Situational Awareness for a Navy Unmanned Undersea Vehicle
By
Andrew Peter Mierisch
B.S. Systems Engineering, United States Naval Academy, 2001
SUBMITTED TO THE DEPARTMENT OF ELECTRICAL ENGINEERING AND
COMPUTER SCIENCE IN PARTIAL FULFILMENT OF THE REQUIREMENTS
FOR THE DEGREE OF
MASTER OF SCIENCE IN ELECTRICAL ENGINEERING
AND COMPUTER SCIENCE
AT THE
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
JUNE 2003
COPYRIGHT @2003 Andrew Peter Mierisch. All rights reserved.
The author hereby grants to MIT permission to reproduce and to distribute publicly paper
and electronic copies of this thesis document in whole or in part.
Author .................................
Department of Electrical Engineering and Computer Science
'a
e
JMay 23,,2003
Certified by .................................
William Kreamer
The Charles Stark Draper Laboratory, Inc.
Technical Supervisor
Certified by .............................
Accepted by ............
.
........................
Professor Leslie Pack Kaelbling
Prfessor of Coputer Science and Engineering, MIT
Thesis Supervisor
.......
.
Arthur C. Smit1
Chairman, Department Committee on Graduate Students
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
BARKER
JUL 0 7 2003
LIBRARIES
2
Situational Awareness for a Navy Unmanned Undersea Vehicle
by
Andrew Peter Mierisch
Submitted to the Department of Electrical Engineering and Computer Science
On May 23, 2003, in partial fulfillment of the
Requirements for the degree of
Master of Science in Electrical Engineering and Computer Science
Abstract
This thesis presents a new approach for Navy unmanned undersea vehicles (UUVs) to
determine if a contact in the vehicle's operating environment has counter-detected it.
This approach uses contact tracking information from an Extended Kalman Filter to
analyze the behavior of a single contact and determine if the contact has counter-detected
the UUV. Basic laws of probability and probabilistic models of potential contact
behavior are used to determine the likelihood that counter-detection has occurred at fixed
time intervals. Different scenarios are tested through computer-simulations to
demonstrate how various behaviors that the contact displayed affected the UUV's belief
that it had been counter-detected. Also, a maneuver decision aid will be presented that
can enable the UUV to make intelligent decisions with respect to quickly determining
that a contact has been alerted to the presence of the UUV. Various scenarios are tested
to show the effectiveness of the maneuver decision aid and its potential value to a UUV
during an intelligence/surveillance/reconnaissance (ISR) mission.
Technical Supervisor: William Kreamer
Title: Member of the Technical Staff, The Charles Stark Draper Laboratory, Inc.
Thesis Supervisor: Leslie Pack Kaelbling
Title: Professor of Computer Science and Engineering, MIT
3
Acknowledgments
I would like to thank all of those who have helped me throughout my life and
particularly here at MIT to attain my goals, including teachers, family, and friends.
Without their support this thesis would not be possible.
My parents and the rest of my family have always provided constant support in
all my endeavors. Because of that support and encouragement to do my best I have been
able to accomplish so much.
I also owe a debt of gratitude to my advisor at Draper Laboratory, William
Kreamer, for his enthusiasm and wisdom in guiding me throughout my research for this
thesis. He always seemed to have the time to answer any sort of question I had for him.
Thank you also to everyone I have encountered at Draper for providing me with the
opportunity to continue my education at MIT.
Professor Leslie Kaelbling also deserves a big thank you for her role in the
completion of this thesis. Her expertise and the time she took from her busy schedule to
advise me were such a valuable source of guidance.
To all of my friends that I met here in Boston at MIT and Draper Laboratory that
made my two years here so enjoyable, I also say thank you. To Jenny, I say the time I've
spent with you here has been incredible and the greatest source of happiness and joy for
me. Also, thank you to all the other Draper Fellows for their friendship and support.
Lunch breaks with them were always a great break from work.
4
Acknowledgment
This thesis was prepared at The Charles Stark Draper Laboratory, Inc. under Contract
N00014-02-C-0191, sponsored by the Office of Naval Research.
Publication of this thesis does not constitute approval by Draper or the sponsoring agency
of the findings or conclusions contained herein. It is published for the exchange and
stimulation of ideas.
Andrew Mierisch
5
May 23, 2003
6
Contents
1 Introduction
19
1.1
Problem Motivation .........................................
19
1.2
Problem Statem ent ..........................................
23
1.2.1 Concept of Operation ....................................
23
1.2.2 Situational Awareness and Assessment .....................
24
1.3
Problem Approach ..........................................
24
1.4
Contributions .............................................
26
1.5
Organization ..............................................
27
2 Extended Kalman Filter Tracking Techniques
2.1 Sonar .......................................................
2.2
2.3
EKF Contact Tracking with Active Sonar ........................
29
29
30
2.2.1
Contact Motion M odel ................................
30
2.2.2
Observation M odel ...................................
32
2.2.3
EKF Algorithm ......................................
34
2.2.4
EKF Example with Active Sonar .........................
38
EKF Contact Tracking with Passive Sonar .......................
2.3.1
EKF Contact Tracking with Passive Sonar and Cartesian
Coordinates ........................................
7
41
41
2.3.2
EKF Contact Tracking with Passive SonarAnd Modified
Polar Coordinates ....................................
3 Counter-Detection Recognition
45
51
3.1
Defining Behaviors .........................................
51
3.2
Counter-Detection Belief State Update ...........................
54
3.3
Hostile PDF M odel
55
3.4
3.5
.........................................
3.3.1
Hostile Course Change PDF .............................
57
3.3.2
Hostile Random Course Change PDF ......................
59
3.3.3
Position Distribution ...................................
60
3.3.4
Example Plot of Hostile PDF ............................
66
Passive PDF Model ........................................
67
3.4.1
Passive Random Course Change PDF .....................
69
3.4.2
Non-Significant Course Change PDF ......................
70
3.4.3
Example Plot of Passive PDF .............................
71
Example Cases .............................................
72
3.5.1
Example Simulation 3.1 (Passive Contact) ..................
73
3.5.2
Example Simulation 3.2 (Hostile Contact) ..................
75
3.5.3
Example Simulation 3.3 (Initially Passive Contact With
Counter-Detection Occurring During Simulation) ............
76
3.5.4 Example Simulation 3.4 (Hostile Contact) ..................
78
3.5.5
Example Simulation 3.5 (Passive Contact) ..................
80
3.5.6
Example Simulation 3.6 (Hostile Contact) ..................
82
3.6 Sum m ary ....................................................
8
84
85
4 Counter-Detection Maneuver Decision Aids
4.1
E ntropy .....................................................
85
4.2
Maneuver Decision Aid Formulation ..............................
87
4.2.1 Decision Space ........................
..............
4.2.2 Objective Function .........................
4.3
One Time Horizon Look Ahead Example Simulations ................
4.3.1
4.3.2
4.3.3
4.3.4
4.3.5
4.3.6
4.4
..............
4.4.2
4.4.3
4.4.4
87
92
Example Simulation 4.1 (Hostile Contact, 1 Time Horizon
M aneuver Decision Aid) ................................
93
Example Simulation 4.2 (Hostile Contact, I Time Horizon
M aneuver Decision Aid) ................................
94
Example Simulation 4.3 (Passive Contact, I Time Horizon
M aneuver Decision Aid) ................................
96
Example Simulation 4.4 (Hostile Contact, 1 Time Horizon
M aneuver Decision Aid) ................................
97
Example Simulation 4.5 (Hostile Contact, 1 Time Horizon
M aneuver Decision Aid) ................................
99
Example Simulation 4.6 (Passive Contact, 1 Time Horizon
Maneuver Decision Aid) ...... ..................................................
1-5 Time Horizon Look Ahead Example Simulations ...
4.4.1
87
............
1u
101
Example Simulation 4.7 (Hostile Contact, 1-5 Time Horizon
Maneuver Decision Aid) ...... .................................................
102
Example Simulation 4.8 (Passive Contact, 1-5 Time Horizon
Maneuver Decision Aid) ...... .................................................
103
Example Simulation 4.9 (Hostile Contact, 1-5 Time Horizon
Maneuver Decision Aid) ......
1iu5
Example Simulation 4.10 (Passive Contact, 1-5 Time
Horizon Maneuver Decision Aid) ..........................
106
4.5 Sum m ary ....................................................
9
107
5 Intelligence/Surveillance/Reconnaissance (ISR) Simulation Results
109
5.1
ISR Concept of Operations .....................................
109
5.2
ISR M aneuver Decision Aid ....................................
110
5.1
1 Time Horizon ISR Maneuver Decision Aid Simulation Results ......
116
5.3.1
5.3.2
5.3.3
5.3.4
Example Simulation Set 5.1 (Hostile Contact and 1 Time
Horizon ISR Maneuver Decision Aid) .....................
117
Example Simulation Set 5.2 (Hostile Contact and 1 Time
Horizon ISR Maneuve Decision Aid) .....................
119
Example Simulation Set 5.3 (Passive Contact and 1 Time
Horizon ISR Maneuver Decision Aid) .....................
121
Example Simulation Set 5.4 (Passive Contact and 1 Time
Horizon ISR Maneuver Decision Aid) .....................
124
5.4 Sum m ary .....................................................
6 Conclusions and Future Research
126
127
6.1
Thesis Contributions .........................................
127
6.2
Future Research .............................................
128
Bibliography
131
10
List of Figures
1.1
UUV Developed by Draper Laboratory ..............................
21
1.2
O O D A Loop ..................................................
25
2.1
Illustration of Observation M odel..................................
33
2.2
UUV, True Contact, and Estimated Contact Positions With Active Sonar . .
39
2.3
X-Position Error and 3 Sigma Confidence Bound ...
40
2.4
Y-Position Error and 3 Sigma Confidence Bound ...
40
2.5
Heading Error and 3 Sigma Confidence Bound .....
40
2.6
Speed Error and 3 Sigma Confidence Bound .......
41
2.7
UUV, True Contact, and Estimated Contact Positions With Passive Sonar
and Cartesian Coordinates ......................
.. ..... ....
42
2.8
X-Position Error and 3 Sigma Confidence Bound ...
42
2.9
Y-Position Error and 3 Sigma Confidence Bound ...
43
2.10
Heading Error and 3 Sigma Confidence Bound .......................
43
2.11
Speed Error and 3 Sigma Confidence Bound .........................
43
2.12
Difficulties in Bearings-Only EKF .................................
44
2.13
UUV, True Contact, and Estimated Contact Positions With Passive Sonar
and M odified Polar Coordinates ...................................
47
2.14
Bearing Rate Error and 3 Sigma Confidence Bound ....................
47
2.15
Range Rate/Range Error and 3 Sigma Confidence Bound ...............
48
11
2.16
Bearing Error and 3 Sigma Confidence Bound .........................
48
2.17
1/Range Error and 3 Sigma Confidence Bound ........................
48
2.18
X-Position Error ...............................................
49
2.19
Y-Position Error ................
2.20
Heading Error ................................................
49
2.21
Speed Error ..................................................
50
3.1
Example #1 of Hostile Behavior ..................................
52
3.2
Example #2 of Hostile Behavior ..........
53
3.3
Example of Passive Behavior .....................................
54
3.4
Illustration of OE for Hostile Course Change PDF ......................
57
3.5
Plot of Hostile Course Change PDF ................................
58
3.6
Distribution of Hostile Random Course Change .......................
59
3.7
Illustration of Covariance ........................................
63
3.8
Illustration of Covariance ........................................
63
3.9
Illustration of Covariance ........................................
63
3.10
Illustration of Covariance ........................................
63
3.11
Illustration of Covariance ........................................
64
3.12
Plot of Likelihood of Resulting Cartesian Coordinates of Hostile Contact ...
67
3.13
Distribution of Passive Random Course Change ........................
70
3.14
Plot of Likelihood of Resulting Cartesian Coordinates of Passive Contact ...
71
3.15
Example Simulation 3.1 UUV and Contact Positions ....................
74
3.16
Example Simulation 3.1 Belief States ................................
74
3.17
Example Simulation 3.1 Passive PDF Values .........................
74
..............................
12
.........................
49
3.18
Example Simulation 3.1 Hostile PDF Values ....
74
3.19
Example Simulation 3.2 UUV and Contact Positions ...................
75
3.20
Example Simulation 3.2 Belief States ...............................
76
3.21
Example Simulation 3.2 Passive PDF Values ........................
76
3.22
Example Simulation 3.2 Hostile PDF Values ..........................
76
3.23
Example Simulation 3.3 UUV and Contact Positions ...................
77
3.24
Example Simulation 3.3 Belief States ...............................
77
3.25
Example Simulation 3.3 Passive PDF Values ........................
77
3.26
Example Simulation 3.3 Hostile PDF Values .........................
77
3.27
Example Simulation 3.4 UUV and Contact Positions ...................
79
3.28
Example Simulation 3.4 Belief States ..............................
79
3.29
Example Simulation 3.4 Passive PDF Values .........................
79
3.30
Example Simulation 3.4 Hostile PDF Values .........................
79
3.31
Example Simulation 3.5 UUV and Contact Positions ...................
81
3.32
Example Simulation 3.5 Belief States ...............................
81
3.33
Example Simulation 3.5 Passive PDF Values .........................
81
3.34
Example Simulation 3.5 Hostile PDF Values .........................
81
3.35
Example Simulation 3.6 UUV and Contact Positions ...................
83
3.36
Example Simulation 3.6 Belief States ...............................
83
3.37
Example Simulation 3.6 Passive PDF Values .........................
83
3.38
Example Simulation 3.6 Hostile PDF Values .........................
83
4.1
Plot of Entropy Vs. Belief State ...................................
86
13
............
91
Example Simulation 4.1 UUV and Contact Positions ...
.. ...........
93
4.4
Example Simulation 4.1 Belief States ................
.....
........
93
4.5
Example Simulation 3.2 UUV and Contact Positions ....
.....
........
93
4.6
Example Simulation 3.2 Belief States ...............
. ............
93
4.7
Example Simulation 4.2 UUV and Contact Positions ....
. ............
95
4.8
Example Simulation 4.2 Belief States ................
....
.........
95
4.9
Example Simulation 3.4 UJV and Contact Positions .
. ............
95
4.10
Example Simulation 3.4 Belief States ............
. ............
95
4.11
Example Simulation 4.3 UUV and Contact Positions .
. ............
96
4.12
Example Simulation 4.3 Belief States .............
.....
........
96
4.13
Example Simulation 3.5 UUV and Contact Positions .
. ............
96
4.14
Example Simulation 3.5 Belief States ............
. ............
96
4.15
Example Simulation 4.4 UUV and Contact Positions .
. ............
98
4.16
Example Simulation 4.4 Belief States ............
....
.........
98
4.17
Example Simulation 3.6 UUV and Contact Positions .
. ............
98
4.18
Example Simulation 3.6 Belief States .............
98
4.19
Example Simulation 4.5 UUV and Contact Positions
99
4.20
Example Simulation 4.5 Belief States .............
99
4.21
Example Simulation 4.6 UUV and Contact Positions
100
4.22
Example Simulation 4.6 Belief States .............
100
4.23
Example Simulation 4.7 UUV and Contact Positions ,
102
4.24
Example Simulation 4.7 Belief States .............
102
4.2
Illustration of fENT
4.3
--
--...............................
-..-
14
4.25
Example Simulation 4.2 UUV and Contact Positions ...................
102
4.26
Example Simulation 4.2 Belief States ...............................
102
4.27
Example Simulation 4.8 UUV and Contact Positions ...................
104
4.28
Example Simulation 4.8 Belief States ..............................
104
4.29
Example Simulation 4.3 UUV and Contact Positions ...................
104
4.30
Example Simulation 4.3 Belief States
104
4.31
Example Simulation 4.9 UUV and Contact Positions ...................
106
4.32
Example Simulation 4.9 Belief States ..............................
106
4.33
Example Simulation 4.4 UUV and Contact Positions ...................
106
4.34
Example Simulation 4.4 Belief States ...............................
106
4.35
Example Simulation 4.10 UUV and Contact Positions
4.36
Example Simulation 4.10 Belief States .............................
107
4.37
Example Simulation 4.6 UUV and Contact Positions ...................
107
4.38
Example Simulation 4.6 Belief States ..............
5.1
Plot of ftime ....
5.2
Plot of ISR Maneuver Decision Aid Objective Function with p=.l ........
115
5.3
Plot of ISR Maneuver Decision Aid Objective Function with p=.5 ........
115
5.4
Example Simulation Set 5.1 UUV and Contact Positions with p=.1 ........
117
5.5
Example Simulation Set 5.1 Belief States with p=.1 ...................
117
5.6
Example Simulation Set 5.1 UUV and Contact Positions with p=.5 ........
117
5.7
Example Simulation Set 5.1 Belief States with p=.5 ....................
117
5.8
Example Simulation Set 5.1 UUV and Contact Positions with p=1
..............................
107
.................
......
112
....................................................
15
107
........
118
5.9
Example Simulation Set 5.1 Belief States with p=1 ....
. . . . . . . . . . ......
118
5.10
Example Simulation Set 5.1 UUV and Contact Positions with p=2 .. ......
118
5.11
Example Simulation Set 5.1 Belief States with p=2 ....
......
118
5.12
Example Simulation Set 5.1 UUV and Contact Positions with p=10 . ......
118
5.13
Example Simulation Set 5.1 Belief States with p=10 ...
......
118
5.14
Example Simulation Set 5.2 UV and Contact Positions with p=.l . . ......
119
5.15
Example Simulation Set 5.2 Belief States with p=.1 ...
......
119
5.16
Example Simulation Set 5.2 UUV and Contact Positions with p=.5 .. ......
120
5.17
Example Simulation Set 5.2 Belief States with p=.5 ...
......
120
5.18
Example Simulation Set 5.2 UUV and Contact Positions with p=1 . . ......
120
5.19
Example Simulation Set 5.2 Belief States with p=1 ...................
120
5.20
Example Simulation Set 5.2 UUV and Contact Positions with p=2 ........
120
5.21
Example Simulation Set 5.2 Belief States with p=2 ...................
120
5.22
Example Simulation Set 5.2 UUV and Contact Positions with p=10 .......
121
5.23
Example Simulation Set 5.2 Belief States with p=10 ...................
121
5.24
Example Simulation Set 5.3 UUV and Contact Positions with p=.1 ........
122
5.25
Example Simulation Set 5.3 Belief States with p=.l
122
5.26
Example Simulation Set 5.3 UUV and Contact Positions with p=.5 ........
122
5.27
Example Simulation Set 5.3 Belief States with p=.5 ..................
122
5.28
Example Simulation Set 5.3 UUV and Contact Positions with p=1 ........
122
5.29
Example Simulation Set 5.3 Belief States with p=1
122
5.30
Example Simulation Set 5.3 UUV and Contact Positions with p=2 ........
123
5.31
Example Simulation Set 5.3 Belief States with p=2 ...................
123
16
...................
...................
5.32
Example Simulation Set 5.3 UUV and Contact Positions with p=10 .......
123
5.33
Example Simulation Set 5.3 Belief States with p=10 ..................
123
5.34
Example Simulation Set 5.4 UUV and Contact Positions with p=.l ........
124
5.35
Example Simulation Set 5.4 Belief States with p=.l
124
5.36
Example Simulation Set 5.4 UUV and Contact Positions with p=.5 ........
124
5.37
Example Simulation Set 5.4 Belief States with p=.5 ...................
124
5.38
Example Simulation Set 5.4 UUV and Contact Positions with p=1 ........
125
5.39
Example Simulation Set 5.4 Belief States with p=1 ...................
125
5.40
Example Simulation Set 5.4 UUV and Contact Positions with p=2 ........
125
5.41
Example Simulation Set 5.4 Belief States with p=2 ...................
125
5.42
Example Simulation Set 5.4 UUV and Contact Positions with p=10 .......
125
5.43
Example Simulation Set 5.4 Belief States with p=10 ...................
125
6.1
Example Simulation UUV and Contact Locations ......................
128
6.2
Example Simulation B elief States ..................................
128
17
...................
18
Chapter 1
Introduction
The objective of this thesis is to develop maneuver decision aid algorithms that
will enable a United States Navy unmanned undersea vehicle (UJV) to track contacts in
its operating environment and determine if counter-detection has occurred while
performing intelligence/surveillance/reconnaissance (ISR) missions. The work in this
thesis will focus on simulating a realistic tracking algorithm that can be used by the
maneuver decision aid algorithms to estimate the state of surrounding contacts and using
the laws of probability to develop an effective maneuver decision aid. Various computersimulated test cases will be investigated to determine the usefulness of the maneuver
decision aid algorithms.
1.1
Problem Motivation
Recently, rapid advances in technology have helped produce a budding industry
of autonomous vehicles. An autonomous system can perform its designated tasks
without the help of a human or other intelligent operator. The appeal of autonomous
systems to the military is that they provide humans with a safer and more efficient
existence by performing tasks that are either too dangerous, difficult, or tedious for
humans. A recent and popular example of an unmanned system is the United States
government's use of the Predator, an unmanned airplane. It has been used to perform
extended surveillance missions over hostile territory. The Predator has performed well in
recent years by providing valuable data and services to commanders in the field of
operations, while performing many missions that are very dangerous and too long for a
human pilot to carry out. Although it requires a human to remotely operate it, the
Predator shows the impact that unmanned vehicles can have on the military [2].
19
The Navy needs stealthy and unmanned systems to gather information and engage
targets in areas where traditional forces are denied entry. Various military threats, as well
as diplomatic constraints or rules of engagement may prevent the early entry of overt
maritime forces into important areas. Assets are needed that avoid counter detection by
the enemy to allow sustained independent operations in these denied regions. With these
types of options, military commanders can keep other forces out of harm's way during
the beginning phases of conflict while still being able to prepare and shape the battle
space [1].
In the future the United States Navy plans on using unmanned undersea vehicles
(UUVs) to play important roles in the battle space. Vital missions including intelligence,
surveillance, reconnaissance, mine countermeasures, tactical oceanography,
communications, navigation, and anti-submarine warfare can be addressed with UUVs.
UUVs are advantageous in these types of missions because they can increase
performance, lower cost, and reduce risk to manned systems. UUVs are able to provide
these improvements because of their ability to put the following operational advantages
to use:
-
Sensor Deployment. UUVs have the ability to put sensors in an excellent
position in both the vertical and horizontal dimensions.
-
Autonomy. The ability of a UUV to operate independently for extended periods
creates a force multiplier that allows manned systems to extend their reach and
focus on more difficult tasks. Reduced costs are also a result when sensors and
weapons are operated from smaller platforms like UUVs.
-
Risk. Since UVs are unmanned, there is a reduced threat to personnel from a
harsh sea or enemy combatants.
-
Deployability. UUVs can be designed as flyaway packages or pre-positioned in
forward areas. They can be launched from a wide variety of platforms including
ships, submarines, aircraft, and shore facilities. Also, they do not have to be
recovered from the same craft they were launched from. Recoveries may be
delayed or abandoned because of the expendability created by a UUV's low cost.
20
Environmental Adaptability. UUVs can operate in a diverse range of
-
environments including deep to shallow water, foul weather and seas, and
tropical or arctic conditions [1].
An unmanned undersea vehicle is defined as a self-propelled submersible whose
operation is either fully autonomous or under minimal supervisory control and is
untethered except for data links such as a fiber optic cable [1]. The work in this thesis
will contribute to the autonomy of an untethered UUV. Humans have traditionally
performed tasks such as mission planning in the Navy. The difficulty in creating an
effective
'a
Figure 1.1 UUV Developed by Draper Laboratory
autonomous system is to translate the thoughts and actions of a human operator into a set
of rules and behaviors for an autonomous system to follow.
21
A study team for the Navy UUV Master Plan [1] established a long-term vision
for UUVs by listing priorities for near-term acquisition programs and technology
investment while laying a foundation for long-term applications. There was no initial
consideration of technical, operational, or fiscal constraints. They first generated a
comprehensive list of potential UUV missions by taking advantage of a wide range of
current and potential UUV users through field surveys, expert panels, and analysis.
The study team found the number one priority mission to be
Intelligence/Surveillance/Reconnaissance (ISR). ISR missions include collection and
delivery of various types of data products such as intelligence collection, target detection
and mapping data. UUVs are well suited for these types of missions because of their
ability to operate at long standoff distances, remain on station for long periods of time,
operate independently, and provide a level of stealth unparalleled by other naval
platforms. As mentioned previously, UUVs can provide the ability to access previously
denied areas and provide information at a lower risk to personnel or higher valued units.
Potential ISR missions include:
- Intelligence collection
- Battle damage assessment
- Bio-chemical or nuclear detection and defense
- Ship escort: extended "eyes and ears"
- Search and recover to full ocean depth
- Deployment of leave-behind sensors or sensor arrays
- Underwater security: divers, mines, etc.[1]
After the list of missions was put together, each mission was analyzed for
technical feasibility, political acceptability, and operational desirability. Table 1.1 shows
the missions that were prioritized by the study team. The feasible and appropriate
missions were then grouped according to operational and technological requirements to
four signature capabilities. The maritime reconnaissance signature capability, which
addresses the ISR missions, was determined to be the top priority capability [1]. The
focus of this thesis will be on Maritime Reconnaissance.
22
Missions
Signature Capability
Intelligence/Surveillance/Reconnaissance
Maritime Reconnaissance
Mine Countermeasures
Oceanography
Communication/Navigation Aids
Anti-Submarine Warfare
Weapons Platform
Logistics Supply and Support
Undersea Search and Survey
Communication/Navigation Aids
Submarine Track & Trail
Table 1.1 Prioritized Missions and Signature Capabilities
1.2 Problem Statement
The objective of Maritime Reconnaissance is to collect multidisciplinary
intelligence data across the entire electromagnetic spectrum while remaining undetected
by the enemy. UUVs can be launched from safe distances to accomplish these missions
in high-risk areas or water that is too shallow for larger, more conventional platforms.
Using UUVs for intelligence/surveillance/reconnaissance (ISR) collection can offset
reduction in the overall size of the fleet, increase coverage rate and tactical reach, and
provide acceptable risk in a hostile area with dynamic threats [1].
1.2.1 Concept of Operations
A general ISR mission for a Navy UUV would begin with the vehicle being
launched from a platform such as a submarine, surface ship, aircraft or shore facility. It
would then maneuver to a designated observation area, to collect information over a
predetermined amount of time. During the mission the UUV could reposition itself to
collect additional data or avoid threats. The information collected could immediately be
transmitted to appropriate parties or the vehicle could carry the information back to its
host platform. While transiting to and from the observation area and collecting data, the
23
UUV must be able to track, recognize, and avoid mobile threats such as enemy ships in
its operating environment [1].
1.2.2 Situational Awareness and Assessment
Charles Stark Draper Laboratory is currently developing a Maritime
Reconnaissance Demonstration system. This system will integrate automated situation
awareness (SA) with closed-loop planning and control aboard in-water systems. The
system will maintain SA for use by the vehicle's planning system and use it as data that
can be drawn upon as needed. This thesis will concentrate on an element of this situation
awareness and assessment subsystem known as the SA Assessor (SAA). Situation
assessment is the process of determining the impact of the current situation as provided
by situation awareness on the vehicle's plans. The SAA must interpret data received
from it sensors to track dynamic obstacles, perform tactical maneuvers, detect hostile
actions by dynamic objects, and manage and improve the SA picture it maintains. Of
particular interest in this thesis is the UUV's ability to track contacts in its area and
determine if any potentially hostile contacts have been alerted to its presence there (also
called counter-detection).
1.3 Problem Approach
A common approach used by military units to accomplish many of its missions is
known as the OODA loop. The OODA loop organizes actions performed by the unit into
one of four blocks: observe, orient, decide, and act. The "observe" block refers to the
ability of the unit to correctly understand information about its state and environment.
The "orient" block determines how possible future actions will affect the current situation
given by the "observe" block. The "decide" block, using information given by the
"orient" block, creates a plan of action. Once the "decide" block generates a plan, the
24
"act" block executes the details of the plan. As the plan is executed and the situation
changes, the "observe" block can detect the need for a new plan and initiate the entire
loop again [21 [3].
The OODA loop can also be applied to a UUV maintaining its SA picture with
respect to determining if counter-detection has occurred. The UUV must be able to
observe the states of contacts in its area and analyze their behavior to detect potentially
hostile intent. Using this information, the UUJV must then be able to understand what
impact various actions will have on its situational awareness and mission
accomplishment, decide which actions are best, and then execute its plans. This thesis
Figure 1.2 OODA Loop
will concentrate on contributing to the observe, orient, and decide phases of Charles Stark
Draper Laboratory's Maritime Reconnaissance Demonstration Situational Awareness
Assessor.
The work of this thesis will contribute to the Situational Awareness Assessor by
developing a maneuver decision aid that will enable the UUV to determine if it has been
counter-detected by contacts in its operating environment. The aid will determine
appropriate course and speed commands for the UUV to determine more accurately if
surrounding contacts have been alerted to its presence. First a realistic tracking system
will be simulated that will estimate the position, course, and speed of contacts in the area
25
of the UUV. Then maneuver decision aids will be developed that will use probabilistic
models of hostile and non-hostile contacts to determine if counter-detection has occurred
and make course recommendations to improve the SAA knowledge of any possible
counter-detection occurrences, while also taking into the consideration the need to arrive
at the designated observation area in a timely fashion. This approach potentially can
enable the UUV to behave in an intelligent matter. Various test cases will be investigated
through computer simulation to evaluate the performance of the maneuver decision aid in
combination with realistic tracking algorithms.
1.4 Contributions
The work in this thesis will contribute to future advancement of the maritime
reconnaissance signature capability discussed earlier. These contributions are listed
below:
1)
Artificial Intelligence in Navy UUVs
* Very little work has been done with any sort of artificial intelligence for
Navy UUVs
* Probabilistic framework will enable UUV to make intelligent decisions for
situational awareness.
2)
Potential Use in Fleet
* Using realistic tracking algorithms in simulated test cases will show that the
maneuver decision aid algorithms developed in this thesis have the potential
to be used in real world applications.
3)
Contribution to Draper Laboratory Maritime Reconnaissance Demonstration
Project
eThe ideas behind the developed maneuver decision aid algorithms can be
used in future integration of automated situation awareness with closed-loop
planning and control aboard in-water systems.
26
1.5 Organization
The remainder of this thesis is divided into five chapters. Chapter 2 investigates
the use of Extended Kalman Filtering techniques to develop realistic tracking algorithms.
Chapter 3 introduces an algorithm that analyzes the motion of a contact in a UUV's
operating environment to determine if it has been counter-detected. Chapter 4 presents
UUV maneuver-decision aids that enable the UUV to quickly determine if a contact has
counter-detected it. Chapters 3 and 4 will be the major contributions made by this thesis.
Chapter 5 will provide examples of how the algorithm presented in Chapter 3 and the
maneuver-decisions aids of Chapter 4 can be used by a UUV during an ISR mission.
Chapter 6 provides conclusions and suggestions for future work.
27
28
Chapter 2
Extended Kalman Filter Tracking Techniques
In the Maritime Reconnaissance role a UUV must be able to estimate the states of
surrounding contacts and know how accurate those estimates are before any maneuver
decision aid can determine if the UUV has been counter-detected by any potentially
hostile contacts. In the context of this chapter a contact's state is its position, heading,
and speed. Many successful implementations of target tracking algorithms use an
Extended Kalman Filter (EKF) for state estimation. This chapter will review EKF
algorithms with sonar measurements that will be used in computer-simulated tests of the
maneuver decision aid algorithms discussed later in this thesis. For a more detailed
description of EKFs refer to Bar-Shalom or Gelb [4] [5].
2.1 Sonar
Sonar is the primary tool used by U.S. Navy undersea platforms to detect and
track contacts in their operating environments. The two types of sonar used are active
and passive. Active sonars operate by generating a directed sound pulse and measuring
the travel time of the reflected return off objects in the environment. The distance to the
object, range, can be calculated using knowledge of the speed of sound and the pulse
duration. The direction of the object relative to the sensor, bearing, can be determined by
knowing where the sound pulse was directed. Inaccuracies can occur in these
measurements due to factors such as angular uncertainty created by a finite beam width
or a pulse reflecting off multiple surfaces. In contrast to active sonar, passive sonar does
not generate sound pulses. It relies on being able to detect acoustic signals created by
other objects. Thus passive sonar is only able to produce bearing measurements of
surrounding contacts [6].
29
2.2 EKF Contact Tracking With Active Sonar
An EKF is a minimum mean squared-error recursive filter. The filter uses
linearized mathematical models for both the measurement process and target state
dynamics to maintain an estimated state vector and a covariance matrix that represents
the uncertainties of the estimates and the correlations of those uncertainties. An EKF
attempts to minimize the mean squared estimation error in the target's state [7].
This section will discuss how an Extended Kalman Filter can be used with active
sonar to track a contact. The contact motion model, observation model, and EKF steps
using these models will be reviewed. Work in this section and the remainder of this
thesis will be restricted to two-dimensional motion for convenience.
2.2.1 Contact Motion Model
The state estimate for a contact in this implementation can be represented by
x c[k]
X [k] =
K
,]
(2.1)
Oc [k]
_vc [k]-
storing the Cartesian north and east coordinates in a locally flat earth reference frame as
well as the heading and speed of the contact. There is a command input, uc, at time k,
with time At between updates, such that
S[k]
u [k]=
c
30
C
YC [k]
,
(2.2)
where
#
[k],
is the commanded heading and y C[k] is the commanded speed.
A discrete time dynamic model describes the state transition of the contact from
time k to time k+At. It can be defined as
X c[k + At] = f(X c[k],u, [k], At) + w, [k] .
(2.3)
The function f represents a nonlinear system that takes as input the contact state, Xc[k],
and the command input, uc[k], at time k and produces the contact state at time k+Lt,
Xc[k+Dt], if there was no noise in the contact's motion. This propagation process can be
written as
[k] + (At xy c [k] x cos((p [k]))~
[yc[k]+(Atxyc[k]xsin(<pc[k]))
J
f(Xc [k],uc[k], At)=
(pc[k]
yjck]
X
(2.4)
Noisy components of the contact's motion are included in the random vector, we, which
is a 4 X 1 vector. This vector is assumed to consist of uncorrelated zero mean Gaussian
white noise [6] [7] with covariance
xw
0
0
0
0
0
0
0
0
P
0
Q = E[ww T]=[
31
0
0
o
7w
(2.5)
2.2.2 Observation Model
An observation model is used to describe measurement of the contact's position
relative to the UUV. In this example using active sonar, the measurements are the range
and bearing of the contact relative to the UUV. At time k a measurement of the contact
produces
z[k] =
[r[k]J
,
k
(2.6)
P[k]_
which consists of the relative range and bearing respectively. A model of the observation
function is given by
z[k]=h(Xv[k],Xc[k])+wo,
(2.7)
where Xc[k] is the state of the contact at time k and
xv[k]
Xv [k] =
[k] ,
Ov[k]
Vv,[k]_
is the state of the UUV at time k. The function h gives the range and bearing
32
(2.8)
Illustration of Observation Model
Contact
Position
5000
-
,x-
4000
3000
2000
Range
Relative Bearing
1000
0
UUV
Heading
-1000
-10 00
UUV
Position
I
I
II
2000
3000
4000
II
0
1000
5000
6000
X-Axis (Meters)
Figure 2.1 Illustration of Observation Model
measurements if no noise is in the observation and can be written as
(x,[k] - xc [k]) 2
h(X v[k], Xc [k])=0
k -aca
+(y-
yc[k])2
y c[k]- yj[k]
.
(2.9)
(X c[k] -y, [k]
Noisy components of the measurements are included in the random vector, wo, which is a
2 X 1 vector. This vector is assumed to consist of uncorrelated zero mean Gaussian
white noise with covariance
R =
'
0
0
19W
33
.
(2.10)
R will be referred to as the observation error covariance matrix where
TW
and Q, are
the variances of the range and bearing measurements respectively. These values can be
chosen properly with knowledge of how the modeled sensors perform in real world
environments [6] [7].
2.2.3
EKF Algorithm
Using the contact motion and observation models presented in the sections 2.2.1
and 2.2.2, an EKF can maintain a single state vector representing the estimates of the
contact's position, course, and speed. An associated covariance matrix, which contains
the uncertainties and correlations of the estimated states, is also maintained. This section
will review the EKF algorithm steps in tracking a single contact with active sonar.
We can assume that at time k there is an initial estimate of the contact's state,
A
Xc[k],
and a covariance matrix P[k]. The "hat" denotes that a term is an estimate. The
state estimate is of the form
A
xc
[k]
A
Xc [k]-
.
(2.11)
Oc[k]
A
vc[k]
The estimated covariance matrix can be expanded as
2
GA
Xc Xc
2
A
P[k]
CTAA
Yc
xc
2
GYAA
2
GA A
VCx
2
GA A
XC yc
2
GA
A
YC
2
GA
2
GA
34
2
A
Xc Oc
2
GFA
A
GA
A
GA
GA A
XC V
2
GA
A
2 YY
2
A
vXy;
YC
VCO
2
CFA
A
GA
A
GA
2
A
2
V C
A
VC v
.C
(2.12)
The terms on the main diagonal of the covariance matrix represent the filter's estimate of
the variance of the error in each state variable estimate. Each off-diagonal element is the
covariance of the respective state variable estimation errors. Maintaining estimates of the
covariance values is important for two reasons. First, information gained on one state
variable can improve estimates of other variables. Also, the covariance estimates keep
the state estimates from becoming overconfident. If all the assumptions and models used
in the EKF are valid, it is guaranteed that the estimates of the state estimate errors are
consistent with the actual errors [6]. At time k+At, a new estimate of the current state and
covariance matrix can be formed using the measurement produced at that time and the
previous state estimate and covariance matrix produced at time k. The steps involved in
forming these new estimates are described next.
The first step in updating the state estimate at time k+At is to predict what the
contact state will be based on the previous estimate. In this review, the new state is
predicted assuming the contact will continue on the previously estimated course and
speed. The contact state can be predicted according to the equation
A
A
A
Xc [k + At] = f(X c[k],uc[k], At),
(2.13)
A
where the f is the equation referred to in Equation 2.4 and Uc[k ] is the estimated
contact command input that can be expanded as
A
uc[k]
_
A
c [0k]l
]
(2.14)
_vc [k]J
The superscript
"-"
in Equation 2.13 and later equations will serve as a reminder that the
term is the best estimate prior to a measurement being taken.
A predicted covariance matrix, P-[k+At] is also produced according to the form
35
A
P- k±+At] = Fc[k] P[k](F [k] T )+Q
(2.15)
F[k] is the Jacobian of the discrete time dynamic model function, f (Equation 2.4), with
respect to the estimated contact state at time k, and is defined as
F
Fc [k]=
a Xc
XIk]
A
0
- At x vC[k]xsin(c[k])
= 0
1
At x sin(Oc [k])
0
0
At x vC [k] x cos(Oc [k])
1
0
1
A
[0 0
Q is the contact
A
A
l
At x cos(Oc [k])
A
A
.(2.16)
0
motion noise covariance matrix defined in Equation 2.5.
The last prediction made by the filter is a predicted measurement of the contact's
relative range and bearing with respect to the UUV, z-[k+At]. Equation 2.9 can be used
to predict the measurement according to the form
z~[k + At] = h(Xc [k + At], X,[k + At]) .
(2.17)
Once these predictions are made, the filter can use the actual measurement
produced, z[k+At], according to Equation 2.7 to update the estimated contact state and
covariance matrix. This process starts by computing the measurement residual, r[k+At],
which is the difference between the predicted and actual measurements, according to the
equation
r[k+At]=z[k+At]-z-[k+At]
36
.
(2.18)
The residual covariance, S[k+At], is then found according to the equation
S[k + At] = H[k + At]P-[k + At](H[k + At] T ) + R ,
(2.19)
where R is the observation error covariance matrix defined in Equation 2.10. H[k+At] is
the Jacobian of the observation model function, h (Equation 2.9), with respect to the
A
predicted contact state, Xc [k + At]. It can be computed by the equation
A
A
yc-yV
H[k+At]=
X
aXC I
A
[k+At]=
2X+
(
0 0
. (2.20)
- y,)2
Ac-yv
0 0
Xv)
-XC
2
+(y
-y)
2
(XC
_X)
2
+(y'
-y,)
2
Next the Kalman gain, K[k+At], for the update of the estimated contact state is
defined by
K[k + At] = P- [k + At](H[k + At]T )S- [k + At].
(2. 21)
When updating the contact state estimate and covariance matrix, this Kalman gain will
minimize the mean squared estimation error [7]. Using the Kalman Gain and the
residual, the updated contact state estimate is determined to be
A
A
Xc [k +At] = Xc[k +At] +K[k +At]r[k +At].
37
(2. 22)
The covariance matrix is updated using the Joseph form covariance update to maintain
the symmetric nature of P. It is defined as
P[k + At]= C[k+ At]P[k + At](C[k+ Atf)+K[k+ At]R(K[k+ Atf),
(2.23)
where C[k+At] is
C[k + At] = I - K[k + At]H[k + At] ,
(2.24)
and I is the identity matrix.
2.2.4
EKF Example With Active Sonar
Figures 2.2, 2.3, 2.4, 2.5, and 2.6 show the computer-simulated results of a UUV
using the EKF algorithm discussed in Section 2.2.3 with active sonar measurements to
track a single contact. It was assumed that the state of the UUV was determined by some
shipboard inertial navigation system during the 2500-second simulation. Gaussian noise
was added to the bearing and range measurements with variances of (.034 radians)2 and
(25 meters) 2 respectively. Noise was introduced to the contact's motion by adding
Gaussian noise to the contact's heading and speed with variances of (.017 radians)2 and
(.4 meters/second) 2 respectively. These values were chosen because they seemed to be
reasonable estimates of the noise values. If more was known of the performance of the
contact or UUV's sensors, then these values could modeled to be more consistent with
their actual values. The initial x and y coordinates of the UUV and contact were (0
meters, 0 meters) and (10,000 meters, 5,000 meters) respectively. The filter was given an
accurate initial estimate of the contact's state.
Figure 2.2 shows that the EKF provided an accurate method to estimate the
position of the contact. There was some error when the contact maneuvered, but the
38
estimate quickly converged back to the true state after a maneuver. The errors in the
estimates when the contact maneuvered occur because the filter's model of the contact
motion expected the contact to follow a straight trajectory.
Figures 2.3 - 2.6 show the
errors in the individual contact state variable estimates as well as the 3 sigma or 99%
confidence bound for the errors provided by the covariance matrix. The estimation errors
were within the bounds except for immediately after maneuvers by the contact occurred.
These errors also quickly returned to within the bounds, showing that the covariance
matrix produced by the filter provided good estimates of the accuracy of the contact state
estimates.
True Position of Contact, Estimated Position of Contact, and Position of UUV With Active Sonar
-
4
8000
16000-
*
True Contact
Estimated Contact
UUV
1400012000-
U)
1000080006000 4000 [
20000
-2000---1.5
-1
-0.5
0
X-Axis (Meters)
0.5
1
-I
1.5
x 104
Figure 2.2 Example UUV, True Contact, and Estimated Contact Positions With Active Sonar
39
Plot of 3 Sigma Confidence Bound and x Position Error with Active Sonar
40C
--
---
-
------
----
- -r--
--
- - ---
30C
-------
--
----
-r-
3 Sigma bound
-3 Sigma bound
x Position Error
200
10C
-10
C
-20C
-30C
-40C
0
500
1000
1500
Time (Seconds)
2000
2500
Figure 2.3 Example X-Position Error and 3 Sigma Confidence Bound
Plot
600
of 3 Sigma Confidence
Bound
and y Position
Error
with Active
Sonar
400
200
0
-200
-400
Sia
Bound
-3 Sigma Bound
y P-osition Error
-600
-800
-3
500
0
100
15
2000
Time (Seconds)
2500
Figure 2.4 Example Y-Position Error and 3 Sigma Confidence Bound
-
300
Plot
of 3 Sigma Confidence
Bound
and Heading
Error
Sonar
with Active
-3
-
-
-Sigma Boundt
-3 :SIgma B~ound
eading E-rror
200
100
0
(
-100
-200
-
-300
--
-400
0
-----
500
1000
1500
2000
T-ime (Seconds)
Figure 2.5 Example Heading Error and 3 Sigma Confidence Bound
40
25
00
lot of 3 Sigma Confidence Bound and Speed Error with Active Sonar
10Speed
3nSgm
Bound
3Sima Bo~undJ
Error
10
5
0
500
1000
1500
Time (Seconds)
2000
2500
Figure 2.6 Example Speed Error and 3 Sigma Confidence Bound
2.3 EKF Contact Tracking with Passive Sonar
This section will discuss the difficulties in using passive sonar to track a contact
using Cartesian coordinates. It will also discuss a different approach, using modified
polar coordinates, that provides improved results when compared to Cartesian
coordinates.
2.3.1 EKF Contact Tracking with Passive Sonar and Cartesian
Coordinates
Figures 2.7, 2.8, 2.9, 2.10, and 2.11 show the computer-simulated results of a
UUV using the EKF algorithm discussed in Section 2.2.3, without range measurements,
and passive sonar to track a single contact in Cartesian coordinates. The conditions of the
simulation were the same as the conditions used in Section 2.2.4 with active sonar. The
filter was initially successful in tracking the contact until the contact's first maneuver.
Once the contact maneuvered at 500 seconds, the estimate of the contact's state quickly
diverged. The filter completely failed in providing an accurate estimate of the contact's
state.
41
True Position of Contact, Estimated Position of Contact, and Position of UUV With Passive Sonar
III
x10 4
2 SI
1.5
I
f
(
1
0.5
x
N
N'
0
N'
N
N
N'
N
-0.5
-1
-
True Contact
Estimated Contact
UUV
-
+
-1.5
-
-2
0
-1
1
X-Axis (Meters)
2
3
4
x 104
Figure 2.7 Example UUV, True Contact, and Estimated Contact Positions With Passive Sonar and
Cartesian Coordinates
S1to'
-
Rot
-
-----
of
3 Sigma
Confidence
Bound and X
3 Sigma bound-3 Sigma bound
X PFosition
Error
PcosItion
Error with
Sonar
Passiv'e
- -
I
0
-2
-3
0
500
1000
1oo
Tirne (Seconds)
2000
Figure 2.8 Example X-Position Error and 3 Sigma Confidence Bound
42
2500
-to
1.5
PIot of 3 Sigma
-
-
Confidence
Eound and
Y
Position
with
Error
Sonar
Passive
3 Sigma Bound
-3 Sigma Bound
Y Position Error
I
0.5
0
-0.5
-1
-1.5
0
2000
1500
1000
Tirne (Seconds)
500
2500
Figure 2.9 Example Y-Position Error and 3 Sigma Confidence Bound
Plot
3
of
Sigrna
Confidence
Bound and
Heading Error with
Sanar
Passive
-300-3
--
1
-L
-400
0
S5gma -- und
3 Sigma Bound
-ieading Error
I
-500
2000
1000
1500
Time&(Seccnds)
2500
Figure 2.10 Example Heading Error and 3 Sigma Confidence Bound
Plot
40
3
--
30
-3
of
3 Sigma Confidence Bound
and
Speed
Error
with
Passive
Sonar
SIgma Bound
Sigma Bound
Speed
Error-
20 -
10
0
CX.
V
:39
-10
-20
-30
-40
0
500
1500
1000
Time (Seconds)
2000
Figure 2.11 Example Speed Error and 3 Sigma Confidence Bound
43
2500
This failure by the Cartesian-coordinate bearings-only EKF can easily be
explained. The x-position and y-position components of the contact's state vector are not
observable when only bearing measurements are provided. One bearing observation can
be produced by an infinite number of contact locations. As illustrated in Figure 2.12,
several different contact trajectories can produce the same series of bearing
measurements. This makes it very difficult for the bearings-only EKF to converge on the
correct state estimate.
4500
Bearing 1
Bearing 2
Bearing 3
4000I
3500/
3000Trajectory 3
2500 2000 -
Trajectory 2
1500
I
1000
/1
Trajectory 1
500-
0
-4000
Sensor
-3000
-2000
-1000
0
1000
2000
Figure 2.12 Difficulties In Bearings-Only EKF
44
3000
4000
2.3.2 EKF Contact Tracking with Passive Sonar and Modified Polar
Coordinates
Using modified polar coordinates for the state estimate in an EKF with passive
sonar provides more observability. A modified polar coordinate contact state vector,
B[k]
Xmp[k]=
(2.25)
k
B[k]
consists of bearing rate, range rate divided by range, bearing, and the reciprocal of range.
All of these components, except for the reciprocal of range, are observable in bearing
measurements. The reciprocal of range is observable if the UUV maneuvers while the
contact maintains course and speed. Whenever the contact maneuvers, the UUV must
also maneuver to make the reciprocal of range observable again [8].
The modified polar coordinate state of the contact can be estimated using an EKF
similar to the one discussed in Section 2.2.3. Given the UUV state, defined by Equation
A
2.8, and the MPC state estimate of the Contact, Xmp[k]
,
the Cartesian coordinates of
the contact state can be determined according to the equation
sin(B[k])
A
[k]
r[k]
Ay,
X [k]=
Oc
-k]
Oc [k]
[k]
_
v[k]
[k]
cos(B[k])
(2.26)
r[k]
__k]
tan'
contyspeed
t contxspeed
-(contxspeed)2 + (contyspeed)2
45
where
r[k] xsin(B[k]) + B[klxcos(B[k])
r[k]
contxspeed=
+(v,[kxcos(,[kl))
r[k]
and
{r[k]
r[k] xcos(B[k]) -kB([kkxsin(B[k])
y
contyspeed
)(2.28)
) +(v,[klxsin(O,[k]))
r[k]
Figures 2.13 through 2.21 show the computer-simulated results of a UUV using
the EKF algorithm discussed in Section 2.2.3, without range measurements, and passive
sonar to track a single contact in modified polar coordinates. The conditions of the
simulation were the same as the conditions used in Section 2.3.1, except that the UUV
zigzagged to keep all four components of the MPC contact state vector observable. The
MPC component of the contact state estimates were transformed to Cartesian coordinates
according to Equation 2.26 with the estimated trajectory shown on Figure 2.13. Figures
2.14 through 2.17 show the 3 sigma bounds and estimate errors in modified polar
coordinates, while Figures 2.18 through 2.21 show the state estimate errors in Cartesian
coordinates. These results show an improvement in performance compared to using
Cartesian coordinates in an EKF with passive sonar, but still leave much to be desired in
state estimate accuracy.
46
True Position of Contact, Estimated Position of Contact, and Position of UUV With Passive Sonar
18000
I
True Contact
- -
16000
-UUV
Estimated Contact
14000
12000-
-...
2 10000
8000
-
6000
-
-
-..
-.
4000
. ...
2000
0
-2000
0
4000
2000
6000
8000
10000
12000
16000
14000
18000
X-Axis (Meters)
Figure 2.13 Example UUV, True Contact, and Estimated Contact Positions With Passive Sonar and
Modified Polar Coordinates
<-t
o
Plot<
of 3 Sigma
Confidence
Bound and Bearing Rate Error With Fassve
0.5
...
Sonar
..
0
-1
S-1.5
-2-
-2-5
-3
0
-.
-3 Sigma Bound
-3
Bearing Rate Erro-r
Sigma Bound
500
-
100
1500
2000
Tirne (Seconds
Figure 2.14 Example Bearing Rate Error and 3 Sigma Confidence Bound
47
2500
-f 0-
8
--
-
of 3 Sigrma
Sigma Elnund
FIct
-
-
Range
E-rrr
Rete/Renge
with
Sonmr
Passive
Sigme Bound
Range
----
Bound and
Confidence
3
-3
Rate/Rang
Errcr
6
0
- ---
-- - ---
-
--
-2
500
0
1000
1500
Time (Seconds
2000
2500
Figure 2.15 Example Range Rate/Range Error and 3 Sigma Confidence Bound
P
2,
Iot
of
3
Sigma
Bound
Confidence
end Bearing
Er-or
with
Sonar
Pessive
3
- - --
1.5
-
- --
-
-
--
- ..
-----
Sigma
Bound
BeeSigma Bound
---
ring Brror
0
-0.5
-2
-2.5
500
1500
1000
Time
2000
25OO
(Seconds
Figure 2.16 Example Bearing Error and 3 Sigma Confidence Bound
of 3 Sigrna
Pliot
--
3
0.08
tConlldence end
1/Range
E~ror-
Sigme Bound
3 Sigma Bound
1/Range Error
-
0.06
0.04
-
0.02
-
- --
0
--
-
-0.02
-0.04
...
-0.06
-0.08
-0.
0
-
500
1000
1500
Time (Seconds
2000
Figure 2.17 Example 1/Range Error and 3 Sigma Confidence Bound
48
2500
X(
0O-
-
- - - -
PF-
tM- E=-rror
-
-10000
-12000
-
500
1
005
O O
-2000 -
2
2000
2500
2000
2500
SOO
Figure 2.18 Example X-Position Error
Y
1 000
-
E-rror
Posiv.ticn
-
------
0
-1000
-2000
-3000
-4000
1000
-50001
5
0
0
1500
Figure 2.19 Example Y-Position Error
-
300
HeadIR.n
Error
200
1
00
- - - --- -- - -
-
-
-
-1 -0 0 -
- --
-
-
- 1-- 0
-0
-100
-200
-300
-400
0
50o
Tirn,
(Seconds
Figure 2.20 Example Heading Error
49
Sr eci
1=rr~,r
5-
0
-5
0o
5o
1000
Trime
Se5md
1500
2000
2500
Figure 2.21 Example Speed Error
In conclusion, an EKF can provide a UUV with the capability to estimate the state
of a contact and the uncertainty of those estimates. This information can then be used by
a UUV to analyze the motion of a contact to determine if counter-detection has occurred.
The EKF algorithms discussed in this chapter will be used to provide contact state
estimates for the counter-detection belief state (presented in Chapter 3) and maneuver
decision aids developed later in this thesis.
50
Chapter 3
Counter-Detection Recognition
Many possible Maritime Reconnaissance missions for Navy UUVs will require
them to avoid counter-detection in regions to which an enemy wants to deny them access,
as discussed in Chapter 1. Counter-detection of the UUV by a potentially dangerous
contact may cause the mission to fail if the UUV is required to abort the mission upon
detection or if the contact engages and captures or destroys the UUV. Thus it is
imperative that a UUV is able to accurately determine if counter-detection has occurred
and have an estimate of the confidence in that determination. This chapter will present an
approach that can be used by a maneuver decision aid in tandem with tracking algorithms
like the ones discussed in Chapter 2 to determine if a UUV has been counter-detected by
any potentially dangerous contacts. The work in this chapter will be restricted to constant
speed motion for the contact and UUV.
3.1 Defining Behaviors
Before one can develop algorithms to recognize counter-detection, one must
know what kind of behaviors to look for in contacts that have counter-detected a UUV
(also referred to in this thesis as hostile contacts) and ones that have not been alerted to
the presence of a UUV (passive contact) in their operating environment. Probabilistic
models of hostile and passive contacts can represent these behaviors to determine the
likelihood that counter-detection has occurred.
In the work of this thesis a hostile contact will be viewed as a contact that is
actively tracking the UV and thus presents a threat to the UUV. There are three key
behaviors that a hostile contact can display. They are:
51
-Contact continuously changes course towards UUV in response to UUV's course
changes.
-Contact remains in UUV's baffles (Area directly behind the UUV).
-Contact suddenly changes course towards the UUV after detecting the UUV.
The first two are typical of a contact that has been alerted to a UUV's presence for
some time and is pursuing it. The trajectories of the contact and UUV shown in Figure
3.1 illustrate these two behaviors.
Position of Contact and UUV
9000
- --
8000
Contact
UUV
7000
6000
5000
-N
.0I
4000
(I)
3000
-
UU-trin
on
2000
1000
Contact Starting Point
--
0
00--
100
A ^^^
-8000
-6000
-4000
-2000
0
2000
X-Axis (Meters)
4000
6000
8000
10000
Figure 3.1 Example #1 of Hostile Behavior
In this example the contact turns to in response to the UUV maneuvers so that it can
continue to follow the UUV. Also the contact remains behind the UUV, revealing the
second behavior listed. The behavior illustrated in this example clearly shows that
counter-detection has occurred.
52
The third behavior is typical of a contact that has recently been alerted to a
UUV's presence and has changed course to pursue the UUV. This type of behavior is
illustrated in Figure 3.2. In this example the contact is moving away from the UUV until
it suddenly turns toward the
I-
12000~
I
I
Position of Contact and UUV
I
I
Contact
UUV
/
10000-
7
#7
8000-
An
6000-
4000-
2000-
a
0
Contact Starting F oint
UUV Starting Point
11R)tj -
I
-2.5
I
-2
i
-1.5
-
S
i
-1
-0.5
0
0.5
X-Axis (Meters)
1
2
1.5
x 10,
Figure 3.2 Example #2 of Hostile Behavior
UUV as if it has recently determined there is a suspicious presence in the area.
On the other hand a passive contact in the context of this thesis is a contact that is
not actively tracking the UUV and thus does not present a threat to the UUV. A passive
contact is not likely to make sudden maneuvers towards the UUV or try to follow the
vehicle. It will tend to keep a constant course for extended periods of time, with some
random course changes. These course changes will not be affected by the location of the
UUV, since the contact is unaware of its presence. This type of behavior is illustrated in
Figure 3.3. In this example there is no behavior that indicates that the contact has been
53
Position of Contact and UUV
20000
Contact
UUV
--15000 -
10000-
5000 -
0/
Contact Starting Point
UUV Starting Point
-5000
-1.5
-1
1
1
-0.5
0
1
0.5
X-Axis (Meters)
1I
1
1.5
2
2.5
x 10
Figure 3.3 Example of Passive Behavior
alerted to the UUV's presence. The contact makes several course changes, but never
moves towards the UUV.
3.2 Counter-Detection Belief State Update
The intentions of a contact are not completely observable to the UUV. The UUV
can only make observations of the contact's physical behavior and attempt to estimate the
likelihood that counter-detection has occurred. Using its understanding of how both
hostile and passive contacts behave, the UUV can try to infer the contact's intent. Also,
the UUV needs some method to summarize its counter-detection observations of the
contact that will allow it to define and periodically update the UUV's uncertainty of
whether counter-detection has occurred. Maintaining a counter-detection belief state can
accomplish this.
54
The counter-detection belief state is the UUV's estimate of the probability that a
contact has counter-detected it. It is updated periodically after a fixed time frame passes,
given the estimated physical states of the contact at the beginning and end of the time
frame provided by tracking algorithms such as the ones discussed in Chapter 2. The
belief state can be updated, using Bayes' rule [9], as
bbk+wAAAA
(bk
bk XX= sfi, l(X (k + w)I X (k), X, [k])
Xfhostile (Xc(k + w) IX,(k)))+ ((1-bk )Xfpassive (X: (k + w) IX, (k))
(3.1)
where bk is the belief state at time k and w is the fixed time frame that passes between
updates'. fhostile is the probability density function (PDF for short) of the estimated contact
A
state at time k+w seconds, Xc [k + w] , given the estimated contact state at time k,
Xc[k], the UUV state at time k, X,[k], and assuming the contact is hostile. fpassive is
the probability density function of the estimated contact state at time k+w seconds, given
the estimated contact state at time k and assuming the contact is passive. These
functions will be defined in Sections 3.3 and 3.4. Using Bayes' rule in this application
allows the UUV to infer the contact's counter-detection state, by observing its
movements. If the hostile and passive PDFs are modeled accurately, then this provides
the best method to estimate the likelihood that counter-detection has occurred.
3.3 Hostile PDF Model
The function fhostile provides the joint PDF for a contact's Cartesian coordinates
and heading at time k+wseconds,
x, [k + w] , yc [k + w], and 0[k + w], given the
contact is hostile and the estimates at time k, xc[k], yc[k], and Oc[k]
.
Speed is not
taken into account since we are only dealing with constant speed motion in this thesis.
55
This joint PDF is a nonnegative function that represents the likelihood of the contact's
state at time k+w if the contact is hostile. For a more detailed description of PDFs refer
to [9].
The PDF,
fhostile,
assumes that the contact will make at most one course change
over the two minute interval. It attempts to capture the hostile behaviors discussed in
Section 3.1 by assuming there is a high probability, P(host_man), of maneuvering
towards the UUV. There is also a lower probability, P(randomman), of making some
random maneuver. fhostile is defined according to Equation 3.2 by using the Total
Probability Theorem, to account for the possibility that the contact can maneuver at any
time during the time frame of length w, and conditioning on the event of a hostile or
random maneuver [9].
Xc [k + w]I Xc [k], X,[k])=
fhosile
t=w-1
A
(Xc [k + w] 10 [4k + w], X, [k]) xfA
(fl
Y,
I=--A
+
I
t=O
A
(fl
X[k], X,[k]) x(P(host-man)fw))
(Or k + w] X,
GOdhost-atk+t
x ,y,19,k+t
t=
(3.2)
(Xc[k + w
A
A
A
10,c[k + w], Xc[k])xfA
A
(Oc[k + w] Xc[k], X,[k]) x (P(random-man)/w))
O Irand-at-k+t
e' yc10ck+t
fl A A
xc,ycl|c,k+t
is the PDF of the estimated x and y Cartesian coordinates of the
contact at time k+w seconds, given the contact's estimated state vector at time k, the
contact's estimated heading at time k+w, and assuming the contact changed course at
time k+t.
fA
is the PDF of the contact's estimated heading at time k+w,
0e host-att+k
given that the contact chooses to make a hostile course change at time k+t.
fA
Oc
rand-atk+t
is the PDF of the contact's estimated heading at time k+w, given that the contact
chooses to makes a random course change at time k+t. These distributions are defined in
Sections 3.3.1, 3.3.2, and 3.3.3.
56
3.3.1 Hostile Course Change PDF
The function f,
is the PDF of the estimated contact heading at time
0c1hostat-k+t
k+w, given that the contact maneuvered in an aggressive manner at time k+t. It is a
triangular shaped distribution centered on the course that would point the contact directly
at the UJV at time k, which will also be referred to as the hostile heading or
0 H.
The
distribution is defined such that the contact's heading at time k+w would point the
contact within approximately 3000 meters of the UUV (illustrated in Figure 3.4) or
within an angle with a width of 1.57 radians (90 degrees) centered around 0H,
whichever creates a smaller angular width. This angular distribution width will be
referred to as
0
E.
It can be explicitly defined as
0 E = arctan(3000/Range) if arctan (3000/Rang e) <1.57 Radians (90 Degrees),
1.57 Radians (90 Degrees)
E
otherwise
3000 Meters
,UUV Position
1
/
rb
3000
1
on-
~
3000 Meters
2000
o
Contad Position
.1000
0
1000
Figure 3.4 Illustration of
2000
30
X-Axis (Meters)
400
50,00
6000
OE for Hostile Course Change PDF
57
(33)
The function,, f,
Oc lhost-att +k
s
,
can be defined as
0k[k + w] I X, [k], X,k]
xok + w - OH
=
OcIhost-at-t+k
OH
= 0 radians. This graph shows that
if
0[k + w] - H
OE
(3.4)
otherwise
0
Figure 3.5 shows the shape of f,
j
, with OE = .785 radians (45 degrees) and
OH is
the most likely course for a hostile
contact, with the likelihood of a subsequent course decreasing the further it is from
OH *
Plot of Hostile Course Change PDF
1.2
1
0.8-
0.6 -
0.4 k
0.2
I
-3
-2
-1
0
1
Contact Course at Time k+120 (Radians)
I
2
3
Figure 3.5 Hostile Course Change PDF With 0E = .785 Radians (45 Degrees) and 0 H = 0 Radians
58
3.3.2 Hostile Random Course Change PDF
is the PDF of the estimated contact heading at time
The function f,.
Oc randatk+t
k+w, given that the contact maneuvered in a random manner at time k+t. It accounts for
the possibility of a hostile contact not maneuvering directly towards the UUV over the
fixed time frame.
This PDF can be viewed as a two-tiered uniform distribution,
centered on OH, which is illustrated in Figure 3.6 and defined as
.0796178 if -
(Oc[k
f,
+ w] IXc[k],X,[k])=
3x.0796178 if
Oc[k +1205 (
+
.79rand6a78k+f
.0796178 if
Defining f,
n +OH
(2)+~OH
2 )+OH
OH
+ 12O]
:
fk20]:5
)+ OH
(3.5)
Oc[k +120]:5 n+OH
in this manner means that if a hostile contact decides to make a
Oc rand_at_k+t
maneuver that is not directly pointed at the UUV, it is three times more likely to move
towards the general vicinity of the UUV than away from it during the time interval.
07S
0.2
0.15
W
0.1
0.051-
-3
-2
-1
0
1
2
3
Estimated Contact Heading at Time k+120 (Radians)
Figure 3.6 Distribution of Hostile Random Course Change With OH=O Radians
59
3.3.3 Position Distribution
The contact's expected state at time k+w seconds, Xc [k + w], and expected
covariance matrix, P- [k + w], can be determined given the following values:
-The estimated contact state vector at time k, Xc [k].
-The contact state estimate error covariance matrix, P[k].
-The contact changed course at time k+t.
-The course the contact changed to at time k+t, which is assumed to be
A
The following series of equations will illustrate this. First, the contact's expected
state immediately before changing course at time k+t, Xc [k + t] is determined
according to the equation
A
A
xc
[k+t]=f[
O)c [k] ,9t,
VC
(3.6)
[kQ
where f is the discrete time dynamic model function defined in Section 2.2.1 and
Equation 2.4. The expected covariance matrix at time k+t can be calculated as
P [k + t]= Fc [k, tiP[k (Fc [k, tDT).
(3.7)
F [k, t] is the Jacobian of the discrete time dynamic model function and is similar to the
Jacobian discussed in Section 2.2.3 and Equation 2.16. It can be defined as
60
A
0 - txvc[k]xsin (9,[k])
0
1
txvc[k]xcos (9[k])
txsin(Oc[k])
0
0
1
0
0
0
0
1
A
Fc [k,t]
=
A
A
1
txcos
1
(9c[k])
A
A
.
(3.8)
A
XC I":[k]
In other words, Xc
j
+ t] and P- k+ t] are the expected contact state and state
estimate error covariance matrices at time k+t if it maintains its original course and speed
at time k until time k+t. Next, the expected contact state and state estimate error
covariance matrix at time k+w seconds can be determined, given that the contact changed
A
course to 0, [k + w] at time k+t seconds. The contact's expected state at time k+w
A
seconds , X~Crk + w], can be calculated according to the equation
A
XC [k + w=f X
[k + tj
c[k
+w],w-t).
(3.9)
vc [k]
In equation 3.9, the (w-t) term is used because that is the amount of time between when
the contact changes course at time k+t and reaches its end state in the time frame at time
k+w. The expected covariance matrix at time k+w can be calculated as
P-[k + w]= Fc [(k + w)(w - t)P-[k + tXFj [(k + w)(w - t)DI,
where F [(k + w),(w -t)
is defined as
61
(3.10)
F[(k + w) (w -(3.11)
X)
1 0 (W
'""
A
A
1 0 -(w -t)x vc[k+ w]xsin(Oc[k + w])
0
1
(w-t)xvc[k+w]xcos(Oc[k+w])
0 0
0 0
X [k + wi and P- [k +w
1
0
A
(w - t)xcos(Oc[k + wi)
(w-t)xsin(c[k+w])
0
1
can be described as the expected contact state and state
A
estimate error covariance matrix at time k+w if it maintains course Oc [k + w] and speed
vc [k + w] from time k+t until time k+w.
Figures 3.7 through 3.11 display the expected contact state and 3 sigma or 99%
confidence region for a contact's expected x and y Cartesian coordinates provided by the
expected state estimate error covariance matrix at time k+w given the initial state
estimate error covariance matrix listed below each figure, the time frame, w, is 120
seconds, and that the estimated contact motion follows the shown trajectory. It is worth
noting how various changes in the initial state estimate error covariance matrix at time k
produce corresponding changes in the expected covariance matrix at time k+w. In each
of these figures the initial estimated contact position is at the origin.
62
3-Sigm Confidence Region For
ad Y
0obebwee X
3-St"m.
Posiio
1800
Regio
Confdence
For
X and
%Asetaierd
Y
Po004
1800
Sigm.
E- eed
C-nfidence Regon
FExpected
Y Comoine,
ont
Exected Cot
Positon0A Time k-120
--
Fo
1400
X nd
ExpecedContac
Position ltime
Eopected -0000
Confidence Regon
For ExpectedX and
1600
W120
y oriee,
200
000
K-
1 000
Soo
1
400
400
Tmjedoy
Estimated Contact
PosiionAt imeek
200
I
Estimated Contac
Position Atime
___Estaled
TCectory
Colted
0
-400
-00
0
200
600
1
21000
-400
-20
0
200
Confdence Region For
o
0
1200
1000
Figure 3.8 Expected Contact
X and Y Coordinates 3 Sigma
Confidence Region With
0
0
0
5*4000
0
0
4000
0
2
P[k]=
x
0 ~1xj
0
00
180f
L0
0
0
(4y
Figure 3.7 Expected Contact
X and Y Coordinates 3 Sigma
Confidence Region With
4000
0
0
0
0
4000
0
0
P[k]= 0
0
x0
0
x1
180
0
0
0
)(.4y_
3-Sigmia
600
400
X-AW9 MeWeS)
0 Axis (Meters)
Subsequed
X .nd
3-Sigma.
Y Position
Confidence
Regon For
Subsequen8
X
and
Y
Positon
1800
1800
Epeded
X SgmaC
CoefKW
egion
Foe Expeedwd 4
1600
Y
Expectooed1 Contact
PosOion At00000.k+0
Expected 3-sigee
Confidene Regi00
For Expected
1ODO
X
and
Expected Contact
Poton0At Timek.120
Y CoelrineO
Coorinales
1400
1400
1200
600
800
siae otc
-
80
Estmated Contact
Estimated
0100400(0
E80III4ed Contact
Positon At raw k
Taetr
Position At Timeek
200
-00
-400
-200
0
2
40
X-Axis (Meters)
00
100
-600
1200
0
0
-200
200
400
0-088*8 (Meters)
0
000
B00
1200
1000
Figure 3.10 Expected Contact
X and Y Coordinates 3 Sigma
Confidence Region With
~4000
0
0
0
0 4000
0
0
Figure 3.9 Expected Contact
X and Y Coordinates 3 Sigma
Confidence Region With
4000
0
0
0
0
5x4000
0
0
0
0
1X--0
0
-400
P~J=
0
_0
(.4y_
63
0
0
5x 1x-E)
0 180f
0
0
(.4y_
Cotact
3-Sig
Confidne
Exp.C00 3-0gm
Co
1600
Regin
Fo
Subseqt
-fdme
-
and Y Posin
p
For EO~oOed X 00
I
dCm
E-Ve,0d CoO.
0rn0
14OCO
X
P
TOme
k120
1200/
z 1000
y
400
P
200
600
400
T..jedoy
1s., 0A Tho,6
.2000
200
40
120
)
X-A0(Mrs
Figure 3.11 Expected Contact
X and Y Coordinates 3 Sigma
Confidence Region With
4000 0
0
0
0 4000
0
0
P[k]= 0
0
x0
X'
0
0
0
5x(.4y
The contact's estimated x and y Cartesian coordinates at time k+w can be
assumed to have a bivariate normal (or Gaussian) probability distribution function, given
the contact's estimated state at time k, the contact changed course at time k+t, and the
contact's estimated heading at time k+w. The expected contact state and state estimate
error covariance matrix derived above are also used to define this distribution. For more
information on bivariate normal PDFs refer to [10]. This distribution of the contact's
estimated x and y Cartesian coordinates at time k+w is defined as
f,
A A
xe,yj0_,k+t
(X0 [k + w0c[k+ wXc[k])
W]
I
(X
64
x
FG
p
-x(r)
_r
2
xG
(3 12)
where
G
=
ft
2
2
2
Xc
(3.13)
Y
contains the expected state estimate error covariance values of the contact's x and y
coordinates provided by the expected covariance matrix P-[k + w],
[
= [k + w]
(3.14)
y [k + w]i
is a 2 x 1 vector containing the x and y coordinates of the expected contact state at time
k+w, and
r = xc [k + w]
(3.15)
ye [k + wi
is the 2 x 1 vector containing the x and y coordinates of the estimated contact state at time
k+w.
65
3.3.4 Example Plot of Hostile PDF
Figure 3.12 represents the likely resulting Cartesian coordinates of a hostile
contact at time k+w if w = 120 seconds and the contact's state at time k is
Xc[k]=[
0 Meters
0 Meters
pi/2 Radians
12 Meters/Second_
and the UUV's state at time k is
X,[k]=
8000 Meters
0 Meters
0 Radians
10 Meters/Second_
66
Likelihood of Resulting Cartesian Coordinates for Hostile Contact
lr~itial Contact.
Position (0,0)
and Initial Contact
Course-of.90 Degrees
(Measured CounterClockwise From
Positi ieX-Axis)
6
x 10
1.5-
0.5
0 >2000
-2000
-1000
0
0
Initial UUV
Position
(8000,0)
0
10002000
-2000
Y-Axis (Meters)
X-Axis (Meters)
Figure 3.12 Plot of Likelihood of Resulting Cartesian Coordinates of Hostile Contact
The figure gives some idea of how the hostile pdf, fhostile, developed earlier in this
section is structured. The plot illustrates that the most likely resulting Cartesian
coordinates of a hostile contact in this case, represented by the peak in the graph, are the
ones where the contact turns toward the UUV during the two minute interval.
3.4 Passive PDF Model
The function fpassive provides the joint PDF for a contact's Cartesian coordinates
and heading at time k+w seconds, x, [k + w], y, [k + w], and O [k + w], given the
A
A
A
contact is passive and the estimates at time k, x,[Ik] , y~ k], and Oc [ki. The passive
67
PDF model is built in a similar manner to the hostile PDF model, except that it attempts
to capture the passive behaviors described in Section 3.1. The reason for the similarities
is so that the hostile and passive pdf values determined after each time interval of length
w can easily be compared using Bayes' Rule, as in Equation 3.1, to determine what sort
of behavior the contact is following. Like in the hostile PDF model, speed is not taken
into account since we are only dealing with constant speed motion in this thesis. This
joint PDF is a nonnegative function that represents the likelihood of the contact's
estimated state at time k+w if the contact is passive.
The PDF,
assumes that the contact will make at most one course change
fpassive,
over the time interval. It attempts to model the passive behaviors discussed in Section
3.1 by assuming there is a high probability, P(noqman), of the contact making no
significant course change over the time interval. There is also a relatively lower
probability, P(maneuver), of making some random maneuver.
fpassive
is defined
according to Equation 3.16 by using the Total Probability Theorem, to account for the
possibility that the contact can maneuver at any time during the fixed time frame, and
conditioning on the events of the contact making no maneuver or a random maneuver [9].
A
f passive
t=--
A
(fA,
Y
t=O
+
fl
A
(Xc[k + w]I Xc[k],X,[k])=
A
A
A
A
(Xc[k + w] I0c[k + w], Xc[k])xf,
(AXJk+ w] 0 [k + w],X[k])xfA
fA
A
(Oc[k + w] IXc[k], X,[k])x (P(maneuver)/w))
01jman-atk+t
x yJ0,.k+t
xe,10,,k
(3.16)
,
0 Ino man
0[k + w
1
c[k],X,[k]
x P(noman)
defined in Section 3.3.3, is the PDF of the estimated x and y
Cartesian coordinates of the contact at time k+w seconds, given the contact's estimated
state vector at time k, the contact changes course at time k+t, and the contact's estimated
heading at time k+w. f,
e, man
at t+k
is the PDF of the contact's estimated heading at
time k+w, given that the contact chooses to makes a significant random course change at
68
time k+t. f
OcInofan
is the PDF of the contact's estimated heading at time k+w, given that
the contact chooses not to make a significant course change during the time frame, w.
The latter two distributions are defined in Sections 3.4.1 and 3.4.2.
3.4.1 Passive Random Course Change PDF
The function f,.
Oc Iman-at-k+t
is the PDF of the estimated contact heading at time
k+w, given that the contact maneuvered in a random manner at time k+t. Like the hostile
random course change PDF defined in Section 3.3.2, this PDF can be viewed as a twoA
tiered uniform distribution, centered on the estimated contact heading at time k, Oc[k],
which is illustrated in Figure 3.13 and defined as
.0796178
frAn
O,[k +
if
-t +Oe[k] Oe[k + w
W11 Xe[k],X,[k])= 3x .0796178 if -Y
.0796178 if
Defining f,
+ 0 [k
+
0
[k + w1:
0e [k,5 0,[k
0-%+&II'k
(3.17)
+ 0.[k,
+ WJs I + .[k)
in this manner means that if a passive contact decides to make a
Oe(man-at-k+t
maneuver, it is three times more likely to move in the same direction as before than turn
completely around during the time frame of length w.
69
0.25-
0.2
-
0.15
CO
0
Q_
0.1
0.05 -
0
-3
2
1
0
-1
-2
3
Estimated Contact Heading at Time k+120 (Radians)
A
Figure 3.13 Distribution of Passive Random Course Change With oC
[k]=o Radians
3.4.2 Non-Significant Course Change PDF
The function fA
0e no-man
is the PDF of the estimated contact heading at time k+w,
given that the passive contact did not significantly change course during the time frame
of length w. It is a triangular shaped distribution centered on the contact's estimated
A
course at time k, OC [k]. Similarly to the hostile course change PDF, the width of this
distribution,
OE,
is defined as in Section 3.3.1. The function,
fA
Oc Ino-man
, can be defined
as
f
0[k + W Xc[k],X,[k]
OE
{0
OEx
0[k
+ w] -0
kj
if 0c[k
otherwise
1
70
+ w - ic[k} <OE
(3.18)
3.4.3 Example Plot of Passive PDF
Figure 3.14 represents the likely resulting Cartesian coordinates of a passive contact at
time k+w if w = 120 seconds and the contact's state at time k is
0
Meters
0 Meters
X [k
pi/2 Radians
12 Meters/Second
Likelihood of Resulting Cartesian Coordinates of Passive Contact
x 10
4
Initial Contact
Position (0,0)
arid Course of
90 Degrees
3.5
3
-
Measured Counter-
ClockWise From
Positive X-Axis)-
2.5
2
-
1.5
1
0.5
0
2000
1000
2000
0
1000
-1000
0
-1000
X-Axis (Meters
-2000
_-2000
Y-Axis (Meters)
Figure 3.14 Plot of Likelihood of Resulting Cartesian Coordinates of Passive Contact
71
The figure gives some idea of how the passive pdf, fpassive, is structured. The plot
illustrates that the most likely resulting Cartesian coordinates of a passive contact in this
case are a result of the contact not deviating significantly from his original course at time
k during the two minute interval.
3.5 Example Cases
The following example simulations illustrate how a UUV can maintain the
counter-detection belief state using the formulation presented in Sections 3.2, 3.3, and
3.4. The values in the hostile pdf model, defined in Equation 3.2, for P(host man) and
P(randman) were .73 and .27 respectively. The values in the passive pdf model, defined
in Equation 3.16, for P(maneuver) and P(noman) were .61 and .39 respectively. Each
example simulated the UUV and a single contact maneuvering in an open area for 2500
seconds. The belief state was updated every 120 seconds (w =120) and the initial belief
state was bo =.5. Also, in each simulation the speed of the UUV was 10 meters/second
and the speed of the contact was 12 meters/second. The initial values for the courses of
the UUV and contact are measured counter-clockwise from the positive x-axis. If a
passive contact was simulated, then the contact was free to maneuver randomly. In the
case of a hostile contact, the contact moved towards the position of the UUV. The UUV
was given random course commands in all of the simulations. All of the examples
simulated the UUV tracking the contact with active sonar and the EKF described in
Section 2.2. The asterisks in the trajectory plots are spaced 120 seconds apart to give a
better understanding of the locations of the UUV and contact at various times in the
simulation.
72
3.5.1 Example Simulation 3.1 (Passive Contact)
Figures 3.15 through 3.18 show the results of a simulation with a passive contact
and initial UUV and contact states being
0 Meters
0 Meters
3.14 Radians (180 Degrees)
5000 Meters
5000 Meters
3.92 Radians (225 Degrees)
10 Meters/Second
12 Meters/Second
For the first few 120 second intervals the belief state stayed around .5 because of the
difficulty in determining if counter-detection has occurred in the initial moments of the
simulation. This difficulty arose because the contact's initial course pointed it directly
towards the UUV. In this situation it becomes difficult to differentiate between hostile
and passive behavior because the contact can continue on its original course and still
pursue the UUV. Figure 3.16 shows that as the UUV was able to move away from the
contact, the belief state dropped when the contact did not maneuver to follow the UUV.
The belief state went to 0 when the contact turned away from the UUV. Quick analysis
of Figure 3.17 shows that the passive pdf values were higher when the contact maintained
course and dropped the two times that the contact changed course. Figure 3.18 shows
that the hostile pdf values initially were similar to the passive values as the contact
moved towards the UUV, but dropped as it moved away, causing the belief state to
correctly go to zero.
73
2
Plot of UUV and Contact Position
X10'
-
-
Cont-ct
-UUV
-- Contact
1.5
1
Initial Contact
Position
0.5 k
-0.5 k
Initial UUV
Position
*
-1
-1.5L
-1.5
0
0.5
X-axis (Meters)
-0.5
-1
1
1.5
2
x 10'
Figure 3.15 Example Simulation 3.1 UUV and Contact Positions
07.
0.60.50.4
O3
0.2
0.1
rs1
25M
15 Q00
ds,
Figure 3.16 Example Simulation 3.1 Belief States
of
Pk)1 of
1I
Pass0
PDF Wk.Plot
I0
10'
10
of Host e PDF Valus
10,
So
1000
Time
10
(Smconds)
2o
0
500
So
0 .
00
Tim (Seconds)
206
2500
Figure 3.17 Example Simulation 3.1
Figure 3.18 Example Simulation 3.1
Passive PDF Values
Hostile PDF Values
74
3.5.2 Example Simulation 3.2 (Hostile Contact)
Figures 3.19 through 3.22 show the results of a simulation with a hostile contact
and initial UUV and contact states being
0 Meters
0 Meters
,XC [01=
3.14 Radians (180 Degrees)
10 Meters/Second
xv 1]=[
~
-18000 Meters
2000 Meters
0 Radians
12 Meters/Second _
In this simulation, the belief state gradually rose to approximately .75 as the
contact maneuvered to its right to initially pursue the UUV. Once the contact was forced
to turn completely around to follow the UUV, the belief state quickly rose to 1. Of note
here is how a drastic course change by the hostile contact caused the belief state to move
more quickly to 1 when compared to the initial gradual course changes of the contact.
Plot of UUV and Contact Position
4000 -
uuv
-
Contact
2000
0
Initial Contact
Position
-2000
Initial UUV
Position
-4000-
-.
>-
-.
-.-.
-
.... ..
..
-8000 -2000-10000-12000
-4
-14000 -
-18
000
-16 00
-14000
-12000
-10000
-8000
X-axis (Meters)
-6000
-4000
-2000
Figure 3.19 Example Simulation 3.2 UUV and Contact Positions
75
0
ICWnt.c
T-rsArw
07
0 604
002
0
Figure 3.20 Example Simulation 3.2 Belief States
P10t
of P-sw
PUo
PDF VOl-s
&f H-sile PDF Vahas
I
0
j
I
1
00
50
(S00
00
2000
2500
2WO
2500
in-Oh
Figure 3.21 Example Simulation 3.2
Figure 3.22 Example Simulation 3.2
Passive PDF Values
Hostile PDF Values
3.5.3 Example Simulation 3.3 (Initially Passive Contact With CounterDetection Occurring During Simulation)
Figures 3.23 through 3.26 show the results of a simulation with initial UUV and
contact states being
0 Meters
SLo]=
8000 Meters
0 Meters
K3.144Radians
(180 Degrees)
-
C
10 Meters/Second
4000 Meters
1.57 Radians (90 Degrees)
12 Meters/Second
76
Plot of UUV and Contact Position
10000-
8000 --
I,
*2
1~
6000-
*
I
4000 14)
*
2000k
0
-20005
----
.nitial UUV
UUV
- Contact
-4000-
-2
-2.5
Position
-1.5
-1
-0.5
X-Axis (Meters)
Initial Contact
Position
0.5
0
1
1.5
x 10'
Figure 3.23 Example Simulation 3.3 UUV and Contact Positions
03..
0.708
2500
T-n (S-40nds
Figure 3.24 Example Simulation 3.3 Belief States
le,
P* of ftV
D
10*,
FIX
&f No"b PDF VAMue
1 ~
30'
1 15
W- 6
0
120
Tkft
'0"
0
Tirne
(Seconds
20 6
2500
Figure 3.25 Example Simulation 3.2
Figure 3.26 Example Simulation 3.2
Passive PDF Values
Hostile PDF Values
77
In this simulation the contact was initially passive, until it switched to hostile behavior
after 1000 seconds to simulate the occurrence of counter-detection. The belief state was
given a floor of .01 in this simulation. Initially, the UUV was very confident that the
contact was passive. At the 1000 second mark the belief state jumped to .4 after the
contact suddenly turn toward the UUV. After that point the belief state continued to
gradually rise as the contact continued to maneuver toward the UUV.
3.5.4 Example Simulation 3.4 (Hostile Contact)
Figures 3.27 through 3.30 show the results of a simulation with a hostile contact
and initial UUV and contact states being
0 Meters
4000 Meters
0 Meters
3.14 Radians (180 Degrees)
1000 Meters
3.26 Radians (187 Degrees)
10 Meters/Second
J
[
12 Meters/Second
In this simulation the contact started on a course where it could trail the UUV without
changing course, making it difficult initially for the UUV to determine if counter-
detection had occurred. The belief state remained close to 0.5 until the hostile contact
had to turn sharply to follow the UUV. Once the contact changed course significantly the
belief state jumped to approximately 0.9 and eventually went to 1.
78
Plot of UUV and Contact Position
2000
0
4
-~*
r1
erCnic
-
U-
-k"
""n'I"
-2000
-4000
-6000
X -8000
UUV
-10000
Contact
-N-
-12000
-14000
-14000
-12000
-10000
-8000
-8000
-4000
-2000
I
|
0
2000
1
6000
4000
X-axis (Meters)
Figure 3.27 Example Simulation 3.4 UUV and Contact Positions
NOc of BOOe Thma Conve isHOSOWe
09[
07
0.51
1051
3
02
Fgr Eu
3000
on 3
B
e
t
Figure 3.28 Example Simulation 3.4 Belief States
10',
Plot Of HOSOiW PDF VMlue
Pssv PDF Vlk-e
10
1o[
wo
2W
0
500 -
too
00
0500-
2M
2500
Figure 3.30 Example Simulation 3.4
Figure 3.29 Example Simulation 3.4
Hostile PDF Values
Passive PDF Values
79
3.5.5 Example Simulation 3.5 (Passive Contact)
Figures 3.31 through 3.34 show the results of a simulation with a passive contact
and initial UUV and contact states being
0 Meters
Meters
Xv10=0
3.14 Radians (180 Degrees)
10 Meters/Second
4000 Meters
1000 Meters
Xc, I[]=- 3.26 Radians
(187 Degrees)
12 Meters/Second
The initial conditions for this simulation were the same as for Example Simulation 3.4,
except that the contact was passive in this example. Similarly to Example 3.4 the contact
started on a course where it could trail the UUV without changing course, making it
difficult initially for the UUV to determine if counter-detection had occurred. The belief
state remained close to 0.5 until the UUV turned and the contact did not maneuver to
follow the UUV. As the contact maneuvered away from the UUV, the belief state went
to zero.
80
1.5
Plot of UUV and Contact Position
F
x 10'
-
-
--onta
Contact
0.5 -
........
....-....
..
*
*
+---
COM-
0
4,
4)
-0.5k
--
-
-
-W
-1
-1.51-.
-21
____
-200 00
_
__
I
_
I
-15000
I
II
0
-5000
-10000
X-Axis (Meters)
5000
Figure 3.31 Example Simulation 3.5 UUV and Contact Positions
071
0.
03
S.2
0.1I
I5D
2000
25o0
Figure 3.32 Example Simulation 3.5 Belief States
Plow
&P..
siv
PDFVal
PM
of 1400t1
PMF Va
10I
10,
10'
10,
10,
107
10'
10
10,
10-
10"
10-
10*"
10
10
500
1500
1000
Ti.
2000
25DO
-
(Seconds)
Figure 3.33 Example Simulation 3.5
Passive PDF Values
1000
100
Ten (Seconds
26-
2SOO
Figure 3.34 Example Simulation 3.5
Hostile PDF Values
81
3.5.6 Example Simulation 3.6 (Hostile Contact)
Figures 3.35 through 3.38 show the results of a simulation with a hostile contact
and initial UUV and contact states being
Xv
10]=
0 Meters
0 Meters
10000 Meters
,
XC
3.14 Radians (180 Degrees)
[0]=
10000 Meters
3.93 Radians (225 Degrees)
1~
12 Meters/Second
10 Meters/Second
The contact's initial course was directly towards the UUV. During the simulation the
contact was able to stay on a near constant course with some slight maneuvering towards
the end of the 2500 seconds. Because of this the belief state slowly drifted upwards and
was slightly above .8 after 2500 seconds. Because the contact made no significant course
changes, it was difficult for the UUV to determine whether the contact was passive or
hostile.
82
Plot of UUV and Contact Position
x 10O
1.5 -
Contact
xrscawl
14'
0.5
A'
AF
z)
0
..........
......
.........
Position
-0.5 k
4
-1
-1.5
-1
-0.5
0
X-axis (Meters)
0.5
1.5
X 10,
1
Figure 3.35 Example Simulation 3.6 UUV and Contact Positions
Plot of BeWe Theit Cantedt isHosWil
0.
0.7 -
-
0.7
0.5
-
03
0.1I
0
-
0
-
0-00
- -
-
- -
-
Figure 3.36 Example Simulation 3.6 Belief States
P0--
--
P.F- V.-
----
P--.ot ofHostdi
---
POF Va
1o'
0
500
100
100
20M
2500
0b.. (Se..&0)
Figure 3.37 Example Simulation 3.6
Figure 3.38 Example Simulation 3.6
Passive PDF Values
Hostile PDF Values
83
3.6 Summary
The example simulations shown in Section 3.5 show how the counter-detection
belief state formulation discussed in Sections 3.2 through 3.4 can enable a UUV to
determine if a contact has counter-detected it. How the belief state behaves is determined
by the structure of the passive and hostile probabilistic models. In the context of this
thesis, if a contact does not maneuver towards the UUV, then the belief state will quickly
go to zero. Conversely, the belief state goes towards one when a contact maneuvers
towards the UUV. The behavior of the contact affects how quickly the belief state goes
to zero or one. In situations like Example Simulations 3.4, 3.5, and 3.6, where the
contact could maintain its initial course and still pursue the UUV, the UUV was slow in
determining if counter-detection occurred. The belief state moved quickly to zero or one
only when the UUV turned and forced the contact to significantly change course to
follow or observed a passive contact continue on its original course. In other simulations
where the occurrence of counter-detection was not so ambiguous, the UUV was able to
quickly assess whether counter-detection had occurred or not. The belief state went to
zero quickly for a passive contact that never maneuvered towards the UUV, as in
Example Simulation 3.1. Sudden hostile maneuvers, such as in Example Simulation 3.2
and 3.3, caused the belief state to quickly move to one.
Chapter 4 will present various algorithms that will recommend courses for the
UUV to follow that will enable the UUV to quickly determine whether counter-detection
has occurred. The algorithms will exploit the knowledge of how passive and hostile
contacts behave, as defined in this chapter, to force a contact to display clear hostile or
passive behavior.
84
Chapter 4
Counter-Detection Maneuver Decision Aids
Examples from Chapter 3 showed that there are cases where it can difficult for the
UUV to determine if counter-detection has occurred if the contact can pursue the UJUV
without making significant course changes. This chapter will present maneuver-decision
aids that will enable a UUV to quickly determine if a contact has counter-detected it.
These aids will use the knowledge of how passive and hostile contacts behave, as defined
in Chapter 3, to move the belief state quickly to zero or one.
4.1
Entropy
Before an attempt can be made to develop a maneuver-decision aid to enable a
UUV to quickly determine if counter-detection has occurred, one must be able quantify
uncertainty. A common method to accomplish this in information theory is to determine
a belief state's entropy. In the context of this thesis entropy can be looked at as a
measure of the uncertainty in the UUV's counter-detection belief state. If the belief state
at time t is bt, then its entropy can be defined as
entropy
(bt)= g(bt)= -bt
x n(bt) -(
-b,)xln(I -b,).
(4.1)
The graph of this function is shown in Figure 4.1. In this case the natural logarithm was
used, but it is also common for log2 to be used.
85
Plot
of Entropy vs. Belief State
0.7
0.6 I
/-
0.51
//-
0.4
0
1~-
0.3
-
/
/
,1
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
Belief State
0.6
0.7
0.8
0.9
1
Figure 4.1 Plot of Entropy Vs. Belief State
As can be seen in Figure 4.1 the entropy of the belief state goes to zero as the belief
approaches zero or one. As the belief state approaches .5, the most uncertain value for
the belief state, the entropy approaches a maximum value of .7. The maneuver decision
aids presented in the next section will utilize the entropy function by attempting to
minimize the entropy of the belief state with each action. By minimizing the entropy, a
UUV will also try to push the belief state to zero or one and thus become certain of
whether or not counter-detection has occurred. For a more detailed explanation of
entropy, refer to [1 1],[12].
86
4.2 Maneuver Decision Aid Formulation
4.2.1 Decision Space
Since this thesis works with constant-speed motion, the counter-detection
maneuver decision aids presented in this chapter will give a recommended course and
amount of time to remain on that course. The recommended course and time duration
will be the ones deemed to minimize the objective function fENT, which will be defined in
Section 4.2.2, among all potential course and time durations examined. The suggested
action can also be viewed as the action that the UV believes will minimize the counterdetection belief state's entropy in the future.
4.2.2
Objective Function
A potential action, ai(k), that can be taken by the UUV at time k can be defined as
ai (k) =
,
(4.2)
Iti[k]]
where 0i[k] is the potential course and ti[k] is the potential time duration of that course
command. The expected UUV state after this action is taken,
Xvfuture i, can
according to the equation
Sv,[k]
k], ti [k]) ,(4.3)
Xa
vftr- = f(X, [k], IV
87
be calculated
where f is the discrete time state transition dynamic model function defined in Section
2.2.1 and Equation 2.4, X,[k] is the state of the UUV at time k, and vv[k] is the speed of
the UUV at time k.
At time k the contact can also choose to follow course O and its expected state at
time k+w (w is fixed time frame defined in Chapter 3) given that chosen course can be
calculated as
Xedt
= f(Xc [k], L
,w)
,
(4.4)
Ivc [k]
where Xc [k] is the estimated state of the contact at time k, and vc [k] is the speed of the
contact at time k. Given that the UUV chose action ai(k) and the contact assumes course
Oc at time k, the expected value of the belief state at time k+w can be calculated as
A
E[bk+,
I= bupdate(X
cf ,
bt X c[k], X Vwe_) i
(4.5)
using the expected UUV and contact states given by Equations 4.3 and 4.4 respectively.
bupdate is the counter-detection belief-state update equation defined in Section 3.2 and
equation 3.1. The entropy of the expected belief state can be determined according to the
equation
A
g(buate (X cfutur, bt Xc [k], Xvfure))
where g is the entropy function defined in Section 4.1 and Equation 4.1.
88
(4.6)
Using the above formulations, an objective function can be used to evaluate the
usefulness of the UUV choosing action ai(k) in regards to pushing the belief state to zero
or one. This objective function,
fENT,
can be evaluated according to the equation
(4.7)
fENT ai bt,Xv[k],Xc[k] =
A
b, f bOc
+ (I- b )ff Oc6as
Oc
A
f|Ic[k],Xb
0,
xg(b
Xc [k], Xvfutr_-
1db
((Xc
c[k],X
x g(bupae ((Xcfutr, b,t
k Xe
)dOc
c [k], Xvfutre-dOc
cIpass
where bt is the belief state at time t, f
Oc host
is the distribution of a hostile contact's
course as defined in Section 3.3, and f
is the distribution of a passive contact's
OcIpass
course as defined in Section 3.4. The objective function evaluated for ai can be
interpreted as the expected entropy value at time k+w if the UUV's state was Xvfuturei at
time k. By evaluating fENT for various actions, a maneuver decision aid can analyze how
various positions that it could maneuver to would influence the counter-detection belief
state. The maneuver decision aids presented in the next section will minimize this
objective function over discrete action spaces. By doing this, the decision aids will tell
the UUV to maneuver in such a way as to reduce subsequent belief states' entropy and
thus quickly determine if counter-detection has occurred. It's expected that the objective
function presented above would recommend maneuvers that would enable to the UUV to
force hostile contacts to make significant course changes or quickly see that a passive
contact has no intentions of pursuing the UUV.
89
Figure 4.2 illustrates what the function fENT looks like. In this particular example
the UUV state was
0
0
X, [k]=
n Radians (180 Degrees Measured From Positive X - Axis)
10 Meters / Second
1
the estimated contact state was
4000 Meters
410 Meters
x [k]
Radians (180 Degrees)
12 Meters/Second_
and the belief state was bk=0.5. The function,
fENT,
was evaluated for several values of ai
such that
a(k)
(n x 5 Degrees)
[n x.085 Radians
k x120 Seconds
I
n = 0,1,2,...,72
k =1,2,3,4,5
The contours in the plot correspond to the locations that the sets of actions evaluated
would have moved the UUV to. These locations were plotted against the expected
entropy values, given by the function fENT, for the corresponding action. The innermost
contour represents the shortest time duration of 120 seconds and the outermost contour
represents the longest time duration of 600 seconds. From analyzing the figure one can
see that the highest values for the expected entropy occur if the UUV chooses to continue
90
on its original course. This would be expected since the contact could follow on its
original course and it would still not be clear whether or not it had counterPlot of Expected Entropy Values
0.9
0.8
c.0.7
05
L5 0.6
w
0.4
0.3
0.2
-6000
-4000
-
UUV Ccturse_,
-400Initial
Position.10.
-200UUV
0
2000
niilot
C
Contac
-4000
-2000
0
4000
X-Axis (Meters)
6000
Ion
4000
-6000
6000
Y-Axis (Meters)
Figure 4.2 Illustration of fENT
detected the UUV. The lowest values for the expected entropy occur if the UUV
maneuvers to its right or left or moves to a point directly behind where the contact was
headed. In these cases a hostile contact would have to make significant maneuvers in
order to pursue the UUV or a passive contact would no longer be following the UUV,
thus
reducing the uncertainty in the occurrence of counter-detection. The figure shows that
the more the UUV can cause a hostile contact to deviate from its original course, the
lower the expected entropy will be and the more desirable that action will be for the
UUV.
91
4.3 One Time Horizon Look Ahead Example Simulations
The following sets of example simulations have the same conditions as in the
examples in Chapter 3. The belief state was updated after every 120 second time
window. After each update, the UUV followed the course recommendations of a
maneuver decision aid that minimized the objective function presented in Section 4.2 and
expressed in Equation 4.7 over a discrete set of action choices consisting of a course
command and a length of time to maintain the commanded course such that
n x.085 Radians (n x 5 Degrees) n = 0,1,2,...,72
120 Seconds
_ k = integer
In other words, every 120 seconds the belief state was updated and the maneuver decision
aid provided a course command that was a multiple of 5 degrees and would be
maintained until the next belief state update at the end of the subsequent 120 second time
horizon. The results of these simulations will be compared to some of the simulations
from Chapter 3 to help determine if the maneuver decisions aids provide course
recommendations that are effective in quickly determining if counter-detection has
occurred. One would expect the tested maneuver-decision aids to avoid some of the
ambiguous situations shown in Chapter 3 by forcing a contact to make significant course
changes if the contact wished to pursue the UUV.
92
4.3.1
Example Simulation 4.1 (Hostile Contact, 1 Time Horizon
Maneuver Decision Aid)
Figures 4.3 and 4.4 show the results of a simulation with a hostile contact and
initial UUV and contact states being
XV 0-
0 Meters
-18000 Meters
0 Meters
2000 Meters
0 Radians
3.14 Radians (180 Degrees)
10 Meters/Sec ond
Po of UUV.4
12 Meters/Sec ond
Co 4.0 Posafos
10000
Plot
of UUV .
Cordac Posion
- I--
400
/odw
I
404
I
I
0
00[MW
0I d
i
.1000
-1400
-2000
-14000
.120000
OMel OUV
P00000
420
.0WW0 -M00
X-oos (Moels)
-000
-4000
-2000
8000
0
-16000
-14000
-12000
-10000
-
0
-6000
-4000
-2000
Figure 4.5 Example Simulation 3.2
UUV and Contact Positions
Figure 4.3 Example Simulation 4.1
UUV and Contact Positions
PIOIIBO*IThdCO1**111S !t
-
0.9-
0.30.7
08
-
06
0.
0.6.
0.4 -
011
0.3 -0.2
0.1
0
02
01
500
1000
T"2 (0000.01.
1500
2000-
5W
2500
1000
5W0
2OW0
2250
1
Figure 4.6 Example Simulation 3.2
Belief States
Figure 4.4 Example Simulation 4.1
Belief States
93
This simulation had the same initial conditions as Example Simulation 3.2, which
is shown in Figures 4.5 and 4.6. In Example Simulation 3.2 the UIUV continued on its
original course for several minutes and the belief state never reached .75 before 900
seconds. The belief state went to 1 quickly after 1000 seconds because the contact turned
around to follow the UUV after closing the initial range from the contact to the UUV. In
Example Simulation 4.1 the maneuver decision aid forced the hostile contact to maneuver
to its left, pushing the belief state to .95 after 900 seconds. The belief state went to 1
before 1500 seconds when the contact made a loop in following the UUV. Example
Simulation 4.1 illustrates how the maneuver decision aid forced the contact to show its
hostile intentions early on in the simulation. The contact was forced to significantly
deviate from its original course in order to continue to pursue the UUV. Also of note is
how the UUV zigzagged at the end of the simulation to force the hostile contact to zigzag
with it.
4.3.2
Example Simulation 4.2 (Hostile Contact, 1 Time Horizon
Maneuver Decision Aid)
Figures 4.7 and 4.8 show the results of a simulation with a hostile contact and
initial UUV and contact states being
4000 Meters
1
1000 Meters
xC [o]=
3.26 Radians (187 Degrees)
0 Meters
xv [o]=
0 Meters
3.14 Radians (180 Degrees)
10 Meters/Second
12 Meters/Second
_
This simulation had the same initial conditions as Example Simulation 3.4, which
is shown in Figures 4.9 and 4.10. The maneuver decision aid clearly provided the UUV
with a course of action that avoiding any ambiguity in determining if counter-detection
occurred. The maneuver decision aid quickly determined that if it could maneuver to its
right it would easily be able to identify either clear hostile or passive behavior. If the
94
contact had remained on its original course after the UUV maneuvered, then it would
have easily been determined to be passive. In this case the contact had to basically make
a "u-turn" to follow the UUV, making it clear that the contact was hostile and pushing the
belief state to 1 after approximately 500 seconds, instead of over 2000 seconds in
Example Simulation 3.4.
PlO
of ULNfnd Con0ed Postio
Pd of UUV
3500-
end Corad
Posit
o
20D00
3000-
200
-4000
2000
-
1500
i
1000
j~:..-~In"j
\
Costed
[
-12000
---
-1000_
-14000
_
-14000
X-exi
(eers)
-12000
-10000
-
000
-4000
2000
0
2000
_
4000
f00
0-ass (1001.0)
Figure 4.9 Example Simulation 3.4
UUV and Contact Positions
Figure 4.7 Example Simulation 4.2
UUV and Contact Positions
1
P10 of OeW That C nled M H."
0.7
09g
08a
0.7
0.6
0.6
O's
0.8
,
,PlkA
of Belef Thet Cortedt isHostile
-+-
0.5
05
0.3
0.2
0.1
0.1
0
00
1000
1500
0
200M
T... (0500100)
Figure 4.8 Example Simulation 4.2
Belief States
000
1000
1500
2000
Figure 4.10 Example Simulation 3.4
Belief States
95
2500
4.3.3
Example Simulation 4.3 (Passive Contact, 1 Time Horizon
Maneuver Decision Aid)
Figures 4.11 and 4.12 show the results of a simulation with a passive contact and
initial UUV and contact states being
0 Meters
4000 Meters
0 Meters
3.14 Radians (180 Degrees)
10 Meters/Second
Plot
80DO
[o11=
Meters
3.26 Radians (187 Degrees)
[361000
12 Meters/Second
of UUV OnW Contad Posit
5
1.
I
r
of UUV and Co.4.0
Position
6000
4000
2010
0
in \U
P\\
D
\i
-2000
-4m
-1
-6m
/
~..-
-1.5
-8m
15000
M
-10000
-500
X.4,. (Meter)
5000
0
-2000
Figure 4.11 Example Simulation 4.3
UUV and Contact Positions
* Belie.11 ' Cori.01
'
i
-15000
0
-5000
0-044. 54J.1os)
-10000
5000
Figure 4.13 Example Simulation 3.5
UUV and Contact Positions
ostle
Plot
of Belief
That
Corcts is
Hostile
0.7,
0,9
05-
04 -
0 4 --
--
03 0.30
051-I
02
~
0
20
-
02 0 1---
01
500
1000
1500
Too. (Seonds)l
20DO
-
25M
Figure 4.12 Example Simulation 4.3
Belief States
Figure 4.14 Example Simulation 3.5
Belief States
96
This simulation had the same initial conditions as Example Simulation 3.5, which is
shown in Figures 4.13 and 4.14. Similarly to Example Simulation 4.2, the maneuver
decision aid quickly decided to turn and see if the contact would follow it. In this case
the contact was passive and continued on it original course. By avoiding the unclear
events shown in Example Simulation 3.5, the maneuver decision aid enabled the UUV to
make a much quicker decision that the contact was passive in Example Simulation 4.3
when compared to Example Simulation 3.5.
4.3.4
Example Simulation 4.4 (Hostile Contact, 1 Time Horizon
Maneuver Decision Aid)
Figures 4.15 and 4.16 show the results of a simulation with a hostile contact and
initial UUV and contact states being
10000 Meters
10000 Meters
3.93 Radians (225 Degrees)
12 Meters/Second
0 Meters
0 Meters
3.14 Radians (180 Degrees)
J
10 Meters/Second
This simulation had the same initial conditions as Example Simulation 3.6, which
is shown in Figures 4.17 and 4.18. At the end of Example Simulation 3.6 the UUV was
only approximately 80% certain that the contact was hostile, because the contact made no
significant course changes while pursuing the UUV. The maneuver decision aid in
Example Simulation 4.4 clearly provided better performance by avoiding that situation
and forcing the hostile contact to constantly maneuver to its right while pursuing the
UUV and then zigzag once it closed range with the UUV.
97
Plot
12000,
of UUV and C~
10'
1.5
Psition
Plot
of UUV and Con0.d
Potm4n
x
Posit
05
I
: - -- --
0
Ow00
40-
4.5
0
L7~a]
-1
-05
0
1,5
05
Figure 4.15 Example Simulation 4.4
UUV and Contact Positions
-1
-0.5
0-xsMtr)
0.5
1
1.5
.00e
Figure 4.17 Example Simulation 3.6
UUV and Contact Positions
0.1
07
01
0!
71. S--
Figure 4.16 Example Simulation 4.4
Belief States
Figure 4.18 Example Simulation 3.6
Belief States
98
4.3.5
Example Simulation 4.5 (Hostile Contact, 1 Time Horizon
Maneuver Decision Aid)
Figures 4.19 and 4.20 show the results of a simulation with a hostile contact and
initial UUV and contact states being
0 Meters
-
0 Meters
3.14 Radians (180 Degrees)
10 Meters/Second
PlOt
4000-
XC 10]=
8000 Meters
- 8000
Meters
.785 Radians (45 Degrees)
12 Meters/Second
of UUV
and Conadt Pos~ion
2000-
-2000-
nit
UUV
4505900
-4000
-000
RAWf C00dle
Position
-a"-800
6000
-4000
-2000
0
X-axis (Meters)
2000
4000
6000
Figure 4.19 Example Simulation 4.5 UUV and Contact Positions
hostile contact to show its intentions clearly. The UUV forced the contact to maneuver to
99
its left and pushed the belief state to approximately .85 once the contact closed its range
to the UUV. Then the UUV made the contact turn completely around, pushing the belief
state to one.
4.3.6
Example Simulation 4.6 (Passive Contact, 1 Time Horizon
Maneuver Decision Aid)
Figures 4.21 and 4.22 show the results of a simulation with a passive contact and
initial UUV and contact states being
0 Meters
-8000 Meters
-
0 Meters
3.14 Radians (180 Degrees)
10 Meters/Second
[.785 Radians (43 Degrees)
12 Meters/Second
PWo
20D0O.i7
8000 Meters
of ULIV and Corded Pos tion
\
p~i
~-4000
-2
000
-12000
-
X100
150
006
Figure 4.21 Example Simulation 4.6 UUV and Contact Positions
0.9 -
O.L
0.70.-
03
0.2
0
500
1000
1500
2W
200
Figure 4.22 Example Simulation 4.6 Belief States
100
In this simulation the maneuver decision aid attempted to see if the contact would
maneuver to its left to pursue the UUV. When the contact did not move towards the
UUV and continued on its original course for several minutes the belief state quickly
went to zero. Another interesting behavior shown here is once the contact turned away
from the UUV, the UV always maneuvered to a point directly behind the contact to see
if it would reverse its course and pursue the UUV.
4.4
1-5 Time Horizon Look Ahead Example
Simulations
The following sets of example simulations have the same conditions as the
previous ones, with one difference. After each belief state update, the UUV followed the
course recommendations of a maneuver decision aid that minimized the objective
function presented in Section 4.2 and expressed in Equation 4.7 over a discrete set of
action choices consisting of a course command and a length of time to maintain the
commanded course such that
n x.085 Radians (n x 5 Degrees)
d
ai(120xk)=1
-n =O,1,2,...,72
j=1,2,3,4,5
Sk
=
integer
In other words, every 120 seconds the belief state was updated and the maneuver decision
aid either provided a course command that was a multiple of 5 degrees and would be
maintained for one through five 120 second time horizons or ordered the previous course
command if it was not completed. The results of these simulations will be compared to
some of the previous simulations of this chapter. The reason for running the next set of
simulation with longer time duration commands was to see if allowing the UUV to plan
further into the future would improve the belief state performance. For example, by
looking further ahead the UUV might be able to recognize situations where it could force
101
a hostile contact to make more significant course changes than it could by looking ahead
for only a single 120 second commanded course time duration.
4.4.1
Example Simulation 4.7 (Hostile Contact and
1-5 Time Horizon Maneuver Decision Aid)
Figures 4.23 and 4.24 show the results of a simulation with the one to five time
horizon aid, a hostile contact, and initial UUV and contact states being
0 Meters
0 Meters
3.14 Radians (180 Degrees)
10 Meters/Second
Xc [o]=
4000 Meters
1000 Meters
3.26 Radians (187 Degrees)
12 Meters/Second
Plo o lULN .40 Coftad Po-b-o
Plot
of UUV
wr
Cor01d Posk
400kc
Mal
Lti"O
2504
//b
5200[
,2ha/U
P'Vl
1000
h.0.1 coftecl
0
hbm VUV
000
-2000
-1000
0
I00
20M0
X.. .)M4
0
400
50
00
.0
Figure 4.23 Example Simulation 4.7
UUV and Contact Positions
0
2000
8000
"0
0000
1200
14000
Figure 4.25 Example Simulation 4.2
UUV and Contact Positions
Pl of
0.9
0.6
0.7
Thate
SCos
07-
-0.6
08
0.1
0.30
0.2
0.3
0
00
w1000
150
5001 (0.0951)
2000
2
Figure 4.24 Example Simulation 4.7
Belief States
Figure 4.26 Example Simulation 4.2
Belief States
102
This simulation had the same initial conditions as Example Simulation 4.2, which
is shown in Figures 4.25 and 4.26. In Example Simulation 4.7 the UUV chose to
maneuver to a point directly behind the contact and force it to turn around. This quickly
pushed the belief state to one but there was no improvement over the one time horizon
decision aid. At every decision point in Example Simulation 4.7 the maneuver decision
aid chose the longest commanded course time duration of five time horizons, but the
advantage of maneuvering to locations where the UUV felt it could get a better
understanding of the contact's intentions appear to be negated by the time it took to
transit to those locations. It is still interesting to note how the UUV forced the contact to
turn around twice and then sharply turn twice to continue to pursue it.
4.4.2
Example Simulation 4.8 (Passive Contact and
1-5 Time Horizon Maneuver Decision Aid)
Figures 4.27 and 4.28 show the results with the one to five time horizon aid, a
passive contact, and initial UUV and contact states being
xv []=
0 Meters
0 Meters
3.14 Radians (180 Degrees)
10 Meters/Second
x[0]
1
4000 Meters
1000 Meters
3.26 Radians (187 Degrees)
12 Meters/Second
J
103
PWo
of UUV &ndContsct Poslb n
P00 of
Fi UUV
20DO
UUV
and Cordtct
Positio
---UUV
-Posirtion
-4000-
-000
-4000
-
11-0sM.r
-.
-
-1
-0.5
0
05
1
X-OA6 ("M66
is
.10'
Figure 4.27 Example Simulation 4.8
UUV and Contact Positions
P0160(000910 C,.d
Figure 4.29 Example Simulation 4.3
UUV and Contact Positions
is 00,10
P" Of60Saw Th1 Co.4.d M 400110
09
0-9
08
07
07
068
06
0-5
05
04
0.4
03
013
02
0.2
Oil
01
0
500 -
0
-
1
-
25M
0
500
1000
1500
2000
2500O
TimeSec-ndt)
Figure 4.30 Example Simulation 4.3
Belief States
Figure 4.28 Example Simulation 4.8
Belief States
This simulation had the same initial conditions as Example Simulation 4.3, which
is shown in Figures 4.29 and 4.30. Once again the one to five time horizon look ahead
aid chose the longest possible course command duration at every decision point.
Similarly to Example Simulation 4.7 the one to five time horizon decision aid initially
attempted to maneuver to a point directly behind the contact to see if it would completely
turn around in pursuit of the UUV. In this case the contact moved away from the UUV
and the belief state quickly went to zero. Although the one to five time horizon decision
aid showed effective behavior in quickly determining if counter-detection occurred, there
was once again no improvement over the one time horizon maneuver decision aid.
104
4.4.3 Example Simulation 4.9 (Hostile Contact, 1-5
Time Horizon Maneuver Decision Aid)
Figures 4.31 and 4.32 show the results with the one to five time horizon aid, a
hostile contact, and initial UUV and contact states being
XV
10000 Meters
10000 Meters
0 Meters
0 Meters
=3.14 Radians (180 Degrees)
0
10 Meters/Second
3.93 Radians (225 Degrees)
12 Meters/Second
This simulation had the same initial conditions as Example Simulation 4.4, which used a
one time horizon look ahead decision aid and is shown in Figures 4.33 and 4.34. Once
again there were no improvements in performance provided by the longer look time
horizon aid. For a few time steps the 1-5 time horizon maneuver decision aid showed
poor performance as the belief state dropped significantly. During this time the UUV
was stuck executing a command for an extended amount of time that increased the belief
state's entropy instead of lowering it. Because of the longer time durations, the UUV
was unable to quickly adjust and fix the problem.
105
PRAt
of UUV and Contact Position
Plot of UUV and Cortact Posbon
UL'V
,.., Con,
P-tion
ct
10OW
Initial Cc.4.d
WWF0
"M0
2000~
4
Initial UUV
000
Positim
'4.
0
-200
2M0
400
0 .)
-05
WW
X-aif (Meter~s)
of Ballef
Contact
"1101
0.5
1
1.5
X-as (Meters)
X1
Figure 4.33 Example Simulation 4.4
Figure 4.31 Example Simulation 4.9
UUV and Contact Positions
Plol
0
UUV and Contact Positions
Plot
isHostile
of
Bate
That Contadti Hostile
-
0.9
0.8
0.5
0.7
06
0.7-j
0.6[0o.
0.5
0.4
0.3
0.2 0.1
0.2[
0.1
o -----
-
cO
1
1500
- 1
-
2500
To.(Soo,0
Figure 4.32 Example Simulation 4.9
---
-50
Figure 4.34 Example Simulation 4.4
Belief States
Belief States
4. 4. 4 Example Simulation 4.10 (Passive Contact, 1-5
Time Horizon Maneuver Decision Aid)
Figures 4.35 and 4.36 show the results with the one to five time horizon aid, a
passive contact, and initial UUV and contact states being
0 Meters
-
0 Meters
3.1 4
3.14 Radians (180 Degrees)
10 Meters/Second
xc [o]=
8000 Meters
- 8000 Meters
.785 Radians (43 Degrees)
12 Meters/Second
106
tio
UU
oUUV
and CoMsd Po
Plot 01 UUV
M-
W4d
co
.d
Po~
0
-'I-Fn-
05
-10000[
,U-
"il
.;
09
160
0
00
Figure 4.37 Example Simulation 4.6
UUV and Contact Positions
_.1
Fiue
.6ExmleSmlain
05
.0
D8 -
Figure 4.36 Example Simulation 4.10
Figur 4.3Emle Simultes
41
UU
eliedfCtatePsios
0402
-
0
;;_0200D
ZOO
Figure 4.38 Example Simulation 4.6
Belief States
This simulation had the same initial conditions as Example Simulation 4.6, which
used a one time horizon look ahead decision aid and is shown in Figures 4.37 and 4.38.
As in the other comparisons of this section, there were no improvements in performance
provided the longer look ahead aid.
4.5
Summary
The results shown in this chapter show how the maneuver decision aids presented
can use their knowledge of how hostile and passive contacts behave to intelligently
maneuver and quickly determine if a contact has counter-detected it by attempting to
minimize the entropy of the counter-detection belief state. In the context of this thesis
107
that knowledge was provided by the probabilistic models of both hostile and passive
contacts that were presented in Chapter 3. In all of the simulations of this chapter the
maneuver decision aids enabled the UUV to maneuver in such a way that it quickly
became clear whether the contact was hostile or passive. By minimizing the entropy of
the belief states at each decision point the UUV forced hostile contacts to make
significant course changes while pursuing the UUV or saw that passive contacts that
continued on their original courses were no longer maneuvering towards the UUV.
Contrasting simulations were run where in some cases the commanded durations of the
UUV course commands only lasted for a single 120 second time horizon and in others the
durations lasted from 1 to 5 120 second time horizons. It was found that the simulations
with the longer duration commands offered no improvements over the single time
horizon duration simulations because the advantage of better locations that were farther
away from the UUV were negated by the amount of time it took to reach those locations.
Also, in one case the performance of the longer duration simulations was poorer because
the UUV made a course change that did not lower the belief state entropy and could not
adjust to a better course until the time of the commanded duration passed. Chapter 5 will
present a way that the maneuver decision aid developed in this chapter can be used in a
potential Maritime Reconnaissance mission and provide some examples.
108
Chapter 5
Intelligence/Surveillance/
Reconnaissance (ISR) Simulation Results
This chapter will present a slightly modified version of the maneuver decision
aids presented in Chapter 4 that will show how the counter-detection belief state update
algorithm presented in Chapter 3 and the maneuver decision aids presented in Chapter 4
can be used by a UUV's situational awareness and assessment subsystem (SA Assessor)
during an ISR mission. The results of several simulations will be shown to help illustrate
their usefulness.
5.1
ISR Concept of Operations
Section 1.2.1 discussed a general ISR mission for a Navy UUV. This mission
involved a UUV being launched from a host platform and then maneuvering to a
designated observation area to collect information. While transiting to that area, the
UUV must be able to track and recognize mobile threats. One part of recognizing mobile
threats that pertains to the work in this thesis would be determining if a potentially
dangerous contact has counter-detected the UUV. This would be one of the things
accomplished by the UJV's SA Assessor (SAA). As mentioned in Chapter 1 the SAA
must interpret data received from its sensors to track dynamic obstacles, perform tactical
maneuvers, detect hostile actions, and manage and improve the Situational Awareness
picture it maintains. It must be able to do this while also maneuvering to the designated
observation area. Of particular interest in this chapter is how the SAA can balance the
need to perform tactical maneuvers to determine if counter-detection has occurred with
the requirement to transit to the observation area in a timely manner. The maneuver
decision aid presented in the next section and the simulations using this aid will illustrate
109
how the algorithms presented in Chapters 3 and 4 can be used by the SAA to accomplish
this balance.
5.2
ISR Maneuver Decision Aid
In order to develop a maneuver decision aid that can balance the requirements of
recognizing counter-detection and quickly transiting to an observation point, one must be
able to quantify how potential actions can delay the UUV from reaching the observation
point as quickly as possible. This can be done in the following manner. Given the
Cartesian coordinates of the observation point,
X
OBS
OBS
_
LYOBS
1
(5.1)
I
and the state of the UUV at time k,
y v[k]
Xv [k]=
,v[k]
(5.2)
Ov[k]
Vvy[k]-
the minimum time required to transit to the observation point, given that the UUV
maintains its speed can be defined as
(xOBS
v [k]Y
vv[k]
110
OBS
-
yv[k]y
(5.3)
Similarly to Chapter 4, a potential action, ai(k), that can be taken by the UUV at time k is
defined as
ai(k)=
(5.4)
.[k]
It [k]]
The expected state at time k+ti[k] if the UUV chooses action ai(k) can be calculated
according to the equation
X
X
=
[k + ti [k]f~
- +tik] =futu[k
f(X,[k],
, ti[k])
0 fUt_[k + t [k]]
LV,[k]_
v
[k + t1 [k]]
'[
(5.5)
where f is the discrete-time state-transition dynamic model function defined in Section
2.2.1 and Equation 2.4. The minimum time required to transit to the observation point,
given that the UUV first executes action ai(k) and maintains its speed can be defined as
ta(k)
=
t [k]+
(x OBS
vfutre-i[k+ti[k]
y
+&OBS
v-[k]
,fure
i [k+t[k]])
(5.6)
A function that can quantify how the potential action can delay the UUV from reaching
the observation point as quickly as possible is
ftime(a i(k),XOBSXV[k])=.7x
111
(
I- t"
taj(k)
.i
(5.7)
The reason for multiplying by .7 is to make the possible values of the function range from
0 to .7, similarly to the entropy objective function presented in Chapter 4. Figure 5.1
shows a graph of the function ftime versus possible UUV courses where the Cartesian
coordinates of the UUV and observation point are
x
yV
4000
XOBS
0
0
and the commanded time duration of the action is fixed at 120 seconds. The minimum of
the function occurs at 3.14 radians, which is the course that would move the UUV
directly towards the observation point. The function increases as the potential course of
the UUV takes it further from the observation point.
Plot of ftime Function vs. Course
0.3
0.25
A'
7
/
0.2 U)
E
0.15
/
I
//
/1
/
//
0.1
0.05 k
0
0
1
2
3
Course (Radians)
4
5
6
Figure 5.1 Plot of fe*. for Fixed Time Duration Command of 120 Seconds and UUV and Observation
Point Cartesian Coordinates Being (0,0) and (-4000,0) Respectively
112
An overall objective function that evaluates how potential action ai(k) satisfies the
requirements of recognizing counter-detection and quickly transiting to an observation
point given the estimated contact state at time k, the UUV state at time k, the belief state
at time k, the observation point, and a weighting parameter, p, is defined as
fobi
where
fENT
i(k Xc [k Xv [klbk,XOBs,
=P
:p
xf.
(i (k),XOBS,X [k])+
fENT (
A
ilb,X,[k],Xcjk]
(5.8)
( .8
is the entropy-driven objective function presented in Chapter 4. The
weighting parameter, p, can be interpreted as an adjustable parameter that determines the
importance of each of the goals of recognizing counter-detection and quickly transiting to
the observation point. A maneuver decision aid that minimizes fobj with a low value for p
would weight its decisions more on how they could influence the counter-detection belief
state. Conversely, a high value for p would put more emphasis on quickly moving to the
observation point. This value could be determined by factors such as how important it is
for the UUV to get to the observation point quickly or how much of a threat the contact
poses if it does counter-detect the UUV.
Figures 5.2 and 5.3 show plots of the ISR Maneuver Decision Aid Objective
function, fobj, with p values of .1 and .5 respectively. In this particular example the UUV
state was
0
X v[k]=0
0I
[ Radians
(180 Degrees Measured From Positive X - Axis)
10 Meters / Second
113
the estimated contact state was
4000 Meters
10
A
Xc[k
Meters
Radians (180 Degrees)
12 Meters/Second
and the observation point was
-OBS
4000
L
Meters
10 Meters
J
The commanded time duration of the course command was fixed at 120 seconds. The
contours in the plot correspond to the location that a potential course command would
have moved the UUV to. These locations were plotted against the objective function,
defined in this section. Of note in these plots is how with a lower p value of .1, the
objective function favored recommending that the UUV turn left or right to see if the
contact would follow it. With the higher p value of .5 the objective function favored
heading more towards the observation point.
114
fobj,
Plot of ISR Maneuver Decision Aid With p=.1
0.75
0.7-
0.65
Obseivation
Point
UUV Position
Contact Position
0.6
4000
0.55
-
Initial UUV
Cowse
0.5
-4%
0 0 1000
2000
m
Corse.
3000
tact
4000
2000
-
-2000
2000
-4000
Y-Axis (Meters)
X-Ais (Meters)
Figure 5.2 Plot of ISR Maneuver Decision Aid Objective Function With p=.1
Plot of
ISR
Maneuver Decision Aid With p=.5
0.9
0.85Observation
Point
0.8
UUV Position
4000
Znitial Contact
Course
0.75
2000
0
0.7Initial UUV
Contact Position
Course
0.65
-4000
-3000
-2000
-2000
-4000
-1000
0 1000
2000
3000
Y-Axis
4000
X-Avis /Mftpr%)
Figure 5.3 Plot of ISR Maneuver Decision Aid Objective Function With p=. 5
115
5.3 1 Time Horizon ISR Maneuver Decision Aid Simulation Results
The following sets of example simulations have the same motion conditions,
tracking conditions, and counter-detection belief state update procedures as in the
examples in Chapters 3 and 4, but tests an ISR maneuver decision aid that uses the
objective function developed in this chapter. The belief state was updated after every 120
second time window. After each update, the UUV followed the course recommendations
of the maneuver decision aid that minimized the objective function, f'bj, presented earlier
in Equation 5.8 over a discrete set of action choices consisting of a course command and
a length of time to maintain the commanded course such that
n x.085 Radians (n x 5 Degrees)
120 Seconds
n = 0,1,2,...,72
n k =integer
In other words, every 120 seconds the belief state was updated and the maneuver decision
aid provided a course command that was a multiple of 5 degrees and would be
maintained until the next belief state update at the end of the subsequent 120 second time
horizon. Various weighting parameter values were used to see how the changes affected
the behavior of the ISR maneuver decision aid. For all of the simulations the Cartesian
coordinates of the observation point were (-15000,0). It would be expected that for small
values of the weighting parameter, p, the UUV would pay little attention to moving
towards the observation point until it was certain of whether the contact had counterdetected it or not. As p became larger the UUV would become less concerned with
determining the intentions of the contact and move more directly to the observation point.
For intermediate values of p there should be some balance between the two requirements.
116
5. 3.1 Example Simulation Set 5.1 (Hostile Contact and 1 Time
Horizon ISR Maneuver Decision Aid)
Figures 5.4 through 5.13 show the results of 5 simulations with the one time
horizon ISR aid, a hostile contact, and initial UUV and contact states being
0 Meters
4000 Meters
Meters
3.14 Radians (180 Degrees)
10 Meters/Second
XC[o01=[
xV[o1=0
1000 Meters
-
3.26 Radians (187 Degrees)
12 Meters/Second
The values used for the weighting parameter, p, were .1, .5, 1, 2, and 10.
Plot
3500
of
UUV
and Cortsd
P0t
P.850.
.4
2000
2000
of
Befe
is HOW&.
That Cofte
0.7
Irtal or0ac
0.8
000
00<t
Obsemfvaidn
-1000
2000
1500
---
2-
Poito
-1000
-5=0
X-Wd8 WeM4*4)
1 -I
0
I.
5000
Figure 5.4 Example Simulation Set 5.1
UUV and Contact Positions
1
p =.
0
500
1500'
200M
20
igure 5.5 Example Simulation Set 5.1
Belief States
1
p =.
Plot of
06fT Corit.ot is Hosile
0.9
0.8
Pot
ofUVWIcntdPsbr
07
05
50
3500
-0080
-
1000 P0.41
7
--
8
UUV
-20000
-1500
~
Cntac
-10000
.000
8-a0d. (M~ars)
-
~.
00010
and
Posions A
0
0
00
1000
1500
2000
2500
Tirr. (Seconds)
Figure 5.6 Example Simulation Set 5.1
UUV and Contact Positions
P=. 5
Figure 5.7 Example Simulation Set 5.1
Belief States
p=. 5
117
Plot
of UUV and Contact Position
Plot of Befef That Contad
is Hoste
3500
-- -. Iuv
Contact
3000
0.8-
2000
0.72000
*I
o0ac
0.6-
1500
0.51000
-
500
---
-
0.3
0
0.2
Pi-
~00
-1500
-
Initial
UUjV
Position
Observaion
-5W0
-1000
-50
0
0
5000
500D
1000
Time
X-Ooxs (Meters)
Confbod
SOy .5V6n
2000
2500
Figure 5.9 Example Simulation Set 5.1
Belief States
Figure 5.8 Example Simulation Set 5.1
UUV and Contact Positions
p= 1
P0011
150
(Seconds)
p=
Pil
P01it10n
o0B0.401Th1
Contad
1
iS
Ho01118
3500
0.1 -
3000
0.8
2500
0.7
2 00
+
1000
-
0.5
~
-
.oiton
0.4
03-
0
---
n
-
0-2-
-IUU
--*
0.1
-1"
M0
-15
1000
5000
x--
1
0
0
50
of
UUV
1000
1500
2000
20
Figure 5.11 Example Simulation Set 5.1
Belief States
Figure 5.10 Example Simulation Set 5.1
UUV and Contact Positions
Plot
W0
and Contad Position
Pk8of
b7ef
That
Contsd
Hostile
0s
2000
0.9
1500
2500
0.0
0.7
2000
n
act
0.6
0.5,
1000
0.4
Soo
0.3
-500
0.2
-/
P086 1UV
-15000-1000
.5000
0t500
0
500
1000
1500
2000
2500
X-mxs (M4.6rs)
Figure 5.12 Example Simulation Set 5.1
UUV and Contact Positions
Figure 5.13 Example Simulation Set 5.1
p=10
p =10
Belief States
The results of these simulations are similar to what would be expected of them.
For low values of p, which would cause the UUV to be more concerned with determining
if counter-detection occurred, the UUN seemed to maneuver without regard for where the
118
observation point was initially and tried to force the contact to maneuver and follow it.
Once the UUV was certain that the contact had counter-detected it, the UUV proceeded
to the observation point. As p took on larger values, the UUV became more concerned
with reaching the observation point quickly and it took the belief state longer to reach one
than for the cases with lower p values. One interesting behavior shown when the value of
p was 2, was how the UUV compromised between forcing the contact to make significant
maneuvers to follow and still trying to head to the observation point. The UUV did not
travel as far up the positive y-axis, as in the simulations for lower values of p.
5.3.2 Example Simulation Set 5.2 (Hostile Contact and 1 Time
Horizon ISR Maneuver Decision Aid)
Figures 5.14 through 5.23 show the results of 5 simulations with the one time
horizon ISR aid, a hostile contact, and initial UUV and contact states being
10000 Meters
10000 Meters
xC[0]= 3.97 Radians (225 Degrees)
12 Meters/Second
0 Meters
0 Meters
x [o]= 3
3.14 Radians (180 Degrees)
10 Meters/Second
The values used for the weighting parameter, p, were again .1, .5, 1, 2, and 10.
P101 of UUV and C4044o P010100
Plot of Babel That Cortat is Hostile
II
1000
0.91
math Cotdc
0.8
10000
06 -- Obt.
bo /
46
\ oii
02
I'
Po"0
-1.5
.1
-0.5
0
05
Is1
50
Figure 5.14 Example Simulation Set 5.2
UUV and Contact Positions
p =. 1
100
150o00
250
Figure 5.15 Example Simulation Set 5.2
Belief States
p =. 1
119
Plot of UUV and Contact Position
12000
-
Plot
Inil Contact
Postion
cuuv
of Beief That Contact
is Hoste
09
10000 -
8000-
07
6000
0.5
4000
0.3
0.2
Observation
Initial
-1
-1.5
01
V
V
Position
Point
-0.5
0
500
1.
1
0.5
X10
X-5xis (Meters)
Figure 5.16 Example Simulation Set 5.2
UUV and Contact Positions
p=. 5
1500
1000
2000
2500
Tirne (Seconds)
Figure 5.17 Example Simulation Set 5.2
Belief States
p=. 5
Plot of UUV end Contect Position
12000 -
Plot of 0e.ef That Contact
is Hosile
I. 10000-
Inibel Contact
0.1
Postion
8000-
0.7-
60000-
4000-
04 -
-
200
,7
0.3 -
0
Observation
-1.5
-1
011
Positio
-0.5
0
X-sxs (Mters)
0.5
1
oL
1.5
XW0'
EUV
Plot
000
Toml
Figure 5.18 Example Simulation Set 5.2
UUV and Contact Positions
p= 1
12000
500
2000
100
2500
(Seconds)
Figure 5.19 Example Simulation Set 5.2
Belief States
p= 1
Plot of Belef That Contect is
of UU(V find Cotact Position
Initial Contact
Posir.on
Hostile
0.9
10000
8000
0.
-
0.6
6000-
i
05
4000-
0.4
0,3
2000-
0.2
0 -
-U
01
Obsemton0
Point
-2000
15
-1
-05
0
X-eAxs (Meters)
0,5
1
0
15
x1'
Figure 5.20 Example Simulation Set 5.2
UUV and Contact Positions
p=2
00
1000
1500
Tane (Seconds)
2000
Figure 5.21 Example Simulation Set 5.2
Belief States
p=2
120
2500
Plot
of
UUV
and Contact
Plot
Position
12000
of Belief That Contact is Hostile
09
Conact
10000
0.8
8000
0.7
0.6
6000
0,5
4000
t
Poi0-
0.4
tion
0.3
2000
Inill UUV
Posiion
0.2
0
-2
-I5x
-1
-0.5
0
0.5
X-aa(Mters)
1
0
1.5
aiD
500
000
1500
2500
2000
Tin, (Sanonale
Figure 5.23 Example Simulation Set 5.2
Belief States
Figure 5.22 Example Simulation Set 5.2
UUV and Contact Positions
p=10
p=10
The results of these simulations show the tradeoffs in choosing a value for p. The
lower the value of p, the more the UUV maneuvered away from the observation point to
force the contact to maneuver in response to it. In the case of p = .1 the UUV did not
even reach the observation point by the end of the simulation. As p became higher, it
took less time for the UUV to get to the observation point, but longer for the belief state
to reach one.
5.3.3 Example Simulation Set 5.3 (Passive Contact and 1 Time
Horizon ISR Maneuver Decision Aid)
Figures 5.24 through 5.33 show the results of 5 simulations with the one time
horizon ISR aid, a passive contact, and initial UUV and contact states being
[3.14
0 Meters
0 Meters
Radians (180 Degrees)
10 Meters/Second
XC [o]=
[
4000 Meters
1000 Meters
3.26 Radians (187 Degrees)
12 Meters/Second
The values used for the weighting parameter, p, were again .1, .5, 1, 2, and 10.
121
0
Plot
2.5 -
F-
Conuc
of UUV
"n Contact Position
Plot of Beief That Contet
j
is Hostile
0.9
20.8
1 51--
j/
0,6
05
I
nitial Corted
Position
I'
0.5
04
Obetion
013
point
02
01
Initial UUV
Positio
--20000
-15000
-10000
-5000
X-axis (Meters)
5000
0
500
Tie
Figure 5.24 Example Simulation Set 5.3
UUV and Contact Positions
2.5
1500
(Seconds)
20
2500
Figure 5.25 Example Simulation Set 5.3
Belief States
p =. 1
Plot of UUV and Contact Position
r la
1000
Plot of
1
Belef
That
Contact
is Hote
0.9
2
078
1.5
0.6
KI
1I
I
Honf
0,5
0tial Contad
Position
as
03
0.2
0
0.1
--Posion
-15000
-26000
-10000
X-ads
-5000
(Meters)
0
500
5000
Twne
Figure 5.26 Example Simulation Set 5.3
UUV and Contact Positions
p=.5
Plot
2.5r
1000
-
1500
2000
2500
(Seconds)
Figure 5.27 Example Simulation Set 5.3
Belief States
P=.5
Plot of
of UUV and Coted Position
Beief That
Contact Is Hostile
1
- -- Contect
0.1
2
0.8
0.7
1.5
0.6
I
nitial Contac
Position
0.4
0.5
0.3
Oberndione
Point
0.2
0
-Initial
-20000
-15000
-10000
-5000
0
UUV
0.1
5000
0
500
1000
1500
2000
2500
X-axds (Meters)
Tie (Seconds)
Figure 5.28 Example Simulation Set 5.3
UUV and Contact Positions
p= 1
Figure 5.29 Example Simulation Set 5.3
Belief States
p= 1
122
le
Plot of UUV
2.5-
and Contact
Position
Plot 0f 0060 That Contacts Host
Contact
0.9
2-
0.80.7
1.5-
0.0
0.5
cnal Contact
Position
Obsertim8o
Point
0,5
0.4
intil8
01
-~2ooo
15000
-100M
-(M
5"
X-6*s (Mets)
0
0
5000
Plot
of
UUV
and Contad
000
1500
2000
2500
1rtoe (500088$s)
Figure 5.30 Example Simulation Set 5.3
UUV and Contact Positions
p=2
2.10'e
500
Figure 5.31 Example Simulation Set 5.3
Belief States
p=2
Plot 01 Oe0ef That Contact is
Postion
sonyd
H0t1
0.1
0,8
07
1.5
0A
0.5
0.4
Iitial Cortact
Position
0.5
0,3
otion
0
02
--
0.1
Mal UUV
Position
0
-
00
X-"s
-5000
0
5000
0
500
1000
1500
2000
2500
Too. (Secoods)
Figure 5.32 Example Simulation Set 5.3
UUV and Contact Positions
p=10
Figure 5.33 Example Simulation Set 5.3
Belief States
p = 10
The results of these simulations show similar behavior to the previous set of
simulation. There appears to be no difference in the behaviors of Example Simulation
Set 5.2 for p values of .1, .5, and 1. The UUV turned to see if the contact would follow it.
When the contact continued on its original course and the UUV was certain that counterdetection had not occurred, the UUV proceeded to the observation point. For p = 2 the
UUV initially turned to see if the UUV would follow, but didn't turn as sharply away
from the observation point as for the lower values of p. When p was 10 the UUV
transited straight to the observation point and it took longer for the UUV to determine the
contact was passive.
123
5.3.4 Example Simulation Set 5.4 (Passive Contact and 1 Time
Horizon ISR Maneuver Decision Aid)
Figures 5.34 through 5.43 show the results of 5 simulations with the one time
horizon ISR aid, a passive contact, and initial UUV and contact states being
0 Meters
-8000 Meters
Meters
Meters
L.781
785 Radians
=0
X~, [O]=,
-8000
3.14 Radians (180 Degrees)
(45 Degrees)
10 Meters/Second
12 Meters/Second
The values used for the weighting parameter, p, were again .1, .5, 1, 2, and 10.
Plot
of UUV
and Contact
Position
Plot
2000,
1000
-- UU
Initial UUV
0.9
Positon
0
of Belef That Contact Is Hnstie
0.8
-1000
0.7 -
Observatone
-2000
0,6
-3000
-
-4000
---
- -
0.30.2Contact
-7000 k
-2
-1.5
-1
-0.5
0
0.1 0.5
1
15
X-eras (Meters)
0
02
1500
2000
2500
Figure 5.35 Example Simulation Set 5.4
Belief States
p = .1
p =.1
Plot of
1000
Tirne (Seconals)
Figure 5.34 Example Simulation Set 5.4
UUV and Contact Positions
2000
500
Plot of Beief That Contact is Hostile
UUV and Contt Position
L~
I
1000
0.9
0-
0.0
-1000
0.7
Obsactanon
Point
-2000
0.6
-3000
05
4000
0.4
-5000
0.3
-7000
W"
0.2
Cordd
-7000[
0.1
-2
-15
-1
-0.5
0
05
X-avs (Meters)X10
1
1.5
500
2
Figure 5.36 Example Simulation Set 5.4
UUV and Contact Positions
p=. 5
1000
1500
Terre (Seconds)
2000
2500
Figure 5.37 Example Simulation Set 5.4
Belief States
p=. 5
124
Ptot
of UUIV and Contadt Position
Plot
Contact
1000
of Belef That
Contact
is Hostile
09
0
0.8
-1000
07
Ob,4ewaons
P000
*
-2000
0.6
-3000
0.5
-4000
04
-5000
0.3
0,2
-6000
01
Initial Cortact
-7000
Position
-2
-1.5
-1
-05
0
05
1
1.5
X-av4s (Meters)
500
2
Xto'
and Contact
1500
inie
2000
2500
(Seconds)
Figure 5.39 Example Simulation Set 5.4
Belief States
p= 1
Figure 5.38 Example Simulation Set 5.4
UUV and Contact Positions
p= 1
Plot of UUV
1000
Position
Plot of
1
2000
Begot That
Contact is Hostile
Cott
0.9
lniWia UUV
0.8
0
0.1
-1000
--
Observation
POWI
06
2000
-300
0.5
i/f
-4000
0.4
0.3
-5000
-6000
0.2
Initial Contact
I
-7000
GA,
Position
-2
-15
-
-05
0
X-aois (Meters)
005
1
1.5
2
y 00
500
Figure 5.40 Example Simulation Set 5.4
UUV and Contact Positions
p= 2
Plot
2000-
of
1500
1000
2000
2500
Tine (Seonds)
Figure 5.41 Example Simulation Set 5.4
Belief States
p= 2
UUV and Contact Position
Plot of
B6.ef
That
Contact
is
Hostile
1
:
Initial UUV
1000-
Contact
09
- oito
0-
0.8
0.7
Observorw
Poirt
-0000
I
\'~
0.6
-2000
0.5
-3000
0.
'F
0.4
0.3
-5000
Poito
-6000-
0.2
Initial contact
-7000
-2
-1.5
-1
45
X-ax0s
0
(Meters)
01
0.5
1
1.5
X
0L
2
W0'
500
1000
T0ne
Figure 5.42 Example Simulation Set 5.4
UUV and Contact Positions
p = 10
1500
2000
2500
(Seonds)
Figure 5.43 Example Simulation Set 5.4
Belief States
p = 10
125
These results show similar behaviors to the previous simulations in this chapter.
Although there was little difference in how quickly the belief state went to zero for the
different p values, the UUV initially maneuvered more directly towards the observation
point as the value of p rose.
5.4 Summary
The results in this chapter illustrate how the algorithms developed in Chapters 3
and 4 could be used in a potential ISR mission while transiting to an observation point.
The ISR maneuver decision aid presented in this chapterwas able to balance the need to
transit to the observation point while also looking to see if a contact has counter detected
it. How much balance there is can be determined by the setting the weighting parameter,
p, to an appropriate value. That value could be set according to how urgent it was to
reach the observation point or how much of a threat is posed by the contact.
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Chapter 6
Conclusions and Future Research
This chapter summarizes the contributions of this thesis and presents suggestions
for future research related to this thesis.
6.1 Thesis Contributions
This thesis presented decision-making algorithms that can assist a Navy UUV in
accomplishing an Intelligence/Surveillance/Reconnaissance (ISR) type mission where it
must transit to an observation point to collect intelligence. Chapter 2 reviewed Extended
Kalman Filter (EKF) tracking algorithms that could be used in computer simulated-tests
to model realistic tracking of contacts in the vicinity of a UUV. An algorithm that
periodically updates a UUV's counter-detection belief state was presented in Chapter 3.
The counter-detection belief state is the probability that a contact has detected the
presence of the UUV. The algorithm used basic laws of probability to define
probabilistic models of hostile (counter-detection occurring) and passive (no counterdetection) contact behavior, analyze the behavior of a contact and update the belief state
using Bayes' rule. The algorithm recognized hostile behavior whenever a contact
maneuvered directly towards the UUV and recognized passive behavior whenever the
contact appeared to mind to its own business and not move towards the UUV. Chapter 4
presented a maneuver decision aid that enabled the UUV to make intelligent maneuver
decisions and quickly recognize if counter-detection had occurred. This was
accomplished by recommending maneuvers for the UUV that would force a contact to
quickly show its intentions. Finally, Chapter 5 provided examples of how the maneuver
decision aid presented in Chapter 4 could be used in an ISR mission for a UUV.
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6.2 Future Research
There are several possible avenues for future research related to the work of this
thesis. One is to model other potentially hostile behaviors into the probabilistic model of
a hostile contact presented in Chapter 3. The algorithms presented in this thesis only
looked at whether a contact was moving directly towards it in order to determine if
counter-detection has occurred. A hostile contact can pursue the UUV in other manners
not investigated in this thesis. Figures 6.1 and 6.2 illustrate this shortcoming. In this
example, the contact paralleled the course of the UUV without moving directly towards
the UUV. Quick analysis of this situation would probably lead one to suggest that the
contact was trailing the UUV, but the UUV failed to recognize this as hostile behavior.
Plot
of UUV and ContOd
P-PCs
5000
P
nidi
30D
1000
0
0.25
+-
i
03
,_
2000
*Ban
-f The) Contt
05
Contact
---
- -
--
What UUV
0,15
0.1
-3000
(000
-400.0
-2.5
-2
-1.5
-
0.5
0
O's
X-Axms (Meters)
x
0
1.5
e0
500
1000
1508
20D
2500
Tbme (Sfcond)
Figure 6.1 Example Simulation UUV
Figure 6.2 Example Simulation
And Contact Locations
Belief States
To evaluate the utility of the decision aids presented in this thesis one could interview
people with submarine experience in the Navy to see how the UUV maneuvers in the
simulations compare to what someone with real world experience would do in those
situations if a submarine were involved, instead of a UUV. Other possibilities for future
work include investigating how the algorithms presented in this thesis perform in
simulations in an environment with obstacles instead of the open environments used in
this thesis. One could also look into how the work presented in this thesis could apply to
scenarios with multiple contacts present. Another area of interest would be to apply
128
alternate objective functions in the maneuver decision aids presented in Chapters 4 and 5.
Also, one could look into how combinations of future actions would affect the counterdetection belief state instead of looking at only one leg of a trajectory as was done in this
thesis.
129
130
Bibliography
[1]
P. Dunn. "Navy UUV Master Plan". Proceedingsof the
International
Unmanned Undersea Vehicles Symposium, Newport, RI, April 2000.
[2]
S. McKeever. Path Planningfor an Autonomous Vehicle. Master of Science
thesis, Massachusetts Institute of Technology, June 2000.
[3]
Col. J Boyd, United States Air Force (Ret). Decision Making, January 2003.
(http://www.mindsim.com/MindSim/ corporate/OODA.html/)
[4]
Y. Bar-Shalom and X. Li. Estimation and Tracking: Principles, Techniques, and
Software. Artech House, 1993.
[5]
A. Gelb. Applied Optimal Estimation. MIT Press, 1973.
[6]
J. Fenwick. CollaborativeConcurrentMapping and Localization.Master of
Science thesis, Massachusetts Institute of Technology, May 2001.
[7]
R.G. Brown and P.Y.C. Hwang. Introduction to Random Signals and Applied
Kalman Filtering.John Wiley & Sons, New York, 1997.
[8]
V. Aidala and S. Hammel. Utilization of Modified Polar Coordinates for
Bearings-Only Tracking. IEEE TransactionsOn Automatic Control, 1983.
[9]
D. Bertsekas and J. Tsitsiklis. Introduction to Probability. Athena Scientific,
Belmont, Massachussetts, 2002.
[10]
H. Hsu. Schaum's Outline of Theory and Problems of Probability,Random
Variables, and Random Processes. McGraw-Hill, 1997.
131
[11]
T. Cover and J Thomas. Elements of Information Theory. Wiley, 1991.
[12]
J.N. Kapur and H.K. Kesaven. Entropy Optimization Principles.Academic Press,
Boston, 1992.
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2/~cZ
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