Multi-Robot Foest Coveage Xiaoming Zheng Sonal Jain Sven Koenig David Kempe Depatment of Compute Science Univesity of Southen Califonia Los Angeles, CA 90089-0781, USA {xiaominz, sonaljai, skoenig, dkempe}@usc.edu Abstact One of the main applications of mobile obots is teain coveage: visiting each location in known teain. Teain coveage is cucial fo lawn mowing, cleaning, havesting, seach-and-escue, intusion detection and mine cleaing. Natually, coveage can be sped up with multiple obots. In this pape, we descibe Multi-Robot Foest Coveage, a new multiobot coveage algoithm based on an algoithm by Even et al. fo finding a tee cove with tees of balanced weights. The cove time of Multi-Robot Foest Coveage is at most eight times lage than optimal, and ou expeiments show it to pefom significantly bette than existing multi-obot coveage algoithms. Index Tems Cell Decomposition, Robot Teams, Spanning Tee Coveage, Teain Coveage. STC MSTC I. INTRODUCTION Fig. 1. Example of STC Fig. 2. Example of MSTC One of the main applications of mobile obots is teain coveage: visiting each location in known teain to pefom a task. Teain coveage is cucial fo tasks anging fom mundane lawn mowing, cleaning o havesting to seach-andescue missions, intusion detection o mine cleaing. It is fequently desiable to minimize the time by which coveage is completed. The single-obot coveage poblem is solved essentially optimally by Spanning Tee Coveage (STC), a polynomialtime coveage algoithm that decomposes teain into cells, computes a spanning tee of the esulting gaph, and makes the obot cicumnavigate it [3]. Natually, coveage can be sped up with multiple obots. The multi-obot coveage poblem is to compute a tajectoy fo each obot so that the cove time (that is, lagest tavel cost of any obot) is minimized. As we show in this pape, this poblem is NPcomplete. It thus becomes necessay to conside heuistics fo solving it. Recently, STC was genealized to Multi-Robot Spanning Tee Coveage (MSTC), a polynomial-time multiobot coveage heuistic [5]. While MSTC povably impoves the cove time of STC, it cannot guaantee its cove time to be close to optimal. In this pape, we descibe Multi- Robot Foest Coveage (MFC), a polynomial-time multi-obot coveage heuistic based on an algoithm fo finding a tee cove with tees of balanced weights, one fo each obot [2]. We thank Gal Kaminka fo inteesting discussions about multi-obot coveage and fo making his pape available on the web. This wok is patially suppoted by NSF Gants IIS-0350584 and IIS-0413196 to Sven Koenig. Any opinions, findings, and conclusions o ecommendations expessed in this mateial ae those of the autho(s) and do not necessaily eflect the views of the National Science Foundation. Ou analytical esults pove the cove time of MFC to be at most eight times lage than optimal, and ou expeiments show it to be significantly bette than the wost-case bound, and also supeio to that of MSTC. MFC has the additional benefit that it tends to etun the obots close to thei initial cells, facilitating thei collection and stoage. II. ASSUMPTIONS The teain to be coveed is discetized into lage squae cells, each of which is eithe entiely blocked o entiely unblocked, and contains fou small squae cells. The obots ae of the same size as a small cell and also identical othewise. We assume that the obots always know thei cuent small cell, and can move between any two unblocked (hoizontally o vetically) adjacent small cells without eo. Unless specified othewise, we assume that such moves takes unit time. Fo ease of exposition, we assume that seveal obots ae able to occupy the same small cell simultaneously, and neve block each othe; a bief discussion in Section VI shows that this assumption is not tuly essential. Note that one could potentially use specialized egionbased coveage algoithms on unit-cost gid gaphs instead of the gaph-based algoithms discussed in this pape. Howeve, even though (some of) ou theoems ae esticted to unit-cost gid gaphs, all of the coveage algoithms discussed in this pape wok on geneal gaphs with positive edge costs. III. SINGLE-ROBOT COVERAGE Spanning Tee Coveage (STC) solves the single-obot coveage poblem in polynomial time [3]. It fist computes
a spanning tee of the gaph whose vetices ae the lage cells, and whose edges connect adjacent unblocked lage cells. The obot then cicumnavigates the spanning tee. STC neve visits any small cell twice and thus minimizes the cove time. In addition, the obot essentially etuns to its initial small cell, facilitating its collection and stoage. Figue 1 shows an example of STC in opeation, including the lage cells (squaes), spanning tee (thick lines), obot (cicle), and its tajectoy (aow). IV. COMPLEXITY OF MULTI-ROBOT COVERAGE While the single-obot coveage poblem can be solved in polynomial time, the poblem becomes significantly moe complex when we ty to minimize the cove time using multiple obots. Theoem 1 shows that two natual vaiants of the multi-obot coveage poblem ae NP-complete. We thus do not expect to be able to solve it exactly in polynomial time, and it becomes necessay to conside heuistics fo solving it. Theoem 1: It is NP-complete to detemine whethe the following two multi-obot coveage poblems can be solved with cove times that ae smalle than a given value: 1) multi-obot coveage poblems with two obots, whee the costs of moving fom one small cell to an adjacent one can be non-unifom (and lage); and 2) multi-obot coveage poblems with k obots, whee k is pat of the input, the costs of moving fom one small cell to an adjacent one ae unifom, and all obots must etun to thei initial small cells. Poof. Clealy, both vesions of the multi-obot coveage poblem ae in NP since one can easily guess the tajectoies of the obots and then veify thei costs in polynomial time. To pove thei NP-hadness, we educe fom patitioning poblems, in which one seeks to patition a set of n integes into k sets of equal sums. Fo k = 2, the poblem is NP-had if the integes can be exponential in n, and is known as the PARTITION poblem [4]. If k is pat of the input, then the poblem is NP-had even if the integes have sizes that ae only polynomial in n. We fist give the eduction fo the second vesion. The 3-PARTITION poblem, known to be stongly NP-complete [4], is defined as follows: Given a positive intege B and positive integes a 1,..., a 3n stictly between B/4 and B/2 with 3n i=1 a i = B n, can they be patitioned evenly into n sets? Given an instance of the 3-PARTITION poblem, we constuct a multi-obot coveage poblem with n obots as follows: We stat with a coido consisting of 3n vetically adjacent lage cells, numbeed fom 1 (bottom) to 3n (top). Fo i = 1,..., 3n, thee is a tunnel of a i 6n hoizontally adjacent lage cells. The tunnel is connected to the i th coido cell. The i th tunnel is to the left of the coido fo odd i and to the ight of the coido fo even i. All n obots stat in the lowe left small cell of the fist coido cell. This completes the constuction, which can be done in polynomial time. We claim that the smallest cove time is at most B 24n + 12n if and only if the given integes can be patitioned evenly into n sets. If the given integes can be patitioned evenly into n sets, then we let the j th obot cove the i th tunnel fo each i S j, ending in its initial small cell. It thus taveses the tunnels fo a cost of at most i S j 4 a i 6n = B 24n, and the coido fo a cost of at most 12n. The total cost thus is at most B 24n + 12n, meeting the equiement. Convesely, if the obots cove all small cells with the desied cove time, then let S j be the set of indices i such that the j th obot is the fist obot to cove the small uppe cell of the i th tunnel cell that is fathest away fom the coido. These sets patition the given integes. The total cost of the j th obot is at least 24n i S j a i, since it needs to tavese its tunnels in both diections to etun to its initial small cell, and needs two moves to tavese each lage tunnel cell. By assumption, the total cost of any obot is at most B 24n+12n, which implies that i S j a i B + 1 2. Since both i S j a i and B ae integes, we have that i S j a i B fo all sets S j. Since n j=1 i S j a i = 3n i=1 a i = B n, all inequalities ae equalities, and the sets S j patition the given integes evenly. This eduction has to be slightly adapted to pove the NP-hadness of the fist vesion. When educing fom the PARTITION poblem, the given integes a i can be exponential in n, so building tunnels of length a i 6n cannot necessaily be done in polynomial time. Instead, we collapse each tunnel to a single lage cell, but make the cost of enteing and leaving the lage cell fom the coido a i 10n. Once we use non-unifom costs fo moving between lage cells, we can also avoid the equiement that the obots etun to thei initial small cells, by adding two moe lage destination cells which have vey high cost (36n 2 i a i) to ente and leave. Then, both obots end in destination cells if thei cove time is small, and thus have enteed and left each of thei tunnel cells. A vey simila poof to the one above then shows that the cove time is bounded by a cetain constant if and only if the given integes can be patitioned evenly into two sets. Cuently, we cannot pove that the second vesion of the multi-obot coveage poblem is NP-had without the equiement that the obots etun to thei initial small cells. It is also open whethe the second vesion is NP-had fo fixed k, although we conjectue it to be. V. EXISTING MULTI-ROBOT COVERAGE ALGORITHMS While single-obot coveage algoithms have eceived a lot of attention, thee ae cuently many fewe algoithms fo the multi-obot coveage poblem. An oveview is given in [1]. Many of the multi-obot coveage algoithms ae fo obots that inteact and plan only locally [9], often called ant obots [8]. Natually, global planning can lead to significantly smalle cove times since it allows the obots to coodinate thei tajectoies much bette. Recently, STC was genealized to Multi-Robot Spanning Tee Coveage (MSTC), a polynomial-time multi-obot cov-
eage heuistic [5]. MSTC fist computes the same spanning tee as STC, and consides the tou that cicumnavigates the spanning tee. Each obot follows the tou segment clockwise ahead of it, with one exception: To impove the cove time, the longest segment is divided evenly between the two adjacent obots. A few small adjustments, detailed in [5], then ensue that MSTC educes the cove time of STC by a facto of at least 2 (o 3/2) fo k 3 obots (o two obots, espectively). Each small cell is visited by only one obot, so thee ae neve any collisions o blockages. Figue 2 shows an example of MSTC in opeation. While the impovement in cove time of MSTC ove STC is significant fo two o thee obots, it does not necessaily incease futhe as the numbe of obots gows. Indeed, Figue 2 gives a bad example fo MSTC, showing that the facto emains two even when a much lage speedup is possible. This is due to the fact that the constuction of the spanning tee does not take into account that it will be split up aftewads, esulting in unbalanced tavel costs of the obots. This obsevation motivates ou idea of constucting a tee cove with one tee fo each obot ight away, whee we ensue duing the constuction that the weights of the tees ae balanced. VI. A NEW MULTI-ROBOT COVERAGE ALGORITHM We now descibe Multi-Robot Foest Coveage (MFC), a new polynomial-time multi-obot coveage heuistic. It is based on an algoithm by Even et al. [2] that gives a fouappoximation fo the poblem of finding a tee cove with given oots, minimizing the weight of the heaviest tee. A. Algoithm MFC opeates on the gaph whose vetices ae the lage cells, and whose edges connect adjacent unblocked lage cells. If obots stat in a lage cell, then MFC makes identical copies of that vetex. MFC fist finds a ooted tee cove fo this gaph in polynomial time, whee the oots ae the vetices that contain obots. (The gaph is allowed to be disconnected, so long as each of its components contains at least one obot.) Each obot then cicumnavigates its tee. We now explain what a ooted tee cove is. Let G = (V, E) be a gaph and R V be a set of oots. An R-ooted tee cove of G is a foest of R tees that cove V. The tees can shae vetices and edges, but thei oots have to be distinct vetices fom R. The weight of a ooted tee cove is the lagest weight of any of its tees. The (min-max) ooted tee cove poblem then is to find a weight-minimal ooted tee cove fo a given gaph and given oots. This poblem is NP-complete, which can be poved by educing the NPhad bin-packing poblem to it [2]. Howeve, TREE COVER is a polynomial-time algoithm by Even et al. that finds a ooted tee cove whose weight is at most fou times lage than optimal [2]. It pefoms a binay seach to detemine the smallest value B such that it can find a ooted tee cove of weight at most 4B. If TREE COVER does not succeed in finding such a tee cove fo the given B, then thee is no ooted tee cove of weight at most B. As a esult, TREE COVER gives a fou-appoximation fo the ooted tee-cove poblem. It opeates as follows: 1) Remove all edges with edge costs lage than B. 2) Contact all oots into a single vetex, find a minimum spanning tee fo the esulting gaph, and then uncontact the single vetex again, splitting the spanning tee into R tees. 3) Decompose each tee into subtees that can shae vetices but no edges. The weight of each subtee is in the ange [B, 2B), with the possible exception of a leftove subtee that contains the oot of the tee, and whose weight is less than B. (See [2] fo details on this step.) 4) Find a maximum matching of all non-leftove subtees to the oots, subject to the constaint that a non-leftove subtee can only be matched to a oot if the non-leftove subtee and leftove tee of the oot (o oot itself) ae at distance at most B. If some non-leftove subtees cannot be matched, this is poof that no ooted tee cove of weight at most B exists. 5) Fo each oot, etun a tee consisting of the oot, the leftove subtee of the oot (if any) of weight at most B, the single non-leftove subtee matched to the oot (if any) of weight at most 2B, and a cost-minimal path of weight at most B fom the non-leftove subtee to the leftove subtee (o oot). The weight of each tee is at most 4B, esulting in a ooted tee cove of weight at most 4B. We enhance TREE COVER in two ways. While these impovements do not affect its wost-case guaantee, they can potentially educe the weight of the etuned ooted tee cove: 1) The smallest value of B fo which TREE COVER finds a ooted tee cove may not be the value of B esulting in the ooted tee cove of the smallest weight. Thus, the impoved vesion of TREE COVER stoes all ooted tee coves that ae computed duing the binay seach, and etuns the best one athe than the last one. 2) When TREE COVER computes the maximum matching in Step 4, it does not take the weights of the esulting tees into account. In the impoved vesion of TREE COVER, a non-leftove subtee can theefoe be matched to a oot only if the non-leftove subtee and leftove tee of the oot (o oot itself) ae at distance at most B, and the weight of the esulting tee is at most B. The impoved vesion of TREE COVER then seaches fo the smallest value B fo which such a matching can be found. Figue 3 shows an example of MFC in opeation. The left figue shows the initial spanning tee fo R = 5 afte it was split into one tee fo each obot. The othe figues show the non-leftove subtees (solid thick lines), the cost-minimal paths fom the non-leftove subtees to the oots (dashed thick lines) and the tajectoies of the five obots (aows). In this case, thee ae no leftove subtees. MFC sends all obots though the naow passage and thus utilizes them to cove
MFC (MST) MFC (Robot 1) MFC (Robot 2) MFC (Robot 3) MFC (Robot 4) MFC (Robot R ) Fig. 3. Example of MFC STC and MFC cove time = 18 Fig. 4. MSTC cove time = 9 MFC and MSTC vesus STC the teain on the othe side, esulting in balanced tavel costs of the obots and a small cove time of 45. Figue 2 showed aleady that MSTC sends only two obots though the naow passage, esulting in unbalanced tavel costs of the obots and a cove time of 73. B. Popeties If thee is only one obot, MFC educes to STC and thus minimizes the cove time. If thee is moe than one obot, ecall that MSTC educes the cove time of STC by a facto of at least 2 (o 3/2) fo k 3 obots (o two obots, espectively). MFC cannot make such a stong wost-case guaantee about how good its cove time is with espect to the smallest cove time of a single obot. Theoem 2: The cove time of MFC can be equal to the cove time of STC, but cannot be wose than it. Poof. The cove time of MFC cannot be wose than that of STC because MFC makes evey obot cicumnavigate a tee that can be extended to a spanning tee. On the othe hand, Figue 4 shows an example whee the cove time of MFC is equal to the cove time of STC (whee, in the case of STC, the second obot does not move), no matte how long the coido is, even though the cove time of MSTC is only half the cove time of STC. Howeve, MFC can make a much moe poweful guaantee, namely a wost-case guaantee about how good its cove time is with espect to the smallest cove time fo the numbe of available obots: it is only a constant facto lage than optimal. Theoem 3: The cove time of MFC is at most a facto of eight lage than optimal (plus a small constant). Poof. Let C be the weight of the ooted tee cove found by TREE COVER, Ĉ the weight of the weight-minimal ooted tee cove, T the cove time of MFC, ˆT the smallest cove time, and ˆT ul the smallest cove time if the obots only need to cove the uppe left small cells of all unblocked lage cells. Fist, because cicumnavigating a tee of weight C equies enteing 4C + 4 small cells, we have T 4C + 4. Second, by the appoximation guaantee poved in [2], the tee cove found by TREE COVER is at most fou times lage than optimal, so C 4Ĉ. Thid, because the weight-minimal ooted tee cove (shifted slightly up and to the left) connects exactly all of the uppe left small cells, it povides a lowe bound on the smallest cove time if the obots only need to cove the uppe left small cells, so 2Ĉ ˆT ul. The facto of two esults fom the fact that tavesing each edge between lage cells equies enteing two small cells. Finally, because only a subset of the small cells need to be coveed if the obots only need to cove the uppe left small cells, we have ˆT ul ˆT. Putting all of these inequalities togethe, we obtain T 4C + 4 16Ĉ + 4 8 ˆT ul + 4 8 ˆT + 4. MSTC cannot claim that its cove time is only a constant facto lage than optimal. Conside again Figue 2, but this time fo an abitay numbe of obots R in a teain of size R + 4 by R lage cells instead of nine by five lage cells. The unblocked teain of size R +2 by R lage cells above the wall contains 4 R 2 + 8 R unblocked small cells. MSTC coves this teain with only two obots, and its cove time thus is at least 2 R 2 + 4 R. On the othe hand, the smallest cove time is no lage than the tavel cost needed fo each obot to completely cicumnavigate its tee shown in Figue 3 (left to exteme ight), which is 8 R + 12. Thus, the cove time of MSTC is at least a facto of (2 R 2 +4 R )/(8 R +12) lage than optimal, and this facto gows unboundedly as R inceases. MFC also has disadvantages. Fo example, even if the obots stat in diffeent small cells, it is possible fo seveal obots to occupy the same small cell at the same time. Thus, some obots might have to wait fo othe obots to leave thei cell if ou assumption that seveal obots ae able to occupy the same small cell simultaneously is unjustified.
Empty Teain Outdoo-Like Teain Indoo-Like Teain Fig. 5. Sceenshots of Diffeent Kinds of Teain Teain Robots Clusteing Ideal Max MFC MSTC Optimized MSTC Cove and Retun Cove Cove and Retun Cove Cove and Retun Cove Max (Min) Ratio Max (Min) Ratio Max (Min) Ratio Max (Min) Ratio Max (Min) Ratio Max (Min) Ratio Empty 2 30 4801 4878 (4731) 1.02 4877 (4730) 1.02 10538 (8666) 2.19 5269 (5048) 1.10 5337 (4410) 1.11 5269 (4346) 1.10 2 60 4801 4886 (4720) 1.02 4885 (4719) 1.02 10889 (8315) 2.27 5445 (5095) 1.13 5513 (4241) 1.15 5445 (4180) 1.13 2 none 4801 4888 (4725) 1.02 4886 (4723) 1.02 11057 (8147) 2.30 5529 (5161) 1.15 5602 (4168) 1.17 5529 (4107) 1.15 8 30 1200 1399 (838) 1.17 1396 (837) 1.16 7499 (73) 6.25 3752 (38) 3.13 3817 (45) 3.18 3751 (38) 3.13 8 60 1200 1415 (904) 1.18 1414 (902) 1.18 6923 (154) 5.77 3462 (77) 2.89 3539 (93) 2.95 3462 (77) 2.89 8 none 1200 1394 (956) 1.16 1391 (953) 1.16 6411 (248) 5.34 3210 (127) 2.68 3281 (146) 2.73 3206 (124) 2.67 14 30 685 841 (431) 1.23 836 (431) 1.22 7369 (5) 10.76 3685 (2) 5.38 3756 (5) 5.48 3685 (2) 5.38 14 60 685 819 (522) 1.20 815 (522) 1.19 6774 (17) 9.89 3387 (8) 4.94 3461 (16) 5.05 3387 (8) 4.94 14 none 685 830 (513) 1.21 824 (511) 1.20 6005 (49) 8.77 3002 (25) 4.38 3072 (40) 4.48 3002 (25) 4.38 20 30 479 615 (307) 1.28 609 (307) 1.27 7224 (3) 15.08 3612 (1) 7.54 3685 (3) 7.69 3612 (1) 7.54 20 60 479 604 (332) 1.26 599 (332) 1.25 6728 (9) 14.05 3364 (4) 7.02 3439 (9) 7.18 3364 (4) 7.02 20 none 479 604 (321) 1.26 599 (319) 1.25 5591 (18) 11.67 2796 (9) 5.84 2867 (18) 5.99 2796 (9) 5.84 Outdoo 2 30 4321 4380 (4269) 1.01 4379 (4268) 1.01 9391 (7893) 2.17 4695 (4574) 1.09 4772 (4031) 1.10 4695 (3960) 1.09 2 60 4321 4382 (4266) 1.01 4381 (4265) 1.01 9556 (7728) 2.21 4778 (4627) 1.11 4854 (3957) 1.12 4778 (3890) 1.11 2 none 4321 4377 (4269) 1.01 4376 (4268) 1.01 9683 (7601) 2.24 4842 (4525) 1.12 4923 (3903) 1.14 4842 (3931) 1.12 8 30 1079 1263 (789) 1.17 1260 (788) 1.17 6985 (36) 6.47 3500 (18) 3.24 3561 (26) 3.30 3494 (18) 3.24 8 60 1079 1278 (790) 1.18 1274 (789) 1.18 6314 (113) 5.85 3158 (59) 2.93 3229 (70) 2.99 3157 (58) 2.93 8 none 1079 1247 (873) 1.16 1243 (871) 1.15 6032 (151) 5.59 3016 (76) 2.80 3099 (94) 2.87 3016 (76) 2.80 14 30 616 764 (450) 1.24 760 (451) 1.23 6759 (6) 10.97 3392 (3) 5.51 3452 (6) 5.60 3380 (3) 5.49 14 60 616 750 (482) 1.22 745 (481) 1.21 6311 (27) 10.25 3156 (13) 5.12 3228 (20) 5.24 3156 (13) 5.12 14 none 616 746 (464) 1.21 741 (463) 1.20 5497 (52) 8.92 2748 (26) 4.46 2819 (37) 4.58 2748 (26) 4.46 20 30 431 572 (280) 1.33 567 (281) 1.32 6723 (3) 15.60 3362 (2) 7.80 3437 (3) 7.97 3362 (2) 7.80 20 60 431 557 (285) 1.29 552 (285) 1.28 6131 (10) 14.23 3066 (5) 7.11 3140 (9) 7.29 3065 (5) 7.11 20 none 431 551 (296) 1.28 547 (294) 1.27 5348 (23) 12.40 2674 (12) 6.20 2740 (18) 6.36 2674 (12) 6.20 Indoo 2 30 4090 4172 (4017) 1.02 4171 (4015) 1.02 8937 (7422) 2.19 4468 (4230) 1.09 4539 (3797) 1.11 4468 (3729) 1.09 2 60 4090 4196 (3995) 1.03 4194 (3994) 1.03 9243 (7116) 2.26 4621 (4290) 1.13 4690 (3648) 1.15 4621 (3585) 1.13 2 none 4090 4172 (4015) 1.02 4171 (4014) 1.02 9326 (7033) 2.28 4663 (4166) 1.14 4739 (3615) 1.16 4663 (3549) 1.14 8 30 1022 1232 (849) 1.21 1225 (849) 1.20 6501 (24) 6.36 3262 (12) 3.19 3319 (17) 3.25 3253 (12) 3.18 8 60 1022 1209 (846) 1.18 1202 (846) 1.18 6081 (86) 5.95 3042 (44) 2.98 3114 (55) 3.05 3041 (43) 2.98 8 none 1022 1209 (842) 1.18 1199 (839) 1.17 5815 (180) 5.69 2905 (90) 2.84 2981 (108) 2.92 2907 (90) 2.84 14 30 584 775 (438) 1.33 768 (439) 1.32 6348 (4) 10.86 3192 (2) 5.47 3254 (4) 5.57 3190 (2) 5.46 14 60 584 748 (452) 1.28 741 (452) 1.27 5995 (22) 10.27 2999 (11) 5.14 3071 (16) 5.26 2998 (11) 5.13 14 none 584 732 (448) 1.25 725 (445) 1.24 5033 (46) 8.62 2517 (23) 4.31 2594 (31) 4.44 2517 (23) 4.31 20 30 408 617 (241) 1.51 608 (242) 1.49 6370 (3) 15.61 3188 (1) 7.81 3248 (3) 7.96 3186 (1) 7.81 20 60 408 570 (270) 1.40 566 (271) 1.39 5732 (10) 14.05 2866 (5) 7.02 2939 (8) 7.20 2866 (5) 7.02 20 none 408 547 (279) 1.34 540 (277) 1.32 4696 (22) 11.51 2348 (11) 5.75 2420 (17) 5.93 2348 (11) 5.75 Fig. 6. Expeimental Results fo MFC and MSTC ( Max = Cove Time) VII. EXPERIMENTAL RESULTS We now compae the cove times of MFC and MSTC expeimentally. We implemented the backtacking vesion of MSTC as descibed in [5]. We evaluate them on two diffeent tasks, namely coveage [ cove ], and coveage with the additional equiement that all obots etun to thei initial small cells afte coveage [ cove and etun ]. Both MFC and MSTC can easily be extended to the second task. Fo MFC, each obot simply cicumnavigates its tee until it eaches its initial small cell. Fo MSTC, each obot that has coveed its cells backtacks until it eaches its initial small cell. Thus, in both cases, the obots continue to move aound the tee(s), extending the oiginal multi-obot coveage algoithms in a vey simple way. We evaluate MFC and MSTC fo both tasks in diffeent scenaios, namely diffeent kinds of teain [teain], diffeent numbes of obots [obots], and diffeent clusteing of the obots [clusteing]. The size of the teain is always 49 49 lage cells. Figue 5 shows the thee diffeent kinds of teain used in the expeiments. The fist kind of teain is empty [empty]. The second kind is an outdoo-like teain whee walls ae andomly emoved fom a andom depth-fist maze until the wall density dops to 10 pecent, esulting in teain with andom obstacles [outdoo]. The thid kind is an indoolike teain with walls and doos [indoo]. The position of the walls and doos ae fixed, but doos ae closed with 20 pecent pobability. We vay the numbe of obots fom 2, 8, 14 to 20 obots. A clusteing pecentage paamete x detemines how stongly thei initial small cells ae clusteed. The fist obot is placed unifomly at andom. Subsequent obots ae then placed within an aea centeed at the fist obot, whose height and width ae (appoximately) x% of the height and width of the teain. (We ensue that no two obots will be placed in the same small cell.) Thus, a small value of x esults in a high clusteing of initial small cells, while x = 200 is equivalent to no clusteing at all [none]. Fo each scenaio, we epot data that has been aveaged ove 100 uns with andomly geneated
teain (if applicable) and andomly geneated initial small cells. All cove times have been ounded to the neaest intege. Table 6 epots fo each scenaio a lowe bound that epesents an idealized cove time [ideal max]: it simply divides the numbe of unblocked small cells by the numbe of obots, and subtacts one, since the initial small cells of the obots ae automatically coveed. The ideal cove time would be the cove time if no obot needed to pass though aleady coveed small cells to each othe small cells that it needs to cove. The table also epots the smallest [min] and lagest [max] tavel cost of any obot fo each combination of a multi-obot coveage algoithm, scenaio and task. The lagest tavel cost is the cove time, and the diffeence between the smallest and lagest tavel costs gives an indication of how balanced the tavel costs of the obots ae. In addition, the table also epots the atio of the actual cove time and the ideal cove time [atio], giving an uppe bound on how fa the actual cove time is away fom optimum. The atio is indeed only an uppe bound, since the ideal cove time may not be achievable. Fo instance, seveal cells must be visited by multiple obots in the example of Figue 3. We make the following obsevations: The atio of the cove time and the ideal cove time inceases with the numbe of obots fo both MFC and MSTC since the ovehead (defined as the numbe of aleady coveed cells that a obot passes though) inceases with the numbe of obots. The atio inceases vey slowly with the numbe of obots fo MFC, but much faste fo MSTC, implying that the cove time of MFC emains close to optimal fo lage numbes of obots. The atio changes insignificantly with the amount of clusteing fo MFC, but a lot fo MSTC, implying that the cove time of MSTC emains small if obots stat in neaby cells a common situation since obots ae often deployed o stoed togethe. The atio changes insignificantly fo MFC if the task is changed fom cove to cove and etun, but inceases by about a facto of two fo MSTC (because the obot with the lagest tavel cost has to backtack along most of its tajectoy), implying that all obots ae close to thei initial small cells when coveage is complete fo MFC, which facilitates thei collection o stoage. Oveall, the atio is small fo MFC (at most 1.51) in all tested scenaios, and in fact significantly smalle than the facto of eight guaanteed by Theoem 3. The atio is much lage fo MSTC (7.81 fo cove and 15.61 fo cove and etun ). The eason is that MSTC does not balance the tavel costs of the obots as well, as evidenced by a lage diffeence between the smallest and lagest tavel costs of the obots. When intepeting these esults, howeve, one needs to keep in mind that the cove times of both MFC and MSTC depend on the initial spanning tees. Among the (lage) numbe of spanning tees fo a given unit-cost gid gaph, some may yield significantly bette cove times but we have not yet expeimented with diffeent ways of constucting the initial spanning tees. We now discuss one impotant optimization. One can educe the cove times of both MFC and MSTC by moving obots on cost-minimal paths to thei initial small cells athe than along the tee(s). This applies to cove and etun when the obots etun to thei initial small cells. Fo MSTC, it also applies to cove, when the obots backtack to thei initial small cells duing coveage. We efe to a vesion of MSTC with these impovements as optimized MSTC. We obseve that the impovements make almost no diffeence fo cove but a lage diffeence fo cove and etun, whee the atio is educed by a facto of two and then no longe diffes significantly fom the atio fo cove. Howeve, even without such optimizations, MFC continues to have much smalle cove times than optimized MSTC, fo both tasks in all scenaios. The eason is that MFC takes the objective, minimizing the cove time, aleady into account when finding a tee fo each obot to cicumnavigate, wheeas MSTC takes the objective only into account when it decides how the obots should cicumnavigate the single tee. VIII. CONCLUSIONS AND FUTURE WORK In this pape, we intoduced a new multi-obot coveage algoithm, called Multi-Robot Foest Coveage (MFC). Ou expeimental esults show that the cove time of MFC is smalle than the one of Multi-Robot Spanning-Tee Coveage (MSTC) and close to optimal in all tested scenaios. It is futue wok to make MFC obust in the pesence of failing obots, a popety that MSTC aleady has. We intend to augment MFC to handle obot failues by eplanning tajectoies fo the functional obots that cove the emaining uncoveed small cells. Futhemoe, it looks vey pomising to combine the ideas behind MSTC and MFC, especially if seveal obots stat in neaby small cells. We also intend to investigate ideas fom othe multi-obot coveage algoithms, such as [6] and [7]. REFERENCES [1] H. Choset. Coveage fo obotics a suvey of ecent esults. Annals of Mathematics and Atificial Intelligence, 31:113 126, 2001. [2] G. Even, N. Gag, J. Könemann, R. Ravi, and A. Sinha. Min-max tee coves of gaphs. Opeations Reseach Lettes, 32:309 315, 2004. [3] Y. Gabiely and E. Rimon. Spanning-tee based coveage of continuous aeas by a mobile obot. Annals of Mathematics and Atificial Intelligence, 31:77 98, 2001. [4] M. Gaey and D. Johnson. Computes and Intactability: A Guide to the Theoy of NP-Completeness. Feeman, 1979. [5] N. Hazon and G. Kaminka. Redundancy, efficiency, and obustness in multi-obot coveage (in pint). In Poceedings of the Intenational Confeence on Robotics and Automation, 2005. [6] D. Kuabayashi, J. Ota, T. Aai, and E. Yoshida. 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