University of Southern California Dissertations and Theses

The effect of Xᵤ and T_{theta_2} on autonomous in-flight refueling: a global Hawk RQ-4A approach

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The effect of Xᵤ and T_{theta_2} on autonomous in-flight refueling: a global Hawk RQ-4A approach

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Content THE EFFECT OFX
U
ANDT

2
ON AUTONOMOUS IN-FLIGHT REFUELING:
A GLOBAL HAWK RQ-4A APPROACH
by
Endri Kerci
A Thesis Presented to the
FACULTY OF THE USC VITERBI SCHOOL OF ENGINEERING
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
MASTER OF SCIENCE
(AEROSPACE ENGINEERING)
August 2013
Copyright 2013 Endri Kerci
Tomygrandparents:
Dhimiter&UraniK¨ erc ¸iand,Marko&SijeStefa.
ii
Acknowledgments
First and foremost I would like to thank my advisor, Professor Jerry Lockenour, for
without his help, encouragement, advise, and expertise this project would not have been
possible. Professor Lockneour your kindness, patience, and knowledge helped me greatly
throughout the completion of my thesis and for that I thank you. Thank you!
Second I would like to thank Pablo Gonzalez, for without his industry experience I
would not have been able to confirm the validity of this thesis. Pablo your support and
willingness to help has been unsurmountable. Thank you!
Third I would like to thank Phil Stern, for his industry knowledge helped verify the
moments of inertia estimated in this thesis. Thank You!
Next I would like to thank my family for their unconditional support and admiration.
You are the foundation for which my success is based on and your hard work and ethics
have certainly helped shape mine: Harilla, Tefta, and Artemisa Kerci. Thank you!
Next I would like to thank the thesis committee for their willingness and patience to
help review this thesis: Dr. Bingen Yang, Dr. Petros Ioannou, and Dr. Charles Radovich.
Thank you!
Finally I would like to acknowledge any one person that has had input on the thesis,
be it conversations related and or otherwise, Dr. Giampiero Campa and Dr. Mark Haynes.
Thank you!
iii
Table of Contents
Dedication ii
Acknowledgments iii
List of Tables vi
List of Figures vii
Nomenclature xiii
Abstract xiv
Chapter 1: Introduction 1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Thesis Overview and Approach . . . . . . . . . . . . . . . . . . . . . . . . 2
Chapter 2: Equations of Motion 8
2.1 Nonlinear Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Linearized Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Aerodynamic Forces and Moments . . . . . . . . . . . . . . . . . . . . . . 13
2.3.1 F andM due to change inu . . . . . . . . . . . . . . . . . . . . . 13
2.3.2 F andM due to change inv;w;p;q;r; _ u; _ w . . . . . . . . . . . . . 15
2.4 State Space Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Chapter 3: Unaugmented Aircraft 20
3.1 Moments of Inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.2 Aerodynamic Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.3 Unaugmented Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3.1 Longitudinal Response . . . . . . . . . . . . . . . . . . . . . . . . 26
3.3.2 Lateral Response . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Chapter 4: Feedback Control 31
4.1 Tanker Longitudinal Feedback Control . . . . . . . . . . . . . . . . . . . . 31
4.2 Tanker Lateral Feedback Control . . . . . . . . . . . . . . . . . . . . . . . 39
4.3 Receiver Longitudinal and Lateral Feedback Control . . . . . . . . . . . . 43
iv
Chapter 5: Results 46
5.1 X
u
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.2 T

2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
Chapter 6: Conclusion and Future Work 68
6.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
6.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
Bibliography 70
Appendices 72
Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
Appendix B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
v
List of Tables
3.1 Global Hawk RQ-4A Percentage Mass Breakdown . . . . . . . . . . . . . 21
3.2 Global Hawk RQ-4A Moments of Inertia . . . . . . . . . . . . . . . . . . 21
3.3 Global Hawk RQ-4A: Tanker and Receiver Aerodynamic Coefficients . . . 25
3.4 Tanker Longitudinal Stability: Eigenvalues, Damping, and Frequency . . . 26
3.5 Tanker Longitudinal Stability Revised: Eigenvalues, Damping, and Fre-
quency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.6 Receiver Longitudinal Stability Revised: Eigenvalues, Damping, and Fre-
quency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.7 Tanker Lateral Stability: Eigenvalues, Damping, and Frequency . . . . . . 29
3.8 Receiver Lateral Stability: Eigenvalues, Damping, and Frequency . . . . . 29
vi
List of Figures
1.1 Receiver and Tanker Formation . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Altitude Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Longitudinal Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Y Error and Yaw Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.5 Controller Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1 Aircraft body axis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.1 Global Hawk RQ-4A SolidWorks Model . . . . . . . . . . . . . . . . . . . 21
3.2 Global Hawk RQ-4A Datcom Model . . . . . . . . . . . . . . . . . . . . . 24
3.3 Global Hawk RQ-4AC
m
vs.C
L
Curve. Tanker: left, Receiver: Right . . . 24
4.1

e
Unaugmented Root Locus and Step Response . . . . . . . . . . . . . . . 31
4.2
ref
Augmented Loop Closure . . . . . . . . . . . . . . . . . . . . . . . . 32
4.3
ref
Augmented Bode Plot and Root Locus Plot . . . . . . . . . . . . . . . 32
4.4
ref
Augmented Step Response . . . . . . . . . . . . . . . . . . . . . . . . 33
4.5 Altitude Loop Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.6 Altitude Loop Closure Bode and Root Locus Plot . . . . . . . . . . . . . . 34
4.7 Altitude Loop Closure Step Response . . . . . . . . . . . . . . . . . . . . 34
4.8 Velocity Loop Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.9 Velocity Loop Closure Bode and Root Locus Plot . . . . . . . . . . . . . . 35
4.10 Velocity Loop Closure Step Response . . . . . . . . . . . . . . . . . . . . 36
4.11 X Loop Closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.12 X Loop Closure Bode and Root Locus Plot . . . . . . . . . . . . . . . . . 37
vii
4.13 X Loop Closure Step Response . . . . . . . . . . . . . . . . . . . . . . . . 37
4.14 Overall Tanker Longitudinal Control . . . . . . . . . . . . . . . . . . . . . 38
4.15 Tanker Y Feedback Control . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.16 Lateral Control Unaugmented=
a
Root Locus . . . . . . . . . . . . . . . 40
4.17 Tanker Augmented Step Response . . . . . . . . . . . . . . . . . . . . . 40
4.18 Tanker Y (left) and (right) Step Response . . . . . . . . . . . . . . . . . 41
4.19 Tanker Feedback Control . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.20 Tanker Lateral Feedback Control . . . . . . . . . . . . . . . . . . . . . . . 42
4.21 Overall Receiver Longitudinal Control . . . . . . . . . . . . . . . . . . . . 44
4.22 Overall Receiver Lateral Control . . . . . . . . . . . . . . . . . . . . . . . 45
5.1 Change in Phugoid Roots with Change inX
u
. . . . . . . . . . . . . . . . 48
5.2 Change in Short Period with Change inT

2
. . . . . . . . . . . . . . . . . 48
5.3 Unaugmented Plant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5.4 Augmented Plant to VaryT

2
. . . . . . . . . . . . . . . . . . . . . . . . . 50
5.5 Receiver and Tanker Trubulence Delay . . . . . . . . . . . . . . . . . . . . 51
5.6 Probability of Exceedance at Medium to High Altitude Turbulence Inten-
sities [3] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.7 Probe X, Y , and Z Position: Baseline Profile . . . . . . . . . . . . . . . . . 53
5.8 Response in Z and X with change inX
u
, No Turbulence . . . . . . . . . . 53
5.9 Drogue Contact with Change inX
u
, No Turbulence, Controller One . . . . 54
5.10 Controller One (Left) and Two (Right) Drogue Contact,X
u
=0:0068 . . 55
5.11 Change inX
u
, Controller One: Y and Z Box Plot (Baseline = -0.0068) . . . 57
5.12 Change in X
u
, Controller One: Time Box Plot and Overall Probability
(Baseline = -0.0068) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.13 Change inX
u
, Controller Two: Y and Z Box Plot (Baseline = -0.0068) . . . 59
5.14 Change in X
u
, Controller Two: Time Box Plot and Overall Probability
(Baseline = -0.0068) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5.15 Response in Z and Drogue Contact with Change inT

2
, No Turbulence . . . 61
viii
5.16 Drogue Contact with Change in T

2
, Light Turbulence. Controller One
(Left), and Two (Right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.17 Change inT

2
, Controller One: Y and Z Box Plot (Baseline = 1.4) . . . . . 64
5.18 Change in T

2
, Controller One: Time Box Plot and Overall Probability
(Baseline = 1.4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.19 Change inT

2
, Controller Two: Y and Z Box Plot (Baseline = 1.4) . . . . . 66
5.20 Change in T

2
, Controller Two: Time Box Plot and Overall Probability
(Baseline = 1.4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
1 Controller One: Drogue Contact with Change inX
u
. . . . . . . . . . . . . 76
2 Controller Two: Drogue Contact with Change inX
u
. . . . . . . . . . . . . 77
3 Controller One: Drogue Contact with Change inT

2
. . . . . . . . . . . . . 78
4 Controller Two: Drogue Contact with Change inT

2
. . . . . . . . . . . . 79
ix
Nomenclature
Aircraft angle of attack, page 13

To
Thrust angle of attach with respect to fuselage, page 13
c
w
Mean aerodynamic chord, page 14
Side-slip angle, page 15

a
Aileron deflection, page 10

e
Elevator deflection, page 10

r
Rudder deflection, page 10

s
Spoiler deflection, page 11

t
Throttle position (limit[0,1]), page 10
_ Rate of angle of attack, page 17
Dynamic pressure ratio of vertical stabilizer to free-stream, page 17
C
D
@C
D
@
, change in coefficient of drag with change in angle of attack, page 24
C
Du
@C
D
@u
, change in coefficient of drag with change in velocityu, page 24
C
D
s
@C
D
@s
, change in coefficient of drag with change in spoiler deflection, page 24
C
L
@C
L
@
, change in coefficient of lift with change in angle of attack, page 24
C
l

@C
l
@
, change in rolling moment coefficient with change in sideslip, page 24
C
lq
@C
l
@q
, change in rolling moment coefficient with change in pitch rate, page 24
C
Lu
@C
L
@u
, change in coefficient of lift with change in velocityu, page 24
C
l
a
@C
l
@a
, change in rolling moment coefficient with change in aileron deflection, page 24
C
L
e
@C
L
@e
, change in coefficient of lift with change in elevator deflection, page 24
C
l
r
@C
l
@r
, change in rolling moment coefficient with change in rudder deflection, page 24
x
C
L
s
@C
L
@s
, change in coefficient of lift with change in spoiler deflection, page 24
C
L
_
@C
L
@_
, change in coefficient of lift with change in rate of angle of attack, page 24
C
lp
@C
l
@p
, change in rolling moment coefficient with change in roll rate, page 24
C
lr
@C
l
@r
, change in rolling moment coefficient with change in yaw rate, page 24
C
m
@Cm
@
, change in pitching moment coefficient with change in angle of attack, page 24
C
mq
@Cm
@q
, change in pitching moment coefficient with change in pitch rate, page 24
C
mu
@Cm
@u
, change in pitching moment coefficient with change in velocityu, page 24
C
m
e
@Cm
@e
, change in pitching moment coefficient with change in elevator deflection,
page 24
C
m
s
@Cm
@s
, change in pitching moment coefficient with change in spoiler deflection, page 24
C
m
_
@Cm
@_
, change in pitching moment coefficient with change in rate of angle of attack,
page 24
C
n

@Cn
@
, change in yawing moment coefficient with change in sideslip, page 24
C
n
a
@Cn
@a
, change in yawing moment coefficient with change in aileron deflection, page 24
C
n
r
@Cn
@r
, change in yawing moment coefficient with change in rudder deflection, page 24
C
np
@Cn
@p
, change in yawing moment coefficient with change in roll rate, page 24
C
nr
@Cn
@r
, change in yawing moment coefficient with change in yaw rate, page 24
C
Y

@C
Y
@
, change inC
Y
with change in sideslip, page 24
C
Y
r
@C
Y
@r
, change inC
Y
with change in rudder deflection, page 24
!
nsp
Short period natural frequency, page 27
Euler bang angle, page 9
Euler yaw angle, page 9
Air density at altitude, page 13
Euler pitch angle, page 9

o
Pitch at equilibrium point, page 11

ref
Reference pitch angle, page 32

sp
Short period damping, page 27
xi
b
w
Span of main wing, page 14
C
D
Coefficient of drag, page 13
C
L
Coefficient of lift, page 13
C
l
Rolling moment coefficient, page 14
C
m
Pitching moment coefficient, page 14
C
n
Yawing moment coefficient, page 15
C
X
Body-fixed, aerodynamic coefficient acting in x-direction, page 13
C
Y
Body-fixed, aerodynamic coefficient acting in y-direction, page 13
C
Z
Body-fixed, aerodynamic coefficient acting in z-direction, page 14
C
Lv
Vertical stabilizer lift coefficient, page 17
F Surface force with respect to body axis, page 13
g Gravitational acceleration constant, page 9
L

Change in lift force with change in angle of attach, page 27
M Net moment with respect to body axis, page 13
m Aircraft mass, page 8
m
q
Change in pitching moment with change in pitch rate, page 27
m

Change in pitching moment with change in angle of attack, page 27
m
_
Change in pitching moment with change in rate of angle of attack, page 27
p Roll rotation rate, page 9
q Pitch rotation rate, page 9
r Yaw rotation rate, page 9
S
v
Vertical stabilizer area, page 17
S
w
Area of the wing, page 13
T Thrust as a result of propulsion system, page 13
T

2
Flight path attitude consonance.
1
T

2
, a zero location of

e
transfer function, page xiv
u Body-axis velocity in the x-direction, page 9
v Body-axis velocity in the y-direction, page 9
xii
V
o
Airspeed at equilibrium point, page 10
V
w()
f
Wind velocity in the Earth-fixed x, y, and z axis, page 9
W Aircraft weight, page 9
w Body-axis velocity in the z-direction, page 9
x
f
Earth fixed x-translation, page 9
x
T
x-coordinate center of thrust, page 14
X
u
@X
@u
Change in linear accelerationX with change in velocityu, page xiv
x
bv
Vertical stabilizer aerodynamic center in the x-direction, page 17
y
f
Earth fixed y-translation, page 9
z
f
Earth fixed z-translation, page 9
z
T
z-coordinate center of thrust, page 14
DOF Degree of freedom, page xiv
xiii
Abstract
The effect of two parameters,X
u
andT

2
, are explored for their influence on autonomous
in-flight aerial refueling. It is believed that these two parameters are very influential to lon-
gitudinal control. A Global Hawk (RQ-4A, receiver) to Global Hawk (RQ-4A, tanker)
configuration is considered, using a probe-and-drogue fueling method. The refueling is
simulated via a 6 DOF state space model developed for both the receiver and the tanker.
The receiver and tanker model formulations are presented as well as the probability distri-
bution of a successful connection, probe to drogue, with the variation in each predefined
parameter. For each parameter the study was performed with two unique control law imple-
mentations (linear and nonlinear). The parameterX
u
was varied from -0.15 to 0.0, while
T

2
was varied from 11.3 to 0.33. It was found, with the varying of parameterX
u
, the prob-
ability of a successful connection remained rather unchanged. With the implementation
of control law one, the probability of a drogue contact was between 20% to 30%. While
for control law two the probability increased to between 70% to 80%. With the variation
ofT

2
, it became apparent that the probability of a successful connection quickly dropped
veering away from the value for which the control laws were implemented. With respect
to control law one, the probability of contact with the drogue dropped from about 25%, for
the initialT

2
design, to less than 10% for all other values. With the use of control law two,
the probability increased with respect to control law one; however, veering from the initial
design point probability of contact dropped from about 73% to less than 25% for extreme
values ofT

2
.
xiv
Chapter 1
Introduction
1.1 Background
Throughout the span of flight history aircraft have become more reliable, efficient,
stealthier, and currently can go further than ever before. Although range and flight time has
improved, the necessity for in-flight aerial refueling remains. Current methods of refueling
include either a boom-and-receptacle or a probe-and-drogue technique. The boom-and-
receptacle technique require the receiving aircraft to maintain a fixed position relative to
the tanker, while a human operator on the tanker, guides the boom. The probe-and-drogue
technique requires the release of a hose, with a drogue fixed at the end, from the tanker.
While then it is the job of the receiving aircraft to plug into the drogue. These methods
have been used for manned flight, however unmanned aerial vehicles (UA Vs) are slowly
catching up in their ability to prove the capability of in flight refueling.
Currently two major UA V tests are trying to bridge the gap of autonomous UA V refu-
eling. The first is the Northrop Grumman X-47; due to soon test refueling between an
unmanned vehicle and a manned tanker [2]. The second is a demonstration, already per-
formed, of refueling between two Global Hawk UA Vs. The demonstration has shown on
a number of flight tests, with the Northrop Grumman RQ-4 Global Hawk, that the aircraft
can refuel autonomously, with a success rate of up to 60% [2]. The DARPA funded exper-
iment uses two Global Hawks, one as a tanker and the other as a receiver, while making
use of the probe-and-drogue technique. Unlike conventional methods however, the receiver
stays in steady-level-flight and the tanker closes in on the drogue. The fuel is then pumped
upwards, which amounts to only the need of a hose and drogue fitting on the receiver.
1
1.2 Thesis Overview and Approach
Currently, to date, very little exists on the topic of autonomous UA V refueling. To
provide more insight on the matter, this thesis will conduct a parameter variation study on
an aircrafts stability derivatives and monitor the aircrafts success of performing an aerial
refueling. The setup of the thesis is the same as that mentioned in the preceding section
performed by DARPA [2]. This section will highlight the overall layout of the simulations
performed, as well as any underlying assumptions made. The parameters chosen to be
studied were mainly influenced by reference [14], as Gonzalez’s close experience with the
subject matter led to the questioning of theX
u
andT

2
parameters, to be discussed in the
coming sections.
Figure 1.1: Receiver and Tanker Formation
The basis for this thesis will be the Global Hawk RQ-4A, of which its merit of study
was primarily inspired by the DARPA project. The overall setup of the refueling procedure
is shown in Figure 1.1. To the left of the figure is the receiver, in the center is the drogue,
and to the right is the tanker. The receivers job is to primarily maintain speed, altitude, and
heading while the tanker closes in on the drogue.
For the purposes of this study both the dynamics of the aircraft are modeled to the best
of ones ability. The aerodynamic and weight distribution data for the Global Hawk are
not public, therefore they are derived in Chapter 3 and 4, and checked by industry experts
[14, 22]. The aerodynamics were modeled using the Unites States Air Force Digital Dat-
com, while the moments of inertia are obtained by creating a 3D CAD model of the aircraft.
In the setup of the overall simulation, with exception of those to be mentioned later on,
2
a number of assumptions were made. Seeing as the goal of the thesis was to monitor the
effect of X
u
and T

2
, the following assumptions were made to isolate the effects of the
parameters as opposed to any other coupling that might occur between events. The first
significant assumption made was based on the response of the aircraft. Understanding that
the aircrafts response is nonlinear, the assumption was made, in Chapter 2, that the air-
crafts response about some equilibrium point is linear. This assumption is valid as long as
the aircraft does not veer too far from the equilibrium position, about which it is linearized.
Therefore all changes that will occur from the equilibrium position during the refueling pro-
cess are assumed to remain close to the steady-level-flight values. This assumption holds
true for in-flight refueling due to the closed loop tracking being a fixed operating point and
small perturbation task. The equilibrium position for the tanker is at 10,000 m altitude,
0 deg. angle of attack, and at a speed of 140.6 m/s (Mach 0.47). While the equilibrium
position for the receiver is at 10,000 m altitude, -2.6 deg angle of attack, and a velocity of
140.6 m/s. The receivers negative angle of attack comes from it having to fly at the same
speed as the tanker, while its weight is significantly less.
The second assumption made to the overall refueling scenario was that the position
of the drogue is known at all times. Current methods for detecting the drogue vary from
electro-optical to LIDAR, as such proposed in reference [23] and [12], respectively. In
addition electro-optical methods have also been proposed, in [11], for detecting the relative
position between the tanker and the receiver. Therefore it will not be the purpose of this
thesis to test the accuracy of these methods in delivering accurate positions as it will be
understood that such methods exist and may be implemented in the future. For the purpose
of this thesis, the assumption will remain that the position of the drogue is known.
The third assumption made is with respect to perturbation of the drogue. In turbulence,
the receiver and the drogue will be disturbed. However the assumption is made that the
drogue, relative to the receiver, remains fixed. This assumption is valid, as the overall posi-
tion of the drogue will shift with the appropriate disturbance of the receiver. Realistically
3
there will be another perturbation about this translation, however it will be assumed that
the magnitude of that perturbation is less than the one created by the disturbance of the
receiver.
The last assumption made had to do with the effect of the shed vortices off the receiver.
The assumption was made that these vortices would not have a very significant effect on
the tanker. The wing vortices were investigated and were found not to be of much signifi-
cance, as the tanker approaches the drogue from a fixed distance from beneath. According
to reference [14] the major problem the vortices caused was introducing turbulent flow with
respect to the pitot tubes, as the pitot tubes on the Global Hawk are placed at the tips of the
tail, and in some conditions would be effected by the wake of the lead aircraft. Although
this was understood, ultimately it was kept out of the simulation. The primary factor being
that it is the receivers job to maintain a given speed and the tankers job to match speeds
with the receiver. The relative speed then could be obtained from any other method not
limited to the pitot tubes. Given that the tanker is closing in on a position, the turbulent
flow over the pitot only introduce minor effects.
Now that the major assumptions about the simulation have been presented, its time
go over the simulation. The simulation was designed partially in Matlab and partially in
Simulink. That is, the 6 degree of freedom (DOF) state space models of the aircraft were
calculated in Matlab and all the control laws to be mentioned were designed and imple-
mented in Simulink. The goal of the Simulink model was to calculate the response of the
feedback loops necessary to close in on the drogue. The loop closures are designed to min-
imize three errors, as discussed in the subsequent paragraphs.
The first error is with respect to altitude. The altitude error or Z error, from here on
out as it will be called, can best be seen in Figure 1.2. This figure shows the respective
coordinate system for the aircraft (in this case, the tanker), the probe, and the drogue. The
controller to be designed in Chapter 4, will try to reduce this error. That is, it will try to
minimize the distance betweenZ
Probe
andZ
Drogue
. To do this, the controller will augment
4
pitch via the elevator input in order to ultimately augment altitude and drive theZ error to
zero.
The next error that is minimized can be seen in Figure 1.3. This is the error in the longi-
Figure 1.2: Altitude Error
Figure 1.3: Longitudinal Error
tudinalX direction. The controller, by augmenting velocity, ultimately tries to reduce this
error and drives it to zero. A series of two unique controllers are used for the longitudinal
X error. Their description is to follow.
Figure 1.4:Y Error and Yaw Error
5
The last and final approach for the tanker, is to close in on theY and yaw error; which
can be seen in Figure 1.4. To do this the controller tries to drive the Y error to zero by
banking and turning, while simultaneously trying to drive the yaw angle to 0 deg.
Figure 1.5: Controller Overview
Throughout the course of this thesis two controllers were ultimately tested for the
tanker. They will be referred to here on out as controller one and two. Figure 1.5 may
be a helpful visualization as they are described. As the receiver tries to maintain altitude
and speed, it is the job of the tanker to approach the drogue. When the tanker approaches
the drogue with controller one, it just tries to minimize the errors previously shown without
regard of how the other axis are doing. The ability of this controller under light turbulence
became very apparent as not being very ideal. The Global Hawk ultimately was designed
for endurance and not maneuverability, this became quite apparent with controller one. As
will be discussed later, the lateral response of the aircraft is quite slow. Therefore controller
two was implemented. Controller two, unlike one, waits forZ error andY error to become
less than the area depicted in Figure 1.5 (a square with sides the diameter of the drogue),
before it tries to minimize theX error. Therefore, initiallyX will try to close in on a dis-
tance 2.5 meters in front of the drogue, once it senses that the error inY and altitude are
within the drogue’s diameter it accelerates quickly inX, trying to impact the drogue. In
reality the probe must hit the drogue with some force (or relative velocity of about 1-2 m/s),
once it has done so it needs to push the drogue anywhere between a minimum of 2 m to a
6
maximum of 4 m forward in order to activate the fuel [14]. In the case of this thesis, the
aftermath of the contact with the drogue is not considered, but controller two does provide
a significantly higher performance controller. The amount that controller two accelerates
by when trying to strike the drogue is calculated as such to stop within the allowed distance
of 2-4 m, once contact has been made. If contact has been made, then controller two tries
to keep the probe location at a distance of -2 m from that depicted in Figure 1.5. It can
be imagined that this controller may take longer to close in on the drogue, but as it will
be mentioned in the chapters to come the probability of making contact with the drogue
greatly increases.
Having laid out a general overview of what is to be done, each subsequent chapter and
section elaborates on the general topics already mentioned. Chapter 2 will develop the
necessary equations representative of the dynamics of the aircraft. Chapter 3 will analyze
the unaugmented aircraft. Chapter 4, will go over the controls for the loop closures pre-
viously mentioned. Chapter 5, will go over the results of varying the parametersX
u
and
T

2
in the presence of no turbulence and light turbulence, in combination with the different
controllers mentioned. While finally, the paper will conclude and make sense of the results
obtained.
For the curiosity of the reader, the appendices at the end of this thesis contain the pro-
gram necessary to reproduce the aerodynamic models and some raw, sample, data obtained
from the trials.
7
Chapter 2
Equations of Motion
2.1 Nonlinear Equations of Motion
The basis for analyzing this aircraft, as a rigid body, begins with Newton’s second
law. From a body-fixed coordinate system the aircraft’s motion can be expressed with the
following equations
F
s
+ W =
d
dt
(mV) +! (mV) (2.1)
M
s
=
d
dt
([I]!) +! ([I]!) (2.2)
where F
s
is the net surface force vector, W is the weight vector, M
s
is the net moment
vector about the body-fixed origin,m is the mass, [I] is the inertia tensor, and V and! are
respectively, the translational and rotational velocities. [20, p. 725-726]
Figure 2.1: Aircraft body axis
Expanding equation 2.1 and 2.2 results in six first-order differential equations [20, p. 730],
seen as followed, with respect to the body-fixed frame, as seen in Figure 2.1. Note that the
8
following terms have been set to zero in the inertia tensor due to symmetry about theXZ
plane:I
xy
,I
yx
,I
yz
,I
zy
.
2
6
6
6
6
4
W=g 0 0
0 W=g 0
0 0 W=g
3
7
7
7
7
5
8
>
>
>
>
<
>
>
>
>
:
_ u
_ v
_ w
9
>
>
>
>
=
>
>
>
>
;
=
8
>
>
>
>
<
>
>
>
>
:
F
x
+W
x
+ (rvqw)W=g
F
y
+W
y
+ (pwru)W=g
F
z
+W
z
+ (qupv)W=g
9
>
>
>
>
=
>
>
>
>
;
(2.3)
2
6
6
6
6
4
I
xx
0 I
xz
0 I
yy
0
I
zx
0 I
zz
3
7
7
7
7
5
8
>
>
>
>
<
>
>
>
>
:
_ p
_ q
_ r
9
>
>
>
>
=
>
>
>
>
;
=
8
>
>
>
>
<
>
>
>
>
:
M
x
+ (I
yy
I
zz
)qr +I
xz
pq
M
y
+ (I
zz
I
xx
)pr +I
xz
(r
2
p
2
)
M
z
+ (I
xx
I
yy
)pqI
xz
qr
9
>
>
>
>
=
>
>
>
>
;
(2.4)
Following the procedure in reference [20], and using Euler angles, equation 2.3 and 2.4
can be used in the kinematic transformation equations, of which represent motion of the
aircraft in an Earth-fixed coordinate system.
8
>
>
>
>
<
>
>
>
>
:
_ x
f
_ y
f
_ z
f
9
>
>
>
>
=
>
>
>
>
;
=
2
6
6
6
6
4
cos cos sin sin cos cos sin cos sin cos + sin sin
cos sin sin sin sin + cos cos cos sin sin sin cos
sin sin cos cos cos
3
7
7
7
7
5
8
>
>
>
>
<
>
>
>
>
:
u
v
w
9
>
>
>
>
=
>
>
>
>
;
+
8
>
>
>
>
<
>
>
>
>
:
V
wx
f
V
wy
f
V
wz
f
9
>
>
>
>
=
>
>
>
>
;
(2.5)
8
>
>
>
>
<
>
>
>
>
:
_

_

_

9
>
>
>
>
=
>
>
>
>
;
=
2
6
6
6
6
4
1 sin sin= cos cos sin= cos
0 cos sin
0 sin= cos cos= cos
3
7
7
7
7
5
8
>
>
>
>
<
>
>
>
>
:
p
q
r
9
>
>
>
>
=
>
>
>
>
;
(2.6)
The result of equation 2.3 and 2.4, in addition to equation 2.5 and 2.6, is a set of
12 nonlinear equations that can be integrated to describe the position vector, Euler angles,
translational velocity, and rotational velocity of the aircraft. A word of caution is to be
metioned for equations 2.5 and 2.6 as they contain a singularity at =90 deg. This
however does not impact this study as the aircraft will never be subjected to a pitch of
90 deg.
9
2.2 Linearized Equations of Motion
As viewed in the previous section, the equations that ultimately describe the behavior of
the aircraft are nonlinear. Since a numerical solver is required to solve these equations, 2.3
through 2.6, it is difficult to gain any intuition about the aircraft’s behavior based only on the
numerical solution. Therefore the equations mentioned are linearized about an equilibrium,
assuming that only small-perturbations occur about this point. Assuming that the aircraft’s
motion about the equilibrium point stays within some relative-linear bound.
The process of linearizing equations 2.3 through 2.6 is presented in reference [20, p.
756-763]. In the process of linearizing the equations a set of new variables are defined. The
variables are defined as its value at the equilibrium plus some small disturbance from the
equilibrium. These variables are defined as followed,
u =u
o
+ u v =v
o
+ v w =w
o
+ w
p =p
o
+ p q =q
o
+ q r =r
o
+ r
x
f
=x
o
+ x
f
y
f
=y
o
+ y
f
z
f
=z
o
+ z
f
.
.
.
.
.
.
.
.
.

a
=
ao
+
a

e
=
eo
+
e

r
=
ro
+
r
At equilibrium the following assumptions are made: steady level flight, fixed speed V
o
,
no sideslip, no bank angle, and no angular velocity. Which results to the following values
being defined as such,
u
o
=V
o
x
o
=

V
wx
f
+V
o
cos
o

t y
o
=V
wy
f
t z
o
=

V
wz
f
V
o
sin
o

t
F
xo
=W
xo
F
zo
=W
zo
v
o
=w
o
=p
o
=q
o
=r
o
=
o
=
o
=F
yo
=W
yo
= M
o
= 0
Note, although a equilibrium value may be set to zero, the disturbance about the equilibrium
value is still considered (i.e. v
o
= 0) v = v). Therefore the new set of variables
become,
10
u =V
o
+ u v = v w = w
p = p q = q r = r
x
f
=

V
wx
f
+V
o
cos
o

t + x
f
y
f
=V
wy
f
+ y
f
z
f
=

V
wz
f
V
o
sin
o

t + z
f
= =
o
+ =
F
x
=W
xo
+ F
x
F
y
= F
y
F
z
=W
zo
+ F
z
W
x
=W
xo
+ W
x
W
y
= W
y
W
z
=W
z
o + W
z
M
x
= M
x
M
y
= M
y
M
z
= M
z

a
=
ao
+
a

e
=
eo
+
e

r
=
ro
+
r

s
=
so
+
s

t
=
to
+
t
The result of linearization produces two sets of equations, with respect to a change from
equilibrium,
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
W=gF
x; _ u
F
x; _ w
0 0 0 0
F
z; _ u
W=gF
z; _ w
0 0 0 0
M
y; _ u
M
y; _ w
I
yy
0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
8
>
>
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
>
>
:
_ u
_ w
_ q
_ x
f
_ z
f

_

9
>
>
>
>
>
>
>
>
>
>
>
>
>
>
=
>
>
>
>
>
>
>
>
>
>
>
>
>
>
;
=
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
F
x;e
F
x;s
F
x;t
F
z;e
F
z;s
F
z;t
M
y;e
M
y;s
M
y;t
0 0 0
0 0 0
0 0 0
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
8
>
>
>
>
<
>
>
>
>
:

e

s

t
9
>
>
>
>
=
>
>
>
>
;
+
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
F
x;u
F
x;w
F
x;q
0 0 W cos
o
F
z;u
F
z;w
F
z;q
+V
o
W=g 0 0 W sin
o
M
y;u
M
y;w
M
y;q
0 0 0
cos
o
sin
o
0 0 0 V
o
sin
o
sin
o
cos
o
0 0 0 V
o
cos
o
0 0 1 0 0 0
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
8
>
>
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
>
>
:
u
w
q
x
f
z
f

9
>
>
>
>
>
>
>
>
>
>
>
>
>
>
=
>
>
>
>
>
>
>
>
>
>
>
>
>
>
;
(2.7)
Equation 2.7 represents the linearized longitudinal equations of motion; while equation
2.8 represents the linearized lateral equations of motion. It is to be noticed that the notation
F
x;u
merely represents the partial derivative ofF
x
with respect tou,F
x;u
=@F
x
=@u.
11
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
W=g 0 0 0 0 0
0 I
xx
I
xz
0 0 0
0 I
xz
I
zz
0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
8
>
>
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
>
>
:
_ v
_ p
_ r
_ y
f

_

_

9
>
>
>
>
>
>
>
>
>
>
>
>
>
>
=
>
>
>
>
>
>
>
>
>
>
>
>
>
>
;
=
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
F
y;a
F
y;r
M
x;a
M
x;r
M
z;a
M
Z;r
0 0
0 0
0 0
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
8
<
:

a

r
9
=
;
+
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
F
y;v
F
y;p
F
y;r
V
o
W=g 0 W cos
o
0
M
x;v
M
x;p
M
x;r
0 0 0
M
z;v
M
z;p
M
z;r
0 0 0
1 0 0 0 0 V
o
cos
o
0 1 tan
o
0 0 0
0 0 sec
o
0 0 0
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
8
>
>
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
>
>
:
v
p
r
y
f

9
>
>
>
>
>
>
>
>
>
>
>
>
>
>
=
>
>
>
>
>
>
>
>
>
>
>
>
>
>
;
(2.8)
Equations 2.7 and 2.8 provide insight into the behavior of the aircraft over some small
deviation from the equilibrium. These equations are no longer valid over large variations
due to coupling that is introduced between the two equations. Nonlinearities in the aero-
dynamic forces and moments also make equations 2.7 and 2.8 invalid for large variations
from the equilibrium.
Equation 2.7 and 2.8 will be the basis of this thesis. In order for the equations to be valid
in describing the dynamics of the Global Hawk, it is vital that the terms in the matrices of
the equations be approximated as closely as possible. In addition, from here on forth, the
notation will be dropped from equation 2.7 and 2.8. It will be common knowledge that
all motion from here forth will be added to the terms of the equilibrium position.
12
2.3 Aerodynamic Forces and Moments
As seen in the previous section, by filling in the values for equation 2.7 and 2.8 the
behavior of the aircraft can be predicted for the small perturbations of this in-flight refueling
study. The terms are first approximated for equation 2.7 in subsection 2.3.1, and then for
equation 2.8 in subsection 2.3.2. The approximations are represented in the same fashion
as reference [20].
2.3.1 F andM due to change inu
Each aerodynamic force term is approximated as the dynamic pressure times the cor-
responding reference area times some aerodynamic coefficient, acting about the axis being
evaluated; i.e.,F
()
= 1=2V
2
S
()
C
()
. For the force acting about the x-axis as a result of a
change in the velocity componentu, the derivation is as followed.
F
x;u
=
@F
x
@u
=
@F
x
@V
=
@
@V

1
2
V
2
S
w
C
X
+T cos (
To
)W sin

= VS
w

C
X
+
V
2
@C
X
@V

+
@T
@V
cos
To
= VS
w

C
L
+
V
2
@C
L
@V

sin

C
D
+
V
2
@C
D
@V

cos

+
@T
@V
cos
To
Assuming the aircraft is only subject to small angle of attacks (cos 1), and that the
change in thrust with a change in velocity is negligible

@T
@V
0

, the equation forF
x;u
becomes,
F
x;u
=VS
w

C
D
+
V
2
@C
D
@V

(2.9)
Following the same procedure forF
y;u
, the equation becomes,
F
y;u
=
@F
y
@u
=
@F
y
@V
=
@
@V

1
2
V
2
S
w
C
Y
+W sin () cos ()

= VS
w

C
Y
+
V
2
@C
Y
@V

(2.10)
13
Where as the equation forF
z;u
becomes,
F
z;u
=
@F
z
@u
=
@F
z
@V
=
@
@V

1
2
V
2
S
w
C
Z
T sin (
To
) +W cos () cos ()

= VS
w

C
Z
+
V
2
@C
Z
@V

@T
@V
sin (
To
)
= VS
w

C
L
+
V
2
@C
L
@V

cos

C
D
+
V
2
@C
D
@V

sin

@T
@V
sin
To
Making the same assumptions as for equation 2.9, the equation forF
z;u
becomes,
F
z;u
=VS
w

C
L
+
V
2
@C
L
@V

(2.11)
Following the same procedure for the moment equations, except with a general form of
M = 1=2V
2
S
()
b
()
C
()
, the equation forM
x;u
becomes,
M
x;u
=
@M
x
@u
=
@M
x
@V
=
@
@V

1
2
V
2
S
w
b
w
C
`

= VS
w
b
w

C
`
+
V
2
@C
`
@V

(2.12)
While the equation forM
y;u
becomes,
M
y;u
=
@M
y
@u
=
@M
y
@V
=
@
@V

1
2
V
2
S
w
c
w
C
m
+z
T
T cos
To
+x
T
T sin
To

= VS
w
c
w

C
m
+
V
2
@C
m
@V

+
@T
@V
[z
T
cos
To
+x
T
sin
To
]
Assuming that the thrust change with speed is negligible

@T
@V
0

the equation sim-
plifies to,
M
y;u
=VS
w
c
w

C
m
+
V
2
@C
m
@V

(2.13)
14
The equation forM
z;u
is defined as follows,
M
z;u
=
@M
z
@u
=
@M
z
@V
=
@
@V

1
2
V
2
S
w
b
w
C
n

= VS
w
b
w

C
n
+
V
2
@C
n
@V

(2.14)
It is to be noticed however that a few of the equations mentioned, equation 2.10, 2.12 and,
2.14, can be simplified even further to
F
y;u
=M
x;u
=M
z;u
= 0
as equations 2.7 and 2.8 are decoupled. Therefore a change in velocity in the x-axis does
not effect the force acting in the y-axis and moment acting in the x and z-axis. In actuality
however, this is not true, but for the purpose of simplicity the effect is neglected as only a
2% change inV
o
is expected.
2.3.2 F andM due to change inv;w;p;q;r; _ u; _ w
The terms to be presented in this section are derived in the same fashion as section
2.3.1. For further elaboration please refer to reference [20, p. 772-785].
Withrespecttov
F
y;v
=
@F
y
@v
=
1
V
o
@F
y
@
=
V
o
S
w
2
@C
Y
@
(2.15)
M
x;v
=
@M
x
@v
=
1
V
o
@M
x
@
=
V
o
S
w
b
w
2
@C
l
@
(2.16)
M
z;v
=
@M
z
@v
=
1
V
o
@M
z
@
=
V
o
S
w
b
w
2
@C
n
@
(2.17)
F
x;v
=F
z;v
=M
y;v
= 0
15
Withrespecttow
F
x;w
=
@F
x
@w
=
1
V
o
@F
x
@
=
V
o
S
w
2
@C
X
@
=
V
o
S
w
2

C
L

@C
D
@

(2.18)
F
z;w
=
@F
z
@w
=
1
V
o
@F
z
@
=
V
o
S
w
2
@C
Z
@
=
V
o
S
w
2

@C
L
@
+C
D

(2.19)
M
y;w
=
@M
y
@w
=
1
V
o
@M
y
@
=
V
o
S
w
c
w
2
@C
m
@
(2.20)
F
y;w
=M
z;w
=M
x;w
= 0
Withrespecttop
F
y;p
=
@F
y
@p
=
1
2
V
o
2
S
w
@C
Y
@p

= 0 (2.21)
M
x;p
=
@M
x
@p
=
1
2
V
o
2
S
w
b
w
@C
l
@p
(2.22)
M
z;p
=
@M
y
@p
=
1
2
V
o
2
S
w
b
w
@C
n
@p
(2.23)
F
x;p
=F
z;p
=M
y;p
= 0
Withrespecttoq
F
x;q
=
@F
x
@q
=
1
2
V
o
2
S
w
@C
D
@q

= 0 (2.24)
F
z;q
=
@F
z
@q
=
1
2
V
o
2
S
w
@C
L
@q
(2.25)
M
y;q
=
@M
y
@q
=
1
2
V
o
2
S
w
c
w
@C
m
@q
(2.26)
F
y;q
=M
x;q
=M
z;q
= 0
16
Withrespecttor
F
y;r
=
@F
y
@r
=
1
2
V
o
2
S
w
@C
Y
@r

=
1
2
V
o

v
S
v
x
bv
@C
Lv
@
(2.27)
M
x;r
=
@M
x
@r
=
1
2
V
o
2
S
w
b
w
@C
l
@r
(2.28)
M
z;r
=
@M
z
@r
=
1
2
V
o
2
S
w
b
w
@C
n
@r

=
1
2
V
o

v
S
v
x
2
bv
@C
Lv
@
(2.29)
F
x;r
=F
z;r
=M
y;r
= 0
Withrespectto _ u
F
x;_ u

=F
z;_ u

=M
y;_ u

=F
x; _ w

= 0
Withrespectto _ w
F
z; _ w
=
@F
z
@ _ w
=
1
V
o
@F
z
@ _
=
1
2
V
2
o
S
w
V
o
@C
L
@ _
(2.30)
M
y; _ w
=
@M
y
@ _ w
=
1
V
o
@M
y
@ _
=
1
2
V
2
o
S
w
c
w
V
o
@C
m
@ _
(2.31)
Although not mentioned here, the force and moment equations for the control surfaces
are of the same form as those mentioned in section 2.3.1, with the slight difference of the
appropriate aerodynamic coefficients (C
L
,C
D
,C
m
, etc.).
17
2.4 State Space Analysis
Referring back to equations 2.7 and 2.8, taking the inverse of the left hand side matrix
and multiplying it with the right hand side, the equations become of the familiar state space
form. The general equations being,
_ x(t) = Ax(t) +Bu(t)
y(t) = Cx(t) +Du(t) (2.32)
Where for the longitudinal equations, x(t) is the set of states [u;w;q;x
f
;z
f
;]
0
, u(t) is the
set of inputs [
e
;
s
;
t
]
0
, and y(t) is the output. The set of equations then become,
_ x(t) =
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
X
u
X
w
X
q
0 0 X
g
cos
o
Z
u
Z
w
Z
q
0 0 Z
g
sin
o
M
u
M
w
M
q
0 0 0
cos
o
sin
o
0 0 0 V
o
sin
o
sin
o
cos
o
0 0 0 V
o
cos
o
0 0 1 0 0 0
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
x(t) +
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
X
e
X
s
X
t
Z
e
Z
s
Z
t
M
e
M
s
M
t
0 0 0
0 0 0
0 0 0
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
u(t)
y(t) = I x(t) (2.33)
Where I is the identity matrix andD is the zero matrix. The termsX
u
,X
w
,X
q
,Z
u
,Z
w
,
Z
q
,M
u
,M
w
, andM
q
, are known as the longitudinal stability derivatives. They represent
a linear or angular acceleration per change in state x(t) or input u(t). Particular to this
investigation, X
u
represents the change in acceleration in the x-axis versus a change in
forward velocityu.
18
While for the lateral equations, x(t) is the set of states, [v;p;r;y
f
;; ]
0
and u(t) is the
set of inputs, [
a
;
r
]
0
. The set of equations then become,
_ x(t) =
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
Y
v
Y
p
Y
r
0 Y
g
cos
o
0
L
v
L
p
L
r
0 0 0
N
v
N
p
N
r
0 0 0
1 0 0 0 0 V
o
cos
o
0 1 tan
o
0 0 0
0 0 sec
o
0 0 0
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
x(t) +
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
Y
a
Y
r
L
a
L
r
N
a
N
r
0 0
0 0
0 0
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
u(t)
y(t) = I x(t) (2.34)
Where again I is the identity matrix andD is the zero matrix. The termsY
v
,L
v
,N
v
,Y
p
,
L
p
,N
p
,Y
r
,L
r
, andN
r
, are known as the lateral stability derivatives. Chapter 3, will make
use of these state space models and investigate the characteristic behavior of the aircraft.
19
Chapter 3
Unaugmented Aircraft
In chapter 2 the equations of motion of the aircraft were introduced and linearized. The
linearization led to a set of state space equations, 2.33 and 2.34, on which the analysis
here on forth shall be based. But first, the coefficients of the state space equations must be
evaluated. In order to do this a set of equations, equation 2.7 through 2.31, remain to be
numerically determined. Primarily what is unknown from these equations are the moments
of inertia and the aerodynamic coefficients. The following sections will provide a best
estimate for these parameters, for they are not publicly available.
3.1 Moments of Inertia
The moments of inertia are essential to obtaining the right dynamics of the aircraft.
Their values have a considerable effect on damping and rotation rates. To best approximate
these values a 3D CAD model of the aircraft was constructed.
The model dimensions were based on the general drawings found in reference [1]. The
model has a wingspan of 35.42 m (116.2 ft) and a length of 13.53 m (44.4 ft). Its maximum
takeoff weight is taken to be 12,020 kg (26,500 lbs) and its empty weight is 5,398 kg
(11,900 lbs). The wing airfoil, referred to in reference [4], is taken to be the NASA LRN
1015; while the tail airfoil is approximated as a symmetric airfoil with a 15% thickness,
specifically theNACA0015.
The following figure, Figure 3.1, shows a model of the aircraft. The models known
masses are modeled exactly, i.e. engine and payload (assuming maximum payload being
carried), while other masses are approximated to keep the center of gravity (CG) at 25% of
20
Figure 3.1: Global Hawk RQ-4A SolidWorks Model
the mean aerodynamic chord (MAC), as derived from reference [1]. The aircraft’s empty
weight breakdown can be seen in Table 3.1. The result of this mass distribution puts the
CG of the empty weight plane at 25% of the MAC. While for the maximum takeoff weight
at 30% of the MAC. It is also to be noticed that as the fuel (highlighted in blue in Figure
3.1) from the wings is removed the CG shifts towards 25% MAC. This model was checked
with reference [22], an industry expert working for Northrop Grumman. Although exact
values could not be given, the model’s weights, CG position, and moments of inertia were
given the approval of being representative of the actual aircraft values.
Table 3.1: Global Hawk RQ-4A Percentage Mass Breakdown
Wings 35%
Tail 10%
Fuselage 40%
Engine 15%
The results from the model then lead to the following moment of inertia values,
Table 3.2: Global Hawk RQ-4A Moments of Inertia
Max Takeoff Weight kgm
2
uncertainty Empty Weight kgm
2
uncertainty
I
xx
384,500 50,000 I
xx
158,900 20,000
I
yy
70,000 8,750 I
yy
63,700 8,000
I
zz
447,800 56,000 I
zz
217,400 28,000
I
xz
5,500 700 I
xz
6,300 800
21
3.2 Aerodynamic Coefficients
The derivation of the aerodynamic coefficients is the very essence of determining the
forces and moments acting on the aircraft. General methods for obtaining these coeffi-
cients include wind tunnel testing and computational fluid dynamics, followed by in flight
testing. While these are the preferred methods, classical mechanics still gives a very good
approximation. To derive the aerodynamic coefficients for the Global Hawk, as those seen
in section 2.3.1 and 2.3.2, a Digital Datcom model was created. The full Datcom input file
can be found in Appendix A, while a few highlights will be made to follow.
To setup the Datcom file, the previously created CAD file was used to help create a pro-
file of the geometry. The Datcom file starts off with a declaration of the flight conditions.
The model is set up for an altitude of 10,000 m (32,800 ft) and a Mach number of 0.47, or,
at altitude a velocity of 140.6m=s. The header of the model looks as follows,
$FLTCON LOOP = 2.0$
$FLTCON NMACH = 2.0, MACH(1)= 0.47, 0.48$
$FLTCON NALT = 1.0, ALT(1) = 10000.0$
$FLTCON NALPHA= 17.0,
ALSCHD(1)= -7.0,-6.0,-5.0,-4.0,-3.0,-2.6,-2.0,-1.0,0.0,
1.0,2.0,3.0,4.0,5.0,6.0,7.0,8.0$
Where the altitude, Mach number, and a variation of alphas are defined. The next step is to
define the location of the wings (XW ,ZW ), horizontal tail (XH,ZH), vertical tail (XV ,
ZV ), and the CG of the aircraft, with respect to a positive distance aft of the nose of the
plane. This is done as followed,
$SYNTHS XCG= 7.153, ZCG= 0.0,
XW = 5.8249, ZW =-0.39,
XH = 11.812, ZH =-0.15, XV = 11.812,
ZV = 0.00, ALIH=-7.0, ALIW=0.00$
22
In order to build the aircraft in Datcom, the aircraft geometry must be defined. The aircraft
fuselage must be entered in as circles, or ellipses, at some distanceX aft of the nose. The
Global Hawks fuselage geometry is as follows,
$BODY NX=19.0, ITYPE=1.0,
X(1)=0.000, 0.250, 0.750, 1.250, 1.750, 2.250, 2.660,
3.070, 3.890, 4.300, 4.710, 5.530, 6.350, 7.170,
7.990,11.390, .650, 13.110, 13.540,
R(1)=0.000, 0.399, 0.689, 0.732, 0.750, 0.757, 0.757,
0.757, 0.757, 0.757, 0.757, 0.757, 0.757, 0.757,
0.757, 0.757, 0.757, 0.757, 0.0,
ZU(1)=0.514, 0.772, 1.262, 1.480, 1.613, 1.667, 1.629,
1.540, 1.359, 1.232, 1.111, 0.808, 0.772, 0.760,
1.728, 1.637, 1.576, 0.650, 0.43,
ZL(1)=0.101, -0.450, -0.546, -0.558, -0.613, -0.613,
-0.620, -0.728, -1.030, -1.079, -1.085, -0.988,
-0.794, -0.716, -0.746, -0.649, -0.432, -0.344,
0.416$
It must be noted that the last parameter in ZL and ZU has been tweaked from it’s original
value. This was due to the moment coefficientC
m
. As the Global Hawk has a serpentine
like structure, and the inlet of the engine is located right aft of the nose curvature, this
creates for a low pressure created just aft of the nose. To cancel this effect theC
m
vs.
plot was shifted accordingly, such that at = 0,C
m
is close to zero.
Next the wing and tail geometry, airfoil shapes, and control surfaces are defined. For the
full set of Datcom instructions please refer to Appendix A. However it is to be mentioned
that due to the ’V’ like structure of the tail, Datcom was inconclusive at calculating C
lr
.
Therefore an artificial vertical stabilizer was added, the same height as the projected height
from the V-tail. The final Datcom model can be seen in Figure 3.2.
A number of iterations were made in the creating of the Datcom model. Static and
23
Figure 3.2: Global Hawk RQ-4A Datcom Model
dynamic stability were constantly checked as changes were made. The Global Hawk is
inherently a very stable aircraft, and one parameter that was monitored throughout the
creation of the Datcom model was static margin. By taking a look at theC
m
vs. C
L
plot,
as seen in Figure 3.3, it can be concluded that the tanker has nearly a 15% static margin,
while the receiver has a 19% static margin; based on a linear fit slope whereSM =
@Cm
@C
L
.
According to reference [14], the static margin of the aircrafts are very close to their true
values. Therefore, with some certainty, it can be concluded that the Datcom model closely
represents the real Global Hawk. However, dynamic stability still remains to be checked.
This is presented in section 3.3. As a reference, the results of the aerodynamic model, at
equilibrium, are shown in Table 3.3.
−0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4
−0.1
−0.05
0
0.05
0.1
0.15
C
m
vs. C
L
SM = 0.14604
C
L
C
m

M = 0.47 Alt(m) = 10000
−0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4
−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
C
m
vs. C
L
SM = 0.19098
C
L
C
m

M = 0.47 Alt(m) = 10000
Figure 3.3: Global Hawk RQ-4AC
m
vs.C
L
Curve. Tanker: left, Receiver: Right
24
Table 3.3: Global Hawk RQ-4A: Tanker and Receiver Aerodynamic Coefficients
COEF. TANKER RECEIVER Units
C
L
6.0650 6.0720 1=rad
C
D
0.0573 -0.0573 1=rad
C
m
-0.5000 -0.5000 1=rad
C
lq
0.0528 0.0560 s=rad
C
mq
-0.3756 -0.3751 s=rad
C
L
_
0.0091 0.0095 s=rad
C
m
_
-0.0310 -0.0328 s=rad
C
mu
-0.0003 0.0000 s=m
C
Du
0.0000 0.0000 s=m
C
Lu
-0.0023 -0.0022 s=m
C
D
0.0280 0.0270
C
L
0.5790 0.2987
C
m
0.0001 0.0206
C
Y

-0.5036 -0.5036 1=rad
C
l

0.0047 -0.0067 1=rad
C
n

0.0148 0.0159 1=rad
C
lp
-0.0639 -0.0637 s=rad
C
np
-0.0096 -0.0056 s=rad
C
lr
0.0219 0.0135 s=rad
C
nr
-0.0037 -0.0029 s=rad
C
L
e
0.2113 0.2113 1=rad
C
m
e
-0.9296 -0.9409 1=rad
C
l
a
0.0507 0.0507 1=rad
C
n
a
-0.0015 -0.0009 1=rad
C
l
r
0.0104 0.0103 1=rad
C
n
r
-0.0380 -0.0380 1=rad
C
Y
r
0.1150 0.1150 1=rad
C
L
s
0.2420 0.2420 1=rad
C
D
s
0.1400 0.1400 1=rad
C
m
s
-0.0150 -0.0150 1=rad
It is to be mentioned that some of the terms in Table 3.3 are not in their traditional
unit-less form. This is so they compliment the rest of the equations in this paper. In order
to non-dimentionalize the longitudinal and lateral terms, multiply by 2V
o
= c and 2V
o
=b
w
,
respectively.
25
3.3 Unaugmented Response
As seen in the previous section all of the unknowns of equations 2.9 through 2.31 have
been solved for. The moments of inertia were verified and the derived aerodynamic terms
conclude to a very statically stable aircraft. To refine the model further, the terms of the
state space equations presented in section 2.4 are populated. Once this has been done, the
longitudinal and lateral stability of the aircraft can be investigated.
3.3.1 Longitudinal Response
The longitudinal dynamic stability of the aircraft can easily be observed by looking
at the eigenvalues of the A matrix in equation 2.33, in other-words by investigating the
homogeneous solution to equation 2.33.
Table 3.4: Tanker Longitudinal Stability: Eigenvalues, Damping, and Frequency
Eigenvalue Damping Frequency

rad
sec

Rigid Body 0.00e+0 -1.00e+0 0.00e+0
0.00e+0 -1.00e+0 0.00e+0
Short Period -7.08e-1 + 1.71e+0i 3.82e-1 1.85e+0
-7.08e-1 - 1.71e+0i 3.82e-1 1.85e+0
Phugoid -3.06e-3 + 6.12e-2i 4.99e-2 6.13e-2
-3.06e-3 - 6.12e-2i 4.99e-2 6.13e-2
As seen in Table 3.4, the A matrix has six eigenvalues. From typical rigid aircraft
dynamic longitudinal stability there should be a set of two complex conjugate pairs; the
short period roots, typically highly damped with high natural frequency, and the phugoid
roots, typically with very light damping and and very low natural frequency. In Table 3.4
another set of roots appear, two zeros. These are just the rigid body modes, they simply
state that steady and level flight can be maintained at different latitudes and longitudes [20,
p 844]. From here on forth it will be common knowledge that these roots exist and that
only the short period and phugoid roots will be referred to.
Referring back to Table 3.4, it appears as if the model built in section 3.2 takes after the
26
standard aircraft model. This is a good checking point, but according to reference [14], the
short period damping is a little low. Therefore the coefficients derived in section 3.2 are
slightly modified. Referring to reference [20, p 852], an approximation for the short period
natural frequency and damping are,
!
nsp
=
s

L

W
m
q
I
yy
g
V
o

m

I
yy
(3.1)

sp
=
1
2

L

W
g
V
o

m
q
I
yy

m
_
I
yy

!
nsp
(3.2)
Therefore in order to adjust the short period damping, without affecting the short period
natural frequency, the values ofC
mq
andC
m
were adjusted. According to reference [14],
as a first run, these parameters can be changed, as multiples of two to three, in order to
provide a more realistic pole placement. ThereforeC
mq
is changed toC
mq
= 3C
mq
and
C
m
to -0.5. With these changes made, the new roots of the system become as shown in
table 3.5.
Table 3.5: Tanker Longitudinal Stability Revised: Eigenvalues, Damping, and Frequency
Eigenvalue Damping Frequency

rad
sec

Short Period -1.25e+0 + 1.34e+0i 6.83e-1 1.84e+0
-1.25e+0 - 1.34e+0i 6.83e-1 1.84e+0
Phugoid -2.23e-3 + 4.50e-2i 4.95e-2 4.51e-2
-2.23e-3 - 4.50e-2i 4.95e-2 4.51e-2
Effectively, the short period damping is closer to the aircrafts actual value and the natural
frequency remains relatively the same. Careful observation leads to notices in the change
of the phugoid mode, but their change, being so small, can be regarded as negligible.
The values presented in Table 3.5 are for the tanker, however the same approach
aforementioned is applied to the receiving aircraft. For the receiver C
mq
is changed to
C
mq
= 2:91C
mq
andC
m
to -0.5. Table 3.6 shows the characteristic roots of the receiv-
ing aircraft. Unlike the tanker, it has a more highly damped short period mode, with a
27
Table 3.6: Receiver Longitudinal Stability Revised: Eigenvalues, Damping, and Frequency
Eigenvalue Damping Frequency

rad
sec

Short Period -1.68e+0 + 1.43e+0i 7.60e-1 2.21e+0
-1.68e+0 - 1.43e+0i 7.60e-1 2.21e+0
Phugoid -5.77e-3 + 7.88e-2i 7.31e-2 7.90e-2
-5.77e-3 - 7.88e-2i 7.31e-2 7.90e-2
significantly higher natural frequency. As a result of changingC
mq
andC
m
the final lon-
gitudinal populated state space matrix for the tanker and receiver become,
Tanker
_ x =
2
6
6
6
6
6
6
6
6
6
6
6
6
6
4
0:006745 0:06284 0 0 0 9:807
0:09979 0:7331 139:6 0 0 0
0:001066 0:01484 1:777 0 0 0
1:0 0 0 0 0 0
0 1:0 0 0 0 140:6
0 0 1:0 0 0 0
3
7
7
7
7
7
7
7
7
7
7
7
7
7
5
x+
2
6
6
6
6
6
6
6
6
6
6
6
6
6
4
0 2:371 2:579
3:575 4:094 0
4:062 0:06164 0:5177
0 0 0
0 0 0
0 0 0
3
7
7
7
7
7
7
7
7
7
7
7
7
7
5
u
y(t) = I x(t) (3.3)
Receiver
_ x =
2
6
6
6
6
6
6
6
6
6
6
6
6
6
4
0:01261 0:08312 0 0 0 9:796
0:06642 1:421 138:5 0 0 0:4507
0:001466 0:01532 1:935 0 0 0:0004997
0:9989 0:04606 0 0 0 6:477
0:04606 0:9989 0 0 0 140:5
0 0 1:0 0 0 0
3
7
7
7
7
7
7
7
7
7
7
7
7
7
5
x+
2
6
6
6
6
6
6
6
6
6
6
6
6
6
4
0 4:597 5:0
6:922 7:929 0
4:462 0:06247 0:5624
0 0 0
0 0 0
0 0 0
3
7
7
7
7
7
7
7
7
7
7
7
7
7
5
u
y(t) = I x(t) (3.4)
Where again x is the set of states [u;w;q;x
f
;z
f
;]
0
and u is the set of inputs [
e
;
s
;
t
]
0
.
Equation 3.3 and 3.4 are here on forth inherently representative of the Global Hawk’s lin-
earized longitudinal dynamics about their respective equilibrium.
28
3.3.2 Lateral Response
As observed in section 3.3.1, the stability of the aircraft’s longitudinal motion was
investigated by looking at the set of roots obtained from the homogeneous solution of the
longitudinal state space equations. The same method is used in investigating lateral stabil-
ity. Typical rigid body aircraft lateral dynamic stability investigation results in three sets of
roots: 1) the spiral mode, often unstable and a real root, 2) the dutch roll mode, a complex
conjugate pair typically stable with low frequency and damping, and 3) the roll mode, a
real and stable root. For the tanker, the set of roots obtained can be seen in Table 3.7.
Table 3.7: Tanker Lateral Stability: Eigenvalues, Damping, and Frequency
Eigenvalue Damping Frequency

rad
sec

Roll -1.14e+0 1.00e+0 1.14e+0
Dutch Roll -1.02e-1 + 4.91e-1i 2.04e-1 5.01e-1
-1.02e-1 - 4.91e-1i 2.04e-1 5.01e-1
Spiral 2.52e-2 -1.00e+0 2.52e-2
It is obvious when looking at these roots that the aircraft is unstable in the lateral motion.
This is as a result of the unstable spiral mode, which can easily be stabilized through feed-
back control.
Table 3.8: Receiver Lateral Stability: Eigenvalues, Damping, and Frequency
Eigenvalue Damping Frequency

rad
sec

Roll -2.44e+0 1.00e+0 2.44e+0
Dutch Roll -1.15e-1 + 7.10e-1i 1.61e-1 7.19e-1
-1.15e-1 - 7.10e-1i 1.61e-1 7.19e-1
Spiral 9.22e-3 -1.00e+0 9.22e-3
In Table 3.8 the characteristic roots of the receiver aircraft are shown. Compared to the
tanker, its roots are shifted more to the left on the Real (Re) axis. Its roll and dutch roll
mode natural frequency are higher. While the spiral mode is closer to the Imaginary (Im)
29
axis. According to reference [14], the lateral roots of the aircraft are representative of the
actual aircraft. Therefore, unlike the longitudinal roots, no coefficients need to be changed
to augment the placements of the roots. As a final result, the tanker and receiver lateral
state space equations become,
Tanker
_ x =
2
6
6
6
6
6
6
6
6
6
6
6
6
6
4
0:06066 0 140:6 0 9:807 0
0:0006468 1:2 0:4113 0 0 0
0:001699 0:1695 0:05382 0 0 0
1:0 0 0 0 0 140:6
0 1:0 0 0 0 0
0 0 1:0 0 0 0
3
7
7
7
7
7
7
7
7
7
7
7
7
7
5
x+
2
6
6
6
6
6
6
6
6
6
6
6
6
6
4
0 1:948
0:9511 0:1852
0:01264 0:6101
0 0
0 0
0 0
3
7
7
7
7
7
7
7
7
7
7
7
7
7
5
u
y(t) = I x(t) (3.5)
Receiver
_ x =
2
6
6
6
6
6
6
6
6
6
6
6
6
6
4
0:1176 0 140:6 0 9:796 0
0:001739 2:474 0:5188 0 0 0
0:003286 0:2268 0:07137 0 0 0
1:0 0 0 0 0 140:5
0 1:0 0:04611 0 0 0
0 0 1:001 0 0 0
3
7
7
7
7
7
7
7
7
7
7
7
7
7
5
x+
2
6
6
6
6
6
6
6
6
6
6
6
6
6
4
0 3:776
1:965 0:3636
0:02325 1:109
0 0
0 0
0 0
3
7
7
7
7
7
7
7
7
7
7
7
7
7
5
u
y(t) = I x(t) (3.6)
Where again x is the set of states [v;p;r;y
f
;; ]
0
and u is the set of inputs [
a
;
r
]
0
. Equa-
tion 3.5 and 3.6 are here on forth inherently representative of the Global Hawk’s linearized
lateral dynamics about their respective equilibrium.
30
Chapter 4
Feedback Control
In the previous couple of chapters, a state space model of the Global Hawk was derived.
This model is now used, in conjunction with a series of loop closers, to simulate closing in
on the drogue. The investigation begins with the tanker.
4.1 Tanker Longitudinal Feedback Control
The longitudinal loop closure on the drogue was previously discussed in section 1.2.
Therefore the analysis begins by first investigating the drogueZ error closure. To control
altitude an inner loop pitch controller is used, and inherently pitch is controlled via the
elevator,
e
, input. The unaugmented response however of the output, due to an input
e
,
is rather oscillatory and not very well damped.
−2 −1.5 −1 −0.5 0
−1
−0.5
0
0.5
1
Root Locus
Real Axis (seconds
−1
)
Imaginary Axis (seconds
−1
)
Figure 4.1:

e
Unaugmented Root Locus and Step Response
Figure 4.1 shows the

e
root locus plot and step response. It is clear from Figure 4.1 that
the phugoid mode dominates the behavior of the response over time (note the time scale in
31
the time response). To provide more damping in controlling an inner loop of pitch rate,
q, is used. The overall loop closure can be seen in Figure 4.2. Where the gainqGain is
used to shape the inner loop to set the damping ratio to 0:9 while a PID controller is used
to help shape the response time for. In addition, an actuator model is used to model any
lag caused by the elevator actuator movement. The actuator is chosen to have a damping
ratio of 0.7 and a natural frequency of 15 rad/s.
Figure 4.2:
ref
Augmented Loop Closure
When adjusting the gains seen in Figure 4.2, they were chosen as such to provide an
overshoot of less than 20% and a settling time of less than 5 sec. In addition to having
a gain margin (G.M.) of greater than 6 dB and a phase margin (P.H.) of 45 deg. Even
Figure 4.3:
ref
Augmented Bode Plot and Root Locus Plot
though these were the minimum phase and gain margins chosen, it is to be mentioned that
32
most of what remains to be covered exceeded 9 dB of G.M. and 60 deg of P.M. Figure 4.3
shows the final frequency domain response, as well as the final pole placement of the=
e
controller. While the time response can be seen in Figure 4.4.
Figure 4.4:
ref
Augmented Step Response
As mentioned previously, the loop shown in Figure 4.2 is used to control the error in
altitude (Z error) between the probe and the drogue. Therefore another loop closure is
performed in addition to that shown in Figure 4.2. In this loop, as seen in Figure 4.5, there
is an inner loop for pitch, an outer loop for the altitude derivative, and an outer outer loop
for altitude. The altitude derivative loop is added to control flight path. Since the aircraft
starts at a fixed altitude offset of 10 m below the drogue, the flightpath is an exponential
decay to the drogue; controlled by the time constantZ tau in Figure 4.5.
Figure 4.5: Altitude Loop Closure
33
The general form for the flightpath takes the form,
Z =Z
o
e
t=
(4.1)
whereZ
o
for the remainder of this thesis will be 10 m below the drogue, and will be 0.17.
Therefore, with (Z tau in Figure 4.5) defined, a PID controller is used to control rate of
ascent,
_
Z. Figure 4.6 shows the frequency response and root migration of Figure 4.5.
Figure 4.6: Altitude Loop Closure Bode and Root Locus Plot
It can be seen from Figure 4.6 that the response is quite stable, having a G.M. of 23.2 dB
and a P.M. of 71 deg; the time domain response of which can be seen in Figure 4.7.
Figure 4.7: Altitude Loop Closure Step Response
34
With the altitude loop closure fully completed, the next step was to close in on the lon-
gitudinal errorX. In order to close in onX, a velocity controller was used. Velocity in this
case is controlled by controlling the throttle input. Another possibility is to use the spoil-
ers, which would augment drag and lift, however in this study they were not considered.
Although the stability derivatives for the spoilers are included in Equation 3.3 and 3.4, for
they may be useful in future studies. In Figure 4.8 the velocity loop closure is presented, it
includes an engine delay, with a time constant of 2 seconds, and a first order throttle servo,
with a time constant of 0.10 seconds.
Figure 4.8: Velocity Loop Closure
A PID controller is used to profile the velocity response, of which is quite stable, as
observed in Figure 4.9. The stability margins far exceed the requirements set earlier, having
a G.M. of 20.1 dB and a P.M. of 73.6 deg. The time response is a little slow, as a result of
Figure 4.9: Velocity Loop Closure Bode and Root Locus Plot
35
the engine time constant, but is rather well damped with very little over shoot. Figure 4.10
shows a sample step response.
Figure 4.10: Velocity Loop Closure Step Response
Having designed a controller for velocity, the next step is to use this controller to design
an outer loop closure for the error along theX axis. The approach then is the same as for
theZ error closure. The overall configuration can be seen in Figure 4.11. A PD controller
Figure 4.11: X Loop Closure
is used to control
_
X, for which the time and frequency response can be seen in Figure 4.12
and 4.13, respectively. X is set to close on the distance to the drogue following an expo-
nential flight path from a fixed distance 50 m aft of the drogue, for which the decay rate of
the flight path is controlled by the time constant (orX tau in figure 4.11). Although
the response inX is very stable, with a G.M. of 40.9 dB and a P.M. of 65.2 deg., the step
response, although nicely damped, is still rather slow. However, this is a product due to the
36
Figure 4.12: X Loop Closure Bode and Root Locus Plot
Figure 4.13: X Loop Closure Step Response
slow engine response.
In Figure 4.14, an overall schematic of the tanker longitudinal loop closures is shown.
It features a couple of nonlinearities, caused by limiters on the throttle and elevator input.
In addition, a disturbance (atmospheric turbulence) block is also featured, of which will
be covered in the next chapter. Under the Plant-Longitudinal block a set of state transfor-
mations occur, including a coordinate transformation to calculate the probe’s tip position
relative to the fixed coordinate system.
37
Figure 4.14: Overall Tanker Longitudinal Control
38
4.2 Tanker Lateral Feedback Control
The overall lateral loop closure was previously discussed in Chapter 1.2. Therefore,
the analysis begins by first investigating the closure about the lateralY error. To control
the displacement inY , bank-to-turn is used. Therefore, there is an inner feedback loop for
bank angle (), followed by an outer feedback loop forY . Displacement inY , in general
can be expected to be slow for this aircraft. If not subjected to high roll angles, the aircrafts
lateral motion in Y , if using banking, is limited by the physics of the system. Primarily
the force that is responsible for theY translation will be rather constricted due to it being
a function of the aircrafts mass, forward velocity, and turning radius. As will be presented
later, this becomes a major limitation in the process of refueling in turbulence.
Figure 4.15: Tanker Y Feedback Control
In Figure 4.15 the overallY feedback control loops can be seen. The aileron actuator
present is the same as used before, having a damping ratio of 0.7 and a natural frequency of
15 rad/sec. It is assumed to have the same dynamic characteristics as the elevator actuator
used in the longitudinal model. Looking at the unaugmented=
a
root locus plot in Figure
4.16, it is quite apparent that the response will be unstable (due to the unstable spiral mode
mentioned in section 3.3.2). To help stabilize the response, provide higher damping, and
reduce the settling time a lead compensator was used. The closed loop response for can
be seen in Figure 4.17. The response is very well damped, has little overshoot, and settles
in less than three seconds. In addition the augmented response has a 16.4 dB G.M. and a
67.6 deg P.M.
39
Figure 4.16: Lateral Control Unaugmented=
a
Root Locus
Figure 4.17: Tanker Augmented Step Response
Having built a controller for banking angle, the next step is to add another feedback
loop to controlY . To augmentY , a PD controller was implemented. Overshoot was kept
at a minimum while achieving a low settling time. Unlike the longitudinal control, no
flight path was designed forY , primarily due to the slow time response. Figure 4.18 gives
a sample step response inY .
In using roll to bank and translate inY , a coupling is introduced in yaw ( ). From the
equations derived in Chapter 2, the assumption was made that the aircraft had a straight
heading. In other words, throughout the process of refueling the aircraft must maintain a
yaw angle of zero degrees, with the assumption that no active gusts were present. There-
fore, the only change in yaw ( ) is caused by the coupling of the ailerons. To accomplish
40
Figure 4.18: Tanker Y (left) and (right) Step Response
this, a feedback loop was used, as shown in Figure 4.19. To augment and control the
response of the feedback loop a PID controller was implemented. A sample step response
can be seen in Figure 4.18. The response is a bit oscillatory, but settles rather quickly.
Stability margins are well over the limits initially set, with a G.M. and P.M. of 15.3 dB and
66.2 deg, respectively.
Figure 4.19: Tanker Feedback Control
The overall lateral feedback control loops for the aircraft can be seen in Figure 4.20. As
in the longitudinal case, nonlinearities are introduced by the rudder and aileron limiters. In
addition due to the aileron and rudder coupling, an additional gain parameter is introduced,
theARIGain. The purpose of this gain is to use the ailerons to cancel out any roll induces
by the rudder.
41
Figure 4.20: Tanker Lateral Feedback Control
42
4.3 Receiver Longitudinal and Lateral Feedback Control
The feedback control for the receiver was performed in the same manner as the tanker.
The step by step process will not be presented, however, the overall design for the longitu-
dinal and lateral control are shown in Figure 4.21 and 4.22. The receiver, unlike the tanker,
is a little more maneuverable. It’s mass is about half that of the tanker and it’sI
xx
andI
zz
moments of inertia are considerably lower. Whereas the tanker had to maneuver to close in
on the drogue, the receiver had to remain stationary relative to its start position. In theX
direction, the receiver had to try to maintain the same initial velocity as its start position.
The translational change in X was not a considerable factor, as it was the tankers job to
close on the drogue; however, the receiver did try to maintain a preset altitude. The altitude
was set as such to keep the drogue, at 10,000 m. In the lateral direction, the receiver tried
to maintain yaw at zero degrees, while also trying to prevent any lateral translation about
Y .
43
Figure 4.21: Overall Receiver Longitudinal Control
44
Figure 4.22: Overall Receiver Lateral Control
45
Chapter 5
Results
In the previous chapters, a state space model was developed for the Global Hawk, the
state space model was populated, and a series of feedback loops were designed to control
the tanker’s closure on the drogue. This chapter will make use of the work performed
earlier and vary two aerocharacteristic parameters of the unaugmented airframe believed
to be of importance in autonomous in-flight refueling [14], X
u
and T

2
. From equation
2.33 it was seen thatX
u
was part of theA matrix in the longitudinal state space equations.
HoweverT

2
remains to be defined.
The definition begins with Equation 2.33. Assuming, for the moment, that the only
input is
e
, elevator deflection, the general state space form is as follows, where x has been
replaced with it’s corresponding states.
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
_ u
_ w
_ q
_ y
f
_ z
f
_

3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
=
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
X
u
X
w
X
q
0 0 X
g
cos
o
Z
u
Z
w
Z
q
0 0 Z
g
sin
o
M
u
M
w
M
q
0 0 0
cos
o
sin
o
0 0 0 V
o
sin
o
sin
o
cos
o
0 0 0 V
o
cos
o
0 0 1 0 0 0
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
u
w
q
y
f
z
f

3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
+
2
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
X
e
Z
e
M
e
0
0
0
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5

e
The goal then becomes to solve the set of coupled differential equations for the transfer
function =
e
. By the use of Laplace Transforms and Cramer’s Rule the above equation
can be solved to obtain,
(s)

e
(s)
=
A

s
2
+B

s +C

D
c
s
4
+E
c
s
3
+F
c
s
2
+G
c
s
(5.1)
46
WhereA

,B

,C

,D
c
,E
c
,F
c
, andG
c
are as followed,
A

= M
e
B

= M
e
X
u
M
u
X
e
M
w
Z
e
+ M
e
Z
w
C

= M
u
X
e
Z
w
M
u
X
w
Z
e
+M
w
X
u
Z
e
M
w
X
e
Z
u
M
e
X
u
Z
w
+ M
e
X
w
Z
u
D
c
= 1
E
c
= (M
q
+X
u
+Z
w
)
F
c
= (M
u
X
q
M
q
X
u
M
q
Z
w
+M
w
Z
q
X
u
Z
w
+X
w
Z
u
)
G
c
= (M
q
X
u
Z
w
M
q
X
w
Z
u
M
u
X
q
Z
w
+M
u
X
w
Z
q
+M
w
X
q
Z
u
M
w
X
u
Z
q
:::
::: M
u
X
g
cos
o
M
w
Z
g
sin
o
)
Factoring Equation 5.1 the following is obtained,
(s)

e
(s)
=
A

(s + 1=T

1
) (s + 1=T

2
)
D
c
s
4
+E
c
s
3
+F
c
s
2
+G
c
s
(5.2)
Therefore, it can be seen thatT

2
is a term that influences a zero of the

e
transfer function.
Before going any further it is important to gain a better insight on these two terms, X
u
andT

2
. Remember thatX
u
is the change in the linear acceleration, along the x-axis, due
to a change or perturbation, velocity u. Therefore it can be expected to have an effect on
the phugoid mode; the very low frequency and lightly damped characteristic roots of the
aircraft. T

2
on the other hand is influenced by multiple stability derivatives. An approx-
imation of the main stability derivatives that contribute to the effect of T

2
will now be
investigated.
Taking Equation 5.4 and plotting the root locus for both a variation inX
u
andT

2
will
help to understand the migration of the poles and zeros as a result of changes in the vari-
ables. This change can be observed in Figure 5.1 and 5.2. The figures show that as X
u
becomes more negative, the phugoid roots move further from the imaginary axis until they
become purely real. At which point, one root becomes positive and one more negative.
47
On the other-hand, as 1=T

2
increases the zero moves further and further negative. The
Figure 5.1: Change in Phugoid Roots with Change inX
u
Figure 5.2: Change in Short Period with Change inT

2
effect can be seen in Figure 5.2. From this figure, it’s observed that the short period feed-
back loop is heavily influenced by the change inT

2
. AsT

2
becomes more negative the
closed-loop short period roots get pulled further negative.
It will be important to see how the change in these two terms affect the ability of maneu-
vering the aircraft to plug into the drogue. Understanding the effect of the terms on the
closed loop pole placement gives the ability to approximate the stability derivatives that
have the most affect on T

2
. This can be done by reducing a degree of freedom. In this
case, the assumption is made that any changes inu will have a very limited affect onT

2
.
48
Therefore, referring back to Equation 5.1, all the derivatives with respect to u are set to
zero. Neglecting to show the denominator terms, the numerator becomes,

e
=
M
e
s
2
+ (M
e
Z
w
M
w
Z
e
)s
:::
(5.3)
Rearranging the terms leads to the conclusion that a very good approximation forT

2
is,

e
=
A

s

s +
1
T

2

:::
=) :
1
T

2
=

Z
w
+M
w
Z
e
M
e

(5.4)
Understanding this is important, as it implies that T

2
is a difficult parameter to change,
once an aircrafts geometry has been fixed. From Equation 5.4, it can be inferred thatT

2
changes as a function ofC
L
,C
m
,C
D
, and horizontal stabilizer effectiveness and geome-
try. In the case of the Global Hawk, the approximation in Equation 5.4, when compared to
the actual value, gives an approximation of within 1.25% of the actual.
In the sections to come, the parameters of study will be varied and the effect they have
on the ability of the aircraft to close in on the drogue will be monitored. Two different
controllers will be tested, in two different environments. The controller logic is the same as
that mentioned in section 1.2, and the environment will include of a fixed altitude, no tur-
bulence, and light turbulence condition.X
u
will be varied from -0.15 to 0, while 1=T

2
will
be varied from 0.15 to 3.0. The value ofX
u
, from its original, will be varied by -200% to
100%. While the value of 1=T

2
, from its original, will be varied from -80% to about 300%.
These values are picked by exaggerating maximum values, from other aircraft, observed in
reference [25].
The variation of X
u
may be apparent, as in the state space model it’s only a matter
of changing one value in the A matrix. T

2
on the other-hand is not as clear. Equation
5.4 showed what important values in the model need to vary in order to have the greatest
impact on the parameter; however, if this method is followed, it may involve augmenting
49
Figure 5.3: Unaugmented Plant
parameters that are not of interest. T

2
could be varied by any combination of, Z
w
, M
w
,
Z
e
, andM
e
. Therefore to isolate only the effect of the magnitude ofT

2
, the state space
feedback model presented earlier, for the T

2
variations, will become a hybrid of a state
space model plus a transfer function representation for pitch attitude and pitch rate. The
plant model is changed from that seen in Figure 5.3 to that seen in Figure 5.4.
Figure 5.4: Augmented Plant to VaryT

2
In the plant model presented in Figure 5.4, the input is applied both to the state space model
and the transfer functions. The output of pitch and pitch rate from the state space model is
then terminated and instead the sum of the output from each transfer function is used. It
can easily be shown that ifT

2
is unaugmented in the
1
s
q

e
transfer function, the output of
the system will be the same as that shown in Figure 5.3.
For majority of the results to follow, the aircraft was subjected to turbulence. In order
50
to subject the aircraft to turbulence the state space input matrix had to be changed. Turbu-
lence acts on the velocity and rates of rotation and therefore any changes will be modeled
as an input to the plant. In the state space equations, 2.33 and 2.34, it can be seen that the
first three states are the velocity and or rotation rate states. Therefore, adding the first three
columns of theA matrix to theB matrix, the newB matrix takes after the following form,
B
new
= [B
old
;A(:; 1 : 3)] (5.5)
The new input u then becomes, [
e
,
s
,
t
,
u
,
w
,
q
]
0
for the longitudinal case, and [
a
,
r
,

v
,
p
,
r
]
0
for the lateral case. The turbulence generated comes from the Dryden Model
and the case primarily investigated was light turbulence, with the probability of exceedance
of high-altitude intensity of 10
2
. According to reference [14] the requirements are that the
aircrafts must be able to refuel in moderate turbulence; however, from pilot ratings the
classification was made that from the Dryden Model moderate turbulence lies somewhere
between light and moderate, Figure 5.6 better depicts the turbulence intensity as a func-
tion of altitude. It was this reason that turbulence results were only obtained for the light
turbulence. The turbulence that each aircraft was exposed to ultimately came from the
same model. However, a continuous time delay was applied to the turbulence as the tanker
Figure 5.5: Receiver and Tanker Trubulence Delay
approached the receiver. Therefore, the receiver was exposed to the turbulence and in some
51
Figure 5.6: Probability of Exceedance at Medium to High Altitude Turbulence Intensities
[3]
time, based on speed and distance away, the tanker was exposed to the same turbulence.
A sample turbulence of the receiver and tanker can be seen in Figure 5.5. This model is
consistent with the normal assumption that atmospheric turbulence (for aircraft design) can
be viewed as a spatially fixed time invariant model. The aircraft penetrate the model at
such a high speed, relatively to the turbulence time variation, that the turbulence can be
considered fixed in time.
Before continuing to the parameter variation a baseline test must first be conducted.
Figure 5.7 shows the probe X, Y , and Z position as it closes in on the drogue in the presence
of no turbulence and using control method one. The profiles seen follow the flight paths
determined in the preceding chapter.
In the results to be seen there are three areas of interest when the probe strikes the
drogue. The most obvious is a miss, that is if the probe strikes the drogue outside of the
diameter of the drogue. The next area is between two thirds the diameter of the drogue and
the diameter of the drogue. In this area the strike is uncertain whether or not it would lead
to a connection [14]. Therefore area three is less than two thirds the diameter of the drogue.
52
0 10 20 30 40 50 60
−10
0
10
20
30
40
50
time (sec)
X (m)
Longitudinal Profile
0 10 20 30 40 50 60
−3
−2.5
−2
−1.5
−1
−0.5
0
time (sec)
Y (m)
Lateral Profile
0 10 20 30 40 50 60
−12
−10
−8
−6
−4
−2
0
time (sec)
Z (m)
Altitude Profile
Figure 5.7: Probe X, Y , and Z Position: Baseline Profile
That is, if the probe strikes within this zone the strike is considered as making a successful
connection.
5.1 X
u
In the variation ofX
u
a number of cases will be shown. To start off, a variation ofX
u
is performed without any turbulence. The range of variance, as mentioned earlier, does not
seem to show a very significant change in response. In Figure 5.8 theX andZ closures to
the drogue are shown with a variation inX
u
. The blue line is indicative of the unchanged
0 10 20 30 40 50 60
−10
0
10
20
30
40
50
time (sec)
X (m)
ΔX vs. ΔXu

X
u
= −0.003
X
u
= −0.0068
X
u
= −0.15
0 10 20 30 40 50
−10
−8
−6
−4
−2
0
time (sec)
Z (m)
ΔZ vs. ΔXu

X
u
= −0.003
X
u
= −0.0068
X
u
= −0.15
Figure 5.8: Response in Z and X with change inX
u
, No Turbulence
value, while the green and red lines show the lowest and highest negative values, respec-
tively. While the response remains fairly similar with a change inX
u
, asX
u
becomes more
53
negative the time it takes to make contact with the drogue increase by a few seconds. This
certainly makes sense as X
u
was derived from the retarding force acting on the aircraft.
Therefore, it is apparent that with a small perturbation, and a more negativeX
u
value, ulti-
mately a greater force is applied. In this case the engine response would have to make up
for the change in drag. Being that the response of the engine is quite slow, ultimately it will
take longer for the aircraft to reach its target.
In Figure 5.9 it can be seen, with controller one and a variation in X
u
, the contact
−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
Y (m)
Z (m)
Z & Y Drogue Contact

−3 −2.5 −2 −1.5 −1 −0.5
x 10
−3
−5.5
−5
−4.5
−4
−3.5
x 10
−3
Y (m)
Z (m)
Z & Y Drogue Contact

X
u
= −0.003
X
u
= −0.0068
X
u
= −0.15
Figure 5.9: Drogue Contact with Change inX
u
, No Turbulence, Controller One
made with the drogue shifts by a few millimeters. However, this change can be considered
negligible as the diameter of the drogue is orders of magnitude larger. To see what the
effect of this small change is in turbulence, 151 difference cases were run for each change
inX
u
. It is to be noted that for each variation inX
u
, the turbulence experienced is different
for each trial, but the same for each variation. Therefore, the same seeds of turbulence are
used per variation. This was done to provide uniformity to the results between cases. If the
turbulence seeds are the same for each variation, then any change occurring to the response,
between two values ofX
u
, is the result of the change alone and not a random turbulence.
A sample of the turbulent contact with the drogue can be seen in Figure 5.10. It is clear
here that the attempts made with controller two are far more successful (i.e. 18% contact
with controller one, and 74% with controller two). In the forthcoming four pages, a
54
Figure 5.10: Controller One (Left) and Two (Right) Drogue Contact,X
u
=0:0068
series of four figures will be presented. They highlight the distribution of data and proba-
bility of success for the two controllers. In Figure 5.11, a plot of the distribution ofY andZ
contacts are presented for controller one. Each box plot shows the percentiles for a differ-
entX
u
value in the standard box plot format (the red + being representative of outliers in
the data). To make it easier for the reader, two horizontal lines have been placed to indicate
the diameter (blue) and two thirds diameter (green) of the drogue. It is apparent from the
figure that almost no change inZ occurs with a differentX
u
, and that all trials resulted in
a hit between the inner two thirds of the drogue. InY it can be seen that for a change in
X
u
the distribution does not change from across the range ofX
u
tested. However, it is also
clear that for each variation inX
u
, the probe misses striking the drogue by over 50% of the
time. The number of successful connections can be seen more clearly in Figure 5.12. The
bar graph shows the probability, for a differentX
u
, of the probe striking the drogue within
the inner two thirds (blue) or outer diameter (blue plus red). It is clear then that the success
rate is not very good, with a peak probability of about 32%, but more importantly there is
no significant difference between each variation ofX
u
. Another insightful plot in Figure
5.12 is a box plot of the time it takes for the connection to occur. For this controller the
time distribution should be just about even as it doesn’t consider any other errors (Y andZ)
55
while closing in on theX error. Primarily for this reason controller two was implemented.
Controller two provides a more realistic approach to the problem, as it couples the
errors ofX,Y , andZ. In Figure 5.13, as before, the distribution of the contact is shown for
bothX andY . From looking at theY plot, it is very apparent that the controller has signif-
icantly increased the probability of the probe making contact with the drogue in the lateral
direction. The distribution inZ has mildly shifted, but the significant improvement inY
justifies the use. From Figure 5.14 it can be seen that the range of probability increased
from 18% to 32% with controller one to 68% to 78% with controller two. However, look-
ing at the time plot it becomes clear that the time to a connection occurring increases by 20
to 30 seconds, with ranges going up as high as over two minutes.
With the results of the variations ofX
u
presented, the effect ofX
u
, in the range pro-
posed, on the ability to make contact with the drogue appears to be insignificant. However,
the effect of just implementing a new control law increased the probability of a successful
connection by a near 50%. The observation of a less scattered probability distribution with
controller two is also indicative that the effect ofX
u
can be overcome by a simple control
law implementation.
56
Figure 5.11: Change inX
u
, Controller One: Y and Z Box Plot (Baseline = -0.0068)
57
Figure 5.12: Change inX
u
, Controller One: Time Box Plot and Overall Probability (Base-
line = -0.0068)
58
Figure 5.13: Change inX
u
, Controller Two: Y and Z Box Plot (Baseline = -0.0068)
59
Figure 5.14: Change inX
u
, Controller Two: Time Box Plot and Overall Probability (Base-
line = -0.0068)
60
5.2 T

2
As in the case ofX
u
the study ofT

2
begins with the non-turbulent investigation. Best
seen in Figure 5.15, the effect of T

2
on the non-turbulent response, compared to X
u
, is
rather significant. The figure shows the closure on altitude,Z, and the contact made with
the drogue. It is quite apparent the higher T

2
the greater the overshoot of the response
becomes. Equally the lower the value ofT

2
becomes the greater the undershoot. However,
from the analysis done, at the beginning of this chapter, this is to be expected, asT

2
deter-
mines the zero placement in the

e
transfer function.
Figure 5.15: Response in Z and Drogue Contact with Change inT

2
, No Turbulence
One may speculate as the effect this may have on the ability to track and close in on the
drogue but a thorough analysis should be performed. As in the case ofX
u
, the variations
forT

2
are performed in light turbulence. T

2
is varied in the range specified at the begin-
ning of the chapter and 151 different trials are ran per variation ofT

2
, per controller. In
Figure 5.16 a sample of the data to be tabulated can be found. The figure shows two results
of controller one and controller two, for two of the sameT

2
values. For the images in the
first row the distribution presented should look familiar as it represents the initial values
the controllers were designed around. Row two however shows a wider spread in the Z
61
direction and much more overshoot.
Figure 5.16: Drogue Contact with Change inT

2
, Light Turbulence. Controller One (Left),
and Two (Right).
All of the results, as before have been tabulated in the four pages to come. Figure 5.17
shows the respective distribution in Z and Y as a result of changing T

2
for the imple-
mentation of controller one. From this figure it can be seen that the distribution inZ, per
change in T

2
, is quite significant. The original T

2
value can be seen to fit right within
the drogue diameter lines drawn, while other values from that either bias toward an over-
shoot or undershoot. TheY distribution remains widespread, again as controller one only
minimizes each error respectively without the consideration of errors in the other axis. The
62
probability of a successful connection, as seen in Figure 5.18 remains quite low. It is clear
looking at Figure 5.18 that the best outcome came from theT

2
value for which the initial
design was conducted.
To get a better feel forT

2
, the results of controller two offer a few more clues. From
Figure 5.19 it is seen that the response in Y has improved, with almost 50% of the dis-
tribution fitting within the drogue diameter lines. However, it is still apparent that asT

2
shifts away from the initial design value, the probability of a successful connection in Z
decreases. Although controller two does do a better job at keeping the distributions from
fanning out forT

2
between 1.4 and 0.63, that is for lower values than the initial design.
It appears that for higher values, although the median has changed, the distribution of data
is about the same as controller one. Looking at the probability of a successful contact,
Figure 5.20 shows that the probability of a successful connection increased but compared
to the initialT

2
value all the other variations have about the same profile as controller one.
Looking at the time distribution however provides input to how hard the controller had to
work for the differentT

2
values. With lowerT

2
values the time to attempt a connection is
by far more than the time for aT

2
value of greater than 0.95.
63
Figure 5.17: Change inT

2
, Controller One: Y and Z Box Plot (Baseline = 1.4)
64
Figure 5.18: Change inT

2
, Controller One: Time Box Plot and Overall Probability (Base-
line = 1.4)
65
Figure 5.19: Change inT

2
, Controller Two: Y and Z Box Plot (Baseline = 1.4)
66
Figure 5.20: Change inT

2
, Controller Two: Time Box Plot and Overall Probability (Base-
line = 1.4)
67
Chapter 6
Conclusion and Future Work
6.1 Conclusion
Throughout the course of this thesis the moments of inertia of the Global Hawk RQ-4A
have been found, an aerodynamic model has been developed, and a state space model of the
aircraft was presented. A series of feedback loop closures were then performed to augment
the performance of the tanker and receiver. In the error minimization of closing in on the
drogue, two controllers were investigated. One closed in on the drogue regardless of the
error of the other axis; while, two only advanced in theX direction once theY andZ error
had become less than the diameter of the drogue.
The goal of the thesis was to investigate the effects of the parameters X
u
and T

2
on
the ability of the two aircraft to refuel autonomously. While the receiver tried to maintain
straight-and-level flight it was the task of the tanker to perform the maneuvers necessary to
make contact with the drogue. Therefore, two scenarios were investigated in the presence
of turbulence and no turbulence. With respect to T

2
an approximation was derived and
showed that four main stability derivatives influenced the parameter.
It was found that the termX
u
mainly had an effect on the phugoid roots of the aircraft,
while T

2
affected the closed-loop short period roots. The investigation led to the con-
clusion that the parameterX
u
was not a very big contributer to the ability of the aircraft to
make a successful rendezvous. In the range thatX
u
was varied, -0.15 to 0.0, the probability
of a successful connection, for both controllers, between changes inX
u
stayed relatively
the same, even though the probability of success for controller two far outperformed that
of one. With a variation in X
u
and implementation of controller one, the probability of
68
a drogue contact remained between 20% to 30%; while, with the implementation of con-
troller two, the probability increased to between 70% and 80%.
With respect toT

2
, given a fixed control law, it was found that there is a significant sen-
sitivity in task performance with variation inT

2
. That is, the parameter variation showed
that the highest probability for a successful connection occurred at the value for which
the control laws were initially designed. Any variation inT

2
from the initial design point
led to a significant decrease in the probability of a successful connection with the drogue.
Varying T

2
, 11.3 to 0.33, it was found that for controller one, the probability of contact
with the drogue dropped from about 25%, for the initialT

2
design, to less than 10% for all
other values. With the use of controller two, the probability increased but the change from
the initial design value remains apparent as the probability of contact went from about 73%
to less than 25% for extreme values ofT

2
.
6.2 Future Work
With respect to a further investigation, of the parameters investigated in this thesis, it
is proposed that a further sensitivity study be performed. This thesis presumed and varied
T

2
to a few extreme points. If a sensitivity study is done on the terms that effect T

2
most, then those values could be used to perform another set of in-flight aerial refueling
simulation tests. The hope being that if T

2
could be monitored a control law could be
developed to give the best possible outcome in an attempt to refuel between two UA Vs. For
X
u
a further study is recommended to be performed, primarily in the presence of stronger
turbulence and a variation in turbulence frequency. The use of spoilers is recommended as
they provide drag control at a moderate to high bandwidth. The idea being that for high
frequency velocity control the engine overload is kept at a minimum and the spoilers are
used. The effect of spoiler use then could be monitored for any advantages with change in
turbulence conditions.
69
Bibliography
[1] Engineering technical letter (etl) 09-1: Airfield planning and design criteria for
unmanned aircraft systems (uas), Sep 2009.
[2] Global hawks blaze trail in autonomous refuelling test darpa completes autonomous
high-altitude refueling tests. Flight International, Oct. 2012.
[3] Dryden wind turbulence model. Matlab Aerospace Blockset Dryden Wind Turbulence
Model Description, May 2013.
[4] Uiuc airfoil coordinates database, January 2013.
[5] Richard J. Adams, James M. Buffington, Andrew G. Sparks, and Siva S. Banda.
RobustMultivariableFlightControl. Springer-Verlag London Limited, 1994.
[6] Ralph C. A’Harrah and Jerry L. Lockenour. Approach flying qualities - another chap-
ter. InAIAAGuidance,Control,andFlightMechanicsConference.
[7] Randal W. Beard and Timothy W. McLain. Small Unmanned Aircraft. Princeton
University Press, 2010.
[8] John H. Blakelock. Automatic Control of Aircraft and Missiles. John Wiley & Sons,
Inc., 1965.
[9] A. W. Bloy and K. A. Lea. Directional stability of a large receiver aircraft in air-to-air
refuling. J.Aircraft, 32(2):453–455.
[10] Christopher Bolkcom. Air force aerial refuelinh methods: Flying boom versus hose-
and-drogue. Technical report, june 2006.
[11] Giampiero Campa. Simulation environment for machine vision based aerial refueling
for uavs. IEEETransactionsonAerospaceandElectronicSystems, 45(1), 2009.
[12] Chao-I Chen and Roger Stettner. Drogue tracking using 3d flash lidar for autonomous
aerial refueling.
[13] Bernard Etkin and Lloyd D. Reid. Dynamics of Flight. John Wiley & Sons, Inc.,
1976.
70
[14] Pablo Gonzalez. Industry expert-northrop grumman. Personal Communication, Jan
2013.
[15] Timothy Allen Lewis. Flight data analysis and simulation of wind effects during aerial
refueling. Master’s thesis, University of Texas at Arlington, 2008.
[16] W. Mao and F.O. Eke. A survey of the dynamics and control of aircraft during aerial
refueling. NonlinearDynamicsandSystemsTheory, pages 375–388, 2008.
[17] Barnes W. McCormick. Aerodynamics Aeronautics and Flight Mechanics. John
Wiley & Sons, Inc., 1995.
[18] Duane McRuer, Irving Ashkenas, and Dunstan Graham. AircraftDynamicsandAuto-
maticControl. Princeton University Press, 1973.
[19] Duane T. McRuer, Irving L. Ashkenas, and C. L. Guerre. A systems analysis view of
logitudinal flying qualities. Technical report, Systems Technology, Inc., 1960.
[20] Warren F. Phillips. MechanicsofFlight. John Wiley & Sons, Inc., 2010.
[21] Jan Roskam. Methods for estimating stability and control derivatice of conventional
subsonic airplanes. Technical report, 1973.
[22] Phil Stern. Industry expert-northrop grumman. Personal Communication, Oct 2012.
[23] Monish D. Tandale, Roshawn Bowers, and John Valasek. Trajectory tracking con-
troller for vision-based probe and drogue autonomous aerial refueling. Journal of
Guidance,Control,andDynamics, 2006.
[24] H. H. B. M. Thomas. Estimation of stability derivatives (state of the art). Technical
report, Ministry of Aviation, 1963.
[25] Thomas R. Yechout. Introduction to Aircraft Flight Mechanics. American Institute
of Aeronautics and Astronautics, 2003.
71
Appendix A
**
* File : GH_RQ-4A.dcm
**
*
* Author : Endri Kerci
* University of Southern California
* endri.kerci@gmail.com
*
* Aircraft: GLOBAL HAWK RQ-4A
**
** ALL UNITS -> METRIC ... ALL DERIVATIVE TERMS -> Per RAD
CASEID --- RQ-4A --- Ailerons
** Setup Flight Conditions
$FLTCON LOOP=2.0$
$FLTCON NMACH = 2.0, MACH(1)= 0.47, 0.48$
$FLTCON NALT = 1.0, ALT(1) = 10000.0$
$FLTCON NALPHA= 17.0,
ALSCHD(1)= -7.0,-6.0,-5.0,-4.0,-3.0,-2.6,-2.0,-1.0,0.0,1.0,2.0,3.0,4.0,5.0,6.0,7.0,8.0$
** Define Locations (With Respect to Nose of Aircraft) -> Center of Gravity -> Wing -> Tail
** Change in CG between fully fueled plane and empty plane -> 5% MAC
** XCG (empty) -> 7.0764 (m) ... XCG (fully fueled) -> 7.1526 (m)
** XCG (Receiver) -> 7.0857 (m) ... XCG (Tanker) -> 7.1526 (m)
** PLEASE CHANGE XCG ACCORDINGLY BEFORE RUNNING
$SYNTHS XCG= 7.0857, ZCG= 0.0,
XW = 5.8249, ZW =-0.39,
XH = 11.812, ZH =-0.15, XV= 11.812, ZV= 0.00, ALIH=-7.0, ALIW=0.00$
$OPTINS SREF=49.90, CBARR=1.5047, BLREF=35.4178$
** Fuselage Data -> Checked vs. SolidWorks Model
$BODY NX=19.0, ITYPE=1.0,
X(1)=0.000, 0.250, 0.750, 1.250, 1.750, 2.250, 2.660, 3.070,
3.890, 4.300, 4.710, 5.530, 6.350, 7.170, 7.990, 11.390, 12.650,
13.110, 13.540,
R(1)=0.000, 0.399, 0.689, 0.732, 0.750, 0.757, 0.757,
0.757, 0.757, 0.757, 0.757, 0.757, 0.757, 0.757, 0.757,
0.757, 0.757, 0.757, 0.0,
ZU(1)=0.514, 0.772, 1.262, 1.480, 1.613, 1.667, 1.629, 1.540,
1.359, 1.232, 1.111, 0.808, 0.772, 0.760, 1.728, 1.637, 1.576,
0.650, 0.43,
ZL(1)=0.101, -0.450, -0.546, -0.558, -0.613, -0.613, -0.620,
-0.728, -1.030, -1.079, -1.085, -0.988, -0.794, -0.716, -0.746,
-0.649, -0.432, -0.344, 0.416$
** Wing Data -> Checked
$WGPLNF CHRDR=2.81117, CHRDBP=1.86672, CHRDTP=0.71882, SSPNE=16.9469,
SSPN =17.7089, SSPNOP=15.5702, SAVSI =7.0, SAVSO=7.0,
CHSTAT=0.0, TWISTA=0.0, DHDADI=0.0, DHDADO=0.0,
TYPE=1.0$
** Wing Airfoil (NASA NLF1015) Data -> Checked
$WGSCHR TYPEIN=1.0, NPTS=49.0, DWASH = 1.0,
XCORD= 0.00040, 0.00080, 0.00250, 0.00600, 0.01030, 0.01770,
72
0.02030, 0.02381, 0.04762, 0.07143, 0.09524, 0.11905,
0.14286, 0.16667, 0.19048, 0.21429, 0.23810, 0.26190,
0.28571, 0.30952, 0.33333, 0.35714, 0.38095, 0.40476,
0.42857, 0.45238, 0.47619, 0.50000, 0.52381, 0.54762,
0.57143, 0.59524, 0.61905, 0.64286, 0.66667, 0.69048,
0.71429, 0.73810, 0.76190, 0.78571, 0.80952, 0.83333,
0.85714, 0.88095, 0.90476, 0.92857, 0.95238, 0.97619,
1.00000,
YUPPER= 0.00000, 0.00260, 0.00620, 0.01200, 0.01690, 0.02370,
0.02590, 0.02889, 0.04381, 0.05543, 0.06503, 0.07326,
0.08042, 0.08671, 0.09226, 0.09716, 0.10147, 0.10526,
0.10855, 0.11138, 0.11378, 0.11574, 0.11730, 0.11845,
0.11920, 0.11954, 0.11947, 0.11898, 0.11803, 0.11657,
0.11456, 0.11194, 0.10870, 0.10485, 0.10045, 0.09561,
0.09037, 0.08477, 0.07885, 0.07262, 0.06611, 0.05935,
0.05232, 0.04504, 0.03747, 0.02957, 0.02103, 0.01101,
0.00000,
YLOWER= 0.00000, -0.00110, -0.00560, -0.00790, -0.01070, -0.01350,
-0.01450, -0.01573, -0.02128, -0.02511, -0.02771, -0.02964,
-0.03113, -0.03221, -0.03299, -0.03353, -0.03385, -0.03399,
-0.03396, -0.03377, -0.03343, -0.03295, -0.03232, -0.03154,
-0.03063, -0.02954, -0.02826, -0.02671, -0.02485, -0.02265,
-0.02016, -0.01742, -0.01452, -0.01155, -0.00855, -0.00558,
-0.00268, 0.00010, 0.00271, 0.00509, 0.00719, 0.00894,
0.01026, 0.01107, 0.01125, 0.01065, 0.00901, 0.00577,
0.00000$
* Wing Aileron
$ASYFLP STYPE=4.0, NDELTA=9.0,
DELTAL(1)=-32.0,-20.0,-10.0,-5.0,0.0, 5.0, 10.0, 20.0, 32.0,
DELTAR(1)= 32.0, 20.0, 10.0, 5.0,0.0,-5.0,-10.0,-20.0,-32.0,
SPANFI=8.2296, SPANFO=13.4112, CHRDFI=0.23876, CHRDFO=0.23876$
NACA V 4 0015
$VTPLNF TYPE=1.0,
CHRDR =1.524, CHRDTP=0.762, SSPN =4.2672, SSPNE =3.505,
CHSTAT=0.000, SAVSI =6.203$
DIM M
DERIV RAD
* DAMP
* TRIM
SAVE
NEXT CASE
**************************************************************************************
* Need to comment this CASEID in order to not get #QNB in Cm data for elevator.
* Error is unknown but worked around.
* CASEID --- RQ-4A --- Wing Inner Spoiler
* * Wing Spoiler
* $SYMFLP FTYPE=1.0, NTYPE=1.0, NDELTA=5.0,
* DELTA(1)= 0.0, -10.0, -20.0, -30.0, -45.0,
* CHRDFI=0.355, CHRDFO=0.305, SPANFI=2.286, SPANFO=4.572$
* * DAMP
* SAVE
* NEXT CASE
73
***************************************************************************************
CASEID --- RQ-4A --- Elevator
* Tail Airfoil -> NOT CHECKED -> Estimated from images, symmetric with 15% thickness
NACA H 4 0015
$HTPLNF TYPE=2.0,
CHRDR =1.524, CHRDTP=0.762, SSPN =4.2672, SSPNE =3.5052,
CHSTAT=0.000, SAVSI =0.0, TWISTA=0.0000, DHDADI=0.000,
CHRDBP=1.524, SSPNOP=3.500, SAVSO =6.203, DHDADO=49.0$
$SYMFLP FTYPE=1.0, NTYPE=1.0, NDELTA=9.0,
DELTA(1)= -32.0, -20.0, -10.0, -5.0, 0.0, 0.7, 10.0, 20.0, 32.0,
CHRDFI=0.45, CHRDFO=0.22, SPANFI=1.000, SPANFO=4.2672$
DAMP
TRIM
* SAVE
* NEXT CASE
74
Appendix B
This appendix includes a set of sample trial tests from the simulations ran in this thesis.
For bothX
u
andT

2
a set of values were picked for which the distribution of connection
attempts are shown. These attempts are shown both for controller one and two mentioned
throughout the paper.
75
Figure 1: Controller One: Drogue Contact with Change inX
u
76
Figure 2: Controller Two: Drogue Contact with Change inX
u
77
Figure 3: Controller One: Drogue Contact with Change inT

2
78
Figure 4: Controller Two: Drogue Contact with Change inT

2
79

Abstract (if available)

Abstract The effect of two parameters, Xᵤ and T_{theta_2} , are explored for their influence on autonomous in-flight aerial refueling. It is believed that these two parameters are very influential to longitudinal control. A Global Hawk (RQ-4A, receiver) to Global Hawk (RQ-4A, tanker) configuration is considered, using a probe-and-drogue fueling method. The refueling is simulated via a 6 DOF state space model developed for both the receiver and the tanker. The receiver and tanker model formulations are presented as well as the probability distribution of a successful connection, probe to drogue, with the variation in each predefined parameter. For each parameter the study was performed with two unique control law implementations (linear and nonlinear). The parameter Xᵤ was varied from -0.15 to 0.0, while T_{theta_2} was varied from 11.3 to 0.33. It was found, with the varying of parameter Xᵤ, the probability of a successful connection remained rather unchanged. With the implementation of control law one, the probability of a drogue contact was between 20% to 30%. While for control law two the probability increased to between 70% to 80%. With the variation of T_{theta_2} , it became apparent that the probability of a successful connection quickly dropped veering away from the value for which the control laws were implemented. With respect to control law one, the probability of contact with the drogue dropped from about 25%, for the initial T_{theta_2} design, to less than 10% for all other values. With the use of control law two, the probability increased with respect to control law one

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Asset Metadata

Creator Kerci, Endri (author)

Core Title The effect of Xᵤ and T_{theta_2} on autonomous in-flight refueling: a global Hawk RQ-4A approach

School Andrew and Erna Viterbi School of Engineering

Degree Master of Science

Degree Program Aerospace Engineering

Publication Date 07/30/2013

Defense Date 06/07/2013

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag Autonomous,Global Hawk,in-flight refueling,OAI-PMH Harvest,simulation

Format application/pdf (imt)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Lockenour, Jerry (committee chair), Ioannou, Petros (committee member), Radovich, Charles (committee member), Yang, Bingen (Ben) (committee member)

Creator Email endri.kerci@gmail.com,kerci@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c3-306375

Unique identifier UC11293541

Identifier etd-KerciEndri-1893.pdf (filename),usctheses-c3-306375 (legacy record id)

Legacy Identifier etd-KerciEndri-1893.pdf

Dmrecord 306375

Document Type Thesis

Format application/pdf (imt)

Rights Kerci, Endri

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA