shortest path - capacitated maximum covering problems

UNIVERSITY OF CINCINNATI Date:___________________ I, _________________________________________________________, hereby submit this work as part of the requirements for the degree of: in: It is entitled: This work and its defense approved by: Chair: _______________________________ _______________________________ _______________________________ _______________________________ _______________________________ 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 Master of Science in the Department of Electrical and Computer Engineering and Computer Science of the College of Engineering; June 5 th 2007 by Manohar Balapa BE Computer Science and Engineering 2003 Visveswaraiah Technological University Bangalore India Committee chair: Dr. Emmanuel Fernandez 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 - 1 - 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. The Flight Path Solution…………………………………………………… 44 - 2 - 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 - 3 - CHAPTER 1 INTRODUCTION - 4 - 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. - 5 - 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 sequence. However their approach has some significant disadvantages. The first is the assumption that [...]... 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 -6 - Bertucelli and How [10]... occupancy probabilities can recover from incorrect assumptions and find the target faster than one which -7 - 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... which moves along with the UAV - 11 - 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... in Figure 1(b) - 12 - 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... close to it, although the distance traveled is much larger than the Euclidean distance - 20 - Angle AOB= NAH= k 2 , N'OA= 2 - d turn k d turn tan -1 ( , N'OC=2 yk 1 xk 1 yk xk AOB= min{rturn [ d turn yk 1 3 2 rturn cos d turn k )2 ( xk d turn tan -1 ( k 1 xk rturn sin yk 1 xk 1 yk xk rturn cos rturn sin rturn cos-1 ( yk 1 rturn cos 2 k) ( xk Length of Arc AB=rturn ( AOB ) ( yk yk 1 2 k) rturn sin d... 1 int = xk 1 xk rturn sin k ( yk 1 yk rturn cos k ) 2 ( xk k 1 ) k )]} rturn cos k k )} 2 k) VT [ k d turn tan -1 ( k k Flight time t flightk (min{r 1 ( xk d turn tan -1 ( = Total distance travelled ( yk xk rturn cos-1 ( t flightk rturn cos rturn sin rturn ( yk ( yk .1 -NAH+N'OC-BOC BOC=cos-1 ( int 2 VT g D tan 75o 1 for a right turn and the turn radius rturn 1 for a left turn Let d turn hk 1 hk rturn... was previously scanned unsuccessfully and whether the target is stationary or moving An analytical expression for this probability is obtained in the next chapter - 23 - CHAPTER 3 THE NON-DETECTION PROBABILITIES - 24 - 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... 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... along the decided path Although all the above - 10 - 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... tan -1 ( rturn cos int yk 1 xk 1 )2 yk xk ( xk 1 rturn cos rturn sin xk k rturn ) cos-1 ( ( yk k rturn sin k In order to attain the desired altitude, the UAV pitches up to an angle tan 1 ( int rturn ( yk k rturn sin k ) cos-1 ( rturn sin yk 1 int )2 rturn cos ( hk Because the heading that the UAV maintained along line BC is still maintaomed at C y yk rturn cos k rturn d turn tan -1 ( k 1 ) cos-1 ( . Heading Correction……………………………………………………… 41 4.4. Shortest Path Problem…………………………………………………… 43 5. The Flight Path Solution…………………………………………………… 44 - 2 - 5.1. Standard Methods ……………………………………………………. one which - 8 - 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. 65 8. References………………………………………………………………………66 - 3 - CHAPTER 1 INTRODUCTION - 4 - 1.1 OVERVIEW The search problem is pervasive in many fields, be