Planning Optimal Paths for Multiple Robots on Graphs

Transcription

1 3 IEEE International Conference on Robotic and Automation (ICRA) Karlruhe, Germany, May 6, 3 Planning Optimal Path for Multiple Robot on Graph Jingjin Yu Steven M. LaValle Abtract In thi paper, we tudy the problem of optimal multirobot path planning (MPP) on graph. We propoe two multiflow baed integer linear programming (ILP) model that compute minimum lat arrival time and minimum total ditance olution for our MPP formulation, repectively. The reulting algorithm from thee ILP model are complete and guaranteed to yield true optimal olution. In addition, our flexible framework can eaily accommodate other variant of the MPP problem. Focuing on the time optimal algorithm, we evaluate it performance, both a a tand alone algorithm and a a generic heuritic for quickly olving large problem intance. Computational reult confirm the effectivene of our method. I. INTRODUCTION Planning colliionfree path for multiple robot, an eaily tated yet difficult problem, ha been actively tudied for decade [, 3,,, 3, 5, 6, 9, 3, 33]. The hardne of the problem mainly reide with the coupling between the robot path which lead to an enormou tate pace and branching factor. A uch, algorithm that are both complete and (ditance) optimal, uch a the A [9] algorithm and it variant, do not perform well on tightly coupled problem beyond very mall one. On the other hand, fater algorithm for finding the path generally do not provide optimality guarantee: Sifting through all feaible path et for optimal one greatly increae the earch pace, which often make thee problem intractable. In thi paper, we invetigate the problem of planning optimal path for multiple robot with individual goal. The robot have identical but nonnegligible ize, are confined to ome arbitrary connected graph, and are capable of moving from one vertex to an adjacent vertex in one time tep. Colliion between robot i not allowed, which may occur when two robot attempt to move to the ame vertex or move along the ame edge in different direction. For thi general etting, we propoe a network flow baed integer linear programming (ILP) model for finding robot path that are time optimal or ditance optimal. Our time optimality criterion eek to minimize the number of time tep until the lat robot reache it goal; ditance optimality eek to minimize the total ditance (each edge ha unit ditance) Jingjin Yu i with the Department of Electrical and Computer Engineering, Univerity of Illinoi at UrbanaChampaign, Urbana, IL 68 USA. jyu8@uiuc.edu. Steven M. LaValle i with the Department of Computer Science, Univerity of Illinoi at UrbanaChampaign, Urbana, IL 68 USA. lavalle@uiuc.edu. Thi work wa upported in part by NSF grant 95 (IIS Robotic) and 3535 (Cyberphyical Sytem), DARPA SToMP grant HR58, and MURI/ONR grant N95. We recommend reading the full verion of thi paper at The oftware from thi work with an API interface i available at traveled by the robot. Taking advantage of the tate of the art ILP olver (Gurobi i ued in thi paper), our method can plan time optimal, colliionfree path for everal dozen of robot on graph with hundred of vertice within minute. A a univeral ubroutine, colliionfree path planning for multiple robot find application in tak panning aembly [8, 6], evacuation [9], formation control [, 7,,, 8], localization [6], object tranportation [5,, 7], earch and recue [], and o on. Given it importance, path planning for multirobot ytem ha remained a a ubject of intene tudy for many decade. Given the vat ize of the available literature, we will only mention related reearch on dicrete MPP and refer the reader to [3,, ] and the reference therein for a more comprehenive review of the ubject. From an algorithmic perpective, dicrete MPP i a natural extenion of the ingle robot path planning problem: One may combine the tate pace of all robot and treat the problem a a planning problem for a ingle robot. A algorithm can then be ued to compute ditance optimal olution to thee problem. However, ince naive A cale poorly due to the cure of dimenionality, additional heuritic method were propoed to improve the computational performance. One of the firt uch heuritic, Local Repair A (LRA ) [33], plan robot path imultaneouly and perform local repair when conflict arie. Focuing on fixing the (locality) hortcoming of LRA, Windowed Hierarchical Cooperative A (WHCA ) [3] propoed to ue a pacetime window to allow more choice for reolving local conflict while limiting the earch pace ize at the ame time. For additional heuritic exploring variou pecific local and global feature, ee [,, 6]. Formulation of MPP problem with optimality guarantee have alo been tudied. The mot general optimality criterion i the total path length traveled by all robot, which i conitent with the ditance heuritic ued by the A algorithm. Since A i the bet poible among all uch algorithm for finding ditance optimal olution, one hould not expect complete and true optimal algorithm to exit that perform much better than the baic A algorithm in all cae. Neverthele, thi doe not prevent algorithm from quickly olving certain intance optimally. One uch algorithm that i alo complete, MGSx, i preented in [5] (note that the grid world formulation in [5], which allow diagonal move in general, even in the preence of diagonal obtacle, doe not carry over to general graph or geometric model in robotic). For time optimality, for a verion of the MPP problem that reemble our formulation more cloely, it wa hown that finding a time optimal olution i NPhard [7], implying that our formulation i alo intractable [ 3]. Finally, U.S. Government work not protected by U.S. copyright 36

2 it wa hown that finding the leat number of move for the N Ngeneralization of the 5puzzle i NPhard [ 8]. Here, time optimality equal ditance optimality, which i not the cae in general. The main contribution of thi paper are twofold. Firt, adapting the contruction from [3], we develop ILP model for olving time optimal and ditance optimal MPP problem. The reulting algorithm are hown to be complete. Our approach i quite general and eaily accommodate other formulation of the MPP problem, including that of [ 5]. Second, we provide thorough computational evaluation of our model performance: With a tateoftheart ILP olver, our model are capable of olving large problem intance with few dozen of robot fairly fat. Such a reult i in ome ene the bet we can hope for becaue the bet poible algorithm for uch problem cannot run in polynomial time unle P=NP. A an added bonu, we alo how that the (time optimal) algorithm work well a a ubroutine for quickly olving MPP problem (nonoptimally). The ret of the paper i organized a follow. We provide problem definition in Section II, along with a motivating example. Section III relate MPP to multiflow, etablihing the equivalence between the two problem. In Section IV, ILP model are provided for obtaining time optimal and ditance optimal olution, repectively. Section V i devoted to briefly dicuing baic propertie of the n puzzle, which i an intereting benchmark problem on it own. We evaluate the computational performance of our algorithm in Section VI and conclude in Section VII. II. MULTIROBOT PATH PLANNING ON GRAPHS A. Problem Formulation Let G=(V,E) be a connected, undirected, imple graph (i.e., no multiedge), in which V ={v i } i it vertex et and E = {(v i,v j )} i it edge et. Let R = {r,...,r n } be a et of robot that move with unit peed along the edge of G, with initial and goal location on G given by the injective map x I,x G : R V, repectively. The et R i effectively an index et. A path or cheduled path i a map p i :Z V, in which Z :=N {}. Intuitively, the domain of the path are dicrete time tep. A path p i i feaible for a ingle robot r i if it atifie the following propertie:. p i () = x I (r i );. For each i, there exit a mallet ki min Z uch that for all k ki min, p i (k) x G (r i ); 3. For any k < ki min, (p i (k), p i (k )) E or p i (k)= p i (k ). We ay that two path p i, p j are in colliion if there exit k Z uch that p i (k)= p j (k) (colliion on a vertex, or meet) or(p i (k), p i (k ))=(p j (k), p j (k)) (colliion on an edge, or headon). If p(k)= p(k ), then the robot tay at vertex p(k) between the time tep k and k. Problem (MPP on Graph) Given (G,R,x I,x G ), find a et of path P = {p,..., p n } uch that p i are feaible path for repective robot r i and no two path p i, p j are in colliion. A natural criterion for meauring path et optimality i the number of time tep until the lat robot reache it goal. Thi i ometime called the makepan, which can be computed from {ki min } for a feaible path et P a T P = max i n kmin i. Another frequently ued objective i ditance optimality, which count the total number of edge traveled by the robot. We point out that ditance optimality and time optimality cannot be atified at the ame time in general: In Fig., let the dotted traight line have length t and the dotted arc ha length.5t from ome large even number t. The four olid line egment are edge with unit length. Auming that robot, are to move from the location marked with olid circle to the location marked with gray dotted circle. Time optimal path take.5t time tep with a total ditance of.5t ; ditance optimal path take t 3 time tep with a total ditance of t. Fig.. Time optimality and ditance optimality cannot be atified imultaneouly for thi etup. B. A Motivating Example Fig (a) (b) a) A 9puzzle problem. b) The deired goal tate. To better characterize what we olve in thi paper, look at the example in Fig.. We call thi problem a 9puzzle, which i a variant of the 5puzzle [8]. Given the robot a numbered in Fig. (a), we want to get them into the tate (configuration i alo ued in thi paper to refer to the ame, depending on the context) given in Fig. (b) (uch a configuration i often referred to a row major ordering). Coming up with a feaible olution for uch a highly contrained problem i nontrivial, let alone olving it with an optimality guarantee. The time optimal algorithm we preent in thi paper olve thi problem intance under. econd. The olution i given in Fig. 3. The time optimality of the olution i evident: It take at leat four tep for robot 9 to reach it goal. Fig. 3. A tep olution from our algorithm. The directed edge how the moving direction of the robot at the tail of the edge. III. MULTIROBOT PATH PLANNING AND MULTIFLOW A. Network Flow In thi ubection we provide a ummary of the network flow problem formulation pertinent to the introduction of our 363

3 algorithm. For urvey on network flow, ee [, 5]. A network N = (G,c,c,S) conit of a directed graph G = (V,E) with c,c : E Z a the map defining the capacitie and cot on edge, repectively, and S V a the et of ource and ink. We let S=S S, with S denoting the et of ource and S denoting the et of ink vertice. For a vertex v V, let δ (v) (rep. δ (v)) denote the et of edge of G going to (rep. leaving) v. A feaible (tatic) S,S flow on thi network N i a map f : E Z that atifie edge capacity contraint, e E, f(e) c (e), () the flow conervation contraint at non terminal vertice, v V\S, f(e) f(e)=, () e δ (v) e δ (v) and the flow conervation contraint at terminal vertice, F( f) = ( f(e) f(e)) v S e δ (v) e δ (v) (3) f(e) f(e)). e δ (v) = v S ( e δ (v) The quantity F( f) i called the value of the flow f. The claic (inglecommodity) maximum flow problem ak the quetion: Given a network N, what i the maximum F( f) that can be puhed through the network? The above formulation concern a ingle commodity, which correpond to all robot being inter exchangeable. For MPP, the robot are not inter exchangeable and mut be treated a different commoditie. Multicommodity flow or multiflow capture the problem of flowing different type of commoditie through a network. Intead of having a ingle flow function f, we have a flow function f i for each commodity i. Again, the maximum flow problem can be poed for a multiflow etup. B. Equivalence between MPP and multiflow Viewing robot a commoditie, we may connect MPP and multiflow. Thi relationhip (Theorem ) wa firt tated in [3]. To make the preentation clear, we ue a an example the imple graph G in Fig. (a), with initial location { i },i=, and goal location{ i },i=,. An intance of Problem i given by (G,{r,r },x I : r i i,x G : r i i ). We now convert thi problem to a network flow problem, N =(G,c,c,S S ). Given the graph G and a natural number T, we create T copie of vertice from G, with u(t ) u(t) (a) v(t ) (b) v(t) Fig.. a) A imple G. b) A gadget for plitting an undirected edge through time tep. indice,,,..., a hown in Fig. 5. For each vertex v G, denote thee copie v() = v(),v(),v(),v(),...,v(t). For each edge (u,v) G and time tep t,t, t < T, add the gadget hown in Fig. (b) between u(t),v(t) and u(t ),v(t ) (arrow from the gadget are omitted from Fig. 5 ince they are mall). For the gadget, we aign unit capacity to all edge, unit cot to the horizontal middle edge, and zero cot to the other four edge. Thi gadget enure that two robot cannot travel in oppoite direction on an edge in the ame time tep. To finih the contruction of Fig. 5, for each vertex v G, we add one edge between every two ucceive copie (i.e., we add the edge (v(),v()),(v(),v() ),...,(v(t),v(t) )). Thee correpond to the green and blue edge in Fig. 5. For all green edge, we aign them unit capacity and cot; for all blue edge, we aign them unit capacity and zero cot. e Fig. 5. The timeexpanded network (T = ). Fig. 5 (with the exception of edge e and e, which are not relevant until Section IV), called a timeexpanded network [], i the deired G. For the et S, we may imply let S = {v() : v { i }} and S ={v(t) : v { i }}. The network N =(G,c,c,S S ) i now complete; we have reduced Problem to an integer maximum multiflow problem on N with each robot from R a a ingle type of commodity, ummarized in the following theorem (for complete proof, ee the full verion of thi paper). Theorem Given an intance of Problem with input parameter (G,R,x I,x G ), there i a bijection between it olution (with maximum number of time tep up to T ) and the integer maximum multiflow olution of flow value n on the timeexpanded network N contructed from (G,R,x I,x G ) with T time tep. C. Accommodating other formulation Our network flow baed approach for encoding the MPP problem i fairly general; we illutrate thi uing an example. The grid world formulation from [ 5] allow (ingle) diagonal croing. That i, for vertice v,...,v on the four corner of a quare cell with v,v 3 and v,v diagonal to each other, repectively, it i poible for a robot to move from v to v 3 provided that v 3 i unoccupied and the v v diagonal i not ued in the ame time tep. To include thi contraint in the ILP model, we may imply add the gadget tructure in Fig. 6 to the timeexpanded network contruction. The incluion of the gadget will allow a ingle diagonal croing; the extra path do not create an iue ince no two robot can go through a ingle vertex at the ame time tep (enforced by the blue dotted edge in Fig. 5). e 36

4 Fig v (t ) v (t ) v (t ) 3 v (t ) A gadget for allowing diagonal croing. IV. ALGORITHMIC SOLUTIONS FOR OPTIMAL MULTIROBOT PATH PLANNING Given the timeexpanded network N = (G,c,c,S S ), it i traightforward to create an integer linear programming (ILP) model with different optimality objective. We invetigate two objective in thi ection: Time optimality or makepan (the time when the lat robot reache it goal) and ditance optimality (the total ditance traveled by all robot). A. Time optimality Time optimal olution to Problem can be obtained uing a maximum multiflow formulation. A a firt tep, we introduce a et of n loopback edge to G by connecting each pair of correponding goal and tart vertice in S, from the goal to the tart. For convenience, denote thee loopback edge a {e,...,e n } (e.g., edge e,e in Fig. 5). Thee edge have unit capacity and zero cot. Next. for each edge e j G, create n binary variable x, j,...,x n, j correponding to the flow through that edge, one for each robot. x i, j = if and only if robot r i pae through e j in G. The variable x i, j mut atify two edge capacity contraint and one flow conervation contraint, e j G, n i= x i, j i, j n, i j, x i, j =, () v G and i n, x i, j = x i, j. (5) e j δ (v) e j δ (v) The objective function i max x i,i. (6) i n For each fixed T, the olution to the above ILP problem equaling n mean that a feaible olution to Problem i found. We are to find the minimal T that yield uch a feaible olution. To do thi, we tart with T being the maximum over all robot the hortet poible path length for each robot, ignoring all other robot. We then build the ILP model for thi T and tet for a feaible olution. If the model i not feaible, we increae T and try again. The firt feaible T i the optimal T. The robot path can be extracted baed on the proof of Theorem. The algorithm i complete: Since the problem i dicrete, there i only a finite number of poible tate. Therefore, for ome ufficiently large T, there mut either be a feaible olution or we can pronounce that none can exit. Calling thi algorithm MINMAKESPAN (time optimal MPP), we have hown the following. Propoition 3 Algorithm MINMAKESPAN i complete. B. Ditance optimality Ditance optimality objective can be encoded uing minimum cot maximum multiflow. Contraint () and (5) remain; to force a maximum flow, let x i,i = for i n. The objective i given by min e j G, j>n, i n c (e j ) x i, j. (7) The value given by (7), when feaible, i the total ditance of all robot path. Let T t denote the optimal T produced by MINMAKESPAN (if one exit), then a ditance optimal olution exit in a timeexpanded network with T = nt t tep. Calling thi algorithm MINTOTALDIST (ditance optimal MPP), we have Propoition Algorithm MINTOTALDIST i complete. Due to the large number of tep needed in the timeexpanded network, MINTOTALDIST, in it current form, i not very fat in olving problem with many robot. Therefore, our evaluation in thi paper focue on MINMAKESPAN which, on the other hand, i fairly fat in olving ome very difficult problem. MINTOTALDIST, however, till prove ueful in providing time optimal and near ditance optimal olution uing the output of MINMAKESPAN. V. PROPERTIES OF THE n PUZZLE The example problem from Fig. eaily extend to an n n grid; we call thi cla of problem the n puzzle. Such problem are highly coupled: No robot can move without at leat three other robot moving at the ame time. At each tep, all robot that move mut move ynchronouly in the ame direction (per cycle) on one or more dijoint cycle (ee e.g., Fig. 3). To put into perpective the computational reult on n puzzle that follow, we make a characterization of the tate tructure of the n puzzle for n 3 (the cae of n= i trivial) Fig. 7. A 3tep procedure for exchanging robot 8 and 9. Propoition 5 All tate of the 9puzzle are connected via legal move. PROOF. We how that any tate of a 9puzzle can be moved into the tate hown in Fig. (b). From any tate, robot 5 can be eaily moved into the center of the grid. We are left to how that we can exchange two robot on the border without affecting other robot. Thi i poible due to the procedure illutrated in Fig. 7. Uing imilar technique, it i not hard to how the following (ee full verion of paper for proof)

5 Theorem 6 All tate of an n puzzle, n 3 are connected via legal move. Corollary 7 All intance of the n puzzle, n 3, are olvable. VI. SOLUTIONS AND EVALUATION Our experimentation in thi paper focue on MIN MAKESPAN with the main goal being evaluating the comparative efficiency of our approach rather than puhing for bet computational performance. A uch, our implementation i Java baed and did not directly take advantage of multicore technology. We note that, Gurobi, the ILP olver ued in our implementation, can engage multiple core automatically for hard problem. We ran our code on an Intel Q66 quadcore machine with a GB JavaVM. A. Time optimal olution to n puzzle The firt experiment we performed wa evaluating the efficiency of the algorithm MINMAKESPAN for finding time optimal olution to the n puzzle for n=3,,5, and 6. We ran Algorithm MINMAKESPAN on randomly generated n puzzle intance for n=3,,5. For the 9puzzle, computation on all intance completed uccefully with an average computation time of.36 econd per intance. To compare the computational reult, we implemented a (optimal) BFS algorithm. The BFS algorithm i heavily optimized: For example, cycle of the grid are precomputed and hard coded to ave computation time. Since the tate pace of the 9 puzzle i mall, the BFS algorithm i capable of optimally olving the ame et of 9puzzle intance with an average computation time of about.89 econd per intance. Once we move to the 6puzzle, the power of general ILP olver become evident. MINMAKESPAN olved all randomly generated 6puzzle intance with an average computation time of 8.9 econd. On the other hand, the BFS algorithm with a priority queue that worked for the 9puzzle ran out of memory after a few minute. A our reult how that an optimal olution for the 6puzzle generally require 6 time tep, it eem natural to alo try bidirectional earch, which cut down the total number tate tored in memory. To complete uch a earch, one ide of the bidirectional earch generally mut reach a depth of 3, which require toring about 5 8 tate (the branching factor i over per tep), each taking 6 bit of memory. Thi turn out to be too much for a GB JavaVM: A bidirectional earch ran out of memory after about minute in general. For the 5puzzle, without a good heuritic, bidirectional earch cannot explore a tiny fraction of the fully connected tate pace with about 5 tate. On the other hand, MINMAKESPAN again conitently olve the 5puzzle, with an average computational time under hour over randomly created problem. We alo briefly teted MIN MAKESPAN on the 36puzzle. While we had ome ucce here, MINMAKESPAN generally doe not eem to olve a randomly generated intance of the 36puzzle within hour, which ha 3.7 tate and a branching factor of well over 6. B. Time optimal olution for grid graph For problem in which not all graph vertice are occupied by robot, MINMAKESPAN can handle much larger intance. In a firt et of tet on thi ubject, a grid ize of 5 i ued with varying percentage of obtacle (imulated by removed vertice) and robot for evaluating the effect of thee factor. The computation time (in econd) and the average number of optimal time tep (in parenthei) are lited in Table I. The number are average over randomly TABLE I % ob Number of robot () 7.3(3) 6.7(7) 3.6(6) 7.7(6).() 7.8() 3.(5).(6) 8.6(5) 5 3.9(5) 6.(3) 3.8(6) 3.8(6) 6(7).(3) 7.7(6).9(7) 39.3(5) 73(7) 5.7(7) 8.(7).8(3) 68.(8) 53(9) 3 3.(3) 9.9(33) 9 3() 5 8.6(9) 3 N/A created intance. For each run, a maximum of econd i allowed (uch limit, omewhat arbitrary, were choen to manage the expected running time of the entire et of experiment; our complete algorithm hould terminate eventually). Entrie with upercript number ugget the run did not all finih within the given time. The upercript number repreent the ucceful run on which the tatitic were computed. N/A mean no intance finihed within the allowed time. From the reult, we oberve that the percentage of randomly placed obtacle doe not affect the problem difficulty, a meaured by computational time, in a monotonic way. On one hand, more obtacle remove more vertice from the grid, making the problem ize maller, reducing the computational difficulty. On the other hand, a more obtacle are introduced, the reduced connectivity of the graph make the problem harder. In particular, the 5 grid etting uddenly become a hard problem with 3% obtacle. The difficulty i alo reflected by the average number of tep in an optimal olution: Longer time mean reduced availability of alternative path. TABLE II Number of robot % ob 3 5.() 3.6() 3.7(3) 87.5(6) (8) 9 In a econd tet on even larger problem, 3 3 grid with % obtacle were tried. For between and 5 robot with an increment of, random intance each were created; each intance i allowed to run a maximum of half an hour. The tatitic, imilarly compoed a that in Table I, i lited in Table II. We oberve that the problem i imilar in difficulty to the 5 grid etting with 5% obtacle, but much impler than that with 3% obtacle. C. Uing MINMAKESPAN a a generic heuritic In the lat experiment, we exploit MINMAKESPAN a a generic heuritic for locally reolving path conflict for 366

6 large problem intance. By generic, we mean that the heuritic i not coded to any pecific robot/grid etting. In our algorithm, path are firt planned for ingle robot (ignoring other robot). Afterward, the robot are moved along thee path until no further progre can be made. We then detect on the graph where progre are talled and reolve the conflict locally uing MINMAKESPAN. For every conflict, we apply MINMAKESPAN to it neighborhood of ditance. The above tep are repeated until a olution i found. For evaluation, we ran the above algorithm on a 3 3 grid with % obtacle. We allow each intance to run a maximum of 3 econd. The reult, each a an average over run for a certain number of robot, are lited in Table III While we did not make idebyide comparion TABLE III Number of Robot Running time () Fully olved % goal reached with the literature due to (eemingly mall but) important difference in problem formulation, the computation time and completion rate of our algorithm appear comparable with the tate of the art reult from other author. VII. CONCLUSION In thi paper, we introduced a multiflow baed ILP algorithm for planning optimal, colliionfree path for multiple robot on graph. We provided complete ILP algorithm for olving time optimal and ditance optimal MPP problem. Our experiment confirmed that MINMAKESPAN i a feaible method for planning time optimal path for tightly coupled problem a well a for larger problem with more free pace. Moreover, we howed that MINMAKESPAN can erve a a good heuritic for olving large problem intance efficiently. 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