Deployment Strategy for Mobile Robots with Energy and Timing Constraints

Transcription

1 Proceedings of the 2005 IEEE Interntionl Conference on Robotics nd Automtion Brcelon, Spin, April 2005 Deployment Strtegy for Mobile Robots with Energy nd Timing Constrints Yongguo Mei, Yung-Hsing Lu, Y. Chrlie Hu, nd C. S. George Lee School of Electricl nd Computer Engineering, Purdue University {ymei, yunglu, ychu, Abstrct Mobile robots usully crry limited energy nd hve to ccomplish their tsks before dedlines. Exmples of these tsks include serch nd rescue, lndmine detection, nd crpet clening. Mny reserchers hve been studying control, sensing, nd coordintion for these tsks. However, one mjor problem hs not been fully ddressed: the initil deployment of mobile robots. The deployment problem considers the number of robots needed nd their initil loctions. In this pper, we present solution for the deployment problem when robots hve limited energy nd time to collectively ccomplish coverge tsks. Simultion results show tht our method uses 26% fewer robots compring with two heuristics for covering the sme size of re. I. INTRODUCTION Mobile robots cn be used in mny pplictions, nd usully crry limited energy, such s btteries. Thus, energy constrints limit the opertionl time of mobile robots. Menwhile, mny tsks hve timing constrints. For exmple, serch nd rescue usully hs to find survivors within 24 hours; otherwise, the chnce of survivl diminishes quickly. Another exmple is to detect nd destroy lndmines before troops rrive. Energy nd time cn be conflicting constrints. For exmple, vehicle cn trvel t high speed nd rech the destintion erlier (meeting the timing constrint). However, fuel efficiency (miles per gllon) cn drop drmticlly t high speed nd the vehicle my run out of fuel (filing the energy constrint). It is crucil to consider both constrints together. Existing studies bout mobile robots focus mostly on enhncing individul robots cpbility, such s sensing, obstcle detection nd voidnce, locliztion, motion plnning, or interctions with humn controllers. Few studies hve been conducted for deploying mobile robots to ddress two issues: () the number of robots needed (i.e. fleet size ) to serch n re nd (b) the initil loctions of these robots. We cn use survivor detection fter n erthquke to explin the deployment problem. Smll robots cn 1 This work ws supported in prt by the Ntionl Science Foundtion IIS nd Creer CNS move under the rubble nd find survivors. Ech robot is equipped with sensors to detect survivors. When survivor is found, the robot sends wireless signls to inform rescuers. Before n erthquke, these robots re stored in n emergency response center. After n erthquke, the robots re trnsported by crrier to the erthquke site to help rescuers find survivors. The deployment problem is ffected by ech robot s energy cpcity, the dedline, nd the moving speed. A desirble deployment strtegy should meet the following gols: () It uses the minimum number of robots to cover given re; nmely, it cn cover the mximum re with the sme number of robots. (b) It cn cover the re within the energy nd the timing constrints. Our deployment pproch considers the time spent in unloding the robots from the crrier. This unloding time ccounts for the time to remove the robots from the crrier, put them on the ground, nd instruct the robots to strt moving. Becuse ll robots hve the sme dedline to finish the tsk, robot tht is unloded ltter hs shorter time before the dedline. After the robots re unloded, they disperse from the unloding loction nd rech the individul regions where the robots re responsible for serching. If more robots re unloded t the sme loction, some robots will wste much time nd energy to move from the unloding loction to the strting loctions of their regions. We consider three types of overhed during the deployment: unloding time, dispersing time, nd prtilly overlpped regions (explined lter in the pper). Our method is clled Spce Prtition Are Coverge Algorithm (SPACA). Simultion results show tht our method uses 26% fewer of the robots to cover the sme res when it is compred with two kinds of heuristics. II. PREVIOUS WORK Energy conservtion is n importnt issue for mobile robots. Menwhile, mny tsks hve timing constrints. Aylett [1] points out tht energy constrints re the most importnt chllenge for mobile robots. Brili et l X/05/$ IEEE. 2816

2 [3] control the velocities to sve energy for mobile robot. Mei et l. [9] present n energy model for mobile robots nd discuss the energy properties of three motion ptterns: scnline, spirl nd squre spirl. Multiple robots cn cooperte to ccomplish tsks, such s serch nd rescue, crpet clening, nd lndmine detection. Bltes et l. [2] show flexible spce prtition method for robot rescue. Ds et l. [7] compre communiction schemes mong mobile robots. Zhng et l. [13] use probbilistic method for serching lndmines. To use multiple robots efficiently, they hve to be deployed t the proper loctions. Simmons et l. [12] study the multi-robot coordintion using robot deployment s n exmple. Their pper focuses on the control nd coordintion. Rybski et l. [11] use lrge rnger robots to trnsport nd deploy smll scout robots. The rngers cn trvel up to 20 kilometers, gretly extending the serch rnge of scouts. Chng et l. [5] study the energy nd time properties of different disptching lgorithms for nt-like robot systems. The nt-like robots strt from nest to explore unmpped terrins. Cloqueur et l. [6] discuss the sensor deployment strtegy. Considering both the deployment cost nd the sensor cost, they provide sequentil deployment process; however, they do not consider the timing constrints. An efficient deployment cn decrese the deployment overhed, thus reducing the number of robots needed. Mei et l. [8] present probbilistic model to determine the number of robots for serving rndom pickup-delivery requests with timing nd energy constrints. III. PROBLEM DESCRIPTION Deployment is complex problem. For simplicity, we mke the following ssumptions in this pper. () All robots re the sme; they hve the sme mount of initil energy E. Ech robot s power consumption is ffected only by its speed. All the robots hve the sme dedline; i.e., they hve to finish their jobs before the sme time. (b) Ech robot is equipped with sensors. The sensing rnge is d from the robot s center; d is clled the sensing distnce. The sensed region is squre of =4d 2 from the robot s center. The re covered by one robot is the product of nd the robot s trveling distnce. (c) The re to be covered is two-dimensionl region without obstcles. The robots trvel long scnlines to cover the re; the time nd energy for chnging directions re not considered. A detiled nlysis of the energy consumption of scnlines is presented in our previous study [9]. (d) The crrier trvels t much higher speed thn the robots so the crrier s trveling time is negligible. This ssumption is justified becuse the crrier cn be truck, helicopter, or even n ircrft driven by humn rescuer. The time to unlod n robots t the sme loction is u(n) =u 0 + c n, where c is positive constnt. The item u 0 is the time to stop the crrier even if no robot is unloded. We consider only one crrier in this pper but our method cn be extend to multiple crriers. At time t =0, the crrier strts moving to the first unloding site. The re hs to be covered before the dedline t = τ. The crrier unlods robots t k different loctions. At the i th loction (1 i k), n i robots re unloded. The totl number of robots used is n = k i=1 n i. We orgnize the robots into k groups. The robots tht re unloded t the sme loction belong to the sme group. Let r i,j (1 i k, 1 j n i ) be robot tht is unloded t the i th loction. The re covered by this robot is denoted s i,j. The re covered by the i th group is ni i,j.the totl re covered by ll robots is A = j=1 k n i i=1j=1 i,j. The deployment problem is to find solution tht minimizes the totl number of robots, n, for covering given re of size A under the energy nd timing constrints. IV. DEPLOYMENT STRATEGY This section describes our deployment strtegy. We first present robots power models nd explin three types of overhed during deployment. Our method clcultes the re covered by one group of robots; then it determines the number of groups. A. Mobile Robots Energy Models We consider the robots motion power only the sum of the power consumed by the motors. A DC motor s power consumption is primrily due to the output mechnicl power nd the rmture loss [4]. We use p(v) to represent the power consumption t speed v, within the mximum speed v m. The energy efficiency cn be defined s the distnce trveled with one unit of energy, v nd the energy efficiency t speed v is p(v). Higher efficiency mens less energy for the sme distnce. We use v o (0 <v o <v m ) to represent the speed of the v o p(v v o) highest efficiency: p(v), v, 0 <v<v m.in this pper, we ssume ll the robots move t the optiml speed v o. A detiled discussion of the effects of speed nd energy efficiency is vilble in our previous study [9]. 2817

3 h () A Fig. 1. () A scnline-covering route. (b) Three robots re unloded t A. The strting loctions re A, B, ndc. The segments AB nd AC represent the dispersing overhed. The second robot runs out of energy nd stops t E. B. Overhed in Deployment There re severl wys to cover n re, s discussed in our previous study [9]. This pper considers scnlines only. Figure 1 () shows scnline route of one robot. The height is h; the distnce between two djcent lines is becuse the sensing distnce is d from the robot s center. There re three types of overhed tht increses the number of robots needed to cover n re. () The first type of overhed is the time spent for unloding the robots. (b) The second type of overhed comes from the time nd the energy spent by ech robot to rech its strting loction fter being unloded. (c) The third type of overhed occurs when robot cnnot finish scnline due to energy or timing constrints or both. As result, nother robot hs to cover the rest of this line. We cll this frgmenttion overhed becuse it is similr to frgmenttion of hrd disks. They re clled unloding, dispersing, nd frgmenttion overhed in the rest of this pper. These types of overhed re relted. Figure 1 (b) illustrtes the second nd third types of overhed. Suppose three robots re unloded t loction A. Their strting loctions re A, B, nd C respectively. The first robot hs zero dispersing time. The second robot s dispersing overhed is to trvel through AB; the third robot s dispersing overhed is the distnce AC. Suppose the second robot stops t E when it exhusts energy fter completing three scnlines, shown in solid lines. The third robot hs to cover CE; otherwise, this re is not covered by ny robot. To simplify our nlysis, ech robot finishes only integer numbers of scnlines. In other words, the second robot stops t D becuse its remining energy nd time do not llow the robot to finish nother scnline. B D (b) E C C. The Are Covered by One Group Our method first considers the dispersing nd frgmenttion overhed of single group. To reduce the dispersing overhed, the robots strting loctions should be close to the unloding loction. Since the robots trvel long scnlines nd cover rectngles, our method covers the four qudrnts symmetriclly in two-dimensionl Crtesin coordintes centered from the unloding loction. Figure 2 shows n exmple of n re covered by group of 12 robots. In the figure, point A is the unloding loction of the whole group nd the strting point of the first four robots. The first four robots move in different directions from point A to cover 1, 2, 3, nd 4. Point B is the strting loction of the 5 th nd 6 th robots. Point C is the strting loction of next two robots the 7 th nd the 8 th. Becuse the 5 th robot spends time trveling cross AB, its covered re cnnot be lrger thn 1, i.e Becuse these four qudrnts re symmetric, we cn obtin the reltionship 1 = 2 = 3 = 4 5 = 6 = 7 = 8 9 = 10 = 11 = 12.Inthe rest of this pper, we consider the first qudrnt only w 5 9 E C A B D Fig. 2. The re is covered by group of 12 robots. The res subscribed from 1 to 12 re the res covered by these robots. The res re symmetric to the unloding loction A. Let w be the width of the re covered by one qurter of the robots in the sme group, s shown in Figure 2. The totl dispersing distnce of the three robots covering 1, 5 nd 9 is 0+AB + AD 0+ w 3 + 2w 3 = w. We cn extend this observtion to more robots. If this group contins 4ψ robots (ψ for ech qudrnt), the totl dispersing distnce in one qudrnt is pproximtely w ψ + 2w ψ (ψ 1)w ψ = (ψ 1)w 2. The verge dispersing distnce for ech robot is (ψ 1)w 1 2 ψ w 2. Menwhile, the verge frgmenttion overhed for ech robot is h 2. Using these two vlues of verge overhed, our strtegy chooses vlues of w nd h so tht they re s close s possible. The rtionle is explined below. All robots in the sme group cn trvel the sme mximum distnce becuse ech hs the sme mount h 2818

4 of energy nd time. Let this distnce be l. The vilble time before the dedline is τ. Suppose the robots trvel t speed v nd consume power p. The trveling time of robot is t most E p so the robot cn operte t most min( E p,τ) nd trvel t most v min( E p,τ). Thisisthe vlue of l. Ech robot cn sense region of from its center; hence, to cover the re wh the totl trveled distnce of the whole group is wh. This re is reduced due to the overhed so ψl wh + ψ( w 2 + h 2 ).Using the Lgrnge multiplier method to mximize the totl covered re, we cn obtin the condition w = h. By setting w = h, we obtin the reltionship ψl h2 +ψh. With the vlue of ψ, we cn determine the vlue of h, i.e. the height of the scnlines. The re covered by this group is 4wh (four qudrnt). We will clculte the number of robots in the group, ψ, in the next subsection. D. Number of Groups nd Group Sizes To minimize the number of robots used is equivlent to mximize the verge re covered by ech robot. The re covered by the robots in ech group cn be ordered by the coverge sizes. When more robots re unloded t the sme loction, the minimum size of this group decreses for two resons. First, it tkes longer to unlod the whole group. Second, some robots need to trvel frther to rech their strting loctions. Bsed on these observtions, our method dopts the following rules to enlrge the verge re covered by ech robot: (1) The minimum re covered by the robots in ech group should be close. Let i,ni nd j,nj be the minimum res covered by the i th nd the j th groups. If i,ni is much smller thn j,nj due to the dispersing overhed, then we should use fewer robots in the i th group nd more robots in the j th group. By djusting the group sizes, we cn enlrge the verge re covered by the robots in both groups. (2) An erlier deployed group should hve group size lrger thn or equl to those of ltter deployed groups. According to the first rule, they ll hve similr minimum res. However, lter deployed groups hve less time before the dedline. (3) For two deployments tht stisfy the bove two rules nd cn cover the sme ssigned re, the one tht hs smller size of the first group is better becuse it mkes the minimum re lrger. These three rules together determine the number of groups nd the sizes of groups of deployment. The next section presents our lgorithm generting deployment solutions tht stisfy the bove three rules. E. Deployment Algorithm ROBOT-DEPLOYMENT(E,τ,A) 1 E t E, τ t τ, A t A /* E: energy,τ: dedline, A: totl re to cover */ 2 n n p n 1 4 τ τ u(n p) 5 A A GET-TOTAL-AREA(E,τ,n p) 6 min-re GET-MINIMUM-AREA(E,τ,n p) 7 flg 0 /* no solution yet */ 8 while flg =0 9 do n n p /* n is the fleet size */ 10 diff /* initiliztion */ 11 while A > 0 nd τ>0 12 do for i 1 to n p /* select next group size */ 13 do τ τ u(i) 14 temp GET-MINIMUM-AREA(E,τ,i) 15 if i =1nd temp < min-re 16 then τ 1, A 1 17 brek 18 /* find the closest minimum re */ 19 if min-re temp < diff 20 then diff = min-re temp 21 x i 22 τ τ + u(i) 23 τ τ u(x) 24 A A GET-TOTAL-AREA(E,τ,x) 25 n n + x 26 n p x 27 min-re GET-MINIMUM-AREA(E,τ,x) 28 if τ>0 /* before the dedline */ 29 then flg 1 30 else n 1 n 1 +1/* chnge first group size */ 31 E E t, τ τ t, A A t 32 n p n 1 33 τ τ u(n p) 34 A A GET-TOTAL-AREA(E,τ,n p) 35 min-re GET-MINIMUM-AREA(E,τ,n p) /* djust the size of lst group */ 36 n n x 37 A A+ GET-TOTAL-AREA(E,τ,x) 38 τ τ + u(x) 39 for j 1 to x 40 do τ τ t u(j) 41 if A GET-TOTAL-AREA(E,τ,j) 42 then n n + j 43 brek 44 else τ τ + u(j) 45 return n Our lgorithm sets the size of the first group, nd then determines the sizes of the other groups using greedy pproch. Ech group s size depends on only the sizes of the previous groups. The lgorithm strts from smll size of the first group then increses the size until solution is found. The size of the first group is initilized to one. This corresponds to the third rule in the lst section. The outer while loop finds the size for the first group until solution is found. The inner while loop ssigns the sizes of the other groups until either the re A hs been covered or the dedline hs pssed. The vrible n p keeps the size of the 2819

5 ltest ssigned group. The minimum re covered by the previous group, i.e. np, is clculted, nd it is recorded by the vrible min re. According to rule (2), the next group size is t most n p.inthefor loop from line 12, the lgorithm computes the minimum res of the next groups with sizes from 1 to n p, nd selects the size when the next group hs the closest minimum re with min re. This is required by the first rule. The functions GET-TOTAL-AREA nd GET-MINIMUM-AREA compute the totl re nd the minimum re covered by one group of robots, respectively. There re three possible cses tht the lgorithm my leve the inner while loop. () The first cse hppens in the first if sttement inside the inner while loop. When the next group size is 1 nd the minimum re is less thn the previous minimum re, the first rule is impossible to be fulfilled. Therefore, we ssign negtive vlue to τ to leve the inner while loop. (b) In the second cse, the time left τ is non-positive, while the left re A is still positive. This mens the re hs not been fully covered but no time is left. (c) The third cse hppens when the re is covered nd there is still time. In cses () nd (b), the lgorithm does not find solution. It increses n 1 by 1, renews the prmeters, nd continues the outer while loop. In the third cse, successful deployment is found nd the vrible flg is set to leve the outer while loop. Becuse the determintion of the lst group size depends only on the comprison of minimum res, not the re left before unloding the lst group, we my use more robots thn necessry to cover the whole re in the lst group. The lst steps djust the size bsed on the re left for the lst group. This lgorithm will report the sitution when it is impossible to cover the re under the constrints. A. Simultion Setup V. A CASE STUDY A commercil robot clled PPRK is used for our cse study. The robot is developed t Crnegie Mellon University [10]. It hs three polyurethne omni-directionl wheels driven by three MS492MH DC servo motors. The energy cpcity is 20736J for four AA btteries (1200 mah nd 1.2V). A dt cquisition crd is used to mesure the voltge nd current to clculte the robot s power t different speeds [9]. The power model used in this pper is p(v) =48.31v v The mximum speed is 0.16m/s, nd the optiml speed is 0.12m/s with power consumption 0.98W. The unloding time is u(n) = n seconds for n robots. The sensing distnce used is 0.8m. Are (m 2 ) 2.4 x A B 60 robots 48 robots 36 robots Rtio (height/width) Fig. 3. Are covered by different number of robots with different rtios of height nd width B. Are Covered by One Group Figure 3 shows the size of the re covered by different numbers of robots with different height-width rtios. The robots hve 6 hours before the dedline. When the rtio is less thn one, the covered re increses s the rtio increses. This is becuse the dispersing overhed domintes when the width is lrger nd the overhed decreses s the rtio increses. Point A indictes the re when h =0.1w for 60 robots. Point B hs h = w nd the covered re incresed by more thn 6%. When the rtio of h nd w exceeds one, the covered re becomes unstble, sometimes incresing nd sometimes decresing. The reson is tht frgmenttion overhed is very sensitive to the vlue of h. The figure shows three different group sizes. With 60 robots, n re lrger thn B cn be covered when h is 10w. However, the sme rtio h =10w cn cover smller re with 48 robots. Hence, we choose h w in our lgorithm becuse this provides stbly lrge covered res. C. Simultion Results We compre our method with two heuristics. The first is equl-number deployment. This method unlod equl number of robots ech time until the ssigned re cn be covered. The number of robots in ech group is decided before the deployment. The second unlods ll robots t one loction. We cll this one-unloding method. This method sves the unloding time but increses the dispersing overhed. Figures 4 nd 5 hve dedlines (τ) four nd eight hours, respectively. For equl-number deployment, we choose two different numbers, 4 nd 10. In both figures, our method requires the fewest robots to cover the res. To cover the sme re, more robots re needed when the dedline is erlier (Figure 4). In Figure 4, equlnumber (4) needs the most robots nd it cn cover t 2820

6 Fleet size Equl number (4) One unloding Equl number (10) Our method (SPACA) Group number Group size Minimum re Averge re Totl re TABLE I DEPLOYMENT FOR COVERING m 2 WITHIN FOUR HOURS. Fig. 4. Fleet size Fig Are (m 2 ) x Fleet size verses re, τ =4hours, energy = 20736J Equl number (4) One unloding Equl number (10) Our method (SPACA) Are (m 2 ) x 10 5 Fleet size verses re, τ =8hours, energy = 20736J most m 2. The lst group is deployed very close to the dedline nd no more groups cn be deployed before the dedline. Figure 5 llows longer dedline so unloding time is less importnt. Deployments with smll group size cn sve dispersing nd frgmenttion overhed, thus requiring fewer robots thn the deployments with lrge group size. Equl-number (4) needs lmost the sme number of robots s our method, while one-unloding needs the most robots. Tble I shows the detils of the deployment generted by our method to cover n re of m 2 within 4 hours. This deployment uses 328 robots. With the sme conditions, the one-unloding method hs to use 416 robots, nd the verge re per robot is 1634m 2. Our method hs higher verge re thn tht of the one unloding method, nd sves more thn 26% of the robots. VI. CONCLUSION This pper presents method to deploy mobile robots for covering n re with energy nd timing constrints. Our pproch determines the number of robots in ech group nd the number of groups. The principles in our pproch re mking the width equl to the height in ech group nd mking the minimum re covered by ech group the sme. We use cse study to demonstrte the effectiveness of our method. For future work, we pln to extend this reserch in two spects: (1) considering obstcles, nd (2) including speed vritions. REFERENCES [1] R. Aylett. Robots: Bringing Intelligent Mchines To Life. Brrons, [2] J. Bltes nd J. Anderson. Flexible Binry Spce Prtitioning for Robotic Rescue. In IEEE/RSJ Interntionl Conference on Intelligent Robots nd Systems, pges , [3] A. Brili, M. Ceres, nd C. Prisi. Energy-Sving Motion Control for An Autonomous Mobile Robot. In Interntionl Symposium on Industril Electronics, pges , [4] P. Bertoldi, A. de Almeid, nd H. Flkner. Energy Efficiency Improvements in Electric Motors nd Drives. Springer, [5] H. J. Chng, C. S. G. Lee, Y.-H. Lu, nd Y. C. Hu. Energy- Time-Efficient Adptive Disptching Algorithms for Ant-Like Robot Systems. In Interntionl Conference on Robotics nd Automtion, pges , [6] T. Clouqueur, V. Phiptnsuphorn, P. Rmnthn, nd K. K. Sluj. Sensor Deployment Strtegy for Trget Detection. In Interntionl Workshop on Wireless Sensor Networks nd Applictions, pges 42 48, [7] S. Ds, Y. C. Hu, C. S. G. Lee, nd Y.-H. Lu. Supporting Mny-to-One Communiction in Mobile Multi-Robot Ad Hoc Sensing Networks. In Interntionl Conference on Robotics nd Automtion, pges , [8] Y. Mei, Y.-H. Lu, C. S. G. Lee, nd Y. C. Hu. Determining the Fleet Size of Mobile Robots with Energy Cons trints. In IEEE / RSJ Interntionl Conference on Intelligent Robots nd Systems, pges , [9] Y. Mei, Y.-H. Lu, C. S. G. Lee, nd Y. C. Hu. Energy-Efficient Motion Plnning for Mobile Robots. In Interntionl Conference on Robotics nd Automtion, pges , [10] G. Reshko, M. T. Mson, nd I. R. Nourbkhk. Rpid Prototyping of Smll Robots. Technicl report, CMU-RI-TR-02-11, Crnegie Mellon University, [11] P. E. Rybski, N. P. Ppnikolopoulos, S. A. Stoeter, D. G. Krntz, K. B. Yesin, M. Gini, R. Voyles, D. F. Hougen, B. Nelson, nd M. D. Erickson. Enlisting Rngers nd Scouts for Reconnissnce nd Surveillnce. IEEE Robotics nd Automtion Mgzine, 7(4):14 24, December [12] R. Simmons, D. Apfelbum, D. Fox, R. P. Goldmn, K. Z. High, D. J. Muslineg, M. Pelicn, nd S. Thrun. Coordinted Deployment of Multiple, Heterogeneous Robots. In Interntionl Conf. on Intelligent Robots nd Systems, pges , [13] Y. Zhng, M. Schervish, E. U. Acr, nd H. Choset. Probbilistic Methods for Robotic Lndmine Serch. In IEEE/RSJ Interntionl Conference on Intelligent Robots nd Systems, pges ,