A Progressive Motion-Planning Algorithm and Traffic FlowAnalysis for High-Density 2D TrafficYanchao Liua
aDepartment of Industrial and Systems Engineering, Wayne State University, Detroit, Michigan 48201Contact: [email protected], http://orcid.org/0000-0002-3256-9336 (YL)
Received: May 8, 2018Revised: October 1, 2018; December 9, 2018Accepted: December 19, 2018Published Online in Articles in Advance:May 30, 2019
Abstract. Unmanned aircraft systems (UASs) are a promising newmode of transportation forcargo delivery. Although control, navigation, and communication technologies are becomingavailable on individual flight units, system-level motion management for dense UAS trafficremains an open question. This paper presents a motion-planning model based on nonlinearoptimization techniques to centrally coordinate paths for all vehicles traversing a shared two-dimensional (2D) space. An exact bound is derived to characterize the discrete-time separationconstraints, and a set of new metrics is proposed to measure 2D traffic flow efficiency. Thegrand goal is to make all vehicles that come under dispatch reach their respective destinationsquickly and efficiently while maintaining a safe intervehicle separation at all times. Thisinfinite-horizon operational problem is formulated as a nonlinear nonconvex optimizationmodel thatmust be solved in a progressive, receding-horizon fashion. To ensure feasibility andovercome path deadlocks attributed to local optima, a series of heuristic measures are de-veloped. By pivoting on artfully coined intermediate feasibility, the algorithm is able tocircumvent imminent deadlocks in a predictive manner and progressively construct the so-lution with guaranteed feasibility. Simulation experiments are performed at various trafficdensity levels to generate useful insights for airspace regulators and traffic managers.
Funding: The work was supported by the Wayne State University Faculty Start-up Fund. No externalfunding was received.
Supplemental Material:Online companion data and the online appendices are available at https://doi.org/10.1287/trsc.2019.0903.
1. IntroductionAs a major social function, transportation accountsfor 8.7% of U.S. gross domestic product (Bureau ofTransportation Statistics 2017). It also takes up a lotof precious space, which could alternatively be usedfor farming, mining, housing, providing room formanufacturing, entertainment, and other activities.Nowadays, the growing demand for mobility chal-lenges the traditional modes and systems of trans-portation. People expect to reach places faster andin more comfortable means, and expect goods andservices to be delivered within a short period of timeafter the need arises. Meanwhile, the rapid devel-opment of new types of vehicles, such as autonomousvehicles and unmanned aerial vehicles (UAVs; ordrones), provides great opportunities for a majorrevamp in our transportation infrastructure. There isan impending call for new systems to provide safe,flexible, and space-efficient means of transportation.
For any service system, increasing the capacity with-out expanding its physical size means elevating theservice density, that is, packing more value-addingoperations into a limited space. High density has beena major point of innovation in areas such as material
science, manufacturing, communication, agriculture,and service systems that has created tremendous so-cial benefits. The modern computer central processingunits (CPUs), high-capacity cell phone batteries, high-speed wireless networks, and even high-rise build-ings are all examples of high-density artifacts that packmore value in less space. Thanks to the rapid ad-vancements in control, wireless communication, andartificial intelligence (AI) technologies, high-densitytransportation is also becoming a reality. Various newconcepts andpilot systemshave emerged in recent years,including automated highway systems in which vehi-cles travel in platoons of minimal intervehicle distanceto increase the overall road capacity (Bergenhem et al.2012; Turri, Besselink, and Johansson 2017), hyper-loop systems that shoot transporting pods at high fre-quency and high speed along a vacuum tube connectingtwo metropolitan centers (Musk 2013, Jenkins 2017),andflying car technology (Hetzner 2018) thatmultipleautomakers around theworld are actively exploring as away to escape congested roads in megacities.Shifting more transport activities into low-altitude
airspace is an attractive option given today’s techno-logical readiness. Inparticular, usingUAVs for inner-city
cargo delivery will provide a promising solution to thelast-mile shipment problem that is costly on the ground.Battery-powered delivery drones are quiet, fast, andclean, and unlike ground-based vehicles, they are notslowed by and do not contribute to traffic congestionon roads, allowing for more efficient operations. En-abling technologies including power, control, sensing,navigation, and communication subsystems are welldeveloped, and several prototype aerial delivery sys-tems have already been field tested. For instance,Amazon (2018) has tested its Prime Air program, asmall drone system that can deliver packages up to fivepounds, in England. In 2016, 7-Eleven completed a fullyautonomous drone delivery test in Reno, Nevada, thefirst such test in the U.S. airspace approved by theFederal Aviation Administration (Glaser 2016). InChina, JD.com launched a trial program for dronedelivery services, testing air drop-offs in Beijing andseveral provinces (Parmar 2016). Similar develop-ments and tests are underway at Google’s,Walmart’s,and NASA’s research arms (Barr and Bensinger 2014,Malcolm and Weise 2015, National Aeronautics andSpace Administration 2015).
Given the limited carrying capacity of individualvehicles, the benefit of aerial delivery systems willmaterialize only when a large number of vehicles isdeployed. The capacity for flight for individual unitsis necessary, but far from sufficient for establish-ing a practical aerial delivery system in which fleetsof drones simultaneously traverse a shared three-dimensional (3D) space in all possible directions andat various speeds. Traffic rules and protocols alongwith the corresponding enforcement, monitoring, andcontingency response mechanisms must be set up,understood, and agreed upon by all parties involvedbefore any deployment at scale is possible. Figure 1illustrates the complexity of traffic flow in an openspace. In a dynamic two-dimensional (2D) trafficscenario, as shown in the figure, vehicles’ on-boardAIalone, that is, the ability to sense and avoid obstacles,is insufficient to support a safe and efficient trafficorder. The traffic flow will be chaotic, inefficient, andconflict prone when each vehicle determines its ownmotion without coordination. Because of the high di-mension, high density, and autonomy of this mode oftransportation, neither the roadnetworkmodels nor thetraditional air traffic control procedures can directlyapply in this space. New research is needed.
In this paper, we address the problem of routing alarge fleet of vehicles in highly dense and congestivetraffic. The grand goal is to make all vehicles thatcome under dispatch reach their respective destina-tions quickly and efficiently while maintaining asafe intervehicle separation at all times. This infinite-horizon operational problem is formulated as a non-linear nonconvex optimization model that must be
solved in a progressive, receding-horizon fashion.To ensure feasibility and overcome path deadlocksattributed to local optima, a series of heuristic mea-sures is developed and validated via computationalexperiments. The target application includes, but is notlimited to, an aerial delivery system in which het-erogeneous drones, each taking off at a random timeand having arbitrary destinations, traverse an urbansky at a given altitude. A set of new metrics is alsoproposed to characterize the 2D traffic flow and spaceefficiency. As an integral part of an intelligent dis-patch software, this work will positively contribute tothe technological preparedness for bringing moderntransportation to a higher density, be it airborne,ground based, marine, or submersible.
1.1. Related LiteratureMobility systemplanning has been actively researchedin recent years. On one end of the spectrum is the“free flight” type of path planning, where only a listof waypoints is computed. It relies on the movingobject’s own navigational and mobile ability to movefrom one waypoint to the next. Examples includeGPS-based direction assistance for automobiles, ve-hicle routing for delivery trucks (Crevier, Cordeau,and Laporte 2007; Eksioglu, Vural, andReisman 2009;Drexl 2012), and navigation and piloting practices in
Figure 1. (Color online) A Snapshot of a Simulated 2DTraffic Scene
Notes. The colored dots represent en-route vehicles heading towardtheir respective destinations along the shortest paths. In UAS-basedlogistics systems, there are no fixed flight routes—“airports” can beeverywhere and anywhere. Key challenges include (1) sustaining acontinuous, safe, and efficient traffic flow and (2) quantifying thesystem’s instantaneous and long-term efficiency.
civil aviation (Bilimoria and Lee 2001) and remote-controlled UAV tasks (Murray and Park 2013).In these systems, human operators’ judgment, inter-vention, or direct action usually plays a critical role,althoughmore andmore routine operations are aidedby computer systems (Hoekstra et al. 1998, McNallyand Gong 2007). On the other end of the spectrum isthe closed-loop end-to-end control method, determiningthe control inputs to drive the controlled object from astart configuration (e.g., location, orientation, speed,etc.) to a desired end configuration. Examples in-clude planning the motion paths for robotic arms in amanufacturing process, planning paths for automaticguided vehicles in warehouses, controlling the posi-tions and movements of spacecraft or satellites, etc.Books by Laumond (1998) and LaValle (2006) providecomprehensive technical details in robotics motionplanning (MP) and control.
Conflict resolution, or collision avoidance, is the pri-mary focus in air trafficmanagement (Vranas, Bertsimas,and Odoni 1994; Durand, Alliot, and Chansou 1995;Kuchar and Yang 2000; Visintini et al. 2006; Cellier,Cafieri, and Messine 2013; Wu and Du 2014; Sunilet al. 2017). When multiple aircraft enter the samevolume of airspace, a path conflict may threaten theloss of safe separation among aircraft. In this situa-tion, a resolution strategy or tactic is needed to findtrajectories that satisfy separation constraints whileminimizing the overall trip disruption for all aircraftinvolved. Mathematical programming and optimi-zation is an oft-used technique for modeling and solv-ing this problem. The seminal paper by Pallottino,Feron, and Bicchi (2002) formulated the multiaircraftconflict-avoidance problem as a mixed integer linearprogram (MILP) to minimize the total flight time. In-teger variables were used for modeling the forbiddencone of velocity (due to collision avoidance) by dis-junctive constraints. The model essentially produced aone-shot alteration to aircraft’s speed or heading at thebeginning configuration to make sure the resultingtrajectories are conflict-free for the rest of the time.This strategy relied on abundant free space in theinitial configuration to accommodate themaneuvers, aluxury that is typically unavailable in dense traffic. Asimilar treatment for collision avoidance was used byNy and Pappas (2010), whereas the disjunctive con-straints were handled by geometric programming tech-niques. Christodoulou andCostoulakis (2004) proposeda mixed integer nonlinear programming (MINLP)formulation for multiaircraft conflict resolution. Thismodel was highly nonconvex because of the use oftrigonometric functions and bilinear terms to ex-press the speed and heading angle changes. Small-scale instances involving two and three aircraft weresolved using optimization solvers within the GAMSmodeling software. Frazzoli et al. (2001) employed a
nonconvex quadratically constrained quadratic pro-gram to model the planar, multiaircraft conflict reso-lution problem. The overall scheme was a mix ofcentralized decision making for safety and decen-tralized preference optimization for efficiency, with acost function chosen to minimize the deviation be-tween desired and conflict-avoiding heading for eachaircraft. The authors furthermore presented a semi-definite programming relaxation scheme and a ran-domized search approach for resolving various localconflict patterns. Dell’Olmo and Lulli (2003) proposeda two-level optimization model to plan airway usageschedule over a large area of airspace. The upper levelwas a commodity flow problem in a graph whereairports were modeled as nodes, airways connectingairports were arcs, and aircraft were commodities. Thelower-level model treated a single airway as a tube inwhich four-dimensional collision-free aircraft tra-jectories were computed. Although a novel approach,the model relied on discretization of many parameters.For instance, a binary variable φk
z,v(t)was defined as “1if aircraft k at time t is at flight level zwith velocity v, 0otherwise” (Dell’Olmo and Lulli 2003, p. 184), andlived in a huge combinatorial space formed by theCartesian product of four large discrete sets. Thiswould inevitably present a hurdle to the model’sscalability. By discretizing time into 20-second in-tervals, Islami, Chaimatanan, and Delahaye (2016)formulated a large-scale MILP for planning the four-dimensional trajectories of aircraft in a nationwide andcontinent-scale airspace. The authors proposed a hy-brid metaheuristics consisting of simulated annealingand local search algorithms to find conflict-free tra-jectories for thousands of aircraft. However, the so-lution process was time-consuming; up to 45 hours ofCPU time was needed for a single run.Apart from using centralized optimization techniques
to resolve conflicts, agent-based distributed coordinationmethods have also been widely studied (Eby and Kelly1999; Sislak et al. 2007; Sislak, Samek, and Pechoucek2008; Pechoucek and Sislak 2009). An extensive reviewof this important thread of research is forgone here.Instead, the readers are referred to a recent survey paperon collision avoidance for UAVs by Mahjri, Dhraief,and Belghith (2015), in which the authors organize thecollision-avoidance system into sensing, detection,and resolution functions and provide a review of lit-erature and technologies of each function.Path-planning algorithms are also extensively stud-
ied in the robotics literature under the term of kino-dynamicmotion planning (Donald et al. 1993, LaValleand Kuffner 2001, Hsu et al. 2002). For instance, Earland D’Andrea (2007) formulated a wheeled robotcontrol problem as an MILP, utilizing big-M con-straints to facilitate the collision-avoidance con-straints. Abichandani et al. (2015) proposed anMINLP
formulation to model the spline curve paths for un-derwater vehicles, in which the cubic spline curvesgave rise to nonconvex nonlinear constraints. The so-lution algorithmwas executed sequentially so that eachvehicle planed its ownmovement by taking as input theplanned movements of other vehicles of a higher pri-ority order. Sunil et al. (2017) generated local flightplans for solving real-time geofencing and trafficconstraints based on the rapidly exploring randomtree technique in discrete time. Desaraju and How(2012) studied path planning for multiagent teamsunder complex constraints and introduced a coop-eration strategy that allowed an agent to modify itsteammates’ plans to select paths that reduce theircombined cost. Cohen et al. (1995) proposed an algo-rithm to perform quick collision detection betweenpairs of objects of polyhedral shapes. It was a two-levelapproach based on the idea of pruning pairs of objectsthat were far from each other using bounding boxes.A similar exact method is also used in this paper forpruning unnecessary collision-avoidance constraintsin the motion-planning model.
Most models with kinematics-level details have fo-cused on resolving conflicts in an area separated fromthe rest of the traffic space. This approach is sufficientfor scenarios in which the traffic density is low in mostparts of the system and path conflicts, especially onesinvolving multiple aircraft, occur only sporadically. Inhigh-density scenarios where congestion is a commonoccurrence (as illustrated in Figure 1), a segregatedlocal approach may cause a chain reaction, in whichthe resolution of one conflict may lead to other im-mediate or imminent conflicts (Kuchar and Yang2000, Jardin 2004). In such circumstances, it is impor-tant to adopt a holistic, global optimization frameworkthat coordinates the entire traffic flow across a broadspace. Furthermore, traffic coordinated under a co-herent theme will provide insights into the airspacecapacity and efficiency.
1.2. Main ContributionsIn both optimization-based and agent-based roboticsmotion and aircraft trajectory planning models, tem-poral discretization is almost ubiquitously used(Richards and How 2002; Dell’Olmo and Lulli 2003;Raghunathan et al. 2004; Sislak, Samek, and Pechoucek2008; Desaraju and How 2012; Cellier, Cafieri, andMessine 2013; Islami, Chaimatanan, and Delahaye2016; Sunil et al. 2017). However, the loss of fidelityin using discrete-time models to approximate con-tinuous motions has been largely unattended, per-haps because it is not a predominant concern in low-density applications where maneuvering space isabundant. In fact, collision-avoidance constraintsmodeled in discrete time will translate to a less securesituation in continuous time, and in the worst case,
a head-on collision can happen evenwhen the discrete-time collision-avoidance constraints are perfectlysatisfied. Conversely, if the safety distance is set in anoverly conservative way, the system capacity will besacrificed. To address this issue, analytical boundson the quality of the discrete time approximation arederived, which provide critical guidance for accu-rately modeling vehicle movements in a dense traffic.It is customary in the literature to formulate the
motion-planning problem as an MILP for its supe-rior solvability over the alternative nonlinear formu-lations. However, linearization of kinematic laws inthe Euclidean space, for example, using polygons toapproximate circular sectors, may lead to a loss of ac-curacy, which matters in high-density scenarios but isoften left unquantified. A solution architecture that isboth aligned with the Euclidean distance metric andsuitable for real-time applications is still lacking in theliterature. This paper presents a nonlinear program-ming (NLP) model that authentically models vehiclemotion in a 2D space, and develops an algorithm thatenables uninterrupted motion planning with feasi-bility guarantee and high solution quality. The authoris not aware of any other solution approach for similarformulations that is able to produce attractive feasiblesolutions for large and dense systems.With the advent of the AI era and the prospect
of mass deployment of unmanned aircraft systems(UASs) in the low-altitude airspace (Hall 2016; Gharibi,Boutaba, and Waslander 2016), understanding trafficflow behavior in 2D (and 3D) space is critical for thesafe and efficient use of the airspace. Neither tradi-tional road-based traffic flow metrics nor the existingair traffic models are adequate for this dense, high-dimensional, and unmanned space. In this regard,a set of new metrics is proposed to quantify the tripperformance, traffic flow, and system efficiency forcoordinated air traffic systems. These metrics are al-gorithm agnostic; therefore, they can facilitate perfor-mance comparisons among a wide range of systemdesigns, models, and implementation schemes. Someof these metrics are used for evaluating the algo-rithm’s performance in different density conditions andpresenting insights in system design.The model, algorithm, and metrics developed in
this paper are, in theory, extensible to air traffic in the3D Euclidean space. However, we focus on the 2Dspace here because (1) vertical separation of aircraft ismanaged differently in practice (i.e., aircrafts’ safetybuffer volume is cylindrical rather than spherical;see, e.g., Lehouillier et al. 2017), as horizontal motionaccounts for the dominant transportation activity, and(2) the proposed metrics will be comparable in scopeto the leading air traffic flow literature in which 2Dflow is the predominant focus. Extension to 3D space isbriefly discussed as future work in Section 6.
1.3. OrganizationThe remainder of this paper is organized as follows.Section 2 derives the exact bounds for collision-avoidanceconstraints and proposes a spatial decomposition strat-egy. Section 3 presents the metrics for quantifying2D traffic efficiency. Section 4 develops the mainmathematical model and solution algorithms anddiscusses their properties. Section 5 reports numeri-cal experiments and analyzes the results. Finally,Section 6 concludes this paper and mentions somefuture work.
2. Collision AvoidanceSafety is the top priority when it comes to movingpeople or goods from one place to another, and crashis the biggest threat to transportation safety. To avoidcrashes, vehicles in the field must be sufficiently sepa-rated. A minimum intervehicle distance that allows foran “escape route” in case of a contingency should bemaintained. The choice of the safety distance in atransportation system depends on the vehicles’ ma-neuverability, motion accuracy, and reliability. Set-ting the safety distance prudently tomeet the underlyingsafety requirements without wasting space is an im-portant task in the design of a transit system.
2.1. Exact Bounds for Collision AvoidanceCollision avoidance must be enforced at all times,whereas collision detection can occur only at discretetime points. This calls for a meticulous discretizationscheme along the time dimension so that (1) feasibilityat contiguous time points can guarantee feasibilityat every time point in the interval formed by thecheckpoints and (2) computational efficiency is maxi-mally preserved, meaning that unnecessarily fine dis-cretization should be avoided. This section will derivebounds on the accuracy of temporal discretization andpresent their practical usage. The notion and result willbe used in subsequent sections.
Let ui,t ∈ Rn be the location of vehicle i at time t,where n is 2 or 3 depending on the dimension of spaceunder study. Assume that in any unit interval of time,say, between t and t + 1, each vehicle imoves at a fixedvelocity vi,t ∈ Rn with ||vi,t|| ≤ Ri, where Ri is themaximum distance vehicle i can travel in a unit timeinterval. This gives ui,t+1 � ui,t + vi,t for each i and t. Asimplied by the notation, a vehicle’s velocity is as-sumed to be unchanging (i.e., linearmovement)withinthe same unit interval of time, although it can varyacross different time intervals. Collision avoidance isconcerned with the relative position between twomoving vehicles, say, i and j. Let Δt � ui,t − uj,t be therelative location of i to j at time t, and let δt � vi,t − vj,tbe the relative velocity of i to j; then we haveΔt+1 � Δt + δt. Triangle inequality indicates that||δt|| ≤ R, whereR � Ri + Rj.When time is discretized in
an algebraic modeling system, the collision-avoidanceconstraint takes the form ||Δt|| ≥ S, t � 0, 1, 2, . . ., re-quiring the intervehicle distance to be greater than apreset constant S at each discrete time point t. In theliterature, most work concerning collision avoidanceemployed this constraint or a close variant of it. How-ever, this constraint says nothing about the intervehicledistance during the time interval. In effect, all-timeseparation is approximated by discrete-time separa-tion. It is critical to understand the property of thisapproximation as well as its implications for safety.In this regard, an immediate inquiry is, do ||Δt|| ≥ Sand ||Δt+1|| ≥ S guarantee ||Δt+α|| ≥ S, for any α ∈ [0, 1]?Or, equivalently, what is the minimum value of ||Δt +αδ|| for 0 ≤ α ≤ 1, given ||Δt|| ≥ S, ||Δt+1|| ≥ S, and||δt|| ≤ R? The following theorem establishes boundson ||Δt+α|| for the case when || · || is taken to be theEuclidean norm. A similar result with the Manhat-tan distance is also obtainable, which is outside thescope of this paper.
Theorem 1. Given constants S,R ≥ 0, let Δt, δt ∈ Rn with||δt||2 ≤ R for t � 0, 1, 2, . . ., and defineΔs � Δ�s� + (s − �s�)·δ�s�, for s ∈ R+. Then the algebraic constraints ||Δt||2 ≥ S,t � 0, 1, 2, . . . imply that the continuous-time separationdistance ||Δs||2 satisfies
||Δs||2 ≥�����������������S2 − (1/4)R2
√when R ≤ 2S,
0 when R> 2S,
{
for all s ∈ R+.The proof is given in Online Appendix A. The
theorem shows that the minimum separation dis-tance stipulated in the series of discrete-time algebraicconstraints does not translate to the same level ofseparation in continuous time, but to a reduced andhence less secure level of separation. Moreover, ifthe maximum closing speed between two vehiclesis greater than two times the stipulated separationradius, the discrete-time separation constraints willhave no effect in preventing collision in real time, andthe two vehicles may collide head-on.The loss of fidelity from substituting discrete-time
constraints for continuous-time constraints is notsurprising. Theorem 1 exactly quantifies this loss offidelity and provides useful bounds on the actualseparation distance. The bounds allow for an accuratecalculation for the parameter values (i.e., S and R)needed in discrete-time models. Let us look at a nu-merical example.
Example 1 (Figure 2). Vehicle 1 (red) travels from co-ordinate (100, 100) to (100, 400), whereas vehicle 2(blue) travels from (200, 400) to (200, 100). For conve-nience of exposition, let us assume the distance ismeasured in meters and the natural time unit is thesecond. The all-time separation distance is set to S′ � 140
meters, shown as the diameter of the solid circles in thefigure. Suppose both vehicles have a maximum speedof 10 meters per second. If time is discretized into10-second intervals, then the maximum closing speedR would be 200 per unit of time. Theorem 1 wouldrequire
�������������S − 1/4R2
√ ≥ S′, that is,������������������S − (1/4)2002√ ≥ 140;
hence, S must be set to 172.1 or above to ensure a safeseparation, shown as the diameter of the dotted circlein Figure 2(c). Panel (a) shows a significant loss of sep-aration during time 2 to 3 when only 140 meters ofseparation was enforced on discrete time points. Us-ing a more fine-grained time discretization, for ex-ample, one-second intervals, as shown in panel (b),could mitigate (but not eliminate) the loss of accu-racy, which would come with a higher computationalcost. Panel (d) demonstrates temporal discretizationby four-second intervals with an exact separation bound
of 148.7 (dotted circle), which approximates the realsituation well with security assurance.In practice, it is more intuitive to specify the sep-
aration margin in terms of time, in which case theactual distance of separation is simply a function ofspeed. Let h be the minimum time distance betweenany pair of en-route vehicles i and j with maximumspeeds si and sj, respectively. Then the separation dis-tance Sij to be enforced at discrete time points is given by
Sij � (si + sj)������������h2 + 0.25
√. (1)
This general formula leaves great flexibility for in-stantiating the length and time units to match dif-ferent application scenarios. Given the same set ofvalues for h, si, and sj, for instance, in civil aviation, thelength unit can be set to 0.1 nauticalmiles and the timeunit to 5 seconds, whereas in a UAV delivery system
Figure 2. (Color online) Can a Discrete Time Constraint Ensure Continuous Time Separation?
Notes. Two vehicles start in opposite directions along parallel paths, and the required separation distance is wider than the interpath distance.Panel (a) shows that a significant loss of separation occurs during t � 2 to 3. Panel (b) shows that the loss is mitigated (but not completelyeliminated) by an overly fine-grained temporal discretization. Panel (c) shows that the loss is eliminated by using exact discrete-time separationbound in accordance with the temporal scheme. Panel (d) shows a finer-grained discretization with an exact separation bound.
consisting of small drones, the unit length and time canbe set to 1meter and 1 second, respectively, to adjust forthe physical dimension and operating characteristics ofsystems.
2.2. Spatial Decomposition by Collision-Cautious Clusters
When the airspace under management has an uneventraffic density distribution, it is possible to dynamicallydecompose the motion-planning problem into smallerindependent problems, one for each cluster of vehicleslinked by the collision-avoidance constraints. Let #t0 ⊂! ×! be the set of collision-cautious vehicle pairswhose separation constraint must be enforced in theplanning horizon covering time periods t0 to t0 + T.Intuitively, when two vehicles i and j are sufficientlyfar away from each other at the current time t0, theirseparation is not an immediate concern in a modelthat plans vehicle motion for the next T time steps.Hence, the pair (i, j) does not need to appear in theset #t0 when solving the model. We will derive thedistance threshold below which a vehicle pair must beincluded in the set#t0 for collision-avoidance constraints.
Consider the worst case and assume that the twovehicles travel head-to-head toward each other atmaximum speeds during the entire planning horizonof T time intervals. By time t0 + T, their distance canbe bridged by no more than T(si + sj). If their distanceis still beyond the minimum separation distance S,then it is safe to conclude that their initial distanceis large enough to warrant a safe separation at anytime during the planning horizon. Hence, (i, j) doesnot need to be included in #t0 . The reasoning is for-malized in the following theorem.
Theorem 2. It is sufficient to set #t0 � {(i, j) ∈ ! ×! :
||ui,t0 − uj,t0 || ≤ (T + ������������h2 + 0.25
√ )(si + sj)}. In other words, ifa pair of vehicles is not included in #t0 determined by theabove formula, then the pair is guaranteed to be at least h timeunits apart during the entire time period from t0 to t0 + T.
Theorem 2 is a straightforward corollary of The-orem 1, so its proof will be omitted. It provides aconservative (large enough) distance threshold to ac-count for any possible conflict. It does not indicate,however, that all vehicle pairs included in #t0 willever enter a direct conflict. As observed in numericalexperiments, many collision-avoidance constraints re-main inactive in the solution process, suggesting ahighly sparse problem structure.
One way to exploit the sparse structure is to de-compose the motion-planning problem over the en-tire airspace by clusters of vehicles. Specifically, ifvehicles in the airspace are modeled as nodes in agraph, and an edge is placed between nodes i and jif (i, j) ∈ #t0 , then each connected subgraph willrepresent an independent portion of the problem
that can be solved separately and in parallel. Theidea is illustrated in Figure 3 and in the followingobservation.
Observation 1. The complexityof a system-widemotion-planning problem is of the same order as the complex-ity of planning for the densest region of the entire space.
Spatial decomposition for large air transportationnetworks was also exploited in Wei, Spiers, and Sun(2014), in which airports were clustered based ongeographic distance. This paper focuses on solvingand analyzing the traffic flow in dense and highlycongestive airspace environments; therefore, thedecomposition and parallel computing schemes arenot implemented in the prototyping code.
3. Traffic FlowMetrics for Dense 2D TrafficFrom a traffic manager’s perspective, planning themotion of vehicles that arbitrarily enter and utilizethe space under management is a continuous, open-ended, and (in theory) infinite-horizon control prob-lem, for which a “globally optimal” course of actionis both difficult to describe and not very relevant forpractice. Therefore, algorithmic performance measuresalone, such as the optimality gap and solution time,fall short of providing practical insights. To amendthis shortage, one needs a set of algorithm-agnosticmetrics to characterize the traffic flow.
Figure 3. (Color online) Spatial Decomposition of theTraffic Space at a Given Time t0
Notes. Each dot represents a vehicle, and the black circle marksits safety zone. The gray circular area surrounding each vehiclerepresents the range the vehicle can possibly reach in the next Tperiods. If a pair of vehicles’ gray areas overlap, the pair is in#t0 . Eachconnected subgraph of #t0 forms a cluster, for which motionplanning can be solved independently. Two clusters are presentin this illustration.
3.1. Inadequacy of Existing FrameworksThere are numerous theories, models, and empiricalstudies to characterize traffic flow in road transportsystems; see, for example, Wardrop (1952), GerloughandHuber (1975), Anderson (1978), Ceder (2007), andLieu (1975). However, themodels and findings do notapply to 2D and 3D scenarios. The added dimension(s)of freedom dramatically increase the traffic complex-ity. On a road segment, effective separation amongvehicles is straightforward to implement by simplerules, for instance, keeping the following distanceabove a minimum threshold will do the job, and thiscan be performed instinctively by drivers.Many trafficmodels take this instinctive collision-avoiding be-havior as a fundamental assumption, based on whichtheoretical and observational characterizations ofthe system are then derived. In contrast, when ve-hicles traverse a 2D space in all possible directions,maintaining effective separation becomesmuchmorecomplicated, and no simple rule will work withoutsignificantly sacrificing capacity and efficiency. Fur-thermore, conventional definitions of speed, flow, anddensity for fixed-route systems will no longer workin characterizing high-dimensional systems. For ex-ample, to avoid collision, an aerial vehicle may travelin a direction unaligned with, or even opposite to, thedesired destination-pointing direction. In this case,high speed does not translate to high efficiency. Simi-larly, if most vehicles are constantly maneuvering be-cause of pathway congestion, the overall corridorefficiency is actually quite low even if speed and den-sity are both at high levels.
Existing experience in air traffic management is alsoinadequate to provide a full understanding of the trafficbehavior. First, traditional air traffic management dealswith open-loop, manned systems in which pilots’ skilland judgment, as well as the air-to-ground and air-to-air communication protocols, play a major role in safeoperations (Federal Aviation Administration 2009,Lehouillier et al. 2014). In an unmanned system, theseelements will become obsolete. Second, the trafficdensity is usually quite low for themost part in a long-haul air trip. In contrast, a practical low-altitude, short-range aerial delivery system is expected to havemuchdenser traffic, because of the limited flight rangeand carrying capacity of each vehicle in the system.Third, the airport-to-airport flight routes are almostfixed for bulk air transportation, which reduces theairspace management (two- or three-dimensional)problem to a corridor management (one-dimensional,spatially) problem (Vranas, Bertsimas, and Odoni1994; Dell’Olmo and Lulli 2003; Bertsimas, Lulli,and Odoni 2011) or to a network optimization prob-lem (Ma, Cui, and Cheng 2004; Wei, Andrisani, andSun 2011; Yang, Mao, and Wei 2016). In contrast,the origin and destination (OD) and flight plan of a
delivery drone can be highly variable. For instance, alightweight drone can be launched from the top of adelivery truck, which serves as the moving depot andbattery stop for the drone (Murray and Chu 2015).For air traffic flow modeling, Eulerian network
modeling (Menon, Sweriduk, and Bilimoria 2004)has been a particularly popular approach. One-dimensional Eulerian models aggregate air trafficinto line segments (called links) and use partial dif-ferential equation–based difference equations to drivethe flow dynamics (Sun, Strub, and Bayen 2007). Two-dimensional models (Menon et al. 2006) partition theairspace into cells and study traffic flows on the cel-lular network. By imposing such structures, Eulerianmodels are able to attain remarkable tractability even incontinent-scale traffic analyses, and have been used for,for example, density prediction and flow optimization(Bayen, Raffard, and Tomlin 2006; Work and Bayen2008). Nonetheless, in dense UAS traffic, the vehiclemotion cannot be simply driven by difference equationsbecause of nonconvex collision-avoidance constraintsand arbitrary new vehicle entrances. Therefore, trafficflowmust bemeasured independent of the underlyingequations of motion (EOMs) or planning algorithms.
3.2. New Metrics and Operational ImplicationsWe present below a set of newmetrics for quantifyingtraffic efficiency in a 2D transit space. LetAdenote thetotal size of the transit area under study. Adopting theEuclidean distance, each vehicle i exclusively oc-cupies a circular area of size π( ������������
h2 + 0.25√
si)2, where his the required time of separation between any twovehicles. This area is called the vehicle’s buffer zone.At any given time, no two buffer zones will overlapbecause of the collision-avoidance constraint. Theinstantaneous traffic density ρt at time t is defined asthe ratio between the sum of all vehicles’ buffer zoneareas and the total areaA, ρt � ∑Nt
i�1 π(������������h2 + 0.25
√si)2/A,
whereNt is the number of active vehicles at time t. Foreach vehicle i at time t, the instantaneous speed effi-ciency η
spdi,t is defined as the ratio between its current
speed and its maximum speed, the heading efficiency
ηhdgi,t as the cosine similarity between its current heading
and the destination-pointing direction, and the flow ef-ficiency ηflowi,t as the product of the speed efficiency andthe heading efficiency. The rationale behind thesedefinitions is that congestion effect will exhibit itselfin two forms, to lower a vehicle’s speed, hence causingdelay, or to divert a vehicle’s heading, hence causingdetour. In congestion-free traffic, a vehicle will moveto its destination along the straight-line path at themaximum speed, in which case all efficiency factorswill be 1. This can be used as a baseline for comparisons.Furthermore, the granular instantaneous metrics can beaggregated at the trip level. For instance, the trip delayfactor of a vehicle is defined as the actual trip time
dividedby the ideal trip time, which is the time it wouldtake to travel from origin to destination along astraight line at maximum speed. Similarly, the tripdetour factor is defined as the actual distance traveleddivided by the straight-line distance between theorigin and destination. Road transportation, for ex-ample, typically results in a much higher detourfactor than air transportation because vehicles areconfined to travel on road segments, which do notform the point-to-point shortest path for most trips.Detour factor and flow efficiency both imply the en-ergy efficiency of the system, that is, how much usefultransit per unit of energy consumed by the vehicles.On the system level, throughput, denoted by V, is thetotal effective transit made by all active vehicles perunit of time. It measures the temporal efficiency of thesystem. Themore vehicles the system accommodates,the higher the throughput will be. However, there isclearly a limit to this positive correlation. If densityis too high to be conducive to effective transit, con-gestion will curb the system throughput. The metricsdiscussed above are summarized in Table 1.
System regulators need to beware of the trade-offbetween temporal efficiency and energy efficiency.The optimal operating point of the system should
depend on the time value of delivery and the energycost of the transit method. It is also important to notethe functional dependence of density on h, the timeheadway. Vehicle count and space size held equal,systems with a higher accuracy in sensing, control,and mobility can tolerate a smaller h and henceachieve a higher throughput. The relationships be-tween density and different performance metrics areexplored via simulation in Section 5.4.
4. UAV Motion-Planning Modeland Algorithms
Motion planning involves making decisions aboutvehicles’ kinematic states over time to achieve certaingoals. Our grand goal is to make safe and efficient useof the 2D transit space, which loosely translates toan operational objective of maintaining a feasible yeteffective traffic flow. Here, “feasible”means that safeseparation is maintained at all times, and “effective”means that the traffic keeps moving in a way to even-tually get every vehicle to its destination quickly andwith minimal detour.
4.1. A Nonlinear Formulation and Its ChallengesLet us first formulate the task of directing a set ! ofvehicles to their respective destinations within aset 7 � {1, . . . ,T} of time intervals, though ultimatelyour algorithm is not confined to the scope of any fixedset of vehicles or of a fixed time horizon. Relevantnotations in addition to those defined in Table 1 aresummarized in Table 2.Given initial speed vi,0 � 0 and initial location ui,0 of
each vehicle i at t � 0 [with ||ui,0 − uj,0|| ≥ Sij, for all(i, j) ∈ #], we formulate the MP model as follows:
The objective (2) is to minimize the distance to des-tination summed over all vehicles and all time points.Constraints (3) to (5) are discretized EOMs, treatingeach vehicle as a point mass. Specifically, (3) stipu-lates a vehicle’s locations between successive timeperiods, assuming the vehicle makes uniform accel-eration during each unit time interval. Note that inthe last section, we adopted a slightly simpler EOM(assuming linear uniform-speed motion) for ease ofexposition, although the analytical results obtainedtherefrom are applicable here. Constraints (4) and (5)
Table 1. Definition of System Configuration andPerformance Metrics
Symbol Description
h System-wide minimum time distance (at all times)between any pair of vehicles
Oi,Di Vehicle i’s origin and destination coordinate vectorssi Maximum speed of vehicle iui,t Location vector of vehicle i at time tvi,t Velocity vector of vehicle i at time tνi,t Destination-pointing component of the velocity of
vehicle i at time t; νi,t � 〈vi,t,Di − ui,t〉/||Di − ui,t ||T∗i Actual trip time of vehicle i from Oi to Di
d∗i Actual trip distance of vehicle i; d∗i � ∑T∗i
t�1 ||ui,t − ui,t−1 ||Ti Ideal trip time of vehicle i; Ti � di/sidi Straight-line distance between Oi and Di;
di � ||Di −Oi|| � ||∑Tit�1(ui,t − ui,t−1)||
ηspdi,t Speed effciency of vehicle i at time t; ηspdi,t � ||vi,t||/si
ηhdgi,t Heading efficiency of vehicle i at time t; ηhdgi,t � νi,t/||vi,t ||
ηi,t Flow efficiency of vehicle i at time t; ηi,t � ηspdi,t η
hdgi,t
φdlyi Trip delay factor of vehicle i; φdly
i � T∗i /Ti
φdtri Trip detour factor of vehicle i; φdtr
i � d∗i/diNt Number of vehicles in the field under study at time tηspdt Vehicle-average instantaneous speed efficiency;
limit the acceleration and the top speed of each vehicle.Constraints (6) are the collision-avoidance constraints,imposing a separation distance of at least Sij betweenvehicles i and j for all pairs (i, j) close enough to becollision cautious. These constraints couple differentvehicles together and also make the solution spacenonconvex.
The MP model serves as the baseline model. Thefirst question one might ask is why the objective (2)is a good choice with regard to achieving the grandgoal. Indeed, in a continuous traffic operation, it is hardto tell what constitutes a global optimal motion plan,particularly when new information (e.g., new vehicleentrance) is gradually revealed and has to be accountedfor dynamically over time. We present an indirect ar-gument for the soundness of the objective function andresort to the metrics developed in Section 3 to measurethe practical performance of motion plans.
Letus lookat a reducedmotion planning (RMP)model,defined by (2)–(5). RMP corresponds to congestion-free traffic in which collision avoidance is not a con-straint. In this condition, Theorem 3 says that theobjective function (2) would induce a straight-linepath at full speed from the origin to destination foreach vehicle, which is the best possible result withregard to the grand goal.
Theorem 3. The solution to RMP gives a minimum-distance and minimum-travel-time path from origin Di todestination Oi for each vehicle i.
In congested traffic, however, low travel time andshort travel distance are not necessarily aligned. Toavoid collision, a vehicle may take an action any-where between the following two extremes: (a) takea long detour at the maximum speed to bypass theconflict area and (b) stop and wait until the collisionrisk is resolved by the other vehicle(s) involved in thesituation and then proceed along the shortest pathtoward the destination. Apparently, by staying on theshortest path, option (b) might require more en-routetime than option (a). We argue that in such cases, theobjective function (2) serves to mitigate delay and
detour on the system level, which will be assessedseparately usingmetrics developed in the last section.
4.1.1. Remarks on Simplifying Assumptions. To servemethodological development, the MPmodel embodiesa plethora of assumptions that greatly simplify the realworld. Several practical elements absent from themodel areworthmentioning:wind effects, interactionswith the payload, and energy consumption are sig-nificant factors in UAS control and routing practices;flight plan disruption and maneuvering fuel costsare common considerations in conflict resolution; andanticipating upcoming trip demand could alleviate themyopia in a deterministic planning model.
4.1.2. Remarks on the Implicit Right-of-Way Priority.Because model MP minimizes the total distance todestination, in the case of a two-vehicle conflict, forexample, it will favor the vehicle farther away from itsdestination, because a delay in that vehicle wouldcontribute a greater value (i.e., its distance to desti-nation) to the objective function than would the other.This is a byproduct of using the objective function (2)to induce shortest paths. A simple case illustratingthis phenomenon is provided Online Appendix B.At first sight, this resolution may seem unfairly dis-criminatory against vehicles closer to their destina-tions. Nevertheless, absent further knowledge aboutthe specific system, the best assumptions one can haveare that all zones are equally likely to become con-gested and that all vehicles’ origins and destinationsare uniformly distributed across the space, and there-fore, on average, the collision-avoidance cost (delay ordetour) is shared evenly across all vehicles in the field.In fact, such a discriminatory effect is negligible whenthe look-ahead horizon T is limited, as is the case in thereceding-horizon algorithm developed in Section 4.2.Moreover, the priority order can be easily adjusted byadding different weights to the objective terms, amongother means, without sacrificing other desirable prop-erties of the model.
4.1.3. Challenges in Solving theMPModel. The form ofthe MP model as presented in (2) to (6) is appropriatefor conceptual exposition, but not suitable for com-putational implementation. First, given a set of triprequests (Oi,Di), for i ∈ !, as well as vehicle config-urations ai and si, for i ∈ !, there is no way to exactlydetermine a priori the planning horizon T needed tocomplete all trips. In other words, the end time τ �min{t : u∗i,t � Di, ∀i ∈ !} exists in the solution of MPonly when it is smaller than T, and its value is notknown until a solution that contains it is found. Thesize of the problem grows linearly with the valueof T, so blindly trying an unnecessarily large T iscomputationally nonviable. Second, when it comesto air traffic management, path planning is not a
Table 2. List of Notations for the MP Model
Symbol Description
! Set of vehicles7 Set of time intervals# ⊂ ! ×!, set of collision-cautious vehicle pairsai Maximum acceleration between successive time points
for vehicle iSij Minimum separation distance between vehicles i and jxi,t, yi,t Abscissa and ordinate components of ui,t, respectively,
in 2D spacex′i,t, y′i,t Abscissa and ordinate components of vi,t, respectively,
in 2D spacexi,t, yi,t Abscissa and ordinate components of the destination
one-shot deal, but a dynamic, evolving process with-out a definite time frame. Vehicles that have reachedtheir destinations should be promptly excluded fromthe planner’s scope, so that they will not block thepathways of other vehicles or prevent other vehiclesfrom landing in the same area. It is not possible torecognize whether a vehicle has reached its destina-tion without significantly complicating the modelformulation, for instance, by using integer variables. Inaddition, as existing trips are being optimized, newtrip requests will continue entering the system andmust be promptly accommodated. For these reasons,themotion-planning problem needs to be solved on arolling basis in practice.
4.2. A Receding-Horizon Progressive Heuristic forMotion Planning
Let us cast the finite-horizon MP model into an infinite-horizon traffic management context. For simplicity ofexposition, a planar model (n � 2) is adopted, and itis straightforward to extend the model and algorithm toa three-dimensional space. In a planar area, the locationand speed vectors are specified by the x and y co-ordinates. Thus, let us define ui,t � (xi,t, yi,t), Di � (xi, yi),and vi,t � (x′i,t, y′i,t). At a given time point t0, the systemstate is represented by the following parameters: theset of active (en-route) vehicles!t0 � {i ∈ ! : xi,t0 �� xior yi,t0 �� yi}, the current location (xi,t0 , yi,t0) and ve-locity (x′i,t0 , y′i,t0) of each active vehicle i ∈ !t0 , thelength of the planning horizon T with the corre-sponding time index set 7t0 � {t0, t0 + 1, . . . , t0 + T},and the set of collision-cautious vehicle pairs #t0 .
The algorithm is designed to work as follows. Attime t0, vehicle motions are planned for the next Tperiods. A feasible and nontrivial solution of such amodel will recommend an optimal T-step movementpath for each vehicle that obeys all the physicalconstraints, including the acceleration limits, speedlimits, and the collision-avoidance constraints. In-stead of following the entire T-step paths, the vehicleswill move only M (M ≤ T) steps according to therecommended paths. By treating the time point t0 +Mas the new t0 for the next iteration, the system state isreevaluated and updated. Specifically, during theM-step move, if a vehicle has reached its destination,it will be eliminated from the active set !t0 in theupcoming iteration. The set of collision-cautious ve-hicle pairs#t0 will also be updated, by Theorem 2. Theabove process then repeats itself until !t0 � ∅, that is,all vehicles have reached their respective destina-tions. The whole solution process progresses at a steplength of M time periods.
The success of the algorithm depends critically onhow this question is addressed: Continuing fromlocation (xi,t0 , yi,t0) and velocity (x′i,t0 , y′i,t0), which rep-resent the end state of the M-step move from the
previous iteration, how can one ensure that the nextT-step MP model always has a feasible solution? If atany time the model becomes infeasible, the chain ofprogression will break and the algorithm fail.In the very beginning of time, when all vehicles are
assumed to start from still with acceptable separationdistance, the model MP always has a trivial feasiblesolution which is to dictate all vehicles to remain still,that is, vi,t � vi,0 � 0 and ui,t � ui,0 for all i ∈ ! and t ∈ 7.However, such a convenient feasible solution is nolonger available when vehicles are in motion, as it isnot possible to instantaneously brake moving vehi-cles to a complete stop (as an attempt to preservefeasibility of collision-avoidance constraints for up-coming time periods).To reinstate the availability of a feasible solution in
each progressive solution of the MP model, an aux-iliary constraint is introduced to fix the speed at theend of the look-ahead horizon to zero. It is critical tonote here that this treatment will not really bringvehicles to a stop; see a detailed explanation inSection 4.2.1. With this treatment, a feasible MP so-lution can be extended to a feasible starting point forthe next MP solution (with an M-step shift). Specifi-cally, at each starting time t0, the following motion-planning model is solved:
The parameters αi are used for deadlock resolution, tobe discussed shortly. At αi � 1, the model MP(t0) is anexpanded and equivalent formulation of the modelMP, parameterized by t0. Constraint (13) forces thespeed at the end time t0 + T to be zero. !t denotesthe set of active vehicles at time t, and #t denotes theset of collision-cautious vehicle pairs at time t, fort ∈ 7t0 . From here to Section 4.3 it is convenient to as-sume!t � !t0 and #t � #t0 for all t ∈ 7t0 , for which thechoice of#t0 is prescribed in Theorem 2. In Section 4.4,
where traffic merging is discussed, !t and #t will de-viate from the above setting for certain t ∈ 7t0 . Let ussee how the progressive solution works via an example.
Example 2 (Look-Far-Move-Small Planning Process). Sup-pose we choose T � 10 and M � 2, and suppose modelMP(t0) has a local optimal solution (u∗i,t0 , u∗i,t0+1, . . . ,u∗i,t0+10) and (v∗i,t0 , v∗i,t0+1, . . . , v∗i,t0+10) for i ∈ !. Then, thetime will be advanced by M � 2 steps with the vehiclesfollowing the first two steps in the solution and movingto u∗i,t0+2, i ∈ !. The next MP solution will take t0 + 2 asthe starting time, fix the starting vehicle locations tou∗i,t0+2, and solve for optimal paths for the subsequentT � 10 time steps, from t0 + 2 to t0 + 12. In other words,the model instance MP (t0 + 2) will be solved. In thissolution, (u∗i,t0+2, u∗i,t0+3, . . . , u∗i,t0+10, u∗i,t0+10, u∗i,t0+10) and(v∗i,t0+2, v∗i,t0+3, . . . , v∗i,t0+10, 0, 0) will be supplied as thestarting point for variables {ui,t}t0+12t�t0+2 and {vi,t}t0+12t�t0+2,respectively. This starting point is actually a feasiblesolution for the model MP(t0 + 2). To see this, let usassume, without loss of generality, !t0+2 � !t0 , andalso assume, for now, #t0+2 � #t0 (this will be gen-eralized later). Then the constraints (8)–(12) for t ∈{t0 + 2, . . . , t0 + 10} are identical in both MP(t0) andMP(t0 + 2), and therefore, feasibility of the solutionpoints {u∗i,t}t0+10t�t0+2 and {v∗i,t}t0+10t�t0+2 carries over. Further-more, the starting point essentially “freezes” and ex-tends the still state (recall v∗i,t0+10 � 0) of t � t0 + 10 to thenext M � 2 periods, t0 + 11 and t0 + 12, in which thefeasibility of all constraints are retained. The abovemethod of setting starting points is generalized below.Strategy 1. Given the solution of MP(t0), that is,{u∗i,t}t0+Tt�t0 and {v∗i,t}t0+Tt�t0 for i ∈ !t0 , set the starting point
for variables {ui,t}t0+T+Mt�t0+M and {vi,t}t0+T+Mt�t0+M , i ∈ !t0+M formodel MP(t0 +M) as follows:
• For t ∈ {t0 +M, . . . , t0 + T}, set ui,t ← u∗i,t andvi,t ← v∗i,t.• For t ∈ {t0 + T + 1, . . . , t0 + T +M}, set ui,t ← u∗i,t0+Tand vi,t ← 0.
This strategy ensures that each time the MP(t0)model is solved, the solver is supplied with a feasi-ble starting point, which is of critical importance forthe solvability of nonconvex nonlinear programs.
There are three important technical issues to beaddressed: (1) the side effect of constraint (13), (2) theappropriate choice of T and M, and (3) the determi-nation of #t0 in each iteration.
4.2.1. Removing the Side Effect of (13) by Choosing Tand M. The sole purpose of the requirement that allvehicles come to a stop at the end of each look-aheadhorizon is to provide a feasible starting point for thenext solution, thereby ensuring that each successiveNLP solution in the algorithmic chain starts with afeasible solution. We now analyze the effect of thisconstraint and show that by choosing the parameters
T and M prudently, restrictive effects on the actualvehicle motion can be removed.As reflected in constraint (10), it might take a few
time intervals to reduce a vehicle’s speed from maxi-mum to zero. Therefore, the effect of constraint (13) isnot limited to the time point t0 + T, but may extendbackward along the time line to exert artificial speedlimits [that override the constraint (11)] for time points,for example, t0 + T − 1, t0 + T − 2, etc. How far backthis effect extends todependson the time to full stop frompresumably the maximum speed, which, on the systemlevel, is upper bounded by maxi∈!�si/ai�. Accordingly,the look-ahead step length T and move-ahead steplength M must follow the following two rules:
T ≥ 2maxi∈!
�si/ai�, (14)
M ≤ T −maxi∈!
�si/ai�. (15)
The lower bound on T in (14) ensures that in eachplanning cycle the vehicles have enough time to ac-celerate to the maximum speed as in free traffic evenwhen starting from still. The upper bound on M in(15) stipulates that the move-ahead step should beconservative enough so as not to land in the forceddeceleration stage. Following these rules is a mini-mum requirement that will remove the definite, fore-seeable, and significant restriction on the solutionspace brought about by (13). Going beyond theminimum (i.e., setting a larger T and smaller M thanthe required bounds, respectively) will further expandthe solution space of each NLP solution and might helpimprove the overall solution quality, but the marginalbenefit will be thin compared with the effort made tobarely satisfy the minimum. This argument is corrobo-rated by experimental results in Section 5.1.Intuitively, the more steps one looks and plans into
the future, the sooner one can detect possible dead-locks in future time periods, and thus themore heads-up time one has to start making corrections for them.On the other hand, a large T also results in a large MPmodel to solve in each iteration, which takes morecomputing time. A similar trade-off goes for the choiceof M. Having conservative move steps leaves moretime for adjustments to avoid future deadlocks, at thecost of wasting much of the motion computation bynot fully following the computed paths.
4.3. Avoiding Deadlocks Due to Local OptimaIn each iteration of the progressive algorithm, thenonconvex MP model that drives all vehicles towardtheir final destination is solved. Each MP solutionalways starts with a feasible solution and ends with alocally optimal solution. Part of the solution (vehiclemotion planned for the first M time intervals) is usedfor setting the actual movement of vehicles, and theremaining part of the solution serves as the building block
of a feasible starting point for the next iteration. Timeadvances by M intervals in each iteration. All seems towork well. However, there is one question remaining tobe answered: howandwhendoes this process terminate?
After each iteration, the active set of vehicles !t0 isupdated with the current vehicle location. If a vehiclereaches its destination, it will be excluded from !t0from the next iteration onward. Therefore, a naturaltermination point is when all vehicles have reachedtheir destinations and !t0 � ∅. However, there existsanother possibility: the solution is stalemated andstops changing over time, in which case the feasiblestarting point becomes the only feasible solution (hence,locally optimal) for model MP(t) for all t beyond acertain time point. Such a solution is marked by anunchanging location vector and a zero speed vector.In other words, a cluster of active vehicles stop mov-ing and cannot advance even a tiny bit toward theirrespective destinations. This phenomenon is called adeadlock. Figure 4 demonstrates a deadlock involvingtwo vehicles. The two vehicles’ destinations are closerto each other than the required separation radius, sowhen they reach the vicinity of their destinations andprepare for landing at the same time, they get stuck.
Deadlocks are attributed to the solution gettingtrapped in a local optimum. In a deadlock, any feasiblemoving direction, if followed, will cause an instanta-neous increase in the total distance to the destination.It is possible but very costly to rectify this situation byattempting to solve the MP model to global optimal-ity. Furthermore, as mentioned in the beginning ofSection 4.2, it is also quite complex to determinea large enough planning horizon T so that a globaloptimum of MP will end up being deadlock-free.
In practice, deadlocks are avoided by assigning dif-ferent right-of-way priorities or issuing direct resolu-tion orders to competing vehicles. Such priority ordersmay be simple enough to be articulated in trafficrules, such as the right-of-way rule at a four-way stopintersection, or they may be determined by the trafficcontroller on a case-by-case basis, such as the landingsequence assignment formultiple aircraft attempting toland on the same runway. Long-term planning, such asthe deliberate planning of delayed takeoffs to mini-mizedelay in the landing stage (Vranas, Bertsimas, andOdoni 1994), is also typically employed by airlinesto minimize fuel costs.To automatically circumvent deadlocks in the al-
gorithmic chain, we use the heuristic tactic of assigningdisparate priorities in the objective function to vehiclesdeemed to be forming a deadlock. Fortunately, in thelook-far-move-small algorithmic paradigm, one doesnot need to wait until all involved vehicles have fullymoved into a deadlock to be able to detect and reactto it. As soon as a stalemate arises in the far end of theplanning horizon, actionswill be taken.A stalemate in alook-far solution is recognized by the occurrence thatany two vehicles stop moving and their collision-avoidance constraint becomes active. Specifically, letμ∗i,j,t denote the optimal Lagrangian multiplier asso-
ciated with the constraint (12) in the local optimalsolution of model MP(t0). A positive Lagrangian mul-tiplier indicates that the corresponding constraint is ac-tive and binding at the solution (Nocedal and Wright2000). Given an integer parameter L ≥ 1, which denotesthe number of time points to check, we determine thatan impending deadlock involving at least vehicles iand j has arisen if the following conditions hold:
μ∗i,j,t > 0, for t ∈ {t0 + T − L, . . . , t0 + T}, (16)
||u∗i,t − u∗i,t−1|| ≤ ε, for t ∈ {t0 + T − L, . . . , t0 + T}, (17)
||u∗j,t − u∗j,t−1|| ≤ ε, for t ∈ {t0 + T − L, . . . , t0 + T}, (18)
where ε is a small positive number. In each itera-tion, the algorithm checks in the MP(t0) solution theabove conditions for all vehicle pairs in #t0 . If theconditions are met for a vehicle pair (i, j), αi is elevatedby a factor of γ, and αj is kept unchanged. In numer-ical instances, γ should be set large enough, com-mensurate with the numeric scale of maxi∈! ||Oi −Di||,to unambiguously arbitrate the priorities. In our nu-merical experiments, γ is set to 100. Such a disparityin priority will elicit the lower-priority vehicle j toyield and make way for the higher-priority vehicle i,therefore circumventing a deadlock situation. An in-stance of deadlock-breaking is illustrated in Figure 5.Note that satisfaction of the above conditions does
not indicate strictly a pairwise deadlock involving onlytwo vehicles. A cluster of multiple vehicles may beinvolved in a deadlock. It is clearly a feasible strategy
Figure 4. (Color online) Traces of Two Vehicles with EqualPriority
Notes. The planning horizon is T � 30, but the plot shows the tracesfor t � 1, . . . , 18. Both vehicles are unable to reach their respectivedestinations (marked by the square points), as their movements arestalemated by a local optimumat t � 18. The graydashed lines indicatethe distances to the destinations at different time points. The circlesindicate the separation constraints at t � 1 and t � 18, respectively.
to deal with them one by one, making one of the ve-hicles pass quickly as a first step to alleviate the con-gestion. Correspondingly, one needs to ensure thatonly one vehicle in a deadlock cluster has a higherpriority and all other vehicles in the cluster remain atthe base priority level, to avoid ambiguity and cycling.This can be achieved by a simple bookkeeping rule,that is, maintaining a set !DL ⊂ !t0 for vehicles thatare in the deadlock resolution mode, and a set #DL ⊂#t0 for vehicle pairs that are in the deadlock resolu-tion mode. Elements in these sets are taboo, that is,excluded from consideration for further priority ad-justments. The priority-setting mechanism is sum-marized in Algorithm 1. The algorithm checks forimpending deadlocks and, if any are detected, ele-vates the priority of an arbitrary vehicle (determinedby the lexicographic ordering of the vehicles’ indexnames) involved in the deadlock, and then augmentsthe sets !DL and #DL for global bookkeeping.
Algorithm 1 (Deadlock Detection and Priority Adjustment)1. procedure SETPRIORITY(#t0 , !DL, Solution of
MP(t0), γ)2. for (i, j) ∈ #t0 do3. if i /∈!DL and j /∈ !DL then4. Check (i, j) for conditions (16)–(18)5. if Conditions hold then6. αi ← γαi7. !DL←!DL∪{i, j}, #DL←#DL∪{(i,j)}8. end if9. end if
10. end for11. end procedure
The priority factor in the objective function isintended only for breaking a deadlock that wouldotherwise materialize in the default configuration
where all vehicles have the same priority. As soon asan impending deadlock is successfully circumvented,the elevated priority should be reset to the baselevel. This ensures that the motion-planning modelcontinues to function on an equitable basis, and alsoreleases affected vehicles from the taboo state in casetheir priority needs to be adjusted for breaking thenext impending deadlock. Two vehicles i and j thathave previously entered a deadlock cluster are saidto be completely disengaged if the following conditionshold at the current MP(t0) solution:
μ∗(i,j,t) � 0, for t ∈ {t0, . . . , t0 + T}. (19)
The procedure of scanning for disengagement andresetting priorities is given in Algorithm 2. Alongwith resetting priorities to the base level, the pro-cedure also takes care of the bookkeeping sets !DLand #DL, that is, releasing appropriate vehicles fromthe taboo state.
Algorithm 2 (Postresolution Priority Reset)1. procedure RESETPRIORITY(#DL, Solution of MP(t0))2. for (i, j) ∈ #DL do3. Check (i, j) for conditions (19)4. if Conditions hold then5. αi ← αj6. !DL ← !DL/{i, j}, #DL ← #DL/{(i, j)}7. end if8. end for9. end procedure
We have eliminated all undesired terminationpoints of the algorithm, including the occurrence of aninfeasible MP(t0) caused by the rolling augmentationof #t0 , and the motion stagnation caused by local op-tima of MP(t0). Therefore, only one termination pointis possible, which is attained when all vehicles havereached their respective destinations. The overallprogressive motion planning (PMP) algorithm ispresented in Algorithm 3. Inputs to the algorithminclude the set of vehicles!; each vehicle’s maximumspeed si, maximum acceleration ai, origin coordinateOi, and destination coordinateDi; the look-ahead stepT; the move-ahead step M; and the minimum sepa-ration time h. The outputs consist of each vehicle’slocation and speed at each time point leading toits destination. This algorithm guarantees a feasibleoutput as long as the trips’ origins are feasible for theseparation constraints.
Algorithm 3 (Progressive Motion Planning)1. procedure PMP(!, T, M, h, Oi, Di, si, ai)2. t0 ← 1, ui,t0 ← Oi, vi,t0 ← 0, αi ← 1, ti ← 1 for
i ∈ !3. !t0 � !, !DL ← ∅, #DL ← ∅4. while !t0 �� ∅ do
Figure 5. (Color online) Traces of Two Vehicles withDifferent Priorities, α1 � 20, α2 � 1
Notes. Enjoying a higher priority, vehicle 1 headed directly to itsdestination and lands at t � 16. Vehicle 2 first makes a detour to yieldand finally reaches its destination at t � 25, as the separation con-straint is removed after vehicle 1’s landing.
5. !t0 ←{i∈! :ui,t0 ��Di}, 7t0 ←{t0, . . . , t0+T}6. Set #t0 by Theorem 27. Fix ui,t0 ← u∗i,t0 and vi,t0 ← v∗i,t08. if t0 � 1 then9. Set starting values {ui,t}Tt�t0 ← 0, {vi,t}Tt�t0 ← 0
10. else11. Set starting values {ui,t}Tt�t0 and {vi,t}Tt�t0 by
Strategy 112. end if13. Solve MP(t0) and obtain solution u∗, v∗ and
multipliers μ∗14. Update (α,!DL,#DL) by Algorithm 215. Update (α,!DL,#DL) by Algorithm 116. t0 ← t0 +M17. end while18. ti ← min{t ∈ {1, . . . , t0} : u∗i,t � Di} for i ∈ !19. return (u∗i,t, v∗i,t) for i ∈ !, t ∈ {1, . . . , ti}20. end procedure
4.4. Accommodating New Vehicle EntranceAlgorithm 3 is able to route a given set of vehiclesto their destinations. In real-world operations, how-ever, new trip requests will constantly emerge whileexisting ones are being routed, which requires newvehicles to be merged to the existing traffic flow. Obvi-ously, it is not always feasible to instantaneously inserta new vehicle into busy traffic at an arbitrary location.The process is analogous to a car attempting to mergeinto highway traffic from a ramp road. It must planearly, sync up the speed, and enter the target lane at theright spot and speed. The algorithm can accommodatenew vehicle entrance in a similar way without dis-rupting the chain of feasibility. In the above algo-rithmic framework, the new vehicle will attempt tosecure an available space in the vicinity of its trip originT time steps in advance.
Suppose that a new trip request (Ok,Dk) arises at thecurrent time t0, and assume that the new vehicle’smaximum speed is sk. To accommodate its entrance,the traffic snapshot at time t0 + T [returned by themodel MP(t0)] will be searched for a spot near Ok atwhich adding the vehicle will not violate collision-avoidance constraints with any surrounding vehicle.Specifically, denoting Ok as (xk, yk), the scheduled en-trance point (xk,t0+T,yk,t0+T) for vehicle k will be set tothe solution to the following ENTRANCE (ET) problem:
In ET(k), the constant Ski is equal to (sk + si) ·��������������(h2 + 0.25)√. The set 9k contains vehicles whose
projected locations at time t0 + T are close enoughto Ok so that the separation constraints need to beinspected. It is constructed based on system statesand an outlook as of t0, and thus is an exogenousparameter to the model. The constant Rk is the radiusof the search area. The rationale behind searching aneighborhood of Ok for an entrance point ratherthan fixing the entrance point strictly toOk is rooted inpractical considerations. It is practical to assume thatin preparation for entrance to the main traffic flow, avehicle is able to maneuver freely to arrive at a des-ignated (based on system feasibility) entrance pointnot too far from its launch point Ok. The maneuveringcan occur, for example, in a lower-altitude space re-served for takeoff where the takeoff traffic is man-aged in serial mode. By prudently setting 9k and Rk,ET(k) is practically a small-scale problem and can besolved efficiently by a global solver such as GloMIQO(Misener and Floudas 2013) or BARON (Tawarmalaniand Sahinidis 2005).In dense traffic, ET(k) may be infeasible at t0, mean-
ing that the requested entrance cannot be accom-modated at the moment, and the vehicle must wait onthe ground or hover in the lower-altitude layer untilenough space in the main traffic becomes available.In this case, solution of ET(k) will be attempted againand again in subsequent iterations of the main al-gorithm until a solution is found.When a solution, say, (x∗k, y∗k), to ET(k) is found at
time t0, the following will be done to incorporatevehicle k into the main planning loop: (1) Add k to!t for t ≥ t0 + T and add (k, i), i ∈ 9k to#t for t ≥ t0 + T.(2) Fix (xk,t0+T, yk,t0+T) to (x∗k, y∗k), and fix (x′k,t0+T, y′k,t0+T)to (0, 0). With augmented sets !t and #t at t � t0 + T,MP(t0) remains feasible because all newly added con-straints [new rows of (11) to (13)] at t0 + T are satisfied.Therefore, the chain of feasibility is preserved. Theabove actions inform the main loop of the PMP al-gorithm that vehicle k is scheduled to enter the maintraffic at time t0 + T. Subsequent planning iterationswill incorporate the new vehicle, whose actual en-trance will take place at time t0 + T.Figure 6 demonstrates the new vehicle entrance
process. In this scenario, vehicles 1 and 2 represent theexisting traffic, and vehicle 3 is a new entrant. Simi-lar to the situation when an automobile merges intohighway traffic, the space accommodation, as wellas the postentrance motion planning, starts way aheadof the actual entrance, which occurs at time t � 7.At t � 1, a feasibility check is presumably performedand passed, and vehicle 3 appears in the planning
horizon at its desired entrance point. At t � 3 and 5,vehicle 3’s postentrance path is being planned alongwith the existing traffic even though the vehicle is notphysically in the traffic yet. At t � 7, vehicle 3 entersthe traffic and starts to follow its planned path. Notethat feasibility of vehicle 3’s entrance is ensured at t �1 by an optimal solution to ET(k). If ET(k) was notfeasible at t � 1, it would be solved again at t � 3 andso forth, until a solution became available.
The imposition of a minimum T-period pre-entrydelay may seem to be a stringent constraint in thecontext of the on-demand business model that willpredominantly exist in the system under consider-ation. We acknowledge that in periods or regionsof low traffic density, faster entries are possible. For
instance, an instantaneous entry of a new vehicle inthe white area in Figure 3 will cause no problemunder the current algorithmic framework. The ET(k)approach is designed to work in an arbitrary condi-tion and hence can be construed as a worst-caseresort.
5. ExperimentsThe PMP algorithm was implemented in GAMS(Windows version, release 25.0.2), and the core op-timization problem MP(t0) was solved using theCONOPT solver (Drud 1985, 2018) with default op-tions. The experimental hardware was an HP Z820workstation with two Xeon E5-6290 v2 CPUs and 96 GBof RAM. We first demonstrate the solution to stylized
Figure 6. (Color online) Demonstration of the New Vehicle Entrance Process
Notes. The look-ahead step is T � 6 and the move-ahead step is M � 2. A vehicle’s current location is marked by a solid dot, and its plannedfuture locations are marked by open circles. Vehicle 3 enters the traffic at t � 7, although the feasibility test is performed six steps earlier, at t � 1.As soon as the feasibility test is passed, vehicle 3 is incorporated in the planning algorithm even though its physical presence and movement donot start until t � 7.
instances to understand the solution process, and thenrun a set of computational experiments to gain moreinsights about high-density air traffic.
5.1. Effects of T and MThe PMP algorithm is inherently a heuristic ap-proach, progressively weaving the overall motionplan with segments of locally optimal plans. We use asimple stylized case to compare PMP’s solution to theglobal optimum and to demonstrate the effects ofparameters T and M on the solution quality, partic-ularly in the context of Section 4.2.1.
The case consists of four vehicles whose OD coordi-nates, maximum speeds si, and maximum accelerationsai, as well as the system-level time separation h,
are given in Table 3. For this case and subsequentnumerical cases, the distance unit (du) and time unit (tu)are kept indefinite, so the units for distance, time,speed and acceleration should be construed as du, tu,du/tu and du/tu2, respectively. We first solve MP withT � 40 using a global solver. At the solution, all ve-hicles are able to reach their destinations at time 33,and the total transit distance is 951.2. Themotion traceat this solution is shown in the upper left plot inFigure 7. We also run the same case using PMP atT � 2 to 9 and M � 1 to T, a total of 44 combinations.The solutions, that is, time to completion and thetotal transit distance, are given in Table 4. Note thatthe objective function (2) that couples time and dis-tance is used for inducing good-property solutions,whereas the objective value itself is of little practicalmeaning to traffic managers. In the interest of space,the objective values are not reported here.As indicated by the values of si and ai, the time to full
stop is 3. Thus, according to the rules (14) and (15),the recommended ranges for T and M are T ≥ 6 andM ≤ T − 3. Conforming configurations have theirtime entries in bold in Table 4. If we consider these
experiments as a “random sample” drawn from thespace of all possible data scenarios and algorithmicconfigurations, we can perform aWelch’s t-test for thenull hypothesis that themean time for the conformingconfiguration is equal to the mean time for the non-conforming configuration, where conforming meansfollowing the rules (14) and (15). The p-value for theone-sided test is 0.0006, giving strong evidence thatthe recommended rules are effective at inducingshorter overall delays. Furthermore, the variance intime for the conforming configurations are quitesmall, indicating that going beyond the minimumrecommendation (i.e., further increasing T and de-creasing M) does not yield significant improvementsin transit time. The transit distances, shown in theright part of the table, are not affected as much byartificial speed limits incurred by nonconformance.However, as a general trend, they do benefit from largerlook-aheadandsmallermove-aheadsteps.Three selectedconfigurationsofT andM are plotted in Figure 7, alongwith the global solution of MP. The look-far-move-small strategy as exemplified by the setting T � 9 andM � 1 approximates the global solution very well,whereas the more frugal and aggressive setting(i.e., the onewithT � 6 andM � 3) generates a slightlycurvy motion path. The setting with T � 2 and M � 2not only artificially limits the move speed (hence,prolongs the total transit time), but also exhibits amyopicmotion plan; that is, the vehicles head straighttoward the destination until bumping into a conflictzone, then make sharp detours to avoid collision.
In terms of T’s and M’s effects on computationalcost, the experimental results are aligned with whatis postulated in Section 4.2.1, that is, a setting witha large T and small M takes more computing time.In this four-vehicle instance, the setting with T � 9and M � 1 took 8.48 seconds to execute (on average,8.48/33 � 0.26 seconds per major iteration), and thesetting with T � 6 and M � 3 took 2.85 seconds [onaverage, 2.58/((34 − 1)/3) � 0.23 seconds per majoriteration]. Moreover, if we were to assume that a unittime interval in the model corresponds to one second
of clock time, then the former setting would receive aper-iteration computing budget of one second and thelatter setting a budget of three seconds. The actualcomputing times would be within budget in bothsettings under this assumption.
5.2. Conflict Resolution of 30 VehiclesWecontinue toapply thePMPalgorithmtoa traffic sceneconsisting of 30 vehicles starting in a ringed formation,shown in the upper left of Figure 8. The destination ofeach vehicle is the point symmetric to its origin on thecircle. For example, vehicle 1’s destination coincideswith the origin of vehicle 16 and vice versa. The radiusof the circle is 250; hence, each vehicle’s trip distance is500, and the total transit distance is 15,000. The speedlimit is si � 12, the acceleration limit ai � 4, and theseparation time is h � 2. The small circle around eachvehicle shows the vehicle’s buffer zone.An ad hoc feasible solution onemight come upwith
is to have all vehicles move along the perimeter ofthe circle at the same pace, figuratively rotating theringed formation by 180 degrees. In this solution, allvehicles refrain from pursuing the shortest path inexchange for time efficiency. The corresponding totaltrip distance is approximately 30 × 250 × π � 23,562,which amounts to a detour factor of 1.57. Another adhocsolution manageable by a human dispatcher wouldbe to dispatch vehicles in small groups at a time. Forexample, let vehicle 1 and vehicle 16 traverse first whilekeeping other vehicles waiting. Such a solution wouldrequire less detour but incur much more delay, whichwould correspond to an inefficient use of the airspaceand result in a very low flow efficiency.We used the PMP algorithm at T � 6 and M � 3 to
produce amore efficient solutionwith a total tripdistanceof 19,385. Figure 8 demonstrates snapshots of the traf-fic scene at four time points, whereas a full animateddemonstration is available in the online companion.In the initial stage (t � 1 to 23), all vehicles make
greedy moves toward their destinations, ostensiblygravitating to the center zone. The instantaneousflow efficiency initially ramps up to a high level
Table 4. Time to Route a Four-Drone Case with Different T and M Values
(the ramping effect during times 1 to 3 as shown inFigure 9 is due to acceleration from the still state)and then drops dramatically as a severe traffic jamis formed around time 24. The subsequent trafficmovement demonstrates an interesting visual effect, inwhich the whole cluster of vehicles rotates in co-ordination so that all vehicles get closer to their des-tinations. During time t � 40 to 50, themajor rotation iscompleted, and the compact cluster starts to disin-tegrate into smaller clusters. In the final stage, that is,t � 60 to 80, most vehicles decelerate to exactly reachtheir destination points. This explains why the speedefficiency ramps downduring this period. The overallprocess finishes at time 80, and the total distancetraveled by all vehicles is 19,385.
Figure 9 tracks the speed, heading, and flow effi-ciency metrics over time. It can be seen that apart fromthe ramping stages in the beginning and the end pe-riods, the instantaneous speed andheading efficienciesare for themost part positively correlated. This impliesthat delay and detour concur at proportional in-tensities, both of which are related to the local densityof traffic. In addition, the instantaneous efficiencymetrics vary greatly over time, prompting the use oftime-averagedmetrics tosummarize theoverall systemperformance. We will do this in the subsequent simu-lation analysis.
5.2.1. Computational Efficiency. The CONOPT solver(single thread) was used to solve the model MP(t0) in
Figure 8. (Color online) Four Temporal Snapshots of a Traffic Scene Consisting of 30 Vehicles
Notes. The vehicles start their trips from a ringed formation. Each vehicle’s destination is the corresponding point symmetric to the center of thering. When a vehicle reaches its destination, it disappears from the scene and is excluded from the model. The demonstrated solutionsimultaneously minimizes the total delay and detour.
each iteration of the PMP algorithm, and the wholesolution process took less than 9minutes (2.5 minuteswhen CONOPT4 was used).
For comparison, a solution was also computedby solving the MPmodel (2)–(6) directly using a largeenough T. Given the knowledge that PMP couldcomplete in 80 time steps, T was set to 90 in this runto allow for enough time. Because it was a one-shotsolve, all pairs (i, j) with i< j were included in #. Theresulting NLP model had 47,086 rows, 10,201 col-umns, and 188,341 nonzeroes. It took CONOPT4(utilizing all 20 cores) 8.7 hours to find a local opti-mal solution. The solution converged at time 83when all vehicles had reached their respective des-tinations. The total distance traveled was 20,759. Thisrun could also be construed as the first iteration ofa run by the PMP algorithm configured with T � 90.The motion trace generated by the direct solution ofthe MP looks similar to that obtained by the PMP al-gorithm shown in Figure 8. The full motion trace dataand animation are provided in the online companion.
The progressive solution appears to be superiorto the direct MP solution in all regards: it takes lesstime (80 versus 83) to dispatch all vehicles, it leads toa shorter total distance (19,385 versus 20,759), and ittakes much less computational time (2.5 minutesversus 8.3 hours). Despite being counterintuitive, itis a legitimate result given the subtle difference be-tween what is being optimized and what is beingassessed here as performance metrics. Specifically,the objective function couples time and distance to-gether, whereas the quality of a solution is moreconveniently assessed by transit time and transitdistance separately.
5.3. More Comparisons Between PMP and the One-Shot MP
We also conducted randomized experiments toelaborate the computational comparison betweenthe PMP algorithm and the single-solve MP model.The goal of the comparison is to demonstrate thevalue of the PMP algorithm in materializing the goodproperties of theMPmodel.Wewill show that, absentthe progressive algorithm, the MP model alone isnot very useful in terms of computing time and re-liability. We created 20 data sets each consisting of 20vehicles whose OD coordinates are uniformly sam-pled in a 200 × 200 square area. The vehicle maximumspeeds are randomly picked from the set {8, 12, 16},time headway h is set to 2, and time to full speed is setto 3. To ensure the cases are solvable, we qualified theOD generation process by requiring not only thatthe 20 origins are adequately distanced (this is neededby both MP and PMP), but also that the 20 destina-tions are adequately distanced (this is needed onlyby the one-shot MP method because it is unable toremove completed vehicles, which might cause dead-locks). The data generation (both here and in the se-quel) was performed in R with documented randomseeds, so the data source is reproducible.We first ran all cases using PMPwith T � 6 andM �
1 and observed that all of them can be completed in60 time intervals. We then ran the cases using a single-solve MP by setting T � 60. Dispatch was consid-ered completed if the total residual distance to des-tination summed over all vehicles was less than 1. Thetotal effective transit distance (sum of straight-line
Figure 9. (Color online) Traffic Efficiency over Time in the30-Vehicle Instance
Notes. The flow efficiency is the product of the speed efficiency andthe heading efficiency. The instantaneous efficiencies vary greatlyover the course of the trip because of congestion. The ramp-up andramp-down effects at the beginning and the end periods, respectively,are attributed to the acceleration constraint.
Table 5. Solution of 20 Randomized Cases of 20-VehicleConflicts
origin-to-destination distances) for these 20 casesranged between 2,300 and 2,813, so the above distancetolerance was small enough.
The results are summarized in Table 5. For eachapproach, the time to complete the dispatch, thedetour factor (distance traveled divided by the to-tal origin-to-destination distance), the total residualdistance (R. dist.) upon algorithm termination, andthe computing time in seconds (CPU) are reported.The performances of the two approaches in termsof time and detour are mixed: PMP won 13 cases intime, 13 cases in detour, and 11 cases in both timeand detour. Notably in four cases (Cases 5, 8, 11, and18), the one-shot MP method failed to complete thedispatch by a significant residual distance mar-gin. Case 8 is visualized in Figure 10 to aid the di-agnosis. It turned out that because of MP’s inabil-ity to recognize and remove completed vehiclesfrom the model, the density remained quite high allthe time, significantly slowing down the trafficprogression.
5.4. Simulation Study of Traffic Efficiency5.4.1. Data Set Generation. Each data set consisted ofN vehicles, each having a randomly generated ODpair in the 500 × 500 square area and a speed limitsi randomly picked from the set {8, 12, 16} to createabundant variation in density. Time headway wash � 2, and acceleration limit ai � si/3. The followingacceptance criteria were applied to ensure initialfeasibility and that the trips were nontrivial: (a) thelocation of all origins must satisfy the separationconstraint (12), and (b) the origin and destination ofeach vehicle must be at least 100 distance units apart.
For the number of vehiclesN, the range between 5 and70 with a spacing of 5 was considered, that is, N � 5,10, 15, . . ., 70, and at each level, 20 random instanceswere generated. A similar experimental setup can befound in Sislak, Samek, and Pechoucek (2008) and Ny
Figure 10. (Color online) Demonstration of Case 8
Notes. The left panels shows the residual distance to the destination over time in the two approaches. PMP completed at time 44 and the one-shotMP failed to complete by time 60. The right panel shows a traffic scene at time 60 in the one-shotMP approach. Several vehicles have not reachedtheir destinations and are moving slowly.
Figure 11. (Color online) An Air Traffic Scene Consisting of70 Vehicles
Notes. In this particular instance, the density is 0.51, and the detourfactor is 2.0, meaning that on average, every mile of effective deliveryrequired two miles of actual travel distance because of congestion. Inthis snapshot, many vehicles are not heading in the destination-pointing direction (marked by gray dashed lines) because of collision-avoidance constraints.
and Pappas (2010). Altogether, there were 280 in-stances in the experiment.
Figure 11 demonstrates one of the densest trafficscenarios simulated in the experiment. It is a snapshotof the continuous traffic flow at time t � 5. The trafficdensity as defined in Table 1 is 0.51, meaning thatabout half of the transit space (outlined by the bluedotted and dashed line in the figure) is filled by ve-hicles’ buffer zones, that is, the nonoverlapping cir-cles. The size of a circle reflects the maximum speedof the corresponding vehicle, as indicated in Equa-tion (1).
5.4.2. Result Analysis. The first thing to note is thealgorithm’s reliability: it was able to produce a fea-sible motion plan for all instances regardless of thetraffic density. This ability is attributed to the me-ticulous design of all components of the algorithm.Finding the global optimal solution is computation-ally prohibitive for reasons discussed in Section 4.1.3.Feasibility guarantee accompanied by good solutionquality is what one would practically aim for in amotion-planning algorithm. Figure 12 demonstratesthe solution quality by plotting pairwise scatter plotsover selected metrics. A full summary table as wellas the detailed motion trace log files are available inthe online companion. In Figure 12, each dot repre-sents one of the 280 experimental instances. Densityis the instantaneous traffic density ρt0 in the startingconfiguration, and throughput is the total effective
transit made by all vehicles per unit of time. We cansee that, in general, a higher density corresponds to ahigher system throughput. However, when density istoo high,most vehicles will go a longway and spend along time maneuvering rather than making effective,destination-bound transits, and therefore the systemthroughput will deteriorate, as exhibited by the down-ward trend in the high-density end in the density–throughput plot. Flow efficiency, as expected, is nega-tively correlated with density, and the correlationappears to be quite uniform across the wide range ofdensities examined here.Finding local optimal solutions for large-scale non-
linear programs, even starting with a good, feasible so-lution, is quite time consuming, as shown in Figure 13.The box plot shows the average solution time for eachrun of MP(t0) for cases with different numbers ofvehicles. Note that it is not the time it took to completethe entire traffic simulation, which was actually oneto two orders of magnitude longer. The computingtime seems to increase at an increasing rate with thenumber of vehicles, and becomes more variable forlarger-scale instances. Nonetheless, there are a fewthings to note. First, in these experiments, the size ofthe airspace under study was fixed; thus, by increas-ing the number of vehicles, we were essentially in-creasing the trafficdensity,with the aimof (a)demonstrat-ing the algorithm’s reliability in handling dense casesand (b) presenting the system behavior across awide range of density settings to generate insights on
Figure 12. Empirical Relation Between Density and System Efficiency
Notes. Each point represents an experimental instance. Density and flow efficiency are negatively correlated. Density and throughput arepositively correlated in themoderately low range. At lowdensity,more detour indicates a higher throughput, whereas toomuch detourwill startto reduce throughput.
system response. It is not meant to imply that ex-tremely high densities are common in practice. Asobserved in Section 2.2, the computational complexityof a system consisting of many vehicles is upperbounded by the complexity of its largest collision-cautious cluster. Second, there exists room to shortenthe computing time. For instance, if it was not for ourexperimental assumption that it took up to three timeintervals to accelerate to full speed (which stipulateda minimum planning horizon T of 6), we could adopta shorter horizon in each incremental run to save time,which of course might come with some compro-mise in flow efficiency. Tuning the algorithmic pa-rameters to achieve the best combination of solutionquality and solution time is best performed for aspecific target system, which we will leave for fu-ture work. Moreover, new NLP solvers such as CON-OPT4 are multithreaded, which can dramatically reducethe computing time on a multicore workstation.
6. ConclusionGoinghighdensity is an inevitable route to increasing thetransportation volume without dramatically expandingspace use. When many vehicles simultaneously traversea shared space, intervehicle motion coordination is ofcritical importance for safety and efficiency. Under safetyconstraints, the foremost objective of a motion-planningalgorithm is to ensure a feasible, deadlock-free solutionthat keeps the trafficmoving at any density level. A goodalgorithm should also mitigate delay and detours toachieve high transit efficiency. In this paper, a motion-planning algorithm that meets these expectations hasbeen developed, and its effectiveness has been validatedvia comprehensive simulation experiments. In particular,we have performed the following main contributions:
1. resolved a safety loophole typically found inoptimization-based motion-planning models via pro-viding analytical bounds on the minimum separationdistance between vehicles;
2. proposed a nonlinear optimization model tocentrally coordinate the trajectories and resolve con-flicts and developed a progressive algorithm based ona series of heuristic measures to dispatch vehicles inhigh-density traffic with a feasibility guarantee;
3. coined a set of metrics to measure high-density2D traffic in a general sense, applied these metrics insimulation experiments to evaluate and validate themotion-planning algorithm, and generated usefulinsights regarding air traffic density and flow.
6.1. Future WorkUAS traffic management in low-altitude airspace isa relatively new area of research, and there are nu-merous important and challenging questions to ad-dress. Summarized below are some future directionsthat will extend what has been proposed in this paper.
6.1.1. Dynamic Space Layering and Altitude Assignment.Although the models developed in this paper areapplicable for an arbitrary dimension in the Euclideanspace, the main target application has been traffic ona 2D planar area. Based on the interface for accom-modating new vehicle entrance in the PMP algorithm,future work could include the dynamic assignmentof vehicles into different altitude layers to furtherreduce congestion; see Lehouillier et al. (2017) forreference. This would entail, for instance, a broader-scale traffic optimization that takes the 2D motion-planning algorithm as a critical component. The trafficflow metrics proposed herein will continue to beuseful and can be further refined.
6.1.2. Hybrid and Distributed Schemes for Vehicle Mo-tion Computing. The large volume of UAS traffic inthe future airspace cannot be all managed centrallyby one computer. As discussed in this paper, bothtemporal and spatial decomposition are available inpractical deployments. Future work could investigatedata processing and edge-computing paradigms formotion computation, as well as hybrid navigationschemes utilizing vehicle’s on-board AI to achieve ascalable implementation of UAS traffic management.
6.1.3. Robustness, Resiliency, and Uncertainty Manage-ment. Uncertainty is almost ubiquitous in any com-plex system. Apart from the inherent variance in thesensing, control, and actuating components, manystochastic factors such as changing weather and elec-tromagnetic environments also heavily affect an ae-rial vehicle’s mobile stability and accuracy. Therefore,comprehensivegeographic information system,weather,and geomagnetic data need to be incorporated and
Figure 13. Solution Time of the Motion-Planning Model forDense Traffic
Notes. It takes increasinglymore time to solve for denser cases involvingmore vehicles. The solution time also varies more greatly for largerinstances. For definition of box plot markers, refer to chapter 6 ofMontgomery and Runger (2014).
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