A Data-Driven Approach to Online Flight Capability
Estimation
by
Marc Alain Lecerf
B.S.E., University of Michigan (2012)
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 INS1T ITTE
OF TECHNOLOGY
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2014
JUN
LIBRARIES
@ Massachusetts Institute of Technology 2014. All rights reserved.
Signature redacted
Author...................
-
Department of Aeronautics ankstronautics
May 22, 2014
Signature redacted
C ertified by ........
.....----- ---------.
Karen E. Willcox
Professor of Aeronautics and Astronautics
Thesis Supervisor
Signature redacted
Accepted by.................
-----------Paulo C. Lozano
I
Associate Professor of Aeronautics and Astronautics
Chair, Graduate Program Committee
2
A Data-Driven Approach to Online Flight Capability
Estimation
by
Marc Alain Lecerf
Submitted to the Department of Aeronautics and Astronautics
on May 22, 2014, in partial fulfillment of the
requirements for the degree of
Master of Science in Aeronautics and Astronautics
Abstract
Similar to a living organism, an autonomous vchicle benefits not only from
awareness of its surrounding environment and mission directives, but also from
awareness of its performance capability. Because this degrades over time due
to fatigue and acute damage, onboard logic often uses conservative estimates of
performance from the initial vehicle design to plan feasible mission trajectories.
We develop an approach for dynamically estimating vehicle capability to
enable safer and more efficient mission planning. The approach leverages
multi-level vehicle models in an offline phase to construct a library of information capturing the vehicle behavior in damage scenarios; the behavior
is discovered via data-driven classification techniques. After construction, the
behavioral library is stored for future queries online by an agent making timeconstrained decisions. The research directly links onboard vehicle sensor measurements with an estimate of the current vehicle maneuvering capability using
the stored behavioral library.
The end-to-end process is implemented and demonstrated in an example
flight scenario where an aircraft sustains structural damage to its wing. Safety
is assessed based on composite material failure allowables, representing damage
to the wing via a local loss of material stiffness. Damage scenarios on the wing
are simulated and stored for query during the flight scenario, where knowledge
of the maximum maneuvering load factor is estimated using structural strain
sensor measurements. Results indicate both an increase in probability of success as well as an increase in overall usage of the vehicle capability, compared
to the baseline case that does not dynamically update the capability with
onboard sensor information.
Thesis Supervisor: Karen E. Willcox
Title: Professor of Aeronautics and Astronautics
3
4
Acknowledgments
I would first like to thank all those who supported me in undertaking this
research.
My advisor, Dr. Karen Willcox, has been an inspiration and guiding
force for me through the academic endeavor of engineering research as well as
through the often equally important endeavors toward personal accomplishment, self-confidence, and a sense of earnestness in all of life's travails. My
sincerest thanks and gratitude goes out to her.
I send my heartfelt thanks to Dr. Ella Atkins, who was a mentor and a
source of guidance during my decision to go boldly forth to begin this research
at MIT.
I would like to thank the members of the DDDAS project. At MIT, thank
you to Laura, Demet, and Doug for their support in this work; and at Aurora
Flight Sciences, thank you to David and Jeff for providing their expertise and
guidance in the realm of aircraft structural composites (and all things practical
about aircraft modeling).
I would also like to acknowledge the funding for this research, supported by
AFOSR grant FA9550-11-1-0339 under the Dynamic Data-Driven Application
Systems (DDDAS) Program (Program Manager Dr. Frederica Darema).
An aerospace graduate student's life at MIT is spent often in the labyrinth
of externally concrete, internally leaking, yet somehow destruction-resilient
research laboratory space we call home. I express my warmth for all the
members of the Aerospace Computational Design Laboratory who shared this
abode with me, both during our peaceful hours in the confines of Building 37
as well as during our less peaceful hours climbing stairs to prepare for Tough
Mudding with Karen!
Beyond the laboratory space, my experience will forever remain a gem for
the friends I have made at the Institute. Festivus Miracles, you hold the deed
to a carefully tended vineyard in my heart, continuing to produce rich bottles
of wine that will only ripen and grow in complexity with age. To Margaret,
you are a radiance in my life, and this thesis bears the fruit your dedication
and support has sowed.
Lastly, I want to express the breadth of my love for my family, who has
empowered me to sculpt my identity and to make them proud. To my mother,
father, and my sister Danielle, you are my persistent and guiding role models.
I know your love has been always near and this work is testament to your
presence in my life.
5
6
Contents
1
1.1
1.2
1.3
1.4
2
3
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
13
13
15
16
16
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
19
19
20
21
21
22
23
24
26
28
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
29
29
31
31
32
36
37
37
41
42
Introduction
Origins of Dynamic Flight Capability Estimation
What is Capability Estimation? . . . . . . . . .
Research Objectives. . . . . . . . . . . . . . . .
Thesis Outline. . . . . . . . . . . . . . . . . . .
Methodology
2.1 Offline Phase . . . . . . . . . . . . . . . . . .
2.1.1 Step one: characterize system . . . . .
2.1.2 Step two: classify behavior . . . . . . .
2.1.3 Step three: construct library . . . . . .
2.2 Online Phase . . . . . . . . . . . . . . . . . .
2.2.1 Notation and assumptions . . . . . . .
2.2.2 Inference using the maximum likelihood
2.2.3 Inference using a mixture distribution .
2.3 Sum m ary . . . . . . . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
Aircraft Capability Model
3.1 UAV Aircraft Design . . . . . . . . . .
3.2 Aircraft M odel . . . . . . . . . . . . .
3.2.1 Model configuration . . . . . .
3.2.2 Model validation . . . . . . . .
3.2.3 Lumped damage representation
3.3 Wing Box Model . . . . . . . . . . . .
3.3.1 Local damage representation . .
3.3.2 Integration of VABS with ASWING
3.4 Integrated Aircraft Capability Model .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
4 Classification, Application, and Results
4.1 Discovering the Capability Set Boundary via Cl assification
7
45
45
4.1.1
4.1.2
4.1.3
Support vector machines . . . . . . . . . . . . .
Probabilistic support vector machines . . . . . .
Capability estimation using probabilistic support
machines ........................
. .
4.2 Online Aircraft Capability Estimation . . . . . .
. .
4.2.1 Flight scenario . . . . . . . . . . . . . .
. .
4.2.2 Library damage cases and maneuver bou nds . .
4.2.3 Visualizing the library . . . . . . . . . .
. .
4.2.4 Flight scenario test cases . . . . . . . . .
. .
4.2.5 Online strain gage measurements . . . .
. .
4.3 Results . . . . . . . . . . . . . . . . . . . . . . .
. .
4.3.1 Comparison of estimator outputs . . . .
. .
4.3.2 Flight scenario performance benchmark .
. .
4.3.3 Lim itations . . . . . . . . . . . . . . . .
. .
5 Conclusion
5.1 Summary of Results and Current Work . . . . . . .
5.2 Future W ork . . . . . . . . . . . . . . . . . . . . . .
8
. . . .
. . . .
vector
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
46
50
52
55
55
56
57
61
63
66
68
75
82
85
85
86
List of Figures
2-1
Offline methodology
. . . . . . . . . . . . . . . . . . . . . . .
20
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
. . . . .
(VABS)
. . . . .
. . . . .
. . . . .
. . . . .
30
31
33
34
35
36
UAV concept for capability analysis . . . . . . . . . .
UAV concept represented in ASWING . . . . . . . . . .
ASWING configuration for pull-up maneuver analysis .
ASWING pull-up maneuver analysis varying airspeed .
ASWING pull-up maneuver analysis varying load factor
Lumped representation of damage in ASWING . . . . .
Variational Asymptotic Beam cross-Sectional Analysis
flow chart . . . . . . . . . . . . . . . . . . . . . . . . .
3-8 VABS damage representation demonstration . . . . .
3-9 Wing cross sectional finite element model for VABS . .
3-10 UAV capability model coupling VABS and ASWING . .
3-1
3-2
3-3
3-4
3-5
3-6
3-7
4-1
4-2
4-3
4-4
4-5
4-6
4-7
4-8
4-9
4-10
4-11
4-12
4-13
Description of a linear SVM discriminant . . . . . . . . . . . .
Refinement of PSVM vehicle capability estimate . . . . . . . .
Adaptive SVM convergence history . . . . . . . . . . . . . . .
Schematic of the flight scenario to which we apply our capability
estimation framework. . . . . . . . . . . . . . . . . . . . . . .
Offline library damage cases . . . . . . . . . . . . . . . . . . .
Capability analysis region of aircraft maneuvering V - n envelope
Damage case PSVM countours . . . . . . . . . . . . . . . . . .
Visualization of vehicle behavioral library with representative
................................
cases ........
Choice of example damage cases . . . . . . . . . . . . . . . . .
Strain sensing locations on aircraft wing box . . . . . . . . . .
Explanation of the probability curve plots used for analysis of
qML and qMD- The vehicle state is x = (V, n), however V stays
fixed at 210 ft/s so we need only plot the value of n. . . . . . .
qML output varying sample accumulation NS . . . . . . . . . .
qMD output varying sample accumulation NS . . . . . . . . .
9
38
40
41
42
47
54
54
55
57
58
59
60
62
64
67
69
70
4-14 Library down-sampling procedure controlled by DSR . . . . .
4-15 qML output varying library down-sampling ratio DSR . . . . .
72
73
4-16 qMD output varying library down-sampling ratio DSR
. . . .
74
4-17 How an agent uses the capability estimation output via Pop .
4-18 Samples of decisions made based on the dynamic and capability
76
estimation strategies, varying pp and n'tatic . . . . . . . . . .
78
4-19 p (MS)-n-,,tj trade-off curves for flight scenario decision strategies using the static and dynamic capability estimates . . . . .
81
10
List of Tables
4.1
The five representative damage cases used in the flight scenario
to test capability estimation performance. . . . . . . . . . . .
61
4.2
Truth reference nmax values for five example damage cases
. .
62
4.3
Damage case strain values for the fixed flight scenario maneuver
Numerical results obtained from the p (MS) - nuti, tradeoff
curves for the decision process using the static and dynamic
capability estimation strategies . . . . . . . . . . . . . . . . .
4.4
11
65
82
12
Chapter 1
Introduction
Modern aerospace vehicles are becoming increasingly independent of human
interaction in-flight. Following this trend, a spectrum of technologies exist
ranging from unmanned-vehicles that do not require a human operator in an
airborne position but may require real-time interaction remotely via a pilot on
the ground-to autonomous-vehicles that are able to make in-flight missioncritical decisions and react to stimuli in the environment. A critical method
for furthering autonomy is to produce self-aware systems: not only can these
systems plan and operate independently of human operators, they are also
able to quantify the state of their available internal resources and maintain
knowledge of their current health beyond their initial baseline performance [1].
In this way, the system mimics behavior of a biological organism-it can act
aggressively when it is healthy and in favorable conditions, and can become
more conservative as it ages and degrades.
In order to dynamically assess vehicle capability and use it to support
autonomous operation, we need to develop an estimation process that can
translate measurable quantities directly into capability quantities of interest.
Especially as vehicle dynamics become increasingly complex, modeling this
relationship is non-trivial when the computation needs to run within time
constraints dictated by free flight.
1.1
Origins of Dynamic Flight Capability Estimation
The methods and tools for dynamic capability estimation have emerged from
an intersection of work in both the vehicle damage detection and vehicle design
communities.
13
Disciplines such as Operational Loads Monitoring (OLM) have worked to
improve the detection of damage and fatigue in vehicle structural members.
In OLM, on-board aircraft sensors gather structural loading information to
identify damage and fatigue (most often post-flight) in order to reduce maintenance costs and increase reliability [2, 3]. On a broader systems level, the
Integrated Vehicle Health Management (IVHM) field searches for frameworks
that incorporate multiple sources of operational data, physics-based models,
and prognosis techniques [4, 5]. Modern IVHM architectures have been developed at NASA [6] and the Department of Defense [7]. Damage and fault
tolerance are now becoming an important component of real-time software architectures for monitoring aircraft component health, such as that proposed by
the ONBASS project [8]. IVHM has also begun to enter the initial aerospace
vehicle design where unit costs are high, and optimization techniques have
been explored to improve IVHM architectures [9].
In addition to systems-level health monitoring, there has been active work
on the vehicle component level, particularly in structural composites. Structural Health Monitoring (SHM) using statistical inference techniques has seen
active progress. Ref [10] presents a broad survey of the SHM field up to 2001,
and recent work has approached damage detection problems using pattern
recognition techniques [11]. Damage identification based on structural vibration data has had particular success [12], where changes in vibration modal frequencies often denote acute material degradation. More recently, high-fidelity
modeling can enter into the damage identification and health management
control loop-candidate models of system behavior can be weighted based on
real-time data, and actions can be performed to increase estimation confidence,
as well as actions to "heal" the system given current damage estimates [13].
However, work still remains to connect damage parameter identification
to the online estimation of quantifiable vehicle capability. There is a need
for global metrics used during the design phase-that drive the performance
requirements of the vehicle-to be tracked and updated throughout the vehicle's lifetime. Standard design principles for aircraft operate on systems-level
analyses such as the V - n diagram [14], where large margins of safety (often
based on empirical evidence and experience) are substantial drivers for system
efficiency. Recent work in condition-aware aircraft maneuverability [15] is
advancing this connection, however open questions exist in how to integrate
recent advances in local damage identification with updates to global aircraft
performance metrics-forming this connection will improve usage of assets
through their lifetime and could enable designs that rely on their dynamic
usage in the presence of degradation.
14
1.2
What is Capability Estimation?
The word capability has a broad definition across engineering disciplines. In
our context, we want to develop a quantitative definition so we can apply
mathematics to the process of estimating capability for a given system.
To begin however, we must first define the concept of a system's state.
We parallel the standard approach taken by the control systems community,
where the state is a minimum number of quantities that, when considered
together, can uniquely specify all possible configurations of the system [16].
To illustrate this concept, we describe three example systems and possible
ways to characterize their state spaces:
1. A valve, with a single discrete state variable that designates whether it
is "open" or "closed."
2. An audio speaker "cone" that produces sound waves via linear resonation, with its state quantified by its position and linear velocity.
3. A rigid-body aircraft in free flight and motionless air, with its state quantified by the following degrees of freedom of a reference frame attached
to a fixed point on its body:
e position and velocity with respect to a fixed Earth frame
* rotational orientation and velocity with respect to a fixed Earth
frame
Using this state vector with physical laws, we can predict how the motion
of the aircraft will evolve in time due to inertial and aerodynamic loads;
hence, its current value also uniquely identifies the aircraft configuration
for future time instances
Now, we define the capability of a system quantitatively as the set of state
vectors that satisfy constraints due to system properties or properties of its
surrounding environment. We will also use the term capability set interchangeably to reinforce the notion that the system capability is a set-valued quantity.
For our three examples, the capability set could be interpreted as follows:
1. The capability set for a valve is normally both positions "open" and
"closed"-however due to a malfunction, the valve could be stuck in
one position, and its capability set would shrink to a singleton with this
position as its only element.
2. Due to actuator saturation and structural durability constraints, the
audio speaker cone could have maximum displacement and maximum
speed limitations during operation; the capability set would be the pairs
of displacement and velocity bounded by these constraints.
15
3. Due to airframe structural failure and aerodynamic stall over the wings,
the aircraft could have constraints designating safe airspeeds and angles
of attack with respect to the local air mass. The structural constraints
could vary due to both local damage events, or due to changes in the
environment such as temperature.
While our definition of capability is general, we focus in this research on quasistatic vehicle capability. Our methodology and application aim particularly
at a system with continuous state variables, without explicitly analyzing the
dynamic evolution of said state variables in time. However, the vehicle operates
with time constraints on computation during operation that make high-fidelity
modeling difficult, so we remain cognizant of the computational requirements
of our approach.
1.3
Research Objectives
The broadest goal of this thesis is to present a method for performing online
vehicle capability estimation leveraging computationally-intense, offline vehicle
behavioral modeling. This goal is segmented as follows:
1. Develop a method for computing a library of system behavior using
physics-based models of the vehicle behavior in loss-of-capability scenarios.
2. Develop a computationally efficient technique for estimating vehicle capability directly from noisy sensor information leveraging a pre-computed
library of system behavior
3. Demonstrate the use of online vehicle capability estimation by an agent
in a scenario where the vehicle degrades, and compare its performance
to the case where the agent only knows the nominal capability given by
the vehicle design.
1.4
Thesis Outline
Chapter 2 introduces the methodology for using offline physics-based modeling
to build a library of vehicle behavior, and for leveraging the library online to
estimate vehicle capability.
Chapter 3 develops a representative UAV model that can capture its behavior in the event of structural damage to its wing.
16
Chapter 4 applies the methodology from Chapter 2 to the representative
UAV from Chapter 3, presenting the algorithms used to implement the capability estimation process and results from a relevant decision-making process.
It compares results to a baseline case that uses no active capability estimation,
with detailed discussion.
Chapter 5 provides a summary of results, draws conclusions, and suggests
directions for future improvement.
17
18
Chapter 2
Methodology
Our approach to flight capability estimation relies on a decomposition of computational effort between offline and online phases. The offline phase occurs
before operation of the system of interest, when we assume we are able to
leverage powerful computational environments that have relaxed execution
time and storage constraints. The online phase refers to the real-time (or simply time- and memory-constrained) parts of system operation, when embedded
computation needs to be lightweight. We utilize complex physics-based models, experimental data, and other sources of information about the system in
the offline phase to build approximations of the system behavior; the approximations can then run in the online phase to improve performance, for example
by informing priors on quantities of interest or by enabling reduced-order models of the system trajectory.
2.1
Offline Phase
Figure 2-1 presents a functional decomposition of the offline phase of our
methodology for estimating vehicle capability. The process is broken into three
stages: characterization of the vehicle using models and/or experiments, classification of vehicle behavior based on failure modes, and storage of these classifiers as records in a behavioral library. The following sections step through
these stages in further detail.
19
1. Characterize System
Failure
Metrics
f
State Vector
2~
2. Classify Behavior
True Capability Set
X2
00 0
Models,
ExperimentsI
M
Loss-of-Capability
Parameters
d
(a)
0
(unkn O)
00
Approimation
%(classifier c)
_Observable
Vector
a
3. Construct Library
Observable
Probabilistic
Classifier
j
Vector
Parameters
1
2
3
s1
82
83
cl
C
C
mO 0
*
~
,/0
(b)
(c)
Figure 2-1: The three steps in the offline phase for building a library that can be
queried in the online phase.
2.1.1
Step one: characterize system
Figure 2-la shows the first step-the user begins with vehicle system models
and/or experiments that represent the vehicle behavior. They have two inputs
and two outputs, the definitions of which are as follows:
* a state vector x E X, or any quantities that specify the configuration of
the vehicle before considering changes to capability. For a maneuvering
aircraft, x could be the kinematic state vector-for instance, in Chapter 4
we consider an aircraft in steady flight with a state quantified by an
airspeed and a wing load factor, where X = R2
" loss-of-capability parameters d E D, or any quantities that specify how
the vehicle could become modified such that its capability set would
change-examples could be parameters describing structural damage, or
parameters describing available system resources such as battery levels
or fuel stores.
* A failure metric f : X xD -+ R that measures how close the vehicle is to
undesirable or unpredictable behavior-examples could be closeness of
structural loads to maximum thresholds, or closeness of available system
resources to minimum safe levels.
" an observable vector s : X x D -+ S of quantities that would be available
online to provide information about the vehicle state- for instance, in
Chapter 4 we consider an aircraft with Ns continuous strain measurements provided by embedded wing sensors, where S = RNs.
20
2.1.2
Step two: classify behavior
The models and experiments produced in step one allow us to model the vehicle
capability set as follows: if we represent a constraint on the vehicle behavior as
an upper bound on the value of the failure metric f(x, d) for any input state x
and loss-of-capability parameters d, then for a fixed value of d, the capability
is the set of "safe" x's whose f values lie below this limit.
More precisely, let Kf be the upper bound on f that represents a constraint
on the vehicle behavior. Then the capability set C is a function of d as follows:
C (d) = { E X : f(x, d) < Kf}
(2.1)
This characterization of C as a set-valued quantity looks mathematically simple, but it is difficult to use in practice. To make the problem tractable, we
use a sampling-based classification technique to approximate C; the technique
is represented conceptually in Figure 2-1b for a two dimensional, continuous
state space X characterized by the coordinates (X1 , X 2 ). For each fixed value
of d, we generate samples from X and label them as "safe" (grey) or "unsafe"
(light grey) based on whether they satisfy or do not satisfy, respectively, the
predicate for set membership in C given by expression (2.1). Then, using
the labeled samples, we train a classifier that is used to designate new query
state vectors as "safe" or "unsafe." Because the classifier is trained off a finite
set of samples, it can only approximate the true underlying capability set to
some finite accuracy-but once we have the classifier trained, we could in theory use it to classify every point in the state space; this would produce the
approximation of the capability set as shown (notionally) in Figure 2-1b.
We consider the possible discrepancy between our classifier and the true
capability set a form of model error, and we introduce uncertainty to represent this error. In particular, we train a probabilistic classifier for each value
of d to evaluate the probability that a query state vector belongs to C(d). We
perform the probabilistic classification for a given input x using the "probabilistic capability classifier quantities" c(d). For example, we implement this
form of classification using a Probabilistic Support Vector Machine (PSVM)
in Chapter 4, where c(d) contains quantities such as support vectors, weights,
and distribution hyperparameters.
2.1.3
Step three: construct library
In step two, we approximated the capability set for each value of the loss-ofcapability parameters via sampling-based classification. Now, by combining
21
the samples produced from these runs, we produce a library of records containing the following features:
"
"
"
"
*
x, a value of the system state vector
d, values of the system loss-of-capability parameters
f, a value of the system failure metric
s,a value of the system observable vector
c, values of the probabilistic capability classifier quantities
We let R represent the number of records in this library, and we assign subscripts to denote these features for a given record
j
=
1,
...
,
R as xj, dj,
fj,
sj, and cj.
As a final, third step, we store this library for later queries in the online
phase, as represented in Figure 2-1c. As we will show in the next section,
the only features necessary for queries in the online phase are the observable
vector s and the probabilistic classifier parameters c; the vehicle state and the
loss-of-capability parameters are "hidden data" that were necessary only for
modeling of the vehicle behavior. Essentially, our stored library will contain
records that provide a direct link from vehicle observable quantities to vehicle
capability.
2.2
Online Phase
In the online phase, the user directly infers the vehicle capability from an input
sample of the vehicle observable vector, by use of the stored vehicle behavioral
library. There are two intertwined classification steps involved:
1. The observable vector sample is used to classify the current vehicle behavior into cases represented in the library. We formulate this classification in a Bayesian sense, where the goal is to minimize the probability
of misclassification.
2. Using the probabilistic classifiers that were pre-computed and stored for
each record in the library, the user retrieves the probability that a query
vehicle state lies within the current capability set.
The following sections introduce the process in a mathematical sense. We
begin with a description of relevant notation, and then we present two possible
methods for the inference process: a maximum likelihood formulation, and a
mixture distribution formulation. The performance of these two methods will
be compared later in our aircraft application in Chapter 4. Because the online
22
phase needs to be cognizant of available computational resources, we analyze
the complexity of each method and discuss its implication with respect to the
method's practical usability.
2.2.1
Notation and assumptions
As we will be working with probabilistic quantities, our convention is to denote
random variables or vectors using serifed letters (e.g. a, b, and s), and to use
shorthand, where values taken by random variables are represented by corresponding unserifed letters (e.g. a, b, and c). We represent probability mass
and probability density functions as p (), where the corresponding discrete
or continuous case will be clear from context. We represent the expectation
operator as E [.]. In the event the random variable shorthand is ambiguous, we
will revert to a subscript notation, so for example Pa (a) and p (a) both represent the probability (or probability density, if a is continuous) that random
variable a takes the value a.
We assume quasi-static vehicle behavior, where for any instant in time
the vehicle state takes some value x C R'. By definition (2.1), the vehicle
capability is a set C C R'.
The models and/or experiments from the offline phase allowed us to build
a library of information about the vehicle behavior. Here, we refer to each
library record as representing a vehicle behavioral case, or a distinct shape of
the vehicle capability set; note this is distinct from the vehicle state.
The notation for features of each record in the library follows that from
section 2.1.3, where the Jth record for j = 1, ... R contains
" a value of the vehicle observable vector sj, containing Fs elements, and
" a probabilistic classifier described by a vector cj of FC elements.
Each cj allows us to compute the probability that a query state x' lies in
the capability set corresponding to the jth behavioral case. For notational
convenience, we define an indicator event Dj to designate whether the vehicle
exists currently in the behavioral case represented by the jth library record, so
that this probability can be written as p (' C CI D). The Dj's are mutually
exclusive, i.e. the vehicle can be in at most one behavioral case at any point in
time; however, this does not mean the the vehicle is guaranteed to be in any
of the library behavioral cases.
We assume the vehicle has a means of measuring the values in the observable vector s; we denote the random vector corresponding to these measure23
ments as
s.
Given the vehicle is in the j'"behavioral case,
s
has the form
s = s3 + e,
(2.2)
where e is a random vector representing measurement noise that is independent
of the vehicle behavioral case. We assume the user has knowledge of the
statistics of e (often for physical systems it is characterized using a multivariate
Gaussian with known mean and covariance), i.e. we can compute Pe (e), as well
as
p (s ID ) = Pe (^ - s,)
(2.3)
The goal of our inference process is to evaluate the vehicle capability given a
measurement of the observable vector s. Because the vehicle capability is a set,
one means of performing this task is to evaluate set membership, as introduced
in Section 2.1.2. That is, we desire to evaluate a function q : X x S -+ R that
closely approximates the probability of a query state x' C X lying within C
given we observe S = s, i.e.
q (x', S) ~ p (x'E C Is)
(2.4)
We develop two different formulations for q in the following sections, and
add subscripts to identify them-the maximum likelihood variant qML is described in Section 2.2.2 and the mixture distribution variant qMD is described
in Section 2.2.3.
2.2.2
Inference using the maximum likelihood
A straightforward means of obtaining an estimate of whether a state x' lies
in the vehicle capability set is to find the library record that maximizes the
likelihood of the measured vehicle observable vector, and to then use the probabilistic classifier stored in that record to label x'.
Given our noise model in equation (2.2) for the measured observable vector,
we can form the log-likelihood
f (Dj
^)= log p (AIDj) = log pe (A- sj)
(2.5)
of seeing measurement s given our vehicle is in the jth behavioral case stored in
our library (note that while sj was computed using values of the vehicle state
and loss-of-capability parameters, these need not be known or represented
explicitly here). We then maximize expression (2.5) over all possible values in
24
the library:
(2.6)
jmax = arg max f (Dj IS)
jEl...R
Our maximum likelihood estimator, qML, is then the output from the probabilistic classifier corresponding to the (jmax)th library record:
qML (X, s) = p (x E
CIDjma )
(2.7)
This process is agnostic of any prior over the records in the library, and simply
seeks to find the record that "best explains" the measurements.
Time complexity
The time complexity of the Bayesian classifier can vary significantly depending
on the application. Duda, Hart, and Stork [17] present a detailed analysis for
the case where the noise model is a multivariate Gaussian-we present an abbreviated form here. In our case, the Jh record of the lookup table represents a
distinct class where the output noise model for said class is p (-I s) - K (sj, E)
for some known covariance matrix E.
The complexity follows from equations (2.5)-(2.7):
1. Computing the log-likelihood f (sj Is) (equation (2.5)). For the multivariate Gaussian case, the likelihood of seeing output s from class j takes
the following form:
2(sys)
(j^ =-
1
22 (
-s)
~
P-s)-2
--
d
1
log 2r - -log|Zl
2
(2.8)
Given each sensor vector has Fs elements, computation of s - sj is
o (Fs) and
multiplication by E-- is 0 (Fl) (computation of E-1 only
need be performed once and does not grow with Fs). Overall, the total
complexity is approximately 0 (Fj).
2. Maximization of the log-likelihood (equation (2.6)). We perform this
operation over the list of likelihoods for each record (i.e. each class) in
our library-the worst-case time complexity for maximization over an
unordered list of n elements is 0 (n), so our complexity is 0 (R).
3. Capability parameters lookup (equation (2.7)). Following the maximization, we wish to access cimx of our library. Assuming we can read the jth
capability parameter vector at the same time as reading the Jh sensor
25
vector when computing the likelihood f (sj IS)in the complexity analysis here.
we need not include it
In summary, the time complexity of the Bayes classification process assuming a Gaussian noise model for a single output sample grows as ~ 0 (RFs).
Our particular application assumes an arbitrarily large library, making R an
important component of the time complexity growth.
Space complexity
The storage requirements for the maximum likelihood classifier grow only with
the initial size of the library, i.e. as ~ 0 (R(Fs + Fc)).
2.2.3
Inference using a mixture distribution
As opposed to the maximum likelihood formulation that uses information from
only the most-likely behavioral case in the library, it seems natural to design
an estimator that combines information from each behavioral case, where more
likely cases have more "influence" on the overall estimate than unlikely ones.
We will first derive the mathematical form of the mixture distribution estimator qMD, and then look into the mathematical form as a way of explaining the
reason behind the "mixture distribution" naming convention.
Observing expression (2.4), we can manipulate the right-hand and express
it as a summation using the Law of Total Probability:
R
p (x' E CIS) ~ 1 p (Djls) p (x' E CID, S
(2.9)
j=1
The expression is an approximation because it relies on an assumption that
j_1 p (D [S) = 1, i.e. that our current vehicle behavioral case is contained
somewhere in the R records in our library. This is a rough approximation that
becomes increasingly accurate as our library becomes larger and richer.
Now, when conditioned on Dj, {x' E C} is independent of {S = S} because
our sensor noise is assumed to be independent of the vehicle behavioral case
(see equation (2.2)). So, we can drop the conditioning on S^ on the right-most
term inside the summation on the right-hand side of equation (2.9); applying
Bayes' Rule, we obtain a final expression for our mixture distribution estimator
26
qMD
(XI S):
qID
qMD
R
_1
p(SIDj) p (Dj)
p (x' E: C IDj )
(^_I j ) p (D y)
(sD
XRlp)
(.0
(.
Note that the right-most term in the right-hand side summation is the value
of each behavioral case's probabilistic classifier for the query vehicle state x'; in
addition, the parenthetically-grouped term in front is essentially a normalized
"weight" term. So, qMD can be interpreted as a weighted sum of the predictions that would be made by each record individually in the library were we
to assume the vehicle was in each record's behavioral case. Probability distributions of this form are called "mixture distributions," where they are derived
as a weighted summation of the distributions of distinct, underlying random
variables -these underlying variables are often called "mixture components"
and their weights are often called the "mixture weights." This is what provides
the motivation behind the naming of the qMD estimator.
The "weight" term on the right-hand side of equation (2.10) requires knowledge of p (Dj), the a prioriprobability that the vehicle is in the jth behavioral
case. In Chapter 4, we choose to set p (Dj) as a maximum-entropy, uniform
prior over all j, however the user could choose to use a different distribution
to encode domain-specific prior knowledge about the vehicle behavior.
The power behind the mixture distribution estimator is an ability to closely
approximate p (x' E CIS) despite the assumption that the vehicle behavioral
case lies within our library of records. Intuitively, the capability set of a behavioral case that is similar to several in the library, but not actually recorded
in the library, could be "interpolated" via this weighting of the capability sets
of nearby library records. However, this method comes at added computational cost compared to the maximum likelihood estimate, as is shown in the
following sections.
Time complexity
We can perform a complexity analysis similar to the maximum likelihood classifier complexity as presented previously, using a multi-variate Gaussian noise
model. By re-writing Eq. (2.10) as
Ej=1 p
(SIDj) p (Dj) p (x' E C IDj)
i
(.|I')
=
27
we see the computation involves two parallel summations over 1... R entries.
The process can be broken down as:
1. Computing p (81 Dj). This is of the same order as computing the loglikelihood for the maximum likelihood classifier, 0 (FS).
2. Computing p (x' E CIDj). Unlike for the maximum likelihood classifier,
we must evaluate the capability boundary for each lookup table record.
This will play an important role in the increased overall complexity-for
now, let us suppose this complexity is some function Oc(R, Fc) that is
of reasonable polynomial order.
3. Summation over all lookup table records 1... R. The summation is of
the order 0 (R) similar to the complexity of performing arg max for the
maximum likelihood classifier.
In summary, the time complexity of the mixture model classification process
grows as 0 (RFSOc(R, Fc)), where Oc(R, Fc) the complexity of performing
a single capability boundary evaluation.
Space complexity
Just as for the maximum likelihood classifier, the storage requirements for the
mixture model classifier grow only with the initial size of the lookup table, as
~ 0 (R(Fs + Fc)).
2.3
Summary
We have developed and presented our methodology for estimating vehicle capability in a quasi-static manner, using a measurement of vehicle observableswith a priori noise model assumptions-to estimate the probability that a
query vehicle state lies within the current capability set. The methodology
comprises an offline stage where we can use physics-based models and experiments to build and store a library for use in an online phase. We have
presented two techniques for capability estimation in the online phase, and
quantified their complexity with respect to the dimensionality of the stored
data.
28
Chapter 3
Aircraft Capability Model
We apply our data-driven capability estimation methodology to the case of a
representative UAV with mission performance affected by in-flight structural
degradation. We present the UAV design in Sections 3.1; next, we present
the development and validation of a global medium-fidelity model of the UAV
Section 3.2 and a local high-fidelity wing box model in Section 3.3. Lastly, we
present the top-level integrated model hierarchy in Section 3.4.
3.1
UAV Aircraft Design
The UAV design evolved from a first-principles sizing routine [18] and Federal
Aviation Regulation (FAR) 23 guidelines. As shown in Figure 3-1, the vehicle
has a wing span of 55 ft. It is designed to cruise at 140 kn (240 ft/s) at an
altitude of 25, 000 ft. The fuselage accommodates a payload of 500 lbs. The
estimated range of the aircraft is 2500 nmi, corresponding to a flight duration
of approximately 17.5 hrs. This allows for adequate operational capability to
explore maneuverability as a function of the changing structural state of the
vehicle.
29
......
-F
300
Sweep Angle=4.3deg
Wing Spars
Wing OML
200
Engines
100
Pylons
-
Fuselage OML
0
0
o
o
Front Landing Gear
. ...........
Sweep Angle=5deg
Main Landing Gear
Payload
100|
Wing Spars
-200
Wing OML
Airplane Center of Gravity
-300-
Airplane Neutral Point
0
100
200
300
Fuselage Station (in)
Figure 3-1: A realistic concept unmanned aerial vehicle established to estimate
the effect of structural degradation on capability.
30
3.2
Aircraft Model
Aero-structural loads on the UAV are estimated using ASWING [19], a nonlinear
aero-structural solver written in FORTRAN for flexible-body aircraft configurations of high to moderate aspect ratio. We use ASWING to predict internal
wing stresses and deflections as a function of input aircraft kinematic states
and estimates of damage to the nominal aircraft structure. Figure 3-2 shows
the representation of our concept UAV in the ASWING framework. The ASWING
model is a set of interconnected slender beams-one each for the wing, fuselage, horizontal stabilizer, and vertical stabilizer. Lifting surfaces (the wing
and stabilizers) have additional cross-sectional lifting properties that are prespecified.
Figure 3-2: The representation of our concept UAV within ASWING. The structure
is specified as a set of interconnected slender beams, where lifting surfaces have
additional aerodynamic properties specified along their span.
3.2.1
Model configuration
ASWING is capable of static, dynamic, and modal analyses of airframes; however, we only use its static analysis tools in this work. Specifically, we trim the
aircraft to simulate "pull-up" maneuver conditions, where the nose is pitched
upward to increase the wing angle-of-attack and wing load factor n =
.
This
maneuver is often used as a representative case of maximal structural loading
conditions where, for a fixed airspeed, we can trim the aircraft to increasing
values of n until either stall or structural failure occurs.
Figure 3-3 presents the internal ASWING constraint matrix used to configure
the pull-up flight conditions. The user supplies a target value for the (indicated) airspeed, and then controls the trim load factor directly by specifying
a target lift force in units of the aircraft gross weight.
31
3.2.2
Model validation
We validate the UAV representation in ASWING by trimming it to steady-level
cruise conditions at a load factor n = 1 for a range of airspeeds above and
below the nominal design cruise speed, and observing its response compared
to expected behavior based on the (previously validated) aircraft design.
Figure 3-4 shows the lift-to-drag, engine setting, angle of attack, and elevator setting obtained from this analysis, along with an example of the deflection
profile across the wing surface in the blue-boxed callout (shown facing the front
of the aircraft; the wing profile is drawn in red).
At cruise conditions, the airspeed that maximizes the UAV lift-to-drag is
approximately 250 ft/s ~ 148 kn, which is a less-than- 6% change from the original design cruise speed of 140 kn. We consider this as acceptable agreement
for continuing use of the ASWING model, especially given the UAV representation in ASWING requires simplification of the geometry to interconnected beams
with only 1-D span-wise lumped properties. We also attribute some error to
some changes to the wing structure that were necessary to integrate with our
high-fidelity beam model in VABS, described further on in Section 3.3.
After observing the n = 1 aircraft behavior, we fix the airspeed at the
optimal cruise speed of 250 ft/s and observe the behavior of the model over
increasing load factor n, shown in Figure 3-5. The aircraft is equipped with
inboard high-lift flaps, that are activated to 150 to give an additional "bump"
to the wing lift profile. This allows the aircraft to perform at higher load
factors (in our case, this is quantified as in excess of 3G's) without stalling.
Maneuvers at higher load factors are in the flight regime where the structural
loading constraints become particularly important, so this regime is where we
wanted to place our structural degradation analysis to have a relevant impact
on the vehicle capability.
32
Aa
a,, 0.000
az,, 0. 000
a. -0.000
aw -0.000
az -0.000
V1As - 2438. 000
a a cy az UU
Uz
x
()
$
Oz XE YE ZE j
6
F
6
F2 6FN 6F
1
3
a**.r0. 000
0. -0.000
aw -0.000
Oz =0.000
XE
=0.00
YE
-0.00
ZE -0.00
$*
-0.00
8*
*0
-0.00
0.00
6FI
0.0000
U..
6F2
-0000
0.0000
6r, =0.0000
6
F5 = 0. 0000
AEI -0.0000
6F3
Y-Fx -
0. 000
0F000
- 0.
-=Fz
0. 000
.0.000
=I
IMx-0.000
MZ -0.000
LirL- 3229.
\
5
00
* -0.00
6Y-Ux 0Z - 0. 000
Cuserj-u 0.000
Cuser2'u 0.000
mI n5 6-6, l/w
1
(a.
2
3
(VrAs = VSPEC, U.)
= 0, 0)
(a = 0, F)
E
M = 0, a)
4
(E F =
5
6
(Ln=1 = W, Uz)
(y = 0, Aeng)
6
cc~
Trim for pitch stability
Trim for zero angular acceleration using ailerons, elevator, rudder
Target indicated airspeed
(including inertial
Zero net force and moment for free flight conditions
reactions)
Directly control lift force to set load factor
Use engine to set a level flight path
Figure 3-3: Snapshot of the ASWING constraint configuration menu for a static
pull-up maneuver analysis. ASWING attempts to satisfy the constraints given in
the matrix to the left when trimming the aircraft. A turquoise box indicates the
flight state variable in the corresponding row is constrained to its assigned value by
the variable in the corresponding column. The table to the right gives additional
explanation for select groups of constraints in the matrix.
33
45
160
100
40
700
140
600
120
500
35
25
400
60
300
460
20
200
VTAS (ftM)
250
350
VIAS (fts)
40
100
150
10
15
In
50
15-
I -30
0
700
10
600
-20
300
450
5-
200
400
500
VTAS "a)
250
350
ViAS (ftfs)
-20
100
150
0
~
50
0.8
2r/cv
0.2-
-0.2-0.4.
0.0
01
4.0
2.
0.0o
ASWING
S. 96
max
Figure 3-4: Output from ASWING for a static pull-up maneuver analysis, varying over airspeed at a constant n = 1 load
factor and 25, 000 ft altitude. The boxed plot to the right shows aerodynamic properties across the wing surface (in
red) for the airspeed that maximizes the lift-to-drag ratio.
."1
-5
OF
1.4
1.8
2.6
Load Factor
2.2
3
3.4
-30
-25
-20
-15
0
5
10
..
L0_
0t5
0.0
0.0.
10.0-
lb/In
2D.0
30.0
(C
2r/c.
ASWING
SAG
due to
max
------------- -- ----------
activation of flaps
Lift "bump
I
Figure 3-5: Output from ASWING for a static pull-up maneuver analysis, varying over load factor at a constant airspeed
of 250 ft/s and a constant altitude of 25, 000 ft. The boxed plot to the right shows aerodynamic properties across the
wing surface (in red) for n = 2, i.e., for a maneuver that loads the wing with twice the gross weight of the aircraft.
II
5
10
15
2.0I
3.2.3
Lumped damage representation
To modify the maneuvering capability of the aircraft, we represent structural degradation due to damage on the aircraft wing. The ASWING structural members, as slender beams, consist of span-wise beam stiffness property
distributions-one means of representing damage is by a reduction in these
beam stiffness properties. Figure 3-6 demonstrates this in the case of the wing
bending stiffness along an axis parallel to the wing chord, EIce (i.e. the wing
bending stiffness that is related to "flapping" or the most visible U-shaped
deflection of the wing); the reduction is greatly exaggerated in the figure for
explanatory purposes. We leave the aerodynamic properties of the model untouched by our damage representation; hence for a given trim condition the
aerodynamic loading will change purely via coupling with the modified structural properties.
(exaggerated)
109 1.0
0.0
-4.0
-3.0
-2.0
-1.0
0.0
1.0
2.0
3.0
102x 4.0
Spanwise coordinate t
Figure 3-6: Illustration of how "lumped" damage effects could be represented in
ASWING by reducing one-dimensional beam stiffness properties along their span.
This simplified representation of damage captures how the damage ham-
36
pers the wing's ability to carry structural loads through the affected region.
However, it is too simplified-local damage to a wing structure will produce
local stress raisers that are not visible to this lumped representation in ASWING.
These local stress raisers are what we can directly compare to material allowables, to determine whether the damage causes unsustainable local stresses in
the surrounding undamaged material. To account for these local stress raisers,
we couple our ASWING model with a high-fidelity wing box representation. The
next section describes the high-fidelity modeling technique and presents how
we couple it with ASWING.
3.3
Wing Box Model
To resolve stiffness loss due to local damage on the aircraft wing, we need
another technique to interface with the global ASWING aircraft model. In lieu
of forming a three-dimensional finite element representation of the wing, we
use Variational Asymptotic Beam Cross-Sectional Analysis (VABS) [20, 21],
a powerful dimension reduction technique used in industry practice. A visual representation of the VABS methodology is shown in Figure 3-7. To
model a beam in VABS, the user first represents the beam as an array of
two-dimensional cross-sectional finite element models. The cross-sections capture the details of a multi-ply composite wing skin and local damage effects.
VABS then computes lumped stiffness and inertial properties at a reference
point in each cross section, forming a "global" line representation of the beam.
A one-dimensional beam solver can then find the global force and moment
distribution along this reference line given input forces and moments. Lastly,
the user recovers the internal stress and strain fields using the reference line
solutions and influence coefficients initially computed by VABS-the influence
coefficients are linear transformations relating the reference line solution to the
3-dimensional stress and strain tensors in each element of the cross section.
In this work, we use the UM/VABS ri.31 implementation developed in
FORTRAN by R. Palacios and C. Cesnik at the University of Michigan [22]
(referred to hereon as VABS). Our beam of interest is the aircraft wing box,
and ASWING manages the one-dimensional beam solution, computing loads in
the wing box for specific flight conditions.
3.3.1
Local damage representation
Using VABS, we can represent structural modifications due to damage on a
cross-sectional, two-dimensional finite element level. Each element in the
37
model represents a portion of composite material viewed through its thickness,
containing material stiffness and angular orientation (i.e., ply angle) properties. We represent damage as a reduction in the material stiffness properties of
affected elements-the physical nature of this damage is general, as we do not
do further detailed damage modeling (e.g., for crack propagation or composite
delamination). However, our implementation is left open-ended to allow for
future addition of higher-fidelity damage models that are compatible with the
VABS technique.
As a first-order approximation, this damage representation allows us to
observe the impact of local stiffness reduction on surrounding un-modified elements. The loss of stiffness in one region causes local stress raisers in nearby
"healthy" elements, potentially causing premature failure in these nearby elements. We define whether a healthy element would fail under given loading
conditions by looking at its internal failure indices, of the form cj /,qlowable
for i, j = 1, ...
, 3;
eij is the strain in the direction of material axis
j
on a sur-
face within the element whose normal points in the direction of material axis
i, and 6 qlowable is the corresponding allowable limit imposed by the material
properties. Thus, for each element there are 6 failure indices corresponding to
failure in
* 3 directions due to normal strain failure (extensional or shear), and
* 3 directions due to shear strain failure.
We then take a maximum over all the failure indices in every element of the
cross section to obtain one single scalar value representing whether a failure
would occur somewhere within the cross section. This is repeated for every
cross section in the wing, and by taking another maximum we can obtain a
U M/V AB 5
Influence coefficients
Stiff ness
properties
*
-
Reference line
forces and moments
Figure 3-7: Variational Asymptotic Beam Cross-Sectional Analysis (VABS) allows
for dimensional reduction of an expensive three-dimensional beam solution into
two-dimensional finite element models coupled with an external beam solver.
38
single scalar failure index representing whether failure would occur somewhere
in the entire wing structure.
The material axes 1,2, and 3 are defined with respect to the fiber orientations in each of the material plies of the cross section, and hence they may
not necessarily coincide with the axes system of the cross section (which by
definition in UM/VABS ri.31 has its 1-axis point outward along the beam span,
its 2-axis point to the right in the plane of the cross section, and its 3-axis
point up in the plane of the cross section).
Figure 3-8 demonstrates this failure analysis on a cross section that is
representative of a symmetric aircraft wing box (note that most wing boxes
would have much thinner walls; this example has thicker ones for illustrative
purposes). A damage event on the top surface causes an increase in the crosssectional maximum failure index for a fixed loading condition (in this case, the
loading is a pure moment about the Y-axis).
39
Undamaged
1
2
0.9
1
0.8
0
0.7
-1
REF
I
-2
0.6
0
Damaged
0.5
0.4
2
6
4
1
0.2
0
R:0.975
L
Z
REF
0.1
8
Local increase in
failure indices in
nearby undamaged
material
-1
0"
-2
il r
CI UII
W
Index
Higher failure indices
on top surface because
material allowable is
lower in compression
than in tension
dan mage region
2
0.3
F:
0.887
IE
,ftlffi
i
0
2
4
6
8
Figure 3-8: Output from VABS for an example cross section (with three plies
oriented in a [00, 900, 00] stack) when under a pure Y-axis moment. Failure indices
are plotted for both the undamaged and damaged cases-note that although no
failure indices are computed in the damaged region, the material is still present in
the FEM.
40
3.3.2
Integration of VABS with ASWING
For integration with our aircraft model, we use a VABS cross-sectional FEM
that matches the same airfoil shape as the ASWING model. For a given damage
case, we run VABS to update the lumped mass and structural stiffness properties for the wing representation in ASWING. Figure 3-9 shows the geometry
of our VABS model for an undamaged configuration with a table of relevant
properties. We only form a model of the wing box, as this carries the majority
of the wing's structural loads (the ribs are at fixed locations with respect to
the chord, and the top and bottom surfaces follow the airfoil outer contour).
Detail A
10-
5-
A
0
LY
-5 -
Airfoil
DA-01
Chord
Rib locations
50 [in]
0.2C, 0.7C
Ply orientations
[00, 450, 900, 450, 00]
[inner,...,outer]
Ply material
MTM45-1/AS4
with respect to leading edge
-10-15
-10
-5
0
5
10
15
x (In)
Figure 3-9: Constant cross section finite element model used for VABS analysis,
allowing refinement of stress raisers caused by local stiffness loss due to damage.
In addition, we use a constant cross section for our analysis to make the
number of function calls to VABS computationally tractable. The chord length
in the VABS model is an averaged value of the tapered chord distribution from
the original ASWING model, while the tapered chord distribution was still retained in the ASWING model to provide realistic, efficient aerodynamic washout.
However, this means the ASWING model uses a wing that (when undamaged)
has constant structural properties but varying aerodynamic properties. The
constant structural properties cause only a small change to the nominal aerostructural performance of the vehicle when compared to the initial design, as
shown previously in Section 3.2.2.
Note that without the VABS technique, the aircraft model in ASWING would
not be able to elicit a high-resolution material failure metric for a given damage
condition, as the elevated material stresses are of a local nature. An alternative
41
is to develop a more simplified structural failure metric (e.g., a maximum sustainable moment at the wing root), but by coupling ASWING and VABS we offer
flexibility for the addition of higher-fidelity damage models in the future, and
we demonstrate the ability of our method to handle medium-to-high fidelity
models in the offline phase.
3.4
Integrated Aircraft Capability Model
The overarching purpose of the aircraft model is to provide the functionality
outlined in Section 2.1.1. After coupling the global ASWING aircraft model and
the local VABS wing model, we have the integrated model as shown conceptually
in Figure 3-10; the model input/output follows the structure as outlined in
Figure 2-la. Each of the model inputs and outputs is summarized below.
State Vector
Maneuver velocity,
Coupled ASWINGNABS Aircraft Model
load factor
Sensed wing
strain
Commandsbox
Airrf
Aircraft Model
BaeieGenerate
Deae.
Damage location,
ASWING
ASSri
Recovery
Cross Section
Loss-of-Capability
Parameters
Observables
Vector
BaselCn
AIrcraft
Parameterl
size
Compare to
Material
Allowables
Failure Metric
Maximum
failure index in
wing box
Figure 3-10: Modeling toolchain for analyzing UAV loss-of-capability due to structural damage, using ASWING coupled with VABS. The 1/O format follows from the
methodology in Chapter 2.
The state space of our vehicle model consists of an airspeed V and load
factor n characterizing a pull-up maneuver-we make the simplifying assumption that the aircraft is in steady flight, where an airspeed and load factor are
sufficient to identify the lift distribution over the aircraft wing surface. We
identify the aircraft capability in this state space.
The loss-of-capability parameters characterize structural degradation, where
we capture the physical (location, size, and severity) properties of a rectangular
42
damage event on the wing surface.
The failure metric for a given state vector and damage parametrization is a
maximum failure index, taken over all possible modes of failure in all elements
of the local VABS wing model. This acts as a single scalar failure metric which
we can use to classify a set of model inputs as "safe" or "unsafe."
The main components of the observable vector are structural strain readings measured by strain sensors on the wing box surface. However, we also
augment these strain values with the vehicle state x in order to produce the
full observable vector s. Intuitively, because the damage representation modifies stiffness properties of the aircraft structure, we must record both the loads
(i.e., the vehicle state), and the deflections (i.e., the strain sensor readings)
in order to have a well-posed inference problem. Thus, a behavioral case (as
defined in Section 2.2.1) in our offline library will contain both the vehicle
state and the strain sensor readings as features in the observable vector; we
will require measurements of both these quantities in the online phase.
43
44
Chapter 4
Classification, Application, and
Results
Once we have constructed the integrated UAV model as presented in Chapter 3, we continue to follow the methodology laid out in Chapter 2, using the
model to discover the vehicle capability set for varying cases of damage. This
allows us to build, offline, the library of damage cases that can be queried
online in an example scenario.
We implement the capability discovery process using classification in Section 4.1. We consider an example flight scenario in Section 4.2; based on the
scenario, we build a library of expected damage cases offline and then use
them in a simulation of the online capability estimation process. We present
results in Section 4.3, comparing the maximum likelihood and mixture distribution formulations of the online estimator. We compare both estimators to
a baseline that uses only knowledge of the vehicle capability from design.
4.1
Discovering the Capability Set Boundary
via Classification
Given a damage case applied to our aircraft model, we can sample in the maneuver state space, labeling each sample as being "safe" or "unsafe" according
to the value of their output maximum failure index. As we perform this task,
we save the observables vector corresponding to each sample for storage and
use later in the online phase.
To perform this classification process, we use a technique from the machine
learning community called the Support Vector Machine (SVM). Section 4.1.1
introduces the mathematical framework behind this technique. Our implemen45
tation is cognizant of inherent uncertainty in the classification process given
a large but finite offline sampling phase; we include a means of extending
the SVM technique to directly represent uncertainty in its output-the probabilistic support vector machine, or PSVM-in Section 4.1.2. Lastly, we apply
an intelligent state vector sampling strategy to reduce the number of model
function calls, building off prior work in the literature, in Section 4.1.3.
4.1.1
Support vector machines
Our explanation here of the SVM is a summary of relevant points from the
standard formulation in the literature, and we refer the reader to related
material in Duda, Hart, and Stork [17] or the original article from 1995 on the
subject by Cortes and Vapnik [23] for further detail.
The purpose of the SVM is to perform binary classification of unlabeled
test samples-i.e. classification into one of two classes- based on trends seen
in a labeled set of training samples.
More formally, let our collection of N labeled training samples take the form
Z = {(xj,yj)
:
E Rn,yj E {-1, 1}, j = 1,..., N}. For our application to
Cj
the aircraft capability model in Chapter 3, each xj is a state vector consisting
of an airspeed V and load factor n; in general, each xj consists of attributes
that describe the sample. Each yj is the corresponding binary label for the
sample, which in our case is the indicator representing whether the failure
metric f(xj, d) (given the fixed damage parameters d) exceeds a nominal safe
threshold value.
We want to create a classifier, C :R" {-1, 1}, that can label a new test
sample x as belonging to either class 1 or class -1. We hope this classifier
can also perform well on (i.e., correctly label) the original training samples in
Z, although this performance might be slightly sacrificed in order to obtain
better performance outside of the training data.
The SVM is one means of implementing the classifier C. It evaluates a
discriminant, S :R" -+ R, such that for some input test sample x,
-
Y
C(x)
{ -1
ifS (x)<w, and
1
otherwise.
(4.1)
The value of this discriminant for a sample x is often called its score.
The simplest SVM uses a linear discriminant S (x) = wTX + b (that is, a
'A nice introduction is given by Andrew Ng in course notes for CS229 (Machine Learning) at Stanford University; they can be found at http://cs229.stanford.edu/notes/
cs229-notes3.pdf.
46
hyperplane with normal vector w and offset b) where C maps elements on one
side to 1, and on the other side to -1. Figure 4-1 illustrates this (notionally)
for our aircraft capability case where the data samples have two attributes V
and n.
n
margin =
0
T\WII S (X) = +1
S (x)
=
-1
0 Class -1 sample
0
0*
0 Class +1 sample
X
x. =
(Vi, nj)
o
Support vector
0
0:
V
Figure 4-1: Visualization of a linear SVM discriminant S applied to our aircraft
capability model, where the data x = (V, n) are the airspeed V and load factor n
as in our aircraft capability application. Samples in class -1 have a negative value
of S, whereas samples in class +1 have a positive value.
Given our set of training samples xj with corresponding labels yj, we seek
to "tune" the SVM discriminant parameters w and b so that the SVM reflects
as much information as possible from our training set. The standard SVM
tuning process finds the hyperplane that lies equally close to the nearest points
in each training class, while being as far as possible from these same points.
The nearest points are called the support vectors, and are the namesake of
the SVM. The distance to the support vectors is also often called the margin.
Often the SVM scores are scaled so the support vectors all have scores of +1 or
-1 (depending on which class they are in), so the resulting magnitude of w is
inversely related to the margin-this is shown in Figure 4-1. The optimization
problem for choosing w and b takes the following form:
*1
II12 s.t.
w,b 2
min -
(4.2)
Yj (wTxj + b) > 1 Vj
However, we may not want a perfect discrimination between the two classes of
training samples, as this can be prone to overfitting and outliers. A regularized
47
formulation of problem (4.2) uses the slack variables j to relax each of the
problem constraints:
1
min - |)wJf 2 + C Z j s.t.
',t 2
(4.3)
Y, (wTXj + b) > 1 - cy Vj
Here, the training samples can lie arbitrarily close to (or even on the wrong side
of) the separating hyperplane, with the downside of incurring a larger penalty
value in the objective via the slack variables j-the value of C controls the
relative weighting of this penalty term.
Now, the dual 2 of the regularized problem (4.3) is more tractable, especially
when extended to the non-linear SVM described later in this section. It can
be shown to have the following form:
max
-aZaZkYjYkXT
0
Xk
<O
+
ail
s.t.
< CVj
(4.4)
ayy, = 0
The optimal aj's can then be used to recover the weights w as
w=E ajyjx,
(4.5)
and the bias b as
b
maxjyj-, w T
+ minj,yj= 1 w xJ
2
(4.6)
Each of the dual variables a are directly related to the original primal constraints; non-zero aj correspond to the support vectors, whose constraints are
active. We also note that the penalty constant C is often called the "box
constraint," as it sets the size of the "box" that the support vector weights aj
must lie within.
2
Here, we refer to the Lagrangian dual that is derived by forming the generalized Lagrangian including the constraints, and minimizing it with respect to the original design
variables. The dual is essentially a "mirrored" optimization problem with respect to the
Lagrange multipliers. We refer the reader to established convex optimization literature such
as by Boyd and Vandenberghe [24] for further explanation.
48
The "Kernel Trick" and the extension to nonlinear SVMs
The dual problem (4.4) can incorporate a smooth "pre-processing" function #
to X, producing an identical SVM training problem except now the separating
hyperplane lies in the image of # as opposed to the original space of x:
max
[
S S ajakyjyk#(X)
c#(Xk)
+
Sa
s.t.
0 <% < C Vj
(4.7)
Eajy, = 0
The function 4 is often called a feature map. A careful distinction is often
made between the "attribute space" of the original sample vector x and the
"feature space" it is mapped to by #. The feature space can be of much higher
dimension than the attribute space, where the original decision boundary in
attribute space can correspond to a linear decision boundary in the feature
space.
Now, the SVM training problem in feature space relies only on computing
and not on the exact value of the feature vector
an inner product #(Xj)T#O(X)
O(xj). In general, an inner product of this form is often called a kernel, denoted
here as K: Rn x R n -+ R, where
K(x, z) = O(x)T#(z).
(4.8)
Often, the kernel can be much easier to compute than #, especially when the
output feature space is high dimensional. For instance a polynomial kernel of
order d,
(4.9)
K(x, z) = (XTz + I)d,
can be computed in 0 (n) time despite the corresponding feature map
# living
n + d -dimensional space (which grows as ~ ( (id).
Using a kernel in this way is often called the kernel trick, and it was popularized initially in the machine learning community by Aizerman, Braverman,
and Rozoner [25]. In most practical applications, it is a key enabler of the
construction of non-linear SVMs.
The non-linear, relaxed SVM optimization problem in equation (4.7) is the
formulation we use as part of discovering the vehicle capability boundary. The
next section explains how we extend the SVM training process to incorporate
uncertainty.
in a
49
4.1.2
Probabilistic support vector machines
The SVM is a powerful classification tool for binary classification. It performs
well on datasets with non-linear decision boundaries, and it effectively minimizes computation by relying on only the samples that lie closest to the decision boundary (i.e. the "support vectors"). However, instead of the SVM's
binary, deterministic classification, we wish to make a "soft" classification,
where a sample is assigned a probability of belonging to one of the two classes.
Using a "soft" classification, samples that lie far from the decision boundary
are more likely to belong to their respective class. As mentioned previously
in Section 2.1.2, this is the behavior we desire from our capability boundary
classifier, where our implementation should be able account for uncertainty in
the offline vehicle behavioral model.
We therefore use an extension to the support vector machine, called a
Probabilistic Support Vector Machine (PSVM), where a trained SVM is postprocessed and fitted with a suitable probability distribution. We follow the
original technique as proposed by Platt [26]; several other implementations
exist, including a modification proposed by Basudhar [27]. Given a sample
x E R' of attributes, a corresponding label y C {-1, 1}, and a support vector
discriminant S (x) characterizing a decision boundary as described previously
in Section 4.1.1, we can look at a probabilistic classifier C : R' - R that
evaluates p (y = 1IS (x)), the probability that the sample x lies in class y = 1
given the output of our support vector machine discriminant.
Platt fits C with a sigmoid
a(s)
1
= p (y = 1S(x))
=
1 + e0is(X)+ 32
(4.10)
where 3 < 0 and /2 are suitable distribution parameters. Given the restriction
on 01, C is monotonic in S, ranging from 0 when S -± -oo to 1 when S -+ oo.
This reflects the fact that C ought to become confident (i.e. a certain 0 or
1) far from the SVM decision boundary at S (x) = 0. To find the values of
#1 and 02, we can use maximum likelihood estimation from a training set of
N independent and identically distributed (iid) samples (xj, yj) where j
1, ... , N. Let
t
Yj
fyj
2
0 if yj = -1
be indicators of whether each training sample
Pj
+ e3
50
j
(4.11)
belongs to class 1, and let
1
1(x)+f2
(4.12)
be the probability that sample j belongs to class 1 given a particular parametrization 01, /2. The likelihood of the training set given i1, /2 is then
(4.13)
f ?(1 - p 3 )I-
We can then find the values of /1, /2 that maximize this likelihood. Equivalently, we can minimize the negative log-likelihood, obtaining a simpler optimization problem:
mmn {
tj log(pj) + (1 - t)log(1
whr pj where Pj
=
1
1+
(4.14)
(4I4
ef1s(xi)+02
However, one disadvantage of this PSVM formulation is its tendency to overfit
the training set. Platt suggests using a modified value for each target tj to
capture a small (but finite) probability of seeing the opposite class label at the
same location in hypothetical "out-of-sample" data. Instead of tj c {0, 1}, we
use
(.5
N+2 if Yj -1
t-
-f
[ N++2 if Yj
=
-1(4.15)
where N+ and N_ are the number of samples in class 1 and the number
of samples in class -1, respectively. As described by Platt, the modified tj
correspond to a MAP estimate of the target probability for each class assuming
a uniform, uninformative prior over the probability of all our training samples
having the correct label.
Implementation and numerical stability
Platt's original implementation of problem (4.14) in pseudo-code had numerical instabilities that were addressed by Lin, Lin, and Weng [28]:
1. Platt's original pseudo-code using the log(.) and exp(.) functions was
susceptible to overflows. However, Lin et al. note the IEEE floatingpoint standard-which is used by our MATLAB implementation-mitigates
these issues.
2. Lin et al. note the computation of I-p is numerically unstable when p
is close to one, and suggest the following reformulation of the objective
51
function:
-
(tj log(p3 )+ (1- t3 ) log(1 - p))
(ti - 1)( 1 S (Xi)
=
+
42)
+ log(1 + ef1S(xj)+02)
t (,AS (xi) + 02) + log(1 + e-1S(Xi)-2)
(4.16)
(4.17)
(4.18)
(4.19)
Here, 1 - pj does not appear, and we can compute log(1 +..) stably for
small operands in MATLAB using the special function loglp(1 + x) that
is stable for small x.
4.1.3
Capability estimation using probabilistic support
vector machines
Using the technique described in Sections 4.1.1 and 4.1.2 -training a support
vector machine to identify a boundary between samples from two classes in
a given feature space, and then fitting a sigmoid probability distribution to
create a probabilistic support vector machine-we can identify the vehicle
capability set for a given damage case.
Adaptive state space sampling technique
Our method requires intelligent sampling of the vehicle state space so that we
can provide our PSVM training process with a rich set of training samples
while minimizing uninformative calls to our aircraft capability model, which is
of non-trivial computational complexity. We have two competing qualitative
goals during the sampling process:
1. We want to sample along the boundary of the vehicle capability set to
provide an accurate description of its limits.
2. We want to sample within the vehicle capability set uniformly in order
to capture the vehicle behavior we expect to see during operation (so
that our online classification process sees library records that are similar
to the observed vehicle behavior).
Techniques exist to provide a space-filling set of samples for goal 2, such as
Latin Hypercube Sampling [29] or a Centroidal Voronoi Tessellation (CVT) [30].
Refinement of the boundary itself for goal 1 can be implemented using adaptive sampling; we use a technique developed by Basudhar and Missoum [27].
52
The algorithm begins with a well-spaced set of samples (here, we start with
a CVT produced using Lloyd's algorithm [31]) and then chooses samples at
each iteration that lie along the boundary of an SVM approximation of the
true capability set. A summary of the algorithm steps is as follows:
1. Begin with an initial set of training samples that has at least one member
from each of the two classes
2. Train a SVM on the initial set of samples
3. Generate a sample on the SVM boundary that lies as far as possible from
all current training samples
4. Generate a second sample nearby the SVM boundary to prevent "SVMlocking" (see [32] for further explanation)
5. Re-train the SVM using the two new samples
6. Repeat from Step 3 until converged
As a choice for a convergence criteria, Basudhar and Missoum suggest using
a polynomial kernel to construct the SVM for each iteration, and looking for a
stabilization of the change in polynomial coefficient values between iterations.
We chose a different criteria that makes use of the computation of the sample
from step 3. Here, we seek to maximize the minimum distance from the new
sample to any other training sample, while constraining the new sample to lie
along the SVM boundary-this distance itself can be used as a convergence
metric, and we stop the algorithm when it decreases below a nominal value
(scaled with respect to the bounds of the whole sample space). The intuition
behind this metric is that as the sampling converges onto a SVM boundary in
the sample space, new samples will begin to "crowd" along the SVM boundary
line until the distance metric settles to a small value.
Using this sampling technique, we can find a SVM representation of the
vehicle capability boundary to within a desired level of sampling accuracy, and
then fit a PSVM model to capture the uncertainty in the boundary location
due to the finite sampling accuracy. Figure 4-2 shows the evolution of the
computed PSVM for a single vehicle damage case and increasing numbers of
state space samples. The first plot includes only samples from the initial CVT,
whereas the next two include samples generated using the adaptive sampling
technique. Figure 4-3 shows the convergence plot for the adaptive sampling
technique when applied to the case shown in Figure 4-2-the pink line is the
value of the convergence metric, and the blue dashed line is the fixed tolerance
value used for the stopping criterion.
53
N =21
N=41
0.2 0.4 0.6 0.8
0.2 0.4 0.6 0.8
3.5
N
=81
0.2 0.4 0.6 0.8
3.5
3.5
3
0
0
15 2
0
u 2.5
U-
CO
U-
0
~02
-60
2.
U0
1.5
1
180
200
220
1
180
240
Equivalent Airspeed (ft/s)
200
220
1
180
240
Equivalent Airspeed (ft/s)
200
220
240
Equivalent Airspeed (ft/s)
Figure 4-2: Refinement of the PSVM vehicle capability estimate for a single vehicle
damage case, using adaptive sampling for increasing sample size N. The horizontal
and vertical axes indicate respectively the airspeed in ft/s and the load factor for
each state space point. White samples are classified as "safe" by the vehicle
model (i.e. they lie within the capability set), whereas grey samples are classified
as "unsafe". Samples outlined in black are support vectors for the SVM whose
boundary is designated by a black line. The colormap indicates the fitted PSVM,
the probability that a point in state space lies within the current vehicle capability
set.
V.V10.
I
I
0.016
C 0.014
4)
C)
0.012
0.01
--- --- -- -
---
- - ---
0.008
0 0.006
C)
0.004
0.002
2
4
6
8
10
Iteration
12
14
16
Figure 4-3: Plot of the convergence history for one run of Basudhar and Missoum's
adaptive SVM-based sampling algorithm using the modified convergence criterion
as described in Section 4.1.3.
54
4.2
Online Aircraft Capability Estimation
The example aircraft we have modeled could be used in a variety of mission
scenarios. We describe a representative mission here, and demonstrate how to
use the offline-online capability estimation framework to improve performance
in this scenario.
4.2.1
Flight scenario
One use of our representative aircraft is for missions in contested environments,
where threats to the vehicle due to hostile agents require a fast, defensive reaction to avoid dangerous regions of the flight zone. In addition, the vehicle
may sustain damage on the wing surface that impedes its ability to operate at
its initial design capability (however, we assume the damage is minor enough
such that continuing operation of the vehicle is still a possibility). Figure 4-4
presents a schematic of this scenario, where the vehicle initiates the evasive
action at an airspeed of 210 ft/s and an initial load factor of 1.3 (representative of a nominal maneuvering speed and an upper bound on the nominal
maneuvering load factor during normal operation while navigating a sequence
of waypoints).
We assume the loads on the wing structure imparted during the maneuver
can be approximated using the pull-up maneuver framework as described in
Section 3.2.1.
Evasive
maneuver
nmax
?
VTAS =210 ft/s
Figure 4-4: Schematic of the flight scenario to which we apply our capability
estimation framework.
These sense-and-avoid conditions push the UAV to operate near its structural limitations on maneuverability. The UAV requires knowledge of its maximum maneuvering capability; in practice, onboard planning algorithms often
55
use static estimates of the vehicle capability (i.e. of the maximum allowable
load factor at a maneuver airspeed), with an added safety factor to be conservative. However, the possibility of damage complicates and renders uncertain
the vehicle capability; this could be mitigated by an even larger safety factor,
however then the vehicle would operate well below its true performance limits
in an overly-conservative fashion.
We will compare the performance of the static estimate to ours that dynamically incorporates sensor information. First, we need to explore the expected
damage cases and vehicle state space in the offline phase.
4.2.2
Library damage cases and maneuver bounds
The capability estimation framework from Section 4.1.3 allows adaptive sampling of the vehicle state space to discover the capability set given a fixed
damage case. However, we still have yet to decide which damage cases to include in the library, as well as what are intelligent boundaries to place on the
vehicle state space sampling algorithms.
We use a full factorial technique to construct damage cases to include in our
offline library; given a fixed location along the wing top surface, we analyzed
wing damage cases of varying extent, depth, and severity; we use the following
symbols to represent each damage factor:
* is, Ic: Damage location in the span-wise and chord-wise directions (normalized with respect to the wingspan and local chord length, respectively)
* WS, wC: Damage extent in the span-wise and chord-wise directions (normalized with respect to the wingspan and local chord length, respectively)
" dt: Damage depth, uniform across the entire damage region (normalized
with respect to the wing box thickness)
* df: Damage "severity," a fraction that is applied to each of the material
stiffness properties of the damaged elements
Figure 4-5 defines the damage case factors visually with respect to the aircraft
planform, and lists the levels for each factor that we included in our offline
library. The purpose of this study was to demonstrate functionality of the
library; in a more realistic scenario, a prior over expected damage cases could
be utilized so that the library cases would better reflect those the user would
expect to see during operation.
We derived bounds for the state space sampling from our mission scenario.
The mission requires the aircraft to perform a constant-velocity evasive ma56
--
lc
WaA
A
Symbol
1,
WS
dt Section A -A-
1e
we
dt
df
Name
Span-wise Location
Span-wise Extent
Chord-wise Location
Chord-wise Extent
Depth
Severity
Levels
0.15
0.05
0.35
0.1,0.2,0.3,0.4,0.5
0.5,0.6,0.7,0.8,0.9
0.95 0.96 0.97
(not to scale)
Figure 4-5: Visualization of the damage cases included in the offline library. We
performed a full factorial enumeration over the given damage factors and levels.
w., s are non-dimensional with respect to the span; we, c are non-dimensional with
respect to the chord; d is non-dimensional with respect to the wing skin thickness;
f is a fractional loss between 0 and 1 across all material stiffness properties in the
damaged region.
neuver at V = 210 ft/s with as high a load factor n as possible. We chose
a range of the V - n space around the maneuver velocity that contained the
expected range of load factors, V E [180 ft/s, 240 ft/s] and n E [1, 3.5]. Figure 4-6 shows the region of the aircraft's entire maneuvering V - n space in
which our rectangular region resides-this is in the region where the vehicle
stall limit becomes preceded by limits due to structural failure, and it is where
we expect to see the most reduction in the maximum load factor due to damage. We define the maximum safe load factor in this region as nmax, where it
should remain fairly constant across airspeed with the region. Note however
that because our technique produces a probabilisticestimate of the vehicle capability, nrmax will now be modeled with underlying randomness, as opposed
to the notional deterministic value as shown in Figure 4-6. We will define 7n max
more precisely in the next section as resulting from a threshold applied to a
PSVM output.
4.2.3
Visualizing the library
To add each damage case to the behavioral library, we perform the capability
boundary estimation process using the technique from Section 4.1.3; this creates a set of PSVMs representing the vehicle capability in each damage case.
It would be uninformative to plot the PSVM for each damage case here (and
in general this methodology is meant to handle hundreds or even thousands
57
n
3.5 ------------nmax
------
-----
1.-------------
0-
180
240
(ft/s)
Figure 4-6: The light blue region is conceptually where the capability analysis
lies with respect to the entire aircraft maneuvering envelope. Note that in our
analysis, rmax becomes a random quantity; the deterministic plot shown here is
for qualitative explanation.
of damage cases, so it would be difficult in any realistic example). Instead, we
find a means of sorting them by a measure of severity, and then choose several representatives to obtain a qualitative understanding of how the library
damage cases modify the nominal vehicle capability.
First, it will be useful to define the variable Pthresh to refer to the value
of a minimum "cutoff" probability, so that any state having a higher probability than Pthresh will be deemed to lie within the capability set. A value of
Pthresh corresponds to a contour of a damage case PSVM output; this is shown
graphically in Figure 4-7.
Next, we can look at the intersection of the Pthresh contour with the vertical
line corresponding to a fixed airspeed V. The value of the load factor n at this
intersection is the maximum load factor predicted for the vehicle, denoted as
nmax; it is a function of the given damage case Dj, the given value of Pthresh,
and the given flight airspeed V as follows:
nmax (Dj, Pthresh,V) =
sup
nE[1,3.5]: p((Vn)ECIDj)>Pthresh
n
(4.20)
As highlighted in Figure 4-7, a reasonable cutoff probability is Pthresh = 0.95
because it seems to correspond well with the sharp drop in PSVM output
that approximates the capability boundary. If we fix V at the flight scenario
airspeed 210 ft/s, then we can use nmax(Dj, 210, 0.95) as a single number representing the severity of the jth damage case; in general, less severe cases will
58
0.2 0.4 0.6 0.8
Detail A
3.5
3
2.5
2
1.5
180
200 220
V (ft/s)
1
240
Pthresh
Contours of PSVM output
Figure 4-7: Plot showing how a value of Pthresh corresponds with a contour of
constant probability of the PSVM output of a damage case. The value pth.es =
0.95 was chosen to develop representative cases for the damage case library.
have higher values of nmax(-, 210,0.95), and the undamaged case should have
the highest value.
With this in mind, we sort the damage cases in the library by their values
of n ,max(.,
210, 0.95)- Figure 4-8 shows this ordering of the damage cases, as
well as the PSVM output from three of the damage cases (corresponding to
the minimum, the median, and the maximum) as representatives of the entire
library. For reference, the physical dimensions of each damage case are shown
above the corresponding PSVM, where the notation follows the damage factor
description in Figure 4-5.
It is important to note that we focus here on ordering the damage cases
based on how they affect the vehicle capability, and not by their geometric dimensions or material stiffness reduction; hence, the words "mild" and "severe"
when used in this context refer to the degree to which a damage case impacts
the vehicle capability, and not to its geometric parameters.
It is clear that the damage cases have a limited impact on the vehicle
capability set; states at n = 1 are likely to always be safe according to the
behavioral library, and states over n = 3 are likely to always be unsafe.
Figure 4-8 suggests the potential increase in performance that could be
obtained by performing dynamic capability estimation. In theory, a UAV
relying on a static capability assessment from design would either need to
operate below the cases in the damage library (i.e., below n = 1.5), or operate
in the full nominal envelope but run a significant risk of mission failure once
damaged. Dynamic estimation could allow for operation in the full range of n
59
we=0.4
0
'V
w= 0.3
0
df = 0.97
0
df = 0.98
dt
1
0
-
lC=0.35
1
0
1
1
lc =0.45
0.2 0.4 0.6 0.8
no damage
0
Wc
0.2 0.4 0.6 0.8
3.5
0.2 0.4 0.6 0.8
3.5
3.5
2.5
2
1
1
3
2.5
C
2
1.5
180
200 220 240
V (f/s) V
11
3.5
180
200
220
V (ft/s)
1M
180
240
'200V (ft/s)220
240
3
2.5
C
E
2
1
C
9
-
20
40
60
80
100
120
Behavioral case index in library
-0
140
160
Figure 4-8: Ordering of the library damage cases by nmax evaluated at the flight
scenario airspeed V = 210 ft/s and the cutoff probability Pthresh = 0.95. Three
representative cases-a minimum, median, and maximum (from left to right)are shown in detail, including both their PSVM evaluated over the entire state
space region as well as a schematic of their damage level occurring on the vehicle
(following the damage factor notation from Figure 4-5).
up to 3.0 while the vehicle is not damaged, and still maintain safe operation
by limiting the vehicle behavior once damage is incurred.
At this point, we have analyzed the behavior of the vehicle capability set
in a broad sense over all the damage cases in our library by assigning a reason60
able cutoff probability of Pthresh = 0.95, look specifically at the flight scenario
velocity V = 210 ft/s, and analyzing representative cases sorted by their resulting maximum load factor nmax (Dj, Pthresh, V). Because this library will be
used by our online estimation process, this gives us intuition into the range of
capability loss we expect to see later on when analyzing estimator performance.
4.2.4
Flight scenario test cases
During the test process, we have two main questions we desire to answer:
1. What role does damage severity play in the estimation performance?
2. What role does damage that is outside of what is represented in the
library play in the estimation performance?
We build five test cases that allow us to draw observations to answer these
questions, summarized in Table 4.1. They are identified by the following characteristics:
" Damage Level: qualitative measure of how much the damage case affects
the vehicle capability
" Included in Library: Whether we store the damage case in the online
library. Although we have a full set of damage cases modeled offline, we
restrict the online data set to see how our estimation process performs
on out-of-sample damage cases.
C1
Damage Level
No Damage
Mild
Included in Library?
Yes
Yes
C2
Severe
Yes
C3
Mild
No
C4
Severe
No
Label
Co
Table 4.1: The five representative damage cases used in the flight scenario to test
capability estimation performance.
Each of the five damage cases labeled according to Table 4.1 is shown in
Figure 4-9, where the full set of cases is sorted identically to Figure 4-8. The
"mild" cases to lie within the upper half of the full sorted set, and the "severe"
cases to lie in the lower half.
61
3.5r
3-
CO
C2
C
E
0
03
........
1.15
0
50
100
Behavioral case index in library
150
Figure 4-9: The 5 damage cases we use to analyze performance of our capability
estimation process, labeled according to Table 4.1.
As a baseline truth reference, we find the value of the maximum load factor
at V = 210 ft/s for each of the 5 example cases cases by using Algorithm 1,
a simple bisection routine that uses the aircraft model failure metric function
f (x, dj) (where dj is the vector of damage parameters corresponding to the
damage case Dj). To distinguish this from the value nmax that is related to
the PSVM built for each damage case, we call this truth reference nmat
3(D)
where it is only a function of the damage case Dj. We could shrink the stopping tolerance Tol in Algorithm 1 to achieve much higher accuracy than the
adaptive sampling algorithm, so it serves well as a truth reference (however this
is only for one airspeed V, whereas the adaptive sampling algorithm obtains
an approximation for all airspeeds and thus is more flexible).
The final nir
computed by Algorithm 1 for each damage case is listed in
Table 4.2. We assumed these values to be of sufficient accuracy to be taken
as deterministic; they are representative of the true location of the vehicle
capability boundary for the one specific flight velocity 210 ft/s.
Damage Case
ntuth
Co
C1
C2
C3
C4
2.94
2.59
2.02
2.53
1.80
Table 4.2: "Truth" reference nmax values obtained by applying Algorithm 1 to
each of the example damage cases.
62
Algorithm 1 Bisection routine for obtaining truth reference values of nmax
for damage case DJ.
nhigh +- 3.5, niow <- 1, V <- 210 ft/s, Tol <- 10-6
repeat
n <- (nhigh
njo.2
-
x <- (V, n)
if f (x, dj) = 1 then
njow
n
else
nhigh
n
end if
until |nhigh - nIow| < Tol
(nhigh - now)/2
maxrth
4.2.5
Online strain gage measurements
The observable vector values stored for each damage case during the model
sampling process in Section 4.1.3 are unprocessed strain values at select locations on the wing surface corresponding to the pre-determined locations of
strain sensors. During the online phase we gather measurements and then
need to post-process them in order to compare them to the strain values in
the library.
The field of strain sensing technologies is broad, with solutions ranging
from optical devices such as Fibre Bragg Grating Sensors [3] to recent advances
in flexible "skin-like" devices using conductive liquid embedded in elastomer
material [33]. We assert our methodology is capable of handling data from any
strain sensing technology where the number of sensing features is essentially
the dominant limiting factor, and not the nature of the sensor itself.
For our purpose and to facilitate potential integration with hardware prototypes, we model the behavior of strain gage rosettes mounted on the surface
of the aircraft wing box to obtain plane-strain measurements, shown in Figure 4-10. As suggested by the proximity of the gages to the representative
orange damage region, we make the assumption that we would have our strain
sensing technology nearby the damaged region to sense local changes to the
strain field. This allows us to validate the capability estimation process knowing our sensor measurements will show noticeable changes due to damage; the
optimal choice of sensing technology and placement is outside the scope of this
work.
63
24
Figure 4-10: Schematic showing the locations of the strain gages on the wing box
top surface in the aircraft model. The inset on the right shows the "rectangular"
variant of strain gage rosette, where three in-plane gage readings can be used to
obtain extensional strains and in-plane shear strain with respect to material axes
1 and 2.
Many varieties of strain gage rosettes exist3 ; for example here, we consider
a rectangular configuration where two gages placed on the principal composite
material axes can obtain extensional strains directly, and a third placed offaxis at 45 degrees may be used to compute indirectly the in-plane shear strain.
Let the strain gage rosette readings at the kth location for k = 1, 2, 3, 4 be
represented as the vector ek = IE
T where 1 is the gage strain along
axis 1, _ is the gage strain along axis 2, and Ek is the gage strain along the
45-degree axis. Assuming our material coordinate 1 and 2 axes align perfectly
with the rosette 1 and 2 axes, respectively (i.e., ignoring possible angular
misalignment of the gages), ck is related to the 3-element plane strain vector
at location k, Ek = [Ek 2Ek2 E
8 2 Tby
(4.21)
Ek = Hek,
where
H
1
0
-1
-1
0
1
0
21.
(4.22)
0
As mentioned in Section 4.2.1, we assume the aircraft maneuver state is
3
For further reference, see documentation from Micro-Measurements, Vishay Precision
Grp. TN-515 "Strain Gage Rosettes: Selection, Application and Data Reduction."
64
known with certainty to be (V, n) = (210 ft/s, 1.3). For this fixed maneuver,
each damage case has a set of nominal strain values as predicted by our aircraft model; these are presented for reference in Table 4.3. We make strong
Damage Case
Sensor
ID
1
2
Component
C2
C3
C4
-677
El
-1306
-1200
-464
2612
-0
340
46
-84
-0
622
622
23
-400
-84
-1192
337
46
-675
46
-675
-0
-461
46
-1303
-0
-1303
-0
23
-401
-84
-1197
339
-1198
339
46
-672
-0
-463
46
-465
46
-1298
-0
23
-402
-84
Eli
-0
23
46
-1297
-1191
2E12
-460
46
-401
-83
-672
-0
-0
23
337
46
Eli
2612
E1
4
C1
-403
622
3
CO
2E12
E22
Table 4.3: Nominal plane strain values (in units of pstrain) computed by the
aircraft model for each strain gage location in the wing, for the fixed maneuver
(210 ft/s, 1.3) and 5 example damage cases.
simplifying independence assumptions that each strain gage has noise that is
independent of all other gages (so we have 12 independent strain gage readings). High accuracy strain gages often have a 2 - 5% accuracy range when
properly calibrated 4 , and by applying this to the range of values in Table 4.3
we estimate that the noise on each strain gage measurement can be modeled
reasonably as Gaussian with zero mean and standard deviation - = 10 pstrain.
This noise assessment is first-order and is meant to be a place-holder for more
accurate, application-specific knowledge.
For completeness, we connect the strain gage sensor framework with our
notation for the observable vector noise model from Equation (2.2). For damage case Dj, 8 is a measurement of the plane strain values at all four sensor
locations; the observable vector sj is the concatenation of the plane strain
values Ek for all four sensor locations. If we let s§ be the three components of
4
See Vishay TN-505-4, "Strain Gage Selection: Criteria, Procedures, and Recommendations," available publicly at http://www.vishaypg.com/docs/11055/tn505.pdf
65
s corresponding
to the kth sensor location, then
^k
-k
+e,
(4.23)
where ek is noise in the measured plane strain values at the kth sensor location
due to strain gage measurement error. We know that the error in each strain
gage reading is zero-mean with variance c21, so by Equation (4.21),
e ~ A (0,
It follows that
(s|)
U2HHT)
~ K (g , u2HHT).
(4.24)
(4.25)
Because each sensor is assumed independent, we can assemble the full measurement noise model p (1 Dj) for damage case j as the product
p (^IDj)
=jp
(^kl JDj).
(4.26)
k
Equations (4.26) and (4.25) can then be used to generate observable vector
samples given the vehicle is in damage case j, retrieving the necessary values
of 4 from the behavioral library.
4.3
Results
The following section presents results and accompanying discussion for a series
of tests of the maximum likelihood estimator qML and mixture distribution
estimator qMD, using the flight scenario from Section 4.2.1, the damage case
library from Section 4.2.2, and the damage test cases from Section 4.2.4.
First, we compare the outputs of the two estimators with respect to the
truth references in Table 4.2, varying the number of accumulated sensor samples and the number of records used in the online library to see their impact
on the nominal estimator outputs. Second, we benchmark the estimators compared to a baseline case that uses a static capability limit based on the vehicle
design, by looking at performance in an onboard decision process.
Our first exploration compares the outputs of qML and qMD for each damage
case D E {Co, C 1 , C 2 , C 3 , C 4 } assuming the vehicle is in the flight scenario
maneuver (210 ft/s, 1.3). The procedure is as follows:
1. Using the nominal strain values in Table 4.3 for damage case D, and
the noise model for the observable vector measurement S from Equation (4.26), generate an observable vector sample S.
66
2. Compute and plot qML (X, 8^) and qMD (x,
8)
for
X E {(V, n) : V = 210 ft/s, n E [1, 3.5]}.
3. For comparison, plot the damage case's PSVM output, p (x E CID), for
the same values of x as in step (3); plot as well the true maximum load
factor nirh(D) (described previously in Section 4.2.4).
4. Repeat 10 times from step 1.
It is easiest to see this process with a visual example-one repetition of steps
1-3 is shown in Figure 4-11, which shows the qML output for the C 2 damage
case. The dotted line shows the damage case's true maximum load factor
n~max'
truth and the black curve is the PSVM
fit p(X
PIMft
( E C|C2)
j 2 constructed in the
offline phase (the two should, and do, agree well). The green line is the qML
output, that is it is a "slice" of the most likely damage case's PSVM output
given s = S across the velocity V = 210 ft/s, and it reports the probability that
the label for the query state (V, n) is +1 (i.e., that the label for x is "safe").
Intuitively, a well-performing qML will be close to 1 for low values of n, and
drop to 0 as n crosses the maximum load factor for the damage case. We will
continue to refer to these curves as "PSVM outputs" or "estimator outputs"
in the rest of the discussion.
ntruth (Dj)
(found via bisection)
9
1
0.8
0.6
o
0.4
PSVM contour of damage case Dj
0.2
0
1
'
1.5
2
-
n
2.5
-
-
3
3.5
Figure 4-11: Explanation of the probability curve plots used for analysis of qML
and qMD. The vehicle state is x = (V, n), however V stays fixed at 210 ft/s so
we need only plot the value of n.
67
4.3.1
Comparison of estimator outputs
In the example case shown previously in Figure 4-11, the estimator agrees
reasonably well with the truth reference. However, in general the estimator
output varies between sensor samples. Our first exploration seeks to mitigate
the sensor noise to see the relative improvement in the estimation quality.
Although the strain gage noise level was fixed at 10 pstrain, a simple means
of improving the estimate would be to "augment" the observable vector to
include NS concatenated samples, assumed to be iid. The modified likelihood
is an extension of Equation (2.5)),
NS
S(D-
$
=
logp (k IDj) ,
(4.27)
k=1
and it can act as a "drop-and-replace" expression for the original single-sample
likelihood throughout the formulation of qML and qMD. We will continue to
use the original expressions qML (X, ^) and qMD (x, $) as shorthand for what
are now ML (x, $1,.--- , Ns) and qMD (x, $1,--- , Ns), where the relevant value
of NS will be clear from context.
In a real-time situation, we could accumulate samples over a period of
time-for instance, if NS = 10, a sampling rate of 10 Hz would allow a capability estimate every 1 second (note that a parameter such as the sensor
sampling rate is highly system-dependent, and so our exploration is more to
compare in a relative sense the impact of NS). From hereon, we will refer to
NS as the sample accumulation.
We ran four cases of the sample accumulation NS E {101, 102, 103, i04}
over each of the five damage cases, and repeated each experiment 10 times.
The resulting 20-element plot matrix is shown in Figure 4-12 for qML and in
Figure 4-13 for qMD. Each plot corresponds to one experiment and has its 10
estimator runs plotted on top of each other, so the spread in the estimator
outputs can be analyzed qualitatively.
We can make the following five observations.
1. The qML estimator output has a consistently S-shaped output.
This is because qML directly uses the PSVM from the most likely damage
case, and all the damage cases were fitted with the S-shaped sigmoid;
the qMD estimator uses a weighted sum of these sigmoids, and thus is
not guaranteed to have any particular shape. We note as a result that
the qMD estimator shows a more consistent output as it is able to blend
the outputs of all the damage case PSVMs together.
68
2. Several estimator outputs show large right-hand "tails" (especially pronounced for qML, for instance in experiments (C1 , 101) and
(C 2 , 101)). This is because the underlying PSVM training process used
for each damage case is approximate; some damage cases have PSVMs
that assign an artificially high probability to the vehicle state being "safe"
at high n, due to lack of data near the boundary of the rectangular state
= 101
NS
C101
0.5
=102
-104
-103
1,
1
1
0.5
0.5
0.5
Co
I-,~
1
2
3
1
L
0.5.
01
2
3
1
0.5E
0VI
1
2
3
0
2
1
0
3
6
3
2
1
1
0.5
0.5
0.5
0
2
1 2
01
3
01
2
1
1
1
0.5
0.5
0.5
L
0
0
1
2
3
2
1
0.5
2
n
1
3
3
2
01
3
0
1
2
3
01
1
0.5
0.5
0.5
2
1
3
2
2
3
C2
2
3
C3
1
2
n
n
3
0.5
1
1
2
Cl
01
1
-3
0.5
11
01
06
1
1
0.5
01
1
3
2
3
C4
1
2
3
n
Figure 4-12: Output from the qML estimator for x = (V, n) over the load factor
range n E [1,3.5] and the fixed flight airspeed V = 210ft/s, for four sample accumulations NS E {101, 102, 103, 104} and the five example damage cases
CO, C 1 , C 2 , C 3 , C 4 . Each experiment is repeated for 10 independent trials, and
the outputs are stacked together to show the spread in the estimator behavior.
69
region. This is confirmed by the mirrored case of occasional left-hand
tails, appearing when the PSVM is not confident the vehicle state is safe
at very low values of n.
3. Both estimators improve in performance by accumulating more
samples in the observable vector. The NS
103 case matches the
NS
=101
=103
=102
0.5
0.5
1
2
000
3
0.5
0.5
1
2
3
20
2
3
12
A
0.5
1
2
<Ci
0.5
2
6
3
2
0.5
1
2
3
0
1
2
6
3
0
1
1
1
0.5
0.5
0.5
1
2
01
3
0LL
3
2
3
C
1
3
C
0.5
1
0.5
01
0.5
1
0.5
01
104
2
3
1
1
1
0.5
0.5
00
1
2
3
1
2
2
2
2
3
2
n
3
1
2
3
1
n
2
n
3
1
2
2
C
3
C
4
3
3
0.50.5
1
C
3
n
Figure 4-13: Output from the qMD estimator for x = (V,n) over the load factor
range n E [1, 3.5] and the fixed flight airspeed V = 210 ft/s, for four sample accumulations NS E {101, 102, 103 10} and the five example damage cases
CO, C1 , C2 , C3 , C 4 . Each experiment is repeated for 10 independent trials, and
the outputs are stacked together to show the spread in the estimator behavior.
70
true PSVM curve well for both qML and qMD in all damage cases, and
the NS = 10' case has all 10 runs line up almost identically over ntruth.
We note however that the NS = 104 case is unrealistic for many real-time
systems onboard an aircraft 5, and is more for comparison purposes
the 10' case shows both high performance and plausibility, where the
majority of both estimator's outputs overlay the true nmax location.
4. There is little difference in the outputs for damage cases that
are in the library (CO, C 1 , C 2 ) versus those that are not (C 3 , C 4 ).
We hypothesize this is because the library still contains almost its full
population, and there are nearby damage cases to C3 and C4 that have
similar values of n truth
5. The mixture distribution estimator qMD performs more precisely at low values of NS. Specifically for the 101 and 102 cases, its
outputs over the 10 trials for each experiment cluster closer than those
of the qML estimator. However, it comes at added computational cost,
because we must query the PSVM from each damage case every time we
call qMD, as opposed the qML estimator which only needs to query the
most likely damage case's PSVM.
Now that we have observed the impact of increasing the sample accumulation NS -essentially decreasing the impact of sensor noise on the estimation
process-we test to see how the damage library itself affects the quality of the
estimation. Although the two damage cases C3 and C4 are not included in the
library, we have still been using all other ~ 150 cases. By down-sampling the
number of library cases, we obtain a sparser set to use in the online phase; we
hypothesize this will make our estimators more precise in the consistency of
their predictions, but also less accurate when compared to the true maximum
load factor.
Figure 4-14 shows an example of the down-sampling process. We define
the down-sampling ratio, or DSR, to be the ratio of the number of original
library records to the number of down-sampled library records. A given value
of DSR works as follows:
1. The full set of offline library records are ordered from least- to mostsevere according to their prediction of nmax at the fixed flight speed
V = 210 ft/s and cutoff probability Pthresh= 0.95 (as in Figure 4-9)
5
For example, a high-performance benchtop strain gage Data Acquisition (DAQ) system
produced by National Instruments is able to supply 8-channel, 24-bit samples at 25 kHz; see
the white paper at www. ni. com/white-paper/3642/en for further details
71
2. Iterating sequentially along the sorted list of records, 1 out of every DSR
records is retained for storage in the online library.
3. The three example damage cases CO, C 1 , C 2 are always included in the
library after the execution of step 2.
3
DSR = 1
-
.
c 2
1
0
DSR =10
-
3
c2%
50
100
Behavioral case index in library
C
150
2
/0
0
0
I
. --
CO
000a
o1010
5 C1
150
50
0
100
Behavioral case index in library
3
DSR = 20
*
E
00
1'
0
50
100
Behavioral case index in library
150
Figure 4-14: The damage case down-sampling procedure produces a sparser set
of records for the online library used by qML and qMD. The down-sampling ratio
DSR controls the rate at which samples are retained from the original complete
library; the "included in library" damage cases CO,C 1 , and C 2 are always retained
after the initial down-sampling process as exceptions.
We perform the same analysis as was done in Figure 4-13, except we now
fix the sample accumulation NS at 103, and analyze the effect of changing the
down-sampling ratio DSR over the values {1, 10, 20, 40} (where 1 is the original
fully-populated library to act as a reference case). The resulting outputs from
qML are shown in Figure 4-15 and the resulting outputs from qMD are shown
in Figure 4-16. Several observations are apparent:
1. The estimators become more consistent while at the same time
more inaccurate as the down-sampling ratio DSR increases
that is, the results confirm our initial hypothesis.
For instance, the
qML output for the (C,1) damage case begins with a strip of possible
72
DS R
11
11
03
0.5
0.5
3
2
1
0.5
0.5
2
1
3
1
3
2
1
0
3
2
07
1
1
2
3
0.5
3
1
n
C
0.5
2
3
2
3
C
0
1
3
2
n
n
2
1
0
-
0
2
11
1
1
1
(CIn
0.5
0
0.5
.5 0.5
0.501
3
2
1
1
0
0
0.5
3
2
1
3
2
10W
1
1
C.5
1
1
1
1
3
2
1
0.5
.50.5
<C
3
2
1
1
3
2
1
0.5
0.5
0.5
0.5
=40
=20
=10
=1
3
1
n
Figure 4-15: Output from the qML estimator for x = (V, n) over the load
factor range n E [1,3.5] and the fixed flight airspeed V = 210ft/s, for four
down-sampling ratios DSR E {1, 10, 20, 40} and the five example damage cases
Co, C1, C 2 , C 3 , C 4 . Each experiment is repeated for 10 independent trials, and
the outputs are stacked together to show the spread in the estimator behavior.
outputs surrounding n',,
where it proceeds to degrade in quality for
the (C 3 ,40) case; at this point, it is consistently reading a single damage
case PSVM output that is incorrect. The qMD output for the C, damage
case shows comparable behavior.
2. At higher DSR values, there is a distinction between the estimator outputs for the in-library and out-of-library damage
73
DSR
=10
=1
0.5
0.5
<cI
1
1
0.5
0.5
01
2
1
0
0.5
0.5
1
3
2
1
0
2
0
1
3
2
0
C
0.5
01
3
Co
0001
0.5
1
3
1
0
3
2
1
=40
=20
1
2
3
01
1
1
1
0.5
0.5
0.5
2
3
(.5
1
2
3
10
0
1
2
1
0
3
0
1
3
2
3
3
2
1
0
0
1
2
60
0
2
3
1
1
1
0.5
0.5
0.5
0.5
1
2
n
0
3
1
2
0
3
n
1
2
n
3
C
0
1
1
0
3
2
0.5
1
3
0
1
01
0.5
0.5
0.5
1
C2
2
3
3
C4
0
2
1
3
n
Figure 4-16: Output from the qMD estimator for x = (V, n) over the load
factor range n E [1,3.5] and the fixed flight airspeed V = 210ft/s, for four
down-sampling ratios DSR E {1, 10, 20, 40} and the five example damage cases
CO, C 1 , C 2 , C 3 , C 4 . Each experiment is repeated for 10 independent trials, and
the outputs are stacked together to show the spread in the estimator behavior.
cases; the in-library cases Co, MI, SI tend to have higher precision,
with varying accuracy (i.e., the estimators are more "certain" but not
always correct in their output), whereas the out-of-library damage cases
C 3 , C 4 show a bi-modal trend, where both estimators seem to vary be-
tween two likely outputs. The bi-modal behavior is clear especially in the
C 4 cases for both estimators; this is the most severe of the out-of-library
cases, and it is a difficult case to handle for the estimators.
74
This concludes the first part of the test cases, where we qualitatively compared the behavior of the qML and qMD estimation strategies.
4.3.2
Flight scenario performance benchmark
Now that we understand better the behavior of our capability estimation algorithms, we compare their performance to a baseline case. For the rest of
this section we continue to assume the vehicle is operating at V = 210 ft/s, so
in the flight scenario we look to discover the maximum load factor at which
an agent can operate the vehicle in a safe manner. In general, we will denote
the agent's choice of maximum load factor as no,, with superscripts to identify
specifically whether the agent uses a static or dynamic capability estimate.
We benchmark against a case where the agent uses a static capability estimate based off the known maximum load factor from design, no. We assume
no is equivalent to nr h (Co), as the case Co has no damage to the vehicle structure. The agent then chooses to operate the vehicle at a maximum
load factor nstatic c [1, no). A ntatic value near 1 would indicate conservative
behavior and a nstatic value near no would indicate aggressive behavior. We
assume that operating at n'tatic = no would always fail because the maximum
load factor represents a point of structural failure.
By comparison, an agent using our dynamic capability estimate operates
the vehicle at a maximum load factor that changes depending on the current
sensor readings. We denote the maximum load factor the agent chooses to
operate the vehicle at as nML for the case that they use estimator qML, and
n MD for the case that they use estimator qMD. In order to choose a value for
ML orMDa
nop
or nMp1D the agent picks the largest load factor that has an acceptable
probability of belonging to the vehicle capability set; we denote this acceptable probability as pop. A value of pop near 1 indicates conservative behavior,
whereas a value of pop near 0 indicates aggressive behavior.
Figure 4-17 presents an example of how an agent would use the qMD Output
to choose a value of nMD _setting pop = 0.9 would cause the agent to operate
the vehicle at a maximum load factor of noMD ~ 1.75. In the case shown, nMD
belongs to the capability set because it is less than the true maximum load
factor n Iru; however in general this may not be the case, and the agent could
cause the vehicle to fail by operating at too high a load factor.
Now, we compare the behavior of the static estimate with the dynamic
estimates from qML and qMD by varying pop and n'tic. We perform the
following procedure:
1. Choose values for nstatic and pop.
75
~truth(j
(found via bisection)
1
9
Pop = 0.9
0.8
0.6
0.4
0.2
0
1
1.5
2
2.5
3
3.5
nMDop~ 1.75
Figure 4-17: Definition of the quantity pop as a choice an agent makes as to
their degree of conservativeness when using the capability estimator output. pop
determines the maximum load factor at which the agent should operate the vehicle
given the current estimator output. In the case shown, the agent would operate
the vehicle successfully because the chosen maximum load factor is less than the
true value.
2. Set the vehicle library down-sampling ratio DSR and the sample accumulation NS to nominal values of 10 and 100, respectively (see the
previous section Section 4.3.1 for explanation). Down-sample the full
vehicle library according to DSR and store it for continuing use.
3. For each example damage case Dj, j = 1, 2,. . . , R from the original full
set of damage cases (prior to down-sampling), accumulate NS observable
vector samples. Use the qML output curve and pop to compute nop; use
the qMD output curve and pp to compute n MD (as shown in Figure 4-17).
4. Repeat steps 2-3 for 10 trials, and save the resulting values of
nfaoc nL MD, as well as the true maximum load factor for the current
damage case, nmax (D).a
,D, overlaying all R damage case
5. Plot the nmah versus sp
L , MOP
a
samples on the same plot.
This was repeated over the values nstatic c {1, 1.5, 2, 2.5, 3} for the static estimation case, and over the values pop E {0.01, 0.26, 0.50, 0.74, 0.99} for the
76
dynamic cases.
The results are shown in figure 4-18. Each subplot corresponds to a choice
by the agent; the dynamic cases are labeled by the choice of p, and estimator
qMD or qML, whereas the static cases correspond to the agent's choice of nstatiC
Within each subplot, a sample corresponds to an operation of the vehicle given
a vehicle damage case and a realization of the observable vector sample. The
x-axis is the true maximum load factor ntruth and the y-axis is the maximum
load factor chosen by the agent no
0 . The black lines with slope 1 indicate the
boundary between successful and unsuccessful operations: the blue samples are
successful because n
< ntruth, and the red are unsuccessful because n
truth
nmax
The static case is the most simple; the agent chooses one value of nstatic and
then sticks with it regardless of any sensor data indicating changes to nt'ut.
to decrease is
Hence, many trials fail because any damage that causes n
ignored by the agent.
The dynamic cases use our capability estimation strategy; depending on
the value of pop, the agent is either less or more conservative. In the less
conservative cases (e.g., when pop = 0.01), the agent operates at higher values
of no,, at the cost of having many more failed operations. More conservative
cases (e.g., when pop = 0.99) have many more successful operations, at the
cost of operating at a lower average value of nop.
It is not clear from Figure 4-18 which of nL or MD performs better out of
the two. But in general, it appears that for a higher value of nop, the dynamic
estimates are able to out-perform the static estimate because there are more
blue (i.e., "safe") instances of the vehicle operation. We want to explore this
behavior in more detail.
This leads us to the question: How does the decision strategy, informed by
the capability estimate, trade off reliability with full utilization of the vehicle
capability? We can quantify this question by looking, for each value of pop (or
for the static estimation case), at
n static
OP
1. the probability of mission success over all possible damage cases to quantify reliability, and
2. the average ratio between the vehicle's operational load factor and the
vehicle's true maximum load factor, to quantify full utilization of the
vehicle capability.
We can compute these metrics by looking at the data in each pop (or natiic)
subplot in Figure 4-18; the following is a derivation of how we mathematically
define and compute these metrics. For this discussion, we use no, to refer to
all of nML ,
MD
and n'tatic where the computations remain the same for all
77
= 0.01
pop
2
3
2
= 0.26
3
2
2
2
2
3
2
23
Static
,2
1 23
1
truth
max
3
2
-
1
3
2
3
-
2
1
truth
nmax
3
= 0.99
31
1
3
= 0.74
31
3
123
1 2
0.50
3
12
Dynamic
=
2
3
truth
1
3
3
2
2
3
truth
1
Umax
3
11
3
2
2
1
truth
3
2
3
2
3
1
truth
Umax
nmax
12
2
41
-
2
2
2 3
1
2 3
truth
Umax
2
3
Umax
3
truth
3
truth
Umax
3
Umax
2
1
-
2 3
truth
Umax
Figure 4-18: Plot of the maximum load factor chosen by an agent in the flight
scenario, using each of the three estimation strategies (nratic using a static choice
of load factor, nrL using the qML estimator, and n D using the qMD estimator)
for 10 realizations of the observable vector in each of the R library damage cases.
The decision made in the dynamic cases depend on the choice of pop as explained
in Figure 4-17; The decision in the static case is simply the chosen value of nstatic
Red samples indicate the decision led to failure, because the chosen maximum load
factor exceeds the true value nIruth for the sampled damage case; blue samples
indicate a safe decision was made.
three cases. For a fixed value of pp (or SF), nop is a function of the following
random quantities:
" D: the current vehicle damage case, taking one of the values D , D ,... , DR
1
2
where R is the number of damage cases in our library. For this analysis,
we make the approximation that the vehicle can only be in one of these
damage cases, i.e. that the approximation made in Equation (2.9) holds.
" s: the current vehicle observable vector measurement.
The true vehicle maximum load factor n Irh
vehicle damage state D.
78
is a function of only the current
The two metrics we compute for each value of po,
(or
nstatic
are:
1. The probability of mission success: this is the probability that the agent
decides on a n that is less than the maximum vehicle load factor n"'ut.
We denote the event that the agent succeeds as MS, where
MS
= {nO(D,
§) <
(4.28)
truth (D)}.
The probability of mission success p (MS) is computed as
R
p (MS)
=
(4.29)
1 p (MSIDj) p (Dj) .
j=1
The quantity p (MSI D) is the probability the agent succeeds given the
vehicle is in damage state D = Dj; it is approximated by the fraction of
the 10 trials for damage case Dj that are successful (obtained in step 4
of the previous procedure). For this analysis, we assume a uniform prior
over all the damage cases so p (Dj) = 1/R for all D.
2. The average ratio between the vehicle's operational load factor and the
vehicle's true maximum load factor: this is the expected value of
nOP(D, S)/ntuth(D) conditioned on the event MS. The conditioning is
because the vehicle capability is only utilized when it does not fail; only
the cases where the agent chooses a safe maximum load factor contribute
positively to the utilization of the vehicle capability.
We denote this metric as hutiI, and compute it as
na
= E[
uti
n (D S)
(D, S)
MSDII
truth
E
=E
MS
ruth (D)n
..
Lmax max
R
n (ys
I, SW Dj p (D ) .
P
=EE
max (Dj
j=1
n_
(4.30)
(4.31)
Now, we can simplify Equation (4.31) by recognizing that
no (D,
E ntuhj
s)
E [nop(Dj, S) MS, Di
1
ms' Dj -
The quantity E [n0 (Dy,
s) MS,
.tut
(4.32)
Dj1 is the mean of the agent's chosen
nop given that the vehicle is in damage case Dj and that no
0 is less than
We approximate this value using the sample mean of all successful
ntrt,.
79
trials out of the 10 total that were conducted for D = D (the same set of
samples from the p (MS) computation above). As mentioned, we assume
a uniform prior over all the damage cases so p (Dj) = 1/R for all D,.
Once we have computed iut1 and p (MS) for each value of pop (or nsatic)
we can plot them together on a single graph. We extended the process to a
finer resolution than just the five values of pop and ntajtc shown in Figure 4-18,
instead refining to 500 equally spaced values of pop from 0.01 to 0.99, and 500
equally spaced values of nstatic from 1 to 3. We can then plot all (nui, p (MS))
pairs together for each of the three decision strategies, as shown in Figure 419. The ideal decision strategy would have both perfect usage of available
capability (i.e., huti1 = 1) and certain mission succss (i.e., p (MS) = 1); this is
marked as the "Utopia" point in the upper right corner. For a sample not at
the Utopia point, it is considered to be non-dominated if no other sample has
a higher p (MS) or hutil value without a low value of the other one; the non-
dominated combinations of (nutil, p (MS)) for each estimator are connected by
a dotted line of the corresponding color.
There are several immediate observations to make about the data in Figure 4-19:
1. The static capability case has a probability of mission success equal to 1
at values of hutil < 0.75-this is because if the agent sets a low enough
static load factor, it becomes less than nruh(Dj) for all j = 1,... R,
and so within the scope of this analysis it appears as though the vehicle
would never fail.
2. The static capability case has a long trail of samples near p (MS) = 0 at
high values of hutiI-this is because at values of n-atic close to 2.9, the
vehicle almost certainly fails unless it is in the pristine case, which has
a small probability (< 0.01) of occurring.
3. Of the two dynamic capability estimators, the qMD estimate has the
most even spread of points across its non-dominated front, whereas the
qML estimate has apparent "fibers" of points that terminate on the nondominated front.
4. The "fibers" of the qML estimate all tend to have a positive slope, with
discrete jumps down in p (MS) to the next fiber as hutil increases to the
right. This is most likely because qML only outputs the PSVM of the
most likely damage case, so its output is restricted to only R possible
forms; whenever an increase in the pop value causes the nML value to
change from "safe" to "unsafe" for a particularly frequent output curve
80
"Utopia'N r
1
0.9
0.8
0.7
0.6
0.5 - Static capability
0.4
0.3
Dynamic capability from MD
0.2
Dynamic capability from ML
0.1
'V
I
.7
0.75
0.8
__
MMEDE=
0.85
0.9
0.95
1
nutil
Figure 4-19: Plot showing the average fraction of the vehicle capability utilized
(Auti1) versus the probability of success (p (MS)) for an onboard decision process
performed by an agent using one of the three capability estimation strategies. The
non-dominated points for each capability estimation strategy are connected by a
dotted line of the corresponding color.
of qML, the value of p (MS) drops sharply as all these outputs suddenly
cross the safe threshold. By comparison, the qMD estimate has a more
gradual transition.
In addition to these observations about the point samples, we can use the
non-dominated fronts as a measure of performance of each capability estimate
when used for decision-making in the flight scenario. For instance, if the agent
wants to utilize 95% of the maximum vehicle load factor on average, then
they would have a 80% chance of success using the qMD estimate of the load
factor as opposed to a 40% chance of success when operating at a static load
factor. On the other hand, if the agent can accept operating at less than 80%
81
of the maximum capability on average, then all three estimators show similar
performance-we note this is most likely because the damage cases in the
library cause a limited reduction in the vehicle capability, and simulating more
severe damage cases would continue to emphasize the improvement gained by
the dynamic capability estimate.
Table 4.4 summarizes important values from Figure 4-19. The qMD estimator displays improvement over both other strategies until niitil < 0.82,
when the qML strategy begins to perform within 5% of the qMD probability of
success.
p (MS)
nutil
Static
Dynamic: qMD
Dynamic: qML
0.95
0.89
0.82
0.40
0.72
0.80
0.77
0.72
0.87
0.64
0.85
0.91
Table 4.4: Numerical results obtained from the p (MS) - hutil tradeoff curves in
Figure 4-19 to compare the three estimation strategies based on performance in
the onboard decision process.
4.3.3
Limitations
We make note of limitations of results obtained previously in Sections 4.3.1
and 4.3.2 as to their applicability and underlying assumptions.
First, the damage case library was generated using a full factorial over a
reasonable range of damage geometries at a fixed span-wise location on the
aircraft wing. A full analysis would perform a sampling of damage regions
across the entire wing structure, or would steer the choice of cases according
to a given prior distribution. As a result, we saw only a limited impact on the
vehicle capability (as quantified by the maximum load factor nma at the fixed
airspeed V = 210 ft/s and threshold probability Pthresh = 0.95) and potentially
missed interesting regions of the damage geometry parameter space.
Second, we used a uniform prior over the damage cases during our analysis.
In a realistic scenario, severe damage cases are more unlikely than the pristine
case; use of a better damage prior would lead to more realistic results. We
hypothesize the average performance of the static capability estimate would
improve, because the usefulness of the dynamic estimate is magnified by a
higher probability of damage, and reduced if damage events are unlikely.
82
Third, the discussion in Section 4.3.2 relied on sample averages to obtain
the trade-off curves in Figure 4-19. This is acceptable for our case that uses
a Gaussian sensor noise model, however in general this may not be a reliable
means of condensing statistical information from the decision process.
Lastly, the results for the flight scenario were obtained assuming full knowledge of the current vehicle maneuver; the full aircraft capability estimation
framework is designed to handle uncertainty in the maneuver, however our
first analysis here makes the simplification to ease explanation and presentation of numerical results; future work will seek to incorporate maneuver
uncertainty, given it is a significant factor in real vehicle systems that identify
the vehicle state using imperfect attitude sensors and air data systems.
83
84
Chapter 5
Conclusion
5.1
Summary of Results and Current Work
Dynamic flight capability estimation has the potential to dramatically increase
the relative usage of the vehicle maneuvering envelope when changes occur to
the nominal design capability. This is particularly relevant when compared to
prevalent planning techniques that rely on the nominal design capability, and
when modern high-power computing can be leveraged before the vehicle is in
operation to pre-compute and store information about its behavior.
This thesis explored the concept of dynamic capability estimation from
a data-driven perspective, where models and experimentation can both be
sources of information and where the vehicle behavior is analyzed cognizant
of model uncertainty. We formulated a quantitative definition of vehicle capability in Chapter 1, and then developed a general framework in Chapter 2
for estimating vehicle capability using offline models and experimentation. We
developed an example of the offline framework using a model of a UAV sustaining wing structural damage in Chapter 3, applying this model to the capability
boundary identification process in Chapter 4. This process enabled the construction of the vehicle behavioral library that was used further in Chapter 4
in a capability-informed decision-making process in an example flight scenario.
Results from Chapter 4 demonstrated the benefit of incorporating vehicle sensor information into an updated estimate of the structural capability.
The baseline case was a static estimate based on the known vehicle structural
design limits; the dynamic estimate outperformed in the case that damage
to the vehicle was a likely event. Two dynamic estimation processes were
used-a maximum likelihood estimator and an estimate based on a mixture
distribution-and results indicated the mixture distribution to perform the
85
most consistently, with the drawback of additional required computation for
querying the classifier from each record in the behavioral library. Recommendations from this thesis are for the continuing use of the mixture distribution
estimator for dynamic vehicle capability estimation, and to look into speedups to the behavioral library queries, with the end goal in mind of performing
time-constrained capability estimation for the onboard decision maker.
Although the aircraft application presented throughout Chapters 3 and 4
relied on computational models, the methodology in Chapter 2 is open-ended
to allow incorporation of experimental data. Recent fabrication of structural
coupon samples meant to mimic the behavior of the aircraft wing box will allow
experimental data to be "interleaved" with outputs from the aircraft model
when constructing the offline behavioral library. This has the potential to
provide further real-world validation of the integrated aircraft capability model
described in Chapter 3, in particular the merits of using the simplified damage
representation in Section 3.3 to capture the effects of damage on capability
when identification of the exact damage parameters are not the end goal.
5.2
Future Work
In addition to work in the aircraft application part of this research, inference
methods employed to formulate the maximum likelihood estimator qML and
the mixture distribution estimator qMD are being improved. In particular,
accumulating samples according to NS (see Section 4.3.1) could be improved
in the case of the qMD estimator by using a recursive Bayes' formulation. In
Equation (2.9), the posterior p (D IS) computed for sample k could be fed
into the Bayesian update in Equation (2.10) by taking the place of the prior
p (D). Specifically, assuming
" that when conditioned on the vehicle damage case, all samples are independent, and
" that we have observed k sensor samples and computed the posterior
p (D
IS,,...,ISW),
then the Bayesian update in Equation (2.10) for adding the information from
the newest observable vector sample k + 1 takes the form:
P (Dilsi, .
.. , sk, sk+1)
--
p(4k+1|IDj) P (Djjsi, . .. , k)
S
=
j/y __ P (S IDj ) p (Dy
Js41,. ..
, s^)
(5.1)
The first sample would still be computed based upon a suitable prior p (Dj)
over the damage cases in the behavioral library, however accumulation of many
86
sensor readings would mitigate the effect of the choice of prior.
Continuing the inspection of the current implementation of the online inference process, a direction for future work is to improve on the strong assumption
that the Law of Total Probability can be applied in Equation (2.9)-this is an
approximation requiring more careful evaluation, and it reflects the inherent
error created by storing only a subset of the possible vehicle damage cases.
A bound on the error induced by this approximation that does not require
knowledge of the true vehicle capability could be beneficial to an agent using
the estimator.
87
88
Bibliography
[1] Brintrup, A. M., Ranasinghe, D. C., Kwan, S., Parlikad, A., Owens,
K., and Company, T. B., "Roadmap to Self-Serving Assets in Civil
Aerospace," Proceedings of the 1st CIRP Industrial Product-Service Systems (IPS2) Conference, Cranfield University, April 2009.
[2] Willis, S., "OLM: A Hands-On Approach," ICAF 2009, Bridging the Gap
between Theory and OperationalPractice, Springer Netherlands, Rotterdam, May 2009, pp. 1199-1214.
[3] Staszewski, W., Tomlinson, G., and Boller, C., Health Monitoring of
Aerospace Structures Smart Sensor Technologies and Signal Processing,
John Wiley & Sons Ltd, 1st ed., 2004.
[4] Benedettini, 0., Baines, T. S., Lightfoot, H. W., and Greenough, R. M.,
"State-of-the-art in integrated vehicle health management," Proceedings
of the Institution of Mechanical Engineers, Part G: Journal of Aerospace
Engineering, Vol. 223, No. 2, March 2009, pp. 157-170.
[5] Ortiz, E. M., Babbar, A., Syrmos, V. L., Clark, G. J., Vian, J. L., and
Arita, M. M., "Multi Source Data Integration for Aircraft Health Management," Fourth IEEE International Workshop on Engineering of Autonomic and Autonomous Systems, Tuscon, AZ, 2007.
[6] Gorinevsky, D., Mah, R., Srivastava, A., Smotrich, A., Keller, K., and
Felke, T., "Open Architecture for Integrated Vehicle Health Management," AIAA InfotechLA erospace 2010, No. 2010-3434, American Institute of Aeronautics and Astronautics, Reston, Virigina, April 2010.
[7] Fox, J. and Glass, B., "Impact of integrated vehicle health management
(IVHM) technologies on ground operations for reusable launch vehicles
(RLVs) and spacecraft," 2000 IEEE Aerospace Conference. Proceedings,
Vol. 2, IEEE, 2000, pp. 179-186.
89
[8] Kirk, B., Schagaev, P. I., Wittig, T., Kintis, A., Kaegi, T., Friedrich, F.,
Ag, E. T., Spirit, S. A., Avenue, S., and Faliro, P., "Active Safety for
Aviation," 6th INO Workshop, EUROCONTROL Experimental Centre
(EEC), 2007.
[9] Mehr, A. F., Tumer, I., and Barszcz, E., "Optimal Design of Integrated Systems Health Management (ISHM) for Improving the Safety
of NASA's Exploration Missions: A Multidisciplinary Design Approach,"
6th World Congresses on Structural and Multidisciplinary Optimization,
Rio de Janeiro, June 2005.
[10] Sohn, H. and Farrar, C. R., "Damage diagnosis using time series analysis
of vibration signals," Smart Materials and Structures, Vol. 10, No. 3, June
2001, pp. 446-451.
[11] Farrar, C. R. and Lieven, N. a. J., "Damage prognosis: the future of structural health monitoring." Philosophicaltransactions. Series A, Mathematical, physical, and engineering sciences, Vol. 365, No. 1851, Feb. 2007,
pp. 623-32.
[12] Farrar, C. R., Doebling, S. W., and Nix, D. a., "Vibration-based structural
damage identification," Philosophical Transactions of the Royal Society
A: Mathematical, Physical and Engineering Sciences, Vol. 359, No. 1778,
Jan. 2001, pp. 131-149.
[13] Prudencio, E., Bauman, P., Williams, S., Faghihi, D., Ravi-Chandar, K.,
and Oden, J., "A Dynamic Data Driven Application System for Real-time
Monitoring of Stochastic Damage," Procedia Computer Science, Vol. 18,
Jan. 2013, pp. 2056-2065.
[14] Raymer, D., Aircraft Design: A Conceptual Approach, American Institute
of Aeronautics and Astronautics, 3rd ed., 1996.
[15] Kordonowy, D. and Toupet, 0., "Composite Airframe Condition-Aware
Maneuverability and Survivability for Unmanned Aerial Vehicles," Infotech#Aerospace 2011, No. 2011-1496, American Institute of Aeronautics and Astronautics, Reston, VA, March 2011.
[16] Nise, N. S., Control Systems Engineering, John Wiley & Sons Ltd, 6th
ed., 2011.
[17] Duda, R. 0., Hart, P. E., and Stork, D. G., Pattern Classification, John
Wiley & Sons Ltd, 2nd ed., 2000.
90
[18] Greitzer, E. M., Bonnefoy, P. A., Blanco, E. D. 1. R., Dorbian, C. S., Drela,
M., Hall, D. K., Hansman, R. J., Hileman, J. I., Liebeck, R. H., Lovegren,
J., Mody, P., Pertuze, J. A., Sato, S., Spakovszky, Z. S., Tan, C. S.,
Hollman, J. S., and D, K., "N + 3 Aircraft Concept Designs and Trade
Studies , Final Report Volume 1," Tech. rep., Massachusetts Institute of
Technology, Dec. 2010.
[19] Drela, M., "Integrated Simulation Model for Preliminary Aerodynamic
Structural , and Control-Law Design of Aircraft," Proceedings of the 40th
AIAA SDM Conference, No. 99-1394, American Institute of Aeronautics
and Astronautics, St. Louis, MO, April 1999.
[20] Cesnik, C. E. S. and Hodges, D. H., "VABS: A New Concept for Composite Rotor Blade Cross-Sectional Modeling," Journal of the American
Helicopter Society, Vol. 42, No. 1, 1997, pp. 27-38.
[21] Hodges, D. H., Atilgan, A. R., Cesnik, C. E., and Fulton, M. V., "On a
simplified strain energy function for geometrically nonlinear behaviour of
anisotropic beams," Composites Engineering,Vol. 2, No. 5, 1992, pp. 513526.
[22] Palacios, R. and Cesnik, C. E., "Cross-sectional analysis of nonhomogeneous anisotropic active slender structures," AIAA Journal, Vol. 43,
No. 12, 2005, pp. 2624-2638.
[23] Cortes, C. and Vapnik, V., "Support-vector networks," Machine Learning,
Vol. 20, No. 3, Sept. 1995, pp. 273-297.
[24] Boyd, S. and Vandenberghe, L., Convex Optimization, Cambridge University Press, Cambridge, 2004.
[25] Aizerman, A., Braverman, E. M., and Rozoner, L. I., "Theoretical foundations of the potential function method in pattern recognition learning,"
Automation and Remote Control, Vol. 25, 1964, pp. 821-837.
[26] Platt, J. C., "Probabilistic Outputs for Support Vector Machines and
Comparisons to Regularized Likelihood Methods," Advances in Large
Margin Classifiers, MIT Press, 1999, pp. 61--74.
[27] Basudhar, A., Computational Optimal Design and Uncertainty Quantification of Complex Systems Using Explicit Decision Boundaries, Doctor
of philosophy, University of Arizona, 2011.
91
[28] Lin, H.-T., Lin, C.-J., and Weng, R. C., "A note on Platt's probabilistic
outputs for support vector machines," Machine Learning, Vol. 68, No. 3,
Aug. 2007, pp. 267-276.
[29] McKay, M. D., Beckman, R. J., and Conover, W. J., "A Comparison of
Three Methods for Selecting Values of Input Variables in the Analysis
of Output from a Computer Code," Technometrics, Vol. 21, No. 2, May
1979, pp. 239.
[30] Du, Q., Faber, V., and Gunzburger, M., "Centroidal Voronoi Tessellations: Applications and Algorithms," SIAM Review, Vol. 41, No. 4, Jan.
1999, pp. 637-676.
[31] Lloyd, S., "Least squares quantization in PCM," IEEE Transactions on
Information Theory, Vol. 28, No. 2, March 1982, pp. 129-137.
[32] Basudhar, A. and Missoum, S., "An improved adaptive sampling scheme
for the construction of explicit boundaries," Structural and Multidisciplinary Optimization, Vol. 42, No. 4, May 2010, pp. 517-529.
[33] Park, Y.-L., Chen, B.-R., and Wood, R. J., "Design and Fabrication of
Soft Artificial Skin Using Embedded Microchannels and Liquid Conductors," IEEE Sensors Journal, Vol. 12, No. 8, Aug. 2012, pp. 2711-2718.
92