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Maneuvering Control and Configuration Adaptation of a Biologically Inspired Morphing Aircraft

Permanent Link: http://ufdc.ufl.edu/UFE0016721/00001

Material Information

Title: Maneuvering Control and Configuration Adaptation of a Biologically Inspired Morphing Aircraft
Physical Description: 1 online resource (147 p.)
Language: english
Creator: Abdul-Rahim, Mujahid
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: adaptation, adaptive, alfred, avian, biological, bird, chicken, configuration, control, flight, inspiration, maneuvering, metric, model, morphing, optimal, reference, rubber, trajectory
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Aerospace Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Natural flight as a source of inspiration for aircraft design was prominent with early aircraft but became marginalized as aircraft became larger and faster. With recent interest in small unmanned air vehicles, biological inspiration is a possible technology to enhance mission performance of aircraft that are dimensionally similar to gliding birds. Serial wing joints, loosely modeling the avian skeletal structure, are used in the current study to allow significant reconfiguration of the wing shape. The wings are reconfigured to optimize aerodynamic performance and maneuvering metrics related to specific mission tasks. Wing shapes for each mission are determined and related to the seagulls, falcons, albatrosses, and non-migratory African swallows on which the aircraft are based. Variable wing geometry changes the vehicle dynamics, affording versatility in flight behavior but also requiring appropriate compensation to maintain stability and controllability. Time-varying compensation is in the form of a baseline controller which adapts to both the variable vehicle dynamics and to the changing mission requirements.Wing shape is adapted in flight to minimize a cost function which represents energy, temporal, and spatial efficiency. An optimal control architecture unifies the control and adaptation tasks.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Mujahid Abdul-Rahim.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Lind, Richard C.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0016721:00001

Permanent Link: http://ufdc.ufl.edu/UFE0016721/00001

Material Information

Title: Maneuvering Control and Configuration Adaptation of a Biologically Inspired Morphing Aircraft
Physical Description: 1 online resource (147 p.)
Language: english
Creator: Abdul-Rahim, Mujahid
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: adaptation, adaptive, alfred, avian, biological, bird, chicken, configuration, control, flight, inspiration, maneuvering, metric, model, morphing, optimal, reference, rubber, trajectory
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Aerospace Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: Natural flight as a source of inspiration for aircraft design was prominent with early aircraft but became marginalized as aircraft became larger and faster. With recent interest in small unmanned air vehicles, biological inspiration is a possible technology to enhance mission performance of aircraft that are dimensionally similar to gliding birds. Serial wing joints, loosely modeling the avian skeletal structure, are used in the current study to allow significant reconfiguration of the wing shape. The wings are reconfigured to optimize aerodynamic performance and maneuvering metrics related to specific mission tasks. Wing shapes for each mission are determined and related to the seagulls, falcons, albatrosses, and non-migratory African swallows on which the aircraft are based. Variable wing geometry changes the vehicle dynamics, affording versatility in flight behavior but also requiring appropriate compensation to maintain stability and controllability. Time-varying compensation is in the form of a baseline controller which adapts to both the variable vehicle dynamics and to the changing mission requirements.Wing shape is adapted in flight to minimize a cost function which represents energy, temporal, and spatial efficiency. An optimal control architecture unifies the control and adaptation tasks.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Mujahid Abdul-Rahim.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Lind, Richard C.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0016721:00001


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MANEUVERING CONTROL AND CONFIGURATION ADAPTATION OF A
BIOLOGICALLY INSPIRED MORPHING AIRCRAFT



















By

MUJAHID ABDULRAHIM


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2007




























( 2007 Mi ,_ii-l Abdulrahim























In the Name of Allah, the Most Gracious, the Most Merciful



To those with whom I am hopelessly infatuated.

C'!i Ipter 1, Introduction, is dedicated to my parents, Arifa Garman and Abdulhamid

Abdulrahim, who first to introduce me to the world of creativity by encouraging me to

design, build, and explore all things mechanical.

C'!i Ipter 2, Literature Review, is dedicated to my mother, Arifa Garman, for spending

untold hours teaching me how to read and appreciate a good book.

C'!i Ipter 3, Biological Inspiration, is dedicated to my wife, Tasneem Koleilat, who is as

enamored with biology as I am with airplanes.

C'!i Ipter 4, Dynamics, is dedicated to my brother, Obaida Abdul-rahim, who was my

first co-pilot and will .J il be the one I choose whenever I need a long drive.

C'!i Ipter 5, Maneuvering Control, is dedicated to my sister, R i, Abdulrahim, with

whom I have many adventures in both vehicular and conversational maneuvering.

C'!i Ipter 6, Optimal Control, is also dedicated to my mother, who gently reminded me

of the verse "God loves a servant, who when he performs an action, perfects it".









ACKNOWLEDGMENTS

Thank you to my advisor, my committee, my colleagues, and my family. All have

been quite supportive in v--,v- that make me feel as though I uphold the fine values of the

scientific method, even during times when the MATLAB rand function produces cleaner

data and Windows Paint produces crisper images.

Thank you to Dr. David Bloomquist, for being my undergraduate honors advisor, for

taking my flying all over the US in the Cessna "N337P" Skymaster, and for mentoring

me on an untold number of research projects. Hopefully he will one d-i, forgive me for

crashing the Telemaster.

Thank you to Dr. Peter Ifju, who invited me to work in his research lab even before

I started freshman classes. Working with Dr. Ifju on the MAV competition team and

research projects was truly a delightful experience. AT ivbe one d-,- our patent will be

approved.

Thank you to my advisor, Dr. Rick Lind, for showing me the wonderful, yet often

violently-turbulent world of flight dynamics. Under his guidance and direction, I have

partly satiated my ongoing passion for conducting meaningful, significant, and delightful

research. Perhaps one day I will be respected as a scientist.

A final thank you to Adam Watkins, Daniel "Tex" Grant, Joe Kehoe, and Ryan

Causey for being very bemusing colleagues. It is doubtful that any of us would have

survived the PhD program without our communal gum-olympics, fi.-li iii- and

helicopter-breaks.









TABLE OF CONTENTS
page

ACKNOW LEDGMENTS ................................. 4

LIST O F TABLES . . . . . . . . . 8

LIST OF FIGURES .. .. .. ... .. .. .. .. ... .. .. .. ... .. .. .. 9

A B ST R A C T . . . . . . . . . . 11

CHAPTER

1 INTRODUCTION .................................. 12

1.1 M otivation . . . . . . . . . .12
1.2 Problem Statem ent ............................... 17
1.3 Dissertation Outline . . . . . . . 18
1.4 C contributions . . . . . . . . 19

2 LITERATURE REVIEW .............................. 20

3 BIOLOGICAL INSPIRATION ........................... 32

3.1 M otivation . . . . . . . . . 32
3.2 Observations of Bird Flight .......................... 33
3.3 Desired M aneuvers ............................... 37
3.3.1 Efficient Cruise ... .. .. .. ... .. .. .. .. ... .. .. 38
3.3.2 Minimum Sink Soaring . . . . . . 38
3.3.3 Direction Reversal . . . . . . . 39
3.3.4 Minimum Radius Turn . . . . . . 39
3.3.5 Steepest Descent . . . . . . . 39
3.3.6 Maximum Speed Dash . . . . . . 40
3.4 Morphing Degrees of Freedom . . . . . . 40
3.5 M orphing M options . . . . . . . 41
3.5.1 Fore-Aft Sweep of Inboard Wings . . . . 41
3.5.2 Fore-Aft Sweep of Outboard Wings ... . . 42
3.5.3 Up-Down Inclination of Inboard Wings . . . . 43
3.5.4 Up-Down Inclination of Outboard Wings . . . 43
3.5.5 Twist of Outboard Wings . . . . . ... 43
3.6 W ing Morphing Model . . . . . . . 44
3.6.1 Pigeon W ing Configuration . . . . . 45
3.6.2 Avian Morphology Studies . . . . . 46
3.7 Aircraft M orphology . . . . . . . 47

4 DY NAM ICS . . . . . . . . . 50

4.1 Aircraft Equations of Motion . . . . . . 50
4.2 Parametric Variations of EOM . . . . . .... 53









4.3 Flight Metrics and Cost Functions . . . . . 56
4.3.1 Cruise . . . . . . . . 58
4.3.2 Maneuvering Metrics . . . . . . 59
4.3.2.1 A gility . . . . . . 60
4.3.2.2 Maneuverability .... . . . 61
4.3.2.3 Aggregate maneuverability and agility metrics . ... 65
4.4 Coupling Morphing Parameters . . . . . . 69
4.5 Stability Criteria . . . . . . . 74
4.5.1 Handling Qualities Criteria (HQC) . . . . 74
4.5.2 Stable Dynamics Criteria (SDC) . . . . . 75
4.5.3 Unrestricted Dynamics Criteria (UDC) . . . . 76

5 MANEUVERING CONTROL . . . . . . . 77

5.1 Overview . . . . . . . . . 77
5.2 Single Degree-of-Freedom Morphing Systems .... . . 80
5.2.1 Robust Control Design . . . . . . 81
5.2.2 Simulation Results . . . . . . 84
5.2.2.1 Overview . . . . . . 84
5.2.2.2 Cruise flight . . . . . . 85
5.2.2.3 Maneuvering flight . . . . . 87
5.2.2.4 Steep descent flight . . . . . 88
5.2.2.5 Sensor-pointing flight . . . . . 89
5.3 Multiple Degree-of-Freedom System . . . . . 90
5.3.1 Optimal-Baseline Adaptive Control ... . . 91
5.3.2 Reference Model Design . . . . . . 96
5.3.3 Design Point Gridding . . . . . . 99
5.3.4 Simulation Results . . . . . . 104

6 OPTIMAL CONTROL . . . . . . . . 111

6.1 Rate Trajectory Generation . . . . . . 113
6.2 Trajectory and Adaptation . . . . . . 115
6.3 Trajectory, Adaptation, and Control . . . . . 117

7 MISSION RECONFIGURATION . . . . . . 119

7.1 M mission Tasks . . . . . . . . 119
7.1.1 Deployment and Recovery . . . . . 119
7.1.2 Long Range Cruise . . . . . . 20
7.1.3 Endurance Loiter . . . . . . . 122
7.1.4 Direction Reversal . . . . . . . 124
7.1.5 Steep Descent . . . . . . . 125
7.1.6 Sensor Pointing . . . . . . . 127
7.2 Mission Profile . . . . . . . . 131
7.2.1 Performance Improvement . . . . . 131
7.2.2 Morphing Joint Trajectories . . . . . 132









7.3 Summ ary . . . . . . . . . 135

8 CONCLUSIONS . . . . . . . . 137

REFEREN CES . . . . . . . . . 139

BIOGRAPHICAL SKETCH ................................ 147









LIST OF TABLES


Table page

4-1 Standard aircraft states describing vehicle motion ..... . . 51

4-2 Handling qualities criteria for morphing configuration ... . . 75

4-3 Stable dynamics criteria for morphing configurations . . . 76

7-1 Maximum sideslip and control deflections for several aircraft configurations. A)
HQC Aircraft Fig.7-1 B) HQC ming J Fig.7-5(a) C) SDC ming J Fig.7-
5(b) D)HQC maxa J Fig.7-6 . . . . . . . 130

7-2 Mission tasks and associated performance metrics relative to launch configuration 131












LIST OF FIGURES


Figure


1-1 UAVs designed for disparate tasks . . . .

1-2 Gull wing variations . . . . . .


3-1 Effect of planform shape on gliding speed . . .


3-2 Effect of wing sweep on yaw stiffness . . . .

3-3 Gliding birds articulating wing joints about longitudinal axes .


3-4 Complex wing morphing shapes achieved in gliding flight .

3-5 Asymmetric morphing in seagull wings . . . .


3-6 Rapid variations in flight path using .,.-.-ressive maneuvering .


3-7 Wing articulation about longitudinal axes . . .

3-8 Physiological representation of avian wing . . .


3-9 Composite photo of pigeon in gliding approach to landing .


3-10 Morphing joint articulations for 4-degree-of-freedom wing .

4-1 Drag polar and lift to drag ratio . . . . .


4-2 Maximum maneuverability configuration . . .

4-3 Maximum agility configuration . . . . .


4-4 4D maneuverability index . . . . .


4-5 Maneuverability metric histogram . . . .

4-6 4D agility index . . . . . . .


4-7 Agility metric histogram . . . . . .


4-8 Histogram comparison for reduced-dimension space . .

4-9 Response surface of L/D metric in reduced-dimension space .


5-1 Lateral Ho controller synthesis model . . . .

5-2 Longitudinal Ho controller synthesis model . . .


5-3 Simulated responses in cruise flight . . . .


5-4 Simulated responses in maneuvering flight . . .

5-5 Simulated responses in steep-descent flight . . .


page

. . . 13

. . . 16


. . . 34


. . . 34

. . . 35


. . . 36

. . . 37


. . . 38


. . . 44

. . . 45


. . . 46


. . . 48

. . . 58


. . . 66

. . . 67


. . . 68


. . . 69

. . . 70


. . . 70


. . . 72

. . . 72


. . . 82

. . . 82


. . . 86


. . . 88

. . . 89









Simulated responses in sensor-pointing flight . . . . .

Model reference adaptive control (\! RAC) architecture for lateral dynamics .

Linear quadratic regulator (LQR) controller used for initialization of gain ma-
trix K . . . . . . . . . . .


5-9

5-10

5-11

5-12

5-13

5-14

5-15

5-16

5-17

5-18

6-1

6-2

7-1

7-2

7-3

7-4

7-5

7-6

7-7


[rations .





1 conditions


103

103

104

104

105

106

107

108

109


Design point grid distribution based on L/D . . .

Design point grid distribution based on stability derivatives .

L/D surface for single parameter segmentation . . .

L/D surface for multiple-parameter segmentation . . .

Permissible 4D actuator trajectory between three disparate configu

Control during morphing with sinusoidal command . ...

Control during morphing with chirp command . . .

Control during morphing with sinusoidal command and poor initial

Variable reference model control with sinusoidal trajectory .

Variable reference model control with a chirp command . .

Optimal rate trajectory framework . . . . .

Optimal rate trajectory and shape adaptation framework . .

Manually piloted deployment and recovery configuration . .

Maximum L/D configuration . . . . .

Maximum endurance and minimum rate of descent configuration .

Steep descent angle configuration . . . . .

Maximum trimmable sideslip . . . . .

Maximized sideslip cost function configuration . . .

Morphing joint angle trajectories for mission tasks subject to SDC


7-8 Morphing joint angle trajectories for mission tasks subject to HQC .


. . 110

. . 114

. . 117

. . 20

. . 22

. . 123

. . 26

. . 29

. . 29

. . 134


. 135









Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

MANEUVERING CONTROL AND CONFIGURATION ADAPTATION OF A
BIOLOGICALLY INSPIRED MORPHING AIRCRAFT

By

,lti .id1 Abdulrahim

August 2007

C!i i': Rick Lind
Major: Aerospace Engineering

Natural flight as a source of inspiration for aircraft design was prominent with early

aircraft but became marginalized as aircraft became larger and faster. With recent interest

in small unmanned air vehicles, biological inspiration is a possible technology to enhance

mission performance of aircraft that are dimensionally similar to gliding birds. Serial

wing joints, loosely modeling the avian skeletal structure, are used in the current study to

allow significant reconfiguration of the wing shape. The wings are reconfigured to optimize

aerodynamic performance and maneuvering metrics related to specific mission tasks. Wing

shapes for each mission are determined and related to the seagulls, falcons, albatrosses,

and non-migratory African swallows on which the aircraft are based. Variable wing

geometry changes the vehicle dynamics, affording versatility in flight behavior but also

requiring appropriate compensation to maintain stability and controllability. Time-varying

compensation is in the form of a baseline controller which adapts to both the variable

vehicle dynamics and to the changing mission requirements. Wing shape is adapted in

flight to minimize a cost function which represents energy, temporal, and spatial efficiency.

An optimal control architecture unifies the control and adaptation tasks.









CHAPTER 1
INTRODUCTION

1.1 Motivation

Research in some areas of atmospheric sciences is facilitated by the proliferation

of unmanned air vehicle (UAV) technology. UAVs are able to safely support high-risk

experiments with lower costs relative to a manned flight program. Such experiments

can yield highly beneficial results that support further technological advancements.

The usefulness of UAVs becomes evident with a cursory survey of disparate fields in

science and application. Flight dynamics and control engineers are increasingly turning

toward UAVs for testing advanced system identification and control techniques that

cannot be safely conducted on a manned vehicle. Wildlife biologists are using UAVs to

conduct population surveys on species in remote areas where the cost, noise, and required

infrastructure of manned vehicles are excessive. Non-science uses of UAVs include military

reconnaissance, police crime-scene investigations, border patrol, and fire fighting.

The widespread use of UAVs underscores a fundamental limitation in the aircraft

design process. The wide range of airspeeds, altitudes, and maneuverability requirements

encountered in UAV missions can be beyond the design range of the UAV. The result is

a flight system that performs at a lower level of maneuverability or efficiency than the

operator may desire during certain portions of a mission. The design process of a UAV

involves finding a fixed configuration that best compromises between the imagined mission

scenarios. Such a compromised design prevents the UAV from operating under varied

flight conditions with a constantly high efficiency.

Consider a UAV that is required to engage in a two-part mission consisting of a

maximum-endurance persistence phase and a maximum-speed pursuit phase. An aircraft

optimized for the former would achieve a high-lift to drag ratio at a specific angle of attack

and may feature high-aspect ratio wings. In transitioning to the pursuit phase, the aircraft

may operate at an inefficient angle of attack and be subject to undesirable drag and struc-

tural flexibility due to the wing design. Furthermore, a fixed aircraft design may be a poor









compromise between the stability and maneuverability requirements of different mission

segments. Such a compromise may result in an aircraft that has diluted performance at all

extremes of the flight envelope. Figure 1.1 shows a geometry comparison of two UAVs, one

designed for endurance and the other for combat engagement.










Figure 1-1. Two dissimilar UAVs designed for disparate tasks, Predator variant (left) and
X-45 variant (right) [18]


A possible solution to the limitations of fixed-configuration aircraft is the use of mor-

phing to dramatically vary the vehicle shape for different mission segments. Morphing is

envisioned for many types of vehicles that engage in multi-part missions. The technology

is particularly attractive for UAVs because possibilities for endurance and maneuverability

are greatly expanded over manned vehicles.

The appeal of morphing is that the vehicle design is adapted in flight to achieve

different mission objectives. A morphing aircraft might morph into distinct shapes, where

each configuration is locally optimal for a particular task of a desired mission. Addition-

ally, shape deformations of morphing aircraft can provide high levels of maneuverability

compared to conventional control surfaces.

The substantial benefits of a morphing aircraft come with considerable technological

challenges. DARPA and NASA envision future aircraft as having highly reconfigurable,

continuous-moldline control systems composed of hundreds or thousands of shape-

affecting actuators. Actuator and structural technologies needed to achieve such a goal

are still relatively immature and confined to laboratory studies. The compliant shape

of a morphing airplane may result in increasingly flexible structures that are prone to









aeroelastic problems or even excessive deformation under aerodynamic loads. Morphing

aircraft also pose challenges in the area of sensing and instrumentation due to the large

number of sensors required and the effect of the shape change on sensitive measurements

such as accelerations, angle of attack, and sideslip.

Perhaps the most under-served area of morphing technology in the literature is

the area of dynamics and control. Significant contributions are made in actuators and

aeroelasticity while the rigid-body motion and stabilization of morphing aircraft is largely

unaddressed. Complications in the vehicle model such as time-dependent dynamics and

highly nonlinear, coupled control pose considerable challenges.

Another open area of research addresses the challenge in identifying how to use the

morphing effectively to satisfy or improve upon current mission profiles. The possibility

of a highly reconfigurable aircraft is quite appealing and poses difficult questions related

to how shape-optimization can be performed in-flight and how the vehicle can be made to

autonomously morph into different shapes.

As with most other problems in aerospace, morphing developments are usually

performed with respect to a specific size and weight application. For instance, a morphing

actuator designed for small UAVs is unlikely to be useful for transport-category aircraft. It

is with this limitation that the current work proposes to address open areas of research in

the application of morphing to small UAVs or Micro Air Vehicles ( \ AV) that may engage

in a broad range of missions.

UAVs are primarily tasked with reconnaissance missions, which require surveillance

of targets of interest using a variety of sensors. Large UAVs, such as the Predator, fly for

extended periods at high altitudes to provide persistent observation of a broad area[18].

The aircraft operate in far-and-away environments largely free of obstacles and traffic. The

data gathered from such operations affords excellent regional awareness, but lacks local

detail that may be obscured by buildings and structures.









Small UAVs are uniquely qualified to observe targets which lie within environments

populated with buildings, alleys, bridges, and trees. The small size and low speed of the

vehicles permits operations within an urban environment, within line-of-sight of targets

underneath canopies or otherwise obscured from observation from above. The flight path

for a small vehicle depends not only on the location of the target, but also on obstacles

which affect visibility and trajectory permissibility. Such a path affords a perspective of

the environment which enhances local awareness compared to a remote observer.

A typical mission for a small, urban reconnaissance UAV may involve such disparate

segments as cruise, steep descent, obstacle avoidance, target tracking, and high-speed

dash. Each phase of the mission may be performed in regions where atmospheric distur-

bances are very large in comparison to the flight speed, thus requiring significant control

power for disturbance rejection. The mission phases are also sufficiently distinct so as to

require very different vehicle geometries for successful completion. A current UAV, specif-

ically the AeroVironment Pointer, features a long, slender wing with rudder and elevator

control surfaces and no wing-borne control surfaces. Cruise, loiter, and broad surveillance

missions are easily performed with such an aircraft but the fixed configuration prevents

the vehicle from performing steep descents, .,-.--ressive maneuvers, and large airspeed

variations. Conversely, a different vehicle designed for high-speed pursuit flight may excel

at speed and maneuvering but lack the persistence necessary for loiter tasks.

Operating a UAV in urban environments depends upon the ability to safely perform

the required flight path maneuvers and also on contingencies involving loss-of-control

incidents. Both the maneuvering and the safety requirements promote a vehicle design

which is small, lightweight, and operates at slow speeds. Acceleration forces due to

intentional maneuvers or unintended collisions are small, which reduce the risk of damage

in the latter case. Flight among buildings and other obstacles favors a small vehicle, which

can maneuver through narrow corridors without impinging on the surroundings.









In the size and weight range of interest for small UAVs, the most prominent example

of morphing is the various species of gliding birds, such as gulls, vultures, and albatrosses.

Of these, the gulls arguably make the most use of wing-shape changes to effect flight

path in different maneuvers. All gliding birds have been observed to vary the span,

sweep, twist, gull-angle, and wing feather orientation over the course of a flight. The gull-

angle appears to have a correlation with the steepness of descent, while different feather

orientations likely correspond to .r: -ressive maneuvering or soaring.










Figure 1-2. Different wing shapes used by a gliding gull through joint articulation


The motivation for using biological systems as inspiration for morphing flight vehicles

lies in the similarity between the two with respect to size, weight, airspeed range, and

operating environment. A Laughing Gull, for instance, has a wingspan of 1 meter, a flying

mass of 322 grams, and a cruise airspeed of 13 meters per second. The gull specifications

are quite similar to many small UAVs, with the possible exception of mass, which is

usually higher on UAVs.

Given the similarities between birds and small UAVs, adapting ideas from biological

systems for conventional flight is quite appropriate both in aerodynamic shape and

mechanical structure. Significant shape changes on a UAV wing can be accomplished by

simplifying the wing skeleton of a bird into a serial joint linkage mechanism. A flexible,

membrane wing attached to the linkage allows the aerodynamic surface to smoothly

deform into several different positions. The aerodynamic benefits of such a morphable

wing are easily accomplished using simple, readily available electro-mechanical actuators.

At the small scales of interest to MAVs, the strength of actuators is quite high compared









to structural stiffness or aerodynamic forces; thus, morphing can be achieved without

immediate need for advanced actuation and structural technology. Although existing

actuators provide a basic level of operation, improved shape-change effectors can be

incorporated into the designs to enhance performance. In particular, combined structures

and actuators can expand the wing configurability compared to a mechanical solution

relying on hinges, bending, and twisting.

1.2 Problem Statement

The design of an aircraft capable of operating in an urban environment involves

numerous challenges. The aircraft must have sufficient endurance for moderate-length

missions yet must also be maneuverable to avoid collisions with obstacles and navigate

through narrow corridors. The UAV is expected to follow a complex trajectory, with each

segment having different requirements for endurance, maneuvering, and airspeed. The

design requirements for each mission task are often conflicting which motivates the use of

wing morphing to change the vehicle shape and performance in flight.

Several smaller challenges exist within the broad problem of a versatile aircraft design.

The simplest question is one having morphological considerations. Into what shapes

should an aircraft morph in order to achieve the range of performance required by the

mission? How should biological-inspiration, taken from gliding birds, be used to influence

the wing design and morphing action?

What control strategy is appropriate for an aircraft that may incur large changes

in the stability and control over the range of morphing? Is there a relation between

the maneuvering rates and the allowable morphing actuation speed? Can stability and

controllability be guaranteed for all mission segments?

An additional problem is present in the control of the morphing configurations. How

can the vehicle shape adapt efficiently to a complex, unknown trajectory? What metrics

can be used in flight to estimate the optimal morphing configuration?









1.3 Dissertation Outline

I present a brief overview of the current research related to morphing aircraft design

and dynamics. Discussion of biological flight studies is given to motivate the use of gliding

birds as a source of inspiration for aircraft design. Simulation and wind tunnel studies of

several morphing vehicles shows the significant performance benefits achievable through

shape change. Robust control and adaptation literature are used to build the framework

for control of a morphing aircraft.

My research discusses in greater detail the effect of wing and feather motions on

the aerodynamics of gliding birds. Disparate wing shapes are compared to observed

flight paths to hypothesize the function of each type of morphing. The musculo-skeletal

structure of birds is then compared to mechanical components to show how biologically

inspired morphing can be implemented on a flight vehicle.

The mission profile is segmented into individual maneuvers. The aerodynamic and

dynamic characteristics needed for each maneuver are given. These characteristics provide

the metrics for which the vehicle shape may be optimized. The equations of motions are

derived for a morphing aircraft, including nonlinear and coupled terms that may become

important for certain morphing configurations.

I use a robust-control framework to develop a time-varying controller that is able to

compensate for the changing dynamics of the aircraft. Output and actuation weighting

are used to specify a desired level of performance for each morphing condition. The

different conditions are then linearly interpolated using methods based on linear parameter

varying theory. Desired performance is specified by a mission-specific reference model. An

adaptive architecture responds to errors in the tracking performance of the aircraft relative

to the reference model and adapts controller gains.

I discuss application of optimal control theory to morphing aircraft control and

adaptation problem. An indirect method is used initially to demonstrate the effect of

morphing parametric dependence on the optimality of trajectory and aircraft shape









solutions. A direct, numerical method demonstrates solutions for the rate trajectory,

configuration adaptation, and maneuvering control of the vehicle.

1.4 Contributions

1. Increase maneuverability and endurance for a morphing MAV

2. Identify a desired mission profile to expand range of functionality by using morphing

3. Identify requirements of mission segments on aircraft dynamics and performance

4. Develop morphing micro air vehicle with biologically inspired morphing wings

5. Parameterize flight dynamics with respect to wing morphology

6. Develop closed-loop performance requirements over parameter space

7. Develop design point criteria for linear parameter varying framework

8. Maneuver aircraft using hybrid robust/optimal baseline adaptive controller

9. Develop adaptation algorithm for automatic control of morphing

10. Apply optimal control to rate command, shape adaptation, and maneuvering









CHAPTER 2
LITERATURE REVIEW

The first national programs designed to advance technologies to facilitate morphing

aircraft were the AFRL Adaptive Flexible Wing and NASA Aircraft Morphing Programs.

In the 1998 NASA paper[95], Wlezien et.al, described programmatic goals related to the

use of -in irt materials and actuators to improve efficiency, reduce noise, and decrease

weight on a conventionally flown a I ll The scope of the morphing is relatively

conservative, considering that the authors primarily intend to apply the technology

to a limited class of air vehicles to achieve modest improvements in performance. The

paper presents an overview of the disparate technologies needed to implement morphing,

including unconventional materials, optimal aerodynamic analysis, aeroelastic modeling,

time-varying dynamic modeling, and learning control systems. The authors propose the

use of adaptive-predictive controllers that are able to "learn the system dynamics on-line

and accommodate changes in the system dyi_ ,iii .

The NASA morphing initiative has since produced a number of excellent studies[22] [95] [79]

on the aerodynamics and aeroelasticity of configuration morphing but has had a relatively

small contribution to understanding the dynamics and control of morphing vehicles. Addi-

tionally, none of the concepts presented has matured to point of flight testing on a manned

or unmanned air vehicle. The program, although limited, has successfully motivated

subsequent progress by many researchers, perhaps including the DARPA Morphing Air-

craft Structures program. A paper by Padula summarizes some of the developments after

several years of work on the NASA morphing program[67]. The main contributions are

cited as multidisciplinary design optimization of the vehicle configuration and advanced

flow control. A novel aircraft control system is designed by using distributed shape change

devices to achieve maneuvering control for a tailless vehicle.

A more recent DARPA initiative seeks to develop and field morphing technology to

enable a broader range of functionality than that envisioned by the NASA program. In

particular, the DARPA Morphing Aircraft Structures has specified a goal of morphing









the aircraft geometry such as wingspan, chord, twist, sweep, and planform shape by as

much as 50'. of the nominal dimension. The gross variations in shape are envisioned

for morphing the aircraft between two flight conditions with different airframe geometry

requirements. In particular, a wing may morph from a long, slender configuration for

cruise to a short, low-aspect ratio planform for high-speed flight. The transition of a wing

from the high-aspect ratio, high wing area to a much smaller wing presents considerable

structural challenges.

The DARPA morphing program has established goals to bring various shape-change

strategies to flight maturity. In particular, three aerospace companies are each pursuing

a form of wing morphing that varies a particular aspect of the geometry. The shape-

changing mechanism are each based on biological systems or design parameters that afford

freedom to change flight condition.

Lockheed Martin is developing a morphing unmanned air vehicle that uses a set of

chordwise hinges to extend and retract the wing into a cruise and dash configuration,

respectively. The morphing causing significant variations in the wingspan, aspect ratio,

wing planform, wing position, and roll control effectiveness. The hinge mechanism opens

new challenges in design, as the engineers try to find v--- to articulate the joint without

opening gaps in the wing surface and causing excessive drag [58].

NextGen Aeronautics is working on a similarly inspired concept of sliding skins to

allow the wing to smoothly change in area[46]. The skins appear loosely based on bird

feathers, where changing the overlap of .,.i ient feathers can be used to increase or

decrease the wing surface area. For both birds and conventional aircraft, the ability to

change wing area affords freedom to optimize aircraft lift-to-drag ratio for different flight

conditions. The NextGen concept is able to address some of the concerns raised in the

Lockheed design; namely, the sliding skins are able to preserve aerodynamic shape between

morphing conditions. Furthermore, the concept is amenable to many possibilities for

underlying structure and actuator combinations.









A special case of sliding skin morphing is under consideration by Raytheon, who is

developing telescoping wings to vary wing span and aspect ratio. The possible uses of

the Raytheon morphing strategy are very similar to the other two DARPA initiatives,

where the aircraft transitions from a high-endurance to a high-speed configuration. The

telescoping wings are relatively simple compared to some other morphing ideas, but offer

benefits across a wide range of operating conditions. Perhaps most notable for civilian

aircraft, telescoping wings allow for large wingspans to reduce stall speed and assist pilots

in making safe landings. The beneficial characteristics on landing do not hamper cruise

operation, where the wing can be retracted to allow for a high cruise airspeed.

The DARPA initiative in general has been an important development in morphing

aircraft research. However, because much of the work is conducted in proprietary settings,

some of the results are not available to the research community. The programs have

proposed to solve many of the challenges facing morphing aircraft development, although

promise of a flight demonstration has thus far not been fulfilled.

Morphing represents a dramatic change in design philosophy relative to recent

aircraft. The seamless shape changing envisioned for morphing aircraft is in direct

contradiction to the stiff, nominally rigid structures found on most modern aircraft.

The compromise between flexibility required for shape change and stiffness required for

aerodynamic load bearing represents one of the i, i. ir challenges and perhaps limitations

of morphing aircraft. Love et. al [57] of Lockheed have studied the effect of the DARPA-

funded folding wing morphing on the ability to withstand loads from different maneuvers.

The authors found that the the structural properties vary quite significantly between

morphing conditions. They have proposed metrics by which the structures of such vehicles

can be assessed in terms of airspeed, maneuvering, and actuator response.

Aircraft morphing presents challenges not only in technological hurdles, but also in

accurately representing the result of the shape-change in a design, control, or structural

sense. The challenge is illustrated by considering the difference between a conventionally









hinged wing and a freely deformable wing. The geometry of the former can be quite

accurately represented by several fixed geometries that are rotated or translated relative

to each other. Fowler flaps, for instance, rotate about a hinge point that itself moves

aft. Conversely, a morphing wing may have a large number of actuators that allow

deformations in span, chord, twist, and camber, where each type of deformation may

be non-uniform over the wing. Even in the case of a single actuator that twists a part

of the wing, describing such a geometry using conventional means is inadequate. The

distribution of twist from the actuation point to the rest of the wing may be represented

by a linear distance relationship or may be a complex function of the structure. In

either case, accurate representation of the wing geometry requires more than one or two

measurements.

Samareh (2000) [79] has proposed a method by which morphing geometries can be

represented using techniques from computer graphics. Soft-object animation algorithms

are used to parametrically represent the shape perturbation, as opposed to the geometry

directly. The framework supports parametric variations in planform, twist, dihedral,

thickness, camber, and free-form surfaces. Such a description may be useful in conjunction

with conventional metrics to describe morphing wings that are combinations of surface

deformations and hinged movements. The biological equivalent is that of a bird wing,

which articulates at shoulder, elbow, and wrist joints but also undergoes motion in the

muscles, skin, and feathers.

The technique proposed by Samareh can be used to compute grids for both com-

putational fluid dynamics and finite element method codes, each having both high and

low fidelity variations. The applicability to both aerodynamic and structural disciplines

allows the morphing wing to be studied in a broad sense. C'!i ''teristics in other areas,

such as control effectiveness, varying dynamics, aerodynamics, and aeroelasticity, can be

determined as a direct outcome of the method.









Beyond the national programs, research in morphing is quite diverse in both subject

and scope. Individual contributions are usually related to a specific application or disci-

pline, which serve as advancements in the general field of morphing aircraft technology.

The innovation in literature is quite impressive. Gano and Renaud [30] in 2002 presented

a concept for slowly morphing the airfoil shape of wings over the course of a flight using

variable-volume fuel bladders. The proposed morphing has been shown to result in im-

proved range and endurance. The use of passive fuel depletion to change the airfoil from

a high-lift to a low-drag configuration illustrates how morphing mechanisms can use inno-

vative effectors to achieve the desired shape change. The paper stresses the importance of

advanced optimization techniques to ensure the desired lift-to-drag optimization holds in

the presence of uncertainty.

Sanders et. al. [82] have shown through simulation possible benefits of camber or

control surface morphing. The 2003 study in the Journal of Aircraft showed conformal

control surfaces, essentially flaps with no hingeline, are able to achieve a higher lifting

force at a lower dynamic pressure than conventional control surfaces. Such results are

encouraging for applications to UAVs, where high-lift and large roll control power are

desired at relatively low dynamic pressures. The conformal control surfaces are shown to

produce a higher maximum roll rate at certain dynamic pressures.

The benefits of such an approach are perhaps mitigated by the increase in nose-down

pitching moment occurring with conformal control surfaces. The larger moment produces

undesired aeroelastic effects at high dynamic pressures, leading to excessive structural

twisting and eventually aileron reversal as dynamic pressure is increased. This limitation

is not significant compared to the benefits for slow UAVs, provided that the wing is

sufficiently stiff in torsion.

Bae et. al presented a variable-span concept [8] for a cruise missile in order to

reduce drag over the airspeed envelope and increase range and endurance. The authors

consider both aerodynamic and aeroelastic effects associated with a sliding-wing. In









particular, they showed a decrease in total drag with the wings extended for certain flight

condition. As with Sanders work, Bae showed some undesirable aeroelastic characteristics

associated with the span morphing. At high dynamic pressures, the wing tip of the

extended configuration incurs large static aeroelastic divergence as a result of the increased

wing root bending moment.

The variable-span wing has been shown in simulation to have control over the

spanwise lift distribution. If combined with a seamless trailing edge effector, such as the

model proposed in Ref. [82], the morphing wing would possess a large degree of control

over the lift distribution and would be able to achieve the aerodynamic, stability, and

stall-progression benefits of a variety of wing planform. Combined morphing mechanisms

are somewhat rare in the literature, considering the immaturity of actuators and materials

that are able to achieve complex shape changes.

Wind tunnels tests of a morphing aircraft with three degrees of freedom showed

significant aerodynamic benefit from span, sweep, and twist morphing [63]. The vehicle

uses a telescoping wing to achieve wing span variations of 4 !'. allowing for increases in

lift coefficient relative to the nominal condition. Hinges at the wing root allow the sweep

angle to range from 0 to 40 aft, delaying stall and improving pitch stability. The authors

have shown that the wing sweep causes a change in both center of gravity position and

aerodynamic center position. The aerodynamic center on the wind tunnel vehicle moves

aft farther than the center of gravity, thus increasing the magnitude of the longitudinal

static margin.

One benefit of the shape-adaptive aircraft is said to be in drag reduction, where three

distinct combinations of wingspan and sweep are required to achieve the minimum drag

over the full range of lift coefficients. The result is important for mission considerations,

where maneuverability and airspeed requirements may change considerably. The paper

- i-.-1 -I that a morphing vehicle is appropriate for reducing drag during several portions

of a mission.









Valasek et. al published a series of studies on morphing adaptation and trajectory

following for an abstracted system[90] [85] [24]. A variable-geometry vehicle is represented

by a blimp-like shape which has various performance capabilities related to the shape. The

approach uses reinforcement learning to control the shape adaptation as the vehicle moves

through the environment. The vehicle morphs to optimally follow a defined trajectory.

Adaptive dynamic inversion is used to control the vehicle and follow trajectory commands.

Although the study is physically unrelated to an aircraft, the authors present a unique

approach for coupling trajectory requirements and vehicle performance capability. The

automatic control of the shape for different environments represents a capability highly

sought for a morphing air vehicle.

Optimality in morphing is addressed by Rusnell, who together with colleagues,

published studies on the use of a buckle-wing concept to achieve enhanced performance in

both cruise and maneuvering flight -1.- [76] [77]. The morphing aircraft in the study uses

a biplane configuration where the top wing conforms to the lower wing to form a single

wing in some configurations, and buckles to form a joined-wing biplane in others. Analysis

of the Pareto curve is used to compare the optimal configurations of the morphing wing

against a conventional wing for the different missions. Vortex-lattice method is used to

compute airfoil and planform aerodynamics of the simulated aircraft. The authors -,1.-.- -1

that the buckle wing morphing can enhance mission performance relative to an aircraft

with fixed configuration.

Boothe et. al[11][12] proposed an approach for disturbance rejection control on a

morphing aircraft. Linear input-varying formulation is used to allow the controller to

account for the dynamic variations of the morphing, which itself is the control input.

The simulated examples include several types of variable wing geometries, including

span, chord, and camber varying. Stability lemmas and associated closed-loop simulation

responses show that the morphing can be used in some instances to adequately reject









disturbances, although transient effects from the morphing action cause performance

limitations.

Bowman[13] studied the mission suitability of morphing and adaptive systems com-

pared to conventional technology. The systems are evaluated on the basis of flight metrics

describing the aircraft efficiency and maneuverability in conjunction with logistical issues

such as cost, production, and maintenance. Performance for adaptive and conventional

vehicles are considered in a multi-role mission scenario, requiring pursuit and engagement

tasks. Bowman -, -::. -I that the performance of morphing systems is worse than existing

capability, except when considering multifunctional structures and decreased dependence

on support vehicle, in which case the adaptive systems offer substantial improvement. The

cost associated with developing the adaptive vehicle causes poor affordability in general,

but the reduced life cycle cost result in long-term gains. The study offers intriguing insight

into the ultimate usefulness of a morphing vehicle in the context of a realistic, multi-task

mission.

Maneuvers for a morphing aircraft can be computed using optimal control tech-

niques. Trajectories generated for each mission task depend intrinsically on the variable

dynamics. Studies relating fighter aircraft agility to allowable flight paths provide a use-

ful architecture on which to develop morphing control techniques[78]. Direct numerical

optimization[49][41] can simultaneously optimize the aircraft shape and the trajectory to

successfully complete the mission requirements. Aircraft agility and maneuverability for

each morphing configuration can be assessed using standard performance metrics[93][61].

These metrics are used within the optimal control framework to morph the aircraft

favorably for each task.

The aerodynamic benefits of morphing are usually cited in studies as the leading

impetus for shape adaptation, although researchers are also considering morphing for

a reduction in actuator energy costs. Johnston et. al have developed a model through

which the energy requirements for flight control of a distributed actuation system can be









compared to conventional control surfaces [45]. Their study uses vortex-lattice method and

beam theory to a' i1v. -. the energy required to deform the wing and produce a sufficiently

large aerodynamic force. Distributed actuators are shown to have potential improvements

in energy requirements by morphing the wing section rather than deflecting a trailing edge

flap.

Biological inspiration as a technology for a multi-role aircraft is discussed by

Bowman[14] and his co-authors. The varied wing shapes of the bald eagle during soar-

ing, cruising, and steep approach to landing are considered as the basis for aerodynamic

evaluation of morphing. Geometric parameters such as planform and airfoil are said to be

particularly important for varying the lift and drag characteristics. The variable perfor-

mance capability of the aircraft is contextualized in an example mission, which describes

the role of morphing in the conceptual aircraft design.

Livne recently published survey papers on the state of flexibility, both active and

passive, in aircraft design[55][56]. The aeroelastic and aeroservoelastic behavior of con-

ventional and morphing airplanes are described both in terms of design challenges and

potential for performance improvement. Livne describes the relation of biological observa-

tion to the design of micro air vehicles, in that such small aircraft commonly use flexible

and flapping wings in order to achieve flight. Morphing for large aircraft is also discussed,

including variable wing sweep on the F-14 and F-ill and the variable wing tip droop

of the XB-70. The author contends that a reconfigurable morphing UAV can operate

efficiently in a diverse set of operating environments.

A study by Cabell [15] describes applications of morphing to an aircraft subsystem

for noise reduction. A variable geometry chevron is used for noise abatement in the nozzle

of a turbofan engine. The authors describe both open-loop and closed-loop techniques to

adapt the chevron shape to achieve the desired reduction in noise without excessive thrust

reduction for a v i,, iv I, of conditions. Shape memory alloy actuators are used to generate a

smooth, seamless actuation of the chevron structure.









Morphing is used in place of conventional control surfaces for a wing which is inflated

and rigidized, and thus cannot sustain a conventional hin-,j. iinl [16] [17]. Differential wing

twisting of the inflated wing is used to provide roll control for maneuvering. The authors

of the study considered several actuation methods, along with the cost and performance

associated with each. Morphing is successfully implemented as a control solution to a wing

which has desirable packaging and deployment properties.

Cesnik et. al have developed a framework for assessing the capabilities of a morphing

aircraft performing a specified mission[19]. The authors describe the desired mission

scenario and identify the necessary vehicle performance for success. The technology

required to achieve the morphing is also considered. The evaluation is used to formulate

the critical component needs for a morphing aircraft. Scoring metrics for various mission

segments are .-regated to determine the relative improvement in performance.

Researchers at Purdue have been studying morphing as fundamental aircraft design

element for multi-role missions [74] [27] [28] [69] [75]. Their approach considers morphing

as an independent design variable and determine the contexts in which various forms

of shape change can be used advantageously. The studies consider both a single vehicle

performing disparate flight tasks and a fleet of vehicles operating cooperatively. Critical

need technology areas are identified by identifying applications for specific types of

morphing. Actuation and structural devices needed to enable the morphing are readily

identified from the analysis.

Morphing the wings of an aircraft in twist is under study for use as both a control

effector and as a method to increase efficiency. A West Virginia University study is

considering applications of wing twist to a swept-wing tailless aircraft, which is itself

inspired by natural flight [39]. The conventional control mechanisms used on such a vehicle

compromise the beneficial aerodynamics of the wing configuration. A twist mechanism

has been shown to have sufficient control power while improving lift-to-drag ratio by 15'.

and decreasing drag by up ot 2-' relative to conventional hinged surfaces. The significant









gains are achieved by adapting the quasi-static wing twist distribution according to the

required flight condition.

Stanford conducted a similar study using a twist actuator to deform a flexible mem-

brane wing for roll control[84]. Multi-objective optimality is addressed by finding the

torque-rod configuration which simultaneously achieves the highest roll rate and the high-

est lift-to-drag ratio during a roll maneuver. A Pareto front describes the configurations

which achieve various combinations of lift and roll optimality. The curve is compared to

the configuration on the experimental flight test vehicle, which can achieve performance

improvements through modification of the actuator.

A wind tunnel investigation has demonstrated the performance and dynamic effects

achieved by quasi-static variations of the wing aspect ratio and wing span[42]. An exper-

imental model uses telescoping wing segments to morph between retracted and extended

configurations. The variable wing area allows large changes in the lift and drag magnitude,

but also has large affects on the roll, pitch, and yaw dynamics. Open-loop dynamics are

shown to vary considerably with symmetric wing extentions. Each of the roll, dutch roll,

and spiral poles show tend toward more rapid responses as the span is increased. The

authors depict a highly unstable spiral mode and are investigating differential span ex-

tensions as a means of stabilizing and controlling the divergent behavior. An additional

study by the authors presents the coupled, time-varying aircraft equations of motion and

explores the use of .i-vmmetric span morphing for roll control[43].

Researchers from Georgia Tech are exploring configuration optimization for a vehicle

that must perform the disparate tasks of subsonic loiter and supersonic cruise[66]. The

novel aircraft configuration requires the use of response surface methodology and several

stages of analysis to estimate the performance of various shapes in the design space.

Aerodynamic characteristics are optimized for each mission by parameterizing the design

elements and finding the set of shapes that achieves the optimal compromise between the

two mission modes.









A new concept for aircraft morphing structure is proposed by researchers at Penn-

sylvania State University, who are studying an aircraft structure composed of compliant,

cellular ti, i-- [72]. The structure is actuated by tendons joining various points of the

trusses and can achieve many types of aircraft shapes through continuous deformation.

Many tendons, distributed throughout the trusses are used to generate global shape

variations through ., -i--regation of local actuation. Actuation of the wing can be achieved

using relatively low forces. The weight of a proposed morphing truss wing is similar to a

conventional wing, except that aeroelastic concerns may require the use of active control of

the tendons.









CHAPTER 3
BIOLOGICAL INSPIRATION

3.1 Motivation

Autonomous control of air vehicles is typically performed using very benign maneu-

vers. Limited operating range of fixed-wing aircraft and the simple control algorithms

used preclude significant variations in flight condition. Maneuvering is thus limited to

small perturbations around level, cruise flight, which may involve shallow-bank turns and

gradual changes in altitude. A notable exception to the maneuverability restriction is an

autonomous, aerobatic helicopter in development at MIT [23], [70], [31],[32].

For fixed-wing unmanned flight, the difficulty of a broad flight envelope lies in the

limitation of a static geometry in providing efficient flight over a range of airspeeds and

angles of attack. Such a range is desired for an aircraft expected to operate in an urban

environment. Specifically, the vehicle must maneuver safely in a complex, 3D environment

where obstacles are not aiv- -, known a priori. Rapid changes in direction require large

aerodynamic forces which are generated at high slip angles and high dynamic pressures.

The motivation for studying bird physiology for insight into the design of a highly

maneuverable air vehicle stems from observations of gliding birds. Rapid accelerations and

flight path variations are achieved through articulation of the wing structure to promote

favorable aerodynamic or stability properties. Birds in general, and laughing gulls in

particular, have been observed using different wing configurations for disparate phases

of flight. Windhovering, a.. i ii.- and steep descents, for instance, each require a specific

trim airspeed, glide ratio, and maneuverability afforded by a different wing configuration.

The present work seeks to identify the dominant wing shapes that are portable to a

conventional aircraft philosophy for the purpose of expanding the achievable range of

flight.

Birds are used as a source of design inspiration for urban-flight vehicles because

of strong similarities in dimension and operating condition. They already achieve a

remarkable range of maneuvers along complex trajectories in urban environments. The









size, weight, and airspeed of a large bird closely resemble the specifications of a small

UAV. Highly dynamic geometries allow birds to continually reconfigure their aerodynamic

shape in response to changing flight condition. A small UAV could mimic some of the

shape-changes and exploit similar structural features as birds to expand the conventional

flight envelope. The design challenge is finding the morphological operations which

improve the vehicle aerodynamics, rather than those intended for physiological or flapping

purposes.

3.2 Observations of Bird Flight

Gliding birds are able to achieve a wide range of maneuvers and flight paths, pre-

sumably resulting from aerodynamic effects of the wing, tail, and body shape changes.

Observations of gliding flight are made to help understand the motivation for each wing

shape incurred during flight. The bird species of interest is the laughing gull, Larus

atricilla, because of the similarities in size, weight, and airspeed to small unmanned air

vehicles. Perhaps most importantly, gulls spend a considerable duration of each flight in

various gliding maneuvers.

Gliding birds use fore-aft wing morphing to control glide speed[88], longitudinal pitch

stability[63] and lateral yaw stiffness[80]. Figure 1.1 shows three planform variations of a

gliding gull, where each of the shoulder, elbow, and wrist joints are modified to change the

overall wing shapes. The planforms differ markedly in wing area, span, and sweep, yet are

accomplished relatively easily through skeletal joints and overlapping feathers.

Figure 3-1 shows results from Tucker establishing a relation between planform shape

and gliding speed for similar glide anis [ ]. A significant reduction in wing span and

wing area is achieved by articulating the elbow and wrist joints such that the inboard wing

is swept forward and the outboard wing is swept aft. The aft rotation of the outboard or

hand wing is such that the aerodynamic center typically moves aft, increasing the pitch

stiffness for operation at higher airspeed.











14


^ o -....
212




.. ..... ,1 .


A B C
Wing Shape

Figure 3-1. Effect of planform shape on gliding speed. See Tucker, 1970 for wing shapes
A,B, and C[88]


Yaw stiffness can be achieved in bird wings with aft sweep of the outboard wings.

Sachs [80] showed the magnitude of the lateral stiffness, C,,, increases with aft sweep

angle and is sufficiently high such that birds do not require vertical stabilizers. Figure 3-2

shows a bird with aft wing sweep and positive yaw stability[81], which is compared by

Sachs to an aircraft with similar characteristics. The aft sweep of the outboard wings has

already been shown to increase pitch stability. The simultaneous increase in yaw stability

allows the bird to glide at rapid speeds while maintaining a desired flight path.




















Figure 3-2. Increasing yaw stiffness through outboard wing sweep









Figure 3-3 shows two gulls using different wing joint angles to deform their wings dur-

ing gliding flight. The purpose of this motion has been the subject of some speculation in

the literature, with some researchers arguing the benefits are predominantly aerodynamic

while others cite stability improvements. In the case of the upper photograph, the bird

uses outboard wings tips angled downward. The configuration has been shown to yield

significant gains in L/D[22], perhaps at the expense of adverse lateral stability[87]. The

lower photograph shows a bird with an I' wing shape when viewed from the front. The

shape has been observed in use in gliding birds to vary the glide angle. A wide M-shape

is used in situations requiring a shallow glide path and a correspondingly large L/D ratio.

Conversely, birds in steep descents often use a narrow M-shape, presumably to reduce

L/D ratio and maintain stability at the high angles of attack incurred during slow descent.


















Figure 3-3. Gliding birds articulating wing joints about longitudinal axes


Figure 3-4 shows two gulls using complex morphing of the shoulder, elbow, and wrist

joints about both longitudinal and vertical axes. The aerodynamics during such motion

likely involve stability and performance elements from each of the individual motions.

More complex flow interactions may also yield improvements in lift through leading edge

vortices, as si --- -I 1 by Videler[91].
































Figure 3-4. Complex wing morphing shapes achieved in gliding flight


Most morphological operations are conducted symmetrically about the body, such

that left and right wings generate similar aerodynamic forces. Asymmetric motions are

used in certain instances of gliding flight, presumably to generate favorable moments for

maneuv, ,iin,. reject disturbances, or allow the bird to trim at large angles of sideslip.

Figure 3-5 shows two birds using .,i-vii ic l wingspan, wing area, dihedral angle, and

sweep. The upper photograph shows different articulations of the wrist joint cause the

outboard wings to extend and sweep to different extents. The lower photograph shows

the wrist joints of another bird at different outboard dihedral angles and extension. The

configurations may be well suited for crosswind operations, considering the spanwise lift

distribution and aerodynamic coupling (C1,, for instance) can be selectively varied.

Gliding birds are capable of rapid changes in flight path using .,.-- ressive maneuvers.

Figure 3-6 shows two gulls near bank angles of 90 performing pull-up maneuvers to

rapidly change directions in small radius turns. Both birds have deploy, .1 the tail feathers,




























Figure 3-5. Gulls using .,i-ii,,, I ii,. wing morphing about vertical (top) and longitudinal
(bottom) axes


presumably for increased pitching moment. The left photograph shows a bird in a

descending, helical path at relatively high airspeed. The orientation of the head shows

that the bank angle is quite large. Aft sweep of the hand wings is likely used to maintain

pitch stability at the high airspeed. The bird shown in the right photograph is performing

a similar maneuver, except at a lower airspeed. Both the tail and wing feathers are

deploy, -1 further than the left bird, perhaps to compensate for the reduced dynamic

pressure with increased surface area. The wings are also swept forward slightly, reducing

the pitch stability and possibly allowing the bird to perform an .,.--ressive turn.

3.3 Desired Maneuvers

Several types of maneuvers are identified for a prospective urban-flight vehicle. The

maneuvers are designed to allow realistic operations from deployment to recovery, with

interim missions of target tracking, obstacle avoidance, maneuvering, and loitering. Each

flight condition can be expressed in terms of performance metrics and modeled by an

equivalent maneuver.



























Figure 3-6. Rapid variations in flight path using .,--:ressive maneuvering


3.3.1 Efficient Cruise

Cruise requirement in general favors the efficiency of the wing in terms of lift to (d .-

the maximum of which occurs at a specific angle of attack for a given configuration. The

cruise segment requires a vehicle configuration which achieves that maximum lift to drag

ratio to minimize energy expenditure from the propulsion system. Factors affecting the

efficiency may include actuator requirements for active stabilization and propulsive effi-

ciency at various airspeeds. However, airspeed and stability considerations are secondary

considerations to flight efficiency. Cruise flight is anticipated for local or long-distance

flights where time enroute is not an overriding concern.

3.3.2 Minimum Sink Soaring

A soaring configuration is necessary to achieve the minimum altitude-loss rate in

gliding flight. Vehicles may benefit from rising air currents as a mechanism to aid in

-I iii-; aloft. The minimum sink rate of an aircraft determines the minimum vertical wind

needed to maintain altitude in gliding flight. Under powered conditions, smaller thrust

forces can be used compared to higher sink rate conditions.









In determining the minimum sink configuration and flight condition, the airspeed

and efficiency are left as variable parameters. For autonomous flight, the usefulness of

such a configuration depends partly on known areas of vertical currents. Otherwise, the

configuration may be used to maximize glide duration.

3.3.3 Direction Reversal

The constraints of flight in an urban environment may require direction reversal

maneuvers. Such maneuvers may be necessary to maintain flight in the presence of

obstacles such as a1 v-,v vs, walls, and buildings were conventional maneuvers would

result in a collision. A dead-end alley between two buildings, for instance, may require

a maneuver similar to an immelman-turn or split-s, depending upon the availability of

space above and below the vehicle, respectively. A configuration is needed to generate

large aerodynamic forces necessary to achieve the linear and angular accelerations in the

maneuvers.

3.3.4 Minimum Radius Turn

A sustained, minimum radius turn configuration is useful for loitering or as an

alternative to the direction reversal maneuver. The maneuver must produce a ground-

track small enough to orbit continuously over a road or small target, but must also afford

sufficient controllability to recover to level flight quickly and on any heading.

3.3.5 Steepest Descent

A variation on the minimum sink configuration adds additional constraints on

the desired flight path to achieve a steep descent at a slow airspeed or rate of descent.

The maneuver is desired for descending between tall obstacles where horizontal area is

limited. Such a maneuver is useful for terminal 1 ,,liir:; but is also important for transient

maneuvers. In the latter case, it is necessary that the aircraft have sufficient control power

to quickly recover to level flight.









3.3.6 Maximum Speed Dash

A final configuration is necessary to allow the vehicle to dash at maximum airspeed

between waypoints. Maneuverability and energy consumption are secondary considerations

to the achievable speed. A low-drag configuration is thus necessary to offset drag increase

with dynamic pressure.

3.4 Morphing Degrees of Freedom

A morphing wing model is proposed based on the gross kinematics of a gull wing.

The model is not intended to emulate the function of feathers or flapping, but rather

seeks to identify the quasi-static wing geometry variation that contribute to the maneu-

verability and aerodynamic performance of the aircraft. Dimensional parameters are fully

represented in the model, although not all are considered to be actively variable in flight.

The configurability of bird wings is achieved using nominally rigid bone sections

connected with flexible joints. Muscles in the wing articulate the joint angles through

tendons connecting contracting muscles with bones. The rotation of the joints moves the

relative position of the bones and changes the gross geometry of the wing.

The bones, joints, and muscles of a bird wing can be roughly emulated using spars,

hinges, and actuators in a morphing wing. Obviously the elegance of the bird movement

is lost in the mechanical system, but the overall shape change is readily achieved. The

biologically inspired mechanical wing uses two primary joint locations, a shoulder and an

elbow, omitting the wrist articulation from bird wings.

Shoulder joint articulation is achieved on the roll and yaw axes, with respect to the

body-axis conventions. The roll movement allows the wing dihedral angle to change,

raising and lowering the vertical position of the outer wing. The wing on the inboard

section is at a fixed incidence, although the wing surface is taken to be flexible and free to

deform due to airloads.

Elbow joint articulation is similarly achieved although adds a degree of freedom in

pitch to allow variations in incidence angle. The shoulder and joint articulations are both









assumed to be measured with respect to a lateral-axis fixed to the aircraft body. Thus, for

a deflected inboard wing, the outboard wing is assumed to remain in a fixed orientation

but translated due to the link geometry.

The respective spans of the inboard and outboard sections are considered variable in

the analysis to examine different effects of relative length differences. For flight however,

each link length is assumed to be fixed due to the mechanical complexity of span morph-

ing. Additionally, there is little biological precedence, at least in gliding birds, for variable

bone length.

3.5 Morphing Motions

The structure of bird wings allows a large range of motions combining rotations

about the different joints. The freedom of motion is quite large, although birds typically

use a subset of joint movements that yield to favorable aerodynamic characteristics. The

combined joint motions are composed of simultaneous actuation of individual joints, which

together produce a desirable effect. The coupled motion of the elbow and wrist joints

is described by Videler [91] as the result of parallel arm bones interacting during wing

extension.

Birds engage in several types of combined motions, with each having an apparent

benefit in flight. Some of the dominant functions are described and related to conventional

flight. The motions are, in general, easily reproduced using a mechanized aircraft wing.

3.5.1 Fore-Aft Sweep of Inboard Wings

The sweep angle of the inboard wing has the primary effect of changing the aero-

dynamic center position relative to the aircraft body. The center of gravity also changes

due to the moving wing mass, although this change is expected to be relatively small due

to light wing structures. The change in aerodynamic center position is due to both the

forward rotation of the inner wing and the forward translation of the outer wing, which is

connected at the mid-wing joint.









The movement of both the aerodynamic and mass centers has an important implica-

tion to the longitudinal stability of the aircraft. A conventionally configured aircraft with

a stabilizing horizontal tail requires a center of gravity position ahead of the aerodynamic

center for stability. The degree of stability is expressed by the static margin, which nor-

malizes the difference in mass and aerodynamic center positions by the reference chord

length.


X.p Xac
static margin = x (3 -1)
c c

The fore-aft motion of the inboard wing thus affects longitudinal stability by changing

the relative positions of the mass and aerodynamic centers. Assuming the mass of the

fuselage is much larger than the mass of the wings, the effect will tend towards instability

as the wings are swept forward and increased stability as the wings are swept aft. The

inboard wing sweep can thus be used to command a desired level of performance, where a

forward sweep results in increased maneuverability relative to a more stable aft sweep.

Sweep of the inboard wing also affects the angle of intersection between the wing

surface and the fuselage. The effects of such a junction in interference drag are neglected

due to the complexity of simulation or measurement.

3.5.2 Fore-Aft Sweep of Outboard Wings

The outboard wing is capable of a range of motion similar to that of the inboard

wing. The longitudinal sweeping motion about a vertical joint axis is assumed to be

coupled passively to the inboard wing using parallel linkages. Sweep of the inboard wing

produces an equal and opposite sweep of the outboard wing, resulting in no net rotation

for the latter relative to the fuselage. Outboard wing sweep is accomplished by actively

varying the length of the trailing parallel linkage.

The sweep of the outboard wing pl il a role similar to the inboard wing in affecting

the positions of the mass and aerodynamic centers, although to a lesser degree. Addition-

ally, the outboard wing sweep angle has a substantial affect on both the dynamics and









aerodynamics. Longitudinal dynamics are affected through an influence of the sweep angle

on the static margin. Lateral dynamics are also affected through an increase in yaw static

stability with aft outboard wing sweep. One study identifies favorable mass scaling effects

that allow birds and small aircraft to fly without vertical tails, using only wingtip sweep to

generate yaw stiffness and d ,ipii.-O[<0].

Aft sweep of the outboard wing is said to generate a leading-edge vortex lift [91],

although this is not modeled explicitly in the current considerations.

3.5.3 Up-Down Inclination of Inboard Wings

Vertical rotation of the inboard wing about a longitudinal axis produces an effect

on the lateral stability through interactions between the wing and the vertical center of

gravity. This pendulum stability is a significant source of both lateral and longitudinal

stability in many species of gliding ciiiin i- [i7]. Upward rotation of the inboard wing

produces both a dihedral effect and elevates the outer wing, in both cases increasing the

wing-level stability[4].

3.5.4 Up-Down Inclination of Outboard Wings

The outboard wing similarly affects the lateral stability through changes in the

vertical location of the center of pressure and angular inclination. Both factors interact

with sideslip to produce stabilizing roll moments. The inclination of the outboard wing

also affects the control effectiveness of the twisting action. The twisting or flap deflection

produces an increase or decrease in lifting force which subsequently produces a rolling mo-

ment. The roll moment decreases as the cosine of the wing inclination angle. Meanwhile,

the yaw moment produced by the twist deflection increases by the sine of the wing angle.

3.5.5 Twist of Outboard Wings

The roll moment necessary for control is generated by differential twisting of the

outboard wings. The actuation rate of the twist actuator is considered sufficiently fast for

stabilizing tasks. The twist can also be morphed quasi-statically, both for roll and pitch

trim using differential and collective twisting, respectively.









3.6 Wing Morphing Model

A mechanical model emulating the basic gull morphological operations is devel-

oped using composite materials and electro-mechanical actuators. Figure 3-7 shows an

annotated view of the left wing compared to two gliding gulls using a similar form of

wing morphing. It is clear that conventional aircraft metrics such as dihedral are in-

sufficient to describe the wing articulation. Instead, the wing is treated as a serial-link

robot manipulator [21], where each joint is represented by appropriate length and angle

dimensions.

























Figure 3-7. Wing articulation about longitudinal axis showing two joint angles. Gliding
gulls are shown for reference.


The morphing wing is similarly equipped with joints by which to articulate the

wing sections in the horizontal plane. Figure 3-8 shows the underside of a gull wing with

annotations to indicate approximate positions of bones and joints. The mechanical model

is of reduced complexity, having only the equivalent of a shoulder and a wrist joint.


























Figure 3-8. Approximate physiological diagram of a gull wing with articulations in the
horizontal plane


3.6.1 Pigeon Wing Configuration

Pigeons are observed using large dihedral angles during steep glides on approach to

landing. Figure 3-9 shows four consecutive frames during such a descent, where the pigeon

uses a turning flight in addition to the inclined wing geometry to control the approach

path. Each frame is photographed using a focal length of 300mm, making the relative size

of the pigeon in the image an indication of its proximity to the camera. The arrangement

of the birds in the image is artificial and does not reflect the actual flight path. However,

the sequence illustrates the wing shape used to command a steep descent and also shows

some maneuvering capability during the unusual joint configuration.

Tail feathers are fully extended during the steep gliding maneuver, ostensibly to

increase drag and reduce lift-to-drag ratio. The wing configuration may also provide

stability benefits at large trim angles of attack such that the glides can be effectively

controlled for precision landings. The head orientation of the pigeon gives some indication

of the angle of descent relative to the body, assuming it is looking in the direction of flight.

The wing morphing shown depicts vertical articulations of the shoulder joint to achieve

the dihedral angles of nearly 450 per wing side. It is difficult to discern from the photos






















Figure 3-9. Composite photo of pigeon in gliding approach to landing


the amount of sweep used, but they appear to be in the neutral or even slightly forward

swept configuration.

Tennekes reports in his 1996 book on the wing morphing used by pigeons to control

glide speed[86]. Fully extended wings are used during low airspeed glides while forward

swept inboards and highly aft swept outboard wings are used for high airspeed dives. The

change in wingspan, wing area, and aspect ratio allow the bird to alter the trim glide

speed by nearly a factor of three.

3.6.2 Avian Morphology Studies

Recent research on the aerodynamic and dynamic characteristics of birds have

quantified the motivation behind some of the wing shape changes observed in nature.

Tucker reported in 1970 on the effect of wing planform shape on the glide angle and glide

speed of f .1- ,-[;][89]. Forward sweep of the shoulder or elbow joints and aft sweep of

the wrist joint, as in Figure 3-8, are used in varying degrees to control the planform area

and wingspan. The subsequent trim airspeed varies substantially, affording control over

loitering or attack speeds.

Sachs reported findings on the role of aft wingtip sweep in providing yaw stiffness and

yaw stability on the aerodynamic and inertial scales of bird flight [80]. The aft wing sweep

also affects the location of the aerodynamic center and contributes to pitch st ii i' [47].









The results partly explain the stability motivations for swept shapes adopted by birds

during high speed dives.

Joint angle articulation along longitudinal axes has been shown by Davidson to

increase lift to drag ratio substantially on a seagull-inspired wing model[22]. Alternate

joint configurations have been shown to have the opposite effect and reduce lift to di

allowing steep dives at moderate airspeeds [3].

Videler 1- --., -I the aerodynamics of complex avian wing shapes benefit from the

deployment of individual feathers, such as the alula, in increasing the lift magnitude by

promoting a leading-edge vortex[91]. Figure 3-8 shows the location of the alula feather,

which is deploy, -1 forward at the wrist joint during certain wing shapes and maneuvers.

The feather is said to have a profound impact in increasing maneuverability and possibly

delaying stall.

The wing geometry of several bird species are presented by Liu et. al in 2006[54].

Three-dimensional scanners are used to generate models of bird wings throughout the

flapping cycle. The wing geometries are presented as time-dependent Fourier series, which

represent the change in the aerodynamic shape resulting from skeletal articulation. The

authors use a two-jointed arm model as a simplification of bird bone structure. The

arm model is characterized by three angles and achieves a sufficient range of motion to

represent the flapping cycle. The wing surface is assumed to be fixed to one of two spars

at the quarter-chord position and maintains proper orientation relative to the flow for all

configurations. Although the research focused on flapping flight, the identified wing shapes

may also be useful for gliding operations.

3.7 Aircraft Morphology

A simplified UAV wing geometry is developed to study the implications of avian-

inspired shape morphology. The wing is based on an existing planform[2] which serves

as the nominal configuration and also provides a baseline for performance comparison.

Figure 3-10 shows four composite views of the morphing degrees of freedom and range.









The nominal configuration, a straight, elliptical wing, is shown in addition to the extreme

positive and negative positions for each joint axis. The morphing is assumed symmetric

about the fuselage centerline.


+








(a) Hi (-30, 0, 30), Positive Up (b) P3 (-30, 0, 30), Positive Up

+












(c) P2 (-30) 0', 30'), Positive Forward (d) 4 (-30, 0, 30), Positive Forward

Figure 3-10. Morphing joint articulations for 4-degree-of-freedom wing


Wing configurations are determined by four joint axes, two at the wing root and two

at the mid-span position. Inboard joint angles, pi and P2, control the rotations about

longitudinal and vertical axes, respectively. Outboard joint angles, 3 and P4, similarly

control the longitudinal and vertical axis rotations, respectively. The views shown in

Figure 3-10 depict variations to one joint axis for each subfigure. Inboard joints produce

rotations of the inboard wing section and translations of the outboard wing section. The

orientation of the outboard sections is controlled strictly by the outboard joints.









A mechanical model uses spars, hinges, and composite skins in place of the bones,

joints, and feathers of a bird. The mechanization allows sufficient range of motion and

structural stiffness to study the effectiveness of biologically inspired wing shapes. The

model is reduced in complexity relative to a bird wing with a shoulder, elbow, and wrist

joint, each having multiple degrees of freedom. Two joint locations, each with two degrees

of freedom, are used per wing side. The variable area function of feathers is replaced by

extensible or sliding -liii~- [: ;].

For simulation, the aircraft is modeled with a similar range of motion but without

the mechanical detail. A vortex-lattice computational aerodynamics 1' 1. [25] is used to

simulate the flight characteristics of various aircraft configurations. Simulations are per-

formed for each combination of joint angles within the operating range. Angular resolution

for each joint is 5, giving 13 unique positions over the 30 range and 28,561 combi-

nations of joint angles. Output data from the simulations are saved to a 4-dimensional

matrix for post-processing. Airspeed, angle of attack, and sideslip are fixed for each simu-

lation at 15 m/s, 6 and, 0, respectively. Each run is trimmed for zero pitching moment

using the elevator control surface.









CHAPTER 4
DYNAMICS

4.1 Aircraft Equations of Motion

The motion of a rigid body through space is determined by the forces and moments

acting on the body and by the inertial properties of the body. A body undergoing linear

acceleration and angular rotation is subject to specific forces and moments in order to

sustain the motion. Equations 4-1-4-3 [64] [33] describe the three orthogonal forces in

terms of the mass, velocity, and acceleration. The force contributions include several

factors, among them body mass, m, times the linear accelerations, 5, b, and tb. Cross-

axis terms also contribute to the forces, where angular velocities are multiplied by the

orthogonal linear velocities. A gravitational component is also included in the force

equations, which accounts for the changing body attitude in a fixed gravitational field.



X = m(5 + qw rv) + mg sin 0 (4-1)

Y = m(t + ru pw) mg cos 0 sin (4-2)

Z = m(W + pv qu) mg cos 0 cos (4-3)

The moments sustained by a rigid body depend upon inertial properties, angular

velocities, and angular accelerations. Equations 4-4 4-6 show the formulation for three

orthogonal moments. Each moment depends on the corresponding moment of inertia

multiplied by the angular acceleration, a cross-axis angular acceleration multiplied by the

product of inertia, and coupled terms relating angular velocities and inertial properties.

The given moments equations are independent of the body orientation and are valid for

aircraft of fixed, laterally symmetric configuration. For aircraft with rapidly time-varying

and .,-viii i. I ic geometi [ -,] [36], the moment equations require inclusion of all the

inertial terms, many of which are zero for conventional aircraft.












L = Ilp Itj + qr(I, Iy) I.pq (4-4)

M = 1,9 + rqI, I, + 11 (p2 r2 (4-5)

N -I + I + pq9I ( ) +I1qr (4-6)


While the general equations express forces and moments in terms of body motion,

the dynamics of an aircraft represent the forces and moments required to achieve a

desired body motion. Furthermore, since the force and moments acting on an aircraft

are composed of contributions from the various aerodynamic appendages, an alternate

representation is used to account for the effect of vehicle motion on the forces and

moments. The aircraft equations of motion are written in terms of the aircraft states

and controls, .,.: --regating individual force and moment contributions to determine the

resulting dynamics.

Aircraft motion can be described by twelve states, which represent position, orienta-

tion, angular velocity, and linear velocity along each of three axes. The standard aircraft

states are described by Table 4.1.

Table 4-1. Standard aircraft states describing vehicle motion
Axis Position Orientation Linear velocity Angular velocity
Roll x 0 u p
Pitch y 0 v q
Yaw z b w r


Linearized aircraft dynamics are written in a form where the effect of each state

or control on each force or moment can be readily identified. Stability derivatives are

coefficients describing a linear relation between a state, such as roll rate, and a force or

moment, such as roll moment. Subscript notation is used for stability derivatives, such as

Ci to describe the linearized influence of roll rate on roll moment. Control derivatives

are similarly expressed, relating control surface deflection to forces and moments such

that Cl, describes the contribution of aileron deflection to roll moment. The individual









contribution of states and controls to forces and moments are dimensionalized by the

vehicle geometric parameters, compared to inertial properties, then linearly summed to

determine state accelerations.

The magnitude and direction of the stability and control derivatives are determined

primarily by the vehicle geometry. Factors such as wing size and shape, tail geometry,

airfoil, weight distribution, and fuselage shape directly affect the coefficients. For a

variable-geometry morphing aircraft, the assumption of derivative linearity may hold

for a fixed configuration, but the coefficient value is expected to change with aircraft

shape. Thus, the standard derivative representation is insufficient to express the change

incurred with morphing. A proper description includes functional dependence of each

coefficient on the morphing parameters, f/, which represent the change in airframe shape.

The nondimensional roll convergence derivative would thus be expressed as C, (f/). The

actual functional dependence may differ for each coefficient and is computed individually

during modeling and simulation.

Dimensional derivatives likewise include dependence on the morphing configuration,

such as,




L( = ---)S(f)b-- (4-7)
21,V


where Q is the dynamic pressure, S is the reference wing area, b is the reference

wingspan, and V is the airspeed. The reference dimensions, S and b, both vary with

respect to morphing due to the changing wing geometery.

The linearized state-space equations of motion including the effects of quasi-static

morphing are shown in Equations 4-8 and 4-9. Lateral and longitudinal dynamics are

assumed uncoupled and are shown independently. The dynamics are affected only by

eight states, four longitudinal and four lateral states. The remaining four states, including

positions, x, y, and z, and yaw angle, i), are strictly for kinematic considerations.











SXW() X (/7) 0 -g u
o Zu() Z.(i7)uo 0 0 w

ST () + M()z() Ma(7) + M(7)z(7) 3(7) +M(7)Uo 0 q
6 0 0 1 0 0



+ Ze


0

(4-8)


Y **** Yp) (1 Yr ) g cos 0 0 Y6r A
O 80 O uoo OI
Lp 3(l) Lp(f) L,(f) 0 p L6a () L6 ,(l) 6

r N3(f) N, (f) Nr(fl) 0 r Ns6 () N6, () 6[

o 0 1 0 0 0 0

(4-9)


4.2 Parametric Variations of EOM

The aircraft geometry is considered to be in the nominal configuration when all

four morphing joint angles (/1, P2, [3, and p4) are 0 neutral. The nominal aircraft

is a straight taper with no dihedral or sweep in either the inboard or outboard wing

sections. The root joint articulates about longitudinal and vertical joint axes as morphing

parameters, pi and P2 are changed, respectively. Under such actuation, the inboard wing

panel changes orientation, while the outboard wing panel changes only position. The

outboard wing panel orientation is decoupled from root joint morphing and is affected

only by morphing parameters, p3 and p4, which cause rotations about longitudinal and

vertical joint axes, respectively. The available configurations combine dihedral/anhedral









and fore/aft sweep of the inboard and outboard wing sections. The wing shapes resulting

from morphing roughly represent observed natural configurations.

The morphing configuration space spans the hypervolume created by all combinations

of morphing parameters, P1,2,3,4 over the range of -30 to +30. A relatively coarse angular

resolution of 5 limits the number of configurations to 134 = 28, 561. Each of these possible

configurations yields a unique value for each of the stability and control derivatives.

The equations of motion for a morphing aircraft depend on the configuration and vary

according to changes in the stability and control derivatives. In addition to the dynamics,

the variable shape also changes the aerodynamic performance and maneuverability of the

flight vehicle.

C'!i ,is,. in the dynamic characteristics during morphing can be quantified at the

derivative coefficient level or can be abstracted to modal characteristics or performance

metrics as functions of one or more aircraft parameters. Such metrics are necessary to

exploit the morphing capability to perform a desired mission, since they can be used to

generate cost functions describing the task-effectiveness of a particular configuration.

Although the vehicle achieves significant variation in wing shape during niip1 iii.-

the relative geometry between the wing and tail remains similar for all configurations. The

wing is used as the primary source for lift and remains distinct from the tail surface, which

is used to oppose the pitching moment from the wing. Furthermore, the shape-change is

assumed to be symmetric about a vertical plane on the fuselage centerline. The left and

right wings morph collectively for all operations, apart from the differential twist action on

the wing tips for roll control.

Shape change occurs at a rate much slower than the vehicle dynamic rates. Time-

varying inertial terms and unsteady aerodynamics associated with the shape change are

not included in the dynamic formulation due to the slow rate of morphing. These assump-

tions are reasonable and allow the structure of the equations of motion to be constant for









all configurations with changes occurring only in the value of the coefficients. The dynamic

properties may still change substantially as a result of the variable coefficients.

The large number of possible configurations results in excessively large tables and

matrices required to store the variations in each parameter. Since independent variations

are available in all four joint angles, the required matrix is four-dimensional and is

difficult to visualize or process. The configuration space can be simplified by fitting a line,

curve, plane, hyperplane, or hypersurface through one or more dimensions. Since each

configuration data is based on aerodynamic configuration modeling, it is unlikely that

a simplistic fit will accurately represent the parameter variations over the entire space.

However, the fit can be used to identify the local variations in parameter values with

respect to a particular configuration.

Reducing the dimensionality of the 4D configuration space may afford access to many

of the desired wing shapes while reducing the complexity of commanding morphing. For

two dimensional configuration subspaces, the surface representing parameter variations

can be quite complex and subject to significant coupling between the two morphing joint

directions. A surface regression fit would require a very large number of parameters in

order to achieve small errors. However, since the rate at which morphing is allowed is

limited due to quasi-static assumptions, modeling of the entire configuration space is per-

haps unnecessary. Local knowledge of the parameter variations may be sufficient to morph

the aircraft into a shape appropriate for the mission task. Gradient descent methods are

used to morph toward shapes with improved performance. Such morphing based on local

knowledge is not guaranteed to converge to the globally optimal configuration.

Local, singular-actuation trends are modeled for the full 4D configuration by isolating

morphing functions. A 4th-order polynomial is used to represent the parametric variations

due to actuation of a single morphing joint. For the nominal configuration, the effect of i1

changes is modeled by constraining P2, [3, and p4 to neutral (0) and tabulating the 13

parameter values for -30 <= /i <= 30 in 50 increments. The polynomial is of the form,












C(pi) = aI + a2/4 + a3/1 + a4/1 + a5/1


where coefficients al,2,3,4,5 are determined by regression and C(pi) is the modeled

parameter, which may be a stability derivative or a performance metric.

The process is repeated for P2, P3, and 4 such that four 4th-order polynomials

describe the parametric variation in all directions through the configuration space and

intersect at the chosen configuration. Linear independence of /7 components is required

to achieve good results using the polynomial representation. For configuration spaces

which are linearly coupled, combined morphing operations are not accurately represented

at the extremes of the space, but are reasonably well defined for local variations in all

morphing directions. In some cases, subspaces of the actuator space can be found where

the morphing is linearly independent and this approach achieves small error over the

permitted space.

4.3 Flight Metrics and Cost Functions

Morphing technology allows an air vehicle to adapt its design during flight, but that

adaptation should be to satisfy some mission objective. As mission tasks change through-

out a flight, the vehicle achieves a high level of performance in each role. Retasking during

a flight may involve disparate objectives and require a substantial change in vehicle geom-

etry. In order to effectively change the aircraft shape, both the capabilities of the aircraft

and the requirements of the mission must be clearly understood.

Quantitative performance metrics are used to evaluate the capability of an aircraft to

achieve a desired level of performance, maneuverability, agility, or efficiency. These metrics

establish the effectiveness of a particular morphing shape at achieving the current mission

objectives and can thus be used to motivate a change in shape, if a more appropriate

configuration is available. As the aircraft is retasked, the metrics by which the aircraft is


(4-10)









evaluated also change. The metrics are used to generate cost functions which relate the

mission requirements to vehicle capability within the configuration space.

A common measure of aerodynamic performance is the ratio of lift to drag. Lift

to drag ratio, L/D, is the fundamental component of determining energy efficiency,

endurance, range, glide angle, soaring capability, and descent profile. The desired L/D

for a given task may be a maximum, minimum, or a power ratio of lift and di ,. such as

L/D2.

For a fixed configuration, lift and drag vary with angle of attack and airspeed.

Optimizing a morphing aircraft for L/D requires a change in configuration in addition to

variation in flight condition. The optimal flight condition for one configuration may be

different from another, -i-:. -I i i; that morphing capability may be used to find efficient

configurations across a range of angles of attack and airspeeds.

The flight conditions for which the aircraft configuration is optimized are segments

of a desired UAV mission. The most common condition is maximum lift to di ,- which

achieves both the longest range cruise[7] and the highest glide ratio[9]. Configuring a

powered UAV for range allows the vehicle to travel the maximum distance for a given

quantity of fuel. In the unpowered case, maximizing lift to drag allows the vehicle to glide

the maximum horizontal distance for a given initial altitude.

Inviscid aerodynamic modeling is used to generate data for morphing configurations,

neglecting sources of drag such as skin friction and interference drag. Modeling results are

supplemented by a simple drag model based on a minimum, zero-lift drag with a quadratic

approximation of lift-induced drag. The drag coefficient is given by,




CD + c (4 )
T ARe L

where CDo is the minimum, zero-lift drag, AR is the wing aspect ratio, and e is the

wing efficiency factor.










CDo is crudely estimated for each configuration using the projected frontal area to

determine pressure drag. The wing efficiency factor is then computed by solving for the

quadratic curve that simultaneously satisfies CDo and the lift and drag output of the

modeling. The drag model is used to estimate aircraft performance during cruise, loiter,

descent, and maneuvering flight phases. Figure 4-1 shows variations of drag with lift for an

inviscid, symmetrical wing. The location of the maximum lift to drag occurs where a line

originating at the origin is tangent to the quadratic drag polar. The lift to drag ratio is

plotted to show show the wing efficiency changes with lift coefficient. The minimum L/D

occurs symmetrically about the CL axis, at a negative angle of attack.


-1.5 -1 -0.5 0 0.5 1 1 %5
0.16 L/D 10
o 0.14 "
0.12 drag / 5
polar
S 0.1
0) 0
) 0.08
o 0.06 -5
O 0.04 *
0"04.'" g tangent -10
0.02 line
0 -15
-1.5 -1 -0.5 0 0.5 1 1.5
Lift Coefficient, CL


Figure 4-1. Drag polar and lift to drag ratio showing graphically location of max L/D


4.3.1 Cruise

The requirements of cruise flight are primarily related to minimizing energy expen-

diture, unless time enroute is additionally restricted. Propulsive and actuator energy

demands are required to be low during cruise flight. Maneuvering is generally not required

during cruise, so actuator usage is not expected to be high, provided the aircraft system

is open-loop stable. Thrust requirements are related to the aerodynamic efficiency of the

aircraft as measured by lift to drag ratio (L/D).

Equation 4-12 from Anderson [7] shows the inverse relationship between L/D and

thrust-required to weight ratio (Tr/W). Tr decreases as L/D increases for fixed W.









Assuming the magnitude of thrust is related to energy consumption, the minimum energy

condition occurs at the maximum L/D for the aircraft.



S (T) (4-12)


Equation 4-13[7] shows the factors that affect L/D for an aircraft in level flight,

where the lift is equal to the weight. The equation includes terms from the drag polar

and the wing loading, both of which are expected to change with different morphed

configurations.


L pV/2 CD,o 2K W(1 (4 13)
D 2 W/S) +p S)

4.3.2 Maneuvering Metrics

Maneuvering and agility metrics describe the ability of an aircraft to achieve desired

angular velocities and angular rates [10], respectively. A change in course requires an

initial acceleration about one or more axis, followed by a sustained angular velocity, and

concluded by accelerating to a terminal flight condition. The desired accelerations and

velocities are large when the maneuver is subject to space and time considerations. A tar-

get engagement, for instance, may require rapid rolling and pitching to acquire the target

in the sensor field of view, followed by precise flight path tracking to maintain a desired

position relative to the target. The aerodynamic requirements for such maneuvers are

different than those for a benign flight segment such as cruise or loiter. The corresponding

geometry that achieves the maximum agility and maneuverability is also expected to differ

from the optimal efficiency configuration.

Several metrics for aircraft maneuverability exist and have been used extensively to

quantify fighter aircraft performance [61] [78]. The basic measure of agility in the current

study is the magnitude of angular acceleration about the three body-axes, t, j, and r.









Similarly, the measure for maneuverability is the magnitude of sustained angular velocity

about the three body-axes, p, q, and r.

The physical limitations which constrain both agility and maneuverability for manned

aircraft include structural load limits, stall boundaries, pilot or occupant G-tolerance,

control surface deflection and rate limits, and available thrust. Linearized morphing

models lack the fidelity to reflect each of these factors, so the limitations will be primarily

assessed from control surface limits, angle of attack limits, and sideslip angle limits. The

structural and occupant factors are not expected to be concerns for MAVs, considering the

high structural strength and pilotless flight, respectively.

4.3.2.1 Agility

Roll acceleration agility is determined from a steady, level flight trim condition where

p, r, and 3 are initially zero.

The roll acceleration dynamics from the lateral equations of motion (Equation 4-8)

are,


SL/3 + Lpp + Lrr + LJa( + L6,6, (4- 14)


In level flight, the sideslip roll coupling, roll damping, yaw-roll coupling terms are

zero. Assuming rudder primarily affects yaw moment, the roll acceleration is,




p = L6a (4-15)


Maximum roll acceleration occurs when the ailerons or wing twist deflect at the travel

limits, 6a,max.




Pmnax = L6aa,max (4- 16)









For uniform deflection limits for all configurations, the roll agility is determined

strictly by the aileron or wing twist roll moment control effectiveness. The magnitude of

Cl, changes significantly due to articulation of the outboard wing with respect to the

flow direction. The dimensional derivative, L6a, changes further due to variations in the

reference wing area and reference span due to morphing.

Yaw agility is derived with similar trim assumptions, namely that all states pertur-

bations are zero. Maximum yaw acceleration occurs when the rudder is deflected at the

travel limits, rax.




max N- = N r,,max ^ (4- 1 7)


Pitch agility is computed about a linearized flight condition, where angle of attack,

velocity, and pitch rate perturbations are zero. The maximum pitch acceleration is

determined simply by the elevator pitching moment effectiveness with the elevator at the

deflection limits, Se,max.




qwmax 6e,,max (4- 18)


4.3.2.2 Maneuverability

The agility obviously changes with flight condition, reaching zero acceleration when

the aircraft reaches the maximum maneuvering rate. Maneuvering limits determine the

capacity to achieve angular velocity along a single axis while maintaining a trim value

along orthogonal axis. Maximum rate is thus determined by cross-axis coupling terms in

addition to rate damping and control effectiveness terms.

The maximum roll maneuverability is derived by first considering the roll acceleration

dynamics.












p = L/3 + Lpp + Lrr + L6a + L,6,r (4-19)


For a simple case that ignores yaw coupling, roll maneuverability is computed

for 3 = r = p = 0. Pitch coupling is already ignored due to separation of lateral

and longitudinal dynamics. Without coupling, unaccelerated roll dynamics are used to

compute the single-axis maximum roll rate, given by,



0 Lpp + L (4-20)

Pmax -Ls.,max (4-21)
Lp

Without coupling, the maximum roll rate is simply the maximum aileron or twist

deflection scaled by the ratio of aileron effectiveness to roll damping.

Single-axis pitch and yaw rate limits are similarly computed. In all cases, rate limits

are simple expressions relating the control power to the rate damping. The stabilized roll,

pitch, or yaw rate is achieved when the moment generated by aileron, elevator, or rudder

deflection is equally opposed by the damping moment opposing aircraft rotation.




qmax -j-(4-22)

rmax (4 23)


Realistically, coupling exists between the aircraft axes and sustained angular rates

are affected by both primary and secondary effects. An aileron deflection primarily affects

roll moment, although it may also cause yaw moment coupling for some configurations.

The yaw moment consequently causes a non-zero yaw rate, which is a secondary and

undesirable effect for a roll-only command. Thus, the actual rate performance must

account for cross-axis coupling and use all applicable control surfaces to achieve maximum









rate in only the desired axis. The roll rate command may require rudder deflection to

oppose the aileron-generated yaw moment. This additional deflection has an effect on the

roll moment which will cause a change in sustained performance relative to the single-axis

case.

The roll rate maneuverability with coupled dynamics is readily derived using the

expressions for roll and yaw acceleration.


=L/3 + Lpp + Lrr + Ls 6a + L6,6r

t =Ng3 + Np + Nrr + NJa& + N,6,r


(4-24)

(4-25)


For the desired maneuver, the aircraft achieves maximum, unaccelerated roll rate with

no coupling to the other states such that,


P -Pmax

r =0

S=0

p=0

-=0


(4-26)

(4-27)

(4-28)

(4-29)

(4-30)


Substituting the state values into the yaw acceleration expression and solving for roll

rate, p, yields,


- N6a N6
P & Np


(4-31)


Substituting this expression into the roll acceleration equation yields an expression

that combines the roll and yaw dynamics and compares the relative magnitude of control

power required to maintain the maneuver conditions.










L N6a a- Ns,6r)
Lp N2 N,, + La6. + L6r, = 0 (4-32)
Lp

P (-N6a N6,) = -L L6, = 0 (4-34)


Np 6, 6,
L No L Nsr Lsa Ls (4-35)

+ (-LPNa6 + L+ -lL + LP N) (4-36)

L --N6 (4-37)


The expression for j gives the ratio between aileron deflection and rudder deflection
required to maintain a roll rate without yaw rate coupling. A ratio greater than 1 requires
larger aileron deflection than rudder deflection. A positive value indicates aileron and
rudder in the same direction, while a negative value indicates opposing control directions.
Maximum roll performance is computed at the deflection limit of one of the control
surfaces. The performance for each configuration depends on the deflection ratio and the
deflection limits for each surface.
In the case of yaw rate maneuverability, the rudder-aileron control ratio is similarly
computed, with the final result a function of control effectiveness and damping along both
roll and yaw axes.





Pitch rate maneuverability with respect to coupling in the longitudinal states cannot
be solved algebraically. The final value theorem [65] and some bounding assumptions
on the states are needed in order to determine the maximum pitch rate. The expression









includes velocity and angle of attack perturbations, which are expected to depart from the

trim values during pitch maneuvering.



-I.'V M'a 31. 6,
q = M^ -3.J (4-39)


4.3.2.3 Aggregate maneuverability and agility metrics

Maneuverability and agility metrics are computed for individual morphing configura-

tions along each of the three aircraft body axes. Aggregate metrics for both maneuverabil-

ity and agility can be computed by combining performance along individual axes.

The maneuverability cost function is given by,



Jm (fl) Wp \Pmax (ji)I + Wq I qmax (ji)I + W, I r.max( )I (4-40)

where axis weights Wp, Wq, and Wr are arbitrary scale factors that are used to bias

maneuverability function value, Jm, to emphasize performance along a particular axis.

The agility cost function is given by,




Juf) Wp \pmax(f)| + Wq \max f7\ + W, \ (4-41)

The configuration which achieves the highest level of performance for a mission

designated by a set of axis weights yields the maximum value for J, and Ja-

Typical aircraft maneuvering requires significantly higher roll rates than either pitch

rates or yaw rates. Roll maneuvering is necessary for changing bank angle to initiate

turns, reject gust disturbances, and perform axial rolls. Pitch maneuvering is required

during climb and dive transitions as well as during loop segments. Both pitch and yaw

maneuvers occur in conjunction with roll during coordinated turns. A standard set

of axis weights is used to reflect the emphasis on roll maneuverability, with secondary










considerations for pitch maneuvering, and tertiary account for yaw maneuvering. The

standard axis weight values are,


Wp =

W, 2
Wq~


(4-42)

(4-43)

(4-44)


Morphing configurations which maximize Jm and Ja are determined using the given

axis weights and stable dynamics criteria. The criteria stipulates that all dynamic modes

must be stable with small unstable allowances for divergent spiral and phugoid modes.

Figure 4-2 shows the most maneuverable configuration under the simulated condi-

tions. The wing assumes an unusual shape near the boundaries of the configuration space,

with limit dihedral and aft sweep on the inboard wing combined with limit anhedral and

large forward sweep on the outboard wing.

Simulating mu,=30, mu,=-30, mu,=-30, mu,=20


05


03


01


Figure 4-2. Maximum maneuverability configuration using stable dynamics criteria
(pi 30, 2 = -30, 3 = -30, and p4 = 20)


The configuration achieves a roll rate of 13.9 rad/s, a pitch rate of 0.05 rad/s, and

yaw rate of 13.1 rad/s. The low pitch rate magnitude is a result of the low elevator

control effectiveness in the simulation model in addition to the large pitch moment of

inertia. The pitch maneuverability may also improve when perturbed velocity and angle










of attack are included in the metric formulation. The large yaw rate is achieved by the

forward swept wingtips, which reduce the sideslip and yaw rate damping.

Figure 4-3 shows the most agile configuration, which assumes a much more nominal

shape than the maneuverable shape. The wing shape uses moderate anhedral on the

inboard wing and slight forward sweep for both inboard and outboard wings. It is

interesting to note the disparity between the two shapes that results from the addition of

damping terms in the maneuverability cost.


Simulating mu =-15, mu2=10, mu =0, mu4=10




01-
005

004
03 0 05


-0 2 01
-03 0


Figure 4-3. Maximum agility configuration using stable dynamics criteria (i = -15,
P2 100, /13 0, and4 = 10)


The agility of the configuration is quite substantial, considering the large roll rate

acceleration of 211.2 rad/s2. Acceleration along the other axis are much smaller, with 70.1

rad/s2 and 20.8 rad/s2 for the pitch and yaw axes, respectively.

While the shapes do not appear directly inspired by nature, certain aspects of the

wing shape reflect observations made of bird flight. The forward swept wingtips of the

maneuvering configuration are common to seagulls performing rapid changes in glide path

and bank angle. The anhedral of the agile configuration are similar to down-turned wings

seen on several species of birds while in accelerating flight.

The morphing configuration space is hyper-dimensional and thus cannot easily be

visualized. Metrics or parameters that depend on each of the four morphing joint angles

may incur highly complex changes through the configuration space. These changes are








represented using slices of the 4-dimensional morphing space hypervolume. Figure 4-4
shows six views of the configuration space. Plots on the top row are fixed at 4 = 15
and show variations in the maneuverability metric due to changes in Pi, P2, and p3. The
color of each square in the plots represents the magnitude of the metric, while its location
indicates the morphing angles. Slices are shown at the space extremes (pi = 30, 2 = 30,
and p3 = -300). A fourth slice is shown at values of pi = (-15, 0, 15). The four plots
show that maneuverable configurations are concentrated at negative values of p3 with
positive values of P2. An additional concentration occurs at positive values of /i with
negative values of P2*






I ii










Figure 4-4. Maneuverability index shown as slices of 4-dimensional hypervolume at
4 =15 (top) and p -15 (bottom)










morphing of 4.
The0 50to 50 fposi iue44so anueaiiymti aafr^













morphing Of /14.









Figure 4-5 represents the maneuverability index for each morphing configuration

as a histogram. Although actuator contextualization is lost, the distribution of values

shows insight into the relative scarcity of highly maneuverable configurations. The average

metric value is 37.3 rad/s, yet the highest value is 57.1 rad/s along a positive skew.


500

S400

S300

o 200

100


01
25 30 35 40 45 50 55
Maneuverability Metric (rad/s)

Figure 4-5. Histogram of maneuverability metric for all morphing configurations
(P1,2,3,4 [-30, 30])


Figure 4-6 shows the agility metric for configurations with 4 = 15 (top row)

and p4 -15 (bottom row). For both cases, the agile configurations exist at positive

values of /i and near nominal values of P2 and p3. The top right plot indicates that the

maximum agility shown occurs at a moderate value of pi, rather than at the extreme.

Relatively non-agile shapes exist along several corner extremes of the space.


Figure 4-7 shows the histogram for the agility metric associated with all configu-

rations. The distribution shows negative skew, with an average agility metric of 599.3

rad/s2. Variations in metric magnitude are large through the configuration space, ranging

from a minimum of 298.2 rad/s2 to a maximum of 807.9 rad/s2.


4.4 Coupling Morphing Parameters

Ci ,i:ig aircraft shape between missions usually requires a trajectory through the

configuration space which invokes multiple joint actuations. Individual joint actuation

has limited usefulness in mission adaptation, whereas variations to combined actuation









L




(


1%


(r


Figure 4-6. Agility index shown as slices of 4-dimensional
and /4 = -15 (bottom)


1


hypervolume at 4 = 15 (top)


600
500

200
100
0
300 400 500 600 700 800
Agility Metric (rad/s2)
Figure 4-7. Histogram of agility metric for all morphing configurations (p1,2,3,4


[-30,30])


can accomplish simultaneous objectives, such as varying aerodynamic performance while
preserving handling qualities.
Biologists have reported observing birds morph wing shape with significant coupling
between joint articulation [91]. Although a bird wing is highly configurable, often only
a subset of the possible shapes are used in flight. The skeletal structure in the wing
promotes coupling through mechanical interaction between the elbow and wrist joint [87].
Two parallel bones between the elbow and wrist allow simultaneous extension of the









joints[91]. The dynamics of the resulting shapes are alvb .-, controllable [54], yet allow the

bird to perform a diverse set of flight tasks.

Similarly, a biologically inspired aircraft may not require the entire configuration

space and may benefit from coupling _i, wr morphing motions. Combining morphing

operations simplifies the shape command by reducing dimensionality and also removes

portions of the configuration space offering limited usefulness.

Desired combinations are identified from 4-dimensional performance or maneuvering

metrics by applying principle component analysis (PCA) to a set of desired configurations.

For individual metrics, such as lift to drag ratio, the shapes which achieve minimum

and maximum values are sampled to generate point clouds in the morphing space.

Both minimum and maximum L/D conditions are desired for descent and cruise flight,

respectively. Many configurations are sampled in the upper and lower percentages, such as

21i '- of the metric range. Combined morphing operations are found by performing PCA

on the configuration points to find the dominant, orthogonal axes through the space which

affect L/D. Reducing the dimension to 2 generates a transformation matrix which relates

morphing operations on the new axes, /PA,B, to the original axes of P1,2,3,4. Morphing

commands to PA,B directly affect the L/D performance and can be commanded directly

by the controller or adaptation system. Transformation to P1,2,3,4 becomes a lower-level

operation that is performed automatically by the actuation system.

Figure 4-8 shows a comparison between the lift to drag performance metric for

the original space and for a reduced-order space with dimension of two. Sampling the

configurations having L/D metrics in the upper and lower 2"1'- at 8.;;' density (1 of 12

shapes) yields 953 sets of actuator positions. All configurations were subject to sampling,

regardless of dynamic characteristics. The distribution of the reduced-dimension set shows

a maximum value close to the actual maximum L/D of 15.1. The minimum L/D value,

however, is 0.61 higher than the actual minimum of 10.5.













0 0

aoc

0
11 12 13 14 15

(0
"Z I2400


0.9
11 12 13 14 15
L/D Ratio

Figure 4-8. Comparison of L/D metric histograms for reduced-dimension space (top) and
original configuration space (bottom)



The L/D values for points in the transformed 2D space can be visualized using a

response surface. The space consists of orthogonal axes, PA and PB, which represent

combined morphing operations. Figure 4-9 shows the variations in L/D with respect to

changes in PA and PB. Primary axis, PA, has a dominant effect on the lift to drag ratio,

while the secondary axis, PI has a somewhat smaller effect. The response surface is still

somewhat complicated and shows significant coupling between the axes and trend reversal

along individual axes. For negative values of PA, for instance, increasing PB decreases

the L/D ratio to the minimum value. Conversely, at positive values of PA, increasing PB

increases L/D to the maximum value.










"2 13



30 -30
20 -20
10 10
0 0
-10 10
-20 20
B -3o0 A


Figure 4-9. Response surface of L/D metric in reduced-dimension space









The general form of the transformation from the 2D combined morphing space to the

4D joint-angle space is,



Pi

P2 T FA
=2 TPP2,3,4 A (4-45)
PA,B
p13 1 B

14

The transformation matrix has the form,



PA,1 IB,1 0.57 -0.41

PA,2 /B,2 0.70 0.27
TP=,2,3, (4-46)
PA,3 /B,3 0.40 -0.24

PA,4 B,4 0.17 0.84

The numerical values of the transformation matrix show the magnitude and con-

vention of the joint angle coupling. For instance, commands to PB result predominantly

in changes to joint P4, with roughly half the deflection in p/, but in the opposite direc-

tion. Smaller commands are issued to P2 and p3 in the same and opposite directions,

respectively, to p4.

Reducing the dimensionality of the configuration space based on a single metric

offers limited benefit compared to a transformation which accounts for multiple mission

metrics. Point clouds are generated using desired values for maneuverability, agility,

L/D, power required, and other metrics. The points are sampled from minimum metric

values, in the case of power required and turn radius, from maximum values, in the case of

maneuverability and agility, or from both extremum values in the case of L/D or airspeed.

Applying PCA to the heterogeneous metric point cloud finds the transformation which

allows the vehicle to transform between various mission-related shapes.









A linear transformation is a relatively simplistic method of reducing the dimensional-

ity of a complex point cloud distribution in the 4-dimensional morphing space. The result

may be inadequate to achieve desired performance in one or more metric. Furthermore,

the shapes transformed space are not guaranteed to comply with dynamic criteria used to

generate the original points.

4.5 Stability Criteria

Several levels of stability criteria are implemented to achieve the desired dynamics in

the optimized morphing configuration. The criteria are based on the operating condition

and the operator, each of which can change over the course of the mission. Mission tasks

requiring a human operator may seek only wing configurations which achieve excellent

handling qualities. Other tasks using an autopilot for routine or long-term operations

may allow any stable configuration. For agile maneuvering, the criteria may allow even

open-loop unstable dynamics, provided that sufficient sensing and actuation capability

exist to stabilize the system.

4.5.1 Handling Qualities Criteria (HQC)

The stability criteria used for remotely piloted tasks must ensure the dynamics are

appropriate for the human operator. The modes must, in general, be stable and lie within

a range of natural frequencies, damping ratios, and/or time constants. Requiring good

handling qualities allows the operator to perform maneuvers manually, without having to

rely on stability augmentation or autopilot. For mission tasks such as take-off or 1 ,idin:

the pilot must maneuver the aircraft along a precise flight path. Manual operations may

also be required for testing new instrumentation or control systems. In each case, the

vehicle must respond predictably to the control inputs and maintain a reasonable level of

pilot workload during the task.

Table 4-2 shows the criteria used to assess handling qualities from the modal charac-

teristics. The criteria corresponds to Class 1 aircraft operating in Category A, which is

applied to light aircraft engaged in precision maneuvers[64]. The numerical values for the









upper or lower modal parameter limits have been determined by qualitatitive pilot eval-

uation over years of flight testing[20]. The use of manned-flight criteria for an unmanned

aircraft may result in conservative performance, considering that the remote pilot is not

subject to the oscillations and loading of the vehicle. However, it will ensure a minimum

level of pilotability in areas where handling criteria for unmanned flight vehicles are not

available.

Mode Time to double Time constant Natural Frequency Damping Ratio
Spiral Divergence T2,s > 12 seconds
Roll Convergence TR < 1.0 second
Dutch Roll Mode* u,DR >1.0 rad/s (DR > 0.19*
Short Period Mode 0.35 < (sp < 2.0
Phugoid Mode T2,PH > 55 seconds (PH > 0.04
Table 4-2. Handling qualities criteria for morphing configuration.
*Dutch roll damping ratio must also satisfy constraint (DRWz ,DR > 0.35 rad/s


4.5.2 Stable Dynamics Criteria (SDC)

Most of the UAV mission will use an autopilot to perform relatively benign maneu-

vers. The presence of a controller relaxes the handling qualities requirement placed on

human-operated flight tasks. The stability criteria for morphing configurations can be

changed to requiring stable poles for all modes. The control gains can be used to obtain

desirable closed-loop dynamics. Requiring stable dynamics prevents excessive actuation

usage resulting from stabilizing an open-loop unstable plant.

Table 4-3 shows the allowable range of eigenvalues for the stable dynamics criteria.

Modes that are classically unstable or marginally stable such as the spiral divergence and

the phugoid mode are allowed to have slightly positive real values. The remaining modes

are allowed to lie anywhere in the left-half plane. The criteria also allows the modes to

deviate from classical form, in that oscillatory modes such as dutch roll and short period

may consist of two negative real poles.









Mode Criteria
Spiral Divergence R(As) < 0.5
Roll Convergence R(AR) < 0.0
Dutch Roll Mode R(ADR) < 0.0
Short Period Mode R(Asp) < 0.0
Phugoid Mode R(ApH) < 0.5
Table 4-3. Stable dynamics criteria for morphing configurations


4.5.3 Unrestricted Dynamics Criteria (UDC)

The final criteria allows unrestricted variation in the dynamics in the morphing

configuration space. Highly unstable dynamics such as forward-swept wings are permitted

to exploit the aerodynamic performance afforded by such configurations. The criteria

makes a number of assumptions about the flight control system that may be unrealistic

for an implementable system. The size of the control surfaces are assumed to be large

enough to trim and perturb the aircraft for all configurations. Actuator rate and motion

are also assumed to have sufficient bandwidth and operating range for stabilization and

control. Finally, it is assumed that the controller can adequately sense the aircraft state

and feedback appropriate commands to the control effectors. In practice, the final criteria

may need modification to allow only a limited range of unstable dynamics. For simulation

purposes, the criteria remains unconstrained.









CHAPTER 5
MANEUVERING CONTROL

5.1 Overview

A morphing aircraft controller must achieve stabilization and command tracking

for all shape configurations and all mission tasks. The highly variable vehicle dynamics

necessitate a change in the controller with changes to the wing shape. The controller

must achieve desired closed-loop performance in the presence of large changes to the

control surface effectiveness and open-loop stability characteristics. The desired controller

performance relates to the mission-task, which mandates rapid responses for accelerating

flight phases and benign, conservative responses for cruise and loitering phases. Such

variations in the performance change as a result of dynamic variations, which must be

properly represented in the control design.

Mission flight commands generate navigation and guidance trajectories which should

be followed for satisfactory task completion. The commands, such as turn, dive, and

pursue, represent outer-loop commands which invoke the maneuvering actions of the

aircraft. The outer control loop accounts for the aircraft dynamics to determine the

allowable accelerations, which may be different than the commanded accelerations.

Disparities between allowable and commanded trajectories motivate shape changes which

seek to reduce flight path errors and improve mission performance [48]. As the dynamics

change, the maneuvering controller compensates to achieve stabilization and tracking of

the inner-loop rate and angular states.

At the most basic level, the purpose of the inner-loop maneuvering controller is to

stabilize and track commands to the angular rates, p, q, and r, and the angles of attack

and sideslip, a and f3. Tracking control objectives for each of these states is determined by

the relative .,.;-* -ressiveness of the mission task. Time- and space-critical flight tasks require

rapid response and allow large actuator usage, whereas other flight tasks emphasize energy

conservation and slower accelerations.









Attitude, force, velocity, and position control are considered as part of the outer-

loop control, which produces commands driving the inner-loop tracking. Each of these

functions relies directly on the inner-loop control of angular rates and flight angles. The

design of the outer-loop controllers are motivated by mission tasks but are independent of

the configuration. Such a decoupling will produce large tracking errors for configurations

not suited for certain mission tasks. These errors are used to drive shape adaptation which

seeks to find a new configuration to improve performance.

Each morphing configuration is assumed to be represented by a slowly time-varying

linear model. Corresponding control designs are linear and do not include time-dependent

aerodynamic or inertial terms. For fixed configurations, the dynamics are represented by

linearized, small-perturbation theory.

As the aircraft varies morphing configuration, the dynamics are known to vary

substantially from the nominal aerodynamics. Given the large configuration space and the

corresponding complex wing shapes, the dynamics and performance vary over a wide range

and can be adapted to achieve high-performance in a variety of mission scenarios.

Adaptation or scheduling of the control gains is required to achieve the desired con-

trollability and stability for all intermediate and extreme configurations. The change in

the controller gains is directly motivated from the change in the open-loop dynamics,

which in turn change due to aircraft shape. However, even a coarsely discretized con-

figuration space results in an excessively large number of possible shapes. Computing a

controller for each of the thousands of configurations is impractical in time and memory

constraints.

One approach is to exploit the smooth variations in the dynamic parameters to reduce

the number of required controllers. Design points are chosen within the configuration

space to provide a simplified representation of the parametric variations due to morphing.

Controllers are developed for these design points and interpolated in the enclosed spaces

using a linear parameter varying (LPV) or quasi-LPV approach. The latter method allows









straightforward interpolation of the controller gains for configurations lying between design

points. Although shown effective in studies[83], the methodology lacks the theoretically

satisfying proof of stability or robustness existing at the design points. The simplistic

gain interpolation can be replaced by a properly designed LPV system, which accounts for

closed-loop performance stability guarantees for all possible configurations.

The choice of design point density relates to the allowable parametric variations in

the baseline control design. A control synthesized to be robust to 10'A. uncertainty in

the parameters requires a new design point whenever the change in the parameter values

exceeds 10'. In a large configuration space that defines a complicated change of wing

shape, the aerodynamic parameters vary over large ranges. Small tolerance to uncertainty

necessitates extremely dense design point grids that approach the density of the original

space. Such a large number of design points trivializes the computational benefits of

interpolation and fails to provide a satisfactory solution to the control challenge.

The 134 wing configurations achievable by the morphing aircraft each require an

appropriate compensator for closed-loop stability and controllability. Designing controllers

for each configuration is computationally unreasonable and requires significant memory for

storage. An alternative approach of linearly interpolating controllers over subspaces of the

configuration space is computationally tractable only for a coarse distribution of design

points. Such a coarse distribution results in significant errors in modeling the complex

parameter response surface resulting from morphing. The scheme linearly interpolates

controller gains, even for cases where the dynamics are not well represented by piecewise

linearity. The errors lead to intermediate configurations using highly off-design controllers.

The desired approach to controlling a highly configurable morphing aircraft is by

allowing the controller to adapt to changes in the vehicle dynamics. Since the morphing is

assumed to be quasi-static, the system does not violate the constant/slowly time varying

assumption necessary for convergence and stability of adaptive controllers. Robust and

optimal control techniques are used to synthesize nominal and design point controllers,









which provide a coarse grid of closed-loop systems with known behavior. At intermediate

points, an adaptive component is added to the baseline interpolated controller to allow

the system to adapt to unmodeled dynamics. The technique can also be used without

interpolation by simply allowing the controller gains to slowly adapt with the dynamic

variations.

The control augmentation uses the form of Model Reference Adaptive Control

(I!IAC), which also requires definition of an additional system to provide the desired or

reference system response [50]. This reference system changes as a function of mission task

to represent varying control objectives. The baseline controller gains are adapted until the

system output closely matches the reference model response.

The MRAC controller structure uses a feedback gain matrix to relate sensor output

to control surface input and a feedforward matrix to relate reference commands to control

input. Both gain matrices are adapted in response to error between the actual and

desired system response. The feedback component is identical to a regulating/tracking

control task. LQR and robust synthesis are used to identify the initial feedback gains,

which provide the desired level of performance and robustness at the design points. The

feedforward component is initialized to relate reference commands to logical control

surfaces, such as using ailerons primarily for roll commands and rudder for sideslip and

yaw commands.

5.2 Single Degree-of-Freedom Morphing Systems

A single degree-of-freedom morphing system uses coupled joint articulation to

limit reconfiguration to a subset of the configuration space. A single allowable actuator

trajectory is defined through several desirable points in the space. Each point may

correspond to a particular mission or metric optimization. The trajectory is in the form

of a 4-dimensional spline to include all desired configurations and guarantee dynamic and

performance requirements along intermediate shapes. While the trajectory is allowed to

take complex shape, it is mostly monotonically varying through the space. Commanding









morphing operation is greatly simplified, while the geometric wing variation remains quite

complex.

While the wing shapes of the single-degree-of-freedom system are complex, the

actuator trajectory is defined at a relatively small number of configurations. A straight

line in the actuator space passing through the nominal configuration encompasses 13

points. The 4D spline trajectory defines less than 20 points, which are easily stored in

memory and recalled during morphing.

Dynamic models for the small number of allowable configurations are used to design

individual controllers. The control design uses either MRAC-based gain scheduling

and Ho robust control. In the latter case, interpolation of the controller gain matrix

for changes in a parameterized dynamic model have been shown in simulation to yield

reasonable closed-loop results[83]. Ho controllers have also been used in switching

schemes[44], where the dynamic model varies over a large range and one controller is used

to control a subset of the dynamic variations.

An effective switching-controller approach[44] switches between .i.11 'ent configu-

rations by using a set of phantom controller. Both .,i.i went controllers are continually

reinitialized using the states and actuator positions of the current dynamics and con-

troller. When the morphing is commanded, the .,.11] ent controller assumes control of

the modified dynamics while preserving continuity of the actuator command. The rate of

morphing is assumed to be slow enough that no transient effects are caused by the shape

reconfiguration.

5.2.1 Robust Control Design

Robust controllers for each configuration are designed with a synthesis model which

uses frecq I ii .--dependent weighting functions to penalize tracking errors and actuator

usage. Figures 5-1 and 5-2 show the synthesis models for the lateral and longitudinal con-

trollers, respectively. The dynamic model and the weights are functions of the wing shape









and mission, respectively. For each configuration in the reduced space, the components of

the synthesis model are assumed to be static.


Piat


. . . .


Figure 5-1. Lateral Ho controller synthesis model








tPlon
65,,--W^L--^ f/^ '---


Figure 5-2. Longitudinal Ho controller synthesis model


Performance synthesis weights are penalties on the tracking or regulating errors of

each state output. The weight magnitude at a particular frequency is inversely related

to the allowable error. This error penalty generally varies with frequency such that low

frequencies near steady-state are tracked with very small errors. At high frequencies,

beyond the point at which the vehicle can physically respond, the penalty is reduced so

that large errors are allowed. The transition from high to low penalty occurs at different

frequencies for the various missions. Cruise flight involves benign maneuvers mostly near









steady-state trim. Thus, the frequency threshold for the performance weight is low, since

tracking acquisition is not critical during trim perturbations. For ..'-.-ressive flight with

rapid maneuvering, the tracking penalty continues to higher frequencies, forcing small

tracking errors.

Actuator synthesis weights are penalties on the actuator usage. The magnitude of the

actuator weight is related to the allowable motion. The weight is typically small at low

frequency and large at high frequency, allowing large, slow motions and only small, rapid

motions. The weights are often used to define an actuator model in the controller. For a

multi-mission aircraft, the weights are also used to promote conservative actuator use for

benign mission tasks and allow full authority control during critical flight regimes.

The variation of the actuator weights between mission segments can allow higher

frequency usage for maneuvering tasks compared to cruise segments. Alternatively, the

magnitude of the actuator deflection can be varied, with only small movements allowed

for cruise while actuation at the deflection limits is allowed for maneuvering. Both the

rate at which the control surfaces are actuated and the deflection angle affect energy

consumption. Thus, the restrictive actuation during benign flight is intended to promote

energy conservation at the expense of transient tracking performance.

Inputs to the lateral synthesis model are reference commands to sideslip, roll rate,

and yaw rate. The sideslip command is typically zero, promoting a regulating rather

than tracking function. Roll rate is commanded by the roll angle error during turns

and roll maneuvers. Yaw rate is commanded during turns in conjunction to the roll

rate and roll angle command to achieve coordinated flight. Aileron and rudder control

surface actuations generated by the controller also input to the synthesis model directly

to the plant. Noise is added in general to the plant outputs, although is not used in the

simulations.

Inputs to the longitudinal synthesis model are reference commands to angle of attack

and pitch rate. The angle of attack is commanded to maintain the required lift force for









steady or maneuvering flight. Pitch rate is commanded from pitch angle error during

looping flight. Elevator is the sole control input to the longitudinal synthesis model.

The outputs from both the lateral and longitudinal synthesis models include sensor

outputs to the controller, and scaled performance and actuator errors. Performance errors

are tracking or regulating errors scaled by the synthesis weights, while actuator errors are

control inputs scaled by the actuation weights.

The Ho, design computes controller gains that minimize the performance and actua-

tor errors from the synthesis model. Robustness of the resulting controller is expressed in

terms of the minimum parametric perturbation that can result in instability. The desired

value for this measure of perturbation, 7, is less than 1 for normalized weights. For certain

dynamic systems, penalties, and actuator constraints, this desired value is unrealistic and

the controller achieves a 7 of greater than 1. Reports in the literature indicate that such

a controller is still both effective and robust over small uncer- ,iii y[29]. Controller robust-

ness despite high 7 is due to conservatism due to unrealistic constructs such as complex

uncertainty and infinitely fast signals[97].

5.2.2 Simulation Results

5.2.2.1 Overview

The simulations are based on the computational models generated for the variable

gull-wing aircraft. The range of morphing is assumed to be identical to the modeling

section, namely inboard vertical angle pi 30 and outboard vertical angle p3 = 30.

Both sweep joints remain in the neutral position, P2 / 4 = 00. The model is assumed

to be single-degree of freedom since the morphing occurs only between the identified

configurations, although two wing joints articulate simultaneously.

Simulated responses are generated for each mission phase, where the aircraft model is

assumed to be morphed at the optimum configuration specified by Equation 5-1. Simple

inner-loop control responses are used to show the different closed-loop objectives for each

mission. The worthiness of a given controller is determined by the ability to track roll









rate, pitch rate, or sideslip commands. Any of the outer-loop control efforts are ultimately

dependent on one or more of these inner-loop controller. Thus, the controller performance

shown here, although simplistic, is indicative of the general applicability of the morphing

and control design to a desired flight regime.



mmin J = Wat Pat) Plat,desired + Wo || ( ) Pln,desired|| (5-1)

Where Wiat and Wiln are design weights, Plat,desired is a reference system exhibiting

the desired lateral response, Plon,desired is a reference system exhibiting the desired lon-

gitudinal response, and Piat(fl) and Plro(fl) are the lateral and longitudinal dynamics,

respectively.

5.2.2.2 Cruise flight

The optimal morphing angles predicted by Equation 5-1 for the cruise condition with

weights Wiat = 0.0 and Won = 1.0 are inboard angle pi = -10 and outboard angle

P13 5- 0

The ultimate requirement of cruise flight is simply to maintain a straight-and-

level attitude at a desired airspeed, altitude, and heading. The airspeed chosen will

balance between energy efficiency (endurance), range, and time-enroute. Cruise altitude

is also related to efficiency, but for small unmanned vehicles, the desired altitude is most

likely influenced by ground-obstacle clearance and observability. Finally, heading is

chosen primarily to be toward the area of interest, but may be varied during the flight to

circumnavigate obstacles.

Level airspeed and angle of attack, a, can be considered as the parameters that most

directly affect L/D and energy efficiency, and thus the objective of the cruise controller

will be to pitch to achieve and maintain a desired airspeed and a. The performance of

the controller in tracking a pitch rate doublet is shown in Figure 5-3-left. The transient

response of the controller is somewhat slow, although the pitch rate converges to the

desired value with little steady-state error. The slow response is necessitated by the low










actuator rate condition. High actuator usage necessarily expends energy, since cruise

flight does not require particularly fast response. The lower plot of Figure 5-3-left shows

the elevator position for the both the desired model and the actual morphing plant. These

do not need to correspond, since they are attempting to control different plant models.

The solid line shows the actual elevator deflection required to achieve the pitch rate shown

in the upper plot. The actuator moves slowly and smoothly, yet is able to stabilize and

track pitch rate satisfactorily.




So1. 1...... a...
................. ..... ................ -1
-cU

0 1 2 3 4 0 1 2 3 4
4
C2-

-2-
0 1 2 3 4 0 1 2 3 4
Time (sec) Time (sec)

Figure 5-3. Pitch rate pulse (left) and roll rate pulse (right) command simulation for
cruise flight. Linetypes: actual responses and elevator/aileron, rudder,
... command


Figure 5-3-right shows the roll rate response and associated aileron and rudder

deflections during a roll rate doublet. As with the pitch response, the roll rate achieves

the desired value within 0.25 s and maintains little error for the duration of the pulse.

The control surface deflections are shown in the lower plot. Both aileron and rudder

are used to track the roll rate command. The rudder input results from adverse yaw

coupling from the ailerons and the penalty on incidental yaw in the controller formulation.

Thus, the rudder actuates to reduce the yaw rate and sideslip fluctuations during the roll

doublet. The rudder also has proverse roll effects, which assists the ailerons in achieving

the commanded roll rate.









5.2.2.3 Maneuvering flight

The optimal morphing angles predicted by Equation 5-1 for the maneuver flight

with weights Wiat 1.0 and Wlo = 0.5 are inboard angle p = 50 and outboard angle

P3 -10.

The requirements of maneuvering flight are dissimilar to the cruise flight objectives

in that the aircraft is assumed to be constantly accelerating in pitch, roll, or heading.

Maneuvers may require large, rapid control deflections to achieve high rise times and

good tracking performance. A maneuvering mode may be appropriate for tasks such

as following a target of interest, avoiding an unexpected obstacle, or maneuvering in a

densely populated environment.

The maneuvering target model has been specified to emphasize fast dynamics and

large force response. The control synthesis weights on the plant outputs and actuators are

determined to exploit the high rates and achieve fast maneuvers. The actuation weight

penalties have been modified relative to the cruise controller to allow movement at higher

frequencies. Additionally, the performance weights are adjusted so that a penalty is

assessed on tracking errors, even at high frequencies.

The results of the maneuvering simulation are shown in Figure 5-4. The response to

a pitch rate doublet is shown on the left and the response to a roll rate doublet is shown

on the right. In both cases, the morphed model achieves a rise time of roughly 0.15 s. The

required actuation for such performance is notably faster than the cruise actuation. Rapid

elevator deflection is required at each step, including a small direction reversal at the peak.

The aileron actuation required to achieve the roll performance is also quite rapid,

although without the reversal necessary in the elevator actuation. Only a small amount

of rudder deflection is necessary to complete the maneuver. The differences between the

levels of control actuation in the cruise and maneuvering models is related to the different

B-matrices for each condition. The maneuvering model incurs less coupling from aileron

actuation to yaw rate and sideforce, and thus requires less corrective rudder.













-, .....o....... ...... ..
I1 / ------- | --1 / ---

2 -1
0 1 2 3 4 0 1 2 3 4
4
5-


|-5-
-4
0 1 2 3 4 0 1 2 3 4
Time (sec) Time (sec)

Figure 5-4. Pitch rate pulse (left) and roll rate pulse (right) command simulation for
maneuvering flight. Linetypes: actual responses and elevator/aileron, -
rudder, ... command


5.2.2.4 Steep descent flight

The optimal morphing angles predicted by Equation 5-1 for the steep descent flight

with weights Wat = .0 and Won= 1.0 are inboard angle p = -25 and outboard angle

/3 25.

Flight conditions at high angles of attack cannot be effectively simulated using vortex-

lattice method codes or linear models. As a result, the control results presented here are

for simple maneuvers at morphing conditions that have been shown in flight tests to be

suitable for steep descents. The objectives are simply to track pitch and roll commands

fast enough to be useful in maneuvering during a steep dive and recovering to level flight.

Figure 5-5 shows the simulated response of the dive-aircraft to pitch and roll rate

doublets. The responses show relatively fast rise times achieved by reasonable levels

of elevator, aileron, and rudder actuation. The pitch rate response, shown on the left,

exhibits little steady state error. The roll rate rise time is slightly faster than the pitch

rate rise time. Both aileron and rudder are used to achieve the roll tracking, especially in

the transient phase of the response. The use of rudder is an indication of the increased

levels of rudder to roll moment coupling and sideslip to roll moment coupling, both of

which will improve the roll rate response of the rudder.













S. \ ,.......... o
2 -10
0 1 2 3 4 0 1 2 3 4
4
-5
0) 0)



-5- -4
0 1 2 3 4 0 1 2 3 4
Time (sec) Time (sec)

Figure 5-5. Pitch rate pulse (left) and roll rate pulse (right) command simulation for
steep-descent flight. Linetypes: actual responses and elevator/aileron, -
rudder, ... command


5.2.2.5 Sensor-pointing flight

The optimal morphing angles predicted by Equation 5-1 for the sensor-pointing flight

with weights Wiat = .0 and Wlo = 0.1 are inboard angle pi = 25 and outboard angle

/L3 -20.

The final phase of the mission consists of a sensor-pointing task, where the vehicle

must decouple the velocity and attitude in order to favorably direct the field of view of

a sensor while maintaining a favorable flight path. The example used here of sideslip

commands assumes that a target of interest has moved laterally in the field of view. The

aircraft will command a sideslip in return the target to the center of the sensor footprint.

A generic unit-step doublet is used to evaluate the sideslip tracking. Figure 5-6 shows

the sideslip, roll rate, and yaw rate responses to the sideslip command. The aircraft is

constrained to maintain low roll rates while tracking sideslip. The morphing optimization

emphasized small sideslip to roll coupling, although some opposite aileron deflection is

necessary to counteract the small amount of coupling remaining in the model.

The desired sideslip is achieved in just over 0.3 seconds. The steady-state error in

sideslip is very small, indicating good tracking. Some roll rate oscillation is evident in the

transient phase of the maneuver, although it is not excessive. Large rudder deflections










are necessary to achieve the desired sideslip. The magnitude of the rudder deflections are

perhaps overpredicted due to the small B-matrix values estimated by the aerodynamic

code.



S5
I 0 0- -S-3S^^^ -^--
-5-
< 0 1 2 3 4
100
2t "
I' -- -- --




100---
0 1 2 3 4
Time (sec)

Figure 5-6. Sideslip pulse command simulation for sensor-pointing flight. Linetypes: -
sideslip and aileron, roll rate and rudder, -.-. yaw rate, ... command


5.3 Multiple Degree-of-Freedom System

Morphing systems with multiple degrees-of-freedom are more flexible in adapting

to disparate mission scenarios. The improved configurability of the wing spans a larger

subset of the configuration space at the expense of an increased number of possible shapes.

The approach is more versatile in adapting to shapes that are far apart in the space

and cannot be easily contained in a single actuator trajectory. However, the increased

dimensionality precludes an approach relying on dynamic model and controller storage for

each shape, as used in the single degree-of-freedom case.

A linear parameter varying approach can be used to solve the multiple degree

of freedom morphing control problem while preserving reasonable memory storage

requirements. The approach assumes that the dynamics vary linearly with the morphing

parameters. The parameters may be the actuator joint angles for a 4D case or the

transformed morphing commands for reduced-dimension cases. A two-dimensional case

will be used in developing the control approach, although it may be extended to higher

dimensions with some increased complexity and memory capacity.









Controllers are computed at a few points where the dynamics are known and in-

terpolated elsewhere. For dynamics that do not vary linearly, this form of interpolation

will introduce modeling errors and may degrade the performance of the controller. How-

ever, for small allowances of parametric variation, the approach has been successfully

demonstrated, despite having no guarantee of robustness[83].

5.3.1 Optimal-Baseline Adaptive Control

The general architecture used for control of morphing vehicles is model reference

adaptive control. The framework allows sufficient flexibility to achieve a high level of

performance in the presence of uncertain, shape-dependent dynamics, and varying mission

objectives. Optimal control techniques can be used with the MRAC design to achieve

known closed-loop characteristics at select design points. For off-design points, the

controller adapts to reduce tracking errors and is shown to have Lyapunov stability [94].

Figure 5-7 shows a block diagram of the MRAC architecture for a lateral controller[94].

The input to the system is a vector of external commands, r, for each of the lateral states.

A reference model, PA, represents a set of desirable dynamics which produce favorable

state output, Xm, when subject to the external command, r. The state response from

the vehicle dynamics, P, is given by vector x, which is used in feedback to the adaptive

controller to produce a vector of actuator inputs, u. The difference between the desired

and actual responses is given by error, e, and is used to drive the adaptation law. The

controller parameters, K, are adapted in order to change the closed-loop dynamics and

reduce the model-following error.

The MRAC structure is extended to encompass a plant model with dynamics that

vary with morphing actuator positions, f/. The reference model dynamics also change with

mission task, [, allowing the desired response to match disparate mission objectives. The

adaptation function of the controller is preserved, except that a nominal controller, K, is

designed and known for several points in the actuator space, /.





















Figure 5-7. Model reference adaptive control (! I RAC) architecture for lateral dynamics


MRAC can readily accommodate systems with arbitrary uncertainty in the param-

eters of the plant model. For the simple case where the plant structure and parameters

are known and the reference model is given, the control task reduces to gain schedul-

ing. In such a case, the desired controller can be directly computed by algebraic matrix

manipulation.

Systems with known dynamics for all configurations are considered as a separate

and somewhat trivial case of the morphing control problem. In general, the aircraft is

considered free to change shape and achieve dynamics that are not exactly known to the

controller during operation. The general form of the MRAC problem for lateral control

requires computing a desired reference model, estimating the dynamics of the aircraft, and

adapting the control to reduce disparities between the reference and actual responses.

The dynamics of a general morphing system are represented by,



-(t) A(7()x(t) + B(fl)u(t) (5-2)


Both lateral and longitudinal open-loop dynamics are given in this form in Equations

4-8 and 4-9. The state vector, x, can represent the lateral, longitudinal, or combined

states. Control input vector, u, is the output of a controller for the closed-loop system.

System matrices, A and B, are dependent on the geometry of the morphing wing.









Equation 5-4 shows the form of the lateral reference model. The structure is similar

to the lateral vehicle dynamics in the A-matrix. The B-matrix represents acceleration

sensitivities with respect to a series of external commands, r, rather than actuator inputs.

The desired states f3, Pm, and rm are direct analogs of actual states. The numeric value

for each of the reference model terms is computed using modal or derivative shaping

procedures.




m = AImxm + Bmr (5 -3)

m m (1 0 m Y ) 0
Lam Lpm LrmO 0

Pm Pm + L3, Lp, Lr, Pr (5-4)
NF m Np_ N_ 0
m rm N3, Np, Nr, r
0 1 0 0

The output of the adaptive controller depends on both the external command, r, and

on the system states, x. Control input to the plant, u, with gain matrices for both the

feedforward and feedback components is given by,




u = Kx + Kr (5-5)


Where K1 is the adaptive feedback gain and Kr is the adaptive feedforward gain.

The control law is included in the open-loop dynamics to yield the closed-loop system

given by,




S= (A + BKY)x + BK'r (5-6)


The convergence of the system output to the desired response depends on the

existence of gains, Kx and Kr, to satisfy the model matching condition,












A, =A + BK) (5-7)

Bm =BKT (5-8)


If A, B, Am, and Bm are known, K, and K~ can be determined algebraically by

solving a system of linear equations. Such a method provides a simple gain scheduling

approach that can be used to rapidly compute a controller.

For uncertain dynamics, controller adaptation is necessary to achieve the desired

performance. The closed-loop system response is compared with the reference model to

generate a tracking error, given by,



e = x Xm (5-9)


Tracking errors in the response drive adaptation of gain matrices, Kx and Kr. Adap-

tation laws are designed to guarantee Lyapunov stability in the response and guarantee

that all the signals and inputs are bounded. The standard form for the adaptation laws

includes step-size and directional parameters, which are used to regulate and tune adapta-

tion behavior. The adaptation laws are given by Wise and L i,. i-:y[94] as,



x= -xe PB (5-10)

4 = -PreTPB (5-11)


Where Q, rP and F, are symmetric, positive definite matrices. P is computed by

solving the algebraic Lyapunov equation [94]



PAm + AP =-Q (5-12)









The feedback gain matrix, K,, is initialized as an optimal LQR controller[47] for a

design point in the configuration space. K, is a solution to the LQR cost function[34]

and achieves desired performance at the design point, although it does not guarantee

robustness to parametric uncertainty.


r --L---IP----- X






Figure 5-8. Linear quadratic regulator (LQR) controller used for initialization of gain
matrix, K1.


Equation 5-13 gives the LQR cost functional, which is minimized by finding the

optimal control input, u.



J= [x'(t)Qx(t) + u'(t)Ru(t)] dt + x'tfQfx(tf) (5-13)
to

Where Q and R are penalty matrices affecting the state error and control actuation

power, respectively.

The controller assumes full state feedback and is of the form,


u(-) -Kx(.) (5-14)


Where K is the controller gain matrix.

The quadratic regulator problem is easily extended to tracking systems by augment-

ing the dynamics to include a tracking error state. The tracking LQR system includes a

feedforward component from the external command and a feedback component from the

state tracking error. The structure is similar to MRAC and is readily used to generate

initial gain values for design point configurations.









Gains K, and Kr are subsequently adapted as the aircraft is reconfigured and the

dynamics diverge from the design point model. The gains are reinitialized as the wing

configuration returns to a design point. The interpolated gains can also be used as a

backup controller in a switching sense. The interpolated controller operates as a phantom

system, computing the input and state errors without affecting the actuators. If the state

errors are less than the adapted gains for a specified time threshold, the system can switch

to the interpolated gains to prevent adaptation divergence and poor performance.

5.3.2 Reference Model Design

The MRAC architecture is based on adapting the closed-loop dynamics to match a

desired response. The design method readily accommodates a multi-role mission scenario,

since the reference model can be changed for each flight task. Designing an appropriate

model response is critical to achieving the desired level of mission performance. Reference

model design considers two broad flight conditions of an accelerating phase demanding

rapid response and small errors and a benign, non-critical phase tolerant of a slower

response and larger errors.

The lateral reference model is designed to achieve pure responses in the roll and

dutch roll modes. The two are uncoupled in the design to promote precise control over roll

rate, yaw rate, and sideslip. Modal approximations[64] are used to formulate the desired

reference dynamic model. The approximations simplify the design process and allow for

direct solution of the model in terms of the transient requirements of the mission.

Roll dynamics are approximated by a first-order function relating roll rate and

aileron deflection to roll acceleration through roll damping and aileron effectiveness,

respectively. The damping term resists roll rate and causes the mode to converge for

most conventional aircraft geometries. Roll response dynamics are characterized by a

time-constant of the exponential response. A small time-constant indicates large damping

and rapid convergence, while a large time-constant indicates the opposite. The first-order

roll approximation[64] is given by,












S=Lpp + L6a (5 15)


The desired roll damping term, Lpm, is related to the mission-specified roll time-

constant, Tr,m, and roll mode eigenvalue, Ar,m, by,



1
Lp,m = = Ar,m (5-16)
m-r,m

The magnitude of the roll time constant lies between the respective requirements for

the fastest, most time-critical missions and the slowest, least-time-critical missions.




Tfast < 'Tr,m < Tslow (5 17)

Dutch roll dynamics are more complex than the simple first-order roll response.

The dynamics are often oscillatory and heavily coupled between the lateral states. An

approximation to dutch roll given by Nelson[64] removes the roll rate component and

describes the motion as a coupled response in sideslip, 3, and yaw rate, r. The dutch roll

dynamics of the reference model are given by,



[4] Y 3, ( 1 Yrym [j [y ],
ST Y[o Y r (5-18)
n N[N,n Nr,m r Nr Nr, jr

The second-order response can be designed to achieve a desired natural frequency

and damping ratio, each depending upon the time-rate and tracking requirements of the

mission task. Time-critical missions use a high-frequency, highly damped dutch roll model

while benign missions use a lower-frequency model.

The desired natural frequency, un,m and damping ratio, (m,n can be readily related to

the stability derivatives of the reference model using the relations from Nelson [64],











Y=3,m,,Nr,m N3,mYr,m, + UoN0,,, (5-19)
Wn,m ------------ (5 19)

-1 (Y + N )(520)

The reference model stability derivatives are grouped and scaled relative to the

parameters from a nominal model to achieve the desired modal characteristics. Damping

derivatives, Y3,, and Nr,m, are grouped and collectively scaled in the computation of

the damping ratio. To achieve the desired natural frequency, N3 is varied while the

remaining derivatives are held fixed. Y, remains at the nominal value due to a small

relative magnitude.

The expressions relating Y3,, and Nr,m to nominal parameter values Y3,o and Nr,0 are

given by,




Y 3,m Z =Z,,. Y3,0 (5-21)

Nr,,m =Z ,,, Nr,o (5-22)


Where scale factor Z,, is given by,



-3,0o Yo + uNr3,0 (5- 23)
.. -NsOflro + oNfo

Similarly, the expression relating Nm,.n to the nominal derivative, N,0, is given by,



N3,rn Z-,, N3,o (5-24)


Where the scale factor Z, is given by,












Z(m n,mUO (5-25)
YU3,0 + toNr,o

The scaled stability derivatives are used to compute the desired set of lateral dynam-

ics, where the roll and dutch roll modes are distinct. Since the derivatives are computed

using simple algebraic scaling, the modal frequency and damping ratio are direct results of

the desired values previously set.

The simplistic desired model does not account for coupling between the modes that

is likely to exist in realistic dynamics. This discrepancy can lead to poor adaptation per-

formance, as the system attempts adaptation to an unachievable model. Including some

coupling between the modes in the reference model may improve adaptation performance

while mostly preserving the desired, uncoupled response.

An extreme logical extension of the reference model design process is individual

stability and control derivative -ih pi'- as studied in [1]. Each term is shaped individually

to the mission requirements and ., .: --regated in the equations of motion to yield the desired

dynamics. This process yields favorable results for simple response, although does not

guarantee modal characteristics. Shaping the derivatives directly may cause undesirable

oscillatory or unstable responses to occur in the reference model. Conversely, designing

the modes directly may command unrealistic stability derivatives. A successful solution

may be some combination of both techniques, as in the case of adding light coupling to the

designed reference model.

5.3.3 Design Point Gridding

Design points provide anchors of dynamic truth within a sea of parametric uncer-

tainty. They are select locations in the configuration space with known dynamics. The

controllers designed at these points will be interpolated for intermediate morphing shapes.

The locations and density of the design points affect the accuracy of the linear dynamics

assumption, the performance of the resulting closed-loop system, and the memory storage









requirement. Relatively few points must be selected from within the large configuration

space to make the interpolation problem tractable.

The locations of the points are chosen such that modeling errors are minimized. A

simplistic rectangular grid, for instance, may result in many design points in an area of

sparse parametric change or few design points in an area of rapid variation. In the former

case, the system uses excessive memory while the latter case causes unacceptable modeling

errors and control results.

The use of design points and interpolation is useful even if adaptive control is used.

The interpolated controller can provide the baseline compensator which is subsequently

adapted to reduce errors. The system can also revert to the interpolated controller if the

adaptive system begins to diverge and produce larger errors than the baseline case.

The design point gridding operation can be performed on spaces of arbitrary shape

and dimension, including the full, 4D actuator space, a 2D transformed subspace, or a 1D

spline trajectory. For illustration, the process is described with respect to a square, 2D

grid with 13 configurations along each dimension and 169 possible shapes. The nominal

configuration exists at the center of the space, although this is not necessarily true in

general.

The gridding procedure identifies points that are parametrically dissimilar as candi-

date design points. The configuration space is initially seeded with design points at the

nominal configuration, the boundary intersections, and the boundary mid-points. These

points are included in the design space regardless of parameteric similarity to ensure that

boundary conditions are properly modeled.

Starting at the nominal configuration, an outward concentric search is performed to

determine the extent of the parametric variation. Configurations immediately .I11i went to

the nominal configuration are evaluated relatively using,











P dj Pnom (5-26)


Where PnomT is generally a matrix of parameters from the nominal configuration. The

parameters can be stability derivatives or can be a set of performance metrics, in which

case Pnom is a vector. Pdj is a vector or matrix of similar parameters from the neighboring

configuration under evaluation, and 6p is a matrix of relative differences in the parameters.

The parameters by which the space is segmented can be either performance metrics,

modal characteristics, stability derivatives, or any other means by which configurations are

differentiated. A single parameter results in a segmented space that is easy to visualize

and is effectively populated with design points. For multiple, simultaneous parameters,

where PTon and Padj are matrices, the variation trends and resulting design spacing are

less obvious.

The design point threshold, Tdp, limits the allowable parametric variation. The

threshold is related to the degree of robustness of the control design. For instance,

Tdp =0.1 for a controller that is robust to 10'. uncertainty in the parameters. If the 6p

for points .,.1i 'ent to the nominal configuration exceed Tdp, then the point is selected as a

design point. An upper limit, Tdp,,max on the threshold can be used to prevents the points

from being excessively dissimilar.

The design point criteria is,


6p > Tp (5-27)

6p < Tdp,mTax (5-28)

If both conditions cannot be satisfied, then the grid is refined to reduce the paramet-

ric variation.

As the configuration search continues concentrically beyond the first 1ivr of shapes

around the nominal configuration, the parametric variation metric is changed slightly to,












Padj Pnea (5-29)


Where Pnea is a matrix of parameters for the nearest design point.

The Euclidean distance between the current point and each of the design points,

whether seeded or assigned, is computed throughout the search. The variation metric and

threshold then apply to the current and nearest design point, building the grid outward

from the center of the space.

As the search reaches the boundaries, the configuration is gridded such that all points

are parametrically dissimilar from the nearest design point by a magnitude less than or

equal to the chosen threshold. The method guarantees a maximum amount of uncertainty

in the interpolated points, which is used in conjunction with known robustness properties

of the controllers to infer stability results.

The configuration space is segmented by Delaunay triangles[60], whose vertices are

at the design points and whose interiors are spatially most similar to the corresponding

vertices. Figures 5-9 and 5-10 show the identified design points for single and multiple

parameter cases. Design points are connected to form the Delaunay triangles. The design

point grids are shown for several values of threshold parameter, Tdp, increasing from left to

right. A large threshold for the single parameter case eliminates the need for any design

points apart from the pre-seeded nominal and extremal points. The single parameter

case shows a relatively sparse point distribution over much of the space. Clustered points

indicates areas where the parameter is changing rapidly and exceeding the difference

threshold. The multiple parameter case shows a much denser grid. For many parameters

whose dependence on the space varies arbitrarily, the design point grid approaches the

density of the original configuration space. In such a case, the interpolation offers little

computational advantage over a large array of stored dynamic models.














130 20 -0 10 20 30 30 20 -0 0 20 30 30 20 -10 0 0 20 30


Figure 5-9. Design point grid distribution and triangular segmentation based on lift to
drag value. Threshold, Tap is 5', (left), 10', (center), and 25, (right)




.. .. .. 30o o. .. ...



PB PB PB

Figure 5-10. Design point grid distribution and triangular segmentation based on lateral
stability derivatives. Threshold, Tdp is 25'. (left), 5(0' (center), and 2 iig'.
(right)

The triangle distribution is used to generate an interpolation schedule between the

arbitrarily located design points. Vertices of each triangle are co-planar or co-hyperplanar

for higher dimensions. Values of points lying in the interior are easily determined by

finding the intersection point between a line normal to the configuration space plane and

the plane occupied by the three vertices.

Figures 5-11 and 5-12 show surface plots of the triangularly interpolated parameter

response compared to the original parameter response surface. The modeling errors, Sp,

are zero at the design point and bounded by Tdp,max elsewhere.

The faceting of the surface is a result of the triangular interpolation between the

design points. Improper point spacing generates facets that fail to properly represent the

parameter variations. Surfaces for the large single-parameter segmentation show deviations

from the original data in the region of negative PA and positive PB. Geometric artifacts,

such as false peaks and valleys, stem from the inability to recreate a complex surface
















20 20 20 20 20 20
PB A PB A PB A

Figure 5-11. Interpolated values for lift to drag based on single parameter segmentation.
Threshold, Tdp is 5'. (left), 10,'. (center), and 25'. (right)


from sparse point spacing. The errors are thus much larger for large values of threshold,

Tdp. The balance between acceptable errors and computational tractability is ultimately

determined with the design of this threshold.








2 0 0 20 -0 _220 0

PB A PB A PB A

Figure 5-12. Interpolated values for lift to drag based on multiple, simultaneous parameter
segmentation. Threshold, Tdp is 25''. (left), 5(0'. (center), and 21II '. (right)


Narrow threshold of the multiple parameter segmentation achieves accurate surface

reproduction. The number of required design points may be prohibitively large, especially

with limited error tolerance. The simultaneous variation of nearly 30 parameters, each

having a unique dependence on the morphing, creates a significant challenge in segmenting

the configuration space based on interpolated errors. Selective use of parameters may offer

improvements in storage and computation requirements by emphasizing the terms which

have the largest affect on the stability and performance.

5.3.4 Simulation Results

Simulations of the maneuvering controller are performed with various commanded

trajectories and reference models. Each system is morphed from the optimal cruise










configuration to the optimal maneuvering shape along a predetermined path through the

4-dimensional actuator space. Feedback control gains are initialized for each simulation

using LQR synthesis and subsequently adapted, along with the feedforward control gains.

Figure 5-13 shows the permissible trajectories through the 4-dimensional configuration

space for three morphing operations. The trajectories permit morphing between configu-

rations optimized for cruise, maneuverability, and agility. P/, P2, and p3 are represented

spatially with projections shown on the space boundaries for clarity. p4 is represented

in colorspace as hue and is indicated by a colored marker at each intermediate trajec-

tory. Dynamic criteria is applied in the computation of the actuator paths and yields

intermediate configurations with satisfactory stability characteristics.



30
30- 20
20 Agile 10
10 ,Maneuverable f
-10

-1010
20- -20
-30 Cruis-30
20....... 30

-30 1
1 -10 1 -10 0 10




Figure 5-13. Permissible 4D actuator trajectory between three disparate configurations


Figure5-14 shows a set of simulation results for a morphing system tracking the

response of a fixed reference system. Histories for states 3, p, and r are shown in the left

plots for the morphing aircraft (top) and the reference model (bottom). A low-frequency,

small-amplitude roll rate sinusoid is used as a reference input, which causes some coupling

to the remaining lateral states due to off-diagonal terms in the reference model control-

effectiveness matrix.











05 0 1


o 5- 1
0 10 20 30 40 50 60 0 10 20 30 40 50 60



0 0 5


0 10 20 30 40 50 60 0 10 20 30 40 50 60
Time (sec) Time (sec)

Figure 5-14. Control results during cruise to maneuvering reconfiguration with a sinusoidal
trajectory command. A) Angular rate response of morphing and reference
model systems. B) Gain adaptation and tracking errors


The morphing aircraft has an initial response that generates small error relative to the

reference response. Gains adapt slightly until the first morphing increment is commanded

at t = 13.2seconds. The morphing is commanded discontinuously, where the dynamics

are switched in a discrete step between time intervals. The relatively large 5 angular

resolution of the morphing joints contributes to a large change in the dynamics and

contributes to tracking errors as the dynamics switch.

Discrete morphing events are indicated by vertical dotted lines and occur every 4

seconds until the aircraft has morphed into the maneuverable configuration. Shape change

contributes to variations in the closed loop system and causes tracking errors. The errors

drive gain adaptation using the gradient descent method, which subsequently improves

tracking performance. Gains continue adapting until the aircraft reaches the final shape.

Figure 5-15 shows simulation results for a similar morphing trajectory and constant

reference system subject to a variable-frequency roll rate command. Gain adaptation

occurs rapidly following morphing operations, but applies selectively to certain gains.

The morphing model achieves good tracking performance throughout the shape-change,

although incurs some cyclical error in the final configuration. The error contributes

to oscillatory gain adaptation. Such undesirable behavior may be reduced with the

application of deadband modification to the adaptation I 1v- [' 11].












05-
_5 o0
05-

0


1 0
0 -
Figure 5-15.
0


Figure 5-15.


10 20 3


0 40 50 60


0

05__
0 10 20 30 40 50 60
0 10 20 30 40 50 60


0
LU


-05
V V-1
10 20 30 40 50 60 0 10 20 30 40 50 60
Time (sec) Time (sec)

Control results during cruise to maneuvering reconfiguration with a
variable-frequency trajectory command. A) Angular rate response of
morphing and reference model systems. B) Gain adaptation and tracking
errors


The adaptive morphing controller can tolerate large initial errors in the baseline

controller and large variations in the dynamics. Figure 5-16 shows time histories for state,

error, and gain trajectories for a model with a poor initial controller design. Initial re-

sponse incurs large errors are rapid adaptation in the controller gains. The first morphing

operation at t = 13.2s involves a large discontinuity with joint actuations exceeding 5.

The gains adapt further to stabilize the new dynamics and reduce the large transient

error. The controller achieves reasonable tracking performance for subsequent morphing

and fixed configurations.


Although the controller successfully stabilizes a model with severe initial errors, this

strategy is not recommended for use in a flight test vehicle. Poor baseline controllers

and rapid dynamic variations both cause tracking errors that require several seconds of

adaptation to subside. The criticality of this adaptation time depends greatly on the

vehicle and flight condition. Adaptation times greater than several seconds may lead to

loss of control or unintended interaction with surrounding obstacles.

The success of a morphing aircraft in tracking a fixed reference system demonstrates

the versatility of the control system. The constant reference, however, may violate

disparate mission constraints and reduce the effectiveness and function of morphing. A












. 05


EL 05
1 -25 3V V V

0 10 20 30 40 50 60


c~)
0)
0)
0)~


I-
05
0
-05
0


Figure 5-1


0 10 20 30 40 50 60


0




10 20 30 40 50 60 0 10 20 30 40 50 60
Time (sec) Time (sec)

6. Control results during cruise to maneuvering reconfiguration with a sinusoidal
trajectory command and poor initial control design. A) Angular rate
response of morphing and reference model systems. B) Gain adaptation and
tracking errors


more realistic approach uses a mission-dependent reference model in order to emphasize

either conservatism or performance.

Figure 5-17 shows the responses of an aircraft morphing from a cruise configuration

to a maneuvering configuration. The reference model initially uses a long time constant

in both the roll and dutch roll modes. At t = 30s, the reference model switches discretely

to shorter time constants in both modes and slightly higher damping in the dutch roll

mode. The change in the reference model occurs in near the midpoint of the morphing

operation between discrete shape changes. A vertical dashed line indicates the reference

model change, which is accompanied by tracking errors and rapid gain adaptation. The

controller summarily adapts to the higher-performance model and achieves good tracking

results. Controller gains continue to adapt with each subsequent morphing and achieve

relatively constant values for the maneuvering configuration.


The transient behavior observed during the reference model switching is partially a

result of discontinuities in the aileron and rudder deflections. A rate filter on the aircraft

controls and an improved switching technique would prevent large errors from inciting

rapid control and state responses. Despite the error, the system recovers quickly and

stabilizes the aircraft.












C 05 0

0 10 20 30 40 50 60 0 10 20 30 40 50 60



-05-

0 10 20 30 40 50 60 0 10 20 30 40 50 60
Time (sec) Time (sec)

Figure 5-17. Control results during cruise to maneuvering reconfiguration with a sinusoidal
trajectory command. A) Angular rate response of morphing and variable
reference model systems. B) Gain adaptation and tracking errors


The multiple-reference model, morphing system is subject to a chirp command,

which is a sinusoid whose frequency increases with time. The chirp simulates benign

motions of the cruise condition initially and develops into rapid motions for maneuvering

flight. Figure 5-18 shows simulated control results for the system. Roll rate command is

slowly increased, which allows the morphing aircraft to easily achieve tracking even in the

presence of controller errors. The gentle command violates the persistence of excitation

condtion[50][94] and fails to adapt the controller gains. Morphing changes generate

tracking errors due to the switch dynamics and invoke gain adaptation. The adaptation

rate increases somewhat with the increasing frequency of the commanded roll.

Tracking errors occur during the reference model switching, causing the gains to

adapt rapidly in response. Adaptation continues as the aircraft completes the morphing

operation and tracks the chirp command in the maneuvering configuration. Closed-loop

performance is good throughout the simulation, apart from the error excursion during the

reference model switch and cyclical errors in the high-frequency rate tracking.





































05-


-0 5
-1 -
0 10 20 30 40 50 60


05



-1
0 10 20 30 40 50 60
Time (sec)


0 10 20 30 40 50 60


0 05


0 5
o-1

0


10 20 30 40
Time (sec)


50 60


Figure 5-18. Control results during cruise to maneuvering reconfiguration with a
variable-frequency trajectory command. A) Angular rate response of
morphing and variable reference model systems. B) Gain adaptation and
tracking errors


CL







wrf


-^-n-P


/\A/A









CHAPTER 6
OPTIMAL CONTROL

The morphing aircraft adaptation and control problem can be solved partly or en-

tirely using optimal control techniques. Optimal control finds the desired state trajectories

and control inputs to minimize a performance index when subject to boundary and path

constraints. The framework is well suited for computing optimal solutions for systems

with variable dynamics, control objectives, and multiple phases. The formulation allows as

inputs the states, vehicle shape, and control surface deflections.

In computing the optimal inputs, the technique can be used to determine the state

trajectories required for a desired maneuver[41] [49], the optimal shape variation through-

out one or more missions, or the required inner-loop compensation for the changing vehicle

dynamics. Solving the optimal control problem becomes progressively more complicated as

the number of inputs and states increases.

This chapter considers several variations of the problem, where the optimal control

is used either in conjunction with the developments from previous chapters or in place

of them. For instance, the section on rate trajectory generation strictly commands the

inner-loop rates to perform part of a mission-task maneuver. The inner-loop stabilization

and tracking is achieved by an independent controller while the vehicle shape is adapted

by an independent adaptation law.

Adaptation can be included in the problem such that the trajectory and morphing

shape are solved simultaneously. In such a case, the trajectories are modified according to

the performance capabilities of the permissible shape. Maneuvering controllers are used

independently to stabilize the shape-changing vehicle and track trajectory commands.

Complete consideration of the problem commands states, morphing, and control

surfaces simultaneously. The unified framework for path planning, shape adaptation, and

inner-loop control can achieve the optimal solution for a particular set of cost functions,

although at considerable computational expense. The performance of the unified controller









can be compared to that of individual control components to examine possible benefits

and complications.

Optimality is not guaranteed, since a direct numerical method is used with no costate

information. Convergence is also not guaranteed with the unified scheme. The optimal

control methodology may be most useful in a mission-planning context, while separate

systems may be implemented using the optimal solution as guidance. Isolating the system

control and adaptation with multiple time-scales may provide the most dependable

performance.

Solutions to optimal control problems are evaluated relative to a performance index,

which is a cost functional relating state and input trajectories to the performance metric.

The performance index is minimized for the optimum set of trajectories. The performance

index is also subject to the boundary and path constraints of the problem, such that only

physically realizable trajectories are admissible as possible solutions.

Performance and maneuvering metrics by which morphing aircraft are adapted are

readily used in the optimal control performance indices. Additional spatial and temporal

metrics are established from mission environments. The combined cost functional relates

aircraft capability and mission requirements and is used to find solutions for objectives

such as,

Minimum power/energy consumption

Minimum time enroute

Maximum maneuverability

Minimum area maneuvering

Minimum path deviation

The general form for the performance index[73] is,


J = (x(to),to, X(tf),tf)+ L[x(t), u(t),t]dt (6-1)









where to is the initial time, tf is the final time, x is the state vector, u is the control

vector, +) is the constraints vector, and L is the Laplacian of the system parameters.

The system dynamics are first-order and quasi-static with parametric dependence on

morphing configuration.



x= f(x,u, fl,t) (6-2)

The rate of morphing is limited to isolate the adaptation time-scales from the aircraft

and controller dynamics. Rate limits on the morphing actuation are established using

a design threshold, Tp. The allowable value of this threshold is determined by iterative

simulation. Adaptability constraints are necessary to preserve the quasi-static assumptions

used in formulating the dynamics and control strategies.



7< T, (6-3)

The initial and terminal constraints must satisfy,



(?x(to), to, x(tf), tf) 0 (6-4)

The trajectories are subject to path constraints



g(x(t),u(t), f(t),t) < 0 (6-5)

6.1 Rate Trajectory Generation

A straightforward application of optimal control to morphing is generating the

inner-loop rate commands required to follow a desired flight path. Neither the vehicle

shape nor the actuated surfaces are directly controlled. Rather, the state trajectories are









idealized and force the aircraft to morph in order to sustain tracking. C'!hi ,iiiig aircraft

shape subsequently changes the dynamics, which requires adaptation in the controller to

maintain stability and control. Each of the processes invoke disparate control systems,

which operate independently. The optimality exists only in the state trajectory, which is

subject to mission-specific temporal and spatial constraints.


Pm (M ) xm


[ R Optimal Inner-Loop x



Generic
Performance m-- K a
Model /[V R]T

SConfiguration
Adaptation +


Figure 6-1. Optimal rate trajectory framework with independent flight control and shape
adaptation processes


A fictitious dynamic model is used in the formulation of the optimal trajectory.

The model achieves maximum performance in all performance and maneuvering metrics.

Although knowingly unrealistic for a single configuration, the model permits an .,.-. --ressive

trajectory that is achievable for an aircraft that changes shape appropriately. Tracking

errors resulting from configurations unable to follow the trajectory drive adaptation and

allow subsequent improvements in tracking performance.

The optimal control solves for trajectories of angle of attack, sideslip, and roll angle to

produce accelerations necessary to follow the desired flight path. Vertical and lateral forces

are used to determine the flight path based on a 3 degree-of-freedom model[78] given in

Equations 6-6, 6-7, and 6-8.











V =[X(cos a cos 3) + Y(sin/3) + Z(sin a cos3) g sin 7] (6-6)

7 =(1/V) [X(cos a sin f sin Q + sin a cos 0)

Y(cos fsin ) + Z(sin a sin 3 sin cos a cos ) gcos 7] (6-7)

'p =(1/Vcos7) + [-X(cos a sin 3cosQ sin a sin Q)

+ Y(cos 3 cos Q) Z(sin a sin f cos Q + cos a sin Q)] (6-8)

Accelerations X, Y, and Z are generated from the fictitious performance model

through variations of angle of attack and angle of sideslip. The stability coefficients

used to relate flight angles and forces are the maximum performance values among all

configurations.

The commanded trajectories for roll angle, angle of attack, and sideslip are used to

generate inner-loop commands to roll rate, pitch rate, and yaw rate. A rate filter acts as

on outer-loop and uses linear gains to command each of the rates from angular tracking

errors. Saturation is implemented in the filter to limit the magnitude of the command.

A baseline controller tracks inner-loop rate commands with reference to a mission-

specific dynamic model. The baseline controller is scheduled on morphing configuration,

which is externally adapted. The controller gains adapt to reduce errors between the

reference model response and the current model response.

Control deflection limits bounds the vehicle response rates and will introduce flight

path errors when the morphing configuration is unsuitable for the desired mission.

Tracking errors in airspeed, flight path angle and turn radius drive the external adaptation

mechanism, which seeks configurations that achieve higher mission performance.

6.2 Trajectory and Adaptation

An extended implementation of optimal control theory includes the nonlinear dynamic

variation due to morphing in the problem formulation. The controller solves for both the

trajectory and the wing shape. The distinct advantage over the earlier method is that the









effects of morphing are used explicitly in determining the desired trajectory. Thus, the

versatility of the shape change is exploited along the flight path to achieve a high level of

performance. Maneuvers, whether .,: '-ressive or benign, are commanded simultaneously

with the appropriate configuration.

The vehicle dynamics can include morphing throughout all or a subset of the config-

uration space. Dimensionality reduction techniques can be applied to reduce the problem

complexity. With appropriate design, the subspace permits a high level of performance

with considerably lower computational cost than the full space.

Morphing configuration is considered as an input and is optimized by the controller

with respect to the performance index and constraints. Including effects such as lift, di ,.

and energy consumption in the cost allows the controller to achieve find efficient shape and

flight path solutions. Adding elements such as temporal and spatial costs yields solutions

that are both efficient in time and energy. In seeking the lowest cost, the controller will

automatically seek maneuverable configurations for the transient flight phases and efficient

configurations for steady phases.

Although the optimal solution is determined using a direct, numerical method, an

indirect method is used to gain insight into the problem structure. Specifically, the first

order optimality conditions are formulated to show the dependence of the problem on the

morphing configuration. A 1-dimensional configuration space is used for simplicity, where

the vehicle dynamic parameters vary along separate 4th-order functions with respect to

the singular morphing actuation.

The functional-dependence of the parameters on morphing are used explicitly in the

optimility conditions. The desired state and input trajectories and associated performance

index relate heavily to the choice of morphing shape. Since the shape is controlled by the

optimal system, it can be reconfigured along the trajectory such that the solution achieves

a minimal cost.









Figure 6-2 shows the architecture of an optimal controller for rate trajectory and

shape adaptation. The external morphing command block is replaced with an expanded

optimal system that adapts the wing shape directly. The inner-loop rate filter operates

as before, operating on the flight angle output of the optimal controller and generating

limited rate commands to the aircraft. The rate commands are used in the adaptive,

model-reference design to generate the desired system response. The controller gains

subsequently adapt to reduce errors in the tracking performance of the reconfigured

aircraft.


--- p~-^-

[Vc, ", RcT Optimal Inner-Loop r
Trajectory Rate FilterKo) ( e T
and IIV-, -, R
Adaptation





Figure 6-2. Optimal rate trajectory and shape adaptation framework with independent
flight control


6.3 Trajectory, Adaptation, and Control

A complete implementation of optimal control for a morphing aircraft replaces both

the control and adaptation functions. The system yields the desired flight path, morphing

configuration, and control surface deflection throughout the mission. The addition of

stabilizing tasks requires the controller to operate at a rate equivalent to the vehicle dy-

namics. An increase in the controller rate significantly increases the computational burden

relative to the sporadic control allowed by the quasi-static morphing. The unified optimal

control architecture is presented for completeness, although with critical limitations in

practicality acknowledged by the author.

Flight dynamics linearized for each configuration are included in the problem formula-

tion. Variations in the lateral and longitudinal parameters are modeled over the allowable









configuration space, which may be the original actuator space or a subspace. Estimated

functional dependencies are given for low-dimensionality spaces while smoothed look-up

tables are given for higher-order space.

The state and input orders increase markedly due to the inclusion of angular rates,

linear velocities, and control surface deflections. The stability and control derivatives

each are affected by the morphing. Additional parameters are needed to establish the

dependencies of these derivatives on morphing.

The complete problem structure becomes somewhat cascaded due to the disparate

time scales affecting the states and inputs. Bang-bang behavior may be observed in the

lower-level controls, such as the elevator, when the system is subject to extreme actuator

commands. A rate filter may alleviate actuator excitability, although this may undermine

the optimality of the solution.









CHAPTER 7
MISSION RECONFIGURATION

7.1 Mission Tasks

An urban mission may consist of disparate maneuvering tasks with each having

unique dynamic and aerodynamic requirements. Each task requires the vehicle to optimize

one or more metrics in order to achieve a high level of performance. The maximization of

each metric occurs at a particular geometric configuration and flight condition.

The mission tasks selected for simulation reflect the shift in UAV use towards a

multi-role reconnaissance vehicle which is capable of efficient loitering, target pursuit,

and maneuvering in a complex urban environment. The most basic requirement is that

the vehicle must be easily operable by a remote pilot for deployment and recovery. Once

aloft, the vehicle must cruise efficiently to an area of interest or loiter efficiently awaiting

commands. An .,.-. --ressive maneuvering mode is also required to avoid obstacles or pursue

elusive targets. Other requirements include relatively inefficient modes that enable steep

descents or large sideslip trim angles for sensor pointing.

7.1.1 Deployment and Recovery

The beginning and end of every UAV mission requires vehicle deployment and

recovery, respectively. The ground station may be situated in an environment with a small

clearing surrounded by trees, buildings, poles, or other obstacles generally hazardous to

UAVs. The deployment of small UAVs is mostly performed without a runway by hand-

launching or catapult-launching. Either method initiates the flight in an often-precarious

situation of low airspeed and unusual attitude. The aircraft must be quickly stabilized

into a climb to clear surrounding obstacles and accelerate to normal flying speed. Even

operations in unobstructed areas require care in piloting during the first few moments

after launch.

The complexity of the launch environment often mandates that the vehicle is manu-

ally piloted by a remote operator for the deployment and initial maneuvers. Such remote

piloting requires that the aircraft respond predictably to control inputs and achieve a










high level of performance to enable maneuvers at low airspeeds. The handling qualities

requirement can be represented by a range of allowable time constants, damping ratios,

or natural frequencies for each of the dynamic modes. The performance requirement can

be represented by an aerodynamic metric such as lift-to-drag ratio. A suitable deployment

configuration is then one that achieves the highest lift to drag ratio while possessing

acceptable dynamic characteristics.

Figure 7-1 shows the simulated configuration for piloted takeoff and landing. The

shape has a relatively high lift-to-drag ratio of 14.6 in addition to good handling qualities

for both lateral and longitudinal modes. The wing has increasing aft sweep toward

the tip, with -5 on the inboard section and -100 on the outboard section. It uses an

inverted gull-wing shape with -5 anhedral on the inboard section and 100 dihedral on the

outboard section.






01,
005


/ 05
01 04
0 03
-01 02
-02 01
-03
0
HQC: p/i --5, t2 = -5, 3 = 10, 4 = -100

Figure 7-1. Manually piloted deployment and recovery configuration



7.1.2 Long Range Cruise

A non-critical phase of most missions is a cruise segment where the vehicle travels

from one area of interest to another. The metric by which cruise flight is assessed is typi-

cally energy efficiency per unit distance traveled, assuming that this task does not include

reconnaissance or time-critical objectives. The most efficient cruise for an aircraft is

achieved at the maximum lift to drag ratio, which is given by Equation 7-1 [7]. Equation









7-2 shows the Breguet Range formula, which gives an estimate for the range based on

aircraft aerodynamic, propulsion, and structural characteristics [40].


L P( V2 CDO 2K W (7
D \ 2(W/S) p. S)V

Where L is lifting force or coefficient, D is the drag force or coefficient, V is the

velocity, CD,O is the zero-lift drag coefficient, W is the weight, S is the wing area, pois the

dynamic pressure, and K is the drag-polar coefficient.



Range = V, s,p In inal (7 2)
D \ Wfinal /

Where Isp is the specific impulse, Winitial is the initial weight and, Wfinal is the final

weight. For battery-powered aircraft with no deplov-, h-I p .,load or ordinance, the initial

and final weights are expected to be identical, since no fuel is exhausted by the propulsion

system.

Figure 7-2(b) shows the configuration that achieves the maximum lift-to-drag ratio

using the stable dynamics criteria. The wing uses spanwise-increasing anhedral and

slight aft sweep to produce an open-downward crescent shape. The configuration closely

resembles that of birds such as albatrosses, pelicans, or seagulls gliding for extended

periods or close to the water surface. Figure 7-2(a) shows a seagull gliding for an extended

length in ground effect. It is also similar to the NASA Langley Hyper-Elliptical Cambered

Span wing, which was reported to achieve a 15'. greater lift-to-drag ratio than a similarly

scaled conventional v ii. [22]

The configuration which achieves the most distance-efficient cruise also serves an

important function for gliding operations. Both maximum range and shallowest angle

of glide are achieved by maximizing the lift to drag ratio. The metric by which glide

performance is gauged is typically glide ratio, which compares the horizontal distance

traveled to the altitude loss during a portion of the glide. The glide ratio is equal to















01


01
03 05
02

-01 02
-02 01
-03 0
(a) Seagull gliding in vicinity of ground (b) SDC: p = -150,2 = -50,3 = -250,4 = 0

Figure 7-2. High lift seagull wing shape (left), Maximum lift to drag configuration (right)


the lift to drag ratio during unaccelerated gliding flight, so an aerodynamically efficient

configuration allows the vehicle to glide for a long distance.

Gliding flight is not normally invoked during either manned or unmanned operations,

although it may become useful for exploiting atmospheric currents to improve the range

and endurance of a UAV[6]. Researchers are currently evaluating the performance of an

autonomous glider which seeks rising currents of air on which to stay aloft. Extended

flight durations may be possible by alternating between powered and gliding modes.

Glides are frequently used by larger birds to fly without flapping. Both thermal and

deflected winds are exploited by birds of prey and sea-borne birds as sources of energy for

extended periods of gliding flight [91]. The range configuration and others can be used for

different phases of gliding flight for a UAV.

7.1.3 Endurance Loiter

A variation on the range cruise is the endurance loiter, which maximizes energy

efficiency relative to time rather than distance traveled. The flight condition is appropriate

for persistence operations, where the vehicle must maximize time aloft or minimize energy

consumption during the mission segment.










Equation 7-3 [40] shows an estimate for power required in level flight as a function of

aerodynamic characteristics and airframe geometry. The configuration which achieves the

minimum power required is also the highest endurance and most energy-efficient.




Preq PV3SCDO + i (e (7 3)


The endurance flight task is generally employ, ,1 when the vehicle is not required to

traverse large distances or perform specific maneuvers. It may be used when loitering

over an area while awaiting commands or providing persistent surveillance of a region of

interest. It can also be used if the aircraft operates as a communications repeater, which

requires that the aircraft stay within a confined region for long periods of time.

Figure 7-3(a) shows the configuration for endurance cruise and minimum unpowered

rate of descent. The shape is very similar to the nominal configuration apart from 100

dihedral on the wingtips. The non-uniform dihedral is similar to competition sailplane

wings, which are also optimized for minimum power required in gliding flight with

handling qualities constraints [52]. The configuration requires 2.503 watts to maintain

flight.







01 01
0 0
0 3 W 05 00 3 5
02 0 02 1
01 04 01 04
0 03 0 03
-01 02 -01 02
-02 01 -02 01
-03 -03
0 0
(a) HQC: p 0, P2 0, [P3 10, 4 00 (b) SDC: = -50, P2 00, P3 00, [4 00

Figure 7-3. Maximum endurance and minimum rate of descent configuration









Using both the stable and unstable dynamics criteria, the endurance configuration

changes slightly to anhedral on the inboard wing and nominal joint angles elsewhere, as

shown in Figure 7-3(b). The relaxed criteria require 2.497 watts for level flight, marginally

less power than the shape generated based on handling qualities. The improvement is

not significant and comes at the expense of pilotability. For an automatically controlled

aircraft, however, this issue is not an important consideration.

The minimum-power-required configuration is also the best shape for maximizing

glide duration. An unpowered aircraft achieves equilibrium by using the loss of potential

energy to oppose the power required to maintain flight[9]. Minimizing the power required

thus minimizes the loss of potential energy and minimizes the rate of descent.

The lowest rate of descent configuration allows an unpowered aircraft to stay airborne

for the longest time. The condition is useful for exploiting atmospheric currents as energy

sources or increasing the decision time for contingencies in the event of powerplant failure.

The configuration are most frequently seen in birds using thermals to loiter over an area in

gliding flight.

7.1.4 Direction Reversal

Successful operation in a crowded environment depends on the ability of a flight

vehicle to change direction in a confined area. The minimum radius turn performance

is an appropriate metric for gauging the ability of a vehicle to avoid obstacles or reverse

course. Equation 7-4 shows the radius of turn is reduced for low airspeeds, large lifting

forces, and steep angles of bank. Equation 7-5 shows that wing l .,iii(- W/S, and drag

must be low to achieve a small radius turn. Thrust force, T, must be also be maximized

to compensate for the increased drag as the aircraft banks to perform a turn. Thrust

is constant for all simulations to focus on the role of morphing in effecting the flight

performance.











mV2 V2
R = (7-4)
L sin g tan 0
4K(W/S)
gp (T/W)/1 4KCDo(T/W)2

The configuration which achieves the minimum radius turn when subject to the han-

dling qualities criteria is identical to the HQC maximum endurance wing shape shown in

Figure 7-3(a). The weight and thrust forces are assumed constant for all configurations.

The large wing planform area achieves a low wing loading while the dihedral wingtips pre-

serve desirable lateral dynamic response. The stable dynamics configuration is simply the

nominal wing shape with all joint angles zero. Maximum wing area is achieved without

morphing, producing the largest lift magnitude. Minimum turn radii are assessed at the

simulated airspeed and thus the configurations are considered based on the simplified drag

estimates and the available wing area.

7.1.5 Steep Descent

Flight maneuvers performed in the vicinity of obstacles can require aerodynamic char-

acteristics very different from flight in open environments. Steep descent maneuvers favor

relatively inefficient configurations to stabilize the aircraft in a steep descent angle or high

rate of descent. The lowest-lift-to-drag and the maximum-power-required configurations

achieve the steepest descent and fastest rate of descent maneuvers, respectively.

The steepest angle of descent flight mode is required when large changes in elevation

are commanded in areas with limited horizontal space. A descent from building-top level

to street level presents such a scenario, where the vehicle must lose considerable altitude

within the hard limits of the horizontal distance between buildings. A steep spiraling flight

path can be used to descend quickly without requiring a large flying area.

Figure 7-4(a) shows one possible configuration for a steep descent mode. The ve-

hicle uses a gull-wing shape with large dihedral on the inboard and large anhedral on










the outboard. Wing sweep decreases outward, with the maximum aft sweep on the in-

board section and moderate aft sweep on the outboard section. Such a configuration is

similar to observed seagull wing shapes used to regulate glide ratio, where increasing

dihedral/anhedral angles decreases glide ratio[68]. Handling qualities criteria is used in the

selection of the wing shape.








005
0005

S05 005
02 04 02 04
01 03 0 03
0 0




-300 -200


Figure 7-4. Steep descent angle configuration


The aerodynamically inefficient steep descent mode is favored over a simple dive

due to airspeed considerations. A configuration optimized for cruise will significantly

gain airspeed during a dive, as opposed to the slow descent of high-drag geometry. The

slow descent of the current configuration also facilitates recovery to level flight at the

termination of the dive.

An alternate form of diving is the maximum rate of descent mode, which achieves

a change of elevation in minimum time. The maneuver is appropriate for time-critical

descents where horizontal flight space is afforded. The best configuration for rate of

descent occurs at the maximum power required condition, which is opposite to the

requirement for soaring flight. The large power requirement for trimmed flight is provided

by the loss of potential energy during the descent. When the potential energy loss is

maximized for a fixed airspeed, the rate of descent is also maximized.









The minimum lift-to-drag wing shape is also the maximum power-required configu-

ration. Figure 7-4(a) thus shows the geometry that is optimal for both descent angle and

descent rate. This result is expected given the constraint on fixed airspeed.

The configuration that achieves the steepest descent angle and maximum rate of

descent using the stable dynamics criteria is shown in Figure 7-4(b). The wing uses the

maximum dihedral angle for both inboard and outboard wings along with maximum aft

sweep for the inboard and moderate aft sweep for the outboard wing. The wing shape

is similar to the form used by homing pigeons, as in Figure 3-9 during the steep descent

phase preceding landing. The configuration generated using the unstable dynamics criteria

is similar to the stable shape, except that the outboard wing uses the maximum aft sweep.

7.1.6 Sensor Pointing

A UAV engaged in reconnaissance of a moving object or general area may find

difficulty in maintaining the target in the sensor field of view. Vision sensors are typically

fixed to the aircraft body, which must fly through the air in a particular attitude to

maintain appropriate angle of attack and sideslip. A surveillance mission targeting the face

of a building would require that the aircraft fly parallel to the building side where only one

part of the sensor field of view is providing useful information. Flying the aircraft toward

the building can offer a better perspective, but only allows surveillance for brief periods

between circling maneuvers to fly away from and re-acquire the target area in the image.

An alternative approach to the mission is to provide sensor pointing capability by

partially decoupling between attitude and velocity. The vehicle would then operate at

large sideslip angles in order to fly parallel to the building side while directing the sensor

footprint towards the area of interest. The technique would also allow the aircraft to track

a moving road vehicle while flying to the side of the roadway.

Trimmed flight at large sideslip requires relatively weak stability derivatives and

strong control derivatives. Stiffness and coupled derivatives such as C", and QC,, re-

spectively, should be low such that the vehicle is not subject to large yawing and rolling









moments. Equation 7-6 shows a simple cost function to find the configuration which

minimizes the combined, squared stiffness and coupled derivatives.




min J= C +c2 (7-6)
pi,P2,P3,P4

The minimization is also subject to additional constraints with respect to dynamic

characteristics and handling qualities. For example, although the directional stiffness

will be reduced, Cr, can remain stable, preventing divergence from a stabilized sideslip

condition.

Figure 7-5(a) shows the configuration which achieves the minimum cost, J, with

good handling qualities. The shape is non-conventional by both biological and aviation

standards in that the inboard wings are swept aft while the outboard wings are swept

forward. The wing also uses a gull-wing configuration with large dihedral on the inboard

and moderate anhedral on the outboard. The unusual orientation of the outboard wings

are expected to contribute to the large sideslip constraint. Anhedral wing tips reduce the

vehicle tendency to produce a roll moment in response to sideslip[64] while the forward

sweep reduces the directional stiffness, C,, [80] [9]. The opposite attitude of the inboard

wings are used to maintain an appropriate position of the aerodynamic center relative to

the vehicle body for the desired dynamic response.

Relaxing the stability criteria to allow unstable dynamics produces a qualitatively

similar shape except that the wingtips are not angled downward and are swept forward.

Figure 7-5(b) shows the resulting wing configuration. The cost for the unstable criteria

is J = 0.0005 whereas for the handling qualities criteria J = 0.0263. In both cases, the

desired sideslip characteristics are mostly achieved. The unstable configuration produces a

moderately divergent spiral mode and a highly divergent short period mode with a time to

double of T2 = 0.5 seconds. The configuration determined by the stable dynamics criteria



















01
0 05 005
00 0

03 05 03 0 5
02 04 02 04
001 0 03 01 0

-0 1 0 2 -0 1 >.
-02 01 _02 1
-03 0 -03

(a) HQC: i 25, P2 -10, P3 -10, P4 (b) UDC: i 30, 2 25, 3 = 0, p4 30
300


Figure 7-5. Maximum trimmable sideslip



is similar to the unstable shape with joint angles of pi = 30, 2 = 15, /3 = -5 and,


/4 = 30.


Interestingly, a very conventional wing configuration is found when the cost function


in Equation 7-6 is maximized. Figure 7-6 shows the wing shape that achieves the highest


value of C2 + C2 is similar to the swept-back design of airliners and other aircraft that


seek to reduce sideslip divergence, among many other factors.









01
0 050


03 / 05
02
01
003
0 1 02
-02 01
-03 0

HQC: t = -5, 2 = -250, P3 5, P4 -300


Figure 7-6. Maximized sideslip cost function configuration









Maximum trimmed sideslip angle simulations are performed for the identified air-

craft by constraining the control surfaces to each move a maximum of 15. Table 7-1

compares the maximum sideslip achieved by four aircraft. The movement of the aileron,

elevator, and rudder control surfaces are constrained to trim to zero the rolling, pitching,

and yawing moments, respectively. The sideslip is increased until the limit deflection is

achieved in one of the surfaces.

Parameter A B C D
Sideslip (/3 a) 22.6 36.1 45.0 11.5
Aileron (6a) 15.0 15.0 5.050 15.0
Elevator (6e) 6.30 7.80 -7.980 18.40
Rudder (6,) -11.10 -8.360 -7.570 -3.50
Table 7-1. Maximum sideslip and control deflections for several aircraft configurations. A)
HQC Aircraft Fig.7-1 B) HQC ming J Fig.7-5(a) C) SDC ming J -
Fig.7-5(b) D)HQC maxa J Fig.7-6


The nominal configuration used for takeoff is listed in the first aircraft column. The

outboard wings have shallow dihedral and the aircraft achieves a moderate value for the

cost function, J. A maximum sideslip angle of 22.60 is achieved with maximum right

aileron and 7!' left rudder deflection. Similar aileron and rudder deflections stabilize

the HQC-sideslip airplane in a 36.1 sideslip. Both aircraft exhibit desirable dynamic

characteristics, yet the optimized aircraft achieves significantly larger sideslip angles,

affording a wider range of sensor pointing. The SDC-sideslip aircraft achieves the sideslip

angle limit of 450 with conservative control surface deflections. The aircraft is able to trim

at yet larger angles, although sustained flight in such attitudes may be unrealistic without

unconventional propulsion or descending flight to counteract the large sideforce drag.

The jet-like wing planform in contrast achieves a maximum sideslip of 11.50 with the

aileron saturated to the right and the rudder deflected left mildly. The elevator exceeds

allowable deflection for all sideslip ranges in order to stabilize the wing in pitch due to the

aft neutral point position. The large elevator deflection underscores an important aspect









of morphing aircraft requiring large wing and tail control effectors in order to stabilize and

control the large forces and moments generated by the wing.

7.2 Mission Profile

A simulated mission is presented to illustrate the function of a morphing vehicle in a

multi-role scenario. The mission consists of each maneuvering task performed for an equal

length of time. The success of each task is determined by the aerodynamic or dynamic

metric governing the maneuver. Long range cruise flight performance, for instance, is

gauged by the lift to drag ratio of the chosen morphing configuration. The performance

of each maneuver is normalized by that of the launch configuration, which is taken as the

nominal or baseline aircraft. Overall mission performance is determined by averaging the

improvement percentage from each task, where the mission segment durations are assumed

equal.

7.2.1 Performance Improvement

Table 7-2 shows two example missions for aircraft flown with with different stability

criteria regulating the morphing configurations. Each aircraft is optimized within the

configuration space to achieve best value for the each maneuver metric. The handling-

qualities criteria restricts the solutions to those that achieve good handling qualities,

while the stable dynamics criteria is less restrictive and allows any open-loop stable

configuration.

Mission Task Metric Nominal Value HQC ( Ci' ,,.) SDC ( Change)
Launch () 14.6 14.6 (0',) 14.6 (0',)
Cruise (L) 14.6 14.6 (0', ) 15.1 (3.21 .)
Loiter Preq,min 2.52W 2.50W (0.>'-.) 2.50W (1.0',)
Turn Rmin 17.8m 17.1m (4.0' ) 16.9m (5.2'",)
Descent (1) 14.6 11.7 (19.'.',) 11.3 (22 -' )
Sideslip max,trim 22.60 36.1 (59.7'.) 450 (99.1 ,)
Recovery (), 14.6 14.6 ((0'.) 14.6 (0' .)
Improvement 12.0' 18 t .
Table 7-2. Mission tasks and associated performance metrics relative to launch
configuration









Most of the performance improvement for both stability criteria comes in the later

mission tasks where the aircraft must achieve unconventional aerodynamic and dynamic

properties. The baseline aircraft is optimized to achieve a high lift-to-drag ratio with good

handling qualities, so it is expected to perform well in cruise and maneuvering tasks. The

performance improvement achieved through morphing is thus relatively small for such

tasks. Reconfiguration for the descent maneuver achieves large gains for both HQC and

SDC missions. The lower lift to drag ratio of both biologically inspired configurations

allows steeper and faster dives. Actual gains may be larger if angle of attack is allowed to

vary, although the current results are presented at a constant angle. Maximum trimmable

sideslip angle improves with morphing due to the unconventional wing configuration

with low sideslip coupling to roll and yaw moments. The HQC sideslip vehicle achieves

a 59.7'. improvement in the maximum sideslip with the same control deflection limits as

the nominal aircraft. The SDC vehicle achieves a greater improvement and reaches the

3 = 45 simulation limit without saturating control surface deflection.

7.2.2 Morphing Joint Trajectories

A morphing aircraft must obviously change shape in order to be useful in flight. The

previous discussion considers only the optimal morphed configurations, whereas a realistic

mission would require intermediate wing shapes as the vehicle morphs from one mission

task to another. The transient morphing shapes are subject to the same constraints as

the initial and final configurations. Namely, the dynamic characteristics must satisfy the

stability criteria by which one or both of the shapes are limited. For instance, an aircraft

morphing from a launch configuration to a manually piloted steep-descent mode must

preserve good handling qualities at each intermediate shape. Thus, the joint trajectory

through the configuration space is limited to those paths which achieve specified damping

ratios, natural frequencies, and time constants.









Equation 7-7 shows the desired morphing direction in configuration space is simply a

direct line from the current joint angles to the desired joint angles. The desired direction is

normalized by the distance between the initial and final shapes.


M-end Ucurrent (7 7)
11 d-dend U- current I I
Where current < Pl,current, P2,current P3,current 4,current > is the current joint

configuration of the wing and /lend < plM,end, M2,end, /3,end, L4,end > is the final or desired

joint vector.

Each morphing joint is allowed to move by one increment at each time step indepen-

dently of the remaining three joints. The angle increment is taken as the 50 minimum joint

resolution used in the simulation and the resulting 4-dimensional data matrix. Increment

vectors i, j, k, and I represent incremental changes to the joint angles Pi, p2 M3, and p4,

respectively. Each vector may take the value of +5 for increasing the joint angle, -5 for

decreasing the joint angle, or 0 for no change. Thus, 34 = 81 possible morphing opera-

tions exist at each time step. The number of possible morphing shapes is decreased when

the aircraft is at the boundary of the configuration space where one or more joint angles is

saturated at the extreme position.

The reconfiguration options are sorted according to Equation 7-8, which determines

the desirability of the reconfiguration command, < i,j, k, I >, based on the similarity to

the desired morphing direction, desired.


S
Error M -desired K (7 8)

The reconfiguration command which achieves the lowest magnitude of direction error,

Error, is checked against the applicable stability criteria to determine the suitability of

the dynamics. If the configuration meets the criteria, the joint angles are morphed and

Current is updated until lend is reached. Should the configuration violate the dynamic










requirements, less direct < i, j, k, I > are tested until an alternate joint angle path is found.

The aircraft is thus able to morph around unstable or undesirable configurations.

Figure 7-7 shows the four joint angle trajectories as the aircraft morphs between

mission task shapes. Each mission task shape is identified by a vertical dashed line and a

text annotation. The duration of the mission task is not represented in the plot. Instead,

closed circles show the optimized morphing shapes which are constant for each maneuver.

The initial configuration shown at time increment 1 corresponds to the manually piloted

launch shape from Figure 7-1. The aircraft is then morphed to the anhedral cruise

configuration from Figure 7-2(b) by changing all joint angles apart from P2, which remains

constant at -5.


SI i 2 3 4
I I I i I I I
30

20 'I
I I II -0 -
1 0 I+ I I, I



-10I I
I W : I ) '-I t I :


I I I I I I

00
I II I .
-30 -

0 5 10 15 20 25 30 35 40
Time increments

Figure 7-7. Morphing joint angle trajectories for mission tasks subject to SDC



The following two missions, endurance loiter and minimum radius turn, use joint

configurations close or equal to the unmorphed shape from Figure 7-3. The next mission

uses the large dihedral, aft swept wing shape (Figure 7-4(b)) to perform steep descents

and scatters the joint angles to the extremities of the plot. The vehicle then transforms to

the sensor-pointing configuration from Figure 7-5 by increasing P2 and p4 while reducing

/13-










Figure 7-8 shows a similar profile for a mission using HQC to limit both mission task

and intermediate configurations. The restricted configuration set causes several maneuvers

to use identical wing shapes. Launch and cruise are both optimized for lift-to-drag ratio

so both use the configuration from Figure 7-1. The aircraft then morphs into the shape

used for both endurance loitering and minimum radius turn. Slight dihedral is used on the

outboard section while the remaining angles are zero, as in Figure 7-3(a).


P i i 2 P3 1l 4

30
I I r I I

-20 o




< I I I I -
.I 10.' I* '

I I I I I
-20 ..I 4 ,

-30- I I I

0 5 10 15 20 25 30
Time increments

Figure 7-8. Morphing joint angle trajectories for mission tasks subject to HQC


Steep descent is performed by morphing into the seagull-inspired gull-wing shape

of 7-4(a). The joint trajectories follow nearly direct paths for all four angles. Morphing

into the sensor-pointing shape of Figure 7-5(a) uses a direct path for p4 and -I ,.--, red

paths for the remaining joints. The final shape change transforms the vehicle back into the

launch configuration for a manually piloted recovery.

7.3 Summary

Morphing continues to be an attractive option for small aircraft designed to operate

in urban environments. Shape change allows the vehicles to reconfigure for missions with

disparate aerodynamic requirements, such as cruise, maneuvering, steep descent, and

sensor-pointing. Biological design elements such as a shoulder and elbow joint are used









to vary the wing geometry in sweep and dihedral. Birds using such morphing are able to

rapidly change direction, achieve high lift-to-drag ratios, descend at steep angles, and soar

in updrafts during different phases of flight.

Simulations of a small UAV with avian-inspired morphology show that performance

improvements are achieved for each task of an urban mission scenario. Optimizing the

vehicle shape for each of these tasks often converges to the biological solution, matching

observations made of gliding birds. The identified vehicle configurations also closely relate

to findings in the literature regarding useful shapes for different flight tasks.

The shapes into which a vehicle may morph can be regulated by use of a stability

criteria which sets an allowable range for the dynamic characteristics. Pilotability can

be guaranteed for any configuration by requiring the morphed shapes to meet classical

handling qualities. The vehicle then achieves versatile performance while allowing a human

operator to manually control the flight path. Further improvements in performance are

achieved by permitting any stable or even unstable configurations. The use of a controller

is then required, but the vehicles may achieve task performance significantly higher than a

fixed-geometry vehicle expected to perform all missions.









CHAPTER 8
CONCLUSIONS

Natural inspiration may transform aircraft design from fixed configurations to amor-

phous shapes that can reconfigure optimally for many disparate flight conditions. In-flight

shape-change extends the design process to real-time, where a vehicle can undergo sub-

stantial variations in shape and change roles. Evidence from the gliding flight of seagulls,

falcons, albatrosses, and other bird species -ir--:. -I that such wing shape change can be

used to control flight modes and maneuverability with improved performance over fixed

aircraft. The potential areas for improvement in energy conservation, maneuverability, and

versatility are significant and have justifiably received substantial research attention.

Avian skeletal structures are similar to those of humans in the joint and bone configu-

ration. The shoulder, elbow, wrist, and digit joints of both allow a wide range of motions.

Unlike a human arm, however, the avian wing is used as an aerodynamic surface rather

than a serial manipulator. As such, the versatility of the skeleton and .....' i .,ving skin

and feathers affords a variety of wing shapes for various flight tasks. Articulation of spe-

cific joints, such as sweep in the wrist joint, provide beneficial aerodynamic effects that are

reported in the literature. Combined, complex articulation of the wing joints permit birds

to achieve configurations that have favorable performance and dynamics. Quasi-static con-

figurations are easily observed in species which spend large duration gliding in atmospheric

currents.

The application of biological inspiration to aircraft design is arguably most relevant

to vehicles that are dimensionally similar to the birds on which they are based. Micro

air vehicles designed to reconnoiter in urban environments are ideal candidates for such

design inspiration. Complicated maneuvers and varied missions envisioned for such aircraft

will require a combination of performance and versatility not available with conventional,

fixed-configuration designs. Morphing the shape of MAV wings will enable the same type

of in-flight redesign versatility that are observed in birds. Multiple mission objectives,









such as endurance, pursuit, tracking, and maneuvering, can be achieved by morphing the

aircraft into a shape appropriate for each task.

Quasi-static articulations of wing joints at the root and midboard positions of the

wing permit a wide range of aerodynamic shapes. Aerodynamic modeling using vortex-

lattice method predicts performance and dynamic characteristics for the morphed wings.

Cost functions associated with each mission task are used to find optimal wing shapes in

the configuration space. For several tasks, the optimal wing shape is similar to the natural

solution. The performance of the adapted aircraft exceeds that of an aircraft with fixed

configuration.

Morphing wing joints allows significant changes to the wing shape which subsequently

cause large variations in the dynamic characteristics. Modeling these changes for any

permissible wing configuration may be computationally prohibit. Reducing the allowable

wing shapes to a subset of the configuration space retains the beneficial effects of the

morphing while reducing the modeling burden.

Controlling the morphing aircraft requires some knowledge of the dynamic variation

that occurs with shape change. With full knowledge of the dynamics, optimal LQR or

robust Ho controllers can be designed for each configuration and switched or scheduled.

Alternatively, the dynamics may be approximately modeled using linear interpolation

between a set of design points. The linear interpolation reduces computational and storage

burden considerably, although introduces modeling errors for nonlinear variation in the

dynamic parameters. Adaptive control is used to allow the control to compensate for

the modeling error. A reference model associated with each mission defines the expected

response for the model and drives the controller adaptation.









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BIOGRAPHICAL SKETCH

Mli ii,1 Abdulrahim was born and raised among the rocky mountains of Calgary,

Alberta, Canada. He proudly descends from a line of educators, nobles, and small-time

inventors in his ethnic Syria, from the cities of Aleppo (Halab) and Al-Rahaab. His

professional maturation began in the 2nd grade with his notable development of a Micro

Machines stunt track from construction paper. He continued his interest in all things

mechanical during his cross-continent move to Panama City, FL, where he firmly decided

his career path as an aeronautical engineer by referencing an 8th-grade math-class poster.

The poster listed various professions on the vertical axis with the corresponding math

course requirement on the horizontal axis. Aeronautical engineering was one of the few

careers that required virtually all the listed mathematics.

Since joining the University of Florida in 1999, A1lui i,1. has been active in micro

air vehicle, dynamics and control, and morphing research. He has also participated in

numerous academic and sporting competitions including regional and national paper

competitions, MAVs, cross-country li. -, ling, photography, autocross, drifting, and

aerobatics. His success varied. 1\ti li6id earned his aerospace engineering B.S. in 2003,

his M.S. in 2004, and his Ph.D. in 2007. Noting, with some sadness, that the University

of Florida offered no further relevant degrees, AM1\i ii.1 promptly packed his belongings

and headed west. At the time of this writing, he is driving in an RV, lost on a winding

mountain road somewhere in Arizona. Somed I', he hopes to get new batteries for his GPS

and continue his trip to California, where he will start R&D work in UAV controls.





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TothosewithwhomIamhopelesslyinfatuated.Chapter1,Introduction,isdedicatedtomyparents,ArifaGarmanandAbdulhamidAbdulrahim,whorsttointroducemetotheworldofcreativitybyencouragingmetodesign,build,andexploreallthingsmechanical.Chapter2,LiteratureReview,isdedicatedtomymother,ArifaGarman,forspendinguntoldhoursteachingmehowtoreadandappreciateagoodbook.Chapter3,BiologicalInspiration,isdedicatedtomywife,TasneemKoleilat,whoisasenamoredwithbiologyasIamwithairplanes.Chapter4,Dynamics,isdedicatedtomybrother,ObaidaAbdul-rahim,whowasmyrstco-pilotandwillalwaysbetheoneIchoosewheneverIneedalongdrive.Chapter5,ManeuveringControl,isdedicatedtomysister,RajaAbdulrahim,withwhomIhavemanyadventuresinbothvehicularandconversationalmaneuvering.Chapter6,OptimalControl,isalsodedicatedtomymother,whogentlyremindedmeoftheverse\Godlovesaservant,whowhenheperformsanaction,perfectsit". 3

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Thankyoutomyadvisor,mycommittee,mycolleagues,andmyfamily.AllhavebeenquitesupportiveinwaysthatmakemefeelasthoughIupholdthenevaluesofthescienticmethod,evenduringtimeswhentheMATLABrandfunctionproducescleanerdataandWindowsPaintproducescrisperimages.ThankyoutoDr.DavidBloomquist,forbeingmyundergraduatehonorsadvisor,fortakingmyyingallovertheUSintheCessna\N337P"Skymaster,andformentoringmeonanuntoldnumberofresearchprojects.HopefullyhewillonedayforgivemeforcrashingtheTelemaster.ThankyoutoDr.PeterIfju,whoinvitedmetoworkinhisresearchlabevenbeforeIstartedfreshmanclasses.WorkingwithDr.IfjuontheMAVcompetitionteamandresearchprojectswastrulyadelightfulexperience.Maybeonedayourpatentwillbeapproved.Thankyoutomyadvisor,Dr.RickLind,forshowingmethewonderful,yetoftenviolently-turbulentworldofightdynamics.Underhisguidanceanddirection,Ihavepartlysatiatedmyongoingpassionforconductingmeaningful,signicant,anddelightfulresearch.PerhapsonedayIwillberespectedasascientist.AnalthankyoutoAdamWatkins,Daniel"Tex"Grant,JoeKehoe,andRyanCauseyforbeingverybemusingcolleagues.ItisdoubtfulthatanyofuswouldhavesurvivedthePhDprogramwithoutourcommunalgum-olympics,frosty-times,andhelicopter-breaks. 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 8 LISTOFFIGURES .................................... 9 ABSTRACT ........................................ 11 CHAPTER 1INTRODUCTION .................................. 12 1.1Motivation .................................... 12 1.2ProblemStatement ............................... 17 1.3DissertationOutline .............................. 18 1.4Contributions .................................. 19 2LITERATUREREVIEW .............................. 20 3BIOLOGICALINSPIRATION ........................... 32 3.1Motivation .................................... 32 3.2ObservationsofBirdFlight .......................... 33 3.3DesiredManeuvers ............................... 37 3.3.1EcientCruise ............................. 38 3.3.2MinimumSinkSoaring ......................... 38 3.3.3DirectionReversal ............................ 39 3.3.4MinimumRadiusTurn ......................... 39 3.3.5SteepestDescent ............................ 39 3.3.6MaximumSpeedDash ......................... 40 3.4MorphingDegreesofFreedom ......................... 40 3.5MorphingMotions ............................... 41 3.5.1Fore-AftSweepofInboardWings ................... 41 3.5.2Fore-AftSweepofOutboardWings .................. 42 3.5.3Up-DownInclinationofInboardWings ................ 43 3.5.4Up-DownInclinationofOutboardWings ............... 43 3.5.5TwistofOutboardWings ........................ 43 3.6WingMorphingModel ............................. 44 3.6.1PigeonWingConguration ....................... 45 3.6.2AvianMorphologyStudies ....................... 46 3.7AircraftMorphology .............................. 47 4DYNAMICS ...................................... 50 4.1AircraftEquationsofMotion ......................... 50 4.2ParametricVariationsofEOM ......................... 53 5

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....................... 56 4.3.1Cruise .................................. 58 4.3.2ManeuveringMetrics .......................... 59 4.3.2.1Agility ............................. 60 4.3.2.2Maneuverability ........................ 61 4.3.2.3Aggregatemaneuverabilityandagilitymetrics ....... 65 4.4CouplingMorphingParameters ........................ 69 4.5StabilityCriteria ................................ 74 4.5.1HandlingQualitiesCriteria(HQC) .................. 74 4.5.2StableDynamicsCriteria(SDC) .................... 75 4.5.3UnrestrictedDynamicsCriteria(UDC) ................ 76 5MANEUVERINGCONTROL ............................ 77 5.1Overview .................................... 77 5.2SingleDegree-of-FreedomMorphingSystems ................. 80 5.2.1RobustControlDesign ......................... 81 5.2.2SimulationResults ........................... 84 5.2.2.1Overview ........................... 84 5.2.2.2Cruiseight .......................... 85 5.2.2.3Maneuveringight ...................... 87 5.2.2.4Steepdescentight ...................... 88 5.2.2.5Sensor-pointingight ..................... 89 5.3MultipleDegree-of-FreedomSystem ...................... 90 5.3.1Optimal-BaselineAdaptiveControl .................. 91 5.3.2ReferenceModelDesign ......................... 96 5.3.3DesignPointGridding ......................... 99 5.3.4SimulationResults ........................... 104 6OPTIMALCONTROL ................................ 111 6.1RateTrajectoryGeneration .......................... 113 6.2TrajectoryandAdaptation ........................... 115 6.3Trajectory,Adaptation,andControl ..................... 117 7MISSIONRECONFIGURATION .......................... 119 7.1MissionTasks .................................. 119 7.1.1DeploymentandRecovery ....................... 119 7.1.2LongRangeCruise ........................... 120 7.1.3EnduranceLoiter ............................ 122 7.1.4DirectionReversal ............................ 124 7.1.5SteepDescent .............................. 125 7.1.6SensorPointing ............................. 127 7.2MissionProle ................................. 131 7.2.1PerformanceImprovement ....................... 131 7.2.2MorphingJointTrajectories ...................... 132 6

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.................................... 135 8CONCLUSIONS ................................... 137 REFERENCES ....................................... 139 BIOGRAPHICALSKETCH ................................ 147 7

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Table page 4-1Standardaircraftstatesdescribingvehiclemotion ................. 51 4-2Handlingqualitiescriteriaformorphingconguration ............... 75 4-3Stabledynamicscriteriaformorphingcongurations ............... 76 7-1Maximumsideslipandcontroldeectionsforseveralaircraftcongurations.A)HQCAircraft-Fig. 7-1 B)HQCmin~J-Fig. 7-5(a) C)SDCmin~J-Fig. 7-5(b) D)HQCmax~J-Fig. 7-6 ............................ 130 7-2Missiontasksandassociatedperformancemetricsrelativetolaunchconguration 131 8

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Figure page 1-1UAVsdesignedfordisparatetasks .......................... 13 1-2Gullwingvariations ................................. 16 3-1Eectofplanformshapeonglidingspeed ...................... 34 3-2Eectofwingsweeponyawstiness ........................ 34 3-3Glidingbirdsarticulatingwingjointsaboutlongitudinalaxes ........... 35 3-4Complexwingmorphingshapesachievedinglidingight ............. 36 3-5Asymmetricmorphinginseagullwings ....................... 37 3-6Rapidvariationsinightpathusingaggressivemaneuvering ........... 38 3-7Wingarticulationaboutlongitudinalaxes ..................... 44 3-8Physiologicalrepresentationofavianwing ..................... 45 3-9Compositephotoofpigeoninglidingapproachtolanding ............ 46 3-10Morphingjointarticulationsfor4-degree-of-freedomwing ............. 48 4-1Dragpolarandlifttodragratio ........................... 58 4-2Maximummaneuverabilityconguration ...................... 66 4-3Maximumagilityconguration ........................... 67 4-44Dmaneuverabilityindex .............................. 68 4-5Maneuverabilitymetrichistogram .......................... 69 4-64Dagilityindex .................................... 70 4-7Agilitymetrichistogram ............................... 70 4-8Histogramcomparisonforreduced-dimensionspace ................ 72 4-9ResponsesurfaceofL/Dmetricinreduced-dimensionspace ........... 72 5-1LateralH1controllersynthesismodel ....................... 82 5-2LongitudinalH1controllersynthesismodel .................... 82 5-3Simulatedresponsesincruiseight ......................... 86 5-4Simulatedresponsesinmaneuveringight ..................... 88 5-5Simulatedresponsesinsteep-descentight ..................... 89 9

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.................... 90 5-7Modelreferenceadaptivecontrol(MRAC)architectureforlateraldynamics ... 92 5-8Linearquadraticregulator(LQR)controllerusedforinitializationofgainma-trix,^Kx. ........................................ 95 5-9DesignpointgriddistributionbasedonL/D .................... 103 5-10Designpointgriddistributionbasedonstabilityderivatives ........... 103 5-11L/Dsurfaceforsingleparametersegmentation ................... 104 5-12L/Dsurfaceformultiple-parametersegmentation ................. 104 5-13Permissible4Dactuatortrajectorybetweenthreedisparatecongurations ... 105 5-14Controlduringmorphingwithsinusoidalcommand ................ 106 5-15Controlduringmorphingwithchirpcommand ................... 107 5-16Controlduringmorphingwithsinusoidalcommandandpoorinitialconditions 108 5-17Variablereferencemodelcontrolwithsinusoidaltrajectory ............ 109 5-18Variablereferencemodelcontrolwithachirpcommand .............. 110 6-1Optimalratetrajectoryframework ......................... 114 6-2Optimalratetrajectoryandshapeadaptationframework ............. 117 7-1Manuallypiloteddeploymentandrecoveryconguration ............. 120 7-2MaximumL/Dconguration ............................ 122 7-3Maximumenduranceandminimumrateofdescentconguration ......... 123 7-4Steepdescentangleconguration .......................... 126 7-5Maximumtrimmablesideslip ............................ 129 7-6Maximizedsideslipcostfunctionconguration ................... 129 7-7MorphingjointangletrajectoriesformissiontaskssubjecttoSDC ........ 134 7-8MorphingjointangletrajectoriesformissiontaskssubjecttoHQC ....... 135 10

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1.1 showsageometrycomparisonoftwoUAVs,onedesignedforenduranceandtheotherforcombatengagement. Figure1-1. TwodissimilarUAVsdesignedfordisparatetasks,Predatorvariant(left)andX-45variant(right)[ 18 ] Apossiblesolutiontothelimitationsofxed-congurationaircraftistheuseofmor-phingtodramaticallyvarythevehicleshapefordierentmissionsegments.Morphingisenvisionedformanytypesofvehiclesthatengageinmulti-partmissions.ThetechnologyisparticularlyattractiveforUAVsbecausepossibilitiesforenduranceandmaneuverabilityaregreatlyexpandedovermannedvehicles.Theappealofmorphingisthatthevehicledesignisadaptedinighttoachievedierentmissionobjectives.Amorphingaircraftmightmorphintodistinctshapes,whereeachcongurationislocallyoptimalforaparticulartaskofadesiredmission.Addition-ally,shapedeformationsofmorphingaircraftcanprovidehighlevelsofmaneuverabilitycomparedtoconventionalcontrolsurfaces.Thesubstantialbenetsofamorphingaircraftcomewithconsiderabletechnologicalchallenges.DARPAandNASAenvisionfutureaircraftashavinghighlyrecongurable,continuous-moldlinecontrolsystemscomposedofhundredsorthousandsofshape-aectingactuators.Actuatorandstructuraltechnologiesneededtoachievesuchagoalarestillrelativelyimmatureandconnedtolaboratorystudies.Thecompliantshapeofamorphingairplanemayresultinincreasinglyexiblestructuresthatareproneto 13

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18 ].Theaircraftoperateinfar-and-awayenvironmentslargelyfreeofobstaclesandtrac.Thedatagatheredfromsuchoperationsaordsexcellentregionalawareness,butlackslocaldetailthatmaybeobscuredbybuildingsandstructures. 14

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Figure1-2. Dierentwingshapesusedbyaglidinggullthroughjointarticulation Themotivationforusingbiologicalsystemsasinspirationformorphingightvehiclesliesinthesimilaritybetweenthetwowithrespecttosize,weight,airspeedrange,andoperatingenvironment.ALaughingGull,forinstance,hasawingspanof1meter,ayingmassof322grams,andacruiseairspeedof13meterspersecond.ThegullspecicationsarequitesimilartomanysmallUAVs,withthepossibleexceptionofmass,whichisusuallyhigheronUAVs.GiventhesimilaritiesbetweenbirdsandsmallUAVs,adaptingideasfrombiologicalsystemsforconventionalightisquiteappropriatebothinaerodynamicshapeandmechanicalstructure.SignicantshapechangesonaUAVwingcanbeaccomplishedbysimplifyingthewingskeletonofabirdintoaserialjointlinkagemechanism.Aexible,membranewingattachedtothelinkageallowstheaerodynamicsurfacetosmoothlydeformintoseveraldierentpositions.Theaerodynamicbenetsofsuchamorphablewingareeasilyaccomplishedusingsimple,readilyavailableelectro-mechanicalactuators.AtthesmallscalesofinteresttoMAVs,thestrengthofactuatorsisquitehighcompared 16

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IncreasemaneuverabilityandenduranceforamorphingMAV 2. Identifyadesiredmissionproletoexpandrangeoffunctionalitybyusingmorphing 3. Identifyrequirementsofmissionsegmentsonaircraftdynamicsandperformance 4. Developmorphingmicroairvehiclewithbiologicallyinspiredmorphingwings 5. Parameterizeightdynamicswithrespecttowingmorphology 6. Developclosed-loopperformancerequirementsoverparameterspace 7. Developdesignpointcriteriaforlinearparametervaryingframework 8. Maneuveraircraftusinghybridrobust/optimalbaselineadaptivecontroller 9. Developadaptationalgorithmforautomaticcontrolofmorphing 10. Applyoptimalcontroltoratecommand,shapeadaptation,andmaneuvering 19

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95 ],Wlezienet.al,describedprogrammaticgoalsrelatedtotheuseof\smartmaterialsandactuatorstoimproveeciency,reducenoise,anddecreaseweightonaconventionallyownaircraft".Thescopeofthemorphingisrelativelyconservative,consideringthattheauthorsprimarilyintendtoapplythetechnologytoalimitedclassofairvehiclestoachievemodestimprovementsinperformance.Thepaperpresentsanoverviewofthedisparatetechnologiesneededtoimplementmorphing,includingunconventionalmaterials,optimalaerodynamicanalysis,aeroelasticmodeling,time-varyingdynamicmodeling,andlearningcontrolsystems.Theauthorsproposetheuseofadaptive-predictivecontrollersthatareableto\learnthesystemdynamicson-lineandaccommodatechangesinthesystemdynamics".TheNASAmorphinginitiativehassinceproducedanumberofexcellentstudies[ 22 ][ 95 ][ 79 ]ontheaerodynamicsandaeroelasticityofcongurationmorphingbuthashadarelativelysmallcontributiontounderstandingthedynamicsandcontrolofmorphingvehicles.Addi-tionally,noneoftheconceptspresentedhasmaturedtopointofighttestingonamannedorunmannedairvehicle.Theprogram,althoughlimited,hassuccessfullymotivatedsubsequentprogressbymanyresearchers,perhapsincludingtheDARPAMorphingAir-craftStructuresprogram.ApaperbyPadulasummarizessomeofthedevelopmentsafterseveralyearsofworkontheNASAmorphingprogram[ 67 ].Themaincontributionsarecitedasmultidisciplinarydesignoptimizationofthevehiclecongurationandadvancedowcontrol.Anovelaircraftcontrolsystemisdesignedbyusingdistributedshapechangedevicestoachievemaneuveringcontrolforataillessvehicle.AmorerecentDARPAinitiativeseekstodevelopandeldmorphingtechnologytoenableabroaderrangeoffunctionalitythanthatenvisionedbytheNASAprogram.Inparticular,theDARPAMorphingAircraftStructureshasspeciedagoalofmorphing 20

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58 ].NextGenAeronauticsisworkingonasimilarlyinspiredconceptofslidingskinstoallowthewingtosmoothlychangeinarea[ 46 ].Theskinsappearlooselybasedonbirdfeathers,wherechangingtheoverlapofadjacentfeatherscanbeusedtoincreaseordecreasethewingsurfacearea.Forbothbirdsandconventionalaircraft,theabilitytochangewingareaaordsfreedomtooptimizeaircraftlift-to-dragratiofordierentightconditions.TheNextGenconceptisabletoaddresssomeoftheconcernsraisedintheLockheeddesign;namely,theslidingskinsareabletopreserveaerodynamicshapebetweenmorphingconditions.Furthermore,theconceptisamenabletomanypossibilitiesforunderlyingstructureandactuatorcombinations. 21

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57 ]ofLockheedhavestudiedtheeectoftheDARPA-fundedfoldingwingmorphingontheabilitytowithstandloadsfromdierentmaneuvers.Theauthorsfoundthatthethestructuralpropertiesvaryquitesignicantlybetweenmorphingconditions.Theyhaveproposedmetricsbywhichthestructuresofsuchvehiclescanbeassessedintermsofairspeed,maneuvering,andactuatorresponse.Aircraftmorphingpresentschallengesnotonlyintechnologicalhurdles,butalsoinaccuratelyrepresentingtheresultoftheshape-changeinadesign,control,orstructuralsense.Thechallengeisillustratedbyconsideringthedierencebetweenaconventionally 22

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79 ]hasproposedamethodbywhichmorphinggeometriescanberepresentedusingtechniquesfromcomputergraphics.Soft-objectanimationalgorithmsareusedtoparametricallyrepresenttheshapeperturbation,asopposedtothegeometrydirectly.Theframeworksupportsparametricvariationsinplanform,twist,dihedral,thickness,camber,andfree-formsurfaces.Suchadescriptionmaybeusefulinconjunctionwithconventionalmetricstodescribemorphingwingsthatarecombinationsofsurfacedeformationsandhingedmovements.Thebiologicalequivalentisthatofabirdwing,whicharticulatesatshoulder,elbow,andwristjointsbutalsoundergoesmotioninthemuscles,skin,andfeathers.ThetechniqueproposedbySamarehcanbeusedtocomputegridsforbothcom-putationaluiddynamicsandniteelementmethodcodes,eachhavingbothhighandlowdelityvariations.Theapplicabilitytobothaerodynamicandstructuraldisciplinesallowsthemorphingwingtobestudiedinabroadsense.Characteristicsinotherareas,suchascontroleectiveness,varyingdynamics,aerodynamics,andaeroelasticity,canbedeterminedasadirectoutcomeofthemethod. 23

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30 ]in2002presentedaconceptforslowlymorphingtheairfoilshapeofwingsoverthecourseofaightusingvariable-volumefuelbladders.Theproposedmorphinghasbeenshowntoresultinim-provedrangeandendurance.Theuseofpassivefueldepletiontochangetheairfoilfromahigh-lifttoalow-dragcongurationillustrateshowmorphingmechanismscanuseinno-vativeeectorstoachievethedesiredshapechange.Thepaperstressestheimportanceofadvancedoptimizationtechniquestoensurethedesiredlift-to-dragoptimizationholdsinthepresenceofuncertainty.Sanderset.al.[ 82 ]haveshownthroughsimulationpossiblebenetsofcamberorcontrolsurfacemorphing.The2003studyintheJournalofAircraftshowedconformalcontrolsurfaces,essentiallyapswithnohingeline,areabletoachieveahigherliftingforceatalowerdynamicpressurethanconventionalcontrolsurfaces.SuchresultsareencouragingforapplicationstoUAVs,wherehigh-liftandlargerollcontrolpoweraredesiredatrelativelylowdynamicpressures.Theconformalcontrolsurfacesareshowntoproduceahighermaximumrollrateatcertaindynamicpressures.Thebenetsofsuchanapproachareperhapsmitigatedbytheincreaseinnose-downpitchingmomentoccurringwithconformalcontrolsurfaces.Thelargermomentproducesundesiredaeroelasticeectsathighdynamicpressures,leadingtoexcessivestructuraltwistingandeventuallyaileronreversalasdynamicpressureisincreased.ThislimitationisnotsignicantcomparedtothebenetsforslowUAVs,providedthatthewingissucientlystiintorsion.Baeet.alpresentedavariable-spanconcept[ 8 ]foracruisemissileinordertoreducedragovertheairspeedenvelopeandincreaserangeandendurance.Theauthorsconsiderbothaerodynamicandaeroelasticeectsassociatedwithasliding-wing.In 24

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82 ],themorphingwingwouldpossessalargedegreeofcontrolovertheliftdistributionandwouldbeabletoachievetheaerodynamic,stability,andstall-progressionbenetsofavarietyofwingplanform.Combinedmorphingmechanismsaresomewhatrareintheliterature,consideringtheimmaturityofactuatorsandmaterialsthatareabletoachievecomplexshapechanges.Windtunnelstestsofamorphingaircraftwiththreedegreesoffreedomshowedsignicantaerodynamicbenetfromspan,sweep,andtwistmorphing[ 63 ].Thevehicleusesatelescopingwingtoachievewingspanvariationsof44%,allowingforincreasesinliftcoecientrelativetothenominalcondition.Hingesatthewingrootallowthesweepangletorangefrom0oto40oaft,delayingstallandimprovingpitchstability.Theauthorshaveshownthatthewingsweepcausesachangeinbothcenterofgravitypositionandaerodynamiccenterposition.Theaerodynamiccenteronthewindtunnelvehiclemovesaftfartherthanthecenterofgravity,thusincreasingthemagnitudeofthelongitudinalstaticmargin.Onebenetoftheshape-adaptiveaircraftissaidtobeindragreduction,wherethreedistinctcombinationsofwingspanandsweeparerequiredtoachievetheminimumdragoverthefullrangeofliftcoecients.Theresultisimportantformissionconsiderations,wheremaneuverabilityandairspeedrequirementsmaychangeconsiderably.Thepapersuggeststhatamorphingvehicleisappropriateforreducingdragduringseveralportionsofamission. 25

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90 ][ 85 ][ 24 ].Avariable-geometryvehicleisrepresentedbyablimp-likeshapewhichhasvariousperformancecapabilitiesrelatedtotheshape.Theapproachusesreinforcementlearningtocontroltheshapeadaptationasthevehiclemovesthroughtheenvironment.Thevehiclemorphstooptimallyfollowadenedtrajectory.Adaptivedynamicinversionisusedtocontrolthevehicleandfollowtrajectorycommands.Althoughthestudyisphysicallyunrelatedtoanaircraft,theauthorspresentauniqueapproachforcouplingtrajectoryrequirementsandvehicleperformancecapability.Theautomaticcontroloftheshapefordierentenvironmentsrepresentsacapabilityhighlysoughtforamorphingairvehicle.OptimalityinmorphingisaddressedbyRusnell,whotogetherwithcolleagues,publishedstudiesontheuseofabuckle-wingconcepttoachieveenhancedperformanceinbothcruiseandmaneuveringighttasks[ 76 ][ 77 ].Themorphingaircraftinthestudyusesabiplanecongurationwherethetopwingconformstothelowerwingtoformasinglewinginsomecongurations,andbucklestoformajoined-wingbiplaneinothers.AnalysisoftheParetocurveisusedtocomparetheoptimalcongurationsofthemorphingwingagainstaconventionalwingforthedierentmissions.Vortex-latticemethodisusedtocomputeairfoilandplanformaerodynamicsofthesimulatedaircraft.Theauthorssuggestthatthebucklewingmorphingcanenhancemissionperformancerelativetoanaircraftwithxedconguration.Bootheet.al[ 11 ][ 12 ]proposedanapproachfordisturbancerejectioncontrolonamorphingaircraft.Linearinput-varyingformulationisusedtoallowthecontrollertoaccountforthedynamicvariationsofthemorphing,whichitselfisthecontrolinput.Thesimulatedexamplesincludeseveraltypesofvariablewinggeometries,includingspan,chord,andcambervarying.Stabilitylemmasandassociatedclosed-loopsimulationresponsesshowthatthemorphingcanbeusedinsomeinstancestoadequatelyreject 26

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13 ]studiedthemissionsuitabilityofmorphingandadaptivesystemscom-paredtoconventionaltechnology.Thesystemsareevaluatedonthebasisofightmetricsdescribingtheaircrafteciencyandmaneuverabilityinconjunctionwithlogisticalissuessuchascost,production,andmaintenance.Performanceforadaptiveandconventionalvehiclesareconsideredinamulti-rolemissionscenario,requiringpursuitandengagementtasks.Bowmansuggeststhattheperformanceofmorphingsystemsisworsethanexistingcapability,exceptwhenconsideringmultifunctionalstructuresanddecreaseddependenceonsupportvehicle,inwhichcasetheadaptivesystemsoersubstantialimprovement.Thecostassociatedwithdevelopingtheadaptivevehiclecausespooraordabilityingeneral,butthereducedlifecyclecostresultinlong-termgains.Thestudyoersintriguinginsightintotheultimateusefulnessofamorphingvehicleinthecontextofarealistic,multi-taskmission.Maneuversforamorphingaircraftcanbecomputedusingoptimalcontroltech-niques.Trajectoriesgeneratedforeachmissiontaskdependintrinsicallyonthevariabledynamics.Studiesrelatingghteraircraftagilitytoallowableightpathsprovideause-fularchitectureonwhichtodevelopmorphingcontroltechniques[ 78 ].Directnumericaloptimization[ 49 ][ 41 ]cansimultaneouslyoptimizetheaircraftshapeandthetrajectorytosuccessfullycompletethemissionrequirements.Aircraftagilityandmaneuverabilityforeachmorphingcongurationcanbeassessedusingstandardperformancemetrics[ 93 ][ 61 ].Thesemetricsareusedwithintheoptimalcontrolframeworktomorphtheaircraftfavorablyforeachtask.Theaerodynamicbenetsofmorphingareusuallycitedinstudiesastheleadingimpetusforshapeadaptation,althoughresearchersarealsoconsideringmorphingforareductioninactuatorenergycosts.Johnstonet.alhavedevelopedamodelthroughwhichtheenergyrequirementsforightcontrolofadistributedactuationsystemcanbe 27

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45 ].Theirstudyusesvortex-latticemethodandbeamtheorytoanalyzetheenergyrequiredtodeformthewingandproduceasucientlylargeaerodynamicforce.Distributedactuatorsareshowntohavepotentialimprovementsinenergyrequirementsbymorphingthewingsectionratherthandeectingatrailingedgeap.Biologicalinspirationasatechnologyforamulti-roleaircraftisdiscussedbyBowman[ 14 ]andhisco-authors.Thevariedwingshapesofthebaldeagleduringsoar-ing,cruising,andsteepapproachtolandingareconsideredasthebasisforaerodynamicevaluationofmorphing.Geometricparameterssuchasplanformandairfoilaresaidtobeparticularlyimportantforvaryingtheliftanddragcharacteristics.Thevariableperfor-mancecapabilityoftheaircraftiscontextualizedinanexamplemission,whichdescribestheroleofmorphingintheconceptualaircraftdesign.Livnerecentlypublishedsurveypapersonthestateofexibility,bothactiveandpassive,inaircraftdesign[ 55 ][ 56 ].Theaeroelasticandaeroservoelasticbehaviorofcon-ventionalandmorphingairplanesaredescribedbothintermsofdesignchallengesandpotentialforperformanceimprovement.Livnedescribestherelationofbiologicalobserva-tiontothedesignofmicroairvehicles,inthatsuchsmallaircraftcommonlyuseexibleandappingwingsinordertoachieveight.Morphingforlargeaircraftisalsodiscussed,includingvariablewingsweepontheF-14andF-111andthevariablewingtipdroopoftheXB-70.TheauthorcontendsthatarecongurablemorphingUAVcanoperateecientlyinadiversesetofoperatingenvironments.AstudybyCabell[ 15 ]describesapplicationsofmorphingtoanaircraftsubsystemfornoisereduction.Avariablegeometrychevronisusedfornoiseabatementinthenozzleofaturbofanengine.Theauthorsdescribebothopen-loopandclosed-looptechniquestoadaptthechevronshapetoachievethedesiredreductioninnoisewithoutexcessivethrustreductionforavarietyofconditions.Shapememoryalloyactuatorsareusedtogenerateasmooth,seamlessactuationofthechevronstructure. 28

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16 ][ 17 ].Dierentialwingtwistingoftheinatedwingisusedtoproviderollcontrolformaneuvering.Theauthorsofthestudyconsideredseveralactuationmethods,alongwiththecostandperformanceassociatedwitheach.Morphingissuccessfullyimplementedasacontrolsolutiontoawingwhichhasdesirablepackaginganddeploymentproperties.Cesniket.alhavedevelopedaframeworkforassessingthecapabilitiesofamorphingaircraftperformingaspeciedmission[ 19 ].Theauthorsdescribethedesiredmissionscenarioandidentifythenecessaryvehicleperformanceforsuccess.Thetechnologyrequiredtoachievethemorphingisalsoconsidered.Theevaluationisusedtoformulatethecriticalcomponentneedsforamorphingaircraft.Scoringmetricsforvariousmissionsegmentsareaggregatedtodeterminetherelativeimprovementinperformance.ResearchersatPurduehavebeenstudyingmorphingasfundamentalaircraftdesignelementformulti-rolemissions[ 74 ][ 27 ][ 28 ][ 69 ][ 75 ].Theirapproachconsidersmorphingasanindependentdesignvariableanddeterminethecontextsinwhichvariousformsofshapechangecanbeusedadvantageously.Thestudiesconsiderbothasinglevehicleperformingdisparateighttasksandaeetofvehiclesoperatingcooperatively.Criticalneedtechnologyareasareidentiedbyidentifyingapplicationsforspecictypesofmorphing.Actuationandstructuraldevicesneededtoenablethemorphingarereadilyidentiedfromtheanalysis.Morphingthewingsofanaircraftintwistisunderstudyforuseasbothacontroleectorandasamethodtoincreaseeciency.AWestVirginiaUniversitystudyisconsideringapplicationsofwingtwisttoaswept-wingtaillessaircraft,whichisitselfinspiredbynaturalight[ 39 ].Theconventionalcontrolmechanismsusedonsuchavehiclecompromisethebenecialaerodynamicsofthewingconguration.Atwistmechanismhasbeenshowntohavesucientcontrolpowerwhileimprovinglift-to-dragratioby15%anddecreasingdragbyupot28%relativetoconventionalhingedsurfaces.Thesignicant 29

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84 ].Multi-objectiveoptimalityisaddressedbyndingthetorque-rodcongurationwhichsimultaneouslyachievesthehighestrollrateandthehigh-estlift-to-dragratioduringarollmaneuver.AParetofrontdescribesthecongurationswhichachievevariouscombinationsofliftandrolloptimality.Thecurveiscomparedtothecongurationontheexperimentalighttestvehicle,whichcanachieveperformanceimprovementsthroughmodicationoftheactuator.Awindtunnelinvestigationhasdemonstratedtheperformanceanddynamiceectsachievedbyquasi-staticvariationsofthewingaspectratioandwingspan[ 42 ].Anexper-imentalmodelusestelescopingwingsegmentstomorphbetweenretractedandextendedcongurations.Thevariablewingareaallowslargechangesintheliftanddragmagnitude,butalsohaslargeaectsontheroll,pitch,andyawdynamics.Open-loopdynamicsareshowntovaryconsiderablywithsymmetricwingextentions.Eachoftheroll,dutchroll,andspiralpolesshowtendtowardmorerapidresponsesasthespanisincreased.Theauthorsdepictahighlyunstablespiralmodeandareinvestigatingdierentialspanex-tensionsasameansofstabilizingandcontrollingthedivergentbehavior.Anadditionalstudybytheauthorspresentsthecoupled,time-varyingaircraftequationsofmotionandexplorestheuseofasymmetricspanmorphingforrollcontrol[ 43 ].ResearchersfromGeorgiaTechareexploringcongurationoptimizationforavehiclethatmustperformthedisparatetasksofsubsonicloiterandsupersoniccruise[ 66 ].Thenovelaircraftcongurationrequirestheuseofresponsesurfacemethodologyandseveralstagesofanalysistoestimatetheperformanceofvariousshapesinthedesignspace.Aerodynamiccharacteristicsareoptimizedforeachmissionbyparameterizingthedesignelementsandndingthesetofshapesthatachievestheoptimalcompromisebetweenthetwomissionmodes. 30

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72 ].Thestructureisactuatedbytendonsjoiningvariouspointsofthetrussesandcanachievemanytypesofaircraftshapesthroughcontinuousdeformation.Manytendons,distributedthroughoutthetrussesareusedtogenerateglobalshapevariationsthroughaggregationoflocalactuation.Actuationofthewingcanbeachievedusingrelativelylowforces.Theweightofaproposedmorphingtrusswingissimilartoaconventionalwing,exceptthataeroelasticconcernsmayrequiretheuseofactivecontrolofthetendons. 31

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23 ],[ 70 ],[ 31 ],[ 32 ].Forxed-wingunmannedight,thedicultyofabroadightenvelopeliesinthelimitationofastaticgeometryinprovidingecientightoverarangeofairspeedsandanglesofattack.Sucharangeisdesiredforanaircraftexpectedtooperateinanurbanenvironment.Specically,thevehiclemustmaneuversafelyinacomplex,3Denvironmentwhereobstaclesarenotalwaysknownapriori.Rapidchangesindirectionrequirelargeaerodynamicforceswhicharegeneratedathighslipanglesandhighdynamicpressures.Themotivationforstudyingbirdphysiologyforinsightintothedesignofahighlymaneuverableairvehiclestemsfromobservationsofglidingbirds.Rapidaccelerationsandightpathvariationsareachievedthrougharticulationofthewingstructuretopromotefavorableaerodynamicorstabilityproperties.Birdsingeneral,andlaughinggullsinparticular,havebeenobservedusingdierentwingcongurationsfordisparatephasesofight.Windhovering,soaring,andsteepdescents,forinstance,eachrequireaspecictrimairspeed,glideratio,andmaneuverabilityaordedbyadierentwingconguration.Thepresentworkseekstoidentifythedominantwingshapesthatareportabletoaconventionalaircraftphilosophyforthepurposeofexpandingtheachievablerangeofight.Birdsareusedasasourceofdesigninspirationforurban-ightvehiclesbecauseofstrongsimilaritiesindimensionandoperatingcondition.Theyalreadyachievearemarkablerangeofmaneuversalongcomplextrajectoriesinurbanenvironments.The 32

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88 ],longitudinalpitchstability[ 63 ]andlateralyawstiness[ 80 ].Figure 1.1 showsthreeplanformvariationsofaglidinggull,whereeachoftheshoulder,elbow,andwristjointsaremodiedtochangetheoverallwingshapes.Theplanformsdiermarkedlyinwingarea,span,andsweep,yetareaccomplishedrelativelyeasilythroughskeletaljointsandoverlappingfeathers.Figure 3-1 showsresultsfromTuckerestablishingarelationbetweenplanformshapeandglidingspeedforsimilarglideangles[ 88 ].Asignicantreductioninwingspanandwingareaisachievedbyarticulatingtheelbowandwristjointssuchthattheinboardwingissweptforwardandtheoutboardwingissweptaft.Theaftrotationoftheoutboardorhandwingissuchthattheaerodynamiccentertypicallymovesaft,increasingthepitchstinessforoperationathigherairspeed. 33

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Eectofplanformshapeonglidingspeed.SeeTucker,1970forwingshapesA,B,andC[ 88 ] Yawstinesscanbeachievedinbirdwingswithaftsweepoftheoutboardwings.Sachs[ 80 ]showedthemagnitudeofthelateralstiness,Cn,increaseswithaftsweepangleandissucientlyhighsuchthatbirdsdonotrequireverticalstabilizers.Figure 3-2 showsabirdwithaftwingsweepandpositiveyawstability[ 81 ],whichiscomparedbySachstoanaircraftwithsimilarcharacteristics.Theaftsweepoftheoutboardwingshasalreadybeenshowntoincreasepitchstability.Thesimultaneousincreaseinyawstabilityallowsthebirdtoglideatrapidspeedswhilemaintainingadesiredightpath. Figure3-2. Increasingyawstinessthroughoutboardwingsweep 34

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3-3 showstwogullsusingdierentwingjointanglestodeformtheirwingsdur-ingglidingight.Thepurposeofthismotionhasbeenthesubjectofsomespeculationintheliterature,withsomeresearchersarguingthebenetsarepredominantlyaerodynamicwhileotherscitestabilityimprovements.Inthecaseoftheupperphotograph,thebirdusesoutboardwingstipsangleddownward.ThecongurationhasbeenshowntoyieldsignicantgainsinL=D[ 22 ],perhapsattheexpenseofadverselateralstability[ 87 ].Thelowerphotographshowsabirdwithan'M'wingshapewhenviewedfromthefront.Theshapehasbeenobservedinuseinglidingbirdstovarytheglideangle.AwideM-shapeisusedinsituationsrequiringashallowglidepathandacorrespondinglylargeL=Dratio.Conversely,birdsinsteepdescentsoftenuseanarrowM-shape,presumablytoreduceL=Dratioandmaintainstabilityatthehighanglesofattackincurredduringslowdescent. Figure3-3. Glidingbirdsarticulatingwingjointsaboutlongitudinalaxes Figure 3-4 showstwogullsusingcomplexmorphingoftheshoulder,elbow,andwristjointsaboutbothlongitudinalandverticalaxes.Theaerodynamicsduringsuchmotionlikelyinvolvestabilityandperformanceelementsfromeachoftheindividualmotions.Morecomplexowinteractionsmayalsoyieldimprovementsinliftthroughleadingedgevortices,assuggestedbyVideler[ 91 ]. 35

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Complexwingmorphingshapesachievedinglidingight Mostmorphologicaloperationsareconductedsymmetricallyaboutthebody,suchthatleftandrightwingsgeneratesimilaraerodynamicforces.Asymmetricmotionsareusedincertaininstancesofglidingight,presumablytogeneratefavorablemomentsformaneuvering,rejectdisturbances,orallowthebirdtotrimatlargeanglesofsideslip.Figure 3-5 showstwobirdsusingasymmetricwingspan,wingarea,dihedralangle,andsweep.Theupperphotographshowsdierentarticulationsofthewristjointcausetheoutboardwingstoextendandsweeptodierentextents.Thelowerphotographshowsthewristjointsofanotherbirdatdierentoutboarddihedralanglesandextension.Thecongurationsmaybewellsuitedforcrosswindoperations,consideringthespanwiseliftdistributionandaerodynamiccoupling(Cl,forinstance)canbeselectivelyvaried.Glidingbirdsarecapableofrapidchangesinightpathusingaggressivemaneuvers.Figure 3-6 showstwogullsnearbankanglesof90operformingpull-upmaneuverstorapidlychangedirectionsinsmallradiusturns.Bothbirdshavedeployedthetailfeathers, 36

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Gullsusingasymmetricwingmorphingaboutvertical(top)andlongitudinal(bottom)axes presumablyforincreasedpitchingmoment.Theleftphotographshowsabirdinadescending,helicalpathatrelativelyhighairspeed.Theorientationoftheheadshowsthatthebankangleisquitelarge.Aftsweepofthehandwingsislikelyusedtomaintainpitchstabilityatthehighairspeed.Thebirdshownintherightphotographisperformingasimilarmaneuver,exceptatalowerairspeed.Boththetailandwingfeathersaredeployedfurtherthantheleftbird,perhapstocompensateforthereduceddynamicpressurewithincreasedsurfacearea.Thewingsarealsosweptforwardslightly,reducingthepitchstabilityandpossiblyallowingthebirdtoperformanaggressiveturn. 37

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Rapidvariationsinightpathusingaggressivemaneuvering 38

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91 ]astheresultofparallelarmbonesinteractingduringwingextension.Birdsengageinseveraltypesofcombinedmotions,witheachhavinganapparentbenetinight.Someofthedominantfunctionsaredescribedandrelatedtoconventionalight.Themotionsare,ingeneral,easilyreproducedusingamechanizedaircraftwing. 41

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80 ].Aftsweepoftheoutboardwingissaidtogeneratealeading-edgevortexlift[ 91 ],althoughthisisnotmodeledexplicitlyinthecurrentconsiderations. 87 ].Upwardrotationoftheinboardwingproducesbothadihedraleectandelevatestheouterwing,inbothcasesincreasingthewing-levelstability[ 4 ]. 43

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3-7 showsanannotatedviewoftheleftwingcomparedtotwoglidinggullsusingasimilarformofwingmorphing.Itisclearthatconventionalaircraftmetricssuchasdihedralarein-sucienttodescribethewingarticulation.Instead,thewingistreatedasaserial-linkrobotmanipulator[ 21 ],whereeachjointisrepresentedbyappropriatelengthandangledimensions. Figure3-7. Wingarticulationaboutlongitudinalaxisshowingtwojointangles.Glidinggullsareshownforreference. Themorphingwingissimilarlyequippedwithjointsbywhichtoarticulatethewingsectionsinthehorizontalplane.Figure 3-8 showstheundersideofagullwingwithannotationstoindicateapproximatepositionsofbonesandjoints.Themechanicalmodelisofreducedcomplexity,havingonlytheequivalentofashoulderandawristjoint. 44

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Approximatephysiologicaldiagramofagullwingwitharticulationsinthehorizontalplane 3-9 showsfourconsecutiveframesduringsuchadescent,wherethepigeonusesaturningightinadditiontotheinclinedwinggeometrytocontroltheapproachpath.Eachframeisphotographedusingafocallengthof300mm,makingtherelativesizeofthepigeonintheimageanindicationofitsproximitytothecamera.Thearrangementofthebirdsintheimageisarticialanddoesnotreecttheactualightpath.However,thesequenceillustratesthewingshapeusedtocommandasteepdescentandalsoshowssomemaneuveringcapabilityduringtheunusualjointconguration.Tailfeathersarefullyextendedduringthesteepglidingmaneuver,ostensiblytoincreasedragandreducelift-to-dragratio.Thewingcongurationmayalsoprovidestabilitybenetsatlargetrimanglesofattacksuchthattheglidescanbeeectivelycontrolledforprecisionlandings.Theheadorientationofthepigeongivessomeindicationoftheangleofdescentrelativetothebody,assumingitislookinginthedirectionofight.Thewingmorphingshowndepictsverticalarticulationsoftheshoulderjointtoachievethedihedralanglesofnearly45operwingside.Itisdiculttodiscernfromthephotos 45

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Compositephotoofpigeoninglidingapproachtolanding theamountofsweepused,buttheyappeartobeintheneutralorevenslightlyforwardsweptconguration.Tennekesreportsinhis1996bookonthewingmorphingusedbypigeonstocontrolglidespeed[ 86 ].Fullyextendedwingsareusedduringlowairspeedglideswhileforwardsweptinboardsandhighlyaftsweptoutboardwingsareusedforhighairspeeddives.Thechangeinwingspan,wingarea,andaspectratioallowthebirdtoalterthetrimglidespeedbynearlyafactorofthree. 88 ][ 89 ].Forwardsweepoftheshoulderorelbowjointsandaftsweepofthewristjoint,asinFigure 3-8 ,areusedinvaryingdegreestocontroltheplanformareaandwingspan.Thesubsequenttrimairspeedvariessubstantially,aordingcontroloverloiteringorattackspeeds.Sachsreportedndingsontheroleofaftwingtipsweepinprovidingyawstinessandyawstabilityontheaerodynamicandinertialscalesofbirdight[ 80 ].Theaftwingsweepalsoaectsthelocationoftheaerodynamiccenterandcontributestopitchstability[ 87 ]. 46

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22 ].Alternatejointcongurationshavebeenshowntohavetheoppositeeectandreducelifttodrag,allowingsteepdivesatmoderateairspeeds[ 3 ].Videlersuggeststheaerodynamicsofcomplexavianwingshapesbenetfromthedeploymentofindividualfeathers,suchasthealula,inincreasingtheliftmagnitudebypromotingaleading-edgevortex[ 91 ].Figure 3-8 showsthelocationofthealulafeather,whichisdeployedforwardatthewristjointduringcertainwingshapesandmaneuvers.Thefeatherissaidtohaveaprofoundimpactinincreasingmaneuverabilityandpossiblydelayingstall.ThewinggeometryofseveralbirdspeciesarepresentedbyLiuet.alin2006[ 54 ].Three-dimensionalscannersareusedtogeneratemodelsofbirdwingsthroughouttheappingcycle.Thewinggeometriesarepresentedastime-dependentFourierseries,whichrepresentthechangeintheaerodynamicshaperesultingfromskeletalarticulation.Theauthorsuseatwo-jointedarmmodelasasimplicationofbirdbonestructure.Thearmmodelischaracterizedbythreeanglesandachievesasucientrangeofmotiontorepresenttheappingcycle.Thewingsurfaceisassumedtobexedtooneoftwosparsatthequarter-chordpositionandmaintainsproperorientationrelativetotheowforallcongurations.Althoughtheresearchfocusedonappingight,theidentiedwingshapesmayalsobeusefulforglidingoperations. 2 ]whichservesasthenominalcongurationandalsoprovidesabaselineforperformancecomparison.Figure 3-10 showsfourcompositeviewsofthemorphingdegreesoffreedomandrange. 47

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(b)3=(30o;0o;30o),PositiveUp (c)2=(30o;0o;30o),PositiveForward (d)4=(30o;0o;30o),PositiveForwardFigure3-10. Morphingjointarticulationsfor4-degree-of-freedomwing Wingcongurationsaredeterminedbyfourjointaxes,twoatthewingrootandtwoatthemid-spanposition.Inboardjointangles,1and2,controltherotationsaboutlongitudinalandverticalaxes,respectively.Outboardjointangles,3and4,similarlycontrolthelongitudinalandverticalaxisrotations,respectively.TheviewsshowninFigure 3-10 depictvariationstoonejointaxisforeachsubgure.Inboardjointsproducerotationsoftheinboardwingsectionandtranslationsoftheoutboardwingsection.Theorientationoftheoutboardsectionsiscontrolledstrictlybytheoutboardjoints. 48

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36 ].Forsimulation,theaircraftismodeledwithasimilarrangeofmotionbutwithoutthemechanicaldetail.Avortex-latticecomputationalaerodynamicspackage[ 25 ]isusedtosimulatetheightcharacteristicsofvariousaircraftcongurations.Simulationsareper-formedforeachcombinationofjointangleswithintheoperatingrange.Angularresolutionforeachjointis5o,giving13uniquepositionsoverthe30orangeand28,561combi-nationsofjointangles.Outputdatafromthesimulationsaresavedtoa4-dimensionalmatrixforpost-processing.Airspeed,angleofattack,andsidesliparexedforeachsimu-lationat15m/s,6oand,0o,respectively.Eachrunistrimmedforzeropitchingmomentusingtheelevatorcontrolsurface. 49

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4{1 4{3 [ 64 ][ 33 ]describethethreeorthogonalforcesintermsofthemass,velocity,andacceleration.Theforcecontributionsincludeseveralfactors,amongthembodymass,m,timesthelinearaccelerations,_u,_v,and_w.Cross-axistermsalsocontributetotheforces,whereangularvelocitiesaremultipliedbytheorthogonallinearvelocities.Agravitationalcomponentisalsoincludedintheforceequations,whichaccountsforthechangingbodyattitudeinaxedgravitationaleld.X=m(_u+qwrv)+mgsin 4{4 4{6 showtheformulationforthreeorthogonalmoments.Eachmomentdependsonthecorrespondingmomentofinertiamultipliedbytheangularacceleration,across-axisangularaccelerationmultipliedbytheproductofinertia,andcoupledtermsrelatingangularvelocitiesandinertialproperties.Thegivenmomentsequationsareindependentofthebodyorientationandarevalidforaircraftofxed,laterallysymmetricconguration.Foraircraftwithrapidlytime-varyingandasymmetricgeometries[ 35 ][ 36 ],themomentequationsrequireinclusionofalltheinertialterms,manyofwhicharezeroforconventionalaircraft. 50

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(4{5)N=Ixz_p+Iz_r+pq(IyIx)+Ixzqr 4.1 Table4-1. Standardaircraftstatesdescribingvehiclemotion AxisPositionOrientationLinearvelocityAngularvelocity RollxupPitchyvqYawzwr Linearizedaircraftdynamicsarewritteninaformwheretheeectofeachstateorcontroloneachforceormomentcanbereadilyidentied.Stabilityderivativesarecoecientsdescribingalinearrelationbetweenastate,suchasrollrate,andaforceormoment,suchasrollmoment.Subscriptnotationisusedforstabilityderivatives,suchasClp,todescribethelinearizedinuenceofrollrateonrollmoment.Controlderivativesaresimilarlyexpressed,relatingcontrolsurfacedeectiontoforcesandmomentssuchthatCladescribesthecontributionofailerondeectiontorollmoment.Theindividual 51

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4{8 and 4{9 .Lateralandlongitudinaldynamicsareassumeduncoupledandareshownindependently.Thedynamicsareaectedonlybyeightstates,fourlongitudinalandfourlateralstates.Theremainingfourstates,includingpositions,x,y,andz,andyawangle,,arestrictlyforkinematicconsiderations. 52

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7 ]andthehighestglideratio[ 9 ].ConguringapoweredUAVforrangeallowsthevehicletotravelthemaximumdistanceforagivenquantityoffuel.Intheunpoweredcase,maximizinglifttodragallowsthevehicletoglidethemaximumhorizontaldistanceforagiveninitialaltitude.Inviscidaerodynamicmodelingisusedtogeneratedataformorphingcongurations,neglectingsourcesofdragsuchasskinfrictionandinterferencedrag.Modelingresultsaresupplementedbyasimpledragmodelbasedonaminimum,zero-liftdragwithaquadraticapproximationoflift-induceddrag.Thedragcoecientisgivenby,CD=CDo+1 57

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4-1 showsvariationsofdragwithliftforaninviscid,symmetricalwing.Thelocationofthemaximumlifttodragoccurswherealineoriginatingattheoriginistangenttothequadraticdragpolar.Thelifttodragratioisplottedtoshowshowthewingeciencychangeswithliftcoecient.TheminimumL/DoccurssymmetricallyabouttheCLaxis,atanegativeangleofattack. Figure4-1. DragpolarandlifttodragratioshowinggraphicallylocationofmaxL/D 4{12 fromAnderson[ 7 ]showstheinverserelationshipbetweenL/Dandthrust-requiredtoweightratio(Tr=W).TrdecreasesasL=DincreasesforxedW. 58

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D=Tr Equation 4{13 [ 7 ]showsthefactorsthataectL=Dforanaircraftinlevelight,wheretheliftisequaltotheweight.Theequationincludestermsfromthedragpolarandthewingloading,bothofwhichareexpectedtochangewithdierentmorphedcongurations. D=1V12CD;0 1V12W S1(4{13) 10 ],respectively.Achangeincourserequiresaninitialaccelerationaboutoneormoreaxis,followedbyasustainedangularvelocity,andconcludedbyacceleratingtoaterminalightcondition.Thedesiredaccelerationsandvelocitiesarelargewhenthemaneuverissubjecttospaceandtimeconsiderations.Atar-getengagement,forinstance,mayrequirerapidrollingandpitchingtoacquirethetargetinthesensoreldofview,followedbypreciseightpathtrackingtomaintainadesiredpositionrelativetothetarget.Theaerodynamicrequirementsforsuchmaneuversaredierentthanthoseforabenignightsegmentsuchascruiseorloiter.Thecorrespondinggeometrythatachievesthemaximumagilityandmaneuverabilityisalsoexpectedtodierfromtheoptimaleciencyconguration.Severalmetricsforaircraftmaneuverabilityexistandhavebeenusedextensivelytoquantifyghteraircraftperformance[ 61 ][ 78 ].Thebasicmeasureofagilityinthecurrentstudyisthemagnitudeofangularaccelerationaboutthethreebody-axes,_p,_q,and_r. 59

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4{8 )are,_p=L+Lpp+Lrr+Laa+Lrr 60

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(4{27)=0 (4{28)_p=0 (4{29)_r=0 (4{30)Substitutingthestatevaluesintotheyawaccelerationexpressionandsolvingforrollrate,p,yields,p=NaaNrr 63

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(4{32)Lp (4{33)Lp 65 ]andsomeboundingassumptionsonthestatesareneededinordertodeterminethemaximumpitchrate.Theexpression 64

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(4{42)Wq=2 (4{43)Wr=1 (4{44)MorphingcongurationswhichmaximizeJmandJaaredeterminedusingthegivenaxisweightsandstabledynamicscriteria.Thecriteriastipulatesthatalldynamicmodesmustbestablewithsmallunstableallowancesfordivergentspiralandphugoidmodes.Figure 4-2 showsthemostmaneuverablecongurationunderthesimulatedcondi-tions.Thewingassumesanunusualshapeneartheboundariesofthecongurationspace,withlimitdihedralandaftsweepontheinboardwingcombinedwithlimitanhedralandlargeforwardsweepontheoutboardwing. Figure4-2. Maximummaneuverabilitycongurationusingstabledynamicscriteria(1=30o,2=30o,3=30o,and4=20o) Thecongurationachievesarollrateof13.9rad=s,apitchrateof0.05rad=s,andyawrateof13.1rad=s.Thelowpitchratemagnitudeisaresultofthelowelevatorcontroleectivenessinthesimulationmodelinadditiontothelargepitchmomentofinertia.Thepitchmaneuverabilitymayalsoimprovewhenperturbedvelocityandangle 66

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4-3 showsthemostagileconguration,whichassumesamuchmorenominalshapethanthemaneuverableshape.Thewingshapeusesmoderateanhedralontheinboardwingandslightforwardsweepforbothinboardandoutboardwings.Itisinterestingtonotethedisparitybetweenthetwoshapesthatresultsfromtheadditionofdampingtermsinthemaneuverabilitycost. Figure4-3. Maximumagilitycongurationusingstabledynamicscriteria(1=15o,2=10o,3=0o,and4=10o) Theagilityofthecongurationisquitesubstantial,consideringthelargerollrateaccelerationof211.2rad=s2.Accelerationalongtheotheraxisaremuchsmaller,with70.1rad=s2and20.8rad=s2forthepitchandyawaxes,respectively.Whiletheshapesdonotappeardirectlyinspiredbynature,certainaspectsofthewingshapereectobservationsmadeofbirdight.Theforwardsweptwingtipsofthemaneuveringcongurationarecommontoseagullsperformingrapidchangesinglidepathandbankangle.Theanhedraloftheagilecongurationaresimilartodown-turnedwingsseenonseveralspeciesofbirdswhileinacceleratingight.Themorphingcongurationspaceishyper-dimensionalandthuscannoteasilybevisualized.Metricsorparametersthatdependoneachofthefourmorphingjointanglesmayincurhighlycomplexchangesthroughthecongurationspace.Thesechangesare 67

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4-4 showssixviewsofthecongurationspace.Plotsonthetoprowarexedat4=15oandshowvariationsinthemaneuverabilitymetricduetochangesin1,2,and3.Thecolorofeachsquareintheplotsrepresentsthemagnitudeofthemetric,whileitslocationindicatesthemorphingangles.Slicesareshownatthespaceextremes(1=30o,2=30o,and3=30o).Afourthsliceisshownatvaluesof1=(15o;0o;15o).Thefourplotsshowthatmaneuverablecongurationsareconcentratedatnegativevaluesof3withpositivevaluesof2.Anadditionalconcentrationoccursatpositivevaluesof1withnegativevaluesof2. Figure4-4. Maneuverabilityindexshownasslicesof4-dimensionalhypervolumeat4=15o(top)and4=15o(bottom) ThebottomrowofplotsinFigure 4-4 showmaneuverabilitymetricdatafor4=15o.Thedistributionofmaneuverabilityconditionshaschangedmarkedly,withmostoccuringatlargepositivevaluesof1withnegativevaluesofeither2or3.Thedistributionofmaneuverableshapesiswiderthanthecasefor4=15o,althoughthedisparityinthecolorbarindexshowthemostdesirableconditionsexistwithpositivemorphingof4. 68

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4-5 representsthemaneuverabilityindexforeachmorphingcongurationasahistogram.Althoughactuatorcontextualizationislost,thedistributionofvaluesshowsinsightintotherelativescarcityofhighlymaneuverablecongurations.Theaveragemetricvalueis37.3rad=s,yetthehighestvalueis57.1rad=salongapositiveskew. Figure4-5. Histogramofmaneuverabilitymetricforallmorphingcongurations(1;2;3;4=[30;30]) Figure 4-6 showstheagilitymetricforcongurationswith4=15o(toprow)and4=15o(bottomrow).Forbothcases,theagilecongurationsexistatpositivevaluesof1andnearnominalvaluesof2and3.Thetoprightplotindicatesthatthemaximumagilityshownoccursatamoderatevalueof1,ratherthanattheextreme.Relativelynon-agileshapesexistalongseveralcornerextremesofthespace.Figure 4-7 showsthehistogramfortheagilitymetricassociatedwithallcongu-rations.Thedistributionshowsnegativeskew,withanaverageagilitymetricof599.3rad=s2.Variationsinmetricmagnitudearelargethroughthecongurationspace,rangingfromaminimumof298.2rad=s2toamaximumof807.9rad=s2. 69

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Agilityindexshownasslicesof4-dimensionalhypervolumeat4=15o(top)and4=15o(bottom) Figure4-7. Histogramofagilitymetricforallmorphingcongurations(1;2;3;4=[30;30]) canaccomplishsimultaneousobjectives,suchasvaryingaerodynamicperformancewhilepreservinghandlingqualities.Biologistshavereportedobservingbirdsmorphwingshapewithsignicantcouplingbetweenjointarticulation[ 91 ].Althoughabirdwingishighlycongurable,oftenonlyasubsetofthepossibleshapesareusedinight.Theskeletalstructureinthewingpromotescouplingthroughmechanicalinteractionbetweentheelbowandwristjoint[ 87 ].Twoparallelbonesbetweentheelbowandwristallowsimultaneousextensionofthe 70

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91 ].Thedynamicsoftheresultingshapesarealwayscontrollable[ 54 ],yetallowthebirdtoperformadiversesetofighttasks.Similarly,abiologicallyinspiredaircraftmaynotrequiretheentirecongurationspaceandmaybenetfromcouplingmajormorphingmotions.Combiningmorphingoperationssimpliestheshapecommandbyreducingdimensionalityandalsoremovesportionsofthecongurationspaceoeringlimitedusefulness.Desiredcombinationsareidentiedfrom4-dimensionalperformanceormaneuveringmetricsbyapplyingprinciplecomponentanalysis(PCA)toasetofdesiredcongurations.Forindividualmetrics,suchaslifttodragratio,theshapeswhichachieveminimumandmaximumvaluesaresampledtogeneratepointcloudsinthemorphingspace.BothminimumandmaximumL/Dconditionsaredesiredfordescentandcruiseight,respectively.Manycongurationsaresampledintheupperandlowerpercentages,suchas20%,ofthemetricrange.CombinedmorphingoperationsarefoundbyperformingPCAonthecongurationpointstondthedominant,orthogonalaxesthroughthespacewhichaectL/D.Reducingthedimensionto2generatesatransformationmatrixwhichrelatesmorphingoperationsonthenewaxes,A;B,totheoriginalaxesof1;2;3;4.MorphingcommandstoA;BdirectlyaecttheL/Dperformanceandcanbecommandeddirectlybythecontrolleroradaptationsystem.Transformationto1;2;3;4becomesalower-leveloperationthatisperformedautomaticallybytheactuationsystem.Figure 4-8 showsacomparisonbetweenthelifttodragperformancemetricfortheoriginalspaceandforareduced-orderspacewithdimensionoftwo.SamplingthecongurationshavingL/Dmetricsintheupperandlower20%at8.3%density(1of12shapes)yields953setsofactuatorpositions.Allcongurationsweresubjecttosampling,regardlessofdynamiccharacteristics.Thedistributionofthereduced-dimensionsetshowsamaximumvalueclosetotheactualmaximumL/Dof15.1.TheminimumL/Dvalue,however,is0.61higherthantheactualminimumof10.5. 71

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ComparisonofL/Dmetrichistogramsforreduced-dimensionspace(top)andoriginalcongurationspace(bottom) TheL/Dvaluesforpointsinthetransformed2Dspacecanbevisualizedusingaresponsesurface.Thespaceconsistsoforthogonalaxes,AandB,whichrepresentcombinedmorphingoperations.Figure 4-9 showsthevariationsinL/DwithrespecttochangesinAandB.Primaryaxis,A,hasadominanteectonthelifttodragratio,whilethesecondaryaxis,Bhasasomewhatsmallereect.Theresponsesurfaceisstillsomwhatcomplicatedandshowssignicantcouplingbetweentheaxesandtrendreversalalongindividualaxes.FornegativevaluesofA,forinstance,increasingBdecreasestheL/Dratiototheminimumvalue.Conversely,atpositivevaluesofA,increasingBincreasesL/Dtothemaximumvalue. Figure4-9. ResponsesurfaceofL/Dmetricinreduced-dimensionspace 72

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4-2 showsthecriteriausedtoassesshandlingqualitiesfromthemodalcharac-teristics.ThecriteriacorrespondstoClass1aircraftoperatinginCategoryA,whichisappliedtolightaircraftengagedinprecisionmaneuvers[ 64 ].Thenumericalvaluesforthe 74

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20 ].Theuseofmanned-ightcriteriaforanunmannedaircraftmayresultinconservativeperformance,consideringthattheremotepilotisnotsubjecttotheoscillationsandloadingofthevehicle.However,itwillensureaminimumlevelofpilotabilityinareaswherehandlingcriteriaforunmannedightvehiclesarenotavailable. ModeTimetodoubleTimeconstantNaturalFrequencyDampingRatio SpiralDivergenceT2;S>12seconds|||RollConvergence|R<1:0second||DutchRollMode*||!n;DR>1.0rad/sDR>0.19*ShortPeriodMode||0.3555seconds||PH>0.04 Table4-2. Handlingqualitiescriteriaformorphingconguration.*DutchrolldampingratiomustalsosatisfyconstraintDR!n;DR>0.35rad/s 4-3 showstheallowablerangeofeigenvaluesforthestabledynamicscriteria.Modesthatareclassicallyunstableormarginallystablesuchasthespiraldivergenceandthephugoidmodeareallowedtohaveslightlypositiverealvalues.Theremainingmodesareallowedtolieanywhereintheleft-halfplane.Thecriteriaalsoallowsthemodestodeviatefromclassicalform,inthatoscillatorymodessuchasdutchrollandshortperiodmayconsistoftwonegativerealpoles. 75

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SpiralDivergence<(S)<0.5RollConvergence<(R)<0.0DutchRollMode<(DR)<0.0ShortPeriodMode<(SP)<0.0PhugoidMode<(PH)<0.5 Table4-3. Stabledynamicscriteriaformorphingcongurations 76

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48 ].Asthedynamicschange,themaneuveringcontrollercompensatestoachievestabilizationandtrackingoftheinner-looprateandangularstates.Atthemostbasiclevel,thepurposeoftheinner-loopmaneuveringcontrolleristostabilizeandtrackcommandstotheangularrates,p,q,andr,andtheanglesofattackandsideslip,and.Trackingcontrolobjectivesforeachofthesestatesisdeterminedbytherelativeaggressivenessofthemissiontask.Time-andspace-criticalighttasksrequirerapidresponseandallowlargeactuatorusage,whereasotherighttasksemphasizeenergyconservationandsloweraccelerations. 77

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83 ],themethodologylacksthetheoreticallysatisfyingproofofstabilityorrobustnessexistingatthedesignpoints.ThesimplisticgaininterpolationcanbereplacedbyaproperlydesignedLPVsystem,whichaccountsforclosed-loopperformancestabilityguaranteesforallpossiblecongurations.Thechoiceofdesignpointdensityrelatestotheallowableparametricvariationsinthebaselinecontroldesign.Acontrolsynthesizedtoberobustto10%uncertaintyintheparametersrequiresanewdesignpointwheneverthechangeintheparametervaluesexceeds10%.Inalargecongurationspacethatdenesacomplicatedchangeofwingshape,theaerodynamicparametersvaryoverlargeranges.Smalltolerancetouncertaintynecessitatesextremelydensedesignpointgridsthatapproachthedensityoftheoriginalspace.Suchalargenumberofdesignpointstrivializesthecomputationalbenetsofinterpolationandfailstoprovideasatisfactorysolutiontothecontrolchallenge.The134wingcongurationsachievablebythemorphingaircrafteachrequireanappropriatecompensatorforclosed-loopstabilityandcontrollability.Designingcontrollersforeachcongurationiscomputationallyunreasonableandrequiressignicantmemoryforstorage.Analternativeapproachoflinearlyinterpolatingcontrollersoversubspacesofthecongurationspaceiscomputationallytractableonlyforacoarsedistributionofdesignpoints.Suchacoarsedistributionresultsinsignicanterrorsinmodelingthecomplexparameterresponsesurfaceresultingfrommorphing.Theschemelinearlyinterpolatescontrollergains,evenforcaseswherethedynamicsarenotwellrepresentedbypiecewiselinearity.Theerrorsleadtointermediatecongurationsusinghighlyo-designcontrollers.Thedesiredapproachtocontrollingahighlycongurablemorphingaircraftisbyallowingthecontrollertoadapttochangesinthevehicledynamics.Sincethemorphingisassumedtobequasi-static,thesystemdoesnotviolatetheconstant/slowlytimevaryingassumptionnecessaryforconvergenceandstabilityofadaptivecontrollers.Robustandoptimalcontroltechniquesareusedtosynthesizenominalanddesignpointcontrollers, 79

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50 ].Thisreferencesystemchangesasafunctionofmissiontasktorepresentvaryingcontrolobjectives.Thebaselinecontrollergainsareadapteduntilthesystemoutputcloselymatchesthereferencemodelresponse.TheMRACcontrollerstructureusesafeedbackgainmatrixtorelatesensoroutputtocontrolsurfaceinputandafeedforwardmatrixtorelatereferencecommandstocontrolinput.Bothgainmatricesareadaptedinresponsetoerrorbetweentheactualanddesiredsystemresponse.Thefeedbackcomponentisidenticaltoaregulating/trackingcontroltask.LQRandrobustsynthesisareusedtoidentifytheinitialfeedbackgains,whichprovidethedesiredlevelofperformanceandrobustnessatthedesignpoints.Thefeedforwardcomponentisinitializedtorelatereferencecommandstologicalcontrolsurfaces,suchasusingaileronsprimarilyforrollcommandsandrudderforsideslipandyawcommands. 80

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83 ].H1controllershavealsobeenusedinswitchingschemes[ 44 ],wherethedynamicmodelvariesoveralargerangeandonecontrollerisusedtocontrolasubsetofthedynamicvariations.Aneectiveswitching-controllerapproach[ 44 ]switchesbetweenadjacentcongu-rationsbyusingasetofphantomcontroller.Bothadjacentcontrollersarecontinuallyreinitializedusingthestatesandactuatorpositionsofthecurrentdynamicsandcon-troller.Whenthemorphingiscommanded,theadjacentcontrollerassumescontrolofthemodieddynamicswhilepreservingcontinuityoftheactuatorcommand.Therateofmorphingisassumedtobeslowenoughthatnotransienteectsarecausedbytheshapereconguration. 5-1 and 5-2 showthesynthesismodelsforthelateralandlongitudinalcon-trollers,respectively.Thedynamicmodelandtheweightsarefunctionsofthewingshape 81

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LateralH1controllersynthesismodel LongitudinalH1controllersynthesismodel Performancesynthesisweightsarepenaltiesonthetrackingorregulatingerrorsofeachstateoutput.Theweightmagnitudeataparticularfrequencyisinverselyrelatedtotheallowableerror.Thiserrorpenaltygenerallyvarieswithfrequencysuchthatlowfrequenciesnearsteady-statearetrackedwithverysmallerrors.Athighfrequencies,beyondthepointatwhichthevehiclecanphysicallyrespond,thepenaltyisreducedsothatlargeerrorsareallowed.Thetransitionfromhightolowpenaltyoccursatdierentfrequenciesforthevariousmissions.Cruiseightinvolvesbenignmaneuversmostlynear 82

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29 ].Controllerrobust-nessdespitehighisduetoconservatismduetounrealisticconstructssuchascomplexuncertaintyandinnitelyfastsignals[ 97 ]. 5.2.2.1OverviewThesimulationsarebasedonthecomputationalmodelsgeneratedforthevariablegull-wingaircraft.Therangeofmorphingisassumedtobeidenticaltothemodelingsection,namelyinboardverticalangle1=30oandoutboardverticalangle3=30o.Bothsweepjointsremainintheneutralposition,2=4=0o.Themodelisassumedtobesingle-degreeoffreedomsincethemorphingoccursonlybetweentheidentiedcongurations,althoughtwowingjointsarticulatesimultaneously.Simulatedresponsesaregeneratedforeachmissionphase,wheretheaircraftmodelisassumedtobemorphedattheoptimumcongurationspeciedbyEquation 5{1 .Simpleinner-loopcontrolresponsesareusedtoshowthedierentclosed-loopobjectivesforeachmission.Theworthinessofagivencontrollerisdeterminedbytheabilitytotrackroll 84

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min~J=WlatkPlat(~)Plat;desiredk+WlonkPlon(~)Plon;desiredk(5{1)WhereWlatandWlonaredesignweights,Plat;desiredisareferencesystemexhibitingthedesiredlateralresponse,Plon;desiredisareferencesystemexhibitingthedesiredlon-gitudinalresponse,andPlat(~)andPlon(~)arethelateralandlongitudinaldynamics,respectively. 5{1 forthecruiseconditionwithweightsWlat=0:0andWlon=1:0areinboardangle1=10oandoutboardangle3=5o.Theultimaterequirementofcruiseightissimplytomaintainastraight-and-levelattitudeatadesiredairspeed,altitude,andheading.Theairspeedchosenwillbalancebetweenenergyeciency(endurance),range,andtime-enroute.Cruisealtitudeisalsorelatedtoeciency,butforsmallunmannedvehicles,thedesiredaltitudeismostlikelyinuencedbyground-obstacleclearanceandobservability.Finally,headingischosenprimarilytobetowardtheareaofinterest,butmaybevariedduringtheighttocircumnavigateobstacles.Levelairspeedandangleofattack,,canbeconsideredastheparametersthatmostdirectlyaectL=Dandenergyeciency,andthustheobjectiveofthecruisecontrollerwillbetopitchtoachieveandmaintainadesiredairspeedand.TheperformanceofthecontrollerintrackingapitchratedoubletisshowninFigure 5-3 -left.Thetransientresponseofthecontrollerissomewhatslow,althoughthepitchrateconvergestothedesiredvaluewithlittlesteady-stateerror.Theslowresponseisnecessitatedbythelow 85

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5-3 -leftshowstheelevatorpositionfortheboththedesiredmodelandtheactualmorphingplant.Thesedonotneedtocorrespond,sincetheyareattemptingtocontroldierentplantmodels.Thesolidlineshowstheactualelevatordeectionrequiredtoachievethepitchrateshownintheupperplot.Theactuatormovesslowlyandsmoothly,yetisabletostabilizeandtrackpitchratesatisfactorily. Figure5-3. Pitchratepulse(left)androllratepulse(right)commandsimulationforcruiseight.Linetypes:|actualresponsesandelevator/aileron,---rudder,...command Figure 5-3 -rightshowstherollrateresponseandassociatedaileronandrudderdeectionsduringarollratedoublet.Aswiththepitchresponse,therollrateachievesthedesiredvaluewithin0.25sandmaintainslittleerrorforthedurationofthepulse.Thecontrolsurfacedeectionsareshowninthelowerplot.Bothaileronandrudderareusedtotracktherollratecommand.Therudderinputresultsfromadverseyawcouplingfromtheaileronsandthepenaltyonincidentalyawinthecontrollerformulation.Thus,therudderactuatestoreducetheyawrateandsideslipuctuationsduringtherolldoublet.Therudderalsohasproverserolleects,whichassiststheaileronsinachievingthecommandedrollrate. 86

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5{1 forthemaneuverightwithweightsWlat=1:0andWlon=0:5areinboardangle1=5oandoutboardangle3=10o.Therequirementsofmaneuveringightaredissimilartothecruiseightobjectivesinthattheaircraftisassumedtobeconstantlyacceleratinginpitch,roll,orheading.Maneuversmayrequirelarge,rapidcontroldeectionstoachievehighrisetimesandgoodtrackingperformance.Amaneuveringmodemaybeappropriatefortaskssuchasfollowingatargetofinterest,avoidinganunexpectedobstacle,ormaneuveringinadenselypopulatedenvironment.Themaneuveringtargetmodelhasbeenspeciedtoemphasizefastdynamicsandlargeforceresponse.Thecontrolsynthesisweightsontheplantoutputsandactuatorsaredeterminedtoexploitthehighratesandachievefastmaneuvers.Theactuationweightpenaltieshavebeenmodiedrelativetothecruisecontrollertoallowmovementathigherfrequencies.Additionally,theperformanceweightsareadjustedsothatapenaltyisassessedontrackingerrors,evenathighfrequencies.TheresultsofthemaneuveringsimulationareshowninFigure 5-4 .Theresponsetoapitchratedoubletisshownontheleftandtheresponsetoarollratedoubletisshownontheright.Inbothcases,themorphedmodelachievesarisetimeofroughly0.15s.Therequiredactuationforsuchperformanceisnotablyfasterthanthecruiseactuation.Rapidelevatordeectionisrequiredateachstep,includingasmalldirectionreversalatthepeak.Theaileronactuationrequiredtoachievetherollperformanceisalsoquiterapid,althoughwithoutthereversalnecessaryintheelevatoractuation.Onlyasmallamountofrudderdeectionisnecessarytocompletethemaneuver.ThedierencesbetweenthelevelsofcontrolactuationinthecruiseandmaneuveringmodelsisrelatedtothedierentB-matricesforeachcondition.Themaneuveringmodelincurslesscouplingfromaileronactuationtoyawrateandsideforce,andthusrequireslesscorrectiverudder. 87

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Pitchratepulse(left)androllratepulse(right)commandsimulationformaneuveringight.Linetypes:|actualresponsesandelevator/aileron,---rudder,...command 5{1 forthesteepdescentightwithweightsWlat=1:0andWlon=1:0areinboardangle1=25oandoutboardangle3=25o.Flightconditionsathighanglesofattackcannotbeeectivelysimulatedusingvortex-latticemethodcodesorlinearmodels.Asaresult,thecontrolresultspresentedhereareforsimplemaneuversatmorphingconditionsthathavebeenshowninightteststobesuitableforsteepdescents.Theobjectivesaresimplytotrackpitchandrollcommandsfastenoughtobeusefulinmaneuveringduringasteepdiveandrecoveringtolevelight.Figure 5-5 showsthesimulatedresponseofthedive-aircrafttopitchandrollratedoublets.Theresponsesshowrelativelyfastrisetimesachievedbyreasonablelevelsofelevator,aileron,andrudderactuation.Thepitchrateresponse,shownontheleft,exhibitslittlesteadystateerror.Therollraterisetimeisslightlyfasterthanthepitchraterisetime.Bothaileronandrudderareusedtoachievetherolltracking,especiallyinthetransientphaseoftheresponse.Theuseofrudderisanindicationoftheincreasedlevelsofruddertorollmomentcouplingandsidesliptorollmomentcoupling,bothofwhichwillimprovetherollrateresponseoftherudder. 88

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Pitchratepulse(left)androllratepulse(right)commandsimulationforsteep-descentight.Linetypes:|actualresponsesandelevator/aileron,---rudder,...command 5{1 forthesensor-pointingightwithweightsWlat=1:0andWlon=0:1areinboardangle1=25oandoutboardangle3=20o.Thenalphaseofthemissionconsistsofasensor-pointingtask,wherethevehiclemustdecouplethevelocityandattitudeinordertofavorablydirecttheeldofviewofasensorwhilemaintainingafavorableightpath.Theexampleusedhereofsideslipcommandsassumesthatatargetofinteresthasmovedlaterallyintheeldofview.Theaircraftwillcommandasideslipinreturnthetargettothecenterofthesensorfootprint.Agenericunit-stepdoubletisusedtoevaluatethesidesliptracking.Figure 5-6 showsthesideslip,rollrate,andyawrateresponsestothesideslipcommand.Theaircraftisconstrainedtomaintainlowrollrateswhiletrackingsideslip.Themorphingoptimizationemphasizedsmallsidesliptorollcoupling,althoughsomeoppositeailerondeectionisnecessarytocounteractthesmallamountofcouplingremaininginthemodel.Thedesiredsideslipisachievedinjustover0.3seconds.Thesteady-stateerrorinsideslipisverysmall,indicatinggoodtracking.Somerollrateoscillationisevidentinthetransientphaseofthemaneuver,althoughitisnotexcessive.Largerudderdeections 89

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Figure5-6. Sideslippulsecommandsimulationforsensor-pointingight.Linetypes:|sideslipandaileron,---rollrateandrudder,-.-.yawrate,...command 90

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83 ]. 94 ].Figure 5-7 showsablockdiagramoftheMRACarchitectureforalateralcontroller[ 94 ].Theinputtothesystemisavectorofexternalcommands,r,foreachofthelateralstates.Areferencemodel,Pm,representsasetofdesirabledynamicswhichproducefavorablestateoutput,xm,whensubjecttotheexternalcommand,r.Thestateresponsefromthevehicledynamics,P,isgivenbyvectorx,whichisusedinfeedbacktotheadaptivecontrollertoproduceavectorofactuatorinputs,u.Thedierencebetweenthedesiredandactualresponsesisgivenbyerror,e,andisusedtodrivetheadaptationlaw.Thecontrollerparameters,^K,areadaptedinordertochangetheclosed-loopdynamicsandreducethemodel-followingerror.TheMRACstructureisextendedtoencompassaplantmodelwithdynamicsthatvarywithmorphingactuatorpositions,~.Thereferencemodeldynamicsalsochangewithmissiontask,Mi,allowingthedesiredresponsetomatchdisparatemissionobjectives.Theadaptationfunctionofthecontrollerispreserved,exceptthatanominalcontroller,Koisdesignedandknownforseveralpointsintheactuatorspace,~. 91

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Figure5-7. Modelreferenceadaptivecontrol(MRAC)architectureforlateraldynamics MRACcanreadilyaccommodatesystemswitharbitraryuncertaintyintheparam-etersoftheplantmodel.Forthesimplecasewheretheplantstructureandparametersareknownandthereferencemodelisgiven,thecontroltaskreducestogainschedul-ing.Insuchacase,thedesiredcontrollercanbedirectlycomputedbyalgebraicmatrixmanipulation.Systemswithknowndynamicsforallcongurationsareconsideredasaseparateandsomewhattrivialcaseofthemorphingcontrolproblem.Ingeneral,theaircraftisconsideredfreetochangeshapeandachievedynamicsthatarenotexactlyknowntothecontrollerduringoperation.ThegeneralformoftheMRACproblemforlateralcontrolrequirescomputingadesiredreferencemodel,estimatingthedynamicsoftheaircraft,andadaptingthecontroltoreducedisparitiesbetweenthereferenceandactualresponses.Thedynamicsofageneralmorphingsystemarerepresentedby,_x(t)=A(~)x(t)+B(~)u(t) (5{2)Bothlateralandlongitudinalopen-loopdynamicsaregiveninthisforminEquations 4{8 and 4{9 .Thestatevector,x,canrepresentthelateral,longitudinal,orcombinedstates.Controlinputvector,u,istheoutputofacontrollerfortheclosed-loopsystem.Systemmatrices,AandB,aredependentonthegeometryofthemorphingwing. 92

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5{4 showstheformofthelateralreferencemodel.ThestructureissimilartothelateralvehicledynamicsintheA-matrix.TheB-matrixrepresentsaccelerationsensitivitieswithrespecttoaseriesofexternalcommands,r,ratherthanactuatorinputs.Thedesiredstatesm,pm,andrmaredirectanalogsofactualstates.Thenumericvalueforeachofthereferencemodeltermsiscomputedusingmodalorderivativeshapingprocedures._xm=Amxm+Bmr 93

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94 ]as,_^KTx=xxeTPB 94 ]PAm+ATmP=Q 94

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47 ]foradesignpointinthecongurationspace.KxisasolutiontotheLQRcostfunction[ 34 ]andachievesdesiredperformanceatthedesignpoint,althoughitdoesnotguaranteerobustnesstoparametricuncertainty. P^Kx;irxFigure5-8. Linearquadraticregulator(LQR)controllerusedforinitializationofgainmatrix,^Kx. Equation 5{13 givestheLQRcostfunctional,whichisminimizedbyndingtheoptimalcontrolinput,u.J=Ztfto[x0(t)Qx(t)+u0(t)Ru(t)]dt+x0tfQfx(tf) (5{13)WhereQandRarepenaltymatricesaectingthestateerrorandcontrolactuationpower,respectively.Thecontrollerassumesfullstatefeedbackandisoftheform,u()=Kx() (5{14)WhereKisthecontrollergainmatrix.Thequadraticregulatorproblemiseasilyextendedtotrackingsystemsbyaugment-ingthedynamicstoincludeatrackingerrorstate.ThetrackingLQRsystemincludesafeedforwardcomponentfromtheexternalcommandandafeedbackcomponentfromthestatetrackingerror.ThestructureissimilartoMRACandisreadilyusedtogenerateinitialgainvaluesfordesignpointcongurations. 95

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64 ]areusedtoformulatethedesiredreferencedynamicmodel.Theapproximationssimplifythedesignprocessandallowfordirectsolutionofthemodelintermsofthetransientrequirementsofthemission.Rolldynamicsareapproximatedbyarst-orderfunctionrelatingrollrateandailerondeectiontorollaccelerationthroughrolldampingandaileroneectiveness,respectively.Thedampingtermresistsrollrateandcausesthemodetoconvergeformostconventionalaircraftgeometries.Rollresponsedynamicsarecharacterizedbyatime-constantoftheexponentialresponse.Asmalltime-constantindicateslargedampingandrapidconvergence,whilealargetime-constantindicatestheopposite.Therst-orderrollapproximation[ 64 ]isgivenby, 96

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64 ]removestherollratecomponentanddescribesthemotionasacoupledresponseinsideslip,,andyawrate,r.Thedutchrolldynamicsofthereferencemodelaregivenby,264_m_rm375=264Y;m 64 ], 97

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2!n;mY;m+u0Nr;m 98

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1 ].Eachtermisshapedindividuallytothemissionrequirementsandaggregatedintheequationsofmotiontoyieldthedesireddynamics.Thisprocessyieldsfavorableresultsforsimpleresponse,althoughdoesnotguaranteemodalcharacteristics.Shapingthederivativesdirectlymaycauseundesirableoscillatoryorunstableresponsestooccurinthereferencemodel.Conversely,designingthemodesdirectlymaycommandunrealisticstabilityderivatives.Asuccessfulsolutionmaybesomecombinationofbothtechniques,asinthecaseofaddinglightcouplingtothedesignedreferencemodel. 99

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60 ],whoseverticesareatthedesignpointsandwhoseinteriorsarespatiallymostsimilartothecorrespondingvertices.Figures 5-9 and 5-10 showtheidentieddesignpointsforsingleandmultipleparametercases.DesignpointsareconnectedtoformtheDelaunaytriangles.Thedesignpointgridsareshownforseveralvaluesofthresholdparameter,Tdp,increasingfromlefttoright.Alargethresholdforthesingleparametercaseeliminatestheneedforanydesignpointsapartfromthepre-seedednominalandextremalpoints.Thesingleparametercaseshowsarelativelysparsepointdistributionovermuchofthespace.Clusteredpointsindicatesareaswheretheparameterischangingrapidlyandexceedingthedierencethreshold.Themultipleparametercaseshowsamuchdensergrid.Formanyparameterswhosedependenceonthespacevariesarbitrarily,thedesignpointgridapproachesthedensityoftheoriginalcongurationspace.Insuchacase,theinterpolationoerslittlecomputationaladvantageoveralargearrayofstoreddynamicmodels. 102

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Designpointgriddistributionandtriangularsegmentationbasedonlifttodragvalue.Threshold,Tdpis5%(left),10%(center),and25%(right) Figure5-10. Designpointgriddistributionandtriangularsegmentationbasedonlateralstabilityderivatives.Threshold,Tdpis25%(left),50%(center),and200%(right) Thetriangledistributionisusedtogenerateaninterpolationschedulebetweenthearbitrarilylocateddesignpoints.Verticesofeachtriangleareco-planarorco-hyperplanarforhigherdimensions.Valuesofpointslyingintheinteriorareeasilydeterminedbyndingtheintersectionpointbetweenalinenormaltothecongurationspaceplaneandtheplaneoccupiedbythethreevertices.Figures 5-11 and 5-12 showsurfaceplotsofthetriangularlyinterpolatedparameterresponsecomparedtotheoriginalparameterresponsesurface.Themodelingerrors,p,arezeroatthedesignpointandboundedbyTdp;maxelsewhere.Thefacetingofthesurfaceisaresultofthetriangularinterpolationbetweenthedesignpoints.Improperpointspacinggeneratesfacetsthatfailtoproperlyrepresenttheparametervariations.Surfacesforthelargesingle-parametersegmentationshowdeviationsfromtheoriginaldataintheregionofnegativeAandpositiveB.Geometricartifacts,suchasfalsepeaksandvalleys,stemfromtheinabilitytorecreateacomplexsurface 103

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Interpolatedvaluesforlifttodragbasedonsingleparametersegmentation.Threshold,Tdpis5%(left),10%(center),and25%(right) fromsparsepointspacing.Theerrorsarethusmuchlargerforlargevaluesofthreshold,Tdp.Thebalancebetweenacceptableerrorsandcomputationaltractabilityisultimatelydeterminedwiththedesignofthisthreshold. Figure5-12. Interpolatedvaluesforlifttodragbasedonmultiple,simultaneousparametersegmentation.Threshold,Tdpis25%(left),50%(center),and200%(right) Narrowthresholdofthemultipleparametersegmentationachievesaccuratesurfacereproduction.Thenumberofrequireddesignpointsmaybeprohibitivelylarge,especiallywithlimitederrortolerance.Thesimultaneousvariationofnearly30parameters,eachhavingauniquedependenceonthemorphing,createsasignicantchallengeinsegmentingthecongurationspacebasedoninterpolatederrors.Selectiveuseofparametersmayoerimprovementsinstorageandcomputationrequirementsbyemphasizingthetermswhichhavethelargestaectonthestabilityandperformance. 104

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5-13 showsthepermissibletrajectoriesthroughthe4-dimensionalcongurationspaceforthreemorphingoperations.Thetrajectoriespermitmorphingbetweencongu-rationsoptimizedforcruise,maneuverability,andagility.1,2,and3arerepresentedspatiallywithprojectionsshownonthespaceboundariesforclarity.4isrepresentedincolorspaceashueandisindicatedbyacoloredmarkerateachintermediatetrajec-tory.Dynamiccriteriaisappliedinthecomputationoftheactuatorpathsandyieldsintermediatecongurationswithsatisfactorystabilitycharacteristics. Figure5-13. Permissible4Dactuatortrajectorybetweenthreedisparatecongurations Figure 5-14 showsasetofsimulationresultsforamorphingsystemtrackingtheresponseofaxedreferencesystem.Historiesforstates,p,andrareshownintheleftplotsforthemorphingaircraft(top)andthereferencemodel(bottom).Alow-frequency,small-amplituderollratesinusoidisusedasareferenceinput,whichcausessomecouplingtotheremaininglateralstatesduetoo-diagonaltermsinthereferencemodelcontrol-eectivenessmatrix. 105

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Controlresultsduringcruisetomaneuveringrecongurationwithasinusoidaltrajectorycommand.A)Angularrateresponseofmorphingandreferencemodelsystems.B)Gainadaptationandtrackingerrors Themorphingaircrafthasaninitialresponsethatgeneratessmallerrorrelativetothereferenceresponse.Gainsadaptslightlyuntiltherstmorphingincrementiscommandedatt=13:2seconds.Themorphingiscommandeddiscontinuously,wherethedynamicsareswitchedinadiscretestepbetweentimeintervals.Therelativelylarge5oangularresolutionofthemorphingjointscontributestoalargechangeinthedynamicsandcontributestotrackingerrorsasthedynamicsswitch.Discretemorphingeventsareindicatedbyverticaldottedlinesandoccurevery4secondsuntiltheaircrafthasmorphedintothemaneuverableconguration.Shapechangecontributestovariationsintheclosedloopsystemandcausestrackingerrors.Theerrorsdrivegainadaptationusingthegradientdescentmethod,whichsubsequentlyimprovestrackingperformance.Gainscontinueadaptinguntiltheaircraftreachesthenalshape.Figure 5-15 showssimulationresultsforasimilarmorphingtrajectoryandconstantreferencesystemsubjecttoavariable-frequencyrollratecommand.Gainadaptationoccursrapidlyfollowingmorphingoperations,butappliesselectivelytocertaingains.Themorphingmodelachievesgoodtrackingperformancethroughouttheshape-change,althoughincurssomecyclicalerrorinthenalconguration.Theerrorcontributestooscillatorygainadaptation.Suchundesirablebehaviormaybereducedwiththeapplicationofdeadbandmodicationtotheadaptationlaws[ 94 ]. 106

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Controlresultsduringcruisetomaneuveringrecongurationwithavariable-frequencytrajectorycommand.A)Angularrateresponseofmorphingandreferencemodelsystems.B)Gainadaptationandtrackingerrors Theadaptivemorphingcontrollercantoleratelargeinitialerrorsinthebaselinecontrollerandlargevariationsinthedynamics.Figure 5-16 showstimehistoriesforstate,error,andgaintrajectoriesforamodelwithapoorinitialcontrollerdesign.Initialre-sponseincurslargeerrorsarerapidadaptationinthecontrollergains.Therstmorphingoperationatt=13:2sinvolvesalargediscontinuitywithjointactuationsexceeding5o.Thegainsadaptfurthertostabilizethenewdynamicsandreducethelargetransienterror.Thecontrollerachievesreasonabletrackingperformanceforsubsequentmorphingandxedcongurations.Althoughthecontrollersuccessfullystabilizesamodelwithsevereinitialerrors,thisstrategyisnotrecommendedforuseinaighttestvehicle.Poorbaselinecontrollersandrapiddynamicvariationsbothcausetrackingerrorsthatrequireseveralsecondsofadaptationtosubside.Thecriticalityofthisadaptationtimedependsgreatlyonthevehicleandightcondition.Adaptationtimesgreaterthanseveralsecondsmayleadtolossofcontrolorunintendedinteractionwithsurroundingobstacles.Thesuccessofamorphingaircraftintrackingaxedreferencesystemdemonstratestheversatilityofthecontrolsystem.Theconstantreference,however,mayviolatedisparatemissionconstraintsandreducetheeectivenessandfunctionofmorphing.A 107

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Controlresultsduringcruisetomaneuveringrecongurationwithasinusoidaltrajectorycommandandpoorinitialcontroldesign.A)Angularrateresponseofmorphingandreferencemodelsystems.B)Gainadaptationandtrackingerrors morerealisticapproachusesamission-dependentreferencemodelinordertoemphasizeeitherconservatismorperformance.Figure 5-17 showstheresponsesofanaircraftmorphingfromacruisecongurationtoamaneuveringconguration.Thereferencemodelinitiallyusesalongtimeconstantinboththerollanddutchrollmodes.Att=30s,thereferencemodelswitchesdiscretelytoshortertimeconstantsinbothmodesandslightlyhigherdampinginthedutchrollmode.Thechangeinthereferencemodeloccursinnearthemidpointofthemorphingoperationbetweendiscreteshapechanges.Averticaldashedlineindicatesthereferencemodelchange,whichisaccompaniedbytrackingerrorsandrapidgainadaptation.Thecontrollersummarilyadaptstothehigher-performancemodelandachievesgoodtrackingresults.Controllergainscontinuetoadaptwitheachsubsequentmorphingandachieverelativelyconstantvaluesforthemaneuveringconguration.Thetransientbehaviorobservedduringthereferencemodelswitchingispartiallyaresultofdiscontinuitiesintheaileronandrudderdeections.Aratelterontheaircraftcontrolsandanimprovedswitchingtechniquewouldpreventlargeerrorsfromincitingrapidcontrolandstateresponses.Despitetheerror,thesystemrecoversquicklyandstabilizestheaircraft. 108

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Controlresultsduringcruisetomaneuveringrecongurationwithasinusoidaltrajectorycommand.A)Angularrateresponseofmorphingandvariablereferencemodelsystems.B)Gainadaptationandtrackingerrors Themultiple-referencemodel,morphingsystemissubjecttoachirpcommand,whichisasinusoidwhosefrequencyincreaseswithtime.Thechirpsimulatesbenignmotionsofthecruiseconditioninitiallyanddevelopsintorapidmotionsformaneuveringight.Figure 5-18 showssimulatedcontrolresultsforthesystem.Rollratecommandisslowlyincreased,whichallowsthemorphingaircrafttoeasilyachievetrackingeveninthepresenceofcontrollererrors.Thegentlecommandviolatesthepersistenceofexcitationcondtion[ 50 ][ 94 ]andfailstoadaptthecontrollergains.Morphingchangesgeneratetrackingerrorsduetotheswitchdynamicsandinvokegainadaptation.Theadaptationrateincreasessomewhatwiththeincreasingfrequencyofthecommandedroll.Trackingerrorsoccurduringthereferencemodelswitching,causingthegainstoadaptrapidlyinresponse.Adaptationcontinuesastheaircraftcompletesthemorphingoperationandtracksthechirpcommandinthemaneuveringconguration.Closed-loopperformanceisgoodthroughoutthesimulation,apartfromtheerrorexcursionduringthereferencemodelswitchandcyclicalerrorsinthehigh-frequencyratetracking. 109

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Controlresultsduringcruisetomaneuveringrecongurationwithavariable-frequencytrajectorycommand.A)Angularrateresponseofmorphingandvariablereferencemodelsystems.B)Gainadaptationandtrackingerrors 110

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41 ][ 49 ],theoptimalshapevariationthrough-outoneormoremissions,ortherequiredinner-loopcompensationforthechangingvehicledynamics.Solvingtheoptimalcontrolproblembecomesprogressivelymorecomplicatedasthenumberofinputsandstatesincreases.Thischapterconsidersseveralvariationsoftheproblem,wheretheoptimalcontrolisusedeitherinconjunctionwiththedevelopmentsfrompreviouschaptersorinplaceofthem.Forinstance,thesectiononratetrajectorygenerationstrictlycommandstheinner-loopratestoperformpartofamission-taskmaneuver.Theinner-loopstabilizationandtrackingisachievedbyanindependentcontrollerwhilethevehicleshapeisadaptedbyanindependentadaptationlaw.Adaptationcanbeincludedintheproblemsuchthatthetrajectoryandmorphingshapearesolvedsimultaneously.Insuchacase,thetrajectoriesaremodiedaccordingtotheperformancecapabilitiesofthepermissibleshape.Maneuveringcontrollersareusedindependentlytostabilizetheshape-changingvehicleandtracktrajectorycommands.Completeconsiderationoftheproblemcommandsstates,morphing,andcontrolsurfacessimultaneously.Theuniedframeworkforpathplanning,shapeadaptation,andinner-loopcontrolcanachievetheoptimalsolutionforaparticularsetofcostfunctions,althoughatconsiderablecomputationalexpense.Theperformanceoftheuniedcontroller 111

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73 ]is,J=(x(t0);t0;x(tf);tf)+Ztft0L[x(t);u(t);t]dt 112

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(6{2)Therateofmorphingislimitedtoisolatetheadaptationtime-scalesfromtheaircraftandcontrollerdynamics.Ratelimitsonthemorphingactuationareestablishedusingadesignthreshold,T_~.Theallowablevalueofthisthresholdisdeterminedbyiterativesimulation.Adaptabilityconstraintsarenecessarytopreservethequasi-staticassumptionsusedinformulatingthedynamicsandcontrolstrategies._~T_~ (6{4)Thetrajectoriesaresubjecttopathconstraintsg(x(t);u(t);~(t);t)0 (6{5) 113

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_~ exm+-rInner-LoopRateFilterTrajectoryGenericPerformanceModel-+x[;;]TPm(Mi)^Ko(~)P(~)_^K[V;;R]TCongurationAdaptationOptimal[Vc;c;Rc]T Optimalratetrajectoryframeworkwithindependentightcontrolandshapeadaptationprocesses Actitiousdynamicmodelisusedintheformulationoftheoptimaltrajectory.Themodelachievesmaximumperformanceinallperformanceandmaneuveringmetrics.Althoughknowinglyunrealisticforasingleconguration,themodelpermitsanaggressivetrajectorythatisachievableforanaircraftthatchangesshapeappropriately.Trackingerrorsresultingfromcongurationsunabletofollowthetrajectorydriveadaptationandallowsubsequentimprovementsintrackingperformance.Theoptimalcontrolsolvesfortrajectoriesofangleofattack,sideslip,androllangletoproduceaccelerationsnecessarytofollowthedesiredightpath.Verticalandlateralforcesareusedtodeterminetheightpathbasedona3degree-of-freedommodel[ 78 ]giveninEquations 6{6 6{7 ,and 6{8 114

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(6{6)_=(1=V)[X(cossinsin+sincos)Y(cossin)+Z(sinsinsincoscos)gcos] (6{7)_=(1=Vcos)+[X(cossincossinsin)+Y(coscos)Z(sinsincos+cossin)] (6{8)AccelerationsX,Y,andZaregeneratedfromthectitiousperformancemodelthroughvariationsofangleofattackandangleofsideslip.Thestabilitycoecientsusedtorelateightanglesandforcesarethemaximumperformancevaluesamongallcongurations.Thecommandedtrajectoriesforrollangle,angleofattack,andsideslipareusedtogenerateinner-loopcommandstorollrate,pitchrate,andyawrate.Aratelteractsasonouter-loopanduseslineargainstocommandeachoftheratesfromangulartrackingerrors.Saturationisimplementedintheltertolimitthemagnitudeofthecommand.Abaselinecontrollertracksinner-loopratecommandswithreferencetoamission-specicdynamicmodel.Thebaselinecontrollerisscheduledonmorphingconguration,whichisexternallyadapted.Thecontrollergainsadapttoreduceerrorsbetweenthereferencemodelresponseandthecurrentmodelresponse.Controldeectionlimitsboundsthevehicleresponseratesandwillintroduceightpatherrorswhenthemorphingcongurationisunsuitableforthedesiredmission.Trackingerrorsinairspeed,ightpathangleandturnradiusdrivetheexternaladaptationmechanism,whichseekscongurationsthatachievehighermissionperformance. 115

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6-2 showsthearchitectureofanoptimalcontrollerforratetrajectoryandshapeadaptation.Theexternalmorphingcommandblockisreplacedwithanexpandedoptimalsystemthatadaptsthewingshapedirectly.Theinner-loopratelteroperatesasbefore,operatingontheightangleoutputoftheoptimalcontrollerandgeneratinglimitedratecommandstotheaircraft.Theratecommandsareusedintheadaptive,model-referencedesigntogeneratethedesiredsystemresponse.Thecontrollergainssubsequentlyadapttoreduceerrorsinthetrackingperformanceofthereconguredaircraft. Optimalratetrajectoryandshapeadaptationframeworkwithindependentightcontrol 117

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7-1 showsthesimulatedcongurationforpilotedtakeoandlanding.Theshapehasarelativelyhighlift-to-dragratioof14.6inadditiontogoodhandlingqualitiesforbothlateralandlongitudinalmodes.Thewinghasincreasingaftsweeptowardthetip,with5oontheinboardsectionand10oontheoutboardsection.Itusesaninvertedgull-wingshapewith5oanhedralontheinboardsectionand10odihedralontheoutboardsection. Manuallypiloteddeploymentandrecoveryconguration 7{1 [ 7 ].Equation 120

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showstheBreguetRangeformula,whichgivesanestimatefortherangebasedonaircraftaerodynamic,propulsion,andstructuralcharacteristics[ 40 ]. D=1V12CD;0 1V12W S1(7{1)WhereLisliftingforceorcoecient,Disthedragforceorcoecient,Visthevelocity,CD;0isthezero-liftdragcoecient,Wistheweight,Sisthewingarea,1isthedynamicpressure,andKisthedrag-polarcoecient.Range=V1L DIsplnWinitial 7-2(b) showsthecongurationthatachievesthemaximumlift-to-dragratiousingthestabledynamicscriteria.Thewingusesspanwise-increasinganhedralandslightaftsweeptoproduceanopen-downwardcrescentshape.Thecongurationcloselyresemblesthatofbirdssuchasalbatrosses,pelicans,orseagullsglidingforextendedperiodsorclosetothewatersurface.Figure 7-2(a) showsaseagullglidingforanextendedlengthingroundeect.ItisalsosimilartotheNASALangleyHyper-EllipticalCamberedSpanwing,whichwasreportedtoachievea15%greaterlift-to-dragratiothanasimilarlyscaledconventionalwing[ 22 ].Thecongurationwhichachievesthemostdistance-ecientcruisealsoservesanimportantfunctionforglidingoperations.Bothmaximumrangeandshallowestangleofglideareachievedbymaximizingthelifttodragratio.Themetricbywhichglideperformanceisgaugedistypicallyglideratio,whichcomparesthehorizontaldistancetraveledtothealtitudelossduringaportionoftheglide.Theglideratioisequalto 121

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(b)SDC:1=15o,2=5o,3=25o,4=0oFigure7-2. Highliftseagullwingshape(left),Maximumlifttodragconguration(right) thelifttodragratioduringunacceleratedglidingight,soanaerodynamicallyecientcongurationallowsthevehicletoglideforalongdistance.Glidingightisnotnormallyinvokedduringeithermannedorunmannedoperations,althoughitmaybecomeusefulforexploitingatmosphericcurrentstoimprovetherangeandenduranceofaUAV[ 6 ].Researchersarecurrentlyevaluatingtheperformanceofanautonomousgliderwhichseeksrisingcurrentsofaironwhichtostayaloft.Extendedightdurationsmaybepossiblebyalternatingbetweenpoweredandglidingmodes.Glidesarefrequentlyusedbylargerbirdstoywithoutapping.Boththermalanddeectedwindsareexploitedbybirdsofpreyandsea-bornebirdsassourcesofenergyforextendedperiodsofglidingight[ 91 ].TherangecongurationandotherscanbeusedfordierentphasesofglidingightforaUAV. 122

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7{3 [ 40 ]showsanestimateforpowerrequiredinlevelightasafunctionofaerodynamiccharacteristicsandairframegeometry.Thecongurationwhichachievestheminimumpowerrequiredisalsothehighestenduranceandmostenergy-ecient.Preq=1 2V3SCD0+W2 1 2VS1 7-3(a) showsthecongurationforendurancecruiseandminimumunpoweredrateofdescent.Theshapeisverysimilartothenominalcongurationapartfrom10odihedralonthewingtips.Thenon-uniformdihedralissimilartocompetitionsailplanewings,whicharealsooptimizedforminimumpowerrequiredinglidingightwithhandlingqualitiesconstraints[ 52 ].Thecongurationrequires2.503wattstomaintainight. Maximumenduranceandminimumrateofdescentconguration 123

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7-3(b) .Therelaxedcriteriarequire2.497wattsforlevelight,marginallylesspowerthantheshapegeneratedbasedonhandlingqualities.Theimprovementisnotsignicantandcomesattheexpenseofpilotability.Foranautomaticallycontrolledaircraft,however,thisissueisnotanimportantconsideration.Theminimum-power-requiredcongurationisalsothebestshapeformaximizingglideduration.Anunpoweredaircraftachievesequilibriumbyusingthelossofpotentialenergytoopposethepowerrequiredtomaintainight[ 9 ].Minimizingthepowerrequiredthusminimizesthelossofpotentialenergyandminimizestherateofdescent.Thelowestrateofdescentcongurationallowsanunpoweredaircrafttostayairborneforthelongesttime.Theconditionisusefulforexploitingatmosphericcurrentsasenergysourcesorincreasingthedecisiontimeforcontingenciesintheeventofpowerplantfailure.Thecongurationaremostfrequentlyseeninbirdsusingthermalstoloiteroveranareainglidingight. 7{4 showstheradiusofturnisreducedforlowairspeeds,largeliftingforces,andsteepanglesofbank.Equation 7{5 showsthatwingloading,W=S,anddragmustbelowtoachieveasmallradiusturn.Thrustforce,T,mustbealsobemaximizedtocompensatefortheincreaseddragastheaircraftbankstoperformaturn.Thrustisconstantforallsimulationstofocusontheroleofmorphingineectingtheightperformance. 124

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7-3(a) .Theweightandthrustforcesareassumedconstantforallcongurations.Thelargewingplanformareaachievesalowwingloadingwhilethedihedralwingtipspre-servedesirablelateraldynamicresponse.Thestabledynamicscongurationissimplythenominalwingshapewithalljointangleszero.Maximumwingareaisachievedwithoutmorphing,producingthelargestliftmagnitude.Minimumturnradiiareassessedatthesimulatedairspeedandthusthecongurationsareconsideredbasedonthesimplieddragestimatesandtheavailablewingarea. 7-4(a) showsonepossiblecongurationforasteepdescentmode.Theve-hicleusesagull-wingshapewithlargedihedralontheinboardandlargeanhedralon 125

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68 ].Handlingqualitiescriteriaisusedintheselectionofthewingshape. Steepdescentangleconguration Theaerodynamicallyinecientsteepdescentmodeisfavoredoverasimplediveduetoairspeedconsiderations.Acongurationoptimizedforcruisewillsignicantlygainairspeedduringadive,asopposedtotheslowdescentofhigh-draggeometry.Theslowdescentofthecurrentcongurationalsofacilitatesrecoverytolevelightattheterminationofthedive.Analternateformofdivingisthemaximumrateofdescentmode,whichachievesachangeofelevationinminimumtime.Themaneuverisappropriatefortime-criticaldescentswherehorizontalightspaceisaorded.Thebestcongurationforrateofdescentoccursatthemaximumpowerrequiredcondition,whichisoppositetotherequirementforsoaringight.Thelargepowerrequirementfortrimmedightisprovidedbythelossofpotentialenergyduringthedescent.Whenthepotentialenergylossismaximizedforaxedairspeed,therateofdescentisalsomaximized. 126

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7-4(a) thusshowsthegeometrythatisoptimalforbothdescentangleanddescentrate.Thisresultisexpectedgiventheconstraintonxedairspeed.ThecongurationthatachievesthesteepestdescentangleandmaximumrateofdescentusingthestabledynamicscriteriaisshowninFigure 7-4(b) .Thewingusesthemaximumdihedralangleforbothinboardandoutboardwingsalongwithmaximumaftsweepfortheinboardandmoderateaftsweepfortheoutboardwing.Thewingshapeissimilartotheformusedbyhomingpigeons,asinFigure 3-9 duringthesteepdescentphaseprecedinglanding.Thecongurationgeneratedusingtheunstabledynamicscriteriaissimilartothestableshape,exceptthattheoutboardwingusesthemaximumaftsweep. 127

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7{6 showsasimplecostfunctiontondthecongurationwhichminimizesthecombined,squaredstinessandcoupledderivatives.min1;2;3;4J=C2n+C2l 7-5(a) showsthecongurationwhichachievestheminimumcost,J,withgoodhandlingqualities.Theshapeisnon-conventionalbybothbiologicalandaviationstandardsinthattheinboardwingsaresweptaftwhiletheoutboardwingsaresweptforward.Thewingalsousesagull-wingcongurationwithlargedihedralontheinboardandmoderateanhedralontheoutboard.Theunusualorientationoftheoutboardwingsareexpectedtocontributetothelargesideslipconstraint.Anhedralwingtipsreducethevehicletendencytoproducearollmomentinresponsetosideslip[ 64 ]whiletheforwardsweepreducesthedirectionalstiness,Cn[ 80 ][ 9 ].Theoppositeattitudeoftheinboardwingsareusedtomaintainanappropriatepositionoftheaerodynamiccenterrelativetothevehiclebodyforthedesireddynamicresponse.Relaxingthestabilitycriteriatoallowunstabledynamicsproducesaqualitativelysimilarshapeexceptthatthewingtipsarenotangleddownwardandaresweptforward.Figure 7-5(b) showstheresultingwingconguration.ThecostfortheunstablecriteriaisJ=0:0005whereasforthehandlingqualitiescriteriaJ=0:0263.Inbothcases,thedesiredsideslipcharacteristicsaremostlyachieved.TheunstablecongurationproducesamoderatelydivergentspiralmodeandahighlydivergentshortperiodmodewithatimetodoubleofT2=0:5seconds.Thecongurationdeterminedbythestabledynamicscriteria 128

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Maximumtrimmablesideslip issimilartotheunstableshapewithjointanglesof1=30o,2=15o,3=5oand,4=30o.Interestingly,averyconventionalwingcongurationisfoundwhenthecostfunctioninEquation 7{6 ismaximized.Figure 7-6 showsthewingshapethatachievesthehighestvalueofC2l+C2nissimilartotheswept-backdesignofairlinersandotheraircraftthatseektoreducesideslipdivergence,amongmanyotherfactors. Maximizedsideslipcostfunctionconguration 129

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7-1 comparesthemaximumsideslipachievedbyfouraircraft.Themovementoftheaileron,elevator,andruddercontrolsurfacesareconstrainedtotrimtozerotherolling,pitching,andyawingmoments,respectively.Thesideslipisincreaseduntilthelimitdeectionisachievedinoneofthesurfaces. ParameterABCD Sideslip(max)22:6o36:1o45:0o11:5oAileron(a)15:0o15:0o5:05o15:0oElevator(e)6:30o7:8o7:98o18:4oRudder(r)11:1o8:36o7:57o3:50o Maximumsideslipandcontroldeectionsforseveralaircraftcongurations.A)HQCAircraft-Fig. 7-1 B)HQCmin~J-Fig. 7-5(a) C)SDCmin~J-Fig. 7-5(b) D)HQCmax~J-Fig. 7-6 Thenominalcongurationusedfortakeoislistedintherstaircraftcolumn.Theoutboardwingshaveshallowdihedralandtheaircraftachievesamoderatevalueforthecostfunction,J.Amaximumsideslipangleof22:6oisachievedwithmaximumrightaileronand74%leftrudderdeection.SimilaraileronandrudderdeectionsstabilizetheHQC-sideslipairplaneina36:1osideslip.Bothaircraftexhibitdesirabledynamiccharacteristics,yettheoptimizedaircraftachievessignicantlylargersideslipangles,aordingawiderrangeofsensorpointing.TheSDC-sideslipaircraftachievesthesideslipanglelimitof45owithconservativecontrolsurfacedeections.Theaircraftisabletotrimatyetlargerangles,althoughsustainedightinsuchattitudesmaybeunrealisticwithoutunconventionalpropulsionordescendingighttocounteractthelargesideforcedrag.Thejet-likewingplanformincontrastachievesamaximumsideslipof11:5owiththeaileronsaturatedtotherightandtherudderdeectedleftmildly.Theelevatorexceedsallowabledeectionforallsidesliprangesinordertostabilizethewinginpitchduetotheaftneutralpointposition.Thelargeelevatordeectionunderscoresanimportantaspect 130

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7-2 showstwoexamplemissionsforaircraftownwithwithdierentstabilitycriteriaregulatingthemorphingcongurations.Eachaircraftisoptimizedwithinthecongurationspacetoachievebestvaluefortheeachmaneuvermetric.Thehandling-qualitiescriteriarestrictsthesolutionstothosethatachievegoodhandlingqualities,whilethestabledynamicscriteriaislessrestrictiveandallowsanyopen-loopstableconguration. MissionTaskMetricNominalValueHQC(%Change)SDC(%Change) LaunchL Dmax14.614.6(0%)14.6(0%)CruiseL Dmax14.614.6(0%)15.1(3.21%)LoiterPreq;min2.52W2.50W(0.8%)2.50W(1.0%)TurnRmin17.8m17.1m(4.0%)16.9m(5.2%)DescentL Dmin14.611.7(19.6%)11.3(22.8%)Sideslipmax;trim22:6o36:1o(59.7%)45o(99.1%)RecoveryL Dmax14.614.6(0%)14.6(0%) Improvement||12.0%18.8% Table7-2. Missiontasksandassociatedperformancemetricsrelativetolaunchconguration 131

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7{7 showsthedesiredmorphingdirectionincongurationspaceissimplyadirectlinefromthecurrentjointanglestothedesiredjointangles.Thedesireddirectionisnormalizedbythedistancebetweentheinitialandnalshapes. 7{8 ,whichdeterminesthedesirabilityoftherecongurationcommand,,basedonthesimilaritytothedesiredmorphingdirection,~desired. 133

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7-7 showsthefourjointangletrajectoriesastheaircraftmorphsbetweenmissiontaskshapes.Eachmissiontaskshapeisidentiedbyaverticaldashedlineandatextannotation.Thedurationofthemissiontaskisnotrepresentedintheplot.Instead,closedcirclesshowtheoptimizedmorphingshapeswhichareconstantforeachmaneuver.Theinitialcongurationshownattimeincrement1correspondstothemanuallypilotedlaunchshapefromFigure 7-1 .TheaircraftisthenmorphedtotheanhedralcruisecongurationfromFigure 7-2(b) bychangingalljointanglesapartfrom2,whichremainsconstantat5o. Figure7-7. MorphingjointangletrajectoriesformissiontaskssubjecttoSDC Thefollowingtwomissions,enduranceloiterandminimumradiusturn,usejointcongurationscloseorequaltotheunmorphedshapefromFigure 7-3 .Thenextmissionusesthelargedihedral,aftsweptwingshape(Figure 7-4(b) )toperformsteepdescentsandscattersthejointanglestotheextremitiesoftheplot.Thevehiclethentransformstothesensor-pointingcongurationfromFigure 7-5 byincreasing2and4whilereducing3. 134

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7-8 showsasimilarproleforamissionusingHQCtolimitbothmissiontaskandintermediatecongurations.Therestrictedcongurationsetcausesseveralmaneuverstouseidenticalwingshapes.Launchandcruisearebothoptimizedforlift-to-dragratiosobothusethecongurationfromFigure 7-1 .Theaircraftthenmorphsintotheshapeusedforbothenduranceloiteringandminimumradiusturn.Slightdihedralisusedontheoutboardsectionwhiletheremaininganglesarezero,asinFigure 7-3(a) Figure7-8. MorphingjointangletrajectoriesformissiontaskssubjecttoHQC Steepdescentisperformedbymorphingintotheseagull-inspiredgull-wingshapeof 7-4(a) .Thejointtrajectoriesfollownearlydirectpathsforallfourangles.Morphingintothesensor-pointingshapeofFigure 7-5(a) usesadirectpathfor4andstaggeredpathsfortheremainingjoints.Thenalshapechangetransformsthevehiclebackintothelaunchcongurationforamanuallypilotedrecovery. 135

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MujahidAbdulrahimwasbornandraisedamongtherockymountainsofCalgary,Alberta,Canada.Heproudlydescendsfromalineofeducators,nobles,andsmall-timeinventorsinhisethnicSyria,fromthecitiesofAleppo(Halab)andAl-Rahaab.Hisprofessionalmaturationbeganinthe2ndgradewithhisnotabledevelopmentofaMicroMachinesstunttrackfromconstructionpaper.Hecontinuedhisinterestinallthingsmechanicalduringhiscross-continentmovetoPanamaCity,FL,wherehermlydecidedhiscareerpathasanaeronauticalengineerbyreferencingan8th-grademath-classposter.Theposterlistedvariousprofessionsontheverticalaxiswiththecorrespondingmathcourserequirementonthehorizontalaxis.Aeronauticalengineeringwasoneofthefewcareersthatrequiredvirtuallyallthelistedmathematics.SincejoiningtheUniversityofFloridain1999,Mujahidhasbeenactiveinmicroairvehicle,dynamicsandcontrol,andmorphingresearch.Hehasalsoparticipatedinnumerousacademicandsportingcompetitionsincludingregionalandnationalpapercompetitions,MAVs,cross-countrybicycling,photography,autocross,drifting,andaerobatics.Hissuccessvaried.MujahidearnedhisaerospaceengineeringB.S.in2003,hisM.S.in2004,andhisPh.D.in2007.Noting,withsomesadness,thattheUniversityofFloridaoerednofurtherrelevantdegrees,Mujahidpromptlypackedhisbelongingsandheadedwest.Atthetimeofthiswriting,heisdrivinginanRV,lostonawindingmountainroadsomewhereinArizona.Someday,hehopestogetnewbatteriesforhisGPSandcontinuehistriptoCalifornia,wherehewillstartR&DworkinUAVcontrols. 147