concepts of operations for the side scan sonar autonomous underwater vehicles ... · 2012. 8....

Defence R&D Canada – Atlantic

DEFENCE DÉFENSE&

Concepts of Operations for the Side Scan

Sonar Autonomous Underwater Vehicles

Developed at DRDC Atlantic

Bao Nguyen and David Hopkin

Technical Memorandum

DRDC Atlantic TM 2005-213

October 2005

Copy No.________

Defence Research andDevelopment Canada

Recherche et développementpour la défense Canada

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Concepts of Operations for the Side Scan Sonar Autonomous Underwater Vehicles Developed at DRDC Atlantic Bao Nguyen and David Hopkin

Defence R&D Canada – Atlantic Technical Memorandum DRDC Atlantic TM 2005-213 October 2005

Abstract

This study reports findings on novel concepts of operations for Autonomous Underwater Vehicles (AUVs) conducting mine hunting missions. The AUVs that we consider are those carrying a side scan sonar and have been developed at DRDC Atlantic. We examine the use of one and two AUVs each sweeping a lawn mowing pattern or a zigzag pattern. Two independent probabilistic models were built in order to assess and compare the Measures Of Effectiveness (MOEs) of these configurations. Through execution of these models and the nature of the Measures of Performance (MOPs) of the side scan sonar, the optimal configuration was found as well as a heuristic proof showing why it is optimal. The paper also describes variational calculus methodologies on how to obtain the optimal search path based on target density distribution in general. Exact equations of motion for the optimal path are found for a specific type of mine density distributions along the vertical axis alone, and along both the horizontal and vertical axis. The optimal paths are then compared to the proposed operations in terms of MOEs.

Résumé

L’objet de cette étude est de présenter les nouveaux concepts opérationnels du AUVs (the Autonomous Underwater Vehicles) au DRDC Atlantic. Nous considérons 1/ l'usage d'un AUV et 2/ celui de deux AUVs , l'un selon le modèle "tonte de pelouse", l'autre selon celui du "zigzag ". L'évaluation et la comparaison des Mesures d'Efficacité (Measures of Effectiveness, MOEs) de ces configurations sont faites selon deux modèles probabilistiques indépendants. L’exécution de ces modèles et l’utilisation des Mesures de Performance (MOPs) du sonar nous donnent la configuration optimale. Notre étude décrit également la méthode d’obtenir la voie de recherche optimale basée sur la distribution de la densité des cibles en genéral. Les équations precises des mouvements pour la voie optimale sont établies pour une spécifique distribution selon l'axe vertical seulement, et selon les deux axes vertical et horizontal ensemble. Les voies optimales sont comparées aux concepts opérationnels en terme de MOEs.

DRDC Atlantic TM 2005-213 i


ii DRDC Atlantic TM 2005-213

Executive Summary

As part of the Underwater Warfare Program, DRDC Atlantic is suggesting several activities on maritime force protection. One objective will be to examine the role of Autonomous Underwater Vehicles (AUVs) in detecting, classifying and identifying underwater threats such as mines. In support of this project, we have conducted this operational research study to examine the utility of AUVs. To improve the detection probability, we have proposed a number of novel concepts of operations for AUVs.

A stochastic simulation and a deterministic model were built to determine the effectiveness of these concepts and hence AUVs’ operational value. To do that, we consider the use of one and two AUVs each sweeping a lawn mowing pattern or a zigzag pattern. It was found that the configurations based on two AUVs improve the Measures Of Effectiveness (MOEs) substantially relative to those based on a single AUV. This is so as the two AUV configurations offer a better probability of detection by collecting data on a mine like object from more than one aspect angle. It was also found that the configuration of two AUVs each sweeping a mowing pattern that is perpendicular to one another provides the best MOEs. This optimality is mainly due to the fact that the angular probability of detection of a mine is maximal if that mine is detected by two AUV legs that are orthogonal to one another. This result is true in general if the shape and symmetry of the probability of detection as a function of angle remains the same as the one we have assumed in this paper.

We have also developed an analytical methodology to generate the optimal path of an AUV based on a priori mine density distribution in the search area. Exact solutions to this problem were derived using variational calculus. We have determined that a specific type of mowing pattern optimizes the probability of detection under a general class of mine density distributions. The improvement in effectiveness between the regular mowing path and the optimal mowing path was obtained. We have also derived the exact solutions for the optimal path based on a more complex mine density distribution. It is a non-trivial solution whose significance is still under investigation and will be reported in the near future.

The main contribution of this study is to provide the Canadian Navy and NATO allies a robust concept of operations for AUVs: one that achieves a high level performance and is implementable in real life. The scope of this study was limited to a number of features for AUVs such as the use of a side scan sonar and the assumption of an ideal environment. In the future, it will certainly be valuable to review these assumptions and further the capacity of our modelling and simulation. This could include modelling of the seabed, investigations of other types of sonar and mine size, material, and cross section.

Bao Nguyen, and David Hopkin. 2005. Concepts of Operations for the Side Scan Sonar Autonomous Underwater Vehicles Developed at DRDC Atlantic. DRDC Atlantic TM 2005-213. Defence R&D Canada – Atlantic.

DRDC Atlantic TM 2005-213 iii

Sommaire Comme partie du Projet de guerre sous-marine, le RDDC suggère plusieurs activités de protection de la force maritime. Un des objectifs sera d’étudier le rôle des AUVs (Autonomous Underwater Vehicles) dans la détection, la classification et l’identification des menaces sous-marines telles que les mines. Pour soutenir ce projet, nous avons développé de nouveaux concepts d’opération pour les AUVs. Ces concepts permettent de déterminer l’efficacité des AUVs, et de là leur valeur opérationnelle. Pour ce faire, nous avons considéré l’utilisation d’un (et de deux) AUVs selon les modèles de « tonte de pelouses » et du « zigzag ». On a trouvé que les configurations de deux AUVs augmentaient les mesures de l’efficacité (MOEs) de façon significative du fait qu’elles offrent une meilleure probabilité de détection en recueillant les données sur les mines à partir de plusieurs angles. On a également découvert que la configuration de deux AUVs , dont les modèles de « tonte » sont perpendiculaires l’un à l’autre, fournissait la meilleure mesure de l’efficacité (MOE). Cette optimalité est due principalement au fait que la probabilité angulaire de détection d’une mine est maximale quand celle-ci est détectée par les AUVs dont les trajets forment un angle droit. Ce résultat est valable en général si la probabilité angulaire de détection garde la même symétrie et la même forme que celle qu’on analyse. Nous avons construit les outils en vue de produire la voie optimale de l’AUV basée sur une distribution de densités de mines dans le secteur de recherche. Les solutions précises de ce problème proviennent de l’utilisation du calcul des variations. Nous avons déterminé qu’un type spécifique de modèle de ‘tonte’ optimisait la probabilité de détection dans le cadre d’une classe générale de distribution de densités de mines. Ces densités sont une fonction d’une coordonnée verticale y et ne s’élèvent pas quand y s’éloigne de la ligne de demi-largeur du secteur de recherche. On obtient l’amélioration de l’efficacité en passant de la voie de la tonte régulière à celle de la tonte optimale. Nous avons aussi dérivé les solutions précises d’une voie optimale en nous basant sur les distributions de densités exponentielles suivant les axes horizontal et vertical. Il s’agit là d’une solution significative d’une équation différentielle du second degré non linéaire. L’importance de cette solution est encore à l’étude et fera l’objet d’un exposé à venir.

Cette étude apporte des concepts concrets qui décrivent comment la marine Canadienne et les pays d’OTAN peuvent se servir des AUVs pour chercher les mines. Ces concepts produisent d’excellents MOEs et peuvent être implémentés dans les opérations réelles. Cependant, ces résultants sont obtenus basant sur quelques hypothèses tels que les MOPs du sonar et un environnement idéal. Dans le futur, nous allons révisiter ces hypothèses pour élargir la portée de nos modèles. Ceci comprend la modélisation du terrain, les différent types de sonar, et les charactéristiques des mines.

Bao Nguyen and David Hopkin. 2005. Concepts of Operations for the Side Scan Sonar Autonomous Underwater Vehicles Developed at DRDC Atlantic. DRDC Atlantic TM 2005-213. Defence R&D Canada – Atlantic.

iv DRDC Atlantic TM 2005-213

Table of contents

Abstract........................................................................................................................................ i

Executive Summary................................................................................................................... iii

Sommaire................................................................................................................................... iv

Table of contents ........................................................................................................................ v

List of figures ........................................................................................................................... vii

List of tables .............................................................................................................................. ix

Acknowledgements .................................................................................................................... x

1. Background.................................................................................................................... 1 1.1 The first phase .................................................................................................. 2 1.2 The second phase.............................................................................................. 3

2. Measures Of Performances (MOPs).............................................................................. 5

3. Search Patterns .............................................................................................................. 7

4. Modelling and Algorithms .......................................................................................... 13 4.1 Stochastic Model ............................................................................................ 13 4.2 Deterministic Model....................................................................................... 14 4.3 MOEs.............................................................................................................. 16

5. Numerical Results ....................................................................................................... 17

6. Optimal Search Paths .................................................................................................. 26 6.1 Uniform Distribution ...................................................................................... 26 6.2 Exponential Distribution ................................................................................ 26

6.2.1 Assumptions ...................................................................................... 27 6.2.2 Along the vertical axis....................................................................... 27 6.2.3 Along the horizontal axis and the vertical axis.................................. 36

DRDC Atlantic TM 2005-213 v

7. Conclusions ................................................................................................................. 38

8. References ................................................................................................................... 39

Annex A – Optimal Observation Angle Between Two AUV Legs.......................................... 40

Annex B – Vertical Exponential Distribution .......................................................................... 44

Annex C – Vertical and Horizontal Exponential Distributions ................................................ 49

List of symbols/abbreviations/acronyms/initialisms ................................................................ 53

Glossary.................................................................................................................................... 54

Internal distribution list ............................................................................................................ 55

External distribution list ........................................................................................................... 56

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List of figures

Figure 1. The Dorado.................................................................................................................. 2

Figure 2. Coordinate system. ...................................................................................................... 3

Figure 3. Probability of detection as a function of range. .......................................................... 6

Figure 4. Probability of detection as a function of aspect angle................................................. 6

Figure 5. Lawn Mowing search pattern. ..................................................................................... 7

Figure 6. Zigzag search pattern. ................................................................................................. 7

Figure 7. Mine observed at 85 degrees....................................................................................... 9

Figure 8. Mine observed at 0 degrees.. ....................................................................................... 9

Figure 9. Mine aspect angles. ................................................................................................... 10

Figure 10. Mine aspect symmetrical angles. ............................................................................ 10

Figure 11. Two AUVs employing perpendicular mowing search patterns. ............................. 11

Figure 12. Two AUVs employing weaving patterns. ............................................................... 12

Figure 13. Stochastic simulation. ............................................................................................. 14

Figure 14. Detection Level. ...................................................................................................... 15

Figure 15. MOEs of an AUV employing the mowing pattern as a function of search time..... 18

Figure 16. Anomaly due to minimal detection range. .............................................................. 20

Figure 17. Probability of detection for four AUV configurations as a function of search time.21

Figure 18. The detection rate (R) for four AUV configurations as a function of search time.. 22

Figure 19. Coverage for four AUV configurations as a function of search time...................... 23

Figure 20. Mean angular probability of detection. ................................................................... 24

Figure 21. Geometry for maximal angular probability of detection......................................... 25

Figure 22. Target density distribution as a function of (the vertical position in the search area).............................................................................................................. 27

y

DRDC Atlantic TM 2005-213 vii

Figure 23. Exponential density distribution along the axis. ................................................. 28 y

Figure 24. AUV path in a sub box area. ................................................................................... 29

Figure 25. Optimal path (dotted lines) in a sub box area.......................................................... 30

Figure 26. Asymptotic solutions to the Euler-Lagrange equation. ........................................... 32

Figure 27. Bounded asymptotic solutions to the Euler-Lagrange equation.............................. 33

Figure 28. Exponential density distribution along the x axis and the axis........................... 34 y

Figure 29. Probability of detection as a function of search time (exponential density along ). ........................................................................................................................... 35 y

Figure 30. Detection rate as a function of search time (exponential density along ). ............ 36 y

Figure 31. Exponential density distribution along the x axis and the axis........................... 36 y

Figure 32. Mean angular probability of detection. ................................................................... 40

Figure 33. Probability of detection as a function of redefined angle........................................ 41

Figure 34. AUV path in a sub box area. ................................................................................... 44

Figure 35. Target density distribution as a function of (the vertical position in the search area)........................................................................................................................ 47

y

Figure 36. Optimal path (dotted lines) in a sub box area.......................................................... 48

viii DRDC Atlantic TM 2005-213

List of tables

Table 1. Configurations. ........................................................................................................... 17

Table 2. Scenario parameters and side scan sonar MOPs. ....................................................... 17

DRDC Atlantic TM 2005-213 ix

Acknowledgements

The authors would like to thank Peter A. Smith of MARLANT/DCOS OR for providing his surveillance model upon which our stochastic model was built. We would also like to thank Dr. James L. Kennedy, Chief Scientist of DRDC Atlantic, for his detailed review, which improves substantially the quality of this Technical Memorandum.

x DRDC Atlantic TM 2005-213

1. Background Since the early 1980s, the Defence Research and Development Canada (DRDC) Agency has provided research and development support to the Canadian Forces in the area of mine countermeasures. This work has primarily focused on the use of high resolution side scan sonars to image the sea bottom, and through subsequent analysis techniques, determine the presence of mine like objects. Much of this early work was the foundation for the development and acquisition of the current high performance AN/SQS-511 multi-beam side scan sonar that is currently employed by the Canadian Forces for route survey operations. Since this early work, DRDC has continued to leverage improvements in sonar technology and developments in computer based detection and classification algorithms to develop a state-of-the-art Remote Minehunting System (RMS) that provides a capability that previously could only be provided by a very expensive, dedicated mine hunting vessel. Shown in Figure 1, this system employs the Dorado semi-submersible vehicle (top) to tow a variable depth towfish (bottom) that carries a high performance multi-beam side scan sonar. Using pre-programmed mission plans, this system is able to survey a mine danger area and transmit the images of the sea bottom to a tactical control centre located many kilometers outside of the mine danger area. The use of a side scan sonar in this type of mine hunting scenario contrasts the use of traditional hull mounted forward looking sonars employed by a conventional mine hunting vessel. Systems like the RMS that employ side scan sonars typically have the advantage of producing a very high resolution image of the sea bottom for subsequent target analysis. However, each pass of the side scan sonar images the targets from only one aspect angle. Since the images of most mine shapes are very aspect dependant, conventional forward looking hull mounted sonar operations will typically circle completely around a potential mine sized target once detected. To address this limitation of side scan sonar based mine hunting operations, recent work has focused on evaluating the influence of aspect angle on the probability of detection for side scan sonar images, and how this aspect dependence influences the mission planning to obtain the best probability of clearance for a mine hunting operation. This document will describe a two-phased approach to examine the influence of aspect angle on the mission planning for AUV based mine hunting operations. We will define several measures of effectiveness (MOEs), and will show how these MOEs are affected by time, number of AUVs, detection range density, and detection angular density. This concept of operations will guide DRDC Atlantic in preparing trials, in collection of data and analysis of them at a later time. Note that some of the materials in this paper have appeared in [1].

DRDC Atlantic TM 2005-213 1

Figure 1. The Dorado.

1.1 The first phase In order to determine a given systems MOE, we have developed a high-level simulation environment. This environment allows the operator to specify a mission scenario and a vehicle and sensor configuration, either graphically or through the use of configuration files. Then, based on the results of the simulation, the operator is provided with a report that shows the systems MOEs, including cost effectiveness, for the selected mission. This high-level simulation environment looks at issues such as the probability of detection/classification/identification for a variety of mines, the time for detection/classification/identification, the cost, the number of AUVs required, and trade-off among these MOEs. Our objective functions can be multiple, constrained or unconstrained. At this development stage, the simulation models the following:

a. Mine – cylindrical shape;

b. Search pattern – speed, endurance, swath width, random walk, lawn mowing pattern, and weaving pattern (for more than one AUV);

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c. Probability of detection – target distance from AUVs, multiple overlapping coverages, and data fusion.

In addition to the above, we consider only ideal environments. However, it has been known to us that the NATO Undersea Research Centre (NURC) is developing a probabilistic model of the seabed, [2]. In the future, this will be integrated in our model of mine hunting.

1.2 The second phase The first approach provides MOEs based on simple search paths consisting of several straight-line segments. However, it is fortunate that, in mine hunting scenarios, we can develop tools to generate the optimal search path that maximizes for example the probability of detection of a mine. We do that by deriving the Euler’s differential equation from the Lagrangian representing the probability density of detection as a function of the search path. This density in turn depends on an assumed a priori distribution. In many simple cases, we can determine the exact equations of motion governing the optimal path of an AUV. For example, assuming an exponential distribution along the vertical axis leads to the mowing pattern while assuming exponential distributions both along the horizontal axis and along the vertical axis lead to a solution of a second order and highly non-linear differential equation. In general, we have developed a numerical methodology that can provide the optimal search path based on an assumed density distribution.

Figure 2. Coordinate system.


Figure 2 defines the coordinate system where the horizontal axis is called x and the vertical axis is called y. The blue lines represent the AUV path while the red circles are mines and the interior of the rectangle frame is the search area. The arrow in the search area indicates the general direction of the AUV. The vehicle starts at the bottom left, then travels from left to right, then travels upward, then travels right to left, then travels upwards and repeats.


2. Measures Of Performances (MOPs) In order to assess the effectiveness of the concept of operations, we need to make use of the side scan sonar Measures Of Performances (MOPs): the probability of detection as a function of range as shown in Figure 3 and the probability of detection as a function of aspect angle as shown in Figure 4. Note that the aspect angle is defined as the angle between the sonar beams and the axis of symmetry of a cylindrical mine. We assume that these two probabilities are independent. The probability of detection as a function of range represents a characteristic of the side scan sonar while the probability of detection as a function of angle represents a characteristic of the mine. These probability curves are modelled using the Johnson distribution [3] as it is fairly versatile in the sense that we can make it unimodal or bimodal, skewed to the left, symmetric, or skewed to the right as well as controlling how narrow each peak is. The scale ( )λ shown on each curve is a factor compounded to Johnson’s distribution such that the maximal value of each curve is equal to 1.

The Johnson’s curve can be expressed as:

( )

( )( ) ( )

21

1 22

1 ln22 2 1

1 2

1 2 2

0 o

x xx xx x

therwise

x x xex x x xf x

α αλ απ

⎛ ⎞⎛ ⎞−− ⋅ + ⋅⎜ ⎟⎜ ⎟⎜ ⎟−⎝ ⎠⎝ ⎠

⎧ ⋅ ⋅ −⎪ < <⎪ − ⋅ − ⋅= ⎨⎪⎪⎩

(1)

The range probability curve implies that if a mine lies between the minimal range and the maximal range then it will be detected with a probability close to 100 percent. The angular probability curve implies that the detection of the mine reaches a maximum when its aspect angle is perpendicular to the side scan sonar beam and its value is decreased symmetrically with respect to (wrt) this case. It is important to observe that these MOPs are the ones assumed in this paper. Their shapes and symmetries come from intuition and experience. However, they will be measured experimentally in the near future. The outcomes of this experiment will be used to perform the same analysis as the one presented here. In addition, these probabilities are independent of the AUV speed when it is less than 1 . For this reason, we assume that the AUV speed is constantly equal to .

0 knots9 knots


Probability of detection as a function of range

0

0.2

0.4

0.6

0.8

1

0 10 20 30 40 50 60 70 80

Range

PD

et

Figure 3. Probability of detection as a function of range.

Probability of detection as a function of angle

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100

Angle

PD

et

Figure 4. Probability of det

6

Mean

Mean

120 140 160 180

ection as a function of aspect angle

DRDC Atlantic TM 2005-213

3. Search Patterns The MOPs presented in the previous section allow us to assess the measures of effectiveness (MOEs) of the AUV based on its search pattern. These include the probability of detection, coverage, overlap and search time. Two typical paths in search operations are the lawn-mowing pattern shown in Figure 5, and the zigzag pattern shown in Figure 6. Both start in the lower left corner and proceed upwards as shown.

Figure 5. Lawn Mowing search pattern.

Figure 6. Zigzag search pattern.


It is very clear from Figure 4 that the probability of detection is substantially degraded if the aspect angle differs from 90 degrees. The impact of this degradation is shown in Figure 7 and 8. In Figure 7, a mine is observed at angle of 85 degrees while, in Figure 8, the same mine is observed at an angle of 0 degrees. The mine is without a doubt identified in Figure 7. However, the same mine in Figure 8 is not recognizable at all due to the aspect angle of 0 degrees. On average, the detection probability of a mine is approximately 40 percent based on a single observation. This value comes from the product of the mean range detection probability (Figure 3) and the mean angular probability of detection (Figure 4). This degradation can be mitigated by fusing several observations of a mine at different aspect angles, [4]. To obtain multiple mine aspects, we propose a novel and synergistic employment of two AUVs such that the aspect angles of a mine yield two different values. This offers a great potential to increase the angular probability. For example, Figure 9 displays a cylindrical mine (in green) observed from two angles: 30 degrees and 60 degrees. The fusion of these two observations improves the angular probability of detection by six percent based on the probability curve shown in Figure 4. We can see this improvement by observing that the angular probability of detection by the first AUV leg is while the angular probability of detection by the second AUV leg is equal to . Combining the two results of the two AUV legs, the angular probability of detection becomes

1 0.77P =

2 0.24P =

( )1 2 1 21 1 (1 ) 0.83P P P− = − − − = – an improvement of six percent wrt . Note that this improvement stems from the fact that the aspect angles of a mine wrt

the AUVs are different. The improvement would be marginal if the angles lie in zero probability areas such as 0 degrees or 180 degrees. This can happen if the two AUVs follow the same path or the same AUV carries out repeatedly the same path. This presumes that when a mine is seen at a different angle and/or a different range, the sonar picture will be different due to the surrounding of that mine. For example, if a mine lies on a slope then observations at different angles yield different information. Hence, multiple angular observations provide additional information. This is how we define an ideal environment. An ideal environment does not imply that there is no noise, since if there was no noise then a mine will be detected at the first opportunity. This clarification is essential in understanding the benefits of multiple images of a mine. In reality, there is likely a resolution in angle such that when two distinct observations are obtained from two different angles, they provide complementary information only if the two angles differ by more than the resolution. A similar resolution in range is also required for two distinct observations obtained from two different ranges.

1 0.77P =


Figure 7. Mine observed at 85 degrees.

Figure 8. Mine observed at 0 degrees.


Figure 9. Mine aspect angles.

Mine

30 deg

60 deg

Figure 10. Mine aspect symmetrical angles.

45 deg 45 deg


Here is an example that could be ambiguous without the notion of an ideal environment. Assume that a cylindrical mine is viewed both at 45 degrees and 135 degrees along its axis of symmetry, Figure 10. If there was no noise surrounding that mine, then the two images collected at 45 degrees and 135 degrees will be identical due to the cylindrical symmetry. However, if the surrounding is composed of ripples, or different size rocks then these two images will be different. Hence, data fusion makes sense in this context. So far, we have discussed the improvement to the angular probability of detection only. In reality, an additional observation not only improves the angular probability of detection but also the range detection probability. However, Figure 3 shows that for most ranges between the minimal and maximal range, the range detection probability is close 100 percent. As a result, the improvement due to range detection probability is marginal relative to that due to the angular probability. Figure 11 displays the search paths of two AUVs each employing the mowing pattern. The first AUV starts from the lower left corner and proceeds upwards, while the second AUV also starts from the lower left corner and proceeds rightwards. Figure 12 displays the search paths of two AUVs each employing the zigzag pattern. The first AUV starts from the lower left corner and proceeds upwards while the second AUV starts from the lower right corner and also proceeds upwards.

Figure 11. Two AUVs employing perpendicular mowing search patterns.


Figure 12. Two AUVs employing weaving patterns.


4. Modelling and Algorithms In order to assess the utility of these proposed search patterns, we have developed a stochastic simulation coded in Visual Basic (VB) and an equivalent deterministic model coded in MathCad.

4.1 Stochastic Model

This simulation allows the user to input the size of the search area, the speed of the AUV, the search paths of the AUVs as shown in Figures 5-6-11-12, as well as the density distribution of the mines. Most of the time, a uniform density distribution of mines is used for simplicity. However, to investigate the optimal search path described later, we also simulate a mine density distribution along the horizontal and the vertical axes through the exponential distribution and the unimodal Johnson distribution. This is a Monte Carlo simulation. The underlying algorithm is shown in Figure 13. For each run, the AUV employs the same search path (blue lines) while the positions (red circles) and angles (not shown) of the mines are generated based on the a priori distribution. If a mine lies between the minimal and maximal range of the AUV then a green line is drawn connecting that mine to the AUV. When multiple AUV legs cover a mine, a green line is drawn from each leg to that mine e.g. the mine laying inside the black circle is covered by two legs, one above and one below. The simulation then computes the probability of detection using the range probability curve and the angular probability curve. It also fuses data when a mine is detected by more than one leg and/or more than one AUV as discussed earlier.


Figure 13. Stochastic simulation.

The program collects the results of each run, performs the statistics of the MOEs and outputs the average probability of detection in addition to the mean number of mines detected. This simulation has the advantage that it displays the dynamics of the search paths, which is useful when designing for an experiment.

4.2 Deterministic Model

We approach the same problem with a different strategy compared to that of the stochastic model described above. The area is divided into a number of cells typically 200 by 200 cells. For each cell and for each leg of the AUVs, we determine whether that cell lies between the minimal and maximal range of detection of the AUV path. If it does then we compute the probability of detection using the probability curves and the probability density of the mine. For example, if the mines are uniformly random then the probability density of a cell is equal to the area size of the cell divided by the size of the search area.


Figure 14. Detection Level.

Figure 14 provides a glimpse of how the computation is carried out. Each colour corresponds to the number of times (or legs) that a cell lies between the minimal and the maximal range of the AUV path(s). The purple areas are not covered i.e. the associated detection level is zero. Similarly the blue areas are covered once, while the green areas are covered twice, and the orange areas are covered three times. The detection level of a cell is defined as the number of AUV legs that cover that cell. In general, the probability of detection increases as the detection level increases. This is so as the probability of detection of a cell can be

expressed as where is the detection level and

cP

(1

d

ci

P 1 M=

= −∏ )i d 1iM ≤ is the probability that

the ith leg misses the target. Since iM is a function of range and aspect angle then so is . cP Assuming a uniform target density, the contribution of each cell to the probability of detection is simply equal to ( )/cP A∆ ⋅ where A is the size of the search area while ∆ is the size of the cell and is the probability of detection based on range, aspect angle and detection level as shown above.

cP

This methodology allows for the collection of all the MOEs provided by the stochastic model. We use it to validate our results by comparing those of the stochastic model to the ones of the deterministic model. They are consistent as shown later in Figure 15.


4.3 MOEs The simulations above provide a number of primary MOEs:

a. Probability of detection;

b. Coverage (the percentage of the search area that lie between the minimal and maximal range of the AUV path);

c. Overlap (the percentage of the search area that is covered more than once by the AUV);

d. Search time and

e. Confidence intervals. There are also a number of secondary MOEs:

a. Ratio of the probability of detection to the search time and

b. Distribution of the aspect angle.

Depending on the mission, we can assign different priorities to the above MOEs. For instance, if the searcher is limited in time then he might want to maximize the ratio of the probability of detection to the search time. Doing so will give him the highest probability of detection in a fixed time. However, if the search area is to be sanitized, then the searcher will do his best in order to maximize the probability of detection. This can be done by searching the area exhaustively to minimize any gaps, and by using more than one AUV with different paths to maximize the angular probability of detection. Note that search time is a function of number of AUV legs. For example, to increase search time we decrease the spacing between the parallel horizontal legs of the search pattern shown in Figure 5.


5. Numerical Results

To examine and compare the MOEs, the following configurations are defined in Table 1.

Table 1. Configurations.

CONFIGURATION PATH DESCRIPTION

1M Figure 5 One AUV mowing

1Z Figure 6 One AUV zigzaging

2M Figure 11 Two AUVs mowing perpendicularly

2Z Figure 12 Two AUVs zigzagging in a weaving pattern

Table 2 displays the parameters and MOPs used in the above configurations. For the range probability curve: 1x represents the minimal range of detection while 2x represents the maximal range. Similarly, for the angular probability curve: 1x represents the minimal angle of detection while 2x represents the maximal angle of detection. Note that for the angular probability curve, we make use of symmetry to reduce 2 2x π= ⋅ to 2x π= . 1α controls the symmetry while 2α controls the shape of the probability curves. Currently, these parameters are estimates. However, in the future, these will be experimentally measured and the resulting data will be implemented in our models.

Table 2. Scenario parameters and side scan sonar MOPs.

DEFINITION PARAMETERS Area 3 km by 3 km

Speed 9 knots

Endurance 30 hours

Range probability curve

1 0α = , 2 0.75α = 1 11.5 mx = , 2 75 mx =

Angular probability curve

1 0α = , 2 1.25α = 1 0x = , 2x π=


MOEs of an AUV employing the mowing pattern as a function of search time

0

0.2

0.4

0.6

0.8

1

2 4 6 8 10 12 14

Search time in hours

MO

Es

CoverageOverlapP(range only)P(stochastic)P(deterministic)

Figure 15. MOEs of an AUV employing the mowing pattern as a function of search time.

To allow consistent comparisons, all MOEs are plotted as a function of search time. Beside the Coverage and the Overlap, the remaining MOEs shown in Figure 15 require some descriptions. P(range only) is the probability of detection based on the range probability curve only. P(stochastic) is the probability of detection based on both the range and the angular probability curves. And P(deterministic) is the same as P(stochastic) with the difference that P(deterministic) is obtained from an exact calculation done in the deterministic model while P(stochastic) is obtained through statistics generated by the stochastic simulation. As shown in Figure 15, Coverage is greater than Overlap. This is true by definition as all areas that have been covered by the AUV more than once i.e. Overlap, are necessarily covered at least once, i.e. Coverage. As also shown in Figure 15, Coverage is greater than but close to P(range only). This is true as the range probability curve is not exactly 100 percent for all values between the minimal and maximal range. And so, P(range only) must be slightly less than Coverage.


Figure 15 also reveals that P(range only) is significantly greater than P(deterministic) and P(stochastic). This is true since P(deterministic) and P(stochastic) are obtained by compounding P(range only) to the angular probability whose values decrease rapidly when the aspect angle differs from 90 degrees. This means that the effective angular probability is substantially less than 100 percent and therefore P(deterministic) and P(stochastic) are significantly less than P(range only). This is why we consider the employment of multiple AUVs. When more than one path is used, a mine can be observed at several different angles, and hence its angular probability of detection can be substantially improved. Figure 15 also reveals that P(deterministic) and P(stochastic) are very similar. P(deterministic) is obtained from the deterministic model. It is essentially a triple integral over the search area and over the aspect angle of the mine. P(stochastic) is obtained from running the stochastic simulation 100 times and then collecting the statistics. This shows that our calculations are consistent since the two models are equivalent and yet are designed differently. In what follows, we will refer to P as the probability of detection: one that accounts for both range and aspect angle. We observe that there is a common anomaly shown in Figure 15, 17 and 18. The coverages and corresponding probabilities of detection decrease slightly when search time is approximately between six and ten hours. This is not an error. It is due to the fact that as search time increases the spacing between the long legs (horizontal legs in the case of the mowing pattern) decreases. Hence, the number of long legs increases as search time increases. Associated with each long leg there is a gap due to the minimal range of detection of the sonar. Thus, the gaps increase when search time increases. As a result, the coverages and probabilities of detection decrease as a function of search time until the spacing is sufficiently small that these gaps are covered by the detection ranges of the legs above and below. From this point on, the coverages and probabilities of detection resume their increasing trends as a function of search time. This anomalous phenomenon is illustrated in Figure 16. The semi-transparent rectangles represent the gaps due to the minimal range of detection. On the left hand side, there are eight such rectangles while on the right hand side there are ten such rectangles. This implies that there are more gaps in the right hand side scenario. As a result, the coverages and corresponding probabilities of detection are lower in the right hand side scenario than those in the left hand side unless the maximal detection range is large enough so that these gaps are covered by the legs above and below. However, as the number of horizontal legs on the right hand side is greater than that on the left hand side, this means that search time is greater in the former case. This example shows how coverages and probabilities of detection can decrease as search time increases.


Figure 16. Anomaly due to minimal detection range.


Probability of detection for four AUV configurations as a function of search time

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

2 4 6 8 10 12 14


Prob

abili

ty o

f det

ectio

n

P2MP2ZP1MP1Z

Figure 17. Probability of detection for four AUV configurations as a function of search time.

Figure 17 compares the probability of detection among the four configurations. To alleviate the text, we will use this notation: P is the probability of detection, R is the detection rate defined as the ratio of P to the search time, C is the coverage and O is the overlap. Furthermore P1M means the probability of detection based on one (1) AUV employing a mowing (M) pattern. All other MOEs are defined similarly. As shown in Figure 17, P2M is higher than P1M. This is true as P2M is based on the employment of two AUVs while P1M is based on the employment of only one AUV. Similarly, P2Z is higher than P1Z. As also shown in Figure 17, P2M is clearly the highest probability of detection followed by P2Z, while P1M and P1Z are similar. We can therefore conclude that in this scenario the employment of two AUVs in a mowing pattern provides a better probability of detection than that of two AUVs in a zigzag pattern. However, there is virtually no difference between the employment of one AUV in a mowing pattern and that of one AUV in a zigzag pattern. This is due to the angular probability. The zigzag generates different aspect angles when compared to those of the mowing pattern. Although the probability of detection is an important MOE, there are other MOEs that can play an important role in designing the configurations of the AUVs. Figure 18 below displays another useful MOE: the detection rate (R) as a function of search time.


Detection rate for four AUV configurations as a function of search time

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

2 4 6 8 10 12 14


Det

ectio

n ra

te R1MR2MR1ZR2Z

Figure 18. The detection rate (R) for four AUV configurations as a function of search time.

Figure 18 compares the ratio of the probability of detection to the search time among the four configurations. As shown in Figure 18, in each case, either zigzag or mowing, the detection rate is higher for one AUV than for two AUVs. This is so as the employment of one AUV accounts for most of the probability of detection while the employment of a second AUV only enhances the probability of detection. So even though, P2M is greater than P1M the rankings in terms of R is reversed wrt the rankings in terms of P. This reversal is also valid for the zigzag pattern. As also shown in Figure 18, R1M curve and R1Z curve are similar. This shows the complexity of the results. Depending on the search time and based on the employment of a single AUV, most of the time the mowing pattern is better than the zigzag pattern but in some cases the reverse is true. This means that the choice between the mowing and the zigzag pattern depends on the scenario. Note that this is also true when we compare the probability of detection. As also shown in Figure 18, R2M is clearly greater than R2Z. This means that we detect more mines in the same time interval when we employ two AUVs in a mowing pattern than when we employ two AUVs in a zigzag pattern.


Coverage for four AUV configurations as a function of search time

0

0.2

0.4

0.6

0.8

1

2 4 6 8 10 12 14


Cov

erag

e C2MC2ZC1MC1Z

Figure 19. Coverage for four AUV configurations as a function of search time.

Figure 19 displays the coverage based on the four configurations. It can be seen that C2M and C2Z are similar. This shows that configuration 2M provides better angular probability than that of configuration 2Z. This is so as configuration 2M and configuration 2Z provide similar coverage, as shown in Figure 19, (which depend only on the minimal and maximal range) yet the configuration 2M provides a better probability of detection than that of configuration 2Z.


Mean angular probability of detection as a function of the angle between two AUV legs

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 20 40 60 80 100 120 140 160 180

Angle between two legs

Pmea

n Pmean2Pmean1

Figure 20. Mean angular probability of detection.

Figure 20 displays the mean angular probability of detection. The mean value is the average in the probabilistic sense. Pmean1 represents the mean angular probability of detection by one AUV leg. Pmean2 represents the mean angular probability of detection by two AUV legs plotted as a function of the angle between them ( )θ . An example of this is shown in Figure 9. To obtain Pmean1 or Pmean2, we assume that a mine lies between the minimal and the maximal range of an AUV leg or two AUV legs respectively. Pmean1 is essentially computed by integrating the angular probability of detection, shown in Figure 4, over all possible angles and dividing by π . Pmean2 is also computed by integrating the angular probability of detection, but based on two AUV legs, over all possible angles of a mine and dividing by π . Figure 20 shows that Pmean2 is clearly higher than Pmean1 implying that there is substantial improvement in the probability of detection if a mine is observed by two AUV legs. This improvement is maximal when θ is equal to 90 degrees. This corresponds to the case of two AUVs employing perpendicular mowing search paths as shown in Figure 21. Although this is not a proof it provides insight on why the configuration 2M provides superior MOEs to those of configuration 2Z. This result is true in general for all angular probability curves that have the shape shown in Figure 4, are symmetric about 90 degrees and have a period of 180 degrees. The mathematical proof of this is described in Annex A. It is based on the expansion of the angular probability curve as a Fourier series. We can understand this by examining the


angular probability of detection curve, Figure 4. First, it is maximal at 90 degrees. This means that if a mine is observed at both zero and 90 degrees, we will achieve the maximal angular probability of detection due to the 90 degrees observation. Second, if a mine is observed at an aspect angle and later at the same aspect angle plus 90 degrees, then the fusion of these two observations will substantially increase the angular probability of detection. On average an angle of 90 degrees (Figure 20) between two AUV legs provides the optimal angular probability of detection.

Figure 21. Geometry for maximal angular probability of detection.

Mine

This simple analysis allows us to make a few deductions. Configuration 1M and 1Z provide similar MOEs. This tells us that there is no difference in merit between a mowing pattern and a zigzag pattern when the search is conducted by a single AUV. However, configuration 2M is clearly better than configuration 2Z in terms of probability of detection and detection rate. Yet they provide similar coverage. This exercise shows that when employing two AUVs, it is better to conduct perpendicular mowing paths than to conduct two weaving zigzag paths. This is not a surprise but it is an appealing result in the sense that the mowing pattern is better and this is how search operations are normally conducted in practice. In addition, the perpendicular mowing patterns are simpler and more robust to conduct than the zigzag pattern in the sense that a zigzag pattern requires fine-tuning of the zigzag angle.


6. Optimal Search Paths Questions that arise naturally are: What is the optimal search path? is it feasible? and why do we do what we do? Although the range probability of detection is independent of the angular probability of detection, the overlaps between different legs, and the minimal and maximal range characteristics of the side scan sonar makes it difficult to determine the optimal search path. Therefore, we propose to look at a simpler case where the optimal search path is a continuous curve emulating the AUV’s path. As it turns out, this will give us a lot of useful information. We enforce the solution to the optimal path such that it doesn’t intersect itself in the sense that there is no loop: a condition described later in the assumptions. For the sake of simplicity, we remove the dependence on the angular aspect of the mine wrt the sonar. This can be reinstated at the end once we determine the optimal search path and implement this into our simulations. We will analyze two different mine density distributions below. The first is a uniform distribution, which is the one we have assumed so far. The second is an exponential distribution.

6.1 Uniform Distribution The uniform distribution is a distribution where each point in the search area has a density of 1/ A where A X Y= ⋅ is the size of the rectangular search area defined by X the length and

the width. This means that the probability that a mine lies at a given point in the search area is 1/Y

A . Clearly, if the density is a constant then the probability of detection is linearly proportional to the coverage. If we further assumed that the probability of detection as a function of range is perfect and there is no angular dependence then the probability of detection and coverage are one and the same. Therefore, to optimize the probability of detection we need to cover the area as much as possible. This means that we need to avoid gaps in the sonar coverage. This can be achieved by using more than one AUV, as described earlier. Overlaps must also be avoided so that the search time is not increased unnecessarily.

6.2 Exponential Distribution Before we proceed in deriving the optimal search path, we will make the following assumption that is needed to solve for the optimal search path generally.


6.2.1 Assumptions Let ( )y s x= be a function of x that describes the optimal search path then we assume that ( )x t y= is also a function of . This innocuous assumption will clarify our definition of an optimal search path. For reasons explained later, we also consider a vertical line and a horizontal line as if they satisfy this assumption.

y

6.2.2 Along the vertical axis

While, it is very clear how to optimize in the case of a uniform distribution, it’s not obvious what to do in the case of a non-uniform distribution. We will illustrate this by analyzing an exponential distribution. The argument is general however and as a result it can be applied to any distribution. We assume an exponential distribution along (the vertical axis). y

Target density as a function of y

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 0.2 0.4 0.6 0.8 1

y

Targ

et d

ensi

ty

Density

Figure 22. Target density distribution as a function of (the vertical position in the search area).

y

Figure 22 displays a density distribution of a mine as a function of – a scaled vertical position in the search area i.e. 1 corresponds to the total width of the search area while 0.5 corresponds to the half width – dashed line in Figure 23. We simulate this distribution by repeating the Monte Carlo runs 100 times.

y


Figure 23. Exponential density distribution along the axis. y

Figure 23 shows such a case where the density is high if the position is close to the half-width line (dashed). Intuitively, we believe that the AUV path should be as close as possible to the half-width line since the density is highest there. The mathematical proof of this is described in Annex B. It is based on the functional expression of the probability of detection and the assumptions above.

y

As shown in Figure 24, it is typical in search operations to divide the search area into sub box areas (e.g. shaded rectangular area) where the AUV employs the mowing pattern. Figure 24 shows the AUV path in a sub box area beneath the semi transparent ellipse. Note that the AUV starting point is a magenta dot on the lower left and ending point is a magenta dot on the upper right. We will answer the following question: given a sub box area like the one just described what is the optimal path starting from the lower left dot to the upper right dot. This is a boundary value problem and hence can be solved once we determine the differential equation governing the optimal path.


Figure 24. AUV path in a sub box area.

x∆ x∆

X

Y∆

Based on the proof in Annex B, the assumption on as a function of y x and x as a function of leads to the following constraints for the optimal path: y

a. The optimal path must lie inside each sub box area defined by the two dots.

b. The optimal path length must be less than or equal to the sum of the length of a horizontal leg and a vertical leg.

c. The optimal path length must be greater than or equal to the length of the diagonal joining the two dots.

The optimal path satisfying all of the above constraints must in addition carry the highest weight. In this context, we define weight as the length of the AUV path compounded to the density distribution. The probability of detection associated with this path is the weight compounded to a thickness of the path. This ensures that the optimal path covers the highest density of mines. Note that the constraints a, b, and c imply that the optimal path is the optimal path only in a sub box area and therefore it might not be the globally optimal path.


The weight of a curve in the search area can be expressed as:

[ ] ( ) ( )2

1X x

x

dyW y dx f x g ydx

−∆

∆

⎛ ⎞= ⋅ + ⋅ ⋅⎜ ⎟⎝ ⎠∫ (2)

where ( ) 1/f x = X is the uniform density distribution along x while

( ) ( )( )/ 2 /

/ 22 1

y Y b

Y b

eg yb e

− −

− ⋅=

⋅ ⋅ −is the exponential density distribution along . y

Annex B shows that the path consisting of the two dotted legs in Figure 25 optimizes the weight of detection. The dashed legs are similar to the dotted legs. However, they are not optimal as the horizontal dashed leg is further from the half-width line than the horizontal dotted line. There are two preferably useful features to this optimal solution. First, it shows that the optimal path is the mowing pattern, which is what we normally do. Second, this result is independent of the distribution of mines as long as the distribution is non-increasing as the distance away from the half-width line increases. Recall that optimality here means optimality satisfying constraints a, b and c.

Figure 25. Optimal path (dotted lines) in a sub box area.


This result assumes that the optimal path is a curve at the sea bottom. This result does not incorporate the detection range but it could be with the following analysis. The optimal AUV path would have the same shape as the dotted lines i.e. the AUV would move upward in a way that its left arm (maximal detection range) traces the vertical leg (dotted) and then rotate then move to the right in a way that its upper left arm traces the horizontal leg (dotted), Figure 25. Note that the optimal path consisting of two dotted legs technically does not satisfy the assumption that is a function of y x and x is a function of . This is so as a horizontal line can be represented by

yconstanty = . Thus, is a trivial function of y

x . However, a horizontal line cannot be represented by x as a function of . Similarly a vertical line can be represented by

yconstantx = . But a vertical line

cannot be represented by as a function of y x . However, in reality the two dotted legs form the asymptotic solution to the Euler-Lagrange equation, which satisfy the above assumption. By adjusting the free parameters of this solution, we can make it as close to the two dotted legs as we would like. Therefore, the two dotted legs represent the optimal solution. We describe below the details of this asymptotic solution.

Assuming that the sub box area is below the half width line, we obtain the following Euler-Lagrange equation from the functional [ ]W y above:

( )( )21'' 1 'y y

b= ⋅ +⎛ ⎞⎜ ⎟⎝ ⎠

The solution to the above differential equation can be written as:

( )1

1

22

2cos

ln2 cos

Cby

C

Cxb

= ⋅ +

⎛ ⎞⎜ ⎟⎜ ⎟

⎛ ⎞⎜ ⎟+⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

where and are constants. We pick so that 1C 2C 2 300C = 300y = when . 0x =


Asymptotic solutions to the Euler-Lagrange equation

0

500

1000

1500

2000

2500

3000

0 500 1000 1500 2000 2500 3000

x

yyhghylowy4y3y2y1

Figure 26. Asymptotic solutions to the Euler-Lagrange equation.

Figure 26 displays the paths such that they correspond to 1 2 3 4, , ,y y y y

10.3 , 0.35 , 0.40 , 0.45C π π π= ⋅ ⋅ ⋅ ⋅π respectively. Note that the slope of each path

increases as approaches 1C ( 0902

π ) . That is, these paths approach more and more to

a vertical line. However, they do not represent the complete solutions. In order to obtain the complete solutions, we need to account for the size of the sub box area (constraint “a” above), which is bounded by the blue lines denoted as and

. For illustration purpose, we have chosen ylow

yhgh 2100b = , 300ylow = and . Any portions that lie above are replaced by . The resulting

solutions after these replacements are shown in Figure 27. Taking the limit as 900yhgh = yhgh yhgh

1 2C

π

−

→ ⎛ ⎞⎜ ⎟⎝ ⎠

, we obtain the optimal solution shown in Figure 25. Of course, this is the

hard way to derive the optimal path. A more succinct derivation is described in Annex B.


Bounded asymptotic solutions to the Euler-Lagrange equation

0

100

200

300

400

500

600

700

800

900

1000

0 500 1000 1500 2000 2500 3000

x

yyhghylowy4y3y2y1

Figure 27. Bounded asymptotic solutions to the Euler-Lagrange equation.

When including all sub areas this means that the vertical size of each sub box area must be chosen so that the AUV in that sub area covers it as much as possible. The AUV must also give higher priority to the sub areas that are closer to half-width line if it is limited in time. For example, the sub areas below the half-width line are pushed vertically toward the centre such that each sub area is as close to the half-width line as possible without overlaps. This ensures that the areas with a higher mine density (close to the half-width line) are given priority over those with lower mine density (away from the half-width line).


Figure 28. Exponential density distribution along the x axis and the axis. y

Figure 28 shows the optimal path in black (dashed) while the regular mowing path is in blue. Note that the spacing between the optimal horizontal segments is less than that of the regular path. This corresponds to the case where the maximal range of detection is less than the regular spacing. By adjusting the spacing to the maximal range of detection, the optimal path avoids gaps in the higher density area, as described above. It may be argued that a normal lawn mowing search path is normally carried out in a way that there is no gap. However, if we were provided with an a priori mine density distribution such as the exponential density then it will determine the size of the area to be searched. Figure 28 shows that the optimal path in black (dashed) does not start from the bottom of the search area as in the case of the regular path, and also does not end at the top of the search area. The optimal search area is hence smaller. Hence, it provides a better rate of detection than the one of regular mowing path.


Probability of detection as a function of search time

0

0.2

0.4

0.6

0.8

1

2 4 6 8 10 12 14


P de

tect

ion

Optimally spacedEqually spaced

Figure 29. Probability of detection as a function of search time (exponential density along ). y

The improvement in the probability of detection due to the optimal path wrt a regular mowing path is shown in Figure 29. It is assumed that the characteristic parameter

of the exponential density is equal to while all other MOPs remain unchanged. The improvement can be as much as twenty percent if the search time is less than five hours. However, when the search time exceeds five hours, the optimal path becomes the regular mowing path and hence the optimal probability of detection is identical to that of the regular mowing path. This is so, as the search time increases the vertical leg size of the optimal path decreases. Eventually this vertical leg size becomes less than twice the maximal range of the AUV. When this happens, we cannot stack the sub box areas of the optimal path any closer to one another without compromising the comparison with the regular mowing pattern.

b 200

Similar improvement is seen in terms of detection rate R, Figure 30.


Detection rate as a function of search time

0

0.05

0.1

0.15

0.2

0.25

0.3

2 4 6 8 10 12 14

Search time

Det

ectio

n ra

te Optimally spacedEqually spaced

Figure 30. Detection rate as a function of search time (exponential density along ). y

6.2.3 Along the horizontal axis and the vertical axis

In this section, we consider the same boundary value problem as the one in the previous section and assume that the density along x is also an exponential density that is similar to the one assumed along . We simulate this distribution by repeating 100 Monte Carlo runs as shown in Figure 31 below.

y

Figure 31. Exponential density distribution along the x axis and the axis. y


Therefore, the weight of a search path in the search area can be expressed as:

[ ] ( ) ( )2

1X x

a bx

dyW y dx g x g ydx

−∆

∆

⎛ ⎞= ⋅ + ⋅ ⋅⎜ ⎟⎝ ⎠∫ (3)

where ( )g x and ( )g y are the density distributions along x and along respectively: y

( ) ( )( ) ( ) ( )( )/ 2 / / 2 /

/ 2 / 2;

2 1 2 1

x X a y Y b

a bX a Y b

e eg x g ya e b e

− − − −

− ⋅ − ⋅= =

⋅ ⋅ − ⋅ ⋅ − (4)

The functional form of W is a special case discussed in [5]. In this specific case, the

Euler’s equation – 0'

dy d∂∂

F Fx y

∂− =

∂ where F is the integrand of W and ' dyy

dx= is

the first derivative of with respect to y x – can be simplified to:

( )21 ''' 1 '

y byb a+ ⎛= ⋅ − ⋅⎜

⎝ ⎠y ⎞⎟ (5)

This is a second order non-linear differential equation, which does not have a solution in general. However, an implicit solution of and y x was found. The derivation is described in Annex C. This optimal path is still under investigation and the results will be reported in the near future.


7. Conclusions In terms of deliverables, we have developed a stochastic mine hunting model written in Visual Basic with graphics displaying the dynamics of the AUVs. We have also developed a deterministic mine hunting model written in MathCad which is equivalent to the stochastic model. We have also proposed and examined a number of concepts of operations for AUVs surveying for mines with a side scan sonar through these two models. These concepts involve the use of one or two AUVs sweeping a lawn mowing pattern or a zigzag pattern. Based on the MOPs, it was found that the configuration of two AUVs each sweeping a mowing pattern that is perpendicular to one another provides the highest probability of detection as well as the highest detection rate. Even though we haven’t proved rigorously that this configuration is always optimal, we show that the average angular probability of detection of a mine is maximal when detected by two orthogonal AUV legs. Evidently, this result may change if the MOPs are modified. However, if the angular probability curve preserves the same shape and symmetry then this result will remain true. Perhaps, the true contribution of this report is that these concepts of operations can all be implemented and assessed in real life scenarios. Further, we have also derived the optimal search path based on mine densities. If an exponential density along the vertical axis is assumed then we show that a specific type of the lawn mowing pattern is optimal. This specific pattern is the same as the regular mowing pattern with the exception that each sub box area is chosen such that they are stacked as close as possible to the half-width line and to one another. In addition to this, the optimal path must have a general direction along the vertical axis. If an exponential density along both the horizontal axis and the vertical axis are assumed then the optimal path is slightly different from the mowing path. This latter result is currently under investigation and will be reported in the future.


8. References

1. Bao Nguyen and Dave Hopkin, “Modelling Autonomous Underwater Vehicles (AUVs) in Mine Hunting”, IEEE Oceans 05 Conference Proceedings, 20-23 Jun 2005, Brest, France.

2. Vincent Myers private communication, NATO Undersea Research Centre, 16 Mar 2005.

3. Averill M. Law and W. David Kelton, Simulation Modeling and Analysis, 3rd edition, McGraw-Hill series in industrial engineering and management science, 2000, pp. 314-315.

4. B. Zerr, E. Bovio and B. Stage, “Automatic Mine Classification Approach Based on AUV Manoeuvrability and COTS Side Scan Sonar”, Autonomous Underwater Vehicle and Ocean Modelling Networks: GOAT2 2000 Conference Proceedings, pp. 315-322.

5. I. M. Gelfand and S. V. Fomin, Calculus of Variations, 13th edition, Prentice-Hall, 1964, p 19.

6. David L. Powers, Boundary Value Problems, 3rd edition, Harcourt Brace Jovanovich Inc, 1987, pp. 46-57.


Annex A – Optimal Observation Angle Between Two AUV Legs

In this Annex, we prove that the angle between two AUV legs, which optimizes the angular probability of detection, is equal to 90 degrees, a generalization of the result shown in Figure 32. The proof is based on the symmetry and shape of the angular probability of detection about 90 degrees.

Mean angular probability of detection as a function of the angle between two AUV legs

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 20 40 60 80 100 120 140 160 180

Angle between two legs

Pmea

n Pmean2Pmean1

Figure 32. Mean angular probability of detection.

To do this, we expand the angular probability of detection as a Fourier series and show that the mean angular probability of detection derivative, wrt the angle φ between two AUV legs,

is equal to zero at 2πφ = . It is convenient to redefine the aspect angle θ by shifting it to the

left by 2π such that the maximal angular probability of detection occurs at zero degree, Figure

33.


Probability of detection as a function of redefined aspect angle

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-90 -60 -30 0 30 60 90

Redefined aspect angle

Prob

abili

ty o

f det

ectio

n

PanglePmean

Figure 33. Probability of detection as a function of redefined angle.

The angular probability of detection shown in Figure 33 is an even function of the aspect angle θ and has a period equal to π (180 degrees). Hence it can be expressed as follows, [6]:

( ) (0

1

cos 22 n

n

af a )nθ θ∞

=

= + ⋅ ⋅ ⋅∑ (A.1)

where are real coefficients obtained by expanding the function na ( )f θ as a Fourier series. The mean angular probability of detection based on two AUV legs is a function of the angle φ :

( ) ( )( ) ( )( )2

2

11 1 1P d f f

π

π

φ θ θ θπ

−

= − ⋅ ⋅ − ⋅ − +∫ φ (A.2)


whose derivative wrt φ is:

( ) ( )( ) ( )2

2

1 1d P d f fd d

π

π

dφ θ θ θ φφ π φ

−

⎛= − ⋅ ⋅ − ⋅ +⎜

⎝ ⎠∫

⎞⎟ (A.3)

Note that the right hand side above integrates from 2π

− to 2π due to the redefinition of the

aspect angle θ . Using equation (A.1):

( )( ) ( ) ( ) ( )1

2

2 cos sin 2mm

d f m a md πφ

mθ φφ

∞

==

⎤+ = − ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅⎥

⎦∑ π θ⋅ (A.4)

Substitute this expression into the derivative of P wrt φ :

( ) ( ) ( ) (2

0

1 12 2

1 1 cos 2 2 cos sin 22 n m

n m

ad P d a n a m md

π

π πφ

)mφ θ θ πφ π

∞ ∞

= == −

⎤ ⎛ ⎞ ⎛= ⋅ ⋅ − − ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅⎜ ⎟ ⎜⎥⎝ ⎠ ⎝⎦

∑ ∑∫ θ ⎞⎟⎠

(A.5)

Make a change of variable: 2ϑ θ= ⋅ , we get:

( ) ( ) ( ) ( )0

1 12

1 1 cos 2 cos sin2 2 n m

n m

ad P d a n a m md

π

π πφ

nφ ϑ ϑ πφ π

∞ ∞

= =−=

⎛ ⎞⎤ ⎛ ⎞ ⎛= ⋅ ⋅ − − ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅⎜ ⎟⎜ ⎟ ⎜⎥ ⋅ ⎝ ⎠ ⎝⎦ ⎝ ⎠∑ ∑∫ ϑ ⎞

⎟⎠

(A.6)


The following identities are provided by [6]:

( )sin 0nπ

π

ϑ−

⋅ =∫ (A.7)

( ) ( )cos sin 0d n mπ

π

ϑ ϑ ϑ−

⋅ ⋅ ⋅ ⋅ =∫ (A.8)

Integrating the right hand side of equation (A.5) term by term and making use of the two identities above, we get:

( )2

0d Pd πφ

φφ =

⎤=⎥

⎦ (A.9)

This tells us that the mean angular probability of detection as a function φ reaches an

optimum when 2πφ = . In principle, we need to show that the second derivative of ( )P φ is

negative at 2πφ = . However, our intuition already tells us that this is a maximum. It is not

hard to see that. Say an observation at 2πθ = − provides an angular probability of detection

equal to zero percent. A second observation at 02πθ φ= = − + provides an angular probability

of detection equal to 100 percent. The two combines an angular probability of detection equal

to 100 percent. Clearly in this example an angle 2πφ = maximizes the angular probability of

detection. For our purpose, this is sufficient.


Annex B – Vertical Exponential Distribution

In this Annex, we will derive the optimal search path when assuming a target exponential density distribution along the vertical axis of the search area. To do that, we will make the following assumption. Assumption Let ( )y s x= be a function of x that describes the optimal search path then we assume that ( )x t y= is also a function of . This innocuous assumption will clarify our definition of an optimal search path. We also consider a vertical line and a horizontal line as if they satisfy this assumption.

y

Figure 34. AUV path in a sub box area.

x∆ x∆

X

Y∆

Derivation As shown in Figure 34, it is typical in search operations to divide the search area into sub box areas (e.g. shaded area) where the AUV employs the mowing pattern. Figure 34 shows the AUV path in a sub box area beneath the semi transparent ellipse. Note that the AUV starting point is a magenta dot on the lower left and ending point is a


magenta dot on the upper right. We will answer the following question: given a sub box area like the one just described what is the optimal path starting from the lower left magenta dot to the upper right magenta dot. This is a boundary value problem and hence can be solved once we determine the differential equation governing the optimal path. The length of a curve joining these two dots can be written as:

2

1X x

x

dyS dxdx

−∆

∆

⎛ ⎞= ⋅ + ⎜ ⎟⎝ ⎠∫ (B.1)

where x∆ is the margin on the left and the right of the AUV path and X is the horizontal side of the search area as shown in Figure 34. By assuming that is a function of

yx and x is also a function of , this implies that is a monotonic

function of y y

x . In this specific case, it is a non-decreasing function from the lower left

dot to the upper right dot. This means that ' dyydx

0= ≥ . Therefore:

( ) ( ) ( )21 ' 1 ' 2X x X x

x x

S dx y dx y X x−∆ −∆

∆ ∆

= ⋅ + ≤ ⋅ + = − ⋅∆ + ∆∫ ∫ Y (B.2)

where is the length of a vertical leg as shown in Figure 34. Note that is an input parameter. Changing

Y∆ Y∆Y∆ leads to changes in search time. This tells us that the

length of an AUV path in a sub box area is less than or equal to the length of the mowing path (dashed line) based on our assumption. Furthermore, the shortest path between two points is a straight line implying that:

( ) ( ) ( )2 22 2X x Y S X x− ⋅∆ + ∆ ≤ ≤ − ⋅∆ + ∆Y (B.3) Note that the upper inequality above, equation (B.3), is also due to the boundary. This can be seen by the following argument. The value of the path at is greater than or equal to the value of the starting point i.e.

y / 2x X=y / 2X xy y∆≥ since for

. Now for , the derivative ' 0y ≥

/ 2x X≤ / 2x X> 'y must also be either non-negative or non-positive in order for x to be a function of . If it is negative then the value of the path will decrease and eventually increase to satisfy the boundary on the right (the

y y


upper right corner dot). When this happens there are two values of for a same yx and hence x is not a function of . This implies that for . Knowing that for , we conclude that for all

y ' 0y ≥ / 2x X>

' 0y ≥ / 2x X≤ ' 0y ≥ ( )x x X x∆ ≤ ≤ − ∆ confirming the inequalities above.

Now that we have established the lower and upper bound of the search path length, we can clearly state the strategy on how to derive the optimal path. The optimal search path that we are looking for must satisfy the following constraints:

a. The optimal path must lie inside each sub box area defined by the two dots.

b. The optimal path length must be less than or equal to the sum of the length of a horizontal leg and a vertical leg.

c. The optimal path length must be greater than or equal to the length of the diagonal joining the two dots.

The optimal path satisfying all of the above constraints must, in addition, carry the highest weight. In this context, we define weight as the length of the optimal path multiplied to the density distribution. Multiplying this weight to an infinitesimal thickness of the curve gives the probability of detection of a mine lying in the area swept out by this curve and the thickness of the curve. Hence, the optimal path is one that provides the highest probability of detection. Note however that the constraints a, b, and c imply that the optimal path is the optimal path only in a sub box area and therefore it might not be the globally optimal path.

The weight of a line in the search area can be expressed as:

( ) ( )2

1X x

x

dyW dx f x gdx

−∆

∆

⎛ ⎞= ⋅ + ⋅ ⋅⎜ ⎟⎝ ⎠∫ y (B.4)

where ( ) 1/f x = X is the uniform density distribution along x while

( ) ( )( )/ 2 /

/ 22 1

y Y b

Y b

eg yb e

− −

− ⋅=

⋅ ⋅ −is the exponential density distribution along . y

The general shape of ( )g y is shown in Figure 35. It reaches the maximum when

/ 2y Y= and decreases symmetrically when moves away from this value. The parameter controls how narrow the density is around the maximal point. The

yb


boundary is such that: ( ) sy x y∆ = i.e. is equal to that of the lower left dot while y

( ) ey X x y− ∆ = i.e. is equal to that of the upper right dot. To solve for the optimal search path, we will assume that the sub box area lies below the half-width horizontal line.

y

Target density as a function of y

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 0.2 0.4 0.6 0.8 1

y

Targ

et d

ensi

ty

Density

Figure 35. Target density distribution as a function of (the vertical position in the search area).

y

Again, we make use of the monotonicity of as a function of y x :

( )

( )( )( )

( )( )( )

( )( )

/ 2 /

/ 2

/ 2 / / 2 /

/ 2 / 2

12 1

2 1 2 1

e

s

X x Y y b

Y bx

yX x Y y b Y y b

Y b Y bx y

X Y

dx dy eWX dx b e

dx e dy eX Xb e b e

P P

−∆ − −

− ⋅∆

−∆ − − − −

− ⋅ − ⋅∆

⎛ ⎞≤ ⋅ + ⋅⎜ ⎟⎝ ⎠ ⋅ ⋅ −

≤ ⋅ + ⋅⋅ ⋅ − ⋅ ⋅ −

≤ +

∫

∫ ∫ (B.5)

where


( )

( )( )( )

( )( )/ 2 / / 2 /

/ 2 / 2;

2 1 2 1

e

s

yX x Y y b Y y b

X YY b Y bx y

dx e dy eP PX Xb e b e

−∆ − − − −

− ⋅ − ⋅∆

= ⋅ = ⋅⋅ ⋅ − ⋅ ⋅ −∫ ∫ (B.6)

Examining the expressions of XP and closely, we can realize YP XP is the weight of the horizontal line (dotted) while is the weight of the vertical line (dotted) shown in Figure 36. Since we have just shown that the weight of any curve satisfying constraints a, b and c is less than the sum of

YP

X YP P+ , this shows that the pattern in dotted line is the optimal search path. This also confirms our intuition discussed earlier, that is, to optimize the weight of a search path, we need to let it be as close as possible to the half-width line since this line corresponds to the highest mine density. Note that we obtain the same solution by solving Euler’s equation through the functional expression of W , [5].

Figure 36. Optimal path (dotted lines) in a sub box area.


Annex C – Vertical and Horizontal Exponential Distributions

In this Annex, we provide a full derivation of the optimal search path when the density distribution in the search area is exponential along x and separately along , denoted as

. The aim is to obtain the optimal search path that maximizes the weight defined as: y

( ,G x y)

( ) (21 ' ,X x

ox

W W dx y G x y−∆

∆

= ⋅ ⋅ + ⋅∫ ) (C.1)

Recall that the corresponding probability of detection is equal to the weight W multiplied by the infinitesimal thickness of the optimal search path. The functional form of the weight W displayed above is a specific case where the Euler’s equation can be simplified to a convenient expression, [5]:

2

'''1 'y x

yG G y Gy

− ⋅ − ⋅ =+

0 (C.2)

xG is the partial derivative of G with respect to x while is the partial derivative of with respect to . Substitute

yG G

y / /x a y be= ⋅G e into the simplified Euler’s equation above:

2

'''1 '

G G yy Gb a y− ⋅ − ⋅ =

+0 (C.3)

Removing the common factor G and perform some simple algebra:

( ) ( )2 1'' 1 '

yy y

bβ− ⋅

= + ⋅' (C.4)

where /b aβ = and for convenience we also define /a bα = . Now that we have obtained the differential equation governing the optimal search path, we will derive its solution. The strategy is to use the above equation to both obtain a solution of as a function of y 'y and separately obtain a solution of x as a function of 'y . These two solutions have similar


expressions that cancel when we compare one to the other and which leads to a first order differential equation ( )' tany z= . To do that, let 'u y= . y as a function of u

Rewrite the above equation as:

( )21 1du ub u udy α

⎛⋅ ⋅ = + ⋅ −⎜⎝ ⎠

⎞⎟ (C.5)

Isolating dy :

( )21 1

b u dudyuuα

⋅ ⋅=

⎛ ⎞+ ⋅ −⎜ ⎟⎝ ⎠

Using partial fractions:

( )2 2

2

1 11 1 11 1

b du udyuu uα α

α α

⎛ ⎞⎜ ⎟⋅ ⎜ ⎟= ⋅ − +

+⎛ ⎞ ⎛⋅ +⎜ ⎟+ ⋅⎜ ⎟ ⎜⎜ ⎟⎝ ⎠ ⎝⎝ ⎠⎞− ⎟⎠

Integrating both sides:

( ) ( )2 1

2

1 1ln 1 tan ln 11 21

by B u uα

α

−⎛ ⎛ ⎞+ = ⋅ ⋅ + − ⋅ − −⎜ ⎟⎜ ⎝ ⎠⎝ ⎠+

uα

⎞⎟ (C.6)

where is a constant. B x as a function of u Rewrite the Euler’s equation as followed:


( )21 1du ub udx α

⎛⋅ = + ⋅ −⎜⎝ ⎠

⎞⎟ (C.7)

Isolating dx :

( )21 1

b dudxuuα

⋅=

⎛ ⎞+ ⋅ −⎜ ⎟⎝ ⎠

Using partial fractions:

2 2

2

11 1 1

1 1 11 1

b udx uu uα

α αα α

⎛ ⎞⎜ ⎟

= ⋅ ⋅ + + ⋅⎜ ⎟+ +⎜ ⎟+ −⎝ ⎠


( ) ( )2 1

2

1 1ln 1 tan ln 11 21

bx D u uα α

α

−⎛ ⎛ ⎞+ = ⋅ ⋅ + + − ⋅ −⎜ ⎟⎜ ⋅ ⎝ ⎠⎝ ⎠+

uα

⎞⎟ (C.8)

where is a constant. Note that this is why we assume that D x is a function of in the main text.

y

First order differential equation The two solutions above lead to a first order differential equation in x . This can be seen by multiplying the factor 1/α to equation (C.6) and subtracting equation (C.8) from the resulting equation:

( )1tanyx C b uα

−− + = ⋅ (C.9)

Isolating u :

1tandy yu xdx b α

⎛ ⎞⎛ ⎞= = ⋅ − +⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠C

This is a first order differential equation, which can be solved by defining:

1 yz xb α⎛ ⎞= ⋅ − +⎜ ⎟⎝ ⎠

C (C.10)


Differentiate both sides with respect to x , and perform some simple algebra: dy dzbdx dx

α α= − ⋅ ⋅

Recall that ( )tandy zdx

= , isolating dx :

( )tandzdx b

zα

α= ⋅ ⋅

−


( ) ( )((2' ln sin c1

a ))osx C z zα αα

+ = ⋅ ⋅ − − + ⋅+

z (C.11)

where is a constant. Perform some simple algebra and take the exponential of each side lead to the implicit solution:

'C

( ) ( )(21

sin cosx z

aE e z zα α

α+

− ⋅ + ⋅⋅ = − + ⋅ ) (C.12)

where is a constant. E


List of symbols/abbreviations/acronyms/initialisms

AUV Autonomous Underwater Vehicle

DCOS OR Deputy Chief Of Staff – Operations Research

DND Department of National Defence

DRDC Defence Research and Development Canada

MARLANT Maritime Forces Atlantic

RMS Remote Mine hunting System

VB Visual Basic

WRT With Respect To


Glossary

1M Configuration where one AUV sweeps a lawn mowing pattern

2M Configuration where two AUVs each sweeping a lawn mowing pattern that is orthogonal to one another

1Z Configuration where one AUV sweeps a zigzag pattern

2Z Configuration where two AUVs each sweeping a zigzag pattern in a weaving way

C1M, C2M, C1Z, C2Z Coverage for configurations 1M, 2M, 1Z and 2Z respectively

P1M, P2M, P1Z, P2Z Probability of detection for configurations 1M, 2M, 1Z and 2Z respectively

R1M, R2M, R1Z, R2Z Ratio of probability of detection to search time for configurations 1M, 2M, 1Z and 2Z respectively


Internal distribution list Library (3 hard copies and 3 CDs) Dr. Bao Nguyen (2 hard copies and 1 CD) Mr. David Hopkin (1 hard copy and 1 CD) Dr. Ron Kuwahara (1) Dr. Ross Graham/Dr. Jim L. Kennedy/Dr. Jim S. Kennedy/Dr. Marcel Lefrancois/Dr. Dan Hutt/Dr. John Fawcett/Dr. Garry Heard (1)


External distribution list NDHQ/DRDKIM (1) NDHQ/DRDC CORA/Library (1) NDHQ/DRDC CORA/DG Ms. Maria Rey/DOR(Corp) /DOR(Joint) /DOR(MLA) (1) NDHQ/DRDC CORA/DOR(Corp)/Dr. Paul Desmier NDHQ/DRDC CORA/DOR(Joint)/Dr. Murray Dixson NDHQ/DRDC CORA/DOR(Joint)/Dr. Greg Van Bavel NDHQ/DRDC CORA/DOR(Joint)/EXORT/Dr. Dave Allen MARLANTHQ/N3OR MARPAC/DCOSOR/Ms. Isabelle Julien 1CAD/CANR/CORA/Dr. John Steele DRDC Valcartier/SORA/Ms. Luminita Stemate NORAD/AN/Col. B. Streett/Mr. Glen Roussos/Mr. Tom Denesia/Dr. Sean Bourdon /Dr. Van Fong (1) NATO Undersea Research Centre (NURC) – Viale San Bartolomeo 400, 19138 La Spezia (SP), Italy

NURC/Dr. Handson Yip (1) NURC/Dr. Ron Kessel/Mr. Vince Myers (1)

NATO C3 Agency – Oude Waalsdorperweg 61, 2597 AK The Hague, Netherlands (Visitors) P.O. Box 174, 2501 CD The Hague, Netherlands

NC3A/Ms. Sylvie Martel/Mr. Jason Offiong (1)


DRDC Atlantic mod. May 02

DOCUMENT CONTROL DATA(Security classification of title, body of abstract and indexing annotation must be entered when the overall document is classified)

1. ORIGINATOR (the name and address of the organization preparing the document.Organizations for whom the document was prepared, e.g. Centre sponsoring acontractor's report, or tasking agency, are entered in section 8.)

DRDC Atlantic

2. SECURITY CLASSIFICATION !!(overall security classification of the document including special warning terms if applicable).

UNCLASSIFIED

3. TITLE (the complete document title as indicated on the title page. Its classification should be indicated by the appropriate abbreviation (S,C,R or U) in parentheses after the title).

Concepts of Operations for the Side Scan Sonar Autonomous Underwater Vehicles Developedat DRDC Atlantic

4. AUTHORS (Last name, first name, middle initial. If military, show rank, e.g. Doe, Maj. John E.)

Bao Nguyen and David Hopkin

5. DATE OF PUBLICATION (month and year of publication ofdocument)

October 05

6a. NO. OF PAGES (totalcontaining information IncludeAnnexes, Appendices, etc). 70

6b. NO. OF REFS (total citedin document)

6

7. DESCRIPTIVE NOTES (the category of the document, e.g. technical report, technical note or memorandum. If appropriate, enter thetype of report, e.g. interim, progress, summary, annual or final. Give the inclusive dates when a specific reporting period is covered).

Technical Memorandum 8. SPONSORING ACTIVITY (the name of the department project office or laboratory sponsoring the research and development. Include address).

Defence R&D Canada – AtlanticPO Box 1012Dartmouth, NS, Canada B2Y 3Z7

9a. PROJECT OR GRANT NO. (if appropriate, the applicable researchand development project or grant number under which the documentwas written. Please specify whether project or grant).

11CL21

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TM 2005-213

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DRDC Atlantic 11. DOCUMENT AVAILABILITY (any limitations on further dissemination of the document, other than those imposed

by security classification)( x ) Unlimited distribution( ) Defence departments and defence contractors; further distribution only as approved( ) Defence departments and Canadian defence contractors; further distribution only as approved( ) Government departments and agencies; further distribution only as approved( ) Defence departments; further distribution only as approved( ) Other (please specify):

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DRDC Atlantic mod. May 02

13. ABSTRACT (a brief and factual summary of the document. It may also appear elsewhere in the body of the document itself. Itis highly desirable that the abstract of classified documents be unclassified. Each paragraph of the abstract shall begin with anindication of the security classification of the information in the paragraph (unless the document itself is unclassified) representedas (S), (C), (R), or (U). It is not necessary to include here abstracts in both official languages unless the text is bilingual).

This study reports findings on novel concepts of operations for Autonomous UnderwaterVehicles (AUVs) conducting mine hunting missions. The AUVs that we consider are thosecarrying a side scan sonar and have been developed at DRDC Atlantic. We examine the useof one and two AUVs each sweeping a lawn mowing pattern or a zigzag pattern. Twoindependent probabilistic models were built in order to assess and compare the Measures OfEffectiveness (MOEs) of these configurations. Through execution of these models and thenature of the Measures of Performance (MOPs) of the side scan sonar, the optimalconfiguration was found as well as a heuristic proof showing why it is optimal. The paperalso describes variational calculus methodologies on how to obtain the optimal search pathbased on target density distribution in general. Exact equations of motion for the optimalpath are found for a specific type of mine density distributions along the vertical axisalone, and along both the horizontal and vertical axis. The optimal paths are thencompared to the proposed operations in terms of MOEs.

14. KEYWORDS, DESCRIPTORS or IDENTIFIERS (technically meaningful terms or short phrases that characterize adocument and could be helpful in cataloguing the document. They should be selected so that no security classification isrequired. Identifiers, such as equipment model designation, trade name, military project code name, geographic location mayalso be included. If possible keywords should be selected from a published thesaurus. e.g. Thesaurus of Engineering andScientific Terms (TEST) and that thesaurus-identified. If it not possible to select indexing terms which are Unclassified, theclassification of each should be indicated as with the title).

Autonomous underwater vehiclesRemotely operating vehiclesMine huntingSearch patternsProbability of detectionCoverageMine density distributionOptimizationVariational calculusMonte Carlo simulation

concepts of operations for the side scan sonar autonomous underwater vehicles ... · 2012. 8....

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