shortest path - capacitated maximum covering problems

A Heuristic Flight Path Planner for a Small UAV Attempting to Find a Single Target in Minimum Time A thesis submitted to the Division of Research and Advanced Studies of the University

UNIVERSITY OF CINCINNATI

Date: _

I, _, hereby submit this work as part of the requirements for the degree of:

Manohar Balapa

Master of Science

Electrical and Computer Engineering and Computer Science

A Heuristic Flight Path Planner for a Small UAV Attempting to Find a Single Target in Minimum Time

Dr Emmanuel Fernandez

Dr Raj K Bhatnagar

Dr Bruce Walker

A Heuristic Flight Path Planner for a Small UAV Attempting to Find a Single Target in Minimum Time

A thesis submitted to the

Division of Research and Advanced Studies

of the University of Cincinnati

in partial fulfillment of the requirements for the degree of

Abstract

The work presented here is a part of ongoing research into flight path planning for autonomous fixed wing aircraft The problem of a single Unmanned Aerial Vehicle attempting to find a single stationary target in minimum time is examined The problem is tailored to a specific type of UAV i.e a low cost high maneuverability propeller driven fixed wing aircraft of wingspan 2-20 meters The objective area

to be searched for the target is split into cells the size of the area occupied by the target and occupancy probabilities for each cell are assigned A methodology for extending the problem to the case of a target that moves but is constrained to move only in the objective area is presented The search problem is set up as an infinite horizon optimization problem The problem is converted to a finite horizon shortest path problem The NP hard shortest path problem is then solved using a two step look ahead heuristic algorithm, which ultimately yields an approximately optimal solution of the search problem itself Finally simulation results are presented which show the heuristic algorithm generated path’s efficacy compared to a zigzag path and a path generated by a greedy algorithm

Acknowledgements

First and foremost, I would like to thank my advisor Dr Emmanuel Fernandez for noticing my enthusiasm for Autonomous Aerial Vehicles, encouraging me to do a research project in that field, giving me highly valuable advice on various technical and other matters pertaining to my research, and most importantly for patiently keeping me focused and motivated during the time I worked on this thesis I would like to thank Dr Raj Bhatnagar for agreeing to serve on the thesis committee and for his excellent courses on artificial intelligence, from which I learned a great deal I would also like to thank committee member Dr Bruce Walker of the Department of Aerospace Engineering for introducing me to the complex but interesting field of flight control systems, and for continually giving me valuable insights into aircraft autopilot design, which proved to be very useful during my research I would like to thank Dr Ernest Hall of the Department of Mechanical, Industrial and Nuclear Engineering and Dr Albert Bosse of the Department of Aerospace Engineering for all their suggestions on path planning for various types of autonomous vehicles

I would like to thank my parents for their unwavering belief in me and for all the support they have provided me before and during my graduate study at the University of Cincinnati I would like to thank my good friend and former workplace mentor Sumit Sharma for inculcating in me a methodical approach to error anticipation, that served me very well during my research I would also like to thank all the new friends I have made here for making my UC experiences a memorable one

Finally, I would like to acknowledge my late grandfather Dr Chennakesavan Balapa for igniting an interest in science, math and engineering in me at a very early age

CONTENTS

1 Introduction……… 3

1.1.Overview……… 4

1.2.The Search Scenario……….4

1.3.Previous Results ……….….5

1.4.The Heuristic Path Planner ……….….7

2 Problem Definition……… 9

2.1.Objective of the Search………10

2.2.The Vehicle …… ……… …10

2.3.Coordinate System and Euler Angles……… 12

2.4.The Sensor Footprint………13

2.5.Constraints on UAV Dynamics………17

2.6.Flight Time ……… ………18

2.7.Minimum Time Formulation……… 22

3 The Non-detection Probability ……… 24

3.1.Relevance of Non-detection Probability……… 25

3.2.Computation of Non-Detection Probability……….26

3.2.1 Initial Scan……… 27

3.2.2 Multiple Previous Scans……….28

3.2.3 Clipping Cells……….32

3.3.Moving target……… 34

4 The Decision Model……… 35

4.1.The Infinite Horizon Optimization Problem………36

4.2.The Finite Horizon Equivalent……….38

4.3.Heading Correction……… 41

4.4.Shortest Path Problem……… 43

5.1 Standard Methods ……… ….45

5.2.The Heuristic Function……….46

5.3. The Heuristic Look Ahead Algorithm……….47

6 Simulation Results………50

6.1.The Baseline Flight Paths……….52

6.2.Non-Informative Probability Map………53

6.3.Informative Probability Map………54

6.3.1 Case 1……… 54

6.3.2 Case 2……… 56

6.3.3 Case 3……… 58

6.4.Moving Target……… 60

7 Conclusion and Future Work……… 63

7.1.Summary………64

7.2.Future Work……… 65

8 References………66

CHAPTER

1

INTRODUCTION

1.1

OVERVIEW

The search problem is pervasive in many fields, be it specialized areas such as the military, emergency rescue, law enforcement etc or more mundane tasks such as finding a street address, looking for a misplaced pair of glasses etc The problem itself has existed for as long as there have been predatory animals, but the principle is still the same – keep traversing an area until the desired object is found The development of faster vehicles and better sensors has significantly affected the efficiency of a search Vehicles equipped with sensors can travel faster over larger areas and see farther than humans can This is especially true of aircraft However searches were still carried out by human operators and thus search plans had to incorporate human safety and fatigue It wasn‟t until the notion of artificial

intelligence and robotic unmanned vehicles was postulated that the idea of autonomous search began

to form Autonomous search is particularly attractive as now the searcher isn‟t constrained to a human operator‟s abilities any more Since then the autonomous search problem has received quite a bit of attention from the Operations Research community [6], the intelligent robotics community [7],[8], and

in recent years from Unmanned Aerial Vehicle researchers [9],[10],[11],[12],[13],[18] The large share

of attention paid to UAVs as autonomous searchers, is due to the proven effectiveness of aircraft as search vehicles

Although sea, air and land based vehicles have been and continue to be used in searches, Unmanned Aerial Vehicles (UAVs) are the most effective vehicles for searches as they can operate over land or sea in rough or level terrain, in turbulent as well as calm seas, and in most kinds of weather Also, because they are airborne, they can see more than other vehicles can

1.2

THE SEARCH SCENARIO

Quite a bit of the literature on searches [9],[10],[11],[12],[13],[18] assume that the objective area is divided into grid squares and that each square has a target occupancy probability assigned to it The same assumption will be made here The occupancy probabilities for all cells can be based on intuition, previous observations etc, but it is immaterial how those probabilities were assigned

Search problems can be broadly classified as single searcher and multiple searchers by single target and multiple targets as well as by whether the targets are moving or stationary The autonomous vehicle search literature has in recent years focused heavily on multiple searchers Examples include Tan [7], Rubio et al [8], Flint et al [9], Vincent and Rubin [13] and Polycarpou et al [18] The advantages of multiple searchers over single searchers are redundancy [13], the ability to share information [7], the ability to interpret information in different manners [9],[18] and increased efficiency through numbers [8] Multiple searchers that reinforced each other have been shown to be able to efficiently locate a target [7] as well as to maintain pursuit of the target [13] The use of single vehicles for searches has not been abandoned though with works like Lum [12] and Sarmiento [15] The reason that single agent search is still researched is because of economic concerns Vehicles in general are expensive; with autonomous vehicles being particularly expensive as the automation of a vehicle‟s functions is itself a complex procedure Obtaining multiple autonomous vehicles might be difficult especially for a lower budget organization such as a county law enforcement agency, a rescue agency or a National Guard unit In such cases, although multiple vehicles are available, there might not be enough of them to allocate a team of vehicles to every search problem Hence some cases might arise where only one autonomous vehicle is available to carry out the search In such a scenario the lone searching vehicle must still perform well without the help of other cooperating vehicles Hence a single agent search problem will be examined and solved in this thesis

All single agent searches be it single or multiple, moving or stationary targets need to find the first target as quickly as possible Once the initial target is found, the rest of the overall problem can be solved In the case of multiple targets, the procedure used to find the initial target can now be repeated

to find the remaining targets Hence if the optimal solution to the problem of finding a single target using a single searcher is found, it will be an important building block to solving all types of single searcher multiple target problems too

1.3

PREVIOUS RESULTS

Sarmiento et al [15] first examined the problem of a robot trying to find a target in minimum time The approach used was to divide the search area into a grid of square cells and assign a target occupancy probability to each cell The search plan used these occupancy probabilities to generate a cell visitation

the robot stops and then makes a 0 radius turns to go to the next point it examines Stopping to make 0 radius turns isn‟t possible for certain types of vehicles such as fixed wing UAVs Hence the calculation of traversal time between the two points is much more complicated than the simple Euclidean distance based expression used in that paper The second disadvantage is that the sensor is omnidirectional and unlimited in range or resolution in that paper Such an assumption about a sensor

is impractical as many sensors have a limited field of view and a limited range This will impact the calculation of the probability of successful detection given the vehicle‟s current location Most importantly, Sarmiento et al [15] considered their target recognition algorithm to be perfect The target recognition algorithm in this thesis is assumed to be imperfect, which is a more practical assumption This has far reaching implications The first one is that once a cell has been unsuccessfully scanned, it cannot be ignored as was done in [15] A balance must be found instead in examining scanned cells again and examining un-scanned cells The second implication is implied by the first – multiple passes over the entire objective area might be needed Hence the strategy of finding a Hamiltonian Path is not viable here as finding such a path depends strongly on the initial position of the vehicle Since the terminal position of a Hamiltonian path is not the same as its initial position, the UAV might have to recalculate the shortest Hamiltonian path from its current position every time that the coverage of the objective area is unsuccessful Finding a shortest Hamiltonian path is NP hard, and would therefore place an impractically heavy computational load on the UAV Instead if a Hamiltonian circuit is found that returns to the initial position, the problem needs to be solved just once offline after which the path found is just repeatedly followed by the UAV until the target is found

Flint et al [9] updated the occupancy probabilities of all cells based on success or failure in detecting a target in one look of a particular cell, using Bayesian updates These updated probabilities were then used in a rolling horizon stochastic decision model, which was solved online using a Best First Label Correction algorithm [2] However, Flint et al placed severe turning constraints on the UAV in order

to limit the choices of cells that the UAV can visit in order to make the rolling horizon problem more tractable to solve online Such constraints cause an aircraft to avoid corners as navigating those require sharp turns to be made This results in „holes‟ in the objective area that are never explored If multiple aircraft are used as in [9], there will be no „holes‟ as a „hole‟ in one aircraft‟s scanning path will lie on another aircraft‟s path and hence is scanned However, multiple aircraft are assumed to be unavailable

in this thesis If the turning constraint were to be relaxed, there will be no holes, but the shortest path problem becomes intractable to compute online Hence a different method that allows the corners to be scanned is required

Bertucelli and How [10] incorporated uncertainty into the initial occupancy probability estimate and calculated the number of „looks‟ that a sensor must have of a cell in order to determine with a desired degree of certainty whether a target is present there or not This approach was extended in [11] to dynamic targets with transition probabilities between cells If the sensor is fixed and has a footprint that moves with the UAV, a cell must be as small as possible in order to maximize the number of looks The smallest possible size of a cell is the size of the bounding rectangle of the target Even with this assumption, the number of looks is still directly proportional to the desired certainty, hence the UAV must choose between time spent in repeatedly scanning a cell and the desired certainty of target existence If only a single target is present in one among all cells, the UAV might spend too much time convincing itself of target‟s absence in other cells, before finding the target This might yield a highly sub-optimal search strategy Hence the single scan update strategy of Flint et al [9] might be more conducive to use with a fixed sensor

Han [7] examined the usefulness of reinforcement learning in finding a single target Although his work did show that two cooperating searchers can reinforce each other and ultimately learn to find a target quickly, it also found that his reinforcement learning procedure worked only for a stationary target with a perfect sensor Furthermore it required that the searcher repeat the search many times after it had found the target, to look for the same target again Thus the initial search will always take a very long time A similar method i.e Evolutionary Computing was used by Rubio et al [8], but his approach was geared toward optimizing aircraft performance in bad weather Hence Rubio‟s method can be used as a precursor to the search plan to account for bad weather but not the search itself

Lum et al [12] solved a single aircraft single target search problem by framing the cell visitation decision problem as a convex optimization problem However this approach doesn‟t update an occupation probability map based on observations It instead adds Gaussian noise to linear target dynamics and then uses that to predict the next possible location of the target The method used guarantees only a quasi-optimal solution, not an optimal one Rubin [13] proposed the use of terrain covering paths such as the ones developed by Chosin et al [4] and Lunnelsky et al [5] to be used to find targets His claim was that in the absence of informative prior probabilities such paths are the only way to find the target However, uninformative prior probabilities are rarely assumed Instead informative but incorrect prior probabilities are the norm An algorithm that updates occupancy probabilities can recover from incorrect assumptions and find the target faster than one which

exhaustively searches the entire area This thesis will show that the Bayesian update based heuristic flight path planner does outperform a terrain covering path

1.4

THE HEURISTIC FLIGHT PATH PLANNER

The thesis solves the minimum time search problem using a heuristic algorithm The search problem is defined, modeled and solved as follows

In Chapter 2, the search problem is defined in detail, following which the use of what Orr et al [1] describe as a Small UAV is justified, and other implications of that choice are presented The infinite horizon dynamics of the search process are developed, constraints on those dynamics are defined, and then an expression for the search time to optimize is developed A detailed explanation of the UAV‟s method of traversing from cell to cell is provided, which in turn is used to derive an expression for the single hop flight time

Chapter 3 states the need for calculating the target non-detection probability and then outlines the detailed procedure used to update this probability based on successive non-detection of the target The moving targets case is explored and a far sighted method initially presented in [19] to incorporate moving target transition probabilities into the non-detection probabilities is explained

Chapter 4 consolidates the search problem, dynamics and constraints and single hop flight time expression developed in Chapter 2 and the non-detection probability expression developed in Chapter

3 to define the problem as an infinite horizon Bellman equation to be solved It then reframes the problem into a finite horizon equivalent problem solvable by a shortest path algorithm

Chapter 5 states the reasons behind choosing a heuristic approach and then defines the heuristic function to be used Finally it lays out the entire algorithm used to find an offline solution to the search problem Chapter 6 tests the algorithm against a baseline zigzag path and a greedy suboptimal solution Results are shown for three cases i.e uninformed prior probabilities, informed prior probabilities and finally for a moving target The thesis is concluded and future avenues for research are discussed in Chapter 7

CHAPTER

2

PROBLEM DEFINITION

2.1

OBJECTIVE OF THE SEARCH

A single fixed wing Small UAV [1] given a square objective area defined by the coordinates xmin, xmax,

ymin, and ymax must find a single target in minimum time The presence of the target in the objective area is confirmed beyond doubt; only its exact location is unknown thus necessitating the search The target is assumed to be either stationary or is allowed to move, but will not leave the UAV‟s objective area What needs to be done once the target has been found is immaterial The characteristics of the target are known beforehand by the UAV

The UAV‟s sensor is coupled to an image recognition algorithm that is assumed to be imperfect just as

it was in [9],[10],[11],[12],[13],[18] Therefore if the target is physically present in a cell inside the sensor footprint, it will be detected with probability є, which is constant for the sensor‟s algorithm It

is assumed that the algorithm does not return false positives i.e detect a „target‟ when one isn‟t present Therefore it is possible that the UAV might have to pass over the target multiple times before successfully detecting it This will have a significant impact on the total time till detection Any search with a perfect sensor such as the one in [15] is guaranteed to find the target in its first pass over the entire area However, if an imperfect sensor is used, the aircraft could search for the target for an infinite amount of time because it could pass over the target many times but not detect it due to the sensor accuracy The sensor is also assumed to have a limited field of view and a limited range This is also important as it will constrain the size of the sensor footprint and hence limit the maneuverability

to and from a command post or other aircraft, and most importantly a Flight Path Planner that decides

mentioned components are important and are still topics of much research, it is the Flight Path Planner that is most relevant to searches It is the Flight Path Planner which must decide where to look for the target and where to fly to after that [8] Hence the search strategy must be programmed into the flight path planner

UAVs are further categorized into fixed wing aircraft and rotorcraft The lift necessary to keep the fixed wing aircraft airborne is generated by the aircraft‟s motion, while rotorcrafts obtain their lift by rotating airfoils Hence the rotorcraft itself need not be in motion in order to fly At low velocities, rotorcraft are capable of maneuvers that conventional fixed wing aircraft cannot perform, such as flying backwards, flying sideways without turning, hovering and making 0 radius turns Such maneuvers especially the low radius turns are highly useful in searches, as path curvature constraints significantly impact search strategies as shown by Flint et al [9] and Polycarpou et al [18] However at these low velocity conditions, the flight dynamics of rotorcraft are highly non-linear and hence implementing a flight control system and autopilot for these vehicles is a very difficult control problem [16] On the other hand fixed wing aircraft operate for a majority of time in flight conditions

in which it is easy to obtain linearized flight dynamics [17] Building autopilots for these linear dynamics is easy as standard control system design techniques can be used [17]

Fixed wing unmanned aircraft are further classified into UAVs, Small UAVs and Micro Air Vehicles [1] According to Orr et al [1], Small UAVs typically weigh about 9.5 kg, have a wingspan of about 2 meters and have a typical cruising velocity of 13 m/s Due to their ability to operate at such low speeds, such vehicles are capable of sharper turns than larger UAVs such as the Predator, and hence and their low speed allows them to be almost as maneuverable in a search as helicopters operating at low velocities normally are These small UAVs are also very inexpensive compared to UAVs such as the Predator, Micro Air Vehicles are however too small and too slow, and hence cannot be used to search larger areas This makes Small UAVs the right kind of low cost vehicle to use for an autonomous large area search One trade-off however is the low mass as that would prevent the UAV from having a gimbals stabilized sensor as gimbals are usually heavy and produce a significant moment that will cause the lightweight aircraft to depart from a stable flight condition Hence any sensor used on a small UAV has to be hard mounted In order to point the sensor in a desired direction, the aircraft must pitch, roll and yaw This will significantly impact the search strategy, because the UAV cannot focus on and scan an arbitrary area The UAV has to simultaneously scan the entire

2.3

AIRCRAFT COORDINATE SYSTEM

Throughout the thesis the NEU (North East Up) coordinate system will be used The system is shown

in Figure 1(a)

Figure 1(a)

The origin is at the bottom-left corner of the objective area The positive X-axis is in the northerly direction, the positive Y-axis is in the easterly direction and the positive Z-axis is pointed upward This is the same coordinate system used by the Global Positioning System [17] and is thus used by all aircraft based systems too

Since the pitch, roll and yaw of the aircraft is a significant factor, these angles will need to be clearly defined The angles are shown in Figure 1(b)

Figure 1(b)

Before the angles can be defined, the axes of rotation of the aircraft must be specified The origin is at the Center of Gravity (CG) of the aircraft The X-axis of rotation lies along the nose of the aircraft The Y-axis of rotation lies along the wings of the aircraft and is positive to the right The Z-axis of

rotation points up just like the world Z-axis The pitch angle is the angle made by the aircraft‟s body with respect to the X-axis, when the aircraft body is rotated about the Y-axis The roll angle is the

angle between the aircraft‟s body and the Y-axis when the aircraft body is rotated around the X-axis

The heading is the angle made by the aircraft‟s body with respect to the northerly direction when the

body of the aircraft is rotated along the Z-axis The heading describes the direction that the aircraft is flying in

2.4

SENSOR FOOTPRINT

The UAV‟s sensor is mounted so that it points 90 degrees downwards relative to the UAV‟s body x axis The sensor‟s field of view angle is 90 degrees vertically and horizontally (Figure 2)

The scanner footprint is defined as the area of the terrain that is currently under the scanner This

footprint has its lowest area when the UAV is in level flight, and increases its area when the UAV pitches or rolls However, when the UAV‟s pitch or roll angle is 45 degrees, the footprint will not exist (Figures 3(a) to 3(d))

The UAV operates by taking a snapshot of its sensor footprint and scanning the snapshot image to try

to positively identify the target in it The image is assumed to be processed pixel by pixel proceeding

in a top-left to bottom right manner Therefore, in order to be able to perceive all relevant features necessary for making a positive identification of the target, the UAV must be within a certain

maximum distance ρ to the sensor footprint If the UAV exceeds this distance, the image recognition algorithm coupled to the sensor will fail This is illustrated in Figures 3(e) and 3(f)

Using Figure 3(g), the bounding points and the area of the sensor‟s footprint are calculated as follows

Previous lattitude coordinate

Previous longitude coordinate

Previous altitude above sea level

Current lattitude coordinate

Current longitude coordinate

Current roll angle

Current pitch angle

Current heading with respect to North

From Figure 3(g) the points defining the rectangular footprint are

TL =Top left point=

From Figure 3(h) the points defining the rectangular footprint are

BL =Bottom left point=

k=Bottom right point=

2.5

CONSTRAINTS ON AIRCRAFT DYNAMICS

As seen in the previous section, the UAV‟s sensor footprint depends on the UAV‟s attitude i.e roll, pitch and yaw angles, altitude and GPS location, all of which the aircraft can change at any time However, the sensor does have a limited range and hence the UAV must select a position, altitude and attitude that allows the sensor to be able to scan a valid footprint This places constraints on the UAV‟s position, altitude and attitude These constraints are defined as follows

Although the UAV can pitch and roll up to 45 degrees, a high pitch or roll angle will result in a large but out of range footprint This is shown in Figures 3(e) and 3(f) To ensure that this situation does not occur, the UAV always calculates the point of the footprint rectangle that is farthest to it, and tries to stay within range to that point This leads to the following constraints

ed up and rolling right TR is the farthest point

When the UAV is pitched down and rolling left BL is the farthest point

When the UAV is pitched down and rolling right BR is the farthest point

In all the cases distance k h k cot (45 |2 k |) cot (45 |2 k|) 1

The UAV scans the region only when its entire footprint is inside the objective area The four points

bounding the footprint rectangle are never allowed to lie outside the area objective

2

2 1

In carrying out the search for the target, an aircraft flies from one grid point to another The aircraft

might also change its altitude in order to be able to scan a larger or smaller footprint An aircraft does

this by changing its pitch and heading angles

The pitch angle can be changed almost instantly in place by using the aircraft‟s elevator [17] The time

taken to do this is quite low and can be neglected Hence the aircraft will not drift away from the line

by just pitching However, the same isn‟t the case for yawing Although the aircraft‟s rudder can be

used to yaw the aircraft to the desired heading, the rudder has very low control authority compared to

the elevator, and will therefore take a very long time to effect a turn A more effective way to turn the

aircraft is to roll the aircraft‟s wings until the aircraft is banked at a desired roll angle When the

aircraft is banked, the lift force keeping the aircraft airborne has a horizontal and vertical component

as shown in Fig 4(a)

Figure 4(a)

The horizontal component LcosΦ causes the aircraft to turn, while the vertical component LsinΦ acts against W the gravitational force If LsinΦ=W the aircraft will execute the turn keeping its altitude and

velocity constant [21] This type of turn is called a coordinated turn

Once the aircraft rolls back to level flight, it will now fly straight along a heading in the same direction

of the tangent to the circle at that point where the aircraft rolled back to level This is illustrated in Figure 4(b)

Let the UAV be at point A, let ψk =NAH be its heading at that time and let C be the point it must get

to Although the line AC (not shown) would be the actual shortest distance between A and C, a fixed wing aircraft will be unable to fly along this line Hence as indicated in Figure 4(b), the UAV must first turn along arc AB until it attains a heading that will bring it to the commanded point The turn can

be a right or left turn as indicated in Figure 4(b) Once the UAV attains its desired heading it flies along the straight line BC tangential to arc AB, which will bring it to the commanded point C At C, the UAV‟s heading will be the heading that it maintained while flying along BC

If the UAV makes a right turn, it can reach any point except those that lie inside the turning circle

centered to the right of point A Likewise if the UAV turns left, it can‟t reach points lying inside the

turning circle with the centre to the left of A However, the opposite is possible i.e the UAV can turn

left and reach points lying inside the circle to the right and vise versa The direction that the UAV

chooses to turn will be based on this reachability criterion If point C is reachable with both a right and

left turn, the UAV chooses that turn which will minimize the total distance traveled

The radius of arc AB is called the turn radius The rate of change of the UAV‟s heading with respect

to time as it flies along arc AB is the angular velocity The turn radius must be as low as possible and

angular velocity must be as high as possible too These goals must be met without stalling the aircraft

The velocity must be maintained above a critical velocity called stall speed, which is designated as

Vstall. The expression for turn radius and the angular velocity of the aircraft executing the turn is given

r roll angle will bring the aircraft dangerously close to structural failure [21]

V is chosen to be the V 3.8 which is the velocity at which maximum deflection ofcontrol sur

manneuvering velocity

faces will cause structural damage It is proven in [21] that using this velocity along with a bank angle

of 75 will ensure a minimum possible turn radius, maximum angular velocity and safety from sta

int

lling

Let be the aircraft's heading at point B It has the same heading at point C

If current position and orientation are

BC has to be perpendicular to OB where O is the centre of arc AB

Point O=( sin( ),

As seen in Figure 4(b), a left turn allows the UAV to get to a point very close to it, although the

distance traveled is much larger than the Euclidean distance

T o

1

2 1

) for a left turn sin

Hence out of all the 6 degrees of freedom available to the UAV i.e x, y and z axis translation and roll,

pitch and yaw, only 4 are needed by the UAV i.e x, y and z axis translation and roll Pitch and yaw

now depend on the 4 degrees of freedom available in order to consume minimum time flying from one

point to another

This method of turning has a significant impact on the search strategy The mission planners developed by Flint et al [9] and Polycarpou et al [18] carried out contiguous scans i.e the aircraft was perennially scanning the footprint underneath it Curvature constraints imposed in these papers enabled such scans to be made However a 75 degree bank angle coordinated turn will result in an invalid footprint and it is possible that the pitch angle of the aircraft required to attain the commanded altitude might also cause an invalid footprint Hence scans in this case are not contiguous Scanning is suspended until the UAV reaches the commanded x and y coordinate and altitude Non-contiguous scans mean that the aircraft can scan any arbitrary region in the objective area at any time This will add a great deal of dimensional complexity to the search

2.7

MINIMUM TIME FORMULATION

All searches are carried out in the following sequence The UAV flies to the initial location from the origin and takes a snapshot of its sensor footprint at that location Snapshots taken by the UAV of its sensor footprint are enumerated as k=1,2,3,… If the target is found in the k‟th snapshot, the search is over If it fails to find the target, the UAV chooses a different location to fly to and scan Scans are not contiguous i.e the region lying between the k‟th snapshot and the k+1‟th snapshot that the UAV flies

over is not scanned The UAV will use up some time flying to the new location This time is the flight

time that was calculated previously This procedure continues until the target is found

Once the UAV reaches its new location, the sensor‟s image recognition algorithm runs a pixel by pixel analysis of the entire footprint in real time using a real time procedure such as the one used by Mahlknecht [20] and returns success or failure in detection These algorithms require only a few milliseconds to run, and hence can be neglected compared to the flight time calculated above, just as the scanning time was neglected by Sarmiento [15]

detects the target in the initial scan and

1 if the target is not detected

1

If the target was detected, total search time=

Else in seconds the UAV flies to a new location and thus the sensor footprint is different

This new footprint is scanned for the target o

flight flight

t t

nly if the previos scan was unsuccesful The UAV is physically incapable of rescanning the previous footprint, as the UAV must stay above a

minimum required velocity in order to remain airborne This

minimum velocity is called

If detection is succesful in the second scan, total search time G= +

This can also be represented for both cases as G=

Where

flight flight flight flight

If this scan is also unsuccesful, the UAV repeats the procedure of flying to a different

location and scanning it untill succesful detection If the detection was succesful in the

nth scan, total searc

0

h time G=

Where is the time taken to fly to the kth scan location

Since the value of n is unknown but can be in the worst case, the total search time

is given by the following e

k

k

n flight k flight

t t

0

1

xpressionG= 5

1 if the target has not been detected in all scans including Where

0 if the target has been detected in any scan prior to anf including the kth scan

Thus is initially set to 1 and retains this value until succesful detection, after which it

remains at value 0 infinitely

Since the total cost depends upon a random variable, only its expected

P D bility that the target is undetected until and including the kth scan

This probability depends on whether a cell is inside the kth footprint, the target detection algorithm‟s efficiency, the number of times a cell inside the footprint was previously scanned unsuccessfully and whether the target is stationary or moving An analytical expression for this probability is obtained in the next chapter

CHAPTER

3

THE NON-DETECTION PROBABILITIES

3.1

RELEVANCE OF NON-DETECTION PROBABILITY

If the UAV is at a certain location and its scan of the sensor footprint was unsuccessful, the autopilot

must decide which location to fly to next and what bank angle to assume once there The flight time

can help it decide which cell to fly to next and the UAV could roll to an angle that maximizes the

footprint area However, this kind of search strategy is extremely myopic as it doesn‟t anticipate the

possibility of non-detection and deal with it This kind of myopic search is an uninformed search

because it ignores information available to it in the form of the occupancy probability map

On the other hand an informed search uses the occupancy probability map available to it These

probabilities act as the prior probabilities with which to carry out a well known and highly effective

search strategy called Bayesian Search [6] In this methodology, the occupancy probability for each

cell is updated using Bayes‟ Theorem whenever a certain cell is examined A typical Bayesian search

proceeds as follows

Let ( ) be the probability that the target is found in location Its complement= ( )

with no overlap between them, (

because it is updated from the prior value based on the observation made.

flight flight

t P S t P S P S t P S P S P S t P S P S P S t

0 1 1 1 2 1 2 1 2 1 3 1 2

If Bayesian updates were being performed to obtain posterior probabilities for every prospective location

Expected total time ( ) = E G t flight P S t( ) flight P S( )P(S /S t) flight P S( )P(S /S P S) ( /S S

This probability ( ) is the probability that the target w

An informed search will have to multiply this non-detection probability P D( k 1) to an estimate of

remaining search time for each location and then decide which cell to visit

3.2

CALCULATING THE NON_DETECTION PROBABILITY

With the sensor footprint known, the non-detection probability can be calculated When calculating

the non-detection probability, two scenarios arise The first is at the initial state when only one scan is

made (Fig 5(a)) The second scenario is when some area has been scanned and the target has not been

found yet (Fig 5(b)) Thus some prior observations are available with which to estimate non-detection

probability The entire area could have been scanned before the current coverage of the objective, but

the target wasn‟t detected because of the imperfect sensor, and hence the UAV is re-covering the

entire region (Fig 5(c))

Thus information from scans made in the current coverage and all previous unsuccessful covers is

available to estimate the non-detection probability This particular situation can repeat over and over

The formulation of non-detection probability for both scenarios is developed below

3.2.1 INITIAL SCAN

The first scenario that requires the calculation of target non-detection probability is the scan of the first region that the UAV flies to At this stage, there are no prior observations with which a Bayesian update can be made Hence the probability depends entirely on the prior cell probabilities and upon which cells lie in the initial footprint It is calculated as follows

Let be the current footprint of the UAV

Let D be the event that the UAV detects the target from its initial position and sensor orientation.Let T be the event that targe

S

t i is physically present in the sensor footprint region

Let V be the event that target i is visible to the UAV

Let R be the event that visible target i is detected by the UAV

the target is present but not visible, or the target is visible but undetected

1

(x,y,h, , , )

(x,y,h, , , )

(V / T ) 0 always because there is nothing preventing the sensor from seeing the target

if the target lies in the sensor's footprint Likewise (V / T ) 1

(R / V) =1- is the prob

P

P P

(T ) is the probability that the target is physically absent in the footprint

It is evaluated by summing the occupancy probabilities of all cells outside the footprint

Likewise (T

P

P

1 , , , ) ) is found by summing the occupancy probabilities of all cells inside the sensor footprint.Let all cells be ennumerated by j=1 to , which is the total number of cells

Each cell will have

cell

n

1 1

1

1

(x,y,h, , , )

(x,y,h, , , )

two values associated with it -:

0 if the cell isn't in the initial footprint

1 if the cell is in the initial footprint

The value of is obtained by clipping the

j

j

S s

S s

Occupancy probability of cell j

All are combined into the vector and all are combined into the vector

The second possibility is that at some point, the UAV has flown over all areas in the rectangular

region many times, but has still not detected the target, due to its imperfect sensor

Let c be the number of times the entire area has been previously searched In the current coverage,

some part of the entire rectangular area has been examined, but there are still some remaining areas

The already examined region is denoted the covered region while the yet to be examined region is

denoted the uncovered region The footprint of the k+1th snapshot might or might not have some

overlap with footprints at previous locations (Figure 6(a) For simplicity it is assumed that the UAV

was always in range previously

Because the sensor footprint is sliding and can change its area as well as the orientation of its square boundary, either the entire footprint or a part of it might lie in the covered region, and will be

designated as an overlap region Depending on what the UAV‟s exact position and orientation is, the

overlap region can be either the entire covered region or just a portion of it The part of the covered

region that is not part of the overlap region is called the scanned region The part of the sensor footprint that isn‟t in the overlap region will lie in the uncovered region This is called the new region The part of the uncovered region that isn‟t in the new region is called the unscanned region If the

UAV flies for a longer period of time it might find itself in a region with absolutely no overlap This is illustrated in Figure 6(b)

If prior scans were unsuccessful, it is because of only two reasons The first is that the target isn‟t present in the covered region, and the second one is that the target is present in the covered region but the inaccurate sensor failed to recognize it Because the only observation made by the sensor is success or failure to detect, it is not possible for the system to know exactly which of the above stated possibilities is true It must consider both possibilities likely

k k

k

Let F be the event that the target is present in the unscanned region and let C be the event that the target

is present in the scanned region Let T be the event that the target is present in the o

k

verlap region and let Z be the event that the target is in the new region Let W be the event that the target is present in the covered region, and let U be the event that the target is in the unco

D

new k

the target is detected in the new region.