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# Trajectory Generation for Effective Sensing of a Close Proximity Environment

## Material Information

Title: Trajectory Generation for Effective Sensing of a Close Proximity Environment
Physical Description: 1 online resource (175 p.)
Language: english
Creator: Kehoe, Joseph John
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

## Subjects

Subjects / Keywords: mav, motion, path, planning, randomized, sensor, uav, unmanned
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Aerospace Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

## Notes

Abstract: Unmanned systems stand to play a significant role in future sensing and information gathering applications. The scope of these scenarios is expanding to include those missions for which the sensor and carrier vehicle are forced to operate in close proximity to the surrounding environment. Several unique challenges are introduced for this class of sensing problems. First, the sensor projects a small footprint on environment surfaces relative to the scale of vehicle motion. Consequently, coupling exhibited between the motion of the vehicle and the direction of sensor pointing has a significant effect on gathered data. Second, view characteristics of sensed objects, such as resolution and aspect, can vary appreciably over the sensor field-of-view. These variations can have an adverse effect on data quality. Finally, vehicle maneuvering in close proximity to obstacles imposes significant risks to safe navigation. As such, consideration of vehicle dynamics within any motion planning strategy is critical to ensure that accurate and precise trajectories are generated. This dissertation addresses the problem of planning sensing trajectories through complex and cluttered environments. A randomized planning technique is developed which utilizes a hybrid modeling strategy to efficiently plan dynamically-admissible trajectories. Then, a generalized measure of sensing effectiveness is formulated to quantify the application-specific effects of sensor operation in close proximity to sensing targets. Finally, these elements are integrated into a randomized trajectory planner that ensures quality data collection regarding a specified set of targets while satisfying the system dynamics. The utility of the planner is demonstrated through several simulated examples.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Joseph John Kehoe.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.

## Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021389:00001

## Material Information

Title: Trajectory Generation for Effective Sensing of a Close Proximity Environment
Physical Description: 1 online resource (175 p.)
Language: english
Creator: Kehoe, Joseph John
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

## Subjects

Subjects / Keywords: mav, motion, path, planning, randomized, sensor, uav, unmanned
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Aerospace Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

## Notes

Abstract: Unmanned systems stand to play a significant role in future sensing and information gathering applications. The scope of these scenarios is expanding to include those missions for which the sensor and carrier vehicle are forced to operate in close proximity to the surrounding environment. Several unique challenges are introduced for this class of sensing problems. First, the sensor projects a small footprint on environment surfaces relative to the scale of vehicle motion. Consequently, coupling exhibited between the motion of the vehicle and the direction of sensor pointing has a significant effect on gathered data. Second, view characteristics of sensed objects, such as resolution and aspect, can vary appreciably over the sensor field-of-view. These variations can have an adverse effect on data quality. Finally, vehicle maneuvering in close proximity to obstacles imposes significant risks to safe navigation. As such, consideration of vehicle dynamics within any motion planning strategy is critical to ensure that accurate and precise trajectories are generated. This dissertation addresses the problem of planning sensing trajectories through complex and cluttered environments. A randomized planning technique is developed which utilizes a hybrid modeling strategy to efficiently plan dynamically-admissible trajectories. Then, a generalized measure of sensing effectiveness is formulated to quantify the application-specific effects of sensor operation in close proximity to sensing targets. Finally, these elements are integrated into a randomized trajectory planner that ensures quality data collection regarding a specified set of targets while satisfying the system dynamics. The utility of the planner is demonstrated through several simulated examples.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Joseph John Kehoe.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.

## Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021389:00001

Full Text
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TRAJECTORY GENERATION FOR EFFECTIVE SENSING OF A CLOSE
PROXIMITY ENVIRONMENT

By

JOSEPH J. KEHOE

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 Joseph J. Kehoe

Dedicated to Sarah

ACKNOWLEDGMENTS

I would first like to acknowledge Dr. Rick Lind for providing steady guidance

have ah--,v- kept my career interests in mind, and have prepared me to (finally) leave the

safe confines of the education system for the i I! world." I thank the other members of

my committee, Dr. Carl Crane, Dr. Warren Dixon, Dr. Peter Ifju, and Dr. Clint Slatton

for taking the time to review my research and provide comments on the directions of my

ideas.

Additionally, thanks go out to the members of flight control lab for providing alh--v -

needed diversions from work, and for the countless in-depth discussions related to nothing

in particular. Sometimes steps in wrong direction are needed to make leaps in the right

direction. I have learned things in the past four years that I never would have imagined

learning in graduate school. Specifically, I thank M1uli !d Abdulrahim for greatly expand-

ing my knowledge regarding rally racing, rubber chickens, and unicycles. Adam Watkins

deserves credit for teaching me the joy that can come from the right combination of cyn-

icism and sarcasm. I would also like to thank Daniel "Tex" Grant for teaching me that

meatloaf made from venison tastes surprisingly just like regular meatloaf.

I would also like to acknowledge Air Force Reasearch Labs, Munitions Directorate,

specifically Johnny Evers, Neal Glassman, Sharon Heise, and Robert Sierakowski, for

continued financial support. Thanks go to Dr. Jason Stack for providing additional

funding as well as for his guidance during the two summers I spent as an intern at the

N i,1 I Surface Warfare Center in Panama City, FL.

Finally, I would like to thank my family and friends for their patience in dealing with

me during these busy past few years. This work would not have been possible without the

love and support of my parents, Joseph and Linda, and my sister, Kathleen. I would like

to thank my Father-in-law, Bob Williams, for being .i.v- li- willing to talk shop, and my

Mother-in-law, Pat Williams, for alv--,v- putting up with it. I owe my wife, Sarah, a great

debt of gratitude for her patience, ( ,iii, and understanding throughout this process. On

this note, her equine companion, Sundance, deserves my thanks for keeping her occupied

during the times that I was unable.

page

ACKNOW LEDGMENTS .................................

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

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

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

CHAPTER

1 INTRODUCTION ..................................

Motivation ......
Problem Description
Problem Statement
Approach Overview
Contributions ..

2 M OTION PLANNING ................................

2.1 Introduction . . . . . . . . .
2.2 Path Constraints . . . . . . . .
2.2.1 Differential Constraints .........................
2.2.2 Obstacle Constraints ..........................
2.3 O ptim al Control . . . . . . . .
2.3.1 Indirect Optimal Control .. ....................
2.3.2 Direct Optimal Control .. .....................
2.4 Hybrid M otion M odels .. ........................

2.4.1
2.4.2
2.4.3
2.4.4

Modeling with Motion
Motion Planning with
Kinematic Example
Dynamic Example .

Primitives
Primitives.

3 RANDOMIZED SAMPLING-BASED MOTION PLANNING

Introduction . . . . .
Random Dense Tree Methods (RDT)......
3.3.1 Rapidly-Exploring Random Trees (RRT)
3.3.2 Expansive-Spaces Trees (EST) ......
3.3.3 Discussion . . . . .

..............................:

3.4 RDT-Based Dynamic Planner for a Planar-Motion Vehicle . ... 68
3.4.1 Model ................... . . .. 69
3.4.2 Overview ................... . 73
3.4.3 Node Selection ............... ........... .. 74
3.4.4 Node Expansion ............... .......... .. 76
3.4.5 Solution C'!I .: ............................. .78
3.5 Exam ple . . . . . . . . .... 79

4 SENSING EFFECTIVENESS .................. ......... .. 85

4.1 Introduction .................. ................ .. 85
4.2 Remote Sensor Technologies ............... ..... .. 86
4.2.1 Computer Vision ............... ......... 87
4.2.2 Radar ................ ........... .. .. 87
4.2.3 Sonar . ............... ............ .. .. 88
4.2.4 Ladar ................... . . .. 89
4.3 Modeling the Sensing Task ............... ........ .. 90
4.3.1 Sensing Geometry ............... ......... .. 90
4.3.2 Visibility .. ... .. .. .. .. ... .. . . . 92
4.3.3 The Visibility Set ............... ......... .. 95
4.3.4 Proximity Effects .................. ......... .. 97
4.4 Effectiveness Metric ............... ........... .. 101
4.4.1 Formulation ............... ........... 102
4.4.2 The Quality Set ............... .......... 104
4.4.3 Sensing Mission Effectiveness .................. ... 104
4.4.4 Example: A Contrived Metric ............... .. 106
4.4.5 Example: Image Area .................. ..... 110

5 RANDOMIZED SENSOR PLANNING .................. .. .. 115

5.1 Introduction ................... . . 115
5.2 Environment Representation ................... . .. 117
5.3 A Randomized Sensor-Planning Algorithm ..... . . ..... 120
5.3.1 Overview .................. .............. 120
5.3.2 Node Selection ... .. .. .. ... .. .. .. ... ... .. 121
5.3.3 Vantage Point Selection .................. ..... 123
5.3.4 Local Planning and Expansion .................. .. 125
5.3.5 Evaluation .................. ............. 126
5.4 Exam ples .. .. ... .. .. .. .. .. .. .. .. ... .. ... .. 127
5.4.1 M odel .. ... .. .. .. .. ... .. .. ... .. ..... 127
5.4.2 Effectiveness M etric .................. ........ 129
5.4.3 Obstacle-Free Examples .................. .... 130
5.4.4 Examples with Obstacles .................. .. ... 136

6 OPTIMAL SENSOR PLANNING ................... ...... 143

6.1 Introduction .................. ................ .. 143
6.2 System .................. ................... .. 144
6.3 Problem Formulation .................. ........... 146
6.4 Variational Approach .................. ........... 149
6.4.1 Necessary Conditions .................. .. ..... 150
6.4.2 Boundary Value Problem .................. .. ... 153
6.5 Direct Transcription Approach .................. .... 155
6.5.1 Numerical Example .................. ........ 156

7 CONCLUSION .................. ................. .. 161

REFERENCES ........................... . . 165

BIOGRAPHICAL SKETCH ........... ........ . ... 175

LIST OF TABLES
Table page

2-1 /AM components for each maneuver .................. ..... .. 57

4-1 Statistics for visibility parameter comparison ................ 101

LIST OF FIGURES

Figure page

1-1 Prevalent examples of advancements in unmanned technology . .... 16

1-2 Comparison of sensing data for different mission scales . ..... 18

1-3 Touring a series of cells for a large-scale sensing mission . 19

1-4 Touring a series of cells for a small-scale sensing mission . . 20

1-5 Motion planning to view a sequence of targets .............. .. 21

1-6 Sensing mission tasks .................... . 24

2-1 Obstacle boundary approximations ................ ...... 34

2-2 Vertex-angle sum collision detection method ............. .. 35

2-3 Vertex edge-vector collision detection method .................. 35

2-4 Obstacle expansion for pointwise safety .................. .. 36

2-5 Automaton model representation as a directed graph .............. ..45

2-6 Special reference frame, D for Dubins path solution ............... .51

2-7 Sample Dubins paths .................. ............ .. .. 54

2-8 Example state trajectory showing maneuver dynamics ............. ..56

2-9 Top view of environment for dynamic example .................. 58

2-10 Top view of solution for M {MSL, MLS, MSL} ..... . . 59

2-11 Heading and turn rate trajectories for M = {MSL, MLS, MSL} . ... 59

2-12 Top view of solution for M {MSL, MLS, MSL, MLS, MSL} . . ... 60

2-13 Heading and turn rate trajectories for M = {MSL, MLS, MSL, MLS, MSL} . 60

2-14 Top view of solution for M {MSL, MLS, MSR, MRS, MSR, MRs} ........ 61

2-15 Heading, turn rate trajectories for M = {MSL, MLS, MSR, MR, MSR, MRS} 61

3-1 The PRM algorithm ............... ............. 63

3-2 RRT algorithm .................. ................. .. 66

3-3 EST algorithm .................. ................. .. 67

3-4 Differences in exploration strategy for the RRT algorithm vs. the EST algorithm 68

3-5 Automaton structure for vehicle model used with current planning strategy

3-6 Distance function comparison . ........

3-7 Distance function computation . .

3-8 Node expansion step . .............

3-9 The new branch is subdivided to a set of nodes. .

3-10 Collision-free solutions . ............

3-11 Unique solution families for vehicle used in example .

3-12 Turn-straight solution sequences . ......

3-13

3-14

3-15

3-16

4-1

4-2

4-3

4-4

Example planning environment .

Incremental tree expansion . .

Nodes in the final solution tree .

Solution path refinement . .

Sensing problem geometry . .

Visibility parameters . ...

Inverted visibility cone . .

Construction of V . ....

4-5 Motion coupling effects for different problem scales

4-6

4-7

4-8

4-9

4-10

4-11

4-12

4-13

4-14

5-1

Range and incidence variation for different problem scales .

Representative trajectories for sensor effectiveness metric .

Quality parameter efficiency functions . ........

Simulated trajectory showing snapshots of effectiveness metric

Max-value mission effectiveness . ............

Environment for image area sensing metric simulation . .

Sensing effectiveness as a function of image area . ..

Effectiveness trajectories . ...............

Simulated mission effectiveness . ............

TSP problem comparison . ...............

.. . 75

.. . 76

.. . 77

.. . 78

.. . 79

. 81

. 81

. 82

. 82

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. 84

. 91

. 94

. 96

. 97

. 99

. 100

. 04

. 07

values . .. 109

. 110

. 112

. 113

. 114

. 114

. 115

5-2

5-3

5-4

5-5

5-6

5-7

5-8

5-9

5-10

5-11

5-12

5-13

5-14

5-15

5-16

5-17

5-18

5-19

5-20

5-21

5-22

5-23

5-24

5-25

6-1

6-2

6-3

Discretization approaches for area coverage . .....

Sensing secondary targets . ...............

Example node weighting function . ...........

Vantage point reachability test . ...

Quality parameter efficiency functions for examples . .

Environments for obstacle-free sensor planning example .

Trajectory refinement for obstacle-free reconnaissance . .

Simulated effectiveness for obstacle-free reconnaissance .

Trajectory refinement for obstacle-free surveillance . .

Simulated effectiveness for obstacle-free surveillance . .

Trajectory refinement for obstacle-free coverage . ...

Simulated effectiveness for obstacle-free coverage . ..

Environments for sensor planning example with obstacles .

Initial tree for reconnaissance with obstacles . ....

Trajectory refinement for reconnaissance with obstacles .

Simulated effectiveness for reconnaissance with obstacles .

Initial tree for surveillance with obstacles . ......

Trajectory refinement for surveillance with obstacles . .

Simulated effectiveness for surveillance with obstacles . .

Initial tree for coverage with obstacles . .

Trajectory refinement for coverage with obstacles . ..

Simulated effectiveness for coverage with obstacles . .

Force coefficients . . . . . . .

Environment for optimal sensing numerical example . .

Solution position and velocity trajectories . .

. 118

. 118

. 119

. 22

. 123

. 25

. 130

. . 131

. . 133

.. . 134

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. 37

. . 138

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. . 139

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. 46

. 57

. 59

6-4 Solution a and o reference trajectories .................. ...... 159

6-5 Solution visibility parameter trajectories for each target . . ..... 160

6-6 Aircraft trajectory and sensor footprint over each target ............ .160

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

TRAJECTORY GENERATION FOR EFFECTIVE SENSING OF A CLOSE
PROXIMITY ENVIRONMENT

By

Joseph J. Kehoe

August 2007

('C! i: Richard C. Lind, Jr.
Major: Aerospace Engineering

Unmanned systems stand to p1 iv a significant role in future sensing and information

gathering applications. The scope of these scenarios is expanding to include those missions

for which the sensor and carrier vehicle are forced to operate in close proximity to the

surrounding environment. Several unique challenges are introduced for this class of sensing

problems. First, the sensor projects a small footprint on environment surfaces relative

to the scale of vehicle motion. Consequently, coupling exhibited between the motion of

the vehicle and the direction of sensor pointing has a significant effect on gathered data.

Second, view characteristics of sensed objects, such as resolution and aspect, can vary

appreciably over the sensor field-of-view. These variations can have an adverse effect

on data quality. Finally, vehicle maneuvering in close proximity to obstacles imposes

significant risks to safe navigation. As such, consideration of vehicle dynamics within any

motion planning strategy is critical to ensure that accurate and precise trajectories are

generated.

This dissertation addresses the problem of planning sensing trajectories through

complex and cluttered environments. A randomized planning technique is developed

which utilizes a hybrid modeling strategy to efficiently plan dynamically-admissible

trajectories. Then, a generalized measure of sensing effectiveness is formulated to quantify

the application-specific effects of sensor operation in close proximity to sensing targets.

Finally, these elements are integrated into a randomized trajectory planner that ensures

quality data collection regarding a specified set of targets while satisfying the system

dynamics. The utility of the planner is demonstrated through several simulated examples.

CHAPTER 1
INTRODUCTION

1.1 Motivation

Unmanned systems technology has become an increasingly common presence in

the world around us. In recent years, CNN has broadcast images and video that were

captured by the Predator drone from behind enemy lines in Iraq and Afghanistan. An

unexpectedly-rr_--_, 1 pair of robotic explorers have provided breathtaking views and

unprecedented scientific data from the fourth planet from the Sun. A winner was declared

in a race across 130 miles of desert in which no human drivers participated. These

examples, shown in Figures 1-1A to 1-1C, represent a small cross-section of the diverse

and growing field of unmanned vehicles.

A B C

Figure 1-1. Prevalent examples of advancements in unmanned technology. A) The RQ-1
Predator. B) A mars rover. C) Stanford Racing's "Stanley"

Of the expanding roles p1. i,, d by existing unmanned technologies, future vehicles

stand to prove especially useful for applications that involve the gathering of information.

Envisioned civilian-domain scenarios include border patrol, traffic monitoring, tactical law

enforcement, maritime surveillance, and environmental sensing [1]. Likewise, intelligence,

surveillance, and reconnaissance (ISR) missions are emerging as the dominant theme

among projected military applications for future unmanned systems. The future combat

paradigm is likely to include the use of cooperating teams of unmanned aerial vehicles

(UAVs), unmanned ground vehicles (UGVs), and unmanned underwater vehicles (UUVs)

equipped with various sensors that autonomc;ro;, navigate complex environments for the

collection of ISR data [2-5].

Existing platforms typically p'1 ,i an ISR role from a standoff range. Such missions

are characterized by sensing the environment from a significantly large distance relative

to the dimensions of typical target surfaces. As such, a remote sensor with even a limited

field-of-view (FOV) can cover a potentially large area from a single vantage point. For

example, the Predator drone typically collects data from altitudes ranging from 15,000-

20,000 ft. Figure 1-2A depicts the University of Florida campus as viewed from an altitude

of approximately 15, 000 ft. The entire campus and surrounding area can be seen in a

single image captured from a single vantage point.

Advances in hardware and control technologies will soon enable mission descriptions

on a much smaller scale in increasingly-complex environments. For example, researchers

foresee cooperative teams of UGVs and UAVs patrolling an urban peace-keeping environ-

ment [6]. N ,i i;, ii. capabilities for UAVs operating in such environments are already

emerging through the development of miniature autopilot systems [7], vision-based control

technologies [8-11], and agility-enhancing morphing strategies [12-14]. Sensing missions in

such a scenario are characterized by vehicle and sensor operation within close proximity to

the environment. Figure 1-2B shows a simulated image of the University of Florida cam-

pus as viewed from a UAV operating at an altitude of approximately 50 ft. Typical data

collected during such missions is obviously different in nature than that collected during

standoff sensing missions. Smaller portions of the environment are covered from a partic-

ular viewpoint; however, the data is of much higher resolution and exhibits richer content.

Moreover, additional sensor modalities, such as audio, gain relevance at close range. These

key benefits expand possible mission scenarios to include tasks such as sensing under and

inside structures.

1.2 Problem Description

This dissertation deals with generating motion plans for small-scale unmanned sensing

missions. Comparison of the images in Figure 1-2 shows that fundamental differences

can be expected in the way the two depicted classes of sensing missions are approached.

A B

Figure 1-2. Comparison of sensing data for different mission scales. A) Images captured
from standoff range. B) Images captured within close proximity to a cluttered
environment.

One aim of the work presented here is to identify and address some the unique challenges

encountered for sensor-planning in close-proximity environments.

Motion planning methods for standoff missions commonly make simplifying as-

sumptions that are consistent with the problem scale. For example, target touring is

often considered sufficient to ensure sensing of a set of targets. Previous efforts have

adopted this technique, which involves simply visiting each target location (or a location

directly above, for the case of UAVs) [15-18]. Similarly, area coverage in such cases is

often achieved through decomposition of the area of interest into cells and then simply

visiting each cell to meet sensing objectives [19-21]. Figure 1-3 shows a representative

example of a sensor motion plan that operates under such assumptions. In the depicted

scenario, the sensor FOV is considered large enough such that cell visitation is sufficient

to sense any targets contained within the cell. These approaches also typically assume

that local motion constraints due to vehicle dynamics are negligible relative to the scale

of the motion planning problem. For example, the vehicle turn radius in Figure 1-3 is

small enough relative to the cell resolution and to the scale of maneuvering that such a

constraint can be neglected in motion planning. Numerous technical challenges are intro-

duced by operation of the vehicle and sensor in close proximity to the environment. First,

Figure 1-2B shows drastically disparate data in different areas of the image as compared

to Figure 1-2A. Some objects are much closer to the sensor than others, and hence appear

Figure 1-3. Touring a series of cells for a large-scale sensing mission.

in the foreground of the image with comparatively-high data resolution. Additionally,

objects are seen at aspects that yield data with varying resolution on a single surface.

The extent of these effects varies as the vehicle moves through the environment and the

viewpoint changes. Data quality with respect to a particular sensed object is seen to vary

as a function of both the vantage point and the location within the FOV. Further, sensor

operation amongst obstacles results in the occlusion of large regions. As such, the view

quality of a particular target could change drastically with small changes in vehicle and

sensor position and orientation.

Another issue is related to the comparatively small spatial area contained within the

FOV for close-proximity cases. A small projected footprint relative to vehicle velocity

causes target surfaces to pass through the FOV quickly as the vehicle moves through

the environment. Resulting data losses are exacerbated by cluttered environments that

require significant maneuvering for safe navigation. As such, a motion plan to view a

particular target must be timed appropriately and tracked with enough precision to ensure

that the sensor is pointing in the right direction at the right time. Several researchers

have expanded the target touring approach to account for a particular sensor footprint

shape and size; however, these efforts fail to account for motion coupling that results from

maneuvering [17, 20, 22-24].

Figure 1-4 shows a representative motion plan that di-pli- data quality and motion

coupling effects related to proximity. In the figure, the sensor footprint is not significantly

large relative to cell resolution. As such, sensor coverage is seen to cover some cells

partially and non-uniformly due to the variable resolution issue discussed previously.

Additionally, the spatial effects of vehicle dynamics are significant relative to the cell

resolution and therefore must be considered in motion planning. The resulting motion

coupling effects are seen in Figure 1-4 as sensor coverage that varies relative to the vehicle

trajectory.

Figure 1-4. Touring a series of cells for a small-scale sensing mission.

Planning a trajectory to view a sequence of targets presents a difficult problem

regardless of sensing proximity effects. Generally, an optimal solution to this problem

requires that:

1. The optimal order of target visitation is determined

2. The optimal viewpoint for each target is determined

3. The optimal trajectory through the optimal viewpoints is computed

These requirements are depicted in Figure 1-5. Efforts to address such problems

typically involve the application of assumptions and the solution of sub-optimal ap-

proximations. For example, the target-touring approach described previously essentially

assumes that the optimal viewpoint for each target is the location of the target itself.

When dynamics are neglected, this problem reverts to the well-known traveling sales-

man problem (TSP) from graph theory, for which numerous solution and approximation

techniques exist [25, 26]. Several efforts achieve an approximation to the optimal set of

viewpoints to sense an area through random sampling, and then proceed to solve the en-

suing TSP [27-29]. These efforts do not consider differentially constrained motion, which,

while some approximation techniques do exist [30], can greatly complicate the solution of

the TSP.

Figure 1-5. Motion planning to view a sequence of targets.

Finally, the close proximity sensing problem requires that planned trajectories

accurately represent feasible vehicle motions such that trajectory tracking will ensure safe

feasible motions are subject to tracking error upon mission execution that could result

in either failure to reach a precisely-defined viewpoint or in collision with an obstacle.

Generally, this requirement implies that planned trajectories should be constrained by

some representation of the vehicle dynamics.

Inclusion of dynamically-feasible motions in a planned trajectory is typically treated

in either a direct or a decoupled fashion [31]. Direct planning methods, such as optimal

control, consider a representation of the vehicle dynamics in the formulation of the

planning problem and directly solve for optimal system inputs. Alternatively, indirect

methods use a simplified model of vehicle motion to plan a reference path and then

"smooth" the path to satisfy dynamics using methods such as feedback control. Direct

methods compute optimal trajectories but are often intractable for realistic problem

descriptions. Indirect methods often exhibit tractable complexity properties that comes at

the expense of optimality.

Researchers have found clever v--i- to manipulate this tradeoff through a v ii I

of techniques such that dynamics can be directly included in the planning process. For

example, some researchers have recognized that systems which exhibit differential flatness

properties admit solutions that can be represented parametrically in terms of a set of

flat outputs and their derivatives [32-34]. Others have applied mixed-integer linear

programming (\!I .IP) to model dynamic constraints as a set of switching bounds on

system velocities and accelerations [35-37]. Frazzoli et al introduced a planning technique

that utilizes a iiipl!'d-dyin ,ii. model which employs a set of dynamically-consistent

the use of any of these techniques as local trajectory generation methods for growing a

probabilistic tree of actions to explore the solution space [40-42].

1.3 Problem Statement

This dissertation considers the problem of planning effective sensing trajectories for a

vehicle-carried sensor operating in a close-proximity environment. Specifically, consider a

class of remote sensors that rely on a clear line-of-sight (LOS) to a target as a necessary

condition for visibility. This class of sensors is assumed to exhibit variations in data

quality resulting from geometric factors related to the relative position and orientation of

the sensor with respect to the target. Consequently, these variations, and hence sensing

effectiveness, can be quantified as a function of sensing geometry. In addition, consider a

vehicle that is subject to differential motion constraints. This vehicle carries a sensor that

belongs to the aforementioned class of LOS remote sensors. The focus of this dissertation

is to develop a method for generating vehicle guidance trajectories that ensure effective

sensing of pre-specified locations embedded within a known operating environment.

Further, the placement of these sensing locations is such that operation of the vehicle and

the sensor are necessarily in close proximity to this environment.

"Close proximity" is loosely defined such that the characteristic dimensions of

the environment, characteristic dimensions of the vehicle dynamics, and typical sensor

operating range are all of the same order of magnitude. Formally, this definition is

intended to reflect several specific requirements that must be imposed upon computed

motion plans:

1. The proximity of vehicle trajectories to obstacles in the environment requires
that differential motion constraints are considered in motion planning. As such,
trajectories must be generated that are consistent with the vehicle dynamics in order
to ensure safe and accurate navigation and tracking.

2. The proximity of targets to sensor vantage points along planned trajectories results
in significant variations in view quality with respect to changes in vantage character-
istics. As such, a metric must be formulated to quantify these variations and must
be subsequently utilized by the motion planning algorithm to ensure effective sensing
trajectories.

3. The proximity of the sensor to the surrounding environment results in a small
projected footprint relative to the characteristic dimensions of the vehicle dynamics.
As such, motion planning must consider effects related to the coupling of sensor
pointing with vehicle maneuvering. Planned trajectories must therefore include
constraints on vehicle/sensor orientation to ensure proper sensor pointing that
accounts for FOV constraints.

Three specific sensing task categories are selected based on their relevance to ISR

applications. These tasks can be described as reconnaissance, surveillance, and area

coverage. Each of these tasks focuses on slightly different mission criteria.

The reconnaissance task assumes that a target or series of targets has been identified

and located. The sensing system is required to visit each target and collect data prior

to either returning home or continuing to some other specified goal. This mission task

is depicted in Figure 1-6A. The purposes of this type of mission task include target

identification, verification, classification, and other 'single-look' reconnaissance objectives.

The surveillance task also assumes that a target or series of targets has been recog-

nized and located; however, the goal has changed to persistent observation. The vehicle

must patrol the series of targets and repeat the sequence until a prespecified termination

condition is met. This mission task is depicted in Figure 1-6B. The purposes of this type

of mission include target verification, classification, monitoring, change detection, and

collection of modeling data.

The coverage task implies no assumptions about target locations. The vehicle must

pass the sensor footprint over the entire area of interest in such a way that some level

of data quality is achieved over the entire task space. This mission task is depicted in

Figure 1-6C. The purposes of this type of mission task include search, explore, target

detection, change detection and wide area surveillance.

A B

Figure 1-6. Sensing mission tasks. A) Reconnaissance mission. B) Surveillance mission.
C) Coverage mission.

Finally, several assumptions are considered in formulating the solution approach.

These main assumptions are as follows:

* Obstacles in the environment are static

* Obstacle and target locations are known a prior

* A vehicle model is available that is capable of representing dynamics at various points
in the operating envelope

* Planned motions are designed as open-loop guidance trajectories

1.4 Approach Overview

The problem stated in Section 1.3 is addressed through three main steps. First, a

method is developed for dynamic point-to-point motion planning in environments that

meet the definition of close proximity as discussed previously. Then, a framework for

quantifying the effects of sensor operation in such environments is developed for use as a

performance metric. Finally, a sensor planning method is developed which integrates this

metric with the point-to-point planning method into a randomized scheme designed to

achieve the tasks specified in Section 1.3.

Motion planning for differentially constrained vehicle systems is examined in Chap-

ter 2. A hybrid representation of the vehicle dynamics which utilizes the concept of

motion primitives is modeled after previous efforts. This hybrid representation is used

to formulate the trajectory planning problem as a hierarchical optimization problem. A

special class of feasible, though suboptimal, explicit solutions are identified and utilized to

approximate optimal solutions. The benefits and drawbacks of these approximations are

demonstrated through simple examples.

The concept of randomized sampling-based motion planning is introduced in ('!C ip-

ter 3 through discussion of several existing algorithms. A randomized planning algorithm

is developed to compute approximate solutions to the optimization problem posed in

C'! Ilpter 2. This method makes effective use of the explicit solution approximations intro-

duced in the previous chapter such that some of the 1n I 'r drawbacks are avoided. This

planning method is demonstrated through a simulated example.

('!i ipter 4 discusses modeling of the sensing task. Specifically, a geometric sensing

model is developed and used to formulate a definition of visibility. Then, the effects of

sensor operation in close proximity to the environment are discussed. A general framework

is introduced to quantify these effects as a measure of data quality. Examples demonstrate

how this framework can be tailored to specific sensing applications.

The sensor-planning task is addressed in ('!C ipter 5. First, a multi-resolution environ-

ment representation is developed to represent the coverage task as a target-centric task.

Consequently, all three of the tasks listed in Section 1.3 can be treated using variations of

one unified method. Then, the randomized planning and sensor effectiveness concepts from

previous chapters are integrated to form a new randomized planning approach to sensing a

series of targets in a close-proximity environment.

The benefits and challenges associated with the application of optimal control

principles to the effective sensing problem are discussed in ('! iplter 6. The necessary

conditions for an optimal solution are derived for a simplified example. Optimal solutions

to a simplified version of the sensing problem are computed using a numerical approach.

Results are then demonstrated through a simulated example.

Finally, C'!I ipter 7 summarizes the project and discusses the utility of the methods

introduced throughout the dissertation. Additionally, recommendations for future work are

provided.

1.5 Contributions

This dissertation considers the problem of planning effective sensing trajectories for a

vehicle-carried sensor operating in a close-proximity environment. Three main technologies

are required to enable solutions to such problems: visibility and sensing effectiveness

concepts for a point in the environment must be characterized, a point-to-point motion

planning method that considers vehicle dynamics must be developed, and a scheme that

allows the application of this motion planning method to sensor planning problems must

be contrived. Aspects of these sub-problems have been researched independently in the

past. The present effort pl-1i- the role of integrating technologies into a functional system.

A variety of modifications are made to existing techniques in addition to the development

of new concepts to enable this integration. As such, several contributions are made in the

process.

* An approximation method to solving both kinematic and dynamic maneuver-based
motion planning problems is developed

* A randomized planning algorithm utilizes this approximation to generate dynamic
goal-based trajectories in complex environments

* A metric of sensing effectiveness is developed to quantify the geometric effects of
sensing at close range

* Concepts are adopted from robotic manipulator-based sensor planning to formulate
the sensing problem as a goal-based planning problem where an effective-sensing goal
set is defined in terms of vehicle configuration

* A randomized approach is developed to sense a series of targets for which a number of
issues related to operation within close-proximity environments are considered

* A sampling-based multi-resolution scheme allows an area-based sensing problem to be
treated as a target-based sensing problem

* A novel application of a recently developed optimal control technique results in
optimal sensing trajectories for a simplified case of the general problem

CHAPTER 2
MOTION PLANNING

2.1 Introduction

Motion planning describes the process of determining the transition of a system from

an initial configuration to a terminal configuration. System configurations are defined as

a vector, xY, on the configuration space, or C-space. For robot and vehicle systems, the

C-space is spanned by the variables that describe the position and orientation of the body

coordinate frame in addition to any other degrees of freedom the system may have. The

terminal or goal configuration is typically specified as a point in the C-space that achieves

some objective such as to arrive at a base location or to point a sensor at a target. Motion

planning determines the function that guides the system from the initial configuration to

this goal.

Motion planning problems for vehicle systems are typically classified according to

three main categories [43]: point-to-point motion, path following, and trajectory tracking.

All three of these classes require that the vehicle move from an initial configuration to a

goal configuration; however, the function describing this motion is constrained differently

for each. The point-to-point motion task places no restrictions or specifications on the

intermediate motions occurring between two configurations as long as the goal is reached.

The path-following class of problems requires that the vehicle follow some specified

continuous path in C-space that satisfies any system differential constraints and which has

the initial and goal configurations as endpoints. Similarly, a continuous path constraint is

imposed for the trajectory tracking problem class; but, the additional requirement of an

associated timing law is included as well. The result of these definitions is that point-to-

point motion plans are typically specified as a series of waypoints, path following motion

plans are specified as a function defined on the C-space, and trajectory plans are specified

as either time-parameterized functions on the C-space or a system input function, il, that

is a function of time.

A particular motion plan can be evaluated according to performance criteria in ad-

dition to the goal configuration endpoint constraint. Such criteria, such as path length,

can be used to characterize a given motion plan as better or worse than another. Con-

sequently, the motion planning problem can be considered as a constrained optimization

problem. An optimal motion plan minimizes (or maximizes) the performance criteria while

satisfying the initial and terminal constraints in addition to any path constraints, such as

those imposed by obstacles in the environment.

This dissertation is concerned with generating guidance trajectories for a vehicle

system given a model of the vehicle motion and a known environment. The environment

is assumed to contain polygonal obstacles with known vertex locations. Further, all

computed trajectories are understood to be open-loop reference trajectories. Practical

implementation of such methods would require some level of feedback control to effectively

deal with disturbances and uncertainties; however, such issues are typically addressed

using a separate control l---r and are beyond the current scope.

The remainder of this chapter discusses deterministic motion planning for a

differentially-constrained system. First, typical path constraints that might be imposed by

obstacles or through the system dynamics are defined and discussed in Section 2.2. Next,

optimal control is examined in the context of benefits and drawbacks to practical planning

problems involving realistic systems. Section 2.4 then introduces a modeling alternative

that provides solutions through lower-dimensional optimization problems than those typ-

ically encountered when using direct optimal control methods. This method utilizes the

concept of motion primitives and even admits closed-form algebraic solutions to special

cases of the planning problem formulation.

2.2 Path Constraints

Path constraints impose restrictions on the values of the configuration variables,

Yx E C, and/or their derivatives along the trajectory. These constraints can be written

in general form as Equation 2-1. The constraint function, c zc, c, c,u,t) represents a

vector of constraints such that the inequality is applied component-wise.

c xcx, X ,, t) < 0 (2-1)

Equation 2-1 can be used to require avoidance of obstacles, to enforce differential

system behavior, to bound allowable input signals, and to restrict the operational envelope

of the system. A path constraint can be applied to a particular point along a trajectory

or to a continuum of points along a trajectory segment. The former case is required for

finite-dimensional trajectory optimization problems while either usage is acceptable for

infinite-dimensional functional optimization problems.

2.2.1 Differential Constraints

The vehicle systems considered in this dissertation are described by systems of

differential equations and are therefore subject to motion constraints imposed by the

dynamic behavior of solutions to these equations. The system dynamics impose an

important class of constraints that restrict allowable system velocities and allow kinematic

and dynamic vehicle behavior to be considered in the motion plan. Integration of these

differential behaviors into the planning process ensures enhanced accuracy of a given

motion plan with respect to trajectories that are executable by the physical system.

Differential motion constraints act on the time derivatives of the configuration

variables, which can be described in terms of the state-space, X. The state-space is

spanned by the minimum set of variables required to completely determine the motion

of the system at a given point in time. The state vector, xs, therefore includes all of the

configuration variables in addition to relevant velocities and accelerations. The C-space is

thus a subspace of X for a particular system, such that C C X, and where the dimension

of X exhibits an increase of at least a two-fold over the dimension of C.

Differential constraints on system motion can be classified as kinematic or dynamic

constraints. Typically, kinematics deals with descriptions of system motions independently

from the forces and moments required to generate these motions. Physically speaking,

dynamics relate the interaction of forces and moments with system inertial properties

to generate accelerations; however, in general the term is often used in reference to any

time-varying behavior. Further, the definition of a dynamic system model as compared to

a kinematic system model is blurred by the direct relationship of the kinematic concept

of acceleration to the dynamic concept of the forces that affect acceleration through

Newton's second law. In other words, a "dyri .i... constraint regarding allowable system

forces is directly related to a 1:.ii 11i iI ,:" constraint on system accelerations.

These technicalities are the source of a great deal of confusion in motion planning lit-

erature, which lies at the intersection of robotics, control theory, and applied mathematics.

The remainder of this dissertation considers first-order differential constraints on the C-

space as kinematic constraints (Equation 2-2). First-order constraints on the state-space,

which include first and second-order differential constraints on the C-space, are considered

dynamic constraints (Equation 2-3). Motion planning under this definition of differential

constraints is referred to in the robotics literature as i .:..,li, :. motion planning [44].

X = f (XY ), C (2-2)

Xs = f(xs, ), s E X (2-3)

This dissertation considers two broadly-defined approaches for generating a trajectory

that satisfies system dynamics:

1. Directly plan a trajectory constrained by the system dynamics

2. Decouple the problem by first planning a trajectory with relaxed constraints, then
applyingg the dynamics to the result

The first approach is enabled using tools from optimal control theory and can yield

optimal solutions, but is plagued by problem complexity for practical applications. The

decoupled approach can be effective to a varying degree depending on the choice of

planning method for the relaxed-constraint problem as well as the choice of method for

application of the dynamics to the relaxed-constraint solution. For example, a simple

point-to-point planning approach can quickly generate a set of desired waypoints. Dynam-

ics can be applied through feedback control that steers the system toward the appropriate

waypoint. Alternatively, solution of an optimal control problem that is constrained to pass

through the waypoints would successfully introduce dynamics as well. Non-traditional

methods exist as well, such as smoothing with primitives that satisfy the system dynamics,

as emploi-, 1 by Bottasso et al [45]. Each of these subproblems are easier than the global

problem; however, some level of performance tradeoff is likely.

As a matter of notation, x will be used interchangeably for the remainder of the

dissertation to represent the state vector, Yx E X, and the configuration vector, Yx E C.

Many of the described techniques operate in a similar fashion when a system is classified

as either dynamic or kinematic. The definition will be stated explicitly when the correct

usage is not apparent from the context.

2.2.2 Obstacle Constraints

This dissertation considers polygonal obstacles for the case of a two-dimensional

environment and extruded p" ..li-ons for the case of a three-dimensional environment. This

definition allows three-dimensional obstacles to be treated as planar polygonal obstacles

at particular locations along the extrusion axis. As such, the subsequent discussion

of collision-checking considers the two-dimensional case of testing a planar point for

intersection with a polygonal obstacle. Without loss of generality, the extrusion axis is

presently assumed parallel to the vertical direction such that planar position is described

by the position vector, AD = [ p Py ]T. The planar positions of the n vertices of a

particular obstacle are given by i,, = [ PT,,, Py,, ], i =1 ..- n. These position vectors

locate points relative to the origin of an inertial reference frame.

Safe motion plans amongst obstacles require that planned trajectories never intersect

any boundary of any obstacle, or equivalently, that the configuration variables describing

position never reach a value on the boundary or inside any obstacle. This condition can be

expressed in several different v--v for the type of obstacles considered here.

One method involves expressing the obstacle boundary as a radius function defined in

cylindrical coordinates centered at an arbitrary point, c = [ pI,c Pv,c ]', that is located

within the obstacle p" ..Il on. This obstacle coordinate frame is simply a translated <" iv of

the inertial reference frame. The obstacle boundary is expressed as rebound( obs), where Oobs

represents a bearing from the obstacle relative to the xob8-axis, which is oriented parallel

to the inertial x-axis. The path constraint restricting PD to values that lie outside the

boundary of a particular obstacle can be expressed as Equation 2-4, where rp represents

the range of AD from ic and 0obs,p represents the bearing of AD relative to ;c. These

relations are given by Equations 2-5 and 2-6, respectively.

rp rbound(Oobs,p)) < 0 (2-4)

rp = || 2D Pc || (2-5)

0obs,p = arctan Py ) (2-6)

Computation of rbound(6obs) requires parameterization of the edges that represent

the obstacle boundary with respect to Oobs. The inertial position of a point along the

obstacle edge connecting the ith obstacle vertex to the jth obstacle vertex is expressed as

Pe, = [ pxe Py,ej ]T The components of this vector can be parameterized as shown
in Equations 2-7 and 2-8. The boundary of the obstacle is then computed using the

piecewise-continuous radius function shown in Equation 2-9, where Oobs,vi and Oobs,vj

represent the bearings of the ith and jLh obstacle vertices relative to c, respectively.

This radius function, rbound,act is depicted as a solid black line in Figure 2-1 for a square

obstacle. The graphic shown in Figure 2-1A shows fe, relative to the reference frame

centered at c = [ 0 0 ]T. The graphic shown in Figure 2-1B shows rbound(obs).

S ( Py,vi + Py,c +pz,c +e Pvi tan obs
Pxe[ (obs) Px,vi (Px,vj n+ + Px, +,)-- tanOobs_ (2-7)
( px,v, pz,Vi) tan Oobs Py,vj Py,vi)
( y,vi + Py,c + Px,c + Px,vi tan Oobs (
Py,e (Oobs) = Py,v (Py, Py, (p,vj p ) tan -obs p-- ,v) (2-8)
( \Px,V Px,Vi) tan Bobos {Py,vj ~ Py,Vi)

\/(Px,eI2 P,c)2 + (Py,e12 Py,c)2 Oobs,v1 < Oobs < 0obs,v2

00xe3 Px c)~ + (Pge03 Pgc)2 obs,vs no "ons K Hobsnvs
rbound,act (Oobs) P < (2-9)

\/(Px,en Pc)2 + (Py,e Py,c) 2 Oobs,v, < Oobs < Oobs,v1
Equation 2-9 accurately represents the boundary of a polygon as a radius function

for use with constraints of the form of Equation 2-4. Unfortunately, the non-smooth

nature of this function can present problems for gradient-based optimization techniques.

Alternatively, a smooth radius function that conservatively approximates the boundary

can be computed. An example of such an approximation is shown in Equation 2-10. This

function, rbound,circ, simply approximates the obstacle boundary as a circle with radius

equal to the maximum value of rbound,act. This approximation is shown in Figure 2-1 as a

dotted red line. The figure shows that the use of such a conservative approximation can

restrict a significant portion of the obstacle-free space within the proximity of the obstacle

being approximated. The number of feasible trajectories could be severely limited by

restricting this space, especially for the case of closely-spaced obstacles.

rbound,circ (Oobs) rbound,act oo (2-10)

Another conservative approximation utilizes the concept of superellipses [46]. This

family of curves represents a generalization of the ellipse that can be used to approximate

rectangular shapes, among others. The obstacle boundary can be represented as a param-

eterized radius function given by a superellipse as shown in Equation 2-11, where as and

b,, represent the maximum radius in the Xobs and yobs directions, respectively. The radius

is thus given as a function of the relative bearing, Oobs. Equation 2-11 approximates a rect-

angle when the exponent, 7e, is a positive, even integer. A superelliptical approximation

to the obstacle boundary is shown in Figure 2-1 as a dashed blue line. The curve shown in

the figure is computed using as = 1.2, be, = 1.2, and 7,s = 8. The figure shows that this

range function provides a more reasonable conservative estimate as compared to the radius

function expressed in Equation 2-10.

rboundse (Oobs) = os + ob, (2-11)
a. se b() ( )i ) 1

2 2

1 1.5 ,

0 :1

-1 0.5
.- ---- -.-,-: 7 '"

2 0 2 0 45 90 135 180
Yobs 0obs (deg)
A B

Figure 2-1. Obstacle boundary approximations. Approximation functions are shown as
approximated boundary. B) Radius approximations parameterized on Oobs

Alternative methods for collision detection avoid the need to explicitly represent the

obstacle boundary. As such, these methods do not give a measure of the proximity of AD

to a given obstacle. Rather, these methods simply provide a binary flag indicating the

collision status of a test point with respect to each obstacle. These methods are therefore

not suitable for use with gradient-based optimization methods.

One such method requires that the set of vectors, ivpV, = v, AD, i = 1 n, is

computed. These vectors represent the vectors from the test point to each obstacle vertex,

and are shown in Figure 2-2. The interior angles between each .,li i,:ent pair of ffpvi,j ,

denoted aobs,ij can be computed using the dot-product operation. A necessary and

sufficient condition for a collision-free point is given by the angle-sum inequality shown in

Equation 2-12.

Saobs,ij < 2 i 1 n (2-12)
i

U obs, y

Ul "--- U,

Figure 2-2. Vertex-angle sum collision detection method.

Another method that requires computation of ipv,, is depicted in Figure 2-3. This

method additionally requires computation of the set of edge vectors. Each of these

vectors corresponds to a polygon edge and indicates edge direction in a clockwise sense.

The edge vector based at the ith vertex points to the jth vertex and can be expressed

as Zfe, = fj iv,. The unit normal to iec, is denoted nfj and is also based at the ith

vertex. This normal vector can be computed via a rotation through r/2 rad, as shown

in Equation 2-13. A necessary and sufficient condition for a collision-free point is met if

Equation 2-14 is satisfied for any (i, j) pair. This condition essentially states that if the

test point is located in the same direction relative to each edge vector, the point lies inside

the ]" Iv.-_on.
---------- ---------------------(

U, hAUi'

Figure 2-3. Vertex edge-vector collision detection method.

1 I cos(7r/2) -sin (7r/2) (213)
I 9 9I sin (7/2) cos (7/2) )

sign(nie, (-4vJ)) < 0

As discussed previously, planning algorithms that rely upon a finite-dimensional func-

tion optimization must enforce path constraints pointwise at a finite set of points placed

along the trajectory. Enforcement of obstacle constraints therefore cannot guarantee a

collision-free trajectory between these test points. Example trajectories can be constructed

that collide with an obstacle, yet technically satisfy all obstacle constraints, as shown

in Figure 2-4A. Obstacle boundaries can be expanded as shown in Figure 2-4B to avoid

such situations. The extent of expansion must be carefully selected to ensure that unsafe

trajectories cannot feasibly satisfy obstacle constraints.

A B

Figure 2-4. Obstacle expansion for pointwise safety. A) Pointwise satisfaction of obstacle
constraints. B) Pointwise satisfaction of expanded obstacle constraints.

2.3 Optimal Control

Optimal control theory provides a rigorous approach to trajectory design for

differentially-constrained systems. The objective of optimal control is to determine

the vector-valued input function, u(t), that drives the state trajectory, x(t), such that a

scalar-valued performance index, J(x(t), u(t), t), is minimized within a time interval of

interest. The state trajectory must satisfy differential constraints specified by the system

dynamics in addition to any algebraic path constraints, such as obstacle regions or state

and control bounds. Endpoint constraints are also imposed at the initial and terminal

conditions of the trajectory. Mathematically, the optimal control problem can be posed as

(2-14)

an optimization problem as shown by Problem 2-15.

(x(t)*, (t)*) arg min J((t), (t),t) (2-15)
x(t),u(t)
s.t.

x(t)- f(x(t),u(t)) = 0

S(xo,to,xf,tf) < 0

c(Q(t), u(t)) < 0

Time arguments are retained in Problem 2-15 to emphasize that the problem seeks

to determine continuous functions, and therefore has infinite dimension. The differential

path constraints are enforced in equality; therefore, they are written separately from the

algebraic path constraints, c(x(t), u(t)). The function, '(xYo, to, Xf, tf), relates endpoint

constraints on the trajectory. Finally, the functional performance index, J(7(t), i(t), t) is

typically written in integral form as in Equation 2-16.

J(M(t), (t)) = ((tf),tf) + t L(Y(t), (t))dt (216)

2.3.1 Indirect Optimal Control

Indirect solution methods to optimal control problems seek the input function and

corresponding state trajectory that satisfy the necessary conditions for an extremal of the

performance index, subject to the constraints. These extremals occur at points where the

performance index is considered stationary. Identification of stationary points requires the

use of variational calculus concepts, as the performance index is described by a functional.

Consider an instance of Problem 2-15 for which there are no algebraic path con-

straints and the endpoint constraint is applied in equality. The cost functional can be

augmented with the differential and endpoint constraints through the introduction of a

vector of Lagrange multipliers, 1i, and a vector of multiplier functions, X(t). Each ele-

ment of 7 is associated with an endpoint constraint while each element of X(t), denoted

the costate, is associated with a differential constraint. The augmented cost functional
is written as J, which is shown in Equation 2-17. In this expression, is denoted the
Hamiltonian of the system and is shown as Equation 2-18. Function arguments have been

omitted here to maintain clarity in the expressions. The constrained optimization problem
in J is now re-expressed as an unconstrained optimization problem in J.

J [- tf + A tjdt (2-17)

= L +Tf(x, u) (2-18)

A set of necessary conditions for a stationary point of J can be found by taking the
first variation and setting it equal to zero, as 6J = 0. A full derivation of these conditions

is beyond the scope of this dissertation but and is available in the literature [47]. The first
set of conditions are given by the costate dynamics, as shown in Equation 2-19. These
differential equations have boundary conditions given by the transversality conditions, as

shown in Equations 2-20 and 2-21.

A (2 19)

XT(to) + T (2-20)
aY(to) aO(to)
(tf) (2 21)
ax9tf) ax;tf)

The optimal control is related by Pontryagin's minimum principle, which states that the

optimal control minimizes the Hamiltonian [48]. The weak form of this condition is shown
as Equation 2-22.
S0 (2-22)

Equation 2-22 relates the optimal control in terms of the state and costate. This
control function can be recovered if the optimal state and costate trajectories are deter-

mined. As such, the optimal control problem can be solved indirectly through solution of

the boundary-value problem (BVP) specified by the state dynamics, the costate dynamics,

the state endpoint constraints, and the transversality conditions. When the state vector

has a dimension of n, solution of this BVP involves solving a coupled differential system

of 2n equations with split-end boundary conditions. Further, some of these boundary

conditions may only be known as a function of the unknown Lagrange multipliers, i.

Consequently, iterative numerical techniques often must be employ, ,1 to determine these

boundary conditions.

Further challenges are introduced when optimal control theory is applied to problems

containing realistic constraints. Inclusion of algebraic path constraints, such as those

imposed by obstacles or system operating restrictions, result in a differential algebraic

system that introduces a number of additional complications [49]. Path or endpoint

constraints that are enforced as inequalities can cause an increase in computational burden

which results from the need to identify the active and inactive elements of these constraint

functions. Further, the optimality condition does not alv--, yield a closed-form expression

for il(t) in terms of the state and costate. Sometimes this condition does not even uniquely

determine the input function. While the indirect approach to solving optimal control

problems utilizes a rigorous mathematical framework, all but trivial problems are often

rendered impractical.

2.3.2 Direct Optimal Control

Various direct numerical methods for solving optimal control problems have emerged

as an alternative to the burdensome indirect approach [49-52]. Such methods involve

transcription of the infinite-dimensional functional optimization shown in Problem 2-15

to a finite-dimensional function optimization. As such, derivation of necessary conditions

and the use of variational calculus is unnecessary; instead, numerical optimization tech-

niques are used to solve the nonlinear program (NLP) which results from the problem

transcription.

The first step in the transcription process is to split the time interval, t E [to, tf], into

a finite set of fixed sub-intervals, as shown by Equation 2-23. The N points at which the

time interval is discretized are denoted nodes.

to < ti < t2 < < tN-1 tf (2-23)

The continuous functions that represent the state and control trajectories are dis-

cretized, as well. These functions are replaced by finite sets of values comprised of each

function evaluated at each of the N nodes. The combined set of these values make up

the decision variables for the subsequent NLP formulation. The vector of these values is

denoted X, as shown in Equation 2-24.

S [ ((to0))T ((to0))T ((t1)) (UT(ti))7 ... ((tN- 1)) ((tN- 1)) (2 24)

The cost functional, J, must be replaced by a finite-dimensional counterpart, denoted

F. The first term in a performance index of the form shown in Equation 2-16 remains

unchanged through transcription if the terminal time, tf, coincides with one of the nodes

in Equation 2-23. The integral term can be approximated as a finite sum using numerical

integration techniques. Similarly, numerical integration is used to represent the differential

constraints as a vector of defects, denoted as (. Each defect enforces the integration

rule between the corresponding pair of nodes. The defect corresponding to the kt node

is shown in Equation 2-25, where f represents the integration rule. Dependence of f

is shown with respect to state and control values at the current and next time step to

maintain generality; however, rn- in: integration techniques only require information at

the current time step. Examples of such techniques include Euler integration and the

Runge-Kutta method.

(k (t+1) k(t) f ((tk), 1(tk),i(tk+l), (tk+l)) (2-25)

The endpoint constraints from Problem 2-15 can be applied directly within the

transcribed formulation if to and tf are included as nodes. Additionally, algebraic path

constraints are applied to the state and control pointwise at each node. These constraints

can be combined with the vector of defect constraints into a single vector of constraint

functions, as shown in Equation 2-26.

S(xAo,to, xf,tf)
C(X = c( (2-26)

4(X)
Finally, the pieces can be assembled as the NLP shown in Problem 2-27. This prob-

lem contains at least N(n + m) variables for a system with n states and m controls, and

thus is very large for even a coarse time-discretization. Fortunately, the relevant matrices

involved in solution of the NLP take on a sparse form as a result of the transcription

method [49]. This sparsity can be exploited to greatly reduce both storage and computa-

tion time. Several NLP solvers that exhibit this capability have been shown to perform

quite well in handling problems of this nature [53].

X* argminF ) (2-27)

s.t.

c(s

Direct transcription provides a practical alternative to indirect methods when consid-

ering realistic optimal control problems; however, the method does have some disadvan-

tages. First, constraints are only applied at the nodes. As such, behavior of the trajectory,

including constraint satisfaction, cannot be guaranteed between nodes. Additionally, NLP

solvers are typically susceptible to local minima and often exhibit dependence on the

choice of initial condition. More importantly, Problem 2-27 is fundamentally a different

problem than Problem 2-15. An interpolated function through the discrete solution values

to the finite-dimensional NLP may not coincide with the optimal solution to the infinite-

dimensional problem. Further, as the necessary conditions are not computed, there is no

way to validate solutions to Problem 2-27. Progress has been made in this area recently

through the development of costate mapping techniques that utilize pseudospectral col-

location methods [51, 52]. Such methods represent an active area of research in optimal

control theory.

2.4 Hybrid Motion Models

2.4.1 Modeling with Motion Primitives

A fundamental property that is common to many vehicle systems of interest is related

to the concept of symmetry. Specifically, the trajectories of a certain subset of the state

or configuration variables are seen to exhibit invariance with respect to certain classes of

transformations. Let x represent either the state or configuration vector of a system whose

dynamics are represented in the form of Equation 2-2 or 2-3, respectively. A system

trajectory can be determined through integration of the dynamics with respect to time.

Now consider a transformation function, H : C -+ C (or H : X -+ X). System trajectories

are said to exhibit invariance with respect to the transformation, H, if Equation 2-28

holds true [39].

(t) =H( f (Fo, )dt) = f(H (fo) u)dt (2-28)

Now, let two trajectories be considered equivalent if they can be exactly superimposed

through time translation and application of H [39]. Satisfying Equation 2-28 thus implies

that a trajectory generated from a particular initial condition, xo, through the application

of an input function, u, is equivalent to all trajectories that result from the application of

u and that are initialized at any point in the range, H(xo). Trajectories that satisfy this

property are denoted motion primitives.

Physically, H represents the class of transformations that do not affect the external

forces acting on the vehicle. For example, consider a car-like vehicle operating on an

expansive, flat, isotropic surface. Forces on the vehicle result mainly from gravity and

friction. Relative displacements in position and heading that result from a particular input

trajectory are invariant to absolute position and heading. Therefore, translations and

rotations in the plane comprise the class of symmetric transformations, H. Alternatively,

vehicle operation on a surface with variable material properties, such as transitions from

concrete to gravel, exhibits variable friction forces as a function of position. Application

of a particular input function results in different relative displacements depending on

the position on the surface. For the former case, any trajectory on the surface can be

reproduced at a different location and heading through application of H. Such a trajectory

can be described as a motion primitive. Conversely, the latter case requires full integration

of the dynamics for different initial conditions. Note that the invariance properties of some

system variables depend on the operating environment. As such, the operating conditions

must be considered when constructing motion primitives for a particular system.

The concept of motion primitives leads to a useful framework for the simplification of

complicated dynamic models. This framework involves combining sequences of compatible

primitives to represent complicated trajectories. A set of compatibility conditions are

detailed in the literature [39]. For the systems considered in this dissertation, the state

displacement resulting from a finite-duration motion primitive can be represented by the

transformation, GM, as shown in Equations 2-29 and 2-30. In Equation 2-30, the relative

state displacement due to the motion primitive, ZA, is translated by xo and rotated

by R(xo). A discrete set of reachable configurations can be achieved through successive

application of compatible transformations of the form, GM.

S= GM (o) (2-29)

GM (o) = o + R(o)-m (2 30)

In addition to these finite-duration primitive solutions, the concept of trim motion

can be used to generate continuously-parameterized families of system trajectories. Trim

motion is characterized by steady-state motions with fixed controls for the systems

considered here [38]. A trim primitive then describes the time-evolution of non-invariant

states resulting from constant values of the invariant states. Continuous state trajectories

are represented by the time-parameterized transformation, GT, as shown in Equations 2

31 and 2-32. This transformation describes the state displacement from the initial

condition, xo, along a trim trajectory after a time duration, r. As before, the relative

displacement due to the motion primitive, Ar(T) is translated by 1o and is rotated by

the transformation, R(xo).

(r) = GT (xo,T) (2-31)

GT (o, ') = Yo + R(xo) AT(r) (2-32)

While executing a trim motion, the reachable set of the system lies along the curve

described by Equation 2-32. This set can be greatly expanded by implementing a finite

number of trim trajectories in a hybrid switching scheme that employs unsteady, finite-

duration primitives to transition from one steady-state trim motion to another. This

model can be represented as a finite-state automaton, as depicted in Figure 2-5. Each

state of the automaton, depicted as nodes of the directed graph in Figure 2-5, represents

a trim primitive, Ti, and is defined by the steady trim velocities that characterize each

particular trim. Motion associated with Ti is governed by a transformation of the form

shown by Equation 2-31. The state transitions of the automaton are depicted as directed

edges of the graph in Figure 2-5 and represent the finite-duration, unsteady transitions,

if between each pair of trim states, Ti and Tj. These unsteady motions are denoted

maneuvers and must originate and terminate in steady-state motion as a condition for

inclusion in the automaton framework. Motion resulting from the execution of a maneuver

is governed by a finite displacement of the form shown in Equation 2-29. The graph

structure represents allowable switching behavior between primitives as required by

compatibility conditions.

Maneuvers can be realized as an instantaneous switch or a smooth dynamic motion,

depending on the extent to which the system is differentially-constrained. Kinematic

systems, as defined previously, allow instantaneous changes in configuration velocities and

Figure 2-5. Automaton model representation as a directed graph

can therefore transition between trims with no configuration changes. Dynamic systems

exhibit bounded accelerations and, as such, smooth system configuration changes will

accompany any transitions between trim states.

The automaton modeling scheme depicted in Figure 2-5 can represent complicated

system trajectories through specification of a sequence of maneuvers and the duration of

the trim states between each consecutive maneuver. This model exhibits both continuous

behavior and discrete switching behavior in the trim states and finite-duration maneuvers,

respectively, and is therefore considered a hybrid system. Given a maneuver sequence

of length n, M = {M1, M., }, and a corresponding length n + 1 sequence of

trim durations, r {-ri, 7 Tn+l }, system trajectories in the C-space are computed

via a series of transformations. Such trajectories alv--i initiate and terminate in a

trim configuration. Equations 2-33 and 2-34 show a general example of this process for

maneuver and time sequences of M = {MI, M.}, and T {Tri, T2, 73}, respectively, where g

represents the total state displacement resulting from the sequence.

x= (x0o,M,r) (2-33)

(fo,, T) GT3 (GM2 (GT2 (GM2 (GT, (fo, Ti)) ,72)) ,T3) (2-34)

2.4.2 Motion Planning with Primitives

Planning a trajectory with motion primitives requires that a sequence of maneuvers,

M, and the associated trim durations, r, are selected such that the system is driven from

the initial configuration, xo, to a terminal configuration, xf, with favorable performance as

evaluated by a performance function, J(M, r, xo). The trajectory planning problem for a

system described by motion primitives can be described by the optimization problem 2-35.

(M*,*) argmin J(M, r7,o) (2-35)
M,T
s.t.

f Qo, M,) 0

(M,7,fYo) < 0

i I E M Vi

ri > 0 Vi

Problem 2-35 presents a difficult mixed-integer nonlinear program (\I NLP); however,

the problem structure does lend itself to a hierarchical decomposition [39]. Given a

maneuver sequence, M, Problem 2-35 reduces to a smooth NLP in the trim durations,

7, that is essentially analogous to an inverse kinematics problem. Such problems are a

common and well-studied class of problems in robot geometry [54]. Hence, a combinatorial

search through the set of all possible M accompanied by the solution of a smooth NLP

for each choice of M is generally required to solve Problem 2-35. The length, n*, of

the optimal sequence, M*, is not known in general. As such, the set of all possible M

is countably infinite; however, it can be shown that a finite-length optimal sequence

exists and can be determined explicitly in some special cases. Additionally, the literature

-,-.-.- that pruning and branch-and-bound techniques can be applied to simplify and

expedite the combinatorial search [39, 55].

Alternatively, consider a family of sequences where each member sequence consists of

a fixed number of motion primitives. Each trim primitive in each member sequence differs

in the magnitudes of the trim velocities; however,the general shape of the state trajectories

remains unchanged. Maneuver primitives differ as needed between member sequences to

accommodate the changes in trim states. As with the trim states, the general shape of

these finite-duration trajectories should remain unchanged. A particular automaton model

can be associated with a finite number of primitive sequences for a given family, as the

model is comprised of finite number of primitives.

Certain sequence families can be shown to admit unique solutions to the endpoint

constraint of Problem 2-35. This constraint is repeated as Equation 2-36, where f

represents the constrained terminal state and g(xo, M, r) represents the .. .- regate state

displacement of the primitive sequence characterized by M and r. At most one feasible

solution exists to the continuous subproblem corresponding to each member sequence

of families exhibiting this property. Further, Equation 2-36 can often be manipulated

to yield a closed-form expression for this family of solutions. As such, these potential

solutions can be efficiently computed for each member sequence and evaluated for both

performance and for satisfaction of path constraints. Note that differential path con-

straints are implicitly handled by the automaton representation of the dynamics. Feasible

solutions can be enumerated for each sequence in a particular family and used to deter-

mine an approximate solution to Problem 2-35. Solutions obtained in this fashion can be

shown to be optimal for some special cases, as will be demonstrated in Section 2.4.3.

Xf g(X, M, 7) = 0 (2-36)

This solution approximation method is imperfect, but efficient and effective for

problems with few path constraints and with unique solution families consisting of short

primitive sequences. Problems that contain more restrictive path constraints are less

likely to yield feasible solutions via this method. Additionally, families that consist of

many primitives admit a prohibitive number of sequence enumerations for all but trivial

automaton models. The true utility of this method for the purposes of this dissertation

comes as a local solution component within a randomized planning scheme, which is

discussed in the next chapter.

2.4.3 Kinematic Example

The Dubins car has served as a frequently-used kinematic model to approximate

solutions to a v ,ii I of non-holonomic motion planning problems [17, 18, 23, 30, 56, 57].

Further, the typical formulation of the model lends itself well to representation in terms of

motion primitives. As such, this model serves as an appropriate and relevant model for use

in a kinematic planning example.

This example demonstrates the unique solution family approach to find approximate

solutions to a problem of the form of Problem 2-35. Minimum time trajectories between

two configurations are sought for the Dubins car model in an environment with no

obstacles and, therefore, no path constraints. The Dubins car is a simple car-like vehicle

model that operates in a C-space spanned by two Euclidean position variables, p, and py,

and an angle describing the heading, Q. The car moves with unit forward velocity and

changes direction by assuming a unit turn rate, w, in either direction. As such, the motion

of the Dubins car is described by the differential system shown in Equation 2-37, where

the discrete set of values assumed by w is shown by Equation 2-38.

9p cos Q

py sin (2-37)

where:

wE {-1, 0, +1} /sec (2-38)

A result of the limited set of turn rate commands is that the Dubins car moves

motion. These three basic motions exhibit constant body-fixed forward velocity, V

2 + p2, and body-fixed turn rate, p, and therefore satisfy the definition of a trim

primitive introduced in Section 2.4.1. These motions are invariant with respect to position

and I. i1,i- and therefore can be translated and rotated to originate from any point

within the C-space of the Dubins car. The transformations that describe evolution of

the configuration variables along trim trajectories are described by Equations 2-39 to 2

41 for turning motions and Equations 2-42 to 2-44 for straight motion. Note that

Equation 2-39 can represent motion along either a right or left turn based on the sign of w

in Equation 2-41.

Gtu (1xo, T-) = xo + Rt,.(xfo) Aurn(r) (2-39)

cos', -sin',, 0

Rturn(io) = sin,',, cos ,',, 0 (240)

0 0 1

1 2(1 cos(Twr)cos ()

Atun(T) 1 2(1 cos(wr) sin () (2-41)

aPT

Gstraight(YO, 7') = + Rstraight(YO) straight(T) (2-42)

cos ', sin 0

Rstraight(XY) = sin ',, cos ',, 0 (2-43)

0 0 1
T
Astraight(T) = 0 0 (2-44)

These three trim primitives can be integrated into a simple three-state automaton

switching structure like that described in Section 2.4.1. The trim states are denoted L ,

S and R which correspond to the left, straight, and right trim motions, respectively.

The six maneuvers that interconnect the trim states consist of instantaneous switches cor-

responding to an instantaneous change of turn rate, w. Recall that such an instantaneous

switch is permissible for a kinematic model.

In general, solution of a particular motion planning problem would require the

solution of the MINLP expressed in Problem 2-35. A potentially infinite set of automaton

sequences could be enumerated and the resulting constrained NLP solved for each.

Alternatively, two families of solution sequences exist that uniquely satisfy Equation 2

36 when turning motion is restricted to heading changes of less than 27 rad. These

length-three sequence families consist of the set of all turn-straight-turn sequences and

the set of all turn-turn-turn sequences. The member sequences of these families can

be enumerated for the current automaton model in terms of trim motions as D1 =

{LSL, RSR, RSL, LSR} and D2 = {RLR, LRL}. The solution to Problem 2-35 can

then be approximated by computing the solutions corresponding to each of the six

sequences contained in D = (D1 U D2). Specifically, Equation 2-36 can be solved to

obtain feasible trim durations for each sequence. The resulting set of six trajectories are

then evaluated for time performance, where the minimum time trajectory is kept as the

solution. Dubins showed in 1957 that the optimal minimum time trajectory for the Dubins

car consists of motions described by one of the sequences in the set, ED [56]. Consequently,

the approximate solution computed using the described method corresponds to the actual

optimal solution for this special case.

T ii:y sequences that are not contained in iD also satisfy the endpoint constraint;

however, these solution sequences generally do not yield a unique solution. Feasible

sequences of length-four or greater overdetermine Equation 2-36 in the trim durations

and therefore generally admit an infinite number of solutions. While an optimal solution

with respect to a particular performance criterion can be found through the solution of

an NLP, these sub-optimization problems do not necessarily guarantee desirable convexity

and conditioning properties, ie., local and poorly conditioned solution cases may exist.

Actual computation of the trim durations that satisfy the configuration endpoint

constraint for sequences contained in ED can be achieved using several different tech-

niques from the literature [58-63]. Here, a particular method which utilizes algebraic

solutions is adopted [61]. This method requires that the initial and goal configurations

are transformed to a special canonical coordinate frame, D, as shown in Figure 2-6. This

coordinate frame has (px,oPyo) located at the origin and the goal configuration located a

distance, d, along the XD-axis. Initial and final headings are measured from the line con-

necting the initial and final positions, as shown in Figure 2-6. Therefore the transformed

initial and final configurations are given as Equations 2-45 and 2-46, respectively.

YD

1 2

IF ------ W _ _ -- - -_ .... __
(0,0) XD (d, 0)

Figure 2-6. Special reference frame, D for Dubins path solution.

X,D = 0 0 (2-45)
T
Xf,D = d 0 1)2 (246)

The transformations describing the left turn, right turn, and straight ahead motions

are rewritten in an alternate form as Equations 2-47, 2-48, and 2-49, respectively. The

resulting motions are unchanged as compared to the primitives defined previously.

L(px,py, r) = (px + sin(p + r) sin py cos(Q + r) + cos + + r) (2-47)

R(ppy,, r) = (px sin( r) + sin py + cos( r) cos b r) (2-48)

S(px, py, ) = (px + cos y + Trsin Q) (2-49)

For each sequence in D, the transformations in Equations 2-47, 2-48, and 2-49

can be applied to the initial configuration consecutively in the proper order through the

composition operation. The result in each case is then equated to final configuration. A

series of algebraic and trigonometric substitutions are then used to achieve closed-form
expressions for the trim durations, Ti, T2, T3. Infeasible solutions return complex values
for at least one trim duration. The resulting closed-form expressions for each sequence
are [61]:

1. L(S(L(po,Ppyo,', 71), T2), T3)) (d, 0, 2)

Ti -61 + arctan CS si-nCOS {mod 2w} (2-50a)
Sd + sin "1 sinS2 2
72 /2 + d2 -2 cos(i 2) + 2d(sin sin,' .) (2-50b)
( COS' COS~ 7
73 2 w/- arctan CS" {mod 271} (2-50c)
d + sin 1 sin s2

2. R (S(R(p., o, '71), 72), 73)) -(d, 0, 2)

1 = "1 arctan S {od 2} (2 51a)
d sini + sin2 mod 2
T2 /2 + d2 2cos(2i "b2) + 2d(sin '. sin 1) (2-51b)
( COSI COS d r
r73 -w 2 + arctan -- sinC-- S I {rmod 27} (2-51c)
3 d sin 6 + sin' _. {

3. R(S(L(p,Ppyo,',, 71), 72),73)) (d,0, 2)

( cos i cos P2 -2
T = -' + arctan sin CsOS'- 2 -arctan {mod 27} (2-52a)
d + sin + sin 2 72
72 \-2 + d2+ 2 cos(Q1i b2) + 2d(sin' i + sin 2) (2-52b)
(- cosin cos '(r. (2 -25
73 -'+ arctan -OS 6 c-CSI arctan mod 2}) (2-52c)
( d + sin 6 + sin ,- 72

4. L(S(R(pz, 'o,, 71), 72),73)) -(d,0, 2)
( cos, +cos, (2)
Ti i arctan -sin '- sin'2 + arctan ( ) {mod 27} (2-53a)
( d sin sin b2 )72
72 --2 + d2 + 2 cos( 2) 2d(sin + sin 2) (2-53b)
( cos ~1i +cos ~2 2('2
73 arctan d in1 + COsin + arctan ( 2- {mod 27} (2-53c)
d sin6 sin,'-. 72

5. R(L(R(p ,py,',, 71,),T72),73)) (d,0, 2)

( cos 1 cos 2 72
T arctan --sIin + ) {mod 27} (2-54a)
d sin Q1 + sin ]2 2/
72 -arccos ((1/8) (6d2 + 2 cos( 2) + 2d (sin sin 2))) (2 54b)
73 2 71+ 72 (2-54c)

6. L(R(L(peo po',, T1),,T2),,T3)) (d, 0, 2)

Ti 6 + arctan i( si + COS ) + ( {)mod 24} (2-55a)
Sd + sin sin 22 2/
T2 = arTCOS ((1/8) (6d2 + 2 cos (Qi 2) + 2d (- sin i + sin b2))) (2-55b)
73 + 2 71 72 (2-55c)

Figures 2-7A to 2-7F show sample solutions to the configuration endpoint constraint

that have been generated using the explicit expressions for the Dubins car trim durations.

Figures 2-7A to 2-7D show solutions from the turn-straight-turn family of sequences while

Figures 2-7E and 2-7F show solutions from the turn-turn-turn family sequences. The

examples for each of these two families are generated using different goal configurations.

Configurations were selected to show interesting trajectories for each sequence class.

Generally, the optimal trajectory between two configurations is found by enumerating the

six possibilities for a given configuration and then comparing the feasible results to find

the optimum.

Infeasible solutions could occur for a particular sequence and goal configuration,

particularly if the goal is located very close to the initial condition. Additionally, infeasible

solutions could occur for paths that intersect an obstacle region. As such, use of this

technique as the primary approximation method to solve the full version of Problem 2

35 with many obstacle path constraints is not likely to yield a feasible trajectory. This

technique demonstrates the most utility for obstacle-free problems or, alternatively, as

a local method integrated into a randomized scheme, such as the methods introduced in

C'!i ,l>ter 3.

3
2
1
0
-1
-2

01
0
-1

2

-1

2

-2

0 2 4 6

0 2 4 6
XD

0 2 4 6 -2 0 2 -2 -1 0 1
XD XD XD
D E F

Figure 2-7. Sample Dubins paths. A) Sample LSL trajectory. B) Sample RSR trajectory.
C) Sample LSR trajectory. D) Sample RSL trajectory. E) Sample RLR
trajectory. F) Sample LRL trajectory.

2.4.4 Dynamic Example

This example considers the dynamic system described by Equation 2-56, which is an

extension of the simple Dubins car model examined in Section 2.4.3. The vehicle described

by Equation 2-56 moves with a constant forward velocity, V, and is controlled via the

commanded turn rate, wc. Dynamics are introduced into the system in the form of a

heading acceleration constraint. This acceleration constraint results in simple first-order

lag behavior of turn rate with respect to commanded turn rate. The lag filter exhibits

a time constant of 7, = 0.25 sec. As such, the turn-rate response to a step command

achieves steady-state in approximately 1-1.25 sec, or 4-5 time constants following the

command.

V cos b

V sin

a)c

(2-56)

The system described by Equation 2-56 admits trim trajectories that behave accord-

ing to the kinematic conditions shown in Equations 2-57 to 2-60. A motion primitive

model is formed through quantization of the dynamics according to a discrete set of turn

rates. Specifically, this set is chosen such that the resulting trim trajectories are character-

ized by turn rates given by cE {-Qa, 0, +max where max =30 deg/sec represents

the maximum achievable turn rate.

V = const. (2-57)

= const. (2-58)

= 0 (2-59)

c = (2-60)

The resulting hybrid system trim trajectories corresponding to this discrete set

of motions consist of a maximum-rate left turn, TL, straight-ahead motion, Ts, and a

maximum-rate right turn, TR. State evolution along the turning trims, TL and TR, is

characterized by the transformation given in Equations 2-61 to 2-63. Similarly, state

evolution along the straight-ahead trim trajectory, Ts, is characterized by Equations 2

64 to 2-66.

Gturn (o, 7) = xo + Rturn(xo) aturn(r) (2-61)

cos ',, -sin ',, 0 0

sin, ,, cos ',, 0 0
Rturn(xo)= (2-62)
0 0 1 0

0 0 0 0

/2(1- cos(wr)cos(T)

S V 2(1 cos(wr) sin ) ()
turn(T) =-2 (2-63)
aPT

0

Gstraight('o, 7) -= F + Rstraight(ro) Astraight(T) (2-64)

cos ,, -sin',, 0 0

sin,',, cos ',, 0 0
Rstraight(Xo) = s (2-65)
0 0 1 0

0 0 0 0
T
Astraight(T) = V 0 0 0 (2-66)

A set of six maneuvers are computed such that a feasible transition exists between

each pair of trim trajectories. Each of these maneuvers consists of the systems's dynamic

response to a step command in wc. The initial and final values of this step command

are determined by the initial and final trim trajectories connected by the maneuver. An

example maneuver is shown in Figure 2-8, which shows the time history of each state

for evolution along the maneuver connecting Ts and TR. The step input is shown as the

dashed red line in the u plot.

0.5 1 1.5 2 0.5 1 1.5 2
t (sec) t (sec)

0.5 1 1.5 2 0.5 1 1.5 2
t (sec) t (sec)

Figure 2-8. Example state trajectory showing maneuver dynamics.

The time history for each state is carried out to t = 5 ,, such that w has reached

99.;:'. of its final value. The state-displacement over each finite-duration maneuver

is given by the transformation shown in Equations 2-67 to 2-69. The time-invariant

components of the displacement vector, AM, take on different values for each maneuver.

The values for these components are di-p, i,' .1 for all maneuvers in Table 2-1, where the

maneuver connecting Ti to Tj is designated as .1 ;.

GM(Yo) = o + RM(YO)AM (2-67)

cos',, -sin', 0 0

sin '1,, cos ',, 0 0
RM(xo)= (2-68)
0 0 1 0

0 0 0 1

AM [A, A A^ A ,] (2-69)

Table 2-1. AM components for each maneuver.
MSL MSR MLS MI R MLR MRL
Az (ft) +54.1487 +54.1487 +55.9118 +55.9118 +55.3357 +55.3357
Ap, (ft) -12.2331 +12.2331 -5.8870 +5.8870 +6.5181 -6.5181
A, (deg) -30.0505 +30.0505 -7.4495 +7.4495 +22.6011 -22.6011
A, (deg/sec) -30.0000 +30.0000 +30.000 -30.0000 +60.0000 -60.0000

The effect of inserting maneuvers between each pair of trim trajectories is to cause a

slight state displacement at the end of the first trim trajectory before the point at which

evolution along the second trim trajectory begins. Consequently, closed-form solutions

derived in a similar manner to that described for the Dubins car in Section 2.4.3 are the-

oretically possible. However, solutions computed in this manner do not explicitly consider

path constraints beyond merely checking for constraint satisfaction after generating the

trajectory, as discussed previously. As an alternative, this example demonstrates a partial

solution to the full MINLP shown in Problem 2-35 for which obstacle constraints are

considered.

The subsequent solution is denoted a Ip itial" solution because only a select few

sequences are enumerated for the combinatorial sequence selection step. Enumerating the

full set of all feasible sequences is impractical, as this set is countably-infinite. Solving this

problem therefore involves enumerating some of these sequences and comparing the local

solutions that result from solving the smooth NLP associated with each. As such, any

solution of this problem could be considered a Ip itial" solution.

The example considers the environment shown in Figure 2-9. This environment

contains two polygonal obstacles with a maximum dimension of 200 ft. The initial and

goal vehicle states are given by Equations 2-70 and 2-71, respectively, where (*) indicates

that the value is unconstrained. Both of these states are assumed to be along a trim

trajectory. The trajectory optimization requires that a sequence of maneuvers, M, and

a corresponding set of trim durations, 7, are found that take the vehicle from xo to Yx

in minimum time while satisfying the differential path constraints given by Equation 2

56 and obstacle path constraints of the form shown in Equation 2-4. The differential

constraints are encoded as part of the endpoint constraints through G(xo, M, r). The

obstacle path constraints utilize the superellipse formulation of the obstacle boundary, as

described in Section 2.2.2. Also note that each maneuver sequence, M, must have either

MSL or MSR as its first element because the initial state lies along the trim trajectory, Ts.

xo = 0 ft Oft 90deg 0 deg/sec (2-70)

Y9 = 100 ft 650 ft 180 deg (*) deg/sec (2-71)

400

200

0

-200 0 200 400 600 800 1000
Y (ft)

Figure 2-9. Top view of environment for dynamic example.
Trajectory generation proceeds by selecting a primitive sequence and solving the

resulting smooth NLP subject to the endpoint constraints and path constraints. Recall

that for the finite-dimensional optimization considered here, the obstacle path constraints

are enforced at a finite number of points along the trajectory. As such, the state must be

computed using g(xo, M, r) at various intermediate points along each trim and maneuver

for the NLP associated with each maneuver sequence, M. Optimization is performed using

the TOMLAB/SNOPT optimization software package [53, 64].

The first sequence examined is M = {MSL, MLS, MSLt}, which requires solution of four

trim durations. An overhead view of the solution is shown in Figure 2-10. Intermediate

points at which obstacle constraints are enforced are depicted as circles along the solution

trajectory. Blue circles indicate a trim segment while red circles indicate a maneuver

segment. The corresponding heading angle and turn rate trajectories are shown in

Figures 2-11A and 2-11B, respectively. In each of these figures, trim segments are depicted

as solid blue lines and dynamic maneuver segments are depicted as dashed red lines. The

solution trajectory requires a total of 29.7140 sec to reach x, from xo.

400

200 -

0 -

-200 0 200 400 600 800 1000
y(ft)

Figure 2-10. Top view of solution for M = {MSL, MLS, MSL}.

5 10 15 20 25 30 5 10 15 20 25 30
t (sec) t (sec)
A B

Figure 2-11. Heading and turn rate trajectories for M {MSL, MLS, MSL}. A) Heading
angle trajectory. B) Turn rate trajectory.

The next sequence examined is M = {MSL, MLS, MSL, MLS, MSL}, which requires

solution of a set of six trim durations. An overhead view of the solution is shown in

Figure 2-12. As before, intermediate points at which obstacle constraints are enforced are

depicted as blue and red circles along the solution trajectory. The corresponding heading

angle and turn rate trajectories are shown in Figures 2-13A and 2-13B, respectively. The

solution trajectory requires a total of 29.3593 sec to reach xY from xo.

400

20

-200 0 200 400 600 800 1000
Y(ft)

Figure 2-12. Top view of solution for M = {MSL, MLS, MSL, MLS, MSL}

5 10 15 20 25 30 5 10 15 20 25 30
t(sec) t (sec)
A B

Figure 2-13. Heading and turn rate trajectories for M {MMSL, MLS, MSL, MLS, MSL}
A) Heading angle trajectory. B) Turn rate trajectory.

The final sequence examined is M = {MSL, MLS, MSR, MRS, MSR, MRs}, which

requires solution of a set of seven trim durations. An overhead view of the solution

is shown in Figure 2-14. As before, intermediate points at which obstacle constraints

are enforced are depicted as blue and red circles along the solution trajectory. The

corresponding heading angle and turn rate trajectories are shown in Figures 2-15A and 2-

15B, respectively. The solution trajectory requires a total of 21.7796 sec to reach Xg from

xo.

400

200

0-

-200 0 200 400 600 800 1000
y(ft)

Figure 2-14. Top view of solution for M = {MSL, MLS, MSR, MRS, MSR, MRS}.

5 10 15 20 25 5 10 15 20 25
t (sec) t (sec)
A B

Figure 2-15. H. ilii:- turn rate trajectories for M = {MSL, MLS, MSR, MRs, MSR, MRS}.
A) Heading angle trajectory. B) Turn rate trajectory.

Several relevant observations and conclusions can be drawn upon examination and

comparison of the presented solution sequences. First, the longest sequence is seen to yield

the minimum-time solution. As such, it is apparent that short sequences do not necessarily

correlate to low cost trajectories. Second, the first two sequences yield similar solutions

while the third sequence follows a markedly different trajectory through the environment.

After enumeration of the first two, there is no indication that a better solution exists. The

third solution sequence demonstrates a significant improvement in performance over the

previous two solutions; however, there is no way to determine if this solution is optimal

or if more sequences should be enumerated. Further enumeration of additional sequences

can introduce combinatorial issues, as there are potentially an infinite number of sequences

that yield a feasible solution. An automaton containing m maneuvers can be shown to

yield m (m 1)"-1 different sequences of length n. As such, termination criteria must be

carefully selected based on the specific application.

CHAPTER 3
RANDOMIZED SAMPLING-BASED MOTION PLANNING

3.1 Introduction

Randomized motion planning has recently emerged as a powerful technique for solving

complex problems. These algorithms proceed through a probabilistic search of the solution

space, and are characterized by subdividing a difficult global problem into a set of easier

local planning problems. Essentially, a large number of points are sampled from the C-

space and are connected to neighboring samples through a planning method that is locally

valid. These connections form a i ... hi 'Ip that provides access to sampled areas of the

C-space and connectivity to other similarly sampled areas. Initial and goal configurations

can be connected to the roadmap network if the space has been sampled effectively.

This chapter discusses Probabilistic Roadmap Methods (PRM) and Rapidly-exploring

Dense Tree (RDT) methods. PRM planners form a roadmap network to map out the

configuration space for a particular system and can be used to solve planning problems

in multitude. The approach adopted by PRM planners can become impractical for

differentially-constrained systems, however. RDT methods generate a tree of feasible

trajectories that is rooted at the initial condition of the system and expand in a depth-first

fashion such that the reachable set of the system is explored rapidly. Such planners have

been effectively demonstrated for systems that are subject to differential constraints.

Finally, Section 3.4 describes a specific implementation of a randomized planning approach

that uses motion primitives to plan trajectories for a differentially-constrained system.

The PRM planning algorithm was introduced by Kavraki as a method to find

feasible solutions to problems with a complicated C-space [41]. PRM planning algorithms

probabilistically construct a roadmap of the C-space of a system through sampling

and connecting configurations. The PRM planner has been shown to be complete in a

probabilistic sense [65, 66]. A planning algorithm is said to be complete if it is guaranteed

to find a solution, provided one exists, and to terminate if one does not. Planning with

PRM-based algorithms typically proceeds in four main steps:

1. Sampling: A large but finite number of configurations are sampled randomly
from the C-space. The distribution from which these points are sampled is a design
choice and can drastically affect the performance of the motion planner [31, 44, 67].
Complex systems that operate in complex environments often require the use
of heuristic-based sampling methods that bias sampling near obstacles, inside
narrow passages, or in other areas of the C-space that are deemed important for the

2. Nearest Neighbor: Each sample is grouped with a set of samples that are deter-
mined to be within a specified neighborhood. This neighborhood is determined using
a distance function that is a valid metric or at least an approximation to a valid
metric on the C-space.

3. Local Planning: A path or trajectory is planned locally from each sample to each
sample within its neighborhood. These interconnections form a network of local
solutions.

4. Query: The initial configuration and goal configuration are added to the existing
roadmap network by planning a path or trajectory between each of these configu-
rations and existing nodes located within a specified neighborhood. Once added,
well-developed graph search techniques such as Dijksta's algorithm or the A*-
algorithm can be applied to search the roadmap network for feasible and/or favorable
solutions [25, 68].

Figure 3-1 depicts an example of a roadmap network that has been formed in a simple

two-dimensional C-space that contains three obstacles. Each node in the graph depicted

in Figure 3-1 represents a system configuration and each edge of the graph represents a

feasible trajectory between the two configurations at its endpoints.

.0

A B C

Figure 3-1. The PRM algorithm. A) Sampling step. B) Local planning step. C) Query
and solution steps.

PRM algorithms can simplify a planning problem on a complex C-space to a simple

graph search problem; however, computational issues arise for differentially-constrained

systems and for high-dimensional C-spaces. Sampling of the C-space is not sufficient for

systems with differential constraints, as local planning solutions between samples must

consider configuration velocities as well. As such, the state-space must be sampled in lieu

of the C-space, thus doubling the dimension. Further, edges added through local planning

will be necessarily unidirectional in many cases due to non-holonomic constraints present

in many vehicle systems. As a result, twice as many local plans must be generated to

achieve the same network connectivity as a system without these constraints.

3.3 Random Dense Tree Methods (RDT)

RDT-based planners provide an alternative to the basic PRM paradigm that enables

efficient solutions to differentially-constrained problems. While PRM generates a roadmap

that describes the connectivity of many configurations to many other configurations, RDT

methods generate a tree that is rooted at a specific initial condition and which describes

connectivity of this initial condition to as many reachable configurations as possible.

Algorithmic details ensure efficient and rapid exploration of the space. A drawback of

RDT methods is that they are designed to solve a single planning problem at a time. This

limitation is in contrast to PRM planning algorithms, which establish a network that

spans the C-space and can be used many times for many different planning tasks. A major

benefit in this tradeoff is that RDT methods can often handle problems involving dynamic

systems. In general, RDT methods incrementally build a search tree from an initial node

in three main steps:

1. Node Selection: A node from the existing tree is selected as a location to add a
branch. Selection of a particular node is usually based on probabilistic criteria that
may require use of a valid distance metric.

2. Node Expansion: A local planning method is used to extend a feasible trajectory
from the selected node. The local goal for this trajectory branch is determined
probabilistically.

3. Evaluation: The new branch is evaluated according to performance criteria and
often for connection to the goal configuration. Additionally, the new branch may be
subdivided into multiple segments, thus adding several new nodes to the existing tree.

A variety of RDT-based planners have been developed with numerous variations on

the main steps listed previously, often to optimize performance for a specific application

or to address a pathological case [69-74]. Two algorithms, the Rapidly-exploring Random

Tree (RRT) algorithm and the Expansive Spaces Tree (EST) algorithms, demonstrate

different core exploration philosophies through the manner in which nodes are selected and

expanded. These algorithms also serve as a basis for many of the existing variations on the

general method, and hence prove useful as demonstrative examples.

3.3.1 Rapidly-Exploring Random Trees (RRT)

The RRT algorithm was developed by Lavalle and Kuffner specifically to handle

problems that involve dynamics and differential constraints [42, 71]. The algorithm biases

tree growth toward unexplored areas of the space and hence focuses on rapid exploration.

The node selection step is initiated with a sampled configuration that is chosen from

a uniform distribution of the C-space. A distance metric is then used to determine the

closest point in the existing tree. During the expansion step, the selected node is extended

incrementally "tcv ,iil- the sampled configuration using a local planning method. This

incremental extension is performed to varying degree in different versions of the algorithm

and is ultimately a design parameter. Some versions use a fixed step size, others use a

step size proportional to the distance from the sample, while others attempt to completely

connect the sampled configuration to the existing tree.

Figures 3-2A and 3-2B depict the RRT expansion process. Both images show a tree

grown from the root node, No, in a two-dimensional C-space that contains obstacles.

Figure 3-2A depicts the sampling step, in which a random configuration, Nrand, is selected

and the nearest node in the existing tree, Nne,,, is determined. Figure 3-2B shows the

expansion step, where a branch is incrementally extended from Nnear toward Nr"nd along

the trajectory connecting the two configurations. A new node, Nnew, is added at the

end of the new branch. The algorithm proceeds in this fashion until a branch of the tree

reaches the goal within some specified tolerance.

Nrand Nrand
-0 ne0

A B
Figure 3-2. RRT algorithm. A) Sampling step. B) Expansion step.

3.3.2 Expansive-Spaces Trees (EST)

The EST algorithm was developed by Hsu et al as a planning method to address

problems involving high-dimensional C-spaces and was later adopted to handle kinody-

namic planning problems [40, 75]. The EST algorithm explores space in a fundamentally

different way than the RRT algorithm. Specifically, node selection occurs through the

random selection of an existing node according to a probability distribution that is left

as a design choice. This node is expanded within a local neighborhood that is defined by

a valid distance metric. A configuration is sampled randomly from within this neighbor-

hood and a local planning method is used to connect the selected node to the sampled

configuration.

Figures 3-3A and 3-3B depict the EST expansion process. Both images show a tree

grown from the root node, No, in a two-dimensional C-space that contains obstacles.

Figure 3-3A depicts the node selection step, in which the expansion node, Ngxp, is selected

from the existing nodes. The neighborhood of Nexp is defined here using a Euclidean

distance metric and is shown as the area within the dashed circle in Figure 3-3A. Figure 3-

3B shows the expansion step, where a random configuration, Nrand is selected from the

neighborhood of Nexp and then a trajectory is planned from Nexp to Nrand. The algorithm

proceeds in this fashion until a branch of the tree reaches the goal within some specified

tolerance.

o 0
' 'Nrand

Nrand

A B
Figure 3-3. EST algorithm. A) Node selection. B) Sampling and expansion.

3.3.3 Discussion

It is important to note the fundamental difference between the v--v- in which the

RRT and EST explore the space. Samples from empty space have a tendency to "pull"

branches off of the tree built in the RRT algorithm. Thus, the space is rapidly spanned

with coarse resolution. Continued sampling has the effect of improving the resolution

of this exploration without appreciably changing the form of the solution. This concept

is depicted in Figure 3-4A. Conversely, the EST selects a node randomly and tends to

"push" branches from the selected node toward empty space as shown in Figure 3-4B. A

benefit to this "pi-I!hi;, tendency is that the shape of the tree is continually evolving

such that expansion is guided by the node sampling distribution. A wise choice of this

distribution can favorably affect solution performance qualities; however, care must be

taken to avoid biasing exploration toward previously explored areas.

The performance qualities of the resulting trajectories planned using these algorithms

can be affected by altering the various degrees of freedom for each planner. The explo-

ration behavior of the RRT is clearly dependent upon the distribution from which random

configurations are sampled. For example, a non-uniform distribution can be chosen to

bias exploration toward the goal; however, care must be taken to avoid pitfalls such as

local minima. Additionally, RRT performance has been shown to exhibit sensitivity to the

choice of a distance metric [42, 76]. C('! i..-, of the exact distance metric, which is the op-

timal cost-to-go, may be impractical for many planning problems. Hence, an appropriate

1000 1000 1000
800 800 800
600 / 600 600
> 400 >< 400 >< 400
200 200 200
0 0 0
0 500 1000 0 500 1000 0 500 1000
Y (ft) Y (ft) Y (ft)
A

1000 1000 1000
800 800 800
600 600 00
400 A 400 < 400
200 200 200
0 0 0
0 500 1000 0 500 1000 0 500 1000
Y (ft) Y (ft) Y (ft)
B

Figure 3-4. Differences in exploration strategy for the RRT algorithm vs. the EST
algorithm. A) RRT expansion. B) EST expansion.

approximation to this optimal distance metric must be chosen. Selection and computation

of such an approximation can be non-trivial for complicated systems.

As discussed previously, the EST provides a particularly useful design choice in

the freedom to select the distribution from which expansion nodes are determined. A

modified node selection scheme, denoted the Guided EST (GEST) assigns each node

a probabilistic weight that reflects proximity to other nodes spatially and temporally

and, more importantly, reflects a measure of performance [73, 77]. This explicit inclusion

of performance cost in the sampling process enables the GEST to consider solution

quality while exploring space. Moreover, the "push" tendency described previously allows

potential solutions to continually improve. This improvement tendency results from the

increased likelihood that as branches exhibiting desirable performance will be selected for

expansion under the GEST framework.

3.4 RDT-Based Dynamic Planner for a Planar-Motion Vehicle

This section considers an RDT-based planner to generate approximate minimum-time

trajectories for a differentially-constrained system, such as an aircraft. The planning

problem considers a cluttered environment that forces the vehicle to operate within close

proximity to obstacles. Consequently, differential motion constraints must be considered

to ensure safe tracking of trajectories. An RDT planner is used as a practical alternative

to optimal solutions offered by optimal control or solutions to the MINLP posed in

Problem 2-35. The method presented here utilizes algebraic solutions for planning with

motion-primitive models as described in Section 2.4.2. These algebraic solutions are used

to generate local, obstacle-free trajectories as part of the tree expansion process.

3.4.1 Model

This planner considers a hybrid motion primitive model that behaves according to

the dynamics described by Equation 3-1. The single input to the differential system is the

turn-rate, w. The constant translational velocity, V, is constrained differentially to act in

the direction of the vehicle heading. The system described by Equation 3-1 admits trim

trajectories that belong to two families: constant rate turns and straight forward motion.

S V cos Q

p4 V sin Q (3-1)

Here, trim trajectories behave according to the kinematic conditions shown in

Equations 3-2 and 3-3. A motion primitive model can be formed by selecting a set of trim

primitives that behave according to these conditions.

V = const. (3-2)

= = const. (3-3)

A set of 2n + 1 trim primitives are selected which consist of constant-rate turns

at n different turn rates in each direction along with a straight-ahead primitive that

corresponds to w = 0. This set is shown as Equation 3-4, where Qmax =30 deg/sec. The

velocity, V, is held fixed over the set of all primitives.

S{E fO, w1L, 2- ,.n} wil < Qmax,i = I, (34)

The allowable switching behavior between these primitives is characterized by the

automaton structure shown as a directed graph in Figure 3-5. Each trim trajectory is

depicted in the figure as a state of the automaton and is designated by the turn rate

corresponding to that trim state.

1 1

Figure 3-5. Automaton structure for vehicle model used with current planning strategy.

State evolution along each turning trim characterized by wi is described by the time-

parameterized transformation given in Equations 3-5 to 3-7. Similarly, state evolution

along the straight-ahead trim trajectory characterized by w = 0 is characterized by

Equations 3-8 to 3-10.

Gturn, i(fo, ) = Fo+ R (iurnYo) tAurn,&ir) (3-5)

cos' ,, -sin ',, 0 0

sin,',, cos ',, 0 0
Return (xo) = (3-6)
0 0 1 0

0 0 0 0

vV 2(1 cos(wrT) cos (j)

S2(1 cos(w,) sin ()
0PT

0

Gstraight(Yo, 7) Yo + Rstracight(o) A straight()

cos ',, -sin',, 0

sin,',, cos ',, 0 0
Rstraight (Xo) = c (3-9)
0 0 1 0

0 0 0 0
T
Astraight(T) = VT 0 0 0 (3-10)

As presented, the model is classified as kinematic, ie. uw can change value instanta-

neously. Therefore, maneuvers are considered as instantaneous switches between trim

trajectories, and the corresponding transformation which describes the configuration

transition for maneuvers is simply an identity matrix.

As described for the Dubins car model, the hybrid model presented here admits

unique solutions between any two configurations. These solutions exist as members of up

to two families of sequences: a turn followed by a straight segment followed by another

turn, or three consecutive turns. If the model consists of three trim primitives, a straight

forward motion at constant velocity and steady, maximum rate turns in each direction, the

set of unique solutions is equivalent to the Dubins set, D, which has six members. When

2n turning trim primitives are considered which consist of opposite direction turns at

rates of n different magnitudes, the set of turn-straight-turn sequences is expanded to size

4n2 and the set of turn-turn-turn sequences is expanded to size 2n3. The trim durations

corresponding to each of these sequences can be expressed in closed form, where infeasible

solutions yield complex-valued trim durations. Thus, computation of an approximation

to the optimal trajectory requires that 4n2 + 2n3 expressions are evaluated and are then

compared for cost performance.

Only turn-straight-turn sequences are considered with variable turn rate to reduce

computational burden, and only turn-turn-turn sequences at the nominal (maximum) turn

(3-8)

rate are considered. Thus, consideration of variable turn rate Dubins sequences results in

4n2 + 2 unique solutions for the trajectory between any two configurations. The equations

that determine the trim durations are similar to those presented in Section 2.4.3; however,

the added consideration of variable turn rates makes explicit presentation somewhat

cumbersome. While the actual expressions are not shown here, the solution procedure is

identical and is easily performed using symbolic algebra software.

In addition to the unique solution families between two points in the obstacle-free

C-space, unique solution families also exist on a useful subspace. This subspace consists

of the position variables, px and py, and is denoted, P. Motion plans ending at a point

on P are not subject to constrained heading, ), at the endpoint. These solution families

consist of turn-straight member sequences. Hence, for a model containing 2n turning

trim primitives comprised of n rates in opposite directions, there are 2n turn-straight

members in the unique solution set. Solutions for the trim durations are shown for the

left-straight family in Equations 3-1 a and 3-1 b and for the right-straight family in

Equations 3-12a and 3-12b.

V aVd
cos + sin b +
7,i = + arctan )- {mod 2} (3-11a)
72 Sin +d) +( COS
2Vd
-r2i sin + d2 (311 b)
Wi

7V2 Vcosi (()2si l Vd
7r1i = i+ arctan si n I- {mrnod 2w} (3-12a)
72 -Sin +d) cos 1
2Vd
2,i = Sin 1i + d2 (3-12b)
LWi

While the model considered here exhibits kinematically-constrained motion, the

addition of simple dynamics in the form of monotonic transitions between trim states does

not appreciably change the behavior of the model. Actually, the procedure for determining

unique solutions for trim durations remains identical: a sequence of transformations are

equated to the desired terminal configuration and the expressions are solved in terms of

the trim durations. The dynamic motions appear in the string of transformations as a

finite displacement in each configuration variable accompanied by a finite shift in time.

Thus, generality is not affected in the subsequent development.

3.4.2 Overview

The algorithm presented here proceeds with the "pull" expansion ]i1i -.. .l hi,- exhib-

ited by the RRT. A point from the C-space is selected and a new branch is "pulled" from

an existing branch that is determined to be nearby. While the EST "pui-l! approach

demonstrates a number of benefits, especially in achieving improved performance of

resulting trajectories, the RRT approach is selected for its efficiency and robustness in

finding feasible solutions. Experience with the EST approach found significant sensitivity

of convergence properties with respect to the various degrees of freedom allowed by the

algorithm.

The main steps of the algorithm are detailed in the next three sections. These three

main steps can be summarized as follows:

1. Select a Node: A point is selected from the subspace of the C-space, P, which is
spanned by the position variables. An approximate, obstacle-free distance metric is
used to determine the nearest node in the existing tree.

2. Extend a Branch: The set of 2n unique solutions on the position subspace are
enumerated, evaluated, and pruned. Selection criteria is used to choose a branch from
the set for addition to the solution tree.

3. Check for Solutions: The new branch is split into an intermediate set of nodes.
Each node is tested for obstacle-free connection to the goal configuration on the full
C-space. If appropriate, new solutions are added to a solution list and the current
upper bound is updated.

Finally, a significant difference between the nominal RRT algorithm discussed in

Section 3.3.1 and the algorithm presented here is related to algorithm termination. The

nominal RRT algorithm seeks a single feasible solution. Here, the first feasible solution

is adopted as a performance upper bound. The algorithm continues to generate new

solutions and update this upper bound in an attempt to arrive at a better solution.

Thus, an independent termination condition must be set that reflects either a measure of

convergence or a measure of computational resource usage.

3.4.3 Node Selection

The first step in each iteration is to select a node, Nexp, from the existing tree for

expansion. This expansion node is selected as the closest node to a randomly sampled

point, psamp E P, where P represents the two-dimensional position subspace. The

reduction in search dimension that is achieved through sampling on P is allowable

because there are no attitude-based constraints on the trajectory besides those enforced

at the initial and goal configurations. Planning on this subspace is sufficient to generate

trajectories that navigate the obstacle field. Moreover, there is a substantial complexity

benefit to the upcoming node expansion step in that the number of explicit solution

sequences that must be evaluated is reduced by an order of magnitude from 4n2 + 2 to 2n,

as described in Section 3.4.1.

The distribution on P from which psamp is chosen pl i, a role in planner perfor-

mance.A uniform distribution over P maximizes space exploration such that trajectories

might 'wander' about the space before achieving a feasible solution. Conversely, a distribu-

tion that heavily emphasizes the goal configuration di pl'-v' greedy goal-seeking behavior

that is subject to local minima [31]. Ideally a balance should be achieved between these

two behaviors. As such, samples are selected from one of two distributions, each of which

is assigned a probability that it will be used to generate the current sample. The two

distributions consist of a uniform distribution on P and a Gaussian distribution centered

at the goal position on P. Thus, sampling proceeds by first selecting a distribution and

then sampling a position from the selected distribution.

Once a point is selected, nearby nodes in the existing tree must be determined. An

intuitive distance metric such as the Euclidean 2-norm does not capture the dependence

of path length on differentially-constrained motions. Specifically, the vehicle is generally

required to change direction from the configuration at Nexp before proceeding towards

Psamp. This behavior results in .i-vmmetric distance measures, eg. a point immediately

in front of the vehicle is closer than a point immediately to the side of the vehicle, even

if the two points are the same Euclidean distance from the current position. The ideal

distance metric is the optimal cost-to-go; however, computation of this value is equally

as difficult as the original planning problem in many cases [42]. Therefore, the minimum

obstacle-free turn-straight trajectory from a configuration to a point on P is considered

as an approximate distance metric. This minimum occurs when the turning segment is

executed at maximum turn rate. A plot of distances measured using this metric is shown

in Figure 3-6A. The vehicle has a configuration of (px,Py, 0) (0,0,0) in Figure 3-6A,

such that distance is measured from the center of the plot where the vehicle is initially

pointing along the p1-axis. For comparison, a plot of the Euclidean 2-norm is shown in

Figure 3-6B for the same vehicle configuration. Inspection of the figures clearly shows

discrepancies between the two metrics.

O!t ..600ft

2Ea0t ( ) oEast (f)f

A B
1. I.I.... 400 ft

East (ft) East (ft)
A B

Figure 3-6. Distance function comparison. A) Approximation of obstacle-free cost-to-go.
B) Euclidean distance function.

Actual computation of the distance from each node to Psamp using the metric depicted

in Figure 3-6A is accomplished by computing the minimum turn-straight trim durations

to reach Psamp from each node in the tree. The minimum for each of these primitives

individually occurs at the maximum turn rate. As such, Equations 3-11a, 3-11b, 3-12a,

and 3-12b with wi = max are used to compute the candidate sequences. The global

minimum for the turn-straight family is then found through direct comparison of the

minimum left-straight and minimum right-straight sequences. This process is shown in

Figure 3-7.

400
300
300
200
200
100 100
10 0

-100
-100
-200
-200 0 200 400 -200 0 200 400
Y (ft) Y (ft)
A B
Figure 3-7. Distance Function Computation. A) Sampled point. B) Turn-straight paths
from each node.

The Node Selection step exposes a troubling practical reality associated with imple-

mentation of RRT algorithms: that a distance computation must be performed at every

node at each iteration. While this operation is often computationally cheap, as is the

case here, new nodes are added to the tree at each step. In the limit, an infinite number

of nodes are involved in a distance measurement operation. Fortunately, in practice, the

algorithm tends to converge before this issue becomes problematic; however, such an

observation is difficult in theory to prove nor guarantee.

3.4.4 Node Expansion

After Nexp is selected, a branch is extended from the configuration at Nexp to the

sampled position, psamp. Unique solution sequences for the trajectory from a configuration

to a point on P are members of the family of 2n turn-straight sequences, as discussed in

Section 3.4.1. Trim duration times are computed using Equations 3-11a, 3-11b, 3-12a,

and 3-12b. Some values of psamp yield infeasible solutions to certain sequences. These

infeasibilities are reflected as complex-valued trim durations. Thus, enumerating the 2n

members of the turn-straight family of sequences in fact yields a maximum of 2n candidate

branches.

Each candidate branch must be evaluated for both cost performance and safety. The

minimum-time cost performance is trivially evaluated by summing the trim durations

and switching maneuver time for each candidate.Maneuver time evaluates to zero for the

instantaneous switch case considered here. Safety is evaluated by checking points along

each candidate trajectory for collisions. Points are generated at intermediate time values

and each is checked using a collision detection method as described in Section 2.2.2. Care

must be taken in selection of a time step in order to ensure that the points are spaced

closely enough to adequately represent the continuous trajectory without imposing undue

computational burden.

Upon detection of a collision, a candidate branch is pruned such that the end of the

branch lies a distance of one minimum turn radius short of the first collision point along

the original candidate branch. This step ensures that the branch can be safely extended

beyond the current endpoint without a collision. This process is depicted in Figure 3-8.

After pruning, a candidate is selected according to Euclidean distance from psamp and

minimum time properties.

400 400

200 ._ 200

0 0

-200 0 200 400 600 -200 0 200 400 600
Y (ft) Y (ft)
A B

Figure 3-8. Node expansion step. A) The new branch is checked for pointwise collisions.
B) The new branch is pruned beyond the first collision.

3.4.5 Solution Check

After the addition of a new branch to the tree, new nodes, N,,, must be added. The

new branch is subdivided by a fixed time step as shown in Figure 3-9. The time step for

subdivision is left as a design choice. More nodes per branch generate a greater v ,i, I, i of

starting points for future node expansion, Ngxp, but induce greater computational burden

in terms of number of operations required for the nearest node operation as well as with

respect to memory capacity required to store tree data. Information regarding total cost,

automaton sequence, trim durations, and node precedence is computed and stored at each

node as it is added to the tree.

600

400

X 200

0

-200 0 200 400 600
Y (ft)

Figure 3-9. The new branch is subdivided to a set of nodes

After computing and adding nodes, each node is tested for connection to the goal

configuration. This connection test requires that trim durations for the 4n2 + 2 variable-

rate sequences are computed for each new node, N,,w,i, as shown in Figure 3-10. The

length-three sequences must be used to satisfy the heading constraint at the goal config-

uration. The solution for each of these sequences is expressed in closed form and as such,

complexity does not exceed O(n2) operations. The cost for feasible sequences originating

at Ne,,i is added to the total cost for Ne,,i. This value is compared to the current upper

bound for the minimum safe and feasible solution. Each feasible sequence that represents

an improvement to the current upper bound is checked for safety with respect to obstacle

collisions. Sequences that improve the upper bound while maintaining feasibility and

safety then replace the current best trajectory. Other sequences are disregarded.

600

400
400

200200

0
0

-200 0 200 400 600 -200 0 200 400 600
Y (ft) Y (ft)
A B

Figure 3-10. Collision-free solutions. A) Right-straight-right family. B) Left-straight-right
family.

After each new node has been tested for feasible and safe reachability of the goal

configuration, the termination condition is checked. As discussed previously, this termi-

nation condition can vary according to the specific restrictions on current operation of

the planner. Termination can depend on runtime, memory usage, or even after a certain

number of valid solutions have been achieved. Upon termination, the planner returns a

precedence list of nodes that lie along the solution trajectory from the initial configuration

to the goal configuration. If the termination condition has not been reached at the end of

a particular iteration, execution returns to the Node Selection step.

3.5 Example

A sample planning problem is solved in this section to demonstrate the utility of

the RDT-based planning algorithm introduced in Section 3.4. This problem considers a

vehicle whose motion is governed by the continuous differential system shown previously

as Equation 3-1. As such, the vehicle moves with forward velocity, V, and is subject to

a bounded turn rate, |i\ < 30 deg/sec. The single system input is the commanded turn

rate, Cu. As previously, the model is characterized as kinematic. The motion of this vehicle

is modeled using a hybrid motion model with seven trim states. Formulation of this model

requires the selection of trim and maneuver primitives. Maneuvers between trim states

occur instantaneously due to the kinematic description of vehicle motion. The trim states

are characterized by constant system velocities as follows:

To =(Vo,wo) = (45 ft/sec, 0 deg/sec)

T, = (Via, wa) = (45 ft/sec, + 30 deg/sec)

T1b = (V1b, Ulb) = (45 ft/sec, 30 deg/sec)

T2a = (V2a, 2a) = (45 ft/sec, + 20 deg/sec)

T2b = (V2b, 2b) = (45 ft/sec, 20 deg/sec)

T3a = (V3a, 13a) = (45 ft/sec, + 15 deg/sec)

T3b = (V3b, W3b) = (45 ft/sec, 15 deg/sec)

The problem of planning a trajectory between two configurations admits two families

of unique solution sequences for the obstacle-free case. These families consist of turn-

straight-turn sequences and turn-turn-turn sequences, respectively. Sequences exhibiting

variable turn rate are considered for the turn-straight-turn family. Here, n = 3 giving a

total of 4n2 = 36 sequences in the turn-straight-turn family. Example solutions are plotted

in Figure 3-11A, which shows trajectories that drive the system from an initial condition

of (px,, Py,, '',,) = (0, 0, 0) to a terminal configuration of (Pf,Pyf, f) = (0, 800, 0).

Only maximum rate turns are considered for the turn-turn-turn family of sequences, as

discussed in Section 3.4.1. Examples of the two resulting solution trajectories are shown in

Figure 3-11B, which shows trajectories that drive the system from an initial condition of

(Px,o,Py,o, ',') = (0,0,0) to a terminal configuration of (p,f,Pyf, f) = (0,100,0).
In addition to the solution sequence families for planning between two configurations,

a solution family exist for planning from a configuration to a point on the subspace, 2,

which consists of the position variables, p, and py. Variable rate turns are considered

for this solution family, which is characterized by turn-straight sequences. There are

500

100

0 0

-100

-500 -200
0 500 1000 -200 0 200
Y (ft) Y (ft)
A B

Figure 3-11. Unique solution families for vehicle used in example. A) Turn-straight-turn
solution sequences. B) Turn-turn-turn solution sequences.

a total of 2n = 6 members in this solution family. Example solutions are plotted in

Figure 3-12, which shows trajectories that drive the system from an initial condition of

(px,o,Py,o,',) = (0,0,0) to a terminal point on P located at (Px,f,Py,f) = (200,500).

200

-200

-200 0 200 400
Y (ft)

Figure 3-12. Turn-straight solution sequences.

The planning environment considered for the example problem is shown in Figure 3-

13. For this example, the inertial reference frame is oriented such that the x-axis points

in the North direction and the y-axis points in the East direction. In the figure, the

bounds on the extent of the environment are shown as the blue square that contains

0 < x < 1000 ft and 0 < y < 1000 ft. All sampling steps draw samples from

within these bounds. Also, p" Iv._aonal obstacles are randomly placed within this region.

200

The example problem considers eight square obstacles with sides of length 150 ft, as

shown in Figure 3-13. Sampling distributions are not defined within obstacle regions.

Finally, the initial and goal configurations are specified as (px,o,py,o ,,) (0, 0, 0) and

(px,fPf, P f) = (1000, 1000, 0), respectively.

1000

S500
7 U

0

0 500 1000
East (ft)

Figure 3-13. Example planning environment.

The algorithm described in Section 3.4 is executed for sixty iterations. Figure 3-14

shows the expansion of the search tree at various intermediate stages progressing from

left to right. Samples on P are drawn from either a uniform distribution defined over the

obstacle-free portions of the planning environment or a Gaussian distribution centered

at the goal location with a standard deviation of 250 ft that is also only valid over the

obstacle-free portions of the environment. The uniform distribution is chosen with a

probability of 0.75 and the Gaussian distribution is chosen with a probability of 0.25.

1000 1000 1000

500 500 500
o o ,

0 0 0

0 500 1000 0 500 1000 0 500 1000
East (ft) East (ft) East (ft)
A B C

Figure 3-14. Incremental tree expansion. A) After 1 iteration. B) After 25 iterations. C)
After 60 iterations.

The image at the left of Figure 3-14 shows the turn-straight branch extended from the

initial configuration to the first sample. The center image di-pl '-, the tree after twenty-

five iterations. Note that branches are not necessarily added during every iteration, as

certain obstacle collisions can prune an entire branch. The image at the right of Figure 3-

14 shows tree expansion after sixty iterations.

Nodes are added at intermediate time steps along each branch as it is appended to

the tree, as discussed in Section 3.4.5. Nodes in the final tree after sixty iterations are

shown in Figure 3-15. These nodes serve as starting points for additional branches as well

as for solution trajectories.

1000

500

0-

0 500 1000
East (ft)

Figure 3-15. Nodes in the final solution tree.

Each node is tested for feasible connection to the goal configuration as it is generated.

Successful connections that improve the current-best solution replace this solution for

future iterations. Figure 3-16 shows the refined solution for the present example as

tree expansion proceeds. In each of the plots shown in Figure 3-16, the precedent path

from the initial configuration along the tree branches to the tested node is depicted as a

solid blue line. The solution trajectory from the tested node to the goal configuration is

depicted as a solid red line.

0 500 1000 0 500 1000 0 500 1000
East (ft) East (ft) East (ft)
A B C

Figure 3-16. Solution path refinement. A) After 7 iterations. B) After 14 iterations. C)
After 37 iterations.

The image at the left of Figure 3-16 depicts the first valid solution found by the

algorithm. This solution was computed during the seventh iteration. Hence, this example

demonstrates the ability of the planner to a find a feasible, but sub-optimal, solution very

quickly. The solution shown in the center image of Figure 3-16 is computed during the

fourteenth iteration. Finally, the minimum-time solution found for the example run is

shown at the right of Figure 3-16. This solution was computed during the thirty-seventh

iteration. As such, the final twenty-three iterations did not contribute to the solution and

could be considered a waste of computational resources. Also note that the improvement

of each solution over previous solutions is fairly benign, due to the nature of the RRT

expansion procedure. Unfortunately, there is no way to determine if a particular solution

is potentially-improvable or how long such potential improvement will take. Therefore, the

termination condition remains a problem-specific parameter.

1000

1000

1000

CHAPTER 4
SENSING EFFECTIVENESS

4.1 Introduction

Sensing tasks are identified in C'! ipter 1 as one of the primary application domains

for unmanned systems. Guidance trajectory design methods that ensure desirable sensor

placement represent a required technology to enable such missions. Before trajectory

design methods can be discussed, a metric to define desirable viewpoints must be estab-

lished. The formulation of such a metric is seen to be especially relevant for the so-called

-i i!!11-- .., sensing missions discussed in this dissertation.

This dissertation considers the class of LOS remote sensors. These sensors collect

data in the form of reflected energy from remote surfaces that exhibit clear LOS from

the sensor and which are contained within the sensor FOV. Data collection can be either

active or passive in nature, depending on the specific sensor. In either case, the sensing

operation can be characterized geometrically. The geometric relationships that relate

energy reflections seen by the sensor are dependent mainly on the relative position and

orientation of the reflecting surface with respect to the sensor.

Sensor performance is affected by variations in the energy that is reflected from each

particular target surface. Additionally, variations in the relative orientation of the target

surface with respect to the sensor can result in data distortions and degraded resolution.

The perspective from which targets are seen continually varies while the sensor moves

through the environment. As such, sensing performance with respect to a particular target

surface can vary appreciably as the sensor is moved relative to that target's position.

Furthermore, .idi i:ent objects are seen from a slightly different perspective even for a

stationary sensor. Hence, sensing performance can show significant variation over the

sensor FOV.

This latter effect is especially pronounced for sensor operation in close proximity to

the environment. Such a scenario involves target surfaces appearing at various relative

ranges and aspects with respect to the sensor. Close proximity operation also increases

the severity of effects resulting from time-varying behavior of these relative geometric

parameters. These effects indicate that the data product can vary significantly with

respect to small variations in guidance trajectory planning and tracking. As such, these

variations should be quantified for integration into the motion planning process to ensure

that planned trajectories yield the desired data product. AT !ii: current research efforts

that involve vehicle-based sensor planning treat visibility in a binary sense and therefore

fail to model these variations effectively [17, 23, 78-80].

This chapter discusses a geometric approach to quantify the effectiveness of a partic-

ular sensor-target configuration. First, a brief discussion of some of the relevant remote

sensor technologies to the applications considered provides context to the subsequent pre-

sentation. Then, a geometric model of LOS sensor operation is introduced in Section 4.3.

A parameterization of the relative geometry is used to define the concept of visibility,

which is discussed in the context of sensing tasks in close-proximity environments. Finally,

a generalized sensing effectiveness metric is formulated that quantifies the quality with

which a particular point in the environment has been sensed. Several specific examples

demonstrate how this concept can be used to evaluate sensing mission performance.

4.2 Remote Sensor Technologies

This section introduces some of the general characteristics of the sensors relevant

to this dissertation. These sensors fall into the class of vehicle-carried remote sensors

which gather information about the environment using reflected electromagnetic energy.

Hence, contact is not required for data collection and sensing tasks can be performed

from a distance. The sensors discussed in this section include video cameras, radar, sonar,

and laser sensors. The quality of data collected by each of these sensors is affected by

generally-similar geometric factors. Sensor-specific quality factors will not be considered

in order to maintain generality. Hence, the detailed physical processes that enable sensor

functionality are considered as beyond the scope of this dissertation.

4.2.1 Computer Vision

Computer vision is a passive, information-rich sensor technology. Sequences of images

collected using a video camera provide a two-dimensional representation of the three-

dimensional environment. The mapping of the environment to a two-dimensional image is

described by projective geometry. A point in the environment is projected onto the focal

plane along the LOS. As such, the depth dimension is lost in the mapping [81].

Despite this loss of depth information, a great deal of information can be gathered

from two dimensional images using vision processing techniques. Points of special signif-

icance denoted 'feature points' can be identified and extracted from a two-dimensional

image. Overlapping views allow feature points in sequential images to be correlated and

tracked. The motion of these tracked features within the image plane can be used with

structure from motion (SFM) techniques to reconstruct the depth dimension in addition

to estimating camera motion [82]. Such techniques have received increasing attention in

recent years for the purposes of vehicle navigation and control [83-87].

Full reconstruction of the three-dimensional geometry is not necessarily required

to render two-dimensional images useful. A variety of applications in both the civil and

military domains employ visible-light imagery. Color and shape information related to

objects within the camera FOV can be used for identification and tracking tasks.

Radar is an active sensor that uses electromagnetic energy in the microwave band.

Pulses of electromagnetic energy are emitted from the transmitting antenna and are

scattered by objects that lie within the antenna footprint. Some of the scattered energy

is reflected back to a receiving antenna. The nature of this scattering is affected by the

material properties of the targets and by problem geometry. The time lapse between these

events is used to compute a range measurement.

Resolution of radar data can present issues for mapping and imaging applications.

Essentially, the range resolution is dictated by the pulse wavelength and incidence angle

while the bearing resolution is dictated by the physical size of the aperture, or an-

tenna [88]. The range resolution has been addressed in recent years through the use of ul-

tra wideband (UWB) radar systems. These systems use a wide range of energy frequencies

to produce better range resolution than standard radar. The bearing resolution depen-

dency presents a difficult tradeoff in that high-resolution data requires unrealistically-large

antennae.

Synthetic-aperture radar (SAR) processing is a technique that uses the vehicle motion

to enhance the b. ,iii.- or azimuth, resolution. The concept behind SAR processing is

that successive measurements from a small, moving radar antenna can be synthesized to a

much larger beamwidth. Thus, when a particular point in the environment is illuminated

by multiple radar pulses, the returns can be processed as if the point is illuminated by a

single pulse from a large synthetic aperture.

Advances in UWB radar systems and SAR processing techniques have begun to allow

new capabilities and applications for radar. The information content of radar data is

dependent on the frequency band in which the system operates. Pulsed UWB systems

utilize a short-duration pulse with support at frequencies that are both low enough to

penetrate foliage and soil and that are high enough to resolve small objects [89-92]. Use

of SAR processing techniques enables high spatial resolution. Predicted advances in

computational and processing techniques -i-'-.- -1 that these radar systems will eventually

exhibit imaging capabilities at frame rates comparable to visible-light imaging sensors [92,

93]. Hence, UWB radar poses great benefit to a variety of applications dealing with

the detection of concealed objects such as buried landmines or enemy forces in deep

cover [89, 92, 94, 95].

4.2.3 Sonar

Sonar operates on principles similar to radar; however, acoustic energy is transmitted

into the environment and reflected back to the sensor as opposed to electromagnetic

energy. The return is range-gated based on the time-lapse since the pulse was transmitted.

Sonar arrays can be implemented to yield bearing measurements in addition to the range

measurements.

Sonar range sensors are used in a variety of applications. Acoustic energy travels

more efficiently through water than it does through the less-dense atmosphere. Therefore,

sonar is often the remote sensor of choice for underwater applications. Sonar sensors are

also frequently used for short-distance ranging in mobile robotics applications [96-99].

The resulting data can be used for navigation, target detection, and even for mapping

and imaging applications. The mapping and imaging applications require the use of

synthetic-aperture sonar (SAS) processing, which is analogous to SAR processing of radar

data [100, 101].

Scannerless imaging ladar (LAser Detection And Ranging) is an emerging sensor

technology that exhibits three-dimensional imaging capability. Ladar systems transmit

pulses of coherent light in the visible, near-infrared (NIR), and infrared (IR) energy bands.

These energy pulses are reflected back to the sensor by objects in the environment. The

time lapse between transmission and reflection is used along with knowledge of the pulse

physical properties to determine a range measurement to the reflecting object. Complex

mechanical scanner systems are typically used to cover a wide swath of the environment

with laser pulses. Accurate range and bearing measurements are possible using such a

system; however, extensive processing is often required to resolve the three-dimensional

measurements.

Recent efforts have applied standard optics to project the laser reflections to a focal

plane array (FPA) [102, 103]. Essentially, a single transmitted laser pulse can result in

a two-dimensional image that contains both range and reflection intensity information

for each pixel. Ladar systems are currently under development with the expectation that

these range images will be produced at frame-rates comparable to standard video cameras.

These research efforts have resulted in several prototype sensors that exhibit much

smaller size, lighter weight, and lower power requirements than typical laser scanning

systems [102-104].

The potential applications for a sensor with three-dimensional imaging capabilities are

numerous and varied. The reduction in size, complexity, and power compared to scanning

systems allows a much broader class of vehicle to consider imaging ladar as a sensor sys-

tem. Scannerless imaging ladar technology has been investigated for use with autonomous

navigation systems and could be extended to other applications, such as remote detection

of concealed objects or targets [104]. The NATO Research and Technology Organization

predicts that ii. i, -pixel arrays making real-time multi-dimensional measurements at

10 km with frame rates of 30 Hz or f ,- I will be a reality within a decade [2]. Addi-

tionally, the energy properties of these future advanced ladar systems could be used to

identify chemicals, pollutants, and organisms along with characteristic vibration properties

of living or man-made objects [2].

4.3.1 Sensing Geometry

The geometry for the general case of a vehicle carrying a single LOS sensor is

described by the vector diagram of Figure 4-1. The vehicle body basis, B, is fixed at

the vehicle center of gravity and is located relative to the inertial frame, E, by the time-

varying vector iB. The orientation of B is described relative to E through the typical

Euler angle parameterization, which is expressed as KB as in Equation 4-1. The Euler

angles are all time-varying as well. The time-evolution of both JB and KB is governed by

the vehicle dynamics.

B B OB OB QB (4-1)

The time-varying transformation from E to B is denoted TEB and is a function of IB.

This transformation consists of the standard 3-2-1 sequence of single-axis rotations shown

in Equation 4-2.

PB

At

Figure 4-1. Sensing problem geometry.

1 0 0 cos OB 0 sin OB cos 5B sin fB 0
TEB B) 0 cos B sin fB 0 1 0 -sin aB COs B 0

0 sin (B cos B sin OB 0 cos 0B 0 0 1
(4-2)

The sensor reference frame is denoted S. The basis vector, 83, is denoted the sensor
axis and is oriented such that it points along the LOS at the center of the sensor FOV.
Without loss of generality, the sensor is assumed to be fixed to the vehicle and is located
relative to B by the vector p1. The inertial position of S relative to E can then be com-
puted as given by Equation 4-3. The sensor is assumed to have a known orientation with
respect to B, which is given by the constant transformation, TBS. The transformation
from E to S is then given by Equation 4-4. The orientation of S with respect to E can

be expressed using an Euler angle parameterization, as well. The time-varying vector of
sensor Euler angles is shown in Equation 4-5. The transformation, TES, can be expressed
relative to these parameters as in Equation 4-6. The sensor Euler angles, 4I, can then be
determined in terms of KB by equating the right hand sides of Equations 4-4 and 4-6 then
solving the resulting system.

PA

Ps P= B + (TEB) l (4 3)

TES (B) TBS -TEB B) (4 4)

s = s IT s (4-5)

1 0 0 cos O 0 sin Os cos ~ sin ~ 0

TES (s) 0 cos, sing 0 1 0 -sing cos&, 0 (4-6)

0 sin ( cos sin 0O 0 cos Os 0 0 1

The ith point-feature, or target, in the environment is located relative to E by ;t,i.

This point is considered to lie on a surface with an associated normal direction at pt,i

designated by the unit vector, hi. The components of these vectors are expressed relative

to E. Here, ~,i and hi are considered constant in time, ie. the environment is static

relative to E. The relative position of the ith target with respect to S can be expressed as

the vector sum shown in Equation 4-7. Relative orientation is described by the orientation

of ii relative to the sensor axis, S3. This relationship is characterized through expression

of ii relative to the orientation of S. The required transformation is shown in Equation 4

8, where is,i represents ii in terms of the basis vectors of S.

P(s-ti) Pt,i- Ps (4 7)

s,i = TES ( i (4 8)

4.3.2 Visibility

The characterization of visibility has been extensively studied in the field of robotics

for computer vision tasks related to industrial automation and manipulation [27, 105, 106].

A viewpoint is typically defined on a '-,-l 'liiy space." This parameter space is generally

spanned by a set of variables that affect visibility and view quality. These parameters

often include optical specifications of the camera and other sensor specific factors in

addition to the geometric definitions discussed in Section 4.3.1. Various definitions of

visibility can then be constructed through the application of sensor and task specific

constraints on this parameter space.

Here, the relative position, ips(ti), sensor orientation, tI, and target orientation, hi,

are projected onto a limited geometric parameter space to maintain generality with respect

to sensor type and environment characteristics. In addition, the specific parameterization

allows a simple and intuitive definition of visibility through bound constraints on the

parameters.

The three-dimensional parameterization with respect to the ith target is defined

by range, ri, incidence angle, a7, and FOV angle, Ofi. The range is simply the norm

of the relative position vector, as described by Equation 4-9. Variations in range are

associated with the intensity and resolution of data. The incidence angle measures the

degree to which the sensor LOS i. the target surface. This effect is characterized

by the angle between the LOS along As(ti) and the target surface normal hi, as shown

in Equation 4-10. High values for ai can result in a distorted data product. Finally, the

FOV angle measures the angular discrepancy between the LOS along t(s.ti) and the LOS

along s3, which is the sensor-frame basis vector that indicates the center of the FOV. This

relationship is described by Equation 4-11. Variations in Oft are associated with angular

position within the FOV. Figure 4-2 depicts a cross section of the FOV that shows the

visibility parameters for a particular target location.

r P = ||A(s-*) (4-9)

aJ = arccos ()) (4-10)

Of7i = arccos ((TEs- ) (4-11)

Once expressed in terms of the range, incidence, and FOV angle, visibility can be

defined through the application of simple bounds. These bounds rely on several basic

assumptions concerning the limitations of the sensor. First, the sensor is assumed to

Figure 4-2. Visibility parameters.

have a finite FOV, ie., the sensor is not omnidirectional. Second, the existence of a range

beyond which objects within the FOV cannot be resolved is assumed. These conditions

result in the visibility constraints given by Equations 4-12 and 4-13.

ri < rmax (4-12)

Of,i < Of,max (4-13)

In physical space, the simultaneous application of these constraints are realized as

the intersection of a sphere that is centered at the origin of S and and infinite circular

cone that originates from this same origin. The sphere has radius rmax. The axis of the

cone is collinear with the sensor axis, s3, and has a half-angle of Of,max. This intersection

thus forms a spherical cone that is aligned with the sensor axis. A necessary condition

for visibility of the ith target is that i(s.ti) must locate the target within this spherical

visibility cone for the current sensor orientation, s,.

Another visibility constraint results from the relative orientation of the target

surface with respect to the location of the sensor. This condition constrains the incidence

angle, ai, as shown in Equation 4-14. This constraint reflects the notion that a point

on a surface cannot be seen when the view is parallel to the surface. A parallel view

occurs when ai = 7/2, or when the LOS is orthogonal to the target surface normal,

hii. Equation 4-14 also restricts incidence angles greater than r/2, as this range of ai

corresponds to viewpoints behind the target surface. A more conservative upper bound of

,max can be considered as well to account for sensor-specific incidence requirements.

a < rmax < (4-14)
2

Physically, the incidence constraint shown in Equation 4-14 is realized as an infinite

cone originating at the target location. The axis of this cone lies along the surface normal,

hi, and the half-angle is related by ,max. In the limiting case where cmax = 7/2, this cone

approaches an open half-space that is bounded by a parallel plane to the target surface. A

necessary condition for visibility is that f(s.ti) must locate the target within this infinite

cone.

A final condition requires that a viewpoint must not be occluded for visibility. This

condition requires that the LOS from the sensor to the ith target does not intersect any

obstacles or surfaces between the sensor and the target. As such, this constraint depends

on the specific environment in which the target and sensor are located and must therefore

be applied in a task-specific fashion, as needed.

4.3.3 The Visibility Set

The visibility conditions discussed in Section 4.3.2 provide a set of necessary condi-

tions for visibility of a particular point from a particular sensor location and orientation.

The range and FOV angle conditions describe a set of points relative to a specific sensor

configuration that may be visible if other conditions are met. Conversely, the incidence

angle condition describes a set of sensor configurations relative to a specific Inl, / con-

figuration from which this target may be visible if other conditions are met. A unified

representation of these constraints in terms of sensor configurations is desirable from a

sensor planning standpoint. Such a representation would allow the definition of a target

set of sensor configurations from which a particular target is visible.

A set of sensor configurations is constructed from which the ith target meets visibility

conditions for any configuration within the set. This set is denoted the iv- l'!ly set"

with respect to i, or Vi. Construction of this set requires inversion of the constraints that

define the visibility cone such that these conditions describe a set of sensor configurations

relative to a particular i instead of a set of target locations relative to a particular sensor

configuration. The visibility set is then defined as the intersection of the inverted visibility

cone with the infinite cone describing the incidence angle constraint.

Inversion of the visibility cone is achieved by recognizing that the sensor must be

visible from a target for the target to be visible by the sensor. Consider a scenario which

has an identical sensor to that carried by the vehicle located at the ith target. This sensor

is described by the reference frame S', which is basically a copy of S that is translated

to the target location. The sensor axis corresponding to the sensor at S' points in the

direction of -s'. This direction is therefore .1- li, exactly opposite to the direction in

which the sensor at S points. The sensor at S' now has a visibility cone like that depicted

in Figure 4-3. This cone describes a set of sensor positions, ps, that are visible from S'.

Therefore the ith target is also visible from sensor positions within the S' visibility cone.

ni, S
83

Figure 4-3. Inverted visibility cone.

As discussed previously, Vi now consists of the intersection of this inverted spherical

visibility cone with the infinite cone that relates the incidence constraint for the ith target.

This set describes the sensor positions and orientations that satisfy the geometric visibility

conditions for the ith target, as described by Equation 4-15. Valid sensor positions are

defined by inclusion within the intersection of these cones while valid sensor orientations

are described implicitly by the orientation of the inverted visibility cone. As such, the

boundaries of this set vary with sensor orientation. A visualization of Vi for a particular

target for a particular sensor orientation at some instant in time is depicted in Figure 4-4.

The visualization shows the full set intersection resulting from the bounds applied to ri,

oi, and Ofi for the inverted visibility cone. In the figure, Vi is shown as the shaded region.

Vi -{ Q(p ,ts) ri < rmx, o7 < 7/2, Oft < Oimax (4-15)

Figure 4-4. Construction of Vi. The visibility set (shaded) is constructed through
application of set intersections applied to an inverted visibility cone

4.3.4 Proximity Effects

Operating a sensor in close proximity to the environment introduces several issues

that are not present for the case of sensing from standoff range. Standoff data collection

is characterized by a large footprint projected by a sensor that is significantly removed

from sensed surfaces. Conversely, close proximity data collection involves a small projected

footprint from a sensor that must navigate amongst and potentially avoid sensed surfaces.

These fundamental differences in mission scale can fundamentally change the nature of

sensor data.

One issue deals with occluded views, as introduced in Section 4.3.2. When a surface

intersects a particular LOS within the sensor FOV, nothing beyond the point of inter-

section is visible along this LOS. As such, obstacles that fall within the visibility cone

effectively cast a shadow of non-visible points. For the case where obstacles are close to

the origin of S, and therefore close to the apex of the visibility cone associated with S, this

shadow can prevent visibility of a large portion of the environment.Cluttered environments

may severely limit Vi for certain targets such that sensor placement must be performed

with extreme care.

Another significant issue is related to the coupling of sensor position and orientation

to vehicle motions. C '! .i;, in vehicle configuration directly affect the location and

pointing direction of the visibility cone regardless of problem scale; however, the effect on

collected data is significantly different for standoff sensing as compared to close proximity

sensing.

For the standoff case, a large footprint is projected onto the sensed environment

even for a narrow FOV sensor. This effect results from the angular nature of the FOV

constraint combined with the large distance from sensed surfaces. For this case, even

large translations of the sensor maintain significant overlap in projected footprint between

the start and end of the translation, as shown in Figure 4-5A. Consequently, angular

motions that occur along such translations do not have a great effect on the -. .r'egate

data collected as long as the visibility cone points in roughly the same direction at the

end of the motion as it does at the beginning. An example of such a case is a downward-

pointing sensor on an aircraft that must change its course. The aircraft must bank to

turn and may temporarily point the sensor away from sensing objectives; however, if a

large portion of the sensor footprint upon completion of the maneuver overlaps with the

footprint preceding the maneuver, there is no significant loss of data.

Alternatively, sensing at close proximity results in a much smaller projected footprint,

even for a fairly wide FOV sensor. As such, there is a high turnover of covered area as

the sensor translates with the vehicle. The sensor footprint following a translation may

be projected onto a completely different set of surfaces than were visible prior to the

translation, as shown in Figure 4-5B. The example of a turning aircraft with a downward-

pointing sensor now exhibits significant data loss as a result of the previously benign

maneuver. Moreover, navigation through close proximity environments often requires

frequent and .-.- ressive maneuvering to avoid obstacles, thus causing motion coupling

effects to be even more pronounced.

l \

A B

Figure 4-5. Motion coupling effects for different problem scales. A) Translation at standoff
range. B) Translation at close range.

Finally, significant relative variations in range and incidence are seen over the

FOV as a result of the close proximity of the sensor to the environment. When the

sensor is located a large distance from target surfaces, the range of all visible points

varies very little relative to the standoff distance. Objects appear within the FOV at

disparate distances and aspects when sensing at close range. The incidence angle can vary

significantly over a surface that would appear at a nearly constant incidence from standoff

range.

Figure 4-6 shows this effect as compared to the variations seen in typical standoff

sensing missions. The two images on the left of Figure 4-6 show range and incidence angle

data over a simulated urban environment for a sensor located at an altitude 10,000 ft.

This altitude is typical for current airborne sensor platforms such as the RQ-1 Predator

drone [3]. The images on the right of Figure 4-6 show range and incidence data when

the sensor is located at an altitude of 200 ft along the same LOS as the left images. This

altitude is typical of projected urban operations for small and micro UAVs.

I90
4000

I0 I_ -6 / I
4Woo -iDO / ao

410 0 400 La0 1 000
180ooo ,00 East (fi loo' 'L 00 East (t)

North (R) North ()
C D

from the two cases depicted in Figure 4-6 are -di it in Table 4-1. The table shows

that the two cases exhibit a similar absolute spread of range data, as the values for
Fthe standard dev and iniiation similar magnitude for each case. However, a
standoff sensing. D) Incidence for close proximity sensing

The statistics describing the variations in range and incidence for all visible points

from the two cases depicted in Figure 4-6 are di 1,v ,-1 in Table 4-1. The table shows

that the two cases exhibit a similar absolute spread of range data, as the values for

the standard deviation of this data is of similar magnitude for each case. However, a

discrepancy of a full order in magnitude is seen when these values are normalized on the

mean range. This discrepancy is reflected by the color variations in the top right image of

Figure 4-6 as compared to those seen in the top left image.

Table 4-1. Statistics for visibility parameter comparison.

Mean ri ri Std. Normalized ri Avg. ri Max ri
Deviation Std. Deviation Std. Deviation Std. Deviation
Standoff 20,023 ft 205.74 ft 0.0103 0.1471 deg 0.3673 deg
Close Proximity 543.87 ft 181.84 ft 0.3343 4.2563 deg 12.0783 deg

Incidence angle must be examined on a surface-to-surface basis, as target surfaces

with different normal directions would artificially skew the statistics. The values in the

last two columns of Table 4-1 di-pl wi average values for each visible surface over all visible

points. Very little absolute variation in incidence angle is seen for the standoff case as

compared to the close proximity case, as indicated by the average ri standard deviation

over all surfaces. Further, an increase in standard deviation by two orders of magnitude

is seen by at least one sensed surface. These results are reflected by the significant color

variations on each surface of the lower right image of Figure 4-6 as compared to those seen

on surfaces in the lower left image.

Frequent occlusions, motion coupling, and large relative variations in the visibility

parameters are demonstrated for sensing missions in close proximity to the sensed environ-

ment. These effects can adversely affect data quality if not properly considered in mission

planning. Proper consideration requires that the effects of these issues be quantified for

inclusion within sensor planning optimization algorithms. The next section describes a

sensor effectiveness metric that is designed to address this problem.

4.4 Effectiveness Metric

The developments of Section 4.3.2 and 4.3.3 allow a geometric definition of visibility

for a generic LOS remote sensor. The ith target is visible for sensor positions and orien-

tations that lie within the bounds of Vi, while for all other configurations the target is

not visible. This relationship can be characterized by a simple binary flag as expressed

in Equation 4 16. This definition of visibility is limiting in terms of describing sensor

performance, or equivalently, quality of the resulting data product. Many cases exist for

which this definition is not sufficient to describe the utility of a particular set of collected

data. Realistically, data quality can be expected to vary substantially over Vi. Moreover,

regions within Vi may exist that correspond to unusable or poor quality data due to in-

sufficient resolution or distortions. While the characterization of these effects is unique to

specific sensor modalities, a framework can be established that allows their consideration

in various sensing problems.

f {-j {.; s 0, (p^,

4.4.1 Formulation

Consider a scalar function, qj, that is associated with the ith target in the environ-

ment. This function takes as arguments the relative position of the sensor with respect

to the target, pst,, sensor orientation, ts, and the target surface normal, ii. These

arguments are mapped to a unit interval on the real number line, as expressed in Equa-

tion 4-17. This function is considered as a metric of the effectiveness with which target i

has been sensed.

qi : ( ) v- [0,1] (4-17)

The function q, necessarily evaluates to zero for all isti and s, that fall outside

Vi. As such, expression of qj in terms of visibility parameters is desirable for convenient

inclusion of this constraint. Additionally, these parameters are also often directly measur-

able from the sensor data itself. Expression of qi in terms of these parameters then allows

direct computation of sensing effectiveness from raw sensor data without requiring access

to the vehicle and sensor configurations. This parameterized form of qi is expressed in

Equation 4-18, where the visibility parameters take on functional dependencies as shown

in Equations 4-19 to 4-21. Note that the explicit form of qi is specific to both the sensor

and the application.

qi : (ri, ,o, O,) -) [0, 1] (4-18)

r ,: (,ti) R+ (4-19)

a : (sti ht,i) [0, 7/2] (4-20)

Of,i" (: s, e, ht+) [0, Ofi]x (4-21)

In addition to the dependencies expressed in Equations 4-18 to 4-21, qi has an

implicit dependence on time, such that qi = q(t). This implicit dependence results

from the time-varying nature of the relative position and sensor orientation, ,sti and

ts, and consequently the visibility parameters, ri, a, and Of. As the sensor frame, S, is

fixed relative to the vehicle body-basis, B, these time-varying quantities are affected by

the vehicle dynamics. As such, qi can be considered as an output function of the vehicle

dynamics for all i.

A representative trajectory of q, might appear as shown in Figure 4-7A. Notice

that the function exhibits apparent discontinuities along the trajectory shown. These

discontinuities result from target i leaving the sensor FOV during these time segments.

As such, the visibility criterion is not met and the effectiveness metric evaluates to zero

during these time segments.

S!,il:/ remote sensor modalities actually operate in discrete-time where data is cap-

tured at an instant in time at a finite rate. The effectiveness metric should be considered

a function of the time-discretized dynamics for these cases, and should be expressed as a

discrete-time function such that qi = qi[k]. Likewise, a representative trajectory of qi for

these cases might appear as in Figure 4-7B.

While necessary to maximize realism, this discrete-time formulation could introduce

a number of problems in gradient-based trajectory optimization schemes. Thus, the

continuous-time form of qi is primarily used throughout the remainder of this dissertation,

unless otherwise indicated. This assumption requires that the sampling rate of the sensor

is fast relative to the dynamics of the vehicle. This condition is met for the 1i i .ri y of the

sensors considered here.

1 1

0.8 0.8

30.6 0.6

S0.4 0.4

0.2 0.2

o c
0 10 20 30 10 20 30
t (sec) t[k] (sec)
A B

Figure 4-7. Representative trajectories for sensor effectiveness metric. A) qi(t). B) qi(k).

4.4.2 The Quality Set

The sensor effectiveness metric formulated in Section 4.4.1 can be used to define a

subset of Vi. This subset, Qi C Vi, represents the sensor positions and orientations that

view target i with an effectiveness of at least qi,desired, as shown by Equation 4-22. The

actual makeup of Qi depends on the specific form of qi, and is not necessarily convex with

respect to the visibility parameters, ri, ai, and Ofi.

i ( s-t, s) I qi > 9i,desired} (4-22)

The benefit to constructing a set of viewpoints that achieve a desired value of qi

is similar to the purpose of constructing V. Such a set provides a target set for motion

planning problems that can be used to formulate constraints and boundary conditions on

allowable sensor configurations.

4.4.3 Sensing Mission Effectiveness

The metric formulated in the Section 4.4.1 provides an instantaneous measure of

the quality with which a remote sensor is gathering data with respect to a particular

point in space. Realistic use of this metric for mission planning purposes would likely

require evaluation of qi at many different points in addition to a method for tracking

accumulated effectiveness. Perhaps viewing a target multiple times adds utility to the

data, even if sensed with somewhat low values for qi. Alternatively, perhaps data can

become "outdated" if a certain period of time passes between views.

Here, the function Qi is introduced to reflect the ..-':-regate sensing effectiveness over a

specified time segment. This function is related in general by Equation 4-23.

Qi : (i,te[o,t]) [0, 1] (4-23)

As with qi, explicit representation of this function is sensor and application specific;

however, several forms commonly appear for realistic sensing applications. Two useful

examples that are considered here relate the i: .::-value" effectiveness and a binary

probability-based effectiveness.

The max-value version of the mission effectiveness metric is shown in Equation 4-24.

This function keeps the maximum value of the instantaneous quality metric, qi, over

the time interval, t E [0, t]. The value of Qmax,i indicates that target i has been sensed

with that value at least once since the beginning of the mission segment. This version of

mission effectiveness can be useful for applications involving imaging sensors that collect

data subject to human analysis. One good view of a sensing objective can often provide a

great deal of information to a human analyst.

Qmax,i = max (i,te[o,t]) (4-24)

Another form of the mission effectiveness metric can be expressed using basic prob-

ability theory. Many active sensors, such as radar and sonar, pl.l a role of detecting and

classifying objects. Mine detection is an example of such an application. The probability

of detecting or classifying such objects can often be characterized in terms of visibility

parameters such as range and aspect. Equation 4-25 shows the case for which qi rep-

resents the probability of an event, such as detection, occurring at the sensing instance

corresponding to qi. Then the probability of that event not occurring can be expressed

as the complement of qi, as shown in Equation 4-26. The probability of the event not

occurring for ,:,;1 of k sensing instances can be expressed as the product in Equation 4-27.

Finally, the probability that the event occurs at least once is given by the complement of

the probability that the event does not occur at all, as shown by Equation 4-28.

P (event) = q (4-25)

Pi(- event) = 1 q, (4-26)

Pi,tot(- event) P(- event)k (4-27)
k
Pi,once (event) 1 Pi,ta( event) (4-28)

The probability that the event occurs at least once can then be implemented in

recursion to express sensing mission effectiveness for detection probability, QpD,i. This

recursion is shown as in Equation 4-29. The use of a recursive formula requires that the

discrete-time form of the sensing effectiveness metric is used.

Qp,, [k] 1 (1 Qpi[k 1])(1 q [k]) (4-29)

4.4.4 Example: A Contrived Metric

This example examines a specific formulation of qi that is contrived to reflect the

sensing performance of a generic airborne imaging sensor. This formulation uses the

concept of "efficiency functions," f,ei, for each of the visibility parameters that affect

sensor performance and data quality. The efficiency function corresponding to a particular

visibility parameter tracks the loss in data quality as a function of that parameter. Sensing

effectiveness at the ith target is then determined by multiplying these functions together as

in Equation 4-30.

qc,i= ffe,i (4-30)

Here, efficiency functions are formulated for the range, incidence angle, and FOV

angle, as shown in Equations 4-31 to 4-33. These functions necessarily evaluate to zero

outside the sensor FOV. Within the FOV, each efficiency function is shaped to reflect

data quality losses within appropriate ranges for the associated visibility parameter. The

effectiveness metric for this generic sensor is then given by Equation 4-34.

1+(r ..)4 ) 0.7exp (-400 1, ri < rmax
f m)i,ax /(4-31)
0, else

2 ni',, ) J1 i 4 f ,i1+((/ma t i ) (4 32)

0, else

f (1+(0.750, f l,maz)4O) f, < Of,max
fefi =i/Of~max ) (4-33)
0, else

q,i = fr". f, f0e, (4-34)

The efficiency functions, f,i fe1, and foe,i, are plotted in Figure 4-8. The efficiency

function corresponding to range, fri, is shown in the left image. This function is seen to

exhibit losses at close range and long range. This shaping reflects poor spatial coverage at

close range and poor resolution at long range.

0.8 0.8 0.8
0.6 \ 0.6 \ 0.6
0.4 0.4 0.4
0.2 0.2 0.2

0 500 1000 1500 2000 2500 0 50 100 20 40 60
rft o (deg) f (deg)
A B C

Figure 4-8. Quality parameter efficiency functions. A) Range efficiency. B) Incidence
efficiency. C) FOV angle efficiency.

The function corresponding to incidence angle, fi, is shown in the middle image of

Figure 4-8. This function exhibits losses as the view direction approaches tangency with

the target surface. Increasing -i has the effect of distorting surface data and limiting

visibility. The figure shows that data quality is not considered to be substantially affected

for ai < 50 deg.

The function corresponding to FOV angle, foe,i, is shown in the right image of

Figure 4-8. This function is seen to exhibit minimal losses over all FOV angles, with the

greatest effects appearing near Of,max. The maximum FOV angle corresponds to the FOV

boundary. As such, the losses are incorporated here to reflect the radial distortion effects

common to camera sensors [82].

Use of the metric described by Equation 4-34 is demonstrated through a simple

simulation. The simulation consists of a kinematic aircraft model flying at a constant

altitude of 200 ft above ground level over an urban environment. This environment

consists of three buildings arranged such that they partially enclose an area. Each building

has a maximum height of 50 ft. A sample mission scenario might require data collection

along the rooftop edges that face the partially-enclosed area to check for snipers prior to a

troop movement.

The aircraft carries an imaging sensor that is located at the origin of the aircraft

body-basis such that fx = 6. The sensor points forward and down at an angle of 60 deg.

As such, the transformation from B to S is given by Equation 4-35. The visibility

bounds for this sensor are given by Equations 4-36 to 4-38. These bounds are used with

Equations 4-31 to 4-34 to formulate the quality metric.

0.866 0 -0.5

TBS 0 1 0 (4-35)

0.5 0 0.866

ra = 2500 ft (4-36)

mrax = 90 deg (4-37)

Of,ma = 35 deg (4-38)

The simulated trajectory is depicted by the series of images in Figure 4-9. For display

purposes, targets are generated by gridding the environment surfaces at fixed resolution

of 25 ft. This selection of targets allows the metric value to be shown at all points in the

projected footprint with the specified resolution. The images in Figure 4-9 show snapshots

of instantaneous metric values within the sensor footprint at several points along the

trajectory. Notice the change in shape and effectiveness makeup of the footprint as the

aircraft banks to turn in the top right and bottom left images. Also, notice the stark

differences in metric value on different surfaces that are perpendicular to each other as the

sensor views building edges.

-300 -300
000
S0\-1000,

500 500

1000\ 1500 1000 1500
1000 1 000
500 500
East (f)150 0 North (f) East (f) 1500 North (n )

-300, i-30o
-20\ --. ------- ---_
S-00 oo -10
0 0 o
500 \ -/ 500 ,

1000 1500 1000 1500
--- 1000 \ -- 1000
) 500 0 5o0 500
East () 1500 1 North (ft) East (ft) 1500 0North (ft)

Figure 4-9. Simulated trajectory showing snapshots of effectiveness metric values.

The max-value mission effectiveness metric is used to track .,-. -egate sensing ef-

fectiveness with respect to each target for the present example. Recall that this metric

maintains the maximum effectiveness value over the entire trajectory, as given by Equa-

tion 4-24. The final values for mission effectiveness are shown in Figure 4-10 for the

trajectory depicted in Figure 4-9.

1000

Z
500

O 500 1000 1500
East (ft)

Figure 4-10. Max-value mission effectiveness.

4.4.5 Example: Image Area

This example constructs a version of qi for use with a video camera sensor. This

function, qA,i, is related to the image area occupied by a reference object. A metric

formulated in this fashion could be used to ensure that a target object appears large

enough in the image to extract desired information. For example, a certain number of

pixels might be required to resolve the license plate number on a car.

The spatial dimensions of the ith reference object must be transformed to pixel

coordinates in order to quantify image area. This projective transformation is achieved

through normalizing lengths using the distance along the sensor axis, d, as described

by the standard pinhole camera model [82]. Simple geometric relationships yield an

expression for di in terms of the visibility parameters, ri and Ofi. This expression is given

as Equation 4-39.

di = ri cos Of,i (4-39)

The physical area of a rectangular reference object is given by the product of the

lengths of its two sides, Wrf and Href, as shown in Equation 4-40. The corresponding

area in image coordinates is given by Equation 4-41, where N represents the number

of pixels in a single line of the image. The length of each side of reference object is

transformed to canonical image coordinates through a pinhole projection and is then
scaled to pixel units. The dimensions of the reference object in pixel units are given by

Wref and Hrf. Similarly, the image area in pixels2 is denoted A.

A = WrefHef (4-40)

A = Wre fHtef (4-41)

( (W,,f N H,,f N
(((rcos Of) (2 tanOf,,,max) )(( (r cosf) (2 tan Of,,,x N

When the image area is constrained to fall between two values, Amin and Amax,
allowable values for sensor range, ri, are restricted. The range must fall between two
values that are determined as a function of FOV angle, f7,i, and characteristics specific to
the sensor, N and Of,max. These bounds result from algebraic manipulation of Equation 4

41 and are shown in Equation 4-42.

(WfH f < r 2 cos Of tan Of,Tmax V Amax )- 2 cos Of tan Of,max Amrin )

The effectiveness metric is formulated as a function of image area as shown in

Equation 4-43. This function exponentially approaches qA,i 1 as A Amax. The
numerical parameters are chosen here such that a rapid rise occurs for A that are slightly

larger that Amin, followed by a large interval for which there is little gain. Such behavior

is chosen to reflect the notion that increasing image area of the reference object does
not yield additional relevant information beyond a certain size. For the aforementioned
example of a license plate, little is to be gained by increasing A once the numbers are
clearly legible. Conversely, a significant amount of information is gained closer to Amin

as slight increases in resolution help to differentiate the characters from each other. The
formulation in Equation 4-43 also includes a scaling efficiency function, f,, to account for

distortions resulting from grazing incidence angles. The same formulation of f, that was

emploiv, in the previous example is used here as well (Equation 4-32).

qA,i f exp -50 0.99(A A o) +0.)) (4-43)

The image area formulation of the sensing metric is demonstrated through a simple

simulation. As before, the simulation consists of a kinematic aircraft model flying through

an urban environment. This environment consists of one building .i.1i i:ent to five cars fac-

ing the North direction located along a roadway, as shown by Figure 4-11. The simulated

aircraft trajectory is designed to perform a diving turn from its initial configuration near

the building and then follow along the roadway. A target is designated on the rear surface

of each car near the likely location of the license plate. As such, target labels are given by

i 1, ,5.

0- )
I -5"- -- n
0

20(0 ,

150
North (ft) 0 0 50 100 50
East (ft)

Figure 4-11. Environment for image area sensing metric simulation.

The aircraft carries a video camera that is located at the origin of the aircraft body-

basis such that fi = 6. The camera points forward and down at an angle of 30 deg. As

such, the transformation from B to S is given by Equation 4-44. The camera used here

has N = 1280 lines of resolution and a FOV of 30 deg, such that Of,max = 15 deg.

0.5 0 -0.866

TBS= 0 1 0 (4-44)

0.866 0 0.5

The reference object for this example is a 1 ft by 0.5 ft license plate. The minimum

image area to resolve the numbers is chosen as Ai = 100 pixels2. The maximum is

chosen as the area that would fill the FOV, A,,x = N2 pixels2. The metric formulated in

Equation 4-43 is plotted in Figure 4-12 after substitution of these numbers. The x-axis of

the plot in Figure 4-12 is shown on a logarithmic scale.

1
0.8

5 0.6
0.4
0.2
0 5
10 10
Image Area (pixels)2

Figure 4-12. Sensing effectiveness as a function of image area.

Execution of the aircraft trajectory results in the camera FOV passing over four of

the five target surfaces. The second target is not viewed as a result of motion coupling.

The sensing effectiveness trajectories for each of the remaining four targets are shown in

Figure 4-13. These trajectories are zero for all time segments during which the target is

outside the camera FOV.

The max-value mission effectiveness metric is used here to track .I.-:-regate sensing

effectiveness for each target. Recall that this metric assumes the maximum effectiveness

value over the entire trajectory, as given by Equation 4-24. The final values for mission

effectiveness are shown in Figure 4-14. Mission Effectiveness values are shown for all

visible surfaces in addition to the five specified targets.

u0 1 2
t (sec)

1 2
t (sec)

3 4

3 4

0t 1 2
t (sec)

J0 1 2
t (sec)

Figure 4-13. Effectiveness trajectories.

North (ft)

100
East (ii)

Figure 4-14. Simulated mission effectiveness.

3 4

3 4

0.8
II

*0.6

0.2

0

CHAPTER 5
RANDOMIZED SENSOR PLANNING

5.1 Introduction

Generation of minimum time, dynamically-constrained trajectories that view multiple

targets poses a challenging problem. Most motion planning methods, such as those

described in C'! lpters 2 and 3, are generally tailored to drive the system from one

specific configuration to another. The problem of finding minimum time trajectories

that visit multiple points represents an instance of the well-known traveling salesman

problem (TSP). While this problem is well-studied in the literature [25, 26], solutions

and approximations typically rely on the assumption that optimal trajectory consists

of optimal segments, as shown in Figure 5-1A. Optimal solutions do not generally take

this form for dynamic systems. Figures 5-1B and 5-1C each show a three-point tour for

a curvature-constrained system starting from the same initial condition. Each point-to-

point segment shown in Figure 5-1B is locally optimal. Figure 5-1C shows the optimal

trajectory for the entire tour, which includes point-to-point segments that are locally

sub-optimal. Generally, both the optimal sequence of points and the optimal trajectory

must be determined; however, these solution elements are usually coupled for systems with

non-trivial dynamics.

700 700 700

600 600 600

S500 = 500 500

400 400 400

300200 300 400 500 300200 300 400 500 300200 300 400 500
East (ft) East (ft) East (ft)
A B C

Figure 5-1. TSP problem comparison. A) Standard TSP. B) Curvature-constrained TSP
with locally-optimal segments. C) Optimal curvature-constrained TSP

The curvature constrained TSP has been addressed to some extent by the research

community [15, 17, 18, 30, 57]. These efforts provide a good start to the multi-target

sensing problem, but they all equate target visitation with target sensing. Close proximity

sensing problems realistically need to consider the full sensor FOV and the coupling of

vehicle motion to sensor pointing, as discussed in C'! lpter 4. In actuality, targets can be

viewed from a set of configurations. Consideration of this property adds another l-v r of

complexity to the planning problem. The sensing problem now requires that the target

sequence is determined, the viewpoints within the visibility set, V, or the quality set, Q,

are determined for each target in the sequence, and the optimal trajectory through these

viewpoint configurations is determined.

This dynamically-constrained, set-visitation TSP has been addressed to a far lesser

extent in the literature. Researchers in the field of manipulator robotics have made efforts

to define ;ood" viewpoints, but typically have not been concerned with generating opti-

mal trajectories. Those that have investigated optimal tours do not treat nonholonomic

dynamics, as manipulator arms are usually fully-actuated systems [27-29]. These efforts

decouple the problem by generating a set of acceptable viewpoints and subsequently

solving a standard TSP through these viewpoints. References [23] and [107] consider FOV

dimensions in their study of a nonholonomic vehicle with a downward-pointing sensor.

Near-optimal trajectories are generated to view a series of targets using a simple three-

element discrete motion model and an heuristic-based enumerative search. The method

performs well for the presented examples, but the enumerative nature of the search would

introduce severe computational issues for problems that involve more than a few isolated

targets or even a slightly more complicated motion model. Additionally, the restriction

that the sensor ah--,i-i points downward neglects motion coupling effects.

This chapter develops a randomized motion planning approach to generate good,

though sub-optimal, trajectories that sense a specified set of targets with a specified

measure of effectiveness in minimum time. Concepts from previous chapters are integrated

to generate a search tree of sensing trajectories that implicitly determines a target

sequence and an associated set of viewpoints. Specifically, branches are added by planning

local trajectories to viewpoint configurations that are sampled from the quality set, Q,

of individual targets. These local motion plans must negotiate a typical close-proximity

environment that is cluttered with obstacles. The nature of this environment and that of

the embedded targets are discussed in Section 5.2. The actual algorithm is then detailed in

Section 5.3. Finally, Section 5.4 demonstrates application of the planner to the core ISR

missions described in ('! plter 1. Recall that these missions consisted of a multi-target,

single view reconnaissance task, a multi-target, multi-view surveillance task, and an area

5.2 Environment Representation

The planning environment is bounded to a finite, three dimensional operating region

in the vicinity of the sensing objectives. This region contains extruded p" .Iv-.onal obstacles

which are each defined by a set of vertex coordinates, as described in Chapter 2. Specific

locations in the operating region are designated as targets, which essentially serve as

output locations for measurement of sensing performance throughout the mission. In other

words, these targets represent the environment locations for which sensing performance is

critical, and are therefore the only locations at which sensing performance is measured.

Each target is fixed in the environment and is described by a three-dimensional

position and unit normal vector. Visibility and sensing performance at these target

locations are characterized by geometric relationships between the sensor and the target.

Specifically, the relative position and orientation are projected onto a parameter space

spanned by the range, r, the incidence angle, o, and the FOV angle, Of, as described in

('!i lpter 4. As before, a set of configurations, V, can be defined for each target from which

that target is visible. Similarly, the set, Q C V, is defined for configurations that sense

the target with at least some specified value of the effectiveness criterion formulated in

C'!h ,pter 4.

Additionally, occlusion regions can be associated with each target. Occluded view-

points are described in C'! lpter 4 as as points that do not have a clear LOS to the target,

such as when an obstacle lies between the sensor and the target. For the purposes of this

dissertation, stationary targets are embedded in a stationary environment. Hence, regions

of occlusion are time-invariant and are defined for each target-obstacle pair. These regions

can each be conservatively approximated as a convex polygon that corresponds to the

maximum -!i ,n[.-v cast by a point source located at the target, as depicted in Figure 5-2.

A more exact approximation can be achieved by parameterizing the vertex locations of

these p" v.. -ons on the vertical position, as shown in Figure 5-3.

1000

-" 500
7

1000

U
UI

500

0 500 1000
East (ft)
A

0 500 1000
East (ft)
B

Figure 5-2. Occlusion shadows. A) A target in a three-obstacle environment.
B) Conservative occlusion -! I iI...--- in the target plane.

parameterization of occlusion polygon vertices on vertical position

Finally, measurement of sensing effectiveness must be addressed for the special case

of area coverage missions. C'! lpter 4 formulates an effectiveness metric as a function

of the geometric relationship between the sensor and a single, discrete point in space.

Evaluation of sensing performance over an area generates a continuum of values and

therefore complicates some of the definitions that prove useful for motion p11 .iiiiili such as

the concept of the visibility set, V, and the quality set, Q.

Alternatively, the continuum of points in the coverage area can be discretized into

a set of "virtual t i1, I- that represent the area with a desired resolution. Such a dis-

cretization could be achieved by simply dividing the entire environment into a grid, as

depicted in the left image of Figure 5-4A; however, this discretization scheme introduces

severe computational inefficiencies. Realistically, a multi-resolution scheme is required to

emphasize the coverage area without devoting unnecessary resources to non-critical areas

of the environment. An example of such a scheme is depicted by the oct-tree-like approach

shown in the right image of Figure 5-4A. This approach involves discretizing the environ-

ment with a coarse resolution and then subdividing cells in the coverage area to increase

emphasis. Alternatively, virtual targets can be sampled from an importance-weighted

distribution that emphasizes the coverage area. The resulting resolution from choosing

virtual targets in this fashion is depicted as a vornoi plot in Figure 5-4B. The voronoi plot

consists of polygons with edges that are equidistant from .,.i ,i:ent virtual targets, and

hence reflect resolution of the discretization.

08 08
06 : 06
04 04
02 02
0 0
0 05 1 0 05 1
Y Y
A B

Figure 5-4. Discretization approaches for area coverage. A) Octree-like approach.
B) Virtual targets approach.

5.3 A Randomized Sensor-Planning Algorithm

This section develops a randomized motion planning approach to achieve the ob-

jective of effectively sensing a specified set of targets in minimum time. This planner

generates trajectories for a dynamically-constrained vehicle which carries a remote sensor

having fixed position and orientation relative to the body frame of the vehicle. The motion

plan is also subject to endpoint constraints, ie. a specified initial and terminal configura-

tion. Additionally, path constraints imposed by obstacles must also be satisfied. Sensing

criteria for the motion plan is reflected by a set of endpoint inequality constraints imposed

on the sensing mission effectiveness metric, which is defined with respect to the ith target

as Qi, as described in ('!i lpter 4. These constraints are expressed as in Equation 5-1,

where Qd, represents the desired sensing effectiveness for the ith target.

Qd, Qi(tf) < 0 (5-1)

5.3.1 Overview

As described in ('!i lpter 3, random dense tree methods generate a tree of feasible

trajectories that is designed to rapidly explore the reachable set of the system. Here,

this expansion is biased toward achieving sensing objectives. Specifically, each branch

added the tree is designed to incrementally satisfy the mission sensing constraints. As

such, each branch satisfies at least one sensing constraint in addition to those satisfied

at the expansion node. Nodes that have satisfied all sensing objectives are expanded

to the mission terminal configuration. An upper bound on the time-performance of

feasible solutions is maintained while additional solutions are generated. Feasible solutions

are generated until an independent termination condition is met, such as a limit on

computational resources or on maximum allowable planning time.

The tree structure is typically initialized as a single node at the initial condition.

An alternate option involves using a standard randomized planning technique, such as

the RRT algorithm, to generate an exploratory initial tree. Use of this option provides a

greater variety of starting locations throughout the environment prior to considering any

sensing constraints.

Following tree initialization, each iteration of the algorithm proceeds in four main

steps which are detailed in the following sections. These steps are summarized here as

follows:

1. Node Selection: A node from the existing tree is selected from a weighted dis-
tribution. Node weights can be chosen to reflect path performance such that tree
expansion is biased towards better-performing solutions.

2. Vantage Point Selection: A set of vantage points are sampled from the set, Ui Qi,
where each Qi represents the quality set associated with the ith target for which
sensing objectives have not yet been achieved.

3. Branch Extension: Candidate branches are generated by computing local trajecto-
ries from the selected node to each of the sampled vantage points. A branch is then
selected from among the candidates for addition to the tree.

4. Solution Check: The new branch is split into nodes and evaluated for possible
solutions. If a segment of the new branch has achieved all sensing objectives, a
trajectory is planned to the terminal configuration and the solution upper bound is
updated, if necessary.

A key concept that contributes to the efficiency of this algorithm is that the

inherently-close target spacing in considered environments often results in the satisfac-

tion of multiple sensing constraints along a given trajectory. This concept is depicted

in Figure 5-5. The left image depicts a scenario where six targets along a road must be

sensed. Generally, there are 6! = 720 possible sequences in which these targets can be

sensed; however, planning trajectories to view these targets individually yields a trajectory

that senses all six, as shown in the right image of Figure 5-5. A sufficient condition for

optimality of this implicitly-determined sequence is optimality of the trajectory to view

the individual target.

5.3.2 Node Selection

The node selection step determines which node is to be expanded for the current

iteration. The choice of a node selection strategy can be an important factor in biasing

-AND.

Figure 5-5. Sensing secondary targets. Planning a trajectory to sense a single target often
results in secondary sensing of other targets.

tree expansion in directions that favor better-performing solutions. Previous research ef-
forts have achieved such a bias by randomly sampling an expansion node from a weighted
probability distribution [73, 77]. Each node is assigned a weight that is dependent on
performance characteristics.

A weighting scheme that reflects the tradeoff between sensing performance and cost
performance is shown in Equation 5-2, where #vis represents the number of targets that
have been successfully sensed and L represents the path length in seconds between the
root node and N ,,,. Additionally, #tgts represents the total number of targets, Lmax is
the maximum path length over all nodes in the tree, and 6 is a shift factor included to
vary the emphasis on unsensed targets. A representative plot of this weighting function is
shown in Figure 5-6. The figure shows that nodes which have achieved sensing objectives
in a short period of time are favored. The formulation of this weighting function allows
for nodes that have sensed more targets to exhibit longer paths yet still maintain a high
weight value. High weights are also seen for branches that have not seen any targets and
have short length.

WN = (#vis)2 + (#tgts) ( ) )2 (5-2)
V \ \l max / /

~~ 3

jr

0U Lmax
Paill Length

Figure 5-6. Example node weighting function.
5.3.3 Vantage Point Selection
At least one vantage point is selected that will incrementally step the current node,
N,,,,, towards achieving sensing objectives. As such, this vantage point is selected such
that sensing requirements are met for a target that has not yet been sensed along the
path between the root and N,,,. These requirements are typically a function of the
visibility parameters associated with the ith unsensed target, ri, ai, and Of,, as described
in ('!i lpter 4. Specifically, a vantage point characterized by the sensor position and
attitude must lie in the quality set, Qi, that is associated with the ith unsensed target.
Additionally, the view from the selected vantage point must not be occluded and must be
reachable from Nr,,,.
The actual process which determines how a particular vantage point is selected de-
pends on how the vehicle is modeled and how the sensor is mounted relative to the vehicle.
For example, a vehicle modeled using motion primitives, as discussed in ('! Ilpter 2, has
a continuum of allowable positions and heading angles but a finite set of body-axis ori-
entations. These quantized elements of the allowable vehicle motions must be considered
in viewpoint selection for this case. Generally, viewpoints are selected pseudo-randomly
through sampling from the allowable vehicle configurations that meet the sensing criteria
for a selected target. Sampling is pseudo-random in that heuristics can be incorporated

to yield favorable viewpoints. For example, a position can be sampled randomly from a

projection of Qi and a favorable heading can be determined based on the relative position

of the sample to the target along with the orientation of the sensor relative to the vehicle.

Prior to acceptance, a sampled vantage point must be checked for occlusions and

reachability. As discussed in Section 5.2, each target has an associated set of p]" Iv.-onal

regions for which the LOS is obstructed. Therefore, a sampled vantage point can be

tested for occlusion by verifying that it does not lie in any of the occlusion p]" Iv.-ons

associated with the selected target. This condition can be checked using one of the

methods described in Section 2.2.2.

Verification of reachability poses a slightly more difficult task. This task involves the

concept of forward and backwards reachable sets. The forward reachable set consists of all

points reachable from the sampled vantage configuration while the backwards reachable

set consists of all points from which the sampled vantage configuration can be reached.

For the present development, a vantage point is considered reachable if the goal location is

contained in the forward reachable set and the current node is contained in the backwards

reachable set.

Unfortunately, characterization of these sets is highly non-trivial. A vantage point

sampled inside an obstacle region can be easily identified and rejected; however, selection

of vantage points that cannot be reached without an obstacle collision or that will result

in an imminent collision in the next expansion are difficult to identify without actually

planning a trajectory to or from the sample. An approximation is obtained by computing

and collision-checking several short trajectory segments near the boundaries of the

forward and backwards reachable sets, as depicted in Figure 5-7. The figure shows short

turning and straight segments into and out from a sampled position and heading. When

these representative trajectories are collision free, a degree of short-time reachability is

indicated. While this approach cannot guarantee full reachability, it can serve as a quick

test to rule out the most obvious cases.

1000

S500 U

0

0 500 1000
East (ft)

Figure 5-7. Vantage point reachability test. Vantage points for several targets are tested
for reachability using representative trajectories from the forward (green) and
backwards (blue) reachable sets.

5.3.4 Local Planning and Expansion

After vantage point selection, a set of candidate branches are generated by planning

trajectories from the current node, N,,,, to each of the sampled vantage configurations.

Each of these candidate branches must reach their respective local goals to ensure that

at least one additional sensing objective is satisfied beyond N,,r,. Consequently, a local

planning method that is capable of effectively dealing with path constraints related to

both obstacles and vehicle dynamics is required.

Recall that the randomized method developed in C'! plter 3 exhibits the capability

to efficiently generate feasible trajectories that meet these criteria. As such, this method

is appropriate for use as the local planning method here; however, use of a randomized

planner as the local planning method for use with another randomized planner requires

that some caution is taken in tree and node data representation. Specifically, the tree

structure used in the local planner should be consistent with the main planning tree to

facilitate integration of local solution nodes into the main solution. Further, a termination

condition for the local planner should be carefully selected. Recall that the planner

described in ('!i plter 3 is capable of generating a feasible solution quickly, but can offer

better approximations to the optimal solution given additional iterations. A reasonable

tradeoff between performance and efficiency must be made based on the application.

Note also that while a randomized approach is chosen here, other local planning

options, such as optimal control, are allowable. Different choices of a local planning

method may yield better solutions at the cost of computational resources. Use of an

approximation technique, such as the randomized approach which is selected here, may

require a high number of computationally cheap iterations to achieve a desirable solution.

Specific scenarios that favor one method over another can be contrived. One 1i i'. i

benefit to using a randomized local planning method is that the nodes along the solution

branch can be directly incorporated into the main solution tree. These intermediate nodes

provide a greater degree of versatility to the main solution tree in that these nodes may be

selected for expansion in the next iteration.

5.3.5 Evaluation

Candidate branches must be evaluated prior to tree addition, as only a single branch

is added at each iteration. Branches are evaluated based on sensing performance as well

as path length. Intuitively, a branch is considered desirable based on two main criteria:

1) the number of sensing constraints that are newly-satisfied along the branch, and 2) the

time duration of the trajectory along the branch.

A variety of methods are suitable for evaluating the candidate branches based on

these competing criteria. For example, a function such as that described by Equation 5-2

provides a measure of the tradeoff seen in these path properties. Alternatively, these

criteria could be applied independently and consecutively. The number of newly satisfied

sensing constraints could be used to reduce the set of candidates followed by a selection of

the minimum time candidate from those that remain.

Regardless of the exact criteria formulation, this approach to evaluating candidate

branches rewards trajectories that satisfy the sensing constraints in "chunks" while using

as little time as possible. Selection of such branches essentially amounts to gradient fol-

lowing, which is not ahv--, the most effective optimization approach for highly nonlinear

and nonconvex problems. Gradient following approaches are notoriously attracted to local

minima. Scenarios can be contrived for the present application where the optimal solution

contains a segment of long duration along which few targets are sensed. Therefore, it is

critical that the vantage point selection step retains a random element to ensure that the

planning solution can escape these local minima with non-zero probability.

Finally, any newly added nodes that have satisfied all of the sensing constraints for

a reconnaissance or coverage mission are expanded toward the mission goal. Surveillance

missions require that all targets are viewed multiple times, and, as such, the mission goal

is considered as the configuration from which the first viewpoint in the sequence is sensed.

The local planner is used for this purpose, as there are no additional sensing requirements

imposed on the trajectory. Once the planner reaches the goal, the path length is compared

to the current best upper bound and is added to a list of completed paths. The upper

bound is replaced, if necessary, and the algorithm continues on to the next iteration.

5.4 Examples

The examples in this section demonstrate an implementation of the algorithm de-

scribed in Section 5.3. These examples employ a simple vehicle model that is constrained

to operate in two dimensions yet exhibits three-dimensional sensing capabilities. Mission

scenarios are constructed such that this vehicle is required to collect data regarding targets

in a close-proximity environment. As such, a sensing metric is selected that appropri-

ately characterizes the effects of the environment proximity on sensing capabilities. The

mission scenarios reflect the three sensing tasks identified in Chapter 1: reconnaissance,

surveillance, and coverage. These tasks are first addressed in an obstacle-free environment

such that occlusions and trajectory safety are not considered. This simplified example is

followed by an example that includes polygonal obstacles.

5.4.1 Model

The examples presented here consider a simple model of a small, fixed-wing aircraft

that flies at a constant altitude of 200 ft. The motion of this vehicle can be described

by the same model used for the example in C'!i pter 3. As such, the vehicle moves with

forward velocity, V, and is subject to a bounded turn rate, |I < 30 deg/sec. The single

system input is the commanded turn rate, w. An additional requirement imposed for the

present example is that the vehicle must bank to turn. The assumption is made here that

the bank angle, o, is coupled directly to the turn rate, Q.

A hybrid motion-primitive model is formulated with seven trim states, as in C'! ,p-

ter 2. As previously, these trim states consist of steady turns in each direction at three

different turn rates and a steady, straight-ahead motion. All trim primitives maintain

constant forward velocity. The model is characterized as kinematic such that transition

maneuvers between trim states occur instantaneously. The trim states are characterized by

constant system velocities and bank-angles as follows:

To = (Vo, wo, o) = (40 ft/sec, 0 deg/sec, 0 deg)

Ta = (Via, Wia, ~ia) = (40 ft/sec, + 30 deg/sec, + 60 deg)

Tb = (VIb, blb, ylb) = (40 ft/sec, 30 deg/sec, 60 deg)

T2a (Via, W2a, 2a)o) (40 ft/sec, + 20 deg/sec, + 45 deg)

T2b = (V2b, 2b, 2b) (40 ft/sec, 20 deg/sec, 45 deg)

T3a = (V3a, 3a, 3a) = (40 ft/sec, + 15 deg/sec, + 30 deg)

T3b = (V3b, L3b, 3b) = (40 ft/sec, 15 deg/sec, 30 deg)

The aircraft carries a downward-pointing video camera. The sensing capabilities

of this camera are characterized by a maximum FOV half-angle of Of,max = 35 deg.

Maximum range for a video camera is generally dependent on the desired image resolution.

A maximum range of rma = 750 ft is chosen arbitrarily as an appropriate value for the

present example. Also, note that the orientation of the camera is fixed with respect to

the aircraft body reference frame. As such, motion coupling effects are non-negligible and

must be considered in the motion plan. These effects are seen for the present example as

sensor-pointing direction varying with respect to bank angle.

5.4.2 Effectiveness Metric

Imperfect tracking performance can be expected in realistic situations, even with

accurate models and well-designed controllers. Deviations from the planned trajectory in

either position or orientation can result in diminished sensing performance. Specifically,

targets that are viewed from the planned reference trajectory may not be visible from the

perturbed actual trajectory. Therefore, the effectiveness metric used for the example is

formulated as a "robust visibility" parameter that essentially builds an allowable margin of

error into the visibility criteria considered by the planner.

The contrived metric example discussed in Section 4.4.4 essentially quantifies this no-

tion of robust visibility. Specifically, a sensing effectiveness metric is constructed using the

product of a series of functions that exhibit dependence on the visibility parameters, as

shown by Equation 5-3. The functions, fri, fci, and fo,,,, take values that reflect the detri-

mental effects on data quality resulting from variations in the visibility parameters, r, a,

and Of, relative to the ith target. These functions take the same form as in Section 4.4.4,

with the exception of f,,i, which is related by the binary condition shown in Equation 5-4.

The parameter values that specify the exact expressions are given by ra = 750 ft,

rmax = 70 deg, Of,max = 35 deg.

qrv,i = fr fo, fof,, (5-3)

1, Ji  Jmax
fU = <(5-4)
0, else

The examples presented here require that the max-value effectiveness criteria cor-

responding to each target achieve a value of at least Qi = 0.75. This requirement is

incorporated into the motion-planning problem through a series of inequality constraints

of the form of Equation 5 1, where Qd = 0.75. The max-value criteria require that each

target is viewed with an effectiveness metric value of qr = Qd at least once along the

trajectory. Satisfaction of these constraints is guaranteed through a conservative defi-

nition of the quality set, Qi C Vi. Specifically, a valid veiewpoint must achieve a value

of Vd = 0.866 for each of the two continuously-defined efficiency functions, f,, and

fe,o. This value is shown oval -i, d1 as a dashed red line on plots of the three efficiency

functions in Figure 5-8. The plots show that simple bounds on the visibility parameters

for a particular target ensure that sensing criteria are met. These bounds are seen as:

49 < ri < 385 ft, Ofi < 29 deg, and, from Equation 5-4, ai < 70 deg. These conservative

requirements allow a non-varying definition of Qi with respect to the visibility parameters,

as shown in Equation 5-5.

--S --- --- --- -- -^ - ----- -- -- -- -- -- -- -
0.8 0.8
0.6 e- 0.6
0.4- 0.4
0.2 \0.2
0 0 20 40 60
0 200 400 60 0 0 (deg)
rft Of(deg)
A B

Figure 5-8. Quality parameter efficiency functions for examples. A) Range efficiency
function. B) FOV angle efficiency function.

Qi,r { s (psti, s) 49 < r, < 385 ft, ci < 70 deg, Of,i < 29 deg} (5-5)

5.4.3 Obstacle-Free Examples

Motion plans for effective sensing are generated in an obstacle-free environment. An

inertial reference-frame is embedded within the environment such that the x-axis points

in the North direction and the y-axis points in the East direction. This environment

is bounded such that all targets are located within an area defined by the bounds,

0 < x < 1000 ft and 0 < y < 1000 ft. All sampling steps of the algorithm draw

from locations that satisfy these bounds. Additionally, targets are placed throughout this

environment. For the present example, the normal vector associated with each target is

orthogonal to the ground plane, and can be described by Equation 5-6.

i = 0 0 Vi (5-6)

This example actually considers two sets of specified targets. The first set of targets

is shown in Figure 5-9A. The figure shows the environment boundary as a solid blue line

and twelve randomly-selected target locations as red asterisks. These targets are used for

the reconnaissance and surveillance mission descriptions. Figure 5-9B shows the second

set of targets, which is used for the coverage mission description. The figure shows a series

of roads passing through an open area. An area to the right of the plot in Figure 5-9B

is enclosed by the roads and is designated for sensor coverage. This sensing requirement

is represented by a set of eighty virtual targets which are randomly-selected from the

coverage region. These targets are depicted as red asterisks in Figure 5-9B.

1000 1000

S500 500 t"
-o 5 00* "
*O O L .*** 4

0 0

0 500 1000 0 500 1000
East (ft) East (ft)
A B

Figure 5-9. Environments for obstacle-free sensor planning example. A) Environment for
reconnaissance and surveillance missions. B) Environment for coverage
mission.

Several degrees of freedom in the algorithm described in Section 5.3 are set specif-

ically for the present example. First, each iteration of the algorithm begins with the

selection of an expansion node from a weighted distribution, where node weights are given

by the weighting function described as previously by Equation 5-2.

Viewpoint selection proceeds by selecting an unsensed target randomly from a

uniform distribution, then sampling an inertial point that satisfies the range condition

associated with Q for the selected target. A heading is computed that places the target

directly broadside the vehicle. This choice of heading increases the likelihood that the

downward-pointing sensor will view the target while in a banked turn. Finally, each bank

angle corresponding to the finite set of primitives is evaluated for sensing effectiveness.

Those primitives that satisfy the sensing constraint for the selected target are retained as

allowable terminal configurations for the local planning method.

Next, candidate branches are generated from the current node to each of the ten

viewpoint configurations. The goal configuration consists of a position, a heading angle,

and a finite set of allowable terminal trim states. For the present example, the randomized

planning method described in C'! lpter 3 is used to generate the candidate branches;

however, other approaches such as optimal control are certainly valid.

Finally, each candidate branch is evaluated for sensing and time performance.

Branches that have newly-satisfied the most sensing constraints are set aside. The

minimum-time branch from this subset is selected for addition to the trajectory tree. If all

sensing constraints have been satisfied, a trajectory is planned to the goal corresponding

to the sensing task. The initial condition serves as a goal for the reconnaissance and

coverage tasks. The first point along the trajectory for which a sensing constraint is

satisfied is used as a goal location for the surveillance task.

The first mission description examined is the reconnaissance task. Recall that this

task requires that each target is viewed at least once with a desired value for the sensing

metric. The trajectory tree is initialized by a single node at the origin of the inertial

frame. The initial automaton state at this node is represented by the trim state, To. The

initial configuration is given by (px, p, ) = (0, 0, 0). For the twelve targets shown in

Figure 5-9A, there are 12! a 479 million possible sequences in which to visit targets. Here,

the algorithm described in Section 5.3 is run for 100 iterations.

As the algorithm proceeds, a minimum upper bound is maintained for computed

solutions. Trajectories corresponding to several of these refinements are shown in Figure 5-

10. In the figure, solution trajectories are depicted as a red solid line. Additionally, the

projection of the restricted sensor footprint on the environment surface is shown for the

point along the trajectory which first satisfies the sensing constraint associated with each

target. These projections restrict the sensor FOV to the values that satisfy the definition

of Q, and are shown in Figure 5-10 as dashed contours. Solutions are shown after, 1, 10,

and 64 iterations and correspond to trajectories of 87.91 sec, 63.30 sec, and 56.95 sec,

respectively.

1000 1000 1000 ......

S500 .. 500 v 500 _}
.... ......

0 0 0

0 500 1000 0 500 1000 0 500 1000
East (ft) East (ft) East (ft)

Figure 5-10. Trajectory refinement for obstacle-free reconnaissance.

A simulation is run to plot the max-value effectiveness metric over the entire environ-

ment for the minimum-time sensing trajectory, which corresponds to the solution found

in the 64'' iteration. The result of this simulation is shown in Figure 5-11. The solution

trajectory is shown as a solid black line and the targets are shown as black circles. Notice

that the value of the effectiveness metric is at least Q = Qd = 0.75 at all target locations.

Also, note that the planner makes use of the motion coupling properties exhibited in

turning flight to sense targets from a distance. As such, the solution trajectory, while

suboptimal, is shorter than any trajectory that passes directly over all twelve targets.

Application of the planner to the surveillance task proceeds in similar fashion. The

same set of targets in the same environment is used for this demonstration. As previously,

the tree is initialized with a single node at (px, py, Q) = (0, 0, 0) in the trim state, To.

S-300(
S-200,
- -100.
0

1000

500

North (ft)

4

1000

East (fR)

1
0.9
0.8
07
0.6
05
04

0.3
0.2
0.1
0

Figure 5-11. Simulated effectiveness for obstacle-free reconnaissance.

The algorithm is run for 100 iterations. As branches are added, a list of viewpoints is

maintained that tracks the first target sensed along a trajectory. Viewpoint configurations

from this list are used as goal locations when a child branch has satisfied all of the sensing

constraints. Use of these configurations as goal locations ensures that the surveillance loop

is closed once all targets have been sensed once. This loop is then traversed repeatedly to

view each target multiple times.

A minimum upper bound on time performance is maintained as surveillance loops

are closed. The upper bound is revised as better solutions are computed. Several of these

improving solution trajectories are shown in Figure 5-12. As before, solution trajectories

are depicted as a red solid line and the "first-v, '.v projected sensor footprints are

depicted as dashed black lines. Solutions are shown after, 1, 7, and 10 iterations and

correspond to trajectories of 80.44 sec, 59.66 sec, and 54.80 sec, respectively. These values

reflect the time required to the reach the end of the first loop from the initial condition.

Figure 5-13 shows the result of a simulation that plots the max-value effectiveness

metric for the minimum-time surveillance loop. As previously, the solution trajectory is

shown as a solid black line and the targets are shown as black circles. Notice that the

sensing criteria are achieved for all target locations.

-----------

IUUU --- IUUU ..-- IUUU
c.K- *? ,, ,."* .., *.' 4, .'....*.. '"C:i ... ... ... -... .....
500 -t. 500 500

o .... .. ..
0 0 0

0 500 1000 0 500 1000 0 500 1000
East (ft) East (ft) East (ft)

Figure 5-12. Trajectory refinement for obstacle-free surveillance.

0.9
0.8
0 .7
g -300

0.5

500 0.2
100.1

North (ft) East (ft)

Figure 5-13. Simulated effectiveness for obstacle-free surveillance.

The algorithm is applied to the coverage task in the environment depicted by Fig-

ure 5-9B. The coverage area is represented by a set of eighty virtual targets which are

sampled from a uniform distribution. The planner and tree are initialized to the same val-

ues as used for the reconnaissance and surveillance tasks. The algorithm proceeds through

100 iterations in the same fashion as the other sensing tasks. Despite the larger set of tar-

gets, the order of computational complexity is the same as for the other mission tasks due

to selection of the same number of viewpoints at each iteration. The algorithm benefits

significantly from the fact that many targets are often seen from a single viewpoint given

the target-spacing that results from the sampling approach. A rise in storage complexity

does result from the need to store effectiveness metric values for each target at each node

in the trajectory tree.

A r r t\

A r r t\

A r r t\

Upon satisfaction of all sensing constraints, a trajectory is planned from a branch

node to the initial condition. The best upper bound is maintained as in the previous

cases. Several solutions that refine this upper bound are depicted in Figure 5-14. Solution

trajectories are shown as a black solid line in the figure. Projected sensor footprints are

not shown to avoid clutter resulting from the large number of virtual targets. Solutions are

shown after, 1, 15, and 84 iterations and correspond to trajectories of 93.19 sec, 78.89 sec,

and 59.02 sec, respectively.

1000 1000 1000

500 '. 500 500 -.

0 0 0

0 500 1000 0 500 1000 0 500 1000
East (ft) East (ft) East (ft)

Figure 5-14. Trajectory refinement for obstacle-free coverage.

Figure 5-15 shows a simulation of the final trajectory that plots the effectiveness

metric over the entire environment. As before, the trajectory is depicted as a solid black

line and the virtual targets are depicted as black circles. Notice from the figure that the

final trajectory does not actually result in perfect coverage. This deficiency demonstrates

the sub-optimality of the planner, and silr-.- -I that a larger set of virtual targets are

needed to adequately represent the considered coverage area.

5.4.4 Examples with Obstacles

The present example considers an environment with identical bounds as those

considered for the examples in Section 5.4.3. Additionally, four polygonal obstacles

are placed randomly within these bounds. These obstacles might represent buildings

in an urban setting. As previously, two different sets of targets are embedded within

this environment: one to demonstrate reconnaissance and surveillance mission tasks

1
0.9
0.8
-300,
-200 .0 7

)0 ---- -0- 4

500 1o
*0.2
1000
0 500 .1
North (ft) 0 0
East (f)

Figure 5-15. Simulated effectiveness for obstacle-free coverage.

and another to demonstrate area coverage. These targets have the same surface normal

properties, as expressed in Equation 5-6.

Unlike the previous case which had targets randomly scattered throughout the

area, the present case considers target locations that are strategically placed close to the

obstacles. Figure 5-16A shows the environment used for reconnaissance and surveillance

missions. Nine targets are seen in locations that could represent entrances to buildings.

Information gathered from such locations could provide valuable intelligence regarding

people or vehicles that enter and leave each building. Figure 5-16B shows the environment

used for coverage missions. One-hundred virtual targets are sampled from the region

surrounding one of the buildings. An example application for coverage of such an area is

to identify the presence of any unauthorized persons or even an explosive device within the

vicinity of the building.

The majority of the algorithm parameters are set identically to those described for

the examples in Section 5.4.3. Nodes are selected from a weighted distribution, a set of ten

viewpoints are sampled from Ui Qi, and candidate branches are planned and evaluated to

these viewpoints. Contrary to the previous example, the tree is initialized here using an

exploratory run of an RRT algorithm. Execution of this algorithm is concerned primarily

with spanning the environment and does not consider sensing objectives. Use of this

S
U.

0 500 1000
East (ft)
A

1000

500

0 500 1000
East (ft)
B

Figure 5-16. Environments for sensor planning example with obstacles. A) Environment
for reconnaissance and surveillance missions. B) Environment for coverage
mission.

technique to initialize the tree generates configuration nodes that might not have been

computed using the sensor planning approach.

The first mission description examined is the reconnaissance task. Each target must

be viewed at least once with a desired value for the sensing metric before a trajectory is

planned back to the initial condition. The trajectory tree is initialized with an exploratory

run of an RRT algorithm resulting in 50 initial nodes. This tree is rooted at the vehicle

initial configuration, (px,p,, ) = (0, 0, 0), and is shown in Figure 5-17. As previously,

the initial automaton state at the root node is represented by the trim state, To. The

algorithm is run for 100 iterations.

1000

500

0 500
East (ft)

1000

Figure 5-17. Initial tree for reconnaissance with obstacles.

1000

- 500
o
.

As the algorithm proceeds, a minimum upper bound is maintained for computed

solutions. Trajectories corresponding to several of these refinements are shown in Figure 5-

18. In the figure, solution trajectories are depicted as a red solid line. As in the previous

set of examples, the projection of the restricted sensor footprint on the environment

surface is shown for a point along the trajectory which satisfies the sensing constraint

associated with each target. These projections are shown as dashed contours. Solutions

are shown after, 1, 20, and 98 iterations and correspond to trajectories of 68.56 sec,

67.06 sec, and 62.97 sec, respectively.

1000

S500

0
0

1000

500

0
1=

1000

500

0
10

0 500 1000
East (ft)

0 500 1000
East (ft)

0 500 1000
East (ft)

Figure 5-18. Trajectory refinement for reconnaissance with obstacles.

The results of a simulation showing the sensing effectiveness over the entire environ-

ment is shown in Figure 5-19. The simulated trajectory in the figure corresponds to the

solution shown in Figure 5-18C. Two views are shown to demonstrate that all targets have

been sensed with desired effectiveness.

-200-

1000
500

North (ft)

500

East (ft)

1200|

z
1000 200

0
-200

500
East (ft)

Figure 5-19. Simulated effectiveness for reconnaissance with obstacles.

Next, the surveillance task requires that a closed orbit is generated such that all

targets can be viewed multiple times. The trajectory tree is again initialized with an

RRT containing fifty nodes and which is rooted at (py, Py, ) = (0, 0, 0) in a trim state

corresponding to To. This initial tree is shown in Figure 5-20. During the 100 iterations

for which the algorithm is run, the first point along each solution path to satisfy a sensing

constraint must be stored. As before, this point serves as a goal location for child branches

that have achieved all sensing objectives.

1000

500
0

0 500 1000
East (ft)

Figure 5-20. Initial tree for surveillance with obstacles.

As surveillance loops are closed, a minimum upper bound on trajectory time per-

formance is maintained. The upper bound is revised as better solutions are computed.

Several of these improving solution trajectories are shown in Figure 5-21. As before,

solution trajectories are depicted as a red solid line and projected sensor footprints are

depicted as dashed black lines. Solutions are shown after 4, 33, and 83 iterations and

correspond to trajectories of 71.40 sec, 66.01 sec, and 61.43 sec, respectively. These values

reflect the time required to the reach the end of the first loop from the initial condition.

Figure 5-22 shows the result of a simulation that plots the max-value effectiveness

metric for the minimum-time surveillance loop shown by Figure 5-21C. As previously, the

solution trajectory is shown as a solid black line and the targets are shown as black circles.

Notice that the sensing criteria are achieved for all target locations.

1000

- 500

0

500

0
0

0 500 1000
East (ft)

0 500 1000
East (ft)

0 500 1000
East (ft)

Figure 5-21. Trajectory refinement for surveillance with obstacles.

1200
.. A

-6011-
6-400-
o 200
0-

North (ft)

500
East (ft)

500 1000
East (ft)

Figure 5-22. Simulated effectiveness for surveillance with obstacles.

The algorithm is then applied to the coverage task in the environment depicted by

Figure 5-16B. The coverage area is represented by a set of one-hundred virtual targets

which are sampled from a uniform distribution. The trajectory tree is again initialized

with an RRT containing fifty nodes and which is rooted at (px,py, ) =(0,0,0) in a trim

state corresponding to To. This initial tree is shown in Figure 5-23.

1000

500

0

0 500 1000
East (ft)

Figure 5-23. Initial tree for coverage with obstacles.

S500

0

1000

1000

:
::..~j~~7

The algorithm proceeds through 100 iterations in the same fashion as the other

sensing tasks. Upon satisfaction of all sensing constraints, a trajectory is planned to the

initial condition from the most recently added node. The best upper bound is maintained

as in the previous cases. Several solutions that refine this upper bound are depicted in

Figure 5-24. Solution trajectories are shown as a black solid line in the figure. Projected

sensor footprints are not shown to avoid clutter resulting from the large number of virtual

targets. Solutions are shown after, 1 and 5 iterations that correspond to trajectories of

106.38 sec and 87.30 sec, respectively. Notice that the best solution out of all 100 itera-

tions was found very early in the process.

1000 -- 1000

500 500

0 0

0 500 1000 0 500 1000
East (ft) East (ft)

Figure 5-24. Trajectory refinement for coverage with obstacles.

Finally, Figure 5-25 shows the result of a simulation that plots the max-value effec-

tiveness metric for the minimum-time coverage trajectory shown by Figure 5-24B. As

previously, the solution trajectory is shown as a solid black line and the targets are shown

as black circles. Notice that the sensing criteria are achieved for all target locations.

0 00 0 08
a- 600 06

East (f) 00 100
East (f) East (ft)

Figure 5-25. Simulated effectiveness for coverage with obstacles.

CHAPTER 6
OPTIMAL SENSOR PLANNING

6.1 Introduction

Optimal control theory was presented in C'!i pter 2 as an approach for the design

of optimal trajectories for systems constrained by dynamics. While tools from optimal

control can provide solutions to many problems of interest, consideration of realistic

constraints and mission descriptions often introduces complications which render optimal

control solutions intractable. Several such complications are seen for the sensor-planning

problem considered in this dissertation. In particular, the discontinuous nature of visibility

and sensing effectiveness conditions violate smoothness requirements. These conditions can

be treated as endpoint constraints in a series of subproblems; however, such an approach

requires knowledge of the optimal sequence in which to view targets. Further, obtacles

and occlusion regions add path constraints that can increase computational burden and

introduce local minima.

This chapter examines the application of optimal control methods to a simplified

version of the close-proximity effective sensing problem. First, the problem of sensing

a single target in an obstacle-free environment is considered. This scenario allows the

formulation of sensing requirements as endpoint constraints and avoids the necessity

to consider path constraints resulting from obstacles and occlusions. The necessary

conditions for an optimal solution are derived to demonstrate some of the difficulties

associated with the use of indirect methods to solve realistic problems. Next, numerically-

based direct optimal control methods are used to transcribe the problem to a finite-

dimensional NLP. The problem is extended to consider the sensing of a small set of targets

in an obstacle-free environment. This scenario allows enumeration of all possible target

sequences such that each can be formulated as a multi-phase optimal control problem.

The optimal solution for each sequence can be computed and compared to solutions which

correspond to all other sequences to determine the optimum.

Each of these problems utilizes concepts from C'! lpter 4 to define a goal set of

vehicle configurations that ensure effective sensing. Specifically, sensing a particular

target does not require visitation. Conversely, the visibility-set concept is employ' 1 to

use the entire sensor FOV and thus account for motion coupling effects that result from

vehicle dynamics. The specific dynamic model used with these examples is described in

Section 6.2. The problem of determining a minimum-time trajectory that terminates

with a desired view of a specified target is then formulated in Section 6.3. The indirect,

variational approach to solving this problem is discussed in Section 6.4 and is followed by

the application of a direct numerical method to the problem in Section 6.5.

6.2 System

The sensing problems considered here utilize a dynamic aircraft model represented in

state-space form as = f(x, i). The state and control vectors are given by Equations 6

1 and 6-2, respectively. The states that comprise Y are: total velocity, flight path angle,

heading angle, altitude, North position, East position, angle-of-attack, angle-of-attack rate,

and roll angle. The controls are commanded angle-of-attack and commanded roll angle.

These control variables represent non-traditional system inputs and can be considered as

generators of reference commands for lower-level control systems. Design of inner-loop a

and o controllers is typical in the field of aircraft control [108].

x= V 7 Q ph px p, a l y (61)

u= U 1 (6-2)

The equations of motion for this nine-state system are given by Equations 6-3 to 6

11, where c = 2 (7/180) and the force coefficients, T, CL, and CD, are taken from

curve-fitted data [109]. Each of these equations represents an element in the vector-valued

function, x= f(x, i). This model is based on a six-state guidance level model from the

literature [109]. The original model used a and o as controls; however, these variables

are related to the aircraft/sensor attitude and are needed as states to formulate boundary

conditions for the visibility problem. Therefore, second-order angle-of-attack dynamics and

first-order roll dynamics are simulated through the addition of the three states, a, Q, and

o, along with the two new controls, u, and u,. The parameters that specify the transient

V = T cos (a + c) CDV2 sin 7 (6-3)

7 = ((T sin (a + c) + CLV2) cos O cos 7) (6-4)
1
S (T sin (a + e) + CV2) sin o (6-5)
V cos 7

Ph =V sin7 (6-6)

x = V cos 7 cos Q (6-7)

y = V cos 7 sin Q (6-8)

& = Q (6-9)

2 ,a 2-Qwn,a + w ,L (6 10)
1
S-1 (u 0) (6-11)

The polynomial curves that are used to approximate the aerodynamic force coeffi-

cients are shown by Equations 6-12 to 6-14, where a1 = (127/180). The data corresponds

to a model of a Boeing 727 [109]. Such an aircraft would not typically be involved in a

close-proximity sensing mission; consequently, a scaled-up version of the problem must be

envisioned. The force coefficient curves are are plotted in Figure 6-1.

T(V) = 0.2476 0.04312V + 0.008392V2 (6-12)

CD(a) = 0.07351 0.086170 + 1.996c 2 (6-13)

0.1667 + 6.231c, a < ca
CL(a) (6-14)
0.1667+ 6.231a- 21.65 (a a)2, else

025 07, 2-
06 15
024
05 1

022 0 30
02 0
021
0 1 1
0 100 200 300 400 500 -15 10 0 5 10 5 20 25 30 '-1 10 5 0 5 10 15 20 25 30
V (ft/sec) a (deg) a (deg)
A B C

Figure 6-1. Force coefficients. A) Thrust. B) Drag. C) Lift.

In addition to scaling the problem to match the maneuvering capabilities of the

aircraft, the model is scaled to facilitate numerical optimization performance [109].

Velocity is expressed in units of /g lc, where g is the acceleration due to gravity and 1c

is a characteristic length. This length is given by Equation 6-15, where W is the aircraft

weight and S is the planform area of the wing. All measures of distance are expressed in

units of lc and time is scaled by vg lc/lc for optimization purposes.

2W
Ic (6-15)

6.3 Problem Formulation

Consider an aircraft whose motion is described by Equations 6-3 to 6-11 and which

carries a LOS remote sensor. Motion is described relative to an inertial frame that is

oriented such that the x-axis points in the North direction, the y-axis points in the East

direction, and the z-axis points downward. The geometry of this vehicle-sensor system

is described as in ('! Ilpter 4. Here, the sensor reference frame, S, is placed to coincide

with the vehicle body reference frame, B. As such, the sensor position and orientation are

equivalently described by the vehicle position and orientation, as shown by Equations 6

16 and 6-17, where 0 represents the aircraft pitch angle. Additionally, this sensor has a

FOV characterized by a maximum range of rm, = 10, 000 ft and a FOV half-angle of

Of,max = 7/4. An incidence angle of a < r/2 is also required for visibility.

1T
Ps =PB = Py -Ph (6-16)

Cs = = 0 (6-17)

A target is located relative to an inertial coordinate frame by jt with an associated

unit normal direction indicated by n, as shown by Equations 6-18 and 6-19, respectively.

This target has an associated visibility set, V, which is defined in terms of the visibility

parameters, r, o, and Of as described in Chapter 4.

t= Pt,x Py -Pt,h (6-18)
T T
= 0 0 o-1 (6-19)

Explicit forms of r and Of are expressed in Equations 6-20 and 6-21, respectively.

The transformation term in Equation 6-21 can be written as Equation 6-22 for the

sensor configuration considered here. Notice that this expression requires knowledge of

the aircraft pitch angle, 0, which is not explicitly modeled in the equations of motion

presented in Section 6.2. The pitch angle can be recovered through the kinematic rate-

of-climb constraint, which is expressed in Equation 6-23 under the assumption that the

sideslip angle, 3 is regulated to zero [108].

r = (px Pt,x)2 + (Py Pt,)2 + (Ph Pt,h)2 (6-20)
P(s-t (TBES
Of = arccos ) (T 3) (6-21)

sin p sin u + cos p sin 0 cos u

TBES3 sin O cos ) + cos cp sin 0 sin ) (6-22)

cos p cos 0

sin 7 = cos a sin 0 cos 9p sin a cos 0 (6-23)

The objective of the problem considered in this chapter is to plan a minimum-time

trajectory from a specified initial condition at t = to to a terminal condition at t = tf

which is characterized by (ps(tf), ((tf)) V. The minimum-time performance objective,

J, is given by Equation 6-24.

(6-24)

The system is subjected to state equality and inequality constraints at the trajectory

endpoints. These constraints are expressed as Teq = 0 and Tineq < 0, where the constraint

vectors are given by Equations 6-25 and 6-26, respectively. Each state is constrained

at t = to to take the value of a specified initial condition. The flight path angle, 7, is

constrained for level flight at t = tf and the final altitude is constrained to match the

initial altitude. Note that the inequality constraint vector is comprised of the conditions

required for visibility, as shown in Equation 6-26. The incidence condition is not included

here because the terminal constraint on altitude ensures a < 7r/2.

S'(7(to), to, x(tf), tf)eq

I(x(to), to, x(tf), tf) )ineq

V(to) Vo/lc

7(to) 7o
p(to) ',o/

ph(to) Ph,o/lc

Px (to) Pol/lc

p,(to) Py,0o/I
a(to) ao

(to) Qo

p(to) o

7(tf)
Ph(tf) Ph,o

O(X(tf))- Omax
Of (2?(^)) Of"...

(6-25)

(6-26)

J= (x(tf),tf) = tf

The resulting optimal control problem is described by Equations 6-27 to 6-32. The

operating conditions have been restricted by placing bounds on allowable values for

the elements of the control vector, u, and u,. These control limits are represented as

Equations 6-31 and 6-32.

min J (6-27)

x- f(, i) = 0 (6-28)

'(I'(to),to, (tf),tf)eq 0 (6-29)

(x(to), to, (tf),tf)ineq < 0 (6-30)

Ua,min U < Ua,max (6-31)

Up,min < Up < Uyp,max (6-32)

6.4 Variational Approach

This section applies the variational calculus approach to determining the optimal

state and control trajectories for the problem described by Equations 6-27 to 6-32. Such

an approach is often called an indirect method because the solution is obtained indirectly

through derivation of the first-order necessary conditions for an optimal solution. State

and control trajectories that satisfy these conditions are determined through the solution

of boundary-value problem that results from the necessary conditions. Unfortunately,

solution of such problems have numerous associated difficulties, including small radii

of convergence, a good initial guess for the costate, and prior knowledge of trajectory

segments for which path constraints are active [51, 52].

As a result of these difficulties, the current presentation formulates, but does not

solve, the boundary-value problem. The primary purpose of this section is to demonstrate

some of the practical issues that arise in applying optimal control theory to even a

simplified version of the sensing problem. As such, the subsequent development actually

considers a reduced-form of the problem formulated in the previous section. Specifically,

the constraints restricting allowable control values, given by Equations 6-31 and 6-32,
are neglected. This assumption removes the need for prior knowledge of the constrained
trajectory segments. Additionally, the endpoint inequality constraints represented by
'(x0(to), to, x(tf), tf)ineq are treated as active qualities for derivation purposes.
6.4.1 Necessary Conditions

The first-order necessary conditions for a stationary point of the augmented cost
functional, J, are derived using the first variation, 6J. The general form of each of these
conditions was introduced in C'i plter 2. Here, these conditions are derived for the problem
described by Equations 6-27 to 6-32. The Hamiltonian of the system must be computed
to in order to form J prior to derivation of the necessary conditions. Recall that the
Hamiltonian is given by = L + ATf(x, ui), where A is the vector of costate variables.
The costates are time-dependent and match the state vector in dimension, as shown by
Equation 6-33. Here, -H is expressed as in Equation 6-34.

A -Av A, A APh Apx py Aa AQ A T

(6-33)

H Av (T cos (a + c) CDV2 sin) +

+A7 ((T sin (a + )+CLV2)cos -cos)) +

S(T sin (a + ) + CV2 sin +APh (V sin) + (6-34)
S v cos 7

+ Ap (V cos 7 cos ) + Ap, (V cos 7 sin )) + A () +

+ AQ (-LL^0a w(aL2n,aQ + wua~a) + A, (1-- ( )

The first condition considered specifies the differential behavior of the costate.
Specifically, the costate dynamics are given by A = For the present system,
the elements of this vector are expressed as shown in Equations 6-36 to 6-44. These
expressions require partial derivatives of the force-coefficient curve fits, Tv, CL., and CD.
These derivatives are shown in Equations 6-45 to 6-47.

Av -((Tvcos (a + )- 2CDV)AV + (6-35)

+((sin (a + e)(VTv T) + CL) cos V + cos 7)
+(sin ( + (VTv T) A, L sin i
+(sin (a + e) CL) + sin ph +
V2 cos y
+ cos 7 cos bApX + cos 7 sin bAp,)
sin 7A. sin 7
A -(- cos Xv + + s (T sin(a + ) +CLV2)sin p + (6-36)
V V cos2 7
+V cos 7Ah V sin 7 cos bApX V sin 7 sin b )

A -(-Vcos7sin mAp + Vcos7cos ,Ay) (6-37)

Aph 0 (6-38)

AP, 0 (6-39)

Ay = 0 (6-40)

A -((-T sin (a + ) CD V2)Av + (6-41)

+(T cos (a + c) + CL V2) ( + ) A) (6-42)

An = -(, 2(,a,,An) (6-43)

A, -((Tsin(a + )+CLV2)( co A.sin ) (6-44)
Vcos 7 V 7r1

Tv(V) = -0.04312 + 2(0.008392)V (6-45)

CD, (a) -0.08617 + 2(1.996)a (6-46)

6.231, a < ca
CL, (a) (6-47)
6.231 2(21.65) (a ac) else

The boundary conditions on the costate are set by the transversality conditions.
These conditions were described in C'!I pter 2 as Equations 2-20 and 2-21, and require the
introduction of a vector of Lagrange multipliers, given here as Equation 6-48.

T
V Vv V-o V,0 VPh0 VP,0 Vp V0 V00 0 Vf Vhf Vf VOf 1 (6-48)

Evaluation of the transversality conditions also requires that partial derivatives

are taken of the endpoint term in the cost functional, f(x(tf), tf), and of the endpoint
T
constraints, T = q Fineq j with respect to the state evaluated at the initial

and terminal conditions. The cost functional exhibits no explicit dependence on the

state such that a a= = 0. The partial derivative of the endpoint constraint
ax(to) ax(t1)
vector with respect to the initial state yields the matrix shown in Equation 6-49, where

I represents an identity matrix and 0 represents a zero matrix. The partial derivative of

the endpoint constraint vector with respect to the terminal state yields the matrix shown

in Equation 6-50. The last row of this matrix contains several complicated derivatives

resulting from the appearance of TBE in the expression for Of. The actual expressions for

these derivatives are not shown here to maintain clarity. The costate boundary conditions

are then formed by combining these expressions as specified by Equations 2-20 and 2-21.

To- 19x9 (6-49)
aY(to) [49j
04x9

09x9

a( -) 0 1 0 0 0 0 0 0 0 (650)

0 0 0 1 0 0 0 0 0

0 0 0 ar or or 0 0 0
5h 5x 9y
0 ae, aso, aso, aso, aso, aso 0 a
a- a9i5 aph px Py ap 9

The consideration of open final time requires the introduction of a condition not cov-

ered in C'! lpter 2. This condition is given by Equation 6-51 and is denoted a Hamiltonian

condition [47]. The result of this expression for the present derivation is to fix '-H at = 1

for all time, as here -H is not an explicit function of time.

f- t -f F (tf) 0 (651)

Finally, the condition for the optimal control is given by the Minimum Principle,

which states that the optimal control will minimize the Hamiltonian [48]. Recall from

C'! lpter 2 that this condition can be expressed as H = 0, which identifies a stationary

point of the Hamiltonian. For the present example, the optimality condition is expressed

as in Equations 6-52 and 6-53. Ideally, these expressions could be used to solve for the

optimal control vector in terms of the states, x, and costates, A. The Hamiltonian for the

present example is linear in the control variables. Consequently, Equations 6-52 and 6-53

do not directly yield any relevant information regarding the control.

9H,0 0 A= A (6 52)

Hn, = 0 = A, (6-53)

6.4.2 Boundary Value Problem

The previous section has derived the first-order necessary conditions for an optimal

solution to a simplified version of the problem stated in Equations 6-27 to 6-32. Obtain-

ing the optimal state and control trajectories from these conditions is not necessarily an

intuitive and straightforward process. When the Minimum Principle yields an expression

for the optimal control in terms of the state and costate, the result can be substituted

back into the state and costate dynamics. The augmented system is then described by the

differential equations shown in Equations 6-54 and 6-55.

.a= (6-54)

A= ( (6-55)

The state endpoint constraints and the transversality conditions provide boundary

conditions for the augmented system of Equations 6-54 and 6-55; however, these con-

ditions are split between the endpoints and are not all known values. For example, the

costate boundary conditions given by the transversality conditions derived in the previous

section are seen as:

VVo 0

90f

7T h0 V hfi + row a r n i r
0Pho + Phf a9
( V r os "f

Knowledge of these values would allow forward integration or the augmented dynamics to

an optimal control trajectory. Unfortunately, determination of the set of initial costate
s t ss t o os ipos a t r r a

The case presented here cannot even proceed in this fashion, as Equations 6 52
90f
Vaco VOf O

Vo 0
9O f
L V 0o Josf ao

These expressions are dependent on the vector of unknown Lagrange multipliers, i .

Knowledge of these values would allow forward integration of the augmented dynamics to

determine the optimal state and costate trajectories. The Minimum Principle then yields

an optimal control trajectory. Unfortunately, determination of the set of initial costate

values that satisfy the conditions imposed at t = t typically requires an intensive and

iterative process commonly referred to as "shooting."

The case presented here cannot even proceed in this fashion, as Equations 6-52

and 6-53 do not provide an expression for u in terms of x and A. Methods for treating

such problems are at least as complex as solution of the boundary value problem. The

problem becomes even more complex when the inequality constraints are considered and

path constraints are introduced. Consequently, an alternative approach to variational

methods is desired.

6.5 Direct Transcription Approach

Transcription of Equations 6-27 to 6-32 from a functional optmization problem to

a finite-dimensional parameter optimization problem presents a practical alternative to

the variational approach presented in the previous section. The general process for state

and control parameterization was described in C'! lpter 2. Basically, the time interval is

split into a finite set of sub-intervals and a vector of unknown parameters is formed using

the state and control values at the sub-interval nodes. Dynamic constraints are applied

in a piecewise fashion through the use of numerical methods to ensure that the state and

control values satisfy the dynamics at these nodes. Path constraints can also be applied at

these nodes, as well. Finally, a cost function is formed in terms of the parameterized state

and control and solutions are determined through the use of well-established NLP solvers.

This section applies a pseudospectral transcription method to the effective sensing

problem. Pseudospectral methods parameterize the state and control using a basis of

global orthogonal polynomials as opposed to the typical piecewise approximations [51,

52, 110, 111]. The specific method utilized here is denoted the Gauss pseudospectral

method. This method collocates the dynamics at Legendre-Gauss points using a basis

of Lagrange polynomials to interpolate the state and control trajectories[51]. The key

benefits to pseudospectral methods are related to their efficiency, accuracy, and ease of

implementation [110, 111].

The multi-target effective sensing problem is treated through sub-division of the

problem into a series of phases. Each phase represents a non-overlapping trajectory

segment that involves different problem specifications such as cost, endpoint constraints,

or even dynamics. These segments are linked by a set of conditions that constrain relevant

parameters across the boundaries of sequential phases. Here, the trajectory segments

associated with sensing each target individually are treated as separate phases. Each

of these phases has an initial condition associated with the endpoint of the previous

phase (or the global initial condition) and terminal constraints associated with sensing a

particular target. For a given target sequence, an NLP can be formulated with a global

state and control parameterization that is constrained individually within each phase as

well as across phase boundaries.

6.5.1 Numerical Example

A numerical example was performed using direct transcription with the Gauss

Pseudospectral method. This example considers an aircraft described by the dynamics

given in Section 6.2. This aircraft carries a downward-pointing sensor as described in

Section 6.3. The objective of the example is to sense two targets in a specified sequence

and then to return to the initial position. Simple visibility is considered sufficient to meet

sensing effectiveness requirements. This example considers the environment depicted in

Figure 6-2, which has an embedded inertial reference-frame that is oriented with the x-axis

pointing in the North direction and the y-axis pointing in the East direction. The two

targets are located relative to the inertial frame by jt, and 2,, as shown by Equations 6

56 and 6-57. The targets at these locations are oriented such that each has a unit normal

vector described by hi, shown by Equation 6-58. These targets are shown in Figure 6-2 as

red circles.

t = 0, 000 5,000 0 (6-56)

aP2 10,000 20,000 0 (6-57)

S= 0 0 -1 i 1,2 (6-58)

The state initial conditions at to = 0 are given by Equations 6-59 to 6-67. This initial

condition represents trimmed steady, straight and level flight at an altitude of 5, 000 ft

heading due North. Note that the states are non-dimensionalized for scaling purposes, as

-5000 0 5000 10000
East

15000 20000

Figure 6-2. Environment for optimal sensing numerical example.

described previously.

V(O)

7(0)

p(0)

ph (0)

P.(0)

Py(0)

a (0)

Q (0)

535/lc

0

5000/lc

0

0

1.7882 (7/180)

0

(6-67)

The problem is set up in three sequential phases. The first phase contains the

trajectory segment along which the aircraft must move from the initial condition to a

point where the target located by jp is visible. The second phase contains the trajectory

segment from the endpoint of the first phase to a point where the target located by it2

is visible. Finally, the third phase contains the trajectory segment from the endpoint of

the second phase back to the initial position. Each phase is discretized using 25 nodes at

which the dynamics are collocated and path constraints are enforced.

Terminal constraints are enforced at the end of each phase. These constraints

are consistent with Equations 6-29 and 6-30; however, the inequality constraints that

reflect target visibility are only enforced for the first and second phases. Further, these

visibility constraints are defined relative to the appropriate target for each phase. An

additional path constraint is imposed that restricts altitude to remain constant at ph(0)

5000/lc. This constraint is added to simplify the computation of the visibility parameter,

Of. Generally, computation of Of requires access to the pitch angle, 0, as described by

Equations 6-21 and 6-22. Recall that the pitch angle is recovered from the kinematic

constraint given as Equation 6-23. This expression requires a numerical solution procedure

for nonzero 7. The constant altitude constraint results in constant 7 = 0 over the entire

trajectory. As such, 0 can be computed analytically as Equation 6-68.

0 = arctan (cos p tan a) (6-68)

The NLP is set up and solved using GPOCS optimization software, which utilizes

TOMLAB/SNOPT [53, 64, 112]. The cost function for the NLP is specified for the total

time elapsed at the end of the third phase. An optimal solution was found that achieved

all objectives with a trajectory of 62.0356 sec in duration. The resulting trajectories of

the six states which describe position and velocity are plotted in Figure 6-3. The position
T
relative to the inertial frame is described by JB P= p pi -Ph while the total

velocity is described by V. The direction of total velocity is specified by the flight path

angle, 7, and the heading angle y.

The trajectories corresponding to the angle of attack, a, and the roll angle, p, are

shown in Figure 6-4. These trajectories would serve as the input reference trajectories to a

lower-level controller for a vehicle tasked with executing the solution trajectory.

10000

5000
0 ^^^

0 10 20 30 40 50 60
x104

0 10 20 30 40 50 60
0 10 20 30 40 50 60

C-,6000
4000-
2000
0 10 20 30 40 50 60

Figure 6-3. Solution position and velocity tra

15

5T 10

5

4500

0 10 20 30 40 50 60

0 10 20 30
300
a 200
100

0

jectories.

40 50 60

10 20 30 40 50 60

0 10 20 30 40 50 60

40

20
-e- t''

0 10 20 30 40 50 60

Figure 6-4. Solution a and o reference trajectories.

The trajectories for the visibility parameters associated with the first and second

targets are shown in Figures 6-5A and 6-5B, respectively. The maximum range and FOV

angle are indicated by dashed lines in each plot. Notice that these constraints are not

necessarily all active at the phase boundaries. The plots show that the visibility bounds

are satisfied for the first target at the end of the first phase and are satisfied for the second

target at the end of the second phase.

Finally, Figure 6-6 shows an overhead view of the total trajectory with projected

sensor footprints shown by dashed contours at the end of each of the first two phases. The

figure shows that the full sensor FOV and motion coupling were considered and utilized in

the optimal solution.

M

2 x 10,

- 1
05
0
0 10 20 30 40 50 60

S100

1 50

0 10 20 30 40 50 60

2x 10
2

05
0
0 10 20 30 40 50 60

150

100

0- 50

0 10 20 30 40 50 60

Figure 6-5. Solution visibility parameter trajectories for each target.

0 5000 10000
East

15000 20000

Figure 6-6. Aircraft trajectory and sensor footprint over each target.

16000

14000

12000

10000

8000

6000-

4000-

2000-

0-

-2000-

-4000-

-5000

CHAPTER 7
CONCLUSION

The work presented in this dissertation has addressed the problem of mission planning

for autonomous sensing tasks. More specifically, mission scenarios have been considered

which require operation of a vehicle-carried remote sensor in close proximity to both

obstacles and sensing objectives. Methods were developed to address some of the unique

challenges introduced for sensing missions fitting this description. A point-to-point

motion planning method that accounts for vehicle dynamics was developed to ensure safe

navigation amongst obstacles, where there is little margin for error. A framework was

introduced to quantify and evaluate the general quality of a particular view with respect

to geometric disparities exhibited by various surfaces within the sensor FOV. Finally, a

randomized motion planning method was developed to generate trajectories that utilize

the entire sensor FOV and account for coupling between sensor pointing and vehicle

maneuvering.

Motion planning for differentially constrained vehicle systems was examined in C'i ip-

ter 2. The concept of motion primitives were used to represent vehicle dynamics in a

hybrid modeling framework. The point-to-point motion planning problem was then formu-

lated as a hierarchical optimization problem that contained a combinatorial element and a

continuous NLP element. The combinatorial element was shown to greatly complicate the

solution procedure for realistic scenarios. As such, a special class of feasible, though sub-

optimal, explicit solutions were identified as a means by which approximate solutions can

be computed efficiently. Utility of these solutions was demonstrated for the obstacle-free

case; however, global solutions were seen to be limited when obstacles are considered. As

such, these solutions are tailored more for local planning tasks, such as that required for

branch extension in randomized methods.

Several areas for future work related to the methods described in C'!i pter 2 can be

identified. The examples discussed in C'! lpter 2 utilized very simple dynamic models to

maintain clarity of presentation; however, the concept of modeling with motion primitives

is general enough to handle more complicated and realistic systems. The introduction

of complicated dynamics is likely to be accompanied by new issues not covered in this

presentation. For example, the identification of families of unique solution sequences

was straightforward for the models considered in C'!i lpter 2, but identification of these

sequences has not been treated here for the general case. Additionally, consideration of

more complicated solution sequences could adversely affect the efficiency of these solutions.

Generalization of this approach provides an interesting direction for future research.

The concept of randomized sampling-based motion planning was introduced in ('! i p-

ter 3. A randomized planning algorithm was developed that utilized the solution methods

introduced in C'!i lpter 2. This planner generated feasible solutions by growing a tree

of sub-solutions that search the planning space in a probabilistic fashion. A simulated

example demonstrated the efficiency and effectiveness of the approach; however, several

drawbacks were apparent as well. First, solutions are sub-optimal and not necessarily

repeatable due to the randomized nature of the algorithm. These issues result from the

tradeoff between optimality of over-simplified problems versus feasible, but suboptimal, so-

lutions to complicated problems. Such a tradeoff is often acceptable for many applications.

Future work might include an investigation into the extent to which performance can be

improved by a post-processing trajectory refinement. A main criticism of randomized

approaches is that actual implementation of such algorithms requires a large number of

design parameters to be set. While these planners have been shown to perform well in

practice, this performance is often dependent on the proper selection of problem-specific

parameters and heuristics. The process of selecting the most effective combination can be

lengthy and involves a significant degree of trial and error.

('!i lpter 4 discussed a geometric approach to modeling sensor visibility. The effects

of sensor operation in close proximity to the environment were discussed and a general

framework was introduced to quantify these effects as a measure of data quality. The

work presented in C'!i ipter 4 represents an initial treatment of a topic that should pi. i a

significant role in future missions for autonomous vehicles. Numerous potential extensions

to the concept of sensing effectiveness can be identified. Future work might include

research into the integration of a FOV-based effectiveness metric with existing data quality

metrics. For example, the characterization of views that provide high-quality data related

to certain features could aid in target recognition tasks. Additionally, a logical next step

involves the inclusion of temporal effects in the formulation of the effectiveness metric.

Such a formulation could account for effects related to the velocity with which an object

moves through the FOV, such as motion blur.

C'! Ilpter 5. The randomized planning and sensor effectiveness concepts from previous

chapters are integrated to form a new randomized planning approach to sensing a series

of targets in a close proximity environment. This method generates efficient solutions

to a difficult problem that exhibits combinatorial and dynamic elements that are highly

coupled. Such solutions are enabled by the property that a single-target sensing trajectory

often senses multiple targets as a result of the mission and environment scale. The

randomized approach suffers from similar drawbacks to those discussed for the planner

developed in C'!i lpter 2; however, the simulated results are quite reasonable considering

the difficulty of the problem.

The efficient but imperfect randomized approach was contrasted by a rigorous but

cumbersome application of optimal control to the sensing problem in Chapter 6. This

chapter derived necessary conditions for an optimal solution to a simplified version of the

problem with the purpose of demonstrating the challenges associated with adopting this

solution method. Then, a direct method is implemented to compute a numerical solution

to the simplified problem. While this implementation neglects some of the issues related

to sensing in a close proximity environment, the results are promising. There is significant

potential in combining concepts from randomized planning methods with direct optimal

control methods. Specifically, solutions derived using optimal control could be used to add

branches to the trajectory tree grown in randomized planning schemes. Optimal control

could be used to refine trajectories generated using randomized methods as described

in previous chapters. Alternatively, randomized methods could provide an approximate

solution to the combinatorial element of the sensor-planning problem, namely the target

sequence, followed by application of optimal control methods to determine the optimal

trajectory. Any of these topics provide promising avenues for future research.

Finally, future research is needed to examine various alternate mission scenarios to

those presented here. For example, future sensing missions are likely to include potentially

mobile targets with unknown locations. The research presented in this dissertation could

be expanded to handle problems such as target tracking and estimation. Additionally,

many scenarios are envisioned for unmanned systems which include teams of different

vehicle platforms interacting and cooperating to achieve a unified mission objective. Lim-

ited knowledge of the environment can also expected for realistic cases. These envisioned

scenarios introduce numerous additional challenges to those treated in this dissertation.

These challenges result in numerous research opportunities that extend upon concepts

presented here.

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[111] Elnagar, G., Kazemi, M.A., and Razzaghi, M., "The Pseudospectral Legendre
Method for Discretizing Optimal Control Problems," IEEE Transactions on Auto-
matic Control, Vol. 40, No. 10, October 1995, pp. 1893-1796.

[112] Rao, A.V., "User's Manual for GPOCS Version 1.0/: A MATLAB Package to Solve
Non-Sequential Multuple-Phase Optimal Control Problems," Gainesville, FL, 2007.

BIOGRAPHICAL SKETCH

Joseph John Kehoe was born in Cooperstown, NY, on July 16, 1980. He grew up

in Oneonta, NY, and graduated from Oneonta High School in June of 1998. He then

attended Virginia Polytechnic Institute and State University, where he received a Bachelor

of Science degree in aerospace engineering in May of 2002. He has been a graduate student

at the University of Florida in the Department of Mechanical and Aerospace Engineering

since August of 2003. His research in the Flight Control Laboratory has involved the

development of .L,, .i ii i.:v-enabling guidance and control technologies for small unmanned

aerial vehicles (UAVs). Joseph received a Master of Science degree in aerospace engineer-

ing from the University of Florida in December, 2004. Upon finishing his doctoral degree,

he will remain at UF as a post-doctoral researcher for the fall semester, after which he

plans to pursue a career researching UAV control technologies.

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page ACKNOWLEDGMENTS ................................. 4 LISTOFTABLES ..................................... 9 LISTOFFIGURES .................................... 10 ABSTRACT ........................................ 14 CHAPTER 1INTRODUCTION .................................. 16 1.1Motivation .................................... 16 1.2ProblemDescription .............................. 17 1.3ProblemStatement ............................... 22 1.4ApproachOverview ............................... 24 1.5Contributions .................................. 26 2MOTIONPLANNING ................................ 27 2.1Introduction ................................... 27 2.2PathConstraints ................................ 28 2.2.1DierentialConstraints ......................... 29 2.2.2ObstacleConstraints .......................... 31 2.3OptimalControl ................................ 36 2.3.1IndirectOptimalControl ........................ 37 2.3.2DirectOptimalControl ......................... 39 2.4HybridMotionModels ............................. 42 2.4.1ModelingwithMotionPrimitives ................... 42 2.4.2MotionPlanningwithPrimitives .................... 45 2.4.3KinematicExample ........................... 48 2.4.4DynamicExample ............................ 54 3RANDOMIZEDSAMPLING-BASEDMOTIONPLANNING .......... 62 3.1Introduction ................................... 62 3.2ProbabilisticRoadmapMethods(PRM) ................... 62 3.3RandomDenseTreeMethods(RDT) ..................... 64 3.3.1Rapidly-ExploringRandomTrees(RRT) ............... 65 3.3.2Expansive-SpacesTrees(EST) ..................... 66 3.3.3Discussion ................................ 67 6

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......... 68 3.4.1Model .................................. 69 3.4.2Overview ................................. 73 3.4.3NodeSelection .............................. 74 3.4.4NodeExpansion ............................. 76 3.4.5SolutionCheck ............................. 78 3.5Example ..................................... 79 4SENSINGEFFECTIVENESS ............................ 85 4.1Introduction ................................... 85 4.2RemoteSensorTechnologies .......................... 86 4.2.1ComputerVision ............................ 87 4.2.2Radar .................................. 87 4.2.3Sonar ................................... 88 4.2.4Ladar ................................... 89 4.3ModelingtheSensingTask ........................... 90 4.3.1SensingGeometry ............................ 90 4.3.2Visibility ................................. 92 4.3.3TheVisibilitySet ............................ 95 4.3.4ProximityEects ............................ 97 4.4EectivenessMetric ............................... 101 4.4.1Formulation ............................... 102 4.4.2TheQualitySet ............................. 104 4.4.3SensingMissionEectiveness ...................... 104 4.4.4Example:AContrivedMetric ..................... 106 4.4.5Example:ImageArea .......................... 110 5RANDOMIZEDSENSORPLANNING ....................... 115 5.1Introduction ................................... 115 5.2EnvironmentRepresentation .......................... 117 5.3ARandomizedSensor-PlanningAlgorithm .................. 120 5.3.1Overview ................................. 120 5.3.2NodeSelection .............................. 121 5.3.3VantagePointSelection ......................... 123 5.3.4LocalPlanningandExpansion ..................... 125 5.3.5Evaluation ................................ 126 5.4Examples .................................... 127 5.4.1Model .................................. 127 5.4.2EectivenessMetric ........................... 129 5.4.3Obstacle-FreeExamples ......................... 130 5.4.4ExampleswithObstacles ........................ 136 7

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.......................... 143 6.1Introduction ................................... 143 6.2System ...................................... 144 6.3ProblemFormulation .............................. 146 6.4VariationalApproach .............................. 149 6.4.1NecessaryConditions .......................... 150 6.4.2BoundaryValueProblem ........................ 153 6.5DirectTranscriptionApproach ......................... 155 6.5.1NumericalExample ........................... 156 7CONCLUSION .................................... 161 REFERENCES ....................................... 165 BIOGRAPHICALSKETCH ................................ 175 8

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Table page 2-1~Mcomponentsforeachmaneuver ......................... 57 4-1Statisticsforvisibilityparametercomparison .................... 101 9

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Figure page 1-1Prevalentexamplesofadvancementsinunmannedtechnology .......... 16 1-2Comparisonofsensingdatafordierentmissionscales .............. 18 1-3Touringaseriesofcellsforalarge-scalesensingmission .............. 19 1-4Touringaseriesofcellsforasmall-scalesensingmission ............. 20 1-5Motionplanningtoviewasequenceoftargets ................... 21 1-6Sensingmissiontasks ................................. 24 2-1Obstacleboundaryapproximations ......................... 34 2-2Vertex-anglesumcollisiondetectionmethod .................... 35 2-3Vertexedge-vectorcollisiondetectionmethod ................... 35 2-4Obstacleexpansionforpointwisesafety ....................... 36 2-5Automatonmodelrepresentationasadirectedgraph ............... 45 2-6Specialreferenceframe,DforDubinspathsolution ................ 51 2-7SampleDubinspaths ................................. 54 2-8Examplestatetrajectoryshowingmaneuverdynamics .............. 56 2-9Topviewofenvironmentfordynamicexample ................... 58 2-10TopviewofsolutionforM=fMSL;MLS;MSLg 59 2-11HeadingandturnratetrajectoriesforM=fMSL;MLS;MSLg 59 2-12TopviewofsolutionforM=fMSL;MLS;MSL;MLS;MSLg 60 2-13HeadingandturnratetrajectoriesforM=fMSL;MLS;MSL;MLS;MSLg 60 2-14TopviewofsolutionforM=fMSL;MLS;MSR;MRS;MSR;MRSg 61 2-15Heading,turnratetrajectoriesforM=fMSL;MLS;MSR;MRS;MSR;MRSg 61 3-1ThePRMalgorithm ................................. 63 3-2RRTalgorithm .................................... 66 3-3ESTalgorithm .................................... 67 3-4DierencesinexplorationstrategyfortheRRTalgorithmvs.theESTalgorithm 68 10

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... 70 3-6Distancefunctioncomparison ............................ 75 3-7Distancefunctioncomputation ........................... 76 3-8Nodeexpansionstep ................................. 77 3-9Thenewbranchissubdividedtoasetofnodes. .................. 78 3-10Collision-freesolutions ................................ 79 3-11Uniquesolutionfamiliesforvehicleusedinexample ................ 81 3-12Turn-straightsolutionsequences ........................... 81 3-13Exampleplanningenvironment ........................... 82 3-14Incrementaltreeexpansion .............................. 82 3-15Nodesinthenalsolutiontree ........................... 83 3-16Solutionpathrenement ............................... 84 4-1Sensingproblemgeometry .............................. 91 4-2Visibilityparameters ................................. 94 4-3Invertedvisibilitycone ................................ 96 4-4ConstructionofVi 97 4-5Motioncouplingeectsfordierentproblemscales ................ 99 4-6Rangeandincidencevariationfordierentproblemscales ............ 100 4-7Representativetrajectoriesforsensoreectivenessmetric ............. 104 4-8Qualityparametereciencyfunctions. ....................... 107 4-9Simulatedtrajectoryshowingsnapshotsofeectivenessmetricvalues ...... 109 4-10Max-valuemissioneectiveness ........................... 110 4-11Environmentforimageareasensingmetricsimulation ............... 112 4-12Sensingeectivenessasafunctionofimagearea .................. 113 4-13Eectivenesstrajectories ............................... 114 4-14Simulatedmissioneectiveness ........................... 114 5-1TSPoroblemcomparison ............................... 115 11

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.................................. 118 5-33DOcclusionshadow ................................. 118 5-4Discretizationapproachesforareacoverage ..................... 119 5-5Sensingsecondarytargets .............................. 122 5-6Examplenodeweightingfunction .......................... 123 5-7Vantagepointreachabilitytest ........................... 125 5-8Qualityparametereciencyfunctionsforexamples ................ 130 5-9Environmentsforobstacle-freesensorplanningexample .............. 131 5-10Trajectoryrenementforobstacle-freereconnaissance ............... 133 5-11Simulatedeectivenessforobstacle-freereconnaissance .............. 134 5-12Trajectoryrenementforobstacle-freesurveillance ................ 135 5-13Simulatedeectivenessforobstacle-freesurveillance ................ 135 5-14Trajectoryrenementforobstacle-freecoverage .................. 136 5-15Simulatedeectivenessforobstacle-freecoverage .................. 137 5-16Environmentsforsensorplanningexamplewithobstacles ............. 138 5-17Initialtreeforreconnaissancewithobstacles .................... 138 5-18Trajectoryrenementforreconnaissancewithobstacles .............. 139 5-19Simulatedeectivenessforreconnaissancewithobstacles ............. 139 5-20Initialtreeforsurveillancewithobstacles ...................... 140 5-21Trajectoryrenementforsurveillancewithobstacles ............... 141 5-22Simulatedeectivenessforsurveillancewithobstacles ............... 141 5-23Initialtreeforcoveragewithobstacles ....................... 141 5-24Trajectoryrenementforcoveragewithobstacles ................. 142 5-25Simulatedeectivenessforcoveragewithobstacles ................. 142 6-1Forcecoecients ................................... 146 6-2Environmentforoptimalsensingnumericalexample ................ 157 6-3Solutionpositionandvelocitytrajectories ..................... 159 12

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....................... 159 6-5Solutionvisibilityparametertrajectoriesforeachtarget .............. 160 6-6Aircrafttrajectoryandsensorfootprintovereachtarget ............. 160 13

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1-1A to 1-1C ,representasmallcross-sectionofthediverseandgrowingeldofunmannedvehicles. B CFigure1-1. Prevalentexamplesofadvancementsinunmannedtechnology.A)TheRQ-1Predator.B)Amarsrover.C)StanfordRacing's\Stanley" Oftheexpandingrolesplayedbyexistingunmannedtechnologies,futurevehiclesstandtoproveespeciallyusefulforapplicationsthatinvolvethegatheringofinformation.Envisionedcivilian-domainscenariosincludeborderpatrol,tracmonitoring,tacticallawenforcement,maritimesurveillance,andenvironmentalsensing[ 1 ].Likewise,intelligence,surveillance,andreconnaissance(ISR)missionsareemergingasthedominantthemeamongprojectedmilitaryapplicationsforfutureunmannedsystems.Thefuturecombatparadigmislikelytoincludetheuseofcooperatingteamsofunmannedaerialvehicles(UAVs),unmannedgroundvehicles(UGVs),andunmannedunderwatervehicles(UUVs)equippedwithvarioussensorsthatautonomouslynavigatecomplexenvironmentsforthecollectionofISRdata[ 2 { 5 ]. 16

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BFigure1-2. Comparisonofsensingdatafordierentmissionscales.A)Imagescapturedfromstandorange.B)Imagescapturedwithincloseproximitytoaclutteredenvironment. Oneaimoftheworkpresentedhereistoidentifyandaddresssometheuniquechallengesencounteredforsensor-planninginclose-proximityenvironments.Motionplanningmethodsforstandomissionscommonlymakesimplifyingas-sumptionsthatareconsistentwiththeproblemscale.Forexample,targettouringisoftenconsideredsucienttoensuresensingofasetoftargets.Previouseortshaveadoptedthistechnique,whichinvolvessimplyvisitingeachtargetlocation(oralocationdirectlyabove,forthecaseofUAVs)[ 15 { 18 ].Similarly,areacoverageinsuchcasesisoftenachievedthroughdecompositionoftheareaofinterestintocellsandthensimplyvisitingeachcelltomeetsensingobjectives[ 19 { 21 ].Figure 1-3 showsarepresentativeexampleofasensormotionplanthatoperatesundersuchassumptions.Inthedepictedscenario,thesensorFOVisconsideredlargeenoughsuchthatcellvisitationissucienttosenseanytargetscontainedwithinthecell.Theseapproachesalsotypicallyassumethatlocalmotionconstraintsduetovehicledynamicsarenegligiblerelativetothescaleofthemotionplanningproblem.Forexample,thevehicleturnradiusinFigure 1-3 issmallenoughrelativetothecellresolutionandtothescaleofmaneuveringthatsuchaconstraintcanbeneglectedinmotionplanning.Numeroustechnicalchallengesareintro-ducedbyoperationofthevehicleandsensorincloseproximitytotheenvironment.First,Figure 1-2B showsdrasticallydisparatedataindierentareasoftheimageascomparedtoFigure 1-2A .Someobjectsaremuchclosertothesensorthanothers,andhenceappear 18

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1-4 assensorcoveragethatvariesrelativetothevehicletrajectory. Figure1-4. Touringaseriesofcellsforasmall-scalesensingmission. Planningatrajectorytoviewasequenceoftargetspresentsadicultproblemregardlessofsensingproximityeects.Generally,anoptimalsolutiontothisproblemrequiresthat:1.Theoptimalorderoftargetvisitationisdetermined2.Theoptimalviewpointforeachtargetisdetermined3.TheoptimaltrajectorythroughtheoptimalviewpointsiscomputedTheserequirementsaredepictedinFigure 1-5 .Eortstoaddresssuchproblemstypicallyinvolvetheapplicationofassumptionsandthesolutionofsub-optimalap-proximations.Forexample,thetarget-touringapproachdescribedpreviouslyessentiallyassumesthattheoptimalviewpointforeachtargetisthelocationofthetargetitself.Whendynamicsareneglected,thisproblemrevertstothewell-knowntravelingsales-manproblem(TSP)fromgraphtheory,forwhichnumeroussolutionandapproximationtechniquesexist[ 25 26 ].Severaleortsachieveanapproximationtotheoptimalsetofviewpointstosenseanareathroughrandomsampling,andthenproceedtosolvetheen-suingTSP[ 27 { 29 ].Theseeortsdonotconsiderdierentiallyconstrainedmotion,which,whilesomeapproximationtechniquesdoexist[ 30 ],cangreatlycomplicatethesolutionoftheTSP. 20

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35 { 37 ].Frazzolietalintroducedaplanningtechniquethatutilizesa\sampled-dynamics"modelwhichemploysasetofdynamically-consistentmotionprimitives[ 38 39 ].Additionally,recentadvancesinrandomizedplanningallowtheuseofanyofthesetechniquesaslocaltrajectorygenerationmethodsforgrowingaprobabilistictreeofactionstoexplorethesolutionspace[ 40 { 42 ]. 22

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2.2 .Next,optimalcontrolisexaminedinthecontextofbenetsanddrawbackstopracticalplanningproblemsinvolvingrealisticsystems.Section 2.4 thenintroducesamodelingalternativethatprovidessolutionsthroughlower-dimensionaloptimizationproblemsthanthosetyp-icallyencounteredwhenusingdirectoptimalcontrolmethods.Thismethodutilizestheconceptofmotionprimitivesandevenadmitsclosed-formalgebraicsolutionstospecialcasesoftheplanningproblemformulation. 2{1 .Theconstraintfunction,c~xc;_~xc;~xc;~u;trepresentsa 28

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2{1 canbeusedtorequireavoidanceofobstacles,toenforcedierentialsystembehavior,toboundallowableinputsignals,andtorestricttheoperationalenvelopeofthesystem.Apathconstraintcanbeappliedtoaparticularpointalongatrajectoryortoacontinuumofpointsalongatrajectorysegment.Theformercaseisrequiredfornite-dimensionaltrajectoryoptimizationproblemswhileeitherusageisacceptableforinnite-dimensionalfunctionaloptimizationproblems. 29

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2{2 ).First-orderconstraintsonthestate-space,whichincluderstandsecond-orderdierentialconstraintsontheC-space,areconsidereddynamicconstraints(Equation 2{3 ).Motionplanningunderthisdenitionofdierentialconstraintsisreferredtointheroboticsliteratureaskinodynamicmotionplanning[ 44 ]._~xc=f(~xc;~u);~xc2C 30

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2{4 ,whererprepresentstherangeof~p2Dfrom~pcandobs;prepresentsthebearingof~p2Drelativeto~pc.TheserelationsaregivenbyEquations 2{5 and 2{6 ,respectively. (2{4) Computationofrbound(obs)requiresparameterizationoftheedgesthatrepresenttheobstacleboundarywithrespecttoobs.Theinertialpositionofapointalongtheobstacleedgeconnectingtheithobstaclevertextothejthobstaclevertexisexpressedas~peij=[px;eijpy;eij]T.ThecomponentsofthisvectorcanbeparameterizedasshowninEquations 2{7 and 2{8 .Theboundaryoftheobstacleisthencomputedusingthepiecewise-continuousradiusfunctionshowninEquation 2{9 ,whereobs;viandobs;vjrepresentthebearingsoftheithandjthobstacleverticesrelativeto~pc,respectively.Thisradiusfunction,rbound;act,isdepictedasasolidblacklineinFigure 2-1 forasquareobstacle.ThegraphicshowninFigure 2-1A shows~peijrelativetothereferenceframecenteredat~pc=[00]T.ThegraphicshowninFigure 2-1B showsrbound(obs).px;eij(obs)=px;vi(px;vjpx;vi)py;vi+py;c+px;c+px;vitanobs 32

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Vertex-anglesumcollisiondetectionmethod. Anothermethodthatrequirescomputationof~vp!viisdepictedinFigure 2-3 .Thismethodadditionallyrequirescomputationofthesetofedgevectors.Eachofthesevectorscorrespondstoapolygonedgeandindicatesedgedirectioninaclockwisesense.Theedgevectorbasedattheithvertexpointstothejthvertexandcanbeexpressedas~veij=~pvj~pvi.Theunitnormalto~veijisdenoted^neijandisalsobasedattheithvertex.Thisnormalvectorcanbecomputedviaarotationthrough=2rad,asshowninEquation 2{13 .Anecessaryandsucientconditionforacollision-freepointismetifEquation 2{14 issatisedforany(i;j)pair.Thisconditionessentiallystatesthatifthetestpointislocatedinthesamedirectionrelativetoeachedgevector,thepointliesinsidethepolygon. Figure2-3. Vertexedge-vectorcollisiondetectionmethod. ^neij=1 35

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2-4A .ObstacleboundariescanbeexpandedasshowninFigure 2-4B toavoidsuchsituations.Theextentofexpansionmustbecarefullyselectedtoensurethatunsafetrajectoriescannotfeasiblysatisfyobstacleconstraints. BFigure2-4. Obstacleexpansionforpointwisesafety.A)Pointwisesatisfactionofobstacleconstraints.B)Pointwisesatisfactionofexpandedobstacleconstraints. 36

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2{15 (~x(t);~u(t))=argmin~x(t);~u(t)J(~x(t);~u(t);t) (2{15) 2{15 toemphasizethattheproblemseekstodeterminecontinuousfunctions,andthereforehasinnitedimension.Thedierentialpathconstraintsareenforcedinequality;therefore,theyarewrittenseparatelyfromthealgebraicpathconstraints,c(~x(t);~u(t)).Thefunction,(~x0;t0;~xf;tf),relatesendpointconstraintsonthetrajectory.Finally,thefunctionalperformanceindex,J(~x(t);~u(t);t)istypicallywritteninintegralformasinEquation 2{16 2{15 forwhichtherearenoalgebraicpathcon-straintsandtheendpointconstraintisappliedinequality.ThecostfunctionalcanbeaugmentedwiththedierentialandendpointconstraintsthroughtheintroductionofavectorofLagrangemultipliers,~,andavectorofmultiplierfunctions,~(t).Eachele-mentof~isassociatedwithanendpointconstraintwhileeachelementof~(t),denoted 37

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2{17 .Inthisexpression,HisdenotedtheHamiltonianofthesystemandisshownasEquation 2{18 .Functionargumentshavebeenomittedheretomaintainclarityintheexpressions.TheconstrainedoptimizationprobleminJisnowre-expressedasanunconstrainedoptimizationprobleminJ. J=~Tjtf+Ztft0hH~T_~xidt (2{18) AsetofnecessaryconditionsforastationarypointofJcanbefoundbytakingtherstvariationandsettingitequaltozero,asJ=0.Afullderivationoftheseconditionsisbeyondthescopeofthisdissertationbutandisavailableintheliterature[ 47 ].Therstsetofconditionsaregivenbythecostatedynamics,asshowninEquation 2{19 .Thesedierentialequationshaveboundaryconditionsgivenbythetransversalityconditions,asshowninEquations 2{20 and 2{21 _~=@H @~x(t0)+~T@ (2{20) @~x(tf)~T@ (2{21) TheoptimalcontrolisrelatedbyPontryagin'sminimumprinciple,whichstatesthattheoptimalcontrolminimizestheHamiltonian[ 48 ].TheweakformofthisconditionisshownasEquation 2{22 2{22 relatestheoptimalcontrolintermsofthestateandcostate.Thiscontrolfunctioncanberecoverediftheoptimalstateandcostatetrajectoriesaredeter-mined.Assuch,theoptimalcontrolproblemcanbesolvedindirectlythroughsolutionoftheboundary-valueproblem(BVP)speciedbythestatedynamics,thecostatedynamics, 38

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49 ].Pathorendpointconstraintsthatareenforcedasinequalitiescancauseanincreaseincomputationalburdenwhichresultsfromtheneedtoidentifytheactiveandinactiveelementsoftheseconstraintfunctions.Further,theoptimalityconditiondoesnotalwaysyieldaclosed-formexpressionfor~u(t)intermsofthestateandcostate.Sometimesthisconditiondoesnotevenuniquelydeterminetheinputfunction.Whiletheindirectapproachtosolvingoptimalcontrolproblemsutilizesarigorousmathematicalframework,allbuttrivialproblemsareoftenrenderedimpractical. 49 { 52 ].Suchmethodsinvolvetranscriptionoftheinnite-dimensionalfunctionaloptimizationshowninProblem 2{15 toanite-dimensionalfunctionoptimization.Assuch,derivationofnecessaryconditionsandtheuseofvariationalcalculusisunnecessary;instead,numericaloptimizationtech-niquesareusedtosolvethenonlinearprogram(NLP)whichresultsfromtheproblemtranscription.Therststepinthetranscriptionprocessistosplitthetimeinterval,t2[t0;tf],intoanitesetofxedsub-intervals,asshownbyEquation 2{23 .TheNpointsatwhichthe 39

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2{24 2{16 remainsunchangedthroughtranscriptioniftheterminaltime,tf,coincideswithoneofthenodesinEquation 2{23 .Theintegraltermcanbeapproximatedasanitesumusingnumericalintegrationtechniques.Similarly,numericalintegrationisusedtorepresentthedierentialconstraintsasavectorofdefects,denotedas.Eachdefectenforcestheintegrationrulebetweenthecorrespondingpairofnodes.ThedefectcorrespondingtothekthnodeisshowninEquation 2{25 ,wherefrepresentstheintegrationrule.Dependenceoffisshownwithrespecttostateandcontrolvaluesatthecurrentandnexttimesteptomaintaingenerality;however,manyintegrationtechniquesonlyrequireinformationatthecurrenttimestep.ExamplesofsuchtechniquesincludeEulerintegrationandtheRunge-Kuttamethod. 2{15 canbeapplieddirectlywithinthetranscribedformulationift0andtfareincludedasnodes.Additionally,algebraicpathconstraintsareappliedtothestateandcontrolpointwiseateachnode.Theseconstraints 40

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2{26 2{27 .Thisprob-lemcontainsatleastN(n+m)variablesforasystemwithnstatesandmcontrols,andthusisverylargeforevenacoarsetime-discretization.Fortunately,therelevantmatricesinvolvedinsolutionoftheNLPtakeonasparseformasaresultofthetranscriptionmethod[ 49 ].Thissparsitycanbeexploitedtogreatlyreducebothstorageandcomputa-tiontime.SeveralNLPsolversthatexhibitthiscapabilityhavebeenshowntoperformquitewellinhandlingproblemsofthisnature[ 53 ]. 2{27 isfundamentallyadierentproblemthanProblem 2{15 .Aninterpolatedfunctionthroughthediscretesolutionvaluestothenite-dimensionalNLPmaynotcoincidewiththeoptimalsolutiontotheinnite-dimensionalproblem.Further,asthenecessaryconditionsarenotcomputed,thereisnowaytovalidatesolutionstoProblem 2{27 .Progresshasbeenmadeinthisarearecently 41

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51 52 ].Suchmethodsrepresentanactiveareaofresearchinoptimalcontroltheory. 2.4.1ModelingwithMotionPrimitivesAfundamentalpropertythatiscommontomanyvehiclesystemsofinterestisrelatedtotheconceptofsymmetry.Specically,thetrajectoriesofacertainsubsetofthestateorcongurationvariablesareseentoexhibitinvariancewithrespecttocertainclassesoftransformations.Let~xrepresenteitherthestateorcongurationvectorofasystemwhosedynamicsarerepresentedintheformofEquation 2{2 or 2{3 ,respectively.Asystemtrajectorycanbedeterminedthroughintegrationofthedynamicswithrespecttotime.Nowconsideratransformationfunction,H:C7!C(orH:X7!X).Systemtrajectoriesaresaidtoexhibitinvariancewithrespecttothetransformation,H,ifEquation 2{28 holdstrue[ 39 ]. 39 ].SatisfyingEquation 2{28 thusimpliesthatatrajectorygeneratedfromaparticularinitialcondition,~x0,throughtheapplicationofaninputfunction,~u,isequivalenttoalltrajectoriesthatresultfromtheapplicationof~uandthatareinitializedatanypointintherange,H(~x0).Trajectoriesthatsatisfythispropertyaredenotedmotionprimitives.Physically,Hrepresentstheclassoftransformationsthatdonotaecttheexternalforcesactingonthevehicle.Forexample,consideracar-likevehicleoperatingonanexpansive,at,isotropicsurface.Forcesonthevehicleresultmainlyfromgravityandfriction.Relativedisplacementsinpositionandheadingthatresultfromaparticularinputtrajectoryareinvarianttoabsolutepositionandheading.Therefore,translationsand 42

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Automatonmodelrepresentationasadirectedgraph canthereforetransitionbetweentrimswithnocongurationchanges.Dynamicsystemsexhibitboundedaccelerationsand,assuch,smoothsystemcongurationchangeswillaccompanyanytransitionsbetweentrimstates.TheautomatonmodelingschemedepictedinFigure 2-5 canrepresentcomplicatedsystemtrajectoriesthroughspecicationofasequenceofmaneuversandthedurationofthetrimstatesbetweeneachconsecutivemaneuver.Thismodelexhibitsbothcontinuousbehavioranddiscreteswitchingbehaviorinthetrimstatesandnite-durationmaneuvers,respectively,andisthereforeconsideredahybridsystem.Givenamaneuversequenceoflengthn,M=fM1;M2;;Mng,andacorrespondinglengthn+1sequenceoftrimdurations,=f1;2;;n+1g,systemtrajectoriesintheC-spacearecomputedviaaseriesoftransformations.Suchtrajectoriesalwaysinitiateandterminateinatrimconguration.Equations 2{33 and 2{34 showageneralexampleofthisprocessformaneuverandtimesequencesofM=fM1;M2g,and=f1;2;3g,respectively,whereGrepresentsthetotalstatedisplacementresultingfromthesequence. 45

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2{35 (M;)=argminM;J(M;;~x0) (2{35) 2{35 presentsadicultmixed-integernonlinearprogram(MINLP);however,theproblemstructuredoeslenditselftoahierarchicaldecomposition[ 39 ].Givenamaneuversequence,M,Problem 2{35 reducestoasmoothNLPinthetrimdurations,,thatisessentiallyanalogoustoaninversekinematicsproblem.Suchproblemsareacommonandwell-studiedclassofproblemsinrobotgeometry[ 54 ].Hence,acombinatorialsearchthroughthesetofallpossibleMaccompaniedbythesolutionofasmoothNLPforeachchoiceofMisgenerallyrequiredtosolveProblem 2{35 .Thelength,n,oftheoptimalsequence,M,isnotknowningeneral.Assuch,thesetofallpossibleMiscountablyinnite;however,itcanbeshownthatanite-lengthoptimalsequenceexistsandcanbedeterminedexplicitlyinsomespecialcases.Additionally,theliteraturesuggeststhatpruningandbranch-and-boundtechniquescanbeappliedtosimplifyandexpeditethecombinatorialsearch[ 39 55 ].Alternatively,considerafamilyofsequenceswhereeachmembersequenceconsistsofaxednumberofmotionprimitives.Eachtrimprimitiveineachmembersequencediersinthemagnitudesofthetrimvelocities;however,thegeneralshapeofthestatetrajectoriesremainsunchanged.Maneuverprimitivesdierasneededbetweenmembersequencesto 46

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2{35 .ThisconstraintisrepeatedasEquation 2{36 ,where~xfrepresentstheconstrainedterminalstateandG(~x0;M;)representstheaggregatestatedisplacementoftheprimitivesequencecharacterizedbyMand.Atmostonefeasiblesolutionexiststothecontinuoussubproblemcorrespondingtoeachmembersequenceoffamiliesexhibitingthisproperty.Further,Equation 2{36 canoftenbemanipulatedtoyieldaclosed-formexpressionforthisfamilyofsolutions.Assuch,thesepotentialsolutionscanbeecientlycomputedforeachmembersequenceandevaluatedforbothperformanceandforsatisfactionofpathconstraints.Notethatdierentialpathcon-straintsareimplicitlyhandledbytheautomatonrepresentationofthedynamics.Feasiblesolutionscanbeenumeratedforeachsequenceinaparticularfamilyandusedtodeter-mineanapproximatesolutiontoProblem 2{35 .Solutionsobtainedinthisfasioncanbeshowntobeoptimalforsomespecialcases,aswillbedemonstratedinSection 2.4.3 47

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17 18 23 30 56 57 ].Further,thetypicalformulationofthemodellendsitselfwelltorepresentationintermsofmotionprimitives.Assuch,thismodelservesasanappropriateandrelevantmodelforuseinakinematicplanningexample.ThisexampledemonstratestheuniquesolutionfamilyapproachtondapproximatesolutionstoaproblemoftheformofProblem 2{35 .MinimumtimetrajectoriesbetweentwocongurationsaresoughtfortheDubinscarmodelinanenvironmentwithnoobstaclesand,therefore,nopathconstraints.TheDubinscarisasimplecar-likevehiclemodelthatoperatesinaC-spacespannedbytwoEuclideanpositionvariables,pxandpy,andanangledescribingtheheading,.Thecarmoveswithunitforwardvelocityandchangesdirectionbyassumingaunitturnrate,!,ineitherdirection.Assuch,themotionoftheDubinscarisdescribedbythedierentialsystemshowninEquation 2{37 ,wherethediscretesetofvaluesassumedby!isshownbyEquation 2{38 where: 2.4.1 .Thesemotionsareinvariantwithrespecttopositionandheading,andthereforecanbetranslatedandrotatedtooriginatefromanypoint 48

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2{39 to 2{41 forturningmotionsandEquations 2{42 to 2{44 forstraightmotion.NotethatEquation 2{39 canrepresentmotionalongeitherarightorleftturnbasedonthesignof!inEquation 2{41 (2{39) (2{42) Thesethreetrimprimitivescanbeintegratedintoasimplethree-stateautomatonswitchingstructurelikethatdescribedinSection 2.4.1 .ThetrimstatesaredenotedL,S,andR,whichcorrespondtotheleft,straight,andrighttrimmotions,respectively.Thesixmaneuversthatinterconnectthetrimstatesconsistofinstantaneousswitchescor-respondingtoaninstantaneouschangeofturnrate,!.Recallthatsuchaninstantaneousswitchispermissibleforakinematicmodel. 49

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2{35 .ApotentiallyinnitesetofautomatonsequencescouldbeenumeratedandtheresultingconstrainedNLPsolvedforeach.Alternatively,twofamiliesofsolutionsequencesexistthatuniquelysatisfyEquation 2{36 whenturningmotionisrestrictedtoheadingchangesoflessthan2rad.Theselength-threesequencefamiliesconsistofthesetofallturn-straight-turnsequencesandthesetofallturn-turn-turnsequences.ThemembersequencesofthesefamiliescanbeenumeratedforthecurrentautomatonmodelintermsoftrimmotionsasD1=fLSL;RSR;RSL;LSRgandD2=fRLR;LRLg.ThesolutiontoProblem 2{35 canthenbeapproximatedbycomputingthesolutionscorrespondingtoeachofthesixsequencescontainedinD=(D1[D2).Specically,Equation 2{36 canbesolvedtoobtainfeasibletrimdurationsforeachsequence.Theresultingsetofsixtrajectoriesarethenevaluatedfortimeperformance,wheretheminimumtimetrajectoryiskeptasthesolution.Dubinsshowedin1957thattheoptimalminimumtimetrajectoryfortheDubinscarconsistsofmotionsdescribedbyoneofthesequencesintheset,D[ 56 ].Consequently,theapproximatesolutioncomputedusingthedescribedmethodcorrespondstotheactualoptimalsolutionforthisspecialcase.ManysequencesthatarenotcontainedinDalsosatisfytheendpointconstraint;however,thesesolutionsequencesgenerallydonotyieldauniquesolution.Feasiblesequencesoflength-fourorgreateroverdetermineEquation 2{36 inthetrimdurationsandthereforegenerallyadmitaninnitenumberofsolutions.WhileanoptimalsolutionwithrespecttoaparticularperformancecriterioncanbefoundthroughthesolutionofanNLP,thesesub-optimizationproblemsdonotnecessarilyguaranteedesirableconvexityandconditioningproperties,ie.,localandpoorlyconditionedsolutioncasesmayexist.ActualcomputationofthetrimdurationsthatsatisfythecongurationendpointconstraintforsequencescontainedinDcanbeachievedusingseveraldierenttech-niquesfromtheliterature[ 58 { 63 ].Here,aparticularmethodwhichutilizesalgebraic 50

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61 ].Thismethodrequiresthattheinitialandgoalcongurationsaretransformedtoaspecialcanonicalcoordinateframe,D,asshowninFigure 2-6 .Thiscoordinateframehas(px0;py0)locatedattheoriginandthegoalcongurationlocatedadistance,d,alongtheXD-axis.Initialandnalheadingsaremeasuredfromthelinecon-nectingtheinitialandnalpositions,asshowninFigure 2-6 .ThereforethetransformedinitialandnalcongurationsaregivenasEquations 2{45 and 2{46 ,respectively. Figure2-6. Specialreferenceframe,DforDubinspathsolution. Thetransformationsdescribingtheleftturn,rightturn,andstraightaheadmotionsarerewritteninanalternateformasEquations 2{47 2{48 ,and 2{49 ,respectively.Theresultingmotionsareunchangedascomparedtotheprimitivesdenedpreviously.L(px;py;;)=(px+sin(+)sin;pycos(+)+cos;+) (2{47)R(px;py;;)=(pxsin()+sin;py+cos()cos;) (2{48)S(px;py;;)=(px+cos;y+sin;) (2{49)ForeachsequenceinD,thetransformationsinEquations 2{47 2{48 ,and 2{49 canbeappliedtotheinitialcongurationconsecutivelyintheproperorderthroughthecompositionoperation.Theresultineachcaseisthenequatedtonalconguration.A 51

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61 ]:1.L(S(L(px0;py0;0;1);2);3))=(d;0;2) (2{50b) 2.R(S(R(px0;py0;0;1);2);3))=(d;0;2) (2{51b) 3.R(S(L(px0;py0;0;1);2);3))=(d;0;2) (2{52b) 4.L(S(R(px0;py0;0;1);2);3))=(d;0;2) (2{53b) 52

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6.L(R(L(px0;py0;0;1);2);3))=(d;0;2) Figures 2-7A to 2-7F showsamplesolutionstothecongurationendpointconstraintthathavebeengeneratedusingtheexplicitexpressionsfortheDubinscartrimdurations.Figures 2-7A to 2-7D showsolutionsfromtheturn-straight-turnfamilyofsequenceswhileFigures 2-7E and 2-7F showsolutionsfromtheturn-turn-turnfamilysequences.Theexamplesforeachofthesetwofamiliesaregeneratedusingdierentgoalcongurations.Congurationswereselectedtoshowinterestingtrajectoriesforeachsequenceclass.Generally,theoptimaltrajectorybetweentwocongurationsisfoundbyenumeratingthesixpossibilitiesforagivencongurationandthencomparingthefeasibleresultstondtheoptimum.Infeasiblesolutionscouldoccurforaparticularsequenceandgoalconguration,particularlyifthegoalislocatedveryclosetotheinitialcondition.Additionally,infeasiblesolutionscouldoccurforpathsthatintersectanobstacleregion.Assuch,useofthistechniqueastheprimaryapproximationmethodtosolvethefullversionofProblem 2{35 withmanyobstaclepathconstraintsisnotlikelytoyieldafeasibletrajectory.Thistechniquedemonstratesthemostutilityforobstacle-freeproblemsor,alternatively,asalocalmethodintegratedintoarandomizedscheme,suchasthemethodsintroducedinChapter 3 53

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B C D E FFigure2-7. SampleDubinspaths.A)SampleLSLtrajectory.B)SampleRSRtrajectory.C)SampleLSRtrajectory.D)SampleRSLtrajectory.E)SampleRLRtrajectory.F)SampleLRLtrajectory. 2{56 ,whichisanextensionofthesimpleDubinscarmodelexaminedinSection 2.4.3 .ThevehicledescribedbyEquation 2{56 moveswithaconstantforwardvelocity,V,andiscontrolledviathecommandedturnrate,!c.Dynamicsareintroducedintothesystemintheformofaheadingaccelerationconstraint.Thisaccelerationconstraintresultsinsimplerst-orderlagbehaviorofturnratewithrespecttocommandedturnrate.Thelaglterexhibitsatimeconstantof!=0:25sec.Assuch,theturn-rateresponsetoastepcommandachievessteady-stateinapproximately1-1.25sec,or4-5timeconstantsfollowingthecommand. 54

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2{56 admitstrimtrajectoriesthatbehaveaccord-ingtothekinematicconditionsshowninEquations 2{57 to 2{60 .Amotionprimitivemodelisformedthroughquantizationofthedynamicsaccordingtoadiscretesetofturnrates.Specically,thissetischosensuchthattheresultingtrimtrajectoriesarecharacter-izedbyturnratesgivenby_2n_max;0;+_maxo,where_max=30deg/secrepresentsthemaximumachievableturnrate.V=const. (2{57)_=const. (2{58)_!=0 (2{59)!c=_ 2{61 to 2{63 .Similarly,stateevolutionalongthestraight-aheadtrimtrajectory,TS,ischaracterizedbyEquations 2{64 to 2{66 .Gturn(~x0;)=~x0+Rturn(~x0)~turn() (2{61)Rturn(~x0)=266666664cos0sin000sin0cos00000100000377777775 !p !p 55

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(2{64)Rstraight(~x0)=266666664cos0sin000sin0cos00000100000377777775 2-8 ,whichshowsthetimehistoryofeachstateforevolutionalongthemaneuverconnectingTSandTR.Thestepinputisshownasthedashedredlineinthe!plot. Figure2-8. Examplestatetrajectoryshowingmaneuverdynamics. Thetimehistoryforeachstateiscarriedouttot=5!,suchthat!hasreached99.3%ofitsnalvalue.Thestate-displacementovereachnite-durationmaneuverisgivenbythetransformationshowninEquations 2{67 to 2{69 .Thetime-invariant 56

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2-1 ,wherethemaneuverconnectingTitoTjisdesignatedasMij.GM(~x0)=~x0+RM(~x0)~M Table2-1. +54.1487 +54.1487 +55.9118 +55.9118 +55.3357 +55.3357py(ft) -12.2331 +12.2331 -5.8870 +5.8870 +6.5181 -6.5181(deg) -30.0505 +30.0505 -7.4495 +7.4495 +22.6011 -22.6011!(deg/sec) -30.0000 +30.0000 +30.000 -30.0000 +60.0000 -60.0000 Theeectofinsertingmaneuversbetweeneachpairoftrimtrajectoriesistocauseaslightstatedisplacementattheendofthersttrimtrajectorybeforethepointatwhichevolutionalongthesecondtrimtrajectorybegins.Consequently,closed-formsolutionsderivedinasimilarmannertothatdescribedfortheDubinscarinSection 2.4.3 arethe-oreticallypossible.However,solutionscomputedinthismannerdonotexplicitlyconsiderpathconstraintsbeyondmerelycheckingforconstraintsatisfactionaftergeneratingthetrajectory,asdiscussedpreviously.Asanalternative,thisexampledemonstratesapartialsolutiontothefullMINLPshowninProblem 2{35 forwhichobstacleconstraintsareconsidered.Thesubsequentsolutionisdenoteda\partial"solutionbecauseonlyaselectfewsequencesareenumeratedforthecombinatorialsequenceselectionstep.Enumeratingthefullsetofallfeasiblesequencesisimpractical,asthissetiscountably-innite.Solvingthis 57

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2-9 .Thisenvironmentcontainstwopolygonalobstacleswithamaximumdimensionof200ft.TheinitialandgoalvehiclestatesaregivenbyEquations 2{70 and 2{71 ,respectively,where(?)indicatesthatthevalueisunconstrained.Bothofthesestatesareassumedtobealongatrimtrajectory.Thetrajectoryoptimizationrequiresthatasequenceofmaneuvers,M,andacorrespondingsetoftrimdurations,,arefoundthattakethevehiclefrom~x0to~xginminimumtimewhilesatisfyingthedierentialpathconstraintsgivenbyEquation 2{56 andobstaclepathconstraintsoftheformshowninEquation 2{4 .ThedierentialconstraintsareencodedaspartoftheendpointconstraintsthroughG(~x0;M;).Theobstaclepathconstraintsutilizethesuperellipseformulationoftheobstacleboundary,asdescribedinSection 2.2.2 .Alsonotethateachmaneuversequence,M,musthaveeitherMSLorMSRasitsrstelementbecausetheinitialstateliesalongthetrimtrajectory,TS. Figure2-9. Topviewofenvironmentfordynamicexample. TrajectorygenerationproceedsbyselectingaprimitivesequenceandsolvingtheresultingsmoothNLPsubjecttotheendpointconstraintsandpathconstraints.Recallthatforthenite-dimensionaloptimizationconsideredhere,theobstaclepathconstraints 58

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BFigure3-2. RRTalgorithm.A)Samplingstep.B)Expansionstep. 40 75 ].TheESTalgorithmexploresspaceinafundamentallydierentwaythantheRRTalgorithm.Specically,nodeselectionoccursthroughtherandomselectionofanexistingnodeaccordingtoaprobabilitydistributionthatisleftasadesignchoice.Thisnodeisexpandedwithinalocalneighborhoodthatisdenedbyavaliddistancemetric.Acongurationissampledrandomlyfromwithinthisneighbor-hoodandalocalplanningmethodisusedtoconnecttheselectednodetothesampledconguration.Figures 3-3A and 3-3B depicttheESTexpansionprocess.Bothimagesshowatreegrownfromtherootnode,N0,inatwo-dimensionalC-spacethatcontainsobstacles.Figure 3-3A depictsthenodeselectionstep,inwhichtheexpansionnode,Nexp,isselectedfromtheexistingnodes.TheneighborhoodofNexpisdenedhereusingaEuclideandistancemetricandisshownastheareawithinthedashedcircleinFigure 3-3A .Figure 3-3B showstheexpansionstep,wherearandomconguration,NrandisselectedfromtheneighborhoodofNexpandthenatrajectoryisplannedfromNexptoNrand.Thealgorithmproceedsinthisfashionuntilabranchofthetreereachesthegoalwithinsomespeciedtolerance. 66

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BFigure3-4. DierencesinexplorationstrategyfortheRRTalgorithmvs.theESTalgorithm.A)RRTexpansion.B)ESTexpansion. approximationtothisoptimaldistancemetricmustbechosen.Selectionandcomputationofsuchanapproximationcanbenon-trivialforcomplicatedsystems.Asdiscussedpreviously,theESTprovidesaparticularlyusefuldesignchoiceinthefreedomtoselectthedistributionfromwhichexpansionnodesaredetermined.Amodiednodeselectionscheme,denotedtheGuidedEST(GEST)assignseachnodeaprobabilisticweightthatreectsproximitytoothernodesspatiallyandtemporallyand,moreimportantly,reectsameasureofperformance[ 73 77 ].ThisexplicitinclusionofperformancecostinthesamplingprocessenablestheGESTtoconsidersolutionqualitywhileexploringspace.Moreover,the\push"tendencydescribedpreviouslyallowspotentialsolutionstocontinuallyimprove.ThisimprovementtendencyresultsfromtheincreasedlikelihoodthatasbranchesexhibitingdesirableperformancewillbeselectedforexpansionundertheGESTframework. 68

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2{35 .Themethodpresentedhereutilizesalgebraicsolutionsforplanningwithmotion-primitivemodelsasdescribedinSection 2.4.2 .Thesealgebraicsolutionsareusedtogeneratelocal,obstacle-freetrajectoriesaspartofthetreeexpansionprocess. 3{1 .Thesingleinputtothedierentialsystemistheturn-rate,!.Theconstanttranslationalvelocity,V,isconstraineddierentiallytoactinthedirectionofthevehicleheading.ThesystemdescribedbyEquation 3{1 admitstrimtrajectoriesthatbelongtotwofamilies:constantrateturnsandstraightforwardmotion. 3{2 and 3{3 .Amotionprimitivemodelcanbeformedbyselectingasetoftrimprimitivesthatbehaveaccordingtotheseconditions. (3{2) _=!=const. (3{3) Asetof2n+1trimprimitivesareselectedwhichconsistofconstant-rateturnsatndierentturnratesineachdirectionalongwithastraight-aheadprimitivethatcorrespondsto!=0.ThissetisshownasEquation 3{4 ,where_max=30deg/sec.Thevelocity,V,isheldxedoverthesetofallprimitives. 69

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3-5 .Eachtrimtrajectoryisdepictedinthegureasastateoftheautomatonandisdesignatedbytheturnratecorrespondingtothattrimstate. Figure3-5. Automatonstructureforvehiclemodelusedwithcurrentplanningstrategy. Stateevolutionalongeachturningtrimcharacterizedby!iisdescribedbythetime-parameterizedtransformationgiveninEquations 3{5 to 3{7 .Similarly,stateevolutionalongthestraight-aheadtrimtrajectorycharacterizedby!=0ischaracterizedbyEquations 3{8 to 3{10 (3{5) !ip !ip 70

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3.3.1 andthealgorithmpresentedhereisrelatedtoalgorithmtermination.ThenominalRRTalgorithmseeksasinglefeasiblesolution.Here,therstfeasiblesolution 73

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3.4.1 .ThedistributiononPfromwhichpsampischosenplaysaroleinplannerperfor-mance.AuniformdistributionoverPmaximizesspaceexplorationsuchthattrajectoriesmightwander'aboutthespacebeforeachievingafeasiblesolution.Conversely,adistribu-tionthatheavilyemphasizesthegoalcongurationdisplaysgreedygoal-seekingbehaviorthatissubjecttolocalminima[ 31 ].Ideallyabalanceshouldbeachievedbetweenthesetwobehaviors.Assuch,samplesareselectedfromoneoftwodistributions,eachofwhichisassignedaprobabilitythatitwillbeusedtogeneratethecurrentsample.ThetwodistributionsconsistofauniformdistributiononPandaGaussiandistributioncenteredatthegoalpositiononP.Thus,samplingproceedsbyrstselectingadistributionandthensamplingapositionfromtheselecteddistribution.Onceapointisselected,nearbynodesintheexistingtreemustbedetermined.AnintuitivedistancemetricsuchastheEuclidean2-normdoesnotcapturethedependence 74

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42 ].Therefore,theminimumobstacle-freeturn-straighttrajectoryfromacongurationtoapointonPisconsideredasanapproximatedistancemetric.Thisminimumoccurswhentheturningsegmentisexecutedatmaximumturnrate.AplotofdistancesmeasuredusingthismetricisshowninFigure 3-6A .Thevehiclehasacongurationof(px;py;)=(0;0;0)inFigure 3-6A ,suchthatdistanceismeasuredfromthecenteroftheplotwherethevehicleisinitiallypointingalongthepx-axis.Forcomparison,aplotoftheEuclidean2-normisshowninFigure 3-6B forthesamevehicleconguration.Inspectionoftheguresclearlyshowsdiscrepanciesbetweenthetwometrics. BFigure3-6. Distancefunctioncomparison.A)Approximationofobstacle-freecost-to-go.B)Euclideandistancefunction. ActualcomputationofthedistancefromeachnodetopsampusingthemetricdepictedinFigure 3-6A isaccomplishedbycomputingtheminimumturn-straighttrimdurationstoreachpsampfromeachnodeinthetree.Theminimumforeachoftheseprimitives 75

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3{11a 3{11b 3{12a ,and 3{12b with!i=_maxareusedtocomputethecandidatesequences.Theglobalminimumfortheturn-straightfamilyisthenfoundthroughdirectcomparisonoftheminimumleft-straightandminimumright-straightsequences.ThisprocessisshowninFigure 3-7 BFigure3-7. DistanceFunctionComputation.A)Sampledpoint.B)Turn-straightpathsfromeachnode. TheNodeSelectionstepexposesatroublingpracticalrealityassociatedwithimple-mentationofRRTalgorithms:thatadistancecomputationmustbeperformedateverynodeateachiteration.Whilethisoperationisoftencomputationallycheap,asisthecasehere,newnodesareaddedtothetreeateachstep.Inthelimit,aninnitenumberofnodesareinvolvedinadistancemeasurementoperation.Fortunately,inpractice,thealgorithmtendstoconvergebeforethisissuebecomesproblematic;however,suchanobservationisdicultintheorytoprovenorguarantee. 3.4.1 .TrimdurationtimesarecomputedusingEquations 3{11a 3{11b 3{12a ,and 3{12b .Somevaluesofpsampyieldinfeasiblesolutionstocertainsequences.These 76

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BFigure3-10. Collision-freesolutions.A)Right-straight-rightfamily.B)Left-straight-rightfamily. Aftereachnewnodehasbeentestedforfeasibleandsafereachabilityofthegoalconguration,theterminationconditionischecked.Asdiscussedpreviously,thistermi-nationconditioncanvaryaccordingtothespecicrestrictionsoncurrentoperationoftheplanner.Terminationcandependonruntime,memoryusage,orevenafteracertainnumberofvalidsolutionshavebeenachieved.Upontermination,theplannerreturnsaprecedencelistofnodesthatliealongthesolutiontrajectoryfromtheinitialcongurationtothegoalconguration.Iftheterminationconditionhasnotbeenreachedattheendofaparticulariteration,executionreturnstotheNodeSelectionstep. 3.4 .ThisproblemconsidersavehiclewhosemotionisgovernedbythecontinuousdierentialsystemshownpreviouslyasEquation 3{1 .Assuch,thevehiclemoveswithforwardvelocity,V,andissubjecttoaboundedturnrate,j_j30deg=sec.Thesinglesysteminputisthecommandedturnrate,!.Aspreviously,themodelischaracterizedaskinematic.Themotionofthisvehicleismodeledusingahybridmotionmodelwithseventrimstates.Formulationofthismodel 79

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3-11A ,whichshowstrajectoriesthatdrivethesystemfromaninitialconditionof(px;0;py;0;0)=(0;0;0)toaterminalcongurationof(px;f;py;f;f)=(0;800;0).Onlymaximumrateturnsareconsideredfortheturn-turn-turnfamilyofsequences,asdiscussedinSection 3.4.1 .ExamplesofthetworesultingsolutiontrajectoriesareshowninFigure 3-11B ,whichshowstrajectoriesthatdrivethesystemfromaninitialconditionof(px;0;py;0;0)=(0;0;0)toaterminalcongurationof(px;f;py;f;f)=(0;100;0).Inadditiontothesolutionsequencefamiliesforplanningbetweentwocongurations,asolutionfamilyexistforplanningfromacongurationtoapointonthesubspace,P,whichconsistsofthepositionvariables,pxandpy.Variablerateturnsareconsideredforthissolutionfamily,whichischaracterizedbyturn-straightsequences.Thereare 80

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BFigure3-11. Uniquesolutionfamiliesforvehicleusedinexample.A)Turn-straight-turnsolutionsequences.B)Turn-turn-turnsolutionsequences. atotalof2n=6membersinthissolutionfamily.ExamplesolutionsareplottedinFigure 3-12 ,whichshowstrajectoriesthatdrivethesystemfromaninitialconditionof(px;0;py;0;0)=(0;0;0)toaterminalpointonPlocatedat(px;f;py;f)=(200;500). Figure3-12. Turn-straightsolutionsequences. TheplanningenvironmentconsideredfortheexampleproblemisshowninFigure 3-13 .Forthisexample,theinertialreferenceframeisorientedsuchthatthex-axispointsintheNorthdirectionandthey-axispointsintheEastdirection.Inthegure,theboundsontheextentoftheenvironmentareshownasthebluesquarethatcontains0x1000ftand0y1000ft.Allsamplingstepsdrawsamplesfromwithinthesebounds.Also,polygonalobstaclesarerandomlyplacedwithinthisregion. 81

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3-13 .Samplingdistributionsarenotdenedwithinobstacleregions.Finally,theinitialandgoalcongurationsarespeciedas(px;0;py;0;0)=(0;0;0)and(px;f;py;f;f)=(1000;1000;0),respectively. Figure3-13. Exampleplanningenvironment. ThealgorithmdescribedinSection 3.4 isexecutedforsixtyiterations.Figure 3-14 showstheexpansionofthesearchtreeatvariousintermediatestagesprogressingfromlefttoright.SamplesonParedrawnfromeitherauniformdistributiondenedovertheobstacle-freeportionsoftheplanningenvironmentoraGaussiandistributioncenteredatthegoallocationwithastandarddeviationof250ftthatisalsoonlyvalidovertheobstacle-freeportionsoftheenvironment.Theuniformdistributionischosenwithaprobabilityof0.75andtheGaussiandistributionischosenwithaprobabilityof0.25. B CFigure3-14. Incrementaltreeexpansion.A)After1iteration.B)After25iterations.C)After60iterations. 82

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B CFigure3-16. Solutionpathrenement.A)After7iterations.B)After14iterations.C)After37iterations. TheimageattheleftofFigure 3-16 depictstherstvalidsolutionfoundbythealgorithm.Thissolutionwascomputedduringtheseventhiteration.Hence,thisexampledemonstratestheabilityoftheplannertoandafeasible,butsub-optimal,solutionveryquickly.ThesolutionshowninthecenterimageofFigure 3-16 iscomputedduringthefourteenthiteration.Finally,theminimum-timesolutionfoundfortheexamplerunisshownattherightofFigure 3-16 .Thissolutionwascomputedduringthethirty-seventhiteration.Assuch,thenaltwenty-threeiterationsdidnotcontributetothesolutionandcouldbeconsideredawasteofcomputationalresources.Alsonotethattheimprovementofeachsolutionoverprevioussolutionsisfairlybenign,duetothenatureoftheRRTexpansionprocedure.Unfortunately,thereisnowaytodetermineifaparticularsolutionispotentially-improvableorhowlongsuchpotentialimprovementwilltake.Therefore,theterminationconditionremainsaproblem-specicparameter. 84

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17 23 78 { 80 ].Thischapterdiscussesageometricapproachtoquantifytheeectivenessofapartic-ularsensor-targetconguration.First,abriefdiscussionofsomeoftherelevantremotesensortechnologiestotheapplicationsconsideredprovidescontexttothesubsequentpre-sentation.Then,ageometricmodelofLOSsensoroperationisintroducedinSection 4.3 .Aparameterizationoftherelativegeometryisusedtodenetheconceptofvisibility,whichisdiscussedinthecontextofsensingtasksinclose-proximityenvironments.Finally,ageneralizedsensingeectivenessmetricisformulatedthatquantiesthequalitywithwhichaparticularpointintheenvironmenthasbeensensed.Severalspecicexamplesdemonstratehowthisconceptcanbeusedtoevaluatesensingmissionperformance. 86

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Sensingproblemgeometry. 4{3 .ThesensorisassumedtohaveaknownorientationwithrespecttoB,whichisgivenbytheconstanttransformation,TBS.ThetransformationfromEtoSisthengivenbyEquation 4{4 .TheorientationofSwithrespecttoEcanbeexpressedusinganEulerangleparameterization,aswell.Thetime-varyingvectorofsensorEuleranglesisshowninEquation 4{5 .Thetransformation,TES,canbeexpressedrelativetotheseparametersasinEquation 4{6 .ThesensorEulerangles,~s,canthenbedeterminedintermsof~BbyequatingtherighthandsidesofEquations 4{4 and 4{6 thensolvingtheresultingsystem. 91

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4{7 .Relativeorientationisdescribedbytheorientationof^nirelativetothesensoraxis,^s3.Thisrelationshipischaracterizedthroughexpressionof^nirelativetotheorientationofS.TherequiredtransformationisshowninEquation 4{8 ,where^ns;irepresents^niintermsofthebasisvectorsofS.~p(s!ti)=~pt;i~ps 27 105 106 ].Aviewpointistypicallydenedona\visibilityspace."Thisparameterspaceisgenerallyspannedbyasetofvariablesthataectvisibilityandviewquality.TheseparametersoftenincludeopticalspecicationsofthecameraandothersensorspecicfactorsinadditiontothegeometricdenitionsdiscussedinSection 4.3.1 .Variousdenitionsof 92

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4{9 .Variationsinrangeareassociatedwiththeintensityandresolutionofdata.TheincidenceanglemeasuresthedegreetowhichthesensorLOS\grazes"thetargetsurface.ThiseectischaracterizedbytheanglebetweentheLOSalong~p(s!ti)andthetargetsurfacenormal^ni,asshowninEquation 4{10 .Highvaluesforicanresultinadistorteddataproduct.Finally,theFOVanglemeasurestheangulardiscrepancybetweentheLOSalong~p(s!ti)andtheLOSalong^s3,whichisthesensor-framebasisvectorthatindicatesthecenteroftheFOV.ThisrelationshipisdescribedbyEquation 4{11 .Variationsinf;iareassociatedwithangularpositionwithintheFOV.Figure 4-2 depictsacrosssectionoftheFOVthatshowsthevisibilityparametersforaparticulartargetlocation. Onceexpressedintermsoftherange,incidence,andFOVangle,visibilitycanbedenedthroughtheapplicationofsimplebounds.Theseboundsrelyonseveralbasicassumptionsconcerningthelimitationsofthesensor.First,thesensorisassumedto 93

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Visibilityparameters. haveaniteFOV,ie.,thesensorisnotomnidirectional.Second,theexistenceofarangebeyondwhichobjectswithintheFOVcannotberesolvedisassumed.TheseconditionsresultinthevisibilityconstraintsgivenbyEquations 4{12 and 4{13 Inphysicalspace,thesimultaneousapplicationoftheseconstraintsarerealizedastheintersectionofaspherethatiscenteredattheoriginofSandandinnitecircularconethatoriginatesfromthissameorigin.Thespherehasradiusrmax.Theaxisoftheconeiscollinearwiththesensoraxis,^s3,andhasahalf-angleoff;max.Thisintersectionthusformsasphericalconethatisalignedwiththesensoraxis.Anecessaryconditionforvisibilityoftheithtargetisthat~p(s!ti)mustlocatethetargetwithinthissphericalvisibilityconeforthecurrentsensororientation,~s.Anothervisibilityconstraintresultsfromtherelativeorientationofthetargetsurfacewithrespecttothelocationofthesensor.Thisconditionconstrainstheincidenceangle,i,asshowninEquation 4{14 .Thisconstraintreectsthenotionthatapointonasurfacecannotbeseenwhentheviewisparalleltothesurface.Aparallelviewoccurswheni==2,orwhentheLOSisorthogonaltothetargetsurfacenormal,^ni.Equation 4{14 alsorestrictsincidenceanglesgreaterthan=2,asthisrangeofi

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4-3 .Thisconedescribesasetofsensorpositions,~ps,thatarevisiblefromS0.ThereforetheithtargetisalsovisiblefromsensorpositionswithintheS0visibilitycone. Figure4-3. Invertedvisibilitycone. Asdiscussedpreviously,Vinowconsistsoftheintersectionofthisinvertedsphericalvisibilityconewiththeinniteconethatrelatestheincidenceconstraintfortheithtarget.Thissetdescribesthesensorpositionsandorientationsthatsatisfythegeometricvisibilityconditionsfortheithtarget,asdescribedbyEquation 4{15 .Validsensorpositionsare 96

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4-5B .Theexampleofaturningaircraftwithadownward-pointingsensornowexhibitssignicantdatalossasaresultofthepreviouslybenignmaneuver.Moreover,navigationthroughcloseproximityenvironmentsoftenrequiresfrequentandaggressivemaneuveringtoavoidobstacles,thuscausingmotioncouplingeectstobeevenmorepronounced. BFigure4-5. Motioncouplingeectsfordierentproblemscales.A)Translationatstandorange.B)Translationatcloserange. Finally,signicantrelativevariationsinrangeandincidenceareseenovertheFOVasaresultofthecloseproximityofthesensortotheenvironment.Whenthesensorislocatedalargedistancefromtargetsurfaces,therangeofallvisiblepointsvariesverylittlerelativetothestandodistance.ObjectsappearwithintheFOVatdisparatedistancesandaspectswhensensingatcloserange.Theincidenceanglecanvarysignicantlyoverasurfacethatwouldappearatanearlyconstantincidencefromstandorange.Figure 4-6 showsthiseectascomparedtothevariationsseenintypicalstandosensingmissions.ThetwoimagesontheleftofFigure 4-6 showrangeandincidenceangledataoverasimulatedurbanenvironmentforasensorlocatedatanaltitude10,000ft.ThisaltitudeistypicalforcurrentairbornesensorplatformssuchastheRQ-1Predatordrone[ 3 ].TheimagesontherightofFigure 4-6 showrangeandincidencedatawhen 99

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B C DFigure4-6. Rangeandincidencevariationfordierentproblemscales.A)Rangeforstandosensing.B)Rangeforclose-proximitysensing.C)Incidenceforstandosensing.D)Incidenceforcloseproximitysensing ThestatisticsdescribingthevariationsinrangeandincidenceforallvisiblepointsfromthetwocasesdepictedinFigure 4-6 aredisplayedinTable 4-1 .Thetableshowsthatthetwocasesexhibitasimilarabsolutespreadofrangedata,asthevaluesforthestandarddeviationofthisdataisofsimilarmagnitudeforeachcase.However,adiscrepancyofafullorderinmagnitudeisseenwhenthesevaluesarenormalizedonthemeanrange.ThisdiscrepancyisreectedbythecolorvariationsinthetoprightimageofFigure 4-6 ascomparedtothoseseeninthetopleftimage. 100

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Statisticsforvisibilityparametercomparison. Meanri Normalizedri Std.Deviation Std.Deviation Std.Deviation Stando 20,023ft 0.1471deg 543.87ft 4.2563deg 4-1 displayaveragevaluesforeachvisiblesurfaceoverallvisiblepoints.Verylittleabsolutevariationinincidenceangleisseenforthestandocaseascomparedtothecloseproximitycase,asindicatedbytheaverageistandarddeviationoverallsurfaces.Further,anincreaseinstandarddeviationbytwoordersofmagnitudeisseenbyatleastonesensedsurface.TheseresultsarereectedbythesignicantcolorvariationsoneachsurfaceofthelowerrightimageofFigure 4-6 ascomparedtothoseseenonsurfacesinthelowerleftimage.Frequentocclusions,motioncoupling,andlargerelativevariationsinthevisibilityparametersaredemonstratedforsensingmissionsincloseproximitytothesensedenviron-ment.Theseeectscanadverselyaectdataqualityifnotproperlyconsideredinmissionplanning.Properconsiderationrequiresthattheeectsoftheseissuesbequantiedforinclusionwithinsensorplanningoptimizationalgorithms.Thenextsectiondescribesasensoreectivenessmetricthatisdesignedtoaddressthisproblem. 4.3.2 and 4.3.3 allowageometricdenitionofvisibilityforagenericLOSremotesensor.Theithtargetisvisibleforsensorpositionsandorien-tationsthatliewithintheboundsofVi,whileforallothercongurationsthetargetisnotvisible.ThisrelationshipcanbecharacterizedbyasimplebinaryagasexpressedinEquation 4{16 .Thisdenitionofvisibilityislimitingintermsofdescribingsensorperformance,orequivalently,qualityoftheresultingdataproduct.Manycasesexistforwhichthisdenitionisnotsucienttodescribetheutilityofaparticularsetofcollected 101

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agvis;i=8>><>>:1;~ps!ti;~s2Vi0;~ps!ti;~s=2Vi(4{16) 4{17 .Thisfunctionisconsideredasametricoftheeectivenesswithwhichtargetihasbeensensed. 4{18 ,wherethevisibilityparameterstakeonfunctionaldependenciesasshowninEquations 4{19 to 4{21 .Notethattheexplicitformofqiisspecictoboththesensorandtheapplication. 102

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4{23 4{24 .Thisfunctionkeepsthemaximumvalueoftheinstantaneousqualitymetric,qi,overthetimeinterval,t2[0;t].ThevalueofQmax;iindicatesthattargetihasbeensensedwiththatvalueatleastoncesincethebeginningofthemissionsegment.Thisversionofmissioneectivenesscanbeusefulforapplicationsinvolvingimagingsensorsthatcollectdatasubjecttohumananalysis.Onegoodviewofasensingobjectivecanoftenprovideagreatdealofinformationtoahumananalyst. 4{25 showsthecaseforwhichqirep-resentstheprobabilityofanevent,suchasdetection,occurringatthesensinginstancecorrespondingtoqi.Thentheprobabilityofthateventnotoccurringcanbeexpressedasthecomplementofqi,asshowninEquation 4{26 .Theprobabilityoftheeventnot 105

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4{27 .Finally,theprobabilitythattheeventoccursatleastonceisgivenbythecomplementoftheprobabilitythattheeventdoesnotoccuratall,asshownbyEquation 4{28 (4{28) Theprobabilitythattheeventoccursatleastoncecanthenbeimplementedinrecursiontoexpresssensingmissioneectivenessfordetectionprobability,QPD;i.ThisrecursionisshownasinEquation 4{29 .Theuseofarecursiveformularequiresthatthediscrete-timeformofthesensingeectivenessmetricisused. 4{30 4{31 to 4{33 .ThesefunctionsnecessarilyevaluatetozerooutsidethesensorFOV.WithintheFOV,eacheciencyfunctionisshapedtoreect 106

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4{34 1+(ri=rmax)40:7exp400r2i 1+(i=max)41;imax0;else 1+(0:75f;i=f;max)4;f;if;max0;else 4-8 .Theeciencyfunctioncorrespondingtorange,fri,isshownintheleftimage.Thisfunctionisseentoexhibitlossesatcloserangeandlongrange.Thisshapingreectspoorspatialcoverageatcloserangeandpoorresolutionatlongrange. B CFigure4-8. Qualityparametereciencyfunctions.A)Rangeeciency.B)Incidenceeciency.C)FOVangleeciency. Thefunctioncorrespondingtoincidenceangle,fi,isshowninthemiddleimageofFigure 4-8 .Thisfunctionexhibitslossesastheviewdirectionapproachestangencywiththetargetsurface.Increasingihastheeectofdistortingsurfacedataandlimiting 107

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4-8 .ThisfunctionisseentoexhibitminimallossesoverallFOVangles,withthegreatesteectsappearingnearf;max.ThemaximumFOVanglecorrespondstotheFOVboundary.Assuch,thelossesareincorporatedheretoreecttheradialdistortioneectscommontocamerasensors[ 82 ].UseofthemetricdescribedbyEquation 4{34 isdemonstratedthroughasimplesimulation.Thesimulationconsistsofakinematicaircraftmodelyingataconstantaltitudeof200ftabovegroundleveloveranurbanenvironment.Thisenvironmentconsistsofthreebuildingsarrangedsuchthattheypartiallyencloseanarea.Eachbuildinghasamaximumheightof50ft.Asamplemissionscenariomightrequiredatacollectionalongtherooftopedgesthatfacethepartially-enclosedareatocheckforsniperspriortoatroopmovement.Theaircraftcarriesanimagingsensorthatislocatedattheoriginoftheaircraftbody-basissuchthat~p=~0.Thesensorpointsforwardanddownatanangleof60deg.Assuch,thetransformationfromBtoSisgivenbyEquation 4{35 .ThevisibilityboundsforthissensoraregivenbyEquations 4{36 to 4{38 .TheseboundsareusedwithEquations 4{31 to 4{34 toformulatethequalitymetric. 108

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4-9 .Fordisplaypurposes,targetsaregeneratedbygriddingtheenvironmentsurfacesatxedresolutionof25ft.Thisselectionoftargetsallowsthemetricvaluetobeshownatallpointsintheprojectedfootprintwiththespeciedresolution.TheimagesinFigure 4-9 showsnapshotsofinstantaneousmetricvalueswithinthesensorfootprintatseveralpointsalongthetrajectory.Noticethechangeinshapeandeectivenessmakeupofthefootprintastheaircraftbankstoturninthetoprightandbottomleftimages.Also,noticethestarkdierencesinmetricvalueondierentsurfacesthatareperpendiculartoeachotherasthesensorviewsbuildingedges. Figure4-9. Simulatedtrajectoryshowingsnapshotsofeectivenessmetricvalues. Themax-valuemissioneectivenessmetricisusedtotrackaggregatesensingef-fectivenesswithrespecttoeachtargetforthepresentexample.Recallthatthismetricmaintainsthemaximumeectivenessvalueovertheentiretrajectory,asgivenbyEqua-tion 4{24 .ThenalvaluesformissioneectivenessareshowninFigure 4-10 forthetrajectorydepictedinFigure 4-9 109

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Max-valuemissioneectiveness. 82 ].Simplegeometricrelationshipsyieldanexpressionfordiintermsofthevisibilityparameters,riandf;i.ThisexpressionisgivenasEquation 4{39 4{40 .ThecorrespondingareainimagecoordinatesisgivenbyEquation 4{41 ,whereNrepresentsthenumberofpixelsinasinglelineoftheimage.Thelengthofeachsideofreferenceobjectis 110

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~A=~Wref~Href =Wref 4{41 andareshowninEquation 4{42 4{43 .ThisfunctionexponentiallyapproachesqA;i=1as~A!~Amax.Thenumericalparametersarechosenheresuchthatarapidriseoccursfor~Athatareslightlylargerthat~Amin,followedbyalargeintervalforwhichthereislittlegain.Suchbehaviorischosentoreectthenotionthatincreasingimageareaofthereferenceobjectdoesnotyieldadditionalrelevantinformationbeyondacertainsize.Fortheaforementionedexampleofalicenseplate,littleistobegainedbyincreasing~Aoncethenumbersareclearlylegible.Conversely,asignicantamountofinformationisgainedcloserto~Aminasslightincreasesinresolutionhelptodierentiatethecharactersfromeachother.TheformulationinEquation 4{43 alsoincludesascalingeciencyfunction,f,toaccountfordistortionsresultingfromgrazingincidenceangles.Thesameformulationoffthatwasemployedinthepreviousexampleisusedhereaswell(Equation 4{32 ). 111

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4{43 isplottedinFigure 4-12 aftersubstitutionofthesenumbers.Thex-axisoftheplotinFigure 4-12 isshownonalogarithmicscale. Figure4-12. Sensingeectivenessasafunctionofimagearea. ExecutionoftheaircrafttrajectoryresultsinthecameraFOVpassingoverfourofthevetargetsurfaces.Thesecondtargetisnotviewedasaresultofmotioncoupling.ThesensingeectivenesstrajectoriesforeachoftheremainingfourtargetsareshowninFigure 4-13 .ThesetrajectoriesarezeroforalltimesegmentsduringwhichthetargetisoutsidethecameraFOV.Themax-valuemissioneectivenessmetricisusedheretotrackaggregatesensingeectivenessforeachtarget.Recallthatthismetricassumesthemaximumeectivenessvalueovertheentiretrajectory,asgivenbyEquation 4{24 .ThenalvaluesformissioneectivenessareshowninFigure 4-14 .MissionEectivenessvaluesareshownforallvisiblesurfacesinadditiontothevespeciedtargets. 113

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Eectivenesstrajectories. Figure4-14. Simulatedmissioneectiveness. 114

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2 and 3 ,aregenerallytailoredtodrivethesystemfromonespeciccongurationtoanother.Theproblemofndingminimumtimetrajectoriesthatvisitmultiplepointsrepresentsaninstanceofthewell-knowntravelingsalesmanproblem(TSP).Whilethisproblemiswell-studiedintheliterature[ 25 26 ],solutionsandapproximationstypicallyrelyontheassumptionthatoptimaltrajectoryconsistsofoptimalsegments,asshowninFigure 5-1A .Optimalsolutionsdonotgenerallytakethisformfordynamicsystems.Figures 5-1B and 5-1C eachshowathree-pointtourforacurvature-constrainedsystemstartingfromthesameinitialcondition.Eachpoint-to-pointsegmentshowninFigure 5-1B islocallyoptimal.Figure 5-1C showstheoptimaltrajectoryfortheentiretour,whichincludespoint-to-pointsegmentsthatarelocallysub-optimal.Generally,boththeoptimalsequenceofpointsandtheoptimaltrajectorymustbedetermined;however,thesesolutionelementsareusuallycoupledforsystemswithnon-trivialdynamics. B CFigure5-1. TSPproblemcomparison.A)StandardTSP.B)Curvature-constrainedTSPwithlocally-optimalsegments.C)Optimalcurvature-constrainedTSP 115

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4.4.4 essentiallyquantiesthisno-tionofrobustvisibility.Specically,asensingeectivenessmetricisconstructedusingtheproductofaseriesoffunctionsthatexhibitdependenceonthevisibilityparameters,asshownbyEquation 5{3 .Thefunctions,fri,fi,andff;i,takevaluesthatreectthedetri-mentaleectsondataqualityresultingfromvariationsinthevisibilityparameters,r,,andf,relativetotheithtarget.ThesefunctionstakethesameformasinSection 4.4.4 ,withtheexceptionoff;i,whichisrelatedbythebinaryconditionshowninEquation 5{4 .Theparametervaluesthatspecifytheexactexpressionsaregivenbyrmax=750ft,max=70deg,f;max=35deg. 5{1 ,whereQd=0:75.Themax-valuecriteriarequirethateachtargetisviewedwithaneectivenessmetricvalueofqrv=Qdatleastoncealongthe 129

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5-8 .Theplotsshowthatsimpleboundsonthevisibilityparametersforaparticulartargetensurethatsensingcriteriaaremet.Theseboundsareseenas:49ri385ft,f;i29deg,and,fromEquation 5{4 ,i<70deg.Theseconservativerequirementsallowanon-varyingdenitionofQiwithrespecttothevisibilityparameters,asshowninEquation 5{5 BFigure5-8. Qualityparametereciencyfunctionsforexamples.A)Rangeeciencyfunction.B)FOVangleeciencyfunction. 130

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5{6 ^ni=001T8i(5{6)Thisexampleactuallyconsiderstwosetsofspeciedtargets.TherstsetoftargetsisshowninFigure 5-9A .Thegureshowstheenvironmentboundaryasasolidbluelineandtwelverandomly-selectedtargetlocationsasredasterisks.Thesetargetsareusedforthereconnaissanceandsurveillancemissiondescriptions.Figure 5-9B showsthesecondsetoftargets,whichisusedforthecoveragemissiondescription.Thegureshowsaseriesofroadspassingthroughanopenarea.AnareatotherightoftheplotinFigure 5-9B isenclosedbytheroadsandisdesignatedforsensorcoverage.Thissensingrequirementisrepresentedbyasetofeightyvirtualtargetswhicharerandomly-selectedfromthecoverageregion.ThesetargetsaredepictedasredasterisksinFigure 5-9B BFigure5-9. Environmentsforobstacle-freesensorplanningexample.A)Environmentforreconnaissanceandsurveillancemissions.B)Environmentforcoveragemission. SeveraldegreesoffreedominthealgorithmdescribedinSection 5.3 aresetspecif-icallyforthepresentexample.First,eachiterationofthealgorithmbeginswiththeselectionofanexpansionnodefromaweighteddistribution,wherenodeweightsaregivenbytheweightingfunctiondescribedaspreviouslybyEquation 5{2 131

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Simulatedeectivenessforobstacle-freereconnaissance. Thealgorithmisrunfor100iterations.Asbranchesareadded,alistofviewpointsismaintainedthattracksthersttargetsensedalongatrajectory.Viewpointcongurationsfromthislistareusedasgoallocationswhenachildbranchhassatisedallofthesensingconstraints.Useofthesecongurationsasgoallocationsensuresthatthesurveillanceloopisclosedoncealltargetshavebeensensedonce.Thisloopisthentraversedrepeatedlytovieweachtargetmultipletimes.Aminimumupperboundontimeperformanceismaintainedassurveillanceloopsareclosed.Theupperboundisrevisedasbettersolutionsarecomputed.SeveraloftheseimprovingsolutiontrajectoriesareshowninFigure 5-12 .Asbefore,solutiontrajectoriesaredepictedasaredsolidlineandthe\rst-view"projectedsensorfootprintsaredepictedasdashedblacklines.Solutionsareshownafter,1,7,and10iterationsandcorrespondtotrajectoriesof80:44sec,59:66sec,and54:80sec,respectively.Thesevaluesreectthetimerequiredtothereachtheendoftherstloopfromtheinitialcondition.Figure 5-13 showstheresultofasimulationthatplotsthemax-valueeectivenessmetricfortheminimum-timesurveillanceloop.Aspreviously,thesolutiontrajectoryisshownasasolidblacklineandthetargetsareshownasblackcircles.Noticethatthesensingcriteriaareachievedforalltargetlocations. 134

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Simulatedeectivenessforobstacle-freecoverage. andanothertodemonstrateareacoverage.Thesetargetshavethesamesurfacenormalproperties,asexpressedinEquation 5{6 .Unlikethepreviouscasewhichhadtargetsrandomlyscatteredthroughoutthearea,thepresentcaseconsiderstargetlocationsthatarestrategicallyplacedclosetotheobstacles.Figure 5-16A showstheenvironmentusedforreconnaissanceandsurveillancemissions.Ninetargetsareseeninlocationsthatcouldrepresententrancestobuildings.Informationgatheredfromsuchlocationscouldprovidevaluableintelligenceregardingpeopleorvehiclesthatenterandleaveeachbuilding.Figure 5-16B showstheenvironmentusedforcoveragemissions.One-hundredvirtualtargetsaresampledfromtheregionsurroundingoneofthebuildings.Anexampleapplicationforcoverageofsuchanareaistoidentifythepresenceofanyunauthorizedpersonsorevenanexplosivedevicewithinthevicinityofthebuilding.ThemajorityofthealgorithmparametersaresetidenticallytothosedescribedfortheexamplesinSection 5.4.3 .Nodesareselectedfromaweighteddistribution,asetoftenviewpointsaresampledfromSiQi,andcandidatebranchesareplannedandevaluatedtotheseviewpoints.Contrarytothepreviousexample,thetreeisinitializedhereusinganexploratoryrunofanRRTalgorithm.Executionofthisalgorithmisconcernedprimarilywithspanningtheenvironmentanddoesnotconsidersensingobjectives.Useofthis 137

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5-18 .Inthegure,solutiontrajectoriesaredepictedasaredsolidline.Asintheprevioussetofexamples,theprojectionoftherestrictedsensorfootprintontheenvironmentsurfaceisshownforapointalongthetrajectorywhichsatisesthesensingconstraintassociatedwitheachtarget.Theseprojectionsareshownasdashedcontours.Solutionsareshownafter,1,20,and98iterationsandcorrespondtotrajectoriesof68:56sec,67:06sec,and62:97sec,respectively. Figure5-18. Trajectoryrenementforreconnaissancewithobstacles. Theresultsofasimulationshowingthesensingeectivenessovertheentireenviron-mentisshowninFigure 5-19 .ThesimulatedtrajectoryinthegurecorrespondstothesolutionshowninFigure 5-18C .Twoviewsareshowntodemonstratethatalltargetshavebeensensedwithdesiredeectiveness. Figure5-19. Simulatedeectivenessforreconnaissancewithobstacles. 139

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5-20 .Duringthe100iterationsforwhichthealgorithmisrun,therstpointalongeachsolutionpathtosatisfyasensingconstraintmustbestored.Asbefore,thispointservesasagoallocationforchildbranchesthathaveachievedallsensingobjectives. Figure5-20. Initialtreeforsurveillancewithobstacles. Assurveillanceloopsareclosed,aminimumupperboundontrajectorytimeper-formanceismaintained.Theupperboundisrevisedasbettersolutionsarecomputed.SeveraloftheseimprovingsolutiontrajectoriesareshowninFigure 5-21 .Asbefore,solutiontrajectoriesaredepictedasaredsolidlineandprojectedsensorfootprintsaredepictedasdashedblacklines.Solutionsareshownafter4,33,and83iterationsandcorrespondtotrajectoriesof71:40sec,66:01sec,and61:43sec,respectively.Thesevaluesreectthetimerequiredtothereachtheendoftherstloopfromtheinitialcondition.Figure 5-22 showstheresultofasimulationthatplotsthemax-valueeectivenessmetricfortheminimum-timesurveillanceloopshownbyFigure 5-21C .Aspreviously,thesolutiontrajectoryisshownasasolidblacklineandthetargetsareshownasblackcircles.Noticethatthesensingcriteriaareachievedforalltargetlocations. 140

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Trajectoryrenementforsurveillancewithobstacles. Figure5-22. Simulatedeectivenessforsurveillancewithobstacles. ThealgorithmisthenappliedtothecoveragetaskintheenvironmentdepictedbyFigure 5-16B .Thecoverageareaisrepresentedbyasetofone-hundredvirtualtargetswhicharesampledfromauniformdistribution.ThetrajectorytreeisagaininitializedwithanRRTcontainingftynodesandwhichisrootedat(px;py;)=(0;0;0)inatrimstatecorrespondingtoT0.ThisinitialtreeisshowninFigure 5-23 Figure5-23. Initialtreeforcoveragewithobstacles. 141

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5-24 .Solutiontrajectoriesareshownasablacksolidlineinthegure.Projectedsensorfootprintsarenotshowntoavoidclutterresultingfromthelargenumberofvirtualtargets.Solutionsareshownafter,1and5iterationsthatcorrespondtotrajectoriesof106:38secand87:30sec,respectively.Noticethatthebestsolutionoutofall100itera-tionswasfoundveryearlyintheprocess. Figure5-24. Trajectoryrenementforcoveragewithobstacles. Finally,Figure 5-25 showstheresultofasimulationthatplotsthemax-valueeec-tivenessmetricfortheminimum-timecoveragetrajectoryshownbyFigure 5-24B .Aspreviously,thesolutiontrajectoryisshownasasolidblacklineandthetargetsareshownasblackcircles.Noticethatthesensingcriteriaareachievedforalltargetlocations. Figure5-25. Simulatedeectivenessforcoveragewithobstacles. 142

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4 todeneagoalsetofvehiclecongurationsthatensureeectivesensing.Specically,sensingaparticulartargetdoesnotrequirevisitation.Conversely,thevisibility-setconceptisemployedtousetheentiresensorFOVandthusaccountformotioncouplingeectsthatresultfromvehicledynamics.ThespecicdynamicmodelusedwiththeseexamplesisdescribedinSection 6.2 .Theproblemofdeterminingaminimum-timetrajectorythatterminateswithadesiredviewofaspeciedtargetisthenformulatedinSection 6.3 .Theindirect,variationalapproachtosolvingthisproblemisdiscussedinSection 6.4 andisfollowedbytheapplicationofadirectnumericalmethodtotheprobleminSection 6.5 6{1 and 6{2 ,respectively.Thestatesthatcomprise~xare:totalvelocity,ightpathangle,headingangle,altitude,Northposition,Eastposition,angle-of-attack,angle-of-attackrate,androllangle.Thecontrolsarecommandedangle-of-attackandcommandedrollangle.Thesecontrolvariablesrepresentnon-traditionalsysteminputsandcanbeconsideredasgeneratorsofreferencecommandsforlower-levelcontrolsystems.Designofinner-loopand'controllersistypicalintheeldofaircraftcontrol[ 108 ].~x=Vphpxpy'T 6{3 to 6{11 ,where=2(=180)andtheforcecoecients,T,CL,andCD,aretakenfromcurve-tteddata[ 109 ].Eachoftheseequationsrepresentsanelementinthevector-valuedfunction,_~x=f(~x;~u).Thismodelisbasedonasix-stateguidancelevelmodelfromtheliterature[ 109 ].Theoriginalmodelusedand'ascontrols;however,thesevariables 144

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108 ]._V=Tcos(+)CDV2sin (6{9)_=!2n;2!n;+!2n;u (6{11)Thepolynomialcurvesthatareusedtoapproximatetheaerodynamicforcecoe-cientsareshownbyEquations 6{12 to 6{14 ,where1=(12=180).ThedatacorrespondstoamodelofaBoeing727[ 109 ].Suchanaircraftwouldnottypicallybeinvolvedinaclose-proximitysensingmission;consequently,ascaled-upversionoftheproblemmustbeenvisioned.TheforcecoecientcurvesareareplottedinFigure 6-1 .T(V)=0:24760:04312V+0:008392V2 145

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B CFigure6-1. Forcecoecients.A)Thrust.B)Drag.C)Lift. Inadditiontoscalingtheproblemtomatchthemaneuveringcapabilitiesoftheaircraft,themodelisscaledtofacilitatenumericaloptimizationperformance[ 109 ].Velocityisexpressedinunitsofp 6{15 ,whereWistheaircraftweightandSistheplanformareaofthewing.AllmeasuresofdistanceareexpressedinunitsoflCandtimeisscaledbyp gS(6{15) 6{3 to 6{11 andwhichcarriesaLOSremotesensor.Motionisdescribedrelativetoaninertialframethatisorientedsuchthatthex-axispointsintheNorthdirection,they-axispointsintheEastdirection,andthez-axispointsdownward.Thegeometryofthisvehicle-sensorsystemisdescribedasinChapter 4 .Here,thesensorreferenceframe,S,isplacedtocoincidewiththevehiclebodyreferenceframe,B.Assuch,thesensorpositionandorientationareequivalentlydescribedbythevehiclepositionandorientation,asshownbyEquations 6{16 and 6{17 ,whererepresentstheaircraftpitchangle.Additionally,thissensorhasaFOVcharacterizedbyamaximumrangeofrmax=10;000ftandaFOVhalf-angleoff;max==4.Anincidenceangleof<=2isalsorequiredforvisibility. 146

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Atargetislocatedrelativetoaninertialcoordinateframeby~ptwithanassociatedunitnormaldirectionindicatedby^n,asshownbyEquations 6{18 and 6{19 ,respectively.Thistargethasanassociatedvisibilityset,V,whichisdenedintermsofthevisibilityparameters,r,,andfasdescribedinChapter 4 .~pt=pt;xpt;ypt;hT 6{20 and 6{21 ,respectively.ThetransformationterminEquation 6{21 canbewrittenasEquation 6{22 forthesensorcongurationconsideredhere.Noticethatthisexpressionrequiresknowledgeoftheaircraftpitchangle,,whichisnotexplicitlymodeledintheequationsofmotionpresentedinSection 6.2 .Thepitchanglecanberecoveredthroughthekinematicrate-of-climbconstraint,whichisexpressedinEquation 6{23 undertheassumptionthatthesideslipangle,isregulatedtozero[ 108 ].r=q 147

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6{24 6{25 and 6{26 ,respectively.Eachstateisconstrainedatt=t0totakethevalueofaspeciedinitialcondition.Theightpathangle,,isconstrainedforlevelightatt=tfandthenalaltitudeisconstrainedtomatchtheinitialaltitude.Notethattheinequalityconstraintvectoriscomprisedoftheconditionsrequiredforvisibility,asshowninEquation 6{26 .Theincidenceconditionisnotincludedherebecausetheterminalconstraintonaltitudeensures<=2.(~x(t0);t0;~x(tf);tf)eq=266666666666666666666666666666664V(t0)V0=lC(t0)0(t0)0ph(t0)ph;0=lCpx(t0)px;0=lCpy(t0)py;0=lC(t0)0(t0)0'(t0)'0(tf)ph(tf)ph;0377777777777777777777777777777775 148

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6{27 to 6{32 .Theoperatingconditionshavebeenrestrictedbyplacingboundsonallowablevaluesfortheelementsofthecontrolvector,uandu'.ThesecontrollimitsarerepresentedasEquations 6{31 and 6{32 minuJ _~xf(~x;~u)=0 (6{28) (~x(t0);t0;~x(tf);tf)eq=0 (6{29) (~x(t0);t0;~x(tf);tf)ineq0 (6{30) 6{27 to 6{32 .Suchanapproachisoftencalledanindirectmethodbecausethesolutionisobtainedindirectlythroughderivationoftherst-ordernecessaryconditionsforanoptimalsolution.Stateandcontroltrajectoriesthatsatisfytheseconditionsaredeterminedthroughthesolutionofboundary-valueproblemthatresultsfromthenecessaryconditions.Unfortunately,solutionofsuchproblemshavenumerousassociateddiculties,includingsmallradiiofconvergence,agoodinitialguessforthecostate,andpriorknowledgeoftrajectorysegmentsforwhichpathconstraintsareactive[ 51 52 ].Asaresultofthesediculties,thecurrentpresentationformulates,butdoesnotsolve,theboundary-valueproblem.Theprimarypurposeofthissectionistodemonstratesomeofthepracticalissuesthatariseinapplyingoptimalcontroltheorytoevenasimpliedversionofthesensingproblem.Assuch,thesubsequentdevelopmentactuallyconsidersareduced-formoftheproblemformulatedintheprevioussection.Specically, 149

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6{31 and 6{32 ,areneglected.Thisassumptionremovestheneedforpriorknowledgeoftheconstrainedtrajectorysegments.Additionally,theendpointinequalityconstraintsrepresentedby(~x(t0);t0;~x(tf);tf)ineqaretreatedasactiveequalitiesforderivationpurposes. 2 .Here,theseconditionsarederivedfortheproblemdescribedbyEquations 6{27 to 6{32 .TheHamiltonianofthesystemmustbecomputedtoinordertoformJpriortoderivationofthenecessaryconditions.RecallthattheHamiltonianisgivenbyH=L+Tf(~x;~u),whereisthevectorofcostatevariables.Thecostatesaretime-dependentandmatchthestatevectorindimension,asshownbyEquation 6{33 .Here,HisexpressedasinEquation 6{34 @~x.Forthepresentsystem,theelementsofthisvectorareexpressedasshowninEquations 6{36 to 6{44 .Theseexpressionsrequirepartialderivativesoftheforce-coecientcurvets,TV,CL,andCD.ThesederivativesareshowninEquations 6{45 to 6{47 150

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(6{35) +((sin(+)(VTVT)+CL)cos'+cos) Vcos2(Tsin(+)+CLV2)sin'+ (6{36) +VcosphVsincospxVsinsinpy)_=(Vcossinpx+Vcoscospy) (6{37) _ph=0 (6{38) _px=0 (6{39) _py=0 (6{40) _=((Tsin(+)CDV2)V+ (6{41) +(Tcos(+)+CLV2)cos' V+sin' Vcos!2n;) (6{42) _=(2!n;) (6{43) _'=((Tsin(+)+CLV2)cos' Vcossin' V+1 (6{44) 2 asEquations 2{20 and 2{21 ,andrequiretheintroductionofavectorofLagrangemultipliers,givenhereasEquation 6{48 151

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@~x(t0)=@ @~x(tf)=0.ThepartialderivativeoftheendpointconstraintvectorwithrespecttotheinitialstateyieldsthematrixshowninEquation 6{49 ,whereIrepresentsanidentitymatrixand;representsazeromatrix.ThepartialderivativeoftheendpointconstraintvectorwithrespecttotheterminalstateyieldsthematrixshowninEquation 6{50 .ThelastrowofthismatrixcontainsseveralcomplicatedderivativesresultingfromtheappearanceofTBEintheexpressionforf.Theactualexpressionsforthesederivativesarenotshownheretomaintainclarity.ThecostateboundaryconditionsarethenformedbycombiningtheseexpressionsasspeciedbyEquations 2{20 and 2{21 .@ @h@r @x@r @y0000@f 2 .ThisconditionisgivenbyEquation 6{51 andisdenotedaHamiltonian 152

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47 ].TheresultofthisexpressionforthepresentderivationistoxHatH=1foralltime,ashereHisnotanexplicitfunctionoftime. @tf~T@ 48 ].RecallfromChapter 2 thatthisconditioncanbeexpressedas@H @u=0,whichidentiesastationarypointoftheHamiltonian.Forthepresentexample,theoptimalityconditionisexpressedasinEquations 6{52 and 6{53 .Ideally,theseexpressionscouldbeusedtosolvefortheoptimalcontrolvectorintermsofthestates,~x,andcostates,.TheHamiltonianforthepresentexampleislinearinthecontrolvariables.Consequently,Equations 6{52 and 6{53 donotdirectlyyieldanyrelevantinformationregardingthecontrol.Hu=0=!2n; 6{27 to 6{32 .Obtain-ingtheoptimalstateandcontroltrajectoriesfromtheseconditionsisnotnecessarilyanintuitiveandstraightforwardprocess.WhentheMinimumPrincipleyieldsanexpressionfortheoptimalcontrolintermsofthestateandcostate,theresultcanbesubstitutedbackintothestateandcostatedynamics.TheaugmentedsystemisthendescribedbythedierentialequationsshowninEquations 6{54 and 6{55 ._~x=@H(~x;~) 153

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6{54 and 6{55 ;however,thesecon-ditionsaresplitbetweentheendpointsandarenotallknownvalues.Forexample,thecostateboundaryconditionsgivenbythetransversalityconditionsderivedintheprevioussectionareseenas: @ph+f@f @px+f@f @py+f@f 6{52 and 6{53 donotprovideanexpressionfor~uintermsof~xand~.Methodsfortreatingsuchproblemsareatleastascomplexassolutionoftheboundaryvalueproblem.Theproblembecomesevenmorecomplexwhentheinequalityconstraintsareconsideredandpathconstraintsareintroduced.Consequently,analternativeapproachtovariationalmethodsisdesired. 154

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6{27 to 6{32 fromafunctionaloptmizationproblemtoanite-dimensionalparameteroptimizationproblempresentsapracticalalternativetothevariationalapproachpresentedintheprevioussection.ThegeneralprocessforstateandcontrolparameterizationwasdescribedinChapter 2 .Basically,thetimeintervalissplitintoanitesetofsub-intervalsandavectorofunknownparametersisformedusingthestateandcontrolvaluesatthesub-intervalnodes.Dynamicconstraintsareappliedinapiecewisefashionthroughtheuseofnumericalmethodstoensurethatthestateandcontrolvaluessatisfythedynamicsatthesenodes.Pathconstraintscanalsobeappliedatthesenodes,aswell.Finally,acostfunctionisformedintermsoftheparameterizedstateandcontrolandsolutionsaredeterminedthroughtheuseofwell-establishedNLPsolvers.Thissectionappliesapseudospectraltranscriptionmethodtotheeectivesensingproblem.Pseudospectralmethodsparameterizethestateandcontrolusingabasisofglobalorthogonalpolynomialsasopposedtothetypicalpiecewiseapproximations[ 51 52 110 111 ].ThespecicmethodutilizedhereisdenotedtheGausspseudospectralmethod.ThismethodcollocatesthedynamicsatLegendre-GausspointsusingabasisofLagrangepolynomialstointerpolatethestateandcontroltrajectories[ 51 ].Thekeybenetstopseudospectralmethodsarerelatedtotheireciency,accuracy,andeaseofimplementation[ 110 111 ].Themulti-targeteectivesensingproblemistreatedthroughsub-divisionoftheproblemintoaseriesofphases.Eachphaserepresentsanon-overlappingtrajectorysegmentthatinvolvesdierentproblemspecicationssuchascost,endpointconstraints,orevendynamics.Thesesegmentsarelinkedbyasetofconditionsthatconstrainrelevantparametersacrosstheboundariesofsequentialphases.Here,thetrajectorysegmentsassociatedwithsensingeachtargetindividuallyaretreatedasseparatephases.Eachofthesephaseshasaninitialconditionassociatedwiththeendpointofthepreviousphase(ortheglobalinitialcondition)andterminalconstraintsassociatedwithsensinga 155

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6.2 .Thisaircraftcarriesadownward-pointingsensorasdescribedinSection 6.3 .Theobjectiveoftheexampleistosensetwotargetsinaspeciedsequenceandthentoreturntotheinitialposition.Simplevisibilityisconsideredsucienttomeetsensingeectivenessrequirements.ThisexampleconsiderstheenvironmentdepictedinFigure 6-2 ,whichhasanembeddedinertialreference-framethatisorientedwiththex-axispointingintheNorthdirectionandthey-axispointingintheEastdirection.Thetwotargetsarelocatedrelativetotheinertialframeby~pt1and~pt2,asshownbyEquations 6{56 and 6{57 .Thetargetsattheselocationsareorientedsuchthateachhasaunitnormalvectordescribedby^ni,shownbyEquation 6{58 .ThesetargetsareshowninFigure 6-2 asredcircles.~pt1=10;0005;0000T (6{58)Thestateinitialconditionsatt0=0aregivenbyEquations 6{59 to 6{67 .Thisinitialconditionrepresentstrimmedsteady,straightandlevelightatanaltitudeof5;000ftheadingdueNorth.Notethatthestatesarenon-dimensionalizedforscalingpurposes,as 156

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Environmentforoptimalsensingnumericalexample. describedpreviously. (6{60) (6{61) (6{63) (6{64) (6{65) (0)=0 (6{66) (6{67) Theproblemissetupinthreesequentialphases.Therstphasecontainsthetrajectorysegmentalongwhichtheaircraftmustmovefromtheinitialconditiontoapointwherethetargetlocatedby~pt1isvisible.Thesecondphasecontainsthetrajectorysegmentfromtheendpointoftherstphasetoapointwherethetargetlocatedby~pt2isvisible.Finally,thethirdphasecontainsthetrajectorysegmentfromtheendpointof 157

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Solutionpositionandvelocitytrajectories. Figure6-4. Solutionand'referencetrajectories. ThetrajectoriesforthevisibilityparametersassociatedwiththerstandsecondtargetsareshowninFigures 6-5A and 6-5B ,respectively.ThemaximumrangeandFOVangleareindicatedbydashedlinesineachplot.Noticethattheseconstraintsarenotnecessarilyallactiveatthephaseboundaries.Theplotsshowthatthevisibilityboundsaresatisedforthersttargetattheendoftherstphaseandaresatisedforthesecondtargetattheendofthesecondphase.Finally,Figure 6-6 showsanoverheadviewofthetotaltrajectorywithprojectedsensorfootprintsshownbydashedcontoursattheendofeachofthersttwophases.ThegureshowsthatthefullsensorFOVandmotioncouplingwereconsideredandutilizedintheoptimalsolution. 159

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BFigure6-5. Solutionvisibilityparametertrajectoriesforeachtarget. Figure6-6. Aircrafttrajectoryandsensorfootprintovereachtarget. 160

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2 .Theconceptofmotionprimitiveswereusedtorepresentvehicledynamicsinahybridmodelingframework.Thepoint-to-pointmotionplanningproblemwasthenformu-latedasahierarchicaloptimizationproblemthatcontainedacombinatorialelementandacontinuousNLPelement.Thecombinatorialelementwasshowntogreatlycomplicatethesolutionprocedureforrealisticscenarios.Assuch,aspecialclassoffeasible,thoughsub-optimal,explicitsolutionswereidentiedasameansbywhichapproximatesolutionscanbecomputedeciently.Utilityofthesesolutionswasdemonstratedfortheobstacle-freecase;however,globalsolutionswereseentobelimitedwhenobstaclesareconsidered.Assuch,thesesolutionsaretailoredmoreforlocalplanningtasks,suchasthatrequiredforbranchextensioninrandomizedmethods.SeveralareasforfutureworkrelatedtothemethodsdescribedinChapter 2 canbeidentied.TheexamplesdiscussedinChapter 2 utilizedverysimpledynamicmodelsto 161

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