Complexity in UAV cooperative control

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ComplexityinUAVcooperativecontrol

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AFRL-VA-WP-TP-2004-315 COMPLEXITY IN UAV COOPERATIVE CONTROL Phillip R. Chandler, Dharba Swaroop, Jason K. Howlett, Meir Pachter, Jeffrey M. Fowler, Steve Rasmussen, Kendal Nygard, and Corey Schumacher JULY 2004

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COMPLEXITY IN UAV COOPERATIVE CONTROL

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14. ABSTRACT This paper addresses complexity and coupling issues in cooperative decision and control of distributed autonomous UAV teams. In particular, the recent results obtained by the in-house research team are presented. Hierarchical decomposition is implemented where team vehicles are allocated to subteams using set partition theory. Results are presented for single assignment and multiple assignment using network flow and auction algorithms. Simulation results are presented for wide area search munitions where complexity and coupling are incrementally addressed in the decision system, yielding radically improved team performance.

15. SUBJECT TERMS Unmanned Air Vehicle, Flight Control Systems, Hybrid Flight Control System, UAV Flight Control, UAV Pack Control, UAV Autonomous Control, UAV Collision Avoidance, UAV Cooperative Control

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Complexity in UAV Cooperative Control

Phillip R. ChandlerFlight Control Division

AFRL/VACAWPAFB, OH 45433-7531

Meir PachterDept of Elec & Computer Engr

AFIT/ENGWPAFB, OH 45433-7765

Dharba SwaroopDept of Mech Engr

Texas A&M UniversityCollege Station, TX 77843-3123

Jeffrey M. FowlerSibley School of Mech and Aero Engr

Cornell UniversityIthaca, New York 14853

Jason K. HowlettDept of Mechanical EngineeringBrigham Young University

Provo, Utah 84602

Steven RasmussenVeridian, Inc

5200 Springfield PikeDayton, OH 45431-1289

Corey SchumacherFlight Control Division

AFRL/VACAWPAFB, OH 45433-7531

Kendall NygardDept of Computer Science and Operations Research

North Dakota State UniversityFargo, ND 58105-5164

Abstract

This paper addresses complexity and coupling issuesin cooperative decision and control of distributed au-tonomous UAV teams. In particular, the recent resultsobtained by the inhouse research team are presented.Hierarchical decomposition is implemented where teamvehicles are allocated to subteams using set partitiontheory. Results are presented for single assignmentand multiple assignment using network flow and auc-tion algorithms. Simulation results are presented forwide area search munitions where complexity and cou-pling are incrementally addressed in the decision sys-tem, yielding radically improved team performance.

1 Introduction

In the operations research and weapon target assign-ment literature, fast and efficient static linear alloca-tion algorithms are available for hundreds, even thou-sands of vehicles (n) and tasks (m). These are globallyoptimal algorithms and require that the complete costmatrix be centrally available. The auction algorithm,discussed in a later section, is a distributed form ofthese static linear algorithms. Sheer size is one form ofcomplexity.

However, coupling induced complexity, and not nec-essarily size, dominates the wide area search munitionproblem: The problems addressed to date are of modestsize, n ≤ 8, but there are m = 4 tasks (search, classifi-cation, attack, verification) per target and an arbitrarynumber of targets (< 10 to date). The vehicles eachhave a default task of performing cooperative search,which introduces extensive coupling in their search tra-jectories. When an object is detected it needs to beclassified. Once a target is attacked, the vehicle is de-stroyed. After attack, the target is viewed by anothervehicle to ensure it has been destroyed. The tasks mustbe correctly ordered and sequenced in the shortest timedue to severe fuel constraints.

To address this complexity, a hierarchical decomposi-tion is used [1]. Figure 1 illustrates a general architec-ture for cooperative control and task apportionmentamong multiple vehicles.

Sub-teams collapse the complexity of the overall teamoptimization problem so that only those vehicles thathave benefit in servicing the objects are considered.The partitioning uses a limited horizon minimumweight spanning tree [2]. The smaller single or mul-tiple assignment problem can then be addressed.

Multiple assignment removes much of the myopic

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penalties of single assignment. A longer planning hori-zon is essential for good performance in highly coupledsystems. Iterative network flow [4], binary linear pro-gramming, and auction [5] are addressed in the paperas algorithms for multiple task assignment.

The paper is organized as follows: decomposition intosub-teams is addressed in §2; §3 addresses multiple as-signment; §4 presents simulation results; and §5 theconclusions.

2 Sub-Teams

A set of UAVs can be clustered into a team if theyservice the same target, or a set of targets are close toone another. A finer assignment of which UAV shouldbe assigned to which task may be made by consideringthe maneuvering constraints of the UAVs.

Suppose Si is an ordered subset of tasks {Γi1 , . . . ,Γipi},then we refer to Si in the context of UAV Vj performingthe tasks in Si in the order in which they appear in Si.The first and last tasks performed by Vj are Γi1 andΓipi respectively. One can associate a cost with Vjperforming tasks in Si. A feasible partition of tasks isan allocation of disjoint subset of tasks for each of theUAVs to perform, so that every task is performed bysome UAV and all timing (coordination) constraints onthe tasks are met. The problem of resource allocationmay be posed as finding the minimum cost partitionof the set of tasks, where P is any feasible partition oftasks.

2.1 Graph ApproachThis approach combines the ideas of iterative resourceallocation with those from graph theory.

The resource allocation is performed in two stages:

• In the first stage, a classical assignment is per-formed to allocate resources to m1 tasks thatmust be serviced/performed as soon as possible.To proceed with the classical assignment, targetsrequiring multiple services are replicated an ap-propriate number of times and treated as distincttargets that are collocated. For the purpose ofreplication, classification followed by attack willbe considered a composite task requiring only onevehicle and will be treated as a terminal task. Toavoid distinguishing between the replica and theoriginal target, we refer to the replica by it’s task.The procedure is as follows:

1. Suppose there are m1 tasks to be servicedand n UAVs with (n ≥ m), where binarylinear programming solves the classical as-signment problem:

minn

i=1

m1

j=1

Tijxij , where

n

i=1

xij = 1,

m1

j=1

xij ≤ 1,

2. There is inefficiency in this assignment,since, of the m1 tasks, only m2 are termi-nal. This implies that more than one taskof the set ofm1−m2 tasks can be assigned toa UAV resulting in a faster servicing of thetasks. This leads to multiple assignment.

3. We are concerned with multiple verifica-tion assignments for a UAV that resultsin smaller total service time. Timing con-straints do not appear at this stage, sinceUAVs that are not assigned in the first stagearrive necessarily later at a target than theircounterparts in the first stage.

• In the second stage, some inefficiency in resourceallocation is weeded out using graph theory.

2.2 Graph ConstructionLet Di be the distance traveled by a UAV to arriveat the i th of the m − m1 targets under the classicalassignment.

To specify a graph, one provides the set of nodes or ver-tices and the set of edges/arcs connecting the nodes.The targets (or verification tasks) are nodes of thegraph.

We first construct a fully connected symmetric graphin such a way that the weight of an edge/arc connectingΓi and Γj is the Euclidean distance, dij , between thenodes. We then construct a benefit graph as follows:The benefit bij is the weight of an arc/edge connectingnodes Γi and Γj and is defined to be Dj − dij . Thebenefit, bij represents the saving in distance traveledin having a UAV that visits Γi also visit Γj . Clearly,it will be beneficial to have a UAV that visits Γi alsovisit Γj only if bij > 0. Clearly, the benefit graph isasymmetric, since Di may not necessarily be equal toDj , although dij = dji.

The problem of sub-teaming can be thought of as par-titioning the directed benefit graph into subgraphs sothat: 1) every node is covered by one and only one

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subgraph; 2) no two edges of a subgraph either startfrom the same node or end in the same node; and 3)the sum of the benefits of all edges in all subgraphs ismaximum.

2.3 Preliminary ResultsWe used a minimum weight spanning tree in [2] to solvethis problem. This algorithm decomposes the graphinto isolated nodes and/or directed unary subgraphswhere no two edges in the subgraphs share the samein-node or out-node.

The results are shown in Figure 2 which indicates howthe verification tasks are partitioned and allocated todifferent UAVs. In the plots, the numbers in blackindicate the index of the UAV and its location is givenby its coordinates in the plot. The numbers in blueindicate the index of the target requiring verificationand its location is specified by the coordinates in theplot. There is a line starting from a UAV and joiningdifferent targets. UAVs that are not connected to anytargets are released for search operations.

3 Multiple Assignment

The three criteria to which the chosen assignmentshould conform are: 1) no task should be assigned tomore than one UAV; 2) no task should be assigned un-less any prerequisite tasks are also assigned; and 3) theestimated time at which a task is accomplished shouldnot be before the estimated time of the immediatelyprerequisite task. Also, attacking a target is a terminaltask. The assignment meeting all of these criteria withthe minimum cost (or maximum benefit) is desired. So-lutions using enumeration, BLP, relative benefit, anditerative network flow are discussed below.

3.1 EnumerationA simplified problem may be solved with an exhaustivesearch. The distribution of feasible assignments andtheir values was calculated using a representative ob-jective function. For each combination of two to threetargets with two to four UAVs, the optimal assignmenthad a value about three times the mean value of fea-sible assignments. The distributions were qualitativelynormal with a standard deviation approximately equalto the mean. As this would suggest, many feasible as-signments had negative value. Figure 3 shows the dis-tribution of values for a three-target, three-vehicle sce-nario, and the statistics for similar distributions withtwo targets. These findings demonstrate two ways inwhich the problem is difficult. First, the number of fea-sible assignments is drastically smaller than the num-

ber of possible assignments. Second, most feasible as-signments have far less value than the optimum, so arandom search over feasible assignments is not likely tofind a near-optimal solution with a small sample size.

3.2 Binary Linear ProgrammingIn [3] a Generalized Assignment Program frameworkwas used to assign multiple identical UAVs to targets.A Binary Linear Programming (BLP) framework canencode all of the same information in a much smallerproblem which is faster to set up and to solve. Thegeneral form of a BLP is as follows:

Optimize A· xSubject to (1) xj ∈ {0, 1} ∀ j

(2) (F · x)i ≤ di ∀ i(3) (Feq · x)i = deqi ∀ i

The optimal solution to the Binary Linear Program-ming problem will optimize the objective functionamong all feasible assignments in which no UAV is as-signed more than the predetermined number of tasks.There are general solvers for BLPs, but a specializedsolver is needed. The problem does not scale as effi-ciently as the Generalized Assignment Problem. Thenumber of columns each scale with the product of thenumber of UAVs and the number of possible tours,though with a smaller coefficient. The number ofconstraints scales only with the number of tasks orthe number of UAVs. A formulation with tours ofthree tasks would have more variables than a formu-lation with tours of two tasks, but the same numberof constraints. As before, some possible tours could beheuristically eliminated in the formulation stage.

3.3 Graph MethodsAn intermediate approach, between single assignmentand exhaustive search, is to start with the single as-signment result and look for nearby assignments withrelatively better objective values. When the objectivefunction is Euclidean distance, the marginal benefit ofone UAV assuming the duties of another UAV in ad-dition to its own has a simple form. If Target i isseparated from Target j by a distance dij , and the costassociated with servicing Target j by single assignmentis Dj , the benefit of the UAV assigned to Target i sub-sequently servicing Target j is Dj − dij . A matrix ofbenefits can be readily constructed according to

bij :=0, if i = jDj − dij , otherwise.

Any of several algorithms can operate on this matrix.One method investigated uses an auction mechanism to

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make the temporary assignment resulting in Dj , usesanother auction to produce a permutation of targets,evaluates the cost for each UAV to service each of theresulting loops, then uses one more auction to assignUAVs to these loops. Another method uses a greedyalgorithm. At each stage, Target j, is designated to beserviced by the UAV servicing Target i. The orderedpair of targets chosen at each stage maximizes bij overall ordered pairs of targets satisfying the conditions.The process is repeated until the maximum such bijis negative, or no such bij exists. The assignment byeither of these algorithms could be used directly, or asa guide to partitioning targets into groups for serviceby sub-teams.

Good results were produced by an algorithm that doesnot preserve the order of targets when one UAV as-sumes the assignment of another, but chooses the or-der in a myopic manner as in Section 3.4. The orderedpair of UAVs maximizing the marginal benefit is chosenat each stage until the benefit is negative or only oneUAV is assigned. This heuristic allows for general ob-jective functions of a UAV assigned to a set of targets.Furthermore, inclusion of dummy UAVs will allow theassignment algorithm to leave some targets unserviced,if the benefits do not outweigh the cost.

3.4 Iterative Network FlowTours can be compiled by an iteration of the lineartransshipment algorithm [3]. At each stage, UAVs havea planned position and heading for the end of their as-signed tour of multiple tasks. The transshipment algo-rithm makes a temporary assignment of these UAVs tosubsequent tasks. Of these temporary assignments, theone with the earliest estimated time is fixed, and theplanned UAV position and target state are updated.The process repeats until all target tasks are fully as-signed. Fixing the earliest estimated time at each stagediscourages, but does not prevent, non-feasible assign-ments. The authors also implemented the iterative al-gorithm as an auction, then addressed the issue of con-flicting task order. One method is to associate a largecost with any potential assignment which would createa conflict. The other method is to adjust the cost ofa potential assignment to include loitering, so that thedependent task occurs at the earliest admissible time.The implementation of the latter adjustment currentlyaddresses only the order of the completion of the tasks.

4 Simulation Results

The authors have developed a MatLab Simulink basedmulti-UAV simulation of a wide area search munition

scenario with a hierarchical distributed decision andcontrol system as depicted in Figure 1. The vehiclescooperatively search for and destroy high value targets.The vehicles low maneuverability and endurance arecritical constraints.

The scenario, shown in Figure 4, has 8 vehicles fol-lowing a preset serpentine search pattern. Figure 4 isa snapshot 98sec into the scenario where the vehiclespath and footprint are color-coded. There are 3 highvalue rectangular targets in the search space that havean arbitrary orientation on the ground. When these po-tential targets pass completely through the footprint,the object is declared detected. The object cannot beattacked until it has been classified, with sufficientlyhigh confidence. The probability of classification is afunction of the aspect angle at which it is viewed. Ad-ditional views by the same, or other vehicles, may benecessary. These views are then combined statistically.More details of the cooperative classification are cov-ered in [6].

Figure 4 uses the decision and control structure of Fig-ure 1 where there all the vehicles and objects are al-ways assigned to one sub-team. The sub-team agenthere solves a static binary assignment problem usinga network flow analogy [4]. The n vehicle by m taskmatrix includes the costs for every surviving vehicle toperform a task that transitions each object to the nextstate (detect, classify, attack, verify). The vehicle agentcalculates these costs by generating minimum time tra-jectories that satisfy kinematic constraints with speci-fied terminal position and heading.

Figure 4 shows the typical consequences of a single steplook ahead decision and control system. Following theirpre-specified waypoints, vehicles 1,2 detect targets 1-3.Vehicles 5-7 are assigned to classify targets 1-3 respec-tively. These vehicles have the lowest cost (time) toview the objects at aspect angles that have the high-est probability of classifying the objects. Once classi-fied, the classifying vehicles are assigned to attack thetarget. The 3 “racetracks” are a consequence of thesingle assignment binary optimization and the policyof returning the vehicles to the point of search depar-ture. The optimization is triggered only when an objectchanges state or a task is completed. The near circulartrajectory of vehicle 1 is an example of a vehicle beingreassigned and not completing a task. This “myopic”optimization uses 6 vehicles to service 3 targets.

Iteration 2 of the design process is shown in Figure 5where the full 3 level decision hierarchy in Figure 1 isused. Here, the team agent allocates resources to anattack sub-team or a search sub-team. Initially, all the

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vehicles are assigned to the search subteam. As moreobjects are detected, vehicles are allocated to the at-tack sub-team for a maximum of 4 vehicles, which isthe minimum needed to attack and verify 3 high valuetargets. Which vehicles are assigned is based on timeto target. The graph based set partition with flyabletrajectories yielded vehicles 4-8 as “closest”. At thesub-team level, the binary network flow assignment al-gorithm is sequentially applied as resources are allo-cated to the sub-team. Now, a maximum of 4 vehiclesare considered for each task, yielding a more tractableproblem. Vehicle 8 does the verification for all 3 tar-gets.

Iteration 3 of the design uses multiple assignments toremove the trajectory inefficiency, but at an apprecia-ble increase in complexity. Here, the generalized as-signment, graph based, and the iterative network, allwith tours of maximum length 3, have been used. Theiterative network assignment is by far the most compu-tationally efficient, but though not guaranteed to yieldthe optimal, the results have been very good. Figure 6shows the 3 task planning horizon for the same 4 vehi-cle attack sub-team. Now, vehicle 8 has been assigneda 3 task verification tour. The coupling of target taskswith search tasks is also seen in this scenario.

5 Conclusions

The primary attack upon coupling-induced complex-ity is hierarchical decomposition. This yields a subop-timal solution, but the optimal solution can only befound by direct enumeration, which is computationallyprohibitive. The graph partition technique with fly-able trajectories is promising for determining sub-teamcomposition.

The solution of the binary linear program for the sin-gle assignment problem is the starting point for muchof what has been done. Network flow solvers, auctionimplementations, and binary linear programming al-gorithms all yield the optimal solution for the binarylinear single assignment problem. All are fast, but theauction mechanism is more readily implemented in adistributed fashion.

Multiple assignment extends the planning horizon andyields significant improvements in performance and ro-bustness. The iterative network flow optimization al-gorithm is a short planning horizon heuristic that hasbeen found to work well in practice. Robustness andfeasibility are issues, and the algorithm could occasion-ally give poor performance. Task order can be enforced

in the generalized assignment problem, the graph par-tition, and binary linear programming, but timing isproblematic. Auction algorithms may be devised aswell, but share the same issues. However, all these al-gorithms are more computationally intensive.

References

[1] Chandler, P. and M. Pachter, “Hierarchical Con-trol for Autonomous Teams”, AIAA GNC 2001, Mon-treal, Quebec, Canada, Aug 2001.

[2] Nemhauser, G., and B. Wolsey, “Integer andCombinatorial Optimization”, Wiley, 1999.

[3] Guo, W. and K. Nygard, “Combinatorial Trad-ing Mechanism for Task Allocation”, 13th InternationalConference on Computer Applications in Industry andEngineering, June, 2001.

[4] Nygard, K., P. Chandler, and M. Pachter, “Dy-namic Network Optimization Models for Air VehicleResource Allocation”, ACC 2001, Arlington, VA, June2001.

[5] Bertsekas, D., “Auction algorithms for networkow problems: A tutorial introduction”, ComputationalOptimization and Applications, Vol. 1, pp. 7—66, 1992.

[6] Chandler, P., M. Pachter, K. Nygard, and D.Swaroop, “Cooperative Control for Target Classifica-tion”, Cooperative Control and Optimization, Kluwer,2001.

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Figure 2: Illustration of team decomposition and verifi-cation assignments when UAVs and targets areinterspersed

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Figure 3: Distribution of values of feasible assignmentsfor a particular scenario with three unclassifiedtargets and three UAVs.

Figure 4: Decision Hierarchy with no Sub-Teams

Figure 5: Decision Hierarchy with Sub-Teams

Figure 6: Decision Hierarchy with Sub-Teams and Mul-tiple Assignment

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