A Heuristic Search Algorithm for Maneuvering of UAVs acrossDense Thermal Areas

Nazli E. Kahveci∗ and Petros A. Ioannou†

University of Southern California, Los Angeles, CA, 90089

Maj D. Mirmirani‡

California State University, Los Angeles, CA, 90032

In order to ensure that an aircraft has the potential to meet the assigned performance requirements whichare often mission specific, the particular aerodynamic demands involved must be taken into account duringthe aircraft design phase. Once the design is completed and the parts of the aircraft are assembled, carefullychosen soaring strategies prove an additional source of flight performance enhancements which in turn providefurther feedback for designers. As such, there has been considerable interest in modern glider design andsoaring flight during the last few decades. The Unmanned Aerial Vehicles (UAVs) designed for soaring flightcurrently demand more efficient soaring strategies that would allow them to cover larger flight distances,possibly even faster.

In this paper we discuss the maneuvering of a glider UAV across dense thermal regions. We present aproblem scenario where the objective is to climb the assigned thermals in the area of interest and completethe flight mission in minimum time. Our solution methodology is based on dividing the maneuvering areainto main sectors and applying a minimal spanning tree algorithm to cover the set of thermals detected ineach sector. A parallel savings based heuristic is included in order to improve the path decision process while amaximum distance constraint is also incorporated. An adaptive control scheme is developed for the linear UAVmodel used in simulations through which the performance of the proposed near-optimal soaring algorithm isverified.

I. Introduction

TAKING advantage of the analogies between the aerodynamic design of birds and aircraft is a popular approachin the literature. A summary of work prior to wind tunnel studies on live birds is given in [1] where the main

concern is the aerodynamics of equilibrium gliding, i.e., nonflapping flight without acceleration. A wind tunnel studyis presented in [2] for gliding flight. Large migrant birds are observed to employ soaring methods which are proven tobe flight techniques that aircraft can also benefit from in order to save energy.

The overall performance of gliders in thermal soaring applications is based on two separate missions which are thegliding phase that utilizes energy and the thermal climbing phase that extracts energy using rising air currents. Theaircraft stores energy in the potential form as altitude while soaring thermals, and it aims to use this energy efficientlyduring its glide. The aerodynamic demands of each phase are quite different, and the required compromise of the finaldesign is discussed in [3].

This work was supported in part by the National Science Foundation Grant No. 0510921 and in part by the National Aeronautics and SpaceAdministration under Grant UAS/USC-220843. Any opinions, findings, and conclusions or recommendations expressed in this material are thoseof the authors and do not necessarily reflect the views of the National Science Foundation or the National Aeronautics and Space Administration.

∗Ph.D. Student, Electrical Engineering, University of Southern California, Student Member, AIAA†Professor, Electrical Engineering, University of Southern California, Member, AIAA‡Professor, Mechanical Engineering, California State University, Senior Member, AIAA

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The flight performance optimization has become the focus of soaring research in recent years.4–7 The best per-formance obtained in real applications is still far from optimal whereas efficiently climbing a set of thermals is stillconsidered to involve a luck factor. However, recent studies on autonomous UAVs show that the significance of theluck factor is overrated in general and that it does not necessarily play a determining role. This work is an attempt todemonstrate this point by applying a systemized optimal flight path decision procedure for a glider UAV to cover adense thermal region.

UAVs differ from both transport aircraft and sailplanes in important ways, and these differences may enable UAVsto better exploit the wind energy. In general, UAVs are lighter in weight than transport aircraft, making wind energiespotentially more effective for UAV flights. Because there is no human on board, UAVs can perform more drasticmaneuvers than manned aircraft in flying wind-optimal trajectories.8

Despite the differences in design mentioned between UAVs and transport aircraft, path planning algorithms usedfor transportation vehicles still prove to be useful. A relevant popular problem in transportation research is the VehicleRouting Problem (VRP). VRP calls for the determination of the optimal set of routes to be performed by a fleet ofvehicles to serve a given set of customers, and it is one of the most important and studied combinatorial optimizationproblems.9 The VRP was first introduced by Dantzig and Ramser.10 The models and algorithms proposed for thesolution of vehicle routing and scheduling problems can be used effectively not only for the solution of problemsconcerning the delivery and collection of goods but for the solution of different real-world applications arising intransportation systems as well.9

VRP requires the efficient use of a fleet of vehicles that make a number of stops to pick up and/or deliver passengersor products. Thermal soaring problem in analogy requires the glider to visit a number of thermal locations in the flightregion. The similarities can be listed further if extended versions of VRP are considered. The reader might refer to [7]for a variety of VRP interpretations for thermal soaring problem and a discussion on how to exploit the correspondingVRP solution methodologies for the soaring path decision.

In this paper we consider a glider flying over a region dense with thermals and apply a heuristic search method inorder to solve the thermal soaring problem involving some extra flight constraints.

The glider UAV model used and our adaptive linear quadratic control design are described in Section II. Thermalsoaring is defined as a constrained optimization problem and the mathematical formulation is given in the form ofa VRP in Section III. In Section IV a flight path decision algorithm is developed. Simulation results for a problemscenario are presented in Section V in order to demonstrate the efficiency of the proposed methodology. Finally,Section VI draws conclusions.

II. Glider UAV Model and Adaptive Control Design

We use a linear incremental state space model which describes the longitudinal dynamics of a glider UAV. Thenominal UAV model is provided in [7] where the states of the model are available for measurement and are hencedefined also as the output of the system. The deviation of the glider’s horizontal velocity from its value at the trimmedconditions, δVh is used to define the flight path and is chosen as the controlled output which can be written in termsof the incremental states of the longitudinal model through a performance output matrix, Cp. The aircraft model canthen be summarized as

xs = Axs + Bu + Guth + HδT (1)y = Cxs

δVh = Cpxs

where A,B,G and H are system matrices. The incremental state vector for longitudinal aircraft model is defined asxs = [ δV δα δq δθ δh ]T where V is the aircraft velocity, α is the angle of attack, q is the pitch rate, θ is the pitchangle, and h is the altitude. Note also that C is the identity matrix since the output is the measurable state vector of the

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aircraft. The control input, u is the elevator deflection, and the thermal strength function, uth is the vertical velocityof the air current effective while the aircraft is soaring a thermal. A negative throttle setting, δT =−0.3 is applied tocancel the effect of the thrust input at the trimmed conditions and thus to obtain a zero-thrust aircraft model.

The horizontal flight velocity can be represented by Vh = Vh0 + δVh where Vh0 is the horizontal trimmedvelocity. The control objective is to have Vh track a reference input, Vopt which is to be defined via optimal trajectorygeneration.

A glider always sinks through the still air, and hence its performance can be characterized by its polar curve, agraph that shows its sink rate at different airspeeds, Vz vs Vh . A quadratic approximation to the polar curve of theglider is given as Vz = aV 2

h + bVh + c where the simulations reveal that a = −0.000391 , b = 0.026 , c = −6.3 isa good approximation to the performance of the aircraft model used.

If the reference horizontal flight velocity is defined as Vopt = Vh0 + δVopt , a set of compatible states can beobtained using the left pseudo inverse of the Cp matrix:

δVopt = Cpxr ⇒ xr = CTp (CpC

Tp )−1δVopt (2)

Since the applied negative throttle setting has the form of a constant disturbance throughout our applications, thedesign of a tracking control with constant disturbance rejection is required. We first reduce the tracking problem toan equivalent disturbance rejection problem, and then apply a constant disturbance rejection method by augmentingthe system as discussed in [11]. The linear quadratic control design for the longitudinal glider UAV model can beimplemented if the system is first augmented as

xaug = Aaugxaug + Baugv (3)

δVh = Cpxs = Cpe + δVopt ⇒ zaug = δVh − δVopt = Cpe ⇒ zaug = Caugxaug (4)

where xaug =

[e

zaug

], e=xs − xr , v= u , Aaug =

[A 0Cp 0

], Baug =

[B

0

], Caug =

[0 0 0 0 0 1

].

If (Aaug, Caug) is detectable and (Aaug, Baug) is stabilizable, the unique positive definite solution P to theAlgebraic Riccati Equation (ARE) :

ATaugP + PAaug + Qz − PBaugR

−1z BT

augP = 0 , Qz =

[0 00 I

]×Q , Q > 0 , Rz > 0 (5)

defines the feedback gain, K=R−1z BT

augP . The control input for the augmented system is to be chosen as

v = −Kxaug = − [ K1 K2 ] xaug = −K1 e−K2 zaug , K1 ∈ R1x5, K2 ∈ R (6)

which asymptotically stabilizes the augmented closed loop system. The control input given as

u = −K1e−K2

∫ t

0

Cpe(τ)dτ (7)

then provides tracking control with disturbance rejection for the original system in (1).If the sensor measurement noise and the process noise are considered in the control design phase, one might also

implement the Linear Quadratic Gaussian design given in [12].The control design is combined with an adaptive law in order to compensate for uncertainties and possible changes

in the UAV dynamics. The same robust adaptive law employed in [12] is also used in this paper and is derived forcompleteness as follows:

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The state vector, xs can be measured whereas the throttle setting, δT and the thermal strength function, uth areboth assumed to be known. Consequently, for any fixed flight conditions we obtain

z = θ∗T φ , θ∗T =[A B

], φ =

1s + λ

[ xs1 xs2 xs3 xs4 xs5 u ]T (8)

where φ is the regressor vector, xsiis the ith element of xs ∈ R5, and λ > 0 is a design parameter. The estimation

model can be defined as

zi = θTi φ , θi(t) =

[ai1 ai2 ai3 ai4 ai5 bi

]T

(9)

for i = 1, 2, ..., 5 where θi(t) is the estimate of θ∗i at time t , and θ∗i ∈ R6 is the ith column of θ∗ ∈ R6×5 . Note thataij is the estimate of aij where aij is the ijth element of A ∈ R5×5, and bi is the estimate of bi where bi is the ith

element of B ∈ R5.We use the discrete approximation of the continuous-time online parameter estimator based on the continuous-time

model of the system. Let k = 1, 2, ... be the discrete-time index and T be the sampling period where estimation isdone at time t = kT. The discrete-time estimation model is zi(kT ) = θT

i (kT )φ(kT ) , i = 1, 2, ..., 5 .For the adaptive law we employ the discrete version of the least-squares algorithm modified with robust weighting

given in [13] as

Pi(kT ) = Pi([k − 1]T ) − ci(kT )Pi([k − 1]T )φ(kT )φT (kT )Pi([k − 1]T )m2(kT ) + ci(kT )φT (kT )Pi([k − 1]T )φ(kT )

(10)

εi(kT ) =zi(kT )− θT

i ([k − 1]T )φ(kT )m2(kT )

(11)

θi(kT ) = θi([k − 1]T ) +√

ci(kT )Pi(kT )φ(kT )εi(kT ) (12)

⇒ θi(kT ) = θi([k − 1]T ) +ci(kT )Pi([k − 1]T )φ(kT )[zi(kT )− θT

i ([k − 1]T )φ(kT )]m2(kT ) + ci(kT )φT (kT )Pi([k − 1]T )φ(kT )

(13)

for each i = 1, 2, ..., 5 where

ci(kT ) =

{ci1 , φT (kT )Pi([k − 1]T )φ(kT ) ≥ δi

ci2 , otherwise(14)

is the robust weighting for ci1 >> ci2 > 0 , δi > 0 , and Pi(0) , ci1 , ci2 , δi , i = 1, 2, ..., 5 are design parameters.The estimation error, εi(kT ) is normalized where m2(kT ) = 1 + φT (kT )φ(kT ) is a proper normalization signal.

To eliminate possible parameter drift due to modeling errors, estimates of parameters are forced to lie inside acompact set by using parameter projection so that their boundedness is guaranteed. It is assumed that lower and upperlimits of each element θ∗ij of θ∗i , j = 1, 2, ..., 6 , are known so that Lij ≤ θ∗ij ≤ Uij for some Lij , Uij ∈ R , i =1, ..., 5 , j = 1, 2, ..., 6 . As a result the orthogonal term-by-term projection is given by

θij(kT )=Pr{θij(kT )}=

θij(kT ) , Lij ≤ θij ≤ Uij

Lij , θij(k) < Lij

Uij , θij(k) > Uij

(15)

where θij(kT ) is the estimate of θ∗ij at time t = kT.The described adaptive control design is employed in our simulations where the vehicle is assigned to follow a

given flight path. In order to generate the optimal flight trajectory for a specific set of thermal locations, the corre-sponding thermal soaring problem is required to be formulated. A possible formulation method is discussed in thenext section.

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III. Soaring Assigned Thermals and Mathematical Formulation as a VRP

The flight time optimization for a glider UAV in an autonomous thermal soaring application can be convenientlystated as a graph problem on a two-dimensional area map of thermals detected over a specific area as motivated in [7].This map has two degrees of freedom, x and y , looking down perpendicular on the maneuvering area of the UAV.Each thermal location represents a node, and the underlying thermal network is represented with a graph G(N ,A)where N is the set of thermal locations, and A is the set of edges which are the straight flight arcs between the nodes.Once the optimal thermal soaring problem is defined on a 2D graph with nodes as thermals and arcs as inter-thermalglide distances, the problem exposes its similarities to the well-known generic VRP.

The soaring problem is adapted to VRP with Time Windows (VRPTW) in [7] where an analogy between thetwo problems is drawn. Extra constraints called time windows that correspond to thermal lifespan periods in soaringapplications are considered, and an exact algorithm is developed for the UAV to arrive at the destination node inminimum time. In this paper we remove the thermal lifespan constraints assuming that thermals are available forsoaring flight at all times once they are detected. However, we pose an additional requirement that the glider returnsback to where it starts after climbing at all the assigned thermal locations. This can be interpreted as serving allcustomers in VRP formulation.

Two metrics are defined in [7] on each arc (i, j) in the graph: A spatial metric, lij and a temporal metric, cij . Thespatial metric is defined as the Euclidean norm of the flight distance in between each pair of thermal locations. It isemployed to decide which thermal locations are directly connected on the map. A maximum value, lmax is chosen,and for any arc (i, j) ∈ A where lij > lmax , the temporal metric, cij is assigned to infinity. Consequently, lmax

poses a restriction on the direct flight range as cij = ∞ implies that flight from node i to node j is not allowed unlessother thermal locations are visited in between. As a result, the parameter lmax is used to define the maximum distanceconstraint in our problem.

Each arc cost cij in the graph represents the particular time required to include that arc in the flight route. Thisinvolves both the glide time along the arc (i, j) and the climb time at the arrival thermal location, j . Using the samemathematical relations derived in [7], optimal arc costs and corresponding horizontal flight velocities can be listed as

cij = ttotalij = tglideij+ tthermalij =

l(i, j)Voptij

+hloss(Voptij

, Vzij)

uthj

(16)

hloss(Voptij, Vzij

)= l(i, j)×|Vzij |Voptij

= l(i, j)×−Vzij

Voptij

(17)

Vzij∼=aV 2

optij+ bVoptij

+ c (18)

Voptij=

√c− uthj

a,

c− uthj

a> 0 (19)

where l(i, j) is the flight distance from thermal location i to thermal location j , uthj is the thermal strength at thearrival node j , and a , b , c are the coefficients of the polar curve. Note that the basic requirement posed to keep theproblem tractable is to force the glider UAV recover the altitude loss during inter-thermal glide while soaring at thearrival node.

The thermal soaring problem on a 2D graph G(N ,A) can then be defined in its more general form as a CapacitatedVRP (CVRP) :

min∑i∈N

∑j∈N

cijxij (20)

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subject to ∑i∈N

xij =∑k∈N

xjk ∀ j ∈ N (21)∑j∈N

x0j = K (22)

xij ∈ { 0, 1 } ∀ i , j ∈ N (23)

In the above formulation xij is the flow variable for the flight arc (i, j) ∈ A which takes value 1 if arc (i, j) ∈ Abelongs to the optimal route and 0 otherwise. N is the set of thermals assigned to be soared where i = 0 ∈ N denotesboth the start and the end location.

Constraint (21) guarantees that if the UAV enters a thermal for soaring, it also leaves that thermal after its climb.Constraint (22) defines the number of tours to be included and is a capacity constraint. If the assignment is completedin a single tour as we assume in this paper, the parameter K is set to 1 . A binary constraint is given in (23) which isalso called the integrality constraint.

In the thermal soaring problem scenario in [7] a relatively small number of thermals is considered, and an exactmethod is effectively employed to find the optimal path. It is stated in [9] that the largest VRP instances that can beconsistently solved by the most effective exact algorithms proposed so far contain about 50 customers, whereas largerinstances may be solved to optimality only in particular cases. Many of the proficient VRP solution techniques thatefficiently handle arbitrary numbers of vehicles and waypoints are heuristic which can typically provide high qualitysolutions with moderate computation times.14

Formulating a VRP as a set-covering problem is suggested in [15], and some exact algorithms are given in [16–19].On the other hand, VRP in most instances can be solved approximately using near-optimal algorithms.9, 20, 21

This paper focuses on dense thermal areas where an approximate solution is acceptable. The details of our routegeneration methodology are described in the next section.

IV. A Parallel Savings Based Heuristic as a Solution Methodology

The glider UAV is assigned to visit all thermal locations in the area of interest which might be viewed as a set-covering problem. An instance (N ,F) of a set-covering problem consists of a finite set N and a family F of subsetsof N , such that every element of N belongs to at least one subset in F as stated in [22] where one can write

N =⋃S∈F

S (24)

A minimum-size set cover can be considered a good match for the thermal soaring problem where N is the set ofassigned thermals on the map G(N ,A) . However, we aim to construct a flight route where each thermal location otherthan the start node is visited only once. We start by dividing the thermal map into nonoverlapping and equal-sizedpartitions unless we know in advance that a particular kind of partitioning leads to better solutions. In order to coverall nodes in these sectors, thermal locations are assigned to initial routes which are defined by applying a minimalspanning tree algorithm. The collection of the routes in all sectors forms a feasible solution for the thermal soaringproblem. The problem requirements are then relaxed, and the routes are connected to one another using a heuristicapproach. Finally, a near-optimal route is selected. The details of the procedure are described in the following fivesubsections:

A. Sectorization

The flight region is first partitioned into several sections as part of an initialization procedure. In our applications wedivide the thermal map into four areas which are East, West, North and South regions.

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It should be noted that the number of initial routes in a sector depends on how dense the thermals are and how theyare located relative to each other in that sector.

B. Kruskal’s Algorithm

Once the flight region is divided into sectors, a minimal spanning tree is found for each sector. A minimal spanningtree of a weighted graph is a collection of edges connecting all the vertices such that the sum of the weights of theedges is at least as small as the sum of the weights of any other collection of edges connecting all the vertices.23

It is explained in [23] that one of the approaches to finding the minimal spanning tree is simply to add edges one ata time, at each step using the shortest edge that does not form a cycle. The algorithm starts with an N -tree forest, andfor N steps it combines two trees (using the shortest edge possible) until there is just one tree left. This algorithm isgenerally attributed to J. Kruskal.

Kruskal’s algorithm computes the minimal spanning tree of a graph in O(ElogE) steps. For a discussion on therunning time comparisons of the three minimal spanning tree algorithms, namely the Priority-First Search, Kruskal’sAlgorithm and Prim’s Algorithm, the interested reader might refer to [23]. We implement Kruskal’s Algorithm inMATLAB and employ it for the required minimal spanning tree calculations in our simulations.

For simplicity we assume that the given map consists of a geographically dispersed set of thermals of the samestrength. This assumption results in a symmetric cost definition for the edges connecting the nodes on the thermal mapwhich then represents a symmetric weighted graph.

Thermals with different climb rates would in general result in flight arcs with asymmetric costs on the map. Suchan asymmetric weighted graph is likely to appear in real applications where the flight arc costs can still be definedusing the formulation given in (16). However, the solution algorithm developed in this paper would then require somefurther modifications.

C. Route Construction

The thermal locations in the spanning tree of each sector are examined. The nodes other than the ones connected totwo or more other nodes in the same sector are used to define new edges in that sector connecting them to node 0 .This step constructs a set of loops on the map which collectively visits all nodes.

The Route Construction step in our algorithm presents similarities with the petal heuristics24 in the sense that thereare possibly more than one route in any sector. However, we do not generate a fixed number of petals for each sector,and the number of routes rather depends on the structure of the spanning tree defined by Kruskal’s Algorithm appliedin the previous step.

D. Relaxation and Parallel Savings Based Heuristics

We apply a Relaxation such that the aircraft need not visit the thermal location at node 0 whereas i ∈ N −{ 0 }represents the nodes required to be included in the route of the UAV. We involve a parallel savings criterion in order tomodify and merge the constructed routes into a single flight path which in turn defines a near-optimal solution for thisparticular relaxed problem.

A survey of the classical and modern heuristics for VRP is given in [25]. The classical heuristics for the CapacitatedVRP are classified in [9] as Constructive Methods, Two-Phase Methods, and Improvement Heuristics. ConstructiveMethods are further classifed as Clarke and Wright Savings Algorithm,26 Matching-Based Savings Algorithms,27–29

and Sequential Insertion Heuristics.30, 31

We inspire from the Clarke and Wright Savings Algorithm and apply a heuristic method similar to its parallelversion. Furthermore, we also incorporate some additional but simple improvements to enhance the quality of theresulting solution.

A parallel and a sequential version of the Clarke and Wright Savings Algorithm are available. The parallel ver-sion of the algorithm, as discussed in [9], proposes the best feasible merge such that these savings are checked in a

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nonincreasing fashion, and given a saving sij , it can be determined whether there exist two routes, one containing anedge (0, j) and the other containing an edge (i, 0) that can feasibly be merged. These two routes are then combinedby deleting the edges (0, j) and (i, 0) , and adding the edge (i, j) .

If the two nodes i and j are connected, the saving sij can be computed as

sij = ci0 + c0j − cij , i 6= j , i , j ∈ S (25)

where S in our case is the set of nodes directly connected to node 0 upon completion of the Route Construction step.The nodes i and j in (25) are also implied to be nodes in different routes which are not initially connected other thanat node 0 .

In order to merge the initial routes using such a savings criterion, the edges (i, 0) and (0, j) are interchangedwith (i, j) , and the costs ci0 and c0j are saved no matter which set of new edges (i, j) the Parallel Savings BasedHeuristics step introduces. The performance of the savings algorithm is thus determined solely by the costs cij , wherei 6= j , i , j ∈ S , i and j are nodes in different routes. Our savings strategy reduces to pairing these nodes i andj such that the sum of the costs cij is minimum. It should be noted that the heuristic step in our algorithm is notnecessarily repeated for all nodes on the map, i.e., we do not calculate the savings for all node pairs i and j wherei , j ∈ N , i 6= j .

A further improvement in the algorithm can be straightforwardly achieved in case the solution for the relaxedproblem is self-intersecting. The intersecting edges determine the nodes which might possibly be reconnected inanother order resulting in a flight cost reduction. Therefore, these nodes along with the other nodes visited betweenthem are evaluated for all possible connections.

Such an improvement process can also be considered in order to concurrently reoptimize the routes produced inthe Route Construction phase. Although it is not a sophisticated neighborhood search rule, it can be implemented witha low calculation cost. Since it is basically a deeper exploration of the most promising regions of the solution space,the solution of the problem is likely to be improved.

E. Near-Optimal Route Selection

Node 0 is not included in the route determined by the Parallel Savings Based Heuristics due to the Relaxation phaseinvolved prior to applying the heuristic step. Feasible solutions to the original problem can be generated simply byinserting node 0 between any two nodes in the route determined using the Parallel Savings Based Heuristics.

Let j = ∆+(i) be the node that follows node i along the route proposed for the relaxed problem. The nodes i andj can be disconnected by deleting the edge (i, j) . If node 0 is then inserted between these two nodes, two new edges(i, 0) and (0, j) are added to the route. While guaranteeing the transformation of the solution for the relaxed problemto a feasible route for the original problem, this insertion produces an extra cost of eij which is given by

eij = c0i + cj0 − cij , j = ∆+(i) , i , j ∈ N , i 6= j , i 6= 0 , j 6= 0 (26)

The two nodes i and j are thus chosen to minimize the value of eij defined in (26). The resulting route is feasibleand is an approximate solution to the thermal soaring problem.

Since the thermal map under consideration is not a directed graph, visiting the same nodes in the reverse directionresults in the same overall cost, and is not considered as a different solution for the scenarios discussed in this paper.

The thermal map is initially divided into sectors with no a priori information on the solution of the problem. Thealgorithm therefore might not be able to reach the optimal solution and is in general an approximate method only.However, the paths in the sectors allocating thermal locations are eventually modified and merged so that the sectorsare not isolated. A near-optimal solution is reached which represents an upper bound on the cost of the optimalroute. Yet the proposed methodology is composed of a set of simple steps and is significantly efficient in terms ofcomputations involved.

Our solution algorithm might reach a set of solutions in the end if the overall costs determined for these routes areequally low.

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Figure 1. The thermal map where the direct flight distances between the thermal locations are shown so that the time costs of the corre-sponding flight arcs can be listed.

V. Simulations

A specific problem scenario is chosen to illustrate the performance of our approximate algorithm. A total of twentythermals are detected to be available for soaring within the flight range including the one located at node 0 where theglider UAV starts. The aircraft is assigned to visit all thermal locations and return back to node 0 in minimum possibletime.

The maximum flight arc distance is defined as lmax = 20, 000ft , and the thermals located further away from eachother are not connected by a direct arc on the map. Each thermal is assumed to have a strength of uth = 10 ft/secdefining the optimal horizontal flight velocity of the glider as Vopt = 204ft/sec. The problem scenario is summarizedin Figure 1.

Each flight arc on the map is associated with a cost using the analytical expression given in (16). The numericalvalues of the time costs are evaluated for all arcs in Figure 1 and listed in Table 1 where the calculated values, tcalc

are compared with the measurements from simulations, tmeas . The two are verified to match within an acceptableerror range. The error is vastly due to the polar curve approximation used in (16) to evaluate tcalc and also due to the

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transients observed in the simulations while tracking the desired trajectory which in turn affect the value of tmeas asdiscussed in [7]. The uncertainties in the UAV model used in our simulations possibly play an additional role.

Table 1. uth = 10 ft/sec , Vopt = 204 ft/sec

Distance tglide tthermal tcalc tmeas

(ft) (sec) (sec) (sec) (sec)4,000 19.6 33.9 53.5 49.95,000 24.5 42.3 66.8 61.76,000 29.4 50.8 80.2 71.77,000 34.3 59.3 93.6 85.68,000 39.2 67.7 106.9 98.59,000 44.1 76.2 120.3 109.6

10,000 49.0 84.7 133.7 125.811,000 53.9 93.1 147.0 138.212,000 58.8 101.6 160.4 148.2

Distance tglide tthermal tcalc tmeas

(ft) (sec) (sec) (sec) (sec)13,000 63.7 110.1 173.8 161.214,000 68.6 118.5 187.1 178.415,000 73.5 127.0 200.5 189.416,000 78.4 135.4 213.9 201.317,000 83.3 143.9 227.2 209.718,000 88.2 152.4 240.6 224.019,000 93.1 160.8 253.9 241.420,000 98.0 169.3 267.3 254.2

The flight region is divided into smaller partitions to initialize our solution algorithm, and the resulting sectors areshown in Figure 2. The minimal spanning trees are found for all four sectors separately and are demonstrated by singlesolid lines on the map. The double solid lines in the same figure represent the edges added to connect the end pointsof these spanning trees to node 0 . The solid single and double lines together display the initial flight routes.

A total of seven routes are determined where the North routes are 0−2−1−0 and 0−2−3−4−0 , the South routesare 0−10−11−9−0 and 0−12−16−15−0 , the East route is 0−7−6−5−8−0 , and the West routes are 0−17−19−18−0and 0−17−13−14−0 .

The East route 0−7−6−5−8−0 has two intersecting edges which are (7, 6) and (8, 0), whereas the same situationholds for the edges (17, 19) and (18, 0) of the West route 0−17−19−18−0 . All possible combinations for reconnectingthe set of nodes { 0, 5, 6, 7, 8 } for the East route and { 0, 17, 18, 19 } for the West route are examined. As a result, theEast route is improved into 0−6−5−8−7−0 which provides the most savings whereas the West route is shown to bean optimal connection of the nodes involved and hence kept unchanged. The Route Construction step is completed.

In the Relaxation step the edges connecting the nodes { 1, 2, 4, 6, 7, 9, 10, 12, 14, 15, 17, 18 } to node 0 are deleted.Parallel savings are considered, and the edges (4, 6), (7, 10), (9, 15), (12, 14) and (1, 18) are added. Note that the tworoutes in the North remain connected at node 2 although the edge (0, 2) is deleted. The same holds for the two routesin the West at node 17 . The two edges (10, 11) and (9, 15) can be shown to be intersecting. After relevant calculationsthe set of nodes { 9, 10, 11, 15 } are reconnected as 10−9−11−15 .

In the Near-Optimal Route Selection step two possible solutions are determined to be S1 and S2 where

S1 : 0−8−5−6−4−3−2−1−18−19−17−13−14−12−16−15−11−9−10−7−0S2 : 0−7−8−5−6−4−3−2−1−18−19−17−13−14−12−16−15−11−9−10−0

The solution S1 is demonstrated in Figure 3. For this specific route the overall flight time is calculated as tcalc =2218.9 sec , whereas the total cost is measured to be tmeas = 2044.1 sec through simulations with the uncertain UAVmodel and the adaptive control design.

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Figure 2. The solution algorithm is iterated on the thermal map.

VI. Conclusion

One of the very basic skills employed by large migrant birds is soaring flight from which human-piloted aircraftand UAVs can also benefit. In an attempt to make the most use of rising air currents, and hence maximize range,endurance and cross-country speed of gliders, corresponding optimal flight trajectories are required to be generated.This process might become quite complicated if the UAV is assigned to soar a geographically dispersed and dense setof thermals located over the flight region.

We propose an approximate algorithm based on the VRP interpretation of the thermal soaring problem for anautonomous UAV. We consider problem scenarios where an exact solution algorithm appears to be demanding in com-putation time. Assuming that a near-optimal solution is acceptable, a methodology involving Sectorization, Kruskal’sAlgorithm and Parallel Savings Heuristics is developed. The simulation results confirm the fact that the physical re-quirements and the flight constraints of soaring UAVs can be properly translated into mathematical language whichresults in significant performance improvements for the aircraft if efficient algorithms can be developed to define theflight path.

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Figure 3. A near-optimal flight path is reached using the proposed solution methodology.

Similar heuristic methods can also be useful for powered UAVs in search flight. Our methodology is promising forsurveillance purposes where energy requirements and onboard storage limitations in size and weight of battery/electricsources inevitably determine the flight performance of the aircraft.

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