A Visibility Roadmap Sampling Approach for a Multi-Robot Visibility-Based Pursuit-Evasion Problem
Trevor Olsen, Anne M. Tumlin, Nicholas M. Stiffler, Jason M. O'Kane
AA Visibility Roadmap Sampling Approach for a Multi-RobotVisibility-Based Pursuit-Evasion Problem
Trevor Olsen, Anne M. Tumlin, Nicholas M. Stiffler and Jason M. O’Kane
Abstract — Given a two-dimensional polygonal space, themulti-robot visibility-based pursuit-evasion problem tasks sev-eral pursuer robots with the goal of establishing visibility withan arbitrarily fast evader. The best known complete algorithmfor this problem takes time doubly exponential in the numberof robots. However, sampling-based techniques have shownpromise in generating feasible solutions in these scenarios. Oneof the primary drawbacks to employing existing sampling-basedmethods is that existing algorithms have long execution timesand high failure rates for complex environments. This paperaddresses that limitation by proposing a new algorithm thattakes an environment as its input and returns a joint motionstrategy which ensures that the evader is captured by one ofthe pursuers. Starting with a single pursuer, we sequentiallyconstruct Sample-Generated Pursuit-Evasion Graphs to createsuch a joint motion strategy. This sequential graph structure en-sures that our algorithm will always terminate with a solution,regardless of the complexity of the environment. We describean implementation of this algorithm and present quantitativeresults that show significant improvement in comparison to theexisting algorithm.
I. I
NTRODUCTION
Autonomous reconnaissance tasks, in which robots striveto observe salient features of objects or agents within theirenvironments, remain one of the most active threads ofresearch within the robotics community. Such tasks havewide-ranging application domains such as environmentalmonitoring [1]–[5], surveillance [6]–[9], and search-and-rescue [10]–[12]. Many of these tasks can be framed as two-player games played amongst opposing teams: evaders (whowish to evade capture) and pursuers (who seek to capturethem). This paper is concerned with a specific form of thistwo-player game, wherein a team of pursuers must locate anevader (or group of evaders) in a polygonal environment.Specifically, we address one such problem where a groupof pursuers, each equipped with an omni-directional sensorthat extends to the polygonal boundary, must form a motionplan to locate an arbitrarily fast evader in a polygonalenvironment. Figure 1 illustrates this scenario. The literaturehas a number of results for this problem in the single-pursuercase, including algorithms with strong guarantees such ascompleteness [13] and optimality [14]. However, the casein which multiple pursuers cooperate is not nearly as wellunderstood. A complete algorithm is known, but it runs intime doubly-exponential in the number of pursuers [15].
T. Olsen, A. M. Tumlin, N. M. Stiffler, and J. M. O’Kaneare with the Department of Computer Science and Engineering,University of South Carolina, Columbia, SC 29208, USA. { tvolsen,atumlin } @email.sc.edu { stifflen, jokane } @cse.sc.edu This material is based upon work supported by the National ScienceFoundation under Grant Nos. 1659514 and 1849291. Fig. 1: An example multi-pursuer solution strategy generated by the pro-posed algorithm.
One approach to overcome the computational challengeposed by this multiple-pursuer pursuit-evasion planningtask [16], showed the feasibility of sampling-based tech-niques to generate joint motion strategies. Nevertheless, thatapproach has several important limitations. (i)
The existing algorithm lacks insight into how the sam-pling should be performed, and treats the samplingdistribution as a “black box.” (ii)
The existing algorithm requires a predetermined numberof pursuers as input; it cannot adapt the number ofpursuers to the complexity of the environment. (iii)
The solutions generated by this approach are of poorquality, in the sense that there nearly always are motionsby one or more of the pursuers that do not activelycontribute to the search.After a review of related work (Section II), precise statementof the problem (Section III), and a summary of importantconcepts from prior work on this problem (Section IV),this paper makes three new contributions that address thelimitations of the existing algorithm. (i)
We introduce a new sampling strategy, tailored to thevisibility-based nature of the problem (Section V-A). (ii)
We describe a method that eliminates the need for thenumber of pursuers to be provided as input, insteaditeratively increasing the size of the team (Section V-B). (iii)
We present a post-processing algorithm that improvessolution quality (Section V-C).Finally, Section VI presents quantitative evaluations of thesenew improvements and Section VII previews future work. a r X i v : . [ c s . R O ] F e b I. R
ELATED WORK
This research blends ideas from two vibrant threads ofprior robotics research: visibility-based pursuit-evasion andsampling-based motion planning in high dimensional spaces.In regard to pursuit-evasion, this work is most closelyaligned with the problem first introduced by Suzuki andYamashita [17], in which an evader operating in a geometricenvironment seeks to locate an unpredictable evader capableof moving arbitrarily quickly. This work was later expandedby Guibas, Latombe, LaValle, Lin, and Motwani [13] whoprovide a complete algorithm for the single pursuer scenarioin simply-connected environments for a pursuer with anomnidirectional field-of-view. Park, Lee, and Chwa [18]identified necessary and sufficient conditions for a search tobe feasible for a single pursuer. Other results for the singlepursuer scenario that build upon this foundation provide re-sults such as completeness [13], optimality [14], or considermore restrictive scenarios with respect to pursuer parameterssuch as sensing and actuation [19]–[23].More recently, the community has placed increased em-phasis on the study of richer scenarios where a team ofpursuers cooperate during the search [16], [24]–[26]. Stifflerand O’Kane [15] present an algorithm utilizing a cylindricalalgebraic decomposition that, while complete, relies uponconstructing a graph whose size is doubly exponential inthe complexity of the environment. That work subsequentlyserved as motivation for approaches that utilize heuris-tics [16] that seek to overcome the problem complexity byutilizing sampling techniques.More generally, sampling based techniques have beenemployed in a number of planning contexts where computingan exact solution proves computationally intractable such asmotion planning[27]–[31] and manipulation planning[32]–[34]. One caveat of sampling-based methods is that they quiteoften suffer from the curse of dimensionality [35] whereby,as the number of dimensions increases, the search spacebecomes so vast that the number of samples required foradequate coverage of the space increases dramatically. Anumber of different approaches have been proposed to com-bat this problem. One recent result draws samples in lower-dimensional subspaces to search for a feasible solution,and incrementally reasons about higher dimensions whileutilizing the information gained in the lower dimensionalgraph [36]. For the specific multi-robot case in which theconfiguration space is a Cartesian product of the configu-ration spaces of individual agents in the system, one novelapproach seeks to reason about each agent independently(a subdimension), and only when the agents reach a pointwhere they interact with one another is there a lifting to ahigher-dimensional space [37].III. P
ROBLEM STATEMENT
A. The environment, the evader, and the pursuers
Let E ⊂ R be a bounded, closed, connected polygonalset called the environment . An evader moves within E .We describe the evader’s position via a continuous function p (0) p ( t ) p ( T ) p (0) p ( t ) p ( T ) e (0) e ( t ) e ( T ) Fig. 2: Example evader and pursuer paths. The evader is detected at time t . e ( t ) : [0 , ∞ ) → E to represent the location of the evader attime t . There is no restriction on the speed of the evader, solong as it is finite. Because our goal is to establish visibilityin the worst case, the ability to detect a single evader isequivalent to being able to detect any number of evaders.Thus, we assume there is a single evader.We task n pursuers with the goal of detecting the evader.We consider both scenarios in which the number of pursuersis known and fixed, an in which the number of pursuersto utilize is determined by the algorithm as the plan isgenerated. Let p i : [0 , ∞ ) → E represent the location of the i th pursuer as a function of time. We refer to such functions as motion strategies . Each pursuer is equipped with knowledgeof E and an omnidirectional sensor which extends until thenearest point on the boundary of E in each direction. That is,a pursuer at point q ∈ E can see every point in its visibilitypolygon, V ( q ) = { q ∈ E | q q ⊆ E } . B. Objective
The algorithmic problem we address is to find a vector ofmotion strategies (cid:104) p , p , . . . , p n (cid:105) such that, for some t ≥ and j ∈ { , . . . , n } , e ( t ) ∈ V ( p j ( t )) . Such a collectionof motion strategies is called a solution . Specifically, weconsider two related problems. a) Fixed: Given an environment E and a positiveinteger n , generate a solution using exactly n pursuers.Algorithms for this problem can be evaluated by examiningboth the time needed to generate a solution as well as thetime needed for the pursuers to execute that solution. b) Variable: Given an environment E , generate a so-lution that uses as few pursuers as possible. In addition tothe run time and execution time criteria mentioned above,algorithms for this variant of the problem can also be judgedby the number of pursuers utilized by the computed solution.IV. B ACKGROUND
This section concisely summarizes some essential priorresults upon which our new contributions build. Specifically,we describe how the pursuers’ knowledge about the evader’spossible position can be maintained (Section IV-A) andpresent a data structure that encapsulates the progress of asearch for a complete solution (Section IV-B). ig. 3: An example of a cleared (right) and contaminated (left) shadow,given the pursuer’s movement history.
A. Shadows and shadow events
In general, the pursuers will be able to see only a subset ofthe environment at any particular time, while the remainingparts of the environment are out of view of all of the pursuers.We call these unobserved regions shadows . Definition
The shadow region is the portion of the environ-ment which cannot be seen by any pursuer at time t . Thatis, S ( t ) = E \ (cid:83) ni =1 V ( p i ( t )) . A maximal path-connectedcomponent of S ( t ) is called a shadow .The crucial information about each shadow is whetheror not the evader may be hiding within that shadow. Ashadow s is called cleared at time t if, based on the pursuers’joint motions up to time t , it is not possible for the evaderto be within s without having been seen by one of thepursuers. Analogously, a shadow is said to be contaminated if it is possible for the evader to be hiding within it giventhe pursuers’ motions up to time t . As the pursuers movethrough the environment, the individual shadows changecontinuously. However, the cardinality of the set of shadowschanges only when a shadow event occurs, i.e. a shadow appears , disappears , splits , or merges with another shadow.The shadow events induce the following changes to the statusof a shadow: • Appear : A shadow can appear if the pursuers lose visionof a region within the environment. In this event, thenew shadow has a cleared status. • Disappear : A shadow can disappear if the pursuers gainvision of a region which was previously a shadow. Thelabel for the shadow which disappeared is discarded. • Merge : Two or more shadows can merge into a largershadow. In this event, the new shadow is assigned thelabel cleared only if all merging shadows were cleared;otherwise, the new shadow is labeled contaminated. • Split : If a shadow becomes path disconnected, we sayit was split. The newly formed shadows have the statusof the initial from which they were formed.More detail about these types of shadow events and theirclear/contaminated labels may be found in the work ofGuibas, Latombe, LaValle, Lin, and Motwani [13].
B. Sample-generated pursuit-evasion graphs
To aid in the search for a joint motion strategy for thepursuers, we utilize the Sample-Generated Pursuit-EvasionGraph (SG-PEG) data structure; additional details about the SG-PEG appear in the original paper [16]. An SG-PEG isa directed graph G = ( V G , E G ) , representing a portion ofthe connectivity of the joint configuration space for a fixednumber n of pursuers. Each vertex in V G represents a jointconfiguration for the pursuers ( p , . . . , p n ) ∈ E n . Edges in E G connect pairs of vertices for which it is possible for eachpursuer to make a collision-free straight line motion betweenthe two corresponding configurations. That is, if an edgeexists between two vertices representing joint configurations ( p , . . . , p n ) and ( q , . . . , q n ) , then p i q i ⊂ E for each i ∈ { , . . . , n } . One vertex, corresponding to the startingjoint configuration of the robots, is designated as the root ofthe graph. Each vertex maintains a list of non-dominatedshadow labels (i.e. clear/contaminated statuses) that arereachable by traversing the graph along some (possibly non-simple) path from the root.The primary operation that can be performed on a stan-dard SG-PEG is A DD S AMPLE ( p , . . . , p n ) which, given acollision-free joint configuration, performs three main steps: (i) It inserts a new vertex v at the given joint configuration. (ii) For any other vertex u within a pre-defined connectiondistance in E n , it checks whether the straight-lineconnection between v and u would be collision-free.If so, and if the shortest path in the graph between v and u is sufficiently long, it creates edges vu and uv . (iii) For each new edge thusly added, A DD S AMPLE com-putes the shadow events described in Section IV-Ainduced by motion along that edge. It then propagatesthe reachable shadow label information across the newedge and then recursively across the graph, to determinewhat new reachable shadow labels, if any, are nowpossible at which vertices, due to this new edge.The utility of the SG-PEG is that if any vertex v has areachable shadow label that is fully cleared (i.e. it has areachable shadow label set in which each shadow bears aclear label), then a solution can be extracted from the graphby tracing back via the appropriate edges to the root vertex.V. A LGORITHM DESCRIPTION
Our approach to this problem is based on two significantadditions and one modification to the prior algorithm ofStiffler and O’Kane [16]. The basic idea of that prioralgorithm is to generate random samples in E n and usethem to construct an SG-PEG, continuing until the SG-PEGindicates that it contains a solution. Algorithm 1 shows theenhancements that we propose. New elements in comparisonto the prior algorithm are highlighted: modifications to thesampling strategy in purple text and new additions in bluetext. Note that, n , the number of robots has been removedin our variant. Details about these changes appear below. A. Web sampling
Stiffler and O’Kane proposed a handful of samplingdistributions, but none of them proved significantly moreeffective than simple uniform random sampling of the jointconfiguration space. This approach appears not to be partic-ularly effective in this domain, because the environment is lgorithm 1 S OLVE ( E , n , C , S ) Input: an environment E , a number of pursuers n ,an expansion criterion C and a sampler S G ← empty SG-PEG for n pursuers 1 pursuer p ← S. GET S AMPLE () G. ADD S AMPLE ( p ) G. SET R OOT ( p ) while no solution has been found do p ← S. GET S AMPLE () G. ADD S AMPLE ( p ) if C is met then G. ADD P URSUER () X ← EXTRACT S OLUTION () X (cid:48) ← REFINE S OLUTION ( X ) return X (cid:48) Fig. 4: [left] An example web. Red points are initial points; the blueare intersection points. Green edges connect each intersection point to theinitial points from which it is induced. [right] The distribution induced byweb sampling, based upon 750 generated webs. Note the higher density atjunctions and corners. comprised of regions whose surveyance is essential to findinga solution. To combat this issue, we propose a new strategyfor generating samples, which specifically takes into accountthe visibility component of the problem.The approach is based on a randomized structure calleda web in E , which is constructed in two steps. First,we construct a set of initial points P ⊂ E . Each initialpoint is selected sequentially and randomly, from the regionoutside the union of visibility polygons of its predecessors.The process continues until (cid:83) p ∈ P V ( p ) = E . Then thealgorithm selects a collection of intersection points Q ⊆ E by examining pairs of initial points { p i , p j } ⊆ P , and placinga point uniformly at random within V ( p i ) ∩ V ( p j ) , if thatintersection is non-empty. A web is simply the combination W = P ∪ Q . Figure 4 illustrates this concept.We utilized webs to generate samples from E n in Lines 2and 6 by generating a separate web for each pursuer andsampling, without replacement, from those points. If thepoints in any of the webs are exhausted, we generate anotherset of webs and continue. B. Variable numbers of pursuers
In the prior algorithm, the number of pursuers in thesolution was required as an input. This information wasnecessary because the SG-PEG data structure stores jointconfigurations drawn from E n ; thus n must be known toconstruct an SG-PEG. To alleviate this limitation, we instead propose a sequen-tial process, in which the number of pursuers is graduallyincreased as the algorithm proceeds. Realizing this approachin the planner requires us to resolve two complications.First, the algorithm requires a mechanism to transition,in mid-stream, from an n -pursuer SG-PEG to an ( n + 1) -pursuer SG-PEG (Line 9). One straightforward approach isto simply discard the existing vertices and edges and restartthe search with an additional pursuer. (This is referred to asthe ‘Clear’ option in Section VI.)However, it may be preferable to ensure that our previouseffort is not wasted. To this end, we propose a new methodthat clones the first pursuer in each vertex of the SG-PEG.That is, for each vertex in the SG-PEG, we replace its jointconfiguration ( p , p , . . . , p n ) with ( p , p , . . . , p n , p ) . Thisadds an additional pursuer to the graph, without changing anyof the reachable labels or edges. Thus, the cloning option isextremely efficient, but it leads to an SG-PEG in which thenewly-added pursuer moves in parallel with another pursuer.Future edges added at the ( n + 1) -th layer will correspondto independent motions for these two robots (as they drawfrom there own unique set of webs).Next, we must decide when to expand the number ofpursuers (Line 8). We consider two options. First, we proposea method that devotes fixed effort to each stage of thesearch. The process begins by rapidly generating a trivialsolution by placing pursuers until their visibility fully coversthe environment. This gives a (generally very loose) upperbound N on the number of pursuers required. Then, givena target total run time of T limit seconds, we apportion thetime between the possible numbers of pursuers , . . . , N viaa Poisson distribution. The Poisson distribution was selecteddue to the placement of the mean, as well as its skew andshape. We choose a tunable parameter α which determinesthe fraction of time to spend on the final step that utilizes N pursuers, so that according to the definition of the Poissondistribution, we have α = λ N e − λ /N ! . From this equation,the algorithm numerically computes the Poisson parameter λ and allocates T limit λ i − e − λ / ( i − to the search with i pursuers before proceeding to i +1 . Because of the existenceof the upper bound N , this method will always produce asolution, although it may require a large number of pursuers.As an alternative, we also consider a stalled progress approach, based upon monitoring the minimum sum of thecontaminated shadow area across all vertices of the graph.(n.b. We have a solution if and only if this value reaches .)If this value fails to improve by at least 5% after adding M samples, we add a new pursuer. To enable a fair comparisonto the fixed effort method, we once again return a trivialsolution if no solution is found with fewer pursuers. C. Solution refinement
Because of the sampling-based nature of this algorithm,its outputs are likely to have extraneous motions. This issueis noticeably more severe in our context than for traditionalsampling-based motion planning because the generated so-lutions may travel several times along certain edges in an z a z b z i z z a z b z i c c (cid:48) Fig. 5: A cut of length c during solution refinement.Fig. 6: Refining a solution. [left] Before. [right] After. effort to clear specific shadows. In this section, we introducea post processing method which takes a joint motion strategyand optimizes it by removing unnecessary pursuer motions.Our method is similar to standard shortcut-based pathsmoothing. We select two points z a and z b at distance c along the solution path in E n , and check whether takinga shortcut directly from z a to z b yields a path that isstill a correct solution. Figure 5 depicts the process. Ourcheck features one important difference from traditional pathsmoothing: In addition to ensuring that the refined solutionis collision-free, we must also ensure that it remains acorrect solution, i.e. that the refinement does not allow anyshadows to remain contaminated. We do this by trackingforward through the shortened path, applying the shadowevents experienced along that path —Recall Section IV-A—to update the shadow labels.Given this shortcut operation, we greedily optimize thepath, proceeding systematically over decreasing values of c and increasing positions of z a . (From these two, z b is readilycomputed.) Each time we discover a shortcut yielding acorrect solution, that shortened solution replaces the previoussolution, and the process continues. See Figure 6.VI. E VALUATION
This section evaluates the performance of the proposedapproach. We implemented the algorithm in C++ and exe-cuted it on a computer with an Intel i7-7500U processor and12GB of memory, running Ubuntu 18.04.2 64-bit.We performed simulations in the environments shown inFigures 1 (
Office ), 4 ( H ), and 6 ( Spider ). This selectionof environments is intended to provide a variety commonenvironmental traits. The H environment contains narrowcorridors, the Spider environment possesses numerous hardto reach areas, and the Office environment is somewhatuniformly spread out, with a complex boundary.We generated solution strategies for these environmentsusing both classes of algorithms described thus far: those thatutilize a fixed number of pursuers and those that can varythe number of pursuers. In the former case, we considered both web sampling (WS) and uniform random sampling(SO14) [16] and varied the fixed number of robots between 2and 5. In the latter case, we considered all four combinationsof expansion criterion (fixed effort (FE) or stalled progress(SP)) and graph expansion method (clone pursuer (Clone) orclear progress (Clear)).For each of these scenarios, we executed 25 trials andrecorded the computation time in seconds, number of pur-suers used in the returned solution, as well as the totalnumbers of vertices and edges generated. Each experimentwas allotted 10 minutes of computation time; if no solutionwas produced in that time, we considered the trial to bea failure. Failed trials are excluded from the statistics, butit should be noted that if they were included, they wouldcontribute a run time of at least 10 minutes. The results,summarized via the means ( µ ) and standard deviations ( σ ) ,appear in Tables II, I and III. A number of conclusions maybe drawn from the results. a) WS improves performance over SO14: First, in all ofthe tested environments, our WS method outperformed SO14by a wide margin. The smallest improvement in computationtime occurred in the Office environment, perhaps due to theuniform nature of the environment and the complexity of thegeometry calculations. b) Clone and Clear appear to be complementary:
Forthe simulations with a variable number of pursuers, we lookto compare the performance between the cloning method andthe clear method when expanding a graph. When using FEas the expansion criterion, the results are remarkably similarbetween the two methods. At first glance, one may assumethe cloning method should outperform the clearing methoddue to the larger amount of information the graph contains.This however, is not always the case, because an increasednumber of vertices means an increased amount of time to addeach new sample, due to the propagation of reachable labelsacross the graph. When SP was the expansion criterion,clearing has a faster computation time, but also a muchhigher mean and standard deviation for the number ofpursuers. It was often the case that SP with clearing wouldreturn a trivial solution, as the algorithm cleared a substantialamount of meaningful calculations performed. c) Allowing the number of pursuers to vary incurs onlya modest computational cost:
Lastly, we compare the fixednumber of pursuers using WS, to the variable number ofpursuers methods. For each environment, the run times ofthe variable number of pursuers were within approximately afactor of 2 of that of the fixed number of pursuers. A portionof this additional time comes from the variable number ofpursuers constructing a single pursuer graph, in environmentswhich all require at least 2 pursuers to solve.Finally, for each environment, we refined all 25 solutionsgenerated by WS with 2 pursuers. The results are sum-marized in Table IV, which shows the mean and standarddeviation of the computation time (in seconds) along withthe length of the solution path (in meters) before and after itwas refined. The results show that this approach effectivelyand consistently reduces the path lengths. able I. Simulation results for the Office environment (Figure 1). success comp time (s) num robots num vertices num edgesrate µ σ µ σ µ σ µ σ
WS ( n = 2 ) SO14 ( n = 2 )
96% 96.890 78.231 2.000 0.000 117.875 62.007 219.167 131.542
WS ( n = 3 ) SO14 ( n = 3 ) WS ( n = 4 ) SO14 ( n = 4 ) WS ( n = 5 ) SO14 ( n = 5 )
80% 235.866 129.786 5.000 0.000 4081.000 1361.404 4495.150 2546.067
FE Clone ( α = 0 . FE Clear ( α = 0 . SP Clone ( M = 30) SP Clear ( M = 30) Table II. Simulation results for the H environment (Figure 4). success comp time (s) num robots num vertices num edgesrate µ σ µ σ µ σ µ σ
WS ( n = 2 ) SO14 ( n = 2 ) WS ( n = 3 ) SO14 ( n = 3 ) WS ( n = 4 ) SO14 ( n = 4 )
96% 43.636 42.848 4.000 0.000 639.042 365.724 972.042 704.745
WS ( n = 5 ) SO14 ( n = 5 )
96% 81.949 62.208 5.000 0.000 1743.708 1512.375 2422.125 2732.005
FE Clone ( α = 0 . FE Clear ( α = 0 . SP Clone ( M = 30) SP Clear ( M = 30) Table III. Simulation results for the Spider environment (Figure 6). success comp time (s) num robots num vertices num edgesrate µ σ µ σ µ σ µ σ
WS ( n = 2 ) SO14 ( n = 2 )
84% 312.341 115.955 2.000 0.000 152.952 90.760 296.619 182.206
WS ( n = 3 ) SO14 ( n = 3 )
96% 211.234 89.930 3.000 0.000 90.167 33.983 160.750 65.811
WS ( n = 4 ) SO14 ( n = 4 )
96% 216.941 83.815 4.000 0.000 142.708 283.093 257.083 560.368
WS ( n = 5 ) SO14 ( n = 5 )
92% 222.129 77.185 5.000 0.000 81.261 31.517 123.087 55.083
FE Clone ( α = 0 . FE Clear ( α = 0 . SP Clone ( M = 30)
88% 403.098 91.530 3.727 0.827 163.682 35.301 275.955 50.614
SP Clear ( M = 30) Table IV. Refinement results for the solutions produced by WS ( n = 2 ). comp length lengthtime (s) before (m) after (m) µ σ µ σ µ σ Office H Spider
VII. C
ONCLUSION
This paper presents a sampling-based algorithm for avisibility-based pursuit-evasion problem that generates ajoint motion strategy for a team of robots in a polygonalenvironment. The three primary contributions are a novelsampling strategy for this domain, an iterative algorithmfor generating a joint motion strategy for the pursuers, and a post-processing path-smoothing algorithm that refines thestrategy returned by the main algorithm. The algorithm wasshown to outperform existing techniques.Future work might build upon the results in this paper.First, possibilities remain for enhancing the post processingstep. There remain a number of open questions on how thesekinds of path smoothing algorithms can be best applied tothe pursuit-evasion domain. Second, is the development of ananytime algorithm that begins with an uninteresting solution—for example, one utilizing enough pursuers to ensure thattheir visibility polygons fully cover the environment— andattempts to work backwards, eliminating robots, by searchingfor solutions in the reduced joint sub-space.
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