Constructing Conflict-Free Schedules in Space and Time

Constructing Conflict-Free Schedules in Space and Time

David W. Hildum and Stephen F. SmithIntelligent Coordination and Logistics Laboratory

The Robotics InstituteCarnegie Mellon UniversityPittsburgh PA 15213-3890{hildum,sfs}@cs.cmu.edu

Abstract

This paper addresses the problem of constructing plans andschedules for resources that must obey spatial constraints inaddition to time and capacity constraints. Spatial constraintsare relevant in environments that involve mobile resourceswhose movements must be coordinated to avoid collisionsand near-misses. In air-campaign planning, for example, air-craft are allocated to missions that must be flown concurrentlywithin a localized and often heavily populated environment.Mission routes tend to be generated dynamically and mustensure that sufficient spatial separation is maintained at alltimes among all aircraft.We present a solution to this class of problems that treats thestandard resource-allocation problem as a four-dimensionalone, where the space in which resources must maneuver isitself managed as a capacitated resource. Underlying our ap-proach is a representation of spatial capacity that uses a lin-ear octree structure for indexing vector-based vehicle routes.Generalizing the notion of contention-based search heuris-tics, we present an algorithm that first solves a relaxed versionof the problem to construct a spatial capacity profile (repre-sented as an octree), and then uses spatio-temporal regionswhere demand exceeds capacity to make conflict-avoidingvehicle routing and scheduling decisions. To demonstrate theviability of the approach we present experimental results us-ing data from a realistically sized air-campaign planning do-main.

Introduction

In both civilian and military settings, effective managementof airspace is an important and difficult problem. Air traf-fic controllers must cope with increasing volume and con-gestion around commercial airports when determining cor-ridors and sequences for landing, takeoff and holding. Aircampaign planners must synchronize large numbers of con-current missions within a limited theater airspace, to be exe-cuted using diverse and increasingly autonomous sets of airvehicles. The key challenge in both cases is to ensure appro-priate separation between all vehicles at all times, and to doso without significantly degrading the efficiency of individ-ual movements. Viewed another way, this challenge can beseen as the problem of efficiently allocating shared airspaceto movements over time.Copyright c© 2007, Association for the Advancement of ArtificialIntelligence (www.aaai.org). All rights reserved.

In this paper, we take the view of airspace deconflictionas an extended resource-allocation problem, and considerthe problem of planning and scheduling vehicle movementsto satisfy spatial constraints in additon to time and capacityconstraints. As a first step we propose an efficient, scalablerepresentation of spatial constraints over time based on a lin-ear octree. We then define an extended planning/schedulingalgorithm that utilizes this representation to construct aconflict-free schedule for accomplishing a given set ofmovements with a given set of vehicles. The extended al-gorithm assumes the existence of a base algorithm for com-puting a resource-feasible solution to the problem, whichis used to compute “spatial capacity” profiles and identifysubspaces of likely spatio-temporal contention. Similar toother contention-based scheduling strategies (Sadeh 1991;Smith 1994; Beck 1999; Cesta, Oddi, & Smith 2002), thisprofile is used to direct the search for conflict-resolvingchanges to movement itineraries and/or timing constraints.

Previous research in airspace deconfliction has focusedprimarily on the local problem faced by a particular move-ment activity (Stilman 2000) or on dynamic real-time syn-chronization of multiple movements (OR Concepts Applied2002; Sridhar et al. 2002; Coppenbarger et al. 2004;Sridhar et al. 2005). In both contexts, the principal goalis simply to make local planning and scheduling decisionsthat avoid airspace conflicts. The advantage of consider-ing spatial constraints up front during more global planningand scheduling of a set of movements that must occupy acommon airspace over a given time interval is that it pro-vides the additional opportunity to optimize overall trafficflow by making efficient use of available airspace. Otherrecent work has developed alternative techniques for repre-senting and reasoning about two-dimensional spatial con-straints for purposes of detecting potential conflicts amongsurface movements that overlap in space and time (Yaman,Nau, & Subrahmanian 2005; Parker et al. 2007). However,the goal of this work is to provide deductive machinery foranswering queries relative to the space-time coordinates ofknown movements. To our knowledge, this paper is the firstto address the problem of constructing a global movementplan and schedule that is conflict-free in 3D space and timeand scalable to real-world problems.

The remainder of this paper is organized as follows. Wefirst describe our linear octree approach to representing

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airspace capacity constriants over time. Next we presentan algorithm for constructing conflict-free movement sched-ules. To demonstrate the viability of our approach, we thendescribe results obtained by coupling our algorithm witha previously developed system for air campaign schedul-ing, and using it to generate realistically sized air campaignschedules. We conclude with a discussion of the limitationsof our current model and our current research plans.

Space as a Capacitated Resource

The process of searching for available resource capacityplays an integral part in the scheduling process. As a re-sult, it is extremely important for a capacity representationscheme to enable rapid access to its contents in response toqueries from the scheduler, even as its contents are beingcontinually modified through the allocation of resource ca-pacity. The concern with representing space as a capacitatedresource is that the cost of supporting recurrent and exten-sive multidimensional capacity queries—on top of the usualqueries for resource and related capacity—is potentially pro-hibitive.

As it turns out, one data structure is naturally suited forthis problem. The octree is a hierarchical, three-dimensionalspatial data structure that recursively divides a spatial region(or volume) into smaller subspaces for storing objects ac-cording to [x,y,z] coordinates. The octree is an extensionof the two-dimensional quadtree, which was developed pri-marily for use in image-processing applications, where thelocalization of similarly shaded pixels in an image helps tominimize its storage requirements. Hierarchical data struc-tures rely on the concept of recursive decomposition to re-peatedly divide large heterogenous spaces into smaller andmore manageable homogenous subspaces.1

The use of hierarchical structures for representing multi-dimensional data has been employed across a broad rangeof problem domains, including image processing, computergraphics, geographical information systems and robotics(e.g., for collision detection). The octree is a natural datastructure for representing spatial or geographic data, typi-cally characterized by [latitude, longitude, altitude] triplets.An octree decomposes a three-dimensional volume by recur-sively halving it along each of its three coordinate axes un-til each subspace (called an octant) achieves some domain-specific condition (e.g., commonality among, or thresholdof objects) or cannot be further subdivided (i.e., owing toits having reached a minimal size). In our case, an octantis only subdivided when a conflict between two or more ve-hicle routes is introduced, in which case the area where theconflict occurs will be localized into a single octant or a clus-ter of colocated octants, to enable easy access to that partic-ular region.

To manage space as a capacitated resource using an oc-tree, we map the 3D environment through which vehicles

1See (Samet 1984) and (Guting 1994) for general surveys ofmultidimensional database access methods focusing on the use ofquadtrees, and (Samet 1990) and (Gaede & Gunther 1998) for moregeneral discussions of multidimensional database access.

will be traveling into a single large octant (the initial con-figuration of the octree). An octant designates a 3D volumewhose contents are the vehicle route segments that intersectits area at any point in time. As routes are planned and al-located, they are inserted into the octree to reflect their ex-clusive right to a portion of spatial capacity for a particularperiod of time. Capacity can only be allocated to a vehicleif there are no other vehicles traveling along the same routeor intersecting it at the same time. That is, they must bespatio-temporally independent of one another, or conflict-free, along their entire routes. Sufficient capacity for a ve-hicle route is dependent on there being a 4D path throughspace and time that does not conflict with any other vehi-cle routes. To find capacity for a vehicle, the scheduler mustquery what may be a sequence of octants intersected by eachsegment of its route to ensure that sufficient spatio-temporalcapacity is available for the duration of the route.

In the remainder of this section, we address a number ofissues relevant to the use of octrees for managing spatial ca-pacity. We begin with a discussion of our use of the linearoctree, which delivers additional efficiency in storage andaccess over the standard octree structure. We then discuss ingreater detail, the process by which vehicle routes are rep-resented and stored in octants. Finally, we describe some ofthe key issues involved in detecting conflicts between vehi-cle routes within an octant.

Linear Octrees

In a tree representation of an octree, each octant (except forthose at the leaf level), has eight children (i.e., 2n, where nis the number of dimensions being represented). Its overallstructure can therefore become quite large and unwieldy asthe number of octants increases, and the process of search-ing for a particular octant within the tree can become morecostly. For our purposes, the linear octree (Gargantini 1982)provides a much more efficient solution. The idea behind alinear octree is to assign each octant in the octree a uniquekey derived from its 3D location (i.e., its origin) and thenrepresent the octree as a balanced binary tree. This approachrequires significantly fewer pointers, and the use of the keysfacilitates rapid access to the octants. Given a key, its cor-responding octant node can be located quickly (in O(log N)time, where N is the number of octants) using a fast binarysearch instead of by traversing the possibly extensive pathswithin a regular octree. The keys are generated by translat-ing points in 3D space into binary x, y & z coordinates andthen interleaving the resulting bits to derive an integer value.These keys impose an ordering on the octants by defining apath for traversing the entire space represented by the octree.

Vector-based Vehicle Routes

To represent the continuous movement of a vehicle as it tra-verses its route, we use a contiguous sequence of 4D vectors,�vi:[Δx,Δy,Δz,Δt], that link together a sequence of [x,y,z,t]waypoints. Inside the octree, each vector �vi may intersecta spatially connected sequence of octants based on thesewaypoints and the vectors linking them. Note that time isnot currently represented explicitly in the octree as a sepa-

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rate dimension.2 Instead, the temporal extent of each vectorsegment that intersects an octant is recorded along with thevector inside the octant. Each octant can be thought of asa bucket containing annotated pointers to all of the vectorsthat intersect it, with each annotation indicating the time atwhich the vector enters and exits the octant.

Figure 1 illustrates how vectors are stored in an octree ac-cording to interpolated [x,y,z] coordinates and temporal ex-tent, and how spatio-temporal conflicts between intersectingvectors are identified and localized in an octant. The left-

O11

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Figure 1: Octant division in response to a conflict betweentwo vectors�v1 and �v2: before (l) and after (r); empty octantsare not shown

hand side of the figure shows two intersecting vectors (�v1and �v2) and a third independent vector (�v3). Suppose thetemporal extent of the three vectors (within the octree) is asfollows: �v1:[1..5], �v2:[4..8] and �v3:[3..7]. Recall that two ormore vectors can reside in the same octant if they occupydistinct intervals of time and space. A conflict occurs—indicating that capacity in a region is exceeded—when twoor more vectors spatio-temporally intersect. In the figure,note that �v1 and �v2 intersect at an [x,y,z] coordinate in thefront-lower-left corner of octant O0 at time 4.5. If the tem-poral extent of �v2 was instead [1..5] (i.e., the same as �v1),then no conflict would occur, since the two vehicles travel-ing on those vectors are not at the same [x,y,z] coordinate atany point in time.

In response to the introduction of the conflict between �v1and �v2, the octree will be subdivided twice, until (in thisexample), the octant containing the conflict reaches its min-imal size, which is ideally set to be large enough to con-tain the conflict itself, but not much more than that. In thesubdivision process, octant O0 is first replaced by its eightchildren: octants O1..O8. Octant O1 (in the front-lower-left corner) will subsequently be subdivided and replacedwith octants O9..O16, leaving the conflict localized in oc-tant O9, where its contents can now be rapidly retrieved forthe purposes of deconfliction. The resulting octree is shownon the right-hand side of Figure 1, with the vectors appor-tioned to octants as follows: �v1:[1..3] to O3, �v1:[3..4] toO11 and �v1:[4..5] to O9, �v2:[4..5] to O9, �v2:[5..6] to O10and �v2:[6..8] to O2, and finally �v3:[3..5] to O6 and �v3:[5..7]

2Time could, however, be represented explicitly using a multi-dimensional, or hyper octree.

to O8. As long as conflicts do not occur, there is no limit onthe number of vectors that can travel through an octant.

Assume that �v1 was inserted into the octree first. If theconflict between �v1 and �v2 had been detected before �v2 wasinserted, through a query of the octree to determine whetherthere was sufficient spatio-temporal capacity for �v2’s route(i.e., in the original octant O0) signified by the absence ofany conflicts with other vectors, then the octant subdivisioncould have been avoided by either leaving �v2 out of the oc-tree or modifying it spatially or temporally to resolve theconflict.

Conflict Detection

The linear octree representation facilitates the rapid retrievalof spatial capacity buckets corresponding to [x,y,z] coordi-nate locations. Given a route vector starting at waypointwi:[xi, yi, zi, ti] and ending at waypoint wk:[xk, yk, zk, tk],the process of determining whether there is sufficientcapacity for that vector begins by accessing the octantthat contains its [xi, yi, zi] origin, interpolating the pointwj:[xj, yj, zj, tj] at which it exits from the octant (if it doesso), and then looking for spatio-temporal conflicts betweenthe vector segment connecting wi and wj with any of thevector segments currently occupying that octant. This pro-cess continues with the remaining portion of the vector thatextends beyond the octant (i.e., from wj to wk), and all sub-sequent vectors comprising a vehicle’s route.

Determining whether a spatio-temporal conflict exists be-tween two vectors involves more than just checking to seewhether they intersect in four dimensions. A conflict condi-tion between two vehicles actually occurs during the periodof time when they are merely traveling too close to one an-other, because each vehicle has its own separation buffer(or maneuver avoidance zone (MAZ): see (Ennis & Zhao2004)) that designates a spatial region centered on the vehi-cle, with which no other vehicle—and by extension, its ownseparation buffer—may overlap. Separation buffers help ac-count for the uncertainty inherent in the movement of vehi-cles over time: given the individual characteristics of a ve-hicle, its true location at any point in time may be offset tosome extent from its intended [x,y,z] position. Exclusion ofother vehicles from the separation buffer zone ensures thateach vehicle can travel safely within its own protective spa-tial envelope.

Detecting the overlap of vehicle separation buffers beginswith the calculation of the closest point of approach (CPA)between their corresponding route vectors. Assuming that avehicle travels in a straight line at a constant speed along avector, a CPA calculation can identify both the exact pointin time when two vehicles, traveling along two different vec-tors at different speeds and times, are closest to one another,and what exactly that distance is. From there, it is possibleto determine the temporal extent of any conflict. This con-cept is illustrated in Figure 2. Two aircraft are set to crosspaths. Their closest point of approach is described by thevector (“CPA vector”), occurring at time tCPA. Using thisvector (when there is a conflict), it is also possible to deter-mine the points at which the separation buffers of the two

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CPAvector

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Figure 2: Determining the overlap of vehicle MAZes usinga closest point of approach calculation

aircraft are just about to overlap and just about to separate,which occurs some offset unit of time (i, in the figure) beforeand after tCPA. The temporal extent of the entire conflict istherefore [(tCPA − i)..(tCPA + i)].

Search Among Neighboring Octants

Conflict detection is performed whenever a vector is consid-ered for insertion into an octant, at which point its potentialinteraction with any vectors currently residing in the octantmust be evaluated. An important issue when working withseparation buffers—instead of simple vectors—is the needto support conflict detection across octant boundaries, sinceoverlap can occur between vehicles in adjoining, or neigh-boring octants. As a result, we have extended the conflict-detection mechanism to consider the vectors in neighboringoctants as additional candidates for overlap whenever a vehi-cle is close enough to an octant boundary that its MAZ couldintersect with the MAZ of a vehicle in an adjoining octant.Conveniently, the linear octree structure facilitates rapid ac-cess to neighboring octants (through the manipulation of oc-tant location codes), so access to such vehicle vectors is eas-ily achieved, thereby enabling the conflict-detection processto consider all potential candidates for interference.

A Conflict-Free Scheduling Approach

Building on the representational techniques described in theprevious section, we have developed an algorithm for gen-erating conflict-free schedules in space and time. The al-gorithm accepts as input, a set of resource-feasible vehiclemissions that guarantee sufficient resource capacity over afixed time period. It extends the solution by allocating thenecessary spatial capacity to those missions. This is accom-plished by a route-planning component that considers thecontention that exists throughout the overall spatio-temporal

region, assessed from the distribution of missions within thelinear octree.

Previous research with resource-constrained problems hasemphasized the utility of access to resource contentionprofiles to inform look-ahead analysis techniques in thescheduling process (Sadeh 1991; Smith 1994; Beck 1999;Cesta, Oddi, & Smith 2002). As it turns out, the linear octreemechanism facilitates the generation of a similar assessmentof the spatial resource requirements for a particular problem.Our algorithm utilizes an initial priming scheduling run per-formed ahead of the main scheduling phase, to populate anoctree with all expected missions and identify all anticipatedspatio-temporal areas of contention. This resource capacityprofile is then used to guide the construction of a spatiallyconflict-free schedule during the second primary schedulingrun.

During the first phase of the algorithm, the primingrun generates a schedule that ignores spatial capacity con-straints. Routes are built without any consideration for theirinteraction with one another, and in the process, a primedoctree is populated, which localizes and highlights antici-pated areas of contention for the entire problem. For anygiven mission, the route planner takes its origin and desti-nation location and generates a sequence of five contiguousflight vectors (i.e., ascent, approach, fly-by, return, and de-scent) to get an aircraft from the origin to the destinationand back again, as shown in Figure 3. Note that this methodgenerates very simple air routes: the expectation is that amore sophisticated technique could eventually be employedfor this process.

[2] Approach leg

[1] Ascent leg

[3] Fly-by leg[4] Return leg

[5] Descent leg

Figure 3: The five component legs of a basic aircraft missionroute from origin to destination and back

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Consider the 3D plot of 1000 air missions shown in Fig-ure 4. These missions are sourced from a group of ten air-ports to reach destinations in the center of the environment.The octants of the primed octree that contains these missionsare shown in Figure 5. The conflicted octants (small, red or

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Figure 4: Flight paths for 1000 aircraft missions; colors areassigned randomly to distinguish between missions

lightly shaded) of Figure 5, which indicate the specific areasof contention, are isolated and shown in Figure 6. Blue/gray(darker shaded) octants designate areas that contain one ormore fully deconflicted air routes. Octants without any traf-fic are not shown. All conflicts in the red (small, lightlyshaded) octants must be resolved to achieve a fully decon-flicted solution. The information contained in these octantsis available during the subsequent primary scheduling phaseto inform the route-planning component in the course of re-solving conflicted missions.

During the second and primary scheduling phase, the ac-tual conflict-free schedule is built, and in the process, spa-tial capacity constraints are given the full attention of thescheduling algorithm. The route planner consults the savedprimed octree—now serving as an oracle—and modifiesroutes as necessary to avoid the conflicts with other vehiclesthat originally appeared in the primed octree. The route-modifying component attempts to introduce a horizontallydisplaced passing lane around each conflict, running paral-lel to the original vector and maintaining its original attitudetrajectory. More intuitively, the route modifier tries to shiftthe conflicted portion of a route away from a conflict regionwithin a conflicted octant, or from one conflicted octant to aless-conflicted neighboring octant. The efficiency of the lin-ear octree representation again plays an important role here,enabling rapid access to the conflicted regions traversed by

longitude

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longitude

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Figure 5: Octants in a primed octree resulting from a prim-ing scheduling run for 1000 aircraft missions; red (small,lightly shaded) octants indicate airspace regions containingvehicle conflicts, blue/gray (darker shaded) octants indicateairspace regions free of any conflicts; empty octants are notshown

the route as well as the various neighboring octants to whichconflicted traffic might be diverted.

Note that no attempt is currently made to enforce any ofthe real-world constraints that govern the maneuverability ofa vehicle being routed in this manner. Refinement of theselarge-grained route vectors could be performed later as aniterative post-processing step. Alternatively, the route mod-ifier itself could be enhanced to generate more fine-grainedalterations.

The final octree produced by the algorithm is illustrated inFigure 7. The absence of conflicted octants confirms that alloriginal conflicts from the priming run have been resolvedthrough modifications to their routes.

The Air-Campaign Scheduling Problem

In this section, we present the details of the air-campaignscheduling problem, to which we have applied our conflict-free scheduling algorithm. The general air-campaignscheduling problem involves the allocation of aircraft andmunitions to air strike missions over time in a battlefield do-main. The problem has two broad types of inputs:

1. A set of targets for which individual air strike missions(sorties) must be scheduled. Each target has an associatedtarget type and location. For our purposes, we consider alltargets to have equal priority, and assume that each sortiecan destroy a single target. Each target also has a specifiedtime-on-target window during which it must be struck.

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longitude

latitude

Figure 6: Conflicted octants from the plot in Figure 5

2. A description of resource capabilities, specifying(1) what types and amounts of resource capacity (air-craft, munitions, runways) are available for use, (2) wheresaid resources are positioned in theater and (3) a ta-ble of weaponeering solutions that maps feasible air-craft/munitions pairs to target categories.

The desired output is a schedule indicating the set of sortiesto be flown (i.e., which aircraft and munitions will be used tostrike each target and at what time). The precise routes to beflown by each sortie will be determined by our conflict-freescheduling algorithm.

Experimental Results

To evaluate the performance of our algorithm, we haveconstructed a series of 20 data sets containing from 50 to1000 randomly generated targets (in 50-target increments)spread across two roughly 700-miles-square adjoining re-gions, bracketed by ten bases from which missions can besourced. Each base is equipped with a complete cross-section and supply of the usable aircraft and munitions(and two runways), and each target is classified by one of58 target categories that maps it to a set of feasible air-craft/munition pairs. All targets must be struck within atwo-day time window. Figure 8 illustrates the largest, 1000-target data set and the common base locations for the entireseries.

In our evaluation, we consider two ways to fold spatialconstraints into the scheduling process, both of which uti-lize the linear octree representation. As part of this process,we have integrated our scheduling algorithm with an exist-ing scheduling engine (Myers et al. 2001) to extend the pro-cess of searching for resource capacity to include airspace in

longitude

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Figure 7: Octants resulting from the primary schedulingphase for 1000 aircraft missions; the conflicts in all previ-ously conflicted octants have been resolved through routemodifications

addition to aircraft, munitions and runways. In the analysisbelow, we compare the performance of our profile-based al-gorithm against a baseline approach that minimally extendsthe algorithm implemented by the underlying scheduling en-gine. These two approaches behave as follows:

• Baseline:As implemented by the underlying scheduling engine, theprocess of scheduling a sortie is a multi-tiered searchprocess that at its highest level considers all feasibleweaponeering solutions for the target, and then for eachweaponeering solution, all bases equipped with its spec-ified aircraft/munitions pairs. For each weaponeering-solution/base pair, the attempt is made to secure the nec-essary aircraft, munitions and runway capacity. The base-line extension to this process adds a call to the route plan-ner, which attempts to build a conflict-free route indicat-ing sufficient spatial capacity. If conflicts between thenew route and any scheduled routes are detected in theoctree, a single attempt will be made to deconflict it. Ifthat attempt fails, the route planner will signal a failureand the scheduling engine will continue its search look-ing for available aircraft, munitions and runway capacityat the current base, but farther downstream. If, however,a conflict-free route is obtained, it is returned and the sor-tie is scheduled. When the timeline for a weaponeering-solution/base pair is exhausted, another feasible base isexplored. When all bases for the weaponeering solutionare exhausted, another feasible weaponeering solution isexplored. If this process fails to secure the necessary air-

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Figure 8: Evaluation data series: target and base layout(1000 targets, 10 bases)

craft, munitions, runway and spatial capacity, the sortie ismarked as unschedulable.

• Profile-based:In our profile-based approach, the underlying schedul-ing engine behaves nearly identically to the baseline ap-proach, except that the scheduler is run an additional timeas a preprocessing step to populate the primed octree.During the priming phase, as described earlier, the routeplanner builds routes as usual, but does not check themfor conflicts. Instead, it simply returns the first route itbuilds for a sortie, and the sortie is scheduled and as-signed that route. During the primary scheduling phase,however, the route planner, when searching for airspacecapacity, checks for conflicts by searching through theprimed octree, which maintains an up-to-date summaryof the airspace contention profile that reflects the entire setof sorties, both scheduled and unscheduled. If a conflict-free route is obtained (possibly the one built during thepriming phase), it is returned to the scheduler. If thereare conflicts that can be resolved through route modifica-tion, the modified route is returned. If a conflict-free routecannot be obtained, the route planner will again signal afailure, and the search process of the underlying schedul-ing engine will explore other feasible options (just as inthe baseline case).

The performance results in Figure 9 show the amount ofprocessing time required by each approach in scheduling the20 random data sets. Note that both approaches are able toproduce conflict-free schedules that include all of the mis-sions in each of the data sets.

These results emphasize an important point, which is thatthe additional overhead introduced by generating, maintain-ing and consulting a global spatial-contention profile (i.e.,the primed octree populated with both scheduled and un-scheduled sortie routes), is clearly compensated for by anotable improvement in overall scheduling performance forruns of more than a minute in duration (e.g., in excess of

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Figure 9: Comparison of processing times for Baseline andProfile-based scheduling approaches

500 missions). The ability of our profile-based approachto exploit the primed octree to resolve conflicts with bothscheduled and unscheduled sorties earlier in the schedulingprocess makes it less likely that the eventual attempts to re-solve them will be postponed until there are fewer resolutionoptions available, and the cost of finding alternate solutionsis greater.

Summary

In this paper, we have considered the problem of construct-ing a movement plan for multiple vehicles that is conflict-free in space and time. By integrating spatial constraints di-rectly into the planning and scheduling process, rather thanconsidering airspace deconfliction after the fact from a local,tactical perspective, it is possible to anticipate likely areas ofcongestion and exploit opportunities to optimize traffic flow.Prior research has approached airspace deconfliction from alocal, tactical perspective, and hence has focused rather ex-clusively on the more immediate problem of conflict avoid-ance.

Central to any approach to solving this extended resource-allocation problem is a convenient, scalable basis for de-tecting airspace conflicts. To this end, we introduced a lin-ear octree representation of available airspace capacity. Wethen used this octree representation to define a contention-driven scheduling procedure, wherein advance computationof a spatial capacity profile is used to direct the schedulegeneration process. Experiments performed using realis-tically sized problems from an air campaign planning do-main showed this procedure capable of efficiently gener-ating large-scale, conflict-free schedules, and for problemsover a certain size, to outperform a simpler baseline proce-dure that instead utilized an octree representation withoutany look-ahead analysis.

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Future Work

Our immediate efforts are focused on three general researchdirections. First is the expansion of our spatial constraintmodel to encompass more diverse requirements. This in-cludes adding support for:

• a wider range of vehicle types with different MAZ vol-umes (we currently distinguish between fixed-wing androtor aircraft types only) and different performance char-acteristics (e.g., uninhabited aerial vehicles (UAVs))

• weapons and other environmental threats (e.g., missiles,radar) that exhibit different spatio-temporal footprints,and may pop-up unexpectedly

• special terrain features that constrain the route-planningand route-modification processes, either through inclu-sion (e.g., established or designated traffic corridors) orexclusion (e.g., high-threat, restricted or no-fly zones)

Secondly, we are working to enhance our route-buildingand modification techniques to reflect more sophisticated,real-world constraints on the maneuverability of mobile ve-hicles. Fuel availability strictly defines the range of feasible(re)routing options, while performance characteristics andother relevant flight variables limit the degree to which avehicle’s speed and trajectory can be altered in-flight. Ad-ditionally, we are exploring the opportunity to tailor routemodification strategies to specific conflict scenarios, such aswhen one aircraft overtakes another along the same vectoror when two aircraft are approaching one another head-on.

Finally, we are working to provide capabilities to supportdownstream projection and analysis of the forward conse-quences to local resolution actions that are taken in real-time, and in response to projected problems, generate possi-ble corrective actions (e.g., alternative routes and/or missionexecution windows).

Acknowledgments

The work reported in this paper has been supported in partby the Boeing Company under contract CMU-BA-GTA-1-BOEING and the CMU Robotics Institute.

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