Cooperative system lecture note in economics and mathematical systems

Lecture Notes in Economics and Mathematical Systems Founding Editors: M Beckmann H P Künzi Managing Editors: Prof Dr G Fandel Fachbereich Wirtschaftswissenschaften Fernuniversität Hagen Feithstr 140/AVZ II, 58084 Hagen, Germany Prof Dr W Trockel Institut für Mathematische Wirtschaftsforschung (IMW) Universität Bielefeld Universitätsstr 25, 33615 Bielefeld, Germany Editorial Board: A Basile, A Drexl, H Dawid, K Inderfurth, W Kürsten, U Schittko 588 Don Grundel · Robert Murphey Panos Pardalos · Oleg Prokopyev (Editors) Cooperative Systems Control and Optimization With 173 Figures and 17 Tables 123 Dr Don Grundel AAC/ENA Suite 385 101 W Eglin Blvd Eglin AFB, FL 32542 USA don.grundel@eglin.af.mil Dr Robert Murphey Guidance, Navigation and Controls Branch Munitions Directorate Suite 331 101 W Eglin Blvd Eglin AFB, FL 32542 USA robert.murphey@eglin.af.mil Dr Panos Pardalos University of Florida Department of Industrial and Systems Engineering 303 Weil Hall Gainesville, FL 32611-6595 USA pardalos@ufl.edu Dr Oleg Prokopyev University of Pittsburgh Department of Industrial Engineering 1037 Benedum Hall Pittsburgh, PA 15261 USA prokopyev@engr.pitt.edu Library of Congress Control Number: 2007920269 ISSN 0075-8442 ISBN 978-3-540-48270-3 Springer Berlin Heidelberg New York This work is subject to copyright All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer Violations are liable to prosecution under the German Copyright Law Springer is part of Springer Science+Business Media springer.com © Springer-Verlag Berlin Heidelberg 2007 The use of general descriptive names, registered names, trademarks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use Production: LE-TEX Jelonek, Schmidt & Vă ockler GbR, Leipzig Cover-design: WMX Design GmbH, Heidelberg SPIN 11916222 /3100YL - Printed on acid-free paper Preface Cooperative systems are pervasive in a multitude of environments and at all levels We find them at the microscopic biological level up to complex ecological structures They are found in single organisms and they exist in large sociological organizations Cooperative systems can be found in machine applications and in situations involving man and machine working together While it may be difficult to define to everyone’s satisfaction, we can say that cooperative systems have some common elements: 1) more than one entity, 2) the entities have behaviors that influence the decision space, 3) entities share at least one common objective, and 4) entities share information whether actively or passively Because of the clearly important role cooperative systems play in areas such as military sciences, biology, communications, robotics, and economics, just to name a few, the study of cooperative systems has intensified That being said, they remain notoriously difficult to model and understand Further than that, to fully achieve the benefits of manmade cooperative systems, researchers and practitioners have the goal to optimally control these complex systems However, as if there is some diabolical plot to thwart this goal, a range of challenges remain such as noisy, narrow bandwidth communications, the hard problem of sensor fusion, hierarchical objectives, the existence of hazardous environments, and heterogeneous entities While a wealth of challenges exist, this area of study is exciting because of the continuing cross fertilization of ideas from a broad set of disciplines and creativity from a diverse array of scientific and engineering research The works in this volume are the product of this cross-fertilization and provide fantastic insight in basic understanding, theory, modeling, and applications in cooperative control, optimization and related problems Many of the chapters of this volume were presented at the 5th International Conference on “Cooperative Control and Optimization,” which took place on January 20-22, 2005 in Gainesville, Florida This day event was sponsored by the Air Force Research Laboratory and the Center of Applied Optimization of the University of Florida VI Preface We would like to acknowledge the financial support of the Air Force Research Laboratory and the University of Florida College of Engineering We are especially grateful to the contributing authors, the anonymous referees, and the publisher for making this volume possible Don Grundel Rob Murphey Panos Pardalos Oleg Prokopyev December 2006 Contents Optimally Greedy Control of Team Dispatching Systems Venkatesh G Rao, Pierre T Kabamba Heuristics for Designing the Control of a UAV Fleet With Model Checking Christopher A Bohn 21 Unmanned Helicopter Formation Flight Experiment for the Study of Mesh Stability Elaine Shaw, Hoam Chung, J Karl Hedrick, Shankar Sastry 37 Cooperative Estimation Algorithms Using TDOA Measurements Kenneth A Fisher, John F Raquet, Meir Pachter 57 A Comparative Study of Target Localization Methods for Large GDOP Harold D Gilbert, Daniel J Pack and Jeffrey S McGuirk 67 Leaderless Cooperative Formation Control of Autonomous Mobile Robots Under Limited Communication Range Constraints Zhihua Qu, Jing Wang, Richard A Hull 79 Alternative Control Methodologies for Patrolling Assets With Unmanned Air Vehicles Kendall E Nygard, Karl Altenburg, Jingpeng Tang, Doug Schesvold, Jonathan Pikalek, Michael Hennebry 105 A Grammatical Approach to Cooperative Control John-Michael McNew, Eric Klavins 117 VIII Contents A Distributed System for Collaboration and Control of UAV Groups: Experiments and Analysis Mark F Godwin, Stephen C Spry, J Karl Hedrick 139 Consensus Variable Approach to Decentralized Adaptive Scheduling Kevin L Moore, Dennis Lucarelli 157 A Markov Chain Approach to Analysis of Cooperation in Multi-Agent Search Missions David E Jeffcoat, Pavlo A Krokhmal, Olesya I Zhupanska 171 A Markov Analysis of the Cueing Capability/Detection Rate Trade-space in Search and Rescue Alice M Alexander, David E Jeffcoat 185 Challenges in Building Very Large Teams Paul Scerri, Yang Xu, Jumpol Polvichai, Bin Yu, Steven Okamoto, Mike Lewis, Katia Sycara 197 Model Predictive Path-Space Iteration for Multi-Robot Coordination Omar A.A Orqueda, Rafael Fierro 229 Path Planning for a Collection of Vehicles With Yaw Rate Constraints Sivakumar Rathinam, Raja Sengupta, Swaroop Darbha 255 Estimating the Probability Distributions of Alloy Impact Toughness: a Constrained Quantile Regression Approach Alexandr Golodnikov, Yevgeny Macheret, A Alexandre Trindade, Stan Uryasev, Grigoriy Zrazhevsky 269 A One-Pass Heuristic for Cooperative Communication in Mobile Ad Hoc Networks Clayton W Commander, Carlos A.S Oliveira, Panos M Pardalos, Mauricio G.C Resende 285 Mathematical Modeling and Optimization of Superconducting Sensors with Magnetic Levitation Vitaliy A Yatsenko, Panos M Pardalos 297 Stochastic Optimization and Worst–case Decisions Nalan Gă ulpinar, Berác Rustem, Stanislav Zakovi c 317 Decentralized Estimation for Cooperative Phantom Track Generation Tal Shima, Phillip Chandler, Meir Pachter 339 Contents IX Information Flow Requirements for the Stability of Motion of Vehicles in a Rigid Formation Sai Krishna Yadlapalli, Swaroop Darbha and Kumbakonam R Rajagopal 351 Formation Control of Nonholonomic Mobile Robots Using Graph Theoretical Methods Wenjie Dong, Yi Guo 369 Comparison of Cooperative Search Algorithms for Mobile RF Targets Using Multiple Unmanned Aerial Vehicles George W.P York, Daniel J Pack and Jens Harder 387 Optimally Greedy Control of Team Dispatching Systems Venkatesh G Rao1 and Pierre T Kabamba2 Mechanical and Aerospace Engineering, Cornell University Ithaca, NY 14853 E-mail:vr47@cornell.edu Aerospace Engineering, University of Michigan Ann Arbor 48109 E-mail: kabamba@engin.umich.edu Summary We introduce the team dispatching (TD) problem arising in cooperative control of multiagent systems, such as spacecraft constellations and UAV fleets The problem is formulated as an optimal control problem similar in structure to queuing problems modeled by restless bandits A near-optimality result is derived for greedy dispatching under oversubscription conditions, and used to formulate an approximate deterministic model of greedy scheduling dynamics Necessary conditions for optimal team configuration switching are then derived for restricted TD problems using this deterministic model Explicit construction is provided for a special case, showing that the most-oversubscribed-first (MOF) switching sequence is optimal when team configurations have low overlap in their processing capabilities Simulation results for TD problems in multi-spacecraft interferometric imaging are summarized Introduction In this chapter we address the problem of scheduling multiagent systems that accomplish tasks in teams, where a team is a collection of agents that acts as a single, transient task processor, whose capabilities may partially overlap with the capabilities of other teams When scheduling is accomplished using dispatching [1], or assigning tasks in the temporal order of execution, we refer to the associated problems as TD or team dispatching problems A key characteristic of such problems is that two processes must be controlled in parallel: task sequencing and team configuration switching, with the associated control actions being dispatching and team formation and breakup events respectively In a previous paper [2] we presented the class of MixTeam dispatchers for achieving simultaneous control of both processes, and applied it to a multi-spacecraft interferometric space telescope The simulation results in [2] demonstrated high performance for greedy MixTeam dispatchers, and Comparison of Cooperative Search Algorithms for Mobile RF Targets 391 Fig Three UAVs in optimal geometry for triangulating The target can lie anywhere in the cone shaped areas determined by the accuracy of the sensor’s DF angle and sensors maximum range target’s location The UAVs within the orbit adjust their velocities to form equi-angle distance among the UAVs (or 90 degree separation if only two UAVs) to improve the accuracy of the estimated target location, as shown in Fig The estimated localization error versus the number of UAVs in ideal geometry for triangulation is plotted in Fig This is used as a measure of merit to determine the optimal number of UAVs necessary to locate targets against all other requirements of the global search (i.e., finding other undiscovered emitters) For example, one could assign all UAVs to locate one emitter at the expense of hindering the global search Fig indicates that accuracy improves little with more than three UAVs, thus we limit the number of UAVs cooperating in target localization to three When only one UAV is available and must localize on its own, it triangulates with its prior estimated angles as it flies around the target The target can move during this process, increasing the localization error 2.4 Stage Four: Local Search for Lost Mobile Target The local search stage is initiated for UAVs (one to three) that have committed to locate a target when the target ceases to emit signals after it has been detected but before it is located The committed UAVs then form an orbit with the radius greater than the one designated to locate a target The radius continues to grow over time if the target is not re-detected As the orbit radius grows, the UAVs continue to search for the target Once the radius reaches a pre-defined maximum value, the UAVs engage in a local search pattern similar to the one in the global search stage, only in a smaller area with a maximum 392 George W.P York, Daniel J Pack and Jens Harder Intersecting Area versus Number of UAVs 100 Intersecting Area (units ) 150 50 0 Number of UAVs in Orbit Fig Determining the optimal number of UAVs to locate a target This figure shows how the location accuracy improves as the number of UAVs increases, assuming the UAVs are in ideal geometry of three UAVs If a target is still not detected after a designated time interval, the UAVs return to the global search stage Currently, the radius of a local search orbit is a function of time, existence of signal, and two past trajectory points of the target emitter As the time between emitter signals grow, the radius of the orbit will grow At each point in time, UAVs involved in the local search uses the following two equations to determine the local orbit When no emitter signal is present, the local orbit equation is as follows (x − [ex (t − 1) + cx (ex (t − 1) − ex (t − 2))])2 + (y − [ey (t − 1) + cy (ey (t − 1) − ey (t − 2))])2 = (r(t) + kt)2 (3) ex (t − 1) and ey (t − 1) are the last estimated x and y location values of the emitter; ex (t − 2) and ey (t − 2) are the second last estimated x and y location values of the emitter; r(t) represents the radius of the current orbit; cx and cy are constants chosen to weigh the movement of the emitter location based on the past history; and kt represents the increase in the local orbit radius based on the elapsed time (t) and a constant (k) to accommodate the movement of the emitter location When an emitter signal is detected, the local orbit equation yields to (x − ex (t))2 + (y − ey (t))2 = r(t)2 (4) Comparison of Cooperative Search Algorithms for Mobile RF Targets 393 2.5 Simulation Currently we have implemented our cooperative control algorithms in a MATLAB simulation A simple example of three UAVs going through these four stages versus one target is shown in Fig The initial locations of the three UAVs and three targets are randomly generated Each UAV resides at the center of a large circle The black dot at the center of a circle indicates the current UAV location The radius of the circle represents the UAV sensor detection range as it sweeps 360 degrees Similarly, each target is located at the center of the small circle, again represented with a black dot, and the radius of the small circle represents a circular orbit of the UAV’s maximum turn radius around a target emitter where UAVs must fly to accurately locate the target The trajectory history of UAVs and the targets are recorded and are shown in the figure by dotted lines within gray paths Each UAV keeps track of the status of the search space and maintains a search map where each search location contains a numerical value which varies based on the search history of the particular location Since the emitters we want to detect can be silent for an unspecified duration of time and are mobile, the UAVs not only must record the history of their past trajectories but also have a mechanism to slowly diminish numerical values for visited locations over time to realistically model the decaying intelligence gathered in the past Such a scheme allows our UAVs to revisit the same spot and detect targets if those targets were ‘silent’ during previous flyovers Cooperative Control Algorithm II The first algorithm performs well during the search stage, spreading the UAVs out to minimize search time This approach has a disadvantage during target localization, however, as it takes time for the neighboring UAVs to join on localization orbit and the target may stop emitting in the process This results in either a non-optimum localization due to poor geometry or a loss of the target resulting in more time spent in local search This motivated the second approach of having a set of three UAVs fly in an equilateral triangle formation as shown in Fig Instead of taking time to get on a proper orbit, the formation simply attempts to fly directly over the target, achieving the optimum geometry for triangulation when the centroid of the formation passes over the target During the general search stage each formation leader follows the search strategy developed in the first algorithm, using the same cost function This formation-based algorithm generally takes longer for the global search since the formations can not cover as much area as the individual UAVs when they are spread out [6] 394 George W.P York, Daniel J Pack and Jens Harder (a) (b) (c) (d) (e) (f) Fig Snapshots showing the progression of three UAVs searching, detecting, and localizing a moving target Frame (a): This frame shows the initial locations of three UAVs (large circles) and one target (small circle) Frame (b): This frame shows the cooperative search mode where the three UAVs are spreading themselves across the area based on the search cost function The top UAV detects a target (indicated by the black line showing the estimate angle to the target) and notifies the other two UAVs Frame (c): Upon receiving the target discovery information, the nearby UAVs compute the ‘search versus locate’ cost function to determine whether or not to help localize the emitter The frame shows that two UAVs have decided to join the efforts The closest UAV starts an orbit, refining the location estimate on its own Frame (d): The emitter turns off before the UAVs are in the proper geometry, so they switch to the local search mode Frame (e): While the UAVs are in the local search mode, the target emits again within range of the UAVs, and they resume flying to the best geometry to locate the target Frame (f): The three UAVs orbit around the detected emitter and adjust their positions to be equi-angle apart from each other for optimal triangulation 3.1 Formation We desire an algorithm to have a set of UAVs join in formation and maintain the formation without undue computation or communication requirements Several authors have proposed simple methods such as Monteiro et al’s attractor dynamics [8] and Spears’ physics-based methods [7] Our method is similar as illustrated in Fig Comparison of Cooperative Search Algorithms for Mobile RF Targets 395 Flight Path Leader RF Target Left Right Leader Left Right Fig Illustrating the triangular formation and the path required to ideally locate the RF target The UAVs first determine who their nearest two neighboring UAVs are The one with the highest priority (known a priori) is designated the leader and the other two are the followers The leader flies following the general search rules of Algorithm One The two followers join in formation behind the leader by being attracted or repelled by their neighboring two UAVs as shown in Fig They are attracted up to a predefined range from the other UAVs and then repelled if they get too close The UAVs continually adjust their velocity and trajectory incrementally as necessary to maintain this relationship This simple method allows the UAVs to initially decide which formation to join and maintain the formation The follower first finds two attract/repel points (influenced by their two closest neighbors), then combine these to determine the resultant waypoint An attract/repel point is found by M= (x1 − x2 )2 + (y1 − y2 )2 x1 − x2 xa = x2 + r M (5) (6) 396 George W.P York, Daniel J Pack and Jens Harder Neighboring UAV UAV attract repel ideal range result Neighboring UAV Fig Method to maintain formation Following UAVs gravitate to a predefined range from their closest two neighboring UAVs using attractive and repelling forces y1 − y2 (7) M where r is the ideal formation range, (x1 , y1 ) is the UAV’s location, (x2 , y2 ) is one of the neighboring UAV’s location, and (xa , ya ) is one of the attract/repel points The other attract/repel point (xb , yb ) is found similarly using the third UAV’s location The waypoint goal (xw , yw ) for the UAV is then computed ya = y2 + r ∆x = ∆y = (xa − x1 ) + (xb − x1 ) (8) (ya − y1 ) + (yb − y1 ) xw = x1 + ∆x (10) yw = y1 + ∆y (11) (9) The UAV then attempts to fly to this goal waypoint, constrained by its flight dynamics (current trajectory, allowable velocity range, and the maximum bank angle for the given velocity) The following UAVs continually recompute the waypoint goals and effectively follow the leader in formation With this simple approach, there is no limit to the number of UAVs in a formation; however, the emergent property appears to be triangular formations of three Occasionally a fourth UAV will temporarily join a group When two formations pass closely to each other, occasionally a UAV will transfer to another formation due to the simple priority scheme Comparison of Cooperative Search Algorithms for Mobile RF Targets 397 Fig Simulation using the formation algorithm The squares indicate targets already located 3.2 Formation Simulation Fig is a snapshot from our MATLAB simulation using this formation approach From the traces the snapshot shows that the two formations located three targets (squares) with four remaining The upper-right formation is in the process of turning to fly over a target detected by one of the followers Results In this section we present a comparative study to demonstrate how well the two algorithms compare against the competing requirements of minimizing the global search time and minimizing the target location error We varied the ratio of the number of UAVs to the number of targets to get an indication of the scalability of the two approaches For our experiment, we changed the number of UAVs from to 9, changed the number of targets from to 9, and averaged the results from 100 simulations each The initial target and UAV locations were randomly selected in a 50 km x 75 km area The UAVs flew in a velocity range of 115 to 260 kph (cruise at 115 kph) with a minimum turn radius of 0.5 km The target’s maximum velocity was 37.5 kph, and a target traveled in random directions for random distances A target emitted randomly, on for an average of 6.8 minutes and off for an average of 4.8 minutes The sensor provided estimates every 12 seconds We used both one degree and seven degree directional sensors The maximum sensor range was assumed to be 4.3 km and the UAVs tried to fly an orbit of 2.2 km from the estimated target location for the first algorithm’s localization For the second algorithm, the formation tried to maintain an equal distance of 4.3 km between UAVs The final localization estimate for a 398 George W.P York, Daniel J Pack and Jens Harder specific target was delayed until the target stopped emitting, giving the UAVs the maximum time possible to get in the proper geometry Since the emitters can turn off at any moment, quite often the ideal number of UAVs and proper geometry may not have been achieved before a localization estimate is made When target localization estimates were made, we tracked the number of UAVs cooperating during the localization (instead of cooperating with the global search task) Figures and compare the average localization cooperation for the two algorithms For the second algorithm, the UAVs are working in formations of three, so on average 2.5 UAVs cooperated regardless of the ratio of UAVs to targets The average number of UAVs was not the ideal three since occasionally only two UAVs would be in range when a target stopped emitting and the localization estimate was made For the first algorithm, as shown in Fig 7, we can see the trade-off in the number of UAVs operating in the cooperative target localization stage and the global search stage As expected, the best localization cooperation ( > 2.5 UAVs) was achieved when UAVs faced only target, while the cooperation reduced ( UAV) as the ratio changed down to UAVs versus targets This amount of cooperation had a direct impact on the localization accuracy, as seen in Figures through 12 Average # of UAVs helping locate Average Localization Cooperation − Algorithm 2.5 1.5 9 6 # of UAVs 3 # of Targets Fig Plot of the average number of UAVs cooperating for each localization estimate when the number of UAVs changes from to and the number of Targets changes from to for Algorithm I Figures through 12 compare the average localization error generated using the two algorithms with the one degree accurate sensor and the seven degree accurate sensor For Algorithm II, flying in formation produced a more consistent localization error for all the UAV/target ratios For the degree sensor the average error was around 0.20 km while for the degree sensor Comparison of Cooperative Search Algorithms for Mobile RF Targets 399 Average # of UAVs helping locate Average Localization Cooperation − Algorithm 2.5 1.5 9 # of UAVs 3 # of Targets Fig Plot of the average number of UAVs cooperating for target localization when the number of UAVs change from to and the number of targets change from to for Algorithm II it was around 0.36 km for Algorithm II For Algorithm I, the localization error improved as the amount of localization cooperation improved For the degree sensor, the average error for UAVs versus targets was > 0.4 km, and the error reduced to 0.16 km for UAVs versus target For the degree sensor using the first algorithm with UAVs versus targets, the error was 0.57 km, and for the case with UAVs versus target, the error was 0.38 km Thus, flying in formation appears to consistently improve localization accuracy However, the cost is in the total search/localization time, as shown in Figures 13 and 14 Figures 13 and 14 compare the total search and localization time for each of the cases for both algorithms For Algorithm I, the best time (19 minutes) occurred as expected when UAVs faced only one target; the time increased up to 174 minutes when UAVs faced targets The total time for Algorithm II was much worse as the formations covered less area over time The total times ranged from 60 minutes for UAVs versus target case to 417 minutes for the UAVs versus targets case Conclusion and Future Work In this chapter, we introduced two algorithms for multiple UAVs to cooperatively search, detect, and locate RF mobile targets The authors are not aware of any work that attempts to solve the current problem reported in the literature We showed preliminary analysis work on the optimum number of UAVs required to locate targets; introduced a search cost function used to maximize 400 George W.P York, Daniel J Pack and Jens Harder Average Localization Error − Algorithm − degree sensor 0.6 0.5 0.4 0.3 0.2 0.1 9 # of Targets # of UAVs Fig Plot of the average localization error in kilometers when the number of UAVs changed from to and the number of Targets changed from to for Algorithm I using a degree accurate sensor Average Localization Error − Algorithm − degree sensor 0.6 0.5 0.4 0.3 0.2 0.1 9 6 # of Targets # of UAVs Fig 10 Plot of the average localization error in kilometers when the number of UAVs changed from to and the number of Targets changed from to for Algorithm II using a degree accurate sensor the use of multiple UAVs to individually and in formation search an area; proposed an algorithm to locate targets by cooperatively arranging multiple UAVs into the proper geometry, showed a quicker method of locating by flying a cooperative formation in the ideal geometry for localization; and illustrated a scheme to locally search targets that have stopped emitting but are expected to emit again We demonstrated our proposed algorithms using simulated results Our comparative analysis for the two cooperative control algorithms Comparison of Cooperative Search Algorithms for Mobile RF Targets 401 Average Localization Error − Algorithm − degree sensor Localization Error (km) 0.6 0.5 0.4 0.3 0.2 0.1 9 6 # of Targets # of UAVs Fig 11 Plot of the average localization error in kilometers when the number of UAVs changed from to and the number of Targets changed from to for Algorithm I using a degree accurate sensor Average Localization Error − Algorithm − degree sensor 0.6 0.5 0.4 0.3 0.2 0.1 9 6 # of Targets # of UAVs Fig 12 Plot of the average localization error in kilometers when the number of UAVs changed from to and the number of Targets changed from to for Algorithm II using a degree accurate sensor showed that Algorithm II, flying in formation, produced higher accuracy in target localization, but a much longer total search/localization time On the other hand, Algorithm I, in which the UAVs search independently and use a cost function to determine when to cooperate on localization, had a much reduced total search time at the cost of less accuracy in target localization We have several plans to expand this research Triangulating with directional sensors is known to occasionally produce very inaccurate results, partic- 402 George W.P York, Daniel J Pack and Jens Harder Total Search/Localization Time − Algorithm −(50x75 km) 450 400 Time (minutes) 350 300 250 200 150 100 50 9 # of Targets # of UAVs Fig 13 Plot of the average total search/localization time in minutes when searching a 50x75 km area as the number of UAVs changed from to and the number of Targets changed from to for Algorithm I Total Search/Localization Time − Algorithm − (50x75 km) 450 400 350 300 250 200 150 100 50 9 # of Targets # of UAVs Fig 14 Plot of the average total search/localization time in minutes when searching a 50x75 km area as the number of UAVs changed from to and the number of Targets changed from to for Algorithm II ularly when two UAVs are in an improper geometry close together in angle We are working on improving triangular localization using a Kalman filter technique [9] as well as a rate-change-of-angle approach [10] We plan to increase accuracy of simulator with DOF UAV models and real RF DF sensor models We also plan to experiment with hardware, first using robots in 2D environment to compare the two cooperative control algorithms We plan Comparison of Cooperative Search Algorithms for Mobile RF Targets 403 to follow this with experiments flying UAVs at the United States Air Force Academy test range References Beard, R., Mclain, T., Goodrich, M., and Anderson, E (2002) Coordinated Target Assignment of Intercept for Unmanned Air Vehicles, IEEE Transactions on Robotics and Automation, vol 18, no Chandler, P., Rasmussen, S., and Pachter, M (2000) UAV Cooperative Path Planning, AIAA Guidance, Navigation, and Control Conference and Exhibit, pp 1255-1265 Dunbar, W and Murray, R (2002), Model Predictive Control of Coordinated Multi-vehicle Formations, Proceedings of the 41st IEEE Conference on Decision and Control, pp 4631-4636 Coffey, T and Montgomery, J (2002) The Emergence of Mini UAVs for Military Applications, Defence Horizons, No 22, pp - Pack, D and Mullins, B (2003) Toward Finding an Universal Search Algorithm for Swarm Robots, Proceedings of the 2003 IEEE/RJS Conference on Intelligent Robotic Systems (IROS), pp 1945-1950 York, G W P and Pack, D J (2004) Minimal Formation Based Unmanned Aerial Vehicle Search Method to Detect RF Mobile Targets, Proceedings of the 2nd Annual Swarming Conference: Networked Enabled C4ISR Spears, W., Spears, D., Hamann, J., and Heil, R (2004) Distributed, PhysicsBased Control of Swarms of Vehicles, Autonomous Robots, Volume 17(2-3) Montereiro, S., Vaz, M., and Bicho, E (2004) Attractor Dynamics generates robot formations: from theory to implementation, Proceedings of 2004 IEEE International Conference on Robotics and Automation York, G W P., and Pack, D J (2005) Comparative Study on Time-Varying Target Localization Methods using Multiple Unmanned Aerial Vehicles: Kalman Estimation and Triangulation Techniques, Proceedings of the 2005 IEEE International Conference On Networking, Sensing and Control 10 Gilbert, H D., McGuirk, J S., and Pack, D J (2005) A Comparative Study of Target Localization Methods for Large GDOP, World Scientific, publication pending Lecture Notes in Economics and Mathematical 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for games involving single- and multiple-pursuers, games with rectilinear and hexagonal grids, games with and without terrain features, and games with varying pursuer-sensor... study of cooperative systems has intensified That being said, they remain notoriously difficult to model and understand Further than that, to fully achieve the benefits of manmade cooperative systems,