A Dynamic Path Planning Approach for Multi-Robot Sensor-Based Coverage Considering Energy Constraints

Transcription

1 The 2009 IEEE/RSJ Interntionl Conference on Intelligent Robots nd Systes October -5, 2009 St. Louis, USA A Dynic Pth Plnning Approch for Multi-Robot Sensor-Bsed Coverge Considering Energy Constrints Ahet Yzici, Gohn Kirli, Osn Prltun, Aydin Siphioglu Abstrct In this study, novel dynic pth plnning pproch is proposed for ulti-robot sensor-bsed coverge considering energy cities of the obile robots. The environent is ssued to be nrrow nd prtilly unnown. A Generlized Voronoi digr-bsed networ is used for the sensor-bsed coverge plnning due to nrrow nture of the environent. On the other hnd, prtilly unnown nture is hndled with proposed dynic re-plnning pproch. Initilly, the robots re ssued to be t the se depot with equl initil energy cities. In this cse, n initil coplete coverge route is constructed considering robot energy cities using clssicl citted rc routing proble CARP) pproch with soe inor odifictions relted to coverge proble. But, due to prtilly unnown nture, the robots y fce with blocge on routes, nd fst re-plnning is required which considers reining energy cities nd current positions of the robots. So, new pln is obtined by odifying Ulusoy s lgorith tht ws developed for clssicl CARP. The developed lgorith is coded in C++ nd ipleented on P3-DX obile robots in MobileSi siultion environent. M I. INTRODUCTION ULTI-ROBOT sensor-bsed coverge pth plnning is the deterintion of pths such tht every point in given worspce is covered t lest once by one of the robot s sensor-rnge. In sensor-bsed coverge of nrrow spces, where obstcles lie within detector rnge, Generlized Voronoi Digr GVD) [] cn be used to odel the environent. This odel cn be represented with networ of consisting edges nd vertices. If soe prts of the environent re not required to be covered, then the corresponding edges to these prts re clled non-required edges. For coplete sensor-bsed coverge of the given This wor is supported by the Scientific nd Technicl Reserch Council of Turey TUBITAK), Contrct No 07E064, project title: Mobile Robot Route Plnning for Coplete Coverge of Dynic Indoor Environents. A. Yzici is with the Coputer Engineering Deprtent, Esisehir Osngzi University, Esisehir, 26480, Turey corresponding uthor: phone: ; fx: ; e-il: yzici@ ogu.edu.tr). G. Kirli is with the Industril Engineering Deprtent, Esisehir Osngzi University, Esisehir, 26480, Turey e-il: gohnirli@ gil.co). O. Prltun is with the Electricl Engineering Deprtent, Esisehir Osngzi University, Esisehir, 26480, Turey e-il: oprl@ ogu.edu.tr). A. Siphioglu is with the Industril Engineering Deprtent, Esisehir Osngzi University, Esisehir, 26480, Turey e-il: siphi@ ogu.edu.tr). re, it is sufficient for the robots to follow the required edges of corresponding networ. The ethod in [] cn be used to coplete coverge of n indoor environent using single robot. However, if the p is nown -priori, ore efficient coverge cn be chieved s in [2]. On the other hnd, if one robot s energy is not enough to copletely cover given environent, then using ultiple robots is necessity to increse efficiency. Additionlly, using ultiple robots y reduce the tie required to coplete the coverge ts nd enhnces robustness copred to the single robot. In the obile robot literture, there re soe pproches for ulti-robot coverge pth plnning [3], [4], [5]). In [3], the cost ws evluted in ters of the trveled distnce by using the edges of the configurtion spce nd Voronoi digr. First tour is generted for trversing ll the pths, nd then pproprite prts of the tour re ssigned to ech robot ccording to the cost evlution. But, in this wor, how to prtition the pth ong the robots considering their energy cities is not given. The pproch in [4] n dpttion of the single robot cellulr decoposition pproch to ultiple robots ws presented. In tht wor lso energy cities of the robots were not considered. In nother wor [5], obile robot deployent proble ws considered for specific type of coverge proble. The deployent proble ws described s deterintion of the nuber of groups unloded by crrier, the nuber of robots in ech group, nd the initil loctions of those robots. Both tiing nd energy constrints of robots were considered for vst environents. However, the deployent ws not considered for prtilly unnown nrrow-coplex environents. Constructing individul sensor-bsed coverge tours considering ech robot s energy city becoes chllenging proble in nrrow environents. Fro nother point of view, this proble cn be described s prtitioning edges of the networ ong robots without exceeding their energy cities. This proble resebles citted rc routing proble CARP) [6]. CARP is proble tht is to construct tours with iniu totl distnce for vehicles without exceeding their loding cities. Due to this reseblnce, the required energy to cover n edge cn be considered s n edge dend, siilrly robot s energy city cn be considered s vehicle s city, in CARP /09/$ IEEE 5930

2 However, there re two differences in coverge proble tht ws not ddressed in clssicl CARP. The first one is relted to edge dend tht y vry regrding the edge s sttus. As the robot psses through non-required edge, edge dend is deterined by only trveling energy which is consued by otors, ebedded coputer, icrocontroller crd nd nvigtion sensors sonr) [5]. On the other hnd, for required edge, robot spends energy both trveling nd perforing its coverge ts nd the coverge energy cnnot be clculted by only considering trversl cost due high energy consuption of soe sensors, s in this cse. Becuse of this difference, CARP solution techniques cn not be used directly for the sensor-bsed coverge proble with ultiple robots. In [7], this difference between the clssicl CARP proble nd the coverge proble explined nd solved by soe inor odifiction to Ulusoy s lgorith tht is used for clssicl CARP. The ethod proposed in [7] ws used to construct initil coverge routes for robot te under the ssuption tht the robots strt fro the se depot point nd they hve equl energy cities ll fully chrged). But, if robots fce with locing obstcle due to prtilly unnown nture of the environent, fst re-plnning is required. In this cse, robots y be t different loctions hving different energy cities. This is nother proble tht cn not be hndled using the clssicl CARP. In this pper, we hve done jor odifiction on Ulusoy s lgorith to hndle this sitution tht y be fced in rel pplictions. So, novel dynic pth plnning pproch is proposed for solving ulti-robot sensor-bsed coverge proble considering energy city of ech robot. The rest of this pper is rrnged s follows. In Section 2, definitions nd lgoriths used in the proposed pproch re given. In section 3, proposed pproch is given. Applictions of the proposed ethod re given in Section 4. Conclusions nd discussions re given in the finl section. II. PRELIMINARIES AND ALGORITHMS USED IN THE PROPOSED APPROACH A networ grph) cn be odeled s GV,A) where V is the set of vertices nodes) nd A is the set of edges rcs) connecting the vertices. A networ is clled s directed or undirected if its edges hve direction or not. If it is possible to rech ll of the vertices through existing edges, networ is clled s connected, otherwise disconnected. If there is connection ech pir of vertices, it is clled coplete networ. If the networ is not coplete, the shortest pth between ny two vertices cn be clculted using Floyd lgorith [8] to construct distnce trix. A tour T is defined s Eulerin if it is possible to return the strting vertex by pssing through ech edge exctly once [6]. If n Euler tour does not exist, soe edges ust be visited twice or ore to return the strting point. In this cse, Chinese Postn Proble CPP) occurs. CPP is n edge visiting proble in networ with iniu cost or iniu distnce. Since CPP is not n NP-hrd proble, both theticl odel nd heuristics such s Edonds nd Johnson s Miniu Perfect Mtching lgorith [9] with Hierholzer lgorith [8] cn be used efficiently. If only soe edges of the networ need to be visited, CPP turns into Rurl Postn Proble RPP). Unlie CPP, RPP is n NPhrd proble nd finding n optil RPP tour is relly hrd. If the required edge set is disconnected, Christofides heuristic cn be used to construct RPP tour. Constructing the new tour with only unvisited edges y result severl visits of the se edges, which y result unnecessrily long tour. So, this tour cn be iproved by using Shorten Algorith [0]. In CPP/RPP, there is only one vehicle visitor) to visit ll the edges. However, if there is ore thn one vehicle, the proble is clled s -Chinese proble []. Besides, if vehicles hve city constrint such s loding or energy, the proble turns into CARP. Grbge collecting is good ppliction of CARP. In this proble, ll vehicles hve certin loding city nd ech edge hs n ount of grbge required to be collected which is clled s edge dend. CARP is n NP-hrd proble. Hence, different heuristic lgoriths re developed in the literture such s Siple constructive ethods Construct-Strie, Modified- Construct-Strie, Pth-Scnning, Augent-Merge lgorith, Prllel Insert Algorith, Augent-Insert Algorith), two-phse constructive ethods Ulusoy- Prtitioning lgorith [2], Cut Algorith nd Cycle Assignent lgorith [3]), nd etheuristics Tu serch-bsed lgoriths [4] nd genetic lgoriths [5]). In [7], inor odifiction ws done on Ulusoy s prtitioning lgorith to hndle ulti-robot sensor-bsed coverge pth plnning proble for the se depot point nd equl energy city. Assue tht e denotes energy consuption during the trvel through edge due to otors, icro controller units, etc., nd q denotes required energy for sensor coverge nd trveling on the edge v, v ), R i j denotes the required edge set which ust be serviced, nd initilly ech robot hs liited energy city E ). The lgorith given in [7] is s follows: Modified Ulusoy s Algortih MUA): Inputs: GV,A), R, T, depot, E. Outputs: Tour for ech robot Step : Re-lel the vertices in G so tht the given tour T is equl to v0, v,..., vt = v0 ), where v is the depot. Let r be 0 the lrgest index of vertex incident to serviced edge on T. Construct directed networ G = V ', A ) with vertex set V = { v, v,..., v } nd introduce edge pirs v, v ) for ' 0 r, b =,2,3, K, r in A tht stisfy b >. Reove ll edges v, v ) for, b =,2,3, K, r such tht b > + nd v, ) v + or v, v ) is not serviced edge on T. Consider cses s b b follows. 593

3 If b = + nd v, ) is not required edge on T, then set d ' = 0. v + If b > + or v, ) is required edge nd chin v + P contins the depot on T. P = v,..., depot,..., v ). Then b dd to P shortest chin between v nd b v in G. And cycle C = depot,..., v, SP, vb,..., depot) is held. Where SP denotes shortest pth between v nd v b in G. If b > + or v, ) is required edge nd chin v + P does not contin the depot on T. P = v,..., v ). Then dd to P shortest chin between the depot nd v, nd shortest chin between the depot nd v in G. And cycle b C = SP, v,..., v, SP ) is held. depot depot b Clculte totl energy lod for C using non-required edge dend e nd required edge dend q for ll edges inc. If totl energy lod does not exceed E, clculte distnce cost d' of v, v ) in G, defined s the totl distnce cost of C in G. Else reove v, v ) fro G. Step 2: Solve shortest pth proble fro v to 0 v in r G. Ech edge v, v ) used in the shortest distnce pth corresponds to fesible vehicle route on G. The logic behind the MUA is prtitioning n initil Euler tour into fesible robot tours considering their energy cities. This lgorith constructs these tours into two in steps. In the first step, the lgorith genertes directed grph G ) tht s ech rc represents fesible robot tour. So, the eleents of the distnce trix of this grph re the cost of the fesible tours. In the second step, shortest pth proble is solved on this directed grph fro source vertex to terinl vertex. The result of the shortest pth proble consist of is iniu cost fesible robot tours tht cover ll required edges. The input of the lgorith is single Euler tour T tht cn be generted using CPP/RPP. Then, the lgorith originlly constructs n interedite networ by considering equl vehicle cities. Lter, shortest pth lgorith is used to deterine both the iniu nuber of required vehicles to visit ll the edges nd tour for ech vehicle. Since, the robots y hve different energy cities t soe points especilly during re-plnning), this lgorith cn not hndle this sitution. So, in this pper, new pproch for dynic re-plnning is proposed. The detils of the proposed pproch cobined with initil plnning re given in the next section. III. THE PROPOSED APPROACH In ulti-robot sensor-bsed coverge proble, since the edges re required to be covered nd robots hve liited energy city, this proble resebles CARP. But, there re soe differences entioned in the previous sections. In this pper, new pproch is proposed for dynic replnning of ulti-robot sensor-bsed coverge proble considering robot energy cities. A flow chrt of the proposed pproch is given in Fig.. Fig.. Flowchrt of the proposed pproch In the flow chrt given in Fig., initilly GVD-bsed networ G V, A) of the environent is loded. If there is ny closed edge between ny pir of vertices; Floyd s lgorith is used to find the iniu pth between tht pir of vertices. Using these iniu pths, the distnce trix D is updted such tht ll d > 0. Then, the infortion out the robots: Nuber of ville robots, depot points Depot ) nd energy city E ) for = K,, of robots, re loded. If there is no chnge in the environent, the networ of the initilly loded p is used, otherwise the networ is updted. Then the required edge set R is deterined. If coplete coverge of the environent is required, then the required edge set includes ll edges of the networ G. But, 5932

4 in chnged environent, the required edge set includes the ones tht re not covered yet. In the next step, single Euler tour T is constructed tht covers ll required edges using the proposed pproch in [2]. After deterining n Euler tour T, one of the two cses is possible: ll robots re in the se depot point with equl energy cities or otherwise. The first cse cn be solved using with MUA proposed in [7]. This cse usully cn be fced during the initil pln where robots re fully chrged nd in the se depot. But, if t lest one robot is t different depot Depot ), or hs different energy city E ) then the proble cn be solved using with Dynic Modified Ulusoy Algorith DMUA) s follows: Inputs: GV,A), R, T, Depot, E for = K,,. Output: Tour for ech robot Step: For = : Set depot = Depot, nd E = E.. Using the step of the MUA given in the previous section deterine C nd clculte the d ), construct G for robot. b. Copy G V, A ) into three diensionl networ G V, A ). Noting tht eleents of d ). A is the edge set of G with Step 2: Solve the shortest pth proble on G by using proposed theticl odel nd construct tour for ech robot with odel results. This lgorith is different thn the Modified Ulusoy lgorith MUA) since it repets step of MUA for ech robot nd constructs new networ which is denoted with G V, A ). The vertices of G is se s the vertices of G. But the edge set of G hs third diension where the depth is deterined by the nuber of robots ). So it differs fro the edge set of G. As result, there is different shortest pth proble thn the forer one in step 2. This shortest pth proble is between the depot vertex nd terinl vertex on networ G V, A ) nd is solved with new integer liner odel. The odel is given s follows: Sets nd preters: n The nuber of vertices in the networ The nuber of robots I, J {, 2,, n} denotes the nubers of vertices K {, 2,, } denotes the nubers of robots c Distnce between vertex i nd vertex j for robot Decision vriles: If robot visitsfro vertex i to vertex j x = 0 otherwise Model: n in i subject to, n = j= = i, j, ) n x j = j=, j, ) A n x in = i= i, n, ) A n = i= i, p, ) A n n x i= j= i, j, ) A = = c x A n x ip x pj = j= p, j, ) A = = 0 ) 2) 3) p = { 2,..., n } 4) = {,..., } 5) x {0,} 6) In this odel, the objective function ) iniizes the su of distnces which re chosen depot vertex to terinl vertex. Constrint 2) ensures tht exctly one sortie fro depot vertex. Constrint 3) ensures tht exctly one rrivl to terinl vertex. Constrint set 4) is flow conservtion constrints which ens only one entry nd one exit should be for interedite vertices. Constrint set 5) ensures tht exctly one edge should be chosen for ech robot. In this study, LP-solver GAMS/CPLEX [6] is used to solve the odel. So, the overll proposed lgorith considers not only different energy consuption i.e. during the coverge nd the trvel) but lso hndle dynic re-plnning for ulti robot sensor-bsed coverge considering both different energy cities nd strting points for ech robot in prtilly unnown environents. IV. APPLICATIONS The lgoriths in the proposed pproch re coded in C++ nd tested in vrious environents. Firstly, the pltfor t ESOGU Artificil Intelligence & Robotic lortory is used to show the effectiveness of the lgoriths. This pltfor is lso used to test vrious coverge lgoriths on P3-DX robots [2]. Videos of the live perfornce of P3-DX obile robots re recorded nd cn be reched fro following web site [7]. Lter, lrger indoor environent is used for the tests up to ten robots in siultion. During the siultions, two types of energy consuption re defined: Dedheding energy which is consued while robot is pssing through non-required edge nd covering energy which is consued while robot is perforing coverge ts. In this study lser rnge finder is used for coverge purposes. Assuing fixed velocity ignoring turnings), nd tie to trvel n edge s t = d velocity, the / energy consuption of trvel is odeled with e = velocity/000))) nd Sic LMS-200 t 5933

5 lser rnge finder s energy consuption is odeled s q = 20. During coverge energy is clculted s the su t of both energy types. Mobile robot s velocity is constnt nd 400/s. Therefore, obile robot consues 0.05 joule/ for trversing nd 0.05 joule/ for coverge. A. Appliction in the lortory environent First, the proposed lgorith is pplied to sensor-bsed coverge of n indoor environent which is shown in Fig. 2. In this study, the pltfor is trnsferred to MobileSi siultion environent. A topologicl p of this pltfor with GVD-bsed networ is given in Fig. 3. Covering ll the re is chieved by following ll the edges of this figure. itlic-written edges for pssing dedheding edges). If both robots were finished their tours without fcing n obstcle, they would consue 2953 joules nd 944 joules energy, respectively. Tour lengths would be nd But, Robot 2 detects n obstcle between vertices 7-2 t 40 th second, nd unle to follow the plnned pth. Up to this oent, Robot hs covered the edges -2), 2-6), 6-7) 7-8), so reining uncovered edges -3, 3-4, 4-5, 5-6, 8-9, 8-, 8-4, 9-0, 0-, 2-3, 2-7, 3-4, 3-6, 4-5, 5-6, 7-8, 7-20, 8-9 nd 9-20 should be covered. At this oent, consued energies re different for ech robot. Robot consued 422 joules for trversing nd 43 joules for coverge. Totlly, it consued 835 joules of energy. Robot 2 consued only 367 joules for trversing. Reining cities for robots re 2965 joule nd 3433 joules, respectively. Considering these vlues, new tours re constructed using the proposed pproch s in Fig. 5. Fig. 2. A photogrph of the test environent Fig. 5. Division of the tour ong two robots in dynic cse Fig. 3. Topologicl p of the test environent In siultions, two obile robots, initilly t vertex, re ssued hve initil energy city of 3800 joules. The proposed ethod genertes the pths s in Fig. 4. The re-ssigned required edges to the robots by the proposed lgorith re s follows; Robot tour: 7-8), 8-9), 9-0), 0-), -8), 8-7), 7-6), 6-2), 2-), -3), 3-4), 4-5), 5-6), Robot 2 tour: 7-8), 8-4), 4-5), 5-6), 6-3), 3-2), 2-7), 7-8), 8-9), 9-20), 20-7), 7-2), 2-3), 3-4). Note tht the upper right edges 3-4, 4-5, 5-6, 3-6) re ssigned to Robot 2 in the new tours. For these tours, Robot nd Robot 2 consue 763 joules nd 2038 joules of their reining energies, respectively. Tour lengths re nd Trce of the obile robots in MobileSi siultion environent for full coverge of the given environent is given in Fig. 6. Fig. 4. Division of the tour ong two robots Robot tours re s follows; Robot tour: -2), 2-6), 6-7) 7-8), 8-9), 9-0), 0-), -8), 8-4), 4-5), 5-6), 6-3), 3-4), 4,3), 3-2), 2-7), 7-6), 6-2) nd 2-), nd Robot 2 tour: -2), 2-6), 6-7), 7-2), 2-7), 7-8), 8-9), 9-20), 20-7), 7-2), 2-7), 7-6), 6-5), 5-4), 4-3), 3-). The robots perfors coverge ts while pssing through bold-written edges nd use the Fig. 6.. Trce of robots for coplete sensor-bsed coverge Reining energy cities for Robot nd 2 re 202 nd 395 joules, respectively t the end of the coverge. 5934

6 B. Test of the lgorith on lrger grph In order to test the proposed lgorith on lrge-nubervertices environent, the first floor of the Esisehir Osngzi University Electricl Engineering lortory building is used s the test bed. In this floor, there re 4 lortories. Inside ech lortory, there re three roos, tles, storge cinets, nd coluns. A corridor connects the lortories. Topologicl p of the first floor is given in Fig. 7. Nuber of vertices in the grph is 90. proposed ethod cn lso be used in the cse of robot filures. A dynic re-plnning with existing robot loctions nd energies should be enough to hndle the filure. This pproch cn lso be used to deterine xiu nuber of robots to be ssigned to given coverge ts for efficient coverge which is very iportnt for resource lloction. Another iportnt contribution of this study is tht the proposed lgorith is flexible nd cn be used in other robot ppliction probles. For exple, lthough the proposed lgorith is used for energy citted obile robots, it cn be extended to consider tie-constrint sensor-bsed obile robot coverge proble, s well. Fig. 7. Topologicl p of the first floor Siultions re crried out up to ten robots using the proposed pproch for different rndoly selected robot initil loctions. Fig. 8. shows considerle decrese in the verge tour length by ech robot up to 6 robots. The solution tie is less thn second up to 4 robots, nd 4 seconds up to ten robots. There is liner increse of solution tie in ters of nuber of robots. Fig. 8. The verge tour length versus nuber of robots As seen fro Fig. 8, rginl utility of dditionl robot decreses in the cse of verge tour length per robot fter 6 robots. Therefore, xiu six robots y be ssigned to the ove coverge ts. This ind of nlysis cn be used to deterine xiu nuber of robots to ssign for given coverge ts. V. CONCLUSION In this study, new pproch, bsed on citted rc routing proble CARP) ws proposed nd pplied to ultirobot dynic sensor-bsed coverge plnning for interior environents. It constructs the sensor-bsed coverge pths both in nown nd prtilly unnown environents considering robot energy cities. Prtilly unnown nture of the environent hndled by dding re-plnning which ws not ddressed in clssicl CARP solutions. The REFERENCES [] E. U. Acr, H. Choset nd J. Y. Lee, Sensor-bsed coverge with extended rnge detectors, IEEE Trnsctions on Robotics, vol. 22 ), pp , [2] A. Siphioglu, A. Yzici, G. Kirli, O. Prltun, A sensor-bsed coverge pproch in dynic environents, The 6th Interntionl Syposiu On Intelligent & Mnufcturing Systes, pp , Sry, Turey, [3] D. Kuryshi et l., Coopertive sweeping by ultiple obile robots, Proc IEEE Int. Conf Robotics Autot, vol. 3, pp , 994. [4] D. Ltier et l., Towrds sensor bsed coverge with robot tes, Robotics nd Autotion Proceedings ICRA '02, vol., pp , [5] Y. Mei, Y.H. Lu,, Y.C. Hu, C.S.G Lee, Deployent of obile robots with energy nd tiing constrints, IEEE Trnsctions on Robotics nd Autotion, vol. 223), pp , [6] H.A. Eiselt, A historicl perspective on rc routing. In: Dror M., editor. Arc routing. Theory, solutions nd pplictions. Boston, Kluwer Acdeic Publishers, pp. -6, [7] O. Prltun, A. Siphioglu, G. Kirli, A. Yzici, Multi-robot sensorbsed coverge pth plnning using citted rc routing pproch, the 24th IEEE Interntionl Syposiu on Intelligent Control ISIC), in MSC09, pp. 46-5, Russi, July [8] K. H. Rosen, Hndboo of Grph Theory, New Jersey: CRC Press, [9] J. Edonds, E.L. Johnson, Mtching, Euler tours nd Chinese Postn, Mtheticl Progring vol. 5, pp , 973. [0] A. Hertz, G. Lporte nd P. N. Hugo, Iproveent procedures for the undirected rurl postn proble, INFORMS Journl on Coputing, vol., pp , 999. [] A. Osterhues nd F. Mri, On vrints of the -Chinese postn proble, Opertions Reserch und Wirtschftsinforti, Universitt Dortund, No. 30, [2] G. Ulusoy, The fleet size nd ix proble for citted rc routing, Europen Journl of Opertion Reserch, vol. 22, pp , 985. [3] A. Hertz, M. Mittz, Heuristic lgoriths. In: Dror M., editor. Arc routing. Theory, solutions nd pplictions. Boston, Kluwer Acdeic Publishers, pp , [4] A. Hertz, G. Lporte, G. nd M. Mittz, A tu serch heuristic for the citted rc routing proble, Opertions Reserch, vol. 48, pp , [5] P. Lcoe, C. Prins nd W. Rdne-Chérif, Copetitive eetic lgoriths for rc routing probles, Annls of Opertions Reserch, vol. 3, pp , [6] CPLEX Optiiztion Inc., CPLEX Version.0, [7] Esisehir Osngzi University, Artificil Intelligence & Robotic Lortory,