VISIONBASED CONTROL FOR FLIGHT RELATIVE TO DYNAMIC ENVIRONMENTS
By
RYAN SCOTT CAUSEY
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
2007
@ 2007 Ryan Scott Causey
To my lovely wife, Liza P. Causey, that has supported me every step of the way. Her love and
understanding through the years have brought my passion for life beyond boundaries.
ACKNOWLEDGMENTS
This work was supported jointly by NASA under NNDO4GRR13H with Steve Jacobson
and Joe Pahle as project managers along with the Air Force Research Laboratory and the Air
Force Office of Scientific Research under F496200310381 with Johnny Evers, Neal Glassman,
Sharon Heise, and Robert Sierakowski as project monitors. Additionally, I thank Dr. Rick Lind
for his remarkable guidance and inspiration that will truly last a life time. Finally, I thank my
parents Sandra and James Causey for making this journey possible by providing me the guidance
and discipline needed to be successful.
TABLE OF CONTENTS
page
ACKNOWLEDGMENTS ........ .. .. 4
LIST OF TABLES .......... ............ 8
LISTOFFIGURES ............. .............. 9
LISTOFTERMS .......... ............. 12
ABSTRACT............_ ........ ...... 19
CHAPTER
1 INTRODUCTION . .... ... .. . 21
1.1 Motivation ........ . .. .. 21
1.2 Problem Statement ........ .. .. 27
1.3 Potential Missions . ... .. .. . 27
1.4 System Architecture ......... ... .. 30
1.5 Contributions ......... . ... .. 33
2 LITERATURE REVIEW .......... . ... .. 36
2. 1 Detection of Moving Objects . . .. 36
2.2 State Estimation Using Vision Information .. .. . 38
2.2.1 Localization ....._.. .... .. 39
2.2.2 Mapping ........... ........... 39
2.2.3 TargetMotion Estimation . ... .. .. 40
2.3 Modeling Object Motion ....... ... .. 41
2.4 Uncertainty in Vision Algorithms . ... .. .. 42
2.5 Control Using Visual Feedback in Dynamic Environments .. .. .. .. .. 43
3 IMAGE PROCESSING AND COMPUTER VISION ... .. .. 45
3.1 Camera Geometry ......... . ... .. 45
3.2 Camera Model ......... . ... .. 47
3.2.1 Ideal Perspective ......... .. .. 47
3.2.2 Intrinsic Parameters ........ .. .. 48
3.2.3 Extrinsic Parameters ........ .. .. 49
3.2.4 Radial Distortion . ..... ... . 50
3.3 Feature Point Detection ......... .. .. 51
3.4 Feature Point Tracking ......... .. .. 53
3.5 Optic Flow ............. ............ 56
3.6 TwoView Image Geometry ........ .. .. 56
3.6.1 Epipolar Constraint ........ .. .. 57
3.6.2 EightPoint Algorithm ....... .. .. 59
3.6.3 Planar Homography .. . .... .. 61
3.6.4 Structure from Motion . ...... ... .. 65
4 EFFECTS ON STATE ESTIMATION FROM VISION UNCERTAINTY .. .. .. 67
4. 1 Feature Points ......... . .. .. 67
4.2 Optical Flow ......... ... .. 70
4.3 Epipolar Geometry ......... . .. .. 71
4.4 Homography ........ . ... .. 73
4.5 Structure From Motion ....... ... .. 75
5 SYSTEM DYNAMICS ......... . ... .. 77
5.1 Dyanmic States ......... . ... .. 77
5.1.1 Aircraft ............. ........... 77
5.1.2 Camera ............. ........... 79
5.2 System Geometry ......... . ... .. 81
5.3 Nonlinear Aircraft Equations ....... .. .. 83
5.4 AircraftCamera System ......... ... .. 84
5.4.1 Feature Point Position ....... .. .. 85
5.4.2 Feature Point Velocity ..... ... .. 85
5.5 System Formulation ........ .... .. 86
5.6 Simulating ....._.. . ... .. 89
6 DISCERNING MOVING TARGET FROM STATIONARY TARGETS .. .. .. 90
6. 1 Camera Motion Compensation . ...... ... .. 90
6.2 Classification ......... . ... .. 95
7 HOMOGRAPHY APPROACH TO MOVING TARGETS ... .. .. .. .. 98
7.1 Introduction ......... . ... .. 98
7.2 State Estimation . . ...... .01
7.2.1 System Description .............10
7.2.2 Homography Estimation .............10
8 MODELING TARGET MOTION.............11
8.1 Introduction .............. ............ 111
8.2 Dynamic Modeling of an Object.............11
8.2.1 Motion Models .............12
8.2.2 Stochastic Prediction .............13
9 CONTROL DESIGN ..............17
9. 1 Control Objectives .............17
9.2 Controller Development .. . .... .. 118
9.2.1 Altitude Control . .... ... . 18
9.2.2 Heading Control ......... .. .... .. 119
9.2.3 Depth Control . . .. ..... .21
10 SIMULATIONS ............. ..............123
10.1 Example 1: Feature Point Generation ..... .... .. .. 123
10.2 Example 2: Feature Point Uncertainty .... .... . .. 126
10.2.1 Scenario . .. .... .. .26
10.2.2 Optic Flow ......... ... .. .. 128
10.2.3 The Epipolar Constraint . .... .. .. 130
10.2.4 Structure From Motion . ... .. .. 132
10.3 Example 3: Openloop Ground Vehicle Estimation .. . 133
10.3.1 System Model . ... ..... .. .34
10.3.2 Openloop Results . .. .. .. .. 135
10.4 Example 4: Closedloop Aerial Refueling of a UAV .. .. .. .. .. 138
10.4.1 System Model ... . ..... .. .39
10.4.2 Control Tuning . ..... .. . 140
10.4.3 Closedloop Results . ... .... .14
10.4.4 Uncertainty Analysis ....... ... .. .. 148
11 CONCLUSION ............. ..............151
REFERENCES ......... . ..... .. 154
BIOGRAPHICAL SKETCH ......... .. ... .. 164
LIST OF TABLES
Table
31
101
102
103
104
105
106
107
108
109
1010
page
.. 64
.... . .23
..... . 13
.... . 14
.... . .25
.. .126
..... . 19
.... . 31
... .. . 33
. .. . 10
. .. .. .. . 10
Solutions for homography decomposition ......
States of the cameras ......
Limits on image coordinates ......
States of the feature points ......
Aircraft states ......
Image coordinates of feature points .......
Effects of camera perturbations on optic flow ......
Effects of camera perturbations on epipolar geometry .....
Effects of camera perturbations on structure from motion ...
Maximum variations in position due to parametric uncertainty
Maximum variations in attitude due to parametric uncertainty .
LIST OF FIGURES
Figure page
11 TheUAV fleet ............. ....... ...... 23
12 AeroVironment's MAV: The Black Widow ..... .. .. 23
13 The UF MAV fleet ......... . . .. 24
14 Refueling approach using the probedrogue method .. .. .. .. .. 28
15 Tracking a pursuit vehicle using a vision equipped UAV .. .. . .. 30
16 Closedloop block diagram with visual state estimation .. .. .. .. 31
31 Mapping from environment to image plane ..... .. .. 46
32 Image plane field of view (top view) . ..... .. .. 46
33 Radial distortion effects ......... .. .. .. 51
34 Geometry of the epipolar constraint . ...... ... .. 58
35 Geometry of the planar homography . ..... .. .. 62
41 Feature point dependence on focal length ..... .... .. 68
42 Feature point dependence on radial distortion .... .. . 68
51 Bodyfixed coordinate frame . ..... .. .. .. 78
52 Camerafixed coordinate frame ........ .. .. 80
53 Scenario for visionbased feedback . ...... ... .. 81
61 Epipolar lines across two image frames . .. .. .. 91
62 FOE constraint on translational optic flow for static feature points .. . 94
63 Residual optic flow for dynamic environments ... .. .. 95
71 System vector description . ... .. .. 102
72 Moving target vector description . .... .. .. 103
91 Altitude hold block diagram ......... .. .... .. 118
92 Heading hold block diagram . ..... .. .. 119
101 Virtual environment for example 1 . ..... .. .. 124
102 Feature point measurements for example 1 .... .... . .. 125
103
104
105
106
107
108
109
1010
1011
1012
1013
1014
1015
1016
1017
1018
1019
1020
1021
1022
1023
1024
1025
1026
1027
1028
Optic flow measurements for example 1 .....
Virtual environment for example 2 ......
Feature points across two image frames .....
Uncertainty in feature point .......
Uncertainty results in optic flow ......
Nominal epipolar lines between two image frames
Uncertainty results for epipolar geometry ....
Nominal estimation using structure from motion .
Uncertainty results for structure from motion ..
Vehicle trajectories for example 3 ......
Position states of the UAV with onboard camera
Attitude states of the UAV with onboard camera
Position states of the reference vehicle ......
Attitude states of the reference vehicle ......
Position states of the target vehicle .......
Attitude states of the target vehicle .......
Norm error .......
Relative position states ......
Relative attitude states ......
Virtual environment .......
Innerloop pitch to pitch command Bode plot ..
Pitch angle step response ......
Altitude step response .......
Innerloop roll to roll command Bode plot ....
Roll angle step response ......
Heading response .......
. .126
..... .. .. .127
....... .. .128
........ .. .128
........ .. .129
. .. ... .. .130
. ...... .. .131
.... .. .. .132
. ... .. .. .133
........ .. .134
.... .. .. .135
.... .. .. .135
....... .. .135
....... .. .136
........ .. .136
........ .. .136
......... .. .. 137
.......... .. 137
.......... .. 137
........ .. .138
. ... .. .. .141
........ .. .141
.......... .. 142
. .... .. .. .143
.......... .. 144
......... .. .. 144
1029 Openloop estimation of target's inertial position
1030
1031
1032
1033
1034
1035
1036
1037
Openloop estimation of target's inertial attitude .
Norm error for target state estimates ......
Closedloop target position tracking ......
Position tracking error ......
Target attitude tracking ......
Tracking error in heading angle .......
Target's inertial position with uncertainty bounds
Target's inertial attitude with uncertainty bounds .
.. .. ... . 145
........ .. .146
........ .. .146
.......... .. 147
.......... .. 147
........ .. .147
.... .. .. .149
. .... ... .. .149
LIST OF TERMS
Acceleration of the target in E
Bodyfixed coordinate frame components
Position vector of camera center in camerafixed coordinated frame
Radial distortion
Nominal radial distortion
Earthfixed coordinate frame components
Focal length
Nominal focal length
Altitude state
Stacked column vector of the entries of the planar homography matrix
Altitude command
Nominal entries of the planar homography matrix
Image motion model
Camerafixed coordinate frame components
Proportional gain on altitude error
Proportional gain on pitch rate
Proportional gain to roll rate
Proportional gain on the lateral position error
Integral gain on the lateral position error
Proportional gain on pitch
Proportional gain to roll
Proportional gain to heading error
Epipolar line in image i
Translation from camerafixed to referencefixed coordinates expressed
relative camerafixed coordinates
a (t)
d
do
{&1l, e2, 83}
f
fo
h
h
he
ho
h(x)
k
kg
k,
kY,
kyi
ke
kg
li
mlF
myr Translation from camerafixed to targetfixed coordinates expressed
relative camerafixed coordinates
mVF Translation from virtual to refemcefixed coordinates expressed relative
virtual coordinates
my, Translation from virtual to targetfixed coordinates expressed relative
virtual coordinates
op Vertical image offset from center to upper left corner in pixel units
ov Horizontal image offset from center to upper left corner in pixel units
p (t) Position of the target in E
pVF Image coordinates in the virtual camera of the reference vehicle
pyr Image coordinates in the virtual camera of the target vehicle
q Stacked column vector of the entries of the essential matrix
qo Nominal entries of the essential matrix
sqc Vertical unit length to pixel scaling
sv Horizontal unit length to pixel scaling
so Image skew factor
u Time rate of change of (p, v)
v (t) Velocity of the target in E
Vb = (u, v, w) Velocity of the bodyfixed frame (velocity of the aircraft in bodyfixed
coordinates)
vc = (uc, Ve, Wc) Velocity of the camerafixed frame along {$1,82, 3 } axes
w (t) Random vector
I Subset image specified by W
xrv Translation from camerafixed to virtual coordinates expressed in
camerafixed coordinates
z Depth components in twoview camera geometry
Zo Nominal depth components in twoview camera geometry
A Twoview feature point matrix using structure from motion
Ao Nominal twoview feature point matrix using structure from motion
B Bodyfixed coordinate frame
C Twoview feature point matrix using epiploar methods
Co Nominal twoview feature point matrix
C Classification group of features to a focus of expansion
D Distance from plane to optical center
E Earthfixed inertial coordinate frame
F Fundamental camera matrix
F Referencefixed coordinate frame
Fb = (F ,F,, F ) Aerodynamic forces about {$~1?, 8 3) aXes
G Image outer product summation
G, Altitude compensator
H Planar homography matrix
IJ Velocity vector of a feature in the image plane (optic flow)
IJo Nominal image plane optic flow
K Intrinsic parameter matrix
I Camerafixed coordinate frame
I, Image gradient in the vertical direction
I, Image gradient in the horizontal direction
Mbr = (L, M, N) Aerodynamic moments about {$1, 82 3) aXes
N Normal vector of the plane containing feature points expressed in I
P (x)
R
RBI
REB
REF
RET
REV
RFV
Rry
Rl
T
T
TBI =
TEB =
U
fl
V
W
X
Xo
Y
lk
3 h
Probability density function
Essential matrix
Relative rotation
Rotational transformation from bodyfixed to camerafixed coordinates
Rotational transformation from Earthfixed to bodyfixed coordinates
Rotational transformation from Earthfixed to reference coordinates
Rotational transformation from Earthfixed to target coordinates
Rotational transformation from Earthfixed to virtual coordinates
Rotational transformation from referencefixed to virtual coordinates
Rotational transformation from targetfixed to virtual coordinates
Rotational transformation from camerafixed to virtual coordinates
Relative translation
Targetfixed coordinate frame
Position of camera along {$1,82, ~3} axes
Position of aircraft along {&1l, e^2, 83 } axes
Control input vector
Classification group of features to an independently moving object
Virtual coordinate frame
Search window in the image
Vector of aircraft states
Vector of initial aircraft states
Feature point measurements in the image plane
Camera parameter of the k camera
Horizontal angle for field of view
(xc,Yc,Zc)
 (Xb, Yb ,Zb)
'Yv Vertical angle for field of view
87 A variation in focal length
6d A variation in radial distortion
8,, A variation in pu
6v A variation in v
8 y A variation in optic flow
Sc A variation in the twoview feature point matrix
Sq A variation in the entries of the essential matrix
Sq, A variation in the twoview feature point matrix using the planar
homography matrix
Sh A variation in the entries of the planar homography matrix
6A A variation in the twoview feature point matrix using structure from
motion
8: A variation to the depth components in twoview camera geometry
rl Position vector of feature point relative to and expressed in camera
coordinate frame I
rlF,n Feature point location on reference vehicle realtive and expressed in
camerafixed coordinates
grl,, Feature point location on target vehicle realtive and expressed in
camerafixed coordinates
TIVF,n Feature point location on reference vehicle realtive and expressed in
virtual coordinates
TIV,,, Feature point location on target vehicle realtive and expressed in virtual
coordinates
pu Vertical coordinate in the image plane
pu (x) Mean operator of a vector x
iuo Nominal pu
lufoe Vertical component of the focus of expansion in image coordinates
p'I Vertical coordinate in image plane in pixel units
p't Vertical coordinate in image plane with radial distortion in pixel units
(p, pU) Vertical minimum and maximum coordinates in image plane
A Vertical velocity in the image plane
iit Vertical velocity in the image plane due to moving objects
4; Rotational component of the vertical velocity in the image plane
4, Estimated rotational component of the vertical velocity component
iPRes Residual vertical component of optic flow
At Translational component of the vertical velocity in the image plane
v Horizontal coordinate in the image plane
vo Nominal v
vfoe Horizontal component of the focus of expansion in image coordinates
v' Horizontal coordinate in image plane in pixel units
v't Horizontal image plane coordinate with radial distortion in pixel units
(v, v) Horizontal minimum and maximum coordinates in image plane
9t Horizontal velocity in the image plane
iti Horizontal velocity in the image plane due to moving objects
t, Rotational component of the horizontal velocity in the image plane
t, Estimated rotational component of the horizontal velocity component
9Res Residual horizontal component of optic flow
itr Translational component of the horizontal velocity in the image plane
5 Position vector of feature point relative to and expressed in Earthfixed
coordinate frame E
(F,n
5T,n
Feature point location on reference vehicle realtive and expressed in
Earthfixed coordinates
Feature point location on target vehicle expressed in Earthfixed
coordinates
Variance operator of a vector x
Gradient threshold
Attitude of aircraft about {$1, 82, ~3) aXeS
Attitude of camera about {81l, c^2, 83) aXeS
Roll command
Heading command
Angular rates of aircraft about {$1, 82 3) aXeS
Angular rates of camera about {FIl, c^2, c^3) aXeS
Radial distortion uncertainty bound
Focal length uncertainty bound
Uncertainty bound in the entries of the planar homography matrix
Uncertainty bound in the entries of the essential matrix
Uncertainty bound in depth components
Lateral deviation between vehicle and target
Uncertainty bound in optic flow
Uncertainty bound in pu
Uncertainty bound v
Twoview feature point matrix using the planar homography matrix
Nominal twoview feature point matrix using the planar homography
(#, 6, y)
S= (p, q, r)
me = (Pc, 4c, re)
Ad
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
VISIONBASED CONTROL FOR FLIGHT RELATIVE TO DYNAMIC ENVIRONMENTS
By
Ryan Scott Causey
August 2007
Chair: Richard C. Lind
Major: Aerospace Engineering
The concept of autonomous systems has been considered an enabling technology for a
diverse group of military and civilian applications. The current direction for autonomous systems
is increased capabilities through more advanced systems that are useful for missions that require
autonomous avoidance, navigation, tracking, and docking. To facilitate this level of mission
capability, passive sensors, such as cameras, and complex software are added to the vehicle.
By incorporating an onboard camera, visual information can be processed to interpret the
surroundings. This information allows decision making with increased situational awareness
without the cost of a sensor signature, which is critical in military applications. The concepts
presented in this dissertation facilitate the issues inherent to visionbased state estimation of
moving objects for a monocular camera configuration. The process consists of several stages
involving image processing such as detection, estimation, and modeling. The detection algorithm
segments the motion field through a leastsquares approach and classifies motions not obeying
the dominant trend as independently moving objects. An approach to state estimation of moving
targets is derived using a homography approach. The algorithm requires knowledge of the
camera motion, a reference motion, and additional feature point geometry for both the target and
reference objects. The target state estimates are then observed over time to model the dynamics
using a probabilistic technique. The effects of uncertainty on state estimation due to camera
calibration are considered through a bounded deterministic approach. The system framework
focuses on an aircraft platform of which the system dynamics are derived to relate vehicle states
to image plane quantities. Control designs using standard guidance and navigation schemes
are then applied to the tracking and homing problems using the derived state estimation. Four
simulations are implemented in MATLAB that build on the image concepts present in this
dissertation. The first two simulations deal with feature point computations and the effects of
uncertainty. The third simulation demonstrates the openloop estimation of a target ground
vehicle in pursuit whereas the four implements a homing control design for the Autonomous
Aerial Refueling (AAR) using target estimates as feedback.
CHAPTER 1
INTRODUCTION
1.1 Motivation
Autonomous systems are an enabling technology to facilitate the needs of both military
and civilian applications. The usefulness of autonomous systems ranges from robotic assembly
lines for streamlining an operation to a rover exploring the terrain of a distant planet. The main
motivation behind these types of systems is the removal of a human operator which in many
cases reduces operational cost, human errors, and, most importantly, human risk. In particular,
military missions consistently place soldiers in hazardous environments but in the future could
be performed using an autonomous system. The federal sector is considering autonomous
vehicles, specifically, to play a more prominent role in several missions such as reconnaissance,
surveillance, border patrol, space and planet exploration over the next 30 years [1]. This increase
in capability for such complex tasks requires technology for more advanced systems to further
enhance the situational awareness.
Over the past several years, the interest and demand for autonomous systems has
grown considerably, especially from the Armed Forces. This interest has leveraged funding
opportunities to advance the technology into a state of realizable systems. Some technical
innovations that have emerged from these efforts, from a hardware standpoint, consist mainly
of increasingly capable microprocessors in the sensors, controls, and mission management
computers. The Defense Advanced Research Projects Agency (DARPA) has funded several
projects pertaining to the advancement of electronic devices through size reduction, improved
speed and performance. From these developments, the capability of autonomous system has been
demonstrated on vehicles with strict weight and payload requirements. In essence, the current
technology has matured to a point where autonomous systems are physically achievable for
complex missions but not yet algorithmically capable.
The aerospace community has employed many of the research developed for autonomous
systems and applied it to Unmanned Aerial Vehicles (UAV). Many of these vehicles are currently
operational and have served reconnaissance missions during Operation Iraqi Freedom. The
Department of Defense (DoD) has recorded over 10,000 flight hours performed by UAV in
support of the war in Iraq since September 2004 and that number is expected to increase [1].
Future missions envision UAV to conduct more complex task such as terrain mapping,
surveillance of possible threats, maritime patrol, bomb damage assessment, and eventually
offensive strike. These missions can span over various types of environments and, therefore,
require a wide range of vehicle designs and complex controls to accommodate the associated
tasks.
The requirements and design of UAV are considered to enable a particular mission
capability. Each mission scenario is the driving force of these requirements and are dictated
by range, speed, maneuverability, and operational environment. Current UAV range in size from
less than 1 pound to over 40,000 pounds. Some popular UAV that are operational, in testing
phase, and in the concept phase are depicted in Figure 11 to illustrate the various designs. The
two UAV on the left, Global Hawk and Predator, are currently in operation. Global Hawk is
employed as a high altitude, long endurance reconnaissance vehicle whereas the Predator is
used for surveillance missions at lower altitudes. Meanwhile, the remaining two pictures present
JUCAS, which is a joint collaboration for both the Air Force and Navy. This UAV is described
as a medium altitude flyer with increased maneuverability over Global Hawk and the Predator
and is considered for various missions, some of which have already been demonstrated in flight,
such as weapon delivery and coordinated flight.
The advancements in sensors and computing technology, mentioned earlier, has facilitated
the miniaturization of these UAV, which are referred to as Micro Air Vehicles (MAV). The scale
of these small vehicles ranges from a few feet in wingspan down to a few inches. DARPA has
also funded the first successful MAV project through AeroVironment, as shown in Figure 12,
where basic autonomy was first demonstrated at this scale [2]. These small scales allow highly
agile vehicles that can maneuver in and around obstacles such as buildings and trees. This
capability enables UAV to operate in urban environments, below rooftop levels, to provide
Figure 11. The UAV fleet
the necessary information which cannot be obtained at higher altitudes. Researchers are
currently pursuing MAV technology to accomplish the very same missions stated earlier for
the unique application of operating at low altitudes in cluttered environments. As sensor and
control technologies evolve, these MAV can be equipped with the latest hardware to perform
advanced surveillance operations where the detection, tracking, and classification of threats
are monitored autonomously online. Although a single mirco air vehicle can provide distinct
information, targets may be difficult to monitor due to both flight path and sensor field of view
constraints. This limitation has motivated the idea of a corporative network or a "swarm" of MAV
communicating and working together to accomplish a common task.
Figure 12. AeroVironment's MAV: The Black Widow
..iii.iii ~~.
Currently, several universities have a research facility dedicated to the investigation of
MAV, including Brigham Young University (BYU), Stanford University, Georgia Institute
of Technology, and the University of Florida. The autonomous capabilities demonstrated by
BYU incorporated an autopilot system for waypoint navigation that integrated traditional IMU
sensors [3, 4]. Meanwhile, Stanford has examined motion planning strategies that optimize flight
trajectories to maintain sensor integrity for improved state estimation [5]. The work at Georgia
Tech and BYU has considered corporative control of MAV for autonomous formation flying [6]
and consensus work for distributed task assignment [7]. Alternatively, vision based control has
also been the topic of interest at both Georgia Tech and UF. Control schemes using vision have
been demonstrated on platforms such as a helicopter at Georgia Tech [8], while UF implemented
a MAV that integrated vision based stabilization into a navigation architecture [9, 10]. The
University of Florida has also considered MAV designs that improve the performance and agility
of these vehicles through morphing technology [1113]. Fabrication facilities at UF have enabled
rapid construction of design prototypes useful for both morphing and control testing. The fleet of
MAV produced by UF are illustrated in Figure 13 where the wingspan of these vehicles range
from 24 in down to 4 in.
Figure 13. The UF MAV fleet
There are a number of current difficulties associated with MAV due to their size. For
example, characterizing their dynamics under flight conditions at such low Reynolds numbers
is an extremely challenging task. The consequence of increased agility at this scale also gives
rise to erratic behavior and a severe sensitivity to wind gust and other disturbances. Waszak
et al. [14] performed wind tunnel experiments on 6 inch MAV and obtained the required
stability derivatives for linear and nonlinear simulations. Another critical challenge toward MAV
implementation is their weight restrictions and limited payload capacity. More importantly,
this restriction places constraints on the types and amount of sensors and processors that can be
carried onboard. So sensor selection is a critical process of optimizing size and weight to the
amount of information the sensor provides. This debate has lead researchers toward using vision
as a primary sensor for the guidance and navigation of autonomous UAV and MAY
Vision as a primary sensor is a favorable direction for MAV and even for larger UAV
This sensor provides an enormous amount of information regarding the scene and the way
objects are moving in relation to that scene. Humans rely heavily on visual information, for
instance, being able to navigate through cluttered environments or distinguishing between objects
based on appearance. Although these qualities are to a large percentage visual, humans also
rely on past knowledge for recognition and classification of objects and their motion. So the
challenges that computer vision has faced is how to interpret this information to gain awareness
of the surroundings and, more importantly, perform efficiently for realtime implementation.
Specifically, for autonomous systems to operate in urban environments, vision algorithms are
required to detect objects, track objects, and provide state estimation to make decisions for
navigation.
The features among vision that are most prevalent to UAV and MAV systems are compact
size, passive sensor qualities, low cost, and an abundance of useful data. The compact size of
these sensors enables MAV to carry onboard cameras with little expense toward weight and
payload capacity. The passive nature of these sensors is another desirable quality that increases
the stealth properties of the vehicle by removing emissions. This benefit has obvious advantages
over other sensors, such as sonar and radar, in operations where UAV with a low probability of
detection are required. Additionally, the low cost of vision sensors coupled with the large data
return provides the "more bang for your buck" approach.
The foundation of this dissertation stems from the ability to estimate motion parameters
through output images acquired from a camera system. The amount of information one can
estimate depends on the camera configuration. The two basic types considered in literature are
a single camera setup, known as monocular vision, and a two camera setup, known as stereo
vision. For monocular vision, a sequence of images are taken over time whereas stereo vision
uses two images taken by different cameras at the same time. Motion estimates, using monocular
vision, has been solved for the cases associated with the movement of the camera relative to a
stationary objects and the reverse problem involving movement of objects relative to a stationary
camera. The process of determining camera motion from stationary objects is commonly referred
to as localization. Conversely, determining the motion or position of an object in space from a
pair of images is known as structure from motion. For fixed objects, simultaneous localization
and mapping (SLAM) can be employed to estimate the camera motion in conjunction with the
object's locations. Meanwhile, the use of stereo vision allows one to estimate the motion of
objects while the camera is also moving. Solutions to these methods are well established in
the computer science community and the mathematical details regarding these techniques are
provided in Chapter 3. This dissertation will focus on the monocular camera configuration to
address the state estimation problem regarding moving targets.
The advantage of these techniques becomes more apparent to UAV when applied to
guidance, navigation, and control. By mounting a camera on a vehicle, state estimation of
the vehicle and objects in the environment can be achieved in some instances through vision
processing. Once state estimates are known, they can then be used in feedback. Control
techniques can then be utilized for complex missions that require navigation, path planning,
avoidance, tracking, homing, etc. This general framework of vision processing and control has
been successfully applied to various systems and vehicles including robotic manipulators, ground
vehicles, underwater vehicles, and aerial vehicles but there still exists some critical limitations.
The problematic issues with using vision for state estimation involves camera nonlinearities,
camera calibration, sensitivity to noise, large computational time, limited field of view, and
solving the correspondence problem. A particular set of these image processing issues will be
addressed directly in this dissertation to facilitate the control of autonomous systems in complex
surroundings.
1.2 Problem Statement
The problem addressed in this dissertation consists of target state estimation with unknown
stochastic motion for autonomous systems using a moving monocular camera. The application
for this type of research applies directly to the advancements of autonomous systems by
increasing their mission capabilities. The estimation of 3dimensional points in space given
two perspective views relies heavily on camera configuration, accurate camera calibration,
and perfect image processing, however, the practical realizations in camera systems involve
limitations to configurations and significant uncertainties. In order to estimate the states of a
moving object in the presence of uncertainty, several key issues will be addressed. These include:
1. segmenting moving targets from stationary targets within the scene
2. classifying moving targets into deterministic and stochastic motions
3. coupling the vehicle dynamics into the sensor observations (i.e. images)
4. formulating the homography equations between a moving camera and the viewable targets
5. propagating the effects of uncertainty through the state estimation equations
6. establishing confidence bounds on target state estimation
The design and implementation of a visionbased controller is also presented in this dissertation
to verify and validate many of the concepts pertaining to tracking of moving targets.
1.3 Potential Missions
Various missions involving autonomous navigation will directly benefit from this research.
The estimates of where objects, both stationary and moving, are located in the environment
enable capabilities such as obstacle avoidance, target tracking, object pa ,~pi ng, and even vehicle
docking. NASA has been particularly interested in Autonomous Aerial Refueling (AAR),
where UAV autonomously dock with a tanker aircraft to replenish their fueling supply. This
capability will have enormous benefits to UAV by expanding the potential missions through
increasing range. Employing AAR systems will also have cost benefits in the design due to the
reduced weight in the vehicle caused by fuel. Meanwhile, return to base requests can be reduced
considerably as long as a tanker aircraft is available which will enable a quick and efficient
response to threats around the world.
There are two aerial refueling techniques implemented currently. The first, employed by the
Air Force, involves a remote pilot stationed on the tanker aircraft that manually controls the boom
to the target while the receiver aircraft maintains a fixed attitude and relative position. On the
other hand, the Navy employs the probeanddrogue method. This method involves the receiver
pilot controlling the aircraft to a drogue basket that is attached to the tanker and requires a
relative position accuracy of 0.5 to 1.0 cm [15]. The drogue is designed in an aerodynamic shape
that permits the extension from the tanker without instability. The probeanddrogue method is
considered the preferred method for AAR, mainly due to the high pilot workload in controlling
the boom [16]. Figure 14 illustrates the view observed by receiver aircraft during the refueling
process where feature points have been placed on the drogue.
Figure 14. Refueling approach using the probedrogue method
Vision can be used to facilitate the AAR problem by augmenting traditional aircraft sensors
such as global positioning system (GPS) and inertial measurement unit (IMU). Gigh precision
GPS/IMU sensors can provide relative information between the tanker and the receiver then
vision can be used to provide relative information on the drogue. The advantage to vision in
this case is its passive nature which eliminates sensor emissions during refueling over enemy air
space. Additionally, employing a vision system will prevent from placing sensors, such as GPS,
on the actual drogue itself considering that most sensors are unable to handle the aerodynamic
loads and provide the necessary update rate for refueling. Utilizing vision in the AAR problem
requires accurate estimation of the drogue which entails precise tracking of both vehicles
throughout the mission. This accuracy presents many challenges for a camera system considering
the variations in estimates due to noise, interference, calibration, and feature tracking errors.
Another main challenge is modeling the dynamics of the drogue to estimate a predicted value.
This step is extremely difficult due to the stochastic nature of the drogue's motion during flight
conditions but is needed for control purposes. The issues involved in the AAR problem can
be categorized into several vision processing tasks followed by two control tasks. The vision
processing tasks include detecting, classifying, tracking, and state estimation of moving objects
while the control tasks involve modeling and waypoint navigation.
Additionally, a potential civilian mission that could benefit from this technology is traclong
a highspeed police pursuit. Imagine a scenario where a high speed pursuit is underway through
a city highway. The police typically have several pursuit cars along with a helicopter for air
support. Ordinarily this situation results in a catastrophic accident where police and innocent
civilians are either hurt or killed. Employing visionbased UAV technology can help to avoid
unnecessary fatalities. During the pursuit, an officer several blocks away can release a small UAV
equipped with a camera and the necessary software and communication to provide every police
car with the criminals 3D location along with direction and speed of travel. This technology
allows the police to back off the chase when speeds reach dangerous numbers. By keeping a
safe distance from the chase vehicle, this in turn naturally results in the suspect also reducing
speed which can decrease the chances of fatal accidents. This technique is especially useful in
residential areas where most innocent fatalities occur. The overall mission involves tracking
and maybe even homing off the target vehicle to provide location information to officers on the
ground. Figure 15 illustrates in a simulated environment this scenario where a UAV observes the
pursuit from above to estimate the suspect's vehicle location. Similarly, this technology applies
directly to aiding the officials for border patrol.
Figure 15. Tracking a pursuit vehicle using a vision equipped UAV
1.4 System Architecture
The system architecture presented in this dissertation was designed in a modular fashion
and is amenable to closedloop control. The closedloop block diagram is depicted in Figure 16,
where commands are sent to a vehicle based on the motions observed in the images. The vehicle
considered in this dissertation is predominately assumed an autonomous UAV, but is generalized
for any dynamical system with position and orientation states. The blocks pertaining to this
dissertation are highlighted in Figure 16 in the image processing block and consists of the
moving object detection, state estimation of a moving object, and classifying deterministic versus
stochastic motion. A brief discussion of each topic is described in this section, while the details
are covered in their respective chapters.
Distinguishing moving objects from stationary objects with a moving camera is a
challenging task in vision processing and is the first step in the state estimation process when
considering a dynamic scene. This information is extremely important for guidance, navigation,
and control of autonomous systems because it identifies objects that potentially could be in a
path for collision. For a stationary camera, moving objects in the scene can be extracted using
Camera Feature Point Moving Object State Sohsi
x(0) = 0Model Tracker Distection Estimation Dermnti
LIIIIIImage 13Processing
Motion
Controller Moc eling
Prediction
Figure 16. Closedloop block diagram with visual state estimation
simple image differencing, where the stationary background is segmented out; however, this
approach does not apply to moving cameras. In the case of a moving camera, the background
is no longer stationary and it begins to change over time as the vehicle progresses through the
environment. Therefore, the images taken by a moving camera contain the motion due to the
camera, commonly called egomotion, and the motion of the object. Techniques that involve
camera motion compensation or image registration have been proposed to work well when there
exists no stationary objects close to the camera which cause high parallax. This dissertation
will establish a technique to classify objects in the field of view as moving or stationary while
accounting for stationary objects with high parallax. Therefore, with a series of observations of a
particular scene, one can determine which objects are moving in the environment.
Knowing which objects are moving in the image dictates the type of image processing
required to accurately estimate the object's states. In fact, the estimation problem becomes
infeasible for a monocular system when both the camera and the object are moving. This
unattainable solution is cause by a number of factors including 1) inability to decouple the
motion from the camera and target and 2) failure to triangulate the depth estimate of the object.
For this configuration, relative information can be obtained and fused with additional information
for state estimation. First, decoupling the motion requires known information regarding motion
of the camera or the motion of the object, which could be obtained through other sensors such
as GPS and IMUs. Second, the depth estimate can be acquired if some information is known
regarding the target geometry (e.g. a fixed distance on the target). For the case of stereo vision,
depth estimates can be obtained for each time step which is suitable for estimating the states of a
moving object. Although this particular configuration addresses the depth estimation, additional
issues involving the correspondence solution emerge when introducing multiple cameras [5].
Furthermore, the accuracy of the state estimates becomes poor for small baseline configurations,
which occurs for MAV using stereo vision. These issues regarding target state estimation will be
considered in this dissertation to show both the capabilities and limitations toward autonomous
control and navigation.
Another important task involved with target estimation is to determine a pattern (if
any) in the object's motion based on the time history. The objects can then be classified into
deterministic and stochastic motions according to past behavior. With this information, prediction
models can be made based on previous images to estimate the position of an object at a later time
with some level of confidence. The predicted estimates can then be used in feedback for tracking
or docking purposes. For stochasticly classified objects, further concerns regarding docking or
AAR are imposed on the control problem.
The primary task of state estimation, for both the vehicle and objects in the environment,
relies on accurate knowledge of the image measurements and the associated camera. Such
knowledge is difficult to obtain due to uncertainties in these measurements and the internal
components of the camera itself. For instance, the image measurements contain uncertainties
associated with the detection of objects in the image, in addition to noise corruption. These
drawbacks have prompted many robust algorithms to increase the accuracy of feature detection
while handling noise during the estimation process. Alternatively, many techniques have been
used to accurately estimate the internal parameters of the camera through calibration. The
parameters that describe the internal components of the camera are referred to as intrinsic
parameters and typically consist of focal length, radial distortion, skew factor, pixel size, and
optical center. This calibration process can become cumbersome for a large number of cameras
and incur cost and time delays. These additional expenses add complexity and eliminate the
attractiveness of low cost autonomous systems. Meanwhile, the current appeal of these systems
has been the use of low cost offtheshelf components, such as cameras that are easily replaced.
Maintaining a low cost product is a goal for UAV that can be accomplished by considering a
vision system. If future operations require a stockpile of thousands of UAV or MAV ready to
deploy, then the capability to switch out or replace components in a timely fashion with little cost
is a tremendous functionality. Therefore, this dissertation describes a method that would enable
cameras to be replaced rapidly and without the need for extensive calibration.
1.5 Contributions
The goal of the work presented in this dissertation is to establish a methodology that
estimates the states of a moving object using a monocular camera configuration in the presence
of uncertainty. The estimates will provide not only critical information regarding targetmotion
estimation for autonomous systems but also retain confidence values through a distribution
around a target's estimate. Previous work has investigated many problems and issues related
to this topic but has neglected several key features. In particular, this thesis addresses (i) the
physical effects of camera nonlinearities on state estimates, (ii) a multilayered classification
approach to object motion based on visual sensing that determines the confidence measure in the
estimates, and (iii) the relationships between vehicle and sensor constraints coupled with sensor
fusion in an autonomous system framework.
The main contribution of this dissertation is the development of a state estimation process of
a dynamic object using a monocular vision system for autonomous navigation. In addition to the
main contribution, there exists some secondary contributions solved in the process of facilitating
the main goal. The contributions presented in this dissertation consist of the following:
*A homography approach to state estimation of moving objects is developed through a virtual
camera to estimate the relative pose of the target relative to the true camera. This virtual
camera facilitated the estimation process by maintaining a constant homography relative to a
known reference object.
* A new approach to detecting moving objects in a sequence of images is developed. This
method computes estimates for the focus of expansion (FOE) and then classifies each
feature point into their respective motions through an iterative leastsquares solution. The
decision scheme for classification maintains a cost function, which determines if a feature
point obeys a particular FOE, under a desired threshold. The dominant motion assumption is
then used to determine which FOE class is considered stationary objects in the environment
and which are associated with moving objects.
* The nonlinear dynamics for an aircraftcamera system are derived for a general camera
configuration and model. This structure allows multiple cameras with time varying positions
and orientations within the derivation to compute image plane quantities such as feature
point position and velocity.
* A new method for obtaining error bounds on the target state is established to provide a
region of where the estimate can lie from the effects of uncertainty. This method can be
described as a deterministic framework that computes upper bound uncertainty and was
implemented to describe variations to image plane coordinates and propagated through
vision based algorithms. Although this upper bound or worsecase approach to uncertainty
is a conservative technique, it provides a fast implementation scheme to account for
inaccurate camera calibration.
* The implementation of the homography of a moving target along with a model prediction
scheme will be incorporated into a controls framework to enable closedloop tracking of an
unknown moving object of interest.
The first chapter of this thesis describes the motivation for this research, some current
objectives and limitations to address followed by a summary of the contribution and descriptions
of potential applications for this research.
Chapter 2 describes the related work and literature review that applies to this particular
research topic.
Chapter 3 introduces the foundation of computer vision and image processing. First the
camera geometry is described along with the projection model followed by the constraints used to
facilitate the estimation process. Lastly, traditional algorithms which estimate both the 3D motion
of the camera and the motion of targets are described.
Chapter 4 quantifies the effects of uncertainty in state estimation from variations in feature
point position caused from camera calibration and feature point tracking.
Chapter 5 derives the system dynamics for an aircraftcamera configuration by formulating
the differential equations and observations into a controls framework.
Chapter 6 describes a method that utilizes image processing techniques to detect and
segment moving objects in a sequence of images.
Chapter 7 formulates a homography technique that estimates relative position and
orientation with respect to a moving reference object. The method fuses traditional guidance
and navigation sensors with the computed homography to obtain relative state estimates of the
unknown object with respect to the moving camera. This process applies directly to solving a
significant portion of the AAR problem.
Chapter 8 summaries a modeling technique for moving objects to predict the target's motion
in inertial space.
Chapter 9 discusses a control design scheme that exploits visionbased state estimates to
track and home on a desired target. The control framework will be generalized for many mission
scenarios involving autonomous UAV but will be discussed in the context of the AAR problem.
Chapter 10 will implement in MATLAB the vision algorithms for both open and closedloop
architectures to demonstrate and verify the purposed methods.
Chapter 11 discusses concluding remarks and proposes future research directions for this
work.
CHAPTER 2
LITERATURE REVIEW
The proliferation of autonomous systems is generating a demand for smarter, more complex
vehicles. The motivation behind these concept vehicles is to operate in urban environments
which requires a number of complex systems. Video cameras have been chosen as sensors to
facilitate topics such as obstacle detection and avoidance, target tracking and path planning.
These technologies have stemmed from two communities in the literature: (i) image processing
and computer vision and (ii) performance and control of autonomous vehicles. This chapter
will focus on the research applied to autonomous systems and describe the current state of this
research, problems that have been addressed, some difficulties associated with vision, and some
areas in need of contribution. In particular, the review will cover the topics of most relevance to
this dissertation and highlight the efforts toward autonomous UAV
The block diagram shown in Figure 16 illustrates the components of interest described
in this dissertation for state estimation and tracking control with respect to a moving object
which involves object motion detection, object state estimation, and object motion modeling and
prediction. The literature review of these topics is given in this section.
2.1 Detection of Moving Objects
In order to track and estimate the motion of objects in images using a monocular camera
system, a number of steps are required. A common first step in many image processing
algorithms is feature point detection and tracking. This step determines features of interest,
such as comers, in the image that usually correspond to objects of interest, such as windows, in
the environment. The famous feature point tracker proposed by Lucas and Kanade [17, 18] has
served as a foundation for many algorithms. This technique relies on a smoothness constraint
imposed on the optic flow that maintains a constant intensity across small baseline motion of
the camera. Many techniques have built upon this algorithm to increase robustness to noise and
outliers. Once feature tracking has been obtained, the next process involves segmenting the
image for moving objects. The need for such a classification is due the fact that standard image
processing algorithms such as SFM are not valid for moving objects viewed by a moving camera.
This limitation is caused by the epipolar constraint no longer maintaining a coplanar property
across image sequences; consequently, research has evolved for the detection of moving objects
in a scene viewed by a nonstationary camera.
The detection of moving object in an image sequence is an important step in image analysis.
For cases involving a stationary camera, simple image differencing techniques are sufficient in
determining moving objects [1921]. Techniques for more realistic applications involve Kalman
filtering [22] to account for lighting conditions and background modeling techniques using
statistical approaches, such as expectation maximization and mixture of Gaussian, to account
for other variations in realtime applications [2328]. Although these techniques work well for
stationary cameras, they are insufficient for the case of moving cameras due to the motion of the
stationary background.
Motion detection using a moving camera, as in the case of a camera mounted to a vehicle,
becomes significantly more difficult because the motion viewed in the image could result from
a number of sources. For instance, a camera moving through a scene will view motions in the
image caused by camera induced motion, referred to as egomotion, changes in camera intrinsic
parameters such as zoom, and independently moving objects. There are two classes of problems
considered in literature for addressing this topic. The first considers the scenario where the 3D
camera motion is known a priori then compensation can be made to account for this motion to
determine stationary objects through an appropriate transformation [29, 30]. The second class of
problems does not require knowledge of the camera motion and consists of a two stage approach
to the motion detection. The first stage involves camera motion compensation while the last stage
employs image differencing on the registered image [31] to retrieve nonstatic objects.
The transformation used to account for camera motion is commonly solved by assuming the
majority of image consists of a dominant background that is stationary in Euclidean space [32,
33]. This solution is obtained through a leastsquares minimization process [32] or with the
use of morphological filters [34]. The transformations obtained from these techniques typically
provide poor estimation if the motions of moving objects are not accounted for in the registration
process or if the image contains stationary objects close to the camera that result in high parallax.
A technique presented by Irani et al. [35] proposed a unified method to detect moving
objects. This proposed method handles various levels of parallax in the image through a
segmentation process that is performed in layers. The first layer extracts the background objects
which are far away from the camera and have low parallax through a general transformation
involving camera rotation, translation, and zoom through image differencing. The next layer
contains the object with high parallax consisting of both objects close to the camera and objects
that are moving independently of the camera. The parallax is then computed for the remaining
pixels and compared to one pixel. This process separates the objects within the image based
on their computed parallax. The selection may involve choosing a point on a known stationary
object that contains high parallax so any object not obeying this parallax is classified as a moving
object in the scene.
Optic flow techniques are also used to estimate moving target locations once egomotion
has been estimated. A method that computes the normal image flow has been shown to obtain
motion detection [36]. Coordinate transformations are sometimes used to facilitate this approach
to detecting motion. For instance, a method using complex log mapping was shown to transform
the radial motions into horizontal lines upon which vertical motion indicate independent
motion [37]. Alternatively, spherical mapping was used geometrically to classify moving objects
by segmenting motions which do not radiate from the focus of expansion (FOE) [29].
2.2 State Estimation Using Vision Information
The types of state estimation that can be obtained from an onboard vision system
are (i) localization which estimates the camera motion between image frames from known
stationary feature points, (ii) mapping which estimates the location of 3D feature points using
reconstruction and structure from motion, and (iii) targpetmotion which estimates 3D feature
points that have independent motion. The work related to these topics are described in this
section.
2.2.1 Localization
Localizing the camera position and orientation relative to a stationary surrounding has been
addressed using a number of methods. An early method presented by LonguetHiggins [38, 39]
used the coplanarity constraint also known as the epipolar constraint. Meanwhile, the subspace
constraint has also been employed to localize camera motion [40]. These techniques have
been applied to numerous types of autonomous systems. The mobile robotic community has
applied these techniques for the development of navigation in various scenarios [4145]. The
applications have also extended into the research of UAV for aircraft state estimation. Gurfil
and Rotstein [46] was the first to extend this application in the framework of a nonlinear aircraft
model. This approach used optical flow in conjunction with the subspace constraint to estimate
the angular rates of the aircraft and was extended in [47]. Webb et al. [48, 49] employed the
epipolar constraint to the aircraft dynamics to obtain vehicle states. The foundation for both
of these approaches is a Kalman filter in conjunction with a geometric constraint to estimate
the camera motion. Some applications for aircraft state estimation have involved missions for
autonomous UAV such as autonomous night landing [50] and road following [51].
2.2.2 Mapping
Location estimation of stationary targets using algorithms such as structure from motion
has been extensively researched for nonstatic cameras with successful results. The foundation
of these techniques still rely on the geometric constraints imposed on stationary targets. The
decoupling of structure from the motion has been characterized in a number of papers by
Soatto et al. [5258]. These approaches employ the subspace constraint to reconstruct feature
point position through an extended Kalman filter. Several survey papers have been published
describing the current algorithms while comparing the performance and robustness [5962].
Robust and adaptive techniques have been proposed that use an adaptive extended Kalman filter
to account for model uncertainties [63]. In addition, Qian et al. [64] designed a recursive Hoo filter
to estimate structure from motion in the presence of measurement and model uncertainties while
Weng et al. [65] investigated the optimal approaches to target state estimation and described the
effects of linear solutions on various noise distributions.
2.2.3 TargetMotion Estimation
The topic of targetmotion estimation is described as the process of estimating the states
of a moving object from image sequences obtained from a moving camera system. For a stereo
camera configuration, the solution can be obtained using the standard epipolar constraint as long
as there exist a sufficient amount of baseline and the correspondence problem can be solved
accurately. For a stationary camera, full state estimates were achieved of a rigid object using a
statistically combined feature point/optical flow method with an extended Kalman filter [66].
This method extended the previous work of Broida et al. [67] that only considered a feature
point approach. For the case of moving monocular camera configuration, the problem becomes
extremely difficult due to the additional motion of the camera. One approach used in literature
relevant to monocular camera systems is bearingsonlytracking. In this approach, there are
several assumptions made: (i) the vehicle has knowledge of its position, (ii) an additional range
sensor, such as sonar or laser range finder, is used to provide a bearing measurement, and (iii) an
image measurement is taken for an estimate of lateral position. The initial research has involved
the estimation process and design with improvements to the performance [6872]. This approach
was implement by Flew [5] to estimate the motion of target within a computed covariance.
Guanghui et al. [73] provided a method for estimating the motion of a point target from known
camera motion.
The robotic community has examined the targetmotion estimation problem from a visual
servo control framework. Tracking relative motion of a moving target has been shown using
homographybased methods. These methods have been demonstrated to control an autonomous
ground vehicle to a desired pose defined by a goal image, where the camera was mounted on
the ground vehicle [74]. Chen et al. [75, 76] regulated a ground vehicle to a desired pose using a
stationary overhead camera. Mehta et al. [77] extended this concept for a moving camera, where
a camera was mounted to an UAV and a ground vehicle was controlled to a desired pose.
Targetmotion estimation has been demonstrated in simulation for applications toward the
AAR problem. Kimmett [15] applied a vision navigation algorithm called VisNAV that was
developed by Junkins et al. [78] to estimate the current relative position and orientation of the
target drogue through a Gaussian leastsquares differential correction algorithm. This algorithm
has also been applied to spacecraft formation flying [79].
2.3 Modeling Object Motion
The modeling of objects in motion from position and/or velocity measurements has been a
topic of interest for many applications that employ vision systems. This additional information
can provide systems with knowledge for tracking, collision avoidance, and docking. For instance,
intelligent robots using vision have been considered for industrial and medical applications that
require tracking and graphing of a moving target. Houshangi at al. [80] demonstrated the control
required to grab an unknown moving object with robotic manipulator using an autoregressive
(AR) model. This model predicts a future position of the target based on velocity estimates
computed from image sequences.
For aerial vehicles, detecting other aircraft in the sky is critical for collision avoidance.
NASA has considered vision in this scenario to aid pilots in detecting aircraft on a crossing
trajectory. A technique combining image and navigation data established a prediction method
through a Kalman filter approach to estimate the position and velocity of the target aircraft ahead
in time [34]. Similarly, the AAR problem requires some form of model prediction when docking
to a moving drogue. Kimmett et al. [15] utilized a discrete linear model for the prediction of the
drogue. The predicted states used for control were computed using the discrete model, the current
states, and light turbulence as input to the drogue dynamics. Successful docking was simulated
for only light turbulence and with low frequency dynamics imposed on the drogue. NASA is
extremely interested in AAR problem and currently has a project on this topic. Flight tests have
been conducted by NASA in an attempt to model the drogue dynamics [81]. In this study, the
aerodynamic effects from both the receiver aircraft and the tanker aircraft were examined on the
drogue, especially moments before the docking phase. The aerodynamic data acquired in these
experiments confirmed several dependencies on turbulence, flight conditions, and geometry.
2.4 Uncertainty in Vision Algorithms
The location of environmental features can be obtained using structure from motion. The
basic concepts are mature but their application to complex problems is relatively limited due to
complexities of realtime implementation. In particular, the noise issues involved with camera
calibration and feature tracking cause considerable difficulties in reconstructing 3dimensional
states. A samplingbased representation of uncertainty was introduced to investigate robustness
of state estimation [82]. Robustness was also analyzed using a leastsquare solution to obtain an
expression for the error in terms of the motion variables [83].
The uncertainty in visionbased feedback is often chosen as variations within feature
points; however, uncertainty in the camera model may actually be an underlying source of
those variations. Essentially, the uncertainty may be associated with the image processing
to extract feature points or with the camera parameters that generated the image. The proper
characterization of camera uncertainty may be critical to determine a realistic level of feature
pomnt uncertainty.
The analysis of camera uncertainty is typically addressed in a probabilistic manner. A
linear technique was presented that propagates the covariance matrix of the camera parameters
through the motion equations to obtain the covariance of the desired camera states [84]. An
analysis was also conducted for the epipolar constraint based on the known covariance in the
camera parameters to compute the motion uncertainty [85]. A sequential Monte Carlo technique
demonstrated by Qian et al. [86] proposed a new structure from motion algorithm based on
random sampling to estimate the posterior distributions of motion and structure estimation. The
experimental results in this paper revealed significant challenges toward solving for the structure
in the presence of errors in calibration, feature point tracking, feature occlusion, and structure
ambiguities.
2.5 Control Using Visual Feedback in Dynamic Environments
The control applications considered for autonomous systems include collision avoidance,
target tracking, surveillance, and docking. These missions can be categorized into reactive
control, tracking control, and homing control schemes. The goals of each control scheme are
diverse and rely on vision information in different ways. In reactive control, the purpose is to
make fast decisions based on image measurements of the environment and respond quickly to
a given control. Alternatively, tracking control attempts to maintain a target of interest within
the field of view for a desired amount of time. Strategies involving homing control use vision to
command the vehicle to a desired location either from image location or state estimation.
The ability to avoid obstacles and moving objects in an unfamiliar surrounding is a key
feature for autonomous navigation. A considerable amount of research in the vision community
has been established to facilitate a variety of autonomous vehicles for control purposes. For
instance, a number of detection and avoidance approaches have been applied to scenarios such
as pedestrian avoidance [87] in traffic situations, low altitude flight of a rotorcraft [88], avoiding
obstacles in the flight path of an aircraft [34], and navigating underwater vehicles [89]. Optical
flow techniques have also been utilized as a tool for avoidance by steering away from areas with
high optic flow which indicate regions of close obstacles [90].
Target tracking is another desired capability for autonomous systems. In particular, the
military is interested in this topic for surveillance missions both in the air and on the ground.
The common approaches to target tracking occur in both feature point and optical flow
techniques. The feature point method typically constrains the target motion in the image to
a desired location by controlling the camera motion [91, 92]. Meanwhile, Frezza et al. [93]
imposed a nonholonomic constraint on the camera motion and used a predictive outputfeedback
control strategy based on the recursive tracking of the target with feasible system trajectories.
Alternatively, optical flow based techniques have been presented for robotic handineye
configuration to track targets of unknown 2D velocities where the depth information is
known [94]. Adaptive solutions presented in [91, 9597] have shown control solutions for
target tracking with uncertain camera parameters while estimating depth information.
The homing control problem has numerous applications toward autonomous systems such
as autonomous aerial refueling, spacecraft docking, missile guidance, and object retrieval
using a robtotic manipulator. Kimmett et al. [15, 98] developed a candidate autonomous
probeanddrogue aerial refueling controller that uses a command generator tracker (CGT) to
track timevarying motions of a nonstationary drogue. The CGT is an explicit model following
control technique and was demonstrated in simulation for a moving drogue with known dynamics
subject to light turbulence. Tandale et al. [16] extended the work of Kimmett and Valasek by
developing a reference observer based tracking controller (ROTC) which does not require a
drogue model or presumed knowledge of the drogue position. This system consist of a reference
trajectory generation module that sends commands to an observer that estimates the desired
states and control for the plant. The input to this controller is the relative position between the
receiver aircraft and the drogue measured by the vision system. A similar vision approach to
aerial refueling is also presented in [99], where models of the tanker and drogue are used in
conjunction with an inferred camera. The drogue model used in this paper was taken from [100]
that uses a multisegment approach to deriving the dynamics of the hose. Meanwhile, Houshangi
et al. [80] considered grasping a moving target by adaptively controlling a robot manipulator
using vision interaction. The adaptive control scheme was used to account for modeling errors in
the manipulator. In addition, this paper considered unknown target dynamics. An autoregressive
model approach was used to predict the target's position based on passed visual information
and an estimated target velocity. Experimental test cases are documented that show tracking
convergence.
CHAPTER 3
IMAGE PROCESSING AND COMPUTER VISION
Image processing and computer vision refers to the process of acquiring and interpreting
2dimensional visual data to achieve awareness of the surrounding environment. This information
is used to infer spatial properties of the environment that are necessary to perform essential tasks
such as guidance and navigation through unfamiliar environments. An important breakthrough in
computer vision occurred when algorithms were able to detect, track, and estimate locations of
features in the environment.
This dissertation relies on feature points as the foundation for any visionbased feedback.
The term "features" allows one to establish a relationship between the scene geometry and
the measured image. These points generally correlate to items in the environment of special
significance. Some examples of items that often constitute feature points are corners, edges
and light sources. Such feature points can provide information about the overall object in the
sense that a set of corners can outline a building. Feature points do not necessarily provide
enough information to completely describe an environment but, in practice, they usually provide
sufficient information for target tracking and position estimation. To understand the algorithms
that use feature points, an establishment of the fundamental equations governed by the physics of
a camera will be described.
3.1 Camera Geometry
A camera effectively maps the 3dimensional environment onto a 2dimensional image
plane. This image plane is defined as the plane normal to the camera's central axis located a focal
length, f, away from the origin of the camera basis. The geometry provided by a pinhole camera
lens is described in Figure 31. The vector, rl, represents the vector between the camera and a
feature point in the environment relative to a defined camerafixed coordinate system, as defined
by I. This vector and its components are represented in Equation 31.
The components of this vector are decomposed along the camera's coordinate frame
to compute the image projection. The projection onto the image plane is where this vector
penetrates the focal plane. Here the offset of the feature from the center is given by the image
plane coordinates, pu and v, which will turn out to be a function of the components of rl and the
focal length for a pinhole camera model.
Figure 31. Mapping from environment to image plane
A major constraint placed on this sensor is the camera's field of view (FOV). Here the FOV
can be described as the 3D region for which feature points are visible to the camera; hence,
features outside the FOV will not appear in the image. The three physical parameters that define
this constraint are the field of depth, the horizontal angle and the vertical angle. A top view
illustration of the FOV can be seen in Figure 32, where the horizontal FOV is defined by the
half angle, yh, and the distance to the image plane is of length f. Likewise, a similar plot can be
shown to illustrate the vertical angle, which can be defined as y.
image
plane ,
camera
center
camera
axis
_ Feature point
Figure 32. Image plane field of view (top view)
Expressions for these angles are shown in Equation 32, where rh,v is defined as the largest
spatial extension in the horizontal and vertical directions.
'Y,v = arctan(rh,v/ f) (32)
Feature points within the field of view must have a horizontal coordinate in the image plane
which lies between maximum and minimum values. These values, given as v and v respectively,
are determined by the tangential angle between the depth component, rlz, and the horizontal
component, Try, of the vector between the camera and the feature point. The range of image
coordinates is given in Equation 33 for the horizontal component.
[V, V] = [ f tan(h) f tan(Yh)] (33)
A similar relationship is computed for the vertical component of field of view. The
minimum coordinate, pu, and the maximum coordinate, pu, for the image plane are computed
using the vertical component, rlx, of the vector connecting the camera and the feature point. This
range is given in Equation 34 for the vertical angle.
[p, p = [f tan( ,), ftan(r,)] (34)
3.2 Camera Model
3.2.1 Ideal Perspective
A geometric relationship between the camera properties and a feature point is required
to determine the image plane coordinates. This relationship is made by first separating the
components of rl that are parallel to the image plane into two directions. The image plane
coordinates are then computed from a tangent relationship of similar triangles between the
vertical and horizontal directions and the depth with a scale factor of focal length. This
relationship establishes the standard 2D image plane coordinates referred to as the pinhole
camera model [101, 102]. Equations 35 and 36 represent a general pinhole projection model
in terms of the relative position with a lens offset, c, relative to the camera frame.
p = x x (35
If the origin of the camera frame is placed at the lens, (i.e., c = 0), Equations 35 and 36
reduce to the very common pinhole camera model and is represented by Equations 37 and 38.
p = f Ex(37)
v =f (38)
This projection is commonly written as a map H:
H : RW3 ,2; X x (39)
The ideal perspective projection given in Equations 37 and 38 can be expressed in
homogeneous coordinates and is shown in Equation 310.
pu f 0 0 rlx
Ezv =0 f 0 Try (310)
1 0 0 1 rlz
3.2.2 Intrinsic Parameters
The image plane that is acquired from physical cameras is more complicated than the ideal
projection given in Equation 310. First, the image plane is discretized into a set of pixels,
corresponding to the resolution of the camera. This discretization is based on scale factors that
relate realworld length measures into pixel units for both the horizontal and vertical directions.
These scaling terms are defined as s, and sv which have units of pixels per length, where the
length could be in feet or meters. In general, these terms are different but when the pixels are
square then s, = sy. Second, the origin of the image plane is translated from the center of the
image where the optical axis penetrates the image plane to the upper left hand corner of the
image. This translation is done using the terms o,, and ov, given in units of pixels. The skew
factor is another intrinsic parameter which accounts for pixels that are not rectangular and is
defined as so. The ideal perspective transformation now takes the general form given in Equation
311, where pixel mapping, origin translation, and skewness are all considered.
p' fs,, fse o,, 1 00
The perspective transformation obtained in Equation 311 is rewritten to Equation 312.
rlzx' = Knorl (312)
The 3 x 3 matrix K is called the intrinsic parameter matrix or the calibration matrix while
the 3 x 4 constant matrix Ho defines the perspective projection, and finally x' represents the
homogeneous image coordinates [p', v', 1]' that contain pixel mapping and skew.
3.2.3 Extrinsic Parameters
In order to achieve this transformation to image coordinates, both intrinsic and extrinsic
parameters must be known or estimated a priori through calibration. The extrinsic parameters
of the camera can be described as the geometric relationship between the camera frame and the
inertial frame. This relationship consists of the relative position, T, and orientation, R, of the
camera frame to an inertial frame. By defining the position vector of a feature point relative to
an inertial as 5 = [t, y,(, ]'l, transformations can map the expression found in Equation 312
to obtain a general equation that maps feature points in the inertial frame to coordinates in the
image plane for a calibrated camera.
p' fs,, fse o,, 1 0
Be v' 0 fsv ov 0 0 R (313)
1 0 1 001
3.2.4 Radial Distortion
Other nonlinear camera effects that are not accounted for in the pinhole model, such as
radial distortion, can be addressed through additional terms. A standard lens distortion model is
considered to account for such nonlinearities in the camera. The general distortion term, given in
Equation 314, requires an infinite series of terms to approximate the value.
d = dr r2+td2r4+td3r,6+ HOT (314)
The distortion model, shown in Equations 315 and 316, maps an undistorted image,
(p', v'), which is not measurable on a physical camera, into a distorted image, (p'd d&), which
is observable [104]. This distortion model only considers the first term in the infinite series to
describe radial distortion and excludes tangential distortion. This approximation in distortion has
been used to generate an accurate description of real cameras without additional terms [105],
p'd = v'(1+t dr2) (315)
vid = p'(1+ dr2) (316)
where r2 __ /1 C1)2 /t _V 2)2 and d is the radial distortion parameter of the camera. Assuming
the origin of the camera frame is placed at the lens, then this term becomes r2 __ 1u2 +t v'2.
In addition, the radial distortion parameter, d, which is not described in Figure 31, attempts
to model the curvature of the lens during the image plane mapping. This distortion in the image
plane varies in a nonlinear fashion based on position. This effect demonstrates an axisymmetric
mapping that increases radially from the image center. An example can be seen in Figure 33B
and 33C which illustrates how radial distortion changes feature point locations of a fixed
pattern in the image by comparing it to a typical pinhole model shown in Figure 33A. Notice
the distorted images seem to take on a convex or concave shape depending on the sign of the
distortion.
20 1 1 0 20 20 1 0 2
5 > 5 **5 **0.0
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Thes fritrst stepi thoe estimabutiohn pobemi rqirtes teaiity trcoor dtct intcerestig features,
and curves. These features usually correlate to objects of interest in the environment such as
buildings, vehicles, bridges, etc. Although this gradientbased criterion is good at detecting
these features, it also produces a large number of detections from highly textured surfaces that
are not as interesting. For example, grassy areas, trees, and shrubbery are problematic under
this criterion due to the noisy images they produce. These additional detections can be limited
through simple smoothing filters and thresholding techniques.
The gradientbased corner detection has been a common algorithm for selecting strong
features in an image. These methods require the computation of the image gradient, which can be
done by convolving a 2dimensional derivative filter with the image. This derivative is realized by
approximating the ideal derivative with sampled Guassian filters defined as g [p], g[v]. Therefore,
an approximation of the image gradients are expressed in Equations 317 and 318 [102, 103].
The image coordinates (pu, v) in these expressions are computed using either Eqaution 310 or
Equation 311 depending on the camera model.
I,, [p, v] = IC, v] +g' [p] g[v] (317)
Iv [p, V] = I~p, v] a g [] a g' [v] (318)
Once the image gradients are computed, then the algorithm proceeds to compute the
summation of the outer product of the gradients within a users specified window, W, which is
given in Equation 319 [102, 103]. The pixel values within the search window are defined as Si.
G(x) = VI(i) VIT(i) (9
isw/l(x) EI ~
This computation is performed to check if the gradient is above some specified threshold,
z, that meets a feature point requirement. This criterion is determined if the smallest singular
value of G is greater than the threshold, z, as shown in Equation 320. If Equation 320 is
satisfied then this is a valid feature point based on the users criterion [102, 103]. This selection is
a function of both the window size, W, and the threshold, z.
o(G) > z (320)
A commonly used algorithm that employed these equations with slight variations is the
Harris comer detector [106]. This method can be extended to edge detection by considering
the structure of the singular values of G. An example of this algorithm is the Canny edge
detector [107].
3.4 Feature Point Tracking
Feature tracking or feature correspondence is the next step in the general state estimation
problem. The correspondence problem is described as the association of a feature point between
two or more images. In other words, the solution to this problem determines that a feature point
in two or more images corresponds to the same physical point in 3D space. The most common
approach to discerning how points are moving between images is the use of intensity or color
matching. This brightness matching is typically performed over a small window, W(x), centered
around the point of interest, as opposed to only matching a single brightness value, which could
have numerous false solutions. The vector of brightness values over a small window set, 1,
contained in the image is shown in Equation 321.
1(x) = {I(i) IE W (x)} (321)
This brightness vector can be compared across images, li and I2, and optimized to find
the minimum error. If a feature point of interest is located at xl = [pt, v1] in image 1, li then a
simple translational model of the same scene can be used as an image matching constraint. This
relationship is shown in Equation 322,
11 (xl) = I2(h(x1)) +t n(h(x1)) (322)
where h(x) defines a general motion transformation to the proceeding image and n(xl) is additive
noise caused by ambiguities such as variations in lighting, reflections, and view point.
Therefore, the correspondence solution is cast as a minimization problem that computes the
best intensity match over a small window by minimizing the intensity error. An equation for the
translation estimate can then be found from this minimization process through Equation 323,
subject to Equation 322. One important limitation of this criterion occurs when the window in
both images contains relatively constant intensity values. This results in the aperture problem
where a number of solutions for h are obtained. Therefore, during the feature selection process
it's beneficial to choose features that contain unique information in this window.
hi=argm~in li(iI2((hSi))2 (323)
There are two common techniques to solve Equation 323 for small baseline tracking: (1)
using the brightness consistency constraint and (2) applying the sum of squared differences
(SSD) approach. Each of these techniques employs a translational model to describe the image
motion. Therefore, if one assumes a simple translational model then the general transformation is
shown in Equation 324.
h(x) = x +t Ax (324)
The brightness consistency constraint is derived by substituting Equation 324 into
Equation 322 while initially neglecting the noise term. Applying the Taylor series expansion
to this expression about the point of interest, x, while retaining only the first term in the series
results in Equation 325.
aI dpu aI dv aI
aSp dt av dt tat (5
This equation relates the spatialtemporal gradients to the pixel motion assuming the
brightness remains constant across images. Rewrting Equation 325 in matrix form results in
Equation 326.
AITu +t It = 0 (326)
where u = [f ].
Equation 326 constitutes 1 equation with 2 unknown velocities; therefore, another
constraint is needed to solve this problem. A unique solution for the velocities can be determined
by enforcing an additional constraint on the problem, which entails restraining regions to a local
window that moves at constant velocity. Upon these assumption one can minimize the error
function given in Equation 327.
El(u) = [VI (i, t)u(x) +tIt (1, t)]2 (327)
w(x)
The minimum of this function is obtained by setting VE1 = 0 to obtain Equation 328,
LI,2 LCly + I<^ = 3
I C,tv LIS ut LIvT, O(8
or, rewritten in matrix form results in the following
Gu tb= (329)
where G(x) was derived in Equation 319.
The final solution for the pixel velocity is found through a leastsquares estimate given in
Equation 330. These image velocities are also referred to as the optic flow. Once the optic flow
is computed for a feature point then the image displacement for feature tracking is trivial to find.
u = G b (330)
On the other hand, the method using SSD, shown in Equation 324, attempts to estimate
the Ax while not requiring the computation of image gradients. This approach also employs
the translational model over a windowed region. The method considers the possible range that
window could move, dpu and dv, in the time, dt. This consistency constraint then leads to a
problem of minimizing the error over the possible windows within the described range. This error
function is described mathematically in Equation 331.
E2 (dpdv) = [I(p + dp, lv + dv, t +dt) I0pv, t)]2 (331)
W( c,v)
The solutions obtained are the displacement components, dpu and dv, of the specified
window that correlates to the translation of the center pixel. This techniques is the foundation
for the LucasKanade tracker [17]. For large baseline tracking simple translational models
begin to falter due to the drastic changes in lighting conditions and the large view point change.
Therefore, a more general motion transformation is used such as an affine transformation.
Normalized crosscorrelation techniques are also used for large baseline configurations to handle
considerable changes in lighting conditions.
An extremely important concern is the accuracy of these algorithms and how variations in
feature point tracking can effects the final state estimation. These concerns will be addressed in
detail in Chapter 4.
3.5 Optic Flow
The next metric of interest in the image plane is an expression for the velocity of a feature
point. This expression is found simply by taking the time derivative of the feature point position
defined in Equations 37 and 38. The velocity expressions, shown in Equations 332 and 333,
describe the movement of feature points in the image plane and is commonly referred to in
literature as the optic flow.
Likewise, the feature point velocity with radial distortion can be computed by differentiating
Equations 315 and 316 while assuming c = 0 is as follows
ild = ii(1+ dr2) +t 2pdri (334)
itd = 9j(1+ dr2) +t 2vdrt (335)
where
pu v
r = p + 9 (336)
3.6 TwoView Image Geometry
The twoview image geometry relates the measured image coordinates to the 3D scene.
The camera configuration could be either two images taken over time of the same scene, as
in the monocular case, or two cameras simultaneously capturing two images of the same
scene, as in the stereo vision case. This section will describe the geometry and establish the
mathematical equations for estimating (i) the camera's pose between frames which consists of
relative translation and rotation, and (ii) the position of feature points in 3D space. First, the
geometry of the twoview configuration will generate the epipolar constraint to allow for the
computation of the camera pose from tracked feature points. Second, the 3D scene reconstruction
will be formulated based on the twoview geometry. Lastly, the limitations on feature points will
be discussed based on the type of camera configuration exploited to obtain a feasible solution.
3.6.1 Epipolar Constraint
The implicit relationship between camera and environment throughout this dissertation is
the epipolar constraint or, alternatively, the essential or coplanarity constraint. This constraint
requires position vectors, which describe a feature point relative to the camera at two instants
in time, to be coplanar with the translation vector and the origins of the camera frames. This
geometric relationship is illustrated in Figure 34 where rll and rl2 denote the position vectors
of the feature point, P, in the camera reference frames. Also, the values of xl and xa represent
the position vectors projected onto the focal plane while T indicates the translation vector of the
origin of the camera frames.
A geometric relationship between the vectors in Figure 34 is expressed by introducing R
as a rotation matrix. This rotation matrix includes the roll, pitch and yaw angles that transform
the camera frames between measurements. The resulting epipolar constraint is expressed in
Equation 337.
r12 (T x Rrll) = 0 (337)
The relationship can also be written in terms of coordinates within the image plane. The
relationship, given in Equation 338, assumes a pinhole camera which is colinear with its
projection into the focal plane.
x2 (T x Rxl) = 0 (338)
(R, T)
Figure 34. Geometry of the epipolar constraint
The expressions in Equation 337 and Equation 338 reflect that the scalar triple product
of three coplanar vectors is zero, which forms a plane in space. These relationships can be
expanded using linear algebra [102, 103] to generate a standard form of the epipolar geometry
as in Equation 339. This new form indicates a relationship between the rotation and translation,
written as the essential matrix denoted as Q, to the intrinsic parameters of the camera and
associated feature points. In this case, the equation is derived for a single feature point that is
correlated between the frames,
[#2 V2 fe 91 v1 fT = 0 (339)
where Q = [T] xR and [T] x is defined as the skewsymmetric form of the translation T.
The geometric relationship formed by this triangular plane is also seen in the epipolar lines
of each image. The 3D plane formed through this triangle constrains a feature point in one image
to lie on the epipolar line in the other image. These constraints can be mathematically expressed
as in Equation 340 with 11 and 12 representing the epipolar lines in image 1 and image 2 being
proportional to the essential matrix, respectfully.
12 ~ Qxt, 11 ~ Qx2 (340)
To extend this analysis to the general case of uncalibrated cameras, Equations 338
and 340 are rewritten in terms of the fundamental matrix, F, and are shown in Equations 341
and 342,
xK TxRK x'g = (341)
where F = KTxRK1 and xi = K'xi and the calibration matrix, K, was defined in
Equation 312.
12 = Fx'g, 11 = FTxy (342)
In the uncalibrated case, the ability to decompose F into R and T is infeasible. The matrix
products of F allow an infinite number of matrices that satisfy the solution, and therefore,
contributing to its impracticality.
3.6.2 EightPoint Algorithm
The eightpoint algorithm is a linear solution to Equation 339 which solves for the entries
of the essential matrix. This algorithm was developed by LonguetHiggins [39] and is described
in this section.
The expression in Equation 339 can actually be expressed as in Equation 343 using
additional arguments from linear algebra [102, 103]. The vector, q E (9P, contains the stacked
columns of the essential matrix Q.
Cq = [#1U2. V1U2. U2. U1V2 V1V2. V2. i1 V1 1] q = 0 (343)
Finally, a set of constraints must be formulated that introduce an expression, given in
Equations 343, for each feature point where the entries of the essential matrix are stacked in the
vector q. A set of row vectors are stacked to form a matrix, C, of n matched feature points and
is related to q as in Equation 344. The matricx C, shown in Equation 345, is a nx 9 matrix of
stacked feature points matched between two views.
Cq = 0 (344)
#1l,192,1 V1,19u2,1 #u2,1 91l,192,1 V1,192,1 V2,l #1l,1 V1,1 1
u1,2iu2,2 V1,2iu2,2 #u2,2 #u1,2V2,2 V1,2V2,2 V2,2 #u1,2 V1,2 1 35
i#1,niU2,n V1,niU2,n iU2,n i#1,nV2,n V1,nV2,n V2,n i#1,n V1,n 1
A unique solution for Equation 344 exists using a linear leastsquares approach only if
the number of matched features in each frame is at least 8 such that rank(C) = 8. Additionally,
more feature points will obviously generate more constraints and, presumably, increase accuracy
of the solution due to the residuals of the leastsquares. In practice, the leastsquares solution to
Equation 344 will not exist due to noise, therefore, a minimization is used to find an estimate of
the essential matrix, as shown in Equation 346.
minCq, q =1 (346)
Once an estimate of the essential matrix is found, the next step is to decompose this matrix
into its translational and rotational components. This decomposition is obtained through singular
value decomposition (SVD) of the essential matrix, and is shown in Equation 347.
Q = UEV' (347)
where E = diag {ol, 2, 03 } are the singular values. In general, this solution is corrupted
from noise and needs to be projected onto the essential space. This projection is performed
by normalizing the singular values to C = diag {1, 1,0} and adjusting the corresponding U
and V. The motion decomposition can now be obtained through Equation 348, where the
translation T is found up to a scaling factor. These four solutions, which consist of all possible
combinations of R and Tx, are checked to verify which combination generates a positive depth
during reconstruction.
R = UR ( + VT) T = URz( +\ EUT (348)
0 +1 0
whr Ry"2+0 41 0 0
0 01
The eightpoint algorithm fails with a nonunique solution when all points in 3D space lie on
the same 2D plane [102, 103]. When this situation occurs one must use the planar homography
approach, which is the topic of the next section.
3.6.3 Planar Homography
The homography approach can be used to solve the degenerate cases of the eightpoint
algorithm. For instance, a very common case where the feature points of interest all lie on the
same 2D plane in 3D space causes the algorithm to produce nonunique solutions. This case, in
particular, is a crucial part of enabling autonomous systems to navigate in urban environments.
Manmade structures such as buildings, roads, bridges, etc. all contain planar characteristics
associated with their geometry. This characteristic also applies especially to aerial imagery at
high altitudes where objects on the ground are essentially viewed as planar objects. Therefore,
this section describes the planar case to estimating motion from two images of the same scene as
shown in Ma et al. [102, 103]. Figure 35 depicts the geometry involved with planar homography.
The fundamental relationship expressing a point feature in 3D space across a set of images is
given through a rigid body transformation shown in Equation 349.
r12 = Rrll + T (349)
Recall that rla and rll are relative position vectors describing the same feature point in space
with respect to camera 2 and camera 1, respectfully, and R and T are the relative rotation and
translation motion between frames.
(R, T)
Figure 35. Geometry of the planar homography
If an assumption is made that the feature points are contained on the same plane, then a new
constraint involving the normal vector can be established. Denote N = [121,122,13 T as the normal
vector of the plane containing the feature points relative to camera frame 1. Then the projection
onto the unit normal is shown in Equation 350, where D is the projected distance to the plane.
N rll = nlrl,x+/t 2291,y ft l23T1,z= (350)
Substituting Equation 350 into Equation 349 results in Equation 351,
82=R+TNI qir (351)
where the planar homography matrix is defined to be the following
H = R t _TNT (352)
The relationship shown in Equation 351 can be extended to image coordinates through
Equation 353.
x2 = Hxl (353)
A similar approach as used in the eightpoint algorithm can be used to solve for the entries
of H. Multiplying both sides of Equation 353 with the skew symmetric matrix xi results in the
planar homography constraint shown in Equation 354.
xiHxy = 0 (354)
Since H is linear, linear algebra techniques can be used to stack the entries of H as a column
vector h and, therefore, Equation 354 can be rewritten to Equation 355,
a'h = 0 (355)
where a is the Kronecker product of xi and xl. Each feature point correspondence between
frames provides two constraints in determining the entries of H. Therefore, to solve for a
unique solution of H, Equation 355 requires at least four feature point correspondences.
These additional constraints can be stacked to form a new constraint matrix ?, as shown in
Equation 356.
'Y= aX1,a2,8 3, , nn T (356i)
Rewriting Equation 355 in terms of the new constraint matrix results in Equation 357.
Wh = 0 (357)
The standard leastsquares estimation can be used to recover H up to a scale factor.
Improvements can be made to the solution when more than four feature point correspondences
are used in the leastsquares solution. The scale factor is then determined as the second largest
singular value of the solution H [102, 103], shown in Equation 358 for the unknown scaler h.
1 = G2 (H) (35 8)
The homography solution is then decomposed into its rotational and translational
components through a similar technique used in the eightpoint algorithm. This approach
uses SVD to rewrite the homography matrix, as shown in Equation 359.
HTH = VEVT (359)
The matrix C = diag [o0a~ 21  ] nd ,mthe vetor V, alt~~re nn otn orma nl eigenvector c~~ol~rresodn to
the singular values of E. The columns of the matrix V can be written as V = [vl, v2, v3]. Defining
two other unitlength vectors, shown in Equation 360, that are preserved in the homography
mapping and will facilitate in the decomposition process.
vi +v3 viv3
U11= /2. = (360)
Furthermore, defining the matrices shown in Equation 361 will establish a homography
solution expressed in terms of these known variables.
Ui = [v2,111slT21/2] Wi = [HyI2Hui~H 2Hul
[ ] (361)
U2 = 7Z2:l 821T22 W2 = [Hy2Hu(II2,H 2H2
The four solutions are shown in Table 31 in terms of the matrices given in Equations 361,
360 and the columns of the matrix V. Notice the translation component is estimated up to a 2
scale factor. This is the same scale ambiguity associated with the eightpoint algorithm, which is
caused by the loss of depth during the image plane transformation.
Table 31. Solutions for homography decomposition
R1 = W1 Uz R3 = R1
Solution 1 NI1 = itul Solution 3 N3 = NI1
T = (HR1)Nl ci73 2T1
R 2 = W2 U2 R 4 = R 2
Solution 2 N2~ = Vi2 Solution 4 N4 q
gT2 = (H R2) N2 I f4 = gf2
A unique solution for the homography is then found by imposing the positive depth
constraint, which is associated with the physically possible solution. This imposition involves
checking the condition that NTe3 = n3 > 0, where e3 is in the direction of the optical axis normal
to the image plane.
3.6.4 Structure from Motion
Structure from motion (SFM) is a technique to estimate the location of environmental
features in 3D space. This technique utilizes the epipolar geometry in Figure 34 and assumes
that the rotation, R, and translation, T, between frames is known. Given that, the coordinates of
rll and r12 can be computed. Recall, the fundamental relationship repeated here in Equation 362.
r12 = Rrll + T (362)
The location of environmental features is obtained by first noting the relationships
between feature points and image coordinates given in Equation 37 and Equation 38. These
relationships allow some components of rlx and rly to be written in terms of pu and v which are
known from the images. Thus, the only unknowns are the depth components, rl1,z and rl2,z, fOr
each image. The resulting system can be cast as Equation 363 and solved using a leastsquares
approach.
2 (R11 z +R12" +1IR13) Tx
z (R21+R2 +R23 r2,z =I T (363)
1 (R31 9+tR32" +1IR33)41 Tz
This equation can be written in a compact form as shown in Equation 364 using z =
[r12,z, r1,z] as the desired vector of depths.
Az = T (364)
The leastsquares solution to Equation 364 obtains the depth estimates of a feature point
relative to both camera frames. This information along with the image plane coordinates can be
used to compute (Tll,x,Tl1,y) and (r12,x,r12,y) by substituting these values back into Equations 37
and 38. The resulting components of rll can then be converted to the coordinate frame of the
second image and it should exactly match r12. These values will never match perfectly due to
noise and unknown camera parameters so, in practice, an averaging process is often used to
estimate the feature coordinates.
There are two fundamental issues regarding the obtained solution. First, by relying on
the solution provided by the eightpoint algorithm, then the translation is only determined up
to a scaling factor. The SFM solution will therefore be corrupted from this scale factor unless
an alternative method is used to obtain this scaling. Second, the uncertainty due to intrinsic
parameters, feature detection, feature tracking, along with the uncertainty in the solution of the
eightpoint algorithm contributes to large variations in the SFM solution. The solution obtained
from Equation 364 is very sensitive to these uncertainties. Chapter 4 will discuss a method to
obtain uncertainty bounds on the SFM estimates based on the sources described.
CHAPTER 4
EFFECTS ON STATE ESTIMATION FROM VISION UNCERTAINTY
The image processing techniques commonly used today for aiding navigation require the
detection of feature points in the image to describe the environment. The concept of feature point
detection and tracking fundamentally relies on the accuracy of the camera intrinsic parameters,
as seen in Chapter 3. Once feature points are located and tracked across images, a number state
estimation algorithms, such as optic flow, epipolar constraint, and structure from motion, can be
employed. Although camera calibration techniques have proven to provide accurate estimates
of the intrinsic parameters, the process can be cumbersome and time consuming when using a
large quantity of low quality cameras. This chapter describes quantitatively the effects on feature
point position due to uncertainties in the camera intrinsic parameters and how these variations are
propagated through the state estimation algorithms. This deterministic approach to uncertainty
is an efficient method that determines a level of bounded variations on state estimates and can be
used for camera characterization. In other words, the maximum allowable state variation in the
system will then determine the accuracy required in the camera calibration step.
4.1 Feature Points
The locations of feature points within the image plane are computed using the geometry
of Figure 31. The resulting values are repeated in Equations 41 and 42 as a function of focal
length, f, and radial distortion, d, in terms of the components of rl.
u= fx 1+td fT2 22 n (41)
v =f 1 + d f2 22 ) (42)
The camera is effectively modeled using the focal length and radial distortion. As such,
these parameters are termed the intrinsic parameters and are found through calibration. A
feature point must be analyzed with respect to these intrinsic parameters to ensure proper
state estimation. The radial distance from a feature point to the center of the image, as shown
in Figure 41, is dependent on both the relative positions of camera and the feature. This
radial distance, as shown in Figure 42, is also related via a nonlinear relationship to the radial
distortion. The analysis of the feature points will require estimation of the camera parameters.
20
15 *
10
5
S 0 *
5
10
20
15
10
5
1 0
5
10
15
20

20
***
* **
***
10 0
10 20
10 0
v
10 20
A B
Feature Point Dependence on Focal Length for A) f = 0.5 and B) f
Figure 41.
0.25
15
10
10
20
15
10
10 0 10 20
v
10 0
v
10 20
B
0.0001 and B)
Figure 42.
Feature Point Dependence on Radial Distortion for A) d
d = 0.0005
The intrinsic parameters, given as focal length and radial distortion, can not be exactly
known, instead, they should be considered as uncertain variables. This chapter uses a sector
bounded approach wherein each parameter is constrained to lie within a set. The set is centered
around a nominal value and extends to a desired norm bound. The expression for focal length,
given in Equation 43, shows the range of values that must be considered for a nominal estimate,
fo, and uncertainty bounded by size of Ay E R. A similar expression in Equation 44 presents
the range of values for radial distortion.
f = { fo $ f : If  < Ay} (43)
d = {do+t8d 116 8d ~d} (44)
The variations of feature points due to the camera uncertainties can be directly computed.
The uncertain parameters given in Equation 43 and Equation 44 are substituted into the
camera model of Equation 41 and Equation 42. The resulting expressions for feature points are
presented in Equations 45 and 46.
pl = foil~lx~ 1+d fo24_ r 2872 (45)
+3doffi, +3dlo +doi +fid+3f d + 3 fod8yd
vl = fo 1+off + + xE28(46
+3dof rl +3dfo +di +f id +3f8df+3f yd d
These~~~~~~ eqain deosrt a opiae eaiosi ewe neranyi etr
points and uncrtainty in camra~ ,6 paaees The featur poits3,6, actual vary lieal with
unerait in foa eghfracmr ihu ail itrinoeeteicuino
u=~~~~~~ {u 8, : ,
t~lof2v = {vgo~f +L~ fiv : t 3fi~  < Av } (48)f
The uncertainties of 8,, and 8, are norm bounded but are not simple to describe. The range
of values for 8,, and 8v must be computed by evaluating their nonlinear relationship to 6f and 6d.
This range also depends on the relative position between camera and feature, as given by rlx and
Try, so the range of uncertainty will actually vary for each feature point. The norm bounds can be
expressed for a given vector, rl, using Equation 49 and Equation 410.
'1r 8I' 2 < A d +3 ff 8dOOf +3O fod d
+f36 8t
4. Optica Flow r 8
featre oins i tha motio is no p rely alon th lineof sight J~o~f th caea.Teopi lo a
postios nd recomutd uin Eqaton 3 Oiand Eqaton 3 npatctevlcte
aromputed byo rrsubtracting location of feature points ac rosste a t pai o images taknatdferen
times Scanppro ach assumen es ao th sat af fesatures pointcne straked andt corereated betweentes
frames. The optic flow is then given as IJ using Equation 411 for a feature point at pul and vl in
one frame and #u2 and v2 in another frame.
y 2. 91l (411)
V2. 91
The expressions for features points, given in Equation 47 and Equation 48, can
be substituted into Equation 411 to introduce uncertainty. The resulting expression in
Equation 412 separates the known from unknown elements.
J = +(412)
Vog Votv v
A range of variations are allowed for optic flow due to the uncertainty in feature points.
The expression for IJ can thus be written using nominal, IJo, and uncertain, Sy, terms as in
Equation 413 where the uncertainty is bounded by Ay E R.
g = {$o+8y 6~ : Sy
The amount of uncertainty in optic flow depends on the uncertainty in each feature point.
The maximum variation in velocity for a given point, determined by rl, is given in Equation 414.
The actual bounds on the feature points, as noted in Equation 49 and Equation 410, varies
depending on the location of each feature point so bounds of A,, and A,, are given for each
vertical component and Av, and Av, are given for each horizontal component. As such, the bound
on variation is noted in Equation 414 as specific to the rll and rl2 used to gather feature points in
each image.
Ay= max  (8,,, 8)2+ _6v 8,2  (414)
V1? I a,1
4.3 Epipolar Geometry
State estimation using epipolar geometry, computed as a solution to Equation 344, requires
a pinhole camera whose intrinsic parameters are exactly known. Such a situation is obviously
not realistic so the effect of uncertainty can be determined. A nonideal camera will lose the
colinearity and coplanarity between the images so the computed solution, q, will not agree with
the true value.
Uncertainty in the constraint matrix, C, will result from variations in the feature points,
as noted in Equation 47 and Equation 48, which are actually caused by uncertainty in the
camera parameters as noted in Equation 43 and Equation 44. The constraint matrix from
Equation 344 can then be written as a nominal component, Co, plus some uncertainty, Sc, as in
Equation 415.
C = Co + i (415)
The matrix of Sc can be directly computed in terms of uncertainty in the feature points by
substituting Equation 47 and Equation 48. The ith row of this matrix can then be written as
Equation 416.
Sc = [miP, 2 #U26pi 1 #2~' V12' t#26vl V1~' #2 (416i)
U16v v2 V2.6pi t1 2v V18 2 V26vl 6vl 6v2
8jV2 Cjl Cj 1 0]
A solution to Equation 344, when including the uncertainty matrix in Equation 415,
will exist; however, that solution will differ from the true solution or the nominal solution.
Essentially, the solution can be expressed as the nominal solution, go, and an uncertainty, Sq, as
in Equation 417. This perturbed system can now be solved using a linear leastsquares approach
for the entries of the essential matrix.
(Co +t 6c) (4o +t 6q) = 0 (417)
The solution vector, q = qo +t 6q, for Equation 417 has variation which will be norm
bounded by Aq as in Equation 418 which indicates the worsecase variation imposed on the
entries of q.
q = {qo+t6q 16q q}~ (418)
The size of this uncertainty, which reflects the size of error in the state estimation, can
be bounded using Equation 419. This bound uses the relationship between uncertainties in
Equation 416 through the constraint in Equation 417. Also, the size of this uncertainty depends
on the location of each feature point so the bounds is noted as specific to the rll and rl2 obtained
from Figure 34.
A = ~max  (Co + 6c) 1 c4o (9
Vj12 I a91
The maximum variation of the entries of q = go +t Aq, determined through Equation 419,
can then be used directly to compute the variation in state estimates. The entries of q are first
arranged back into matrix form to construct the new essential matrix that includes parameter
variations. This new essential matrix is then decomposed using SVD techniques described in
Section 3.6.1i.
4.4 Homography
A similar approach can be used to describe the variations to the entries of the homography
matrix, H, where the system equation was shown in Equation 357. Substituting Equation 47
and Equation 48 into Equation 357 results with a variation in the system matrix Y. Likewise,
the new system matrix with uncertain intrinsic paramters can be written as a nominal matrix, Yo
plus some variation, SY, as shown in Equation 420.
Y = Yo +t 8, (420)
As in the epipolar solution, the matrix of 8,u can be directly computed in terms of
uncertainty in the feature points by substituting Equation 47 and Equation 48. correspondingly,
the ith row of this matrix can then be written as Equation 421.
6iu = ~I,;+2tpU 11 pillig V16p, t26v1 8vi,; 8,;,?~ L (421)
U16,v V28pi t ivz V18vz V26vl t vl vz
A solution to Equation 357, when including the uncertain matrix in Equation 420, will
exist, however, that solution will differ from the true solution. Essentially, the solution can be
expressed as the nominal solution, ho, and an uncertainty, Sh, as in Equation 422.
('Fo +t 8,) (ho +t Sh) = 0 (422)
The solution vector, h = ho +t Sh, for Equation 422 has variation which will be norm
bounded by Ah as in Equation 423.
h = {ho+8 h 16 8h ~h} (423)
The size of this uncertainty, which reflects the size of error in the state estimation, can
be bounded using Equation 424. This bound uses the relationship between uncertainties in
Equation 421 through the constraint in Equation 422. Also, the size of this uncertainty depends
on the location of each feature point so the bounds is noted as specific to the rll and rl2 obtained
from Figure 34.
Ahx = I (Fo 6v)1 who (424)
8v, < av,
Iav1 < Av2
The maximum variation of the entries of h = ho +t Ah, determined through Equation 424,
can then be used directly to compute the variation in state estimates. The entries of h are first
arranged back into matrix form to construct the new homography matrix that includes parameter
variations. This new homography matrix is then decomposed using SVD techniques described in
Section 3.6.3.
4.5 Structure From Motion
Any uncertainty in the camera will result in uncertainty in the feature points and,
consequently, create uncertainty in the matrix used in Equation 364 for the structure from
motion relationship. As such, the matrix should be written in terms of a nominal value, Ao, and
an uncertain perturbation, 6A, as in Eq. 425
A = Ao +t 6A (425)
The uncertain perturbation can actually be computed by substituting the uncertain
expression in Equation 47 and Equation 48 into Equation 364. The perturbation is then
written as Equation 426.
fsfi (R"o +R+12 ~~i + R13)
SA = (R21 +R 22 + +R23 (426)
0 (R31 ~~h + R32 + R33);
The solution to Equation 364 when considering Equation 425 will obviously result in a
depth estimate that differs from the correct value. Define zo as the actual depths that would be
computed using the known parameters of the nominal camera and 8z as the corresponding error
in the actual solution. The leastsquares problem can then be written as Equation 427 and solved
using a pseudoinverse approach.
(Ao +t 6A) Zo +t 6z) = T (427)
The solution, zo +t 6z, will have a range of values bounded by Az as in Equation 428. This range
of solutions will lie within the bounded range determined from the worstcase bound.
z = {Zo+t6z : 16z
The bound on error, Az, can be expressed using Equation 429. This bound notes that the
bound on variations in feature points, and ultimately the bound on solutions to structure from
motion, depends on the location of those feature points.
Az n
 (Ao + BA)1( T (Ao + BA) o) 
(429)
m~ax
Is1, P la
8#2 I < 2,
CHAPTER 5
SYSTEM DYNAMICS
The previous chapters described techniques to (i) compute image coordinates, and (ii) the
effects of uncertainty for inertial estimations. Future chapters will discuss (i) detecting and
tracking of moving objects in a scene, (ii) obtain state estimates of moving objects, and (iii)
classify and model these objects into deterministic or stochastic motion. These topics each
build upon a commonality of feature points. As a result, this chapter describes the feature point
dependence on vehicle dynamics for a camera mounted system. The vehiclecamera relationships
presented will focus on the nonlinear aircraft dynamics and how they relate to feature points for a
cameraaircraft system. Although this description concentrates on aircraft dynamics, the modular
form of these equations is compliant to any dynamical system.
5.1 Dyanmic States
The formulation that describes feature points in the image plane starts by considering the
vector geometry involved in an aircraftcamera setup. The geometry can be described through
a number of coordinate frames. This section will utilize the camera geometry described in
Chapter 3 to derive the system equations.
5.1.1 Aircraft
The kinematics of an aircraftcamera system in flight are derived by first defining the
required coordinate frames. The standard measurements for an aircraft are based in either the
Earthfixed coordinate system or the bodyfixed coordinate system. Each of these coordinate
systems use a righthanded axes framework that obeys crossproduct rules. A pictorial
representation of these axes is given in Figure 51 along with the respective origins.
The bodyfixed coordinate system has the origin located at the center of gravity of the
aircraft. The axes are oriented such that 61 aligns out the nose and 62 aligns out the right wing
with $3 pointed out the bottom. The movement of the aircraft, which includes accelerating, will
obviously affect the coordinate system; consequently, the bodyfixed coordinate system is not an
inertial reference frame.
a I
Figure 51. Bodyfixed coordinate frame
The orientation angles of the aircraft are of particular interest for modeling a visionbased
sensor. The roll angle, 4, describes rotation about $1, the pitch angle, 6, describes rotation about
$2. and the yaw angle, W, describes rotation about 63.
The transformation from a vector represented in the Earthfixed coordinate system to
the bodyfixed coordinate system is required to relate onboard measurements to inertial
measurements. This transformation, given in Equation 51, uses REB which are Euler rotations
of roll, pitch and yaw [29, 108],
by 1
b2. = REB e2 (51)
b3 B 3 E
where REB is the relative rotation between frame E and B, respectfully which can be decomposed
as a sequence of singleaxis Euler rotations as seen in Equation 52. The order of this matrix
multiplication needs to be maintain for correct computation.
REB = 1 2>]e(ele 3(w)] (52)
where the individual singleaxis rotations are commonly referred to as 321, or rollpitchyaw,
[el(#) 2 6) e3(w)] respectfully. The full rotation matrix is represented by Equation 53.
cos(0) cos(y) sin( ) sin(0) cos(W) cos( ) sin(W) cos( ) sin(0) cos(W) + sin( ) sin(W)
REB = COS(8) Sin(W) Sin( ) Sin(8) Sin(W) COS( ) COS(W) COS( ) Sin(8) Sin(W) Sin( ) COS(W)
sin(0) sin( ) cos(0) cos( ) cos(0)
(53)
The rates of change of these orientation angles also require a coordinate transformation. The
roll rate, p, is the angular velocity about bl, the pitch rate, q, describes rotation about b2, and the
yaw rate, r, describes rotation about $3. The vector, m,, is given in Equation 54 to represent these
rates.
o,= p61+tq62 t r3 (54)
5.1.2 Camera
The camera is also described using a righthanded coordinate system defined using
orthonormal basis vectors. The axes, as shown in Figure 52, use the traditional choice of i3
aligning through the center of view of the camera. The remaining axes are usually chosen with
12 aligned right of the view and it aligned out the top although some variation in these choices
is allowed as long as the resulting axes retain the righthanded properties. The direction of the
camera basis vectors are defined through the camera's orientation relative to the bodyfixed
frame. This framework is noted as the camerafixed coordinate system because the origin is
always located at a fixed point on the camera and moves in the same motion as the camera.
The camera is allowed to move along the aircraft through a dynamic mounting which
admits both rotation and translation. This functionality enables the tracking of features while
the vehicle moves through an environment. The origin of the camerafixed coordinate system is
attached to this moving camera, consequently, the camerafixed frame is not an inertial reference.
A 6 degreeoffreedom model of the camera is assumed which admits a full range of motion.
Figure 52 also illustrates the camera's sensing cone which describes both the image plane and
the field of view constraint.
b3i
Figure 52. Camerafixed coordinate frame
Similar to the bodyfixed coordinate frame, a transformation can be defined for the mapping
between the bodyfixed frame, B and the camera frame, I as seen in Equation 55
i; by
i2. = RBI b2. (55)
13 b3B
where RBI is the relative rotation between frame B and I, respectfully. This transformation
is analogous to the aircraft's rollpitchyaw, where now these rotation angles define the roll,
pitch and yaw of the camera relative to the aircraft's orientation. The coordinate rotation
transformation, RBI, can be decomposed as a sequence of singleaxis Euler rotations as seen in
Equation 56, similar to the bodyfixed rotation matrix. The orientation angles of the camera are
required to determine the imaging used for visionbased feedback. The roll angle, #c, describes
rotation about ;;3, the pitch angle, 8c, describes rotation about 12 and the yaw angle, c,, describes
rotation about ii.
RBI= l(cOle 2e>le Oc)3c)] (56)
The matrix RBI in Equation 56 will transform a vector in bodyfixed coordinates to
camerafixed coordinates. This transformation is required to relate camera measurements to
onboard vehicle measurements from inertial sensors. The matrix again depends on the angular
differences between the axes in each coordinate system and the sequence of singleaxis rotations.
In particular, the rotation order used for this transformation was a 321 sequence.
cos(e,) cos(Ve) sin(Qc) sin(ec) cos(Ve,) cos(Qc) sin(Ve,) cos(Qc) sin(ec) cos(Ve,) + sin(Qc) sin(Ve,)
RBI = cos(8c) sin(Vec) sin(#c ) sin(8c) sin(Ve,) + cos(#c) cos(Ve,) cos(#c) sin(8c) sin(Ve) sin(#c) cos(Ve,)
sin(ec) sin(Qc) cos(6c) cos(Qc) cos(6c)
(57)
The rates of change of these orientation angles are again required for coordinate frame
transformations. The roll rate, pe, is the angular velocity about 13, the pitch rate, q,, describes
rotation about i2, and the yaw rate, re, described rotation about ii. The vector, me, is given in
Eq. 58 to represent these angles.
OWe = rcil +t qc 2 tPc 3
(58)
5.2 System Geometry
The fundamental scenario involves an aircraftmounted camera and a feature point in the
environment. This scenario, as outlined in Figure 53, thus relates the camera and the aircraft to
the feature point along with some inertial origin.
2, ~
;;;;~e
Feature Point
Figure 53. Scenario for visionbased feedback
The sensor modeling for visionbased feedback has to carefully account for the various
coordinate systems utilized in the scenario. The location of the aircraft and the feature point, as
given in Equation 59 and Equation 510 are typically represented in the inertial reference frame
relative to the Earthaxis origin.
TEB = Xb81 Yb82 +t Zb83 (59)
5 = 5xil +t (vi2 +t 523 (510)
The location of the camera, as given in Equation 511, is typically given with respect to
the bodyaxis origin. This choice of coordinate systems reflects that the camera is intrinsically
affected by any aircraft motion.
TBI = xIc 1t yc62 +t ze63 (511)
The remaining vector, rl, was defined in Equation 31 to describe the relative position
between the camera and the feature point. Recall, this vector was given in the camerafixed
coordinate system to note the resulting image is directly related to properties relative to the
camera. The representation of rl is repeated here in Equation 512 for completeness.
4 = ~vil+ Br2 + :13(512)
Applying the two rotational and translational concepts described in this chapter one
can transform vectors across all three coordinate frames. To fully describe a vector in the
camerafixed frame a transformation defined in Equation 513 is used. This expression
incorporates the translations involved with the origins of each coordinate frame through a
series of singleaxis rotations until the correct frame is reached.
11 e1
[2= RBIREB e2 +t RBITEB +t TBI (513)
13 e3
5.3 Nonlinear Aircraft Equations
The equations of motion of an aircraft can be represented in several different fashions.
The most general form of the aircraft equations are the nonlinear, highly coupled equations of
motion. These equations of motion are the standard equations which have been derived in a
typical aircraft mechanics book [108110] and are repeated in Equation 514 to 526 for overall
completeness.
P' mg sin O = m(ui + qw ry) (514)
F, + mg cos 6 sin = m(v + ru pw) (515)
F, + mg cos ecos = m(wi,+ py qu) (516)
L= A Grqr(z l) 1!4(517)
M =Ir,q+rp(l, I,) + ,(P2_ 2) (518)
N = IeI + Ir~+ pq(1I,) + zqr (519)
p = # isine (520)
q = O cos #+ ~icos 8sin (521)
r = Qicos ecos @ 0sin (522)
0= qcos rsin~ (523)
S= p +q9sin tan O+ rcos ~tan 8 (524)
i = (q sin + r cos #) sec 0 (525)
=xd CeS, S4SecwcS, +CC, CSeS, S4C, v (526)
dZb Se S4Ce C4Ce
The shorthand notation for Sw sinW, Cw cosy, So sin6, Co cos6, and Sq sing,
C4 cos # is used in Equation 526.
The aircraft states of interest for the camera motion system consist of the position and
velocity of the aircraft's center of mass, TEB and Vb, the angular velocity, co,, and the orientation
angles of the aircraft, (#, 6,11). The velocity of the aircraft's center of mass is vb and is defined in
Equation 527. As stated in Equation 527, the aircraft's velocity is expressed in the bodyfixed
coordinate frame. Each of these parameters will appear explicitly in the aircraft camera
equations.
vb = ub1 t +vb2+ twb3 (527)
The first six equations represent the force and moment equations, while the remaining
equations are kinematic relationships. The aerodynamic parameters consist of both the
aerodynamic forces, {K,, F, FZ}", on the aircraft and the aerodynamic moments, {L,1M, N}",
which are all contained in the force and moment equations. Although these equations do not
contain control inputs explicitly, the aerodynamic parameters are directly effected by the position
of the control surfaces on the aircraft. In other words, when the control surface deflections are
changed the flow over that surface also changes. This flow change over a surface results in
changes of the aerodynamic forces, such as lift and drag, which directly produce forces and
moments that roll, pitch, and yaw the aircraft and are described by the stability derivatives for
each aircraft. Therefore, controlled maneuvers are accomplished by changing these aerodynamic
parameters through the control surfaces.
An alternative approach to solving the nonlinear equations is to linearize these equations
about a trim condition using a Taylor series expansion. By linearizing these equations about a
level flight condition, the aircraft equations become decoupled into two planar motions. This set
of equations, although easy to solve, have limitations outside the chosen trim state, especially for
smaller more maneuverable aircraft. The choice of what set of aircraft equations to use depends
primarily on the aircraft and the application.
5.4 AircraftCamera System
The preliminary definitions established in the previous sections will now be used to
formulate the aircraftcamera system by using the systems described in this chapter. Here the
dependence of image plane position and velocity on the aircraft states along with the kinematic
states of the camera are shown. This derivation is shown here for one camera but is easily
extended to multiple cameras at various locations on the aircraft, as shown in the next section.
Meanwhile, this section obtains a result for feature points as a function of camera location and
aircraft states.
5.4.1 Feature Point Position
The fundamental results regarding the aircraftcamera system that relates 3D motion
to image plane motion starts simply by the vector summation of the defined positions. This
relationship is illustrated in Figure 53 for a feature point relative to the inertial frame. Therefore,
the vector sum can be used to solve for the relative position between the camera and a 3D feature
point. After making the proper coordinate transformations by using Equations 55 and 513, this
relative position can be expressed in camera frame, I, as shown in Equation 528.
rl = RBIREB(S TEB) RBITBI (528)
In summary, the resulting expression allows the position of each feature point in space
to be characterized by it's position in the image plane. By substituting the components of
Equation 528 into Equations 37 and 38 an image can be constructed as a function of aircraft
states. The major assumption of these equations is prior knowledge of the feature point location
relative to the inertial frame, which may be provided by GPS maps. Furthermore, the image
results obtained can also be passed through Equations 315 and 316 to add the effects of radial
distortion. The distorted image will provide a more accurate description of an image seen by a
physical camera, assuming the intrinsic parameters of the camera are known.
5.4.2 Feature Point Velocity
A feature point in the focal plane can be further characterized by deriving its velocity vector.
The velocity of a feature point in the image plane can be found by taking the time derivative of
Equation 528 with respect to the inertial frame, as shown in Equation 529.
Ed Ed Ed Ed
(l) = (5) (TEB) (TBI) (529)
dt dt dt dt
For a stationary feature point in space, the position vector, 5, is constant in magnitude and
direction and is expressed in the inertial frame; therefore, this time derivative is zero. Likewise,
the position vector of the aircraft's center of mass, TEB, is also expressed in the inertial basis and,
therefore, the time derivative just becomes ~TEB. Meanwhile, the Derivative Theorem is employed
on such terms as rl and TBI to express these terms in the moving frame. By applying this theorem
and solving for feature point velocity with respect to the camera frame, Equation 529 can now
be rewritten to Equation 530 for a nonstationary feature point.
'd(r) = ( ~TEB Bd(TBI) a~x TBI Eo Ix r (530)
dt dt
This equation can be reduced further if the cameras are constrained to have no translation
relative to the aircraft so 3(TEI) = 0. Alternatively, this term is retained in the derivation to allow
this degree of freedom in the camera setup. The angular velocity, E I, can be further decomposed
using the Addition Theorem. The final step implements Equations 55 and 513 to transform
each term into the camera frame. After some manipulation, the expression for the velocity of a
feature point relative to the camera frame results in Equation 531.
Ti = RBIREB ( TEBR RBITBI RBI( (x TBI) ((RBIO+ Wc) x 8) (531)
The image plane velocity of a feature point relative to the camera frame is finalized by
substituting both equations for position and velocity derived in Equation 528 and 531 into
Equations 332 and 333. This result will provide a description of the optical flow for each
feature point formed by either the camera traveling through the environment or the motion of the
feature points themselves. To incorporate radial distortion effects into the optic flow computation
requires the additional substitution into Equations 334 and 335.
5.5 System Formulation
The derivation of the aircraftcamera equations can be easily extended to systems with
multiple cameras all of which have their own position and orientation relative to the aircraft while
acquiring numerous feature points in each image. Although this adds computational complexity,
typical solutions to most visionbased problems require multiple views of the environment
in addition to having an adequate number feature points. The freedom to translate and rotate
cameras also gives the ability to track a particular target or region which reduces the amount
of aggressive maneuvers required by an aircraft to keep the target in the FOV. This additional
capability is extended by treating each camera and feature point separately when computing
Equations 528, 531, 315, 316, 334, and 335. Arranging the parameters for the kth camera
into a single vector, as shown in Equation 532, results then in the formulation of a generic
aircraftcamera system with k cameras all having independent motion that track n feature points
is obtained.
OCkT (t)= 1~,~2k, 2,,k, kc,k, Oc~kl Wc,k: fk, dk } (532)
The focal length, radial distortion, position, and orientation are now represented for each
camera present in the system, where the parameters of the kth camera are explicitly shown in
Equation 532. This vector can be extended to include other camera features such as CCD array
misalignment, skewness, etc.
The focal plane positions can then be assembled into a vector of observations as shown in
Equation 533, where n number of feature points are obtained. Likewise, the states of the aircraft
can be collected and represented as a state vector as shown in Equation 534. In addition, the
initial states of the vehicle are defined as Xo.
YT = {(p1,v1), (iU2,V2),..., /Pn, Vn> T (533)
XT(t) = {u, v, w, p, q, r, Xc, Yc, Zc, #, 6, W}" (534)
The coupled aircraftcamera system can now be formulated as a control problem by
incorporating the aircraft's equations of motion, the states, observations, and the kinematics of
the camera given in Equations 528 and 531. The observations used in this dissertation consist
of measureable images shown in Equations 315 and 316 which capture nonlinearities such as
radial distortion. This system, which measures image plane position, is described mathematically
through Equation 537
Xi(t) = (X (t), U(t), a(t), t) (535)
X (0) = Ko (536)
Y (t) = g (X (t), a(t), rl, t) (537)
where U(t) is defined as a set of control inputs to the aircraft and u(t) is a vector containing
the camera parameters aT = {ul, a2, ak T for k number of cameras. These equations that
utilized feature position will be referred to as the Control Theoretic Form of the governing
cameraaircraft equations.
Alternatively, if the image plane velocities are employed instead of the image plane
positions, as seen in Equation 537, then a different set of equations can be obtained which will
be referred to as the Optic Flow Form of the governing aircraftcamera equations of motion. This
system is given in Equation 540, which uses the optic flow expression given in Equations 334
and 335 as the observations.
Xi(t) = (X (t), U(t), a(t), t) (538)
X(0) = Ko (539)
J"(t) = 3M(X (t), a(t), rl, t) (540)
The two system equations just described both have applications to missions involving
unmanned aerial vehicles. The Control Theoretic Form primarily applies to missions involving
target tracking and surveillance such as aerial refueling and automated visual landing.
Meanwhile, the Optic Flow Form is useful for guidance and navigation through unknown
environments. The information provided by optic flow reveals magnitude and direction of each
feature point in the image which gives a sense of objects in close proximity. Incorporating
this information, along with some logic, a control system can be designed to avoid unforeseen
obstacles throughout the desired path.
5.6 Simulating
The inclusion of visionbased feedback into a flight simulator is a critical application of
these sensor models. A straightforward procedure is outlined that allows any flight simulator to
be augmented with visionbased feedback. Such a simulator can be augmented with additional
algorithms for image processing and synthetic vision to generate situational awareness.
Several requirements are required to implement this approach. First, a flight simulator, using
either linear or nonlinear equations of motion, must allow access to all vehicle states. Second, a
mounting system with known dynamics must be available to describe the camera states. Third,
a virtual environment must exist as a database of 3dimensional coordinates along with inherent
states of any timevarying features. Given these baseline tools, an algorithm is outlined to
compute visionbased feedback.
Algorithm 1.
for every time value of t {
compute aircraft states
compute camera states
compute environment states
for every feature in environment {
compute rl between camera and feature
compute pu and v
eliminate if outside field of view
eliminate if occluded
assemble image from pairs of (p,V) for features
This algorithm will directly augment a simulator with visionbased feedback; however, it is
not an optimal formulation to minimize computational cost. Various subroutines, such as feature
selection and database reduction, may be included to increase the efficiency of the simulator.
CHAPTER 6
DISCERNING MOVING TARGET FROM STATIONARY TARGETS
Classifying objects in a scene as stationary or moving is an essential task for autonomous
systems navigating through unknown terrain. The need for such a classification is due to the fact
that standard image processing algorithms, such as SFM reconstruction, do not hold for moving
objects since the epipolar constraint is violated when both the camera and the objects in the
scene are moving in space. Although the image processing algorithms for fixed features had been
introduced in Chapter 3 and how these features relate to the states of an aircraft in Chapter 5, the
effects of independently moving objects need to be handled in a different manner.
For cases involving a stationary camera, such as in surveillance applications, simple filtering
and image differencing techniques are employed to determine independently moving objects.
Although these techniques work well for stationary cameras, a direct implementation to moving
cameras will not suffice. For a moving camera, the apparent image motion is caused by a number
of sources, such as camera induced motion (i.e. egomotion) and the motion due to independently
moving objects. A common approach to detecting moving object considers a two stage process
that includes (i) a compensation routine to account for camera motion and (ii) a classification
scheme to detect independently moving objects.
6.1 Camera Motion Compensation
Camera motion estimation is a critical first step in the process of detecting moving objects.
The goal of camera motion estimation is to find a transformation that maps one image into
another given a sequence of images of the same scene. Two common approaches in literature for
this problem (i) employ the epipolar constraint with known feature point correspondence and/or
(ii) reliably compute image flow to decouple the apparent motion. The first technique assumes
feature point correspondence across image frames and uses the epipolar constraint to estimate the
relative rotation and translation between frames, as shown in Chapter 3.6. The second approach
uses the smoothness constraint in attempt to minimize the sum of square differences (SSD) over
either a select number of features or the entire flow field. This approach assumes the stationary
background exists as the "dominant" motion in the image which is not always true for urban
scenarios. For this dissertation a feature point solution will be employed which follows the
material presented in Chapter 3.6.
The epipolar constraint can be used to relate feature points across image sequences through
a rigid body transformation. The epipolar lines of a static environment are computed using
Equation 340 or Equation 342 depending if the essential matrix or the fundamental matrix
is required. An illustration of the computed epipolar lines is depicted in Figure 61 for a static
environment observed by a moving camera. Notice for this static case, the feature points in the
second image (the right image containing the overlaid epipolar lines) are shown to lie directly on
the epipolar lines.
0.4~ 0.4 .
0.2~ *. .0.2F
0.4~ 0.
0.6 0.4 0.2 0 0.2 0.4 0.6 0.6 0.4 0.2 0 0.2 0.4 0.6
v v
A B
Figure 61. Epipolar Lines Across Two Image Frames: A) Initial Feature Points and B) Final
Feature Points with Overlayed Epipolar Lines
Once camera motion estimation has been found, the epipolar lines can be used as an
indication of moving objects in the image. For instance, the feature points corresponding to the
stationary background will lie on the epipolar lines while the feature points corresponding to
moving objects will violate this constraint.
Similarly, the computation of optical flow can also be used for detecting independently
moving objects. In computing the optical flow, the motion induced by the camera along with
moving objects is fused together in the measured image. Recall, the optic flow expressions
are given in Equations 332 and 333 or Equations 334 and 335 for radial distortion.
Decomposing the optical flow into its components of camera rotation (Ar,, rt) and translation
(t, it,) and independently moving objects (A~i, ii) facilitates the detection problem. Therefore, the
components of the optical flow can be written as in Equation 61.
1:1+i +Ii Iiiii (61)
An expression for the optical flow induced by camera motion only can be rewritten in terms
of the aircraft states defined in Chapter 5S: the translational velocity [u,: v, w]T and the angular
velocity [p, q, r] of the camera. The resulting expressions are shown in Equations 62 and 6i3
and applies only to features stationary in the environment. The details describing the substitution
of the camera motion states are described in Chapter 5.
prv (1+p~2) u
= f q (62)
Iir ] u puV (1+tv2)
= z f oz V (63)
It was shown by Kehoe et al. [1 11] that the rotational states [p, q, r] can be estimated
accurately for a static environment through a nonlinear minimization procedure for n features
where n > 6. The approach used a vectorvalued flow field J (x) and is given in Equation 64,
91 "x, + x?2 912 14 23 \115 +12)r6)
J (x) = I (64)
(n xt x3n Un 4~r( np2)X 5 Vn 6)
x(6+n) x6n
where the vector x in Equation 65 is composed of unknown vehicle states and depth parameters.
x = u v w p q r rlz zn 1~(65)
The estimated vector, ^<, is found through solving the optimization problem posed in
Equation 66 that minimizes the magnitude of the cost function.
x =arg min 1  J(x) 2 (66)
The same approach is taken here with caution. Recall that the measured optical flow also
contains motion due to independently moving objects in addition to the induced optical flow
caused by the camera motion. In general, these variations in the measured optical flow will
introduce error into the [pl, q, r estimates. If some assumptions are made regarding the relative
optical flow between the static environment and moving objects, then errors in the state estimates
can have minimal effect. For instance, if the static portion of the scene is assumed to be the
dominant motion in the optical flow then the estimates will contain minimal errors. Employing
this assumption, estimates for the angular velocities [li:P ~1r~ of the camera/vehicle are obtained.
Substituting these estimates into Equation 62 results in estimates for the rotational portion
of the optical flow, as shown in Equation 67.
= f 4(67)
fr pI pv1ti (1+tv2)
Eliminating the rotational effects of the camera motion from the optical flow results in
Equation 68. The residual optical flow j~ess Res,, contains only the components of the camera
translation and independently moving objects. From this expression, constraints can be employed
on the camera induced motion to detect independently moving objects.
PRes it, ii t itr
I II (68)
For feature points that are stationary in the environment, the translational optic flow
induced by the camera motion is constraint to radial lines emanating from the FOE, as shown
in Figure 62. Consequently, feature points that violate this condition can be classified as
independently moving objects. This characteristic observed from static features will be the basis
for the classification scheme.
1 r
c
r
r
I ~JI
r
I I
r ,r
I ( I
I ~
r'r r
*~
2
~, r
/* ir
I
1.
r
E;OE
J'
Figure 62. FOE constraint on translational optic flow for static feature points
The residual optical flow may contain independently moving objects within the environment
that radiate from their own FOE. An example of a simple scenario is illustrated in Figure 63 for
a single moving object on the left and a simulation with synthetic data of two moving vehicles
on the right. Notice the two probable FOEs in picture on the left, one pertaining to the static
environment and the other describing the moving object. In addition, the epipolar lines of the two
distinct FOEs intersect at discrete points in the image. These properties of moving objects are
also verified in the synthetic data shown in the plot on the right. Thus, a classification scheme
must be designed to handle these scenarios to detect independently moving objects. The next
section examines the motion detection problem through the residual optical flow to further
classify static objects from dynamic objects in the field of view.
~0.5
FOE2 '
0.5
SOE 1
10.5 0 0.5 1
Figure 63. Residual optic flow for dynamic environments
6.2 Classification
The classification scheme proposed in this dissertation is an iterative approach to computing
the FOE of the static environment using the residual optical flow given in Equation 68. An
approximation for the potential location of the FOE is found by extending the translational
opticalflow vectors to form the epipolar lines, as illustrated in Figure 63, and obtaining all
possible points of intersection. As mentioned previously, the intersection points obtained will
constitute a number of potential FOEs; however, only one will describe the static background
while the rest are due to moving objects. The approach considered for this classification that
essentially groups the intersection data together through a distance criterion is an iterative
leastsquares solution for the potential FOEs.
The iteration procedure tests all intersection points as additional features are introduced
to the system of equations each of which involves 2 unknown image plane coordinates of the
FOE (Pfoe;, VSoe;). The process starts by considering 2 feature points and their FOE intersection
for the first iteration. It is assumed for the first iteration that the two features are static. The
leastsquares solution is then given in Equation 69 for the FOE coordinates (Pfoetv,Vfoe) (for
the first iteration a leastsquares solution is not necessary because two lines intersect at a single
point) .
= ar mm  b 2 (69)
where
1M= I # ##2l Prz Vi+1 (610a)
1 1 1
b ~ ~ ~ t = z p 2 2 "" Vi+1 A Pi 1 (610b)
The next iteration adds another feature into the system of equations and a new potential FOE
point is obtained. If the new feature point is a static feature, then the new estimated FOE will be
near the static FOE, which is found in the first iteration, causing a small residual. Alternatively,
if the feature is point is due to a moving object then the epipolar line will not intersect the
static FOE and shift the solution causing a large residual. Defining the new FOE coordinates as
(pfoue27 1982). A cost function is then checked to verify if the new feature point contains a similar
motion to that of the static background by checking the residual. This residual is defined as the
Euclidean distance from the two FOE solutions found before and after adding the next feature.
If cost is higher than some maximum threshold Jmax then the feature point is discarded into a set
of points classified as moving, fl; else, the feature point is classified into the static FOE solution,
C. This process is repeated until all n feature points have been checked using this cost function
which is shown in Equation 611 for the ith iteration. Mathematically, the classification scheme
for the ith iteration is given in Equations 612 and 613.
12 PVi Pfei Pfoi1 2 foeiVfei) 2(611)
Ci = {(Oli, Vi) if A (iPi, Vi) < Jmax }
(612)
else
Hi={(ii)if J2 (Pui,Vi) > Jmax} (613)
After all n feature points have been examined under this criterion, a set of m feature points
are classified to the static background, C. Meanwhile, a set of n m feature points are classified
as objects disobeying the static trend, 'L, and are considered moving objects. The class of moving
objects can be further classified into distinct objects through a clustering method. This method
removes all static features and uses the intersections of the epipolar lines pertaining to moving
objects as data points in the clustering algorithm. The resulting data will produce distinct clusters
around the FOEs pertaining to moving objects. The threshold Jax is a design parameter that
segments the feature points into their respective classes and needs to be tuned to account for
measurement noise.
CHAPTER 7
HOMOGRAPHY APPROACH TO MOVING TARGETS
7.1 Introduction
Autonomous vehicles have gained significant roles and assisted the military on the
battlefield over the last decade by performing missions such as reconnaissance, surveillance,
and target tracking with the aid of humans. These vehicles are now being considered for more
complex missions that involve increased autonomy and decision making to operate in cluttered
environments with less human interaction. One critical component that autonomous vehicles
need for a successful mission is the ability to estimate the location and movement of other objects
or vehicles within the scene. This capability, from a controls standpoint, enables autonomous
vehicles to navigate in complex surroundings for tracking or avoidance purposes.
Target state estimation is an attractive capability for many autonomous systems over a
broad range of applications and is the focus of this dissertation. In particular, unmanned aerial
vehicles (UAV) have shown a great need for this technology. With UAV becoming more prevalent
in the aerospace community, researchers are striving to extend their capabilities while making
them more reliable. The key applications of interest for future UAV regarding target estimation
pertain to both civilian and military tasks. These tasks range from police car pursuits and border
patrol to locating and pursuing enemy vehicles during lethal engagement. A major limitation
to small UAV are their range, payload constraints and fuel capacity. These limitations generate
the need for autonomous aerial refueling (AAR) to extend the vehicle's operational area. Target
state estimation facilitates a portion of the AAR problem by estimating the receptacle's current
position and orientation during approach. Therefore, the purpose of this paper is to demonstrate
a method that estimates the motion of a target using an onboard monocular camera system to
address these applications.
Most techniques for visionbased feedback share some commonality; namely, a sequence of
image processing and vision processing are performed on an image or a set of images to extract
information which is then analyzed to make a decision. The basic unit of information from an
image is a feature point which indicates some pixel of particular interest due to, for example,
color or intensity gradient near that pixel. These intensity variations correlate well to physical
features in the environment such as comers and edges which describe the character of buildings
and vehicles within a scene as described in Chapter 3. Among the techniques that utilize feature
points, the approach related to this paper involves epipolar geometry [39, 112]. The purpose of
this technique is to estimate relative motion based on a set of pixel locations. This relative motion
can describe either motion of the camera between two images or the relative distance of two
objects of the same size from a single image.
The 3D scene reconstruction of a moving target can be determined from the epipolar
geometry through the homography approach described in Chapter 3. For the case described in
this chapter, a moving camera attached to a vehicle observes a known moving reference object
along with an unknown moving target object. The goal is to employ a homography visionbased
approach to estimate the relative pose and translation between the two objects. Therefore, a
combination of vision and traditional sensors such as a global positioning system (GPS) and
an inertial measurement unit (IMU) are required to facilitate this problem for a single camera
configuration. For example in the AAR case, GPS and IMU measurements are available for both
the receiver and tanker aircraft.
In general, a single moving camera alone is unable to reconstruct the 3D scene containing
moving objects. This restriction is due to the loss of the epipolar constraint, where the plane
formed by the position vectors relative to two camera positions in time to a point of interest
and the translation vector is no longer valid. Techniques have been formulated to reconstruct
moving objects viewed by a moving camera with various constraints [35, 113116]. For instance,
a homography based method that segments background from moving objects and reconstructs
the target's motion has been achieved [117]. Their reconstruction is done by computing a virtual
camera which fixes the target's position in the image and decomposes the homography solution
into motion of the camera and motion caused by the target. This decomposition is done using a
planar translation constraint which restricts the target's motion to a ground plane. Similarly, Han
and Kanade [115] proposed an algorithm that reconstructs 3D motion of a moving object using
a factorizationbased algorithm with the assumption that the object moves linearly with constant
speeds. A nonlinear filtering method was used to solve the process model which involved both
the kinematics and the image sequences of the target [118] This technique requires knowledge
of the height above the target which was done by assuming the target traveled on the ground
plane. This assumption allowed other sensors, such as GPS, to provide this information. The
previous work of Mehta et al. [77] showed that a moving monocular camera system could
estimate the Euclidean homographies for a moving target in reference to a known stationary
object.
The contribution of this chapter is to cast the formulation shown in Mehta el al. to a
more general problem where both target and reference vehicles have general motion and are
not restricted to planar translations. This proposed approach incorporates a known reference
motion into the homography estimation through a transformation. Estimates of the relative
motion between the target and reference vehicle are computed and related back through known
transformations to the UAV. Relating this information with known measurements from GPS
and IMU, the reconstruction of the target's motion can be achieved regardless of its dynamics;
however, the target must remain in the image at all times. Although the formulation can be
generalized for n cameras with independent position, orientation, translations, and rotation this
chapter describes the derivation of a single camera setup. Meanwhile, cues on both the target
and reference objects are achieved through LED lights or markers placed in a known geometric
pattern of the same size. These markers facilitate the feature detection and tracking process by
placing known features that stand out from the surroundings while the geometry and size of the
pattern allows for the computation of the unknown scale factor that is customary to epiploar and
homography based approaches.
This chapter builds on the theory developed in Chapters 3 and 5 while relying on the moving
object detection algorithm to isolate moving objects within an image. Recall the flow of the
overall block diagram shown in Figure 16. The process started by computing features in the
image relative to an aircraft and then employing the moving object detection algorithm shown in
Chapter 6. Once moving objects in the image are detected, the homography estimation algorithm
proposed in this chapter is implemented for target state estimation.
7.2 State Estimation
7.2.1 System Description
The system described in this paper consists of three independently moving vehicles or
objects containing 6DOF motion. To describe the motion of these vehicles a Euclidean space is
defined with five orthonormal coordinate frames. The first frame is an Earthfixed inertial frame,
denoted as E, which represents the global coordinate frame. The remaining four coordinate
frames are moving frames attached to the vehicles. The first vehicle contains two coordinate
frames, denoted as B and I, to represent the vehicle's body frame and camera frame, as described
in Chapter 5 in Figure 51. This vehicle is referred to as the chase vehicle and is instrumented
with an onboard camera and GPS/IMU sensors for position and orientation. The second vehicle,
denoted as F, is considered a reference vehicle that also contains GPS/IMU sensors and provides
its states to the chase vehicle through a communication link. Lastly, the third vehicle, denoted
as T, is the target vehicle of interest in which unknown state information is to be estimated. In
addition, a fictitious coordinate frame will be used to facilitate the estimation process and is
defined as the virtual coordinate system, V.
The coordinates of this system are related through transformations containing both rotational
and translational components. The rotational component is established using a sequence of
Euler rotations in terms of the orientation angles to map one frame into another. Let the relative
rotation matrices REB, RBI, REF, REV, RIV, RFV, RTy and RET denote the rotation from E to B,
B to I, E to F, E to V, I to V, F to V, T to V, and E to T. Secondly, the translations are defined
as TEB, F, XV, XT, F,n, T,n, TBI, XIV, mFI mIT, TF,n N1T,n, W1VF, FVT, TVF,n, and rlvr,n which
denote the respective translations from E to B, E to F, E to V, E to T, E to the 12th feature point
on the reference vehicle and target vehicles all expressed in E, B to I expressed in B, I to V, I to
F, I to T, I to the 12th feature point on the reference and target vehicles expressed in I, V to F, V
to T, V to the nth feature point on the reference and target vehicles expressed in V. This vector
geometry relating the coordinate frames is illustrated in Figure 71 for a camera on board a UAV
while the vectors relating the feature points to both the real and virtual cameras are depicted in
Figure 72. The estimated quantities computed from the vision algorithm are defined as RTB and
xTB which are the relative rotation and translation from T to B expressed in B.
1 3
02\
Feature Point
Figure 71. System vector description
The camera is modeled through a transformation that maps 3dimensional feature points
onto a 2dimensional image plane as described in Chapter 3. This transformation is a geometric
relationship between the camera properties and the position of a feature point. The image plane
coordinates are computed based on a tangent relationship from the components of rln. The
camera relationship used in this chapter is referred to as the continuous pinhole camera model
and is given in Equations 37 and 38 for a zero lens offset, where f is the focal length of the
camera and rlx,n, Tly,n, rlz,n are the (x, y, z) components of the nth feature point.
This pinhole model is a continuous mapping that can be further extended to characterize
properties of a physical camera. Some common additions to this model include skewness, radial
77VT,n
/IT~n XV
11IT
mlVT
Target
e eFeature
Fly~ syPoin E lve~ sPoint
sF FauePoint sF) Fetur n
IF F
A B
Figure 72. Moving target vector description relative to A) camera I and B) virtual camera V
distortion, discrete mapping into pixels, and field of view constraints which are further also
specified in Chapter 3. Each extension to the model adds another parameter to know for the
estimation problem and each can introduce uncertainty and large errors in the estimation result.
Therefore, this chapter will only consider the field of view constraint and leave the nonlinear
terms and the effects on estimation for future work. Recall the field of view constraints given in
Chapter 3. These constraints can be represented as lower and upper bounds in the image plane
and are dependent on the half angles (yh, t) which are unique to each camera. Mathematically,
these bounds are shown in Equations 71 and 72 for the horizontal and vertical directions.
[pip] = [f tanh, f tanyh (71)
[v, V] = [ f tany,, f tany,] (72)
7.2.2 Homography Estimation
The implicit relationship between camera and environment is known as the epipolar
constraint or, alternatively, the homography constraint. This constraint notes position vectors that
describe a feature point, rln, at two instances in time are coplanar with the camera's translation
vector [102]. The same constraint holds for the image coordinates as well but also introduces an
unknown scale factor. Employing this constraint, estimates of relative motion can be acquired
for both camerainhand and fixed camera configurations. This dissertation deals with the
camerainhand configuration while assuming a perfect feature point detection and tracking
algorithm. This assumption enables the performance of the vision based state estimation to be
tested before introducing measurement errors and noise.
The homography constraint requires a few assumptions based on the quantity and the
structure of the feature points. The algorithm first requires a minimum of four planar feature
points to describe each vehicle. This requirement enables a unique solution to the homography
equation based on the number of unknown quantities. The reference vehicle will have a minimum
of four pixel values in each image which will be defined as pF,n = CUF,n,VF,n] Vn2 feature points.
Likewise, the target vehicle will have four pixel values and will be defined as pr,n = [PT,n, VT,n] Vn2
feature points. This array of feature point positions are computed at 30 Hz which is typical for
standard cameras and the frame count is denoted by i. The final requirement is a known distance
for both the reference and target vehicle. One distance represents the position vector to a feature
on the reference vehicle in Euclidean space relative to the local frame F and the second distance
represents the position vector to a feature on the target vehicle in Euclidean space relative to
the local frame T. In addition, the length of these vectors also must be equal which allows the
unknown scale factor to be determined. The vector describing the reference feature point will be
denoted aS SF expressed in F, while the vector describing the target feature point is referred to as
sT expressed in T. These feature point position vectors are also illustrated in Figure 72.
The feature points are first represented by position vectors relative to the camera frame,
I. The expressions for both the reference and target feature points are given in Equations 73
and 74. These vector components are then used to compute the image coordinates given in
Equations 71 and 72. The computation in Equation 74 requires information regarding the
target which is done solely to produce image measurements that normally would be obtained
from the sensor. Remaining computations, regarding the homography, will only use sensor
information provided only by the camera vehicle, the reference vehicle and the images acquired
from the camera.
T1F,n = RBIREB (F,n TEB) RBITBI +tRBIRFBSF (73)
T1T,n = RBIREB (T,n TEB) RBITBI +tRBIRTBST (74)
The variables RFB and RTB in Equations 73 and 74 are the true rotations matrices from F to B
and T to B, repectfully, and are shown in Equations 75 and 76.
RFB = REBRETF (75)
RTB = REBRETT (76)
For state estimation of a moving target using a moving camera the homography approach
requires the reference vehicle to be stationary in the image [77]. In this case, both the reference
and target vehicles are in motion and are being viewed by a moving camera. Therefore, the next
step is to transform the camera to a virtual configuration that observes the reference vehicle
motionless in the image over two frames. In other words, this approach computes a Euclidean
transformation that maps the camera's states at i 1 to a virtual camera that maintains the relative
position and orientation between frames to fix the feature points of the reference vehicle. This
transformation is done by making use of the previous image frame and state information at
i 1 from both the camera and the reference vehicle. After the virtual camera is established the
homography equations can be employed for state estimation.
To compute the location and pose of the virtual camera at i the relative position and
orientation from I to F at i 1 is required. This relative motion is computed through known
measurements from GPS/IMU and the expressions are shown in Equations 77 and 78 for
translation and rotation at i 1, respectfully.
xlF 1) IF 1) I (i 1)(77)
RIF\D (i1 E i1 R ( )Rg i1 (78)
Once the relative motion is determined, the position and orientation of the virtual camera
relative to E can be computed. These relationships are shown in Equations 79 and 710 for the
current frame i.
xy (i) = xF (i) +t REB (i 1) TBI ( 1) XIF (i 1) (79)
R\EV (i) = RBI( (i 1) REB( (i 1) REF (i I1) EF~ (i) (710)
The virtual camera position and orientation is then used to update the image coordinates for
both the reference and target vehicles. This update requires computing new rln vectors in terms
of the virtual camera's position and orientation followed by a substitution of those components
into the pinhole camera model given in Equations 37 and 38. The expressions for the new
vectors TIVF,n and vrlV,n in terms of the virtual camera are given in Equations 711 and 712 for
the reference and target vehicles.
rlVF,n = REV (F,n XV) +tRFVSF (711)
vrlV,n = REV (T,n XV) +t RTVSy (712)
Equations 711 and 712 are one way to compute image coordinates for the virtual camera,
but there are unknown terms in Equation 712 that aren't measurable or computeable in this
case. Therefore, an alternative method must be used to compute image values of the target
in the virtual camera. Using the position and orientation of the virtual camera, as given in
Equations 79 and 710, the relative motion is computed from camera I to camera V while using
epipolar geometry to compute the new pixel locations. This relative camera motion is given in
Equations 713 and 714 where the translation is expressed in I.
xyV = R\BIREB InV rEB R'EBTBI) (3
RlV = REVREBRBI (714)
Using this relative motion and the pixel locations obtained from camera I, the pixel
coordinates are computed of the target in the image plane seen by the virtual camera V. The
epipolar constraint enables the relationship between image frames to compute the features of the
target through Equation 715
PvT,n (xyTVRIV) pr,n= (715)
where xyjv is the skew symmetric representation of the relative translation from I to V expressed
in I and the new pixel coordinates determined from the virtual camera are denoted as pVF,n =
[PVF,n, VVF,n] Vn2 for the reference vehicle and pyr,n = [PVr,n, Wa~n] Vn2 for the target vehicle. As
a result of the virtual camera, the desired property is obtained regarding pixels of the reference
vehicle computed from the camera at i 1 are equal to the pixels generated by the virtual camera
at i. Mathematically, this property is expressed in Equation 716 which relies on the relative
motion remaining constant to maintain the reference stationary in the image.
PF,n ( 1) = PVF,n (i) (716)
With this virtual camera in place and the reference pixels stationary, the computation of
the homography between the reference and target vehicles is considered. First, the geometric
relationships are established relative to the virtual camera of both the reference and target
vehicles by denoting their feature point positions in Euclidean coordinates. The time varying
position of a feature point on the reference vehicle expressed in V is given in Equation 717.
Likewise, the time varying position of a feature point on the target vehicle expressed in V is given
in Equation 718.
rlVF,n = mvF +tRFVSF (717)
rlvr,n = myr +tRTVSy (718)
The components of these Euclidean coordinates are defined in Equations 719 and 720 and are
relative to the virtual camera frame.
rlVF,n (t) t yt t)(9
vrn()a x t r()L t (720)
After some manipulation, an expression for the relative translation and rotation between the
reference vehicle and the target vehicle can be written as shown in Equation 721.
TIvr,n = X +tRTVF,n (721)
The relative translation, x, expressed in V and rotation, R, are defined in Equations 722 and 723
which describe the relative motion between the reference and target objects.
x = myr R (inVF +t RFV (SF ST)) (722)
R = RTVRFV (723)
By employing some known quantities and assumptions regarding the feature points, the
unknown scale factor in the homography equation can be determined. Recall, the virtual camera
location is known through Equation 79 and the reference vehicle location is known through
GPS along with the feature point locations, therefore, a projected distance can be computed that
scales the depth of the scene. To compute this distance the normal vector, n, that defines the plane
which the reference feature points lie is required and can be computed from known information.
Ultimately, the projective distance can be obtained and is defined in Equation 724 through the
use of the reference position.
D (t) = n"BlVF (724)
Substituting Equation 724 into Equation 721 results in an intermediate expression for the
Euclidean homography and is shown in Equation 725.
,vr.n = R+ onX _VFT (725)
To facilitate the subsequent development, the normalized Euclidean coordinates are used and
defined in Equations 726 and 727.
rlVF,n
TIVF,n (726)
From Equations 725, 726, and 727 the normalized Euclidean homography is established
which relates the translation and rotation between coordinate frames F and T. This homography
expression is shown in Equation 728 in terms of the normalized Euclidean coordinates.
TIvr,n = R h )Vn
(728)
a H
In Equation 728 a (t) denotes the depth ratio, H (t) denotes the Euclidean homo graphy, and
Th (t) denotes the scaled translation and is defined in Equation 729.
Th =) (729)
The Euclidean homography can now be expressed in terms of image coordinates or pixel
values through the ideal pinhole camera model given in Equations 37 and 38. This expressing
is done by first rewriting the camera model into matrix form which is referred to as the camera
calibration matrix, K. Substituting the camera mapping into Equation 728 and using the camera
calibration matrix, K, the homography in terms of pixel coordinates is obtained and given in
Equation 730. This final expression relates the rotation and translation of the two vehicles F
and T in terms of their images coordinates. Therefore, to obtain a solution from this homography
expression both vehicles need to be viewable in the image frame.
pyr..n = a (KHK ) pVF.,n
v (730)
The matrix G (t) is denoted as a projective homo graphy in Equation 730 which are a set
of equations that can be solved up to a scale factor using a linear least squares approach. Once
the components of homography matrix are estimated the matrix needs to be decomposed
into translational and rotational components to obtain xh and R. This decomposition is
accomplished using techniques such as singular value decomposition and generates four possible
solutions [119, 120]. To determine a unique solution some physical characteristics of the problem
can be used. First, two solutions can be eliminated by using the positive depth constraint. The
decision regarding the remaining two solutions is more difficult to decipher unless the normal
vector is known or can be estimated, which in this case is known. Recall the normal vector,
n, describes the plane containing the feature points of the reference vehicle. As a result, the
homography solution is determined uniquely.
The final step in this development is to use the homography solution to solve for the relative
translation and rotation from T to B. The resulting equation for the rotation uses a sequence of
transformations and is shown in Equation 731.
RTB = RBIREBRETVRRBI (i 1) REB (i 1) RETF (i 1) (731)
The translation is found through a series of scalings followed by a vector sum. The relative
translation, xh, is first multiplied by D to scale distance which is given in Equation 729 to obtain
x. Secondly, x is then divided by oc to scale the depth ratio resulting in the final x expressed in I.
This result in conjunction with R is then used in Equation 722 to solve for myr. The next step is
to compute the relative translation from I to V which is given in Equation 732.
my=RE V EB+ETBTI (732)
The relative translation from T to B is then given in Equation 733.
xTB = REBRETV (mvT mIV) (733)
In conclusion, Equations 731 and 733 represent the relative motion between the camera
vehicle and the target vehicle. This information is valuable for the control tasks described earlier
involving both tracking and homing applications. The next section will implement this algorithm
in simulation to verify the state estimator for the noise free case.
CHAPTER 8
MODELING TARGET MOTION
8.1 Introduction
Once state estimation of a moving target has been obtained the next step is to record these
estimates over time to try and leamn the object's general motion. The purpose of understanding
these motions are useful for prediction and allows for closedloop control for applications such as
autonomous docking and AAR. In essence, this prediction step provides the tracking vehicle with
future state information of the target which assists the controller in both the tracking and docking
missions. This chapter describes a probabilistic method that employs the time history estimates
of the target's motion to determine future locations. In addition to providing state predictions, the
modeling scheme also provides position updates when features are outside the field of view.
Linear modeling is not sufficient for prediction in this situation, where the motion is
stochastic. Linear techniques that estimate a transfer function, such as ARX, require that the
inputs and outputs of the system are known. Although this is the case for many systems, it
doesn't apply in this scenario because the inputs (i.e. the forces) on the target are assumed to be
unknown. For example, in the AAR mission the target, or drogue, interacts with a flow field that
is potentially turbulent due to the effects of the surrounding aircraft (i.e. tanker and receiver)
and difficult to model. The drogue is also tethered by a flexible boom that applies reaction forces
which are dictated from the tanker aircraft and the aerodynamic forces on the boom. These
factors make the modeling task challenging to accurately represent the motion of a general target
with unmodeled dynamics and disturbances. Therefore, the method considered in this dissertation
will consist of a probabilistic approach to account for general motions with stochastic behavior.
8.2 Dynamic Modeling of an Object
There are numerous modeling schemes in the research community. The probabilistic
approaches can be separated into two main categories consisting of supervised and unsupervised
learning algorithms. Supervised algorithms require training data that determines trends apriori
and classifies the the motion under consideration to the trends observed during training.
Alternatively, unsupervised learning requires no explicit training. Instead, these algorithms
rely on data clustering to determine the natural trends of the motion. The approach taken in
this dissertation is an unsupervised technique presented by Zhu [121] that employs a Hidden
Markov Model to predict the motion of moving objects. The benefits in using a Hidden Markov
Model include a time dependence framework incorporated into the probabilistic model as well
as the ability to handle stochastic processes. The underlining concept of a Hidden Markov
Model describes the probability of a process to sequentially go from one state to another. This
sequential property provides the necessary framework for time dependence modeling, which is an
attractive approach for the applications considered, where the time history data is a critical piece
of information included in the modeling.
8.2.1 Motion Models
The selection of motion models that can be used for predicting the location of a target
contains infinite possibilities due to the various types motions. A target's motion generally
involves a single model but also can contain various models that comprise the overall motion.
These motions can exist at different periods throughout the trajectory. Incorporating this
logic into the prediction scheme, models are chosen based on the current acceleration of the
target which is determined by the time history of the position estimates. Constant velocity and
stochastic acceleration models are two general types of motions considered.
The constant velocity model is derived by assuming the acceleration of the target is zero, as
shown in Equation 81. Therefore, the velocity and position are updated through Equations 82
and 83. Although this model is limited, it describes a foundation for modeling target motion and
covers the basic model constant velocity.
a (t) = (81)
v (t) =s (82)
p (x, Y, t +t At) = p (x, y, t) +t s~t (83)
The next model considered involves a random motion model. The assumed acceleration is
shown in Equation 84 and is characterized by a random vector, w (t) and is scaled by a constant,
p. The velocity corresponding to this acceleration is described in Equation 85. This model
attempts to capture the stochastic behaviors by utilizing a probabilistic distribution function.
a (t) = pw (t) (84)
v (t) = v (t A) + p w (T) d (85)
Alternatively, the model shown in Equation 84 can be modified to incorporate some
dependence on the previous acceleration value. This dependence is achieved by weighting the
previous acceleration in the model and is shown in Equation 86. The benefit to this type of
model as oppose to Equation 84 requires some knowledge of the target; namely, that the target
cannot achieve large abrupt changing in acceleration. The resulting velocity expression for this
model is given in Equation 87.
a (t) = poa (t At) +t p w (t) (86)
v (t) = v i+ JN(t A) + a( A) w(T) dt (87)
8.2.2 Stochastic Prediction
The image sequence obtained from the camera are processed by the homography to obtain
the position estimates of the target. These position estimates are then used to compute a velocity
profile of the target, as shown in Equation 88 for the ith target and N image frames. The velocity
profile is computed using a backwards difference method and is given in Equation 89.
[vi (t 1) ,vi(t 2) ,...,vi (t N+1 ),vi (t N)] (88)
vi (t j) = pi (t j) pi (t j 1) (89)
Similarly, an acceleration profile, defined in Equation 810, is obtained from the velocity
profile given in Equation 88. The same backwards difference method is used to compute this
profile and is provided in Equation 811. This acceleration time history is computed implicitly
through the position estimates obtained from the homography algorithm
[ai (t 1) ,ai (t 2),...,ai (t N+2),ai (t N+1)] (810)
ai (t j) = vi (t j) vi (t j 1) (811)
The motion profiles given in Equations 88 and 810 provide the initial motion state
description that propagates the Markov transition probability function. The form of the Markov
transition probability function is assumed to be a Gaussian density function that only requires
two parameters for its representation. The parameters needed for this function include the mean
and variance vectors for the acceleration profile given in Equation 812. Note, during this chapter
pu (x) is the mean operator and not the vertical component in the image plane. Likewise, 02 (X) iS
referred to as the variance operator.
[p (ai (t +t j)) G2 (ai (t + j))] j ,1 ,k(812)
The Markov transition function is defined in Equation 813, where the arguments consist of
the mean and variance pertaining to the estimated acceleration.
P (ai (t + j)) = xn Cp (ai (t + j)) 02 (ai (t + j))) (813)
The initial mean and variance for acceleration are computed in Equations 814 and 816
for the transition function. The functions f, and f, are chosen based on the desired weighting of
the time history and can simply be a weighted linear combination of the arguments. These initial
statistical parameters are used in the prediction step and updated once a new measurement is
obtained.
p (ai (t)) = fe (ai (t 1) ai (t 2) ,. .ai (t N)) (814)
a2 Si F)) f (a ir l ) i (t2))2 (815)
Finally, the Markov transition probability function is given explicitly in Equation 816
as threedimensional Gaussian density function and is uniquely determined by the mean and
variance.
1 1 I(ai (t) p (ai (t))) 2
P (ai (t))= ep(6
AGn (ai (t))1 e 2 G2 a t (6
The probability function is then extended over the entire time interval [t, t +t k] to estimate
the prediction probability. Mathematically, this extension is expressed in Equation 817.
P (ai (t +t j)) = P (ai (t +t j 1)) = .. = P (ai (t)) (817)
Employing Equations 816 and 817, the predictive probability for object i at time t +t k
is given as Equation 818. This framework enables the flexibility of computing the predicted
estimates at any desired time in the future with the notion that further out in time the probability
diminishes.
k1
Probe (ai (t + k)) = n P (ai (t. +j) (818)
j= 0
A similar process is considered for computing the Markov transition functions for both
velocity and position. First, the mean and variance vectors for velocity and position are defined in
Equations 819 and 820 for the entire time interval.
[p (vi (t + j)) G2 (Vi (t + j))] j ,1 ,k(819)
[p(pi (t+ j)) ,G2 (pi(t + j))] j=01.k(820)
The initial mean and variance expressions for the velocity are given in Equation 822
and 823.
pu (vi (t)) = pu (vi (t 1) +t ai (t 1)) (821)
= vi (t 1) +t p (ai (t 1))
G2 (Vi (t)) = G2 (Vi (t 1) +t ai (t 1)) (822)
Meanwhile, the expressions for the mean and variance for the position are given in
Equations 824 and 825.
pu (pi (t)) = u (pi (t ) +vi (t ) + a~lai (t 1) (823)
= pi (t 1) + p (vi (t 1)) 2 p (ai (t 1))
$(02 ) 2 (p (t ) Vi (t ) + a t (t ) (824)
= G2 (Vi (t 1)) O2 2a t )
Lastly, the probability functions for velocity and position are used to compute the predictive
probabilities for object i that are given in Equations 825 and 826 for velocity and position,
respectfully.
Probe (vi (t + k)) = g P (vi (t + j)) (825)
j= 0
k1
Probe (pi (t + k)) = gP t(pi (t. + j) (826)
j= 0
Therefore, the probability given in Equation 826 is the probability that target i is located in
position p (x, y, z). So the the overall process is an iterative method that uses the motion models,
given in Section 8.2.1, to provide guesses for position and velocity in attempt to maximize
the probability functions given in Equations 825 and 826. The position that maximizes
Equation 826 is the most likely location of the target at t +t k with a known probability.
CHAPTER 9
CONTROL DESIGN
The control strategy considered in this dissertation uses the computed relative states found
between a moving camera and a moving target of interest as shown in Chapter 7. Effectively,
these quantities are the error signals used for control to track the moving camera toward a desired
location based on the motion of the target. The framework presented here will use aircraft and
UAV navigation schemes for the aerial missions described in Chapter 1. Therefore, the control
design described in this chapter focuses on the homing mission to facilitate the AAR problem,
which involves tracking the position states computed from the homography.
Various types of guidance controllers can be implemented for these types of task once
the relative position and orientation are known. Depending on the control objectives and how
fast the dynamics of the moving target are, low pass filtering or a low gain controller may be
required to avoid high rate commands to the aircraft. In the AAR problem, the success of the
docking controller will directly rely on several components. The first component is the accuracy
of estimated target location which during AAR needs to precise. Secondly, the dynamics
of the drogue are stochastic. This causes the modeling task to be impractical in replicating
real life so the controller is limited to the models considered in the design. In addition, the
drogue's dynamics may not be dynamically feasible for the aircraft to track which may further
reduce performance. Lastly, the controller ideally should make position maneuvers in stages by
considering the altitude as one stage, the lateral position as another stage, and the depth position
as the final stage. In close proximity, the controller should implement only small maneuvers to
help maintain the vehicles in the FOV.
9.1 Control Objectives
The control objectives for the AAR mission is to track and home on the target drogue and
successfully dock with the receptacle. This controller is designed using a tracking methodology
that regulates the relative distance to within a specified tolerance. For example, the tolerance
required for aerial refueling is on the centimeter scale [15].
9.2 Controller Development
The control architecture chosen for this mission consisted of a Proportional, Integral and
Derivative (PID) framework for waypoint tracking given in Stevens and Lewis [110]. The
standard design approach was used by considering the longitudinal and lateral states separately as
in typical waypoint control schemes. This approach separated the control into three segments: 1)
Altitude control, 2) Heading Control and 3) Depth Control.
9.2.1 Altitude Control
The first stage considered in the control design to home on a target is the altitude tracking.
This stage considers the longitudinal states of the aircraft using the elevator as the control
effector. The homography generates the altitude command necessary to track and dock with the
refueling receptacle. The architecture for the altitude tracking system is shown in Figure 91. The
first portion of this system is described as the innerloop where pitch and pitch rate are used in
feedback to stabilize and track a pitch command. Meanwhile, the second portion is referred to as
the outerloop which generates pitch commands for the innerloop based on the current altitude
error. The innerloop design enables the tracking of a pitch command through proportional
Figure 91. Altitude hold block diagram
control. This pitch command in turn will affect altitude through the changes in forces on the
horizontal tail from the elevator position. The two signals used for this innerloop are pitch and
pitch rate. The pitch rate feedback helps with short period damping and allows for rate variations
in the transient response. A lead compensator was designed in Stevens et al. [110] to raise the
loop gain and to achieve good gain and phase margins for the pitch command to pitch transfer
function.
The outerloop design involved closing the loop in altitude. The altitude error signal is
generated by the difference in current altitude and the commanded altitude computed by the
estimation algorithm. The compensator designed for the innerloop pitch is augmented to
maintain the high loop gain and is defined as G, in Figure 91. This structure will provide good
disturbance rejection during turbulent conditions. In addition, bounds were placed on the pitch
command to alleviate any aggressive maneuvers during the refueling process.
9.2.2 Heading Control
The next stage in the control design consist of the turn or heading coordination. This
aspect involves the lateral directional states of the aircraft. The control surfaces that effect these
states are ailerons and rudder. Similar to the altitude controller, the homography estimates a
heading command that steers the aircraft in the desired direction toward the target of interest. The
control architecture that accomplishes this objective is depicted in Figure 92. The innerloop
Figure 92. Heading hold block diagram
component of Figure 92 deals with roll tracking. The feedback signals include both roll and
roll rate through proportional control to command a change in aileron position. The innerloop
stabilization design also included a roll to elevator connect to help counteract the altitude loss
during a tumn.
The outerloop is completed by simply closing the loop around the roll tracker using a
proportional gain to follow to desired heading. In addition, command limits of +600 were
placed on roll to regulate aggressive turns and a yaw damper was also implemented that included
a aileronrudder interconnect which helps a tumn in a number of ways. The aileronrudder
interconnect helps to raise the nose up during a turn. Meanwhile, the yaw damper is employed to
damp oscillations from the Dutchroll mode during a heading maneuver. The design of the yaw
damper is provided in Stevens et al. [110]. Consequently, the tumn smoother and contains less
oscillations.
Tracking heading is not sufficient to track the lateral position with the level of accuracy
needed for refueling task. The final loop was added to account for any lateral deviation
accumulated over time due to the delay in heading from position. This delay is mainly due to the
time delay associated with sending a roll command and producing a heading change. Therefore,
this loop was added to generate more roll for compensation. The loop commanded a change in
aileron based of the error in lateral position. This deviation, referred to as Ay, was computed
based on two successive target locations provided by the estimator. The current and previous
(x, y) positions of the target were used to compute a line in space to provide a reference of the it's
motion. The perpendicular distance from the vehicle's position to this line was considered the
magnitude of the lateral command. In addition, the sign of the command was needed to assign
the correct direction. This direction was determined from the relative y position, expressed in the
bodyfixed frame, that was found during estimation. Once the lateral deviation was determined,
that signal was passed through a PI structure, as shown in Figure 92. The gains corresponding to
the proportional, kyp, and integrator, kyi, were then summed and added to compute the final roll
command. The complete expression for the roll command is shown in Equation 91.
ky
#cmd = kW (Vcmd W) +t ky p~y + ~Yi (91)
9.2.3 Depth Control
The last stage in the homing mission is the depth position or axial position to the tracked
target. Once the altitude and lateral position are aligned using the two previous controllers then
the depth control is engaged. The error to regulate for this last stage is the axial position in
the bodyfixed frame. There are many challenges when approaching this control design due to
the reliance of visionbased feedback. One particular problem associated with state estimation
algorithms that employ vision is the break down when features exit the field of view. For
instance, during approach the objects within the image become larger which makes them harder
to maintain within the field of view. Therefore, to account for this drawback the controller should
be restricted to very slow steady maneuvers and avoid sudden changes in orientation.
The design approach taken for this control loop is to increase velocity while maintaining
altitude and restrict large changes in pitch angle. Once the lateral position and altitude are aligned
then the axial position is regulated to zero. The control architecture chosen for this loop was
proportional where the error is multiplied by a the gain factor, ks, which generates a change in
thrust command. During thrust changes, the aircraft tends to climb or descend due to the change
in airspeed. This resulting altitude change is counteracted by adjusting the elevator through the
altitude controller designed in the beginning of this chapter.
Meanwhile, any adjustments that are made to maintain altitude are governed by the pitch
angle, which directly affects the field of view. As a result, the pitch was limited to maintain
features within the image when the receiver is within a specified distance. This methodology
will still not guarantee features will stay in the image. For example, the homography requires a
minimum of four distinct feature on each vehicle. The closer in proximity the receiver gets the
larger the objects get in the image. At close distances the object can fill the entire image causing
the feature points to leave the field of view even if the object is centered in the image. This
creates a dead zone in the measurable space that can either be fixed by estimating a prediction of
the target over time or by customizing the camera parameters to correspond with camera position
and orientation along with the size of the objects to ensure feature remain in the image.
The modeling scheme presented in Chapter 8 provides a method to estimate targets in
Euclidean space when features do exit the image. This method works well for short periods of
time after the target has left; however, the trust in the predicted value degrades tremendously
as time increases. Consequently, when a feature leaves the image the controller can rely on the
predicted estimates to steer the aircraft initially but may resort to alternative approaches beyond a
specified time. As a last resort, the controller can command the aircraft to slow down and regain a
broader perspective of the scene to recapture the target.
CHAPTER 10
SIMULATIONS
10.1 Example 1: Feature Point Generation
A simulation of visionbased feedback is presented to demonstrate the implementation,
and resulting information, associated with sensor models and aircraft dynamics. This simulation
utilizes a nonlinear model of the flight dynamics of an F16 [110]. A baseline controller is
implemented that allows the vehicle to follow waypoints based entirely on feedback from inertial
sensors.
Images are obtained from a set of cameras mounted on the aircraft. These cameras include
a stationary camera mounted at the nose and pointing along the nose, a translating camera under
the centerline that moves from the right wing to the left wing, and a pitching camera mounted
under the center of gravity. The parameters for these cameras are given in Table 101 in values
relative to the aircraft frame and functions of time given as t in seconds.
Table 101. States of the cameras
position (ft) orientation (deg)
camera xe ye ze Wcc V
1 24 0 0 0 90 0
2 10 153t 0 0 45 0
3 0 0 3 0 459t 0
The camera parameters are chosen as similar to an existing camera that has been flight
tested [111]. The focal length is normalized so f = 1. Also, the field of view for this model
correlates to angles of y;; = 32 deg and y, = 28 deg. The resulting limits on image coordinates are
given in Table 102.
Table 102. Limits on image coordinates
coordinate minimum maximum
pu 0.62 0.62
v 0.53 0.53
A virtual environment is established with some characteristics similar to an urban
environment. This environment includes several buildings along with a moving car and a
moving helicopter. In actuality, these features are simply represented by sets of points and any
motion results from simple kinematic motion. A specific point, positioned as in Table 103, is
associated with each feature for direct identification in the camera images.
Table 103. States of the feature points
position (ft)
feature point north east altitude
1 3500 200 1500
2 1000+200t 500 500
3 6000 200cos( t) + 1000 200sin( t)1000
The flight path through this environment is shown in Figure 101 along with the features.
The aircraft initially flies straight and level toward the North but then turns somewhat towards the
East and begins to descend from a dive maneuver.
3000 600
5000
2000
1L 4000
O ~ 3000
0 2000
1000
0 1000
1000
2000 400 6000
E f) 30 00 _10000 0 1000 2000 3000
N (ft) E (ft)
A B
Figure 101. Virtual Environment for Example 1: A) 3D View and B) Top View
Images are taken at several points throughout the flight as indicated in Figure 101 by
markers along the trajectory. The states of the aircraft at these instances are given in Table 104.
The image plane coordinates (p, v) are plotted in Figure 102 for the three cameras at
t = 2 sec. This computation is accomplished by using Equation 528 in conjunction with
Equations 35 and 36 while applying the field of view constraint shown in Equations 34 and
33. All three cameras contain some portion of the environment along with distinct views of the
feature points of interest. For example, camera 1 contains a forward looking view of a stationary
Table 104. Aircraft states
Time North East Down u v w
(s) (ft) (ft) (ft) (ft/s) (ft/s) (ft/s)
2 1196.9 0.44 2174.8 573.52 56.79 126.46
4 2112.7 143.04 1645.4 527.37 54.94 17.77
6 2989.8 353.63 1100.7 528.30 4.26 45.57
Time # 6 W p q r
(s) (degp) (degp) (degp) (deg/s) (deg/s) (deg/s)
2 13.92 22.43 1.81 13.56 36.82 1.38
4 39.21 37.90 22.79 32.31 28.41 0.04
6 6.98 14.85 11.93 7.63 9.34 1.15
point on the corner of a building as well as the moving helicopter. Meanwhile, cameras 2 and 3
observe a top view of the moving ground vehicle traveling forward down a road. These image
measurements provide a significant amount of data and allow for more advanced algorithms for
state estimation and reconstruction.
" ~
.t
A
B
C
Figure 102. Feature point Mmasurements at t = 2 sec for A) camera 1, B) camera 2, and C)
camera 3
Figure 103 depicts the optic flow computed for the same data set shown in Figure 102.
This image measurement gives a sense of relative motion in magnitude and direction caused
by camera and feature point motion. The expressions required to compute optic flow consisted
of Eqs. 528, 531, 35, 36, 332, 333, 34 and 33. In this example, the optic flow has
many components contributing to the final value. For instance, the aircraft's velocity and angular
rates contribute a large portion of the optic flow because of their large magnitudes. In addition,
the smaller components in this example are caused from vehicle and camera motion which are
smaller in magnitude but have a significant effect on direction. Comparing cameras 1 and 2, there
are slight differences in direction due to the translating camera. Likewise, the optic flow observed
by camera 3 is different due to the camera's orientation.
0 4 02 0 02 04 06 06 04 02 0 02 04 06 06 04 02 0 0 4 0
A B C
Figure 103. Optic flow Measurements at t = 2 sec for A) camera 1, B) camera 2, and C) camera
A summary of the resulting image plane quantities, position and velocity, is given in
Table 105 for the feature points of interest as listed in Table 103. The table is organized by the
time at which the image was taken, which camera took the image, and which feature point is
observed. This type of data enables autonomous vehicles to gain awareness of their surroundings
for more advanced applications involving guidance, navigation and control.
Table 105. Image coordinates of feature points
Time (s) Camera Feature Point pu v p 9
2 1 1 0.157 0.162 0.610 0.044
2 1 3 0.051 0.267 0.563 0.012
2 2 2 0.308 0.075 0.464 0.254
2 3 2 0.011 0.077 0.583 0.235
4 2 2 0.279 0.243 0.823 0.479
4 3 2 0.365 0.248 0.701 0.603
6 1 3 0.265 0.084 0.267 0.015
10.2 Example 2: Feature Point Uncertainty
10.2.1 Scenario
Feature point uncertainty is demonstrated in this section by extending the previous example.
This simulation will examine the uncertainty effects on vision processing algorithms using
simulated feature points and perturbed camera intrinsic parameters.
Similarly to the previous example, visionbased feedback is generated using a flight
simulation. The overall setup of this example is the same where a nonlinear model of an F16 is
used to fly through a cluttered environment while capturing images from an onboard camera.
Camera settings, such as focal length and field of view, are kept the same from the previous
example. The actual environment has been normalized based on the aircraft velocity so units are
not presented.
A set of obstacles, each with a feature point, are randomly distributed throughout the
environment and are not the same same as the previous example. This environment is shown in
Figure 104 along with a pair of points indicating the locations at which images will be captured.
The aircraft is initially straight and level then translates forward while rolling 4.0 deg and yawing
1.5 degp at the final location.
100
*. *
10 00* ** o "
1000 *0 0 00 50 2
0 **
Figure 104. Virtual environment of obstacles (solid circles) and imaging locations (open circles)
A) 3D view and B) top view
A single camera is simulated at the center of gravity of the aircraft with line of sight aligned
to the nose of the aircraft. The intrinsic parameters are chosen such that fo = 1.0 and do = 0.0
for the nominal values. The images for the nominal camera associated with the scenario in
Figure 104 are presented in Figure 105 to show the variation between frames.
The visionbased feedback is computed for a set of perturbed cameras. These perturbations
range as 87 E [0.2, 0.2] and 6d E [0.02, 0.02]. Obviously the feature points in Figure 105 will
vary as the camera parameters are perturbed. The amount of variation will depend on the feature
0.4 ~ '* 0.4
.
0.2 0.2
0.4~ 0.4
0.6~ 0.6
0.6 0.4 0.2 0 0.2 0.4 0.6 0.6 0.4 0.2 0 0.2 0.4 0.6
v v
A B
Figure 105. Feature points for A) initial and B) final images
point, as noted in Equations 49 and 410, but the effect can be normalized. The variation in
feature point given nominal values of Po = vo = 1 is shown in Figure 106 for variation in both
focal length and radial distortion. This surface can be scaled accordingly to consider the variation
at other feature points. The perturbed surface shown in Figure 106 is propagated through three
main image processing techniques for analysis.
d 2 0.
10.2.2 Otic Flo
Figure 106 t foceral lngth and ftrada doistorin ersnaiecmaio fotcfo o
the nominal camera and a set of perturbed cameras is shown in Figure 107.
04, 0 /0
A B C
Figure 107. Optical flow for nominal (black) and perturbed (red) cameras for A) f = 1.1 and
d = 0, B) f = 1.0 and d = 0.01, and C) f = 1.1 and d = 0.01
The vectors in Figure 107 indicate several effects of camera perturbations noted in
Equations 45 and 46. The perturbations to focal length scale the feature points so the
magnitude of optic flow is uniformly scaled. The perturbations to radial distortion have larger
effect as the feature point moves away from the center of the image so the optic flow vectors
are altered in direction. The combination of perturbations clearly changes the optic flow in both
magnitude and direction and demonstrates the feedback variations that can result from camera
variations.
The optic flow is computed for images captured by each of the perturbed cameras. The
change in optic flow for the perturbed cameras as compared to the nominal camera is represented
as 6Sy and is bounded in magnitude, as derived in Equation 414, by Ay. The greatest value of Sy
presented by these camera perturbations is compared to the upper bound in Table 106. These
numbers indicate the variations in optic flow are indeed bounded by the theoretical bound derived
in Chapter 4 and indicate the level of flow variations induced from the variations in camera
parameters.
Table 106. Effects of camera perturbations on optic flow
Perturbation Analyze Analyze Analyze
Set only with 87 only with 8d with 8f and 6d
87 = 0.2 and 6d = 0.02 0.0476 0.0476 0.0040) 0.0040) 0.0496 0.0543
87 = 0. 1 and 8d 0.01 0.0238 0.0476 0.0020 0.0040 0.0252 0.0543
87 = 0.1 and 8d 0.01 0.0238 0.0476 0.0020 0.0040 0.0264 0.0543
87 = 0.2 and 6d = 0.02 0.0476 0.0476 0.0040 0.0040 0.0543 0.0543
10.2.3 The Epipolar Constraint
State estimation is performed by considering the epipolar constraint to relate the pair of
images. The evaluation of images generated using the nominal camera for this simulated case is
able to estimate the correct states. An investigation of the epipolar lines shown in Figure 108
shows the quality of the estimation. Essentially, the epipolar geometry requires a feature point
in one image to lie along the epipolar line. This epipolar line is constructed by the intersection
between the plane formed by the epipolar constraint and the image plane at the last measurement.
The data in Figure 108 show the features in the second image do indeed lie exactly on the
epipolar lines.
0.4~ '* + 0.4~ .
0.2 ~ 0.2
0.4~ 0.4
0.6 0.4 0.2 0 0.2 0.4 0.6 0.6 0.4 0.2 0 0.2 0.4 0.6
v v
A B
Figure 108. Epipolar lines between two image frames: A) initial frame and B) final frame with
overlayed epipolar lines for nominal camera
The introduction of uncertainty into the epipolar constraint will cause variations in the
essential matrix which will also propagate through the computation of the epipolar line. These
variations in the epipolar line are visual clues of the quality of the estimate in the essential
matrix. These variations can occur as changes in the slope and the location of the epipolar line.
Figure 109 illustrates the epipolar variations due to perturbations on 87 = 0.1 and 6d = 0.01 to
the camera parameters. The feature points with uncertainty and the corresponding epipolar line
was plotted along with the nominal case to illustrate the variations. The key point in this figures
is the small variations in the slope of the epipolar lines and the significant variations in feature
Figure 109. Uncertainty results for epipolar geometry: A) initial frame and B) final frame with
overlayed epipolar lines for cameras with f 1.0 and d = 0.0 (black) and f = 1.1
and d = 0.01 (red)
The essential matrix is computed for the images taken using a set of camera models.
Each model is perturbed from the nominal condition using the variations in Figure 106. The
change in estimated states between nominal and perturbed cameras is given by 89 over the
uncertainty range and is bounded, as derived in Equation 419, by Aq. The value of 6q for
a specific perturbation is shown in comparison to the upper bound in Table 107 which also
indicate the variation in entries of the essentail matrix which propagate to the camera states.
Table 107. Effects of camera perturbations on epipolar geometry
Perturbation Analyze Analyze Analyze
Set only with 87 only with 6d with 87 and 6d
87 = 0.2 and 6d = 0.02 293.14 293.14 4.45 4.45 288.75 297.34
87 = 0.1 and 6d 0.01 122.26 293.14 2.19 2. 19 288.75 297.34
87 = 0.1 and 6d 0.01 90.48 293.14 2.11 2. 19 288.75 297.34
87 = 0.2 and 6d = 0.02 159.31 293.14 4.15 4.45 288.75 297.34
point locations that occur along these lines. The small variations in the slope would suggest
reasonable estimation of the rotational component of the essential matrix; however, the variation
along the epipolar line would indicate a sensitivity in focal length variations due to the scaling
effects revealed from the plots.
0.6 0.4 0.2 0 0.2 0.4 0.6
0.4
:.* .* *
Z
0.6 0.4 0.2 0 0.2 0.4
0.2
10.2.4 Structure From Motion
The images taken during the simulation are analyzed using structure from motion to
determine the location of the environmental features. The initial analysis used the nominal
camera to ensure the approach is able to correctly estimate the locations in the absence of
unknown perturbations. The actual and estimated locations are shown in Figure 1010 to indicate
that all errors were less than 106
*Actual
a Estimate
500
0 ..
N. *
500 *
1000
1000\ 
2000
1000
Figure 1010. Nominal estimation using structure from motion
The depths are also estimated using structure from motion to analyze images from the
perturbed cameras. A representative set of these estimates are shown in Figure 1011 as having
clear errors. An interesting feature of the results is the dependence on sign of the perturbation to
focal length. Essentially, the solution tends to estimate a depth larger than actual when using a
positive perturbation and a depth smaller than actual when using a negative perturbation. Such a
relationship is a direct result of the scaling effect that focal length has on the feature points.
Estimates are computed for each of the perturbed cameras and compared to the nominal
estimate. The worstcase errors in estimation are compared to the theoretical bound, given in
Equation 429, to these errors. These numbers shown in Table 108 indicate the variation in
structure from motion depends on the sign of the perturbation. The approach is actually seen
to be less sensitive to positive perturbations, which causes a larger estimate in depth, than to
negative perturbations. Also, the theoretical bound was greater than, or equal to, the error caused
by each camera perturbation.
100 3 c*, 100 100 o*d
A B C
Figure 1011. Estimation using structure from motion for nominal (black) and perturbed (red)
cameras with A) f = 1.1 and d = 0, B) f = 1.0 and d = 0.01, and C)
f = 1.1 and d = 0.01
Table 108. Effects of camera perturbations on structure from motion
Perturbation Analyze Analyze Analyze
Set only with 87 only with 8d with 8f and 6d
87 = 0.2 and 6d = 0.02 4679.8 4679.8 75.02 75.02 4903.5 4903.5
87 = 0.1 and 8d 0.01 1045.6 4679.8 36.90 75.02 1076.6 4903.5
87 = 0.1 and 8d 0.01 485.80 4679.8 35.73 75.02 498.76 4903.5
87 = 0.2 and 6d = 0.02 1092.4 4679.8 70.34 75.02 1092.5 4903.5
10.3 Example 3: Openloop Ground Vehicle Estimation
An openloop simulation was executed in Matlab and replayed in a virtual environment
to test the state estimation algorithm. The scenario envisioned in Chapter 1 involving a police
pursuit is demonstrated through this simulation. The setup consisted of three vehicles: an UAV
flying above with a mounted camera, electronics and communication, a reference ground vehicle
which is considered the police pursuit car, and a target vehicle describing the suspects vehicle.
The goal of this mission is for the UAV to track both vehicles in the image, while receiving
position updates from the reference vehicle, and estimate the target's location using the proposed
estimation algorithm.
The camera setup considered in this problem consist of a single downward pointing camera
attached to the UAV with fixed position and orientation. While in flight the camera measures and
tracks feature points on both the target vehicle and the reference vehicle for use in the estimation
algorithm. This simulation assumes perfect camera calibration, feature point extraction, and
tracking so that the state estimation algorithm can be verified. As stated in Chapter 7 the
geometry of the feature points are predescribed and a known distance is provided for each
vehicle. A further description of this assumption is given in Section 7.2.2. Future work will
examine more realistic aspects of the camera system to reproduce a more practical scenario and
try to alleviate the limitations imposed on the feature points.
10.3.1 System Model
The motion of the vehicles were generated to cover a vast range of situations to test the
algorithm. The UAV's motion was generated in openloop from a nonlinear aircraft model in
trimmed flight. Meanwhile, the reference vehicle and the target vehicle exhibited a standard car
model with similar velocities. Sinusoidal disturbances were added to the target's position and
heading to add some complexity to it's motion and to replicate swerving. The three trajectories
are plotted in Euclidean space, as shown in Figure 1012, for illustration. The initial frame for
this simulation is located at the aircraft's position when the simulation starts. The velocity of the
ground vehicles were scaled up to the aircraft's velocity which resulted in large distances but also
helped to maintain the vehicles in the image.
Aircraft x1
Refernce 25x0
0 Target
0 15
1000 1
/ x140 5  Refernce
2 /1 Target
x 10 1
E (ft) o O N(t E (ft)
A B
Figure 1012. Vehicle trajectories for example 3: A) 3D view and B) top view
The position and orientation states of the three vehicles are plotted in Figures 1013 1018
and all are represented in the inertial frame, E. The positions indicate that all three vehicle
initially travel north until the target vehicles makes a left turn and heads west and is subsequently
followed by the pursuit vehicles.
2 100'""
10000
00 20 40 60 <* 0 00 20 40 60
Time (sec) Time (sec) Time (sec)
A B C
Figure 1013. Position states of the UAV with onboard camera: A) North, B) East, and C) Down
60 10 5
50 0
0 20 40 60 10 20 40 60 0020 40 60
Time (sec) Time (sec) Time (sec)
A B C
Figure 1014. Attitude states of the UAV with onboard camera: A) Roll, B) Pitch, and C) Yaw
1~ 5 800
400
0 5 15000
020
00 20 40 60 <* 0 00 20 40 60
Time (sec) Time (sec) Time (sec)
A B C
Figure 1015. Position states of the reference vehicle (pursuit vehicle): A) North, B) East, and C)
Down
10.3.2 Openloop Results
The homography was computed for this simulation to find the relative rotation and
translation between the ground vehicles. These results are then used to find the relative
135
1 )(
Time (sec) Time (sec)
A B
Time (sec)
Figure 1016. Attitude states of the reference vehicle (pursuit vehicle): A) Roll, B) Pitch, and C)
Yaw
Time (sec)
A
Figure 1017. Position states o
Down
I
100(
80(
Q
20(
0 20 40 60
Time (sec)
I
1000(
15
Time (sec)
f the target vehicle (chase vehicle): A) North, B) East, and C)
10
20 40 60 10
Time (sec)
20 40 60 0
Time (sec)
20 40
Time (sec)
Figure 1018. Attitude states of the target vehicle (chase vehicle): A) Roll, B) Pitch, and C) Yaw
motion from the UAV to the target of interest. The norm error of this motion are depicted in
Figure 1019. These results indicate that with synthetic images and perfect tracking of the
vehicles nearly perfect motion can be extracted. Once noise in the image or tracking is introduced
the estimates of the target deteriorate quickly even with minute noise. In addition, image artifacts
such as interference and drop outs will also have an adverse affect on homography estimation.
136
x 10
4
011
200 400
Index (counts)
200 400
Index (counts)
Figure 1019. Norm error for A) relative translation and B) relative rotation
Figures 1020 and 1021 show the relative translation and rotation decomposed into their
respective components and expressed in the body frame, B. These components reveal the relative
information needed for feedback to track or home in on the target of interest.
4500
5000
5500
600
Index (counts)
Index (counts)
C
Index (counts)
Figure 1020. Relative position states: A) X, B) Y, and C) Z
60
40
20
Index (counts)
B
Figure 1021. Relative attitude states: A) Roll, B) Pitch, and C) Yaw
Jc
100
00O 00 200 400 600
Index (counts)
C
Index (counts)
A
The simulation was then played in a virtual environment to enhance the graphics and
illustrate the application of this algorithm. To add the vehicles within the virtual environment the
velocities of each vehicle had to be scaled down to practical values that fit the scene. Snapshots
are shown in Figure 1022 of the camera view depicting the vehicles and the surrounding scene.
The red vehicle was designated as the reference whereas the grey vehicle was the target vehicle.
The next step in this process is to implement an actual feature tracking algorithm on the synthetic
images that follows the vehicles. This modification alone will degrade the homography results
immensely due to the troublesome characteristics of a feature point tracker.
Figure 1022. Virtual environment
10.4 Example 4: Closedloop Aerial Refueling of a UAV
A closedloop simulation was executed in Matlab to replicate an autonomous aerial
refueling task. As Chapter 1 described the motivation and the benefits of AAR, this section will
demonstrate it by combining the control design given in Chapter 9 with the homography result
in Chapter 7 to form a closedloop visual servo control system. The vehicles involved in this
simulation includes a Receiver UAV instrumented with a single camera, a tanker aircraft also
referred to as the reference vehicle and the target drogue also referred to as the target vehicle.
Ultimately, the goal of this task is to mate in flight the receptacle probe on the receiver aircraft to
the drogue that is tethered from the tanker aircraft.
The camera setup used for this simulation was a forward looking camera located at the
nose of the aircraft with fixed position and orientation. The field of view angles used in this
example were yh = 350 and y, = 350 along with a focal length of f = 1 and radial distortion set
to d = 0. To facilitate feature point tracking cues were painted on both the tanker and drogue
with an identical pattern and size. A square shape was chosen for this simulation with a length
of 4 feet on all sides. The same assumptions given in the previous example regarding feature
point tracking were applied to this example as well, including the assumption that both the
tanker and drogue remain in the field of view at all times. An additional assumption made to
facilitate the estimation was that data communication between the tanker and receiver was in
place to allow transmission of the tanker's position and orientation. Once the homography
estimation was computed, the relative position between the receptacle and the drogue was found.
Finally, the relative position was used in conjunction with the receiver's position to find the
inertial coordinates of the drogue. These inertial coordinates were then used in feedback for the
controller, similar to a waypoint structure except these inertial points are moving.
10.4.1 System Model
The aircraft model used in this development was a high fidelity nonlinear F16 model
constructed by the University of Minnesota. The aircraft was trimmed at sea level traveling 500
ft/s straight and level. The details of the aircraft model are extensive and include aerodynamic
tables, actuator models, leading edge flap models, and position and rate limits on all actuation.
States for this model are the standard aircraft states given in Equations 526 with additional
states such as V, ce, p, the acceleration terms, Mach number, and dynamic pressure. Although
the controller will not use all states, the assumption of full state feedback was made to allow all
states accessible by the controller. The controller uses these states of the aircraft along with the
estimated results to compute actuator commands around the specified trim condition.
The same model was also used for the tanker or reference vehicle. The tanker was exactly
trimmed at the same conditions and airspeed as the receiver aircraft and given a specified
trajectory to follow. Initially the tanker's position was offset from the receiver's position at the
start of the simulation. The values of this offset are described relative to receiver's coordinate
frame and are as follows: 500 ft in front (+tX direction), 20 ft to the side (+tY direction),
and 100 ft above (Z direction). The trajectory generated for the tanker aircraft prior to the
simulation was a straight and level flight with a slight drift toward the East direction. This lateral
variation was added to the trajectory to incorporate all three dimensions into the motion to test in
all directions.
On the other hand, the modeling of the drogue is much more difficult to characterize and is
of much interest in the research community. The stochastic nature of its motion is what makes
the modeling so challenging. The flow field affecting the drogue consist of many nonlinear
excitations including turbulence due to wake effects and vortex shedding from the tanker aircraft.
For this drogue model an offset trajectory of the tanker's motion was used as the drogue's general
motion. The offset of the drogue is initially at 200 ft in front +tX direction), O ft to the side (+tY
direction), and 70 ft above (Z direction) relative to the receiver aircraft. More complicated
motions of the drogue were considered during testing but resulted in a diverging trajectory for
the receiver. This deviation from the desired path was due high rate commands saturating the
actuators. Low passing filtering can be incorporated to alleviate this behavior.
10.4.2 Control Tuning
The control architecture described in Chapter 9 is integrated and tuned for the nonlinear
F16 model to accomplish this simulation. It was assumed that full state feedback of the aircraft
states were measurable including position. The units used in this simulation are given in ft and
deg which means the gains determined in the control loops were also found based on these units.
First, the pitch tracking for altitude controller is considered. The innerloop gains for this
controller are given as ke = 3 and kg = 2.5. The bode diagram for pitch command to pitch
angle is depicted in Figure 1023 for the specified gains. This diagram reveals the damping
achieved in the phugoid mode. In addition, a 12.4 dB gain margin at 8.15 rad/s and a 1570 phase
margin at 0.381 rad/s was achieved. This metric reveals robustness of the loop gain with respect
to increases in gain and phase shifts.
Bode Diagram
Gm = 12.4 dB (at 8.15 rad/sec) Pm = 157 deg (at 0.381 rad/sec)
From Pftch emden 1) To rea2deginemuxl (p 3
50
50
150
m 180
270
10 4 102 100
Frequency (rad/sec)
Figure 1023. Innerloop pitch to pitch command Bode plot
The step response for the pitch controller is given in Figure 1024 and shows acceptable
performance. The outerloop control will now be designed using this controller to track altitude.
Response
Command
20
0 5 10 15 20
Time (sec)
Figure 1024. Pitch angle step response
The outerloop that connects altitude to pitch commands is considered. The gains for the
innerloop pitch tracking remained the same while the gain in altitude error was set to k = 1.25.
The final compensation filter is given in Equation 101 and was designed in Stevens et al. [110].
A step response for this controller is illustrated in Figure 1025 that shows a steady climb with
no overshoot and a steadystate error of 2 ft. This response is realistic for an F16 but not ideal
for autonomous refueling mission where tolerances are on the cm level. The altitude transition is
slow due to the compensator but one may consider more aggressive maneuvers for missions such
as target tracking that may require additional agility.
s2 + 0.35s +t 0.015
Gs2+t2.41s+t0.024 (01
2450
Response
Command
2350
2250
2200
2150
0 20 40 60 80 100
Time (sec)
Figure 1025. Altitude step response
The next stage that was tuned in the control design was the heading controller. The
innerloop gains were chosen to be kg = 5.7 and kp = 1.6 for the roll tracker. The bode
diagram for this controller of roll command to roll angle is shown in Figure 1026 which shows
attenuation in the lower frequency range. This attenuation removes any high frequency response
from the aircraft which is desired during a refueling mission, especially in close proximity.
Meanwhile, the coupling between lateral and longitudinal states during a turn was counteracted
by an aileronelevator connect. This connection involved a proportional gain of k, = 0.35
multiplied to the roll angle and added to the elevator position.
Bode Diagram
Gm = 15.9 dB (at 43.4 rad/sec) Pm = 179 deg (at 0.0583 rad/sec)
From Bankcmd(pt 1) To r2d(pt 1)
50
r 100
150
a, 90
270
104 102 100 102
Frequency (rad/sec)
Figure 1026. Innerloop roll to roll command Bode plot
The step response for this bank controller is illustrated in Figure 1027. The tracking
performance is acceptable based on a rise time of 0.25 see, an overshoot of 6% and less than a
3% steadystate error.
The outerloop tuning for heading controller consisted of first tuning the gain on heading
error. A gain of kw = 1.5 was chosen for this mission which demonstrated acceptable
performance. Figure 1028 shows the heading response using this controller for a right turn.
The response reveal a steady rise time, no overshoot, and a steadystate error of less than 2 deg.
Finally, the loop pertaining to lateral deviation was tuned to k,, = 0.5 and kyi = 0.025 which
produced reasonable tracking and steady error for lateral position.
The final stage of the controller involves the axial position. This stage was designed to
increase thrust based on a velocity command once the lateral and altitude states were aligned.
A proportional gain was tuned based on velocity error to achieve a slow steady approach speed
Response
 Command
U'
0 5 10 15 2(
Time (sec)
step response
120
Response
Command
100
80
60
40
20
20
",15
av
d 10
5
Figure 1027. Roll angle
0 20 40 60
Time (sec)
80 100
Figure 1028. Heading response
to the target. A gain of ks = 3.5 was determined for this loop which generates the desired
approach. Lastly, to help limit the number of times the feature points exit the field of view a
limit was imposed on the pitch angle. This limit was enforced when the approach achieve a
specified distance. For this example, the distance was set to within 75 ft in the axial position of
the bodyfixed frame which was determined experimentally from the target's size.
10.4.3 Closedloop Results
The state estimation performance of the target drogue during this simulation was similar to
the previous simulation regarding the tracking of a ground vehicle. The estimated target states
were plotted against the true values in Figure 1029 for position and Figure 1030 for orientation
and revealed a correct fit. This result demonstrates the functionality of the estimator with an
accuracy on the order of 109. This error was plotted in Figure 1031 for both position and
orientation.
SX 10'
2~" 2300
8~ 1 25
0 10 20 30 40 00 10 20 30 40 200 10 20 30 40
Time (sec) Time (sec) Time (sec)
A B C
Figure 1029. Openloop estimation of target's inertial position: A) North, B) East, and C)
Altitude
Estlmat0 023
~3 0 005o
0o 10 20 30 40 000 10 20 30 40 010 10 20 30 40
Time (sec) Time (sec) Time (sec)
A B C
Figure 1030. Openloop estimation of target's inertial attitude: A) Roll, B) Pitch, and C) Yae
Furthermore, the closedloop results for this simulation were plotted in Figures 1032
and 1034 for position and orientation of both the receiver aircraft and the target drogue relative
to the earthfixed frame. The tracking of this controller showed reasonable performance for the
desired position and heading signals. The remaining orientation angles were not considered in
feedback but estimated for the purpose of making sure the drogue's pitch and roll are within the
desired values before docking. As seen in Figure 1032, the receiver was able to track the gross
motion of drogue while having some difficultly tracking the precise motion.
x01 t 6x10
05
010 20 30 40 oO 10 20 30 40
Time (sec) Time (sec)
A B
Figure 1031. Norm error for target state estimates A) translation and B) rotation
Target 2300are
8/ 1 10i
0 10 20 30 40 50 10 20 30 40 250 10 20 30 40
Time (sec) Time (sec) Time (sec)
A B C
Figure 1032. Closedloop target position tracking: A) North, B) East, and C) Altitude
The components of the position error between the receiver and drogue are shown in
Figure 1033 to illustrate the performance of the tracking controller. These plots depict the initial
offset error decaying over time which indicates the receiver's relatives distance is decreasing. The
altitude showed a quick climb response where as the response in axial position was a slow steady
approach which was desired to limit large changes in altitude and angle of attack. The lateral
position is stable for the time period but contains oscillations due the roll to heading lag.
The orientation angles shown in Figure 1034 indicate the Euler angles for for the
bodyfixed transformations corresponding to the bodyfixed frame of the receiver and the
bodyfixed frame of the drogue. Recall, the only signal being tracked in the control design was
heading. This selection allowed the aircraft to steer and maintain a flight trajectory similar to the
drogue without aligning roll and pitch. The receiver should fly close to a trim condition rather
then matching the full orientation of the drogue, as illustrated in Figure 1034 for pitch angle.
_/1
u0 10 20 30 40
Time (sec)
10 20
Time (sec)
30 40
20
Time (sec)
B
Figure 1033. Position tracking error: A) North, B) East, and C) Altitude
The error in heading is depicted in Figure 1035 which shows acceptable tracking performance
over the time interval.
..il
70
60
50
40
30
20
10
10 20 30 40
Time (sec)
B
Roll, B) Pitch, and C) Yaw
2
0 10 20 30 40
Time (sec)
C
Time (sec)
A
Figure 1034. Target attitude tracking: A)
3.5
3
2.5
o 1.
0.5
0 10 20
Time (sec)
Figure 1035. Tracking error in heading angle
The results shown in these plots indicate that the tracking in the lateral position and altitude
are nearly sufficient for the refueling task. The simulation reveals bounded errors in these
dimensions of 3 ft. The main issues occur in the axial position during the approach stage. The
state estimator seems to have trouble during approach when the vehicles in the image reach the
bounds of the field of view. Without four features on each vehicle the estimator cannot function
and is unable to provide updates of the relatives states. The performance in axial position
achieved during this simulation was a relative distance of 7 ft until the first feature left the image.
Once the features are out of view the estimator no longer provides updates and the controller
commands the aircraft to fly at a straight and level trim condition at a slower airspeed in attempt
to regain features.
Overall the simulation results shown here are inefficient to achieve a successful aerial
refueling mission based on the requirement of cm precision. The reasons that this mission was
not achieved in these results is the control is unable to maintain the vehicle features within the
image in close proximity during approach. The estimation task is able to function with good
accuracy but has the drawback of requiring a minimum of four features on each vehicle in every
frame. Although this drawback is common in vision processing, there has been techniques used
to estimate the position of the feature that has left the image with some variance. This modeling
approach can help serve two purposes 1) to predict where the features are when they leave the
image and 2) help predict where the features might be going in future steps. Implementing the
modeling task presented in Chapter 8 will help to aid the controller, or at least to help determine a
region of where the features most likely have traveled.
10.4.4 Uncertainty Analysis
The simulation result of the estimated target states were computed from the homography
estimation task given in Chapter 7. To see what levels of variations exist in these results an
uncertainty analysis was performed. Chapter 4 derived a method to compute worsecase bounds
on state estimates from the homography approach using visual information. The technique
described in Chapter 4 was used for this uncertainty analysis.
The target estimates for absolute position and orientation along with upper and lower
bounds were computed for this simulation and are shown in Figures 1036 and 1037. These
plots contain error bars computed at 0.5 Hz for three levels of parametric uncertainty. The
three levels consist of 1) focal length uncertainty, 2) radial distortion uncertainty and 3) focal
length and radial distortion uncertainty. The values for the camera parameters were set to fo = 1
and do = 0 for the nominal values and the perturbed set consisted of 87 = [0.1 : 0.1] and
8d = [0.05 : 0.05]. These plots describe the worsecase bounds for each state. The computations
confirm the maximum state variations occur at the maximum level of uncertainty where both
focal length and radial distortion are at their maximum perturbations. The trend observed in these
plots indicates an increase in uncertainty as features move closer to the camera.
Od 0, 232
2  f 5 8 d 20
10 20 30 40 00 10 20 30 40 240 10 20 30 40
Tune (sec) Tune (sec) Tune (sec)
A B C
Figure 1036. Target's inertial position with uncertainty bounds: A) North, B) East, and C)
Altitude
0 08Nomll 6Nonma 3Nonm
d 1CCCC @d
004
1 20 3 4010 0 2 3 4 30 1 0 0 4
Tune (sec) Tune (sec) Tune (sec)
A B C
Figure 1037. Target's inertial attitude with uncertainty bounds: A) Roll, B) Pitch, and C) Yaw
The maximum uncertainties in target position relative to the earthfixed frame are
summarized in Table 109. Meanwhile, Table 1010 contains the maximum uncertainties in
target orientation. The three levels of uncertainty are included in these tables. This comparison
helps to verify that the maximum state variation corresponds to the maximum camera parameter
variation for all states. These state variations indicate levels of uncertainty greater than the
allowable tolerance for autonomous refueling which would indicate an need for improved
performance in camera calibration. Iterations can be made on the uncertainty set involving the
camera parameters to find the range which meets the allowable tolerance for a safe refueling.
This information can provide a method for determining the accuracy needed during camera
calibration for a task that requires such precise estimation.
Table 109. Maximum variations in position due to parametric uncertainty
uncertainty parameter north (ft) east (ft) altitude (ft)
f 2.79 4.10 20.54
d 5.66 10.53 14.40
f and d 9.61 15.09 30.82
Table 1010. Maximum variations in attitude due to parametric uncertainty
uncertainty parameter # (deg) 6 (deg) W (deg)
f 00 0
d 0.06 4.48 2.29
f and d 0. 10 7.94 3.48
CHAPTER 11
CONCLUSION
Visionbased feedback can be an important tool for autonomous systems and is the primary
focus of this dissertation in the context of an unmanned air vehicle. This dissertation describes
a methodology for a vehicle, such as a UAV, to observe features within the environment and
estimate the states of a moving target using various camera configurations. The complete
equation of motion of an aircraftcamera system was derived in its general form that allows
multiple cameras. Camera models were summarized and the effects of uncertainty regarding the
intrinsic parameters was discussed. Expressions for worsecase bounds were derived for varies
vision processing algorithms on a conservative level. A classification scheme was summarized
to discern between stationary and moving objects within the image using a focus of expansion
threshold method. The homography derivation proposed was the main contribution of this
dissertation where the states of a moving target were formulated based on visual information.
Some underlining assumptions were imposed on the features and the system to obtain feasible
estimates. The two critical assumptions imposed on the features were the planar constraint and
the requirement of the distance to a feature on the reference and target vehicles be known and
equal. An additional assumption was placed on the system which involved a communication link
that allows the vehicle to have access to the states of the reference vehicle. The modeling of the
target position attempted to anticipate future locations to enable a predictive capability for the
controller and to provide estimates when the features are outside the field of view. The approach
summarized here consisted of a Hidden Markov method which has limitations for general 6DOF
motion due to incomplete motion models. Lastly, a standard control design is tuned for an aircraft
performing waypoint navigation to use in closedloop control where commands are generated
from the state estimator.
Simulations were presented to validated the proposed algorithms and to demonstrate the
applications for autonomous vehicles. The first simulation verified the feature point and optic
flow computation for a aircraftcamera system containing multiple cameras with time varying
position and orientation. The second simulation illustrated the effects of uncertainty on image
processing algorithms due camera intrinsic parameters showed the conservative nature of this
approach. The next simulation confirmed the homography expressions proposed for target state
estimation. This was demonstrated in a openloop fashion for the scenario involving a fictitious
police pursuit that employed a camera equipped UAV. The results presented in this simulation
revealed accurate estimation under ideal conditions. The final simulation incorporated the
target state estimates in feedback for closedloop control to accomplish a docking task for the
aerial refueling mission. The target state estimator provided commands to a waypoint tracking
controller in attempt to regulate the relative position between the receiver and the basket drogue.
Simulating the system produced results that were reasonable but inadequate for the requirements
necessary for refueling. The system was capable of homing in on the drogue to within 7 ft in
the axial direction and within 3 ft in both the lateral and altitude directions. The main cause
of this shortcoming is due to the difficulty of maintaining features within the image at close
proximity. Even the slightest shift in orientation can eliminate features from the image causing
a break down in target estimation and overall performance. Error bounds on the state estimates
were also computed for this simulation to examine the effects of uncertain camera parameters.
A worsecase bound was found to exist for the case when both focal length and radial distortion
were at their maximum variations. Analyzing the worsecase bounds, one can determine the
accuracy needed during calibration to obtain a level of confidence in the target estimates.
Future work of this project will examine more realistic aspects of the camera system to
reproduce a more practical scenario. Typical artifacts seen in real images including noise, pixel
quantization, and feature point tracking errors should all be incorporated into the simulation.
Although this will inevitably degrade the estimation results, additional filtering techniques may
be used to improve reconstruction. The next step is to alleviate the limitations imposed on the
feature points. For example, the restriction that the distance of a feature on both the reference
vehicle and the target are known and equal may limit the usefulness in certain applications.
Meanwhile, the aerial refueling simulation requires realistic dynamics of a drogue in flight to
test the system under practical conditions. Additionally, incorporating the modeling scheme
presented in Chapter 8 into the refueling simulation will help the controller by providing state
estimate when the target exits the field of view.
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August 1990.
BIOGRAPHICAL SKETCH
Ryan Scott Causey was born in Miami, Florida, on May 10, 1978. He grew up in a stable
family with one brother in a typical suburban home. During his teenage years and into early
adolescence, Ryan built and maintained a small business providing lawn care to the local
neighborhood. The tools acquired from this work carried over into his college career. After
graduating from Miami Killian Senior High School in 1996, Ryan attended Miami Dade
Community College for three years and received an Associate in Arts degree. A transfer student
to the University of Florida, Ryan was prepared to tackle the stresses of a university aside from
the poor statistics on transfer students. A few years later, he received a Bachelor of Science in
Aerospace Engineering with honors in 2002 and was considered in the top three of his class.
Ryan soon after chose to attend graduate school back at the University of Florida under Dr. Rick
Lind in the Dynamics and Controls Laboratory. During the summertime, Ryan interned twice at
Honeywell Space Systems as a Systems Engineer in Clearwater, FL and once at The Air Force
Research Laboratory in Dayton, OH. Visionbased control of autonomous air vehicles became
his interest and he is now pursuing a doctorate degree on this topic. Ryan was awarded a NASA
Graduate Student Research Program (GSRP) fellowship in 2004 for his proposed investigation on
this research.
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ThisworkwassupportedjointlybyNASAunderNND04GR13HwithSteveJacobsonandJoePahleasprojectmanagersalongwiththeAirForceResearchLaboratoryandtheAirForceOfceofScienticResearchunderF496200310381withJohnnyEvers,NealGlassman,SharonHeise,andRobertSierakowskiasprojectmonitors.Additionally,IthankDr.RickLindforhisremarkableguidanceandinspirationthatwilltrulylastalifetime.Finally,IthankmyparentsSandraandJamesCauseyformakingthisjourneypossiblebyprovidingmetheguidanceanddisciplineneededtobesuccessful. 4
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page ACKNOWLEDGMENTS .................................... 4 LISTOFTABLES ....................................... 8 LISTOFFIGURES ....................................... 9 LISTOFTERMS ........................................ 12 ABSTRACT ........................................... 19 CHAPTER 1INTRODUCTION .................................... 21 1.1Motivation ...................................... 21 1.2ProblemStatement ................................. 27 1.3PotentialMissions .................................. 27 1.4SystemArchitecture ................................. 30 1.5Contributions .................................... 33 2LITERATUREREVIEW ................................. 36 2.1DetectionofMovingObjects ............................ 36 2.2StateEstimationUsingVisionInformation ..................... 38 2.2.1Localization ................................. 39 2.2.2Mapping ................................... 39 2.2.3TargetMotionEstimation .......................... 40 2.3ModelingObjectMotion .............................. 41 2.4UncertaintyinVisionAlgorithms .......................... 42 2.5ControlUsingVisualFeedbackinDynamicEnvironments ............ 43 3IMAGEPROCESSINGANDCOMPUTERVISION .................. 45 3.1CameraGeometry .................................. 45 3.2CameraModel .................................... 47 3.2.1IdealPerspective .............................. 47 3.2.2IntrinsicParameters ............................. 48 3.2.3ExtrinsicParameters ............................. 49 3.2.4RadialDistortion .............................. 50 3.3FeaturePointDetection ............................... 51 3.4FeaturePointTracking ............................... 53 3.5OpticFlow ..................................... 56 5
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............................. 56 3.6.1EpipolarConstraint ............................. 57 3.6.2EightPointAlgorithm ............................ 59 3.6.3PlanarHomography ............................. 61 3.6.4StructurefromMotion ............................ 65 4EFFECTSONSTATEESTIMATIONFROMVISIONUNCERTAINTY ........ 67 4.1FeaturePoints .................................... 67 4.2OpticalFlow ..................................... 70 4.3EpipolarGeometry ................................. 71 4.4Homography .................................... 73 4.5StructureFromMotion ............................... 75 5SYSTEMDYNAMICS .................................. 77 5.1DyanmicStates ................................... 77 5.1.1Aircraft ................................... 77 5.1.2Camera ................................... 79 5.2SystemGeometry .................................. 81 5.3NonlinearAircraftEquations ............................ 83 5.4AircraftCameraSystem .............................. 84 5.4.1FeaturePointPosition ............................ 85 5.4.2FeaturePointVelocity ............................ 85 5.5SystemFormulation ................................. 86 5.6Simulating ...................................... 89 6DISCERNINGMOVINGTARGETFROMSTATIONARYTARGETS ........ 90 6.1CameraMotionCompensation ........................... 90 6.2Classication .................................... 95 7HOMOGRAPHYAPPROACHTOMOVINGTARGETS ................ 98 7.1Introduction ..................................... 98 7.2StateEstimation ................................... 101 7.2.1SystemDescription ............................. 101 7.2.2HomographyEstimation .......................... 103 8MODELINGTARGETMOTION ............................ 111 8.1Introduction ..................................... 111 8.2DynamicModelingofanObject .......................... 111 8.2.1MotionModels ............................... 112 8.2.2StochasticPrediction ............................ 113 9CONTROLDESIGN ................................... 117 9.1ControlObjectives ................................. 117 6
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............................... 118 9.2.1AltitudeControl ............................... 118 9.2.2HeadingControl ............................... 119 9.2.3DepthControl ................................ 121 10SIMULATIONS ...................................... 123 10.1Example1:FeaturePointGeneration ........................ 123 10.2Example2:FeaturePointUncertainty ....................... 126 10.2.1Scenario ................................... 126 10.2.2OpticFlow .................................. 128 10.2.3TheEpipolarConstraint ........................... 130 10.2.4StructureFromMotion ........................... 132 10.3Example3:OpenloopGroundVehicleEstimation ................ 133 10.3.1SystemModel ................................ 134 10.3.2OpenloopResults .............................. 135 10.4Example4:ClosedloopAerialRefuelingofaUAV ................ 138 10.4.1SystemModel ................................ 139 10.4.2ControlTuning ............................... 140 10.4.3ClosedloopResults ............................. 144 10.4.4UncertaintyAnalysis ............................ 148 11CONCLUSION ...................................... 151 REFERENCES ......................................... 154 BIOGRAPHICALSKETCH .................................. 164 7
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Table page 31Solutionsforhomographydecomposition ....................... 64 101Statesofthecameras .................................. 123 102Limitsonimagecoordinates .............................. 123 103Statesofthefeaturepoints ............................... 124 104Aircraftstates ...................................... 125 105Imagecoordinatesoffeaturepoints ........................... 126 106Effectsofcameraperturbationsonopticow ...................... 129 107Effectsofcameraperturbationsonepipolargeometry ................. 131 108Effectsofcameraperturbationsonstructurefrommotion ............... 133 109Maximumvariationsinpositionduetoparametricuncertainty ............ 150 1010Maximumvariationsinattitudeduetoparametricuncertainty ............. 150 8
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Figure page 11TheUAVeet ...................................... 23 12AeroVironment'sMAV:TheBlackWidow ....................... 23 13TheUFMAVeet .................................... 24 14Refuelingapproachusingtheprobedroguemethod .................. 28 15TrackingapursuitvehicleusingavisionequippedUAV ................ 30 16Closedloopblockdiagramwithvisualstateestimation ................ 31 31Mappingfromenvironmenttoimageplane ....................... 46 32Imageplaneeldofview(topview) .......................... 46 33Radialdistortioneffects ................................. 51 34Geometryoftheepipolarconstraint ........................... 58 35Geometryoftheplanarhomography .......................... 62 41Featurepointdependenceonfocallength ........................ 68 42Featurepointdependenceonradialdistortion ..................... 68 51Bodyxedcoordinateframe .............................. 78 52Cameraxedcoordinateframe ............................. 80 53Scenarioforvisionbasedfeedback ........................... 81 61Epipolarlinesacrosstwoimageframes ......................... 91 62FOEconstraintontranslationalopticowforstaticfeaturepoints ........... 94 63Residualopticowfordynamicenvironments ..................... 95 71Systemvectordescription ................................ 102 72Movingtargetvectordescription ............................ 103 91Altitudeholdblockdiagram ............................... 118 92Headingholdblockdiagram .............................. 119 101Virtualenvironmentforexample1 ........................... 124 102Featurepointmeasurementsforexample1 ....................... 125 9
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........................ 126 104Virtualenvironmentforexample2 ........................... 127 105Featurepointsacrosstwoimageframes ........................ 128 106Uncertaintyinfeaturepoint ............................... 128 107Uncertaintyresultsinopticow ............................ 129 108Nominalepipolarlinesbetweentwoimageframes ................... 130 109Uncertaintyresultsforepipolargeometry ........................ 131 1010Nominalestimationusingstructurefrommotion .................... 132 1011Uncertaintyresultsforstructurefrommotion ...................... 133 1012Vehicletrajectoriesforexample3 ............................ 134 1013PositionstatesoftheUAVwithonboardcamera ................... 135 1014AttitudestatesoftheUAVwithonboardcamera ................... 135 1015Positionstatesofthereferencevehicle ......................... 135 1016Attitudestatesofthereferencevehicle ......................... 136 1017Positionstatesofthetargetvehicle ........................... 136 1018Attitudestatesofthetargetvehicle ........................... 136 1019Normerror ........................................ 137 1020Relativepositionstates ................................. 137 1021Relativeattitudestates .................................. 137 1022Virtualenvironment ................................... 138 1023InnerlooppitchtopitchcommandBodeplot ..................... 141 1024Pitchanglestepresponse ................................ 141 1025Altitudestepresponse .................................. 142 1026InnerlooprolltorollcommandBodeplot ....................... 143 1027Rollanglestepresponse ................................. 144 1028Headingresponse .................................... 144 1029Openloopestimationoftarget'sinertialposition .................... 145 10
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.................... 145 1031Normerrorfortargetstateestimates .......................... 146 1032Closedlooptargetpositiontracking .......................... 146 1033Positiontrackingerror .................................. 147 1034Targetattitudetracking ................................. 147 1035Trackingerrorinheadingangle ............................. 147 1036Target'sinertialpositionwithuncertaintybounds ................... 149 1037Target'sinertialattitudewithuncertaintybounds .................... 149 11
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1 ].Thisincreaseincapabilityforsuchcomplextasksrequirestechnologyformoreadvancedsystemstofurtherenhancethesituationalawareness.Overthepastseveralyears,theinterestanddemandforautonomoussystemshasgrownconsiderably,especiallyfromtheArmedForces.Thisinteresthasleveragedfundingopportunitiestoadvancethetechnologyintoastateofrealizablesystems.Sometechnicalinnovationsthathaveemergedfromtheseefforts,fromahardwarestandpoint,consistmainlyofincreasinglycapablemicroprocessorsinthesensors,controls,andmissionmanagementcomputers.TheDefenseAdvancedResearchProjectsAgency(DARPA)hasfundedseveralprojectspertainingtotheadvancementofelectronicdevicesthroughsizereduction,improvedspeedandperformance.Fromthesedevelopments,thecapabilityofautonomoussystemhasbeendemonstratedonvehicleswithstrictweightandpayloadrequirements.Inessence,thecurrenttechnologyhasmaturedtoapointwhereautonomoussystemsarephysicallyachievableforcomplexmissionsbutnotyetalgorithmicallycapable.TheaerospacecommunityhasemployedmanyoftheresearchdevelopedforautonomoussystemsandappliedittoUnmannedAerialVehicles(UAV).Manyofthesevehiclesarecurrently 21
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1 ].FuturemissionsenvisionUAVtoconductmorecomplextasksuchasterrainmapping,surveillanceofpossiblethreats,maritimepatrol,bombdamageassessment,andeventuallyoffensivestrike.Thesemissionscanspanovervarioustypesofenvironmentsand,therefore,requireawiderangeofvehicledesignsandcomplexcontrolstoaccommodatetheassociatedtasks.TherequirementsanddesignofUAVareconsideredtoenableaparticularmissioncapability.Eachmissionscenarioisthedrivingforceoftheserequirementsandaredictatedbyrange,speed,maneuverability,andoperationalenvironment.CurrentUAVrangeinsizefromlessthan1poundtoover40,000pounds.SomepopularUAVthatareoperational,intestingphase,andintheconceptphasearedepictedinFigure 11 toillustratethevariousdesigns.ThetwoUAVontheleft,GlobalHawkandPredator,arecurrentlyinoperation.GlobalHawkisemployedasahighaltitude,longendurancereconnaissancevehiclewhereasthePredatorisusedforsurveillancemissionsatloweraltitudes.Meanwhile,theremainingtwopicturespresentJUCAS,whichisajointcollaborationforboththeAirForceandNavy.ThisUAVisdescribedasamediumaltitudeyerwithincreasedmaneuverabilityoverGlobalHawkandthePredatorandisconsideredforvariousmissions,someofwhichhavealreadybeendemonstratedinight,suchasweapondeliveryandcoordinatedight.Theadvancementsinsensorsandcomputingtechnology,mentionedearlier,hasfacilitatedtheminiaturizationoftheseUAV,whicharereferredtoasMicroAirVehicles(MAV).Thescaleofthesesmallvehiclesrangesfromafewfeetinwingspandowntoafewinches.DARPAhasalsofundedtherstsuccessfulMAVprojectthroughAeroVironment,asshowninFigure 12 ,wherebasicautonomywasrstdemonstratedatthisscale[ 2 ].Thesesmallscalesallowhighlyagilevehiclesthatcanmaneuverinandaroundobstaclessuchasbuildingsandtrees.ThiscapabilityenablesUAVtooperateinurbanenvironments,belowrooftoplevels,toprovide 22
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TheUAVeet thenecessaryinformationwhichcannotbeobtainedathigheraltitudes.ResearchersarecurrentlypursuingMAVtechnologytoaccomplishtheverysamemissionsstatedearlierfortheuniqueapplicationofoperatingatlowaltitudesinclutteredenvironments.Assensorandcontroltechnologiesevolve,theseMAVcanbeequippedwiththelatesthardwaretoperformadvancedsurveillanceoperationswherethedetection,tracking,andclassicationofthreatsaremonitoredautonomouslyonline.Althoughasinglemircoairvehiclecanprovidedistinctinformation,targetsmaybedifculttomonitorduetobothightpathandsensoreldofviewconstraints.ThislimitationhasmotivatedtheideaofacorporativenetworkoraswarmofMAVcommunicatingandworkingtogethertoaccomplishacommontask. Figure12. AeroVironment'sMAV:TheBlackWidow 23
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3 4 ].Meanwhile,Stanfordhasexaminedmotionplanningstrategiesthatoptimizeighttrajectoriestomaintainsensorintegrityforimprovedstateestimation[ 5 ].TheworkatGeorgiaTechandBYUhasconsideredcorporativecontrolofMAVforautonomousformationying[ 6 ]andconsensusworkfordistributedtaskassignment[ 7 ].Alternatively,visionbasedcontrolhasalsobeenthetopicofinterestatbothGeorgiaTechandUF.ControlschemesusingvisionhavebeendemonstratedonplatformssuchasahelicopteratGeorgiaTech[ 8 ],whileUFimplementedaMAVthatintegratedvisionbasedstabilizationintoanavigationarchitecture[ 9 10 ].TheUniversityofFloridahasalsoconsideredMAVdesignsthatimprovetheperformanceandagilityofthesevehiclesthroughmorphingtechnology[ 11 13 ].FabricationfacilitiesatUFhaveenabledrapidconstructionofdesignprototypesusefulforbothmorphingandcontroltesting.TheeetofMAVproducedbyUFareillustratedinFigure 13 wherethewingspanofthesevehiclesrangefrom24indownto4in. Figure13. TheUFMAVeet ThereareanumberofcurrentdifcultiesassociatedwithMAVduetotheirsize.Forexample,characterizingtheirdynamicsunderightconditionsatsuchlowReynoldsnumbersisanextremelychallengingtask.Theconsequenceofincreasedagilityatthisscalealsogivesrisetoerraticbehaviorandaseveresensitivitytowindgustandotherdisturbances.Waszaketal.[ 14 ]performedwindtunnelexperimentson6inchMAVandobtainedtherequiredstabilityderivativesforlinearandnonlinearsimulations.AnothercriticalchallengetowardMAV 24
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3 .Thisdissertationwillfocusonthemonocularcameracongurationtoaddressthestateestimationproblemregardingmovingtargets.TheadvantageofthesetechniquesbecomesmoreapparenttoUAVwhenappliedtoguidance,navigation,andcontrol.Bymountingacameraonavehicle,stateestimationofthevehicleandobjectsintheenvironmentcanbeachievedinsomeinstancesthroughvisionprocessing.Oncestateestimatesareknown,theycanthenbeusedinfeedback.Controltechniquescanthenbeutilizedforcomplexmissionsthatrequirenavigation,pathplanning,avoidance,tracking,homing,etc.Thisgeneralframeworkofvisionprocessingandcontrolhasbeensuccessfullyappliedtovarioussystemsandvehiclesincludingroboticmanipulators,groundvehicles,underwatervehicles,andaerialvehiclesbuttherestillexistssomecriticallimitations.Theproblematicissueswithusingvisionforstateestimationinvolvescameranonlinearities,cameracalibration,sensitivitytonoise,largecomputationaltime,limitedeldofview,andsolvingthecorrespondenceproblem.Aparticularsetoftheseimageprocessingissueswillbeaddresseddirectlyinthisdissertationtofacilitatethecontrolofautonomoussystemsincomplexsurroundings. 26
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1. segmentingmovingtargetsfromstationarytargetswithinthescene 2. classifyingmovingtargetsintodeterministicandstochasticmotions 3. couplingthevehicledynamicsintothesensorobservations(i.e.images) 4. formulatingthehomographyequationsbetweenamovingcameraandtheviewabletargets 5. propagatingtheeffectsofuncertaintythroughthestateestimationequations 6. establishingcondenceboundsontargetstateestimationThedesignandimplementationofavisionbasedcontrollerisalsopresentedinthisdissertationtoverifyandvalidatemanyoftheconceptspertainingtotrackingofmovingtargets. 27
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15 ].Thedrogueisdesignedinanaerodynamicshapethatpermitstheextensionfromthetankerwithoutinstability.TheprobeanddroguemethodisconsideredthepreferredmethodforAAR,mainlyduetothehighpilotworkloadincontrollingtheboom[ 16 ].Figure 14 illustratestheviewobservedbyreceiveraircraftduringtherefuelingprocesswherefeaturepointshavebeenplacedonthedrogue. Figure14. Refuelingapproachusingtheprobedroguemethod VisioncanbeusedtofacilitatetheAARproblembyaugmentingtraditionalaircraftsensorssuchasglobalpositioningsystem(GPS)andinertialmeasurementunit(IMU).GighprecisionGPS/IMUsensorscanproviderelativeinformationbetweenthetankerandthereceiverthenvisioncanbeusedtoproviderelativeinformationonthedrogue.Theadvantagetovisioninthiscaseisitspassivenaturewhicheliminatessensoremissionsduringrefuelingoverenemyair 28
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15 illustratesinasimulatedenvironmentthisscenariowhereaUAVobservesthe 29
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Figure15. TrackingapursuitvehicleusingavisionequippedUAV 16 ,wherecommandsaresenttoavehiclebasedonthemotionsobservedintheimages.ThevehicleconsideredinthisdissertationispredominatelyassumedanautonomousUAV,butisgeneralizedforanydynamicalsystemwithpositionandorientationstates.TheblockspertainingtothisdissertationarehighlightedinFigure 16 intheimageprocessingblockandconsistsofthemovingobjectdetection,stateestimationofamovingobject,andclassifyingdeterministicversusstochasticmotion.Abriefdiscussionofeachtopicisdescribedinthissection,whilethedetailsarecoveredintheirrespectivechapters.Distinguishingmovingobjectsfromstationaryobjectswithamovingcameraisachallengingtaskinvisionprocessingandistherststepinthestateestimationprocesswhenconsideringadynamicscene.Thisinformationisextremelyimportantforguidance,navigation,andcontrolofautonomoussystemsbecauseitidentiesobjectsthatpotentiallycouldbeinapathforcollision.Forastationarycamera,movingobjectsinthescenecanbeextractedusing 30
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Closedloopblockdiagramwithvisualstateestimation simpleimagedifferencing,wherethestationarybackgroundissegmentedout;however,thisapproachdoesnotapplytomovingcameras.Inthecaseofamovingcamera,thebackgroundisnolongerstationaryanditbeginstochangeovertimeasthevehicleprogressesthroughtheenvironment.Therefore,theimagestakenbyamovingcameracontainthemotionduetothecamera,commonlycalledegomotion,andthemotionoftheobject.Techniquesthatinvolvecameramotioncompensationorimageregistrationhavebeenproposedtoworkwellwhenthereexistsnostationaryobjectsclosetothecamerawhichcausehighparallax.Thisdissertationwillestablishatechniquetoclassifyobjectsintheeldofviewasmovingorstationarywhileaccountingforstationaryobjectswithhighparallax.Therefore,withaseriesofobservationsofaparticularscene,onecandeterminewhichobjectsaremovingintheenvironment.Knowingwhichobjectsaremovingintheimagedictatesthetypeofimageprocessingrequiredtoaccuratelyestimatetheobject'sstates.Infact,theestimationproblembecomesinfeasibleforamonocularsystemwhenboththecameraandtheobjectaremoving.Thisunattainablesolutioniscausebyanumberoffactorsincluding1)inabilitytodecouplethemotionfromthecameraandtargetand2)failuretotriangulatethedepthestimateoftheobject.Forthisconguration,relativeinformationcanbeobtainedandfusedwithadditionalinformationforstateestimation.First,decouplingthemotionrequiresknowninformationregardingmotionofthecameraorthemotionoftheobject,whichcouldbeobtainedthroughothersensorssuch 31
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5 ].Furthermore,theaccuracyofthestateestimatesbecomespoorforsmallbaselinecongurations,whichoccursforMAVusingstereovision.Theseissuesregardingtargetstateestimationwillbeconsideredinthisdissertationtoshowboththecapabilitiesandlimitationstowardautonomouscontrolandnavigation.Anotherimportanttaskinvolvedwithtargetestimationistodetermineapattern(ifany)intheobject'smotionbasedonthetimehistory.Theobjectscanthenbeclassiedintodeterministicandstochasticmotionsaccordingtopastbehavior.Withthisinformation,predictionmodelscanbemadebasedonpreviousimagestoestimatethepositionofanobjectatalatertimewithsomelevelofcondence.Thepredictedestimatescanthenbeusedinfeedbackfortrackingordockingpurposes.Forstochasticlyclassiedobjects,furtherconcernsregardingdockingorAARareimposedonthecontrolproblem.Theprimarytaskofstateestimation,forboththevehicleandobjectsintheenvironment,reliesonaccurateknowledgeoftheimagemeasurementsandtheassociatedcamera.Suchknowledgeisdifculttoobtainduetouncertaintiesinthesemeasurementsandtheinternalcomponentsofthecameraitself.Forinstance,theimagemeasurementscontainuncertaintiesassociatedwiththedetectionofobjectsintheimage,inadditiontonoisecorruption.Thesedrawbackshavepromptedmanyrobustalgorithmstoincreasetheaccuracyoffeaturedetectionwhilehandlingnoiseduringtheestimationprocess.Alternatively,manytechniqueshavebeenusedtoaccuratelyestimatetheinternalparametersofthecamerathroughcalibration.Theparametersthatdescribetheinternalcomponentsofthecameraarereferredtoasintrinsicparametersandtypicallyconsistoffocallength,radialdistortion,skewfactor,pixelsize,andopticalcenter.Thiscalibrationprocesscanbecomecumbersomeforalargenumberofcameras 32
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16 illustratesthecomponentsofinterestdescribedinthisdissertationforstateestimationandtrackingcontrolwithrespecttoamovingobjectwhichinvolvesobjectmotiondetection,objectstateestimation,andobjectmotionmodelingandprediction.Theliteraturereviewofthesetopicsisgiveninthissection. 17 18 ]hasservedasafoundationformanyalgorithms.Thistechniquereliesonasmoothnessconstraintimposedontheopticowthatmaintainsaconstantintensityacrosssmallbaselinemotionofthecamera.Manytechniqueshavebuiltuponthisalgorithmtoincreaserobustnesstonoiseandoutliers.Oncefeaturetrackinghasbeenobtained,thenextprocessinvolvessegmentingtheimageformovingobjects.Theneedforsuchaclassicationisduethefactthatstandardimage 36
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19 21 ].TechniquesformorerealisticapplicationsinvolveKalmanltering[ 22 ]toaccountforlightingconditionsandbackgroundmodelingtechniquesusingstatisticalapproaches,suchasexpectationmaximizationandmixtureofGaussian,toaccountforothervariationsinrealtimeapplications[ 23 28 ].Althoughthesetechniquesworkwellforstationarycameras,theyareinsufcientforthecaseofmovingcamerasduetothemotionofthestationarybackground.Motiondetectionusingamovingcamera,asinthecaseofacameramountedtoavehicle,becomessignicantlymoredifcultbecausethemotionviewedintheimagecouldresultfromanumberofsources.Forinstance,acameramovingthroughascenewillviewmotionsintheimagecausedbycamerainducedmotion,referredtoasegomotion,changesincameraintrinsicparameterssuchaszoom,andindependentlymovingobjects.Therearetwoclassesofproblemsconsideredinliteratureforaddressingthistopic.Therstconsidersthescenariowherethe3Dcameramotionisknownapriorithencompensationcanbemadetoaccountforthismotiontodeterminestationaryobjectsthroughanappropriatetransformation[ 29 30 ].Thesecondclassofproblemsdoesnotrequireknowledgeofthecameramotionandconsistsofatwostageapproachtothemotiondetection.Therststageinvolvescameramotioncompensationwhilethelaststageemploysimagedifferencingontheregisteredimage[ 31 ]toretrievenonstaticobjects.ThetransformationusedtoaccountforcameramotioniscommonlysolvedbyassumingthemajorityofimageconsistsofadominantbackgroundthatisstationaryinEuclideanspace[ 32 33 ].Thissolutionisobtainedthroughaleastsquaresminimizationprocess[ 32 ]orwiththeuseofmorphologicallters[ 34 ].Thetransformationsobtainedfromthesetechniquestypically 37
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35 ]proposedauniedmethodtodetectmovingobjects.Thisproposedmethodhandlesvariouslevelsofparallaxintheimagethroughasegmentationprocessthatisperformedinlayers.Therstlayerextractsthebackgroundobjectswhicharefarawayfromthecameraandhavelowparallaxthroughageneraltransformationinvolvingcamerarotation,translation,andzoomthroughimagedifferencing.Thenextlayercontainstheobjectwithhighparallaxconsistingofbothobjectsclosetothecameraandobjectsthataremovingindependentlyofthecamera.Theparallaxisthencomputedfortheremainingpixelsandcomparedtoonepixel.Thisprocessseparatestheobjectswithintheimagebasedontheircomputedparallax.Theselectionmayinvolvechoosingapointonaknownstationaryobjectthatcontainshighparallaxsoanyobjectnotobeyingthisparallaxisclassiedasamovingobjectinthescene.Opticowtechniquesarealsousedtoestimatemovingtargetlocationsonceegomotionhasbeenestimated.Amethodthatcomputesthenormalimageowhasbeenshowntoobtainmotiondetection[ 36 ].Coordinatetransformationsaresometimesusedtofacilitatethisapproachtodetectingmotion.Forinstance,amethodusingcomplexlogmappingwasshowntotransformtheradialmotionsintohorizontallinesuponwhichverticalmotionindicateindependentmotion[ 37 ].Alternatively,sphericalmappingwasusedgeometricallytoclassifymovingobjectsbysegmentingmotionswhichdonotradiatefromthefocusofexpansion(FOE)[ 29 ]. 38
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38 39 ]usedthecoplanarityconstraintalsoknownastheepipolarconstraint.Meanwhile,thesubspaceconstrainthasalsobeenemployedtolocalizecameramotion[ 40 ].Thesetechniqueshavebeenappliedtonumeroustypesofautonomoussystems.Themobileroboticcommunityhasappliedthesetechniquesforthedevelopmentofnavigationinvariousscenarios[ 41 45 ].TheapplicationshavealsoextendedintotheresearchofUAVforaircraftstateestimation.GurlandRotstein[ 46 ]wasthersttoextendthisapplicationintheframeworkofanonlinearaircraftmodel.Thisapproachusedopticalowinconjunctionwiththesubspaceconstrainttoestimatetheangularratesoftheaircraftandwasextendedin[ 47 ].Webbetal.[ 48 49 ]employedtheepipolarconstrainttotheaircraftdynamicstoobtainvehiclestates.ThefoundationforbothoftheseapproachesisaKalmanlterinconjunctionwithageometricconstrainttoestimatethecameramotion.SomeapplicationsforaircraftstateestimationhaveinvolvedmissionsforautonomousUAVsuchasautonomousnightlanding[ 50 ]androadfollowing[ 51 ]. 52 58 ].TheseapproachesemploythesubspaceconstrainttoreconstructfeaturepointpositionthroughanextendedKalmanlter.Severalsurveypapershavebeenpublisheddescribingthecurrentalgorithmswhilecomparingtheperformanceandrobustness[ 59 62 ].RobustandadaptivetechniqueshavebeenproposedthatuseanadaptiveextendedKalmanltertoaccountformodeluncertainties[ 63 ].Inaddition,Qianetal.[ 64 ]designedarecursiveHltertoestimatestructurefrommotioninthepresenceofmeasurementandmodeluncertaintieswhile 39
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65 ]investigatedtheoptimalapproachestotargetstateestimationanddescribedtheeffectsoflinearsolutionsonvariousnoisedistributions. 66 ].ThismethodextendedthepreviousworkofBroidaetal.[ 67 ]thatonlyconsideredafeaturepointapproach.Forthecaseofmovingmonocularcameraconguration,theproblembecomesextremelydifcultduetotheadditionalmotionofthecamera.Oneapproachusedinliteraturerelevanttomonocularcamerasystemsisbearingsonlytracking.Inthisapproach,thereareseveralassumptionsmade:(i)thevehiclehasknowledgeofitsposition,(ii)anadditionalrangesensor,suchassonarorlaserrangender,isusedtoprovideabearingmeasurement,and(iii)animagemeasurementistakenforanestimateoflateralposition.Theinitialresearchhasinvolvedtheestimationprocessanddesignwithimprovementstotheperformance[ 68 72 ].ThisapproachwasimplementbyFlew[ 5 ]toestimatethemotionoftargetwithinacomputedcovariance.Guanghuietal.[ 73 ]providedamethodforestimatingthemotionofapointtargetfromknowncameramotion.Theroboticcommunityhasexaminedthetargetmotionestimationproblemfromavisualservocontrolframework.Trackingrelativemotionofamovingtargethasbeenshownusinghomographybasedmethods.Thesemethodshavebeendemonstratedtocontrolanautonomousgroundvehicletoadesiredposedenedbyagoalimage,wherethecamerawasmountedonthegroundvehicle[ 74 ].Chenetal.[ 75 76 ]regulatedagroundvehicletoadesiredposeusingastationaryoverheadcamera.Mehtaetal.[ 77 ]extendedthisconceptforamovingcamera,whereacamerawasmountedtoanUAVandagroundvehiclewascontrolledtoadesiredpose. 40
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15 ]appliedavisionnavigationalgorithmcalledVisNAVthatwasdevelopedbyJunkinsetal.[ 78 ]toestimatethecurrentrelativepositionandorientationofthetargetdroguethroughaGaussianleastsquaresdifferentialcorrectionalgorithm.Thisalgorithmhasalsobeenappliedtospacecraftformationying[ 79 ]. 80 ]demonstratedthecontrolrequiredtograbanunknownmovingobjectwithroboticmanipulatorusinganautoregressive(AR)model.Thismodelpredictsafuturepositionofthetargetbasedonvelocityestimatescomputedfromimagesequences.Foraerialvehicles,detectingotheraircraftintheskyiscriticalforcollisionavoidance.NASAhasconsideredvisioninthisscenariotoaidpilotsindetectingaircraftonacrossingtrajectory.AtechniquecombiningimageandnavigationdataestablishedapredictionmethodthroughaKalmanlterapproachtoestimatethepositionandvelocityofthetargetaircraftaheadintime[ 34 ].Similarly,theAARproblemrequiressomeformofmodelpredictionwhendockingtoamovingdrogue.Kimmettetal.[ 15 ]utilizedadiscretelinearmodelforthepredictionofthedrogue.Thepredictedstatesusedforcontrolwerecomputedusingthediscretemodel,thecurrentstates,andlightturbulenceasinputtothedroguedynamics.Successfuldockingwassimulatedforonlylightturbulenceandwithlowfrequencydynamicsimposedonthedrogue.NASAisextremelyinterestedinAARproblemandcurrentlyhasaprojectonthistopic.FlighttestshavebeenconductedbyNASAinanattempttomodelthedroguedynamics[ 81 ].Inthisstudy,theaerodynamiceffectsfromboththereceiveraircraftandthetankeraircraftwereexaminedonthe 41
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82 ].Robustnesswasalsoanalyzedusingaleastsquaresolutiontoobtainanexpressionfortheerrorintermsofthemotionvariables[ 83 ].Theuncertaintyinvisionbasedfeedbackisoftenchosenasvariationswithinfeaturepoints;however,uncertaintyinthecameramodelmayactuallybeanunderlyingsourceofthosevariations.Essentially,theuncertaintymaybeassociatedwiththeimageprocessingtoextractfeaturepointsorwiththecameraparametersthatgeneratedtheimage.Thepropercharacterizationofcamerauncertaintymaybecriticaltodeterminearealisticleveloffeaturepointuncertainty.Theanalysisofcamerauncertaintyistypicallyaddressedinaprobabilisticmanner.Alineartechniquewaspresentedthatpropagatesthecovariancematrixofthecameraparametersthroughthemotionequationstoobtainthecovarianceofthedesiredcamerastates[ 84 ].Ananalysiswasalsoconductedfortheepipolarconstraintbasedontheknowncovarianceinthecameraparameterstocomputethemotionuncertainty[ 85 ].AsequentialMonteCarlotechniquedemonstratedbyQianetal.[ 86 ]proposedanewstructurefrommotionalgorithmbasedonrandomsamplingtoestimatetheposteriordistributionsofmotionandstructureestimation.Theexperimentalresultsinthispaperrevealedsignicantchallengestowardsolvingforthestructureinthepresenceoferrorsincalibration,featurepointtracking,featureocclusion,andstructureambiguities. 42
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87 ]intrafcsituations,lowaltitudeightofarotorcraft[ 88 ],avoidingobstaclesintheightpathofanaircraft[ 34 ],andnavigatingunderwatervehicles[ 89 ].Opticalowtechniqueshavealsobeenutilizedasatoolforavoidancebysteeringawayfromareaswithhighopticowwhichindicateregionsofcloseobstacles[ 90 ].Targettrackingisanotherdesiredcapabilityforautonomoussystems.Inparticular,themilitaryisinterestedinthistopicforsurveillancemissionsbothintheairandontheground.Thecommonapproachestotargettrackingoccurinbothfeaturepointandopticalowtechniques.Thefeaturepointmethodtypicallyconstrainsthetargetmotionintheimagetoadesiredlocationbycontrollingthecameramotion[ 91 92 ].Meanwhile,Frezzaetal.[ 93 ]imposedanonholonomicconstraintonthecameramotionandusedapredictiveoutputfeedbackcontrolstrategybasedontherecursivetrackingofthetargetwithfeasiblesystemtrajectories.Alternatively,opticalowbasedtechniqueshavebeenpresentedforrobotichandineyecongurationtotracktargetsofunknown2Dvelocitieswherethedepthinformationis 43
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94 ].Adaptivesolutionspresentedin[ 91 95 97 ]haveshowncontrolsolutionsfortargettrackingwithuncertaincameraparameterswhileestimatingdepthinformation.Thehomingcontrolproblemhasnumerousapplicationstowardautonomoussystemssuchasautonomousaerialrefueling,spacecraftdocking,missileguidance,andobjectretrievalusingarobtoticmanipulator.Kimmettetal.[ 15 98 ]developedacandidateautonomousprobeanddrogueaerialrefuelingcontrollerthatusesacommandgeneratortracker(CGT)totracktimevaryingmotionsofanonstationarydrogue.TheCGTisanexplicitmodelfollowingcontroltechniqueandwasdemonstratedinsimulationforamovingdroguewithknowndynamicssubjecttolightturbulence.Tandaleetal.[ 16 ]extendedtheworkofKimmettandValasekbydevelopingareferenceobserverbasedtrackingcontroller(ROTC)whichdoesnotrequireadroguemodelorpresumedknowledgeofthedrogueposition.Thissystemconsistofareferencetrajectorygenerationmodulethatsendscommandstoanobserverthatestimatesthedesiredstatesandcontrolfortheplant.Theinputtothiscontrolleristherelativepositionbetweenthereceiveraircraftandthedroguemeasuredbythevisionsystem.Asimilarvisionapproachtoaerialrefuelingisalsopresentedin[ 99 ],wheremodelsofthetankeranddrogueareusedinconjunctionwithaninferredcamera.Thedroguemodelusedinthispaperwastakenfrom[ 100 ]thatusesamultisegmentapproachtoderivingthedynamicsofthehose.Meanwhile,Houshangietal.[ 80 ]consideredgraspingamovingtargetbyadaptivelycontrollingarobotmanipulatorusingvisioninteraction.Theadaptivecontrolschemewasusedtoaccountformodelingerrorsinthemanipulator.Inaddition,thispaperconsideredunknowntargetdynamics.Anautoregressivemodelapproachwasusedtopredictthetarget'spositionbasedonpassedvisualinformationandanestimatedtargetvelocity.Experimentaltestcasesaredocumentedthatshowtrackingconvergence. 44
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31 .Thevector,h,representsthevectorbetweenthecameraandafeaturepointintheenvironmentrelativetoadenedcameraxedcoordinatesystem,asdenedbyI.ThisvectoranditscomponentsarerepresentedinEquation 3 45
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Mappingfromenvironmenttoimageplane Amajorconstraintplacedonthissensoristhecamera'seldofview(FOV).HeretheFOVcanbedescribedasthe3Dregionforwhichfeaturepointsarevisibletothecamera;hence,featuresoutsidetheFOVwillnotappearintheimage.Thethreephysicalparametersthatdenethisconstraintaretheeldofdepth,thehorizontalangleandtheverticalangle.AtopviewillustrationoftheFOVcanbeseeninFigure 32 ,wherethehorizontalFOVisdenedbythehalfangle,gh,andthedistancetotheimageplaneisoflengthf.Likewise,asimilarplotcanbeshowntoillustratetheverticalangle,whichcanbedenedasgv. Imageplaneeldofview(topview) 46
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3 ,whererh;visdenedasthelargestspatialextensioninthehorizontalandverticaldirections. 3 forthehorizontalcomponent. 3 fortheverticalangle. 3.2.1IdealPerspectiveAgeometricrelationshipbetweenthecamerapropertiesandafeaturepointisrequiredtodeterminetheimageplanecoordinates.Thisrelationshipismadebyrstseparatingthecomponentsofhthatareparalleltotheimageplaneintotwodirections.Theimageplanecoordinatesarethencomputedfromatangentrelationshipofsimilartrianglesbetweentheverticalandhorizontaldirectionsandthedepthwithascalefactoroffocallength.Thisrelationshipestablishesthestandard2Dimageplanecoordinatesreferredtoasthepinholecameramodel[ 101 102 ].Equations 3 and 3 representageneralpinholeprojectionmodel 47
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3 and 3 reducetotheverycommonpinholecameramodelandisrepresentedbyEquations 3 and 3 3 and 3 canbeexpressedinhomogeneouscoordinatesandisshowninEquation 3 3 .First,theimageplaneisdiscretizedintoasetofpixels,correspondingtotheresolutionofthecamera.Thisdiscretizationisbasedonscalefactorsthatrelaterealworldlengthmeasuresintopixelunitsforboththehorizontalandverticaldirections.Thesescalingtermsaredenedassandsnwhichhaveunitsofpixelsperlength,wherethelengthcouldbeinfeetormeters.Ingeneral,thesetermsaredifferentbutwhenthepixelsaresquarethens=sn.Second,theoriginoftheimageplaneistranslatedfromthecenterofthe 48
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3 ,wherepixelmapping,origintranslation,andskewnessareallconsidered. 3 isrewrittentoEquation 3 3 toobtainageneralequationthatmapsfeaturepointsintheinertialframetocoordinatesintheimageplaneforacalibratedcamera. 49
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3 ,requiresaninniteseriesoftermstoapproximatethevalue. 3 and 3 ,mapsanundistortedimage,(0;n0),whichisnotmeasurableonaphysicalcamera,intoadistortedimage,(0d;n0d),whichisobservable[ 104 ].Thisdistortionmodelonlyconsidersthersttermintheinniteseriestodescriberadialdistortionandexcludestangentialdistortion.Thisapproximationindistortionhasbeenusedtogenerateanaccuratedescriptionofrealcameraswithoutadditionalterms[ 105 ], 31 ,attemptstomodelthecurvatureofthelensduringtheimageplanemapping.Thisdistortionintheimageplanevariesinanonlinearfashionbasedonposition.Thiseffectdemonstratesanaxisymmetricmappingthatincreasesradiallyfromtheimagecenter.AnexamplecanbeseeninFigure 33B and 33C whichillustrateshowradialdistortionchangesfeaturepointlocationsofaxedpatternintheimagebycomparingittoatypicalpinholemodelshowninFigure 33A .Noticethedistortedimagesseemtotakeonaconvexorconcaveshapedependingonthesignofthedistortion. 50
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B CFigure33. RadialDistortionEffectsforA)f=0:5d=0,B)f=0:5d=0:0005,andC)f=0:5d=+0:0005 3 .Assuch,theseparametersaretermedtheintrinsicparametersandarefoundthroughcalibration.Afeaturepointmustbeanalyzedwithrespecttotheseintrinsicparameterstoensureproperstateestimation.Theradialdistancefromafeaturepointtothecenteroftheimageisdependentonboththerelativepositionsofcameraandfeaturealongwiththefocallength.Thisradialdistanceisalsorelatedviaanonlinearrelationshiptotheradialdistortion.Clearlyanyanalysisofthefeaturepointswillrequireestimationofthecameraparameters.Chapter 4 willdiscussatechniquethatconsidersboundeduncertaintytowardtheintrinsicparametersandestablishesaboundedconditiononthefeaturepointpositions. 51
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3 and 3 [ 102 103 ].Theimagecoordinates(;n)intheseexpressionsarecomputedusingeitherEqaution 3 orEquation 3 dependingonthecameramodel. 3 [ 102 103 ].Thepixelvalueswithinthesearchwindowaredenedasx. 3 .IfEquation 3 issatisedthenthisisavalidfeaturepointbasedontheuserscriterion[ 102 103 ].Thisselectionisafunctionofboththewindowsize,W,andthethreshold,t. 52
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106 ].ThismethodcanbeextendedtoedgedetectionbyconsideringthestructureofthesingularvaluesofG.AnexampleofthisalgorithmistheCannyedgedetector[ 107 ]. 3 3 3 53
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3 .Oneimportantlimitationofthiscriterionoccurswhenthewindowinbothimagescontainsrelativelyconstantintensityvalues.Thisresultsintheapertureproblemwhereanumberofsolutionsforhareobtained.Therefore,duringthefeatureselectionprocessit'sbenecialtochoosefeaturesthatcontainuniqueinformationinthiswindow. 3 forsmallbaselinetracking:(1)usingthebrightnessconsistencyconstraintand(2)applyingthesumofsquareddifferences(SSD)approach.Eachofthesetechniquesemploysatranslationalmodeltodescribetheimagemotion.Therefore,ifoneassumesasimpletranslationalmodelthenthegeneraltransformationisshowninEquation 3 3 intoEquation 3 whileinitiallyneglectingthenoiseterm.ApplyingtheTaylorseriesexpansiontothisexpressionaboutthepointofinterest,x,whileretainingonlythersttermintheseriesresultsinEquation 3 dt+I 3 inmatrixformresultsinEquation 3 dt;dn 3 constitutes1equationwith2unknownvelocities;therefore,anotherconstraintisneededtosolvethisproblem.Auniquesolutionforthevelocitiescanbedeterminedbyenforcinganadditionalconstraintontheproblem,whichentailsrestrainingregionstoalocalwindowthatmovesatconstantvelocity.Upontheseassumptiononecanminimizetheerror 54
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3 3 3 .ThenalsolutionforthepixelvelocityisfoundthroughaleastsquaresestimategiveninEquation 3 .Theseimagevelocitiesarealsoreferredtoastheopticow.Oncetheopticowiscomputedforafeaturepointthentheimagedisplacementforfeaturetrackingistrivialtond. 3 ,attemptstoestimatetheDxwhilenotrequiringthecomputationofimagegradients.Thisapproachalsoemploysthetranslationalmodeloverawindowedregion.Themethodconsidersthepossiblerangethatwindowcouldmove,danddn,inthetime,dt.Thisconsistencyconstraintthenleadstoaproblemofminimizingtheerroroverthepossiblewindowswithinthedescribedrange.ThiserrorfunctionisdescribedmathematicallyinEquation 3 17 ].Forlargebaselinetrackingsimpletranslationalmodels 55
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4 3 and 3 .Thevelocityexpressions,showninEquations 3 and 3 ,describethemovementoffeaturepointsintheimageplaneandiscommonlyreferredtoinliteratureastheopticow. 3 and 3 whileassumingc=0isasfollows 56
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34 whereh1andh2denotethepositionvectorsofthefeaturepoint,P,inthecamerareferenceframes.Also,thevaluesofx1andx2representthepositionvectorsprojectedontothefocalplanewhileTindicatesthetranslationvectoroftheoriginofthecameraframes.AgeometricrelationshipbetweenthevectorsinFigure 34 isexpressedbyintroducingRasarotationmatrix.Thisrotationmatrixincludestheroll,pitchandyawanglesthattransformthecameraframesbetweenmeasurements.TheresultingepipolarconstraintisexpressedinEquation 3 3 ,assumesapinholecamerawhichiscolinearwithitsprojectionintothefocalplane. 57
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Geometryoftheepipolarconstraint TheexpressionsinEquation 3 andEquation 3 reectthatthescalartripleproductofthreecoplanarvectorsiszero,whichformsaplaneinspace.Theserelationshipscanbeexpandedusinglinearalgebra[ 102 103 ]togenerateastandardformoftheepipolargeometryasinEquation 3 .Thisnewformindicatesarelationshipbetweentherotationandtranslation,writtenastheessentialmatrixdenotedasQ,totheintrinsicparametersofthecameraandassociatedfeaturepoints.Inthiscase,theequationisderivedforasinglefeaturepointthatiscorrelatedbetweentheframes, 58
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3 withl1andl2representingtheepipolarlinesinimage1andimage2beingproportionaltotheessentialmatrix,respectfully. 3 and 3 arerewrittenintermsofthefundamentalmatrix,F,andareshowninEquations 3 and 3 3 3 whichsolvesfortheentriesoftheessentialmatrix.ThisalgorithmwasdevelopedbyLonguetHiggins[ 39 ]andisdescribedinthissection.TheexpressioninEquation 3 canactuallybeexpressedasinEquation 3 usingadditionalargumentsfromlinearalgebra[ 102 103 ].Thevector,q2R9,containsthestackedcolumnsoftheessentialmatrixQ. 3 ,foreachfeaturepointwheretheentriesoftheessemtialmatrixarestackedinthevectorq.Asetofrowvectorsarestackedtoformamatrix,C,ofnmatchedfeaturepointsand 59
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3 .ThematricxC,showninEquation 3 ,isan9matrixofstackedfeaturepointsmatchedbetweentwoviews. 3 existsusingalinearleastsquaresapproachonlyifthenumberofmatchedfeaturesineachframeisatleast8suchthatrank(C)=8.Additionally,morefeaturepointswillobviouslygeneratemoreconstraintsand,presumably,increaseaccuracyofthesolutionduetotheresidualsoftheleastsquares.Inpractice,theleastsquaressolutiontoEquation 3 willnotexistduetonoise;therefore,aminimizationisusedtondanestimateoftheessentialmatrix,asshowninEquation 3 3 3 ,wherethetranslationTisfounduptoascalingfactor.Thesefoursolutions,whichconsistofallpossiblecombinationsofRandT,arecheckedtoverifywhichcombinationgeneratesapositivedepth 60
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102 103 ].Whenthissituationoccursonemustusetheplanarhomographyapproach,whichisthetopicofthenextsection. 102 103 ].Figure 35 depictsthegeometryinvolvedwithplanarhomography.Thefundamentalrelationshipexpressingapointfeaturein3DspaceacrossasetofimagesisgiventhrougharigidbodytransformationshowninEquation 3 61
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Geometryoftheplanarhomography Ifanassumptionismadethatthefeaturepointsarecontainedonthesameplane,thenanewconstraintinvolvingthenormalvectorcanbeestablished.DenoteN=[n1;n2;n3]Tasthenormalvectoroftheplanecontainingthefeaturepointsrelativetocameraframe1.ThentheprojectionontotheunitnormalisshowninEquation 3 ,whereDistheprojecteddistancetotheplane. 3 intoEquation 3 resultsinEquation 3 62
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3 canbeextendedtoimagecoordinatesthroughEquation 3 3 withtheskewsymmetricmatrixbx2resultsintheplanarhomographyconstraintshowninEquation 3 3 canberewrittentoEquation 3 3 requiresatleastfourfeaturepointcorrespondences.TheseadditionalconstraintscanbestackedtoformanewconstraintmatrixY,asshowninEquation 3 3 intermsofthenewconstraintmatrixresultsinEquation 3 102 103 ],showninEquation 3 fortheunknownscalerl. 63
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3 3 ,thatarepreservedinthehomographymappingandwillfacilitateinthedecompositionprocess. 3 willestablishahomographysolutionexpressedintermsoftheseknownvariables. ThefoursolutionsareshowninTable 31 intermsofthematricesgiveninEquations 3 3 andthecolumnsofthematrixV.Noticethetranslationcomponentisestimateduptoa1 Table31. Solutionsforhomographydecomposition Solution1 64
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34 andassumesthattherotation,R,andtranslation,T,betweenframesisknown.Giventhat,thecoordinatesofh1andh2canbecomputed.Recall,thefundamentalrelationshiprepeatedhereinEquation 3 3 andEquation 3 .Theserelationshipsallowsomecomponentsofhxandhytobewrittenintermsofandnwhichareknownfromtheimages.Thus,theonlyunknownsarethedepthcomponents,h1;zandh2;z,foreachimage.TheresultingsystemcanbecastasEquation 3 andsolvedusingaleastsquaresapproach. 3 usingz=[h2;z;h1;z]asthedesiredvectorofdepths. 3 obtainsthedepthestimatesofafeaturepointrelativetobothcameraframes.Thisinformationalongwiththeimageplanecoordinatescanbeusedtocompute(h1;x;h1;y)and(h2;x;h2;y)bysubstitutingthesevaluesbackintoEquations 3 and 3 .Theresultingcomponentsofh1canthenbeconvertedtothecoordinateframeofthesecondimageanditshouldexactlymatchh2.Thesevalueswillnevermatchperfectlydueto 65
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3 isverysensitivetotheseuncertainties.Chapter 4 willdiscussamethodtoobtainuncertaintyboundsontheSFMestimatesbasedonthesourcesdescribed. 66
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3 .Oncefeaturepointsarelocatedandtrackedacrossimages,anumberstateestimationalgorithms,suchasopticow,epipolarconstraint,andstructurefrommotion,canbeemployed.Althoughcameracalibrationtechniqueshaveproventoprovideaccurateestimatesoftheintrinsicparameters,theprocesscanbecumbersomeandtimeconsumingwhenusingalargequantityoflowqualitycameras.Thischapterdescribesquantitativelytheeffectsonfeaturepointpositionduetouncertaintiesinthecameraintrinsicparametersandhowthesevariationsarepropagatedthroughthestateestimationalgorithms.Thisdeterministicapproachtouncertaintyisanefcientmethodthatdeterminesalevelofboundedvariationsonstateestimatesandcanbeusedforcameracharacterization.Inotherwords,themaximumallowablestatevariationinthesystemwillthendeterminetheaccuracyrequiredinthecameracalibrationstep. 31 .TheresultingvaluesarerepeatedinEquations 4 and 4 asafunctionoffocallength,f,andradialdistortion,d,intermsofthecomponentsofh. 67
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41 ,isdependentonboththerelativepositionsofcameraandthefeature.Thisradialdistance,asshowninFigure 42 ,isalsorelatedviaanonlinearrelationshiptotheradialdistortion.Theanalysisofthefeaturepointswillrequireestimationofthecameraparameters. BFigure41. FeaturePointDependenceonFocalLengthforA)f=0:5andB)f=0:25 BFigure42. FeaturePointDependenceonRadialDistortionforA)d=0:0001andB)d=0:0005 4 ,showstherangeofvaluesthatmustbeconsideredforanominalestimate, 68
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4 presentstherangeofvaluesforradialdistortion. 4 andEquation 4 aresubstitutedintothecameramodelofEquation 4 andEquation 4 .TheresultingexpressionsforfeaturepointsarepresentedinEquations 4 and 4 4 andEquation 4 donotdependonuncertaintysotheseportionsrepresentnominalvalues,oandno,whicharethecorrectlocationsoffeaturepoints.Thesecondtermswhichincludedfandddtermsaretheuncertainty,danddn,ineachfeaturepointwhichareboundedinnormbyDandDn.Assuch,thefeaturepointsmaybewrittenasinEquation 4 andEquation 4 toreecttheuncertainty. 69
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4 andEquation 4 3 andEquation 3 .Inpractice,thevelocitiesarecomputedbysubtractinglocationsofafeaturepointacrossapairofimagestakenatdifferenttimes.Suchanapproachassumesthatafeaturepointcanbetrackedandcorrelatedbetweentheseframes.TheopticowisthengivenasJusingEquation 4 forafeaturepointat1andn1inoneframeand2andn2inanotherframe. 70
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4 andEquation 4 ,canbesubstitutedintoEquation 4 tointroduceuncertainty.TheresultingexpressioninEquation 4 separatestheknownfromunknownelements. 4 wheretheuncertaintyisboundedbyDJ2R. 4 .Theactualboundsonthefeaturepoints,asnotedinEquation 4 andEquation 4 ,variesdependingonthelocationofeachfeaturepointsoboundsofD1andD2aregivenforeachverticalcomponentandDn1andDn2aregivenforeachhorizontalcomponent.Assuch,theboundonvariationisnotedinEquation 4 asspecictotheh1andh2usedtogatherfeaturepointsineachimage. 3 ,requiresapinholecamerawhoseintrinsicparametersareexactlyknown.Suchasituationisobviouslynotrealisticsotheeffectofuncertaintycanbedetermined.Anonidealcamerawilllosethe 71
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4 andEquation 4 ,whichareactuallycausedbyuncertaintyinthecameraparametersasnotedinEquation 4 andEquation 4 .TheconstraintmatrixfromEquation 3 canthenbewrittenasanominalcomponent,Co,plussomeuncertainty,dC,asinEquation 4 4 andEquation 4 .TheithrowofthismatrixcanthenbewrittenasEquation 4 3 ,whenincludingtheuncertaintymatrixinEquation 4 ,willexist;however,thatsolutionwilldifferfromthetruesolutionorthenominalsolution.Essentially,thesolutioncanbeexpressedasthenominalsolution,qo,andanuncertainty,dq,asinEquation 4 .Thisperturbedsystemcannowbesolvedusingalinearleastsquaresapproachfortheentriesoftheessentialmatrix. 4 hasvariationwhichwillbenormboundedbyDqasinEquation 4 whichindicatestheworsecasevariationimposedontheentriesofq. 72
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4 .ThisboundusestherelationshipbetweenuncertaintiesinEquation 4 throughtheconstraintinEquation 4 .Also,thesizeofthisuncertaintydependsonthelocationofeachfeaturepointsotheboundsisnotedasspecictotheh1andh2obtainedfromFigure 34 4 ,canthenbeuseddirectlytocomputethevariationinstateestimates.Theentriesofqarerstarrangedbackintomatrixformtoconstructthenewessentialmatrixthatincludesparametervariations.ThisnewessentialmatrixisthendecomposedusingSVDtechniquesdescribedinSection 3.6.1 3 .SubstitutingEquation 4 andEquation 4 intoEquation 3 resultswithavariationinthesystemmatrixY.Likewise,thenewsystemmatrixwithuncertainintrinsicparamterscanbewrittenasanominalmatrix,Y0plussomevariation,dY,asshowninEquation 4 4 andEquation 4 .correspondingly, 73
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4 3 ,whenincludingtheuncertainmatrixinEquation 4 ,willexist;however,thatsolutionwilldifferfromthetruesolution.Essentially,thesolutioncanbeexpressedasthenominalsolution,ho,andanuncertainty,dh,asinEquation 4 4 hasvariationwhichwillbenormboundedbyDhasinEquation 4 4 .ThisboundusestherelationshipbetweenuncertaintiesinEquation 4 throughtheconstraintinEquation 4 .Also,thesizeofthisuncertaintydependsonthelocationofeachfeaturepointsotheboundsisnotedasspecictotheh1andh2obtainedfromFigure 34 4 ,canthenbeuseddirectlytocomputethevariationinstateestimates.Theentriesofharerstarrangedbackintomatrixformtoconstructthenewhomographymatrixthatincludesparameter 74
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3.6.3 3 forthestructurefrommotionrelationship.Assuch,thematrixshouldbewrittenintermsofanominalvalue,Ao,andanuncertainperturbation,dA,asinEq. 4 4 andEquation 4 intoEquation 3 .TheperturbationisthenwrittenasEquation 4 3 whenconsideringEquation 4 willobviouslyresultinadepthestimatethatdiffersfromthecorrectvalue.Denezoastheactualdepthsthatwouldbecomputedusingtheknownparametersofthenominalcameraanddzasthecorrespondingerrorintheactualsolution.TheleastsquaresproblemcanthenbewrittenasEquation 4 andsolvedusingapseudoinverseapproach. 4 .Thisrangeofsolutionswillliewithintheboundedrangedeterminedfromtheworstcasebound. 75
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4 .Thisboundnotesthattheboundonvariationsinfeaturepoints,andultimatelytheboundonsolutionstostructurefrommotion,dependsonthelocationofthosefeaturepoints. 76
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3 toderivethesystemequations. 51 alongwiththerespectiveorigins.Thebodyxedcoordinatesystemhastheoriginlocatedatthecenterofgravityoftheaircraft.Theaxesareorientedsuchthatb1alignsoutthenoseandb2alignsouttherightwingwithb3pointedoutthebottom.Themovementoftheaircraft,whichincludesaccelerating,willobviouslyaffectthecoordinatesystem;consequently,thebodyxedcoordinatesystemisnotaninertialreferenceframe. 77
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Bodyxedcoordinateframe Theorientationanglesoftheaircraftareofparticularinterestformodelingavisionbasedsensor.Therollangle,f,describesrotationaboutb1,thepitchangle,q,describesrotationaboutb2andtheyawangle,y,describesrotationaboutb3.ThetransformationfromavectorrepresentedintheEarthxedcoordinatesystemtothebodyxedcoordinatesystemisrequiredtorelateonboardmeasurementstoinertialmeasurements.Thistransformation,giveninEquation 5 ,usesREBwhichareEulerrotationsofroll,pitchandyaw[ 29 108 ], 5 .Theorderofthismatrixmultiplicationneedstobemaintainforcorrectcomputation. 78
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5 5 torepresenttheserates. 52 ,usethetraditionalchoiceofi3aligningthroughthecenterofviewofthecamera.Theremainingaxesareusuallychosenwithi2alignedrightoftheviewandi1alignedoutthetopalthoughsomevariationinthesechoicesisallowedaslongastheresultingaxesretaintherighthandedproperties.Thedirectionofthecamerabasisvectorsaredenedthroughthecamera'sorientationrelativetothebodyxedframe.Thisframeworkisnotedasthecameraxedcoordinatesystembecausetheoriginisalwayslocatedataxedpointonthecameraandmovesinthesamemotionasthecamera.Thecameraisallowedtomovealongtheaircraftthroughadynamicmountingwhichadmitsbothrotationandtranslation.Thisfunctionalityenablesthetrackingoffeatureswhilethevehiclemovesthroughanenvironment.Theoriginofthecameraxedcoordinatesystemisattachedtothismovingcamera;consequently,thecameraxedframeisnotaninertialreference.A6degreeoffreedommodelofthecameraisassumedwhichadmitsafullrangeofmotion.Figure 52 alsoillustratesthecamera'ssensingconewhichdescribesboththeimageplaneandtheeldofviewconstraint. 79
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Cameraxedcoordinateframe Similartothebodyxedcoordinateframe,atransformationcanbedenedforthemappingbetweenthebodyxedframe,Bandthecameraframe,IasseeninEquation 5 5 ,similartothebodyxedrotationmatrix.Theorientationanglesofthecameraarerequiredtodeterminetheimagingusedforvisionbasedfeedback.Therollangle,fc,describesrotationabouti3,thepitchangle,qc,describesrotationabouti2andtheyawangle,yc,describesrotationabouti1. 5 willtransformavectorinbodyxedcoordinatestocameraxedcoordinates.Thistransformationisrequiredtorelatecamerameasurementstoonboardvehiclemeasurementsfrominertialsensors.Thematrixagaindependsontheangular 80
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5 torepresenttheseangles. 53 ,thusrelatesthecameraandtheaircrafttothefeaturepointalongwithsomeinertialorigin. Figure53. Scenarioforvisionbasedfeedback 81
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5 andEquation 5 aretypicallyrepresentedintheinertialreferenceframerelativetotheEarthaxisorigin. 5 ,istypicallygivenwithrespecttothebodyaxisorigin.Thischoiceofcoordinatesystemsreectsthatthecameraisintrinsicallyaffectedbyanyaircraftmotion. 3 todescribetherelativepositionbetweenthecameraandthefeaturepoint.Recall,thisvectorwasgiveninthecameraxedcoordinatesystemtonotetheresultingimageisdirectlyrelatedtopropertiesrelativetothecamera.TherepresentationofhisrepeatedhereinEquation 5 forcompleteness. 5 isused.Thisexpressionincorporatesthetranslationsinvolvedwiththeoriginsofeachcoordinateframethroughaseriesofsingleaxisrotationsuntilthecorrectframeisreached. 82
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108 110 ]andarerepeatedinEquation 5 to 5 foroverallcompleteness.Fxmgsinq=m(u+qwrv) 5 .Theaircraftstatesofinterestforthecameramotionsystemconsistofthepositionandvelocityoftheaircraft'scenterofmass,TEBandvb,theangularvelocity,w,andtheorientation 83
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5 .AsstatedinEquation 5 ,theaircraft'svelocityisexpressedinthebodyxedcoordinateframe.Eachoftheseparameterswillappearexplicitlyintheaircraftcameraequations. 84
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53 forafeaturepointrelativetotheinertialframe.Therefore,thevectorsumcanbeusedtosolvefortherelativepositionbetweenthecameraanda3Dfeaturepoint.AftermakingthepropercoordinatetransformationsbyusingEquations 5 and 5 ,thisrelativepositioncanbeexpressedincameraframe,I,asshowninEquation 5 5 intoEquations 3 and 3 animagecanbeconstructedasafunctionofaircraftstates.Themajorassumptionoftheseequationsispriorknowledgeofthefeaturepointlocationrelativetotheinertialframe,whichmaybeprovidedbyGPSmaps.Furthermore,theimageresultsobtainedcanalsobepassedthroughEquations 3 and 3 toaddtheeffectsofradialdistortion.Thedistortedimagewillprovideamoreaccuratedescriptionofanimageseenbyaphysicalcamera,assumingtheintrinsicparametersofthecameraareknown. 5 withrespecttotheinertialframe,asshowninEquation 5 dt(h)=Ed dt(x)Ed dt(TEB)Ed dt(TBI)(5) 85
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5 cannowberewrittentoEquation 5 foranonstationaryfeaturepoint. dt(h)=xTEBBd dt(TBI)wTBIEwIh(5)ThisequationcanbereducedfurtherifthecamerasareconstrainedtohavenotranslationrelativetotheaircraftsoBd dt(TEI)=0.Alternatively,thistermisretainedinthederivationtoallowthisdegreeoffreedominthecamerasetup.Theangularvelocity,EwI,canbefurtherdecomposedusingtheAdditionTheorem.ThenalstepimplementsEquations 5 and 5 totransformeachtermintothecameraframe.Aftersomemanipulation,theexpressionforthevelocityofafeaturepointrelativetothecameraframeresultsinEquation 5 5 and 5 intoEquations 3 and 3 .Thisresultwillprovideadescriptionoftheopticalowforeachfeaturepointformedbyeitherthecameratravelingthroughtheenvironmentorthemotionofthefeaturepointsthemselves.ToincorporateradialdistortioneffectsintotheopticowcomputationrequirestheadditionalsubstitutionintoEquations 3 and 3 86
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5 5 3 3 3 ,and 3 .Arrangingtheparametersforthekthcameraintoasinglevector,asshowninEquation 5 ,resultsthenintheformulationofagenericaircraftcamerasystemwithkcamerasallhavingindependentmotionthattracknfeaturepointsisobtained. 5 .ThisvectorcanbeextendedtoincludeothercamerafeaturessuchasCCDarraymisalignment,skewness,etc.ThefocalplanepositionscanthenbeassembledintoavectorofobservationsasshowninEquation 5 ,wherennumberoffeaturepointsareobtained.Likewise,thestatesoftheaircraftcanbecollectedandrepresentedasastatevectorasshowninEquation 5 .Inaddition,theinitialstatesofthevehiclearedenedasX0. 5 and 5 .TheobservationsusedinthisdissertationconsistofmeasureableimagesshowninEquations 3 and 3 whichcapturenonlinearitiessuchasradialdistortion.Thissystem,whichmeasuresimageplaneposition,isdescribedmathematically 87
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5 5 ,thenadifferentsetofequationscanbeobtainedwhichwillbereferredtoastheOpticFlowFormofthegoverningaircraftcameraequationsofmotion.ThissystemisgiveninEquation 5 ,whichusestheopticowexpressiongiveninEquations 3 and 3 astheobservations.X(t)=F(X(t);U(t);a(t);t) 88
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3 andhowthesefeaturesrelatetothestatesofanaircraftinChapter 5 ,theeffectsofindependentlymovingobjectsneedtobehandledinadifferentmanner.Forcasesinvolvingastationarycamera,suchasinsurveillanceapplications,simplelteringandimagedifferencingtechniquesareemployedtodetermineindependentlymovingobjects.Althoughthesetechniquesworkwellforstationarycameras,adirectimplementationtomovingcameraswillnotsufce.Foramovingcamera,theapparentimagemotioniscausedbyanumberofsources,suchascamerainducedmotion(i.e.egomotion)andthemotionduetoindependentlymovingobjects.Acommonapproachtodetectingmovingobjectconsidersatwostageprocessthatincludes(i)acompensationroutinetoaccountforcameramotionand(ii)aclassicationschemetodetectindependentlymovingobjects. 3.6 .Thesecondapproachusesthesmoothnessconstraintinattempttominimizethesumofsquaredifferences(SSD)overeitheraselectnumberoffeaturesortheentireoweld.Thisapproachassumesthestationary 90
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3.6 .Theepipolarconstraintcanbeusedtorelatefeaturepointsacrossimagesequencesthrougharigidbodytransformation.TheepipolarlinesofastaticenvironmentarecomputedusingEquation 3 orEquation 3 dependingiftheessentialmatrixorthefundamentalmatrixisrequired.AnillustrationofthecomputedepipolarlinesisdepictedinFigure 61 forastaticenvironmentobservedbyamovingcamera.Noticeforthisstaticcase,thefeaturepointsinthesecondimage(therightimagecontainingtheoverlaidepipolarlines)areshowntoliedirectlyontheepipolarlines. BFigure61. EpipolarLinesAcrossTwoImageFrames:A)InitialFeaturePointsandB)FinalFeaturePointswithOverlayedEpipolarLines Oncecameramotionestimationhasbeenfound,theepipolarlinescanbeusedasanindicationofmovingobjectsintheimage.Forinstance,thefeaturepointscorrespondingtothestationarybackgroundwilllieontheepipolarlineswhilethefeaturepointscorrespondingtomovingobjectswillviolatethisconstraint.Similarly,thecomputationofopticalowcanalsobeusedfordetectingindependentlymovingobjects.Incomputingtheopticalow,themotioninducedbythecameraalongwithmovingobjectsisfusedtogetherinthemeasuredimage.Recall,theopticowexpressions 91
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3 and 3 orEquations 3 and 3 forradialdistortion.Decomposingtheopticalowintoitscomponentsofcamerarotation(r;nr)andtranslation(t;nt)andindependentlymovingobjects(i;ni)facilitatesthedetectionproblem.Therefore,thecomponentsoftheopticalowcanbewrittenasinEquation 6 5 :thetranslationalvelocity[u;v;w]Tandtheangularvelocity[p;q;r]Tofthecamera.TheresultingexpressionsareshowninEquations 6 and 6 andappliesonlytofeaturesstationaryintheenvironment.ThedetailsdescribingthesubstitutionofthecameramotionstatesaredescribedinChapter 5 hz1 111 ]thattherotationalstates[p;q;r]Tcanbeestimatedaccuratelyforastaticenvironmentthroughanonlinearminimizationprocedurefornfeatureswheren6.TheapproachusedavectorvaluedoweldJ(x)andisgiveninEquation 6 92
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6 iscomposedofunknownvehiclestatesanddepthparameters. 6 thatminimizesthemagnitudeofthecostfunction. 2kJ(x)k2(6)Thesameapproachistakenherewithcaution.Recallthatthemeasuredopticalowalsocontainsmotionduetoindependentlymovingobjectsinadditiontotheinducedopticalowcausedbythecameramotion.Ingeneral,thesevariationsinthemeasuredopticalowwillintroduceerrorintothe[p;q;r]Testimates.Ifsomeassumptionsaremaderegardingtherelativeopticalowbetweenthestaticenvironmentandmovingobjects,thenerrorsinthestateestimatescanhaveminimaleffect.Forinstance,ifthestaticportionofthesceneisassumedtobethedominantmotionintheopticalowthentheestimateswillcontainminimalerrors.Employingthisassumption,estimatesfortheangularvelocities[p;q;r]Tofthecamera/vehicleareobtained.SubstitutingtheseestimatesintoEquation 6 resultsinestimatesfortherotationalportionoftheopticalow,asshowninEquation 6 6 .TheresidualopticalowRes;nRescontainsonlythecomponentsofthecameratranslationandindependentlymovingobjects.Fromthisexpression,constraintscanbeemployedonthecamerainducedmotiontodetectindependentlymovingobjects. 93
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62 .Consequently,featurepointsthatviolatethisconditioncanbeclassiedasindependentlymovingobjects.Thischaracteristicobservedfromstaticfeatureswillbethebasisfortheclassicationscheme. Figure62. FOEconstraintontranslationalopticowforstaticfeaturepoints TheresidualopticalowmaycontainindependentlymovingobjectswithintheenvironmentthatradiatefromtheirownFOE.AnexampleofasimplescenarioisillustratedinFigure 63 forasinglemovingobjectontheleftandasimulationwithsyntheticdataoftwomovingvehiclesontheright.NoticethetwoprobableFOEsinpictureontheleft,onepertainingtothestaticenvironmentandtheotherdescribingthemovingobject.Inaddition,theepipolarlinesofthetwodistinctFOEsintersectatdiscretepointsintheimage.Thesepropertiesofmovingobjectsarealsoveriedinthesyntheticdatashownintheplotontheright.Thus,aclassicationschememustbedesignedtohandlethesescenariostodetectindependentlymovingobjects.Thenext 94
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Residualopticowfordynamicenvironments 6 .AnapproximationforthepotentiallocationoftheFOEisfoundbyextendingthetranslationalopticalowvectorstoformtheepipolarlines,asillustratedinFigure 63 ,andobtainingallpossiblepointsofintersection.Asmentionedpreviously,theintersectionpointsobtainedwillconstituteanumberofpotentialFOEs;however,onlyonewilldescribethestaticbackgroundwhiletherestareduetomovingobjects.TheapproachconsideredforthisclassicationthatessentiallygroupstheintersectiondatatogetherthroughadistancecriterionisaniterativeleastsquaressolutionforthepotentialFOEs.Theiterationproceduretestsallintersectionpointsasadditionalfeaturesareintroducedtothesystemofequationseachofwhichinvolves2unknownimageplanecoordinatesoftheFOEfoei;nfoei.Theprocessstartsbyconsidering2featurepointsandtheirFOEintersection 95
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6 fortheFOEcoordinatesfoe1;nfoe1(fortherstiterationaleastsquaressolutionisnotnecessarybecausetwolinesintersectatasinglepoint). 2kM264n375bk2(6) where 6 fortheithiteration.Mathematically,theclassicationschemefortheithiterationisgiveninEquations 6 and 6 foeifoei12+nfoeinfoei12(6) 96
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3 .Amongthetechniquesthatutilizefeaturepoints,theapproachrelatedtothispaperinvolvesepipolargeometry[ 39 112 ].Thepurposeofthistechniqueistoestimaterelativemotionbasedonasetofpixellocations.Thisrelativemotioncandescribeeithermotionofthecamerabetweentwoimagesortherelativedistanceoftwoobjectsofthesamesizefromasingleimage.The3DscenereconstructionofamovingtargetcanbedeterminedfromtheepipolargeometrythroughthehomographyapproachdescribedinChapter 3 .Forthecasedescribedinthischapter,amovingcameraattachedtoavehicleobservesaknownmovingreferenceobjectalongwithanunknownmovingtargetobject.Thegoalistoemployahomographyvisionbasedapproachtoestimatetherelativeposeandtranslationbetweenthetwoobjects.Therefore,acombinationofvisionandtraditionalsensorssuchasaglobalpositioningsystem(GPS)andaninertialmeasurementunit(IMU)arerequiredtofacilitatethisproblemforasinglecameraconguration.ForexampleintheAARcase,GPSandIMUmeasurementsareavailableforboththereceiverandtankeraircrafts.Ingeneral,asinglemovingcameraaloneisunabletoreconstructthe3Dscenecontainingmovingobjects.Thisrestrictionisduetothelossoftheepipolarconstraint,wheretheplaneformedbythepositionvectorsrelativetotwocamerapositionsintimetoapointofinterestandthetranslationvectorisnolongervalid.Techniqueshavebeenformulatedtoreconstructmovingobjectsviewedbyamovingcamerawithvariousconstraints[ 35 113 116 ].Forinstance,ahomographybasedmethodthatsegmentsbackgroundfrommovingobjectsandreconstructsthetarget'smotionhasbeenachieved[ 117 ].Theirreconstructionisdonebycomputingavirtualcamerawhichxesthetarget'spositionintheimageanddecomposesthehomographysolutionintomotionofthecameraandmotioncausedbythetarget.Thisdecompositionisdoneusingaplanartranslationconstraintwhichrestrictsthetarget'smotiontoagroundplane.Similarly,Han 99
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115 ]proposedanalgorithmthatreconstructs3Dmotionofamovingobjectusingafactorizationbasedalgorithmwiththeassumptionthattheobjectmoveslinearlywithconstantspeeds.Anonlinearlteringmethodwasusedtosolvetheprocessmodelwhichinvolvedboththekinematicsandtheimagesequencesofthetarget[ 118 ].Thistechniquerequiresknowledgeoftheheightabovethetargetwhichwasdonebyassumingthetargettraveledonthegroundplane.Thisassumptionallowedothersensors,suchasGPS,toprovidethisinformation.ThepreviousworkofMehtaetal.[ 77 ]showedthatamovingmonocularcamerasystemcouldestimatetheEuclideanhomographiesforamovingtargetinreferencetoaknownstationaryobject.ThecontributionofthischapteristocasttheformulationshowninMehtaelal.toamoregeneralproblemwherebothtargetandreferencevehicleshavegeneralmotionandarenotrestrictedtoplanartranslations.Thisproposedapproachincorporatesaknownreferencemotionintothehomographyestimationthroughatransformation.EstimatesoftherelativemotionbetweenthetargetandreferencevehiclearecomputedandrelatedbackthroughknowntransformationstotheUAV.RelatingthisinformationwithknownmeasurementsfromGPSandIMU,thereconstructionofthetarget'smotioncanbeachievedregardlessofitsdynamics;however,thetargetmustremainintheimageatalltimes.Althoughtheformulationcanbegeneralizedforncameraswithindependentposition,orientation,translations,androtationthischapterdescribesthederivationofasinglecamerasetup.Meanwhile,cuesonboththetargetandreferenceobjectsareachievedthroughLEDlightsormarkersplacedinaknowngeometricpatternofthesamesize.Thesemarkersfacilitatethefeaturedetectionandtrackingprocessbyplacingknownfeaturesthatstandoutfromthesurroundingswhilethegeometryandsizeofthepatternallowsforthecomputationoftheunknownscalefactorthatiscustomarytoepiploarandhomographybasedapproaches.ThischapterbuildsonthetheorydevelopedinChapters 3 and 5 whilerelyingonthemovingobjectdetectionalgorithmtoisolatemovingobjectswithinanimage.RecalltheowoftheoverallblockdiagramshowninFigure 16 .Theprocessstartedbycomputingfeaturesinthe 100
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6 .Oncemovingobjectsintheimagearedetected,thehomographyestimationalgorithmproposedinthischapterisimplementedfortargetstateestimation. 7.2.1SystemDescriptionThesystemdescribedinthispaperconsistsofthreeindependentlymovingvehiclesorobjectscontaining6DOFmotion.TodescribethemotionofthesevehiclesaEuclideanspaceisdenedwithveorthonormalcoordinateframes.TherstframeisanEarthxedinertialframe,denotedasE,whichrepresentstheglobalcoordinateframe.Theremainingfourcoordinateframesaremovingframesattachedtothevehicles.Therstvehiclecontainstwocoordinateframes,denotedasBandI,torepresentthevehicle'sbodyframeandcameraframe,asdescribedinChapter 5 inFigure 51 .ThisvehicleisreferredtoasthechasevehicleandisinstrumentedwithanonboardcameraandGPS/IMUsensorsforpositionandorientation.Thesecondvehicle,denotedasF,isconsideredareferencevehiclethatalsocontainsGPS/IMUsensorsandprovidesitsstatestothechasevehiclethroughacommunicationlink.Lastly,thethirdvehicle,denotedasT,isthetargetvehicleofinterestinwhichunknownstateinformationistobeestimated.Inaddition,actitiouscoordinateframewillbeusedtofacilitatetheestimationprocessandisdenedasthevirtualcoordinatesystem,V.Thecoordinatesofthissystemarerelatedthroughtransformationscontainingbothrotationalandtranslationalcomponents.TherotationalcomponentisestablishedusingasequenceofEulerrotationsintermsoftheorientationanglestomaponeframeintoanother.LettherelativerotationmatricesREB,RBI,REF,REV,RIV,RFV,RTVandRETdenotetherotationfromEtoB,BtoI,EtoF,EtoV,ItoV,FtoV,TtoV,andEtoT.Secondly,thetranslationsaredenedasTEB,xF,xV,xT,xF;n,xT;n,TBI,xIV,mIF,mIT,hF;n,hT;n,mVF,mVT,hVF;n,andhVT;nwhichdenotetherespectivetranslationsfromEtoB,EtoF,EtoV,EtoT,EtothenthfeaturepointonthereferencevehicleandtargetvehiclesallexpressedinE,BtoIexpressedinB,ItoV,ItoF,ItoT,ItothenthfeaturepointonthereferenceandtargetvehiclesexpressedinI,VtoF,V
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71 foracameraonboardaUAVwhilethevectorsrelatingthefeaturepointstoboththerealandvirtualcamerasaredepictedinFigure 72 .TheestimatedquantitiescomputedfromthevisionalgorithmaredenedasRTBandxTBwhicharetherelativerotationandtranslationfromTtoBexpressedinB. Figure71. Systemvectordescription Thecameraismodeledthroughatransformationthatmaps3dimensionalfeaturepointsontoa2dimensionalimageplaneasdescribedinChapter 3 .Thistransformationisageometricrelationshipbetweenthecamerapropertiesandthepositionofafeaturepoint.Theimageplanecoordinatesarecomputedbasedonatangentrelationshipfromthecomponentsofhn.ThecamerarelationshipusedinthischapterisreferredtoasthecontinuouspinholecameramodelandisgiveninEquations 3 and 3 forazerolensoffset,wherefisthefocallengthofthecameraandhx;n,hy;n,hz;narethe(x;y;z)componentsofthenthfeaturepoint.Thispinholemodelisacontinuousmappingthatcanbefurtherextendedtocharacterizepropertiesofaphysicalcamera.Somecommonadditionstothismodelincludeskewness,radial 102
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BFigure72. MovingtargetvectordescriptionreltivetoA)cameraIandB)virtualcameraV 3 .Eachextensiontothemodeladdsanotherparametertoknowfortheestimationproblemandeachcanintroduceuncertaintyandlargeerrorsintheestimationresult.Therefore,thischapterwillonlyconsidertheeldofviewconstraintandleavethenonlineartermsandtheeffectsonestimationforfuturework.RecalltheeldofviewconstraintsgiveninChapter 3 .Theseconstraintscanberepresentedaslowerandupperboundsintheimageplaneandaredependentonthehalfangles(gh;gv)whichareuniquetoeachcamera.Mathematically,theseboundsareshowninEquations 7 and 7 forthehorizontalandverticaldirections. 103
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102 ].Thesameconstraintholdsfortheimagecoordinatesaswellbutalsointroducesanunknownscalefactor.Employingthisconstraint,estimatesofrelativemotioncanbeacquiredforbothcamerainhandandxedcameracongurations.Thisdissertationdealswiththecamerainhandcongurationwhileassumingaperfectfeaturepointdetectionandtrackingalgorithm.Thisassumptionenablestheperformanceofthevisionbasedstateestimationtobetestedbeforeintroducingmeasurementerrorsandnoise.Thehomographyconstraintrequiresafewassumptionsbasedonthequantityandthestructureofthefeaturepoints.Thealgorithmrstrequiresaminimumoffourplanarfeaturepointstodescribeeachvehicle.Thisrequirementenablesauniquesolutiontothehomographyequationbasedonthenumberofunknownquantities.ThereferencevehiclewillhaveaminimumoffourpixelvaluesineachimagewhichwillbedenedaspF;n=[F;n;nF;n]8nfeaturepoints.Likewise,thetargetvehiclewillhavefourpixelvaluesandwillbedenedaspT;n=[T;n;nT;n]8nfeaturepoints.Thisarrayoffeaturepointpositionsarecomputedat30Hzwhichistypicalforstandardcamerasandtheframecountisdenotedbyi.Thenalrequirementisaknowndistanceforboththereferenceandtargetvehicle.OnedistancerepresentsthepositionvectortoafeatureonthereferencevehicleinEuclideanspacerelativetothelocalframeFandtheseconddistancerepresentsthepositionvectortoafeatureonthetargetvehicleinEuclideanspacerelativetothelocalframeT.Inaddition,thelengthofthesevectorsalsomustbeequalwhichallowstheunknownscalefactortobedetermined.ThevectordescribingthereferencefeaturepointwillbedenotedassFexpressedinF,whilethevectordescribingthetargetfeaturepointisreferredtoassTexpressedinT.ThesefeaturepointpositionvectorsarealsoillustratedinFigure 72 .Thefeaturepointsarerstrepresentedbypositionvectorsrelativetothecameraframe,I.TheexpressionsforboththereferenceandtargetfeaturepointsaregiveninEquations 7 and 7 .ThesevectorcomponentsarethenusedtocomputetheimagecoordinatesgiveninEquations 7 and 7 .ThecomputationinEquation 7 requiresinformationregardingthetargetwhichisdonesolelytoproduceimagemeasurementsthatnormallywouldbeobtainedfromthesensor.Remainingcomputations,regardingthehomography,willonlyusesensor 104
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7 and 7 arethetruerotationsmatricesfromFtoBandTtoB,repectfully,andareshowninEquations 7 and 7 77 ].Inthiscase,boththereferenceandtargetvehiclesareinmotionandarebeingviewedbyamovingcamera.Therefore,thenextstepistotransformthecameratoavirtualcongurationthatobservesthereferencevehiclemotionlessintheimageovertwoframes.Inotherwords,thisapproachcomputesaEuclideantransformationthatmapsthecamera'sstatesati1toavirtualcamerathatmaintainstherelativepositionandorientationbetweenframestoxthefeaturepointsofthereferencevehicle.Thistransformationisdonebymakinguseofthepreviousimageframeandstateinformationati1fromboththecameraandthereferencevehicle.Afterthevirtualcameraisestablishedthehomographyequationscanbeemployedforstateestimation.TocomputethelocationandposeofthevirtualcameraatitherelativepositionandorientationfromItoFati1isrequired.ThisrelativemotioniscomputedthroughknownmeasurementsfromGPS/IMUandtheexpressionsareshowninEquations 7 and 7 fortranslationandrotationati1,respectfully. 105
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7 and 7 forthecurrentframei. 3 and 3 .TheexpressionsforthenewvectorshVF;nandhVT;nintermsofthevirtualcameraaregiveninEquations 7 and 7 forthereferenceandtargetvehicles. 7 and 7 areonewaytocomputeimagecoordinatesforthevirtualcamera,butthereareunknowntermsinEquation 7 thataren'tmeasurableorcomputeableinthiscase.Therefore,analternativemethodmustbeusedtocomputeimagevaluesofthetargetinthevirtualcamera.Usingthepositionandorientationofthevirtualcamera,asgiveninEquations 7 and 7 ,therelativemotioniscomputedfromcameraItocameraVwhileusingepipolargeometrytocomputethenewpixellocations.ThisrelativecameramotionisgiveninEquations 7 and 7 wherethetranslationisexpressedinI. 106
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7 7 whichreliesontherelativemotionremainingconstanttomaintainthereferencestationaryintheimage. 7 .Likewise,thetimevaryingpositionofafeaturepointonthetargetvehicleexpressedinVisgiveninEquation 7 7 and 7 andarerelativetothevirtualcameraframe. 107
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7 7 and 7 whichdescribetherelativemotionbetweenthereferenceandtargetobjects. 7 andthereferencevehiclelocationisknownthroughGPSalongwiththefeaturepointlocations;therefore,aprojecteddistancecanbecomputedthatscalesthedepthofthescene.Tocomputethisdistancethenormalvector,n,thatdenestheplanewhichthereferencefeaturepointslieisrequiredandcanbecomputedfromknowninformation.Ultimately,theprojectivedistancecanbeobtainedandisdenedinEquation 7 throughtheuseofthereferenceposition. 7 intoEquation 7 resultsinanintermediateexpressionfortheEuclideanhomographyandisshowninEquation 7 DnThVF;n(7)Tofacilitatethesubsequentdevelopment,thenormalizedEuclideancoordinatesareusedanddenedinEquations 7 and 7 108
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7 7 ,and 7 thenormalizedEuclideanhomographyisestablishedwhichrelatesthetranslationandrotationbetweencoordinateframesFandT.ThishomographyexpressionisshowninEquation 7 intermsofthenormalizedEuclideancoordinates. {z }hVF;naH(7)InEquation 7 7 D(7)TheEuclideanhomographycannowbeexpressedintermsofimagecoordinatesorpixelvaluesthroughtheidealpinholecameramodelgiveninEquations 3 and 3 .Thisexpressingisdonebyrstrewritingthecameramodelintomatrixformwhichisreferredtoasthecameracalibrationmatrix,K.SubstitutingthecameramappingintoEquation 7 andusingthecameracalibrationmatrix,K,thehomographyintermsofpixelcoordinatesisobtainedandgiveninEquation 7 .ThisnalexpressionrelatestherotationandtranslationofthetwovehiclesFandTintermsoftheirimagescoordinates.Therefore,toobtainasolutionfromthishomographyexpressionbothvehiclesneedtobeviewableintheimageframe. {z }pVF;nG(7)ThematrixG(t)isdenotedasaprojectivehomographyinEquation 7 whichareasetofequationsthatcanbesolveduptoascalefactorusingalinearleastsquaresapproach.OncethecomponentsofhomographymatrixareestimatedthematrixneedstobedecomposedintotranslationalandrotationalcomponentstoobtainxhandR.Thisdecompositionisaccomplishedusingtechniquessuchassingularvaluedecompositionandgeneratesfourpossiblesolutions[ 119 120 ].Todetermineauniquesolutionsomephysicalcharacteristicsoftheproblem 109
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7 7 toobtainx.Secondly,xisthendividedbyatoscalethedepthratioresultinginthenalxexpressedinI.ThisresultinconjunctionwithRisthenusedinEquation 7 tosolveformVT.ThenextstepistocomputetherelativetranslationfromItoVwhichisgiveninEquation 7 7 7 and 7 representtherelativemotionbetweenthecameravehicleandthetargetvehicle.Thisinformationisvaluableforthecontroltasksdescribedearlierinvolvingbothtrackingandhomingapplications.Thenextsectionwillimplementthisalgorithminsimulationtoverifythestateestimatorforthenoisefreecase. 110
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121 ]thatemploysaHiddenMarkovModeltopredictthemotionofmovingobjects.ThebenetsinusingaHiddenMarkovModelincludeatimedependenceframeworkincorporatedintotheprobabilisticmodelaswellastheabilitytohandlestochasticprocesses.TheunderliningconceptofaHiddenMarkovModeldescribestheprobabilityofaprocesstosequentiallygofromonestatetoanother.Thissequentialpropertyprovidesthenecessaryframeworkfortimedependencemodeling,whichisanattractiveapproachfortheapplicationsconsidered,wherethetimehistorydataisacriticalpieceofinformationincludedinthemodeling. 8 .Therefore,thevelocityandpositionareupdatedthroughEquations 8 and 8 .Althoughthismodelislimited,itdescribesafoundationformodelingtargetmotionandcoversthebasicmodelconstantvelocity. 112
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8 andischaracterizedbyarandomvector,w(t)andisscaledbyaconstant,r.ThevelocitycorrespondingtothisaccelerationisdescribedinEquation 8 .Thismodelattemptstocapturethestochasticbehaviorsbyutilizingaprobabilisticdistributionfunction. 8 canbemodiedtoincorporatesomedependenceonthepreviousaccelerationvalue.ThisdependenceisachievedbyweightingthepreviousaccelerationinthemodelandisshowninEquation 8 .ThebenettothistypeofmodelasopposetoEquation 8 requiressomeknowledgeofthetarget;namely,thatthetargetcannotachievelargeabruptchanginginacceleration.TheresultingvelocityexpressionforthismodelisgiveninEquation 8 8 fortheithtargetandNimageframes.ThevelocityproleiscomputedusingabackwardsdifferencemethodandisgiveninEquation 8 8 ,isobtainedfromthevelocityprolegiveninEquation 8 .Thesamebackwardsdifferencemethodisusedtocomputethis 113
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8 .Thisaccelerationtimehistoryiscomputedimplicitlythroughthepositionestimatesobtainedfromthehomographyalgorithm 8 and 8 providetheinitialmotionstatedescriptionthatpropagatestheMarkovtransitionprobabilityfunction.TheformoftheMarkovtransitionprobabilityfunctionisassumedtobeaGaussiandensityfunctionthatonlyrequirestwoparametersforitsrepresentation.TheparametersneededforthisfunctionincludethemeanandvariancevectorsfortheaccelerationprolegiveninEquation 8 .Note,duringthischapter(x)isthemeanoperatorandnottheverticalcomponentintheimageplane.Likewise,s2(x)isreferredtoasthevarianceoperator. 8 ,wheretheargumentsconsistofthemeanandvariancepertainingtotheestimatedacceleration. 8 and 8 forthetransitionfunction.Thefunctionsfandfsarechosenbasedonthedesiredweightingofthetimehistoryandcansimplybeaweightedlinearcombinationofthearguments.Theseinitialstatisticalparametersareusedinthepredictionstepandupdatedonceanewmeasurementisobtained.
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8 asthreedimensionalGaussiandensityfunctionandisuniquelydeterminedbythemeanandvariance. 2(ai(t)(ai(t)))2 8 8 and 8 ,thepredictiveprobabilityforobjectiattimet+kisgivenasEquation 8 .Thisframeworkenablestheexibilityofcomputingthepredictedestimatesatanydesiredtimeinthefuturewiththenotionthatfurtheroutintimetheprobabilitydiminishes. 8 and 8 fortheentiretimeinterval. 8 and 8
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8 and 8 2ai(t1) 2(ai(t1)) 2ai(t1) 2s2(ai(t1))Lastly,theprobabilityfunctionsforvelocityandpositionareusedtocomputethepredictiveprobabilitiesforobjectithataregiveninEquations 8 and 8 forvelocityandposition,respectfully. 8 istheprobabilitythattargetiislocatedinpositionp(x;y;z).Sothetheoverallprocessisaniterativemethodthatusesthemotionmodels,giveninSection 8.2.1 ,toprovideguessesforpositionandvelocityinattempttomaximizetheprobabilityfunctionsgiveninEquations 8 and 8 .ThepositionthatmaximizesEquation 8 isthemostlikelylocationofthetargetatt+kwithaknownprobability. 116
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7 .Effectively,thesequantitiesaretheerrorsignalsusedforcontroltotrackthemovingcameratowardadesiredlocationbasedonthemotionofthetarget.TheframeworkpresentedherewilluseaircraftandUAVnavigationschemesfortheaerialmissionsdescribedinChapter 1 .Therefore,thecontroldesigndescribedinthischapterfocusesonthehomingmissiontofacilitatetheAARproblem,whichinvolvestrackingthepositionstatescomputedfromthehomography.Varioustypesofguidancecontrollerscanbeimplementedforthesetypesoftaskoncetherelativepositionandorientationareknown.Dependingonthecontrolobjectivesandhowfastthedynamicsofthemovingtargetare,lowpasslteringoralowgaincontrollermayberequiredtoavoidhighratecommandstotheaircraft.IntheAARproblem,thesuccessofthedockingcontrollerwilldirectlyrelyonseveralcomponents.TherstcomponentistheaccuracyofestimatedtargetlocationwhichduringAARneedstoprecise.Secondly,thedynamicsofthedroguearestochastic.Thiscausesthemodelingtasktobeimpracticalinreplicatingreallifesothecontrollerislimitedtothemodelsconsideredinthedesign.Inaddition,thedrogue'sdynamicsmaynotbedynamicallyfeasiblefortheaircrafttotrackwhichmayfurtherreduceperformance.Lastly,thecontrollerideallyshouldmakepositionmaneuversinstagesbyconsideringthealtitudeasonestage,thelateralpositionasanotherstage,andthedepthpositionasthenalstage.Incloseproximity,thecontrollershouldimplementonlysmallmaneuverstohelpmaintainthevehiclesintheFOV. 15 ]. 117
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110 ].Thestandarddesignapproachwasusedbyconsideringthelongitudinalandlateralstatesseparatelyasintypicalwaypointcontrolschemes.Thisapproachseparatedthecontrolintothreesegments:1)Altitudecontrol,2)HeadingControland3)DepthControl. 91 .Therstportionofthissystemisdescribedastheinnerloopwherepitchandpitchrateareusedinfeedbacktostabilizeandtrackapitchcommand.Meanwhile,thesecondportionisreferredtoastheouterloopwhichgeneratespitchcommandsfortheinnerloopbasedonthecurrentaltitudeerror.Theinnerloopdesignenablesthetrackingofapitchcommandthroughproportional 6 Altitudeholdblockdiagram control.Thispitchcommandinturnwillaffectaltitudethroughthechangesinforcesonthehorizontaltailfromtheelevatorposition.Thetwosignalsusedforthisinnerlooparepitchandpitchrate.Thepitchratefeedbackhelpswithshortperioddampingandallowsforratevariationsinthetransientresponse.AleadcompensatorwasdesignedinStevensetal.[ 110 ]toraisethe 118
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91 .Thisstructurewillprovidegooddisturbancerejectionduringturbulentconditions.Inaddition,boundswereplacedonthepitchcommandtoalleviateanyaggressivemaneuversduringtherefuelingprocess. 92 .Theinnerloop +fcmd 6y ?Dy 6f Headingholdblockdiagram componentofFigure 92 dealswithrolltracking.Thefeedbacksignalsincludebothrollandrollratethroughproportionalcontroltocommandachangeinaileronposition.Theinnerloop 119
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110 ].Consequently,theturnsmootherandcontainslessoscillations.Trackingheadingisnotsufcienttotrackthelateralpositionwiththelevelofaccuracyneededforrefuelingtask.Thenalloopwasaddedtoaccountforanylateraldeviationaccumulatedovertimeduetothedelayinheadingfromposition.Thisdelayismainlyduetothetimedelayassociatedwithsendingarollcommandandproducingaheadingchange.Therefore,thisloopwasaddedtogeneratemorerollforcompensation.Theloopcommandedachangeinaileronbasedoftheerrorinlateralposition.Thisdeviation,referredtoasDy,wascomputedbasedontwosuccessivetargetlocationsprovidedbytheestimator.Thecurrentandprevious(x;y)positionsofthetargetwereusedtocomputealineinspacetoprovideareferenceoftheit'smotion.Theperpendiculardistancefromthevehicle'spositiontothislinewasconsideredthemagnitudeofthelateralcommand.Inaddition,thesignofthecommandwasneededtoassignthecorrectdirection.Thisdirectionwasdeterminedfromtherelativeyposition,expressedinthebodyxedframe,thatwasfoundduringestimation.Oncethelateraldeviationwasdetermined,thatsignalwaspassedthroughaPIstructure,asshowninFigure 92 .Thegainscorrespondingtotheproportional,kyp,andintegrator,kyi,werethensummedandaddedtocomputethenalrollcommand.ThecompleteexpressionfortherollcommandisshowninEquation 9 120
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121
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8 providesamethodtoestimatetargetsinEuclideanspacewhenfeaturesdoexittheimage.Thismethodworkswellforshortperiodsoftimeafterthetargethasleft;however,thetrustinthepredictedvaluedegradestremendouslyastimeincreases.Consequently,whenafeatureleavestheimagethecontrollercanrelyonthepredictedestimatestosteertheaircraftinitiallybutmayresorttoalternativeapproachesbeyondaspeciedtime.Asalastresort,thecontrollercancommandtheaircrafttoslowdownandregainabroaderperspectiveofthescenetorecapturethetarget. 122
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110 ].Abaselinecontrollerisimplementedthatallowsthevehicletofollowwaypointsbasedentirelyonfeedbackfrominertialsensors.Imagesareobtainedfromasetofcamerasmountedontheaircraft.Thesecamerasincludeastationarycameramountedatthenoseandpointingalongthenose,atranslatingcameraunderthecenterlinethatmovesfromtherightwingtotheleftwing,andapitchingcameramountedunderthecenterofgravity.TheparametersforthesecamerasaregiveninTable 101 invaluesrelativetotheaircraftframeandfunctionsoftimegivenastinseconds. Table101. Statesofthecameras position(ft) orientation(deg) camera 24 0 0 0 90 0 2 10 153t 0 0 45 0 3 0 0 3 0 459t 0 Thecameraparametersarechosenassimilartoanexistingcamerathathasbeenighttested[ 111 ].Thefocallengthisnormalizedsof=1.Also,theeldofviewforthismodelcorrelatestoanglesofgh=32degandgv=28deg.TheresultinglimitsonimagecoordinatesaregiveninTable 102 Table102. Limitsonimagecoordinates coordinate minimum maximum 0.62 0.53 Avirtualenvironmentisestablishedwithsomecharacteristicssimilartoanurbanenvironment.Thisenvironmentincludesseveralbuildingsalongwithamovingcaranda 123
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103 ,isassociatedwitheachfeaturefordirectidenticationinthecameraimages. Table103. Statesofthefeaturepoints position(ft) featurepoint north east altitude 1 3500 200 1500 2 1000+200t 500 500 3 6000 200cos(2p TheightpaththroughthisenvironmentisshowninFigure 101 alongwiththefeatures.TheaircraftinitiallyiesstraightandleveltowardtheNorthbutthenturnssomewhattowardstheEastandbeginstodescendfromadivemaneuver. BFigure101. VirtualEnvironmentforExample1:A)3DViewandB)TopView ImagesaretakenatseveralpointsthroughouttheightasindicatedinFigure 101 bymarkersalongthetrajectory.ThestatesoftheaircraftattheseinstancesaregiveninTable 104 .Theimageplanecoordinates(;n)areplottedinFigure 102 forthethreecamerasatt=2sec.ThiscomputationisaccomplishedbyusingEquation 5 inconjunctionwithEquations 3 and 3 whileapplyingtheeldofviewconstraintshowninEquations 3 and 3 .Allthreecamerascontainsomeportionoftheenvironmentalongwithdistinctviewsofthefeaturepointsofinterest.Forexample,camera1containsaforwardlookingviewofastationary 124
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Aircraftstates Time North East Down v w (ft) (ft) (ft) (ft=s) (ft=s) 1196.9 0.44 2174.8 573.52 56.79 126.46 4 2112.7 143.04 1645.4 527.37 54.94 17.77 6 2989.8 353.63 1100.7 528.30 4.26 45.57 Time q y q r (deg) (deg) (deg) (deg=s) (deg=s) 13.92 22.43 1.81 13.56 36.82 1.38 4 39.21 37.90 22.79 32.31 28.41 0.04 6 6.98 14.85 11.93 7.63 9.34 1.15 pointonthecornerofabuildingaswellasthemovinghelicopter.Meanwhile,cameras2and3observeatopviewofthemovinggroundvehicletravelingforwarddownaroad.Theseimagemeasurementsprovideasignicantamountofdataandallowformoreadvancedalgorithmsforstateestimationandreconstruction. B CFigure102. FeaturepointMmasurementsatt=2secforA)camera1,B)camera2,andC)camera3 Figure 103 depictstheopticowcomputedforthesamedatasetshowninFigure 102 .Thisimagemeasurementgivesasenseofrelativemotioninmagnitudeanddirectioncausedbycameraandfeaturepointmotion.TheexpressionsrequiredtocomputeopticowconsistedofEqs. 5 5 3 3 3 3 3 and 3 .Inthisexample,theopticowhasmanycomponentscontributingtothenalvalue.Forinstance,theaircraft'svelocityandangularratescontributealargeportionoftheopticowbecauseoftheirlargemagnitudes.Inaddition,thesmallercomponentsinthisexamplearecausedfromvehicleandcameramotionwhicharesmallerinmagnitudebuthaveasignicanteffectondirection.Comparingcameras1and2,there 125
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B CFigure103. OpticowMeasurementsatt=2secforA)camera1,B)camera2,andC)camera3 Asummaryoftheresultingimageplanequantities,positionandvelocity,isgiveninTable 105 forthefeaturepointsofinterestaslistedinTable 103 .Thetableisorganizedbythetimeatwhichtheimagewastaken,whichcameratooktheimage,andwhichfeaturepointisobserved.Thistypeofdataenablesautonomousvehiclestogainawarenessoftheirsurroundingsformoreadvancedapplicationsinvolvingguidance,navigationandcontrol. Table105. Imagecoordinatesoffeaturepoints Time(s) Camera FeaturePoint 1 1 0.157 0.162 0.610 0.044 2 1 3 0.051 0.267 0.563 0.012 2 2 2 0.308 0.075 0.464 0.254 2 3 2 0.011 0.077 0.583 0.235 4 2 2 0.279 0.243 0.823 0.479 4 3 2 0.365 0.248 0.701 0.603 6 1 3 0.265 0.084 0.267 0.015 10.2.1ScenarioFeaturepointuncertaintyisdemonstratedinthissectionbyextendingthepreviousexample.Thissimulationwillexaminetheuncertaintyeffectsonvisionprocessingalgorithmsusingsimulatedfeaturepointsandperturbedcameraintrinsicparameters. 126
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104 alongwithapairofpointsindicatingthelocationsatwhichimageswillbecaptured.Theaircraftisinitiallystraightandlevelthentranslatesforwardwhilerolling4.0degandyawing1.5degatthenallocation. BFigure104. Virtualenvironmentofobstacles(solidcircles)andimaginglocations(opencircles)A)3DviewandB)topview Asinglecameraissimulatedatthecenterofgravityoftheaircraftwithlineofsightalignedtothenoseoftheaircraft.Theintrinsicparametersarechosensuchthatfo=1:0anddo=0:0forthenominalvalues.TheimagesforthenominalcameraassociatedwiththescenarioinFigure 104 arepresentedinFigure 105 toshowthevariationbetweenframes.Thevisionbasedfeedbackiscomputedforasetofperturbedcameras.Theseperturbationsrangeasdf2[0:2;0:2]anddd2[0:02;0:02].ObviouslythefeaturepointsinFigure 105 willvaryasthecameraparametersareperturbed.Theamountofvariationwilldependonthefeature 127
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BFigure105. FeaturepointsforA)initialandB)nalimages point,asnotedinEquations 4 and 4 ,buttheeffectcanbenormalized.Thevariationinfeaturepointgivennominalvaluesofo=no=1isshowninFigure 106 forvariationinbothfocallengthandradialdistortion.Thissurfacecanbescaledaccordinglytoconsiderthevariationatotherfeaturepoints.TheperturbedsurfaceshowninFigure 106 ispropagatedthroughthreemainimageprocessingtechniquesforanalysis. Figure106. Uncertaintyinfeaturepoint 106 tofocallengthandradialdistortion.ArepresentativecomparisonofopticowforthenominalcameraandasetofperturbedcamerasisshowninFigure 107 128
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B CFigure107. Opticalowfornominal(black)andperturbed(red)camerasforA)f=1:1andd=0,B)f=1:0andd=0:01,andC)f=1:1andd=0:01 107 indicateseveraleffectsofcameraperturbationsnotedinEquations 4 and 4 .Theperturbationstofocallengthscalethefeaturepointssothemagnitudeofopticowisuniformlyscaled.Theperturbationstoradialdistortionhavelargereffectasthefeaturepointmovesawayfromthecenteroftheimagesotheopticowvectorsarealteredindirection.Thecombinationofperturbationsclearlychangestheopticowinbothmagnitudeanddirectionanddemonstratesthefeedbackvariationsthatcanresultfromcameravariations.Theopticowiscomputedforimagescapturedbyeachoftheperturbedcameras.ThechangeinopticowfortheperturbedcamerasascomparedtothenominalcameraisrepresentedasdJandisboundedinmagnitude,asderivedinEquation 4 ,byDJ.ThegreatestvalueofdJpresentedbythesecameraperturbationsiscomparedtotheupperboundinTable 106 .ThesenumbersindicatethevariationsinopticowareindeedboundedbythetheoreticalboundderivedinChapter 4 andindicatethelevelofowvariationsinducedfromthevariationsincameraparameters. Table106. Effectsofcameraperturbationsonopticow PerturbationSet Analyzeonlywithdf kDJk kdJk kDJk kdJk kDJk 0.0476 0.0040 0.0040 0.0496 0.0543 0.0476 0.0020 0.0040 0.0252 0.0543 0.0476 0.0020 0.0040 0.0264 0.0543 0.0476 0.0040 0.0040 0.0543 0.0543 129
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108 showsthequalityoftheestimation.Essentially,theepipolargeometryrequiresafeaturepointinoneimagetoliealongtheepipolarline.Thisepipolarlineisconstructedbytheintersectionbetweentheplaneformedbytheepipolarconstraintandtheimageplaneatthelastmeasurement.ThedatainFigure 108 showthefeaturesinthesecondimagedoindeedlieexactlyontheepipolarlines. BFigure108. Epipolarlinesbetweentwoimageframes:A)initialframeandB)nalframewithoverlayedepipolarlinesfornominalcamera Theintroductionofuncertaintyintotheepipolarconstraintwillcausevariationsintheessentialmatrixwhichwillalsopropagatethroughthecomputationoftheepipolarline.Thesevariationsintheepipolarlinearevisualcluesofthequalityoftheestimateintheessentialmatrix.Thesevariationscanoccuraschangesintheslopeandthelocationoftheepipolarline.Figure 109 illustratestheepipolarvariationsduetoperturbationsondf=0:1anddd=0:01tothecameraparameters.Thefeaturepointswithuncertaintyandthecorrespondingepipolarlinewasplottedalongwiththenominalcasetoillustratethevariations.Thekeypointinthisguresisthesmallvariationsintheslopeoftheepipolarlinesandthesignicantvariationsinfeature 130
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BFigure109. Uncertaintyresultsforepipolargeometry:A)initialframeandB)nalframewithoverlayedepipolarlinesforcameraswithf1:0andd=0:0(black)andf=1:1andd=0:01(red) Theessentialmatrixiscomputedfortheimagestakenusingasetofcameramodels.EachmodelisperturbedfromthenominalconditionusingthevariationsinFigure 106 .Thechangeinestimatedstatesbetweennominalandperturbedcamerasisgivenbydqovertheuncertaintyrangeandisbounded,asderivedinEquation 4 ,byDq.ThevalueofdqforaspecicperturbationisshownincomparisontotheupperboundinTable 107 whichalsoindicatethevariationinenteriesoftheessentailmatrixwhichpropagatetothecamerastates. Table107. Effectsofcameraperturbationsonepipolargeometry PerturbationSet Analyzeonlywithdf kDqk kdqk kDqk kdqk kDqk 293.14 4.45 4.45 288.75 297.34 293.14 2.19 2.19 288.75 297.34 293.14 2.11 2.19 288.75 297.34 293.14 4.15 4.45 288.75 297.34 131
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1010 toindicatethatallerrorswerelessthan106. Figure1010. Nominalestimationusingstructurefrommotion Thedepthsarealsoestimatedusingstructurefrommotiontoanalyzeimagesfromtheperturbedcameras.ArepresentativesetoftheseestimatesareshowninFigure 1011 ashavingclearerrors.Aninterestingfeatureoftheresultsisthedependenceonsignoftheperturbationtofocallength.Essentially,thesolutiontendstoestimateadepthlargerthanactualwhenusingapositiveperturbationandadepthsmallerthanactualwhenusinganegativeperturbation.Sucharelationshipisadirectresultofthescalingeffectthatfocallengthhasonthefeaturepoints.Estimatesarecomputedforeachoftheperturbedcamerasandcomparedtothenominalestimate.Theworstcaseerrorsinestimationarecomparedtothetheoreticalbound,giveninEquation 4 ,totheseerrors.ThesenumbersshowninTable 108 indicatethevariationinstructurefrommotiondependsonthesignoftheperturbation.Theapproachisactuallyseentobelesssensitivetopositiveperturbations,whichcausesalargerestimateindepth,thantonegativeperturbations.Also,thetheoreticalboundwasgreaterthan,orequalto,theerrorcausedbyeachcameraperturbation. 132
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B CFigure1011. Estimationusingstructurefrommotionfornominal(black)andperturbed(red)cameraswithA)f=1:1andd=0,B)f=1:0andd=0:01,andC)f=1:1andd=0:01 Effectsofcameraperturbationsonstructurefrommotion PerturbationSet Analyzeonlywithdf kDzk kdzk kDzk kdzk kDzk 4679.8 75.02 75.02 4903.5 4903.5 4679.8 36.90 75.02 1076.6 4903.5 4679.8 35.73 75.02 498.76 4903.5 4679.8 70.34 75.02 1092.5 4903.5 1 involvingapolicepursuitisdemonstratedthroughthissimulation.Thesetupconsistedofthreevehicles:anUAVyingabovewithamountedcamera,electronicsandcommunication,areferencegroundvehiclewhichisconsideredthepolicepursuitcar,andatargetvehicledescribingthesuspectsvehicle.ThegoalofthismissionisfortheUAVtotrackbothvehiclesintheimage,whilereceivingpositionupdatesfromthereferencevehicle,andestimatethetarget'slocationusingtheproposedestimationalgorithm.ThecamerasetupconsideredinthisproblemconsistofasingledownwardpointingcameraattachedtotheUAVwithxedpositionandorientation.Whileinightthecamerameasuresandtracksfeaturepointsonboththetargetvehicleandthereferencevehicleforuseintheestimationalgorithm.Thissimulationassumesperfectcameracalibration,featurepointextraction,and 133
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7 thegeometryofthefeaturepointsarepredescribedandaknowndistanceisprovidedforeachvehicle.AfurtherdescriptionofthisassumptionisgiveninSection 7.2.2 .Futureworkwillexaminemorerealisticaspectsofthecamerasystemtoreproduceamorepracticalscenarioandtrytoalleviatethelimitationsimposedonthefeaturepoints. 1012 ,forillustration.Theinitialframeforthissimulationislocatedattheaircraft'spositionwhenthesimulationstarts.Thevelocityofthegroundvehicleswerescaleduptotheaircraft'svelocitywhichresultedinlargedistancesbutalsohelpedtomaintainthevehiclesintheimage. BFigure1012. Vehicletrajectoriesforexample3:A)3DviewandB)topview ThepositionandorientationstatesofthethreevehiclesareplottedinFigures 1013 1018 andallarerepresentedintheinertialframe,E.Thepositionsindicatethatallthreevehicle 134
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B CFigure1013. PositionstatesoftheUAVwithonboardcamera:A)North,B)East,andC)Down B CFigure1014. AttitudestatesoftheUAVwithonboardcamera:A)Roll,B)Pitch,andC)Yaw B CFigure1015. Positionstatesofthereferencevehicle(pursuitvehicle):A)North,B)East,andC)Down 135
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B CFigure1016. Attitudestatesofthereferencevehicle(pursuitvehicle):A)Roll,B)Pitch,andC)Yaw B CFigure1017. Positionstatesofthetargetvehicle(chasevehicle):A)North,B)East,andC)Down B CFigure1018. Attitudestatesofthetargetvehicle(chasevehicle):A)Roll,B)Pitch,andC)Yaw motionfromtheUAVtothetargetofinterest.ThenormerrorofthismotionaredepictedinFigure 1019 .Theseresultsindicatethatwithsyntheticimagesandperfecttrackingofthevehiclesnearlyperfectmotioncanbeextracted.Oncenoiseintheimageortrackingisintroducedtheestimatesofthetargetdeterioratequicklyevenwithminutenoise.Inaddition,imageartifactssuchasinterferenceanddropoutswillalsohaveanadverseaffectonhomographyestimation. 136
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BFigure1019. NormerrorforA)relativetranslationandB)relativerotation Figures 1020 and 1021 showtherelativetranslationandrotationdecomposedintotheirrespectivecomponentsandexpressedinthebodyframe,B.Thesecomponentsrevealtherelativeinformationneededforfeedbacktotrackorhomeinonthetargetofinterest. B CFigure1020. Relativepositionstates:A)X,B)Y,andC)Z B CFigure1021. Relativeattitudestates:A)Roll,B)Pitch,andC)Yaw 137
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1022 ofthecameraviewdepictingthevehiclesandthesurroundingscene.Theredvehiclewasdesignatedasthereferencewhereasthegreyvehiclewasthetargetvehicle.Thenextstepinthisprocessistoimplementanactualfeaturetrackingalgorithmonthesyntheticimagesthatfollowsthevehicles.Thismodicationalonewilldegradethehomographyresultsimmenselyduetothetroublesomecharacteristicsofafeaturepointtracker. Figure1022. Virtualenvironment 1 describedthemotivationandthebenetsofAAR,thissectionwilldemonstrateitbycombiningthecontroldesigngiveninChapter 9 withthehomographyresultinChapter 7 toformaclosedloopvisualservocontrolsystem.ThevehiclesinvolvedinthissimulationincludesaReceiverUAVinstrumentedwithasinglecamera,atankeraircraftalso 138
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5 withadditionalstatessuchasV,a,b,theaccelerationterms,Machnumber,anddynamicpressure.Althoughthecontrollerwillnotuseallstates,theassumptionoffullstatefeedbackwasmadetoallowallstatesaccessiblebythecontroller.Thecontrollerusesthesestatesoftheaircraftalongwiththeestimatedresultstocomputeactuatorcommandsaroundthespeciedtrimcondition. 139
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9 isintegratedandtunedforthenonlinearF16modeltoaccomplishthissimulation.Itwasassumedthatfullstatefeedbackoftheaircraftstatesweremeasurableincludingposition.Theunitsusedinthissimulationaregiveninftanddegwhichmeansthegainsdeterminedinthecontrolloopswerealsofoundbasedontheseunits.First,thepitchtrackingforaltitudecontrollerisconsidered.Theinnerloopgainsforthiscontrolleraregivenaskq=3andkq=2:5.ThebodediagramforpitchcommandtopitchangleisdepictedinFigure 1023 forthespeciedgains.Thisdiagramrevealsthedamping 140
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Figure1023. InnerlooppitchtopitchcommandBodeplot ThestepresponseforthepitchcontrollerisgiveninFigure 1024 andshowsacceptableperformance.Theouterloopcontrolwillnowbedesignedusingthiscontrollertotrackaltitude. Figure1024. Pitchanglestepresponse 141
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10 andwasdesignedinStevensetal.[ 110 ].AstepresponseforthiscontrollerisillustratedinFigure 1025 thatshowsasteadyclimbwithnoovershootandasteadystateerrorof2ft.ThisresponseisrealisticforanF16butnotidealforautonomousrefuelingmissionwheretolerancesareonthecmlevel.Thealtitudetransitionisslowduetothecompensatorbutonemayconsidermoreaggressivemaneuversformissionssuchastargettrackingthatmayrequireadditionalagility. Figure1025. Altitudestepresponse Thenextstagethatwastunedinthecontroldesignwastheheadingcontroller.Theinnerloopgainswerechosentobekf=5:7andkp=1:6fortherolltracker.ThebodediagramforthiscontrollerofrollcommandtorollangleisshowninFigure 1026 whichshowsattenuationinthelowerfrequencyrange.Thisattenuationremovesanyhighfrequencyresponsefromtheaircraftwhichisdesiredduringarefuelingmission,especiallyincloseproximity.Meanwhile,thecouplingbetweenlateralandlongitudinalstatesduringaturnwascounteracted 142
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Figure1026. InnerlooprolltorollcommandBodeplot ThestepresponseforthisbankcontrollerisillustratedinFigure 1027 .Thetrackingperformanceisacceptablebasedonarisetimeof0:25sec,anovershootof6%andlessthana3%steadystateerror.Theouterlooptuningforheadingcontrollerconsistedofrsttuningthegainonheadingerror.Againofky=1:5waschosenforthismissionwhichdemonstratedacceptableperformance.Figure 1028 showstheheadingresponseusingthiscontrollerforarightturn.Theresponserevealasteadyrisetime,noovershoot,andasteadystateerroroflessthan2deg.Finally,thelooppertainingtolateraldeviationwastunedtokyp=0:5andkyi=0:025whichproducedreasonabletrackingandsteadyerrorforlateralposition.Thenalstageofthecontrollerinvolvestheaxialposition.Thisstagewasdesignedtoincreasethrustbasedonavelocitycommandoncethelateralandaltitudestateswerealigned.Aproportionalgainwastunedbasedonvelocityerrortoachieveaslowsteadyapproachspeed 143
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Rollanglestepresponse Figure1028. Headingresponse tothetarget.Againofkx=3:5wasdeterminedforthisloopwhichgeneratesthedesiredapproach.Lastly,tohelplimitthenumberoftimesthefeaturepointsexittheeldofviewalimitwasimposedonthepitchangle.Thislimitwasenforcedwhentheapproachachieveaspecieddistance.Forthisexample,thedistancewassettowithin75ftintheaxialpositionofthebodyxedframewhichwasdeterminedexperimentallyfromthetarget'ssize. 144
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1029 forpositionandFigure 1030 fororientationandrevealedacorrectt.Thisresultdemonstratesthefunctionalityoftheestimatorwithanaccuracyontheorderof109.ThiserrorwasplottedinFigure 1031 forbothpositionandorientation. B CFigure1029. Openloopestimationoftarget'sinertialposition:A)North,B)East,andC)Altitude B CFigure1030. Openloopestimationoftarget'sinertialattitude:A)Roll,B)Pitch,andC)Yae Furthermore,theclosedloopresultsforthissimulationwereplottedinFigures 1032 and 1034 forpositionandorientationofboththereceiveraircraftandthetargetdroguerelativetotheearthxedframe.Thetrackingofthiscontrollershowedreasonableperformanceforthedesiredpositionandheadingsignals.Theremainingorientationangleswerenotconsideredinfeedbackbutestimatedforthepurposeofmakingsurethedrogue'spitchandrollarewithinthedesiredvaluesbeforedocking.AsseeninFigure 1032 ,thereceiverwasabletotrackthegrossmotionofdroguewhilehavingsomedifcultlytrackingtheprecisemotion. 145
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BFigure1031. NormerrorfortargetstateestimatesA)translationandB)rotation B CFigure1032. Closedlooptargetpositiontracking:A)North,B)East,andC)Altitude ThecomponentsofthepositionerrorbetweenthereceiveranddrogueareshowninFigure 1033 toillustratetheperformanceofthetrackingcontroller.Theseplotsdepicttheinitialoffseterrordecayingovertimewhichindicatesthereceiver'srelativesdistanceisdecreasing.Thealtitudeshowedaquickclimbresponsewhereastheresponseinaxialpositionwasaslowsteadyapproachwhichwasdesiredtolimitlargechangesinaltitudeandangleofattack.Thelateralpositionisstableforthetimeperiodbutcontainsoscillationsduetherolltoheadinglag.TheorientationanglesshowninFigure 1034 indicatetheEuleranglesforforthebodyxedtransformationscorrespondingtothebodyxedframeofthereceiverandthebodyxedframeofthedrogue.Recall,theonlysignalbeingtrackedinthecontroldesignwasheading.Thisselectionallowedtheaircrafttosteerandmaintainaighttrajectorysimilartothedroguewithoutaligningrollandpitch.Thereceivershouldyclosetoatrimconditionratherthenmatchingthefullorientationofthedrogue,asillustratedinFigure 1034 forpitchangle. 146
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B CFigure1033. Positiontrackingerror:A)North,B)East,andC)Altitude TheerrorinheadingisdepictedinFigure 1035 whichshowsacceptabletrackingperformanceoverthetimeinterval. B CFigure1034. Targetattitudetracking:A)Roll,B)Pitch,andC)Yaw Figure1035. Trackingerrorinheadingangle Theresultsshownintheseplotsindicatethatthetrackinginthelateralpositionandaltitudearenearlysufcientfortherefuelingtask.Thesimulationrevealsboundederrorsinthese 147
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8 willhelptoaidthecontroller,oratleasttohelpdeterminearegionofwherethefeaturesmostlikelyhavetraveled. 7 .Toseewhatlevelsofvariationsexistintheseresultsanuncertaintyanalysiswasperformed.Chapter 4 derivedamethodtocomputeworsecaseboundsonstateestimatesfromthehomographyapproachusingvisualinformation.ThetechniquedescribedinChapter 4 wasusedforthisuncertaintyanalysis.ThetargetestimatesforabsolutepositionandorientationalongwithupperandlowerboundswerecomputedforthissimulationandareshowninFigures 1036 and 1037 .These 148
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B CFigure1036. Target'sinertialpositionwithuncertaintybounds:A)North,B)East,andC)Altitude B CFigure1037. Target'sinertialattitudewithuncertaintybounds:A)Roll,B)Pitch,andC)Yaw ThemaximumuncertaintiesintargetpositionrelativetotheearthxedframearesummarizedinTable 109 .Meanwhile,Table 1010 containsthemaximumuncertaintiesintargetorientation.Thethreelevelsofuncertaintyareincludedinthesetables.Thiscomparisonhelpstoverifythatthemaximumstatevariationcorrespondstothemaximumcameraparameter 149
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Table109. Maximumvariationsinpositionduetoparametricuncertainty uncertaintyparameter north(ft) 4.10 20.54 10.53 14.40 15.09 30.82 Table1010. Maximumvariationsinattitudeduetoparametricuncertainty uncertaintyparameter 0 0 4.48 2.29 7.94 3.48 150
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8 intotherefuelingsimulationwillhelpthecontrollerbyprovidingstateestimatewhenthetargetexitstheeldofview. 153
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RyanScottCauseywasborninMiami,Florida,onMay10,1978.Hegrewupinastablefamilywithonebrotherinatypicalsuburbanhome.Duringhisteenageyearsandintoearlyadolescence,Ryanbuiltandmaintainedasmallbusinessprovidinglawncaretothelocalneighborhood.Thetoolsacquiredfromthisworkcarriedoverintohiscollegecareer.AftergraduatingfromMiamiKillianSeniorHighSchoolin1996,RyanattendedMiamiDadeCommunityCollegeforthreeyearsandreceivedanAssociateinArtsdegree.AtransferstudenttotheUniversityofFlorida,Ryanwaspreparedtotacklethestressesofauniversityasidefromthepoorstatisticsontransferstudents.Afewyearslater,hereceivedaBachelorofScienceinAerospaceEngineeringwithhonorsin2002andwasconsideredinthetopthreeofhisclass.RyansoonafterchosetoattendgraduateschoolbackattheUniversityofFloridaunderDr.RickLindintheDynamicsandControlsLaboratory.Duringthesummertime,RyaninternedtwiceatHoneywellSpaceSystemsasaSystemsEngineerinClearwater,FLandonceatTheAirForceResearchLaboratoryinDayton,OH.Visionbasedcontrolofautonomousairvehiclesbecamehisinterestandheisnowpursuingadoctoratedegreeonthistopic.RyanwasawardedaNASAGraduateStudentResearchProgram(GSRP)fellowshipin2004forhisproposedinvestigationonthisresearch. 164
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