf(x(tf), tf)
=
0
(3.6)
where ho : R" x R - Re and hf : Rn x R - Rqf.
41
3.2.4
Performance Index
The performance index is the functional (i.e. it is a function of a function) that
is to be optimized in the optimal control problem. The performance index produces a scalar output. Since the goal is to minimize (or maximize) the performance index, an accumulation (or depreciation) in value of the resulting scalar
can be thought of as a cost penalty. Often referred to as the cost functional,
the performance index can be broken into three parts: an initial cost, a terminal
cost, and an integrated cost. As the name implies, the initial cost Jo is associated
with the initial state, x(to), and the initial time, to. Similarly, the terminal cost Jf
is associated with the terminal state, x(tj), and the terminal time, tf. The initial
and terminal cost are given, respectively, as
Jo
= A4(x(to),to)
Jf =
(3.7)
M(X(tf), tf)
where AI : R" x R - R and N : R" x R - R. The integrated cost is a cost that
accumulates throughout the trajectory and is given as
t5
Ji =
.1:
(3.8)
£(x(t), u(t), t)dt
where f : R" x R"I x R - R. The total cost is then given as
tf
J = M(x(to), to) + N (x(tf), tf) +
3.2.5
fto L(x(t), u(t), 0 dt
General Form of an Optimal Control Problem
Using the definitions in Sections 3.2.1-3.2.4, an optimal control problem is now
stated formally as follows. Determine the control u* (-), and the state x* (-) on
the interval t c [to, tf] that minimizes the cost functional
+T
J = _'l(x (to), to) + X (x(ty ), tf ) + toI(x
42
t), u t), t d t
(3.9)
subject to the dynamic constraints
x
=
f(x(t), u(t), t)
(3.10)
the path constraints
gmi n
g(x(t), u(t),t) s gmax
(3.11)
and the boundary conditions
3.3
ho(x(to), to)
=
0
(3.12)
hf(x(tf),tf)
=
0
(3.13)
Methods for Solving Optimal Control Problems
A solution to an optimal control problem is obtained using either an analytic
or a numerical method. Typically, optimal control problems cannot be solved
using analytic methods and thus solutions are obtained via numerical methods.
Nonetheless, it is important to understand both analytic methods and numerical
methods.
3.3.1
Analytic Methods for Solving Optimal Control Problems
Analytic solutions to optimal control problems are generally determined by one
of two approaches: calculus of variations and dynamic programming. Calculus of variations involves setting the first variation of the cost functional (or an
augmented cost functional) equal to zero which leads to a set of first-order necessary conditions for a solution to the optimal control problem. Pontryagin's
Minimum Principle [17] is used to determine the optimal control. Application of
dynamic programming leads to the Hamilton-Jacobi-Bellman (HJB) equation [17].
The HJB equation is a partial differential equation which governs the dynamics
of the optimal cost functional. Calculus of variations, in combination with the
43
principle of optimality, leads to a derivation of the HJB equation. The intention here is to provide the reader with a brief explanation of analytic methods
and the difficulties that hinder the implementation of analytic methods. For a
complete explanation and derivation of both calculus of variations and dynamic
programming please refer to Refs. [31 and [171.
Calculus of Variations
Calculus of variations, in terms of optimal control problems, is motivated by the
desire to determine the feasible control and feasible trajectory that minimizes a
performance index. In the unconstrained case, the optimal control problem simplifies to a functional minimization problem. Consider the functional J(x(t)). A
local minimum of J exists at x* (t) if
(3.14)
J(x(t)) - J(x* (t)) > 0
for all admissible x(t) in some neighborhood around x*(t), (i.e. Ilx(t) - x* (t)I <
c). If the neighborhood can be extended to the entire domain of x(t), then x* (t)
is a global minimum. A necessary condition for x* (t) to be a local minimum of
J is
6J (x*(t), 6x(t)) = 0
for any
6x(t)
where 6J is the first variation of the functional. In order to determine if the
stationary point is indeed a minimum, the second variation of the functional is
considered. By doing so, second order sufficient conditions for a local minima
are established. Please refer to Ref. [31 for a complete explanation and derivation
of the second order sufficient conditions.
By definition, an optimal control problem has constraints and thus the application of functional minimization to an unconstrained problem must be extended to handle a constrained problem. Consider the following optimal control
44
problem: Minimize
J =
tf
L(x(t), u(t), t)dt
T
(x(tf), tj) +
subject to the system equations
x = f(x(t),u(t), t)
and control constraints
u(t) E U(t)
where x(to) and to are fixed, tf is free, and there are simple form terminal constraints. In order to impose the state differential equations, an augmented cost
functional Ja is considered where
Ja = X(x(tf), tj) +
I L(x(t),u(t),
t) + A(t)T[f(x(t),u(t), t)
t0
-
i]
dt
A(t) E R" is the co-state. In taking the first variation in Ja, it is convenient to
define the Hamiltonian,H [171:
H(x(t), u(t), A(t), t) = L(x(t),u(t), t) + AT (t)f(x(t), u(t), t)
(3.15)
In terms of the Hamiltonian, the necessary conditions for an extremal trajectory
are
f(x, u, t)
(3.16)
x(to) = Xo
(3.17)
i
_aH T
(3.18)
H(tj) + at (tf) = 0
(3.19)
x (tf) = xf,i
(3.20)
Ai(tj) =
axi (tj)
45
(3.21)
where Eq. (3.16) is the dynamic constraints, Eq. (3.17) is the initial conditions,
Eq. (3.18) is the co-state equations, Eq. (3.19) is the transversality conditions,
and Eqs. (3.20) and (3.21) are the terminal conditions. In a neighborhood of a
locally optimal solution, where the state and co-state differential equations and
all the boundary conditions are satisfied, the first variation becomes
6]a
f
Hu(t)6u(t)dt
to
H, is the functional gradient of the augmented cost functional with respect to
the control at every point in time. If the extremal solution is a minimum, then
any variation from that point will yield a positive variation.
Hu(t)6u(t) > 0
for all admissible
5u(t)
The goal is to minimize H over the admissible range of u. From Pontryagin's Minimum Principle [17], the admissible control that minimizes H can be determined
and is given as
mn
u* (t) = arg
H(x*(t),u(t), A*(t), t)
(3.22)
u(t)EU(t)
Only in simple cases can a solution that satisfies the necessary conditions stated
in Eqs. (3.16)-(3.23) be obtained. The combination of nonlinear differential equations and split boundary values creates difficulty in finding an analytic solution
to the optimal control problem.
Dynamic Programming
This approach uses calculus of variations and the principle of optimality to develop a partial differential equation which governs the optimal cost functional.
Consider the following nonlinear system with general terminal constraints and
control constraints. Minimize
tJ
J (x (t), U (t), t) = X (x(tj), tf ) + ft I(x (t), u(t), t d t
46
subject to
xkt) = f (X(t), U(t), t)
hf(x(tf), tf) = 0
u(t) E U(t)
Given the initial state and control history, the state history is computable and the
optimal cost is a result of the optimal control history. As a result, the optimal
cost does not depend on the control, J*(x(t), u(t), t) = J*(x(t), t).
Consider a control problem where given an initial state x(to), the goal is to
drive the system to a terminal state, x(tf). Suppose that the optimal solution
passes through some intermediate point x(ti). The principle of optimality states
that the solution to the optimal control problem starting at x(t
1
) and terminating
at x(tf) is a segment of the solution to the optimal control problem that starts
at x(to) and terminates at x(tj)'[17]. In other words, any portion of an optimal
solution is itself an optimal solution.
Using the principle of optimality and assuming that J is twice differentiable
with respect to x(t) and t, a Taylor series expansion of J* about (x(t), t) yields
the Hamilton-Jacobi-Bellman (HJB) equation [17]:
a*(x(t), t) =
at
min I (x (t), U(t), t) + a*(X (t), t f (X(t), U(t), t)
u~U t)
ax
subject to the constraints
J* (x(t), t)
=
N(x(t), t)
on
hf(x(t), t) = 0
The HJB equation is both necessary and sufficient for optimality [17]. If the
above equation can be solved to obtain J* [x(t), t], then the optimal control is
determined as a feedback law and is given as
u* (t) = arg
mn
[L(x(t), u((t, t) +
aj*
(x(t), U(t), t
(3.23)
While the result of Eq. (3.23) applies to problems with general dynamics, a gen47
eral cost functional, and constraints, it is rarely possible to obtain analytic solutions to the HJB equation. HJB theory could be used to generate an optimal
feedback law numerically, but this is not usually done. Instead the HJB equation is used to test the optimality of a control whose form was either guessed or
obtained by some other method 1171.
3.3.2
Numerical Methods for Solving Optimal Control Problems
In general, the optimal control problem of Sec. 3.2 cannot be solved analytically,
so the solution must be attained using a numerical method. Numerical methods
fall under two main categories: indirect methods and direct methods.
In an indirect method, the Hamiltonian boundary-value problem (HBVP) that
arises from the first-order necessary conditions via the calculus of variations is
solved numerically. The general procedure for solving the HBVP [3] begins with
making an initial guess for the unspecified initial (or terminal) conditions. An
iterative procedure is then used to modify the estimate of the unknown initial
(or terminal) conditions, where each modification should improve the solution.
An improvement occurs when the current solution is "closer" to satisfying the
necessary conditions than the previous solution. If the iterative procedure converges, it will produce a solution that satisfies all of the necessary conditions.
Obtaining an initial guess is not a trivial procedure and thus more often than
not the solution that results will violate at least one of the necessary conditions. Examples of such iterative procedures include steepest descent methods,
neighboring extremal methods, and quasilinearization methods. Please refer to
Refs. [3] and [17] for an explanation of each procedure.
An advantage of using indirect methods is that an accurate co-state can be
found [201, which is beneficial because this co-state is then used to compute an
accurate control. Unfortunately, it is often impossible to obtain an initial guess
for the unknown conditions at one end which will produce a solution sufficiently
close to the optimal solution. As a result, it is often difficult to solve the optimal
48
control problem using an indirect method.
In a direct method, the optimal control problem is discretized at particular
time points called nodes. This discretization leads to a nonlinear programming
problem (NLP). The number of nodes is chosen large enough so that the time
steps are small enough to adequately represent the solution characteristics and
the implicit integration of the system equations produce sufficiently accurate
results. Provided that the time steps adequately represent the solution, the accuracy of the implicit integration depends on the specific quadrature rule used
[12]. In terms of parameterizing the problem, there are two approaches taken:
differential inclusion and collocation. Both of these methods involve implicit
integration of the system governing equations. However, differential inclusion
only discretizes the state variable time history while collocation discretizes both
the state and control variable time histories.
Differential inclusion methods replace bounded controls with bounds on admissible values of the state variable time rates of change. Elementary implicit
integration rules are then used to write the time rates of change as functions of
only the state variables. Since the control variables are eliminated, the number
of variables in the resulting NLP is reduced which, in turn, significantly reduces
the computation time required to solve the NLP [4]. Differential inclusion is restricted to problems with linearly appearing controls and the state rate must be
determined by the least accurate quadrature rule, Euler integration [9].
In collocation methods, the state and control are known at the node points
and the system governing equations are satisfied by including nonlinear constraint equations at the node points. The time histories of the state and control
variables are obtained using interpolation and the state differential equations are
satisfied using implicit integration. In most collocation methods, linear or cubic
splines are used as the interpolating polynomial and Gauss-Lobatto quadrature
rules, such as trapezoidal and Hermite-Simpson, are used for implicit integration [10]. The NLP resulting from using a collocation method typically has many
more variables and constraints; however, collocation methods are more accurate
49
than differential inclusion [4]. Collocation methods can use implicit integration
schemes with a higher order of accuracy and the number of nodes needed to
obtain the same level of accuracy as in differential inclusion is much smaller.
Finding a solution to the NLP that results from employing a direct method
is significantly easier than solving a HBVP [19]. As a result, direct methods are
capable of solving complex problems with a relatively poor initial guess. However, the co-state is not as accurate as that obtained via an indirect method.
Consequently, it is difficult to implement a direct method in real time.
3.4
Direct Transcription of Optimal Control Problem
Via Pseudospectral Methods
Spectral collocation methods, also referred to as pseudospectralmethods [12, 27],
combine the advantages of differential inclusion and collocation methods. Pseudospectral methods use differential inclusion, but retain the desired accuracy of
using higher order quadrature rules. Partitioning of the time interval is based
on the Gaussian quadrature formula, which results in an unequal distribution
of time points. The state and control are approximated by global orthogonal
polynomials while the derivative is approximated by a discrete differentiation
operator. Gauss-Lobatto quadrature rules are then used to approximate the integral with a summation. Despite the fact that both methods use essentially
the same technique, pseudospectral methods are faster and more accurate than
traditional collocation methods [6]. The solution to the CAV optimal control
problem considered in this thesis is solved using a pseudospectral method.
3.4.1
Pseudospectral Methods
In pseudospectral methods the time interval is divided into segments where the
nodes correspond to the locations of knots in Gaussian quadrature formulas.
The knots in Gaussian quadrature formulas are chosen such that the approxi50
mation of the function is exact for polynomials of higher order [12]. According
to the approximation theory, nodes that are the roots of orthogonal polynomials
will yield the best approximation [9].
The state and control constraints are satisfied at the nodes using global orthogonal polynomials. Orthogonal polynomials are closely related to Gauss-type
integration rules which yields an easy transformation of the state and control
constraints to algebraic equations [10]. Letting Ti, for i = 0, 1, 2, ... , N represent
the nodes, the function
is approximated as
y((T)
N
y(T) ~yN(T)
=
>y4
1T
i=O
where y is the value of y at
T1
and <pi(T) are the interpolating polynomials such
as Chebyshev or Legendre [121. The set of interpolating functions satisfy
<hi (Tj) = Sij
1i=j
0 i =j
Thus the value of yN(T) at the point Ti, for i = 0, 1, 2, .... ,N is equal to the value
of the function y
(T)
Y (Ti) = yN (i
According to this definition of interpolation, the approximation is exact at the
nodes.
Typically the derivatives are approximated using finite difference or finite element methods. In pseudospectral methods, the state differential constraints
are imposed by collocating the differentiation matrix at the nodes. The differentiation matrix is determined by taking the analytic derivative of the interpolating
polynomials as shown below
N
f(T)
~9N(T)
yN
(Ti)
j=0
51
Denoting the differentiation matrix by D whose elements are Dij = <j (Ti), we
have that
j,N(T)
(3.24)
= DyN(T)
In terms of accuracy, as N increases the convergence rate of finite difference
or finite element methods decreases on the order of N-" where m is a constant
that depends upon the order of the approximation and the smoothness of the
solution. Spectral methods will converge faster than any finite negative power
of N [16].
The integral performance functional is approximated using Gauss-Lobatto
integration rules [9]. Consider the integral of y(T) with respect to a weighting
function 0(r)
Iy
=f
(T)Y(T)dT
N
Iy
IyN
f
o1T)
Y Yi~
1
(T)dT
i=O
Discrete weights wi, for i
0,1, 2,...,N, which correspond to a set of orthogo-
nal polynomials, are defined as
wi
j
(T)pi(T)dr
which results in the following summation approximation to the integral
N
IyN _
Wjy,
i=0
3.4.2
Legendre Pseudospectral Method
The Legendre Pseudospectral Method of Refs. [7, 9, 10, 11] is a direct method
that converts the optimal control problem into a nonlinear programming problem. One of the many available software programs is then used to solve the
resulting nonlinear programming problem (NLP). The collocation points for the
Legendre Pseudospectral Method are the Legendre-Gauss-Lobatto (LGL) points
52
where the state and control parameters are the unknown values of the states
and controls at the LGL points. The continuous time problem is transformed to
a set of algebraic expressions using Nth degree Lagrange interpolating polynomi-
als to approximate the state and control parameters and the performance index
is descretized using the Gauss-Lobatto quadrature rule. The remainder of this
section provides a detailed description of the Legendre Pseudospectral Method
taken from Ref. [9].
Optimal Node Spacing
When determining collocation points it is advantageous to choose a distribution
that gives the best polynomial approximation. LGL points produce the smallest
L2 interpolation error [9] and thus yield better results than approximations obtained using equidistant points [11]. The LGL points lie on the interval [-1,11 and
are defined as
-1
To
TI
= roots of LN(T)
TN
=1
for l = 1, 2,...,N - 1
where LN(t) is the time derivative of the Nth degree Legendre polynomial, LN(t)As depicted in Fig. 3-1, this particular node distribution creates a clustering
of points near the endpoints. Denoting the initial time by to and the terminal time by tj, let t represent actual time and
T
E [To,
TN
T
represent LGL time where
-1, 1]. The actual time is mapped to LGL time by the follow-
ing affine transformation:
2(t - to) - (tf - to)
t5 - to
(3.25)
and conversely, LGL time is obtained from actual time via the inverse affine
transformation:
t - (tf - to)T + (tf + to)
2
53
(3.26)
46
a
3
±.~
35
*4**
0
* .4.4,
*
:4
4-
* *40
-
0
4-,25
0
~20
+4
+
4-. 4,4
.4 . ...
4.
*.4#
44
+-
15|-
1
-1
* -0.8' e
e-0.4'
'
-0.6
's
-0.2
I
*
0
0.2
0.4
0.6
0.8
"
Location of LGL Points
Figure 3-1: Distribution of LGL points for a given number of nodes
Taking the differential of Eq. (3.26), we obtain
(3.27)
dt = (tj - to)
2
d
Subsequently, in terms of LGL time, the optimal control problem of Eqs. (3.9)(3.12) becomes: Minimize
J
=
'M(x(-1), t 0 ) + N(x(1), tj) + tf 2 to
f
L(x(T), u(T), T, to, tj)d r
(3.28)
subject to the dynamic constraints
t5 - to
x = tf2
f(x(T), u(T),
T, to, tj)
(3.29)
the path constraints
gmin : g(x(T), u(T), T, to, tf)
54
Ymax
(3.30)
and the boundary conditions
ho(x(-1), to)
=
0
(3.31)
hf(X(1), tf)
=
0
(3.32)
Lagrange Interpolation with Legendre Polynomials
The state and control functions are approximated at the LGL points using Nth
degree Lagrange interpolating polynomials. Obtained from orthogonal Legendre
polynomials, the Lagrange polynomials are the trial functions while the state
and control variables at the LGL points are the unknown coefficients. Legendre
polynomials have a weight function c(t) = 1 and are orthogonal over the interval
[ -1, 1] [9]. In terms of Lagrange polynomials <q (T) for I = 0, 1, 2,..., N, the state
and control variables are approximated as:
N
x(T)
XN
(3.33)
XT4T
_
1=0
N
u(T)
UN(T)
=
YU Ti
1=0
P(T
(3.34)
where the general equation for the Lagrange interpolation scheme at the LGL
pointsTi,
1=0,1,2,...,Nis
(T - TO)... (T - TI- 1 )(T - T11)... (T - TN)
(Tj - TO) ...(Tj - Tj- 1) (Tj -- Tj+1)... (Tj - TN)
More concisely, Eq. (3.35) can be expressed as
N
<(T) =
T - Tn
M=1 T - Tm
m:j
55
=1,...,N(3.36)
(3.35)
In order to obtain an expression of the Lagrange polynomial in terms of Legendre
polynomials, a function w (T) is defined as
N
w(T)=
7
(T - Tm)
(3.37)
m=O
Evaluating the time derivative of w at Tj,
j = 0,1,2,...,Nwehave that
N
'W)(Tj)
H (-r1
M=1
mij
-
(3.38)
Tm)
Consequently, the Lagrange interpolating function can be rewritten as
w(T)
1
(3.39)
(T - Ti) W (Tj)
Referring to the definition of the LGL points, the derivative of the Nth degree
Legendre polynomial can be expressed as
LN (T)
=
(T
(T - TN-1)
- TI) (T - T2) -.-.
(3.40)
Combining Eqs. (3.37) and (3.40) with the fact that To = -1 and TN = 1, we obtain
N
w(T) = 1
(T - Tm)
=
(T - To)(T - T 1 ) ...
(T -
=
(T - To)LN(T) (T -
TN)
=
(T2 _ 1)LN(T)
TN-1)(T - TN)
m=O
(3.41)
In addition, the Legendre polynomials are the eigenfunctions of the differential
equation
d
dt (1
dt
-
T 2 )LN]
+ N(N + 1)LN(t) = 0
56
(3.42)
Using this property, the following expression shows the relationship between
1W(T) and LN(T).
N(N + 1)LN(Tj)
=
d [(T2_ 1) LNIT=T = W(TI)
(3.43)
The equation for the Nth degree Lagrange interpolating polynomial in terms of
the Legendre polynomial of degree N is
(T2 _ 1)iN (T)
-(3.44)
<pl(T ) =
(T - TI)
1
N(N + 1)LN(TI)
It can be shown that
1 if
1= k
0 if
1
=6ik=
<pI(Tk)
k
(3.45)
which leads to
XN(Tk)
uNTk
X(Tk)
=
k = 0,1,...,N
(3.46)
U(Tk)
Derivative Approximation
To impose the state differential equations at the LGL points, a differentiation
matrix is calculated by taking the analytic derivative of the interpolating polynomial. Since only the derivative at the node is desired, the following expression is
used
N
N(Tk
x(TlOL(Tk
(3.47)
1=0
N
=
ZDkIx(TI)
1=0
57
(3.48)
Dkl is the (N + 1) x (N + 1) differentiation matrix where
LN(Tk)
1
LN(TI) Tk
-
k =
Ti
N(N + 1)
k
1
0
4
DkIl=
(3.49)
N(N +1)
k
4
0
1=N
otherwise
Integral Approximation
Using the Gauss Lobatto integration rule, the cost functional is transformed to
an algebraic expression in terms of the state and control as follows
j
jN
=
Al(XN(-1), to) + N(xN(1), t)
=
'(x(-1),to)
+ t
C(XN
N(T), UN(T), T)dt
-
N
+
+ tft -
(x(1), t)
£(X(Tk), U(Tk), Tk, to, tf)Wk
k=o
where Wk are the weights corresponding to the Legendre polynomials [91 and are
expressed as
2
1
Wk = N(N + 1) LN(Tk ) 2
(3.50)
Nonlinear Programming Problem
The optimal control problem of Equations (3.9)-(3.12) is approximated by the
following nonlinear programming problem. Minimize the cost functional
N
J = 'M(x (-1), to) + X(x(1), t)
+ t2
to
I (X(Tk), U(Tk), Tk, to, tf)Wk (3.51)
k=O
over the variables
R",
k =0, 1,.., N
u(Tk) E RmI,
k =0, 1,..., N
x(TO)
to E R
tE C R
58
(3.52)
(.2
subject to
N
Z DkLX(T)
tf-t
-
2 t
f(x(Tk),U(Tk), Tk, to, tf)
=
0,
k = 0, 1,...N
1=0
gmax,
gmin : g(X(Tk ), u(Tk), Tk, to, tf)
ho(xo, to)
hf(xN, tf)
3.5
=
k=0,1,...,N
0
0
Summary of Optimal Control
According to the definition of an optimal control problem presented in Section
3.2, the CAV optimal mission problem formulated in Chapter 2 is an optimal
control problem. Because of the complexity of the optimal control problem,
it is necessary to obtain a solution numerically. Numerical methods fall into
two main categories: indirect methods and direct methods. Indirect methods
produce an accurate control, but they require an initial guess that produces a
solution close to the optimal solution. It is often difficult to obtain such an
initial guess. Direct methods have a wider range of convergence than indirect
methods, but produce a less accurate control than that which is obtained via
indirect methods. Pseudospectral methods comprise a class of newly developed
direct methods for solving optimal control problems which have a wide range
of convergence and produce an accurate control. In the subsequent chapters,
the Legendre Pseudospectral Method is applied to the Common Aero Vehicle
mission design problem.
59
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Chapter 4
Numerical Optimization Study of the
Common Aero Vehicle Problem Using
the Legendre Pseudospectral Method
4.1
Overview
The purpose of this chapter is to provide a detailed description of the steps involved in obtaining a solution to the CAV optimal control problem via the Legendre Pseudospectral Method. In particular, the discretization of the CAV optimal
control problem is described in detail. Properties of the resulting nonlinear programming problem are discussed in terms of characteristics that have an impact
on the optimization algorithm. The optimization algorithm used to solve the
NLP, SNOPT, is introduced with a brief explanation of how it solves the NLP. The
user inputs required for the implementation of SNOPT are also included in this
discussion about the optimization algorithm. Then, the specific values used for
the vehicle dynamic model and the bounds on the variables and constraints described in the discretization process along with the inputs to SNOPT are listed.
However, the number of nodes required to obtain a sufficiently accurate solution
is unknown a prior. Similarly, the choice of values for the weighting factors in
61
the performance index that will produce the most desirable trajectory and control is also not known a priori. Consequently, the last two sections are devoted
to the analysis used to choose appropriate values for the number of nodes and
the weighting factors. In doing so, the accuracy of the results obtained via the
Legendre Pseudospectral Method is assessed and the desirable characteristics
of the solution are noted. It may be useful to review Appendix A and B before
proceeding.
4.2
Discretization via the Legendre Pseudospectral Method
The Legendre pseudospectral transcription, described in Chapter 3, is applied to
the CAV optimal mission design problem formulated in Chapter 2. Implementing
the Legendre Pseudospectral Method requires discretization of the dynamics,
boundary conditions, path constraints, and performance index. The resulting
NLP is comprised of a bounded optimization vector, a bounded vector of equality
and inequality constraints, and a cost functional.
4.2.1
Optimization Vector
The optimization vector is comprised of the variables manipulated by the NLP
programming solver to determine the optimal solution. These variables are referred to as decision variables and include the state and control variables at
the LGL points as well as any undefined time points.
The augmented state
variables at the M (= N + 1) LGL points are the ECEF Cartesian components
of position (x c RM, y
(vx e RM, vY e
(o- E RM).
E
M Vz E
RM, z
E
RM), the Earth relative velocity components
RM), the angle of attack (o E RM), and the bank angle
The control variables at the M LGL points are the angle of attack
rate (u, e Rm) and the bank angle rate (u, E RM) and the final time, tf E R, is the
only undefined time point. In terms of these decision variables, the optimization
62
vector, xopt E R(f10M+1), for the CAV optimal mission design problem is
xpt =
X
y
Vx
Z
Vy
Of
Vz
o
Ua
Uof-
tf]
(4-1)
Naturally there is a range of admissible values pertaining to each of the decision variables. This leads to a lower bound vector bi c RM and an upper bound
vector b, E RM for each state and control decision variable and a lower bound
bEt5 c R and an upper bound bu,tf c R for the final time. An additional subscript
on b, and b, indicates which decision variable pertains to the respective bound.
Boundary conditions are then imposed by setting the upper and lower bounds
equal to the same value. For instance, initial conditions are imposed by setting
the upper and lower bounds equal to the appropriate initial value at the first LGL
point. The vectors of lower bounds on each decision variable are:
bi,x
=
X0
bi,,
=
YO Ymin
bl,z
=
Zo
bi,vx
Xmin
Zmin
v
V
...
Xmin
...
Ymin
zmin
...
x5
Yj
Z]
x,min
...
Vx,min
Vx,min
...
Vy,min
Vy,min
bi,vy
VyO
Vy,min
bi,vz
Vzo
Vz,min
bi,a
0 0 ...
...
Vz,min
Vz,min
0 0
bi,o,
=
rmin
bi,ux
=
Ucmin
Ucn-min
...
Ua,min
Ua,min
bi,uo-
=
uo,min
Uo-,min
...
U,min
Uo-min
bL,tf
=
o-min
(4.2)
-min
...
0min
0
Recall that the CAV considered has one-sided angle of attack control which is
indicated by a lower bound vector of zeros. Also note that there are no initial
conditions on the angle of attack and bank angle. This means that the optimizer
is free to choose their respective initial values.
63
Combining the lower bound
vectors corresponding to each of the decision variables yields a lower bound
vector for the optimization vector denoted by
BL,xopt
--
bl,x bl,y
bl,z bl,vx
BL,xopt E
R(1OM+1) as shown below.
bl,vy bl,vz bl,of bl,or
bl,uaf bl,uog bl,tf
(4.3)
Similarly, the upper bound vectors for each decision variable are
bu,x
X0
Xmax
...
Xmax
Xf
...
Ymax
YJ
bUy
=
Yo
Ymax
bu,z
=
Zo
Zmax
zmax
...
Zf
bu,vx
VxO
Vx,max
...
Vx,max
Vx,max
bu,vy
vyo vy,max
...
Vy,max
Vy,max
bu,vz
Vzo
Vz,max
...
Vz,max
aXmax
bu,o
=
amax
L(max
...
bua
=
rmax
o-max
...
-
max
(4.4)
Vz,max
0]
0-max
bu,ua
U a,max
U ,max
...
U ,max
U ,max
bu,uo
Uo,max Uo,max
...
U,max
U,max
bu,t
=
tfmax
Notice that the terminal velocity is not fully specified and thus the bounds on the
components of velocity at the last LGL point are simply the minimum and maximum values respectively. The resulting upper bound vector for the optimization
e,xopt
R(OM+l) as shown below
vector is B
Bu,xopt
=
bu,x
bu,y bu,z bu,vx bu,vy bu,vz
bu,t
bu,o
bu,ua
bu, 0-u bu,tf
(4.5)
64
Discretization of the Dynamic Constraints
4.2.2
In terms of the optimization vector, the equations of motion given in Eq. (2.21)
are expressed as
(4.6)
xcem = f (xop)
where
vx
vy
vz
gx + Lx + Dx + 2Qv, + Q 2x
f(xOPt) =
(4.7)
gy +g~±L~±Ly + Dy -2v+0y
2Qvx +Q 2 y
m
gz + Lz + Dz
m
Using the differentiation matrix
DN,
the continuous time equations are trans-
formed into an algebraic expression. The dynamic constraints defined in Eq. (4.7)
are denoted by C E RM and a subscript that indicates which decision variable
corresponds to that particular constraint equation. Rewriting the equations in
constraint form yields
vx
Cx
DNX
Cy
DNY
Cz
DNZ
vy
vz
Dx + Lx + gx
+ 2wvy +
Cvx
DNVx
tf - to
Cvy
DNVY
2
Coz
DNVz
Ca
DNa
Ca
DNO'
m
+
±Y
±Ygy +
m
2wvx + (2
Dz + Lz + gz
m
65
2
= 0
(4.8)
Together, these constraints comprise the dynamic constraint vector, Cde e R8M,
as shown below
Cdc
[ Cx
Cy
Cz
CVy
Cvx
CVz
C
C
]
(4.9)
Since the dynamic constraints are equality constraints, the lower bound vector,
BL,dc
E
4.2.3
R8M, and the upper bound vector, Bu,dc
E
p8M, are each a vector of zeros.
BL,dc
=
0
(4.10)
BU,dc
=
0
(4.11)
Discretization of the Path Constraints and the Terminal
Constraints
Path constraints confine the optimizer to stay within a set region when determining the trajectory. Referring to the path constraints listed in Eq. (2.27), the
path constraints corresponding to decision variables (a, u, u,) are included in
the optimization bound vectors. The remaining constraints are placed on the
radius, (r E RI), speed (v c RM), dynamic pressure (q c Rm), and sensed acceleration (a E RN) at every LGL point. Also included in this discretization is the
terminal constraint on the total heat load (Q c R) at the final LGL point. Recall
from Chapter 2 that the heat load is expressed as an integral. Discretization of
the integral results in the following summation:
Q
=
2 t
Wk
(4.12)
k=O
The terminal boundary condition on the flight path angle has yet to be imposed
and thus a constraint is placed on the sine of the flight path angle (Cy E R) at the
final LGL point. The resulting constraint vectors, denoted by C and a subscript
66
which indicates the corresponding constraint, are:
Cr
r
CV
V
Cq
q(4.13)
Ca
a
CO
Q
CY
S
where
s = sin(y(tj))
(4.14)
These constraints combine to form a constraint vector C1 tc
C R(4M+2)
as shown
below
Ch
CPtC
Cs C,
Ca CQ
Cy
(4.15)
where the subscript ptc is used to indicate path constraints and terminal constraints. Similar to the dynamic constraints, there are lower and upper bounds
on each path constraint and terminal constraint. The bounds on the path constraints imposed throughout flight are denoted by b E RI where the subscript
includes an 1 for lower bound or a u for upper bound and an additional letter to
indicate which path constraint corresponds to the bound vector. The terminal
constraints on the total heat load and the sine of the flight path angle are denoted by b E R and the same subscript notation as described above. The vectors
of lower bounds on the path constraints and terminal constraints are:
bir
bi
Re
=
Re
...
Re
Re
...
Vmin
Vf
qn
q"
Vmin
Vmin
biqc=
qmi
qm1 n ...
bia
0 0 ...
bio
=
0
biy
=
sin(yf)
67
0 0
(4.16)
where the complete vector of lower bounds,
BL,ptc
E R(4M+2), on the path con-
straints is
blr
BL,ptc
ble
blq
blQ
bla
bly
(4.17)
Likewise, the upper bounds on each path constraint are
bur
rmax
rmax
Vmax
Vmax
...
Vmax
V5
buq
qmax qmax
...
qmax
qmax
bua
amax
...
amax
buv
=
buQ =
buy
=
...
amax
rmax
rmax
(4.18)
amax
Qmax
sin(yf)
and the vector of upper bounds on all of the path constraints, BU,ptc
is
[bur
Bu,ptc
bu
bua
buq
buQ buy
E
p(4M+2)
(4.19)
Notice that the remaining terminal conditions pertaining to velocity are imposed
through the bounds on the final speed and the final flight path angle.
4.2.4
Discretization of the Performance Index
The performance index of Eq. (2.28) discussed in Chapter 2 is transcribed to a
summation which produces a scalar F where
F = t5 -k
-
k1
Nm
a
2
)2
+ k2
68
Ua,k
Ua,max/
2
+ k3
uo,m
uomax/
(4.20)
4.3
Common Aero Vehicle Nonlinear Programming Problem
Discretization of the CAV optimal mission design problem results in a nonlinear
progranmming problem. A nonlinear programming problem (NLP) is a problem
where it is desired to minimize or maximize a real-valued nonlinear function of
variables subject to real-valued nonlinear constraints. This section is devoted
to describing the NLP for the CAV mission design problem. First, it is important to understand the components of the NLP in terms of the breakdown of
the constraints. Thus, the NLP is summarized by recognizing both the number
and the type of constraints that comprise the NLP. Second, the structure of the
problem is important in terms of understanding the NLP as well as choosing an
optimization algorithm. Third, the NLP must be scaled properly in order to enhance the performance of the optimizer. In fact, in some cases it is necessary to
appropriately scale the problem in order to even obtain a solution.
4.3.1
Summary of the Common Aero Vehicle Nonlinear Programming Problem
To summarize the NLP resulting from discretization of the CAV optimal control
problem, the dynamic constraints, path constraints, and terminal constraints are
joined to form the following constraint vector C c(12M+2).
C
=
[
Cdc
Cptc
1
(4.21)
Similarly, the lower and upper bound vectors are also combined to form the
vector BL,C E
j(12M+2)
and the vector Bu,c E
69
p(12M+2)
respectively, where the
subscript C is used to denote bounds pertaining to the constraints:
BL,C
=
BL,dc
BL,ptc
(4.22)
Bu,c
=
B,d
BU,ptc
(4.23)
The resulting NLP is to minimize:
N
F
= Ft5
2
-
t
Z wi i
i=0
k1
L
uc,i
+ k2
max
\
amax
± ks
/
"'
\Uu,max
)
2]
(4.24)
over x 0 pt subject to
BL,xopt
s
BL,C
Xopt
s Cs
Bv,xopt
(4.25)
Bu,c
(4.26)
The breakdown of the NLP in terms of the number of optimization variables and
types of constraints is shown below.
# of Optimization Variables
= 10M + 1
# of Linear Equality Constraints
= 10
# of Nonlinear Equality Constraints
= 8M + 2
# of Linear Inequality Constraints
= 3M
# of Nonlinear Inequality Constraints
= 4M + 1
The optimization variables are split as follows: 8M variables correspond to components of the augmented states at the LGL points, 2M variables correspond
to components of the augmented controls at the LGL points, and the remaining
1 variable corresponds to the free terminal time. The 10 linear equality constraints correspond to each of the terminal boundary conditions with the exception of the speed and flight path angle. The nonlinear equality constraints are
comprised of 8M constraints corresponding to the eight discretized differential
equations at the LGL points and the remaining two correspond to the terminal
70
speed and flight path angle. The 3M linear inequality constraints correspond
to the path constraints placed on the angle of attack and the controls (ua, u,)
at the LGL points. The nonlinear constraints correspond to the remaining path
constraints where 4M of the constraints correspond to the radius, speed, sensed
acceleration, and dynamic pressure at the LGL points and the remaining 1 constraint corresponds to the total heat load at the terminal time.
4.3.2
Structure of the Common Aero Vehicle Nonlinear Programming Problem
After defining the NLP, it is useful to define the structure of the NLP. In general,
the more information the optimizer knows about the problem, the better the
optimizer will perform. Define the constraint Jacobian, [Cjac], as
[Cjac]
=
aC
aXopt
(4.27)
where [Cjac] is a (12M + 2) x (10M + 1) matrix. Each column corresponds to
each optimization variable at the LGL points and each row corresponds to each
constraint at the appropriate LGL points. (See Appendix B for a review of vector
differentiation rules used in this thesis.) The structure of the problem is best
described by indicating the dependence of the components of the constraint
Jacobian on the components of the optimization vector using a dependence matrix [Cdep]. If a component of [Cjac] is dependent upon a component of xopt, then
the corresponding element in [Cdep] is assigned the value of unity, otherwise it
is zero. The resulting matrix [Cdep] is a matrix of ones and zeros commonly
referred to as the dependency pattern. In the case of trajectory optimization
problems, the dependence matrix is a sparse matrix, i.e. a large percentage of
the individual derivatives of the nonlinear constraints with respect to the optimization variables are zero. The sparsity pattern for the CAV mission design
problem is shown in Fig. 4-1 where rows Cx-C, correspond to the dynamic con-
71
straints, rows labeled r-a correspond to the path constraints, and Q and y are
the terminal constraints. The sparsity pattern is partitioned into three main sections: the first partition corresponds to the discretized dynamic constraints, the
second partition corresponds to the discretized path constraints, and the third
partition corresponds to the terminal constraints. The dynamic constraints section can be partitioned even further into a block that depends only on the state
decision variables, a block that depends only on the control decision variables,
and a block that depends only on the final time. The block that depends only on
the state decision variables has blocks of size M x M along the main-diagonal
that result due to the dependence of the discretized differential equations on
the differentiation matrix. The off-diagonal blocks are either the M x M zero
matrix or the M x M identity matrix. The block that depends only on the control
decision variables also consists of either the M x M zero matrix or the M x M
identity matrix while the block that depends only on the final time is a column
of ones. The discretized path constraints have dependencies similar to the block
in the discretized dynamic constraint partition that depends only on the control
decision variables. Finally, the terminal constraints depend on their respective
state decision variables at the final LGL point and, in the case of the total heat
load constraint, the final time as well.
4.3.3
Scaling of the Conunon Aero Vehicle Nonlinear Programming Problem
In the extreme situation, a poorly scaled problem may prevent the optimizer
from even obtaining a solution and at the very least, it can negatively affect the
performance of the algorithm. In particular, scaling can change the convergence
rate, termination tests, and numerical conditioning [2]. A well-scaled problem
will be much better behaved numerically. One of the basic guidelines in determining appropriate scaling factors is to make every state and control variable
about the same order of magnitude and as close to unity as possible. One set of
72
x
y
z
V,,
v,
V,
Of
O-
u
u,,
t,
C
CA
CzI
C
CLIv
CUz
C,,
r
V
q
a
Q
'Y
Figure 4-1: Sparsity Pattern of the Common Aero Vehicle Nonlinear Programming
Problem
scale factors that leads to a well-scaled NLP for the CAV mission design problem
are as follows:
Units of Length:
Units of Time:
Units of Density:
Earth Radii
Period of a Spacecraft in Circular Orbit at One Earth Radii
Air Density at Sea Level
Since the flight of the vehicle is restricted to the Earth's atmosphere, scaling the
position by Earth radii makes the scaled position 0(1). The scale factor for time
is chosen such that the scaled velocity 0(1) where the velocity is scaled by the
term that results from dividing the units of length by the units of time. Similarly,
73
scaling the density of the atmosphere by the sea level density results in density
values close to unity. From these three base values, a canonical transformation
is used to convert all other values from one set of consistent units to another
set of consistent units. In particular, the canonical transformation used in this
thesis converts values from SI units to a set of nondimensional values with a
magnitude close to unity. The following useful nondimensionalizing constants
are derived in order to maintain a canonical transformation:
nlength
=
nime
=
ndensity
=
1e(4.28)
(4.29)
Po
(4.30)
(4.31)
-ength
nvelocity
ntime
nmass
nforce
ndensitynlength
4
fdensitynlength
2-
-
(4.33)
ntime
nenergy
nforcenlength
(4.34)
and nondimensional angles are represented in radians. In order to nondimensionalize a quantity, simply multiply it by the corresponding scaling factor. Conversely, in order to dimensionalize a nondimensional quantity, divide by the appropriate nondimensionalizing constant. While nondimensional quantities are
used in the optimization algorithm, the results are scaled to dimensional quantities for analysis purposes.
4.4
Numerical Optimization via SNOPT
There are many available software programs capable of solving the resulting
NLP; however, it is desirable to use a computationally efficient and robust method.
The current problem has both linear and nonlinear inequality and equality constraints. Sequential quadratic programming (SQP) methods are designed to han74
dle optimization problems with linear and nonlinear constraints [131. In addition, it is beneficial to use an optimization algorithm that takes advantage of the
sparsity of this problem.
Three well-known SQP numerical optimizers are NPSOL, SNOPT, and SPRNLP.
Both NPSOL and SNOPT were written by Gill, Murray, and Saunders [13, 14, 15]
while SPRNLP is a Boeing code developed by Betts and Frank [1]. NPSOL is very
similar to SNOPT; however, it does not take advantage of the sparsity of the
Jacobian and is not designed to solve large-scale NLPs. The study conducted
in Ref. [1] demonstrates that while SPNRLP solves larger problems in a shorter
period of time, SNOPT is faster for smaller problems. SPNRLP has the advantage of utilizing first and second derivative information versus SNOPT which
only uses first order information. Nonetheless, if given enough time, SNOPT will
solve large complex problems with the same accuracy as SPRNLP. SNOPT is a
dependable SQP method for solving sparse large-scale NLPs.
4.4.1
Description of SNOPT
SNOPT is a general purpose solver for optimization problems that have many
variables and constraints. It minimizes a linear or nonlinear function subject to
bounds on variables and linear or nonlinear constraints. Using a SQP algorithm,
SNOPT solves the NLP by solving a sequence of quadratic programming problems
(QP subproblems). The basic idea is to iteratively solve the problem, each time
working towards the optimal solution. In doing so, the task becomes to find a
direction in which the function approaches a minimum and to determine how
far to move in that particular direction. A series of major iterations and minor
iterationsare completed to determine a search direction while a merit function
is used to determine the step length.
While a complete analysis of SNOPT is not included in the scope of this thesis,
the following discussion is included to introduce the reader to the work required
to solve major and minor iterations. For a more complete explanation please re-
75
fer to Refs. [131 and [151. Initially, SNOPT converts inequality constraints into
equality constraints by introducing slack variables. Then SNOPT enters a major
iteration which generates an iterate of the optimization variables that satisfy the
linear constraints. The search direction for the next iterate is determined by
solving a QP subproblem. Minor iterations correspond to the iterative process
involved in solving the QP subproblem for each major iteration. In doing so,
the nonlinear constraints are linearized by a Taylor series expansion. The QP
subproblem is to minimize a quadratic approximation of a modified Lagrangian
subject to linear constraints and simple bounds on the variables. A reduced
Hessian algorithm is used to solve the QP subproblem where the Hessian is a
matrix of second derivative information used to create the quadratic approximation. A BFGS quasi-Newton approximation of the Hessian is used versus other
algorithms that utilize a full sparse Hessian. After the QP subproblem is solved,
a new estimate of the solution is obtained by completing a line search on an
augmented Lagrangian merit function. The merit function is used to determine
if and how much progress is being made by the algorithm. The line search determines the step length (how far to go in the search direction) in order to produce
the most significant decrease in the merit function. Eventually, this iterative
process will converge to a point that satisfies the first order conditions for optimality.
4.4.2
User Requirements and Options for SNOPT
User requirements in order to run SNOPT consist of creating two subroutines
and supplying an initial guess. One subroutine defines the objective function
and the other defines the constraints as well as the sparsity of the constraint Jacobian. Each must return their respective function values and, optionally, their
respective gradients. SNOPT provides the user with the option of coding as
many or few of the gradients as desired and the remaining derivatives are approximated with finite differences. In fact, SNOPT has the capability of verifying
76
the analytic gradients by comparing these gradients to finite difference approximations obtained via central differences.
Using this capability, the user can
correct any errors made in coding the analytic derivatives. While coding the ana-
lytic derivatives will enhance the performance and increase the reliability of the
optimization algorithm, analytic derivatives are often inconvenient to compute.
As mentioned earlier, an initial guess must be supplied to the optimizer, which
can be a daunting task depending on the problem at hand. Generally speaking,
a good start is to select any feasible point.
The user can control the performance of SNOPT by choosing various options.
Each option has a default setting chosen based off of the norm for most problems. These options include tolerance levels, derivative verification , level of desired output, and both major and minor iteration limits. More detailed options
include information about the QP subproblem, the SQP method, and the Hessian
approximation. For a complete list and description of the options please see the
SNOPT User's Guide [151.
4.5
Numerical Optimization Study
This study involves the setup for numerical optimization, which includes specifying the values used to describe the CAV mission design problem and the values
corresponding to the discretization of the mission design problem as well as the
inputs necessary for the implementation of SNOPT.
4.5.1
Specification of the Required Inputs
Table 4.1 includes all of the particular values used in the CAV mission design
study. Bounds on the optimization vector corresponding to Eqs. (4.2) and (4.4)
are listed in Table 4.2. The upper bounds on the position and velocity components correspond to 1.5 times their initial values respectively and the value of
the lower bounds are simply the negative of the upper bounds.
77
Table 4.1: Numerical Values Used for Numerical Optimization
Mass of the CAV (kg)
Aerodynamic Reference Area (M2 )
687
0.6
CL,
Zero-Lift Drag Coefficient
Drag Polar Parameter
Density at Sea Level (kg/m 3 )
Density Scale Height (m)
Angular Rotation of the Earth (s1)
Radius of the Earth (m)
Gravitational Parameter (m 3/s 2 )
Heating Rate Constant (W/m 2 )
0.043
1
1.225
6914
7.29x105
6378145
3.986x 1014
1.9987x108
The boundary conditions used in this analysis were taken from Ref. [261 and
are as follows:
to
=
0
x(to)
=
6415145 m (=37 km in altitude)
y(to)
=
0m
z(to)
=
0m
vx(to)
=
Om/s
vy (to)
=
7137.9 m/s
vz(to)
=
0m/s
x(tf)
5773486 m
y(tf)
2710645 m
z(tf)
=
Om
V(tf)
=
1219 m/s
y(tf)
=
-89.9 deg
a(tf)
=
0deg
These boundary conditions correspond to a terminal state approximately 2800
km downrange from the initial position in the initial Earth relative trajectory
plane. While the terminal velocity of the vehicle should actually be orthogonal
78
Table 4.2: Numerical Values for the Bounds on the Optimization Variables
Variable
Lower Bound
(m)
y (m)
z (M)
vx (m/s)
vy (m/s)
vz (m/s)
ot (deg)
6
-9.6227x10
-9.6227x10 6
-9.6227 x106
-10706.85
-10706.85
-10706.85
0
Upper Bound
9.6227x10 6
9.6227x10 6
9.6227x106
10706.85
10706.85
10706.85
25
o (rad)
-67T
6r
ua (deg/s)
u. (deg/s)
-10
-30
10
30
tc (s)
0
5000
x
to the plane tangent to the point of impact, (thus requiring a terminal flight path
angle of -90 deg), the unit lift direction of Eq. (2.18) chosen for this study is
undefined when y = -90 deg. Therefore, a terminal flight path angle of -89.9
deg is chosen in order to obtain results that are similar to those that would be
obtained for yf
=
-90 deg.
Bounds on the path constraints introduced in Eqs. (4.16) and (4.18) are listed
in Table 4.3 where go is the gravitational acceleration at sea level. At this point,
Table 4.3: Numerical Values for the Bounds on the Path Constraints
Path Constraint
r (m)
V (m/s)
Lower Bound
Re
10
Upper Bound
9.6227x10 6
10706.85
q (kPa)
11.97
0o
a (go)
Q (MJ/m 2 )
0
45
-00
00
sin(yf)
-1
-1
the total heat load the vehicle can sustain is unspecified and thus the heat load
is actually unconstrained. However, using the optimizer as a design tool, a parametric study presented in Chapter 5 will analyze the affects on the trajectory
79
and control from varying the maximum allowable heat load.
The choice of weighting factors used in the performance index is not obvious
prior to running the optimizer. Consequently, these values are varied and the
results are compared to determine the values that reflect the most desired characteristics of the trajectory and control. The corresponding study and results
are presented in Section 4.5.3.
In terms of the inputs required to implement SNOPT, the information provided in Section 4.2 along with the specific values listed in Tables 4.1-4.3 are
used to create the subroutines. The initial guess supplied to SNOPT varies depending on the specific case being run. A discussion of the choice of initial guess
is given with the description of each case. In addition to the required inputs, the
user must make three major decisions. First, the user must decide if the gain in
speed and accuracy is worth the work required to compute and input the analytic
constraint Jacobian and objective gradient. In this application, the analytic objective gradient and constraint Jacobian are computed analytically and are given
in Appendix C. The analytic derivatives are verified using SNOPTs derivative verifier (as described earlier). Second, the options must either be tailored to the
problem at hand or left at the default settings. For this study, the options corresponding to the limits on the total number of iterations, the number of major
iterations, and the number of minor iterations were changed from their default
values. Table 4.4 indicates the value to which each of these options were set as
well as the default setting (note that m is the number of constraints in the NLP).
Third, the number of nodes must be determined in order to produce accurate
Table 4.4: Options Set in SNOPT
OPTION
Iteration Limit
Major Iteration Limit
Minor Iteration Limit
SETTING
1000000
1000000
1000000
DEFAULT SETTING
max(10000,20m)
max(1000,m)
max(1000,5m)
results without sacrificing computational effort and time. This value is also un-
80
known prior to running the optimization algorithm and is problem dependent.
Section 4.5.2 includes the analysis used to determine the appropriate number of
nodes for this particular application.
4.5.2
Determination of an Adequate Number of Nodes
The number of nodes used to solve the NLP arising from the Legendre Pseu-
dospectral Method discretization directly affects the accuracy of the discrete approximation to the continuous time optimal control problem. An infinite number
of nodes will theoretically produce the most accurate solution. However, the efficiency of the optimizer decreases as the number of nodes increases. Therefore,
the process of determining an adequate number of nodes for a given problem involves a trade-off between the desired solution accuracy and the time required to
obtain a solution to the NLP. An adequate number of nodes for the CAV mission
design problem is determined by comparing results from using 25, 50, 75, and
100 nodes. In terms of solution accuracy, the smoothness of the control profile
is considered along with the accuracy of the controls. While the time it takes the
optimizer to solve each case is also considered, more emphasis is placed on the
accuracy of the results. Nonetheless, the resulting control profile is weighted
against the solution time in order to determine an appropriate number of nodes
to use for this study. Please see Appendix D for a description of the initial guess
used to obtain these solutions.
Effects of the Number of Nodes on the Control Profile
The smoothness of the control profile is assessed visually by observing the control profile along with the angle of attack and bank angle profiles. Even though
the angular rates are the control in the optimal control problem, the angle of
attack and bank angle are actually used to steer the vehicle. As a result, the
smoothness of the angles is more important than the smoothness of the angular
rates. Figures 4-2 and 4-3 clearly indicate that 25 nodes is not enough to obtain a
81
smooth control. Therefore, 25 nodes is no longer considered and a comparison
is made between 50, 75, and 100 nodes.
Accuracy of the controls is assessed by integrating the equations of motion
to Earth impact (altitude=0) and comparing the resulting error in the terminal
state. The same dynamic model used in the optimization algorithm is used in
the numerical integration and the controls are approximated by Lagrange interpolation. Integration of the equations of motion is carried out using a 4t4
order Runga-Kutta routine with a constant stepsize of h = 0.001s. The position
of the vehicle is completely described in a plot of altitude versus time and the
Earth relative crosstrack distance versus the Earth relative downtrack distance.
The Earth relative crosstrack and downtrack distance are defined in Appendix
E. Since the terminal condition in the integration eliminates the possibility of a
terminal error in altitude, the altitude profile is not shown. The crosstrack versus downtrack plots shown in Figures 4-4-4-6 may mislead the observer to think
that the integrated solution matches the Legendre pseudospectral solution. In
reality, there are differences, but they appear negligible in terms of the distance
the vehicle is traveling. The same holds true in the plot of speed versus time for
all three cases as shown in Figs. 4-7-4-9.
82
4-d
C-4
100
0
200
400
300
500
600
700
Time (s)
Figure 4-2: Angle of Attack vs. Time for M=(25, 50, 75, 100)
S)
to
'0
100
200
300
400
500
600
Time (s)
Figure 4-3: Bank Angle vs. Time for M=(25, 50, 75, 100)
83
700
-. Integrated soln, h=0.(X)1i
35 0 -
-.
--
-
-
- - - -. - -.-
-..
-.-.-.-.-.-.-
-
-.-.- -.-.--
-.-.-
.- .
250-
200 -
- -
---
--
---
U
aV
> 150-
;-4
50
--
--
-
0
500
1(0
1500
2000
2500
Earth Relative Downtrack Distance (km)
---
--
3000
Figure 4-4: Earth Relative Downtrack Distance vs. Earth Relative Crosstrack Distance for 50 Nodes
500
-8-
-.-.
-
400
LPS soin, N= 75
Integrated soln. h-0.001
----
- --
300
200-
100-
0
500
1000
1500
2000
2500
Earth Relative Downtrack Distance (km)
300
Figure 4-5: Earth Relative Downtrack Distance vs. Earth Relative Crosstrack Distance for 75 Nodes
84
CeI
;0
Cn
C
W-
-1001
1
0500
1
1
1
1000
1500
20'00
2500
3oo
Earth Relative Downtrack Distance (km)
Figure 4-6: Earth Relative Downtrack Distance vs. Earth Relative Crosstrack Dis-
tance for 100 Nodes
Ct
S
-0
a)
a)
C.fl
a)
Ce
a)
F-
Ce
0
100
200
30
400
50
60
Time (s)
Figure 4-7: Earth Relative Speed vs. Time for 50 Nodes
85
7(X)
0
100
200
300
40()
500
600
70()
Time (s)
Figure 4-8: Earth Relative Speed vs. Time for 75 Nodes
8000
6000
5000
~4000
3000
0
100
200
300
4(X)
500
600
Time (s)
Figure 4-9: Earth Relative Speed vs. Time for 100 Nodes
86
700
In order to properly assess the accuracy, the error in terminal position and
speed is calculated. The error in position and speed is calculated by taking the
square root of the sum of the squares of the differences between the integrated
solution and the Legendre pseudospectral solution. Let the subscript "LPS" denote the solution obtained from the optimizer at the last LGL point and the
subscript "INT" denote the results from integration to Earth impact. The terminal position error epos and the speed error espeed are then determined using the
following equations:
epos
espeed
=
(XLPS
XINT)
-
(Vx,LPS
-
2
± (YLPS -
Vx,INT)
2
+
y.NT)
2
+
(ZLPS -
(Vy,LPS - Vy,INT)
2
ZINT)2
+ (Vz,LPS
-
(4.35)
Vz,NT)2
(4.36)
Table 4.5 shows the results from integration using the control histories attained
for 50, 75, and 100 nodes. It is seen that the accuracy in position improves
significantly as the number of nodes increases. On the other hand, the speed
accuracy is virtually unaffected by the number of nodes used.
Table 4.5: Terminal errors produced by integration for M = (50, 75, 100)
No. of LGL Points (M)
Position Error (m)
Speed Error (m/s)
50
75
100
220.0763
40.2454
0.6722
9.53003
6.8386
5.9902
Effect of the Number of Nodes on the Solution Time
One last mode of comparison is the solution time, which is highly dependent
upon the type of machine used to run the optimization algorithm. The solution
time for the 50 node case is 768.22 seconds, the 75 node case took 2653.62
seconds, and the 100 node case took 3378.91 seconds. Using 50 nodes significantly reduces the computational time to solve the problem and the difference
in solution times between the 75 and 100 node cases is small in comparison.
87
Summary of the Results from Varying the Number of Nodes
In determining an adequate number of nodes for the CAV mission design problem, the results from using 25, 50, 75, and 100 nodes were compared. The accuracy of the control profile was compared along with the solution time. Looking
at the smoothness of the control profiles, it was immediately apparent that 25
nodes is not adequate for solving this problem. As to be expected, the 100 node
case produced the most accurate results; however, it also required the longest
amount of time to obtain a solution. Similarly, the use of 50 nodes significantly
reduced both the accuracy of the solution and the solution time. While the 75
node case fell in between the 50 node and 100 node cases, the results from this
case were closer to the 100 node case in terms of the control profile and solution time. Recall that the terminal conditions for HDBTs (described in Chapter 1)
require position accuracy to within several meters and speed accuracy to within
500 m/s. While the speed accuracy requirements were satisfied in all three cases,
the position accuracy requirements were only satisfied in the case of 100 nodes.
In this analysis, the solution accuracy was more important than the time it takes
to obtain a solution. Consequently, all the results presented in the remainder of
this chapter will be shown for M = 100 (i. e. 100 Nodes).
4.5.3
Choice of Weighting Factors Used in the Performance Index
It is crucial that the trajectory be robust to environmental perturbations, especially near the end of flight. A robust trajectory is one in which the terminal
conditions are met despite unpredictable conditions experienced during an actual flight. The CAV used in this thesis has limited control authority. Thus, it is
desirable to maintain control flexibility in order to compensate for unexpected
disturbances. For instance, by keeping the angle of attack away from its upper
and lower limits, it is possible to either increase or decrease the angle of attack
in the presence of an uncertainty (e.g. a windgust, a thicker than predicted atmo88
spheric density, or a thinner than predicted atmospheric density). Furthermore,
control flexibility is maintained by keeping the controls in the middle of their
respective corridors. Consequently, the performance index keeps the angle of
attack as close to the middle of its corridor (61) as possible. However, the angle
of attack reaches a maximum near the end of flight in order to meet the terminal
conditions on speed and flight path angle. This dramatic increase arises from
the need to deplete speed over a short period of time and obtain a large and
negative flight path angle. In order to decrease the speed of the vehicle, the drag
must increase and thus the angle of attack increases. The terminal flight path
angle requires that the vehicle approaches the target with negative lift. Since
the angle of attack cannot be negative, the only way to generate negative lift on
the vehicles is to rotate the vehicle 180 degrees. The amount of lift required to
execute this maneuver causes the angle of attack to increase as well. The an-
gle of attack then rapidly decreases to its prescribed terminal condition of zero
degrees. In order to maintain flexibility in the angle of attack near the end of
flight, the maximum angle of attack should be minimized. Redefining the control margin, the goal is to keep a near 61, to minimize 0(max, and to minimize uO
and u,.
Recall that ki corresponds to keeping a near 6z, k 2 corresponds to the angle
of attack rate, and k 3 corresponds to the bank angle rate. The weighting factors
in the performance index are selected according to the goal of maximizing the
control margin and minimizing the control rates. In determining the appropriate
values for k 1 , k 2 , and k 3 , each parameter is varied independently. The parameter
being varied takes on the values of 0.1, 1.0, 10, and 100 while the remaining two
parameters are set to 1.0. The initial guess used to obtain results is the solution
to the 100 node case described is Section 4.5.2.
Effects on the Control Margin due to Variations in ki
Increasing k 1 places more emphasis on keeping the angle of attack near 6(. However, as mentioned earlier the angle of attack increases near the end of flight. In
89
order for the angle of attack to increase while more emphasis is placed on keeping the angle of attack near &, the angle of attack remains near 6 for as long as
possible. By delaying the increase in angle of attack, the controls are forced to
decrease the angle of attack to zero over a shorter time interval. Consequently,
a remains closer to & for a longer period of time and the angle of attack rate
increases as ki increases. In order to generate an adequate amount of lift to rotate the vehicle in a shorter period of time, amax and u, must also increase near
the end of flight as ki increases. Figures 4-10 and 4-11 show that the maximum
angle of attack and the angle of attack rate increases slightly at ki = 10 and
noticeably for ki = 100. The same is true for the bank angle rate as indicated in
Fig. 4-12. Also, the angle of attack rate reaches its minimum value at the end of
flight for the values of ki = 1.0, 10, and 100.
25
20
CZ
15
0410
5
0'
0
100
200
400
300
500
600
700
Time (s)
Figure 4-10: Angle of Attack vs. Time for k1 = (0.1, 1.0, 10, 100), k 2 = k3 = 1.0
90
-2-
U
-4
0
-6-
kl=
kl=
-Ukl=
A8 kl=
1
-10
0
1 30
S-8 -+*
V
200
300
400
500
700
-600
Time (s)
Figure 4-11: Angle of Attack Rate vs. Time for ki = (0.1, 1.0, 10, 100), k 2 = k=
1.0
6
2
_2
-4
--BA-
-6
0
kl=
kl=
kl=
kl=
10 0
200
400
300
500
600
700
Time (s)
Figure 4-12: Bank Angle Rate vs. Time for ki = (0.1, 1.0, 10, 100),
91
k2
= k3
=
1.0
Effects on the Control Margin due to Variations in k 2
Increasing k2 increases the emphasis on minimizing the angle of attack rate. In
order to generate the same amount of lift using a slower angle of attack rate,
the deviation of of from 6 increases as shown in Fig. 4-13. In regards to the
maximum angle of attack, Fig. 4-13 also indicates that the smallest maximum
angle of attack is obtained for k2 = 10 while the largest maximum angle of attack
is obtained for k2 = 100. Figure 4-14 clearly shows that increasing k2 decreases
the magnitude of the maximum angle of attack rate and the angle of attack rate
reaches its minimum value at the end of flight for the values of k2 = 0.1 and
1.0. Fig. 4-15 shows that the bank angle rate appears to be unaffected by the
variation in k2 .
25
20
U
15
0t
4-j
10
100
200
400
300
500
600
700
Time (s)
Figure 4-13: Angle of Attack vs. Time for k2 = (0.1, 1.0, 10, 100), ki = k 3 = 1-0
92
Q-)
4-4
0
a-)
300
700
400
Time (s)
Figure 4-14: Angle of Attack Rate vs. Time k 2 = (0.1, 1.0, 10, 100), ki = k3 = 1.0
-4
a)
0
100
200
300
400
500
600
700
Time (s)
Figure 4-15: Bank Angle Rate vs. Time for k 2
93
=
(0.1, 1.0, 10, 100), ki = k3 = 1-0
Effects on the Control Margin due to Variations in k 3
Increasing k 3 places more emphasis on minimizing the bank angle rate. Looking
at Figure 4-16, the weighting on the bank angle rate appears to only affect the
angle of attack profile in the case where k3
scenario (k3
=
=
100. It is obvious that in this
100) the deviation of the angle of attack from
et
increases and
the angle of attack reaches its upper limit. The angle of attack rate reaches its
maximum value at the end of flight for every value of k3 , as indicated in Fig. 417. As to be expected, Fig. 4-18 confirms that as k3 increases, the bank angle
rate decreases throughout the trajectory.
25
-4-
y
k3= 0.1
k3= 1.0
-UA
k3= 10
k3= 100
20-
-
-
- -
to
0
100
200
300
500
400
600
700
800
Time (s)
Figure 4-16: Angle of Attack vs. Time for k 3
94
=
(0.1, 1.0, 10, 100), ki = k2 = 1-0
-2
-4
4'-
0
-6
-8
0
100
500
400
300
200
600
700
Time (s)
Figure 4-17: Angle of Attack Rate vs. Time for k3 = (0.1, 1.0, 10, 100), ki = k2
1.0
1.5
S 0.5
01
<-0.5
2
-1
-1.5
-2'0
100
200
300
400
500
600
700
800
Time (s)
Figure 4-18: Bank Angle Rate vs. Time for k 3 = (0.1, 1.0, 10, 100), k 1 = k2 = 1.0
95
Summary of the Results from Varying the Weighting Factors in the Performance Index
The performance index is constructed such that the optimal control and trajectory maximize the control margin. The control margin is measured by three
terms. The first term keeps the angle of attack in the middle of its corridor,
the second term minimizes the angle of attack rate, and the third term minimizes the bank angle rate. The desired trajectory and control is such that each
of these terms is minimized. Analyzing these results according to the desired
performance, each term in the cost functional is evaluated separately (without
the weighting factor) as depicted below
N
Term1
=2
i=0
N 1
Term2
=
Term3
=
_
2
Ofmax /2
2
(Ua,)
i=0 Uoa,max
2
N
i=0
20-,max)
(4.37)
Furthermore, the angle of attack profile is such that the angle of attack reaches
a maximum value near the end of flight. Near the end of flight it is crucial to
maximize the control margin which corresponds to minimizing the maximum
angle of attack. Thus, the maximum angle of attack is also considered. The
overall performance of the vehicle resulting from varying the weighting factors
is assessed by considering all three terms defining the control margin as well as
the maximum angle of attack. Consequently, the overall performance is assessed
by summing the values of each term in addition to the maximum angle of attack.
Table 4.6 summarizes the results from varying the parameters in terms of each
of these values for each of the cases and the last column is a summation of the
preceding values in each row. The only undesirable case is the last one where
ki = k2 = 1 and k 3 = 100 because the angle of attack reaches its upper limit.
However, the case where ki = k 2 = 1 and k 3 = 0.1 produces slightly more
96
desirable results. Consequently, these values are used for in the remainder of
this thesis.
Table 4.6: Results from Varying the Weighting Factors (ki, k 2 , k 3)
ki
0.1
k2
k3
Termi
1
1
10
1
10
0.1
10
1
0.0048
0.0025
0.0023
0.0023
0.0024
0.0025
0.0047
0.0146
0.0025
100
1
1
1
1
1
1
1
1
4.6
10
100
1
1
1
1
1
1
1
1
I 0.11
10
100
Term2
0.0014
0.0020
0.0023
0.0026
0.0022
0.0020
0.0014
0.0011
0.0020
Term3
4.3919 x 104
4.6944 x 104
5.894 x 104
0.0017
4.6589 x 104
'I
0.0025
I
0.002
0.0026
0.0084
0.0020
0.0031
4.6944 x 104
4.9231 x 104
5.3504 x 104
5.3961 x 104
4.6944 x 104
4.188 x 104
2.633 x 10-4
amax
Total
19.8101
19.8161
18.9726
20.2372
20.9097
18.9676
20.2320
20.9031
20.1295
18.9676
20.1346
18.9726
19.7476
19.7533
22.0621
22.0783
18.8591
18.8541
18.9676
19.3337
25
4
18.9726
19.3387
25.0118
Summary of the Numerical Optimization Study
Applying the Legendre Pseudospectral Method to the Common Aero Vehicle optimal mission design problem results in a nonlinear programming problem. It is
important to identify the sparsity pattern of the NLP and to scale the NLP properly in order to improve the performance of the optimizer. The CAV optimal
control problem has a sparse nonlinear constraint Jacobian. SNOPT is designed
to handle problems with sparse nonlinear constraint Jacobians and thus it is the
optimization algorithm used to solve the NLP. Once the optimal control problem
is discretized to form the NLP and the optimization algorithm was chosen, the
specific values pertaining to the NLP and the options in SNOPT used in the optimization study were listed. A study was then conducted in order to choose an
appropriate number of nodes in terms of the accuracy of the solution and the
time required to obtain a solution. Results were obtained using 25, 50, 75, and
97
100 nodes and it was shown that 100 nodes was the only case that produced
results which met the accuracy requirements set forth by the CAV mission de-
sign problem. Using 100 nodes, another study was conducted to determine the
choice of weighting factors in the performance index that produced the most desirable trajectory and control. In this case, it was desirable to design a trajectory
that is robust to environmental disturbances. A robust trajectory corresponds
to one in which the vehicle flies in the middle of its control capabilities. Thus,
the control margin was used as a metric for quantifying the desirability of the
trajectory and control. The control margin was assessed in terms of keeping the
angle of attack in the middle of the corridor, minimizing the maximum angle
of attack, and minimizing the control rates. One of the weighting factors was
varied while the remaining two factors were held constant. It was found that
setting ki = k2 = 1 and k 3 = 0.1 maximized the actual control margin and thus
these values are used in the remainder of this thesis. Furthermore, upon the
completion of this analysis it is evident that the value of the performance index
reflects the control margin. In particular, the goal is to minimize the performance index, which maximizes the control margin. Thus, the smaller the value
of the performance index, the larger the control margin and vice versa.
98
Chapter 5
Parametric Optimization Study of the
Common Aero Vehicle Problem
5.1
Overview
This chapter focuses on the general behavior of the Common Aero Vehicle as
well as the response of the solution from the optimization to changes in parameters. Since the CAV is a new concept, it is important to first understand the
motion of the vehicle during flight. Therefore, the first step is to identify the
key features of the trajectory and control. Utilizing the optimization setup as
a design tool, parameters are then varied and the differences in the trajectory
and control profiles are determined. As discussed in Chapter 4, the performance
index is a measure of the amount of control margin. Since the goal is to produce
a solution that maximizes the control margin, the value of the performance in-
dex is used as the metric to compare the quality of the solutions obtained for
different values of the parameters. In particular, the quality of the solution is
compared for different values of minimum allowable dynamic pressure, maximum allowable stagnation point heat load, and maximum lift-to-drag ratio of
the vehicle.
99
5.2
Key Features of the Trajectory and Control
The solution used to identify the key feature of the trajectory and control pertains to 100 nodes, ki = k2 = 1, k 3 = 0.1, qmin = 11.97 kPa, Qmax = oo, and
(L/D)max ~ 2.4. The first key feature of the optimal trajectory is the behavior
of the altitude as shown in Fig. 5-1. It is seen from Fig. 5-1 that the altitude increases twice during flight. Initially, the altitude increases to a region where the
atmospheric density is small, thus allowing the vehicle to achieve the required
range of 2800 km. The subsequent decrease in altitude increases the dynamic
pressure in order to produce enough lift to rotate the vehicle. The final decline
in altitude reduces the speed to meet the specified terminal speed of 1219 m/s
and satisfy the required range. In order to maintain control authority, a constraint is placed on the minimum allowable dynamic pressure. The points where
the altitude attains a local maximum correspond to points where the dynamic
pressure constraint is active as seen in Fig. 5-2.
0
2
4
6
8
10
The second key feature of the
12
14
16
Energy (GJ)
Figure 5-1: Altitude vs. Energy for M=100, ki = k2 = 1,k 3 = 0.1
100
18
10 0
-7
1 0 -Altitude
Attains
Local Maximum
4-0
Dynamic Pressure
Constraint Active
0
100
200
400
300
500
600
700
Time (s)
Figure 5-2: Altitude and Dynamic Pressure vs. Time for M=100, ki
=
1,k 3
0.1
optimal trajectory is the behavior of the in-plane out-of-plane motion. Figure 5-3
shows the Earth relative crosstrack distance versus the Earth relative downtrack
distance traveled by the vehicle (see Appendix E). It is seen that the vehicle steers
out of plane close to 410 km and actually approaches the target slightly from
behind. The third key feature of the results is the behavior of the optimal angle
of attack. It is seen from Fig. 5-4 that, because of the desire to minimize the
performance index, the angle of attack remains near 6 = 11.9 deg throughout a
large portion of the trajectory. The dramatic increase and decrease in angle of
attack at the end of flight arises from the need to meet the terminal conditions
on speed, flight path angle, and angle of attack. The increase in angle of attack
near the end of flight arises from the need to deplete speed over a short period
of time and obtain a large and negative flight path angle. In order to decrease
the speed of the vehicle, the drag must increase and thus the angle of attack
increases. Attaining the terminal flight path angle requires negative lift. Since
101
-~
-
INUU
k 4=
1.0k2=
1.0k3= 0.1
350
Q
300250
-
-
0 200
150-
-
100S500
0
500
1000
1500
2000
2500
3000
3500
Earth Relative Downtrack Distance (km)
Figure 5-3: Earth Relative Crosstrack Distance vs. Earth Relative Downtrack Distance for M=100, k 1 = k2 = 1,k3 = 0.1
the angle of attack must remain positive, the only possible way to generate a
sufficient amount of negative lift is to roll (bank) the vehicle -180 degrees and
increase the angle of attack. Once the vehicle is oriented properly, the angle
of attack decreases rapidly in order to meet the required terminal condition of
zero degrees. The last key feature of the optimal trajectory is the behavior of the
bank angle. Fig. 5-5 shows that the vehicle is banked to -90 deg for roughly 200
seconds of flight. When the bank angle is -90 deg, there is no vertical component
of the lift direction. The vehicle flies with this orientation in order to decrease
the altitude of the vehicle. It is also seen in Fig. 5-5 that the bank angle is -180
deg when the vehicle reaches the target. As mentioned earlier, the restrictions
on the angle of attack in combination with the terminal condition on the flight
path angle (-89.9 deg) require that the vehicle fly upside-down as it approaches
the target. Furthermore, lower in the atmosphere the forces acting on the vehicle are greater. Larger forces acting on the vehicle results in a high bank angle
102
18
161412108
4
20
100
200
300
400
500
700
600
Time (s)
Figure 5-4: Angle of Attack vs. Time for M=100, k 1 = k 2
1,k 3
=
0.1
rate as the vehicle rotates to -180 deg. Since the performance index minimizes
the bank angle rate, the vehicle ascends to a higher altitude before rotating. The
altitude is increased by decreasing the magnitude of the bank angle. In this case,
the bank angle decreases in magnitude to 50 deg from -90 deg before rotating
the vehicle over.
103
0
100
200
300
400
500
600
Time (s)
Figure 5-5: Bank Angle vs. Time for M=100, ki = k2 = 1,k 3 = 0.1
104
700
5.3
Effects of Dynamic Pressure on the Trajectory and
Control
A minimum dynamic pressure constraint is added in order to keep the vehicle
from exiting the Earth's atmosphere thereby maintaining aerodynamic control.
In order to assess the affect of the minimum allowable dynamic pressure on
the resulting trajectory and control, the minimum allowable dynamic pressure
is varied between 11.97 kPa (250 psf) and 47.88 kPa (1000 psf) while Qmax = co
2.4) are held constant. The initial guess is the solution for the
and (L/D)max
case where M
=
100, ki = k2
=
1, k 3 = 0.1, qmin = 11.97 kPa, Qmax = oo, and
(L/D)max ~ 2.4.
In terms of trajectory characteristics, it is known that the dynamic pressure
is a function of density, which is a function of the altitude, and speed. While
the initial and final altitude is specified in the boundary conditions, the altitude
is free to increase and decrease throughout flight. However, the local maximum
in altitude are constrained by the minimum allowable dynamic pressure. The
initial and final speed is specified as well, but contrary to the altitude, the speed
of the vehicle can only decrease during flight. The only way to increase the dynamic pressure without increasing the speed is to decrease the altitude of the
vehicle. However, the vehicle must meet the range requirements for the terminal conditions which forces the altitude to increase in the beginning of flight.
As depicted in Figs. 5-6 and 5-7, these conflicting trends result in the following
trade-off: as the minimum dynamic pressure increases, the initial increase in
altitude decreases and the speed depletes at a slower rate. Note that regardless
of the constraint on the dynamic pressure, the vehicle reaches 20 km in altitude
with nearly the same amount of energy and the remaining energy is dissipated
in the same manner for all values of qmin.
In regards to the lateral motion of
the vehicle, it is seen in Fig. 5-8 that the crosstrack distance varies as the minimum allowable dynamic pressure decreases. Fig. 5-8 also shows that the vehicle
approaches the target further from behind as the dynamic pressure constraint
105
60
50 -
40-
"030 -
20 -
.. . . . .
-..
.
- - qmin= 11.97kPa
...
10-
-UA
0
0
2
4
6
8
10
12
qmin= 23.94 kPa
qmin= 35.91 kPa
qmin= 47.88 kPa
14
16
18
Energy (GJ)
Figure 5-6: Altitude vs. Energy for qmin = (11.97, 23.94, 35.91, 47.88) kPa
is lowered.
In terms of the effects of the dynamic pressure on the control, the control
profile is considered along with the value of the performance index. Referring to
the angle of attack proffle in Figure 5-9, the maximum angle of attack increases
slightly and the deviation of the angle of attack from d&increases as the minimum
dynamic pressure increases. As to be expected, Fig. 5-10 shows that the value
of the performance index increases as the minimum allowable dynamic pressure
increases. Recall that the performance index is a measure of the control margin
and the smaller the value of the performance index, the larger the control margin. Thus, increasing the minimum allowable dynamic pressure decreases the
control margin.
To summarize, the purpose of including a constraint on the minimum allowable dynamic pressure is to prevent the vehicle from skipping out of the atmosphere and to maintain control authority. As to be expected, as the minimum
allowable dynamic pressure was increased, the maximum altitude reached de106
8000
7000
P
..
-
.
6000 -
-
5000 -
-
-..
4000 -..
44
2000 --
-
-
S3000 -
-4-
qmin= 11.97 kPa
qmin= 23.94 kPa
-UA
qmin= 35.91 kPa
qmin= 47.88 kPa
1000
0
100
200
300
-
500
400
600
700
Time (s)
Figure 5-7: Earth Relative Speed vs. Time for qi
kPa
=
(11.97,23.94,35.91,47.88)
creased. However, as the minimum allowable dynamic pressure was increased,
the control margin decreased. Since in each case the vehicle did not exit the
Earth's atmosphere, the case which maximized the control margin is desired.
Thus, it is beneficial to design a vehicle that can be controlled at higher altitudes
(i.e. at a lower minimum dynamic pressure constraint).
107
500
qmin= 11.97 kPa
-*-
V
450
A
400
V
qmin= 23.94 kPa
qmin= 35.91 kPa
qmin= 47.88 kPa
-U-
350-c-I 300
/
250
-
II
/
0
U 200
a)
150
-
.
.
-.
-
--
I
/
-
a)
-
100
...
.
--
50
.. . . .. . . . .. . .
500
1500
1000
2500
2000
- 3000
3500
Earth Relative Downtrack Distance (km)
Figure 5-8: Earth Relative Crosstrack Distance vs. Earth Relative Downtrack Distance for qmin = (11.97,23.94, 35.91,47.88) kPa
25
20(
-4y
-u-A
qmin= 11.97 kPa
qmin= 23.94 kPa
qmin= 35.91 kPa
qmin= 47.88 kPa
0
-.--
-
15
-.
-...
-.
10
-.-.-.-.-
-
-.
to
. .. .. ... .... .
. .. .
. . . ..
.
. ..
--
5
0 O0
I
I
I
100
200
300
I
400
I
I
500
600
700
Time (s)
Figure 5-9: Angle of Attack vs. Time for q
108
=
(11.97,23.94, 35.91,47.88) kPa
5.8
x 10-
5.6 -
5.4-
-
5.2 -
5
- ---.- -
0
.
4.8
Sqmin= 11.97 kPa
V qmin= 23.94 kPa
* qmin= 35.91 kPa
A qmin= 47.88 kPa
4.6 -
4.4 10
15
20
25
30
35
40
45
50
qmin (kPa)
Figure 5-10: Value of the Performance Index vs. Minimum Allowable Dynamic
Pressure for qmi = (11.97,23.94, 35.91,47.88) kPa
109
5.4
Effects of the Stagnation Point Heat Load on the
Trajectory and Control
The maximum allowable stagnation point heat load that the vehicle can withstand depends on the thermal protection system. The total stagnation point
heat load sustained by the vehicle in the unconstrained case is approximately
2300 MJ/m 2 . In this study, the maximum allowable heat load is varied between
1100 MJ/m 2 and 2300 MJ/m
2
while qmin = 11.97 kPa and (L/D)max ~ 2.4 are
held constant. The initial guess is the solution for the case where M = 100,
ki = k 2 = 1, k 3 = 0.1, qmin = 11.97 kPa, Qmax = oo, and (L /D)max ~ 2.4.
It is seen in Fig. 5-11 that this constraint is active in every case. Furthermore,
tightening the constraint on the maximum allowable heat load results in an increase in the maximum heating rate, as shown in Fig. 5-12. In reference to tra2500
Qmax=
Qmax=
Qmax=
Qmax=
Qmax=
2000
1500F
2300
1900
1600
1200
1 100
MJ/m2
MJ/m2
MJ/m2
MJ/m2
MJ/m2
- - -.-
C 1000 -...
500 F I
0
I
100
200
300
400
500
600
700
800
900
Time (s)
Load
Heat
Total
5-11:
Figure
(1100, 1300,1400,1700,2000,2300) MJ/m 2
vs.
Time
for
Qmax
jectory characteristics, it is known that the heat load is a function of density and
110
18
16 -
Qmax= 1600 MJ/m2
. .. ..
..
~
..-.. ...
.......
.
14 12
Qmax= 1200 MJ/m2
Qmax= I1100 M J/m2 -
-
A-,
-..
. .. .
. .. ..
104
-
4
8O
.. .. .
2 -
.
-
-
.
-
.. . .
..
-
-
---
4--.
0
0
100
200
300
400
500
600
700
800
900
Time (s)
Figure
5-12:
Heating
Rate
vs.
(1100, 1300, 1400, 1700,2000,2300) MJ/m 2
Time
for
Qmax
=
altitude (see Eq. (2.25)). Using the same argument as presented in Section 5.3, the
density is used to control the heat load. Consequently, the density is lowered to
decrease the heat load. In order to decrease the density, the vehicle increases in
altitude to a low density region until the dynamic pressure constraint becomes
active. To relieve the dynamic pressure constraint, the vehicle descends to a
lower altitude while depleting speed. Consequently, as the maximum allowable
heat load is lowered, the vehicle undulates through the atmosphere at a higher
frequency and, initially, the speed is depleted at a faster rate. In fact, the faster
the altitude decreases, the faster speed is depleted.The affects on altitude and
speed for different values of Qm ax are shown in Figs. 5-13 and 5-14. Similar to
the affects of loosening the constraint on dynamic pressure, as the maximum
total heat load is increased, the vehicle approaches the target from further behind. This trend is depicted in Fig. 5-15 which also shows that the vehicle takes
a more direct trajectory and approaches the target from the front in the more
111
constrained cases. The crosstrack distance traveled by the vehicle is a maximum
for Qmax = 1900 MJ/m
2
and is a minimum for Qmax = 1200 MJ/m 2 .
-oL
0
100
200
300
400
500
700
600
800
900
Time (s)
Figure 5-13: Altitude vs. Time for Q m ax = (1100, 1300, 1400, 1700,2000,2300)
MJ/m 2
The effects of the maximum allowable heat load on the control are evident
by looking at the angle of attack profile, the bank angle rate, and the value of
the performance index. The angle of attack is not specified in the initial conditions which allows the optimizer to choose the initial value for the angle of
attack. As shown in Fig. 5-16, the angle of attack reaches its upper limit in the
beginning of the trajectory for Qmax = 1200 MJ/m 2 and Qma = 1100 MJ/m 2 .
Furthermore, amax is a minimum for Qmax = 1100 MJ/m
2
and a maximum for
= 1900 MJ/m 2 . Also evident in Fig. 5-16 is that the deviations of a from
& increase in both magnitude and frequency as Qmax decreases. However, as
Qrnax
the vehicle nears the target, it looses the ability to make any necessary corrections. Hence, the primary concern is maintaining control authority near the end
of the trajectory. Figure 5-17 shows that the bank angle rate reaches its upper
112
8000
--- Qmax= 2300 MJ/m2
V
--A
7000
cj~
Qmax=
Qmax=
Qmax=
Qmax=
-*-
1900 MJ/m2
1600 MJ/m2
1200 MJ/m2
1 100 MJ/m2
S 6000
5000
4000
-
3000
.
..
2000|
1AA
I
0
100
200
300
400
600
500
800
700
900
Time (s)
Earth Relative Speed
Figure
5-14:
(1100, 1300,1400,1700,2000,2300) MJ/m 2
vs.
Time
for
Qmax
and lower limits both in the beginning and at the end of flight in the case where
Qmax = 1200 MJ/m 2 and Qma = 1100 MJ/m 2 . It is seen from Fig. 5-18 that the
value of the performance index increases as the maximum allowable heat load
decreases. Thus, the control margin increases as the maximum allowable heat
load increases.
In each case the total heat load experienced by the vehicle is exactly the value
of the maximum allowable heat load. To account for unexpected events encountered during flight, it is beneficial to add a buffer region between the amount of
heat the vehicle will sustain and the amount of heat the vehicle is capable of withstanding. In other words, the trajectory and control should be designed based on
a maximum allowable heat load which is less than what the thermal protection
system is designed to handle. In addition, the maximum rate at which the vehicle can be heated also depends on the thermal system. Thus, the heating rate
and the heat load experienced by the vehicle must both be taken into considera-
113
450
4 400
-I
Qmax= 2300 MJ/m2
Qmax= 1900 MJ/m2
-U-
MJI/m2
A - Qmax =1200 MJ/m2
-0- Qmax = 1100 MJ/m2
350
300
250
c 200
U
150
-
100
- -
-.....
-.....
....
. ...
...-..
-...
-..
. ..
-..
................
-
..
-.
-................
50
0
500
1000
1500
2000
2500
3000
3500
Earth Relative Downtrack Distance (km)
Figure 5-15: Earth Relative Crosstrack Distance vs. Earth Relative Downtrack
Distance for Qmax = (1100, 1300, 1400, 1700,2000,2300) MJ/m 2
tion when designing a trajectory and control. Regardless, it was shown that the
control margin increased as the maximum allowable heat load increased. Thus,
it is desirable to design a vehicle that can withstand as much heat as possible.
114
2
0-2
15,
4-)
0
1
-4-
Qmax=
Qmax=
-o- Qmax=
-A -Qmax=
-0- Qmax=
n
-5
0
100
2300
1900
1600
1200
I 100
|
200
MJ/m2
MJ/m2
MJ/m2
MJ/m2
MJ/m2
|
|
300
400
500
600
700
800
900
Time (s)
of
Attack
Angle
5-16:
Figure
(1100, 1300,1400, 1700, 2000,2300) MJ/m 2
vs.
Time
for
Qmax
r4
0'
100
200
300
400
500
600
800
700
900
Time (s)
Rate
Bank
Angle
5-17:
Figure
(1100, 1300,1400,1700,2000,2300) MJ/m 2
115
vs.
Time
for
Qmax
0.08
*
Qmax=
Qmax=
Qmax=
Qmax=
Qmax=
y
-..
0.07-
A
A
2300
1900
1600
1200
1100
MJ/m2
MJ/m2
MJ/m2 MJ/m2
MJ/m2
0.06a)
0.05a)
C-)
0.04C
I.
0.03-
a)
0.020.01
'
100
1200
1400
1600
1800
2000
4
2200
2400
Qm ax (MJ/m 2 )
Value of the Performance
Figure 5-18:
(1100, 1300,1400, 1700,2000,2300) MJ/m 2
116
Index vs.
Qmax for Qmax
5.5
Effects of the Lift-to-Drag Ratio on the Trajectory
and Control
The maximum lift-to-drag ratio, (LID)max, is determined by the specific design
of the vehicle. The vehicle used in this thesis has a maximum lift-to-drag ratio of
approximately 2.4. This study analyzes the effects on the trajectory and control
of varying (L/D)max between 2.0 and 2.5 while Qmax = oo and qmin = 11.97 kPa
are held constant. The initial guess fed into the optimizer is the solution for the
case where M = 100, ki
k2 = 1, k 3 = 0.1, qmin = 11.97 kPa, Qmax = oo, and
(L/D)max ~ 2.4. In order to vary (L/D)max, it is assumed that the lift coefficient
corresponding to the maximum lift-to-drag ratio is constant. The zero-lift drag
coefficient, CDO, and the drag polar parameter, K, are the parameters that are
The lift-to-drag ratio is computed as
altered as a result of varying (L/D)ma.
follows:
L
_
D
CL
(5.1)
CDo + KC2(
Let CL be the value of CL corresponding to (L/D)max.
Since (L/D)max is the
maximum value of LID, a necessary condition for LID to equal (L/D)max is
(L /D)= 0
aCL
(5.2)
Computing a(LID)/WCL using Eq. (5.1), we have that
CDO - KC
(CDO
and from which we have that CL
0
+ KCLZ)2
CDOIK.
(5.3)
Substituting CL into Eq. (5.1), CDO
and K are given in terms of (L/D)max as follows:
CDO
=
K
=
-(L
CL
2(L/D)max
1
2CL(L/D)max
117
(5.4)
(5.5)
In regards to the effect of the maximum lift-to-drag ratio on the trajectory, as
(LID) decreases the vehicle loses some of its maneuverability. In other words,
if the vehicle is constrained to fly in the downtrack direction during a glide
maneuver, the range of the vehicle will decrease as (LID) decreases. The only
notable distinctions in the trajectory occurs when (L/D)max is reduced to 2.0.
Thus, the comments on the results refer to the differences between a vehicle
with (L/D)max = 2.0 and one with (L/D)max > 2.0. Figure 5-19 shows that during
the initial increase in altitude an (L/D)max of 2.0 depletes more energy while
achieving a slightly lower altitude and the maximum altitude attained near the
end of flight is higher. Looking at the motion of the vehicle in the crosstrackdowntrack plane shown in Fig. 5-20, as the maximum lift-to-drag ratio increases,
the vehicle approaches the target further from behind. The crosstrack distance is
maximized at the lowest maximum lift-to-drag ratio and the vehicle approaches
the target perpendicular to the downtrack direction. Looking at Fig. 5-21, the
speed profile is noticeably different in the case where the maximum lift-to-drag
ratio is the smallest. In the beginning of the trajectory, a vehicle with (L/D)max =
2.0 decreases its speed at a faster rate before reaching a relatively constant speed
while, during the speed depletion phase, it depletes speed at a slower rate. It is
seen in Fig. 5-22 that as the maximum lift-to-drag ratio increases, the total heat
load increases as well.
The effect of the maximum lift-to-drag ratio on the control is minimal. Fig. 5-
23 shows that as (L/D)max is increased the maximum angle of attack decreases
slightly. In terms of the control margin, it is seen in Fig. 5-24 that the performance index increases as (L/D)max decreases. Notice that the increase in the
performance index is relatively constant with the exception of the difference between (L/D)max = 2.0 and (L/D)max = 2.1. In this case, there is a greater increase
in the performance index.
Overall, varying the maximum lift-to-drag ratio has little affect on the trajectory and control margin until (L/D)max is reduced to 2.0. At this point, there is
a clear distinction in the behavior of the vehicle even though the trends are sim118
60
50
0
-
20 -
V
-U--0.
0
0
2
4
6
8
12
10
2.0
2.1
L/Dmax= 2.2
L/Dmax= 2.3
L/Dmax= 2.4
L/Dmax= 2.5
L/Dmax=
L/Dmax=
14
16
18
Energy (GJ)
Figure 5-19: Altitude vs. Energy for (L/D)max
=
(2, 2.1, 2.2, 2.3, 2.4, 2.5)
ilar. In designing the vehicle, it is important to keep in mind that the maximum
lift-to-drag ratio effects the heat load that the vehicle endures throughout flight.
Furthermore, the higher the maximum lift-to-drag ratio, the larger the control
margin. Thus, in this particular application, the more maneuverable the vehicle
the better.
119
500
450
400
L/Dmax= 2.0
L
-O-
A
-0-
I
I
-I
-
=
/DT
L/Dmax=
L/Dmax=
L/Dmax=
L/Dmax=
21
2.2
2.3
2.4
2.5
350
C 300-
-10 x
-/m
-.
250-
- ---- - -U
...-.. ..... .
U 200-
4-0
150-
-......
.... ...
. .
-..
-.-.
. -..........
...... -..
........
10050\500
500
00
1000
1500
2000
3000
2500
3500
Earth Relative Downtrack Distance (km)
Figure 5-20: Earth Relative Crosstrack Distance vs. Earth Relative Downtrack
Distance for (L ID)max = (2, 2.1, 2.2, 2.3, 2.4, 2.5)
a)
w)
4-
L/Dmax=
- L/Dmax=
-U- L/Dmax=
A- L/Dmax=
-0- L/Dmax=
L/Dmax=
1000 "
0
100
Earth
Figure 5-21:
(2, 2.1, 2.2, 2.3, 2.4, 2.5)
2.0
2.1
2.2
2.3
2.4
2.5
200
300
400
600
500
700
Time (s)
Relative
Speed
120
vs.
Time
for
(L/D)max
2500
2000
C
1500
1000*
L/Dmax=
2.
. .....
. .............. .. V L/tb max = 2.
...
....
-U--L/Dmax= 2.
500
-o
A
200
100
0
400
300
L/Dmax=
L/Dmax=
L/Dmax=
2.
2.
2. 5
600
500
700
Time (s)
Stagnation
Figure 5-22:
(2, 2.1, 2.2, 2.3, 2.4, 2.5)
Point Heat
Load
vs.
Time
for
18- .
16
S)
4-d
-
14 121 __
10
. . . . . ......... .
............ .
..........
..
.. -..
-
4-4
..
...-..
8
. ..
-..
..
6
-44
2
V
-UA
--
0,
C
2.0
2.1
2.2
2.3
L/Dmax= 2.4
L/Dmax= 2.5
L/Dmax=
L/Dmax=
L/Dmax=
L/Dmax=
100
200
300
400
500
600
700
Time (s)
Figure 5-23: Angle of Attack vs. Time for (L /D)ma = (2, 2.1, 2.2, 2.3, 2.4, 2.5)
121
=
I17
1
201
(L/D)ma
-3
4.75
4.7
a)
4 .65|
a)
C-)
4.6
C
a) 4.55.
4.5
*
A
A
0
00
4.45'
2
L/Dmax=
L/Dmax=
L/Dmax=
L/Dmax=
L/Dmax=
L/Dmax=
2.05
2.1
2.0
2.1
2.2
2.3
2.4
2.5
2.15
.
2.2
2.25
2.3
2.35
2.4
2.45
2.5
(L/D)max
Figure 5-24: Value of the Performance Index vs. (L/D)max for (L/D)max
(2, 2.1, 2.2, 2.3, 2.4, 2.5)
122
5.6
Summary of the Parametric Study
The key features of the optimal trajectory were described to provide insight to
the behavior of the Common Aero Vehicle. It was seen that the vehicle initially
increased in altitude until the dynamic pressure constraint became active. This
was done to ensure the vehicle can travel the required downtrack distance. The
local maximum in altitude reached near the end of flight resulted from the need
to first increase the speed to generate lift and then the need to deplete speed,
both of which are done in order to meet the terminal conditions. In terms of
lateral motion, the vehicle steered 410 km out of the Earth Relative downtrack
plane. Another important feature was that while the performance index kept the
angle of attack near the middle of its corridor for a majority of the trajectory,
the angle of attack reached a maximum value near the end of flight. This trend
arose from the need to meet the terminal conditions imposed on the speed and
the flight path angle. Furthermore, the vehicle was banked at -90 deg for about
200 seconds of flight in order to decrease the altitude of the vehicle and the
terminal bank angle was -180 degrees, which resulted from the need to meet the
terminal flight path angle.
The minimum allowable dynamic pressure, maximum allowable heat load,
and maximum lift-to-drag ratio were varied in order to assess their effect on
the resulting trajectory and control profiles. The dynamic pressure constraint
maintains control authority by preventing the vehicle from exiting the Earth's
atmosphere. The minimum allowable dynamic pressure was varied and in each
case the constraint became active at each point where the vehicle attained a local
maximum in altitude. As the minimum allowable dynamic pressure increased,
the initial increase in altitude decreased and the speed depleted at a slower rate.
It was found that in terms of maintaining control authority, the lower the min-
imum dynamic pressure constraint the better. The maximum allowable stagnation point heat load that the vehicle can sustain is determined by the thermal
protection system of the vehicle. In each case of varying the maximum total
123
heat load, the optimizer designed a trajectory in which this constraint was active. This is an important characteristic to consider when designing the vehicle
and in determining an optimal trajectory. As the maximum allowable stagnation point heat load was decreased, the vehicle skipped through the atmosphere
more, initially depleted speed at a faster rate, and the vehicle took a more direct
line to the target by approaching it more from the front (versus further downtrack). It was also found that the more heat the vehicle can withstand, the larger
the control margin. Another design parameter considered was the maximum
lift-to-drag ratio which reflects the maneuverability of the vehicle. Of the three
parameters varied, the range of lift-to-drag ratios considered has the least effect
on the trajectory and controls. Even though the differences in the control margin
for each case was smaller, the higher the lift-to-drag ratio the larger the control
margin.
It is evident from this parametric study that the properties of the Common
Aero Vehicle in combination with the required terminal conditions result in four
important features of the trajectory and control. By varying parameters in the
problem, it was found that that the looser the constraints on dynamic pressure
and maximum heat load and the higher the maximum lift-to-drag ratio, the larger
the control margin.
124
Chapter 6
Preliminary Study of the Real-Time
Application of the Legendre
Pseudospectral Method
6.1
Overview
This chapter addresses the potential real-time application of the Legendre Pseudospectral Method. A useful method to consider in the context of real-time is
one which is capable of obtaining a solution in a sufficiently short period of time,
can solve a wide range of problems, and produces an accurate control. The ability to implement a method in real-time depends upon both the amount of computational resources that are available and the computational complexity of the
problem. In particular, the execution time required to solve an optimal control
problem using the Legendre Pseudospectral Method is highly dependent upon
the optimizer and the machine used. A complete assessment of the solution
time involves comparing the execution time of various optimization algorithms.
However, this thesis is concerned with the Legendre Pseudospectral Method, not
the optimization algorithm. Therefore, the execution time required to solve the
CAV optimal control problem using the Legendre Pseudospectral Method is not
125
considered in this preliminary analysis. Furthermore, it is shown from the diversity of problems solved in Refs. [6, 7, 9, 10, 11, 19, 20, 211 that the Legendre
Pseudospectral Method is indeed capable of solving a wide range of problems.
Thus, this preliminary study is restricted to the assessment of the accuracy of
the solution obtained using the Legendre Pseudospectral Method.
The accuracy of the solution obtained via the Legendre Pseudospectral Method
is assessed by simulating the flight of the Common Aero Vehicle. In the simulation, the control is updated periodically based on the current state of the vehicle. However, a point is reached where the control can no longer be updated.
At this point, the motion of the vehicle is simulated using the previous control
to fly the vehicle until Earth impact. The state of the vehicle at Earth impact
is considered to be the actual performance of the CAV while the most recent
solution obtained via the Legendre Pseudospectral Method is considered to be
the predicted performance of the CAV. Thus, the accuracy of the solution obtained using the Legendre Pseudospectral Method is assessed by comparing the
predicted solution to the actual solution. Furthermore, realistic vehicle and environmental dispersions are added to the simulation. The perturbed model is
created with the intention of assessing the accuracy of the solution subject to
"real life" uncertainties in the vehicle performance.
6.2
Common Aero Vehicle Flight Simulation
Fig. 6-1 depicts a typical simulation for the flight of a vehicle, where N, G, and
C are the navigation, guidance, and control systems, respectively, that comprise
the flight software. The navigation system estimates the current state of the
vehicle and provides this information to the guidance system. The guidance system uses the navigation information to determine the commands that steer the
vehicle to the prescribed terminal state. The control system then implements
these control commands and provides the control to the environment model.
The environment model uses this information to simulate the flight of the vehi126
cle given a particular environment model and a vehicle model. The state of the
vehicle at the end of the cycle is predicted by using the control from the flight
software to integrate the equations of motion. The state calculated by the environment model is then fed into the navigation system and the steps described
above are repeated for the duration of the flight.
For simplicity, it is assumed in this simulation that the state is known perfectly and that there is no error associated with implementing the controls. As
a result, the navigation and control systems are not modeled. Thus the simulation consists of the guidance system and the environment model. Furthermore,
the simulation operates under the assumption that the optimizer can instantaneously produce a new set of control commands which are then updated every
10 seconds.
Target Information
Flight Software
N
G
C
1_
_Model
Figure 6-1: Flight Simulation Block Diagram
Guidance System
In this study, the "guidance algorithm" is the iterative procedure that arises
from using SNOPT to solve the NLP that arises from the Legendre Pseudospectral
Method. The guidance law is the steering command that arises from solving the
NLP. The NLP is solved periodically at time intervals called guidance cycles: thus,
each time the NLP is solved, the vehicle is closer to the target and the time of
flight decreases. Since the number of nodes used throughout the simulation is
127
fixed and the duration of flight decreases with each guidance cycle, the absolute
spacing between the LGL points decreases. This creates the effect of increasing
the number of nodes, which increases the accuracy. As a result, instead of using
100 nodes (which was used to compute the trajectories in the previous chapters),
the number of nodes is reduced to 50 for the guidance simulation. Furthermore,
the guidance algorithm requires an initial guess. Prior to flight, an optimal solution to the CAV optimal control problem is generated. This predetermined
optimal solution is supplied to the guidance system as an initial guess. In this
study, the simulation begins with a converged optimal solution that corresponds
to using 50 nodes and the numerical values previously stated in Chapter 4. After
the completion of the first guidance cycle, the most recent solution generated by
SNOPT is used as the initial guess for the current guidance cycle.
Environment Model
The environment model predicts the state of the vehicle after flying with the
current control for one guidance cycle (10 seconds). The state of the vehicle
is predicted by using the current control to integrate the equations of motion.
The equations of motion include models of the vehicle and the environment. In
particular, a 4 th order Runga-Kutta integration scheme with a constant stepsize
of h = 1 s is used to integrate the equations of motion. Furthermore, Lagrange
interpolation is used to compute the controls during the numerical integration.
The integration is carried out in SI units because the information readily avail-
able from the guidance system is usually in dimensional quantities. The state is
then fed into the guidance system which solves the optimal control problem to
determine a new set of control commands based on the most recent estimation
of the state of the vehicle. In order to maintain a continuous control profile from
one cycle to the next, the controls are included in the specification of the initial
state. This process is repeated ideally until the time remaining in the flight is
less than ten seconds. However, there comes a point at which the optimizer is
unable to find an optimal solution. From the point where the optimizer is no
longer able to find a solution, the flight of the vehicle is simulated using the
128
control from the last converged solution. For the remainder of this chapter, the
portion of the simulation in which the control is updated is referred to as the
closed-loop simulation while the portion of the simulation in which the control
can no longer be updated is referred to as the open-loop simulation.
6.3
Assessment of the Accuracy of the Legendre Pseudospectral Method
The simulation described in the Section 6.2 is implemented both with and without perturbations in the environment model. In the perturbed cases, dispersions
in the value of the mass, lift-to-drag ratio, and the density are each added to the
simulation separately. Simulation perturbations are implemented by altering the
vehicle and environmental models used in the environment block. In all cases
the predicted results obtained via the Legendre Pseudospectral Method are compared to the actual results generated by the environment model. In particular,
the terminal error in position and speed is used to assess the accuracy of the
Legendre Pseudospectral Method in terms of the potential for real-time application. These terminal errors are calculated in the same manner described in
the node analysis. Please refer to Section 4.5.2 for a description of the specific
equation used. Since an optimal solution from the optimizer by definition satisfies the terminal conditions, any error in the terminal state results from the
open-loop simulation. Since the integration terminates at a zero altitude, the
integration time may exceed the final time from the solution generated by the
Legendre Pseudospectral Method. When this occurs, the integrator runs out of
control. If the integration time is greater than the predicted final time, the control used in the integration is set equal to the control from the optimal solution
corresponding to the final time and held constant. Thus both the time at which
the open-loop guidance begins and ends along with the final time corresponding
to the optimal solution are considered.
129
CASE I: Accuracy of Simulation Results Without Perturbations
For this study, the model used in the environment model is the same as
the model used in the optimization algorithm. The terminal position error is
14.0 meters and the terminal speed error is 2.77 m/s. While the position error
violates the requirement of position accuracy to within several meters associated
with striking HDBTs, the error is small when considering the distance traveled
by the vehicle. The speed accuracy is well within the prescribed accuracy range
of ±500 m/s. In this case the closed-loop simulation terminates at 610 seconds
into the flight and at this point, the vehicle is flown open-loop for an additional
41 seconds. The integrated solution guides the vehicle to a zero altitude with
a final time of 651 seconds while the last optimal solution from the Legendre
Pseudospectral Method begins at 610 seconds and terminates at 647 seconds.
Thus the actual solution terminates 4 seconds after the predicted final time.
In order to improve the accuracy of the simulation results, the number of
nodes can be increased. However, this will increase the solution time, which is
undesirable when considering real-time. In terms of the open-loop simulation,
there are two factors that effect the accuracy of the solution: the accuracy of
the integration scheme and the length of time for which the vehicle is flown
using the open-loop simulation. To improve the accuracy of the integration
scheme it would be beneficial to conduct an analysis to determine which integrator produces the most accurate results given the control from the Legendre
Pseudospectral Method. To shorten the duration of flight flown in an open-loop
simulation, the point at which the closed-loop simulation terminates may be delayed by loosening the terminal constraints on the speed, flight path angle, and
angle of attack. For example, since the terminal speed must lie within ± 500 m/s
of vf, set the lower bound on speed at the final LGL point equal to (vf - 500)
m/s and the upper bound at the final LGL point equal to (vf + 500) m/s.
CASE II: Accuracy of Simulation Results with Perturbations in the Mass of the
Vehicle
In this case the value of the mass used in the environment model is per130
turbed by ±1%. This accounts for a 6.87 kg difference between the assumed
and actual values of the mass of the vehicle. With a positive perturbation in
mass, the closed-loop simulation terminates at 620 seconds into flight and the
vehicle flies using the open-loop simulation for 84 seconds. The integrated solution terminates 3.04 seconds after the predicted final time from the last optimal
solution. The terminal error in position is 290 m while the speed error is 36.7
m/s. A negative deviation in mass results in a 660 second closed-loop simulation and a 52 second open-loop simulation where the actual flight terminates
2.23 seconds after the predicted flight. The corresponding trajectory hits the
ground 210 m away from the target with a 7.47 m/s error in speed. Comparing
both cases, a negative perturbation in the mass results in a longer closed-loop
simulation, shorter open-loop simulation, and better accuracy in regards to the
terminal error in position and speed than a positive perturbation in mass.
CASE III: Accuracy of Simulation Results with Perturbations in the Lift-to-Drag
Ratio of the Vehicle
In this case the lift-to-drag ratio (LID) is perturbed by ±1%. This is done
by perturbing the lift coefficient and the drag polar parameter. Consider the
situation where oc =
ec.
A ±1% (LID) perturbation corresponds to perturbing
the angle of attack by ±0.119 deg while the drag polar parameter is perturbed
by T0.1. With a positive deviation in the lift-to-drag ratio, the vehicle is flown
using a closed-loop simulation for 570 seconds before switching to an openloop simulation for the remaining 82 seconds of flight. In this case the actual
flight is 3.56 seconds longer than the predicted flight. The resulting terminal
error in position is 287 m and the terminal speed error is 7.92 m/s. A negative
deviation in (LID) has a much greater affect on the accuracy of the solution. In
this case the closed-loop simulation only lasts for 500 seconds while the openloop simulation lasts for 153 seconds. The actual terminal time is 2.38 seconds
greater than the predicted final time. The terminal error in position is 2580 m
and the speed error is 71.4 m/s. A negative deviation in LID leads to a much
shorter closed-loop simulation and a much longer open-loop simulation than a
131
positive deviation. Despite the fact that the difference in final times between the
actual and predicted solutions is shorter, the terminal errors are significantly
larger for the case with a negative perturbation versus a positive perturbation in
the lift-to-drag ratio.
CASE IV: Accuracy of Simulation Results with Perturbations in the Atmospheric Density
In this case the density is perturbed by ±5%. Perturbing the density this
amount results in a deviation of ±0.06125 kg/m 3 , respectively, from the assumed sea level density of 1.225 kg/m 3 . Take the situation where the vehicle
is at a zero altitude. Using the strictly exponential density model of Eq. (2.17)
and taking the vehicle to be at sea level, a ±5%deviation in the density results
in a altitude deviation of roughly ± 345 m. This is an extremely large difference
which will decrease as the density decreases. When the density is perturbed by
+5%, the closed-loop simulation flies the vehicle for 550 seconds and from this
point, the open-loop simulation flies the vehicle for 102 seconds. The final time
from the open-loop simulation is 2.33 seconds longer than the final time predicted by the closed-loop simulation. The resulting terminal error in position is
1990 m and the terminal speed error is 54.2 m/s. Similar results are obtained
from perturbing the density by -5%. The closed-loop simulation terminates at
570 seconds and the duration of the open-loop simulation is 98 seconds. This
leads to a difference of 4.33 seconds between the actual and predicted final time
of flight, where the actual final time is the greater of the two. The terminal error
associated with the negative deviation in density is 1560 m for position and 113
m/s for speed. The closed-loop simulation is 20 seconds longer and the openloop simulation is 2 seconds shorter in the case with a negative perturbation
versus the case with a positive perturbation in density. However, the difference
in the actual final time from the predicted final time is much larger in the case
with a negative perturbation. While the position error is better in the case with a
negative deviation in density, the speed error is roughly double that of the case
with a positive density deviation.
132
Summary of the Results from the Perturbed Simulations (CASE II-IV)
Table 6.1 and 6.2 summarize the results from perturbing the environment
model in terms of the terminal errors and computational performance. Table
6.1 lists the terminal position and speed errors corresponding to the value that
was perturbed. It is evident from these values that while the vehicle does not
hit the specified target, in every case the vehicle impacts the Earth with the required kinetic energy associated with striking HDBTs. Table 6.2 compares the
computational performance of the simulation with perturbations where TCL is
the duration of the closed-loop simulation, TOL is the duration of the open-loop
simulation, and ATj is the difference between the actual final time and the predicted final time. It is seen that a negative perturbation in the mass of the vehicle
results in the longest closed-loop simulation while a negative perturbation in the
lift-to-drag ratio of the vehicle has the shortest closed-loop simulation.' The duration of the open-loop simulation directly relates to the terminal position error
in the fact that the shorter the open-loop simulation, the smaller the position
error. Thus, the case with a negative deviation in mass has the shortest openloop simulation and the case with a negative deviation in the lift-to-drag ratio
has the longest open-loop simulation. The difference in final times between the
actual and predicted solutions are pretty close for all of the cases considered.
The case with a negative perturbation in the density model produces the largest
difference while the case with a negative mass deviation produces the smallest
difference in final times.
Table 6.1: Terminal Errors from the Simulation with Perturbations
Perturbation Position Error (m) Speed Error (m/s)
+1% m
290
36.7
-1%m
+1%L/D
-1%L/D
210
287
2580
1990
1560
7.47
7.92
71.4
54.2
113
+5% p
-5%p
133
Table 6.2: Computational Performance of the Simulation with Perturbations
6.4
Perturbation
TCL (S)
+1%m
-1%m
+1%L/D
-1%L/D
+5%p
-5%p
620
660
570
500
550
570
(s)
84
52
82
153
102
98
TOL
ATf (S)
3.04
2.23
3.56
2.38
2.33
4.33
Summary
The Legendre pseudospectral method was assessed in the context of real-time
application in terms of the solution accuracy. A flight simulation of the Common
Aero Vehicle was constructed where the state of the vehicle was updated periodically throughout flight. A
4 th
order Runga-Kutta integration scheme was used
in the environment model to update the state of the vehicle. The updated state
along with the last control were used as an initial condition while the previous
converged solution was used as the initial guess for the optimization algorithm.
Since the optimization process was repeated with current information regarding
the state of the vehicle, the number of nodes was reduced to 50. It was found
that without perturbations in the environment model, the terminal position error was roughly 14 meters. With this position error, the vehicle will miss the
target; however, it is insignificant in comparison to the distance traveled by the
vehicle. The speed error was well within the allowable bounds and thus, the
vehicle will penetrate the surface of Earth with the required kinetic energy.
The robustness of the solution was then addressed in terms of accuracy by
adding perturbations to the mass, lift-to-drag ratio, and density values used in
the environment model. A deviation in the mass of the vehicle was modeled in
the environment and it was found that in terms of the terminal error in both
position and speed, it is better to over estimate than to under estimate the mass
of the vehicle.
In terms of the lift-to-drag ratio, there was a large difference
134
in accuracy between a positive and negative deviation in the lift-to-drag ratio.
In this case it is much more beneficial to under estimate the lift-to-drag ratio.
With a density perturbation, the vehicle came closer to hitting the target with
a negative deviation versus a positive deviation and the speed error was larger
for the case with a negative deviation in density versus a positive deviation in
density. Overall, the solution was the most sensitive to deviations in the density
and the least sensitive to deviations in the mass. However, a negative deviation
in LID resulted in the largest terminal position error.
The results from the unperturbed simulation indicate that the Legendre Pseudospectral Method shows promise for use in real-time. In terms of the perturbed
cases, the solution is the most sensitive to a negative perturbation in LID as well
as any perturbation in the density model. The resulting position errors directly
correspond to the duration of the flight flown using the open-loop simulation.
In order to improve the accuracy of the solution subject to perturbations in the
model, the duration of the open-loop simulation should be decreased. In order
to improve the robustness of the solution to perturbations, a more accurate vehi-
cle model should be developed and a more accurate atmospheric model should
be used. However, these improvements may increase the solution time. Thus,
in order to continue this analysis on the real-time application of the Legendre
Pseudospectral Method to the Common Aero Vehicle a detailed comparison of
both the execution time and the solution accuracy should be conducted.
135
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Chapter 7
Conclusions
7.1
Summary
The United States desires space-based global strike capabilities. Global strike
refers to the ability to project power anywhere on the globe from the continental United States in short notice. This desire leads to the development of new
vehicles which involve space launch and Earth re-entry. This thesis considered
the use of the Common Aero Vehicle (CAV) as the Earth re-entry vehicle. Furthermore, the Common Aero Vehicle (CAV) considered is an unpowered bank-to-turn
high lift-to-drag ratio Earth penetrating re-entry vehicle. The natural behavior of
the vehicle conflicted with the behavior required to satisfy the terminal conditions for striking HDBTs. As the vehicle neared the target, the demands on the
guidance and control systems increased greatly. In order to maintain control
flexibility, it was desirable to determine a trajectory and control in which the
control margin was maximized. The CAV mission design problem was to steer
the CAV from a completely specified initial condition to a partially specified
terminal state on the surface of the Earth such that a performance index is minimized and the constraints imposed on the vehicle are satisfied. This resulted in
an optimal control problem.
A solution to the optimal control problem was obtained using a direct Legendre Pseudospectral Method. The Legendre Pseudospectral Method discretizes
137
the optimal control problem at the Legendre-Gauss-Lobatto points and the resulting NLP is solved using one of the many available software programs. The
resulting Common Aero Vehicle NLP has both linear and nonlinear inequality
and equality constraints and a sparse Jacobian.
SNOPT is a general purpose
solver that takes advantage of the sparsity of the problem and thus, it was used
to obtain a solution to the NLP. The steps required to obtain a solution to the
Common Aero Vehicle optimal control problem via the Legendre Pseudospectral
Method were explained in detail. Also included was an analysis to determine the
number of nodes to use in order to obtain an accurate solution. Another analysis involved in setting up the numerical optimization problem was the choice of
weighting factors in the performance index that maximize the control margin.
Thus, the generation of a trajectory and control was discussed in terms of the
desired vehicle performance as well as the accuracy of the solution obtained.
Once the optimization setup was completely defined, the key features of the
trajectory and control were noted to better understand the Common Aero Vehicle. A parametric optimization study was then conducted to demonstrate the
application of the Legendre Pseudospectral Method to vehicle design. The minimum allowable dynamic pressure, maximum allowable stagnation point heat
load, and maximum lift-to-drag ratio were varied independently to determine
their effects on the trajectory and control.
Finally, a preliminary study assessed the real-time application of the Legen-
dre Pseudospectral method to the Common Aero Vehicle optimal control problem in terms of the accuracy of the solution. This was done by simulating the
actual flight of a Common Aero Vehicle. An environment model was used to
update the state of the vehicle and the state was then used to update the control
history. This process repeated until the optimizer could no longer find an optimal solution. At this point an open-loop simulation was used to fly the vehicle.
The open-loop simulation integrated the equations of motion using the most recent control until the vehicle impacted the Earth. Included in this last analysis
was the effect of model uncertainties on the ability of the control to steer the
138
vehicle to a specified target on the surface of the Earth. In both cases with and
without uncertainties, the accuracy of the solution was assessed by calculating
the resulting terminal error in position and speed.
7.2
Conclusions
The Legendre Pseudospectral Method is capable of solving a complex optimal
control problem. The trajectory and control generated using 100 nodes satisfied the strict accuracy requirements associated with the Common Aero Vehicle.
Furthermore, the performance of the Common Aero Vehicle was optimized by
maximizing the control margin. Maximizing the control margin refers to keeping
the angle of attack near the middle of its corridor, minimizing the maximum angle of attack, and keeping the control rates small. The value of the performance
index were used as a direct measure of the control margin corresponding to the
optimal solution.
The terminal constraints are the driving force behind the characteristics of
the trajectory and control for the Common Aero Vehicle. The vehicle initially
increased in altitude which resulted from the need to satisfy the range requirements. At each local maximum in altitude the density constraint became active.
A minimum allowable density constraint was imposed to prevent the vehicle
from escaping the Earth's atmosphere and to maintain control authority. Furthermore, the vehicle steered out of plane 410 km and traveled farther downtrack than the target. Thus, it actually approached the target from behind. While
the performance index kept the angle of attack near the middle of its corridor
throughout most of the trajectory, the angle of attack reached a maximum value
near the end of flight. This resulted from the need to meet the terminal conditions on the speed and flight path angle. The terminal flight path angle also
drove the bank angle to -180 deg. In order to obtain a flight path angle of -89.9
deg, the vehicle must approach the target with negative lift. Negative lift was
generated by rotating the vehicle upside-down, which corresponds to a bank
139
angle of -180 deg.
In order to better understand the behavior of the Common Aero Vehicle, the
minimum allowable dynamic pressure, maximum allowable heat load, and maximum lift-to-drag ratio were varied. In each case where the dynamic pressure
constraint was varied the local maxima in altitude corresponded to the points
where the dynamic pressure constraint was active. Since the dynamic pressure
constraint is a function of altitude and speed, as the minimum allowable dynamic pressure increased, the initial increase in altitude decreased and the speed
decreased at a slower rate. The stagnation point heat load is also a function of
altitude and speed. In each case where the maximum allowable heat load constraint was imposed, the optimizer yielded a solution in which the vehicle hit its
upper limit on heat load. As the maximum allowable heat load decreased, the
vehicle skipped through the atmosphere more and approached the vehicle on a
more direct path. The looser the constraint on dynamic pressure and heat load,
the larger the control margin. Thus it is desirable to design a vehicle that can not
only fly in a low density region while still maintaining control authority, but can
withstand a large amount of heat load. The lift-to-drag ratio corresponds to the
maneuverability of the vehicle and had an insignificant effect on the trajectory
and control. Nonetheless, the value of the performance index indicated that the
higher the lift-to-drag ratio (the more maneuverable the vehicle) the larger the
control margin.
In terms of the real-time application of the Legendre Pseudospectral Method
to the Common Aero Vehicle optimal control problem, a preliminary study was
conducted. Without perturbations in the simulation environment, the position
error was roughly 14 m and the speed error was well within the range of the
required accuracy. These results are impressive considering the distance that the
vehicle is traveling and the stressing terminal conditions imposed. This indicates
that the Legendre Pseudospectral Method shows promise for the use in real-time
and that a more detailed analysis involving the optimizer used in the closed-loop
simulation and the integration scheme used in the open-loop simulation must
140
be conducted for a more conclusive assessment. The robustness of the solution
was also considered in terms of the application of the Legendre Pseudospectral
Method to the Common Aero Vehicle optimal control problem. The mass, liftto-drag ratio, and density in the environment model were perturbed and the
resulting accuracy was considered along with the computational performance.
The solution was the most sensitive to a negative perturbation in the lift-to-
drag ratio followed closely behind with any deviation in the density. It was also
seen that the shorter the open-loop simulation, the smaller the terminal error in
position.
141
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Appendix A
Notation
1. If a e R"Y and b E R, the following notation represents term-by-term multiplication on vectors
ab=
ai
a1 bi
a2
a2 b2
a3
a3 b 3
an
anbn
and the same holds true for division.
2. A diagonal matrix is denoted by (-) as shown below.
(1)=
1
0
0
...
0
0
1
0
---
0
0
0
1
-..
0
- - --
0
--- 1
3. Square brackets are used to indicate a matrix. For example, [0] is a matrix
of zeros. However, in some instances, square brackets may represent a
row or column "matrix". The matrix dimensions should be obvious based
on the context.
143
[This page intentionally left blank.]
Appendix B
Matrix Derivatives
The rules listed in this appendix pertain to the matrix manipulations used to
calculate derivatives. These rules are used to calculate the analytic Jacobian and
objective gradient. First, the matrix which results from taking the derivative of
one vector with respect to another vector is defined. Second, the chain rule is
used to generalize the rule for differentiating the multiplication of two vectors
with respect to a common vector. The term common vector refers to the fact
that both vectors in the multiplication are a function of the same vector.
Consider the following two vectors with different lengths: y E Rm and x C R".
Taking the derivative of y with respect to x results in the following m x n matrix:
dy =
dx
dy
1
dy 1
dy 1
dy
dx
1
dx
dx
3
dxn
2
1
dy 2
dy 2
dy 2
dy
dx
dx2
dX 3
dxn
dym
dym
dym
dym
dx
dx
dx
dxn
1
1
2
3
2
(B.1)
Now consider two vectors with the same length, a c R" and b c Rm. Define
the vector y c Rm to be the term-by-term multiplication of a and b. Using the
notation defined in Appendix A, y can be written as:
y = ab
145
(B.2)
Furthermore, assume that a and b are both functions of x E Rn. In order to
determine the derivative of y with respect to x, the chain rule must be used.
da1
db 1
bi+ aid
da2
db 2
b + a2
dx1 2
dx 1
dy
dx
da1 b +
db1
dx2
dx 2
da2
dx
2
dbm dam
dxi dx 2
dam
dx 1
da 1
dxn
da2
dxn
b2 + a 2 db2
dx
2
dam b
dxn
dbm
+am dx
db 1
dxn
db2
dxn
2
+mdbm
+am~~
dxn(I
(B.3)
Rewriting the matrix above as the sum of two matrices, the following is obtained:
-
b2
dy
dx
da1
da1
dx1
dx
b2
dx 1
dxn
b 2 da2
2
dx2
dxn
dam
bmdam
bmdx1
dx
dbi
dx1
-
bi dal
a2
+±
db 2
a 2 dx
dx 1
dbm
bdam
mdx
_
2
_ adx
db1 ~
dxn
db
ai dx 1
dx2
db 2
1
aid
db2
azd
dxn,
2
dbm
adn
dbm
am dx 2
(B.4)
dy
Splitting each matrix in B.4 into the multiplication of two matrices d, can be
dx
rewritten as:
bi
0
dy
0
0
---
0
0
b2
0--
0
0
0
a1
0
0
0
0
a2
0
0
0
0
0
da1
dx 1
da2
dx1
da1
da1
dx
2
dx 3
2
dx
da 2
dx
dx
+±
---
-
bm
am
da 2
3
dam dam dam
dx1 dx2
dx 3
dbi
db1 db 1
dx 1 dx2
dx 3
db 2 db 2
db2
dx 1 dx 2 dx 3
dbm dbm
dbm
dx 1
dx
dx
2
3
da 1
dxn
da 2
dxn
dam
dxn
dbi
dxn
db 2
dxn
(B.5)
dbm
dxn
Using the notation defined in Appendix A, the matrix equation in B.5 can be
146
condensed to the following form:
dx
(b) d + (a)
dx
dx
147
(B.6)
[This page intentionally left blank.]
Appendix C
Constraint Jacobian and Objective
Gradient Derivation
This appendix defines the constraint Jacobian and objective gradient of the CAV
mission design problem used in this thesis. The notation defined in Appendix A
and the rules set forth in Appendix B are used.
149
C.1
Constraint Jacobian
acx
acx
ay
ax
aC
ax
ax
ac,
ax
acx
acx
acx
az
avx
v
acy acy
y
acy
ay
az
vx- avy
acz acz acz acz acz
ay
az avx avy
acx
ac,
acx
acvx
ay
acx
avx
az
acey
ay
acoz
ay
aca
az
acovz
aC
ac
ac, ac,
acr
ax
ax
acV
ay
ay
aCq
ax
aCa
ax
acQ
acoz
avy
aca
Vx
az
ay
acv
aCq
acoz
az
az
acQ
ac,
aCy
acy
acy
ae
acx
ae
aca
avy
acQ
acQ
aca
aCa
avz
ae
aca
aT a0
acr
acr
avz
avz
aca
aCa
acQ
ax
ay
az
acy
acy
acy
acy
avz
acy
ax
ay
z
vx
av-y
avz
v
aCq
v
av
ua
a-
aa
aCa
acr
Toac,
av
aCq
aua
au
acuz
aue
aca
au
ac, ac(7
aua
acr
aua
au,
acr
u,
acv
acV
at5
acz
atj
acvx
atj
at5
acoz
ataca
atf
acJ
atf
acr
atf
ac,
a,
ana uT,
aCq
aCq
atf
aCq
aCa
aca
aCa
aco
ao- aua
aCa
tf
Cvy c acvy acy
aca
Oac( ac,
avz
v
c y
ao-
aa
v
ac,
au
acz
au
ac.x
au
aua
aca
acy
v v
aua
ac x
a
acvz
aCq
ac,
Jo-
ac,,
a0-
avz
acoz
acvy
aU(T
ana
acz
0
acz
acvy
a- aua
acvz acoz
v
aCq
avx
avx
acy
ac
acz
v
ae
avy
ac,
acv
acv
aCq
ac
acx
avz
avx
az
ay
acr
avx
acv
aCq
acQ
avx
acr
az
acv
ay
avy
vx
acr
acx
Dua
ac,
j0-
c
avz
acz
avz
avy
aCa
az
acr
ax
az
aca
ay
avy
acx
a
ac,,y acvy acuz acvy
avx
ax
actz
ax
aca
ax
aJac
ac
acx
T,
DVz
au7
aae
acQ
aacQ
aua
acQ
acQ
au
atj
atf
a-
7-
aua
-u-l
atf
acy
acy
acy
acy
acy
7a
-
auT,
atj
o
ua
_
(C.1)
Partial Derivatives of Cx Constraint
Cx = DNX -
aCx
ax
a_
aCx
acc
= DN
[01
=
ay
acx
az
(C.2)
tfto)vx
ao-
[0]
=
[0]
=
[0]
x
[0]
0acx
[0]
___=
aCx
avx
acx
av,
aCx
___=
-
(tf,-to
au,,
(C)
au,
ac
[0]
atf
[0]
avz
150
_
1
2
= -- [vx
(C.3)
Partial Derivatives of C, Constraint
CY = DNY-
acy
ax
acy
ay
acy
az
acy
avx
=
[0]
=
DN
=
[0]
=
[0]
acy
t2to)V
ac,
aua
acy
=
[0]
=
[0]
=
[0]
(C.5)
au"
acy
0]
-
auf
(t5 - to ()
-
(C.4)
vy
1
2
at5
acy
avz
[0]
=
Partial Derivatives of C, Constraint
Cz = DNZ
acz
acz
az
[0]
-
=
2
t
DN
(C.6)
vz
acz
ac(
acz
ao
acz
= [0]
ax
ay
acz
-
[0]
=
[0]
-
= [0]
(C.7)
acz
avz
-
[0]
=
[0]
au"
acz
atf
0]
-
-I
2[vz]
(t5 -to)()
=
Partial Derivatives of Cx Constraint
Cvx = DNVx
tf
-
( to
Dx + Lx + gx + 2wvy + w2]
151
(C.8)
acvx
ax
acvx
ay
acox
az
acvx
acvx
(t5
_
-
to) (aDx
/\ax
2
(tf
/aDx
2 to
2)
)x
ax
/
C Lx+
agx
ay ay/
a Lx
agx
)z az/
J\az
2
+aLx
_
_
/
( 5 - to
\ 2 /
\ 2
Dx
av
a Dxav
t5-t0
/\avz
_ _ tf-toDx
acvx
am o
)\aa
2
(t5-to)
acvx
a o
2
-
au"
=
atf
agx
\ay
(t5 - to) /aDx
acvx
av
avz
acvx
acvx
aLx
aDx
(\ao
+
agx
av,
v
agx + 2w
)Lx
vy
avy
Lx
agx
z
avz
Lx
Nvz
(C.9)
agx
Of
aa/
aLx
agx
af/
av
[0]
[0]
=-[Dx+Lx+gx+2wvy
+ ozX]
Partial Derivatives of Coy Constraint
Coy = DNVy -
to
D
152
+L + gy
-
2wvx + w2y]
(C.10)
acvy
ax
/
- to
S(t
K
\2
acVY
(tj -to)
ay
acvy
+ a L,
ax
x
aDy
a L,
(tjt~)KaDy
az
2
acvy
avx
\t2 t
acvy
avy
acvy
DN -
az/
agy
} avy
avy
avy
)\aD
a
x
ag, - 2w
avx
L,
ag,
avz
avz/
a L,
agy
z Of
aa/
L,
ar
agy
[D+Ly +gy--2wvx+W
2
aD,
aa
acv
(t-t
atj
ag,
)z
aLy
2
act5
aL,
av
f -
t5 -to)
-
ay
)
aa
acVy
+gy
y
av
- to )D
2
aua
ax/
Kvx
\2
(t f
S
avz
au,
i
ay
2
az
ag,
)K
aDy
or
(c. 11)
au/
[0]
acoy - [0]
=
2y
yY
Partial Derivatives of Coz Constraint
Coz =
DNVz
tf 2 to) [Dz + Lz + gz]
153
(C.12)
acvz
(t5-to)K/aDz
ax
) \ax
\2
acvz
ay
\ 2
acvz
_
2(
}
acz
Jvx
\ 2
)
avy
acvz
_~~~ \ _2( 5- t
}
acvz
au
acvz
_~
\ 2
=
(
ac t
2
ay/
/ Dz
+ agz
\az + aLz
az
az/
/aDz
\avx
+ aLz + agz\
avx
avx
+ aLz + agz\
Dz
avy
avy
avy/
aDz +L
i
t5-t
\2
= DN -
ay
ay
}
( t5- to);
\
avzc
K aDz + aLz + agz
( tf- to);
_
+ agz
+ aLz
ax ax/
) \ iJvz
Jvz
agz
avz/
(C.13)
DzLz + agz
~~ ( 5-oa +
\aa
)
±a
ax/
/aDz + aLz + agz
\ao
Jo
a
Jo/
[0]
=
=
[0 ]
-
[Dz + Lz + gz]
Partial Derivatives of C, Constraint
C,
acx
ax
= DNa -
S [0]
acta S [0]
ay
acaf = [0]
az
acta [0]
avx
ac
actx
av7
S(0]
=
ac
t
-
2
=
ac
a tj
[0]
154
[0]
tf - to
2
aca
au"
aua
(C.14)
DN
=
ac
aci
to u
(C.15)
=
[0]
1 [
2
Partial Derivatives of C, Constraint
t
Ccr= DNO
- to
2
= [0]
[0]
x
acc
a
[0]
=
DN
-
[0]
=[0]
az
3Co,
10]
aucT
[0]
ac
tj - to
2
avx
aCo-
avy
(C.16)
U
(C.17)
1
atf
2
-
[0]
avz
Don
Partial Derivatives of Position Constraint
Cr =
X 2 +y
2
+z
2
(C.18)
aCr
=KX) a~r
aCr
ay
[0]
aCr
[0]
-
aCr
[0]
az
r au"
aCr
aCr
av,
aCr
av2
aCr
=
Cr
ac
at5
-101
[0]
= [0]
=
(C.19)
=
[0]
=
[0]
[0]
Partial Derivatives of Speed Constraint
Co= v2, + v2 + v
155
(C.20)
ax
a f --
[0]
ac
-[0]
C
[0]
acV
0
3C,
ay
azV
u
= [0]
az
/vx
aCV
ac.
(C.21)
aCV
\
avx
[0]
_
av,
\v
3CV
avz
|vz\
au,
-
[0]
a
-
[0]
atf
\v/
Partial Derivatives of Dynamic Pressure Constraint
Cq
(C.22)
= 1pv2
= v2
a
=
V2
=
[0]
_
=
1V)v2
=
[0]
a_
aC4
avx
-
aCq
(pv)
av,
_C
avz
(pv)
=
av
avx
aC4qC.3
au, = [0]
av
at5
av,
0
(pv) av
avz
Partial Derivatives of Sensed Acceleration Constraint
Ca =
L 2 +1D2
156
(C.24)
aL
aCa
ax
aCa
ay
a
az
KD\ 3D
KD)
aK~
a
\a/
aCa
\a
az
aCa
aca
aca
aua
aCa
aua
ay
azL
\a
av,
\a/ avz
avz
aKD)
av,
+
+
aCa
'a
atf
()a
[0]
= [0]
(C.25)
aD
aCa
av,
aca
aL
Ca
[0]
= [0]
aD
D 3D
/D\
\ai/av
Partial Derivatives of Total Heat Load Constraint
Ca =
to) N
k =O
LK
157
()
Pk
Po
3.s51 Wk
/2( Vk ) 3
/2
Ve
I5
J
g
(C.26)
1/2
acQ
ax
aCQ
2
P
Po
12
W
ap
ax
1/2
ay
K
2
aCQ
az
aCQ
Po
p-
2/1
3
1/2
( -1
2
~2
p 1 12
po)
tK - to12
1
)3 is
v
\PO)
v
)
ap
.is
az
v 2.1 5 W
/ v
aCQ
-tv12.poiw
t
t - to
avy
aCQ
aCQ
aua
aCQ
aur
aCQ
1
2.
av
(C.27)
[0]
[0]
[0]
=
=
-
[0]
0]
1/2
a tf
P
2
=
Vk )3.15
7)
(,~L\PO)
jI
Partial Derivatives of Terminal Flight Path Angle Constraint
Cy
rf -Vf
(C.28)
rfvf
XfVx,f + Yf Vy,f + ZfVz,f
rfvf
158
(C.29)
acy
axf
vx,frfvf - (xfvx,f + yfvy,f + zfv z ,f)( rf /axf)vf
(rfvf )2
acy
Vy,5rfvf - (xJvx,f + yfvy,f + zJvz,5)( r/
ayf
yf )vf
(rfvf ) 2
acy
vz,frfvf - (Xfvx,f + Yfvy,f + zfvz,f)(arf /azf)vf
azj
(rfvf )2
acy
avx,5
xfrfv5 - (xJVx,f + Yfvy,f + zfvz,f )rf (arf/avx,f)
(rfvf) 2
acy
yfrfvf - (xfVx,f + YJVy,f + ZfVz,f)rf (aVf/avy,5)
(rfvf )2
avy,f
ZfrfVf
acy
avz,5
acxj
=
[0]
acy
=
[0]
ac
a tf
)rf(aVf5/aVz,f)
(rfvf)z
a5f
acy
aaU
a,f
o-f
(XfVx,f + YfVy,5 + ZfVZ,f
-
(C.30)
= [0]
=
[0]
=
[0]
Partial Derivatives of r
r =
ar
x+
rx
ax
z
y+
ar
o
2
(C.31)
-
(0]
-
[0]
ar
ay
\r /
a0
ar
[0]
r aO
r
ar
[0]
avx
ar
[0]
ar
avy
avz
aua
=
atf
[0]
159
(C.32)
=
[0]
=
[0]
Partial Derivatives of p
p = po exp-
ap
ax
ap
_
ar/ax
-
H
ap
aaf
(p)
ar/ay ()
_
ay
ap
H
ap
avz
(C.34)
ap
= [0]
au
=
[0]
=01at5
at
ap
avy
= [0]
aua
ap
av,
10
air
= ar/az
az
[0]
=
ap
H
ap
(C.33)
(r-Re)IH
= [0]
ap
=[0]
0
0
_
[0]
=_
Partial Derivatives of v
v =
vV+v2,+v
±
S (0]
av
ay
ay
av
=
av
av
30-
= [0]
=
[0]
av
az
aua
av
av
avz
[0]
0]
-
=0]
[
(C.36)
av
Ivx\
v
\v /
av
(C.35)
-
[0]
av
_
Partial Derivatives of CL
CL
CL,O(a
160
(C.37)
aCL
-01
a
aCL
aCL
= [0]
ay
(f
aCL
[0]
acr
aCL
aCL
az
aCL
aua
avx,
-(CL,
a
[0]
[
(C.38)
[0]
=[]
aCL
au,
[0]
[0]
aCL
[0]
aCL
avy
aCL
tx)
at
avz,
Partial Derivatives of CD
aCD
aCD
[0]
ax
[0]
ay
aCL
aCL
[0]
a0-
aCL
aCL
az
aCL=
[0
[0]
aua
=
[0]
a
0
[0]
[
acL
au
avx
a L
(2KCL)
aca
aCL
avy
aCL
(C.39)
CDO + KC
CD
atf
(C.4)
= [0
avz
D
=
12
2
pv2SCD
161
(C.41)
Partial Derivatives of Drag Magnitude
1
D
ax
V2
K1v2SCD9
2
1V2SCD)
ap
K
az
aD
(PSCD)
aD
av,
___
=
(PSCD)
/
1
~
a
az
2
avvx
D
_p
ax
ap
ay
2
aD
ay
\
vSCD)
aD
aor
[0]
aD
[0]
Ju,,
av
aD
a
au,
av
aD
av,
t
2\
pv
aCD
o
(C.42)
[0]
Partial Derivatives of Lift Magnitude
LL=
I
al,
1
2
PV 2 SCL
ap
K
/ax
2
KLI
ap
K2Iv2SCLav/az
ax
2SCL
aL
\2
al,
AL
av,
av
-
avy
aL
av,
a
/
ap
az
/
1
2S
aCL
\ 2 PaV
[0]
-
al
au,
[0]
-[0
(C.44)
aL
au,
(PSCL) av,
(PSCL)
_
ac,
aL
i2SCL
ay
(C.43)
av
aL
av,
atif
av
at
(pSCLav
=v (pSCL)av
[0]
Partial Derivatives of Gravity
g = gxex + gyey + gzez
(C.45)
Partial Derivatives of gx
gx= -
162
x
(C.46)
agx
agx
ax
_ \r3/
agx
ay
agx
_
agx
avy
agx
-
agx
/ay
3px \ ar
Kr4 / az
ao
agx
[0]
agx
aua
au,
_
-
[0]
-
[0]
av2
agf
<3px\ ar
r4
az
ag,
avx
agx
+p( rpx\
ar
4
ax
ax
at5
-
[0]
=
[0]
-
[0]
-
[0]
(C.47)
[0]
Partial Derivatives of gy
gy =
K3piy /ax\ r
3Ay)
K-b
K3y ar r4
ag,
r4
ax
agy
ay
agy
-~~
r4
az
agy
agy
agy
=
[0]
=
[0]
=
[0]
/az
(C.48)
y
agy
a; ag
agy
[0]
[0]
auo
ag
au"
[0]
agy
[0]
atj
[0]
(C.49)
Partial Derivative of gz
gz = -z
163
(C.50)
(yz\
agz
agz
ar
/ ax
/3pz\
ar
4
r lay
ax
r
agz
ay
ag
4
az
K-r3)\rH/az
p + (3pz ar
aa
agz
aug
agz
[0]
agz
avx
agz
avy
au
=
agz
agz
[0]
atf
=
[0]
=
[0]
=
[0]
(C.51)
[0]
-
=
[0]
[0]
avz
Partial Derivatives of Drag
D = Dxex + Dyey + Dzez
(C.52)
Dx = D vx
(C.53)
Partial Derivatives of Dx
vx
aDx
V
aDv
ax
aDx
aDx
avx
aDx
avy
aDx
avz
=
aDx
V - ~~Kvx~a!
azV
au"
vx
D +D
v avx
vx aD
V avy
(vx) D
V avz
[0]
avr
D
vx
_Dvx
av
v
V2
ovx
Dvx)
-- --av--v2
avy
aDx
aua
aDx
a t5
D
\V aa(
aia
aDx
ax
aDx
ay
aDx
az
/vx\
[0]
(C.54)
-
[0]
-
[0]
Dvx) av
v2
avz
Partial De rivatives of Dy
Dy = D
164
Y
(C.55)
aDy
ax
aD,
ay
aD
az
aDy
ax
\v/
Kv
aD,
vy aD
aDv\a
(v y a
\V/ az
aD
vy\ avy
\v/
aDy
avy
aD,
v /avy
avz
V
_
Dv
\v2
avx
/
v
\
-vy- +
/D)
avz
V2 / av
KDvy\
av
\v/
/vy) a
~\v /B
aa
_Dvy
v2
aDc
[0]
aua
aDy
au,
[0]
(C.56)
=
[0]
=
[0]
aD
av
a tf
/avy
Partial Derivatives of Dz
(C.57)
Dz = D V
V
aDz
ax
aDz
ay
aDz
az
aDz
avx
aDz
avy
vz
v
aD
aDz
ax
Ba
aDz
vz aD
a- vz aD-V
az
aD
vz
V avx
- (vz- aDV avy
aa
vz
=
[0]
=
[0]
=
[0]
aDz
aua
Dvz
avx
Dvz
av
avy
v2
aDz
au,
av--
v2
aDz
at
aD
(C.58)
[0]
aDz
avz
vvz
aD
/ avz
D
V
Dvz\
av
v2
avz
Partial Derivatives of Lift
L = Lxex + Lye, + Lzez
Partial Der ivatives
(C.59)
of Lx
Lx = L (sinOw 2 ,x
165
+ cos uw 3 ,x)
(C.60)
Lx
ax
aL (sin
o-w2,x
ALx
ay
aL
Lx
az
Lx
-
ax
ay
-
-
+ COS
avy
aL (sin o-w2x + COS
'Nvz
avz,
aL
aL
(sincow
W3,x ( +
(sin 0-)
'
(L (cos -w 2,x - sinow 3 ,x))
=
[0]
=
[0]
=
[0]
aL
±
aw2,
aw ,x
((sin 0-) 2
(L)
/
(cos o-)
(Cos-)
±(COS
'
a
a'
aw3v )
awsx)
7)
+ (cos o-) a y,
(sin (-)
(L)
,x + cos w 3,x) + (L)
2
+ (cos)
(y
-Ws,x) + (L)
=
aLx
atf
az (sin o-w2,x
aL (sin-w ,x + cos o-w3,x) +
2
-
aua
aua
(sin o) a
(sin o-w2,x + Cos o-w3,x) + (L)
(sinow 2 ,x + cos o-w3,x) + (L) ((sin
av2
aL~
aL
aL,
aL
(sin -) a ' + (cos a)
+ COS -W3,x) + (L)
aw2 ,x + (cos) ac
(sin-)a
(C.61)
Partial Derivatives of Ly
Ly = L (sin Uw 2,y + cos -w3, y)
166
(C.62)
+ cos Uw 3,y) + ( L) ((sin a) a'
=(sin-w2,y
aLy
aL
a-ay
ay
aLy
(sinow2,y + cos o-w 3 ,y) + ( L)
aL (sinow 2 ,y + cos o-w 3,y)
az
=
aL
(sin-w
2 ,y
aLy
± (L)
aLy
((sin
-)
av,
a
+(cos
)
-) ay
+ (cos 0-)
azf
+ (cosa) a
+
(cos 0-)
ay)
+ (cos-) aw 3 y )
+ (cos)
a
)
3 ,y))
[0]
au"
aLy
[0]
au,
aLy
atf
( (sin g-) avy
+ (L) (sin o)
(sinaw 2 ,y + cos o-w 3,y)
(L (cos O-w2 ,y - sin-w
ao-
(L) ((sin a-) av
+ cos w 3,y) + (L)
(sin0-w2 ,y + cos w 3 ,)
A
y
+ ( L) (sin -) az
aw 2y
(sin-w2,y + cosow 3 ,y) +
avy
(sin -)
+ (cos-) a'
=
[0]
(C.63)
Partial Derivatives of Lz
Lz = L (sinUw2,z + cos -w3, z)
167
(C.64)
aLz
aL
ax
aLz
ay
ax
aLz
aL
ay
(sin o-w2 ,2 + cos 0-w3 ,z) +
az
- (cos a-)
axz
+ (cos a) aw3 z)
(sin-w
2 ,2
+ (L) (sin -) aw 2z
+ cos O-w3,z)
(sino-w 2 ,z + cos O-w3,z)
az
(L) (sin-
+ (L)
+-(cos a) az
(sin a-) aw
aLz
az + (cos 0-) awz
-
avx
aLz
aL2
L
-
aLz
aaz
aLz
au
aLz
aL
-
(sin o-)
(sin0-w2,z + cos w 3,z) + (L)
(sin o-)
[0]
au,
=
[0]
at1
=
[0]
avy
+
(cos 0-)
2,z -
sinw
a)z
+ (cos a) aw 3,z
+ Waw 2 z + (cos0-)
(sinow 2 ,z + cos ow3 ,z) +±(L)(sina a-a'
= (L (cos-w
=
aL
(sin-w2,z + cos a-w 3 ,z) + (L)
V
a'
3 ,z))
(C.65)
Partial Derivatives of the Unit Direction wi
v
V
(C.66)
Partial Derivatives of wi,x
wi,x
=
168
vx
v
(C.67)
awlx
ax
[0]
-
"awi,x
ay
awi,x
az
[0]
=
= [0]
awix
awx_
[0]
ao
[0]
aua
vx
=
[0]
av
aw,x
[0
v2
avx
V
Kx)
v
av
au,
awl,x
[0]
V2
av,
atf
Kx)
av
avy
aw,x
aa
awi,x
1
avx
awix
-
avz
(C.68)
\V2 / avz
Partial Derivatives of wi,y
wyy
aiy=
ax
(C.69)
{0]aw~
aa
awi,y = 0
=__
= [0]
wi,y=[0
[0]
=
'w~ = [0]
'w~ = [0]
az
aw)y
Jvx
V2/
awi,y
_1
avy
V
[0]
av
aua
awi,y
vx
au,
vy) av
V2
awi,y
vyy
vz
\ V2/
avy
awi,y
atj
[0]
(C.70)
- [0]
av
avz
Partial Derivatives of wiz
wi,z = Vz
169
(C.71)
aw,
ax
aw,
ay
awl,z
az
aw,
=
[0]
=
[0]
=
[0]
awl,z
acx
awi,z
ao
awl,z
vz
avx
v
vz
avy
v
2
2
aua
av
avx
aw,z
av
avy
awi,z
auf.
at;
=
[0]
=
[0]
=
[0]
(C.72)
[0]
[0]
1\I - vz\ av
Partial Derivatives of Unit Direction w 2
r
x
v
(C.73)
||r x vil
=
(C.74)
w2,xex + w2,yey + w2,zez
Yvz
W2,x =
-
(C.75)
v z )2 + (xvy - yVx)
(ZV
S(Yvz - ZV)
zvy
2
zvx - xvz
w
2 ,y
=
+ (zvx
-
XVz)
2
+ (XVy - Yvx) 2
V(yvz - zvy)2 + (zvx
-
XVz)
2
+ (XvY - yvx) 2
j(Yvz
- zvY)
2
W2,z =
170
(C.76)
(C.77)
Partial derivative Of W2,x
aW2,X,
zvY) [vY (xv
[ (yvz -zV Y)2 + (ZVX
yVZ
_
ax
aW2 ,x
[(yvz
-
zVY) 2+ (ZVX XVZ) 2 ± (XVY - yvx) 2 ]
)2]1/ 2/
VZ) 2 + (XVY -yVx
- zvY) 2 + (ZVX
Vz [(YVz
ay
- xvI)]I
Vz
XV) 2 + (XVy -yVx )2]31/2/
YVx)
-
-
-
-
K(YVz
-zVY)
[(yvz
-VY
[VZ (yvz - ZVY) - VX (xvY - yVx)]
2 3 2
2
2± (ZVX - XVZ) + (XVY - YVx ) ] /
az~
ZVY)
-
2
aW2,x
avx
___x
\
+-(ZVX
-V)
+
-
(ZVX
-
zVY) [z(zvx
~- (
~(YVz - zvY) 2 + (ZVX-
2
XVZ)
-
zVY)[Ivx(zvx
[(yvz
av
2
K(yv
(yv
=
2±
Z _XVZ)
- Y2+
KVy[ (Yvz
aW2,
xvz)
XVZ)
xVz)
-
-
2
-
x
(XVY i
+ (XV y
3
YVx ) ] /
2
-
Vy (YVz
+ (XVY
Y(XV
XIVz) ± (XVY
2
zvY)]I
-
2 3 2
] /
-yVX)
yYvx)]I
2
yVx)
2
]3
1
yVx) 2 ]
XV) 2 ± (XVY
-Z[(YV -zzV Y)2 ± (ZVX
2
2
yVx) 2 ] 31
XVz) + (XVY
[(yv z -ZVY) 2 + (ZV,
/
(C.78)
-( yVz-zv)[X(XVy-YVxVZ(YVz-ZVy)l
2
+ (XVY - yx
zV)2+ (ZVX - XVZ)
-
KY[(vz-zV
avX
-
(YVz
[ (YVz
-
+
(ZVX
-
XVZ)
-ZVy)[Y(YVz
±V)
=
[0]
aW2,x
-
[0]
2
-zvy)
(ZVXXVZ)
aua
aw 2,x
2 ±+(XVY
Y) 2±+(ZVX -XZ)
zvY) 2
K(
[(yvz
au,
/
/
/
[ (yvz
aw,
/
a tf
171
+ (XVY
-x(zv
2 + (XV
-
_x) 2
yVx)
-
v
2
]
3
2
/
xv)]
X) V2]312/
Partial derivative Of w 2 ,y
____y
_
ax
Vz [(yVz -ZVY) +(ZVX -XVZ) 2 +(XV-yVx)2
~ [(yvz - ZV y)2 +±(ZVx - wz) 2 ±+(XVYy-yVX)2]I2/
/(zvx
-
xvz)[Ivy (xv
\(yvZ -zvY)
aW2,y
_
ay
/
az
[(yVz - ZV y)
f\zvx
\(yvz
2
-ZVY)
-
-
2
-
± (zv,
xvz) [V (yvz
[(y~z - ZV y)I +(ZVX
Kvx [(yVz
aW2 ,y
2
-
yvx)
xVi) ]
XV) 2± (XVY - yVx) 2 P/3
avx
-
/
zvy) - vx (xvY - yvx)]
VZ) 2 ±+(xvy
-yv )2]3/2/
+ (ZV
2
XV)
2
+±(ZVx -XVZ)
xvz) IV,,(zvx
±V) (ZVX
+ (XVi - yVx) 2 ]
±+(XV
yv)2]32/
xvz) - Vy (Yvz
-XV) 2 ± (XVi
z [ (YvZ - zvY) 2 + (ZVX ~(yv 2 - ZV )2 ± (ZVX -XVZ)
aW,<y
V Z
-
±V)
2
K(zv
(XVY
±+ (XV
-
-
ZVO)I
YVx
)2 ] 3 /2
/
yVx) 2 ]
/
y yVx) 2] 3 /2
-xVY 2 [z(ZVX XVZ) 2Y+(XV yYVx) 2]3/
-
(V
K
avy(yvz
K
avz
aW2 ,y
aW2,y
±V) (ZVX
-
/(zvx
ZVy ) 2
-
2
XVZ) 2 + (XVy -yVx) 2]
+ (ZVX
+ (ZVX
-
xVz) [Y(YVz
\(YV z- ZVY) 2 + (ZVX
,
[0]
-W
,
[0]
iTou
20]-
10
au"
-W,
[0]
-W,
[0]
au,
at
YX
-z
xII(yvz - zVY)
-W
-
xVz)Ix (xvY - YVx) - z (yv zV
I
- ZV y)2 ± (ZVX - XVZ) 2 + (XV
~yVX) 2 ] 3 /2
I(yv -
act
± (XVY
XVZ)
172
XV)
2
+ (XVY -
zvY)
-XVZ)
-
2
yvx )2]3/2/
x (ZVX - XVI) ]
± (XVY~ - yV
]/
/
(C.79)
(.9
Partial derivative of w2,,z
DW2,z
ax
~[(yvz - zv~) 2 + (ZVX -
K(xvi
[(yVz
aW2,
XV) 2
-
- zVY) + (ZVX -XVZ)
2 ±
XV)
(ZVX -
K(xvY
~~(YVz -ZVX)
3
2 +
az
(I (YvK
2 ±
-zvY)
-
(ZVX
±VY
2
Y I(YV -zVy)
2 +
[(YVz - zvY)
____z
avx
K
aW2z
____
-
XVZ) 2 + (XVi
-
XVZ) 2 + (xviy
XVZ)
(ZVX
+
(ZV,
2 ±
- XVZ)
yx)]/
YVx )2]3/ 2
+ (XVY
(XVX
-
yVx)
2
]
yvx )2]3/2
)/2
-XVZ)
(YVz
K
K
avz
K
aW2,
2
zV ) 2±+(ZVxXVZ)
yv) Iy(xvz
(xvi-Y
(Yvz
zvY)
-
-
2
±
-
yv)
(ZVX - xV)
101(~z
2
z~
a[0]
aor
aux
aW 2,z
2
-
yYVx)I
-
xVz) -y (XVy YVx) Y -Z)2+(V XZ2+(V y)]3/
(xvi - yvx) Iz(zvx
2 + (XVY _ yvx)2 ] 1 /
±VY (ZVX
[(YVz 2
(C.80)
(XVy YVX)l
zV y 2 (ZV -XVZ)
[(YVz
aW 2 ,z
(ZVX
(XVy - yvx) 2 ] 3/ 2/
- xvx
-yvx)[Vz (yvz - zvY)
[(Yvz
yv )2]3/2/
(XVY -
±
XVz)]
YVx)[Vy (Xvy -YVx) -v2 (zvx
2 31 /
2 + (ZVX - XVZ) 2 ± (XV,
]
yVx)
ZVY)
2+ (xv , -yVx) 2]
2
VxK -vI(yvz
ay
2]
Kv[(yVz - ZVY) (~XV,) 2 ±+(XVYyX)
[ 0]
au,
a tj
173
+ (XVY YVx)
- x(z
+
- xzv)]
(XVY - yx
~z~-x)
2 3 2
]
Partial Derivatives of Unit Direction w 3
W3
=
(C.81)
Wi X W 2
(wl,yw2,z - wi,zw 2 ,y)ex
+
(wl,zw2,x -
wi,xw 2 ,z)ey +
(w,xw
2
,y
-
y
Partial Derivatives of W3,x
(C.83)
W3,x = Wl,yW2,z - Wl,zW 2 ,y
aw3,x
ax
-
Ow3,x
-
ay
aw3,x
az
ax
wi,y
W2,z
Wly
ayWi,y
-
-
ax
w2,y W,
-
ay
avx
avy
Ovx
-
aW3,
Ovz
au,
aw3,x
a t;
vx
w2,z + aw 2 ,zvywi,y
w2,z +
w2 ,
vy
aw-
Wl,y ~
avx
avx
W2,y
vx
-
vy
w2,y -
aw
awl
wl,y -
vy
vx
vy
wl,z
wl,z
_w2,y
w
2 ,y
vy
Wl,z
(C.84)
[0]
auo
aw,x
aw3,x
au"
aw3,x
W2,z ±
-wl2y
aws,x
w l ,z
wl__z
___wly
-
wl,z
[0]
=
[0]
[0]
[0]
Partial Derivatives of w 3 ,y
W3,y = Wl,zW2,x - W1,xW2,z
174
(C.85)
aW 2 ,x
aw3,y
aw2,2
ax
ax
ax
aw 2,2
_wzx
ay , 1,z
ay
aW3,y
~ ay
ay
ay
az
W1z-
W2,x
aW3,y
avx
avx
aW3,y
av,
avx
=
W1,x
+
W2,x
aw
2 ,x
1, aW
+ [W1,z
z
'W2,x
+
W2,x
vy
aw,
-
-W1,z
vx
v,
WWx±
aw1
avx
avz
aW2,z
W2,x
-~
aW3,'y
W1,X
vx
_____
NN2,z
-
a
vy
W1,
~
w1,-W2,z
vy
vx
W,
w1,x
W2,z
W1,x
W2,z~
-
w1,x
W,
v,
-
W2,z
vy
w1,x
(C.86)
aW3,y
aua
aw3,y
a o-
aw3,y
au,
=
[0]
= [0]
= [0]
aW3,y
au,
a3,y
= [0]
at5
Partial Derivatives of
W3,z
W3,z
= Wl,xW2,y - W1,yW2,x
175
(C.87)
aW2,y Wx
aW3,z
ax
aW3,z
W2,.x
-
ay
-w1,x
ay
aW3,z
aW 2 ,y
-
az
3
ax
- W2,y Wx
ay
aW
- W2,x W~
ax
-
,z
avx
-aw
=
=
[0]
=
[0]
=
[0]
=
[0]
aw3,
ay
+
1,~~
avxW,
w~
avx
W2,xwy
-
ay
aW1 ,x
W
vx
v
+
w2,y +
w1,x
W2,x
-
-
i~
vi;
WJ~
vx
vx
wi,x
W ,
-
vy
vy
w2,x
-
w 2,x
-
Wi,y
vY
aW2,x W~
(C.88)
vi; wiy
[0]
aty
Objective Gradient
Fgrad
C.2
aF
[
aF
aF
aF
aF
aF
aF
aF
aF
aF
ay aZ avx avy
Ovz
o au" au- atff
aof a&
1 (C.89)
Objective Gradient
N
F = tf 2-to
-
Wk'k
2 k=O
Fk1 0max /2!
/2
2
+
k2 (
Ua,k
(Uamax)
176
±
k3
Uo-,k
\Uo-,max
2
(C.90)
aF
aF
=
[0]
=
[0]
=
[0]
aF
aF
[0]
aF
aF
av,
aF
aua
aF
aua,
aF
=
[0]
=
[0]
=
(tf - to)ki
=
[0]
=
(t5 - to)k
(C.91)
w
Oax&
Uxl
w
2
L
aor
=
(t5 - to)k 3
x,max
w
o-,max
L
1
2 10
Wk
k1
\
a
-
Umax /2!
2]
2
2
+k2(Umk)
177
+
k3
( Uo-,max)
U-,k
[This page intentionally left blank.]
Appendix D
Initial Guess
In order to obtain a solution it is necessary to provide the optimization algorithm
with an initial guess. The closer the guess is to the optimal solution, the less time
it takes the optimizer to find a solution. However, if a poor initial guess is given,
the optimizer may not even converge to an optimal solution. It is often difficult
to generate an initial guess for a problem that is being considered for the first
time, especially for complex problems.
A different formulation of the CAV mission design problem studied in this
thesis was actually solved prior to the work completed in this thesis [201. The
differences between the problem statements stem from the fact that Ref. [20]
formulates the equations of motion in spherical coordinates. The initial guess
was taken to be a converged solution computed by applying the Legendre pseudospectral method described in this thesis and SNOPT. The solution uses the
same constants stated in Chapter 4 and Table D.1 lists the values of the weighting factors used.
Before the converged solution from using spherical coordinates is fed into
ki
k2
k3
.047597
19.828991
17.846091
Table D.1: Values Used to Generate an Initial Guess
179
the optimizer, the solution must be transformed to ECEF Cartesian coordinates.
Coordinate Transformations
Let r = (x, y, z) and v = (vx, vy, vz) denote the ECEF Cartesian position and
velocity, respectively, of a vehicle. In spherical coordinates the position vector
is defined by the geocentric radius r, the Earth relative longitude 0 measured
East from the Prime Meridian, and the geocentric latitude <p measured positively
North from the equatorial plane. Using Fig. D-1 it is evident that the position is
e2e
0
ex
Figure D-1: Spherical Representation of Position with Respect to a Cartesian ECEF
Coordinate System
transformed as
x
= rcos<cos0
(D.1)
y
=
r cos <p sin 0
(D.2)
z
=
r sin <6
(D.3)
The velocity vector is defined by the Earth relative speed v, the Earth relative
flight path angle y, and the heading angle (p. The flight path angle is the angle
between the plane passing through the vehicle that is perpendicular to the position vector (local horizontal) and the velocity vector. When the velocity vector is
above the horizontal, y is positive. The heading angle is positive in the eastward
180
direction. In order to transform the velocity vector, three additional unit vectors
er, eo, and ep are defined as
r
er
r
eo
ee
(D.4)
ez x r
Ilez x r112
(D.5)
x eo
(D.6)
=er
As shown in Fig. D-2 , the velocity vector can then be written as
Y
e,
Figure D-2: Spherical Representation of Velocity with Respect to a Set of Axes
Defined in the Cartesian ECEF Coordinate System
v = v sin yer + v cos y cos peo + v cos y sin qieo
(D.7)
The resulting transformation matrix from spherical coordinates to Cartesian co-
ordinates is denoted by Ts2c where
Ts2c =
[ er eo e+ ]
(D.8)
The velocity is transformed as
L
v sin y
vx
vy
vz
=
J
Ts2c
v cos y cos (P
v cos y sin (p
181
(D.9)
[This page intentionally left blank.]
Appendix E
Earth Relative Downtrack and
Crosstrack
This appendix defines the Earth relative downtrack and crosstrack distances. Let
ro and vo be the initial position and velocity of a vehicle expressed in Cartesian
Earth-centered Earth-fixed (ECEF) coordinates. The three orthogonal unit vectors,
u 1 , u 2 , and u 3 , comprise the downtrack-crosstrack coordinate system and are
defined as:
The u1 -u
2
U1
=
U3
=
U2
=
ro
(E. 1)
ro x vo
(E.2)
Ilrollz
||ro X Vo0ll2
(E.3)
u3 x u 1
plane is the Earth relativedowntrack plane and the u 1 -u
3
is the Earth
relative crosstrack plane. As shown in Fig. E-1, the downtrack angle is denoted
by a while the crosstrack angle is denoted by b. Let r 12 be the projection of the
position vector in the Earth relative downtrack plane and r 3 be the component
of the position vector in the u3 direction. The downtrack and crosstrack angles
183
U3
U2
U1
Figure E-1: Earth Relative Downtrack Plane and Earth Relative Crosstrack Plane
are computed as follows:
(b
riz
a
= arccos
b
=arctan( 11 r31 1 1)
-
rr 2rIIU
2
(E.4)
(E.5)
From these angle, the Earth relative downtrack distance d and Earth relative
crosstrack distance c are then given as
d
=
Rea
(E.6)
c
=
Reb
(E.7)
where Re is the radius of the Earth.
184
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