Genetic Algorithm for Cooperative UAV Task Assignmentand Path Optimization

Eugene Edison∗ and Tal Shima†

Technion – Israel Institute of TechnologyFaculty of Aerospace Engineering

Haifa 32000, Israel

The problem of assigning a group of uninhabited aerial vehicles to cooperatively perform tasks onmultiple stationary ground targets is addressed. A specific set of consecutive tasks needs be performedon each target. A Dubins’ car model is used for motion planning, enabling taking into account eachvehicle’s specific constraint of minimum turn radius. By discretizing the possible heading angle of a vehiclewhile flying over a target we pose the coupled problem of task assignment and path optimization in theform of a graph. This allows obtaining suboptimal trajectory assignments that improve the finer theresolution of the visitation heading angle is. Due to the computational complexity of this coupled problem,we propose a centralized genetic algorithm for the stochastic search of the space of solutions. Results showthat the algorithm quickly provides good feasible solutions and converges toward the optimal one. Theperformance of the genetic algorithm is demonstrated through a Monte Carlo simulation study.

I. Introduction

Autonomous operation of uninhabited aerial vehicles (UAVs) is a problem of much recent interest. Having thiscapability will allow UAVs to perform missions with minimal or no human intervention. Such military and civilianmissions may include intelligence gathering, target tracking, and rescue missions. Moreover, it will enable newoperational paradigms of employing numerous such vehicles in groups, performing tough endurance and complexscenarios. The main motivation for team cooperation stems from the possible synergy, as the group performanceis expected to exceed the sum of the performance of the individual agents.

Task assignment for the cooperative group of nonholonomic vehicles, while optimizing the trajectories ofeach individual, is one of the most challenging problems in cooperative autonomous multi-vehicle operation.Complexity, as one of the main features of cooperative problems, might be posed as the size of the problem(e.g. number of vehicles, tasks, and constraints). Another form of complexity is introduced through the couplingbetween the performance of different tasks.1 For example, if the vehicles each have a default task of searching anarea, then performing it cooperatively introduces extensive coupling in their search trajectories. Once a target hasbeen detected it might require monitoring by the cooperating team, which further imposes coupling between thetrajectories of different team members.

Motion planning2,3 is a long studied problem in robotics that can be tackled for example by using graph andoptimal control theories. It involves the search for a collision free path, taking into consideration the dynamicsof the vehicle and its surroundings, the vehicle’s kinematic constraints, and any other external constraints thatmay affect the planning of a path. Majority of practical applications involving physical machines require smoothfeasible paths. The feasible routes of minimal length for a planar nonholonomic vehicle, constrained to move ata constant speed along paths of bounded curvature without reversing direction, were introduced by Dubins.4 Afeasible trajectory for such a vehicle, denoted as a Dubins’ car, should be defined as a curveγ : [0,T]−→ℜ2 thatis twice differentiable almost everywhere, such that the magnitude of its curvature is bounded from above by1/r,wherer > 0 is the vehicle’s minimum turn radius.

∗Graduate Student. The work of the author was supported in part by the Bernard M. Gordon Center for Systems Engineering, at theTechnion.

†Senior Lecturer; tal.shima@technion.ac.il ; The work of the author was supported in part by a Horev Fellowship through the Taub Founda-tion, and the Bernard M. Gordon Center for Systems Engineering, at the Technion.

1 of 14

American Institute of Aeronautics and Astronautics

AIAA Guidance, Navigation and Control Conference and Exhibit18 - 21 August 2008, Honolulu, Hawaii

AIAA 2008-6317

Copyright © 2008 by Eugene Edison and Tal Shima. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

Task assignment is one of the fundamental combinatorial optimization problems in the branch of operationsresearch in mathematics.5 The basic problem refers to assigning a number of agents to perform a number of tasks.Any agent can be assigned to perform any task, incurring some cost that may vary depending on the agent-taskassignment algorithm. It is required to perform all tasks in such a way that the total cost of the assignment isminimized. An assignment task might be presented as a problem in graph theory.6 Obtaining a good solutionrequires the use of a search algorithm. The basic principle is that a vertex is taken from a data structure, itssuccessors examined, and then added to the data structure. By data structure manipulations the graph is exploredin different orders, for instance level by level (breadth-first search) or reaching leaf node first and backtracking(depth-first search). Other examples of deterministic graph search algorithms include minimum weighted pathsearch like Dijkstra,7 and Bellman-Ford.8

The significant difficulty in solving many assignment tasks, such as the well-known general assignment prob-lem5 is that they are NP-hard combinatorial optimization problems and therefore could not be solved in polynomialtime by deterministic methods. So, due to the prohibitive computational complexity of the problem, the traditionaldeterministic search algorithms provide an optimal solution only for small-sized problems. For large sized prob-lems they may provide a feasible solution and lower / upper bounds on complexity. In contrast, stochastic searchalgorithms9 employ a degree of randomness as part of their logic. An algorithm of this type uses a random inputto guide its behavior in the hope of achieving good performance in the ”average case” and converge to a goodsolution in the expected runtime. One of the efficient ways to solve such a problem by a stochastic approach isevolutionary algorithms.10 The evolution usually starts from a population of randomly generated solutions andevolves in generations. In each generation, the fitness of every individual in the population is evaluated, multipleindividuals are randomly selected from the current population (based on their fitness), and modified (recombinedand possibly mutated) to form a new population. This population is then used in the next iteration of the algorithm;and the process is repeated until some stopping condition is met.

The cooperative multiple task assignment problem (CMTAP),11 associated with the autonomous operationof a group of homogeneous fixed wing UAVs performing multiple tasks on multiple stationary ground targets,is the basis for the current study. Several algorithms of different classes have been proposed for solving such aproblem involving integer assignment decision variables and continuous controllers associated with the specificguidance input of each vehicle. These algorithms are based on customized combinatorial optimization methodsincluding: mixed integer linear programming (MILP),12,13 the capacitated transhipment network solver,14,15 treesearch,16 and genetic algorithm (GA).11 Due to the special characteristics of the problem and the requirement fora tractable solution, all of these proposed algorithms basically decouple the problem of task assignment from thatof trajectory optimization, resulting with a suboptimal solution. For example, the MILP algorithm of Ref. 12uses Euclidean distances instead of computing flyable trajectories; while that of Ref. 13 optimizes the motionplanning only for one task at a time and thus the resulting trajectories are only piecewise optimal. The singletask assignment algorithm in Ref. 14 is only optimal for the current tasks and does not take into account tasksthat will be required when the current tasks are completed. Although iterations on the single task assignmentalgorithm (that utilize in essence a greedy solver) of Ref. 17 provide a solution to the multiple task assignmentrequirement, the approach is heuristic in nature and therefore optimality is not achieved. In a recent paper, Ref.16, a tree search algorithm was developed that produces the suboptimal solution to the assignment problem basedon piecewise optimal trajectories. These trajectories are obtained by setting the initial heading angle conditionon the flight direction of a UAV for each segment based on the final direction obtained in the previous segment.This results with trajectories that are optimal only from one task to the next, but the entire trajectory is generallynot. In another recent paper, Ref.11, the same piecewise optimal trajectories methodology was used and a GA wasproposed for the stochastic search of the space of feasible assignments.

In this paper we concentrate on the cooperative multiple task assignment coupled with the problem of trajectoryoptimization for the UAV group servicing a set of targets. The GA, derived in this study, provides us with amonotonically improving solution for this highly complex coupled problem. The remainder of this manuscriptis organized as follows: in the next section the formulation of the problem is given. It includes denotation ofthe problem components and its graph representation. In the following section, the GA proposed for solvingthe entire combinatorial optimization problem is presented along with the computational complexity calculation.Performance analysis, including the results of sample and Monte Carlo simulation runs are presented in the sequel.Concluding remarks are offered in the last section.

II. Problem Formulation

In this section a formulation of the coupled cooperative task assignment and motion planning problem ispresented.

2 of 14

American Institute of Aeronautics and Astronautics

A. Vehicles

LetU = u1,u2, . . . ,uNu (1)

be a set ofNu cooperating homogeneous fixed wing UAVs.

For the sake of simplicity we will assume that (i) the UAVs’ flight can be confined to a plane in a given altitude,(ii) the involved UAVs have collision free paths achieved by altitude layering, (iii) the UAVs speed is constant,and (iv) there are no constraints on scenario duration, fuel consumption, and required weapons. We also assumethat the vehicles spatial configuration can be defined by three states

(x,y,ψ) (2)

with the following equations of motion

x = vcosψ (3a)

y = vsinψ (3b)

ψ = c Ωmax (3c)

wherex andy are a UAV’s horizontal coordinates in a Cartesian inertial reference frame;ψ is the azimutha flightpath angle;v is the constant speed;Ωmax is the maximum turning rate of the vehicle; andc is the steering commandsuch that|c| ≤ 1. The above model for representing the kinematics of a UAV is commonly denoted as a Dubins’car model.2–4

Following Ref. 4, the minimum length feasible curve for a Dubins’ car, from any arbitrary initial configuration,(xinit ,yinit ,ψinit ) to any arbitrary final configuration,(xf inal ,yf inal ,ψ f inal), is either: (i) an arc of a circle of radiusrwhere

r = v/Ωmax (4)

followed by a line segment, and then an arc of a circle of radiusr, or (ii) a sequence of three arcs of circles ofradiusr, or (iii) a sub-path of a path of type (i) or (ii).

To specify the type of these minimum length feasible curves for a Dubins’ path, we follow the same notationsused in Ref. 4. Three elementary motions are considered: turning to the left (denotedL), turning to the right(denotedR), and straight line motion (denotedS). Note that bothL andR turns are along a circle of radiusr. Thus,we define the Dubins’ setD as

D = LSL,RSR,RSL,LSR,RLR,LRL. (5)

which is the domain for the type of the minimum length feasible curve for a Dubins’ car between given initial andfinal configurations. For example, the notationLSLdenotes a trajectory composed ofL, S, andL segments.

B. Tasks

LetT = t1, t2, . . . , tNt (6)

be the set ofNt ground stationary targets, with known positions, designated to the UAV group, possibly by ahuman operator.

In this assignment problem the UAVs are required to perform a set of tasksMt on each target

Mt ⊆M = C,A,V (7)

whereC, A, andV denote classify, attack, and verify, respectively. Note that there is a precedence requirement, asfor example a target can not be attacked prior to being classified.

LetNmt = ‖Mt‖ (8)

denote the number of tasks that needs be performed on each target. Each taski ∈ Mt on targett ∈ T must beperformed only once. Hence, the number of tasks that all the UAVs are required to perform throughout thescenario is given by

Nc =Nt

∑t=1

Nmt (9)

aThe reference plane is true north and is considered0 azimuth. Moving clockwise, a point due east would have an azimuth of90.

3 of 14

American Institute of Aeronautics and Astronautics

For the simplicity of presentation, when all the targets need be serviced an identical number of times we denotethe number of tasks on each targetNmt asNm. In this case the number of all tasks throughout the scenario, seeEq. (9), merely becomesNc = NtNm.

C. Graph representation

The graph representation of the problem, see Fig. 1, is based on heading angle discretization. The heading anglediscretization set is defined as

H = ψi : ψi = 2πi/Nψ, i = 0,1, . . . ,Nψ−1 (10)

whereNψ > 0 is an integer defining the desired heading angle resolution.

LetVT = (t1,ψ1),(t1,ψ2), . . . ,(t1,ψNψ),(t2,ψ1), . . . ,(tNt ,ψNψ) (11)

be a set of vertices in the graph, where each node defines a targett ∈ T identified by its position and an angledescribing a headingψ ∈ H a UAV flies over it, where‖VT‖= NtNψ.The set of vertices that designate initial positions and headings of the UAVs is

VU = (u1,ψ10),(u2,ψ20), . . . ,(uNu,ψNψ0) (12)

where‖VU‖= Nu.

Thus, the set of all vertices in the graph is given by

V = VT ∪VU (13)

whereNv = ‖V‖= Nu +NtNψ (14)

A directed edge connects between any two nodes inVT and every nodeVU to all nodes inVT . A path in thegraphV is intended to symbolize a transition from an initial configuration(xinit ,yinit ,ψinit ) (a node inVU ) to afinal configuration(xf inal ,yf inal ,ψ f inal) (a node inVT ). The weight of each edgee= (vi ,v j), defined asw(vi ,v j ), is

calculated by using the, publicly released, trajectory optimization subroutine of the MultiUAV2 simulation18 andequals to the Dubins’ distance between two vehicle configurations contained in connected vertices. The set ofedges in such multiple Dubins’ paths optimization graph is given by

E = (vi ,v j) : vi ∈V, v j ∈VT (15)

where‖E‖= NtNψ(Nu +NtNψ) (16)

It should be noted that trajectory obtained for a vehicle transition from any initial configuration,(xinit ,yinit ,ψinit ),to any final configuration,(xf inal ,yf inal ,ψ f inal), is identical to the inverse travel trajectory from(xf inal ,yf inal ,ψ f inal +π) to (xinit ,yinit ,ψinit +π). This feature enables reducing the amount of edge weights precalculations by approxi-mately half.

D. Combinatorial optimization problem

We choose the cumulative distance that all the UAVs travel in order to fulfill the tasks of the coupled problem asthe cost function to be minimized

J =Nu

∑u=1

Nc

∑k=1

Nv

∑i=1

Nv

∑j=1

Xu(vi ,v j ) w(vi ,v j ) (17)

whereXu(vi ,v j )

is a binary decision variable

Xu(vi ,v j ) ∈ 0,1 (18)

It is 1 if the edgee= (vi ,v j) is used to indicate the movement of the vehicleu∈U from any configurationvi toany otherv j , and is 0 otherwise. Note that only weightsw(vi ,v j ) of the existing edgese= (vi ,v j) ∈ E in the graphare counted in the above formulation ofJ.

4 of 14

American Institute of Aeronautics and Astronautics

Figure 1. Multiple targets Dubins’ trajectories graph representation. Target nodes defined by target identity and heading angle(ti ,ψ[deg]); vehicle source nodes identified by UAV and initial heading angle(ui ,ψi0 [deg])

There are two constraints that have to be satisfied throughout the solution process. The first

Nu

∑u=1

Nv

∑i=1

Nψ−1

∑k=0

Xu(vi ,(t j ,ψk))

= Nmt , ∀ t j ∈ T (19)

is referred to the exact number of tasks that have to be performed on each target during the scenario.

The second

The set Xu(vi ,v j )| X

u(vi ,v j ) = 1, ∀u∈U, vi ∈V, v j ∈VT

compose apath in the graph (20)

is posted to verify that the solution constitutes a set of connected traceable vehicle pathsb in the multiple targetsDubins’ trajectories graph, see Fig. 1.

III. Genetic Algorithm

A large number of UAVs, targets, and discretization angles significantly enlarge the multiple Dubins trajecto-ries graph making the problem computationally prohibitive, even for a single vehicle. This leads us to seek forstochastic alternatives to deterministic search algorithms. In this section we present a GA for efficiently searchingthe space of solutions. Being a stochastic search method it is used for obtaining a good solution to the problem,not necessarily the optimal one.

A. Methodology

At first, many individual solutions are randomly generated to form an initial population, covering the range ofpossible solutions (the search space). The population size depends on the nature of the problem. Individualsolutions are selected from the population through a fitness-based process. The selection process has a randomelement, and there is a larger probability to choose more fit solutions. Nonetheless, even less fit solutions havea chance of being selected. This helps in keeping the diversity of the population large, preventing prematureconvergence to a solution (i.e. prevents the algorithm from getting trapped in a local minimum).

bA list of a graph vertices, where each vertex connected by an edge to the next vertex, defines a path in a graph.

5 of 14

American Institute of Aeronautics and Astronautics

The next step is to use the selected solutions to generate a second generation of solutions. For this process weuse the operators: crossover (also called recombination), and/or mutation on each set of two selected solutions,denoted as parents. A new ”child” solution, produced using the above methods of crossover and mutation, typicallyshares many of the characteristics of its ”parents”. New parents are selected for each child, and the processcontinues until a new population of solutions of appropriate size is generated. This generational process is repeateduntil a termination condition is reached.

B. Encoding

The most critical part of a GA is the solution chromosome encoding. For this problem the encoding of thechromosome is based on the targets’ visitation order, by each vehicle, with a heading angle selected from the setH. Thus, every gene corresponds to a specific configuration of an assigned vehicleui ∈U at task execution ontargett j ∈ T. Such a configuration is specified by a vertexvi ∈ VT ⊂ V from the multiple targets Dubins’ pathoptimization graph.

An example chromosome for a scenario that involves one vehicle (identified as1 in the second row of achromosome) performing a single task on each of three targets starting from its initial configuration at(0,0,330)with Nψ = 36 (heading angle discretization of10), is shown in table 1. The candidate solution that is encoded inthe chromosome orders the UAV to visit the target found at(50,300) with ψ = 70, to proceed to the next targetlocated at(100,150) and fly over it withψ = 140, and finish the mission at(150,350) by visiting the last targetwith ψ = 10.

(50,300,70) (100,150,140) (150,350,10)1 1 1

Table 1. Feasible solution encoded as a chromosome;Nψ = 36, Nm = 1.

Another example chromosome that clearly describes a potential solution of coupled multiple task assignmentand trajectory optimization problem for two cooperating UAVs is presented in table 2. Now, the scenario involvesthe vehicles identified as1 and 2, performingNm = 2 tasks on each of two targets starting from their initialconfigurations at(0,0,315) and(200,0,45) respectively withNψ = 36.

Thus, the chromosome corresponds (i) to a path of the first UAV from its initial configuration to a target locatedat (150,350), visiting it atψ = 70, going on to the target at(300,450) with ψ = 50 and revisiting the last targetat ψ = 260, and (ii) to a visitation of the target at(150,350) by the second vehicle with heading angleψ = 230.The chromosome provides neither information about arrival time of any vehicle to any target nor precedence orderof the tasks execution. Thus, during each visitation of a target by a vehicle, the currently needed task is executed.

(150,350,70) (300,450,50) (300,450,260) (150,350,230)1 1 1 2

Table 2. Feasible solution encoded as chromosomes;Nψ = 36, Nm = 2.

Note that, firstly, thei ∈ 0, . . . ,Nc genes representing operation of the first UAV are presented, after thatj ∈ 0, . . . ,Nc− i genes for the second vehicle activity are designated, and so on. By such an arrangementwe achieve unique representation of each candidate in the solutions search space. For instance, a chromosomeobtained by switching of the third gene with the fourth one in the second example, see table 2, represents the samesolution and therefore will be eliminated as an unnecessary one.

C. Computation complexity

To analytically analyze the computational complexity of the problem by terms of combinatorics,19 we computethe cardinality of the feasible solutions space. The number of feasible chromosomes defines the complexity of theproblem.

Each of theNt targets requiresNmt visits of assigned UAVs, as a result the length of a chromosome is equal tothe number of needed tasks throughout a scenario, and given in Eq. (9).

Now, we shall calculate the number of task permutations with repetitions in theNc genes of a chromosome.Each targett ∈ T is presented exactlyNmt times in a chromosome genes and the order of such repetitions is not

6 of 14

American Institute of Aeronautics and Astronautics

important. The number of such generalized permutations19 is defined by Eq. (21). To obtain this equation, first,we setNm1 genes for the tasks on the first target by computing

( NcNm1

)possibilities. Next, we setNm2 tasks for the

second target, out of the remaining(Nc−Nm1) genes, resulting with(Nc−Nm1

Nm2

)possibilities. We then continue the

process for all the remaining targets.

P(Nc;Nm1,Nm2, . . . ,NmNt) =

Nc!Nm1!Nm2!, . . . ,NmNt

!(21)

To consider the number of different vehicle allocations to a given set of tasks, we need to calculate all thepossibilities for distinctive arrangements ofNu vehicles in a chromosome of lengthNc.

During the encoding process the sub-chromosome genes were ordered according to the vehicle identificationnumber, see subsection III.B. The number of such arrangements ofNu vehicles in a chromosome is given by

Na =rmax1

∑r1=1

rmax2

∑r2=1

. . .

rmaxNc−1

∑rNc−1=1

rmaxNc

(22)

wherermaxi = rmax

i−1 − r i−1 +1 ∀ i = 2,3, . . . ,Nc (23)

andrmax1 = Nu (24)

Ther i-th sum in the above result refers to the possible vehicles that can be placed in thei-th gene of a chromosome.

To complete the calculation of the problem complexity, we need also to take into account all the heading anglesin which a UAV can visit each target, resulting with(Nψ)Nc possibilities.

Ultimately, the solutions space size is given by

Nf =Nc!

Nm1!Nm2!, . . . ,NmNt!Na(Nψ)Nc (25)

D. Fitness evaluation

The fitness evaluation is based on the edge weights in the multiple targets Dubins’ paths graph of Fig. 1. Eachedge weight denotes the Dubins’ distance between the destinations with initial and final heading angles.

The precalculation of these distances is accomplished by using the path optimization subroutine of the MultiUAV2simulation.18 A reduction in the number of such precalculations is achieved by efficient use of repeated data, aswas explained in subsection II.C. The resulting weight of each edge is stored in a data table and is pulled outduring the evaluation process.

Every path in the graph, encoded in a chromosome solution, denotes a vehicle trajectory. A trajectory length isequal to the sum of the edge weights representing it. A chromosome fitness is defined as an inverse of all vehicletrajectories lengths sum, i.e.1/J, whereJ is given by Eq. (17).

E. Evolutionary operators

1. Crossover

The single site crossover method has been chosen to create a pair of child chromosomes from a pair of parentchromosomes. The idea behind the crossover operator is that the offspring chromosome may have better fitnessthan both of the parents if it takes the best characteristics from each of the parents.

The recombination example is illustrated in Fig. 2. Firstly, primary progenitor is randomly chosen from thepair of involved parents. Then, the crossover sites is uniformly selected from[1,Nc], ands−1 first genes of theprimary parent are copied to the generated offspring. In order to force a child chromosome to be feasible fromthe aspect of task assignment correctnessc all the target positions are copied to a child chromosome in the sameorder that they appeared in the primary parent. Now, the genes recombination pool is created. It includes all

cEach targeti ∈ T has to be visited exactlyNmt times.

7 of 14

American Institute of Aeronautics and Astronautics

the genes of secondary parent chromosome that encode any task execution on the same target as at the presentlyrecombined gene. One gene is randomly chosen from this pool and copied to a current gene of the child. A newpool from secondary parent candidate genes is generated for each of[s,Nc] child genes. The process of randomselection from a pool is repeated until the offspring construction is completed. Thus, a crossover process affects avehicle allocation and a visitation heading angle in a particular gene. To finalize the creation of the first child, theoffspring is sorted according to the UAV identity.

The production of a second child includes changing the secondary parent to be the primary one and an initiationof the same process like that for the first child.

2. Mutation

Mutation is a genetic operator that alters one or more gene values in a chromosome from its initial state. Mutationis an important part of the search as it helps the convergence of the GA toward an optimal solution by preventingthe population from stagnating at a local minimum.

In our study, the mutation is performed in randomly chosen sites of each chromosome in the population. Weapply four alternative ways to mutate a gene contents:

• heading angle mutation

• mutation in vehicle allocation

• targets swap

• targets and heading angles swap

A mutation of a heading angle is performed by a random choice of a different heading angleψnew∈H \ψoldand inserting it to the mutated gene instead ofψold. A mutation in vehicle allocation is achieved by substitutingvehicleuold with vehicleunew∈U \uold. Swapping of two tasks is done by mutual interchange of two differenttargets from two randomly chosen different genes. An additional used form of mutation is a swap of targets andheading angles of a pair of genes.

Each of the four mutation alternatives is applied with an empirically obtained probabilitypmi , where i ∈1,2,3,4. We define these probabilities such that more than one kind of mutation operator may be applied oneach chromosome.

3. Elitism

The elitism operator is employed to preserve the best solution obtained during each iteration. A solution thatwas found best through the fitness evaluation process is declared as an elite and is kept in a predefined spaceof a chromosome population. This elite solution is substituted in the following iterations with one of the newlygenerated chromosomes only if it was found to have a higher fitness.

IV. Performance Analysis

A. Examples

The GA simulation parameters, for the scenario involving three similar targets and a single vehicle, are presentedin table 3.

Population size 200

Number of iterations 50

Minimum turn radius [ft] 200

Discretization [deg] 10

Table 3. GA Simulation Parameters

A sample of a trajectory obtained with the relevant heading angles is plotted in Fig. 3. The trajectory definesalso the target visitation order. The path length of the achieved solution, after the allotted 50 iterations, is1535.3

8 of 14

American Institute of Aeronautics and Astronautics

Figure 2. Crossover Process

[ft], compared with the minimal length path of1483.4 [ft] obtained by using a deterministic graph search method.

Raising the population size and/or the number of iterations produces better results. As an example we raisedthe number of iterations from50 to 200and obtained a very similar solution with a length of1486.2 [ft] that is

9 of 14

American Institute of Aeronautics and Astronautics

−50 0 50 100 150 200 250 300 350 400 450 500 550−50

0

50

100

150

200

250

300

350

400

450

[ft]

[ft]

Start

Finish

1

2

3

ψ = 0o

ψ = 50o

ψ = 70o

ψ = 290o

Figure 3. Resulting trajectory from GA; rmin = 200[ f t], Nψ = 36, Nm = 1.

only marginally longer than the minimum length solution. The only difference was with the heading angle at thelast target (330 in the deterministic solution instead of290 obtained by the GA).

We now extend the problem to examine a more complex scenario. The new scenario consists of two vehiclesU = 1,2 with initial configurations(0,0,315) and(200,0,45) respectively; and two targetsT = 1,2, locatedat (150,350) and(300,450) respectively, each requiring two visits of a vehicle, i.e.Nm = 2.

We apply the GA simulation with parameters brought up in table 4. The obtained trajectories of the cooperating

Population size 300

Number of iterations 200

Minimum turn radius [ft] 200

Discretization [deg] 10

Table 4. GA Simulation Parameters

vehicles are plotted in Fig. 4 and the relevant heading angles of each vehicle when flying over a target are presented.The cumulative path length of the solution achieved by the GA run is1796.4 [ft].

The simple deterministic graph search methods, like Dijkstra algorithm, cannot deal with the case of coopera-tive multiple task execution because of the need for complex multiple-source task coordinated graph search. Thus,an exhaustive search technique was used to obtain the optimal total path length. A comprehensive examinationof the entire solution space yields the optimal trajectory (under the given discretization) with a length of1709.7[ft]. Here too the visitation order was identical, and the path was very similar, with deviations of up to20 in theheading angles.

B. Monte Carlo Study

A Monte Carlo study, consisting of 100 runs for each set of parameters, provides us with an extended analysis ofthe GA algorithm. The scenarios examined involve three homogeneous UAVs, four targets, three tasks per target,and turn radius of200[ f t],100[ f t], and50[ f t] respectively.

10 of 14

American Institute of Aeronautics and Astronautics

−100 −50 0 50 100 150 200 250 300 350 400 450 500 550 600 650−50

0

50

100

150

200

250

300

350

400

450

500

[ft]

[ft]

ψ = 315o

ψ = 80o

ψ = 40o

ψ = 45o

ψ = 240o

ψ = 230o

Start1

Finish1

Start2

Finish2

1

2

Figure 4. Resulting trajectory from GA; rmin = 200[ f t], Nψ = 36, Nm = 2.

Fig. 5 demonstrates sample trajectories in one of these complex scenarios. All the relevant data is broughtup in tables 5-7. Note that the defined cost criterion of minimum cumulative path length and the assumption onunlimited resources cause one of the vehicles to execute significantly more tasks than the other vehicles.

UAVs’ Initial Configurations

UAV X [ft] Y [ft] ψ [deg]

1 0 0 0

2 200 0 0

3 400 0 0

Table 5. Initial UAVs Configuration.

Targets Positions

Target X [ft] Y [ft]

1 50 300

2 250 350

3 150 450

4 450 250

Table 6. Targets Positions.

Population size 300

Number of GA iterations 300

Minimum turn radius [ft] 200

Discretization [deg] 10

Table 7. GA Simulation Parameters

In Fig. 6 the average over the set of 100 runs of the normalized costsJi/J1 is presented as a function of thegenetic generation, wherei ∈ [1,2, . . . ,300] is the generation. Here, the progressive improvement of an achievedsolution for each case of different turn radius is evident. Note that reducing the turn radius reduces the affect ofthe kinematic constraints on the resulting point-to-point Dubins’ trajectories. E.g., an assumption ofrmin = 0[ f t]causes all of the Dubins’ path lengths between one configuration to another to be identical, i.e. the metric becomesEuclidean. Therefore, the diversity of the search space lessens with the reduction of the turn radius. This canexplain the faster and improved convergence with the reduction of the turn radius.

11 of 14

American Institute of Aeronautics and Astronautics

−400 −300 −200 −100 0 100 200 300 400 500 600

0

100

200

300

400

500

600

700

800

900

[ft]

[ft]

ψ = 0o

ψ = 330o ψ = 210o

ψ = 330o

ψ = 50o

ψ = 0o

ψ = 30oψ = 30o

ψ = 30o

ψ = 0o

ψ = 10o

ψ = 170o

Start1

Finish1

Start2

Finish2

Start3

Finish3

1

2

3

4ψ = 160o

ψ = 70o

ψ = 90o

Figure 5. Resulting trajectory from GA; rmin = 200[ f t], Nψ = 36, Nm = 3.

50 1 100 150 200 250 3000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

i generation

E(J

i/J1)

rmin

= 50 [ft]

rmin

= 100 [ft]

rmin

= 200 [ft]

Figure 6. Mean of100Monte Carlo runs: 4×3×3 scenario;Nψ = 36population size = 300, number of iterations = 300.

V. Conclusions

An integrated task assignment and motion planning algorithm was presented for a problem where multiplehomogeneous UAVs cooperatively perform multiple tasks on multiple targets. The method uses a discretization of

12 of 14

American Institute of Aeronautics and Astronautics

the possible visitation angle of a vehicle at its target. Using a smaller discretization variable produces a solutionthat is closer to the optimal one. However, this enlarges the search space and can render the use of deterministicsearch methods infeasible. For such cases a GA was proposed. The algorithm enables to rapidly derive a goodsolution even to a complex scenario. The tendentious convergence of the GA toward the optimal solution waspresented using a Monte Carlo study.

13 of 14

American Institute of Aeronautics and Astronautics

REFERENCES

1Chandler, P. R., Pachter, M., Swaroop, D., Fowler, J. M., Howlet, J. K., Rasmussen, S., Schumacher, C., and Nygard, K., “Complexityin UAV Cooperative Control,”Proceedings of the American Control Conference, Evanson, IL, 2002.

2Latombe, J.-C.,Robot Motion Planning, Kluwer Academic Publishers, Norwell, MA, 1991.3LaValle, S.,Planning Algorithms, Cambridge University press, New York, NY, 2006.4Dubins, L. E., “On Curves of Minimal Length with a Constraint on Average Curvature, and with Prescribed Initial and Terminal

Position,”American Journal of Math, Vol. 79, 1957, pp. 497–516.5Hillier, F. S. and Lieberman, G. J.,Introduction to Operations Research, McGraw–Hill, New York, NY, eighth ed., 2005.6Cormen, T. H., Leiserson, C. E., and Rivest, R. L.,Introduction to Algorithms., MIT press, Cambridge, MA, 1990.7Dijkstra, E. W., “A Note on Two Problems in Connexion with Graphs,”Numerische Mathematik, Vol. 1, 1959, pp. 269–271.8Bellman, R., “On a Routing Problem,”Quarterly of Applied Mathematics, Vol. 16, No. 1, 1958, pp. 87–90.9Michalewicz, Z. and Fogel, D.,How to Solve It: Modern Heuristics, Springer, Berlin, Germany.

10Goldberg, D. E.,Genetic Algorithms in Search, Optimization, and Machine Learning, Addison-Wesley, Reading, Massachusetts, 1989.11Shima, T., Rasmussen, J., Sparks, A., and Passino, K., “Multiple Task Assignments for Cooperating Uninhabited Aerial Vehicles using

Genetic Algorithms,”Computers and Operations Research, Vol. 33, No. 11, pp. 3252–3269.12Richards, A., Bellingham, J., Tillerson, M., and How, J. P., “Coordination and Control of Multiple UAVs,”Proceedings of the AIAA

Guidance, Navigation, and Control Conference, Washington, D.C, 2002.13Schumacher, C. J., Chandler, P. R., Pachter, M., and Pachter, L., “Constrained Optimization for UAV Task Assignment,”Proceedings

of the AIAA Guidance, Navigation, and Control Conference, Washington, D.C, 2004.14Schumacher, C. J., Chandler, P. R., and Rasmussen, S. J., “Task Allocation for Wide Area Search Munitions,”Proceedings of the

American Control Conference, Evanson, IL, 2002.15Chandler, P. R., Pachter, M., Rasmussen, S., and Schumacher, C., “Multiple Task Assignment for a UAV Team,”Proceedings of the

AIAA Guidance, Navigation, and Control Conference, 2002.16Rasmussen, J. and Shima, T., “Tree Search for Assigning Cooperating UAVs to Multiple Tasks,”Internation Journal of Robust and

Nonlinear Control, Vol. 18, No. 2, pp. 135–153.17Schumacher, C. J., Chandler, P. R., and Rasmussen, S. J., “Task Allocation for Wide Area Search Munitions Via Iterative Network

Flow Optimization,”Proceedings of the AIAA Guidance, Navigation, and Control Conference, Washington, D.C, 2002.18Rasmussen, S. J., Mitchell, J. W., Schulz, C., Schumacher, C. J., and Chandler, P. R., “A Multiple UAV Simulation for Researchers,”

Proceedings of the AIAA Modeling and Simulation Technologies Conference, Washington, D.C, 2003.19Graham, R., Groetschel, M., and Lovasz, L.,Handbook of Combinatorics, Vol. 1,2, Elsevier, Amsterdam, and MIT Press, Cambridge,

MA, 1996.

14 of 14

American Institute of Aeronautics and Astronautics