Shortest Paths for a Two-Robot Rendez-Vous

Transcription

1 Shotest Paths fo a Two-Robot Rendez-Vous Eik L Wyntes Joseph S B Mitchell y Abstact In this pape, we conside an optimal motion planning poblem fo a pai of point obots in a plana envionment with polygonal obstacles We seek a pai of paths upon which the obots can tavel to move fom thei initial positions to positions in which they ae able to communicate In ode to communicate, the obots need to be visible to one anothe We give ecient algoithms fo minimizing eithe the sum o the maximum of the two path lengths 1 Intoduction Imagine that two mobile obots ae woking simultaneously but sepaately on elated tasks Suppose that they must \endez-vous" peiodically to shae infomation which one obot has acquied which may be useful to the othe obot in caying out its assigned task Depending upon the model of communication used, the obots may need to tavel to the same location to shae infomation, o they may only have to move so that they ae within a pescibed distance of one anothe o so that they ae visible to one anothe We assume that the obots use a line-of-sight communication system Theefoe, we seek a pai of endez-vous paths upon which the obots can tavel to become visible to each othe Ou goal is to nd optimal solutions to two vaiations of this poblem In one, we wish to minimize the sum of the lengths Math & Compute Science Dept, Bloomsbug Univesity, Bloomsbug, PA Pats of this eseach wee conducted while the st autho was a gaduate student with the Cente fo Applied Mathematics, Conell Univesity, suppoted by a gant fom Hughes Reseach Laboatoies eik@satunbloomuedu y Applied Math, SUNY, Stony Book, NY Patially suppoted by NSF gant ECSE , by Ai Foce Oce of Scientic Reseach contact AFOSR , and by a gant fom Hughes Reseach Laboatoies jsbm@amssunysbedu of the endez-vous paths; in the othe, we wish to minimize the maximum of the path lengths Given the situation descibed above, ou goal is to nd a pai of paths fo the obots upon which they can tavel to each a conguation (a pai of positions) that pemits them to communicate Futhemoe, we equie the paths to be optimal accoding to one of two distance measues: we seek to minimize eithe the sum of the path lengths o the maximum of the path lengths In addition to its application to obot motion planning, ou wok is applicable to the design of communication netwoks Conside the poblem of establishing a micowave link between two sites that ae not visible to each othe Given that we wish to minimize the total cost of outing cables fom the sites to the bases of the towes and that such costs ae popotional to the total length of cable needed, ou methods can be used to nd optimal locations fo the two towes (the towes must be visible to each othe fo a micowave link to be established) The methods descibed hee complement some of ou ealie wok In [4] we discuss how to nd optimal paths fo two obots that must maintain line-of-sight communication while moving In that pape we stat with obots that ae able to communicate and ensue that communication is not boken duing the motion In this pape, we stat with obots that ae not able to communicate and move them to establish communication Togethe, these esults allow us to nd paths that bing the obots quickly to positions in which they can communicate befoe continuing on to thei destinations, maintaining communication In the next section, we give an algoithm that nds optimal endez-vous paths accoding to the \minsum" citeion In Section 3 we discuss the \minmax" vesion of the poblem Page 1

2 $s 1$ $s 2$ Lemma 21 If an optimal endez-vous (t 1 ; t 2 ) coesponds to a line-of-sight l, then l contains an edge of the visibility gaph $l$ Figue 1: A visible endez-vous coesponds to a lineof-sight though an obstacle vetex 2 The Min-Sum Poblem In this section we dene the Min-Sum vesion of the poblem moe fomally and show how to solve it We assume that the obots ae modeled by points in the plane and that the obstacles to visibility ae disjoint simple polygons We dene feespace to be the set of all points that ae not in the inteio of any obstacle A visible endez-vous is an odeed pai of points that can see one anothe in the sense that the line segment connecting them lies entiely in feespace A line-of-sight though a point p in diection is a line segment obtained by extending ays fom p in diections and? until each cannot be extended futhe without intesecting the inteio of an obstacle We dene a path to be a connected onedimensional subset of feespace Finally, we denote the length of a path by () We wish to solve the following poblem The Min-Sum Visible Rendez-Vous Poblem Instance: An initial conguation (s 1 ; s 2 ), and a set S of disjoint simple polygons with a total of n vetices Requiement: Find a pai of paths ( 1 ; 2 ) fom (s 1 ; s 2 ) to a visible endez-vous (t 1 ; t 2 ) such that the total length of the paths ( 1 ) + ( 2 ) is minimized It is clea that, in ode fo a visible endez-vous to be optimal, it must coespond to a line-of-sight though an obstacle vetex v as shown in Figue 1; othewise, at least one of the two paths could be shotened (we ae assuming that the obots ae not visible to one anothe in the initial conguation) Futhemoe, given an oientation of a line-of-sight l though v, the best paths fom (s 1 ; s 2 ) to a visible endez-vous on l consist of shotest paths fom s 1 and fom s 2 to l The following lemma shows that not evey oientation of a line-of-sight l though v is a candidate line-of-sight fo the optimal endez-vous Poof Suppose that l is a line-of-sight that passes though only one obstacle vetex v Then l is fee to \otate" about v (the length of l is adjusted duing the otation so that it extends as fa as possible at each end without intesecting the inteio of an obstacle) We show that thee exists a diection of otation that impoves the value of the objective function, ie, it deceases the optimal combined distance fom s 1 and fom s 2 to a visible endez-vous on l Suppose that we have an optimal visible endezvous (t 1 ; t 2 ) on l and an optimal pai of paths 1 fom s 1 to t 1 and 2 fom s 2 to t 2 We examine the case in which t 1 and t 2 ae in the elative inteio of l and the distance d 1 fom t 1 to v is geate than the distance d 2 fom t 2 to v The following discussion is illustated in Figue 2 Let x 1 be the last \tuning point" on 1 pio to eaching t 1, and let x 2 be the coesponding tuning point of 2 Fo 1 and 2 to be optimal, segments x 1 t 1 and x 2 t 2 must be pependicula to l Let a 1 and a 2 denote the lengths of these segments If we otate l though a small angle in the diection that educes the distance fom x 1 to l, then we get a new visible endez-vous (t 0 1 ; 2) and new distances a 0 1 and a 0 2 Fom the gue, it is clea that t0 a a0 2 < (a 1? b 1 ) + (a 2 + b 2 ) We also have that b 1 > b 2 since d 1 > d 2 Thus, a a0 2 < a 1 + a 2 Since the topology of the optimal paths will not change in some neighbohood of the cuent oientation of l, small otations in the diection shown will impove the value of the objective function A simila analysis shows that thee exists a diection of otation that will impove the value of the objective function if d 1 < d 2 o d 1 = d 2 The same esult holds if one o both of t 1 and t 2 is an endpoint of l instead of a point in the elative inteio of l Theefoe, l cannot be optimal if it is not \pinned" by two o moe vetices Thus, the optimal line-ofsight must lie along an edge of the visibility gaph and must contain its two endpoints 2 The lemma suggests the following algoithm The Min-Sum Visible Rendez-Vous Algoithm 1 Constuct a shotest path map using s i as the souce fo i 2 f1; 2g 2 Constuct the visibility gaph G = (V; E) of the set of polygons S Page 2

3 $x 1$ $x 2$ $a 1pime$ $a 1$ $t 1pime$ $lpime$ $a 2pime$ $a 2$ $b 1$ $d 2$ $l$ $t 2$ $d 1$ $b 2$ $t 1$ $t 2pime$ Figue 2: Rotating l in the diection that educes the distance fom x 1 to l, educes the total length of the optimal paths to a visible endez-vous on l 3 Fo each extension edge e 2 E, a Find the line-of-sight e 0 obtained by extending e in feespace as fa as possible in both diections b Find the length of the shotest path 1 fom s 1 to e 0 and the length of the shotest path 2 fom s 2 to e 0 using the shotest path maps c Compae ( 1 ) + ( 2 ) to the total length of the best pai of paths found so fa and update if necessay 4 Retun the best pai of paths found The coectness of this algoithm follows fom Lemma 21 The time and space complexity of the algoithm is analyzed below Step 1 takes O(n 2 ) time and O(n) space using the algoithm of Reif and Stoe [7] Step 2 takes O(E + n log n) time and O(E) space using the algoithm of Ghosh and Mount [1] In Step 3 we st nd, fo each edge e of the visibility gaph, the endpoints of the line-of-sight e 0 that we will efe to as the extensionedge of e This takes constant time, using the epesentaion of the visibility gaph obtained in Step 2 Then we nd the lengths of the shotest paths fom s 1 and fom s 2 to e 0 This takes O(n) time as follows Given an extension edge e 0, we locate one endpoint p of e 0 in the shotest path map and calculate its geodesic distance fom s 1 This takes O(log n) time, using standad point location techniques [6] We save p as ou initial guess of the endpoint of the shotest path fom s 1 to e 0 Since p is located on the bounday of an obstacle, it lies on the bounday of a cell c in the shotest path map We continue by tavesing the bounday of c until we each anothe point whee e 0 intesects the bounday of c We calculate the geodesic distance of fom s 1 the same way we did fo p, and compae this distance to the one obtained fo p If is close to s 1, then we have a new candidate fo the endpoint of the shotest path fom s 1 to e 0 Befoe poceeding, we conside the open line segment between p and Since this potion of e 0 lies completely in c, the shotest path fom s 1 to e 0 can end at a point q on this segment only if the shotest path fom the oot of c to e 0 is a segment pependicula to e 0 that intesects e 0 at q If such a segment exists, the geodesic distance of its endpoint q fom s 1 can be calculated in constant time and compaed with that of the cuent candidate fo the endpoint of the shotest path fom s 1 to e 0 If the compaison is favoable to q, it becomes the new candidate endpoint We continue to tavese adjacent cells that ae pieced by e 0 and update ou epesentation of the best candidate path fom s 1 to e 0 When we each the othe endpoint of e 0, we will have detemined the length of the shotest path fom s 1 to e 0 Afte caying out this pocess fo both s 1 and s 2, we will have detemined the sum of the lengths of the shotest pai of paths 1 and 2 that pemit a visible endez-vous on e 0 The total time needed to tavese the boundaies of the cells pieced by e 0 is linea in the numbe of obstacle vetices since the shotest path map is a plana subdivision of feespace Theefoe, Step 3, iteating ove all edges of the visibility gaph, takes O(En) time In summay, we have the following theoem Theoem 21 The Min-Sum Visible Rendez-Vous Poblem can be solved in O(En) time and O(E) space 3 The Min-Max Poblem Now we tun to the Min-Max vesion of the poblem The Min-Max Visible Rendez-Vous Poblem Instance: An initial conguation (s 1 ; s 2 ), and a set S of disjoint simple polygons with a total of n vetices Requiement: Find a pai of paths ( 1 ; 2 ) fom (s 1 ; s 2 ) to a visible endez-vous (t 1 ; t 2 ) such that the maximum length of the paths maxf( 1 ); ( 2 )g is minimized In the Min-Max poblem, Lemma 21 no longe holds In Figue 3, each of the indicated paths has the same length and togethe they fom an optimal minmax pai The line-of-site l containing the optimal visible endez-vous (t 1 ; t 2 ), howeve, is not \pinned" by two vetices Thus we must now conside paths to a line-of-sight l that passes though only a single vetex and calculate the optimal oientation of such a line Page 3

4 $l$ $s 1$ $d$ $t 1$ $s 2$ $d$ $t 2$ Figue 3: An optimal min-max visible endez-vous may occu at an oientation of a line-of-sight l that is not \pinned" by two o moe vetices Given a line l (an innite line, not a line-of-sight) that passes though a vetex v, we nd the optimal oientation of l by solving O(n 2 ) subpoblems In each subpoblem, we choose vetices x 1 and x 2 and assume that they ae the last tuning points on the shotest paths to l fom s 1 and s 2, espectively With this assumption, the length of the shotest path fom s i to l is the sum of the geodesic distance fom s i to x i and the Euclidean distance fom x i to l Only the second tem vaies with the oientation l() of l It is easy but tedious to solve fo the optimal oientation of l using elementay algeba Seveal cases aise because the shotest path fom x i to l may intesect l at a point in the inteio of feespace o it may be detemined by the intesection of l and an obstacle edge The numbe of possible cases is bounded by a xed constant, howeve, so the solution can be found in constant time We ignoe the obstacles when solving fo the optimal oientation of l Thus we must check whethe the solution we nd is feasible Let l = l( ) and let (t 1 ; t 2 ) be the optimal visible endez-vous on l using x 1 and x 2 as nal tuning points Fo feasibility, we must ensue that each of the pais of points (t 1 ; t 2 ), (x 1 ; t 1 ), and (x 2 ; t 2 ) coesponds to a visibility segment, ie, a line segment lying entiely in feespace If the st test fails, ie, if (t 1 ; t 2 ) is not a visible endez-vous, then l intesects an obstacle in the inteval between t 1 and t 2 In this case, the optimal feasible oientation of l occus when t 1 () and t 2 (), the closest points of l() to x 1 and x 2, ae aligned with v and anothe obstacle vetex In othe wods, t 1 () and t 2 () lie on an extension of a visibility gaph edge We nd visible endez-vous paths of this fom sepaately using the method descibed in Section 2 On the othe hand, if t i is not visible to x i fo some i 2 f1; 2g, then ou choice of x i 's was not optimal, ie, x i is not the last tuning point on a shotest path fom to t i In eithe case, if the feasibility test fails, we s i dop the cuent pai of x i 's fom consideation With these discussions as a pelude, we now descibe the algoithm fo the Min-Max poblem The Min-Max Visible Rendez-Vous Algoithm 1 Constuct a shotest path map using s i as the souce fo i 2 f1; 2g 2 Constuct the visibility gaph by nding the visibility polygon of each vetex 3 Find the best visible endez-vous on an extension of a visibility gaph edge as in Step 3 of the Min- Sum Algoithm using the Min-Max optimization citeion 4 Fo each odeed tiple (x 1 ; v; x 2 ), a Find the Min-Max value of the best pai of paths ( 1 ; 2 ) fom s 1 and s 2 to a line l though v such that x 1 and x 2 ae the last tuning points on 1 and 2 b Check the feasibility of the visible endezvous (t 1 ; t 2 ) detemined by ( 1 ; 2 ), using the visibility polygons of x 1, x 2, and v c If the cuent tiple is feasible, compae maxf( 1 ); ( 2 )g to the best value found so fa, and update if necessay 5 Retun the best pai of paths found This algoithm leads to the following theoem Theoem 31 The Min-Max Visible Rendez-Vous Poblem can be solved in O(n 3 log n) time and O(n 2 ) space Poof The algoithm given above solves the poblem coectly as discussed above and can be implemented to un in O(n 3 log n) time using O(n 2 ) space as follows Computation of the visibility polygons and the visibility gaph in Step 2 takes O(n 2 ) time and O(n 2 ) space using the algoithm of Welzl [8] Steps 1 and 3 take O(En) time just as they did in the Min-Sum case In Step 4, we solve O(n 3 ) subpoblems In each subpoblem, the cost of pefoming the feasibility check dominates the cost of othe opeations and can be done in O(log n) time using binay seach on the visibility polygons about x 1, x 2 and v 2 Page 4

5 4 Extensions and Open Poblems We have shown that the Min-Sum and Min-Max vesions of the Rendez-Vous poblem can be solved ef- ciently Now we give seveal extensions and open poblems One extension to the poblems discussed hee is that in which we allow \Steine Points" in the visibility link between two agents In othe wods, we could equie that the agents move to a conguation in which they can be connected by a path of link distance no geate than k fo some xed k This vesion of the poblem models the situation in which we want to connect two stations by micowave towes and seveal towes in succession may be used to elay messages between stations The Min-Sum vesion of this poblem, in which we minimize the sum of the cable lengths needed to connect the stations to the st and last towes, can be solved by combining ou techniques with known methods fo detemining the link distance between two edges of the visibility gaph [3] Anothe extension of ou poblem is that in which we equie that the agents not only see each othe but also be within distance D of each othe in ode to communicate It is not yet clea whethe this exta equiement inceases the complexity of the poblem If we equie that agents need to be within distance D to communicate, without equiing that they be visible to one anothe, the poblem becomes one of nding the best placement of a cicle of adius D so that thee exist optimal obstacle-avoiding paths fom the initial positions of the agents to two points on the cicle In this case we call the desied pai of nal positions an optimal poximity endez-vous When D = 0 the poblem is tivial since any point on the shotest path between the initial positions is optimal fo the min-sum vesion of the poblem, and the \midpoint" of the shotest path is optimal fo the min-max vesion of the poblem We call the optimal meeting place in this vesion an optimal physical endez-vous Finding an optimal physical endez-vous becomes moe inteesting if the numbe of agents is geate than two and we wish to minimize the maximum geodesic distance fom an initial position to the point whee the physical endez-vous takes place In this case, we have a genealization of the well-known 1- cente poblem In the Euclidean plane this poblem is called the smallest-enclosing-cicle poblem [6] since the solution is the cente of a cicle of minimum adius that contains all of the intial locations The solution in the plane can be found in linea time using a vaiation of Megiddo's linea pogamming algoithm [2] In a simple polygon, the 1-cente o \geodesic cente" can be found in O(n log n) time [5] The question of whethe thee exists an ecient algoithm fo nding the geodesic cente of a set of points among polygonal obstacles, howeve, seems to be still open Extending ou visible endez-vous esults to the k- agent case fo k 3 seems to be challenging, as well Even in the 3-agent min-sum vesion, the local optimality citeion that the line-of-sight connecting two agents must be an extension of a visibility gaph edge no longe holds It is not yet clea whethe the poblem is discete o continuous Refeences [1] S K Ghosh and D M Mount An output sensitive algoithm fo computing visibility gaphs In Poceedings of the 28th Annual IEEE Symposium on the Foundations of Compute Science, pages 11{19, 1987 To appea: SIAM Jounal on Computing [2] N Megiddo Linea pogamming in linea time when the dimension is xed Jounal of the ACM, 31:114{127, 1984 [3] J S B Mitchell, G Rote, and G Woeginge Minimum-link paths among obstacles in the plane In Poceedings of the 6th Annual ACM Symposium on Computational Geomety, pages 63{72, 1990 [4] J S B Mitchell and E L Wyntes Optimal Motion of Covisible Points among Obstacles in the Plane Technical Repot 896, School of Opeations Reseach and Industial Engineeing, Conell Univesity, 1990 [5] R Pollack, M Shai, and G Rote Computing the geodesic cente of a simple polygon Discete and Computational Geomety, 4:611{626, 1989 [6] F P Pepaata and M I Shamos Computational Geomety - an Intoduction Spinge Velag, New Yok, 1985 [7] J H Reif and J A Stoe Shotest Paths in Euclidean Spaces with Polyhedal Obstacles Technical Repot CS , Bandeis Univesity, Waltham, Mass, 1985 [8] E Welzl Constucting the visibility gaph fo n line segments in O(n 2 ) time Infomation Pocessing Lettes, 20:167{171, 1985 Page 5