Simulation for Pedestrian Dynamics by Real-Coded Cellular Automata (RCA)

Transcription

1 Smulaton for Pedestran Dynamcs by Real-Coded Cellular Automata (RCA) Kazuhro Yamamoto 1*, Satosh Kokubo 1, Katsuhro Nshnar 2 1 Dep. Mechancal Scence and Engneerng, Nagoya Unversty, Japan * kazuhro@mech.nagoya-u.ac.jp 2 Dep. Aerospace Engneerng, Unversty of Tokyo, Japan Abstract In ths paper, we propose a new approach for pedestran dynamcs. We call t a Realcoded Cellular Automata (RCA). The scheme s based on the Real-coded Lattce Gas (RLG), whch has been developed for flud smulaton. Smlar to RLG, the poston and velocty can be freely gven, ndependent of grd ponts. Our strategy ncludng the procedure for updatng the poston of each pedestran s explaned. It s shown that the movement of pedestrans n an oblque drecton to the grd s successfully smulated by RCA, whch was not taken nto account n the prevous CA models. Moreover, from smulatons of evacuaton from a room wth an ext of varous wdths, we obtan the crtcal number of people beyond whch the cloggng appears at the ext. PACS: Vn; Lm; Ns Keywords: Pedestran dynamcs, real-coded cellular automata, crowd 1. Introducton Snce Cellular Automata (CA) have been proposed by von Neumann n the late 1940s, CA have been appled n a varety of scentfc researches on complex system, ncludng traffc models and bologcal felds. It s an dealzaton of a physcal system n whch space and tme are all dscrete. As one of the examples achevng most remarkable progress s the CA model for pedestran dynamcs. Snce the pedestran flows are caused by collectve crowd behavor, t s dffcult to handle drectly each pedestran by solvng coupled dfferental equatons, although the socal force model has been proposed and could reproduce some basc features of pedestran behavor [1]. CA approach could be more approprate to descrbe pedestran dynamcs n complex stuatons because of ts smplcty, flexblty and effcency. The floor feld CA model has been developed for pedestran dynamcs [2-4], where two knds of floor felds, a statc and a dynamc one, are ntroduced to translate a long-ranged spatal nteracton nto an attractve local nteracton. Dfferent CA models have also been proposed so far to smulate b-drectonal flow and cloggng at the ext [5-8]. It s mportant for the models to reproduce known collectve behavors of

2 pedestrans such as lane formaton n a corrdor, oscllatons of the drecton at bottlenecks and the so-called faster-s-slower effect n evacuaton [9]. It has been confrmed that the socal force model and floor feld model successfully show all of these collectve dynamcs. In our prevous study, the extended floor feld CA model has been presented to consder the complex room of arbtrary geometry [10]. To descrbe the evacuaton dynamcs, the statc floor feld s gven accordng to the mnmum path based on the vsblty graph and Djkstra s algorthm. As seen n Fg. 1, the von Neumann neghborhood was adopted. For each pedestran, the transton probablty, P x,y, where x and y s a move n x and y drectons, respectvely. The pedestran moved to the nearest four cells at next tme step or remaned at the same cell, but he could only move n four drectons: forward, backward, left, and rght. That s, the drecton of each pedestran movement was lmted. Ths mght be a problem f we dscuss the evacuaton tme n detal. Fgure 2 shows the example of evacuaton toward the ext. We consder two paths of A and B. Needless to say, the dstance of path B s much shorter than that of path A n real stuaton (see left fgure), because there are no grds and people can take any paths. Snce there are grds n the CA smulaton (see rght fgure), both are the same dstance. Therefore, f we count the evacuaton tme n CA model, the oblque four drectons n Fg. 1 may be needed as well. However, t should be noted that, because of the longer movement wthn one tme step, the allowance of movement toward the oblque neghbor cells corresponds to the faster moton of the pedestran, whch may also gve unrealstc soluton. Ths s one of the serous common problems n all CA models proposed so far. To mprove the model, t s better to consder any drecton and any velocty of pedestran movement. P-1,0 P0,-1 P0,1 P1,0 Fg. 1 Target cells for a person at the next tme step.

3 Ext Ext A B A B Fg. 2 Example for evacuaton toward the ext, wth two paths of A and B. Left fgure s movement n real stuaton wthout grd ponts, and rght fgure shows one wth CA grds. In the present paper, we propose the real-coded cellular automata (RCA) as a new numercal model for pedestran dynamcs. The poston and the velocty can be freely gven, ndependent of grd ponts. The procedure of updatng rule for each pedestran s explaned n the next secton. 2. Numercal procedure of RCA Here, we explan our new approach for arbtrary velocty and drectons for pedestran dynamcs. It s based on the Real-coded Lattce Gas (RLG), whch has been developed for flud smulaton [11,12]. In RLG model, smlar to the Lattce Gas Automata [13,14], the partcles are used for modelng flud as a fully dscrete molecular dynamcs. The man dfference s that the partcles have contnuous velocty dstrbutons to show Maxwell-Boltzmann dstrbuton n the equlbrum state. Furthermore, collson and streamng schemes do not depend on the explct lattce structure n the dscrete space. That s, the partcle of lattce gas has real number n the velocty, and travel to any drecton. We apply ths scheme to the CA model for pedestran dynamcs. We call t Real-coded Cellular Automata (RCA). The numercal procedure s explaned brefly. The update rule of RCA conssts of 3 steps, and the poston of the pedestran s renewed. The update rules are appled to each pedestran randomly. The unt dscrete tme step of t s used, and the space s dscretzed wth grds. The grd s square and ts length s. Here, t s assumed that the pedestran moves toward the target, for example, the ext n Fg.2. 1) Frst, the streamng process s performed to move the pedestran poston by ts movng velocty. It can be descrbed smply as the sum of poston and velocty vectors of pedestran, x ' = x + v (1)

4 where x and x are the post- and pre-streamng poston for the pedestran, and v s ts movng velocty. In ths method, v can be arbtrary velocty and x s not on at the grd at ths stage. Then, as shown n Eq.2, the velocty components n x- and y- drectons are dvded nto two parts of [ ] v and { v } : the former s the nteger part correspondng to grd number and the latter s the decmal part less than the grd length. v v x, y, = = [ vx, ] / t + { vx, } [ v ] / + { v } y, t y, / t / 2) To keep the pedestran poston rght on the grd pont, the pedestran s repostoned on the grd pont. Ths procedure s shown n Fg. 3. There are four canddates, ponts A, B, C, and D. Whch one s selected s stochastcally determned by each probablty. As shown n Eqs.3-6, the probablty of movement to each pont s p, A p, B p, C p, respectvely. D { v x, }{ v y } ( { v x, } ) { v y } { vx, } ( { vy } ) ( { v }) ( { v }) p A, p B, p C = 1, p D = x, 1 y, = (3) = 1 (4) (5) 1 (6) Needless to say, the sum of these values s 1. However, ths s not the fnal poston. The next thrd step s needed to avod the collson between pedestrans. t (2) B A { vy,} [ v y, ] C v D x [ vx, ] { vx,} Fg. 3 The poston and movement of the pedestran at the step 2 3) The thrd step s needed only when the pedestran attempt to move to the grd pont where someone already exsts. In ths case, he remans at the pre-streamng poston.

5 Instead, he changes the angle of +45 or -45. The choce of +45 or -45 s determned to make the pedestran face the grd where nobody stays, whch corresponds to our natural behavor when we try to avod nstantly the collson durng walkng or runnng. If the pedestran may ht the wall, he also changes the drecton. It could be a corner n the corrdor when people evacuate n the buldng [4]. In the above rule of RCA model, the pedestran only change the drecton and always keeps the same magntude of velocty. In the next secton, some results are shown to demonstrate the capablty of RCA. 3. Results 3.1 Movement along straght lne Frst, to demonstrate the pedestran moton by RCA model, benchmark smulaton s conducted. Here, the smple moton of the pedestran along the lne s smulated. Fgure 4 shows the calculaton doman. In each test run, one pedestran starts to leave from the corner grd. He moves towards to the pont P along the straght lne nclned to the x- or y-axs. The calculaton doman s 16m 20m, and the length of the lne pedestran walks, L, s 20 m. The grd sze s 0.4 m and the tme step s 0.5 s. These values are referred to Ref. 2. He keeps walkng at the speed of V = 1.3 m/s, and hs nclned angle of θ does not change. When one arrves at the end, the next test s conducted, so that the number of pedestran n the calculaton doman s always unty and no collsons between pedestrans occur. We conduct 10,000 test runs and record the tme when each pedestran arrves at the end. Fgure 5 shows the arrval tme n the test. The number of people s counted to obtan the hstogram. It s found that the profle s smlar to the normal dstrbuton, because the poston of the pedestran s stochastcally determned. The averaged value s 15.6 s. Ths value s very close to the estmated tme of 15.4 (= L / V). Therefore we have successfully solved the nclned-path problem by our RCA. P 20m V θ L 12m 16m Fg. 4 Calculaton doman. Number of people Arrval tme(s)

6 Fg. 5 Arrval tme n the movement along the straght lne. 3.2 Evacuaton n a large room Next, we conduct the evacuaton smulaton. Fgure 6 shows the typcal snapshot of evacuaton n a large room. The ntal number of people s 300. The calculaton doman s 16m 16m, and the grd length of s 0.4 m. The tme step of t s 0.5 s. Ther ntal postons are randomly gven, and they start to evacuate towards the ext (e.g. n case of fre). The velocty s 1.3 m/s, and ext wdth s 0.8 m. As seen n Ref. 3, three stages are observed: (a) begnnng (t = 0.5 s), (b) mddle (t = 25 s), and (c) fnal stages (t = 52.5 s). Intally, the pedestran can pass the ext smoothly. In the mddle stage, the bottleneck s thoroughly formed around the ext. Ths stuaton s automatcally formed snce the balance of nflow and outflow of pedestrans at the ext breaks. When the number of people s relatvely small, the evacuaton process s smooth and there are no jams all over the calculaton doman. However, as seen n Fg. 6, when relatvely large number of people evacuate, people becomes less flexble. To examne further, we obtan the correlaton between the number of people n the room and the total evacuaton tme. Fgure 7 shows the total evacuaton tme as functons of the ntal number of people. From ths fgure, two regons are observed. In the regon 1, the total evacuaton tme s constant even f the number of people n the room s ncreased. By checkng the tme-dependent evacuaton dynamcs, no bottlenecks are formed. (a) (b) (c) Fg. 6 Evacuaton smulaton n a large room wth one ext. Three typcal stages are shown; (a) t = 0.5 s, (b) t = 25 s, and (c) t = 52.5 s.

7 Total evacuaton tme (s) 200 Regon 1 Regon N c Intal number of people Fg. 7 Total evacuaton tme at dfferent number of people n the room. On the other hand, n the regon 2, as the ntal number of people s ncreased, more tme s needed to evacuate all people n the room. That s, the evacuaton tme depends on the ntal number of people n ths regon. Expectedly, the bottleneck s observed. Thus, dependng on the ntal number of people, the drastc change of the pedestran dynamcs appears through the formaton of bottleneck. By changng the ext door wdth, W, we examne ths crtcal number of people where the bottleneck process appears, defned as N s. Fgure 8 shows the crtcal number of people n the above room sze as functons of ext door wdth. As seen n ths fgure, as the ext door wdth s larger, N c s larger. That s, more people n the room are needed to observe the bottleneck process. The curve n Fg. 8 could be changed when the room sze s dfferent. Although more benchmark studes are needed, our proposed RCA model could be a good tool to examne the evacuaton dynamcs, especally to count the evacuaton tme n the crowds N c W (m) 4 Fg. 8 Crtcal number of people as functons of ext door wdth. 4. Conclusons

8 We have presented the real-coded cellular automata (RCA) as a new numercal model for pedestran dynamcs. The approach s orgnally based on the real-coded lattce gas (RLG). The procedure for updatng the poston of each pedestran s explaned. As the benchmark study, the movement along the straght lne s smulated. Ths stuaton s rather smple, but n the prevous CA models, the correct evacuaton tme s hard to be obtaned, because the movement n oblque drecton s not consdered. In our model, the pedestran movement at any drecton s gven, and the reasonable evacuaton tme s calculated. In the smulaton of evacuaton n a large room, the so-called bottleneck s observed at the ext. We examne the crtcal number of people causng the bottleneck process, N c. It s found that when the ext door wdth s larger, more people n the room are needed. To predct N c by changng the ext door wdth as well as the room sze, ths smulaton s needed to construct the safety standards. We conclude that our proposed RCA can be a good tool to examne the pedestran dynamcs. References [1] D. Helbng, I. Farkas, and T. Vcsek, Smulatng dynamcal features of escape panc., Nature vo.407 (2000) [2] C. Burstedde, K. Klauck, A. Schadschneder, and J. Zttartz, Smulaton of pedestran dynamcs usng a two-dmensonal cellular automaton., Physca A, vol.295 (2001) [3] A. Krchner and A. Schadschneder, Smulaton of evacuaton processes usng a boncsnspred cellular automata model for pedestran dynamcs., Physca A, vol.312 (2002) [4] A. Krchner, K. Nshnar and A. Schadschneder, "Frcton effect and cloggng n a cellular automaton model for pedestran dynamcs.", Phys. Rev. E, vol.67 (2003) [5] Masakun Muramatsu, Takash Nagatan, Jammng transton n two-dmensonal pedestran traffc, Physca A vol.275 (2000) [6] Vctor J. Blue, Jeffrey L. Adler, Cellular automata mcrosmulaton for modelng bdrectonal pedestran walkways, Transportaton Research Part B vol.35 (2001) [7] Gay Jane Perez, Govann Tapang, May Lm, Caesar Saloma, Streamng, dsruptve nterference and power-law behavor n the ext dynamcs of confned pedestrans, Physca A vol. 312 (2002) [8] L Jan, Yang Lzhong, Zhao Daolang, Smulaton of b-drecton pedestran movement n corrdor, Physca A vol.354 (2005) [9] D. Helbng, Traffc and related self-drven many-partcle systems., Rev. Mod. Phys., vol.73 (2001) [10] K. Nshnar, Extended Floor CA Model for Evacuaton Dynamcs., IEICE TRANS. INF. &SYST., VOL.E87-D (2004) [11] A. Malevanets, R. Kapral, Europhys. Lett. 44 (1998) [12] Y. Hashmoto, Immscble real-coded lattce gas., Computer Physcs Communcatons vol.129 (2000) [13] U. Frsch, B. Hasslacher, and Y. Pomeau, Lattce-Gas Automata for the Naver-Stokes Equaton., Phys. Rev. Lett., vol.56 (1986) [14] U. Frsch, D. D humères, B. Hasslacher, P. Lallemand, Y. Pomeau, J. P. Rvent, Lattce Gas Hydrodynamcs n Two and Three Dmensons., Complex Systems, 1 (1987)