Springer cooperative control and optimization 2002 ISBN1402005490

Contents ixLit-Hsin Loo, Erwei Lin, Moshe Kam & Pramod Varshney Introduction Group formation by autonomous homogeneous agents The noiseless full-information case Limitations on communica

Cooperative Control and Optimization

University of Florida, U.S.A.

The titles published in this series are listed at the end of this volume.

Cooperative Control and Optimization

Edited by

Robert Murphey

Air Force Research Laboratory,

Eglin, Florida, U.S.A.

and

Panos M Pardalos

University of Florida,

Gainesville, Florida, U.S.A.

KLUWER ACADEMIC PUBLISHERS NEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW

Print ISBN: 1-4020-0549-0

©2002 Kluwer Academic Publishers

New York, Boston, Dordrecht, London, Moscow

Print ©2002 Kluwer Academic Publishers

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No part of this eBook may be reproduced or transmitted in any form or by any means, electronic, mechanical, recording, or otherwise, without written consent from the Publisher

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Dordrecht

“ Things taken together are wholes and not wholes, something

is being brought together and brought apart, which is in tune and out of tune; out of all things there comes a unity, and out

of a unity all things.”

- Heraclitus

Preface

1

xi

Cooperative Control for Target Classification

P R Chandler, M Pachter, Kendall E Nygard and Dharba Swaroop

21 21 22 26 30 31 33

35

36 36 39 42

4 9 51 53

2

Guillotine Cut in Approximation Algorithms

Xiuzhen Cheng, Ding-Zhu Du, Joon-Mo Kim and Hung Quang Ngo

How are we approaching these challenges?

Where are we heading from here?

References

Appendix: Notes

vii

Optimal periodic stochastic filtering with GRASP

Paola Festa and Giancarlo Raiconi

4.1

4.2

4.3

4.4

Introduction and problem statement

Two interesting particular cases

Discrete problem formulation

Numerical results

55 56 58 65 68 71

73 73 75 76 84 87 88 91

95

96 99 100 107 112 118 119

121 121 124 127 129 131 140 141

143

8

Cooperative Multi-agent Constellation Formation Under Sensing

and Communication Constraints

Parameter extension of the optimal algorithm

Critical point in the parameter extension: the optimal rithm

algo-Structural complexity of cooperative systems versus tion problems

Applied response surface methodologies

Results and analysis

Conclusions and recommendations

6

Cooperative Behavior Schemes for Improving the Effectiveness of

Autonomous Wide Area Search Munitions

Cooperative Control of Robot Formations

Rafael Fierro, Peng Song, Aveek Das, and Vijay Kumar

Contents ix

Lit-Hsin Loo, Erwei Lin, Moshe Kam & Pramod Varshney

Introduction

Group formation by autonomous homogeneous agents

The noiseless full-information case

Limitations on communications and sensing

Limitation of communications

Oscillations due to sensing limitation

Group formation with partial view

The use of ‘meeting point’ for target assignment

Cooperative Aircraft Control for Minimum Radar Exposure

Meir Pachter and Jeffrey Hebert

Single vehicle radar exposure minimization

Multiple vehicle isochronous rendezvous

Conclusion

References

11

Robust Recursive Bayesian Estimation and Quantum Minimax Strategies

P Pardalos, V Yatsenko and S Butenko

Introduction

Differential geometry of Bayesian estimation

Optimal recursive estimation

Quantum realization of minimax Bayes strategies

Concluding remarks

References

12

Cooperative Control for Autonomous Air Vehicles

Kevin Passino, Marios Polycarpou, David Jacques, Meir Pachter, Yang Liu, Yanli Yang, Matt Flint and Michael Baum

171 172 180 185 193 195

199 200 207 208 211

213 214 215 217 225 229 231

Introduction

Autonomous munition problem

Cooperative control via distributed learning and planning

Stable vehicular swarms

273 276 279 284 293

297 299

References

Appendix

References

13

Optimal Risk Path Algorithms

Michael Zabarankin, Stanislav Uryasev, Panos Pardalos

Model description and setup of the optimization problem

Analytical solution approach for the risk path optimization problem

Discrete optimization approach for optimal risk path tion with a constraint on the length

A cooperative system is defined to be multiple dynamic entities thatshare information or tasks to accomplish a common, though perhaps notsingular, objective Examples of cooperative control systems might in-clude: robots operating within a manufacturing cell, unmanned aircraft

in search and rescue operations or military surveillance and attack sions, arrays of micro satellites that form a distributed large apertureradar, employees operating within an organization, and software agents.The term entity is most often associated with vehicles capable of physicalmotion such as robots, automobiles, ships, and aircraft, but the definitionextends to any entity concept that exhibits a time dependent behavior.Critical to cooperation is communication, which may be accomplishedthrough active message passing or by passive observation It is assumedthat cooperation is being used to accomplish some common purpose that

mis-is greater than the purpose of each individual, but we recognize that theindividual may have other objectives as well, perhaps due to being amember of other caucuses This implies that cooperation may assumehierarchical forms as well The decision-making processes (control) aretypically thought to be distributed or decentralized to some degree For

if not, a cooperative system could always be modeled as a single entity.The level of cooperation may be indicated by the amount of informationexchanged between entities Cooperative systems may involve task shar-ing and can consist of heterogeneous entities Mixed initiative systemsare particularly interesting heterogeneous systems since they are com-posed of humans and machines Finally, one is often interested in howcooperative systems perform under noisy or adversary conditions

In December 2000, the Air Force Research Laboratory and the sity of Florida College of Engineering successfully hosted the first Work-shop on Cooperative Control and Optimization in Gainesville, Florida.About 40 individuals from government, industry, and academia attendedand presented their views on cooperative control, what it means, and how

Univer-it is distinct or related to other fields of research This book contains

se-xi

lected refereed papers summarizing the participants’ research in controland optimization of cooperative systems.

We would like to take the opportunity to thank the authors of thepapers, the Air Force Research Laboratory and the University of FloridaCollege of Engineering for financial support, the anonymous referees, S.Butenko for preparing the camera ready manuscript, and Kluwer Aca-demic Publishers for making the conference successful and the publica-tion of this volume possible

Robert Murphey and Panos M Pardalos

August 2001

Chapter 1

COOPERATIVE CONTROL FOR TARGET CLASSIFICATION

P R Chandler

Flight Control Division

Air Force Research Laboratory (AFRL/VACA)

Wright-Patterson AFB, OH 45433-7531

phillip.chandler@va.afrl.af.mil

M Pachter

Department of Electrical and Computer Engineering

Air Force Institute of Technology (AF1T/ENG)

Wright-Patterson AFB, OH 45433-7765

mpachter@afit.af.mil

Kendall E Nygard

Department of Computer Science and Operations Research

North Dakota State University

Abstract An overview is presented of ongoing work in cooperative control for

unmanned air vehicles, specifically wide area search munitions, which perform search, target classification, attack, and damage assessment The focus of this paper is the cooperative use of multiple vehicles to

1

R Murphey and P.M Pardalos (eds.), Cooperative Control and Optimization, 1–19.

© 2002 Kluwer Academic Publishers Printed in the Netherlands.

maximize the probability of correct target classification Capacitated transhipment and market based bidding are presented as two approaches

to team and vehicle assigment for cooperative classification Templates are developed and views are combined to maximize the probability of correct target classification over various aspect angles Optimal trajec- tories are developed to view the targets A false classification matrix

is used to represent the probability of incorrectly classifying nontargets

as targets A hierarchical distributed decision system is presented that has three levels of decomposition: The top level performs task assign- ment using a market based bidding scheme; the middle subteam level coordinates cooperative tasks; and the lower level executes the elemen- tary tasks, eg path planning Simulations are performed for a team of eight air vehicles that show superior classification performance over that achievable when the vehicles operate independently.

Keywords: cooperative control, autonomous control

The wide area search weapon system, as presently envisioned [1] has anumber of air vehicles operating independently The vehicles are released

in a target area, and follow a set of waypoints that arc preset at launch

If an object is detected in the sensor footprint, the vehicle tries to classifythe object as a target If the classification satisfies the criteria, the vehicleattacks the target and is destroyed To maximize the probability of find-ing high value targets in a short period of time, cooperation among thevehicles has been proposed and work is ongoing in developing cooperativecontrol algorithms Cooperative search algorithms are being pursued in[2] where a cognitive map of threats, targets, and terrain is constructedusing sensor inputs from all the vehicles Cooperative classification al-gorithms are being developed by the authors that combine aspect angledependent views of an object from multiple vehicles to maximize theprobability of correct classification Cooperative attack algorithms arebeing developed in [3] to ensure that sufficient weapons are engaged toensure destruction of the target The weapon target assignment problem

is being addressed in [4] using dynamic stochastic prgramming and in [5]using dynamic network flow optimization models Online optimal tra-jectory generation for cooperative rendezvous has been pursued by theauthors [6] and others [7, 8]

Cooperative classification as discussed in Section 2 is the task of mally and jointly using multiple vehicles to maximize the probability ofcorrect target classification This is shown in Fig 1.1 where the arrowsrepresent the velocity vectors of the vehicles The vehicles can communi-cate over some fixed range, which is represented by the large circles The

opti-Cooperative Control for Target Classification 3

vehicle at 1 has detected a potential target, but in general the probability

of classification or some threshold The vehicle could perform

a loopback maneuver or another vehicle could view the potential target

at a different aspect angle An approach to combining the views tically is given in the next section An important issue is choosing theoptimal aspect angle for the second view Initially, the second view waschosen to be orthogonal to the first

statis-In Fig 1.1 the vehicle at 2 also has detected a potential target Thevehicle at 3 could view either potential target 1 or 2, or the vehicle at 4could view potential target 1 Determining which vehicle could optimallyprovide the second view is the assignment problem The optimizationfunction could be, among other alternatives, to maximize the value oftargets classified, or to minimize the time to classify Two approaches tothe assignment problem are given in Section 3

The mission performance of wide area search munitions is quite sitive to false target attack rate This stems from the sensor used, thecapability of the sensor processing algorithm, the number and type ofobjects in the search area, and whether the objects are partially hidden

sen-in clutter The basic approach is to observe the object at the optimumaspect angle, as discussed in Section 2, as well as over the largest range

of aspect angles, at minimum cost Cost is defined as detraction fromsearch time or attack tasks The cooperative classification uses adjacent

vehicles to maximize aspect angle ranges to achieve high probability ofcorrect classfiication or low false target attack rate.

Section 4 discusses the development of a hierarchical architecture forcooperative search, classification, attack, and assessment While many

of these component functions are under development, the critical nizational theory of how to integrate the disparate and generally con-tradictory functions into a decision system has not been available Thethree level hierarchy allows sub-teams to be formed dynamically at themidlevel The top level uses a market analogy biding procedure to assignvehicles and tasks to the sub-teams

orga-In Section 5, our Matlab/Simulink simulation is discussed This highfidelity simulation has eight vehicles searching an area that has bothtargets and nontargets The sensor processing is emulated, includingfalse classification While decisions are made concerning which task eachvehicle is to perform, the coupling between the tasks is not completelyaccounted for For example, the change in the search pattern if a vehicle

is assigned a classification task

Section 6 discusses many of the issues in cooperative classificationthat have yet to be addressed The classification performance is alsodiscussed Section 7 presents the conclusions

The key technique for achieving a low false target attack rate is to usemultiple views A notional template is shown in Fig 1.2 of probability

of correct classification versus the aspect angle at which the object

is viewed This can also be looked at as a confidence level False fication is addressed later in this section To keep the occurance of falseclassification low, the threshold is set high, in this case, beforethe target can be attacked As can be seen in Fig 1.2, the threshold isachieved only over a narrow range of aspect angles (0 or 180 deg).The objective is to combine the statistics from multiple views; in gen-eral, for two views:

classi-If the views are statistically independent:

Initially it is assumed the views are uncorrelated This assumption will

be relaxed later in the section From eqn 2, one can see that if the object

is viewed at the same increases Intuitively, this is not reasonable,since there is no additional information It is generally true that if the

Cooperative Control for Target Classification 5

aspect angles are separated by 90 deg the views should be uncorrelatedand the information content should be greater Based on this insight,trajectories now need to be derived that result in views orthogonal to

the first

Fig 1.3 shows a sample configuration for two vehicles The X marksthe location of an object and the arrow the velocity vector of the vehiclethat detected the object The other arrow represents the velocity vector

of an adjacent vehicle that could provide a second view As stated earlier,the simplification is that the second vehicle should come in ±90 deg.Because the sensor looks ahead of the vehicle, the vehicle must be onthe orthogonal line at least the distance of the sensor offset The circles

represent a specified minimum turn radius R.

It can be proven that the minimum time trajectory to a target consists

of an initial turn through a circular arc, a straight line dash, and a turnthrough a final circular arc The arcs are on the circles in Fig 1.3 As can

be seen, there are eight possible trajectories for the adjacent vehicle toplace it’s sensor on the object The approach pursued here, is to calculatethe distances traveled for all eight trajectories The shortest, of course,

is the minimum time trajectory It can be shown that this algorithmholds for any configuration of adjacent vehicle and object Once thetrajectories are defined, we return to the target templates

A notional target is shown in Fig 1.4 For simplicity, the target is

rectangular with sides a, b and is the aspect angle at which the target

is viewed The assumption is that the projected line length l is

propor-tional to the probability of classification Views at

Cooperative Control for Target Classification 7contain projections from only one side, so that no estimate of aspect ra-tio can be made The probability, or confidence level, is defined as beingproportional to the length of the side that is viewed The projected line

length is normalized by the length a + b, where the maximum occurs at

The orientation of the target on the ground is defined by andThe probability (projection) is:

The maximum projected line length occurs at (6/a) Themaximum value is:

Fig 1.5 shows the periodic nature of since a rectangle has 2 axes

of symmetry Finally, the plot is scaled by which in the figure is

.8 If the threshold is 9, this means that it is not possible to classify thetarget from one view

If statistically independent views of the target have been taken, theprobability of identifying the target is calculated as:

In the special case of as before

The joint probability calculation given above is overly optimistic whenthe aspect angles are close The exact joint probability for 2 views is notavailable, but it is reasonable that correlation when

and when Therefore, as an approximation, ablending function is defined as:

The modification to the 2 view probability for correlated views is asfollows:

This assumes the views are uncorrelated when

We now have an algorithm for generating classification probabilities,

or more acurately, confidence levels, from two views of an object For the

autonomous munition, False Target Attack Rate is probably the critical

factor in weapon system performance Therefore, a reasonable emulationmust include false classification as well A notional false classificationprobability matrix is given in Table 1.1 When the vehicle detects a

target, the class is selected according to the probabilities in the table.The emulation enters the “selected” class template, not the “true” classtemplate, at to get If then another view is needed If thesecond class is the same as the first, then proceed to combine views asabove If the combined statistic does not exceed the threshold, then the

Cooperative Control for Target Classification 9allocation process will determine if taking another view is cost effective.

If the two classes are not the same, then the view with the highest priorityclass could be retained for the assignment and the other view discarded

An alternative is to retain both views and let the assignment algorithmdetermine from the priorities if an additional view is warrented Cooper-ative target classification is driven by inputs from the upper level of thehierarchical cooperative control system currently under development Inthis overview paper, we outline in the following 2 sections the assignmentalgorithms and hierarchical control architecture

In general, the assignment problem involves not only classification, butalso search, attack, and damage assessment For purposes of illustrationhere, all of the vehicles are considered available to perform cooperativeclassification The assignment algorithm then, is to select the optimalvehicle to provide the second view An assignment method that includesthe other tasks is addressed later in this section

When an object is detected, the location, heading angle probability,and aspect angle is transmitted to all the other vehicles The vehiclesuse Fig 1.3 to determine distances and time to the object at, initially,angles perpendicular to Later on, four angles to the object were used,these represent the best vectors to view the object from the template inFig 1.5 The calculated minimum time, distance, or cost to the object

is then transmitted to the other vehicles This is done for all the objectsthat need classification The result is that all the vehicles have completeinformation and solution of the assignment problem is globally optimal.All the vehicles solve the same problem and therefore arrive at the samesolution – conflicts are avoided and a degree of redundancy is achieved

An example assignment matrix is given in Table 1.2 The columns

are targets, the rows are vehicles, and the entries are costs, for example:time to object; remaining life; distance; or a weighted target value Each

of these types of costs have been used in the simulations discussed in a

later section This is a straightforward linear assignment problem andcan be put in an integer linear programming form This is easily solvable,even on modest hardware, for many targets and vehicles The matrix iscompletely dynamic As new objects are found, all of the vehicles areoptimally reassigned Or, when classified, taken out of the assignmentmatrix The objective could also be to maximize the vehicles remain-ing life or to maximize the value of objects classified The next topicaddresses assignment for all the tasks.

The assignment of vehicles to search, classification, attack, and battledamage assessment is posed as a network flow optimization model Themodel shown in Fig 1.6 is described in terms of supplies and demandsfor a commodity, nodes which model transfer points, and arcs that inter-connect the nodes and along which flow can take place Arcs can havecapacities that limit the flow along them An optimal solution is theglobally greatest benefit set of flows for which supplies flow through thenetwork to meet the demands In the model, vehicles are supplies andthe tasks are demands Since the vehicles are in only one mode at a time,the arcs have a flow of 0 or 1 Fig 1.6 is also known as a Capacitated

Network Trans-shipment Model and reduces to an integer (binary) linear

Cooperative Control for Target Classificat ion 11programming problem The linear program is formulated as follows:

the vector is the binary decision variable, are the benefits to bemaximized, Eqn 4 is the flow balance, and Eqn 5 is the link or flowcapacity

As in the previous assignment problem, the solution is globally timal The LP problem has a specialized structure that is very fast tosolve, is highly flexible, event driven, and dynamic An important issue

op-is the determination of the costs above Determining the utility of tinuing to search may be particularly difficult to calculate Also, output

con-of the nodes are restricted to 1 to maintain linear form, which meansmulti-vehicle attack of a target is not allowed This restriction is relaxed

in the next section, either by augmenting the matrix or by a process of

“bidding"

Agents at the next lower level Based on this information, this agentmay autonomously abandon certain high-level goals in favor of others.The domain of responsibility for an Intra-team Cooperative ControlPlanning Agent involves the division of responsibilities among the ve-hicles working as a configured team Leadership responsibiliites andcoordination mechanisms depend on the mission, the models available

to support accomplishing the mission, available data, and the currentcapabilites (eg fuel status) of the vehicles on the team

The vehicle planning agents function specifically within the domain

of an individual vehicle These on-board planners accept a specific goalthat is approprite for a single vehicle, then invoke path planning andscheduling algorithms aimed at meeting the goal

Finally, the vehicle Regulating Agents provide command sequences forthe vehicle, in order to accomplish such tasks as following trajectories,

Cooperative Control for Target Classification 13activating sensors, executing maneuvers, changing speed, and releasingweapons.

At the Inter-team level is a high-level auction procedure [10] for mining which targets should be assigned to which team We assume that

deter-an initial allocation of targets to vehicles has been made, deter-and that eachteam has solved its own generalized assignment problem to determinewhich vehicles attack which targets, and the total expected value of thechosen decisions Thus, a team derives value through its current assets,which are targets to strike, and vehicles to strike them To potentiallyimprove the overall value among the teams, we now allow targets andvehicles to be “traded" from one team to another, in a way that simu-lates a stock exchange When a team hypothetically gives up an asset,the following computations can be derived:

For a specified target, the reduction in value to the team if thetarget is given to another team This is the target “sell" value.For a specified vehicle, the reduction in value to the team if avehicle is given to another team This is the vehicle “sell" value.Similarly, when a team acquires an asset, viz an additional target orvehicle, the following computations can be derived:

The gain in value to the team if a specified target is received fromanother team This is the target “buy" value

For a specified vehicle, the gain in value to the team if a vehicle isreceived from another team This is the vehicle “buy" value.The advantage of making a trade is guaranteed to be realized only if thetrade is isolated from other trades involving the same teams, becausethe buy and sell values apply at the margin and the assigning of multiplevehicles to a target is inherently nonlinear

The Intra-team level has agents that manage cooperative behavior,including: cooperative search, cooperative classification, cooperative at-tack, damage assessment, and rendezvous Cooperative search consists

of building maps of threats, targets, and terrain As each of the vehiclesuncovers information, it is transmitted to the other vehicles to build themaps An optimization problem is solved to apportion individual vehi-cles to search areas that have the greatest probability of containing highvalue targets, while minimizing fuel and exposure Cooperative classifi-cation has already been discussed Cooperative attack stems from theprobability of kill from an individual munition is less than one

Multiple munitions may be needed to kill the target with sufficient fidence Cooperative damage assessment is to ensure that high value

con-targets have indeed been destroyed by viewing the target after attack.The rendezvous function is the time coordination of vehicles arrival at atarget.

A simulation was developed for up to eight vehicles cooperatively trolled in a wide area search and attack mission The simulation is based

con-on the Ccon-ontrol Automaticon-on and Task Allocaticon-on (CATA) [9] simulaticon-on

in C++ The simulation was converted to run under Matlab Simulink

to expedite algorithm research Much of the software is compiled C++code that is incorporated into Simulink blocks The research algorithmsare coded using graphics or math script

The simulation scenario entails eight vehicles searching a battle spacethat has six targets of various values and up to five nontargets Thevehicles are initially in a echelon formation and following a serpentinepath As targets are detected, vehicles are dynamically assigned to per-form classification and attack The search could be dynamically changed

as vehicles are assigned so as to cover the areas that have the highestprobability of containing a high value target In this simulation, if avehicle is reassigned back to search, it returns to the original sepentinepath All of the targets are found and attacked before the vehicles runout of fuel No nontargets are attacked

Fig 1.8 shows a typical scenario

In Fig 1.8 vehicle 2 detects target 2 first Vehicle 5 is assigned toclassify target 2 Vehicle 2 then detects target 6 and vehicle 3 detectstarget 5 Vehicle 6 is then assigned to classify target 5 and vehicle 7

is assigned to classify target 6 Then vehicle 3 detects target 7, whichresults in vehicle 8 being assigned to classify target 7 At this point,vehicle 2 detects target 4, which results in vehicle 4 being assigned toclassify target 4 Vehicle 5 also detects target 5 on it’s way to classifytarget 2, but this does not trigger a reassignment This is because vehicle

5 does not pass over the target close enough to the specified aspect angle.The fortuitious detections could be more optimally incorporated Finally,vehicle 3 detects target 3, however, vehicle 8 is assigned target 3 Thisresults in vehicle 1 being assigned to target 7, where vehicle 1 had notbeen previously assigned Vehicles 2 and 3 continue on the serpentinepath, while the other vehicles classify and attack their assigned targets.All of the vehicles cross over their assigned targets at the specified aspectangles and the classification threshold is crossed All of the targets are

of a high value, so the targets are attacked as soon as they are classfied– there is no delayed attack

Cooperative Control for Target Classification 15

Not shown, but if a false target attack rate and nontargets are duced, all of the valid targets are attacked, but one of the nontargets isattacked If the probability of correct classification threshold is raised,then the potential targets are viewed at more aspect angles This pre-vents the nontargets from being attacked, but sometimes results in validtargets not being attacked The developed simulation tool allows us toconduct a parametric study and thus optimally address this trade offsituation

1 Aspect angle estimate This estimate could be used to determinethe 2nd optimum viewing angle To date, 2nd view angles arebased on the heading angle of the first view, not the orientation

of the object on the ground The computation of the optimal 2ndview could be done offline for the finite set of templates However,this does not mean that the classification threshold will be crossed.Another offline optimization could be performed to determine thenumber and aspect angles of views to yield classification Giventhat this information were available, the algorithms discussed pre-viously could use it and allocate resources optimally

2 Statistically combining 3 or more views For a possible high valuetarget, an arbitrary number of views may be desirable The sim-plified joint probability approach presented earlier would have to

be much more complex Instead of including all the views at once,the calculation could be recursive Calculate the joint probabilityfor 2 views, use the best add the 3rd view, etc

3 How to account for clutter To date, all the targets are assumedviewable from all angles with no objects obstructing the view Toemulate a target obstructed by a building on one side, one couldscale the template on that side with a squashing function For

has a large impact on the performance of the search algorithm todetect targets

4 Classification mismatch False Target Attack Rate (FTAR) stemsmost directly from the sensor and sensor processing If on the 1stview the classification threshold is crossed for the wrong object,then cooperative classification cannot contribute If the threshold

is not crossed on 1st view, a 2nd view is then taken, and the classesare different, it is of course not possible to combine the statistics.However, the mismatch could be resolved by the optimization algo-rithm Select one of the classes either based on probability of occu-rance or target value The optimization then determines whetherresources should be assigned to classify the selected object If so,then the class from the 3rd view should break the tie The addi-tional views contribute to reducing the FTAR

5 Other issues – registration Previous discussions assume the ple views are of the same object This is a function of the naviga-tion precison versus object density for fixed targets If the objectscan move between views, then registration is more of a concern andcontributes to classification mismatch Finally, if a vehicle is pulledoff of search to perform a classification or other task, what is theimpact on the search strategy? This is especially critical in missionperformance if the objects to classify are ultimately nontargets

High fidelity simulations have been performed of eight vehicles in dom and serpentine search patterns to detect, classify, and attack targets.Sensor processing is emulated using the target templates previously dis-cussed Work to date has focused on orthogonal 2nd views where theviews are combined statistically Minimum time maneuvers are used

ran-ACKNOWLEDGMENTS 17

to view the potential target at the specified heading The optimal signment is based on minimizing the time to classify Other metricsalso used include: maximizing remaining life, and maximizing value oftargets classified The largest difference using these metrics is that max-imizing remaining life resulted in delayed attacks until the vehicles werenearly out of fuel Which of these functions are best in maximizing theprobability of targets killed would come from a systems analysis study.With communications and cooperative classification, fewer loop-backmaneuvers are performed where the same vehicle performs the secondview This implies a more efficient utilization of resources If the vehi-cles are in line formation where there is significant overlap in the sensorfootprints, this results in extensive looping maneuvers to perform classi-fication Placing the vehicles in an echelon or staggered formation yieldsmuch more direct (efficient) classification trajectories

as-The assignment techniques discussed are fast and globally optimal.The market approach to assignment becomes more useful as the number

of vehicles increase; however, the benefit degrades as the transactionsbecome more coupled

Scenarios without coordination frequently result in valid targets notbeing found; cooperative classification successfully addresses this prob-lem Introduction of false classification can be countered with moreemphasis on cooperative classification, but with some increase in theprobability of not classifying valid targets

Hierarchical cooperative control allows for near optimal solution ofthe large scale optimization problem It is compatible with the prevail-ing information pattern in the air to ground attack acenario, and it iscomputationally efficient for dynamic replanning

Acknowledgments

The authors wish to thank Lt Col David Jacques and Dr RobertMurphey for their contributions and support The authors also wish tothank their colleague Steven Rasmussen for his technical support

Gillen, Daniel P., and David R Jacques, “Cooperative BehaviorSchemes for Improving the Effectiveness of Autonomous Wide AreaSearch Munitions", ibid.

Murphy, Robert A., “An Approximate Algorithm for a Weapon get Assignment Stochastic Program", in Approximation and Com-plexity in Numerical Optimization: Continuous and Discrete Prob-lems, Klewer Academic Publishers, 1999

Tar-Nygard, Kendall E., Phillip R Chandler, and Meir Pachter, namic Network Flow Optimization Models for Air Vehicle ResourceAllocation", submitted to American Control Conference 2001.Chandler, P., S Rasmussen, M Pachter, “UAV Cooperative Con-trol", AIAA GNC 2000, Denver, CO, Aug 2000

“Dy-McLain, T., “Cooperative Rendezvous of Multiple Unmanned AirVehicles", AIAA GNC 2000, Denver, CO, Aug 2000

Bortoff, S., “Path Planning for UAVs", AIAA GNC 2000, Denver,

Multia-Chapter 2

GUILLOTINE CUT IN APPROXIMATION ALGORITHMS

Xiuzhen Cheng, Ding-Zhu Du, Joon-Mo Kim and Hung Quang Ngo

Department of Computer Science and Engineering,

University of Minnesota, Minneapolis,

MN 55455, USA.

{cheng,dzd,jkim,hngo}@cs.umn.edu

Abstract The guillotine cut is one of main techniques to design polynomial-time

approximation schemes for geometric optimization problems This cle is a short survey on its history and current developments.

arti-Keywords: approximation algorithms, guillotine cut

Introduction

In 1996, Arora [1] published a surprising result that many geometricoptimization problems, including the Euclidean TSP (traveling salesmanproblem), the Euclidean SMT (Steiner minimum tree), the rectilinearSMT, the degree-restricted-SMT, and have polynomial-time approximation schemes More precisely, for any there exists

an approximation algorithm for those problems, running in time

which produces approximation solution within from optimal It

made Arora’s research be reported in New York Times again. 1 Severalweeks later, Mitchell [19] claimed that his earlier work [17] (its journalversion [18]) already contains an approach which is able to lead to thesimilar results However, one year later, Arora [2] made another bigprogress that he improved running time from to

His new polynomial-time approximation scheme also runs randomly intime Soon later, Mitchell [20] claimed again that his ap-proach can do a similar thing We were curious about this piece ofhistory and hence made a study on these two approaches In this article,

1.

21

R Murphey and P.M Pardalos (eds.), Cooperative Control and Optimization, 21–34.

© 2002 Kluwer Academic Publishers Printed in the Netherlands.

we would like to share with readers the result of our investigation andsomething interesting that we found in their publications.

Rectangular partition and guillotine cut

Let us start from rectangular partition In fact, before prove his maintheorem, Mitchell [17, 18] stated clearly that “Our proof is inspired bythe proof in [7]” where the reference [7] in [17] ([9] in [18]) is actually

a paper of Du, Pan, and Shing [7] on minimum edge-length rectangularpartition This paper initiated the idea of using guillotine cut to designapproximation algorithms

The minimum edge-length rectangular partition (MELRP) was firstproposed by Lingas, Pinter, Rivest, and Shamir [13] It can be stated

as follows: Given a rectilinear polygon possibly with some rectangularholes, partition it into rectangles with minimum total edge-length

titioning into offices) The minimum edge-length partition is a natural

goal for these problems since there is a certain amount of waste (e.g.sawdust) or expense incurred (e.g for dividing walls in the office) which

is proportional to the sum of edge lengths drawn For VLSI design, thiscriterion is used in the MIT ‘PI’ (Placement and Interconnect) System

to divide the routing region up into channels - we find that this produceslarge ‘natural-looking’ channels with a minimum of channel-to-channelinteraction to consider."

Guillotine Cut in Approximation Algorithms

They showed that the holes in the input make difference on the putational complexity While the MELRP in general is NP-hard, the

com-MELRP for hole-free inputs can be solved in time where n is the

number of vertices in the input rectilinear polygon The polynomial gorithm is essentially a dynamic programming based on the followingfact

al-Through each vertex of the input rectilinear polygon, draw a verticalline and a horizontal line Those lines will form a grid in the inside of the

rectilinear polygon Let us call this grid the basic grid for the rectilinear

polygon (Fig 2.2)

Lemma 2.1 There exists an optimal rectangular partition lying in the

basic grid.

Proof Consider an optimal rectangular partition not lying in the basic

grid Then there is an edge not lying in the basic grid Consider themaximal straight segment in the partition, containing the edge Say, it

is a vertical segment ab Suppose there are horizontal segments ing the interior of ab from right and horizontal segments touching the interior of ab from left If then we can move ab to the right with-

touch-out increasing the total length of the rectangular partition Otherwise,

we can move ab to the left We must be able to move ab into the basic grid because, otherwise, ab would be moved to overlapping with another

vertical segment, so that the total length of the rectangular partition isreduced, contradicting the optimality of the partition

A naive idea to design approximation algorithm for general case is

to use a forest connecting all holes to the boundary and then to solvethe resulting hole-free case in time With this idea, Lingas [14]

gave the first constant-bounded approximation; its performance ratio is

41 Later, Du [9, 10] improved the algorithm and obtained a tion with performance ratio 9 Meanwhile, Levcopoulos [15] provided agreedy-type faster approximation with performance ratio 29 and conjec-tured that his approximation may have performance ratio 4.5

approxima-Motivated from a work of Du, Hwang, Shing, and Witbold [6] onapplication of dynamic programming to optimal routing trees, Du, Pan,and Shing [7] initiated an idea which is important not only to the MELRPproblem, but also to many other geometric optimization problems This

idea is about guillotine cut A cut is called a guillotine cut if it breaks a

connected area into at least two parts A rectangular partition is called

a guillotine rectangular partition if it can be performed by a sequence

of guillotine cuts Du et al [7] noticed that there exists a minimum

length guillotine rectangular partition lying in the basic grid, which can

be computed by a dynamic programming in time Therefore, theysuggested to use the minimum length guillotine rectangular partition toapproximate the MELRP and tried to analyze the performance ratio.Unfortunately, they failed to get a constant ratio in general and onlyobtained a result in a special case

In this special case, the input is a rectangle with some points inside.Those points are holes It had been showed (see [11]) that the MELRP

in this case is still NP-hard Du et al [7] showed that the minimum

length guillotine rectangular partition as approximation of the MELRPhas performance rato at most 2 in this special case The following is asimple version of their proof, published in [8]

Theorem 1 The minimum length guillotine rectangular partition is a approximation with performance ratio 2 for the MELGP.

Proof Consider a rectangular partition P Let denote the totallength of segments on a horizontal line covered by vertical projection of

by induction on the number of segments in P.

Guillotine Cut in Approximation Algorithms

If the segment is vertical, then and hence

Now, we consider Suppose that the initial rectangle has each

vertical edge of length a and each horizontal edge of length b Consider

two cases:

Case 1 There exists a vertical segment s having length Apply

a guillotine cut along this segment s Then the remainder of P is divided

into two parts and which form rectangular partition of two resultingsmall rectangles, respectively By induction hypothesis,

Therefore,

Case 2 No vertical segment in P has length Choose a zontal guillotine cut which partitions the rectangle into two equal parts.Let and denote rectangle partitions of the two parts, obtained from

hori-P By induction hypothesis,

for Note that

Therefore,

Gonzalez and Zheng [12] improved the constant 2 in Theorem 1 to1.75 with a very complicated case-by-case analysis Du, Hsu, and Xu [8]extended the idea of guillotine cuts to the convex partition problem

Mitchell [17, 18] gave an approximation with performance ratio 2 forthe MELRP in the general case by extending the idea of guillotine cut.First, he uses a rectangle to cover the input rectangular polygon withholes Then, he extended the guillotine cut to the 1-guillotine cut A

1-guillotine cut is a partition of a rectangle into two rectangles such

that the cut line intersects considered rectangular partition with at mostone segment (Fig 2.4) For simplicity, the length of this segment iscalled the length of the 1-guillotine cut A rectangular partition is 1-

guillotine if it can be realized by a sequence of 1-guillotine cuts (Fig 2.5).The minimum 1-guillotine rectangular partition can also be computed

by dynamic programming in time In fact, at each step, the1-guillotine cut has choices There are possible rectangles

Guillotine Cut in Approximation Algorithms

appearing in the algorithm Each rectangle has possible boundaryconditions

To establish the performance ratio of the minimum 1-guillotine gular partition as an approximation of the MELRP, Mitchell [17] showedthe following

rectan-Theorem 2 For any rectangular partition P, there exists a 1-guillotine rectangular partition covering P such that

Proof It can be proved by an argument similar to the proof of Theorem

1 Let denote the minimum length of a 1-guillotine rectangular

partition covering P and length(P) the length of the rectangular tion P Let denote the total length of segments on

parti-a horizontparti-al (verticparti-al) line covered by verticparti-al (horizontparti-al) projection of

the partition P We will prove

by induction on the number of segments in P.

general-ity, assume that the segment is horizontal Then we have

and Hence

Now, we consider in the following two cases:

Case 1 There exists a 1-guillotine cut Without loss of generality,

as-sume this 1-guillotine cut is vertical with length a Suppose the der of P is divided into two parts and By induction hypothesis,